One Hat Cyber Team
Your IP :
216.73.216.115
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
home
/
vo
/
book-newprint
/
View File Name :
the.ps
%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: the.dvi %%Pages: 265 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -x 1000 -O-1cm-1cm the.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.11.03:1602 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (the.dvi) @start %DVIPSBitmapFont: Fa eufm8 8 1 /Fa 1 75 df<EB07C0EB1FE0EB3FF8497E497E80D801E0EBC0103A03C03FE0F09039001F FFC0000690380FFE00EC03F848903801E1800008EB0003C8EA0700150E151E5D157C5D14 01A681A31400A381A2157EA3157FEA03E0D81FF07F487E12FF12871203A20001143EA215 3C157C6D13785D6C6C485A9038FF83C06DB4C7FC6D5AEB0FF0243780AD26>74 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb msbm8 8 2 /Fb 2 83 df<DA0FFC138091B51201903903FC07E190390F7001FF90393CE0007FD97180 133101C114194848C7120DD80306140712064848140312181601485AA20060150017005B A212C0AA1260A27FA21230A26C7EA2000C16706C6C15F06C6C15E0D80183EC01C06C6C6C EB0380D970C0EB0F00D93C70133E90390F3E01F80103B55A010014C0DA0FFEC7FC2C307E AE21>67 D<B612F015FF3A0C01C07FC03A06030018F0ED0C38ED060C826F7EA2EE0180A6 EE0300A2ED06065EED0C38ED39F091380FFFC0DAFFFEC7FCECF830EC18186E7EA26E7E81 14039138018180ED80C0EC00C01660ED6030A26F7E6F7E82150C82923806018092380300 C0260C01801460B539FC01FFFC812E2E7FAD30>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr6 6 4 /Fc 4 53 df<137013F0120712FFA212F91201B3A6B512E0A313217AA01E>49 D<EA01FC380FFF804813C0383C0FE0387003F0387C01F8EAFE0014FCA2147C127C003813 FCC7FC14F8130114F0EB03E0EB07C01480EB0F00131E13385B3801E01CEA038038070038 120E5A383FFFF85AB512F0A316217CA01E>I<13FE3803FFC04813E0380F03F0381E00F8 003F13FC147C138013006C13FC000C13F8C7FCEB01F0EB03E0EB0FC03801FF00A2380003 E0EB00F8147C147E143E143F127CA212FEA2147E5A007813FC383E01F8381FFFF0000713 C00001130018227DA01E>I<14E0130113031307A2130F131FA2133B137313E3A2EA01C3 EA0383EA0703A2120E121CA21238127012E0B6FCA3380003E0A6EB7FFFA318227DA11E> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd eusm8 8 4 /Fd 4 80 df<140EEC3FF00103B512F090391FFE7FFE90397C3E00FFD801E0EC1F80D807 C0EC0FC0D80F00EC07E0121E003E15035AA212FC17C0A216076C168048ED0F00007C151E C75C16F091383F0FC0EDFFFEEEFFC091393E003FE0EE07F0023CEB03F8EE01FC160017FE 177E5CA31470177C5C17F8495A4948EB01F00107EC03E049C7EA0780013EEC1F00D801FC EB03FC000FB612F016804802F8C7FC2F2F7DAE37>66 D<1538902601FFF81303011F9038 FFC002D97E01EBFF0E2601F00014FCD803C001F813F8D80F80141F90C7EB01F0001E92C7 FC123EA2127EA4127FA2123E121CC8FCA20107B6FC5E90C700F8C7FCAA5DA214015D1403 00785C00FC495A140F6C011EC8FC6C137C387FC1F8383FFFE06C1380D807F8C9FC302F7D AD31>70 D<902601FFF8EC01F0011FEE0FF8017F01E0141F2701FE07C0143FD803E0EE70 78D80780EEE000EA0F00484C5A001E1603003E5F1707A295C7FC003F5EA36C5E120CC7FC 171EA292B512FEA29238C0003EA65D140FA492C7FCA2141E173F021C800078133C007C49 166000FE49ED80C0267E01E0EC0F81267F0780EDC3806CB4C83807FF00D81FFC6F5AD807 E0ED01F83D2F7EAD41>72 D<EC03FC91380FFF809039103E07C09039707801F09039E0E0 00F83801C1C0D80381147CD80783143C0103143E120FD81E0780A2003E7F003C7F6EEB0F 80EA7C03A2387801F090C8FC12F8A9EE1F00A27E127C161E163E127E003E153C167C003F 15786C15F86D495A6C7E6C6C495A6C6CEB0FC0D801FC49C7FC3900FF80FE90383FFFF801 0F13E0D903FEC8FC29307BAD33>79 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmmi6 6 6 /Fe 6 114 df<137CD803FF131C48138048EBC03848EBE070EA3E01397800F0E0007013 7048EB79C01439C71380143FEC1F00A3141EA2141CA4143CA214381478A45CA35C14401E 217F9520>13 D<1238127C12FE12FFA2127F123B1203A31206A2120C121C121812701220 08117B8613>59 D<903A7FFF80FFFCA217F8903A01FC003F00163C6D6C13705E4B5A9138 7F03804BC7FCEC3F8E159CEC1FF85D6E5AA26E7EA24A7E141FEC39FC1471ECE0FEEB01C0 903803807FEB0700010E6D7E5B496D7E5BD803F06D7ED87FFEEB7FFF5B00FF91B5FC2E22 7EA132>88 D<EC01F0EC3FE0A3140715C0A4EC0F80A490381F1F00EBFFDF3801E0FF4848 7E3807807E380F003E121E123E003C5B127CA3485BA215C015E0903801F1C012781303D8 38071380381E1EF3390FF87F003803E03E1C247EA220>100 D<001F13FE393F83FF8039 33C707C03873EC03D863F813E0EAE3F0A213E00007EB07C013C0A2EC0F80EA0F80A2EC1F 0C150ED81F00131C143E15181538003EEB1E70EC0FE0001CEB07C01F177D9526>110 D<EB1F03EBFFCF3801E0FF3803C07F3807807E380F003E121E123E003C137C127CA34813 F8A4EB01F012781303EA3807381E1FE0EA0FFBEA03E3EA0003EB07C0A4EB0F80EBFFF8A2 5A18207E951C>113 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy6 6 1 /Ff 1 4 df<136013F0A30060136000F013F0EAFC63EAFE67383FFFC03807FE00EA01F8 EA07FE383FFFC038FE67F0EAFC63EAF0F00060136000001300A3136014157B9620>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmbx8 8 12 /Fg 12 104 df<EB0FFC90387FFF8048B512E048803907FC0FF848486C7E48486C7EEBE0 01003F80497EA2007F1580A400FF15C0B0007F1580A4003F15006D5A001F5CEBF0036C6C 485A6C6C485A6CB55A6C5C6C6C1380D90FFCC7FC222D7DAB29>48 D<14F01303130F137FB5FCA313BFEA003FB3AE007FB512F0A41C2C7AAB29>I<EB3FF038 01FFFE0007EBFF80001F14C0D83F8013F0007FEB3FF8EBC01F00FFEB0FFC01E013FE1407 15FFA26C487EEA3F80EA1F00C75AA215FEA2EC0FFC15F8EC1FF0EC3FE015C0EC7F80ECFE 00495AEB03F0903807E00FEB0F80EB1F00013E131F49131E13F048B512FE5A5A5A5A5A48 14FCB6FCA3202C7CAB29>I<EB0FFC90383FFF8090B512E03901F81FF03903C00FF80007 80380FF00701F87F121F13FCA34A5AEA0FF8EA07F06C48485AC75B4A5AECFFC0013F5B4A C7FC6E7E15F09038001FF8EC07FC6E7E816E1380EA1F80D83FC014C0EA7FE0EAFFF0A416 8013E06C48481300EA3F804A5A391FF01FFC6CB55A000314E0C61480D91FFCC7FC222D7D AB29>I<15FC1401A214031407140F141F143F147FA214FFEB01F7EB03E7EB07C7130F14 87EB1F07133E137C13F81201EA03F013E0EA07C0EA0F80EA1F00123E127E5AB712F0A4C7 380FFC00A7010FB512F0A4242C7EAB29>I<00181438391FC003F890B5FCA215F015C015 80150014FC14F0148090C8FCA5EB0FF8EB3FFF90B5128015E09038F03FF09038C00FF801 0013FC1407001E14FEC7FCA215FF120C123FEA7F8013C0EAFFE0A215FE13C015FC387F80 0F010013F8003EEB1FF0391FC07FE06CB512C06C1400000113FC38003FE0202D7CAB29> I<ECFF80010713F0011F7F90387FC07C9038FE007E484813FE4848487E48485A120FEA1F E0A2003F6D5A6E5A4848137892C7FC14FE9038C3FFC0D8FFCF13F001DF7F9038FE07FC90 38F801FE01F07F80491480A216C05BA4127FA4003F1580A26C7E16006C6C485A00075C39 03FC07F86CB55A6C5C013F1380D907FCC7FC222D7DAB29>I<123C123F90B612E0A44815 C0168016005D5D5D397C0003F00078495A00F85C48495A141F4AC7FCC7127E5CA2495A13 03A2495AA2130FA2131F5CA2133FA4137FA86D5AA2010FC8FC232E7CAC29>I<EB07FC90 383FFF8090B512E03901F80FF03903C001F800076D7E5B000F147EA2121F7F7F13F801FE 5BEBFF80ECE1F86CEBF3F0ECFFE06C5C7E6C14F06C8048804880D80FF77FEA1FE3D83F80 1480143F267F000F13C0140300FE1300157F153F151FA21680127F16006D5B6C6C137E39 1FF803FC6CB55A000314E0C61480D91FFCC7FC222D7DAB29>I<EB0FFCEB7FFF48B512C0 3903FC0FE03907F003F048486C7E001F80D83FC07F007F130081A200FF1580A416C0A400 7F5BA3003F5B13E0001F5B380FF81E3807FFFE6C5BC601F01380EB1FC090C7FCEA078048 6C481300EA1FE0D83FF05BA24A5A5D49485A391FC01FE0EC7FC06CB55A6C49C7FC000113 F838007FC0222D7DAB29>I<EA03F812FFA4120FABEC1FF0ECFFFC01FB13FF90B6128002 C013C09039FE003FE001F8EB1FF0A2ED0FF8A316FCA816F8A2151F16F06D14E06D133F90 39FF80FFC001E7B5128001C3EBFE00018013F89038003FE0262E7DAD2D>98 D<ED07C090391FF01FE09039FFFE3FF0000390B512F84814F9390FF01FE3391FE00FF33A 3FC007F9F0EDF8E0007FECFC00A5003F5CA26C6C485A6C6C485A6CB55A485C001E49C7FC EB1FF090C9FC123E123F90B512C06C14F815FE6CECFF8016C07E001F15E05A48C7EA3FF0 00FE140F1507A3007FEC0FE001C0133F3A3FF801FFC06CB612800007ECFE00000114F8D8 001F1380252D7E9E29>103 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmti8 8 68 /Fh 68 128 df<92393F8007C0913A01FFE01FE04A9038F03FF0913A07C0787878913A0F 00F8F0F8DA1E0113F116F9023E01F113F0923900E1E0E04A01031300A45C4C5AA4013FB7 FC4916801800903A01F0000F80A3494849C7FCA54948133EA5130F4A5BA549C75AA5013E 495AA35E013C1303137C5E38387838007C90387C078038FCF0FC4BC8FC39F8E0F81E00F1 EBF07C397FC07FF86C486C5A391E000FC0353D81AE2C>11 D<ED1FF8EDFFFE02037F9139 07E00F8091390F0003C04A1307021E130F143E023C1480027CEB070093C7FCA35CA5013F B512FE5BA2903901F0007CA2495AA25EA44948485AA44B5AEB0F80A2EEE180ED07C3A2EB 1F00A2EE8700A216C7013E14CEED03FE6F5AED00F0013C91C7FC137CA213781238EA7CF8 EAFCF0A2EAF9E0EAF1C0127F6C5A001ECAFC2A3D81AE28>I<DB3FE0EB7FE0923AFFF803 FFFC0203D9FC0F7F913B07C01E1F801F913C0F003E3C000780021ED97E7C130F023E0278 131FEE7CF8023CD938F01400027C0101140E96C7FCA24C5A5CA44C5A017FB812FCA3903C 01F00007C000F84C5A495AA24E5AA30107EC1F004A4B5AA4043E495A495AA219C34E5A5E 49C7FCA2198E190E4C148E013E179CF007FC725A4C6D5A013C010191C8FC137C5E137800 38EB3803267CF87C5B3AFCF0FC0780A227F9E0F80FCAFC39F1C0F03E007FEB7FFC393F80 3FF0391E000FC0413D81AE3F>14 D<127012F8A27E127C7EA27E7EEA078012031201090C 6CAD24>18 D<1318137C13FCEA01F8EA03F0EA07E0EA0FC0EA1F00123E5A12F05A12400E 0D69AD24>I<3807803C380FC07E381FE0FFA2EA3FE1A2EA1FE0000F137F3801C00EA338 03801CA238070038000E1370A24813E0383801C038F0078000E01300EA4002181574AD25 >34 D<EA0780EA0FC0EA1FE0A2123FA2121F120FEA01C0A3EA0380A2EA0700120EA25A5A 12F05A12400B1571AD16>39 D<EC01801407EC0F00141E143814785C495A495A495AA249 C7FC131EA25B5BA25BA2485AA212035B12075BA2120F90C8FCA25A121EA2123EA2123CA2 127CA31278A312F8A45AA71278A51238123C121C121E120EA27E6C7EA2194376B11D>I< 14C014E0A2147014781438143CA2141C141EA3140E140FAB141FA3141EA3143EA2143CA2 147CA21478A214F814F0A2130114E0A2EB03C0A213071480130F1400131EA25BA25B5BA2 485A485A485A90C7FC120E5A5A5A5A5A18437EB11D>I<EA0380EA07C0EA0FE0121FA312 0F12071200A2EA01C0A2EA038012071300120E5A5A5A5A12400B157B8716>44 D<387FFFC0A3B512806C130012057A901A>I<121C123E127F12FFA212FE127C12380808 788716>I<1404140E141CA2143C14FCEB01F81307133F13FE3801F9F0EA00C11301A2EB 03E0A4EB07C0A4EB0F80A4EB1F00A4133EA45BA45BA21201B512F0A3172C78AB24>49 D<13F0EA01F812031207A213F01203EA01C0C7FCAD121C123E127F5AA25A127C12380D1D 789C16>58 D<ED01C01503821507A2150F151FA2153FA2157715F715E7EC01C782EC0383 14071503140EA2141CA214381478147002E07F1501EB01C01303148049B5FCA25B90380E 00015B133C1338498015005B12015B1203000F1401D8FFF890381FFFE05D812B2F7BAE35 >65 D<011FB512FCEEFF8017C0903A00FC000FE0EE07F0EE03F81601494814FCA4494814 F8A21603EE07F0494814E0EE0FC0EE1F80EE7F0090390FC001FE91B512F816E016F89039 1F8001FCED007E82178049C7FC161FA3137E163FA21700495C167E16FE4B5A4848495A4B 5AED3FE0007FB61280B648C7FC6C14F82E2D7BAC32>I<DA01FE133091390FFFC070023F EBE0F09138FF80F1903A03FC0039E0D907F0131FD90FC0130F495A013EC7EA07C05B13FC 485A4848158012075B485A001F160082484891C7FCA290CAFC5AA312FEA648153016385E 7E007E5DA24B5A123E003F4A5A6C4AC7FC6D130E6C6C133CD807F013F83903FC03E0C6B5 5A013F90C8FCEB0FF82C2F75AD33>I<011FB512FCEEFF8017C0903A00FC001FE0EE03F0 EE01F8EE00FC4948147C177E173E173F495AA4495AA4495A177FA34948147E17FEA217FC 49C71201A217F8EE03F0137EEE07E0A2EE0FC049EC1F80EE3F00167E5E4848495AED07F0 ED3FC0007FB6C7FCB612FC6C14E0302D7BAC36>I<010FB612FE5BA2903900FC0001EE00 7E173E173C4948141CA4495A16E0A3903A07E001C01817001503A290390FC00F8091B5FC A390261F800FC7FCA490263F000E136017E0A292380C01C0017E1300EE0380A2EE07005B 5E161E163E48485CED01FCED0FF8007FB6FCB75A7E2F2D7CAC30>I<010FB612F85BA290 3900FC00071600A21770495AA4495AA216C016E0903A07E001C0601700A21503D90FC05B 150F91B5FCA24991C7FCEC801F81A290383F000EA4017E130C92C8FCA35BA4485AA3387F FFE0B57E6C5B2D2D7CAC2E>I<03FF1318020FEBC038023FEBF0789139FF80F8F8903A01 FC001CF0D907F0130FD90FC01307495A49C7EA03E0137E5B485A484815C012075B120F48 4815801601484891C7FCA290CAFC5AA312FEA392387FFFC092B512E06F13C0923800FC00 A24B5AA2127EA24B5A7EA26C6C1307000F5DD807E0130F6C6C133D3901FE01F96CB5EAF0 C0013FEB8040D907FCC8FC2D2F75AD37>I<903B1FFFF81FFFF8A203F014F0D900FEC7EA FE004A5CA34948495AA44948495AA44948495AA44948495A91B6FCA3903A1F80001F80A4 49C748C7FCA4017E147EA4495CA44848495AA33B7FFFC07FFFC0B590B5FC6C80352D7BAC 35>I<90380FFFF04913F815F0903800FE005CA3495AA4495AA4495AA4495AA4495AA449 C7FCA4137EA45BA4485AA3387FFFC0B5FC7E1D2D7CAC1B>I<91383FFFE05C80913800FC 00A44A5AA44A5AA44A5AA44A5AA44A5AA44AC7FCA4147EA31238007C5B12FEA248485A5C EAF003495A38781F80D83FFFC8FCEA1FFCEA07F0232E7AAC25>I<90261FFFF8EB7FFCEF FFFE4BEB7FFCD900FEC7EA3FC04A1500173C17704948495A4C5A4CC7FC160E4948133C5E 16E04B5A903907E00780030EC8FC5D153C90380FC0FE14C1ECC3BFECCF3F90391F9E1F80 14B814F04A6C7EEB3FC091380007E0A26F7E137EA26F7EA2496D7EA3167E485A167F833B 7FFFC003FFF8B55B6C6E5B372D7BAC37>I<90381FFFFCA3D900FEC7FC5CA3495AA4495A A4495AA4495AA4495AA449C7120C161CA21638137E1678167016F04914E01501A2ED03C0 4848130F151FEDFF80007FB6FCB712007E262D7BAC2D>I<D91FFFED0FFF60F03FFE0100 17C0F07F8018EFECEF80D901CF913801DF00A2EF039FEF071FD9038F153E170E171CA290 260707C0495AA2177017E0010E5EEE01C0EE0380A2011C91380701F0EC03E0160E161C01 384B5A1638A2167001704A485AEC01F0EDF1C0EDF38001E04B5AEDF700A215FED801C049 49C7FC13E000076D5AD87FFE913807FFF800FF4A5A007F4A7E402D7BAC40>I<D91FFC90 3803FFF88018F0D900FF9038003F00171E6F131CA2D901DF5CECCFC0A214C790260387E0 5BA2EC83F0A2D907035C6E7EA26E7E010E4A5AA2157EA24990383E0380153FA2ED1F8349 0287C7FCA2ED0FC7A249EB07CE16EEA2ED03FE495CA21501A248486D5A7F1207D87FFE14 7800FF1570007F1530352D7BAC35>I<4AB4FC020F13C0023F13F09138FE03FC903903F8 00FED907E0137FD91F807F49C7EA1F80017E140F4915C0485AEE07E0485A485AA2485A12 1F5B123FA290C8120F5AA300FEED1FC0A3EE3F80A217005E167E16FE5E15014B5A007E5D 007F4A5A6C4A5A4B5A6C6C49C7FC6C6C13FE9038E003F83903F80FF06CB512C06C6C90C8 FCEB0FF02B2F75AD37>I<010FB512FC49ECFF8017C0903A00FC000FF0EE03F8160117FC 49481300A449481301A317F84948130317F01607EE0FE04948EB1FC0EE3F80923801FE00 91B512F84914E093C7FC0280C8FCA249C9FCA4137EA45BA4485AA3387FFFC0B5FC7E2E2D 7CAC30>I<4AB4FC020F13C0023F13F09138FE03FC903903F800FED907E0137FD91FC07F 49C7EA1F80137E49EC0FC012014915E04848140712075B485A121FA2485A160F90C8FC5A A300FEED1FC0A3EE3F80A21700485D167E16FE5E6C14014B5A007E01F05B903903FC07E0 6C486C485A90390F0E1F80271F9E073FC7FCD80FDC13FE01FC13F80003495A6CB512C06C 6CEB0018EB0FF7D900071338ED8030167016F0EDC1E015FF5E5E93C7FC6E5A6E5A6E5A2B 3B75AD37>I<011FB512E016FC16FF903A00FC003F80EE0FC0EE07E017F049481303A217 F817F049481307A3EE0FE0495AEE1FC01780EE3F00494813FEED07F891B512E016805B91 38800FC06F7E1503D93F007F1501A282017E495AA4491307A3171848481538A21770277F FFC00313F0B5903801FFE06C6E13C0C9EA3F002D2E7BAC34>I<91380FF00C91383FFC1C ECFFFE903901F81F3C903903E003F89038078001EB0F00011E1300013E14F0133C137CA2 01FC14E0A31600A27FEB7F8014F86DB47E6D13E06D13F86D7F01017FEB001FEC01FF6E7E 81A281A2120C121CA2151E003C143EA25D1578007E5C007F495A6D485A397BF01F8000F1 B5C7FC38E07FFC38C00FF0262F7BAD28>I<0007B712F05AA23A1FC00FC0070100EC01E0 121E5A0038EB1F80A24816C0A2EC3F005AA348017E1480C791C7FCA35CA4495AA4495AA4 495AA4495AA4495AA2133F003FB57EA292C8FC2C2D74AC33>I<3B3FFFF007FFF0A202E0 14E0D801FCC7EA7E0049143C1638A248485CA448485CA44848495AA44848495AA448C748 C7FCA4007E140EA4485CA35D5D127C5D4A5A6C495A4AC8FC6C131E380FC07C6CB45A0001 13E06C6CC9FC2C2E72AC35>I<D87FFFEC7FF8B56CEBFFFC6C90C713F8D807F0EC1F8049 EC1E00161C7F00035D167816705EA24B5A4B5A7F00014AC7FC5D150E5DA25D15786D1370 5D000013015D4A5AA24AC8FC5C140E6D5A137E5C5CA25C137F5C5CA26DC9FCA2133E133C A22E2E72AC35>I<D87FFF903A7FFF803FFEB517FF050013FED807E0D907F0EB07F04CEB 03C019801807030F1500180E4B7E60153B037B5C0373147803E3147018F0DA01C35C4D5A EC03834D5A3903F00703020F4AC7FC020E5C021E140E021C141E0238141C5F027013F803 015B14E001F15D14C001F3ECF9C0028013FBD9F7005C04FFC8FC13FE5E5B00015D5B5E5B 495C5E491300402E72AC47>I<B5EC7FFC6E13FF91C7EA7FF8D807F0EC1FC00003ED1E00 6D5C5E000115706D5C4B5A00004A5A6D13074BC7FC017E131E017F131C5D6D6C5A5D1481 90381FC3C0ECC780010F90C8FC14EE14FC6D5AA25C5CA35C130FA35C131FA391C9FC5BA3 380FFFF85A7E2E2D72AC35>89 D<0107B612C05B1780903A1FF8003F0002C05B91C712FE 013E5C013C495A0138495A0178495A0170130F4B5A49495A93C7FC157E495B90380001F8 14034A5A5D4A5A4A5A4AC8FC147E14FE495A5C4948133049481370495A49485B133F49C7 FC017E1301495C48481303485A00074A5A4848130F4848131F49017FC7FC393F0003FF48 B6FCB65AA22A2D7AAC2C>I<EB4002EBE0073801C00E3803801C38070038000E13704813 E0A2383801C0A238700380A238E0070000FE13F000FF13F8A5387E03F0383801C018156E AD25>92 D<EB07C0EB1FF090387FF9809038FC3FC03801F01FEA03E03807C00FEA0F8015 80EA1F00A25A003EEB1F00127EA348133EA31518EC7C385AA214FC397C01F8701303EA3C 07393E1F7CE0391FFE3FC0380FF81F3903E00F001D1F799D24>97 D<13F8121FA3EA01F0A4485AA4485AA4380F87C0EB9FF0EBBFF8EBF87C381FE03E13C0EB 801F13005A123EA348133FA448137EA3147C14FC14F8130114F0387803E0EB07C0387C0F 80383C1F00EA1FFEEA0FF8EA03E0182F78AD21>I<EB01F8EB0FFEEB3FFF90387E078090 38F803C03801F0073803E00FEA07C0D80F801380001F140090C8FC5A123E127EA35AA65C 007C14801407003CEB0F00003E133E381F01FC380FFFF06C13C0C648C7FC1A1F799D21> I<153EEC07FEA3EC007CA415F8A4EC01F0A4903807C3E0EB1FF3EB7FFBEBFC3F3901F01F C0EA03E03807C00FEA0F801580EA1F00A25A003EEB1F00127EA348133EA31518EC7C385A A214FC397C01F8701303EA3C07393E1F7CE0391FFE3FC0380FF81F3903E00F001F2F79AD 24>I<EB03F8EB0FFCEB3FFEEBFE0F3801F8073903E00380EA07C0120F9038800700EA1F 0048130E147C48B45A5C1480007EC8FC127C12FCA4127C5C1580003C1307003EEB0F0000 1E133E380F81FC3807FFF06C13C0C648C7FC191F799D21>I<15F0EC03F8EC07FCEC0F1E EC1E3E157E143E157CEC3C38EC7C00A45CA590383FFFE04913F015E0903801F000A2495A A6495AA5495AA549C7FCA5133EA4133C137CA213781238EA7CF8EAFCF0A2EAF9E0EAF1C0 127F6C5A001EC8FC1F3D81AE16>I<14F8EB03FE90380FFF3090381F07F8EB3E03EB7C01 13F8EA01F015F0EA03E01207A29038C003E0120FA3391F8007C0A4EC0F80A2000F131FA2 EC3F00000713FFEA03C1EBFFDFC613BEEB7E3E1300A25CA400385B387C01F012FCEB07E0 38F81FC0B5C7FC6C5AEA1FF01D2C7C9D21>I<131FEA03FFA3EA003EA45BA45BA43801F0 7EEBF1FF01F713809038FF83C03903FE03E013F8A213F0EA07E0A213C0A2390F8007C0A3 EC0F80EA1F00A2EC1F001506003E140E143EA2151C48133C147CEC3C38157048EB1FE0EC 0FC00070EB07801F2F7BAD24>I<1306131F1480EB3F007F130C90C7FCA9EA03E0487E48 7EEA1C78EA387CA21270A2485AA2EAE1F012411201485AA3485AA3EA0F831387EA1F07A2 130E121E123EEA1E1C5BEA0FF05BEA03C0112E7AAC16>I<EC0380EC07C0140FA2EC0780 EC030091C7FCA9EB01F0497E497EEB0E1E131CEB381FA21370143E13E0A21340EB007CA4 5CA4495AA4495AA4495AA4495AA349C7FC1238EA7C1EEAFC3E5B485AB45AEA7FC0001FC8 FC1A3B82AC16>I<131FEA03FFA3EA003EA45BA45BA43901F001E0EC07F0EC0FF8EC3C38 3903E07878EC60F814C1EBE1813907C301F09038C700E001CE130013FC485A7F13FF14C0 381F0FE013036D7E1530003E1470A4007C14E0A2ECF1C0A239F800FF80EC7F000070131E 1D2F7BAD21>I<137CEA07FC120FA2EA00F8A4EA01F0A4EA03E0A4EA07C0A4EA0F80A4EA 1F00A4123EA45AA31330EAF870A4EAF0E0A212F1EAF9C0EA7F80EA3F00121E0E2F7AAD12 >I<3B07801FC007F03B0FC07FE01FF83B1FE1FFF07FFC3B39F3E0F8F83E3B38F7807DE0 1F3A70FF007FC0491480491400D8E1F8137EA249137C1241D803E049133EA35F4848485A A25F1830484848481470EE01F0A218E03B1F0007C003E0A2933801E1C0EFE380003E903A 0F8000FF00177E001C6DC7123C341F7A9D3A>I<3907801FC0390FC07FE0391FE1FFF039 39F3E0F83938F7807C3870FF005B5BEAE1F8A25B1241D803E05BA34A5AEA07C0A24A5A16 C0D80F8013E1EC07C1A2EDC380D81F001383140F9138078700158E003EEB03FC6E5A001C 6D5A221F7A9D28>I<EB03F8EB0FFEEB3FFF90387E0F809038F807C03901E003E01203D8 07C013F0380F8001121F1300481303123E127EA348EB07E0A3EC0FC0A21580EC1F00007C 5B143E003C5B383E01F8381F07F0380FFFC06C90C7FCEA01FC1C1F799D24>I<90383C01 F090387E07FC9038FF0FFE3901C79E1F9138B80F80380387F09138E007C014C0EA070F14 80A2120239001F000FA4013EEB1F80A31600495B153E157E157C01FC5B6D485A4A5A6D48 5A3901F3FF80D9F1FEC7FCEBF0F891C8FC485AA4485AA4EA7FFC12FF127F222B7F9D24> I<903807C02090381FF0E0EB7FF9EBFC3D3901F01FC0EA03E03807C00FEA0F801580EA1F 00A25A003EEB1F00127EA348133EA45C5AA214FC387C01F81303EA3C07EA3E1F6CB45AEA 0FF9EA03E1EA0001495AA4495AA43801FFFCA31B2B799D21>I<3807803E390FC0FF80D8 1FE113C03838F3C19038F701E03870FE03EBFC07A2D8E1F813C09038F0038091C7FC1241 EA03E0A4485AA4485AA448C8FCA4123EA2121C1B1F7A9D1E>I<EB0FC0EB3FF0EBFFF838 01F03C3803C01E13800007133EA2000F133C1418140013F06CB4FC14806C13C06C13E06C 13F0130F130113001238127C12FC14E0EAF80100E013C0130338780F80383FFF00EA1FFC EA07F0171F7A9D1D>I<131C133EA35BA45BA4485A387FFFE0B5FC14C03803E000A4485A A4485AA448C7FCA314C0EA3E01A213031480383C0700A2130EEA3E3CEA1FF86C5AEA07C0 132B7AA918>I<EA03C0D807F01338486C137CEA1C78D8387C13F8A21270A239E0F801F0 A2EAE1F012410001EB03E0EA03E0A33907C007C0A315C3390F800F87A3141F150E000713 3FEBC07F3903E1EF9C3901FFC7FC6CEB83F890387E01E0201F7A9D26>I<3903C001C039 07F003E0380FF807D81C7813F038387C03A20070130115E0EAE0F81400EAE1F000411301 000114C0EA03E0A33907C00380A3EC0700EA0F80A2140EA200075BEBC01814383803E0F0 6CB45A6C5B013FC7FC1C1F7A9D21>I<D803C01407D807F09038700F80486CEBF81FD81C 7815C03A387C01F00FA200701507178039E0F803E01603EAE1F0004115070001D907C013 00EA03E0A33A07C00F800EA44948485A120F5EA212076D486C5A027F5B3903E0F7C13A01 FFE3FFC06C01C15B90263F007EC7FC2A1F7A9D2F>I<90383E01E09038FF07F848EB8FFC 3903C3DE1C390703F81ED80E01133EECF07EEA1C03ECE07C0038143815001210C6485AA4 495AA3151890381F00381238127C007E1470485A15E039F87F01C03970EF8780397FE7FF 00383FC3FE380F00F81F1F7C9D21>I<EA03C0D807F01338486C137CEA1C78D8387C13F8 A21270A239E0F801F0A2EAE1F012410001EB03E0EA03E0A33907C007C0A4390F800F80A3 141F150000075B6D5A3803E1FF6CB45A6C13BEEB7E3E13005CA2001E5B123E387E01F05C 387C03C038780780D83C1FC7FCEA1FFEEA0FF8EA07E01E2C7A9D23>I<010F136090383F 80E090387FC1C0EBFFE148EBFB809038E1FF003803807F140E495AC75A5C5C495A495A49 C7FC130E5B5B5B9038E001803801C003EA03803807000701801300380FF00F381FFC3E38 3EFFFC38383FF8486C5A486C5A6D5A1B1F7C9D1D>I<007FB512FEB6FC7E1F037A9224>I< 381C01C0383E03E0387F07F0EAFF0FA200FE13E0387C07C03838038014086EAD24>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi eusm7 7 1 /Fi 1 67 df<1438903801FFF0011FEBFFC09039FE780FF0D803E0EB01F8D80780EB007C D81E00805A0038151E1278A212F8161C163C16385E00705DC7EB0380031FC7FC91387FFF E016F891387801FEED007F0270EB1F80160F17C016075CA35C010115804A130F49C71300 49141E010E5C013C5C01F8EB03F0000FB612C04892C7FC15F02A297BA836>66 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj msbm10 12 1 /Fj 1 68 df<922601FFE01330033F01FC13784AB6FC020715C0021F9039C03FF0F8913A 7FFE000FFF902601FEF81303902603F9F0130090260FE3E0EC7F7890261F87C0143E4948 48141F017C90C8EA0FF8D9F81E1507D801F01603495AD803C0160100075B01801600EA0F 004A1678121EA248485AA2193000781800A3495A12F0AE1278A26D7EA2123CA36C6C7EA2 7EA226078078160601C0170F00037FD801E0171E6D6C163ED800F8177CD97C0F16F8013F 6DEC01F090261F87C0EC03E090260FE3E0EC0FC0902603F9F8EC3F80902601FEFEECFF00 903B007FFFC00FFE021F90B512F802075D02011580DA003F49C7FC030113E040487CC52E >67 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmr12 12 3 /Fk 3 52 df<140E141E143C147814F0EB01E0EB03C0EB0780A2EB0F00131E133E133C13 7C137813F85B12015B1203A2485AA2120F5BA2121F90C7FCA35AA2123EA2127EA5127CA2 12FCB2127CA2127EA5123EA2123FA27EA37F120FA27F1207A26C7EA212017F12007F1378 137C133C133E131E7FEB0780A2EB03C0EB01E0EB00F01478143C141E140E176477CA26> 40 D<12C07E7E12787E7E7E6C7EA26C7E6C7E7F12007F1378137C133C133E131E131FA2 EB0F80A214C01307A214E01303A314F0A21301A214F8A51300A214FCB214F8A21301A514 F0A21303A214E0A3130714C0A2130F1480A2EB1F00A2131E133E133C137C137813F85B12 015B485A485AA248C7FC121E5A5A5A5A5A16647ACA26>I<49B4FC010F13E0013F13F849 13FE3901FE01FF3A03F0007F8001C0EB3FC048C7EA1FE0487E01E014F0486C130F6D14F8 A46C5AA26C5AC813F0151FA216E0A2ED3FC01680ED7F0015FE5DEC03F8EC1FF090380FFF C092C7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0ED07F816FCA216FE1503A2 16FFA2121F487E487E487EA316FE15075B6C4814FC007EC7FC0038EC0FF8003C15F06CEC 1FE06C6CEB3FC0D807E0EB7F803A03FC01FF006CB55A6C6C13F8011F13E0010190C7FC28 447CC131>51 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmsy10 14.4 1 /Fl 1 4 df<140E141F4A7EA76EC8FCA20078ED03C000FCED07E0B4151F0180143F01C0 147F01F0EB01FFD87FF84913C0D81FFC491300D807FFEB1FFCC690388E3FE090393FEEFF 8090260FFFFEC7FC010313F8010013E0EC3F80ECFFE0010313F8010F13FE90393FEEFF80 9039FF8E3FE0000790381F1FFCD81FFCEB07FFD87FF86D13C0D8FFF06D13E001C0EB007F 0180143F0100141F00FC15070078ED03C0C791C7FCA24A7EA76EC8FC140E2B3378B73C> 3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmbx7 7 3 /Fm 3 71 df<EC03E04A7E4A7EA34A7EA24A7EA24A7EA391B57E14F901018014F0010380 ECE07FA2010780ECC03F010F80EC801F011F80EC000FA2498091B5FC4980A290B77E9038 F80001A2000182497F00038249147FB5010FB51280A431297DA838>65 D<B7FC16F016FC823B03FE0003FF801500EE7FC0A2EE3FE0A4167F17C016FF1780030313 00ED1FFC90B612F08216FE903AFE0001FF809238007FC0EE3FE017F0161F17F8A6EE3FF0 A2EEFFE0030313C0B81280170016FC16C02D287DA735>I<B712FCA4000390380007FE15 00167E163EA2161EA2161FED780FA3EDF800A2140314FFA414031400A21578A392C7FCA8 B6FCA428287DA72F>70 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn lcircle10 10 1 /Fn 1 115 df<133F3801FFE0000713F8487F487F487FA2481480A3B612C0A66C1480A3 6C1400A26C5B6C5B6C5B000113E0D8003FC7FC1A1A8D8C19>114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmr12 14.4 1 /Fo 1 49 df<EC1FF891B5FC010314C090390FF81FF090391FC003F849486C7E49C77E01 FE147F4848EC3F8049141F000316C04848EC0FE0A2000F16F0491407A2001F16F8A2003F 16FCA3491403007F16FEA700FF16FFB3A5007F16FEA66D1407003F16FCA4001F16F8A36C 6CEC0FF0A2000716E06D141F000316C06D143F000116806C6CEC7F00017F14FE6D6C485A 90391FE007F890390FF81FF00103B512C0010091C7FCEC1FF830517BCE3B>48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi8 8 24 /Fp 24 118 df<123C127EB4FCA21380A2127F123F1203A312071300A2120EA25A123C12 385A122009157A8714>59 D<167016F8A215011503A21507A2150F151F82153B15791571 15E1A2EC01C11403ED81FEEC07011500140E5CA25C147814704A7FA249487F49B6FCA25B 49C77E130E5B178049143F137813705BA21203000F157FD8FFFC903807FFFE6D5B496D13 FC2F2F7DAE35>65 D<011FB6FC4915E06D15F8903A00FE0003FCEE00FE177FA249488018 80A3494815005FA217FE49485C4C5AEE07F04C5A4948EB7F8091B6C7FC16FC16FF903A1F C0003FC0EE0FE0707E1603494880A21601A249C7FC1603A25F01FE14074C5AA24C5A4848 EC7F804CC7FCED07FCB75A16C04BC8FC312D7DAC35>I<92387FC003913903FFF807021F EBFC0F91397FC03E1F903A01FE00071ED903F8EB03BED90FE0EB01FED91F80130049C812 FC017E157C5B485A48481578A2485A120F491570001F163048481500A348CAFCA312FEA6 16031780EE0700127E160EA26C5D5E6C7E16F06C6C495A6C6CEB07C0D803F8011FC7FC6C B413FE39007FFFF8011F13E0D903FEC8FC302F7CAD32>I<011FB712805B7FD900FEC712 7F171F170F1800494880A4495A1638A34948EB700694C7FC16F0A290390FE003E091B5FC A390391FC007C01503A3D93F80EB800C171CA24C5A49C8FC5FA25F13FE16014C5A160748 484A5A163F4BB4C7FCB8FC5EA2312D7DAC34>69 D<011FB7FC5B7FD900FEC7FC173F171F 171E4948140EA4495AA2166016704948EBE00C1700A21501D90FE05B150791B5FCA2495C ECC00F1507A2D93F8090C7FCA490387F000692C8FCA313FEA4485AA3B512F8805C302D7D AC2D>I<011FB5FC5B6D5B010090C8FC5CA3495AA4495AA4495AA4495AA4495AA44948EB 01801603A2EE070049C7FC5E160E161E01FE141C163C167C5E484813011507ED3FF0B7FC 5EA2292D7DAC30>76 D<D91FFF91383FFF80496D5B6D810100923803F8006FEB01E060EC EFE0D901CF4A5AECC7F0A281D903834AC7FC81148181D90700140E81811680010E013F5B A2ED1FC0A2496D6C5AA2ED07F0A2496E5A150316FC1501496E5A150016FF167F495D163F A2161F48485D6D140FEA07F0B5140794C8FC4980392D7DAC38>78 D<ED7FC0913807FFF8021F13FE91397F80FF80903A01FC001FC0D907F0EB0FE0D90FC0EB 07F04948130349C7EA01F8017E15FC491400485A484815FE120749157E120F485AA24848 15FEA348C9FCA300FEED01FCA3EE03F8A217F0160717E0160F17C0EE1F80007EED3F0002 3E5B6C01FF13FE49EB81FC3A1F83C383F89039C781C7E03A0FE700DFC0D807F701FFC7FC 3903FF03FEC6EBFFF8013F9038E00180EB07FE90C7130317006F5A160EEDF83EEDFFFCA2 5E5E6F5A6F5A6FC7FC2F3B7CAD38>81 D<011FB512F849ECFF806D15E0903A00FE000FF0 EE03F8EE00FC834948147EA4494814FEA34C5A495A4C5A4C5A4C5A4948EB1F8004FFC7FC 91B512FC16E049809138C003F86F7E6F7E495A167FA349C712FEA401FE495AA3EF018048 481503A2EF0700B539F000FE1EEE7FFC4A6D5AC9EA07E0312E7DAC35>I<913807F00691 383FFE0E91B5FC903901F80F9E903903E003FC903807800149C7FC011E147C013E147813 3C137CA201FC1470A316007F7F14F06DB4FC15E06D13F86D7F6D7F01037FD9007F138014 07EC007F153FED1FC0150F1680120C121CA21600003C5C151E153E153C007E5C007F5C90 388003E0397BF00FC000F1B55A26E07FFEC7FC38C00FF0272F7CAD2B>I<0007B8FC5AA2 3B1FE003F8007F0100151E121E5A0038495AA248161CA24A5A5AA34849481318C71500A3 4A5AA44AC8FCA414FEA4495AA4495AA4495AA2130F000FB512F048805D302D7FAC29>I< 263FFFF8EB7FFF4892B5FC6C81D801FEC7EA07F049EC03C01780A24848EC0700A4484814 0EA448485CA448485CA448485CA448C85AA400FE4A5AA24B5AA2007E4AC7FCA2150E003E 5C003F5C6C14F0390FC003E03907F01FC06CB5C8FCC613FCEB1FE0302E7CAC30>I<3E7F FFC01FFFF003FFF0B54916F86C19F02807F80001FEC7EA7F00494A143CA26D1738611203 4B6C5C180103075D030F4A5A150E031C4AC7FC600338140E606D137003F05C000114E04A 486C5B7013F0DA03805C02074A5A1500020E4A5A17074A92C8FCD9FE3C140E1438000049 5C17BC4A14B8EE3FF06D5A5F5C91C75BA2495D94C9FC5B0178143EA20170141C452E7CAC 43>87 D<90260FFFFC90B5FC814B14FE9026003FE0EB1FE04B14006E6C131C5F5F6E6C5B 4C5A913807F8034C5A6E6C48C7FC161E6E6C5A5E5E6E6C5A5E6F5AA26F7EA24B7E15FF5C EDCFF0EC038F91380707F8140E4A6C7E14384A6C7E14E0130149486C7E495A49C76C7E13 1E496E7E13FC00074B7ED87FFF0103B5FCB56C5A6C497E382D7EAC3A>I<157CEC01FE4A 7E9138078780EC0F9FEC1F1F153F143FED1F00150E027EC7FCA55CA390387FFFFCA25DD9 01F8C7FCA5495AA5495AA6495AA5495AA491C8FC5BA3133E121CEA7E7E137C12FE137848 5A485A127FEA3F806CC9FC213D7CAE22>102 D<147CEB03FF49139C90381F83FEEB3F01 EB7E005B4848137E15FC485A1207A29038E001F8120FA3391FC003F0A4EC07E0A2120F14 0FEC1FC00007133F3803E0FF3801FFEF6CEBDF80EB3F1F1300A2EC3F00A31238007C137E 00FE5BA238FC03F8EB0FF0387FFFC06C90C7FCEA0FF81F2C7F9D22>I<1307EB0F80EB1F C0A21480A2EB0E0090C7FCA8EA01E0EA07F8487EEA1E3E1238A2EA707EA212E05BA21240 EA01F8A3485AA2485AA214C0EA0FC1A2EA1F81EB8380A2EB8700A2EA0F8E13FC6C5AEA01 F0122E7EAC18>105 D<137EEA0FFEA3EA00FCA4EA01F8A4EA03F0A4EA07E0A4EA0FC0A4 EA1F80A4EA3F00A4127EA31318EAFC38A4EAF870A3EAFCE0127FEA3FC0EA0F000F2F7DAD 15>108 D<39078007F0390FE03FFC486C487E3938F8F83F9038F9C01F2670FF80138014 005BEAE1FCA25B1241D803F0EB3F00A3157E485AA25D161848481438EC01F8A21670391F 8003F0A2020113E0EDF1C048C7EAFF806CEC7F00000E143E251F7E9D2B>110 D<EB01FE903807FF80011F13C090387F03F0EBFC013901F800F8484813FC485A4848137E 121F5B003F14FE90C7FC5AA300FEEB01FCA3EC03F8A215F0007EEB07E0EC0FC01580003E EB1F006C137E380FC1FC3807FFF06C5BC690C7FC1F1F7E9D22>I<903803E01890381FF8 3890383FFCF8EBFC1F3901F80FF03803F007EA07E013C0000F14E0EA1F80A2123F903800 0FC05AA300FEEB1F80A4EC3F005AA2007C5B147E007E13FEEA3E03EA1F0F6CB45AEA07FC EA01F0C7FC495AA4495AA490B5FCA31D2B7E9D20>113 D<EB07E0EB1FF8EB7FFCEBF81E 3801F00FEBE01F0003133FA20007133E141C140013FEEBFFC06C13E014F06C13F838007F FC131F1301EA1800007C137C12FEA214784813F84813F038F001E0387C07C0383FFF806C 1300EA07F8181F7C9D21>115 D<EA03E0486C130E486C131FD81C7C1480D8387EEB3F00 A21270A24848137EA3EA41F800015C485AA34848485AA316609039C003F0E0120FA30007 903807E1C0EBE00F141F3A03F07BF3803901FFF1FF6C01E0130090383F803C231F7E9D29 >117 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi12 12 6 /Fq 6 116 df<121E123FEA7F80EAFFC0A313E0127F123F121E1200A5EA01C0A4EA0380 A2EA0700A2120E121E121C5A5A12200B1D78891B>59 D<027FB812F091B912F8A21AF002 000180C7127F037F150F4BC81203A21901A24A481500A31AE04A5AA44A5AA218C0844A48 494813C01A00A217034A485CA21707170F4A4849C8FC17FF92B6FCA24A5CA2ED8000173E 4AC7123CA449481438A449485C173094C9FCA2495AA4495AA4495AA4133F137F007FB512 F8B6FCA345447CC33F>70 D<001FB500F80103B512E0485EA24B16C026003FF0C8383FF8 006D48ED0FC049486F5A96C7FCA34948150EA449C95AA448485EA448485EA448485EA448 484B5AA448484B5AA448484BC8FCA44848150EA448C95AA35FA25FA2485E7E4C5A6C4B5A 160794C9FC6C150E6D143C001F5D6C6C5C6DEB03E0D807F8EB1FC06CB401FFCAFCC6EBFF FE6D13F8011F13E0D903FECBFC43467AC342>85 D<EC03FCEC3FFF91B512C0903903FC07 E0903907F001F090390FC000F849487F017EC7127E13FE4848147F498012034848158048 5AA2121F5B003F157F5B127FA348C8EAFF00A35E481401A25E15035E4B5AA24B5A007E5D 4B5A007F4AC7FC6C147E6C5C6D485A390FC007F03907F01FC00001B55A6C01FCC8FCEB1F E0292D7CAB2F>111 D<91380FE00191393FF803809139FFFC0F00903903F83E1F903907 E00F3F90380FC00790391F0003FE5B137E49130100015D485AA2485A4B5A485AA2121F49 495A123FA34848495AA490C7485AA35A6C4A5AA44BC7FC5D6C5BA26CEB07FEEB800F380F C01E3807E0FC3903FFF1FCC613E1EB3F0113004A5AA44A5AA44A5AA4141F010FB57E4980 5EA2293F7DAB2B>113 D<EC0FF0EC7FFE49B5FC903903F00F809039078003C049C712E0 131E49EB03F0017C1307150F49EB1FE0A2ED0FC06DEB078092C7FC7FEBFFC014FEECFFC0 6D13F06D7F6D7F6D7F13039038007FFF14031400ED7F80153F000F141FD83F8014007F12 7F151E4848133E90C7FC485C007C147800705C6C495A003EEB07C0391F803F800007B5C7 FC6C13FC38007FC0242D7BAB2E>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr msam10 10 1 /Fr 1 23 df<12C07EA27E7E7E7E7EEAF780EAF3E0EAF1F0EAF0FC133F130F13031300B3 B3B3A41260104A71B923>22 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs eufm7 7 2 /Fs 2 76 df<D91FE0130CD9FFFC5B00036D1370260703FF13F0D81C00EB83E048137FEC 3FC348131F00F0130FA26C13077E7E127FEA3F801583EA0FC01207EC0F03EA0380A23807 001E0006131C0018133800701370C712E0EB01C0EB0380EB0F00131E013880017E130748 B46C487E48EBC03B000F9039E0F1FC203B1C1FF3C1FDC03B3007FF80FF803B4003FE00FE 00C66C48137CD900F01378026013602B297DA730>65 D<EDFF80020713E0021F13F89138 7803FCECC0004948137C49C7123C0106141C130E491408A2013C013EC7FC4AB4FCEB7C03 020F7FEC1C1F90387E700F4A6C7ED93FC07F91388003FC02005B6DEB01C00307C7FC1538 90380F81FCEC8FFEECF07F903807C01F02807F150FA36F7E1400EA0806D8180E80D83C0C ECF0C0D87E18903803FF80D8FFF01500D87FC05C6C486D5A000EC812F02A2A7EA72D>75 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft line10 10 7 /Ft 7 83 df<1C0C1C1E1C3E1C7C1CF8F301F0F303E0F307C0F30F80F31F001B3E636350 5A505A505A505A50C7FC1A3E62624F5A4F5A4F5A4F5A4FC8FC193E61614E5A4E5A4E5A4E 5A4EC9FC183E60604D5A4D5A4D5A4D5A4DCAFC173E5F5F4C5A4C5A4C5A4C5A4CCBFC163E 5E5E4B5A4B5A4B5A4B5A4BCCFC153E5D5D4A5A4A5A4A5A4A5A4ACDFC143E5C5C495A495A 495A495A49CEFC133E5B5B485A485A485A485A48CFFC123E5A5A5A1260575782D453>0 D<1302130380497E8080497E8080497E15804913C015E015F04913F815FC90B512FE15FF 4814F815C048EBFE0014F84813C091C7FCEA0FFC13F0EA1FC090C8FC123C123012405A20 20809F53>9 D<1506151E157EEC01FE1407143F14FF1307133F48B5FC120FB6FCA2120F 1201EA003F13071300143F14071401EC007E151E15061F187E8B53>27 D<12C012F012FCB4FC13C013F813FEEBFFC014F814FF15E015FEA215E0150014F814C049 C7FC13F813C090C8FC12FC12F012C01F184E8B53>45 D<B6FCA26C13FEA26C13FCA26C13 F8A26C13F0A26C13E0A36C13C0A26C1380A36C1300A3137EA3133CA31318A4181F8CA053 >63 D<126012F07E127C7E7E6C7E6C7E6C7E6C7E6C7E137C7F7F6D7E6D7E6D7E6D7E6D7E 147C80806E7E6E7E6E7E6E7E6E7E157C81816F7E6F7E6F7E6F7E6F7E167C8282707E707E 707E707E707E177C8383717E717E717E717E717E187C8484727E727E727E727E727E197C 8585737E737E737E737E737E1A7C8686747E747E747E747E747E1B7C8787F30F80F307C0 F303E0F301F0F300F81C7C1C3E1C1E1C0C575782D453>I<144014C01301497E1307130F 497E133F137F497E5A487F5A5A487F5A487FB6FC001F14801203C66C13C0131F010313E0 1300EC3FF0140FEC03F81400153C150C1502150120204D9F53>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu msbm7 7 6 /Fu 6 91 df<1410143014381478146C14CC14C6A2EB018314C31303ECC1801461903806 60C01430010E1360EB0C1881EB180CA281EB300681EB700301607FA29038E0018390B5FC 4814C39039C000C180A20003EC60C05B0007156015301630120D1618D818C0EB601C26FF F003B5FCA228297FA81E>65 D<D901FE131090390FFFC03090383F83F89039F6007FF0D8 01CC131FD80318EB0630000614034848EB01B00018EC00F01670485A1630006015101600 5BA212C0AA1260A21360A21230A26C7E160C000C151C6C6C14706C6CEB01E0D801C6EB07 C03A00F3807F0090383FFFFC010F13F0010190C7FC262A7EA820>67 D<B512C0A2380C1800B3B2B512C0A212287DA726>73 D<D8FFF8EB07FF7FD81C06EB00F0 D8060314606D7E1207EB80C03806C0608013606D7E6D7E6D7EA26D7E9038030180903801 80C01560EB00C0EC6030EC3018EC180CA2EC0C06EC060391380301E0A2EC0180913800C0 6015601530A21518150C1506A21503000FEC01E0D87FF013001660C9122028297FA72B> 78 D<B6FC15E0390C1C07FC903818019EEC00C3ED6180ED60C0ED3060A21630A61660A2 ED60C0ED6180EDC700EC03FE90381FFFF81580D918C3C7FCECC18014616E7EEC18606E7E 6E7E6E7EEC01861583913800C180ED60C0ED3060ED1830ED0C1CB538C007FF8128287FA7 2F>82 D<001FB612C05A3A33E00181803A3700030300003C13064814060070495AA20060 495A1430C75B4A5A14C05D9038018180EB03014AC7FCEB0606A2495A13185C495A13605C 495AEA0180D9818013204848C71260A2D8060614E0120C491301484814C0003014034913 064848131C00C014F8B7FCA223287EA737>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmmi5 5 16 /Fv 16 116 df<146014E0A2EB01C0A2EB0380A2EB0700A3130EA25BA35BA25BA25BA348 5AA2485AA248C7FCA3120EA25AA35AA25AA25AA25A13297B9E1F>61 D<3801FFFC14F8A238001F00A3133EA45BA45BA4485AA4485AA3EA7FFEA212FF161C7D9B 1A>73 D<90380FFFC0A39038007C00A35CA4495AA4495AA4495AA21218127C38FC0F80A2 49C7FCEAF81EEA707CEA3FF8EA0FC01A1D7B9B20>I<D803FFEB07FF148014C0D80037EB 00706E1360EB33F0016314C0EB61F81360809039C07E0180A2143F141FD80180EB8300EC 0FC3140715E339030003E615F6EC01FE140000065C157C120ED8FFE0133C1518A2281C7C 9B2D>78 D<D87FFCEB3FF0A200FFEC7FE0D807C0EB0E005D6D13185D0003147015605D6D 485A12014AC7FC1406140EEBF80C00005B5CA25C6D5A137D5C017FC8FCA2137E133C1338 A2241D7C9B22>86 D<9039FFF80FFEA248141F3A0007E00380ED06006D6C5A5D6D6C5A15 E0903800FDC0ECFF806EC7FC80A281147FECEFC0EB01CF90380307E01306496C7E131849 6C7E136048486C7E3A7FF003FFE05C12FF271C7D9B2E>88 D<14F8EB03FCEB073CEB0F7C A2EB0E38EB1E00A53803FFF05AA238003C00A25BA65BA5485AA4EA71C012FB5B00F3C7FC 12FE123C16257B9C1C>102 D<137013F8A213F013E01300A6EA0F80EA1FC0EA31E01261 A2EAC3C01203EA0780A3EA0F001308EA1E18A213301370EA0FE0EA07800D1D7D9C16> 105 D<EB0180EB03C01307A2EB038090C7FCA6137CEA01FEEA038FEA070F1206120C1200 A2131EA45BA45BA4EA70F012F8EAF9E0485AB45A007EC7FC12257E9C18>I<EA1FE0A3EA 03C0A4485AA4380F00F8EB01FCEB070CEB0C1C381E383CEB607CEA1FC0EBE038383FF800 EA3C7C131E14040078130CA21418130F00F013F0386003E0161D7C9C1F>I<EA1FC0123F A2EA0780A4EA0F00A4121EA45AA45AA3138012F1A3EAF300127E123C0A1D7C9C14>I<3A 0F01F807E03A3F87FE1FF83A33CE1F387C3A63D80F603CD8C3F013C001E01380D803C013 00A22607801E5BA3EEF04048484814C0ED01E0EEE18016E3001E90397800FF00000C0130 137C2A127D9133>I<380F03F0383F87FC3833DC1EEA63F8EAC3F013E0EA03C0A248485A A3EC7820D80F00136014F015C014F1001EEB7F80000CEB3E001B127D9125>I<3803C0F8 380FE3FE380CFF0F3918FC0780EA30F813F01200A23801E00FA3150048485A141E6D5AEB F0F83807BFE0EB8F800180C7FCA248C8FCA3EA7FE0A212FF191A7F911F>112 D<380F07E0383F8FF83833D81CEA63F038C3E03CEBC07C1203143838078000A448C7FCA4 121E120C16127D911C>114 D<137E3801FF80EA0381380703C0380E0780EB0300EA0F80 EA07F86CB4FC6C1380EA000FEA3003127812F8EB0700EAF00EEA7FFCEA1FF012127C911C >I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmsy5 5 5 /Fw 5 64 df<B612FEA31F037A8B2D>0 D<13E0A438F0E1E0EAF8E3387EEFC0381FFF00 EA03F8A2EA1FFF387EEFC038F8E3E0EAF0E13800E000A413127B921F>3 D<EA0780EA0FC0A4EA1F80A21300A25A123EA2123CA2127C1278A2127012F0A25A0A157D 9612>48 D<D801FEEB01FC3A07FF8007FF001F9039E01FC7C03B3E1FF03E00E0277803F8 781370276001FCE0133027E0007FC01318486D5A141F6E7E814A6C13380060D939FC1330 0070D9F0FE13F03B3803E07FC3E03B1F1FC03FFFC02707FF000F1300D801FCEB03FC2D12 7B9139>I<14C01301B3A7007FB61280B7FC7E211D7B9C2D>63 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmr5 5 18 /Fx 18 118 df<B612E0A33807C003140015601570A21530A31500AEB5FCA31C1C7E9B22 >0 D<13301360EA01C0EA038013001206120E5AA25AA35AA312F0AB1270A37EA37EA27E 12067E1380EA01C0EA006013300C297B9E16>40 D<12C0126012387E120C7E1207EA0380 A2EA01C0A3EA00E0A313F0AB13E0A3EA01C0A3EA0380A2EA070012065A121C5A12605A0C 297C9E16>I<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D<EA01FCEA07FF380F07 80381E03C0383C01E0A2387800F0A300F813F8AB007813F0A3383C01E0A2381E03C0380F 07803807FF00EA01FC151D7D9B1C>48 D<1360EA01E0120F12FFA212F11201B3387FFF80 A3111C7B9B1C>I<EA03FCEA0FFF003F13C038781FE0EB07F0EAFC0114F8A213001278EA 000114F0A2EB03E0EB07C0EB0F80EB1E005B1370EA01C038038018EA0700120C00381338 387FFFF0B5FCA3151C7D9B1C>I<EA03FCEA0FFF4813C0383C07E0EA7C0314F0EA7E01EA 7C031238000013E0EB07C0EB1F803803FE005B38000F80EB03C014E0EB01F014F8123012 7812FCA2EB03F0007813E0130F383FFF80000F1300EA03FC151D7D9B1C>I<EB01C01303 1307130F131F133B1373136313C3EA01831203EA0703120E121C1238127012E0B512FEA3 380003C0A5EB7FFEA3171C7E9B1C>I<001C13E0EA1FFF14C01480EBFE0013F00018C7FC A413FCEA1BFF381F07C0381C01E0001813F01300C712F8A3127012F8A214F01301006013 E0387C07C0383FFF80000F1300EA03F8151D7D9B1C>I<EB3F80EBFFC0000313E03807C0 F0EA0F01121E383C00E01400127CEA781013FF00FB138038FF03C038FE01E038FC00F0A2 4813F8A41278A214F07EEB01E0381F03C0380FFF806C1300EA01FC151D7D9B1C>I<131F EBFF803801E7C0EA03C7138738078380EB8000A5EAFFFCA3EA0780ACEA3FF8A3121D7F9C 12>102 D<12FEA3121EA8EB1FC0EBFFE0381FC0F0EB0078A2121EAA38FFC3FFA3181D7D 9C1F>104 D<123C127EA4123CC7FCA512FEA3121EACEAFFC0A30A1D7D9C11>I<38FE1FC0 EBFFE038FFC0F0381F0078A2121EAA38FFC3FFA318127D911F>110 D<EA0FE6EA3FFEEA701EEAC00E1306A2EAF800EA7FE0EA3FF8EA07FCEA001EEAC0071303 12E012F0EAF80EEAFFFCEA87F010127D9117>115 D<1203A45AA25AA2EA3FFC12FFA2EA 0F00A71306A5EA078CEA03F8EA01F00F1A7E9916>I<38FE03F8A3381E0078AA14F8A238 0F03FF3807FF7FEA03F818127D911F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmex10 10 63 /Fy 63 126 df<1438147814F0EB01E0EB03C0A2EB0780EB0F005B131E133E5BA25BA248 5AA212035B1207A2485AA3485AA4123F90C7FCA35AA5127EA212FEB3A2127EA2127FA57E A37F121FA46C7EA36C7EA212037F1201A26C7EA2137CA27F131E131F7FEB0780EB03C0A2 EB01E0EB00F0147814381562738226>0 D<126012F012787E7EA27E6C7E7F12037F6C7E A26C7EA2137CA2137E133E133FA2EB1F80A3EB0FC0A414E01307A314F0A51303A214F8B3 A214F0A21307A514E0A3130F14C0A4EB1F80A3EB3F00A2133E137E137CA25BA2485AA248 5A5B12075B48C7FC121EA25A5A5A126015627D8226>I<B51280A400F0C7FCB3B3B3B3B2 B51280A41162708223>I<B51280A4EA0007B3B3B3B3B2B5FCA411627F8223>I<151E15FE 1407EC1FF0EC7FC0ECFF00EB01FC495A495AA2495AB3AC495A5C133F49C7FCEA01FEEA03 F8EA0FF0EA7FC000FEC8FCA2EA7FC0EA0FF0EA03F8EA01FEEA007F6D7E131F806D7EB3AC 6D7EA26D7E6D7E6DB4FCEC7FC0EC1FF0EC07FE1400151E1F62788230>8 D<127012FEEA7FC0EA1FF0EA07FCEA01FEEA007F6D7E6D7EA26D7EB3AC6D7E1303806D7E 6DB4FCEC3F80EC1FE0EC07FCEC00FEA2EC07FCEC1FE0EC3F80ECFF00EB01FC495A5C1307 495AB3AC495AA2495A49C7FCEA01FEEA07FCEA1FF0EA7FC000FEC8FC12701F62788230> I<1406140FA2141EA3143CA21478A214F0A3EB01E0A2EB03C0A2EB0780A3EB0F00A2131E A25BA35BA25BA2485AA3485AA2485AA248C7FCA3121EA25AA25AA35AA21278A37EA27EA2 7EA36C7EA26C7EA26C7EA36C7EA21378A27FA37FA27FA2EB0780A3EB03C0A2EB01E0A2EB 00F0A31478A2143CA2141EA3140FA214061862788227>I<126012F0A21278A37EA27EA2 7EA36C7EA26C7EA26C7EA36C7EA21378A27FA37FA27FA2EB0780A3EB03C0A2EB01E0A2EB 00F0A31478A2143CA2141EA3140FA2141EA3143CA21478A214F0A3EB01E0A2EB03C0A2EB 0780A3EB0F00A2131EA25BA35BA25BA2485AA3485AA2485AA248C7FCA3121EA25AA25AA3 5AA2126018627A8227>I<126012F0B3B3B01260043674811C>I<0060133000F01378B3B3 B000601330153674812E>I<151E153E157C15F8EC01F0EC03E015C0EC0780140FEC1F00 143E143C147C5C13015C495AA2495A130F5C131F91C7FC5BA2137EA25BA3485AA3485AA2 12075BA3120F5BA3121FA25BA2123FA490C8FCA25AA7127EA212FEB3A6127EA2127FA77E A27FA4121FA27FA2120FA37F1207A37F1203A26C7EA36C7EA3137EA27FA27F80130F8013 076D7EA26D7E801300147C143C143E80EC0F801407EC03C015E0EC01F0EC00F8157C153E 151E1F94718232>16 D<127012F8127C7E7E6C7E12076C7E7F6C7E6C7E1378137C7F133F 7F6D7EA26D7E80130380130180A26D7EA2147EA380A3EC1F80A215C0140FA315E01407A3 15F0A21403A215F8A41401A215FCA71400A215FEB3A615FCA21401A715F8A21403A415F0 A21407A215E0A3140F15C0A3141F1580A2EC3F00A3147EA35CA2495AA25C13035C13075C 495AA249C7FC5B133E5B137813F8485A485A5B485A120F48C8FC123E5A5A12701F947D82 32>I<160F161F163E167C16F8ED01F0ED03E016C0ED0780150FED1F00153E5DA25D4A5A 14035D14074A5A5D141F4AC7FCA2147EA25CA2495AA2495AA2495AA2130F5C131F5C133F A291C8FC5BA2137E13FEA3485AA3485AA4485AA4120F5BA3121FA35BA2123FA55BA2127F A990C9FC5AB3AC7E7FA9123FA27FA5121FA27FA3120FA37F1207A46C7EA46C7EA36C7EA3 137E137FA27F80A2131F80130F801307A26D7EA26D7EA26D7EA2147EA280A26E7E140F81 6E7E14038114016E7E157CA28181ED0F801507ED03C016E0ED01F0ED00F8167C163E161F 160F28C66E823D>I<127012F8127C7E7E6C7E6C7E12036C7E7F6C7E137C7FA27F6D7E80 1307806D7E1301806D7EA2147EA280A26E7EA26E7EA26E7EA281140381140181A2140081 A2157E157FA3ED3F80A3ED1FC0A4ED0FE0A416F01507A316F8A31503A216FCA51501A216 FEA9150016FFB3AC16FE1501A916FCA21503A516F8A21507A316F0A3150F16E0A4ED1FC0 A4ED3F80A3ED7F00A3157E15FEA25D1401A25D14035D14075DA24A5AA24A5AA24AC7FCA2 147EA25CA2495A5C1303495A5C130F5C49C8FC133EA25B5B485A5B485A1207485A48C9FC 123E5A5A127028C67E823D>I<161E167EED01FE1503ED0FF8ED1FE0ED3FC0ED7F80EDFE 004A5A4A5A14074A5A4A5AA24A5AA24A5AA44AC7FCB3B3B3A35C1301A3495AA25C130749 5A5C131F495A5C49C8FC13FEEA03FCEA07F0485AEA3FC048C9FC12FCA2127FEA3FC0EA0F E06C7EEA03FCC67E137F6D7E806D7E130F806D7E130380A26D7EA3130080B3B3B3A36E7E A46E7EA26E7EA26E7E6E7E14036E7E6E7EED7F80ED3FC0ED1FE0ED0FF8ED03FE1501ED00 7E161E27C675823E>26 D<127012FCB4FCEA7FC06C7EEA0FF86C7EEA01FE6C7E6D7E6D7E 6D7E6D7EA26D7E6D7EA2130180A36D7EB3B3B3A38081A36E7EA2141F81140F816E7E1403 6E7E6E7E157FED3F80ED1FC0ED0FF0ED03F8ED01FEED007EA2ED01FEED03F8ED0FF0ED1F C0ED3F80ED7F0015FE4A5A4A5A14074A5A5D141F5D143FA24A5AA392C7FC5CB3B3B3A349 5AA35C1303A2495A495AA2495A495A495A49C8FC485AEA07FC485AEA3FE0485A48C9FC12 FC127027C675823E>I<EE01E0A2EE03C0EE0780EE0F00161E163E5E5EA24B5A4B5A1507 5E4B5A151F93C7FC153E157E157C15FC4A5AA24A5AA24A5AA24A5A141F5D143F92C8FC5C A214FEA25C1301A2495AA213075CA2495AA3131F5CA2133F5CA2137FA291C9FC5BA35B12 01A3485AA5485AA5120F5BA4121FA35BA4123FA55BA3127FAA5BA312FFB3B2127FA37FAA 123FA37FA5121FA47FA3120FA47F1207A56C7EA56C7EA312007FA37F80A2133FA280131F A280130FA36D7EA2801303A26D7EA2130080A2147FA28081141F81140F6E7EA26E7EA26E 7EA26E7E157C157E153E8182150F6F7E8215036F7E6F7EA2167C82161E82EE0780EE03C0 EE01E0A22BF86C8242>32 D<127012F012787E7E7E7F6C7E6C7EA26C7E6C7E7F137C7F13 3F7F6D7E801307806D7EA26D7EA26D7EA2147E147F8081141F81A26E7EA2140781A26E7E A2811401A26E7EA38181A282153FA282A2151F82A3150F82A36F7EA56F7EA5821501A482 A381A41780A5167FA317C0AA163FA317E0B3B217C0A3167FAA1780A316FFA51700A45DA3 5EA415035EA54B5AA54B5AA35E151FA35E153FA25EA2157F93C7FCA25D5DA34A5AA21403 5DA24A5AA25D140FA24A5AA25D143F92C8FC5C147E5CA2495AA2495AA2495A5C130F5C49 C9FC5B133E5B13FC5B485A485AA2485A485A90CAFC121E5A5A5A12702BF87E8242>I<EE 03C0160F161F167FEEFF00ED03FEED07F84B5A4B5A4B5A4B5A15FF93C7FC4A5A14035D14 074A5AA25D141FA24A5AA44A5AB3B3B3B25D14FFA392C8FC5BA25C13035C495AA2495A5C 131F495A5C49C9FC13FE485A485A485AEA1FC0485A48CAFC12FCA2127F6C7E6C7EEA07F0 6C7E6C7E6C7E137F6D7E806D7E130F806D7EA26D7E80130180A27F81A3147F81B3B3B3B2 6E7EA46E7EA2140F81A26E7E14038114016E7E82157F6F7E6F7E6F7E6F7EED03FEED00FF EE7FC0161F160F16032AF8748243>40 D<180E181FA3183EA3187CA318F8A2EF01F0A3EF 03E0A3EF07C0A3EF0F80A3EF1F00A2173EA35FA35FA34C5AA34C5AA24C5AA34C5AA34CC7 FCA3163EA35EA25EA34B5AA34B5AA34B5AA34B5AA24BC8FCA3153EA35DA35DA34A5AA24A 5AA34A5AA34A5AA34AC9FCA3143EA25CA35CA3495AA3495AA3495AA2495AA349CAFCA313 3EA35BA35BA2485AA3485AA3485AA3485AA348CBFCA2123EA35AA35AA3127038947B8243 >46 D<177CA217F8EE01F01603EE07E017C0EE0F80161FEE3F00A2167E5EA24B5A15035E 4B5A150F5E151F5E153F93C7FC5D15FEA24A5AA24A5AA214075D140F5D141FA25D143F5D 147FA292C8FC5CA2495AA3495AA3495AA3495AA3131F5CA2133FA25CA2137FA25C13FFA4 4890C9FCA5485AA512075BA4120FA25BA5121FA35BA5123FA65BA3127FAD5BA212FFB3A6 2E95688149>48 D<127812F8127C7E123F6C7E120F6C7E7F6C7EA26C7E6C7EA2137E137F 7F6D7E80130F801307801303806D7EA26D7EA2147FA281143F81141F81A2140F81140781 A2140381A26E7EA36E7EA36F7EA36F7EA382151FA282A2150FA282A2150782A46F7EA56F 7EA5178081A417C0A2167FA517E0A3163FA517F0A6161FA317F8AD160FA217FCB3A62E95 7E8149>I<B612F0A748C8FCB3B3B3B3B3B3B3AF1C94668137>I<B612F0A7C71207B3B3B3 B3B3B3B3AF1C94808137>I<12FEB3B3B3B3B3B3B3AFB612F0A71C94668237>I<EC07F0B3 B3B3B3B3B3B3AFB6FCA71C94808237>I<12FEB3B3B00734668037>I<12FEB3B3B007346B 8037>I<EC01F01407140F143F147FECFFC0491380903803FE00495A495A495A495A495A 13FF485B91C7FC5A5B1207485AA2485AA25B123FA25B127FA4485AB3B3A81C4B607E4A> I<EAFFC0B3B3A86C7EA4123F7FA2121F7FA26C7EA26C7E12037F7E806C7F137F6D7E6D7E 6D7E6D7E6D7E903801FF806D13C0EC7FF0143F140F140714011C4B60804A>58 D<EC1FF8B3B3A715F0143FA415E0147FA215C014FF1580A25B15005B5C495AA2495A5C49 5A133F495A5C49C7FC485A485AEA0FF8EA1FE0485AB45A48C8FC5A7E6C7EEA3FC06C7EEA 0FF8EA03FC6C7E6C7E6D7E806D7E131F6D7E806D7EA26D7E807F15807FA215C0147F15E0 A2143F15F0A4141F15F8B3B3A71D9773804A>60 D<EAFFC0B3A90A1B60804A>62 D<12F0B3B3B00434678037>I<EAFFC0B3A6127FA27FAD123FA37FA6121FA57FA3120FA5 7FA21207A47F1203A56C7EA56C7FA4137F80A2133FA280A2131FA280130FA36D7EA36D7E A36D7EA36D7EA28081A2143F81141F81A2140F811407811403A26E7EA26E7EA2157F8182 151F82150F8215076F7E8215016F7EA2167E82A2EE1F80160FEE07C017E0EE03F01601EE 00F8177CA22E95688349>I<EE0FFCB3A617F8A2161FAD17F0A3163FA617E0A5167FA317 C0A516FFA21780A45D1700A54B5AA54B5AA45E150FA25EA2151FA25EA2153F5EA34B5AA3 4BC7FCA34A5AA34A5AA25D1407A25D140F5D141FA25D143F5D147F92C8FCA214FEA2495A A2495A5C13075C130F5C131F5C49C9FC5B137E5BA2485A485AA2485A5B485A121F48CAFC 123E5A5A12782E957E8349>I<EAFFC0B3B3B00A34688049>I<EAFFC0B3B3B00A345A8049 >I<ED0180ED03C01507A2ED0F80A2ED1F00A3153EA25DA35DA24A5AA34A5AA24A5AA34A 5AA24AC7FCA3143EA25CA35CA2495AA2495AA3495AA2495AA349C8FCA2133EA35BA25BA3 485AA2485AA3485AA2485AA348C9FCA2123EA35AA25AA3127CA27EA37EA26C7EA36C7EA2 6C7EA36C7EA26C7EA3137CA27FA37FA26D7EA36D7EA26D7EA36D7EA26D7EA2147CA380A2 80A36E7EA26E7EA36E7EA26E7EA36E7EA2157CA381A281A3ED0F80A2ED07C0A21503ED01 802295778233>I<127012F07EA2127CA27EA37EA26C7EA36C7EA26C7EA36C7EA26C7EA3 137CA27FA37FA26D7EA36D7EA26D7EA26D7EA36D7EA2147CA380A280A36E7EA26E7EA36E 7EA26E7EA36E7EA2157CA381A281A3ED0F80A2ED07C0A3ED0F80A2ED1F00A3153EA25DA3 5DA24A5AA34A5AA24A5AA34A5AA24AC7FCA3143EA25CA35CA2495AA3495AA2495AA2495A A349C8FCA2133EA35BA25BA3485AA2485AA3485AA2485AA348C9FCA2123EA35AA25AA25A 12702295798233>I<EE3FFE0303B512E0031F14FC92B77E020316E04A82021F16FC4AD9 E7F37F913CFFFE07F03FFF804901F002077F49018002007F4901006F7ED90FFCEE1FF849 48707ED93FE0EE03FE4948707E4948707F91C7167F4848727E4848727EA24848727E4918 07000F86491803001F86491801003F86491800A2007F8690C883A4481B80481A3FA2BDFC A748C8D807F0C8123FA26C1A7F6C1B00A46D61003F62A26D1801001F626D1803000F626D 18070007626D180F6C6C4E5AA26C6C4E5A6C6C4E5A6E17FF6D6C4C90C7FC6D6C4C5AD91F F8EE0FFC6D6C4C5A6DB4EE7FF06D01804B5A6D01F002075B6D01FE023F5B91293FFFE7F3 FFFEC8FC6E90B65A020716F06E5E02001680031F02FCC9FC030314E09226003FFECAFC51 537B7F5C>76 D<95387FFF80050FB512FC057FECFF800403B712F0041F16FE047F707E4B B912E0030718F84BDAC7F880033FD9F807010713FF92B500C0020014C04AD9FE00031F7F 4A01F004037F4A01C004007F021F49EF7FFE4A48C7EE1FFFDA7FF806077F4A48727F4949 727F4949727F92C8173F4948747E4948747E4948747E4A1A03013F884948747F4A864948 757EA24890C9727E491C1F00038A491C0F00078A491C07000F8A491C03001F8A491C01A2 003F8A4988A3007F1F80491D7FA400FF1FC090CA193FA390C0FCA890CAD807F8CA123FA3 6D1D7F007F1F80A46D1DFF003F1F00A36D64001F66A26D1C03000F666D1C070007666D1C 0F0003666D1C1F0001666D1C3F6C6D515AA26D6C515A6E626D6C5090C7FC011F646E1A07 6D6C505A6D6C505A6D6C505A03C019FF6D6D4E5B6D6D4E5B6E6C4E5BDA3FFE061F90C8FC 6E6C6CEF7FFE02076DEFFFF86E01F004035B6E01FE041F5B6ED9FFC092B55A033F01F802 0791C9FC030FD9FFC790B512FC6F91B75A030118E06F6C1780041F4CCAFC040316F0DC00 7F1580050F02FCCBFCDD007F138072747B7F7D>I<EE3FFE0303B512E0031F14FC92B77E 020316E04A82021F16FC4AD9E0037FDAFFFEC7383FFF804901F002077F49018002007F49 90C96C7ED90FFCEE1FF84948707ED93FE0EE03FE496C4C7E496C4C7F6E5E486D4C7F2603 FDFF93387FDFE001FC6DEDFF9F2607F87FEF0FF0496C6C913801FE07000F6D6CDA03FC7F 01E06D02071303001F6D6CDA0FF87F496C6C91381FF001003F6D6CDA3FE07F496C6C9138 7FC0006E6D495A007F6E6C4890C77E90C76C6C4848806F6C485A6F6C485A6F6C485A486E 6C48481580486E6C4848143F6F495A7090C8FC705A705A705A4C7E4C7E4C7E93B57E4B6D 7E6C4A486C6C147F6C4A486C6C15004B486C7E4B486C7E4B486C7E6D49486C6C5C003F4A 486C6D5B4A90C76C7E6D484891383FE001001F4948DA1FF05B6D484891380FF803000F49 48DA07FC5B01F0490203130700074948DA01FE5B6D4848913800FF0F2603FCFFEF9FE001 FD90C9EA7FDF6CB44870B45A6C49705B4A826D487090C7FC6D48705AD91FF8EE0FFC6D6C 4C5A6DB4EE7FF06D6D4B5A6D01F002075B6D01FE023F5B91293FFFE003FFFEC8FC6E90B6 5A020716F06E5E02001680031F02FCC9FC030314E09226003FFECAFC51537B7F5C>I<95 387FFF80050FB512FC057FECFF800403B712F0041F16FE047F707E4BB912E0030718F84B DAC00080033F01F8C7000713FF92B500C0020014C04A49C9001F7F4A01F004037F4A01C0 04007F021F49EF7FFE4A48CBEA1FFFDA7FF806077F4A48727F4949727F4949727FA2496D 4E7F496D4E7F496D4E7F6F60496D4E7F90267FE7FFDE3FF97F02C36DEF7FF0D9FF816D4E 6C7E0280F1FFC048496C6C4C496C7E496D6C4CEB001F00036E6C4C4880496D6C040F140F 00076F4C4880496D6C4C481307000F6E6D4B4880496D6D4B481303001F6E6D4A4980496E 6C4A491301706C4A90C7FC003F6F6C4A4881496E6C4A4880706C4A5A706D495A007F6F6D 49481680496E6D4849157F716C485B716C4890C9FC716C485A00FF706C484817C090C96C 6C4848163F71EBFFF0715C715C725B7290CAFC725AA24E7E4E7F95B57E4D804D804DEB3F F86D4B486C6C167F007F4C486C6C17804D486C7E4D486C7F4D486C7F6D4A496C6D15FF00 3F4B496D6C16004C90C76C7E4C486E7E6D4A486E6C5C001F4B486E6C5D4C486E7F6D4A48 6E6D1303000F4A496E6D5C6D49496F6C130700074A90C96C6C5C6D4948706C130F00034B 706C5C6D49480407141F00014A48706C5C6D494870EB803F6C6D4848706D485A0281F17F E0D97FC34972485A02E790CBEA3FF96DB44872B5C7FC6D49725B4B846D49725B6D49725B 6D49725BA26D6D4E5B6D6D4E5B6E6C4E5BDA3FFE061F90C8FC6E6C6CEF7FFE02076DEFFF F86E01F004035B6E01FE041F5B6ED9FFC092B55A033F01F8020791C9FC030FD9FFC090B5 12FC6F91B75A030118E06F6C1780041F4CCAFC040316F0DC007F1580050F02FCCBFCDD00 7F138072747B7F7D>I<007FBA12FCBB7EA3D87FE0C9001F7F003FEF007F6C6C170F6D05 031380000F18006C6C183F6C6CF01FC06D180F6C19076C6DEF03E06E17016D6C1700013F 19F06D6C1870806D6C1838010719006D7E806D7F7F6E7E81143F6E7E816E7E14076E7E81 6E7F806F7E826F7E151F6F7E826F5A6F5A6F5A4B5A15074B5A4BCBFC151E5D157C5D4A5A 5D4A5A4A5A140F4ACCFC143E143C5C02F8183849481870495A4A18F0494818E049CB1201 491803013EF007C049180F0178181F49F03F80000119FF48485F4848050F130049177F48 CA001FB5FC48BA5A5A5ABB5A7E4D537B7F58>I<BB12E0A4000F0180C8383FFE00000318 F86C606C60B3B3B3AF486D4B7E486D4B7E000F01F8020313FEB66C013FEBFFE0A443537B 7F4E>I<167E923801FF804B13E0923807C1F092380F807892381F00F892383F01FCED3E 03157EA215FCEE01F8EE00F002011400A35D1403A81407A35DA5140FA84A5AAA4A5AA85D A5147FA392C8FCA8147E14FEA3003C5B127EB4FC495AA25CEAFE03007C5B387807C0383E 0F806CB4C9FCEA07FEEA01F82E5C7C7F27>I<00381707007CEF0F8000FEEF1FC0B3B3B3 A36C173F6C1880A26D167F003F1800A26D5E6C6C4B5AA26C6C4B5A6C6C4B5A6D150F6C6C 4B5A6C6C6CEC7FE06C6D4A5AD97FF801075B6DB4013F90C7FC6D90B55A6D5D010315F06D 5D6D6C1480020F01FCC8FC020113E03A537B7F45>I<913801FFE0020F13FC027FEBFF80 49B612E04981010F15FC4981499038003FFFD97FF801077FD9FFC001007F48496E7E4848 C8EA1FF048486F7E49150748486F7E48486F7EA248486F7E4982A2007F188090CA123FA2 4818C048171FB3B3B3A3007CEF0F800038EF07003A537B7F45>I<007FBE12E0BF7EA38A 6C90CCFC6C6D0600806C1B0F6E07007F6C6D1A3F6C6D1A0F6CF403FF6E1A006C6DF37F80 6C6D1B1F017F1C0F6EF307C06D6D1A036D6DF201E0A26D6DF200F06D6D1B707F6F1B386D 6D1B006D7F147F816E7F6E7F80826E7F6E7F80826E7F6F7E81836F7F6F7FA26F7F6F7F81 836F7F707E8284707F707F8284707F707F82A2715A173F715A60715A4D5A4DCDFC177E17 FE5F4C5A4C5A4C5A4C5A161F5F4CCEFC167E5E4B5A4B5A15075E4B5A4B5A4BCFFC157E5D 14015D4A481B384A481B704A5A4A481BF04ACF12E04A1B01027E1B034AF307C049481B0F 4948F31F8049481B3F010F1CFF4A50130049481A0749CE121F017EF3FFFE491A07484897 B5FC000397B65A90BEFC48655A5A48655ABFFC6C656D747B7F78>88 D<BE12FEA5000302E0C9000F1480C66CF2FC00011F1AF06D626D62A26D62B3B3B3B3B3A4 496D4C7FA2496D4C7F496D4C7F017F01FF4BB512FC0003B600E0020FECFF80B8D88003B7 12FEA55F747B7F6A>I<F107C0F11FF0F13FF8F17C3CF1F81E953801F03E953803E0FF18 0719C1180FA295381F80FEA295383F00381A0060A2187E18FEA2601701A44D5AA3170760 A3170FA260A2171FA360173FA44D5AA417FFA295C8FCA35EA35F1603A416075FA4160FA2 5FA2161FA35FA2163FA45F167FA45F16FFA45F5DA494C9FC5DA45E1507A45E150FA45EA2 151FA35EA2153FA25EA4157F5EA45E15FFA393CAFCA35CA25DA44A5AA45D1407A35DA214 0FA25DA3141F5DA34A5AA492CBFC5CA2147E14FEA25C121C387F01F8A238FF83F0A25C13 075C387C0F80D8781FCCFCEA3C3EEA1FFC6C5AEA03E048B87B7F2E>I<003C1A1E007E1A 3FB4F27F80B3B3B3B3A76D19FF007F1B00A46D60003F62A26D1803001F626D1807000F62 6D180F6C6C4E5A6D183F0003626D187F6C6D4D5A6C6D4C5B6D6C4C90C7FC02F8160F6D6C 4C5A6DB4EE7FFC6D6D4B5A6D01F002075B6D01FE023F5B01009026FFE003B512806E90B7 C8FC021F16FC6E5E020316E002001680031F02FCC9FC030314E09226003FFECAFC51747B 7F5C>I<D91FC01420D9FFF81470000301FF14F8489138C003F0489138FC1FC0D81FC190 B51280267E001F140000F801075B0070010013F80020EC1FC02D0A80BB2E>101 D<912607FF80178091B500F81607010702FFEE1FC0013F03F0ED7F0090B700FEEC07FC00 03D98001D9FFE0EB7FF0D80FF8C7001F90B612C0D83F80020393C7FC00FEC9003F14F800 78040714C000409326007FF8C8FC520B80BD53>I<ED03C0151F157F15FF02031300EC07 FCEC0FF0EC1FC04A5A4AC7FC14FE495AA2495AA2495AB3B3A8495AA3495AA2495A49C8FC 137E5B485AEA07F0485AEA3FC048C9FC12FCA2127FEA3FC0EA0FE06C7EEA01F86C7E137E 137F6D7E6D7EA26D7EA36D7EB3B3A86D7EA26D7EA26D7E147F6E7E6E7EEC0FF0EC07FC6E B4FC020013C0157F151F15032294768237>110 D<127012FCB4FCEA7FC06C7EEA0FF8EA 03FC6C7E6C7E6D7E133F6D7EA26D7EA26D7EB3B3A86D7EA36D7E130080147F6E7E6E7E6E 7E6E7EEC03FCEC00FFED3FC0151FA2153FEDFF00EC03FCEC07F04A5A4A5A4A5A4AC7FC14 FE5C1301495AA3495AB3B3A8495AA2495AA2495A137F49C8FC485A485AEA0FF8EA3FE048 5A48C9FC12FC12702294768237>I<1B301B78A21BF0A2F201E0A2F203C0A2F20780A2F2 0F00A21A1EA262A262A262A24F5AA24F5AA24F5AA34FC7FCA2191EA261A261A261A24E5A A24E5AA24E5AA24EC8FCA2181EA260A201085E1318017C5E13FC00014C5AEA07FE000F4C 5A487E007C4C5A486C7E00704CC9FC12006D6C141EA26D6C5CA26D6C5CA25F6D7E4C5A6D 7E4C5A6D7EA24C5A6D7E4CCAFCEC7F80161EEC3FC05EA26E6C5AA26E6C5AA2913807F9E0 A2EDFBC0EC03FF5E8093CBFC805DA2157CA215384D64788353>I<1B301B78A31BF0A3F2 01E0A3F203C0A3F20780A3F20F00A31A1EA362A362A362A44F5AA34F5AA34F5AA34FC7FC A3191EA361A361A361A34E5AA34E5AA44E5AA34EC8FCA3181EA360A301085E1318133801 3C5E137C13FC00014C5A487E1207000F4C5AEA0EFF121E484C5A5A486C7E126000004CC9 FC6D7EA2171EA26D7E5FA26D7E5FA36D6C5CA36D6C495AA34C5A6D7EA24C5A6D7EA24CCA FCA2EC7F80161EA2EC3FC0A25EA2EC1FE05EA2EC0FF05EA3913807F9E0A36EB45AA35E80 A293CBFC80A25DA2157E157CA2153C15384D96788353>I<14F0B3AD6C151000E0157000 F8EC01F000FE1407D87F80EB1FE0D81FC0EB3F80D807F0EBFE003901F8F1F83900FCF3F0 90383EF7C06DB45A6D90C7FC6D5A6D5A6D5A6D5AA214602431777F37>121 D<EE7F80ED0FFF157F4AB5FC140F143F5C49B6FC5B130F4991C7FC4913E04990C8FCEBFF F84813E04813804848C9FCEA0FF85B485A485A485A90CAFC12FE5AA21278291B838925> I<B4FC13F813FF14C014F814FE8015C08115F8C66C7F01037F9038007FFF020F7F02037F 02007FED3FF0ED0FF815076F7E6F7E6F7E82EE3F80161FA2EE0F00291B818925>I<1278 12FCA27E127F7F6C7E6C7E6C7E7FEA07FE6C6C7E6C13E06C13F86DB4FC6D13E06D13FF6D ECFF8013037F6D7E80140F14016E7E150FED007F291B839A25>I<160FEE1F80A2163FEE 7F005E4B5A4B5A4B5A150FED3FF0EDFFE002035B020F5B027F90C7FC903803FFFE017F5B B65A15E05D92C8FC5C14F814C091C9FC13F890CAFC291B819A25>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz eufm10 10 11 /Fz 11 88 df<1840DAFF8014C0010F01F0EB0380013F01FCEB070090B56C130E2701E0 3FFF133E2603800FEB807E260F0003EB81FC000E6D13C1487F003C147F48EC3FE1A200F8 141FA27E6C140F7E7F6C7EA26C6C14C17F121F6C7E000715810003141FA26C481401153F 49133E4848137E49137C48C712F8000EEB01F0003CEB03E00010EB0780C7EA0F00141E5C 5C5C495A494880EB0F80EB1FE0D93FF8EB07FFD9FFFC131F486D013D1380486D13F848DA 81E013C3D81F039039C7C07FCFD83C00D9FF8013FE007090267FFE0013F800406D48EB3F F0C76C4814E0DA0FE0EB1F806E4814004B130E0202C71208383C7EB93C>65 D<EB03FCD90FFFEC0FE0013F01C0EB7FF890B56C48487E486E5A489139F80F87FED807C0 9038FC1E013B0F003FFE7800001E90380FFFF0003E6D81486D497F5E8000FC7F6C188019 836C027FEC3FFF6D17FE19F86C6CEE1FE06D1780003F023F15006D167E001FEE01F86C6C 4B5AEF0FC00007EE3F809338C1FFE06C4802CF13FE93B612804917C04848DAC00F13E0DC 800113F048C7EC007F001EEF1FF848170F0030EF07FCC848C7FC1803157E03FE14015D4A 5AEC03E04A4815F84A5A021EC8FC4A16F002FF15034901F815E0010701FFEC07C04902C0 1480013F02F8EB0F0090B600FE131C2601FC1F9038FFE0782607E001ECFFF048C7003F14 C0000E02075CC949C7FCEE0FF8403C7CB949>I<1604163C16FE1507151FED79FFEC01F1 EC63E0D901FFEB7F84010715FCD91F1FEB3FF8D9783FEB1FE001F0EC0F00D803E091C7FC 3807C01F120FD81F807FA2383F000FA281127E1407A312FEA25DA34A5A5D023FC8FC6C13 7EEB01F8EB03C090CAFC6C7EA37F123F7F7F121F7F6C7E7F6C6C6C14046E141C6C01F014 786C01FEEB01F06C9039FFC007C0013F90B512006D5C010714F801015CD9003F13C0DA01 FCC7FC2E3B7CB833>I<913801FFF8021FEBFFE049B612FC010715FF011F16C04916F049 48C614FCD9FFC001077F4848C87FD803F8031F1380D807E08149010C010313C04848013C 6D13E090C7487F001E4948147F4A48EC3FF0484948141F0038495A021FED0FF84AC8FC5A 4A150781006013FF6F1403A2C77F6E7E143F6E7E6F15F0140F6E7E1403A2140119E05D4B 140719C04A5A15C04A48EC0F80021FC8FC147CD903F0ED1F00D9FFE0151E4801FE5D48D9 FFC05C4802F85C4802FF495A489238E007C0D83E009138FC1F800038011F90B5C7FC4801 0114FC48D9003F5B4802075B48020113C0C9003FC8FC3D3B7FB845>I<EB0FE0EB3FFC90 B514C0489138C00380489138F00700EDFC1E48ECFFFC260FE07F13F0D9800F5B261E0001 1383001C9038007F0E0018EC0C1C0038EC003C00305D007015F800604A5A1503481407C8 120FA3151FA2150FA682A31507AA5EA34B5A133001705C13F0486C495A000392C7FC0007 143E486C5B6D5B391FFF83E0003BEBFFC0007091C8FC38407FFC38001FC02A3B7EB82E> 73 D<EE7FE0923807FFFC031F13FF037F1480912601F80113C0913903C0003F4AC7120F 020E14074A14035C4A140114F01700494815801303041F13000107ECFFC04B7F010F1307 031F7FED3E0FED78079139F0E003F89138F1C001902607F3807F02FFC77E4AECFFE04A14 7F6D481500173C6EEB01F00101EC0780043EC7FC4BB4FCDAFE0F13C00100133F91B67EED E03F9138FE000F027E801607A2160383A2147CA2000401786D7E000C1370001E49ECFE01 4AECFF07D83F0116FED87F83C813FCD8FFFE16F86C48ED7FF06C4816E0D81FE0ED3F80D8 0780ED1E00383A7FB737>75 D<D901FEEF0F8090260FFF80EE7FE0013F01E0903A1FC001 FFF090B56C9038FFF00348DAFC01D9F80F7FD803C00207EC1E0F2807001FFE0F9038FC38 07000E903D0FFF1E0FFE7003FC001E6DD9380713E0486D903BF003FFC001FE007C4B6C01 80EBFFB86E01C017F07190C713E000FE6D1980F37F006C027F6F137C6D19386D047F5C00 7F626D18016C6C013F16036C6C4E5A1A0F6C7E0007191F12036312014917004C153F5B48 5A485A485A001EC7157E4892C7FC00105CC8FC187C157E18FC605DA24B4A80020114014B 5C4A484A803B01E007C00F032A03F80F801FC780ECF0602907FE1F003FF7C7EBF9E0260F FF3ED97FFE91381FFFC04801FC494816004801F04A6E5A4A4817F829707FC003C7F06E5A 26E03F80D901E05D29C00F000200C0EC0380D8000690CA6CC7FC553B7EB857>77 D<D903FCEC01FC90261FFF80EB0FFE017F6DEB3FFF90B500F0491380484BB5FCD803C090 38F803C03D07003FFC07807FC0000E011FEB1E0048902607FE3C133F003C153800380103 5B007801015B03FF15E000F86D4914F36C4B14FF6CEF1FFC6C92C713F86D6DEC0FE0007F EF07C06DEE03806C6CEE07006D161E6C6C163E000F173C6C6C167C18FC120300014C7E5B 4982A248486F1380484817C048C7157F001EEF3FE00038171FC816F0180F18071803A218 01A21800DA3FFF15E091B5FC13034902C014C04902F8130104FF1480D91F039138C00700 90263C007FEBFC0F0138EDFFFE495E01606E13F001E0020F5B4902015B49DA003EC7FC90 C791C8FCA65D82A54BC9FC4A5A15F05D1580404B7EB845>80 D<151EDBFF805B020701E0 1480023F01F8140091B500FE5B903A01F07FFF80902607800FEBE00690260F0003EBF81E 013E0100EBFFFC0178023F5B496E5B000103075B4902015B48486E90C7FC0007161C260F 800691C8FC141E381F007814F848485AEA3E03A2EA7E07EF7F806E90381FFFF0DAFE07B5 7E00FE90B77E6D82846DECC01F9028007FE00003138091C87E6C82A2187FA26D163FA212 7F6D1700A27F003F173E7F001F177E6D167C6D16786C6C16F86C6C5E6E4A5A6C6D4A5A6C 01F04A5A6C6D021FC7FC6DB4147C6D9038E003F86D90B512E0010792C8FC010014F8020F 1380393B7BB845>83 D<14100238154002F0EC01C0D901C04A5A4948140F01074BC7FC49 C8127F011E15FE013E14035B13FC5F1201A212036D1401A37F7E807E80137F80133F131F 80130FA26D7EA21303A21301A45CA25C495A4A80A249C7FC130E494A7E495C494A7FD801 FC141E2607FF8090393C7FC18002F0017813FF4801FC9039F03FFE00489026FF83E05B48 DAFFC05BD8603F9138801FE0D8C00702005BD88001496D5AC7D83FFC010EC7FCDA07F013 04393C82B936>85 D<D901FE0218147E90260FFFC0013C49B47E013F6D017E4913E090B5 00F801FE497F48DB03FF011F13FC2803E07FFC076D48EBFFF82807801FFE0F9026C0783F 13F0260E000F011E9026E0F00F13E0001E902A03FF3C3FF1C00313C0486D9028781FFB80 01138004F001FF9038007F00007C6D496C90C7121E4C6C5A00FCDA7F80495C6C0403157C 637E6D013F010114016C6C18037F6C7E6D1500121F6C6C91C7FC12071203A212015B4913 3EA24848017E835BD80F80017C5C001EC7FC121CC8481301A24A484A1301A24A5A4B495A 4A485C4AC75B023C4AC7485A5CD903F0140ED93FE04A5D4948143C2601FFF0DA7FC0495A 486D902601FFF85C4801FE010701FE495A489026FFC03F9026FFC01EC7FCD81F0790B7EA F07CD83C01DAF80FEBFFF8486C6CD9E0015C486D902680007F13C00060010F90C7001F5B C7D801FCDA03FEC8FC553B7EB857>87 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA eusm10 10 15 /FA 15 90 df<EF03F0173FEFFFE004035BEE07CFEE0F0F161E161C16381678167016F0 16E01501ED03C0A215071680A2ED0F00A25D151E153EA25DA35DA214015DA2140392B6FC 5CA391390F80000FA24AC7FCA2141E143E143C027C8100381378127E00FE5B4A160C0101 EEF03C4948EDF8704AEDFFE0D87E07C814C0D87F1F6F1380D83FFCEEFE006C485ED807E0 ED01F03E3A81B840>65 D<1540EC03E091381FFFE049B67E010F15F8903A3FC7E01FFCD9 7E07EB01FED801F8EC007FD803E0ED3F80EA07C0D80F80ED1FC0EA1F0048160F123E127E A212FE1880A2EF1F007E173E007E163C5F00384B5AC74A5AEE1F8092B5C7FC17F89238FE 3FFFDBE0001380EF3FC04BEB1FE0EF0FF0EF07F8170318FC4B1301140FA392C7FCA3021E 15F8A2021CEC03F0143C023815E04A140702F0EC0FC04948EC1F80D907C0EC3E00010F15 FCD97F80EB0FF80007B712E017804803FCC7FC16C0363B7DBA3F>I<1560EC01E091387F FFFC0107B612E0011F15FC903B7F83F01FFF802601FC0301007FD803F0ED1FF0D807C0ED 07F8D80F806F7ED81F00ED00FE003E177F007E1880007C173FF01FC012FC6CEF0FE0A218 07A219F0A2007C17031238C7FCA819E05DA21807A219C0A24BEC0F80140719004B5C181E 183E92C8123C4A5D020E15F84A5D023C4A5A4AEC07C04A4A5A4948023EC7FCD907C0EB01 FCD93F80EB1FF00003B75A4893C8FC16FC4815C03C3B7DBA45>68 D<DB03801304DA7FFF140C0103B51418011F14F8903B7F803FFFC0309026FC001FEBFFF0 D801F002BF13E0D807E01483499138807FC04848ED0780001F93C7FC90C7FC5AA35AA47F A36CC7FC121EC8FCA349B612F05F490280C7FC90C7121FAA93C8FCA4151E153EA2153C00 7C147C00FC5C6C5C4A5A6C495A6D485A267FC01FC9FC383FF07C381FFFF06C13C0D801FE CAFC363B7CB938>70 D<F101F891263FFFC0EC07FC0103B548141F011F49C8EA3FFE9026 7FC0FCED7FFCD9FC0016F8D801F0EFE038D803C0040113000007604848160390C75E4817 0796C7FC485FA3606D161EA2183E121F90C7FC120EC8157E187CA292B612FCA303FCC712 7C18FCA75D1401A35DA314035D844A48147EA2003C4A1620007E49C81430021E037F1360 B4013E6F13E06C017817C0D9C1F092381F81806CB44816C3028092380FFF00000F90C9EA 07FCD803F8EE01F0473B7EB94B>72 D<ECFFFE011F13FC90B512F03903FE07C0EA07E0EA 0F80EA1F00123E127E127C12FCA37EA3127E127C1238C7FCB3A21580A2140FA21500A314 1EA20078131C00F8133C6C5B14705C38FE03C0387F07806CB4C7FCEA1FFCEA07F01F3A7E B824>I<15FF020713E0021F7F91383E1FF8EC78074A6C7E1301ECE0011303A26F5A4948 137093C8FCA3130FAB5CA2131FA691CAFCA2EF0180133EA2013C16C05B5B485AEA03F848 B4FC4813E04801F8EC03804813FE6D6C6C1307D87C0701F0130FD8780001FEEB1F000070 90393FFFC07F020FEBFFFE48130302005C033F5B030F13E0C800035BDB007FC7FC323B7E B93B>76 D<ED1FC0EDFFF8D9040313FE90390E07E07F903A3C0F801F80903A781F000FC0 D9F03EEB07E03801E03C027CEB03F03803C0780007ED01F8EB80F8120F496CEB00FC5A80 003E7F81177E007E137FA2007C6DC7FC140C91C8FC12FCAA177C17FCA27E127E17F81601 A2007F16F07E16036C6C15E016076D15C0000F150F6C6C15806DEC1F006C6C143E6C6C5C 6C6C5C90397F8003F090391FE01FC06DB5C7FC010313FC9038007FC02F3B7BB83B>79 D<1530EC01F091B512C0010F14FE013FECFFC0903AFF03E07FF0D803F8EC07FCD807C0EC 01FED80F806E7ED81F0081001EEE3F80123E48EE1FC0A2170F12FCA47EA21880A2007C16 1F00381700C7FC173EA25F5F4C5A4C5AEE0F80047EC7FCEDFFF8168003E0C8FCAC1407D8 07805BEA0FC0121F4A5AA24AC9FC5C380FE07E6D5A3807FFF06C5BC66CCAFC323D7DBB37 >I<EC7FF00103B5FC011F14C090397FC01FF09039FE0003F8D801F86D7E48486D7E4848 147E484880A24848EC1F80A248C8EA0FC0A2127EA3EE07E012FE5AAA127C17C0127E160F 123E003F16807E7F000FED1F006C7E6C6C141E6C6C143ED800FC143CD97F80137C010F14 7890C85A4B5A4B5A923807801092381F0018013F01FC13300003B512F04802C013704802 F013E04814FE01079038FFC3C02638003FEBFF80003013039139003FFE0048EC01FC92C8 FC2D3C7AB937>I<1518157891383FFFFE49B612E0010F15FC903A3FE1F807FED97E01EB 007F01F8ED3F80D803F0ED1FC0D807E0150F13C0D80F8016E0D81F001507A35AA31380A2 18C0001F160F130000061780C7151F1800173E5F5FEE03F0EE0FC092B5C7FC16FCA2EDF8 7C167EA2163EEDF03F8283160F15E0830203130703C07F1603834A486C7EA20078D90F00 EBFC01007C923800FE0300FE011E14FF4AEC7F8E6C49EC3FFE267F81F015FCD9FFE0EC1F F86C0180EC0FF06C48C8EA07C0D807F092C7FC383C7EBA3C>I<1810D907FC153890263F FFC0143090B500FC147048DAFFC013E02607807F9038FE03C0260E000790B5FC48D9007F 14806F1400486F5A0078ED03F893C8FCA212F8A27E7E7EA27EA2123E121CC8FCB1153EA5 5DA2137001FC1378120115F85D4A5AA24A5A3900FF0F80D97FFEC9FC6D5AEB0FE0353B7E B934>84 D<D803F01680D80FFC1507D83FFEED0F00173E486C15FC00701501486C4A5A6D 7E00C05EA2131FA34C5AA41200A25F91C7FCA25BA3160F133EA2137EA3137CA213FCA45B A21201A7161F4C7E6D147F04FF130804EF130C6D903901CFF018000091390787F8386D90 390F07FC70913A801E03FFF090267FE0FC14E090273FFFF00113C06D496C13806D0180EB 7E00D901FCC9FC363C7DB93B>I<1A3FD801F0030EECFF80D807FC033C010313E0486C03 7C4913F0486CEF0E1F486D93381C07F8F1180326787FC093381001FCD8703F17006D6C14 FCD8E00F1800130700C07F010314018300006D18F813011603A2010003BFEC01F06E1307 053F15E0717E040E1503027C18C04C6C7EF20780027EEB380772140004305D023E902670 03F0130E0460151E9326E001F8131C1A3C4B486C6C1338A24B485D187E03075E93C77E4B 4B5A150EDA1E1E91381F8380A2DA1F3C0387C7FCF00FC74B15CE19DE4B140719FC5D61A2 4B6E5AA25D6192C8FC021E1680021092C9FC4E3C7EBA52>87 D<01F01406D803F8141E48 6C147E120F121F123F127812E012C01200AD16FE16FC150115031507ED0F7C6D133E017F 137CECC1F090383FFFE06D138090380FFE00EB03F090C8FCED01FC1507ED0F7C153C15F0 EC01C0EC0780EC0F00141C5C5C4A13F81301495A495A010F14F091C7FC491301013E14E0 A249130316C0A201FCEB078016005D151E5D6D5B017F5BEC83E090383FFF806D48C7FCEB 07F027467EB931>89 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FB cmti10 10 65 /FB 65 124 df<047FEB03F0923A03FFE007FC030F9038F01FFE923A1F80F83E0FDB3E01 EB7C3F037C1478DBF803EBF87FA20201D901F1133E923AF000E1F01C020302031300A44A 48495AA54A48495AA4010FB812C0A3903C001F80001F8000A34AC7FC4DC7FCA4147E177E A54A5CA54948495AA54948495AA54948495AA44A5C010F140FA24A5CA2011F141F91C790 C8FC001CEB0380003E903807C03E387E1E0F26FE3E1F5B013C1478D8FC78EB81F03A7870 0F03E03A7FF007FFC0263FC00390C9FC390F8000FC404C82BA33>11 D<EE3FFC4BB51280030714C092390FC003E092391F0001F0033C1303037C13075D18E014 014BEB0380020391C7FCA34A5AA54A5AA4010FB612FE835F903A001F80007EA34AC75AA4 4C5AA2147EA24C5AA35C4C5AA44948495AEFC1C0A393381F8380495AA3EF8700A24948EB 0F8F178EEE07FC705A4AEB00F0010F92C7FCA25CA349CAFC121C123EEA7E1EEAFE3E133C 485A1278EA7FF0EA3FC0EA0F80344C82BA2F>I<DC7FC0EB0FFF922603FFF890B512E04B D9FC0314F0923C1FC03E07F000F892283E007E0F80137C4B021EC712FCDDFE3E13014B5C 0201027C15F8EF38FC4BD900F8EB00E0020302011500A34A48495AA54A48495AA40107BA 12804919C01B80903D001F80000FC0001FA34D48EB3F004AC7FCA31A7E4DC7FC147EA262 A34A147E4F5AA449484A495A1B70A3963807E0E04948495AA3F2E1C0A249484948903803 E380A2963801FF00735A4A4A143C010F020792C7FCA25F5C160F011F5D001CEB0380003E 496C48CAFC267E1E0F131E26FE3E1F133E013C5CD8FC78495A007890380F03F03A7FF007 FFC06C486C5B270F8000FECBFC4E4C82BA49>14 D<3901E001E03903F003F03907F807F8 000F130F391FFC1FFCA4390FF80FF83907B807B839003800380178137801701370A24913 E0A23901C001C03903800380A23907000700000E130E485B485B485B485B004013401E1A 71B92B>34 D<EA01E0EA03F0EA07F8120FEA1FFCA4EA0FF8EA07B8EA0038A21370A213E0 A2EA01C012031380EA0700120E5A5A5A5A12400E1A6FB919>39 D<EA01E0EA03F0EA07F8 120FEA1FFCA4EA0FF8EA07B8EA0038A21370A213E0A2EA01C012031380EA0700120E5A5A 5A5A12400E1A7B8919>44 D<387FFFF8A3B512F0A2150579941E>I<120FEA1F80EA3FC0 127F12FFA3EA7F801300123C0A0A788919>I<EC03F8EC1FFEEC7FFF9138FC1F80903901 F007C0D903E013E0903807C003EB0F80D91F0013F01501133E017E1303137C13FCA2485A A2000314075BA2120716E049130F120FA34848EB1FC0A44848EB3F80A448C7EA7F00A315 7E15FE127E5D00FE13015DA248495AA25D007C13075D4A5A141F007E49C7FC003E137E00 3F5B381F83F8380FFFE06C1380C648C8FC243A77B72A>48 D<151015381570A215F01401 EC03E0140F141F147F903803FFC0EB1FEFEB3F8FEB1C0F9038001F80A4EC3F00A4147EA4 5CA4495AA4495AA4495AA4495AA4495AA449C7FCA25B007FB5FCB6128015001D3877B72A >I<EC03F8EC0FFEEC3FFF91387C0F809138F007C0903901C003E09039038001F0EB0700 010E14F8131EEB1C30D9387813FC14381370A213E0A33A01C07003F8A214E0ED07F0EBC1 C016E09038C7800F9039FF001FC06C4814800178EB3F0090C7127E5D4A5A4A5AEC07C04A 5A023EC7FC5CEB01F0EB07C0495A011EC8FC5B017014C0491301485A485A48C7EA0380A2 000E1407481500D81FE05BD83FFC131E9038FFC07E39787FFFFCD8700F5BD8F0075BD8E0 015B6D5B48011FC7FC263A79B72A>I<EC03FCEC1FFF4A13809138FC07C0903901E001E0 D903C013F049C712F8130E5B4914FC1460EB70F01470A201E0EB01F8A3D971E013F09038 7FC003D93F8013E090381E000790C713C0ED0F80ED1F00153E15FC903801FFF05D495B90 38000FE0EC01F06E7EA281A281A6123C007E495A12FEA248495A5A00E05C4A5AA24A5A6C 495A007049C7FC6C13FE383E03F86CB45A00071380D801FCC8FC263A78B72A>I<ED01C0 ED03E01507A4ED0FC0A3ED1F80A316005DA2153E157E157CA215FC5D14015DA24A5AA24A 5AA24A5A92C7FC5C141E143E143C5C14F8ECF038903801E07C903803C0FCA2EB0780EB0F 0090381E01F8131C1338137849485AEA01E0EA03C0EA0780390FFC07E0383FFF874813F7 26F803FF13803AE0007FFFC048130FC7EBFE0015C04A5AA44AC7FCA4147EA4147C143823 487CB72A>I<010314106E13F89138F007F049B512E016C0168016004913FC15F015C001 0EC8FC5BA45BA45BA2147E903873FF804913E09038FF83F0EBFC01496C7ED801E07F5B49 137CC8127EA315FEA54A5A121C123E127F48495A5A485C48130700E05C4A5AA24A5A4AC7 FC0070133E14FC6C485A383E07F0381FFFC0000790C8FCEA03F8253A77B72A>I<157F91 3803FF80020F13C091383F81E091387E00F002F81370903901F001F0903803E003903807 C007EB0F80011F14E090393F0003C0017E90C7FCA25B1201A2485AA2485AECFF80D80FF1 7F9038E3EFE0391FE701F09038EC00F813F848487FA2497FA2485AA25BA21401EAFF00A3 48495AA44A5A5A5D140F5D5D141F5D007C49C7FC143E6C5B495A381F07F0380FFFC06C5B D801FCC8FC243A76B72A>I<D9607C130CD973FE131C497E01EF143848B5137016F04801 0713E001F813033A07F00387C001C013FF48486C138090C7130048140F001E140E001C14 1E485CA2485CA2485C00601301C75B1403A24A5AA24A5AA2141F92C7FC5C143E147EA25C A213015C1303A25C1307A2495AA3131F5CA2133F5CA2137FA291C8FCA2133E131C263A74 B72A>I<EC01FCEC0FFF023F138091387E07C09138F803E0903901E001F0903803C00049 4813F849C7FC5B131E133EA3017EEB01F0A2ED03E0A2017FEB07C09138800F8090393FC0 1F00ECE03E6E5A6D6C5A90380FFFE06D5B6D5B6D7F01077F011F7F90383E1FF89038780F FC48486C7E3803E0033807C00148486C7E90C77E5A003E80A25AA248143EA3153C157C15 785D140100785C007C495A003CEB0F80003E49C7FC381F80FC3807FFF86C13E0C66CC8FC 253A78B72A>I<EC01FCEC0FFE91383FFF8091387E07C0ECF803D901F013E0D903E013F0 903807C001130FEB1F80013F14F8EB7F0015035B5B1201A21507485AA216F0A249130F12 07A2151F16E049133F1203157F16C0000114FF6D5A0000903803BF809038FC077FEB7FFE D93FFC130090380FF0FF90C75AA24A5AA25D14035D4A5A121E003F495A48495A92C7FC00 FE133E485B00705BEB03F0383C0FE0383FFF80D80FFEC8FCEA03F8253A78B72A>I<131E 137F5B481380A314005B6C5A137890C7FCB0120F487E487E127F12FFA36C5A90C7FC123C 112478A319>I<EB03C0EB0FE0131FEB3FF0A314E014C0EB1F80EB0F0090C7FCB0EA01E0 487E487E120F121FA46C5A1207EA0070A25BA2485AA2485AA248C7FC120EA25A5A5A5A12 4014347BA319>I<EE01C01603A21607160FA2161F83163FA2167F16FF16EF150116CFED 038FA2ED070FA2150E151E151C1538A203707FA2EDE007A2EC01C014031580EC0700A214 0EA25CA25C027FB5FCA291B6FC9139E00007F849481303A2495A130791C7FC5B130E5BA2 5BA25B13F01201D807F84A7EB56C48B512F0A2020015E0343C7BBB3E>65 D<0103B612FC49EDFF806D16C0903B000FF0001FF04BEB0FF81707EF03FC4A5A18FE1701 A24A5A1703A34AC713FC170718F8170F02FE15F0EF1FE0EF3FC0EF7F804948ECFF00EE03 FEEE1FF891B612E0495D17F89139F80003FC707E49486D7EEF7F80A218C04948143FA449 48147FA44948ECFF80A218005E49C7485A5F4C5A160F01FEEC1FE0EE7FC000014A485AB7 48C7FC16F816C037397BB83A>I<DB03FE130492391FFF800C92B5EAE01C913A03FE01F0 3C913A0FF000787CDA1FC0EB1CF8DA7F80130D02FEC7120F49481407494815F0495A4948 1403495A494815E0137F49C8FC5B000117C0485AA2485A1880120F5B001F93C7FCA2485A A3485AA45B12FFA4173890C9FCA25FA25F7E6D4A5AA2003F4B5AA24CC7FC6C6C140E5E6C 6C5C00075D6C6C5C6C6CEB03C0D800FEEB0F8090267FC07EC8FC90381FFFFC010713F001 0090C9FC363D74BA3B>I<0103B8FC5B7F903A000FF000034BEB007F183E181E4A5AA44A 5A181CA34AC8FCEE0380A2181802FE49C7FCA35E4948130E161E167E91B512FE495CA2EC F800167C49481338A4494849136018E0A293386001C049481300EF0380A34948EC0700A2 170E171E49C8FC5F177C5F01FE140116070001ED7FF0B8FC5FA238397BB838>69 D<0103B712FE5B7F903A000FF000074B1300187C183C4A5AA44A5A1838A34AC8FCA21603 EF803002FE49C7FCA35E4948130E161EA216FE49B55AA3ECF80049481378A449481370A4 4948136093C8FCA3495AA449CAFCA413FEA2487EB6FCA25C37397BB836>I<0103B5D8E0 0FB51280496E4814C004E015809026000FF0C7383FC0004B5DA34A484AC7FCA44A4814FE A44AC7485AA402FE4A5AA449484A5AA391B7FC495EA202F8C7120FA249484A5AA449484A 5AA449484AC8FCA4494814FEA449C7485AA401FE4A5AA200011507B5D8FC03B512F0A342 397BB83E>72 D<0103B512F8A216F090390007F8005DA34A5AA44A5AA44A5AA44AC7FCA4 14FEA4495AA4495AA4495AA4495AA4495AA4495AA449C8FCA25B007F13FEB5FC7E25397C B820>I<0103B512F849806D5C9026000FF8C7FC15E0A34A5AA44A5AA44AC8FCA414FEA4 495AA4495AA4495AA44948140C171CA21738495AA21770A2494814F017E01601A249C7EA 03C01607A2EE1F8001FE143FEEFF0000011407B8FC5EA22E397BB834>76 D<902603FFF8923807FFE0494D13F06D4D13E0D9000FEFF0004F5AA21977021D4C5AA2DA 1CFCEC01CFA202384B485AF0071FA2180E02704CC7FC181CA2183802E0ED707EA2037E14 E0A2D901C04A485AEF0380A2EF0700D903804B5A170EA2171CD907004A485AA26F1370A2 010E4B485AEE01C0A2EE0380494C5AEE0700A2160E494A495AA2ED1FB8A24902F049C8FC 5EA201F05C187E00015DD807F816FEB500C09039007FFFFC151EDA800E5D4C397AB84A> I<902603FFF091B51280494B14C06F1680D9000F9139000FF000F007C06F5DA2021D93C7 FCEC1CFEA2814A6C140EA26F7EA202705D6F7EA282DAE00F5CA26F7EA2D901C05D6F7EA3 49486C6C5BA282150049C7495A167FA3010E91383F8380A217C3161F4903C7C8FCEE0FE7 A349EC07FEA31603495DA2160113F0705A1201EA07F8B500C014781770173042397BB83E >I<ED03FE92383FFFC092B512F0913903FC07F891390FE001FC91393F8000FE4AC7127F 02FEEC3F80D901F8EC1FC0EB07F04948EC0FE0131F4A15F04948140749C8FC4916F8485A A2485AA2485AA2120F5B001F160F5B123FA34848ED1FF0A44848ED3FE0A3EF7FC0A21880 17FF18005E5F16035F007F4B5A4C5AA24C5A003F4B5A6D4A5A001F4BC7FC6D495A000F4A 5A6C6CEB07F06C6C495A6C6CEB3F802700FF81FFC8FC90383FFFFC6D13E0010190C9FC35 3D74BA40>I<0103B612F84915FF6D1680903B000FF0007FE04BEB0FF0EF07F817034A48 14FCA318FE4A5AA44AC7EA07FCA318F802FE140F18F0EF1FE0A24948EC3FC0EF7F80EFFF 00EE03FC4948EB1FF891B612E0178004F8C7FCD907F0C9FCA4495AA4495AA4495AA449CA FCA413FEA21201B512FCA25C37397BB838>I<0103B612E04915FC17FF9027000FF00013 804BEB1FE0170FEF07F04A4814F8A318FC4A5AA44AC7EA0FF8A3EF1FF002FE15E0A2EF3F C0EF7F804948ECFF004C5AEE07F8EE3FF049B612C04CC7FC5E9138F8007E49487F828316 0F494880A44948131FA44948133FA283A249C7127F1804180EA201FE161C163F00011738 B500FC011F137093380FF0E0933807FFC0C96C13809338007F00373B7BB83D>82 D<92383FC004913901FFF00C0207EBF81C91390FC07E3C91393E001E7C4AEB07F84A1303 495A4A1301494814F013075C130F91C713E05BA34915C0A36E90C7FCA2806D7E14FCECFF 806D13F06D13FE6D6D7E6D806D80023F7F02077FEC007FED0FFC150315011500A3167C12 06120EA3001E5DA34B5A003E5D15035E003F4A5A48140F6D49C7FC6D133ED879F05B39F0 FC03F039E07FFFE0011F138026C003FCC8FC2E3D7ABA2F>I<0003B812E05AA2260FF800 EB001F01C049EB07C090C71403121E4A5A121C003C178012384A5A12781270EF07004A5A 5AA2481606C7484890C7FCA44A5AA44A5AA44AC9FCA414FEA4495AA4495AA4495AA4495A A2EB3FF0007FB512F8A3333971B83B>I<001FB5903807FFFC486E4813FE030014FC2600 7F809038007F8091C8EA3E00173CA201FE1538A448485DA448485DA448484A5AA448484A 5AA448484AC7FCA44848140EA448C85AA400FE5DA35EA25EA24B5A007E4A5AA24BC8FC00 3E140E003F5C6C5C6C6C13F03907C003E03903F01F806CB5C9FC38007FFCEB1FE0373B70 B83E>I<277FFFF007B590381FFFE0B5495D6C84D803FEC7D83FE0903803FE0001F80380 EB00F86262A24F5A1903621907047F92C7FC190E16FF4B5DA2DB03BF5C7F0001DA073F5C A2030E5D83DB1C1F495A180303385D4EC8FC157003F0140E15E0DA01C05CA2DA03805CA2 DA07005CA2020E5D17C14A5DEFC3805C027802C7C9FC14704A14CE13FE6C6C4814DCA24A 14F8A291C75B160F495D5F5B5F5B4992CAFCA249140E4B3B6FB853>87 D<0110131001381338017813784913F03901C001C039038003800007130701001300000E 130E485BA2485BA2485BA300EF13EF39FF80FF806D13C0A5397F807F8001001300003C13 3C1D1A6CB92B>92 D<14F8EB07FE90380FFF1C90383F07BE90387C03FEEBF801EA01F000 035CEBE0001207485A4A5AEA1F80123FA249485A5AA300FE495AA448495AEDC1C0A39138 1F8380A2143F127C91387F8700007E13FF393E03CF8E381F0F8F390FFF07FC3907FC03F8 3901F000F0222677A42A>97 D<133FEA0FFF5A7EEA007EA45BA4485AA4485AA4485A14F8 EBE3FEEBEFFF390FDF0F809038FC07C001F013E0EBE003EA1FC015F01380A2EA3F00A400 7E1307A448EB0FE0A315C048131FA21580EC3F00A2147E147C14FC007C5B495A383C03E0 381E0FC06CB4C7FC6C5AEA01F01C3B77B926>I<147F903803FFC04913E090381FC1F090 383F0078017C13384913F83801F00100031303EA07E0D80FC013F0EC01E04848C7FCA212 3F90C8FC5AA312FEA55AA315101538007C147015F0007EEB01E0003EEB07C06CEB1F0038 0F80FE3807FFF86C13E0C690C7FC1D2677A426>I<ED01F8157F15FF157FED03F0A4ED07 E0A4ED0FC0A4ED1F80A4ED3F0014F8EB07FEEB0FFF90383F07FEEB7C03EBF801EA01F000 035CEBE0001207485A4A5AEA1F80123FA249485A5AA300FE495AA448495AEDC1C0A39138 1F8380A2143F127C91387F8700007E13FF393E03CF8E381F0F8F390FFF07FC3907FC03F8 3901F000F0253B77B92A>I<147F903803FFC0010F13E090381F81F0EB7E00491378485A 485A485AA2485A001F14F090388001E0003FEB07C0EC3F8048B5120014FC14E090C8FC12 FEA65AA2007C14101538007E147015F0003EEB01E06CEB07C0EC1F00380F80FE3807FFF8 6C13E0C690C7FC1D2677A426>I<ED07C0ED0FF0ED1FF8ED3C3CED78FC15F8EC01F915F1 EDF0F80203137016005D1407A44A5AA54A5AA2010FB5128016C016809039001F80004AC7 FCA5147EA55CA5495AA5495AA5495AA45C130FA35C131FA391C8FCA2EA1C3E123E127EEA FE3C5B12FCEA78F0EA7FE06C5AEA0F80264C82BA19>I<EC07C0EC3FF091387FF8E09039 01F83DF0903803E01F903807C00FEB0F80011F14E090383F0007A2137E01FEEB0FC05B12 01A249EB1F801203A34848EB3F00A449137EA45DA20003130114036C6C485A140F3800F8 3DEB7FF96D485AEB0FC3EB0003A24A5AA44A5A121C003E495A127E00FE49C7FC147E485B 387803F8383FFFE06C1380D803FEC8FC24367CA426>I<EB03F013FF5A7EEB07E0A4495A A4495AA449C8FCA4137EEC07F0EC1FFCEC7FFE9038FCF83F9039FDE01F80EBFF80EC000F 485A16C05B49EB1F8012035BA34848EB3F00A3157E485AA35D485A913801F81CA33A3F00 03F038A3EDE070127E16E0A2EDE3C048903801FF80007C6D13000038143C263B7BB92A> I<EB01C0EB03E0EB07F0A214E014C0EB038090C7FCAB13F0EA03FC487EEA0F1F121CA212 38A2485AA3EAE07EA25B1200A2485AA3485AA3485AA214E0EA0FC0A2381F81C0A3EB0380 A2EB0700A2131EEA0FFC6C5AEA01E0143879B619>I<150E151F153F157FA2153E151C15 00ABEC1F80EC3FC0ECFFE0903801E1F014C190380381F8EB0701130EA2131CA2EC03F013 38A21300EC07E0A4EC0FC0A4EC1F80A4EC3F00A4147EA45CA4495AA4495A121C383E07E0 127E38FE0FC05C4848C7FCEA787EEA7FFCEA3FF0EA0FC0204883B619>I<EB03F013FF5A 7EEB07E0A4495AA4495AA449C8FCA4137EED0F80ED3FC0ED7FE09038FC01F0913803C1F0 EC0703EC0E073A01F80C0FE0141891383007C09138700380484848C7FCEBF1C0EBF38001 FFC8FC485A6D7E14E014F8380FC3FCEBC0FE143FA248486C7EED81C0A33A3F003F0380A3 ED0700127EA2EC1F0E5D48EB0FFC007C6D5A0038EB01E0243B7BB926>I<EB0FC0EA03FF 5A7E38001F80A4EB3F00A4137EA45BA4485AA4485AA4485AA4485AA4485AA448C7FCA412 7E130EA3485AA45BA2EA7C70A2EA3FE06C5A6C5A123B79B915>I<D801E001FEEB07F03C 07F803FF801FFC260FFC0F9038C07FFE3C1E3E1F07E0F83F001C903B3C03F1E01F802638 3F70EBF380913AE001F7000F02C013FE007018C002805B4A4848EB1F80485A017E5CA212 00494948EB3F00A3187E4848495AA3604848495A943801F81CA34848903A3F0003F038A3 F0E0704848137E19E0A2F0E1C0484849903801FF80000F027C6D13006CC70038143C3E26 79A444>I<D801E013FE3A07F803FF80260FFC0F7F3A1E3E1F07E0001C90383C03F03838 3F70ECE00114C000708114804A485A485A137EA2120049495AA34B5A485AA34B5A485A92 383F0380A3484890387E0700A3ED7C0E485A5EA25E4848EB3FF0000F6E5A6CC7EA078029 2679A42F>I<147F903803FFC04913E090381FC1F090383E00F849137C49137E4848133E 12034848133F485AA2485AA2003F147F90C7FC5AA300FE14FEA315FC14014814F8A2EC03 F0A2EC07E0007CEB0FC01580007EEB1F00003E133E6C5B380F83F83807FFE06C5BC648C7 FC202677A42A>I<9039078007C090390FE01FF090391FF07FF8903938F8F87C9138F9E0 3E903970FF803F4B7E495A17805CA23801C1F8A3EA00014948133FA44948EB7F00A3167E 494813FEA25E4B5AEB1F804B5A5EECC007013F5C4B5A6E48C7FCECF07E90387E7FF86E5A EC0F8091C9FC5BA4485AA4485AA3387FFFC0B5FC6C5B293580A42A>I<ECF803903807FE 0790380FFF0F90383F079E90387C03BE9038F801FEEA01F0000314FCEBE0001207485AEC 01F8EA1F80123FA290380003F05AA300FEEB07E0A448EB0FC0A4EC1F80A2143F127CEC7F 00007E5BEA3E03381F0FBF380FFF7EEA07FCEA01F0C7FC5CA4495AA4495AA2130748B512 C0A21580203577A426>I<3903C003F03907F00FFC390FF83FFE391C7C7C1FECF00F3A38 7FC01F80153F00709038807F001400153E017E131C484890C7FCA31200485AA4485AA448 5AA4485AA4485AA448C9FC7E120E212679A423>I<14FE903807FF804913C090381F03E0 90383C00F049137001F813F0EBF00100011303A215E00003EB01C06DC7FC7F3801FFC014 F8806C7F6D7E6D1380130F010013C0141F140FA2123C127E00FE1480A2481400485B00E0 131E00705B0078137C383E03F06CB45A6C5BD801FEC7FC1C267AA422>I<1307EB0F8013 1FA4EB3F00A4137EA45BA2B512FC14FE14FC3801F800A3485AA4485AA4485AA4485AA448 C7FC1438A21470127E14E0A2EB01C0A2EB0380383E0700131EEA1FFC6C5AEA03E0173578 B31C>I<13F8D801FEEB01C0486CEB03E039070F8007120E121C1238ED0FC0EA301F1270 A23AE03F001F80A3EA007EED3F005BA34848137EA448485B160EA3913801F81CA3163814 0300011307020E13702600F83C13F090397FF87FE090393FF03FC090390FC00F00272679 A42D>I<01F0130ED803FC131F486C5BD80F1FEB7F80121C153F0038141FA24848130FA2 1600D8E07E7FA3C6485B150E485AA348485BA35D485AA25DA2156015E05D14014A5A0003 49C7FC6D5A3801F81E3800FFFCEB3FF0EB0FC0212679A426>I<01F01507D803FC903903 800F80486C903807C01FD80E1F010FEB3FC0121C171F0038160FED1F8048481507A21880 D8E07E90383F0003A3C6481507037E1400485AA3484849130EA4484848485BA35FA25FA2 02035C1203D9F007495A000190390EFC03803A00FC1E7E0F90277FFC3FFFC7FC90393FF0 0FFC90390FE003F0322679A437>I<903907E007C090391FF01FF090393FF83FF8903978 3C783C9038E03EE02601C01F137ED80380EBC0FED80700EBC1FC1581000EEC80F8167048 49C7FCA3C7FC147EA45CA4495A1670A3261C03F013E0123E007FEC01C0130700FEEC0380 D8FC0FEB070039780EF80E90383C783C393FF83FF86C486C5A3907C00FC027267CA427> I<13F0D803FCEB01C0486CEB03E0D80F1F1307121CA21238ED0FC0485AA3D8E07EEB1F80 A3C65AED3F00485AA34848137EA448485BA44A5AA314035D00031307EBF00F3801F83F6C B55AEB7FF7EB1FC7EB00074A5AA25D001E131F003F91C7FC485B147E007E137C007C5B38 7001F0387803E0383C0FC0D81FFFC8FC6C5AEA03F0233679A428>I<903903C00380EB0F F090391FF80700133F90387FFC0EECFF1C9038F83FF8EBE00F3901C001F06E5A49485AC7 485A4AC7FC140E5CA25C5C495A495A49C8FC130E5BA249131C5B49133C48481338484813 78D807E01370390FFC01F09038FF07E0391E3FFFC0EA3C0F486C5B007091C7FC38E003FC EB00F021267BA422>I<B712F016F816F0250378972A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmsy8 8 2 /FC 2 50 df<131EA2131F131EA30070EB0380007C130F00FEEB1FC0007FEB3F80EB9C7F 390FEDFC003803FFF0C613C0013FC7FCEBFFC0000313F0380FEDFC397F9C7F80EB1E3F00 FEEB1FC0007CEB0F8000701303000090C7FCA3131F131EA21A1D7C9E23>3 D<D91FC0EC0FF0D9FFF8EC3FFC4801FEECFFFE00076D0103EBFF8048913AC007F007C001 80903AE00FC001E03C1E003FF01F80004890260FF83EC7127000386D6C5A480103017814 386E6C5ADA00FF153C484B141C6F5A153F6F7E6F7E824B7E6C023F15380070EC3DFEED78 FF6C02F86D13704A486C6C13F06C903B07E03FF001E0001E903B0FC01FFC07C0280F803F 800FB5FC6CB5D800031480000101FC6DEBFE006C01F06D6C5AD93FC0EC0FE03E1F7C9D47 >49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FD cmr8 8 86 /FD 86 128 df<91383FC03F903A03FFF0FFC0010F01FB13E0903A1FC03FE3F090393F00 7F87017CEBFF075B0001ED03E0484890387E01C093C7FCA9B712FCA32703F0007EC7FCB3 A4157F3A7FFF8FFFF8A32C2F7FAE29>11 D<EC7F80903803FFE0010F13F890383F807CEB 7E0001F8137E000114FE485A5B0007147C151092C7FCA6153EB612FEA33807E000157EB3 A4B5380FFFF0A3242F7FAE27>I<91393FC007F8903B03FFF07FFFC0010F01F9B5FC903A 1FC07FF81F90393F00FFE0017C14804914000001160F4848137EAAB9FCA33B03F0007E00 0FB3A53C7FFF8FFFF1FFFEA3372F7FAE3B>15 D<EA03E012FFA3120F1207B3A4EAFFFEA3 0F1E7F9D13>I<EA01E0EA03F0A21207120FEA1FE01380EA3F00127E127C12F012600C0C 72AD23>19 D<00C013C0EAF003EAFC0F387F3F80381FFE006C5AEA03F06C5A120878AA23 >I<00E0131CA36C133C00701338007813786C13F0EA3F03381FFFE06C13C000031300EA 00FC160C7AAD23>I<EA0FC0EA3FF0EA7FF8EAF03CEAE01CA3EAF03CEA7FF8EA3FF0EA0F C00E0B6DAE35>23 D<157015F015E0140190381FE1C0EB7FFB48B512803903F03F003907 C00F80D80F8013C0391F001FE0003EEB1DF01439007EEB79F8007C137014F000FCEBE0FC EB01C013031480130714005B130E5BD87C3C13F81338393E7801F01370391FE003E01407 390FC00FC03907F03F803903FFFE004813F8380F1FE0000EC8FC121E121C123C12381E29 7EA223>28 D<123C127EB4FCA21380A2127F123F1203A312071300A2120EA25A123C1238 5A122009157AAD14>39 D<1307130F131E133C1378137013F0EA01E0120313C012071380 120F13005AA2121E123EA35AA612FC5AAD7E127CA67EA3121E121FA27E1380120713C012 0313E01201EA00F013701378133C131E130F130710437AB11B>I<126012F012787E7E12 0E120FEA078013C0120313E0120113F0120013F8A21378137CA3133EA6133F131FAD133F 133EA6137CA3137813F8A213F0120113E0120313C012071380EA0F00120E121E5A5A5A12 6010437CB11B>I<EC0380B3A4B812FCA3C7D80380C7FCB3A42E2F7CA737>43 D<123C127EB4FCA21380A2127F123F1203A312071300A2120EA25A123C12385A12200915 7A8714>I<B512C0A512057F9018>I<123C127E12FFA4127E123C08087A8714>I<15C014 01A2EC0380A3EC0700A2140EA35CA35CA35CA35CA2495AA3495AA349C7FCA3130EA25BA3 5BA35BA35BA3485AA2485AA348C8FCA3120EA35AA35AA25AA35AA25A1A437CB123>I<EB 3FC0EBFFF0000313FC3807F0FE380FC03F497E391F000F80A24814C0003E1307007E14E0 A500FE14F0B0007E14E0A46CEB0FC0A36C1480390F801F006D5A3807F0FE6CB45AC613F0 EB3FC01C2D7DAB23>I<130E131E137EEA01FE12FFA2EAFE7E1200B3AF13FF007F13FFA3 182C7BAB23>I<EB7F803801FFF0000713FC380F81FE381E007F0038EB3F8048EB1FC000 7814E000FC130F7E6C14F01407A2127E123CC7120FA215E0A2EC1FC01580143F1500147E 5C495A495A14C0495A49C7FC131E5B4913705B485A484813E048C7FC120E001FB5FC5A5A B612C0A31C2C7DAB23>I<EB3F803801FFF04813FC3807C0FE380E007F001EEB3F80123F 9038801FC0A4381F003F000E1480C7FCA2EC7F00147E5CEB03F8EBFFF014C014F0EB00FC 147E80EC1F8015C0EC0FE0A215F0A2123C127EB4FCA215E0A248EB1FC0127C0078EB3F80 6C1400381F80FE3807FFFC6C13F038007F801C2D7DAB23>I<140F5CA25C5CA25C5BA25B EB07BF143F130F131E131C1338137813F013E01201EA03C013801207EA0F00120E121E5A 123812785AB612FCA3C7EA3F00A8EC7F8090381FFFFCA31E2D7EAC23>I<000CEB018038 1F801F90B5FC15005C14F85C14C0D81C7CC7FC90C8FCA7EB1FC0EB7FF0381DFFFC381FE0 7EEB801F01001380001EEB0FC0120CC7EA07E0A315F0A3123C127E12FEA215E0A248130F 007014C00078EB1F8000381400001E133E380F80FC6CB45A000113E038007F801C2D7DAB 23>I<EB03F8EB0FFEEB3FFF90387E0780EBF8033901E00FC03803C01F1207EA0F80121F 9038000F804890C7FCA3127EA2EB1FF038FE3FF8EB7FFEEBE03F6C487E9038800F800100 13C0EC07E0A34814F0A4127EA4003E14E0123FA26CEB0FC01580EA0F803907C01F003803 F07E6CB45A38007FF0EB1FC01C2D7DAB23>I<1238123E003FB512F8A34814F015E015C0 38700001EC0380EC070048130EA25CC75A5CA25C495AA21303495AA2130F91C7FCA25BA2 5B133EA2137EA513FEA8137C13381D2E7CAC23>I<EB1FC0EBFFF04813FC3803E07E3807 801F390F000F805A001EEB07C0123EA4123F9038800F80EA1FC09038F01F00380FF81EEB FE7C6C6C5A6C5B7E6C6C7E3801FFFE3803EFFFD807831380D80F0113C0381E007F003EEB 3FE048130FEC07F0481303A21401A315E0127C14036C14C0EC0780391F800F00380FE07E 3803FFFC6C13F038003FC01C2D7DAB23>I<EB3F80EBFFF0487F3807E07C380F803E001F 7F9038000F805A007E14C0A2140700FE14E0A415F0A4007E130FA37E6C131F380F803FEB C0773807FFE7000113C76CEB87E0EB0007A3EC0FC0A2001F1480383F801F1500A2143E49 5A001C5B380F03F06CB45A6C1380C648C7FC1C2D7DAB23>I<123C127E12FFA4127E123C 1200AD123C127E12FFA4127E123C081D7A9C14>I<B812FCA27ECBFCAD007FB712FCB8FC A22E137C9937>61 D<EC03C0A24A7EA34A7EA34A7EA3EC39FCA2EC79FE1470A2ECF0FF4A 7EA249486C7EA349486C7EA2010780EC000FA24980010E1307A2011FB57EA24980903838 0001A201788001701300A249147FA30001ED3F801203D80FF0EC7FC0D8FFFE0107B5FCA3 302F7EAE35>65 D<B612FCEDFF8016E03A03FC000FF80001EC03FC6F7E6F7E821780A617 005E5E4B5AED07F8ED1FF090B612C05E16F09039FC0007F8ED01FE6F7EEE7F80163F17C0 161F17E0A6EE3FC0A2EE7F80EEFF005D0003EC07FCB75A16E093C7FC2B2D7EAC32>I<DA 1FF013C09138FFFE010107EBFF8390390FF807C390393FC001E749C7127F01FC143F4848 141F4848140F00071507485A491403121F123F491401A2127F90C8FC93C7FC5AA97EA26D EC01C0123FA27F121F000FED03807F6C6CEC070012036C6C140E6C6C5C017F5CD93FC013 F090390FF807E06DB51280010049C7FCEC1FF02A2F7CAD33>I<B612FCEDFF8016F03A03 FC000FF80001EC01FCED007FEE3F80EE1FC0160F17E0EE07F0160317F8A217FC1601A317 FEAA17FCA3EE03F8A217F0160717E0160FEE1FC0EE3F80EE7F00ED01FE0003EC0FF8B75A 16C003FCC7FC2F2D7EAC36>I<B712FEA33903FC00030001EC007F828282A282A392381C 0380A493C7FC153CA215FC90B5FCA3EBFC00153CA2151C17E0A3EE01C01500A31603A217 801607160F161F163F0003EC01FFB81200A32B2D7EAC30>I<B712FCA33903FC00030001 EC00FE163EA2161EA2160EA316071538A31600A21578A2EC01F890B5FCA3EBFC01EC0078 A21538A592C7FCA9487EB512FCA3282D7EAC2E>I<DA1FF013C09138FFFE010107EBFF83 90390FF807C390393FC001E749C7127F01FC143F4848141F4848140F00071507485A4914 03121F123F491401A2127F90C8FC93C7FC5AA892383FFFFE7EA26D9038003FC0003F151F A27F121F120F7F6C7E12036C7ED800FE143F137FD93FC013F790390FF803E36DB512C101 001400DA1FF813002F2F7CAD37>I<B5D8F81FB5FCA3D803FEC7EA7FC06C48EC3F80B090 B7FCA301FCC7123FB2486CEC7FC0B5D8F81FB5FCA3302D7EAC35>I<B512F8A33803FE00 6C5AB3B3487EB512F8A3152D7EAC19>I<90387FFFF8A30100130080B3AC123C127EB4FC A2147E4813FE007C5B387801F8383E07F06CB45A00075BD801FEC7FC1D2E7EAC24>I<B5 00F8EB7FFFA3D803FEC7EA3FF06C48EC1F80041EC7FC5E5E5E4B5A4B5A4B5A4BC8FC151C 5D5D4A5A4A5A4A7E140F4A7E4A7EEC7BFC14E19038FDC0FE9038FF80FF4A7E496D7E496D 7E82150F6F7E8215036F7E6F7E8282707E83707E486C4A7EB5D8F801B51280A3312D7EAC 37>I<B512FCA3D803FEC8FC6C5AB3A5161CA41638A41678A216F815011503ED07F00003 143FB7FCA3262D7EAC2C>I<D8FFFE92381FFFC0A3D803FF92383FF0006C5FD9DF801477 A3D9CFC014E7A2D9C7E0EB01C7A3D9C3F0EB0387A2D9C1F8EB0707A3D9C0FC130EA3027E 131CA26E1338A391381F8070A291380FC0E0A3913807E1C0A2913803F380A3913801FF00 A36E5A487ED80FF8017C497EB500800103B512C0A215383A2D7DAC41>I<D8FFFC49B5FC 7FA2D801FF9038001FF06EEB07C06EEB038013DFEBCFE08013C7EBC3F8EBC1FC8013C014 7F1580143FEC1FC0EC0FE015F01407EC03F815FCEC01FE1400157F1683153FED1FC3ED0F E316F31507ED03FB16FF8181167FA2163F161F487ED80FF8140FB56C13071603A2302D7E AC35>I<EC3FF0903801FFFE01076D7E90391FE01FE090393F8007F09039FE0001FC4848 6D7E4848147F4848EC3F8049141F000F16C04848EC0FE0A24848EC07F0A2007F16F8A290 C81203A24816FCAA6C6CEC07F8A3003F16F06D140F001F16E0A26C6CEC1FC06D143F0007 16806C6CEC7F006C6C14FE6C6C495A90397F8007F890391FF03FE00107B51280010149C7 FC9038003FF02E2F7CAD37>I<B612F8EDFF8016E03A03FC001FF00001EC03F86F7E6F7E 82821780A717005E5E4B5A4B5AED1FF090B65A168003F8C7FC01FCC9FCB0487EB512F8A3 292D7EAC30>I<EC3FF0903801FFFE01076D7E90391FE01FE090393F8007F09039FE0001 FC48486D7E4848147F4848EC3F80A24848EC1FC04848EC0FE0A2003F16F0491407007F16 F8A290C81203A24816FCAA6C16F86D1407A2003F16F0A26C6CEC0FE0A23B0FE007801FC0 EC1FE03B07F03FF03F803B03F838387F003A01FC701CFED800FEEB1DFC90397FF00FF890 391FF81FE00107B512800101EC000C9039003FF780EC0007EEC01C1503EEF07CEEFFF8A2 8117F06F13E0167FEE3FC0EE1F002E3B7CAD37>I<B612E015FE6F7E3A03FC003FE00001 EC07F06F7E6F7E82150082A65E15015E4B5A4B5AED3FE090B612804BC8FCA29038FC007F ED1F806F7E82150782A582A4EF038016FC1503486C903901FE0700B500F86D5A9238007F FE705AC9EA07F0312E7EAC34>I<90383F80303901FFF07048EBFCF0380FC07E381F000F 001E1307481303007C1301127800F81300A21570A27EA26C1400127F7FEA3FF8EBFF806C 13F86C13FE6C7F6C14C0C614E0130F010013F0EC0FF814031401EC00FCA20060147C12E0 A46C1478A26C14F07E6CEB01E039FF8003C039F7F00F8000F1B5120038E07FFE38C00FF0 1E2F7CAD27>I<007FB712F8A39039800FE007D87E00140000781678A20070163800F016 3CA348161CA5C71500B3A74A7E011FB512F0A32E2D7EAC33>I<B5D8F801B5FCA3D803FE C7EA1FF06C48EC07C0EE0380B3AB0000ED07007F137E160E6D5C6D7E010F1478D907E05B 903903F807E00100B51280023F90C7FCEC07F8302E7EAC35>I<B500E090381FFFC0A3D8 03FEC73803FC006C48EC01F05F7F00005E6D14036D5DA26D6C49C7FCA26E5B011F140EA2 6D6C5BA26E133C01071438A26D6C5BA26E13F001015CA26D6C485AA2ECFF03027F5BA2DA 3F87C8FCA215CFEC1FCEA2EC0FFCA36E5AA26E5AA36E5AA2322E7FAC35>I<B53CE03FFF F003FFF8A32707FC000190C7EA7F806C486D48EC1F006D171E00016F141C826D173C0000 4B6C1338A26D17786D902601DFC01370A2028016F0013F9026038FE05BA202C01501011F 90260707F05BA202E01503010F90260E03F85BA202F01507010790261C01FC90C7FCA202 F85D010390393800FE0EA202FC151E010149EB7F1CA202FE153C010049EB3FB8A202FF15 F86E486D5AA2023F5D4B130FA2021F5D92C71207A26E5D020E1403452E7FAC48>I<3B7F FFF007FFF8A30001D9800113006C90C712FC6D6C137016F06D6C485AD91FE05B15036D6C 485AD907F890C7FC903803FC0E151E6D6C5A903800FF3815786E5A6E5AA2141F6E7EA281 4A7EEC1DFE143DEC78FF4A6C7E14E001016D7E49486C7E148001076D7E49486C7E010E6D 7E5B013C6D7E496D7EA2D801F8EC7F80D807FCECFFC0B5010713FFA3302D7EAC35>I<B5 00E090381FFFC0A3D803FEC73803FC000001ED01F06C6C15C06E1303017F5D6D6C49C7FC 6E5B011F140E6D6C131E6E5B010714386D6C13786E5B01015C903800FF015EEC7F839138 3FC78093C8FCEC1FEFEC0FFE5D14076E5AAF140749B512E0A3322D7FAC35>I<003FB612 C0A39039F0003F800180EB7F0090C75A003E5C003C495A007C130300785C14074A5A0070 5C141F4A5A5DC7127F4AC7FC5C13015C495A13075C495A131F5C013F14E0495A91C7FC5B 485A4913011203485A5B000FEC03C05B48481307003F140F49133F48C712FFB7FCA3232D 7CAC2B>I<EAFFE0A3EAE000B3B3B3A7EAFFE0A30B4379B114>I<EAFFE0A31200B3B3B3A7 12FFA30B437FB114>93 D<1238127C12FEA4127C123807087AAD14>95 D<EB7F803803FFE0000F13F848C67E143E383F803FEC1F80A2391F000FC0120EC7FCA3EB 07FF137F3803FF8F3807F80FEA1FE0EA3F8013005A00FE14C7A4141F127F1477393FC1E7 FE381FFFC30007EB81FC3901FC00F020207E9E23>97 D<EA03E012FFA3120F1207AB147F 9038E3FFE001E77F9038FF01F89038FC007C497F49133F49EB1F80A2ED0FC0A216E0A816 C0A2151F1680A26DEB3F006D137E01DC5B90388F03F8903887FFF0010313C0C76CC7FC23 2F7FAD27>I<EB1FE0EB7FF83801FFFE3803F01FEA07E0390FC03F80EA1F80130048EB1F 0048130E91C7FC127E12FEA8127E127FA26CEB01C01380001FEB0380EA0FC03907E00700 3803F81E3801FFFC6C6C5AEB1FC01A207E9E1F>I<157CEC1FFCA314011400ABEB1FE0EB 7FF83801FFFE3803F81F3807E007380FC001381F800048C7FCA25A127EA212FEA7127EA3 7EA2381F8001000F13036C6C487E3A03F03EFFE03801FFFC6C13F090391FC0F800232F7E AD27>I<EB1F80EBFFF0487F3803F0FC3807C03E380F803F48487E481480140F127E15C0 A212FEB6FCA348C8FCA4127EA2127F6CEB01C0A2391F800380EA0FC03907E007003803F8 1E3801FFFC6C6C5AEB1FC01A207E9E1F>I<EB01F8EB07FEEB1FFF90383F1F80EB7C3F13 F81201EC1F003803F00E91C7FCA9B512E0A3D803F0C7FCB3A47F387FFFC0A3192F7FAE16 >I<90383F80F89038FFE3FC0003EBFFFED807E0133E390F803E1C1508391F001F00A248 1480A56C1400A2380F803EA23807E0FCEBFFF8000C13E0EB3F80001CC8FCA2121E121F38 0FFFF814FF6C14C015E04814F0391F0007F8003C1301007CEB00FC48147CA5007C14F86C EB01F06CEB03E0390FE01FC06CB512800001EBFE0038003FF01F2D7E9D23>I<EA03E012 FFA3120F1207ABEC3F80ECFFE001E37F9038E781F89038EE00FC13FC49137E5BA25BB2B5 380FFFF0A3242E7FAD27>I<EA0780EA0FC0EA1FE0A4EA0FC0EA0780C7FCA8EA03E012FF A3120F1207B3A4EAFFFEA30F2E7FAD13>I<EB0780EB0FC0EB1FE0A4EB0FC0EB078090C7 FCA8EB07E013FFA3130F1307B3AB1210127C00FE13C0130F1480A2387C1F00EA3FFEEA1F F8EA07E0133C84AD16>I<EA03E012FFA3120F1207ACEC1FFFA3EC0FF015C092C7FC141E 5C5C14E0EBE3C0EBE7E013EFEBFFF0EBFDF8EBF8FCEBE07C147E806E7E140F816E7E6E7E 8181B5380FFFC0A3222E7FAD25>I<EA03E012FFA3120F1207B3B3A2B5FCA3102E7FAD13> I<2607E07FEB0FE03BFFE1FFC03FF801E36D487E903AE783F0F07E3B0FEE01F9C03F3B07 F800FB001F03FF1480495BA2495BB2B53A1FFFE3FFFCA3361E7E9D3B>I<3903E03F8000 FFEBFFE001E37F9038E781F8390FEE00FCEA07FC49137E5BA25BB2B5380FFFF0A3241E7F 9D27>I<EB1FE0EB7FF83801FFFE3803F03F3907C00F80390F8007C0391F0003E04814F0 A2007EEB01F8A300FE14FCA9007E14F8A26CEB03F0A26C14E0EB8007390FC00FC03907F0 3F803901FFFE0038007FF8EB1FE01E207E9E23>I<3803E07F39FFE3FFE001E77F9038FF 03F8390FFC00FC6C48137E497F491480151F16C0A2ED0FE0A816C0151FA21680153F6DEB 7F006D137E6D5B9038EF03F89038E7FFF001E313C0D9E07FC7FC91C8FCA9B5FCA3232B7F 9D27>I<90381FC01C90387FF83C3801FFFC3803F81E3907E0077C390FC003FC001F1301 EA3F8090C7FC5AA2127E12FEA7127E127FA27EEB8001001F1303EA0FC03807E00F3803F8 3E3801FFFC6C13F0EB1FC090C7FCA991381FFFE0A3232B7E9D25>I<3807C1F038FFC7FC EBCFFEEBDE7FEA0FD8EA07F8EBF03E141C14005BB17FB51280A3181E7E9D1C>I<3801FE 183807FFB8001F13F8EA3F01EA7C000078137800F81338A37E6C13006C7EEA7FFC383FFF 806C13E06C13F06C13F8C613FC1307EB00FE0060137E00E0133EA27EA26C133C147C6C13 7838FF01F038F7FFE000E313C000C0130017207E9E1C>I<13E0A41201A31203A2120712 0F121FB512F0A33807E000AE1438A70003137013F03801F8E0EA00FFEB7FC0EB1F00152A 7FA81B>I<D803E0133E00FFEB0FFEA3000F13000007147EB115FEA3000313019038F003 7F2601F80E13F03800FFFCEB7FF890391FE07C00241F7F9D27>I<3AFFFE03FFC0A33A07 F000FE00157800031470A26C6C5BA2EBFC0100005CA290387E0380A2EB7F07013F90C7FC A2148FEB1F8EA2EB0FDCA214FC6D5AA26D5AA36D5AA2221E7F9C25>I<3BFFFE7FFE0FFF A33B0FF007F003F83B07E003E001E015F0D803F015C0A29039F807F80300011680A29039 FC0FFC070000D90E7C1300A290397E1E7E0EEC1C3E153FD93F3C5BEC381F169C90391FF0 0FB8A216F86D486C5AA36D486C5AA36D486C5A301E7F9C33>I<3AFFFE0FFF80A33A07F8 07F8000003EB03E0D801FC138000001307D97E0FC7FCEB7F1EEB3FBCEB1FB8EB0FF85C13 07130380497E497EEB1E7EEB1C3F90383C1F8001787FEBF00F48486C7E00036D7E390FF0 07F83AFFF80FFFC0A3221D7F9C25>I<3AFFFE03FFC0A33A07F000FE00157800031470A2 6C6C5BA2EBFC0100005CEBFE03017E5BA26D48C7FCA2148FEB1F8EA2EB0FDCA214FC6D5A A26D5AA36D5AA25CA213035C1238D87C07C8FC12FE130E131E485AEA7878EA7FF06C5AEA 0F80222B7F9C25>I<003FB51280A39038007F00003C137E00385BEA7801495A00705B49 5A130F495A00005B133F49C7FC9038FE038013FC1201EA03F813F03807E007120FD81FC0 1300495A48485A48137FB6FCA3191D7E9C1F>I<B712C0A32203809223>I<001C13E0383E 01F0387F03F800FF13FCA2007F13F8383E01F0381C00E016087AAD23>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FE cmcsc10 10 21 /FE 21 122 df<B512E0A33807FC006C5AB3B1487EB512E0A3132B7DAA19>16 D<00E014E0A36C1301007014C0A2007813036CEB0780003E130F6CEB1F00380FC07E6CB4 5A6C5BC613E0EB3F801B0F77B92E>21 D<913801FFC0020F13F8027F13FF903A01FF007F C0D903FCEB1FE0D90FF0EB07F8D91FC0EB01FC49486D7E49C8127F01FE6F7E48486F7E00 038349150F48486F7E000F83491503001F83A248486F7EA3007F834981A300FF1880AB00 7F18006D5DA3003F5FA26D1503001F5FA26C6C4B5AA26C6C4B5AA26C6C4B5A6C6C4B5A00 005F017F4BC7FC6D6C14FED91FE0EB03FC6D6C495AD903FCEB1FE06DB46CB45A6D6CB5C8 FC020F13F8020113C0393D7ABA46>79 D<D907F81330D93FFF137090B512C03A01F807F0 F03903E000F9D80780133D48C7120F481407123E1503481401A300FC1400A316707EA27E 6C15007F13E0EA3FF8EBFF806C13FC6CEBFFC06C14F06C14FC6C806C80013F1480010714 C0D9003F13E0020313F0EC007FED1FF81507150316FC1501126012E01500A47E16F87E15 016C15F016E06C14036C6CEB07C0D8FBC0EB0F80D8F9F0EB1F00D8F0FF137E39E03FFFFC 010F13F048C61380263D7ABA33>83 D<B500FE91381FFFF8A3000301C0020313806C90C9 EAFE006C177C6E1578017F1670A26E15F0013F5E80011F4B5AA26E1403010F5EA26D6C4A C7FCA26E5C0103150EA26E141E0101151C6E143C6D1538A26F1378027F147081023F5CA2 EDE001021F5C15F0020F495AA2EDF807020791C8FC15FC0203130EA2EDFE1E0201131C15 FF6E5BA216F86F5AA26F5AA36F5AA26F5AA36FC9FC3D3B7DB844>86 D<B6913807FFFEA3000101E0020013E06C6C48ED7F006E157C013F16786D7E606D6C5D6D 6C1401606D6C4A5A6E14076D93C7FC6D6D5B6F131E027F141C6E6C133C6F5B021F14706E 6C13F06F5B020713016E6C485A03FF5B6E13076E018FC8FC16CEED7FDEED3FFC5E151F6F 5AB3A24B7E021FB512F0A33F397EB844>89 D<14074A7EA34A7EA24A7EA34A7E1477A2EC E3F8A201017F14C1A201037F1480A2903807007FA3010E6D7EA2011E80011C131FA2013F B57EA39039700007F0A201F080491303A20001814913011203486C80D81FF0497ED8FFFC 011F13F8A32D2C7DAB33>97 D<B612F815FF823A07F8003FE00003EC07F06F7EED00FC16 7E167FEE3F80161F17C0A2EE0FE0A417F0AA17E0A3EE1FC0A21780163F1700167E16FE4B 5AED07F00007EC3FE0B75A93C7FC15F82C2B7DAA34>100 D<B712F0A33907F8000F0003 EC01F815001678A21638A3163CED701CA3160015F0A2140390B5FCA3EBF8031400A21570 1607A31500160EA4161EA2163E167E16FE0007EC07FCB7FCA3282B7DAA2E>I<B512E0A3 3807FC006C5AB3B1487EB512E0A3132B7DAA19>105 D<B539E001FFFCA3D807FCC713C0 6C48EC7E00167C16F04B5A4B5A4B5A4BC7FC151C5D15F04A5A4A5A4A5A140F4A7E4A7E14 7FECE7F09038FBC7F89038FF83FC1401496C7E497F497F6F7E6F7E82150F6F7E826F7E15 016F7E821780486C15C0B5D8E00313FEA32F2B7DAA35>107 D<B512F0A3D807FCC8FC6C 5AB3A41670A416E0A31501A21503A21507151F0007EC7FC0B7FCA3242B7DAA2B>I<D8FF FCEDFFFCA3D807FE4A13800003170001BFEC03BFA3D99F80EB073FA2D98FC0130EA3D987 E0131CA3D983F01338A2D981F81370A3D980FC13E0A291387E01C0A391383F0380A39138 1F8700A2EC0FCEA3EC07FCA26E5AA2EA07C0271FF001F0EB7F80D8FFFE91380FFFFCA2EC 00E0362B7CAA3E>I<D8FFF8903803FFF87FA2D803FE9038007FC06DEC1F006E130E13BF EB9FC080EB8FF01387EB83F8801381EB80FE147F1580143FEC1FC0EC0FE015F01407EC03 F815FCEC01FE1400157F168E153FED1FCEED0FEE16FE150715031501A21500167E487ED8 1FF0143ED8FFFE141E160EA22D2B7DAA33>I<EC7FC0903803FFF8010F13FE90393FC07F 8090397F001FC0D801FCEB07F048486D7E49130148486D7E000F814848147FA24848EC3F 80A2007F16C0A290C8121FA24816E0AA6C6CEC3FC0A3003F16806D147F001F1600A26C6C 14FE6C6C495AA26C6C495A6C6C495AD8007FEB1FC090393FC07F8090260FFFFEC7FC0103 13F89038007FC02B2D7BAB35>I<B612C015F815FE3A07F8007F800003EC1FC06F7E6F7E 6F7EA282A55EA24B5A4B5A4B5AED7F8090B500FEC7FC15F8819038F800FE153F6F7E8215 0F82A482A4171C16F81507486C153CB539E003FC38923801FFF09238007FE0C9EA1FC02E 2C7DAA32>114 D<017F13603901FFE0E0000713F9380F80FD381E001F48130714035A14 0112F81400A37E15007EEA7F8013F86CB47E14F06C13FC6C7F00037F6C1480D8003F13C0 13039038003FE0140FEC07F01403A20060130112E0A36C14E0A214036C14C06CEB0780B4 130F39F7E03F0038F3FFFE00E013F838C01FE01C2D7BAB26>I<007FB712C0A39039003F 801F007C15070078150300701501A200F016E0A2481500A5C71500B3A64A7E013FB57EA3 2B2B7DAA31>I<B539E003FFF8A3D807FCC7EA7FC06C48EC1F00160EB3A900015DA27F00 005D137E013E5C013F5C90391FC003C0903907F01F806DB5C7FC010013FCEC1FE02D2C7D AA33>I<B500C0EB7FFEA3D807FEC7EA1FF06C48EC0FC00001ED078017006D5C0000150E A2017F5CA26E133C013F1438A26D6C5BA26E13F0010F5CA2ECF00101075CECF80301035C A2ECFC07010191C7FC6E5A0100130EA2ECFF1EEC7F1C15BCEC3FB8A215F86E5AA26E5AA3 6E5AA26E5A2F2C7EAA33>I<B500C0EB3FFEA3D807FEC7EA1FF00003ED0F80000116006C 6C140E161E6D6C5B6D6C133816786D6C5B6D6C5B1501D907F85B4B5A903803FC07D901FE 90C7FC150E903800FF1EEC7F9C15BCEC3FF86E5AA26E5AAE4A7E0103B57EA32F2B7EAA33 >121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FF cmsy10 12 1 /FF 1 4 df<147014F8A8003815E0007CEC01F000FEEC03F8D8FF80130FD87FC0EB1FF0 01E0133F3A1FF870FFC02607FC7113003901FF77FC39007FFFF0011F13C0010790C7FCEB 01FCEB07FF011F13C0017F13F03901FF77FC3907FC71FF261FF87013C03A7FE0F83FF001 C0131FD8FF80EB0FF8D8FE001303007CEC01F00038EC00E0C71400A81470252B7AAD32> 3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FG cmbx12 12 42 /FG 42 122 df<EA07C0EA1FE0EA3FF8127F13FCEAFFFEA313FFA27EA27E7EEA07CFEA00 0FA2131EA4133CA21378A213F0120113E0EA03C01207EA0F80EA1F00120E12041022788E 1F>44 D<B612F8AA1D0A7F9A25>I<EA07C0EA1FF0EA3FF8EA7FFCA2EAFFFEA5EA7FFCA2 EA3FF8EA1FF0EA07C00F0F788E1F>I<EC01E01407140F143FEB01FF131FB6FCA413E3EA 0003B3B3AD007FB612FEA5274178C038>49 D<ECFFE0010F13FE017F6D7E90B612E00003 15F82607F8037F260FE0007FD81F80EB3FFF48C76C138001E06D13C0486C6D13E07F00FF 6E13F07F8117F8A36C5A6C5A6C5A6C5AC8FC17F05D17E0A24B13C0A24B138017004B5A5E 4B5AED7FE04B5A4A5B93C7FCEC03FC4A5AEC0FE04A5A4A4813F84AC7FC14FC495A4948EB 01F0495AEB0F8049C7FC013E140349140790B7FC4816E05A5A5A5A5A5AB8FC17C0A42D41 7BC038>I<ECFFE0010713FE011FEBFFC0017F14F0D9FF807F3A01FC003FFCD803F06D7E 48486D7E01FC1580487E6D15C05A1480A56C01001480A26C485BD801F81500C85B153F5E 4B5A4B5A4A13C0020F5B902607FFFEC7FC15F815FF16C090C713F0ED3FFC6F7E6F7E6F13 8017C017E08117F0A317F8EA0FE0487E487E487E487EA317F0A36C484913E05B4915C06C 484913806C48491300D80FF0495AD807FEEBFFFC6CB65A6C15E06C6C5C010F49C7FC0101 13E02D427BC038>I<163F5E5EA25D5D5DA25D5D5D5DA292B5FC5C5C15EFEC07CF140FEC 1F8FEC3F0F143E147C14FCEB01F8EB03F014E0EB07C0130FEB1F801400133E137E5B485A 5B485A1207485A5B48C7FC5A127E5AB91280A5C8001F90C7FCAB91B71280A531417DC038 >I<00031503D807E0143F01FFEB07FF91B6FC5E5E5E5E5E5E93C7FC15FC15F015C002FC C8FC01C0C9FCA9EC3FF001C1B5FC01C714C001DF14F09039FFC03FF89039FE000FFC01F8 6D7E496D7E4915804915C06C5AC86C13E0A217F0A317F8A21203EA1FC0487E487E7F12FF 7F17F0A25B17E06C485B4915C05B6CC74813806C6C15006D495AD80FF0EB3FFCD807FEEB FFF86CB65AC615C06D91C7FC010F13FC010113C02D427BC038>I<EE1F80A24C7EA24C7E A34C7EA24B7FA34B7FA24B7FA34B7F16BF031F80163F82033F80ED3E0F037E80157C8203 FC804B7E02018115F0820203814B7E0207815D177F020F814B7F021F8292C7FC834A8202 3E80027E82027FB7FCA291B87EA2498302F0C71203830103834A800107835C187F494882 84011F8491C97E4984133E84B6021FB612F0A54C457CC455>65 D<DCFFF81430031F01FF 14F04AB6EAE0010207EDF803021FEDFC07027F9039F001FF0F494848C7EA3F9F4901F8EC 0FFF010F01E01403490180804948C9FC4948167F4948163F485B191F4849160F485B1907 5A485B190391CAFC5A1901A25A5B96C7FCA212FFAC127FA36DEF01F07EA37E80F103E06C 7F7EF107C06C7F6C6DEE0F80191F6C6D17006D6C163E6D6C5E6D6C6C15FC6D01E0EC03F8 010301F84A5A6D01FFEC3FC06D6C9039F001FF80021F90B6C7FC020715FC020115F0DA00 1F1480030001F8C8FC44467AC451>67 D<BA1280A419C0D8003F90C7123F17031700187F 183F181F180F19E01807A31803A3EE03E0F001F0A495C7FC1607A3160F161F167F92B5FC A5ED007F161F160F1607A31603A693C9FCAFB712F8A53C447CC346>70 D<B712F0A5D8001FEB8000B3B3B3A4B712F0A524447DC32A>73 D<B712F8A5D8003F90CA FCB3B1F00F80A4181F1900A460A360A218FEA2170117031707170F171F177FEE03FFB95A A539447CC343>76 D<B695B612806F5E6F5EA3D8003F6D4C49C7FCA2013E6DEE0FBFA26E 6CEE1F3FA36E6C163EA26E6C167CA26E6C16F8A26E6DEC01F0A36E6DEC03E0A26E6DEC07 C0A26E6DEC0F80A36F6CEC1F00A26F6C143EA26F6C5CA36F6C5CA26F6D485AA26F6D485A A26F6D485AA36F6D485AA2706C48C7FCA293383FFC3EA3706C5AA2706C5AA2705BA3705B A2705BA2705BB66C93B71280A271C7FCA2173E61447CC36A>I<923807FFC092B512FE02 07ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF010701C0010713C04990C70001 7F49486E7F49486F7E49486F7E49486F7E488448496F13804A814819C04A814819E0A248 90C96C13F0A24819F8A348487013FCA500FF19FEAD007F19FCA26D5EA26C19F8A36C6D4B 13F0A36C6D4B13E0A26C6D4B13C06C19806E5D6C19006C6D4B5A6D6C4B5A6D6C4B5A6D6C 6C01035B6D6D495B6D01F0011F5B010101FE90B5C7FC6D90B65A023F15F8020715C00200 4AC8FC030713C047467AC454>79 D<B9FC18F018FE727E19E0D8001F902680001F7F0503 7F05007F727E727E721380A21AC084A21AE0A91AC0A24E1380A21A00604E5A4E5A05035B 051F5B92B712C096C7FC18FC18C00380CAFCB3A7B712F0A543447DC34D>I<B812F8EFFF C018F818FE727ED8001F902680003F13E005077F05017F717F727E727EA28684A286A762 A24E90C7FCA2614E5A18FF4D13F005075B057F138092B7C8FC18F818C018F0DB800013FC EF3FFF717F717F83717F8583A285A685A61B1FA2717FA2726C133EA2B700F06DEB807E72 EBE0FC72EBFFF8060314F0DE007F13E0CC0007130050457DC354>82 D<DAFFE0130C010701FE131C013F9038FF803C49ECE07C48B6EAF0FC489038801FFD3A07 FE0003FFD80FF813004848143F49141F003F150F160748481403A2160112FF1600A27F17 7C7FA27F01FE92C7FC6C6C7E14F8ECFFC06C14FCEDFFC06C15F86C8116FF6C826C826C82 6C82013F81010F811303D9003F801403DA001F7F15016F7E041F13808282127800F881A2 82A27EA218007EA26C4B5AA26D5D01E014076D5D01FC4A5AD9FF80EB3FE0489039F801FF C0D8FC3FB65A486C92C7FCD8F00714FC48C614F0480107138031467AC43E>I<003FBA12 E0A59026FE000FEBC003D87FF09338007FF049173F0180170F190790C7FC007E1803A300 7C1801A400FC19F8481800A5C81700B3B3A20107B87EA545437CC24E>I<B7017FB66C48 B512FEA5C66C48C8003F90C9EA7C006E7115FC6D705FA26F7014016D705F816D724A5A84 6F7014076D4C5FA26F4A6D140F6D64814E6D141F6D043E94C7FC6F705C6D047E163EF07C 7F6F02FC6D147E027F4B6C157C8105016F13FC6E4B6C5D048017016E020303C05B4E7E04 C0EEE0036E4A486C5DA2DCE00FEDF0076E4B6C5D16F0051FEDF80F6E4B6C5D04F8EEFC1F 6E4A94C8FC053E7FDCFC7E6F5A6E027C027F133E16FE05FCEDFF7E037F496E137C04FF17 FC6F604D80A26F496E5BA36F496E5BA26F604D80A26F6094C87EA26F486F90C9FCA36F48 167E047C167C6F457EC374>87 D<903801FFF0011F13FF017F14C048B612F04848C66C7E D807F8EB1FFC486C6D7E6D6D7E486C81818381836C5A6C5A6C5A6C5AC8FCA30203B5FC91 B6FC1307013F13F19038FFFE01000313F0481380481300485A485A485AA2485AA45DA26C 6C5BA26C6C010E13F86C6C013CEBFFC03A0FFF80F87F6CEBFFF06CECE01FC66CEB8007D9 0FFCC9FC322F7DAD36>97 D<EB3FE0B5FCA512037EB1ED0FFC92387FFFC002E3B512F002 EF14FC9139FFE01FFE92380007FF02FC010113804A15C04A6D13E04A147F18F018F8A217 3F18FCA318FEAB18FCA4EF7FF8A218F0A2EFFFE06E15C06E4913806E5B023F90380FFE00 903AFE1FE03FFCD9FC07B55A496C14E049C61480C8D80FF8C7FC37467EC43E>I<EC3FF8 49B57E010F14E04914F890397FF007FC9039FFC001FE4849487E48495A484A1380485AA2 485A123F6F13006F5A48486D5A6F5A93C7FCA212FFAA127FA27FA2123FEE07C06C7EEE0F 806C7E6CED1F006C7F6C6D133E6C01F013FC90397FFC03F86DB55A010F14C0010391C7FC 9038003FF82A2F7CAD32>I<4CB4FC0307B5FCA5ED001F82B1EC3FF0903803FFFE010FEB FF8F013F14EF90267FF807B5FC3901FFC00148496C7E4890C77E49140F120F485AA2123F A2485AA412FFAA127FA4123F7F121FA2000F5D7F6C6C147F6C92B512806CD9C00314FE6C 9038F01FEF013FB512CF6D140F010713FC9026007FC0EBF80037467CC43E>I<EC3FF849 B5FC010F14C0013F14F090397FF01FF89039FFC007FC48496C7E48496C7E48481580000F 80484815C0167F003F16E0A25B127FEE3FF0A212FFA290B7FCA401F8C9FCA5127FA36C7E A2001FED01F0A26C7EEE03E06C6C14076C6DEB0FC06C6DEB1F806C01F0EB3F0090397FFE 01FE011FB55A010714F0010014C0DA1FFCC7FC2C2F7DAD33>I<913801FF80021F13E002 7F13F849B512FC0107138790390FFE0FFED91FFC13FF49485A137F14F013FF14E048EC0F FEED07FCED03F8ED00E01600AAB612F8A5000101E0C7FCB3B0B612E0A528467DC522>I< DAFFE013FC010F9038FE03FF013FD9FF8F138090B812C048D9C07F1307489039001FF87F 4848EB0FFC000F9238FE3F80491307001F9238FF0E0094C7FC003F82A7001F93C7FCA200 0F5D6D130F00075D6C6C495A6C9038C07FF091B55A481580D8078F49C8FC018013E0000F 90CAFCA47FA213F090B612C016FC6CEDFF80836C16F0836C826C821203000F82D81FF0C7 7ED83FC01407007F6F1380498000FF81A56C6C4A1300A26C6C4A5A6D14076C6C4A5AD80F FEEC3FF83B03FFE003FFE06C90B65A6C6C92C7FC010F14F8D9007F90C8FC32427DAC38> I<EB3FE0B5FCA512037EB1ED03FF031F13E04B13F892B57E9139E1F81FFE9139E3C00FFF 9138E78007DAEE008002FE7F4A815CA25CA35CB3A7B600C1B61280A539457DC43E>I<13 7C48B4FC487F487FA2487FA56C5BA26C5B6C90C7FCEA007C90C8FCAAEB3FE0EA7FFFA512 037EB3AFB61280A519467DC51F>I<EC01F0EC07FC4A7E4A7EA24A1380A56E1300A26E5A 6E5AEC01F091C8FCAA913801FF800103B5FCA5EB000F80B3B3A6EA0F80EA3FE0EA7FF0A2 D8FFF814005CA25D4A5AEA7FF049485A393FC07FE06CB55A6C91C7FC000313FC38007FC0 215A87C522>I<EB3FE0B5FCA512037EB3B3B3A3B612C0A51A457DC41F>108 D<90277FC003FFEC07FEB5011F01E090383FFFC04B01F84913F092B56C48B57E913DC1F8 1FFE03F03FFC913DC3C00FFF07801FFE00039026C7800790380F000F6CD9CE00029C8002 DE6D01BC7F02DC03F8158002F85DA24A5DA34A5DB3A7B600C1B60083B6FCA5582D7DAC5D >I<90397FC003FFB5011F13E04B13F892B57E9139C1F81FFE9139C3C00FFF00039038C7 80076CD9CE008002DE7F02DC8114F8A25CA35CB3A7B600C1B61280A5392D7DAC3E>I<EC 1FFC49B512C0010714F0011F14FC90397FF80FFF9026FFC0017F48496C7F4890C76C7E48 486E7E000F8249141F001F82A248486E7EA2007F82A400FF1780AA007F1700A46C6C4A5A A2001F5EA26C6C4A5A00075E6D147F6C6D495A6CD9E0035B27007FF80F90C7FC6DB55A01 0F14F8010114C09026001FFCC8FC312F7DAD38>I<90393FE00FFCB590387FFFC002E3B5 12F002EF14FC9139FFE03FFE9238000FFF000301FC010313806C4915C04A6D13E05C7013 F018F8177FA218FCA2173F18FEAB18FC177FA318F817FF18F0A24C13E06E15C06E491380 6E5B02FF90380FFE009238E07FFC02E7B55A02E314E002E01480DB0FF8C7FC92C9FCADB6 12C0A537407EAC3E>I<90397FC03F80B5EBFFE002C113F802C313FC9138C7C7FEECCF07 00039038CE0FFF6C13DE14FC14F8ED07FEA29138F003FCED00F01600A25CB3A6B612F0A5 282D7EAC2E>114 D<90391FFC038090B5128F000314FF5A380FF003381FC0004848133F 48C7FC151FA248140FA27FA26D90C7FC13F0EBFF806C13FCECFF806C14E015F86C806C80 6C8012016C1580011F14C01301D9000F13E014010078EB007F00F8143FA26C141FA36C15 C0A27E6DEB3F807F6DEBFF009038FC03FE90B55A00F814F0D8F03F5B26E007FEC7FC232F 7CAD2C>I<EB03E0A61307A3130FA3131FA2133F137F13FF5A5A001F90B51280B7FCA400 0101E0C7FCB3A3ED03E0AA6CEC07C014F0017F1480ECF80F90393FFC1F0090381FFFFE6D 5B010313F09038007FC023407EBE2C>I<D93FE0EC7FC0B549B5FCA50003EC00076C81B3 A85EA35E7E5E043B7F6D6C017BEBFF8090393FFC03F36DB512E36D148301031403902600 3FF849C7FC392E7DAC3E>I<B6000FB539C03FFFF0A500019027E0007FF0C7EAF800707E 6E16016C6F6C5C80017F4D5A836E013F1407013F6F5C804C140F011F04805B6E90B5131F 6D02F901C090C7FC158003815D6D02F0EBE03E03C3157E6D9139E07FF07C15E303E715FC 6D4A6C6C5A03FF14F96D9139801FFDF0A218FF6E496C5BA26E486D5BA36E486D5BA26E48 6D90C8FCA36E486D5AA26E48147C4C2C7EAB51>119 D<B690B512F0A5C601F8903807E0 006D6C495A013F141F6E495A6D6C49C7FC6DEB80FE6D6D5AEDC1F86DEBE3F06D13F76DEB FFE06E5B5E6E90C8FC806E7FA26E7F6E7F4A7F5C4A7F4A7F91387F3FFE91387E1FFF14FC 49486C7F01036D7F49486C7F49487E02C08049486C7F49C76C7E496E7EB5D8F003B512FC A5362C7EAB3B>I<B6398007FFFCA5000101F0C7EA7E006C167C8017FC017F5D6E130101 3F5D6E1303011F5D6E13076D5D1580160F6D01C05B161F6D92C7FC6F5A6D143EEDF07E6D 147C15F816FC6E6C5A15FD023F5B15FF6E5BA26E5BA36E5BA26E90C8FCA26E5AA26E5AA3 5D14015D000F1303D83FC05B387FE0075D38FFF00F5D141F92C9FC143E387FE07EEBC1FC 383F87F8EBFFF06C13C000075BD801FCCAFC36407EAB3B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FH cmss8 7 42 /FH 42 124 df<EB07C0EB1FF8497E497E13FFEBFC7F3801F03F6E7E13E01203A3143F92 C7FC6D5A14FEEBF3FCD9FFF813C06C4913F8ECE001028013F0140048481303000715E0D8 1FFF130748018013C09038BFC00FD87F1FEB1F80387E0FE000FC9038F03F00903807F87E 6D6C5AEB01FF6D5B6C6D5A007E6D5A3A7F81FFF8076CB7FC7E6C13F96CEBE07F3A01FE00 0FF8282A7DA82F>38 D<131F133E137C13F8120113F0EA03E0120713C0120F1380121FA2 EA3F00A3123E127EA4127C12FCAF127C127EA4123E123FA3EA1F80A2120F13C0120713E0 1203EA01F013F81200137C133E131F103B7CAB18>40 D<12F8127C7E7E1380120FEA07C0 13E0120313F0120113F8A2EA00FCA3137C137EA4133E133FAF133E137EA4137C13FCA3EA 01F8A213F0120313E0120713C0EA0F80121F1300123E5A5A103B7DAB18>I<127EA6121E 123CA212381278127012F0070D7C8511>44 D<12FCA606067B8511>46 D<130C133C13FC120712FFA312F81200B3A9387FFFFCA416277CA61F>49 D<13FE3807FF804813E04813F04813F8387E07FCEA7C0138F800FE48137E0070137F1260 1220C7123FA2147FA2147EA214FC130114F8EB03F0EB07E0EB0FC0EB3F80EB7E005B485A 485A485A485A485A003EC7FC5AB6FCA518277DA61F>I<EB0FC0EB3FF013FF5A5A3807F8 30380FE000485A5B48C7FCA2127EEB0FC0EB7FE0387CFFF0B512F814FC13C1EB00FE147E 5A143F5AA5127C127EA2143E003E137E123F381F80FCEBC1F8EA0FFF6C13F06C13E06C13 8038007E0018287DA61F>54 D<137E3801FF80000713C04813F05AEBC3F8383F00FC127E 147C007C137E12FC143EA2143FA5147F127E14FF127FEA3F8390B5FC7E6C133E3807FE7E EA03F0C7FC14FCA2EB01F8A2381803F0383C0FE0383FFFC04813806C1300EA0FFEEA03F8 18287DA61F>57 D<EB01F8497EA2497EA214BEEB0FBF143FA2496C7EA2013E7FA2140F01 7E7F137C140701FC7F13F800016D7EA213F000036D7EA213E048B57EA24880A290388000 7F001FEC3F80A290C7FC48EC1FC0A2007E15E0150F127C00FC15F0150724287EA729>65 D<B512F014FEECFF8015C039FE003FE0EC0FF0EC03F8140115FC1400A31401EC03F81407 EC1FF0ECFFE0B61280ECFE008015C015F039FE001FF8EC03FCEC01FE1400157FA2153FA2 157FA215FE1401EC03FCEC1FF8B612F015C0150014F820287BA729>I<ECFF80010713F8 011F13FE017F7F90B6FC48EB807F3903FE001ED807F8130648481302484890C7FC5B123F 5B48C9FCA35A5AA691380FFF80A27E7EA2EC001F6C7E7F121F7F6C7E6C7EEA03FE3901FF 807F6C90B5FC7F011FEBFE00010713F801001380212A7DA829>71 D<00FEEC3F80B1B7FCA548C7123FB221287BA72C>I<B4ED3FE06D147FA36D14FF00FD15 F76D1301A200FC15E76D1303A26D1307017814C7A2017C130F013C1487A2013E131F011E 1407011F133FA26D133EEC807EA2903807C0FCA2010313F814E1A2010113F014F3010013 E0A3EC7FC0A2EC3F80A391C7FC2B287BA736>77 D<B46CEB1F807FA27FA2EAFDF0A2EAFC F8A2137CA2137E7FA2EB1F80A2EB0FC0A2EB07E0A2EB03F0A2EB01F8A2EB00FCA2147EA2 143F141FA2EC0F9FA2EC07DFA2EC03FFA280A28021287BA72C>I<49B4FC010F13E0013F 13F8497F48B6FC48010113803A07FC007FC001F0131F4848EB0FE04848EB07F0A24848EB 03F8A248C7EA01FCA300FEEC00FEA96C14016C15FCA36C6CEB03F8A26C6CEB07F06D130F 6C6CEB1FE06C6CEB3FC06D137F3A03FF01FF806C90B512006C6C13FC6D5B010F13E00101 90C7FC272A7DA82E>I<B512F014FE8015C039FE003FE0EC0FF01407EC03F8140115FCA2 1400A21401A215F814031407EC0FF0EC3FE0B612C01580ECFE0014F848C8FCB01E287BA7 27>I<B512F014FEECFF8015E039FE001FF0EC07F8EC01FC140015FE157EA315FE15FC14 01EC07F8141FB612F015E01580ECFE0014F838FE01FC130080147FA2EC3F80141F15C0EC 0FE0A2EC07F0A2EC03F8A2EC01FCEC00FEA2157F20287BA728>82 D<B712F8A5260001FCC7FCB3B125287EA72A>84 D<00FE15FC7E6CEC01F87F003FEC03F0 A27F001FEC07E0A26C6CEB0FC0A27F0007EC1F80A26C6C14005D7F0001147EA26C6C137C 15FCA2017F5B1401A290383F81F01483011F5B14C314C7010F5B14E7903807EF80A36DB4 C7FCA26D5AA326287FA729>86 D<13FE3803FFC0000F13E04813F014F8EB01FCEA1C0012 10C7127EA5EB7FFEEA03FF121F5A387FE07EEAFE005A14FEA2EAFE01EA7F0713FFA26C13 7EEA1FFC3807E000171D7E9B1E>97 D<12FCADEB0FC0EB7FF038FDFFF8B57E80138148C6 7E487F801580141FA8EC3F00A26C5B14FEEAFF83EBFFFC5C00FD5B00FC5B38001F801929 7CA720>I<EB7F803801FFE04813F84813FC5A381FC07C383F8018EB0008007E1300A35A A8127EA2007F1304383F801C381FC07C380FFFFCA2000313F86C13E038007F80161D7E9B 1B>I<EC1F80ADEA01FC3803FF9F000F13FF5AA2EA3FE0EB803F387F001F127EA25AA812 7EA2007F133F383F807FEBC0FF6CB5FC7E14DF0003131FD801FCC7FC19297EA720>I<13 7E3801FF80000713C04813E04813F013C3383F01F8EA7E001478007C137CB512FCA500F8 C7FC7EA3127C127E6C1304EB801C381FC0FC13FF7E000313F86C13E038007F00161D7E9B 1B>I<EB1F80EB7FE013FF5A5A3807F060EBE000485AA8B5FCA4EA0FC0B3A413297FA813> I<90387E03E03801FF9F48EBFFF05A5A391FC3F8001300487F003E137CA5003F13FC6C5B 13C36CB45A485B5C5CD83E7EC7FC90C8FCA2381FFFF014FEECFF804814C04814E0387E00 1F48EB07F01403A36C1307397FC03FE06CB512C06C14806C1400000313FC38007FE01C28 7F9A1F>I<12FCADEB1FC0EB7FF038FDFFF8B512FCA21381EB00FE48137EA25AB217287C A720>I<12FEA71200A8127EB3A807297DA80F>I<12FCB3B3A406287CA70F>108 D<90391FC00FE03AFC7FF03FF83AFDFFF8FFFCB500FD13FE91B5FC018113C00100EB807F 4890387F003FA248137EB2281B7C9A31>I<EB1FC038FC7FF038FDFFF8B512FCA21381EB 00FE48137EA25AB2171B7C9A20>I<133F3801FFE0487F487F487F381FE1FE383F807F49 7E007EEB1F80A2007C130F00FC14C0A8007EEB1F80A2007F133F393F807F00381FE1FE13 FF6C5B000313F06C5BD8003FC7FC1A1D7E9B1F>I<EB0FC038FC7FF038FDFFF8B57E8013 8348C67E487F801580A2141FA6143F1500A26C5B14FEEAFF83EBFFFC5C00FD5B00FC5BEB 1F8090C8FCAA19267C9A20>I<131EEAF87E13FE12F912FBEAFFF013C0138013005AA25A B00F1B7C9A15>114 D<EA03FE380FFF804813E05A5A387E03C038FC00401400A37EEA7F F013FE6C7E6C13806C13C0000113E038001FF013071303A21240127038FC07E0B5FC14C0 6C1380001F1300EA03FC141D7E9B18>I<EA0FC0A7B512C0A4380FC000B0EBE040EBE1C0 3807FFE0A27E14803800FC0013227FA016>I<00FC137EB314FEA2EAFE03B5FC7EA2383F FC7E381FE000171B7C9920>I<00FCEB1F807E007EEB3F00A2127F6C137EA2381F807C14 FCA26C6C5A13C1A200075B13E3A200035B13F3A23801F7C0A200005B13FFA26DC7FCA219 1A7F991C>I<00FC01FE137E1301007E157C6E13FC14EF1303003F9038CF81F8A2130726 1F87C713F015C31487A2D80F8F14E09038CF83E71403000715C0A21401D803DE1480A2EC 00F701FE13FF6C481400A281271A7F992A>I<00FCEB1F807E007EEB3F00127F6C137EA2 1380001F5BA2EA0FC0EBC1F81207EBE1F0A2EA03E3EBF3E012015CA21200EBF780137713 7F91C7FC7F133EA3133C137C137813F81201EA7FF05BA25B90C8FC19267F991C>121 D<B612FCA51E0580921F>123 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FI msbm10 10 11 /FI 11 127 df<DA03FF1418021F01E0133C027F01F8137C49B500FE13F8903B07FC00FF 81F0D90FE090381FC3E0D91F80903807EFC0013EC73801FF80496E130049157C484815FE 4914014848EC03EFEE0FCF484891381F8780EE3F0748C8387C03C016F8ED01F0001E9139 07E001E0ED0FC0ED1F80ED3E005D5D4A5AEC07E04A5A4A5A6C013EC7EA03C05C5C260783 F0EC0780EB87E02603CFC0EC0F0001DFC8FCD801FE151E49153E6C485DD803FC5D486C4A 5A260FDF80EB07E0261F0FE0EB1FC0263E07FCEBFF80D87C01B548C7FC486C6C13F84801 1F13E00060010390C8FC36307BAF41>63 D<DB3FF01360912607FFFE13E0021FEBFF8091 39FFF01FE101039038C003FF903907EF0001D91F1EEB007ED93E3C143CD97838141E4948 140E2601E0F0140748485A260781C0140301011501120E495A001C1600A24848C9FCA200 7017601800A2130E12E0AD12707FA31238A2121C6D7E120E180C6C6C6C151C0181163C6C 6C6C15786C6C6C15F02600F070EC01E0D978381403D93E3CEC07C0D91F1FEC1F80902607 EF80EB7E00903A03FBF003FC0100B612F0021F5C020791C7FC9138003FF8363D7DBA27> 67 D<007FB512FCB6FC7E3901C00E006C6C5AB3B3AD48487E007FB512FCB6FC7E1E397F B82D>73 D<B500E0903801FFFE6E5B6E7F260F00389038000FC06C6C6CEC07802603C01E 14036C6C7E806D6C7E01F87F6D6C7EEBDC0001CE137001CF13789038C7803C01C3131C90 38C1C00EECE00F9039C0F00780EC700391383801C0023C13E091381E00F06E13706E1338 913803803CEDC01E913801E00E0200130703701383ED780392383C01C3031C13E392380E 00F3030F1373923807803B0303133F923801C01FEEE00F923800F0071670EE3803163C16 1E160E16071783EE03C31601EE00E317F3177B486C153B486C151F387FFFC0B5150F6C16 07CA12031701373B7DB832>78 D<EDFFE0020F13FE027FEBFFC0903A01FFC07FF0903A07 FE000FFCD90FBCEB07BED93E78903803CF80D97C70903801C7C0D9F0E0903800E1E0D801 E0EDE0F02603C1C0EC7078018116384848486E7E000F171E000E170E001C8349C8121C00 38EF0380A30070EF01C0010E150EA300E0EF00E0AD0070EF01C0A36D151C0038EF0380A3 001CEF07006D6C1438000E170E000F171E0007171C260381C04A5A01C116786C6C6C4A5A D800F0EDE1E0D97C70903801C7C0D93E78903803CF80D90FBCD907BEC7FCD907FEEB0FFC 903A01FFC07FF06D6CB512C0020F49C8FC6E5BED001E6E6C7E6F6C7E02016D7E913900E0 01F0923AF000FE01C00378EB3FFF033E130F6F01011300923907C0003E923903FC01FC03 00B512F0043F13C0DC03FEC7FC3B4B7DBA35>81 D<007FB612E0B712FC6CEDFF80270380 1E0313E03B01C03800F3F0EE7078707E83170E707EA2EF0380A7EF0700A25FEE380E173C EE707CEEF3F0923807FFE0023FB5128004FCC7FC1638EC38E0ED701CA26F7EA26F7EA26F 6C7EA26F6C7EA292380380E083923801C07083ED00E083167083EE380FEF0780EE1C03EF 01C0040E13E048486CEC00F0007FB539F007FFFEB6FC6C8137397DB836>I<007FB812C0 B9FCA23BE0FE38071FC1D8E3E0EC01F1D8E7C0EC00F9D8EF00153D00FE161F48160F4816 07A2481603A2481601A400601600C71600B3B14A6C7E011FB6FC5B7F32397DB838>84 D<0007B712FC5AA23B0E1FE0038038D93E005C0178EB0700D80FF0495B49EB0E01018001 1C5B001F4B5A90C71238001EDA7807C7FCED700F001CECE00E5EEC01C002035BC7EB8078 91380700705E140E91381E01C0141C4A485A1507027090C8FC150E14E001015BECC03C90 380380385DEB0700495BEB0E01011C49EB01804A4813031338D97807C7FCD9700F1407EB E00E4A1500EA01C00003495C13804848485C02F05C000E495C494814F7001CED01E7263C 0380EB03C7D83807EC078E007090C7EA1F0E010E14FEB812FEA331397DB83E>90 D<962603FFFE1F030603B600F0F61F8095B8F501FF051F05C01C0F0403B900F899B51200 043F7209075B0303B626E0000301FF097F13F0033F02E0C8001F01C00707B512800203B5 00FCC9000301F0077F01F8C7FC021F02809326007FFC0607B512C049B500F0CBD81FFF95 B500FCC8FC010F91CC000701E0041F14E090B500F0070101FC0303B548C9FC000791CE26 7FFFE00103B612E0003F01F0091F90B848CAFC4801807518E0D8FFF8CF000105FCCBFC01 C0766C168000FCD1000703E0CCFC0060E4003F01E0CDFCC11480CAC2>94 D<0060150600F0150F6C151F007C153F6C157F6C15FF6C6C5B6C6CEB03EF6C6CEB07CF6C 6CEB0F8F6C6CEB1F0F017C133E6D137C6D13F890380F81F0903807C3E0903803E7C09038 01FF806D1300147EA214FF491380903803E7C0903807C3E090380F81F090381F00F8013E 137C49133E49131F4848EB0F8F4848EB07CF4848EB03EF4848EB01FF48C8FC003E157F48 153F48151F48150F00601506282874A841>111 D<EB03F013FF5AA2380007E0A4495AA2 0007B612E0826C15C026001F80C8FCA349C9FCA4137EEC03F8EC1FFE4A6C7E9039FCFC0F C09038FDE007D9FFC07F148048EB00035B5B49130712035BA34848495AA34B5A485AA24B C7FCA2485A92387E0380A348C738FC0700A3EDF80E127E5E5EED787848EC7FF0007C6E5A 0038EC0F80293B7CB92D>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FJ cmsl10 10 39 /FJ 39 122 df<B512FCA616067C941C>45 D<121E123FEA7F80EAFFC0A31380A2EA7F00 123C0A0A798917>I<EC01C014031407143FEB01FF90B51280A23801FE3FC7FCA3EC7F00 A614FEA6495AA6495AA6495AA6495AA6495AA4133F497EB612F0A215E01C3878B72A>49 D<EC07F8EC3FFF91B51280903901F01FE09039078007F090390F0003F8011C14FC491301 16FE5B5B01FC14FF7F487EA35B16FE6C5A137890C7EA03FCA2150716F8ED0FF016E0151F ED3FC01680ED7F0015FE4A5AEC03F04A5A4A5A4A5A4AC7FC147C5C495A495AD9078013E0 49C7FC131E49EB01C05B5BD801C013034848EB078048B6FC5A4815005A5AB65AA228387C B72A>I<EC0FF0EC7FFE49B51280903903F01FC0903907800FE090390E0007F04914F813 3C137F1480A213FFA2EB7F00017E130F011C14F090C7FCED1FE0A2ED3FC01680ED7F0015 FE4A5AEC0FF0903803FFE092C7FC15E090380001F86E7E157E81168016C0151F16E0A316 F0A2121E003FEC3FE0487E12FFA216C090C7127F48158000F8ECFF005A0070495A007849 5A6C5C001FEB0FF0390FE03FC00003B55AC601FCC7FCEB1FE0253A7BB72A>I<16E01501 15031507150F151F16C0153F157F15FF5C15DF913803BF80EC073F140F141E143C143891 38707F0014F0EB01E0EB03C01480EB0700010E13FE131E5B5B13705B0001495AEA03C0EA 07801300120E5A003C495A5A5AB712F8A3C73807F000A64A5AA5EC3FF0011FB512C0A325 397BB82A>I<01081418011E147890391FE007F891B512F016C0491480160015FC15F001 3B13C00138C8FC5BA65BA4147F9038E3FFE0D801CF7F9038DF81F89038FC007C01F0137E 497F5B491480C8FC151F16C0A2153FA4120C123E127F6DEB7F8012FF90C7FC1600485C00 F85C12E06C495A00705C4A5A6C495A003C495A6CEB3F806C6CB4C7FC3807FFFC000113F0 38007F80253A7AB72A>I<160E161EA2163FA25EA25EA24B7FA25DA2ED077FA2030E7F16 3F151CA21538A203707F161F15E0A2EC01C0A2DA03807F160FEC0700A2140EA24A801607 5CA25CA291B67EA25B9138C00003495AA249C77F1601130EA25BA2498182137813F8487E D80FFE02031380B500C0017F13FF93B6FC18FE383C7DBB3E>65 D<017FB612F017FEEFFF 8001009039C0003FE06E48EB0FF0EF07F84AC713FC170318FE1701A3495AA31703A218FC 49481407EF0FF818F0EF1FE0EF3FC0EFFF804948491300EE0FFC91B612F05F17FC9139F8 0001FE49489038007F80EF3FC0A2EF1FE018F0170F495A18F8A5494815F0171FA218E017 3FEF7FC0495AEFFF804C13004C5AEE0FFCD801FFEC3FF0B812C094C7FC16F837397DB83B >I<DB07FC130492397FFF800E0203B5EAC01C913A0FFC03F03C913A1FE000787CDA7F80 EB3CFC02FEC7120ED903FC140FD907F0EC07F849481403495A495A017F150191C8FC01FE 16F000011600485AA2485AA2484816E0A2121F5B003F1700A3485AA5485AA61703007F5E A3170EA2123F6D5DA2001F5E7F000F5E6C6C5D4C5A6C6C4A5A6C6C4AC7FC6C6C140E017F 143CD93FC013F890390FF807F06DB512C0010091C8FCEC1FF0373D77BA3C>I<013FB512 E0A25B9039007FE0006E5AA24A5AA64AC7FCA6495AA6495AA6495AA6495AA6495AA6495A A5EBFFE0007FEBFFC0A2B6FC23397EB81E>73 D<017FB512F0A3010001C0C8FC6E5AA24A C9FCA6495AA6495AA6495AA6495AA5EF03804948EC0700A35F170EA24948141E171CA217 3CA25F494814F8160116034C5A161FD801FFEB01FFB8FC5FA231397DB834>76 D<90267FFFC0923803FFFE6181010094380FFE00027F5F191D02EF4C5AECE7F01973A219 E3A2902601C3F84A485AA2F00387A2F00707DAC1FC140ED903814C5A181CA2DA80FE1438 A21870D907004C5A18E0157FEF01C0EF0380A2010E4B48485AED3F80170EA25FA2496D6C 4849C7FCA25FA26F6C5AEEE1C04917FEEEE380A2DB07F7C7FCA216FE494C5A5E150301F8 5C00015DD807FEEE07FEB500F0D9E003B512FC150116C04F397DB84C>I<ED0FFC92B57E 020314E091390FF80FF891393FC003FC4AC77ED901FC147F4948EC3F804948EC1FC04948 EC0FE0495A4948EC07F049C8FC49ED03F8485A4916FC12035B0007EE01FE120F5B121FA2 5B123F17035B127FA54848ED07FCA4EF0FF8A3EF1FF0127F18E0173F18C06DED7F80123F EFFF004C5A6C7E4C5A6C6C4A5A00074B5A6D4A5A6C6C4A5A6C6C4AC7FC6C6CEB01FE6D6C 485A90391FF01FF06DB512C0010149C8FC9038003FF0373D77BA41>79 D<017FB612E017FC17FF0100D9C00013C06E48EB1FE0EF0FF04AC7EA07F8EF03FCA318FE A2495AA518FC49481407A218F8EF0FF0A2EF1FE04948EC3FC0EF7F80933801FE00EE0FFC 91B612F017C04902FCC7FC02F0C9FCA5495AA6495AA6495AA548487EB67EA337397DB839 >I<017FB6FC17F017FC01009038C003FF913A7F80007FC0EF1FE04AC76C7EA2717EA284 A2495AA54D5A495A60171F604D5A4D5A494802FEC7FCEE03FCEE1FF091B612C094C8FCA2 903A0FF0007F80EE1FC0707E707E831603494880A649481307A6494816C01801A3F00380 48486C1303B60080EC0700933801FE0E92C7EAFFFCCAEA3FF8EF07F03A3B7DB83D>82 D<B5D8FC01B5D8F801B51280A24B5D0007902780000FFEC7383FF8006C48C76C48EC0FE0 494BEC07807F000197C7FC71140EA21A1E040F151C1A3C6D021F15386C1978DC3BFE1470 163904715DA204E14A5A80017F01014B5AEEC1FFDB03C04AC8FC5E190E4B5A61ECC00E01 3FEE803C4B017F133819784B1570A24B5D14E0011F49ECC1C0173F02E1EDC3805D02E303 C7C9FC5D02F715CE010F90C7FC18FC02FE141FA24A5DA24A5D13074A5D170F4A5DA24A5D A295CAFC6D4880513B75B855>87 D<14FF010F13E04913F890383E01FC496C7E01FE137F 0001801680A25B6C5A132090C7FCA4EC3FFF0103B5120090381FFE7FEB7FC03801FF00EA 03FCEA07F0484813FE485A123F4848140EA21401D8FF00EBFC1CA21403387F8006020E13 38003FEB1CFE3A1FE0F8FFF03A0FFFE07FE06C01C013C03A00FE001F0027277CA52A>97 D<EB3F80D81FFFC8FCA312017EA25BA6485AA6485AA2EC0FE0EC7FFC9038F9FFFE9038FB E07F3A07FF801F809039FE000FC049EB07E05B4914F01503484814F8A648481307A516F0 4848130FA216E0151F16C0A248C7EA3F8016006D137E5D6D485A397CE007F039F8781FE0 39F03FFF80486C48C7FC380007F8253B78B92E>I<EC7F80903803FFF0010F13FC90383F C07E90387F003F01FC137F484813FF485A1207484813FE157C484813101500485AA2127F A290C8FC5AAA6C14181538A26C6C137015E0391FC001C0390FE003803907F81F003803FF FEC613F8EB1FC020277AA525>I<EE3F8092381FFF00A3150181A25EA64B5AA64B5AA214 FF010313C3010F13F390383FC0FB90397F003FF001FC131F4848130F485A48481307120F 4848495AA2123F5B127FA290C7485A5AA5484A5AA6007E4AC7FC127F6C5C5C6C6C5A3A0F C00F7F803A07F03EFFFC3803FFFCC601F05B90263F80FEC7FC293B7AB92E>I<147F9038 03FFE0010F7F90383F81F890387E00FC49137E485A48487F485A120F491480121F123F5B A2127F90B6FC1600B7FC90C9FCA65A7EA26C140C151C7E6D5B001F5C6C6C5B3907E003C0 3903F80F806CB5C7FC38007FFCEB1FE021277BA525>I<ED3F80EDFFE0020313F0EC0FE1 91381F83F8EC3F07147E02FE13F014FC0101EB03E04948C7FCA4495AA6495AA30003B512 F0A326001FC0C7FCA6495AA649C8FCA613FEA6485AA5487EB512F8A3253B7FBA19>I<EE 03C091391FE01FF09138FFF83F01039038FEF8F8903907F03FE1903A0FC01F81F0903A1F 800FC0E0D93F001400017E80A213FE5B1201A44B5AA200005D4BC7FC017C133E017E13FC 90387F83F8ECFFF001C713C02601C1FEC8FC0180C9FC1203A37F7F90B512E015FE6C6E7E 6C814881000781390FC0003F4848EB07F848C71203007E1401A25AA34B5AA24B5A007E4A 5A6CEC3F80D81FC001FFC7FC390FF007FC0003B55AC614C0D91FFCC8FC2D387FA52A>I< 14FEEB7FFCA313071303A25CA6495AA6495AA2ED1FC0EDFFF002E17F9138E7C1FC90391F CE00FE14DC02D8137E02F0137F5C167E494813FEA25CA44948485AA601FE495AA6484849 5AA50003140FB500F1B512C0A202E114802A3A7EB92E>I<EB01E0EB03F0EB07F8130F14 FCEB1FF8130F14F0EB07E0EB03C090C7FCA9EB07E0EA03FFA338003FC0131FA5EB3F80A6 EB7F00A613FEA6485AA51203B512E0A316387EB718>I<153C157E15FF5C16804A130080 5D6E5A157892C7FCA9EC01FC14FFA3EC07F81403A54A5AA64A5AA64A5AA64A5AA64AC7FC A614FEA3121C003E5BEA7F0100FF5B495AA2387E07E0387C0FC0D83FFFC8FC6C5AEA07F0 214986B719>I<14FEEB7FFCA313071303A214F8A6EB07F0A6EB0FE0A6EB1FC0A6EB3F80 A6EB7F00A613FEA6485AA51203B512F014E0A2173A7EB918>108 D<90270FE01FE0EB3FC0D803FFD97FF8EBFFF0489027E1FFFC037F913BE3C0FE0781FC3D 003FC7007F0E00FED91FCC141802D86D48137E02F05C4A4A137F197E4948494813FEA24A 91C7FCA449C700FE495AA601FE4948495AA648484948495AA5486C496C497EB500F1B500 E3B512C0A202E102C3148042257EA446>I<90390FE01FC0D803FFEBFFF04801E17F9138 E7C1FC3A003FCE00FEEB1FDC02D8137E02F0137F5C167E494813FEA25CA44948485AA601 FE495AA64848495AA50003140FB500F1B512C0A202E114802A257EA42E>I<EC3FC09038 01FFF801077F90381FC0FF90397F003F8001FCEB0FC0485A4848EB07E0485AED03F0485A 001F15F8A2485AA2127FA290C712075AA416F048140FA216E0A2ED1FC0A2007FEC3F8016 006C147E6D5B6C6C485A390FE007F03907F81FE00001B512806C49C7FCEB1FF025277BA5 2A>I<903901FC07F0017FEB3FFE01FF90B5FC9139FDF03F80903A07FFC00FC06D010013 E04AEB07F05C4AEB03F8A2494814FCA6495AA5EE07F8495AA217F0160F17E0161F494814 C0EE3F806EEB7F0016FE6E485A6E485A90397F3C0FF091381FFFC06E90C7FCEC03FC91C9 FCA213FEA6485AA3487EB512F0A32E3581A42E>I<90380FC0FC3903FFC3FE48EBC7FF91 38CF3F8038003F9C90381FB87F14B09138F03F00ECE03E150C4948C7FCA25CA449C8FCA6 13FEA6485AA5487EB512F8A321257EA421>114 D<903801FE0C90380FFF9C013F13FCEB 7E03EBF0004848137812034913381207A3000F14306D130013F86CB4FC14F86C13FE806C 14806C14C0013F13E0130F9038007FF0140F0018130700381303A2140115E0007C1303A2 15C0007E13071580007FEB1F0038FFC07E38F3FFFC00E013F038C03F801E277DA521>I< 1307A45BA2131EA2133EA2137E13FE485A1207121FB6FCA33803F800A6485AA6485AA638 1FC01CA6383F8038A2121F147013C014E0380FE1C03807FF806C1300EA00FC183479B220 >I<D801FCEB07F0007FEB01FF00FF5BA23A07F8001FE00003140FA54848EB1FC0A64848 EB3F80A64848EB7F00A45DA248485B1401A2001F130314064A7E260FE07913F83807FFF1 6C01E113F03A007F81FC00252679A42E>I<B53A87FFF83FFFA2148F260FFC009038800F F86C4890397F0007E00003EE03C018806F13074B14005F6D49130E0001ED801E4A141C03 DF133CDA039F133817789026FE071F13700000EDC0F0020E5CED0FC1021C5C16C3D9FF38 EBE380017F14E7027091C7FCED07EF02E013EE16FE02C05B133F02805B150302005BA201 3E5C131E011C6D5A382579A33C>119 D<0003B53803FFFC17F85D27001FFC011380D907 F8EBFC00ED00F00103495A02FC5B0101495A6E48C7FC01005BECFF1E6E5AEC3FF85D6E5A A2140F81A24A7E143BEC7BFC14F1903801E0FE49487E49487ED90F007F011E133F498001 7C131FD801FC80D807FE497EB548B51280A32E247FA32C>I<90B538C07FFE5A15803B00 0FFC001FF0D907F0EB0FC01700800103140E5EA25E8001015C16F05EECFE0101005C4B5A A24BC7FC14FFEC7F0E151E151C15BCEC3FB815F0A25DA26E5AA25D92C8FC80140EA25CA2 5C147814705C123E387E01C0EAFE03495A49C9FCEAFC1EEA787CEA7FF8EA3FE0EA0F802F 3580A32C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FK cmr7 7 38 /FK 38 127 df<B7FCA33903F0007FED1F80150F15071503A4ED01C0A392C7FCB3A5B512 E0A322287EA729>0 D<EC03804A7E4A7EA24A7EA24A7E143BEC79FC1471ECF0FE14E001 01137F4A7E010380EC801FD907007F150F010E8015074980013C6D7E133801786D7E1370 01F06D7E5B0001157F5B0003ED3F8049141F000716C090C8120F4816E0000E15074816F0 1603003FB712F84816FCA2B812FEA22F2A7DA937>I<90383FFFFCA3D9007EC7FCA69038 03FF80013F13F83901FC7E7FD807E0EB0FC0D80FC0EB07E0D81F80EB03F0D83F00EB01F8 4815FC00FEEC00FEA6007FEC01FC6C15F8D81F80EB03F0D80FC0EB07E0D807E0EB0FC0D8 01FCEB7F0039003FFFF801031380D9007EC7FCA690383FFFFCA327287DA72F>8 D<0207131CA4020E5BA44A5BA44A5BA44A485AB812F8A27E270001C007C7FCA490380380 0EA349485AA4007FB712F8B8FCA227001C0070C7FC495BA449485AA449485AA4484848C8 FCA42D337CA737>35 D<1306130C13181330136013E0EA01C0EA0380A2EA07005AA2121E A35AA3127C1278A412F8AD1278A4127C123CA37EA37EA27EEA0380A2EA01C0EA00E01360 13301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01C013E0A2 EA00F0A31378A3137C133CA4133EAD133CA4137C1378A313F0A3EA01E0A213C0EA0380A2 EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C7000EC8FCB3A22B2B7D A333>43 D<B5FCA410047F8E16>45 D<EB3F803801FFF03803E0F83807803C48487E001E 7F003E1480A2003C1307007C14C0A400FC14E0AE007C14C0A36CEB0F80A36CEB1F006C13 1E6C6C5A3803E0F86CB45A38003F801B277EA521>48 D<13381378EA01F8121F12FF12FE 12E01200B3AAB512F8A315267BA521>I<13FF000313E0000F7F381E07F8383801FC486C 7E0078137F00FC7F6C1480A2141FA2127CC7123F1500A2147EA25C5C495A495AEB078049 C7FC131E5B13709038E00380EA01C0EA03803907000700120E1218003FB5FC5AB55AA319 267DA521>I<13FF000713E0487F381F01F8383C00FC147E007E137F80A3003C5BC7127E A25C5C495AEB0FE03801FF8091C7FC380003E0EB00F8147C147E80A21580A21238127C12 FEA21500485B0078137E5C383F03F86CB45A000713C0C690C7FC19277DA521>I<143814 7814F8A2130113031307A2130E131C1338A2137013E0A2EA01C0EA0380EA0700A2120E5A 121812385A5AB612E0A3C7EAF800A890383FFFE0A31B277EA621>I<0010130C001F137C EBFFF85C5C148049C7FCEA1DF0001CC8FCA6137F381DFFE0381F81F0381E0078001C7F00 18133EC77EA31580A21230127C12FCA3150000F05B0070133E00785B6C13FC381F03F838 0FFFE000035BC648C7FC19277DA521>I<EB0FE0EB3FF8EBFFFC3801F81E3803E01F3807 803F120FEA1F00121E003E131E91C7FC127E127C1304EB3FC038FCFFF038FDC078B4C67E 143E48131E141FA2481480A4127CA4003C1400123E001E133EA26C5B3807C0F86CB45A6C 13C06C6CC7FC19277DA521>I<1238123E003FB512E0A34814C015803978000700007013 0EA2485B5C5CC7FC5C495A495AA249C7FC5BA2131EA3133EA2133C137CA413FCA7137813 301B287DA621>I<137F3803FFE04813F8380F80FC381E003E48131E0038131F00787FA3 127C007E131EEA3F80EBE03C6C6C5A380FFCF03807FFC06C5BC613E0487F38079FFC380F 07FEEA1E0348C67E48133FEC1F8048130F1407A46C140000785B007C130E6C133C381F80 F86CB45A00035BC66CC7FC19277DA521>I<1238127C12FEA3127C12381200AB1238127C 12FEA3127C123807197B9813>58 D<B812E0A27ECBFCAB007FB712E0B8FCA22B117D9633 >61 D<B512E0A3D803F0C8FCB3A3ED01C0A315031680A31507A2150F151F153F913801FF 00B7FCA322287EA729>76 D<B67E15F015FC3903F000FFED3F806F7E6F7E150782A55EA2 4B5A4B5A4BC7FC15FE90B512F05D9038F001F8EC007C81153F8182A482A317201770ED0F E0A2B5D8C00713E0923803F0C0923801FF80C9EA3F002C297EA730>82 D<EAFFC0A3EAE000B3B3B1EAFFC0A30A3B7AAB13>91 D<EAFFC0A31201B3B3B112FFA30A 3B7FAB13>93 D<13FE3807FFC0380F03E0381C00F0003E1378003F137C143C143E121EC7 FCA3EB3FFEEA01FF3807F03EEA1FC0EA3F00127EA2481438A3147E127E14DE393F838FF0 390FFE0FE03903F807C01D1C7E9A21>97 D<EB3FC0EBFFF83803E03C3807C00E380F801F 381F003F123EA2007E131E007C1300A212FCA7127C127E1407123E6C130EEA0F803807C0 1C3803F0783800FFE0EB3F80181C7E9A1E>99 D<EC03E0143FA314071403A9EB3F833801 FFF33803E03F3807800F380F0007481303123E127E127CA212FCA7127CA2127E123E001E 1307001FEB0FF0390F801FFE3803E07B3801FFE339007F83E01F297EA725>I<133F3801 FFC03803C1F038078078380F003C121E003E133E141E5A141F12FCA2B6FCA200FCC7FCA4 127C127E003E1307A26C130E7E3807801C3803E0783800FFE0EB3F80181C7E9A1E>I<EB 07E0EB3FF0EB7C78EBF0FCEA01E01203EBC078000713301400A8B51280A33807C000B3EA 7FFEA316297FA815>I<EA0F8012FFA3121F120FA9EB81FCEB8FFF90389C0F809038B007 C013E09038C003E0A31380AF39FFF83FFEA31F287EA725>104 D<120E121FEA3F80A3EA 1F00120EC7FCA7EA078012FFA3121F120FB2EAFFF8A30D287EA713>I<260F81FC137F3B FF8FFF03FFC0903A9C0F8703E0903AB007CC01F0D81FE013D83B0FC003F000F8A301805B AF3CFFF83FFE0FFF80A3311A7E9937>109 D<380F81FC38FF8FFF90389C0F809038B007 C0EA1FE0390FC003E0A31380AF39FFF83FFEA31F1A7E9925>I<EB3F80EBFFE03803E0F8 3807803C48487E001E7F003E1480A248EB07C0A300FC14E0A7007C14C0A2007E130F003E 1480001E1400001F5B380F803E3803E0F86CB45A38003F801B1C7E9A21>I<380F83FC38 FF9FFF9038BC0FC09038F007E0391FC003F0380F800115F8EC00FCA2157C157EA7157C15 FCA2EC01F801C013F0EC03E09038F007C09038BC1F8090389FFF00EB83F80180C7FCA7EA FFF8A31F257E9925>I<3803F8C0EA0FFFEA3C0FEA78031270EAF001A312F86CC7FCEA7F C013FC6CB4FC6C1380000713C0C613E0130738E003F0130113007EA214E0EAF80100FC13 C038FE078038E7FF00EAC1FC141C7E9A1A>115 D<EA01C0A51203A31207A2120F383FFF E0B5FCA23807C000AB1470A7000313E013E03801F1C03800FF80EB3E0014257FA31A>I< 390F8003E000FF133FA3001F1307000F1303AF1407A20007EB0FF09038C01BFE3803E073 3801FFE339007F83E01F1B7E9925>I<380F8010381FF038383FFFF04813E038E07FC038 400F8015067BA621>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FL cmmi7 7 65 /FL 65 123 df<EB07F0EB3FFCEB7C1F3A01F00F80703A03C007C0E000071303D80F8013 E0D81F00EBE1C05A003EECF180007EEB01F316004814F715FEA25D485C5DA31403007C13 07003C011E13E0003E90387CF8C0391F01F0793A07FFC03F803A01FE001E00241B7D992B >11 D<EC01FCEC07FF91381E078091383801C002E013E0903901C000F0495A49C7FC1501 130E5B16E049130316C0ED0780491400EC7FDEECFFFCA29038E07FDEEC000FED0780A248 4814C0A4485AA448C7EA0F80A216005D486C131E5D6D5B6D5B391CF003C090387FFF80D9 1FFCC7FCEB00804890C8FCA45AA45A243480A826>I<133F3A01FFC001C04813E0489038 F0038048EC0700381F00F8003CEB7C0E48133C0070EB1C1C48131EEC0E385AC7EA0F7014 075DA25DA35DA292C7FCA45CA2140E141EA45CA31438A2143022267F9923>I<EB07F8EB 1FFF01381380EB303FEC1F00140691C7FC1338133CA27F131F6D7E806D7EA2EB3FF0EB7B F8EA01E13803C1FCEA0780EA0F00001E137C003E137E123C007C137CA25AA4481378A214 F85CA2387801E05C6C485A6C48C7FCEA0FFE6C5A192A7DA81E>I<EB0FF8137F3801FFF0 3803F000EA07C0485A48C7FC5A127E387FFFC014E0B512C000FCC7FCA6127CA2123C003E 13606C13E0EA0F833803FF803800FE00151A7D981C>I<3907801F80390FE07FE03918F1 E0F03938F380F83930FF0078EA70FE5B5B00E114F85B1201A23903E001F0A43907C003E0 A4390F8007C0A4391F000F80A2120EC7FCEC1F00A4143EA4143C14381D267D9922>17 D<EA01F0EA03FCEA007E7FA26D7EA2130F80A26D7EA2130380A26D7EA26D7EA2147C147E A214FF5BEB03DF9038079F80EB0F1F90381E0FC0133C90387807E013F83801F003D803E0 13F0EA07C0390F8001F8EA1F00003E13004814FC5A157E0070143F20297CA727>21 D<137001F81338157CA248485BA44848485AA44848485AA44848485AEDC380A3001F9038 0F8700141FA29038C0778E393FE0E7CE9038FFC3FC393E7F00F090C9FC5AA45AA45A5A21 267D9928>I<D803E01370003F14F8127F123F3907C001F0A3EC03E0EA0F80EC07C01580 140F391F001F00141E5C5C003E5B495A495A495AD87C1FC7FC133C13F0EA7FC048C8FC12 781D1A7C9921>I<48B61280000715C0481580481500263C0E07C7FC0070130EEAE01CA2 1200133CEB381EA21378A213F0A3EA01E0141F1203A2D807C07FA2380F800FA26C486CC7 FC221A7D9827>25 D<0107B51280013F14C049148048B6FC2603F03FC7FC3807C00FD80F 807F1300481307123E140F123C127CA3007849C7FC12F80078131E143E143C5C6C5B381C 01E0381F07C0D807FFC8FCEA01F8221A7E9826>27 D<1407A4140EA45CA45CA2903807FF 80011F13E090387C70F0D801F0133CEA03C0D80780131E390F00E01F48140F123EA2387C 01C0A200F8141FA2EB0380151E153E153C903807007C1578007814F0EC01E0393C0E03C0 001EEB0F80000FEB3E003807FFF8000113E0D8001CC7FCA35BA45BA420347CA728>30 D<1538A35DA45DA44A5AA3D803E014703A07F00380F8D80C78EB81FCEA187C12380030EB 07000070157C13F800E01538140EEA01F0A2167048485AA216E0EA07C091383801C0ED03 80A20003EC07009038E0700E0001143C01F8137039007E71E090383FFF80D907FEC7FCEB 00E0A2495AA4495AA449C8FC26347DA72C>32 D<157C39038001FF4848481380000E010F 13C048140791381C01E0481338EC7000A2485BA2495A00E0EC01C014801303ED038039F0 07000716000070140E0078143CD83E0F137C393F0E01F8391FFE0FE06CB55A00035CC649 C7FCEB3FF0013CC8FCA2137CA21378A213F8A35B5B23267C992C>39 D<1238127C12FEA3127C123807077A8614>58 D<1238127C12FE12FFA2127F123B1203A3 1206A3120C121812381270122008127A8614>I<160E163E16FEED03F8ED0FC0ED3F0015 FCEC03F0EC0FC0023FC7FC14FCEB03F0EB0FC0013FC8FC13FCEA03F0EA0FC0003FC9FC12 FC12F012FC123FEA0FC0EA03F0EA00FC133FEB0FC0EB03F0EB00FC143FEC0FC0EC03F0EC 00FC153FED0FC0ED03F0ED00FE163E160E27277AA134>I<EC01801403A2EC0700A2140E A35CA25CA35CA35CA2495AA3495AA249C7FCA3130EA25BA35BA25BA35BA2485AA3485AA2 48C8FCA3120EA35AA25AA35AA25AA25A193B7CAB22>I<12C012F8127EEA1F80EA07E0EA 01F8EA007EEB1F80EB07E0EB01F8EB007EEC1F80EC07E0EC01F8EC007EED1F80ED07E0ED 01F8ED007E161E167EED01F8ED07E0ED1F80ED7E00EC01F8EC07E0EC1F80027EC7FCEB01 F8EB07E0EB1F80017EC8FCEA01F8EA07E0EA1F80007EC9FC12F812E027277AA134>I<5B 5BA4497EA50040140800FFEB83FC6CB512F8001F14E000071480C6EBFC00EB3FF06D5A49 7EA2EB7CF8EB7878497EEBE01C0001131EEBC00E48487E39070003800002EB01001E1D7E 9C22>I<EB03FCEB0FFF013F13C090387C07E09038F001F09038E000F800011478486C13 3C153EA25B6C5AC8123FA3EB1FF8EB7FFCEBF8063901E0033E3907C001FEEA0F8090C7FC 5A4814FC123E127EEC01F85A15F0A214034814E0EC07C0A2EC0F8000781400007C131E6C 137C6C485A6CB45A6C13C0D801FEC7FC202A7CA823>I<ED0780150FA2151FA2153F157F A24B7E15EFEC01CFEC038FA2EC070F140F140E021C7F150714381478147014E0130114C0 D903807FA249B5FC5BA290381C0003A25B5B8249130112015B1203120FD8FFF890383FFF C0A25B2A2A7CA932>I<017FB512F016FEEEFF80903A03F0003FC0161FEE0FE049481307 17F0A3494814E0160F17C0161F4948EB3F80EE7F0016FEED03F849B512E0A291380003F8 ED00FC017E147E821780A25BA44848EC7F00A216FE4B5A4848495AED0FF0ED3FE0B71280 4BC7FC15F02C287CA732>I<4AB41308020FEBE038027FEBF078903A01FF8078F0903903 F8001DD90FE0130FEB1F8049C7EA07E0137E491403485A484815C0485A120F5B001F1680 5B003F92C7FCA248CAFCA4127E12FEA2160E5E127EA25E007F5D7E5E6C6C495A6C6C495A 6C6C010FC7FCD803F8131E6CB413F86CEBFFF0013F13C0D907FCC8FC2D2A7DA830>I<01 7FB512F816FEEEFF80903A03F0003FC0EE0FF016034948EB01F8EE00FCA2177E495AA449 48147FA3177E49C8FC17FEA3017E15FC160117F816034915F0A2EE07E0EE0FC0485AEE1F 80EE3F0016FE4848EB01F8ED07F0ED3FC0B75A03FCC7FC15E030287CA736>I<017FB612 C017E0A2903903F0000FEE03C0A249481301A4495AED0381A21780494848C7FCA25D5D90 383FFFFEA3EC003E017E131CA4495B151892C8FCA2485AA4485AA3B512C0805C2B287CA7 2A>70 D<4AB41308020FEBE038027FEBF078903A01FF8078F0903903F8001DD90FE0130F EB1F8049C7EA07E0137E491403485A484815C0485A120F5B001F16805B003F92C7FCA248 CAFCA4127E00FE91383FFFE05DA2923800FC00127EA3007F4A5A7EA26C7E6C6C495A6C6C 1307D803F8130F6CB4137E6C9038FFF860013FEBE020010790C8FC2D2A7DA834>I<903B 7FFFC03FFFE0A3903B03F00001F800A34948495AA44948495AA44948495AA449B65AA391 C7121F017E4AC7FCA449147EA448485CA44848495AA3B539807FFFC0A333287CA736>I< 90387FFFC0A3903803F000A3495AA4495AA4495AA449C7FCA4137EA45BA4485AA4485AA3 B57EA31A287CA71D>I<91383FFFE05C16C0913800FC00A34A5AA44A5AA44A5AA44A5AA4 4A5AA44AC7FCA4001C137E123E127F5C12FE387C01F800785B383007E0383C0FC0D80FFF C8FCEA03F823297CA725>I<903B7FFFC001FFF05EA2D903F0C7130017F8EE01E04948EB 07C04CC7FC161C167849485BED01C0ED07804BC8FC90381F803C5D15F8148349487E141E EC3C7E147049487E14C09138001F80137E496D7EA26F7EA248486D7EA26F7EA248486D7E 82A2B5398007FFF05D6F5B34287CA738>I<90387FFFE0815DD903F0C7FCA3495AA4495A A4495AA449C8FCA4137EA216301638491470A216E0A24848130116C01503ED0780484813 0F153F913801FF00B7FC5DA225287CA72E>I<D97FF8EDFFF0805F0103923803FC00A2EF 077CD9077CEC0EF880171C1738010EED39F0177117E180011C4A485AEE0383A2EE070301 3891380E07C0A291380F801CA201704A485A1670A216E001E0902681C01FC7FCEC07C1ED C380EDC700D801C0153E15CE15DCEC03FCD80380495B5DEA0FC0D8FFFC9039E01FFFE015 C014013C287BA73F>I<D97FF0903807FFE0808001039138007E006E1438A2D9077F5C80 81141F010E6D5B140F811407011C6D485A1403A2EC01F801384A5AEC00FCA2157E494AC7 FC153FA2ED1F8749148EED0FCEA2ED07EE484814FC1503A2150148485C1500EA0FC0D8FF FC14781670163033287CA735>I<4AB4FC021F13E091387F01F8903901F8007ED907E07F D90F80EB1F8049C7EA0FC0017E14074915E0484814034915F01203485A120F4915F8121F 5B123F17F048C81207A4EE0FE012FEEE1FC0A21780163F007E1600167E007F15FE6C4A5A 5E6C6C495AED07C06C6CEB1F806C6C49C7FC6C6C13FC3900FC07F090383FFFC0D907FCC8 FC2D2A7DA832>I<017FB512F016FEEEFF80903A03F0003FC0EE0FE01607494814F01603 A349481307A317E04948130F17C0EE1F80EE3F0049C712FEED03F891B512E01680017EC9 FCA45BA4485AA4485AA3B57EA32C287CA72A>I<017FB512C016F882903903F000FFEE3F 80161F4948EB0FC0A44948131FA3EE3F8049481400167E5E4B5A90393F000FE091B5C7FC 5DEC001F017E6D7E6F7E82A25BA44848130FA3173048481570A21760B5D8800713E09238 03E1C0923801FF80C9EA7E002C297CA732>82 D<91381FE0189138FFFC380103EBFE7890 3907E03FF090380F000F011E13035B49EB01E0137013F0A2000115C0A292C7FC7F7F6CB4 7E14F86DB47E6D13F06D7F01077F01007F1407EC00FF153F81A281121CA2003C141EA215 1C153C007C5C007E5C397F8003E09038F00FC000F3B5C7FC00E05B38C01FF0252A7CA829 >I<000FB712C017E04816C09039800FC00FD81E0014035A0038EB1F80A2481680A2EC3F 005AA2481600C7007E90C7FCA45CA4495AA4495AA4495AA4495AA3000FB57E48806C5C2B 287DA727>I<3B7FFFC007FFE0A3D803F0C7EA7E001638A248485CA448485CA44848495A A448C7485AA4007E4AC7FCA448140EA35DA25D5D127C5D6CEB03C06CEB0F80260FC07EC8 FC6CB45A000113F038007F802B297BA72D>I<D87FFFECFFF0B55BA2D807E09038003F00 163C163816786D147000035D15015E4B5A15076D91C7FC150E0001141E151C5D15786D13 705D000013015D4A5A140792C8FCEBFE0EEB7E1E141C5CA25CEB7FE0A26D5A5CA291C9FC A2133E131C2C297BA727>I<D87FFFD9FFFEEB7FFCB580495CD80FE0D90FE0EB0FC04CEB 0780190000076F130E181E031F141C033F5CA203775C18F003E75C02014A5A01F013C7DA 0387495A0003EDF007DA070391C7FC020F140E140E021C5CA202385C0278147802701470 D9F8E05CA22601F9C0EBF9C0ED01FBD9FB805C01FF02FFC8FC1400495CA2495C5E5B6C48 5CA2496D5A3E297BA73E>I<903B1FFFE00FFFC05B6D13C0903B00FE0003F80017E0027F EB078094C7FC6E130E6F5A021F5B6F5A5E91380FE1C0EDE380DA07F7C8FC15FE6E5A5D6E 7EA28114034A7E4A7EEC1E3F023C7FEC781F02F07FEB01E049486C7EEB078049486C7E13 1E496D7E5B496D7E1207D87FFE90380FFFE000FF4A7F6C485D32287DA736>I<B5ECFFF0 4B13F86F13F0D807F0EC3F006D143C00035D5E6C6C495A4B5A6C6C495A4BC7FC017F131E 151C5D6D6C5A5DEB1FC1ECC3C090380FE78002EFC8FC14FE6D5A5C6D5AA25C1307A35C13 0FA35C131FA33807FFFCA25C2D287CA727>I<EB0FC0EB3FE09038F873803801E03F3807 C01F1380EA0F0048130F48EB1F00123E127EA248133EA4485B1538A3ECF870EA78011303 393C0778E0381E1E7C390FFC3FC03903E00F001D1B7D9924>97 D<EA01E0EA3FF05BA212 03A2485AA4485AA4381F1F80EB7FE0EBF0F0EBC078EA3F80EB007C123EA248137EA44813 FCA438F001F8A214F0EB03E0A2EB07C00070138038780F00EA3C3EEA1FF8EA07E017297C A71D>I<EB07F0EB3FFCEBFC1E3801F00F3803C01F3807803F120F381F003E5A003E1300 127EA25AA45AA31404140E0078131E007C133C003C13F8381F03E03807FF803801FC0018 1B7D991E>I<1578EC0FFC15F8A21400A2EC01F0A4EC03E0A490380FC7C0EB3FE7EBF877 3801E03F3907C01F801380EA0F0048130F48EB1F00123E127EA248133EA4485B1538A3EC F870EA78011303393C0778E0381E1E7C390FFC3FC03903E00F001E297DA723>I<EC03E0 EC0FF0EC1E38EC3C3CEC7C7C15FC1478ECF8F815701500A2495AA590B512E0A215C09038 03E000A3495AA5495AA649C7FCA5133EA4133C137C123CEA7C78127EEAFCF85BEA78E0EA 71C0EA3F806CC8FC1E357CA820>102 D<EB01F0EB0FFC90381E1EE0EB7C0FEBF807EA01 F03803E003A20007EB07C013C0120FA2391F800F80A49038001F00A35C6C133E147E3807 80FEEA03C33801FF7CEA007C1300A25CA21238387C01F012FC495A48485A38781F80D83F FEC7FCEA0FF01B267E9920>I<130E131F5BA2133E131C90C7FCA7EA03E0487EEA0C78EA 187C1238123012705B12E0A2EA01F0A3485AA2485AA2EBC380EA0F83A2381F0700A2130E EA0F0C131CEA07F06C5A11287DA617>105 D<1407EC0F80141FA21500140E91C7FCA7EB 03E0EB07F8EB1E3C1318EB303E137013E0A248485AA2C7FCA25CA4495AA4495AA4495AA4 495AA21238D87C1FC7FC12FC133E485AEA70F8EA7FE0EA1F80193380A61B>I<133CEA07 FE5BA2EA007CA25BA4485AA43903E00780EC1FC0EC38E014613807C0C3EBC187EBC30701 C613C0390F8C038001B8C7FC13E07FEA1FFEEB3F80EB0FC06D7EEA3E0315E0A3007CEBC1 C0A2158014C339F801E700EB00FE0070137C1B297CA723>I<1378EA07FCEA0FF8A21200 A2EA01F0A4EA03E0A4EA07C0A4EA0F80A4EA1F00A4123EA45A1338A3EAF870A21360EA78 E013C0EA3F80EA0F000E297EA715>I<3B07801FC007E03B0FE07FF01FF83B18F0E0F878 3C3B38F1807CE03E3B30FF007DC01ED870FEEB3F80491400153ED8E1F8017E133E49137C 1201A24848495BA35F4848485A1870EE01F0A23C0F8003E003E0E0A2EFE1C00401138027 1F0007C013E3933800FF00000E6D48137C341B7D993B>I<3907801FC0390FE07FF03918 F0E0F83938F1807C3830FF00D870FE133C5BA2D8E1F8137C5B1201A248485BA34A5AEA07 C016E0EC03E0A23A0F8007C1C0A2EDC38002031300D81F0013C7EC01FE000EEB00F8231B 7D9929>I<EB07F0EB3FFEEB7C1F3901F00780D803C013C000071303D80F8013E0EA1F00 5A123E007E14F015E0481307A315C048130F1580A2EC1F00143E007C133C5C6C485A381F 07E03807FF80D801FCC7FC1C1B7D9921>I<9038F007E03901FC1FF039031E787C0007EB E03E39061FC01E000EEB801F1400A2D81C3E1480A212001600495BA3153E49137EA2157C 5D12014A5A4A5A6D485A2603EE0FC7FCEBE7FEEBE1F001E0C8FC485AA4485AA2EA7FF812 FFA22125809922>I<90380F8180EB7FE3EBF0F73803E07F3907C03F00EA0F80497E5A48 133E123E127EA2485BA4485BA4495A1303EA7807EA3C0F495AEA1FFBEA07E3EA0003495A A4495AA2EBFFF85AA219257D991E>I<3807807E390FE1FF803818F3C33938F703C03830 FE073870FC0FA201F8138039E1F0070091C7FC1201A2485AA4485AA4485AA448C8FCA212 0E1A1B7D991F>I<EB0FE0EB7FF8EBF03C3801C01E12033807803EA2143C000F1318EBE0 006CB4FC14C06C13E06C13F06C13F813071301EA3C00007E1378A24813F05A387001E0EB 03C0383C0F80381FFE00EA07F8171B7C991F>I<131C133EA25BA45BA4485A387FFFE0B5 FC14C03803E000A4485AA4485AA448C7FC14E0A2EB01C0123EEB0380EB0700EA1E0EEA1F 1CEA0FF8EA03E013267EA419>I<90387C07C03901FF1FE0390787B870390E03F078000C 14F8001C13E1123815F0397007C0E015001200A2495AA449C7FC15701238007C14E0EAFC 3EEC01C012F839F07F03803970EF0F00383FC3FC381F01F81D1B7D9926>120 D<EA03E0486C1370D80C7813F8EA187C0038EB01F01230127013F800E0EB03E0A2EA01F0 A2EC07C0EA03E0A33907C00F80A4EC1F00A25C00035B3801E0FE3800FFBEEB3F3E13005C 121E003F5B5C387E01E0383C03C038380780D81C1FC7FCEA0FFCEA07F01D267D9922>I< 90383E01C0137F9038FF838048EBC7004813FF380781FEEB001C00065BC75A5C495A495A 49C7FC130E133C13705B3901C00380EA03803907000700000E5B381FF81EEBFFFE38383F FC486C5A486C5AEB07C01A1B7D9920>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FM cmsy7 7 21 /FM 21 113 df<B712FEA327037A8F34>0 D<1238127C12FEA3127C123807077A9114>I< 0060140600F0140E0078141E6C143C6C14786C14F039078001E03903C003C03901E00780 3900F00F00EB781E6D5A6D5A6D5A6D5A6D5A497E497EEB1E78497E497E497E3901E00780 3903C003C039078001E048C712F0001E147848143C48141E48140E006014061F1F769D34 >I<1338A50060130C00F8133E00FC137E00FE13FE383FBBF83807FFC000011300EA007C 48B4FC000713C0383FBBF838FE38FE00FC137E00F8133E0060130C00001300A517197B9A 22>I<1406140EB3B812E0A3C7000EC8FCB1B812E0A32B2B7CA834>6 D<EC7FC0903803FFF8010F13FE90393F8E3F809039FC0E07E0D801F0EB01F0D803C0EB00 78D8078080D80F0080000E150E4881A248ED0380A248ED01C0A348ED00E0A3B8FCA326E0 000EC7FCA30070ED01C0A36CED0380A26CED0700A26C150E000F151ED807805CD803C05C D801F0495AD800FCEB07E090393F8E3F8090260FFFFEC7FC010313F89038007FC02B2B7C A334>8 D<EC7FC0903803FFF8010F13FE90393F803F809039FC0007E0D801F0EB01F0D8 03C0EB0078000781486C147ED80EF014FED81C78EB01E76DEB03C7486C90380783806DEB 0F033B7007801E01C0903803C03C903801E0783BE000F0F000E0EC79E0EC3FC06E5A6EC7 FC4A7E4A7EEC79E0ECF0F03B7001E07801C0903803C03C903807801E3B380F000F038001 1EEB07836C48903803C70049EB01E76C48EB00FED80FE0147E6C48143C00035DD801F049 5AD800FCEB07E090393F803F8090260FFFFEC7FC010313F89038007FC02B2B7CA334>10 D<913801FFC0020F13F8027F13FF903A01FE003FC0D907F0EB07F0D90F80EB00F8013EC8 123E498101F0ED078048486F7E48486F7E48486F7E90CA1270000E83001E173C001C171C 4883A34883A348EF0380A90070EF0700A36C170EA36C5F001E173C000E17386C5F6D16F0 6C6C4B5A6C6C4B5A6C6C4B5A017C031FC7FC6D153ED90F8014F8D907F0EB07F0D901FEEB 3FC0D9007FB5C8FC020F13F8020113C039357CA842>13 D<137F3801FFC0000713F0380F 80F8381E003C487F0038130E487FA248EB0380A70070EB0700A26C130E003C131E6C5B38 0F80F86CB45A000113C06C6CC7FC19197C9A22>I<1606161E167EED01F8ED07E0ED1F80 ED7E00EC01F8EC07E0EC1F80027EC7FCEB01F8EB07E0EB1F80017EC8FCEA01F8EA07E0EA 1F80007EC9FC12F87E123FEA0FC0EA03F0EA00FC133FEB0FC0EB03F0EB00FC143FEC0FC0 EC03F0EC00FC153FED0FC0ED03F0ED00FC163E160E1600AB007FB612FCB712FEA227357A A734>20 D<12C012F012FC123FEA0FC0EA03F0EA00FC133FEB0FC0EB03F0EB00FC143FEC 0FC0EC03F0EC00FC153FED0FC0ED03F0ED00FE163E167EED01F8ED07E0ED1F80ED7E00EC 01F8EC07E0EC1F80027EC7FCEB01F8EB07E0EB1F80017EC8FCEA01F8EA07E0EA1F80007E C9FC12F812E0CAFCAB007FB612FCB712FEA227357AA734>I<13E0EA01F0EA03F8A3EA07 F0A313E0A2120F13C0A3EA1F80A21300A25A123EA35AA3127812F8A25A12100D1E7D9F13 >48 D<D97F80147F2601FFE0903803FFC0000701F8010F13F04801FE90381F8038261F01 FF90383E001C281C007F8078130C486D6C487F4890261FE1E07F006090380FF3C06EB45A 486D90C7EA0180806E5A157F6F7E4B7E8200604A6CEB0300913801E7F8912603C3FC5B6C 90260781FE130E6C49C66C5B001C013E90387FC07C6C01FC6DB45A6CB448010F5B000101 E0010313C06C6CC890C7FC391B7C9942>I<49B5FC130F133F01FEC7FCEA01F0EA03C048 5A48C8FC121E121C5AA25AA35AA3B7FCA300E0C8FCA31270A37EA27E121E7E6C7E6C7EEA 01F0EA00FE013FB5FC130F130120277AA12D>I<150C150E151CA21538A21570A215E014 0115C0EC0380A2EC0700A2140EA25C143C14385CA25CA2495AA2495A130791C7FC130EA2 5BA25B137813705BA2485AA2485AA248C8FC5A120E5AA25AA25AA25A12601F3576A800> 54 D<1406140EB3B2007FB712E0B8FC7E2B287CA734>63 D<147EEB03FEEB0FE0EB1F00 133E5BB35BA2485AEA07E0EAFF8000FCC7FCB47EEA07E0EA01F06C7EA2137CB37F7FEB0F E0EB03FEEB007E173B7BAB22>102 D<12FCB47EEA0FE0EA01F06C7E137CB37FA27FEB0F C0EB03FEEB007EEB03FEEB0FC0EB1F00133EA25BB35B485AEA0FE0EAFF8000FCC7FC173B 7BAB22>I<12E0B3B3B3A5033B78AB14>106 D<38C00180EAE003B3B3B3A3EAC001113B78 AB22>I<186018E0170118C0EF0380A2EF0700A2170EA25F173C17385FA25FA24C5AA24C 5AA24CC7FC5E160E5EA25EA26D5CEA03C000075DEA1FE0003F4A5AD863F0130300C35DD8 01F849C8FCA26C6C130EA2017E5BA26D5BA26D6C5A90380FC0F05D903807E1C0A2903803 F380A26DB4C9FCA26D5AA2147CA21438A2333A7B8237>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FN cmmi10 10 82 /FN 82 127 df<EC3FC0ECFFF0010713FC90380FE07E90263F801F131C90397E000F8049 14C0484801071338484814E0485A03031370484814F0001F16F04915E0123F90C7EBF1C0 5AEEF380A200FEEDF70016FEA25E5E485DA382127E150F151F003E027D13186C02F11338 390F8007E03B07E03FC07C706CB538007FE06C01F8EB3FC026003FC0EB0F802E267DA435 >11 D<ED01FC923807FF80031F13C092387E03E09238F800F0DA01E013F8DA0380137C4A C7FC140E4A147EA25C5C177C5C17FC494814F8A24948EB01F0A2EE03E049C7EA07C0EE0F 809238FFBF00020313FED90E0713F86E7F6E133E91C77E49801780A2EE07C05BA317E05B A317C049140FA40001ED1F805B17005E486C143E5E16FC6D5CD80770495A0178EB07E06D 495A011F013FC7FC390E0FF7FE903807FFF8010113C090CAFC5AA45AA45AA45A12602F4B 80BA2F>I<EB0FE0D93FF814C0D9FFFC1301487F486DEB0380481480260FF07FEB070039 1F800FC0263E0007130E003CEB03E04801015B0070130000F014F048EC7038A2485DC812 78ED38E0A2ED39C0A3ED3B80A2033FC7FC81A2151E153EA2153CA31538A31578A2157015 F0A44A5AA44A5AA45D140792C8FC14022A377EA42B>I<EC0F80EC3FF891B512804914C0 14C0903803807F0107EB1F809138000F0092C7FC80A36D7EA2806D7EA2806D7EA2147E14 7F808149B47E01077FEB0FEF90383F0FF0EB7E0701F87F3801F003EA03E01207496C7EEA 0F80121F1300481300A2127EA25D5A1401A35DA24813035DA26C495A127C4A5A003C91C7 FC003E131E6C133E6C6C5A3807FFF06C5BC61380223D7DBB25>I<EC3FE0903801FFF001 0F13E090381FC000017FC7FC13FC485A485A485A485A121F5B123FA2387FFFFE80A290C8 FC127E12FEA5127EA3123E123FA26C14406C6C13E03807C0033903F81F806CB512003800 7FFCEB0FE01C257DA322>I<D803E0137F3A07F803FFC0486C4813F03A1C3E1F81F8EC3C 0026383F7813FC5C007049137C5C5C91C712FC485A137EA2120049EB01F8A44848EB03F0 A44848EB07E0A44848EB0FC0A44848EB1F80A44848EB3F00120F6CC7FCC8FC157EA45DA4 4A5AA44A5AA25D6E5A26377EA429>17 D<EC03F0EC0FF8EC3FFCEC7C1E4A7ED901F01380 EB03E0903907C007C0EB0F80A2EB1F004914E0133E137E5BA2485AA212035B0007140FA2 5B120F16C049131F121FA290B6FC481580A3903880003F48C7EA7F00A3157E007E14FEA2 5D14015D12FE4A5A127C4A5AA24A5A5D141F007E91C7FC003E133E5C1478001E5B381F01 E0380F87C06CB45A6C90C8FCEA00F8233C7EBA27>I<1338017C14F0017EEB07F849EB1F FC153F157B913801E3F83A01F803C1F04AC65A020E90C7FC5C3803F0785CEBF1C0EBF780 48B4C9FC14F814FF15C0390FC03FE0EC07F06E7E140148486C7EEE0380A33B3F0001F807 00A3160E127E0200130C161CED7C3848EC3FF0007C6E5A0038EC07C029267CA430>20 D<133F14C080EB07F06D7E6D7EA2130080A2147E147F80A281141F81A2140F81A2140781 A2140381A2140181A2140081140114034A7E4A7EEC1E3F023C13801478ECF01F010114C0 EB03E0903807C00FD90F8013E0EB1F00013E13074914F001FC1303484814F8485A484813 01484814FC485A003F140048C812FE12FE48157E48153F0070151F283B7CB930>I<EB03 80496C13076EEB0F804948131FA44948EB3F00A449C7127EA4017E5CA449495AA4484849 5A1738A30003913807E070A2150F151F000716E06D137F9238F3E1C09038FF03E33B0FDF FFC1FF8001CF01001300D9C3FC133E01C0C9FC485AA448CAFCA4127EA45AA25A12702D37 7EA432>I<017E1438D81FFE147C003F157E4914FC1200A2ED01F8485AA2ED03F0A24848 EB07E0A2ED0FC0A24848EB1F8016005D157E4848137C5D4A5A1403D81F805B4A5A4A5A02 3EC7FC48485A5CEB03F0EB07C0387E1F8001FEC8FCEA7FF813E0EAFF800078C9FC27257C A429>I<14035CA5EDFFC04A13E0143F91B5FC903903FC7F80D907F0C7FC495A495A495A 91C8FC5B13FEA3485AA512007FA2137F90383FBFF06DB47E6D7F495B90383F7FE0017CC8 FC13F0485A485A485A120F90C9FC5A123EA2123C127CA312FCA3127E127F7FEA3FE0EA1F FC13FF6C13E0000313F86C13FF6C6C13C0010F7F01037F9038007FF8140F14031400A401 1E5B90381F81E06DB45A01035BD9007EC7FC234B7EB924>I<013FB612E090B712F05A5A 4816E0270F80700EC7FCEA1E00001C13E048141E48141C130100E013C012000103133CA2 14801307A2157CEB0F00A25BA2131E133EA25BA201FC137EA25B0001147FA248487FA249 131E6C48130C2C257EA32F>I<15FE913803FF80020F13C091381F83E091383E01F09138 F800F8494813FC4A137C13034948137E130F5C131F91C7FC4914FE133E137EA349EB01FC A316F84848130316F0A2ED07E0485AED0FC06D1480ED1F000007143E6D5B6D5B9038EF03 F0390FC7FFE001C31380D9C0FCC7FC91C8FC485AA448C9FCA4127EA45AA25A127027377E A42B>I<027FB512C00103B612E05B131F4915C090267F81FEC7FC9038FC007E48487FEA 03E048487F000F81485AA248C7FCA2127EA3484AC7FCA3153E157E5A5D5D14015D6C495A 007C495A5D6C011FC8FC6C133E380F81FC3807FFF06C5BC66CC9FC2B257DA32F>I<013F B512FE90B7FC5A4815FE4815FC260F801CC7FCEA1E00001C133C5A48133814785AC7FCA2 5CA31301A25CA21303A3495AA3130FA25CA2131FA349C8FC7F130E28257EA324>I<1503 824BC7FCA4150EA45DA45DA45DA3EC1FFF027F13E00103B57E903907F0E1FC90391F80E0 7E90393F01C01F137C01F8EC0F80D801F015C03A03E0038007EA07C0000F16E0EA1F8090 380007005AA2127E140EA248ED0FC0A25CEE1F80A217004A5B167E007C157C5E007E4948 5A003E4A5A6C4A5AED1F80270FC0E03FC7FC3907F0E1FC0001B55A6C14C0011F90C8FCEB 01C0A3495AA449C9FCA4130EA4130C2B4B7CB931>30 D<13F8D803FE1504486C6C140E26 0E1FC0141C486C6C143817706D6C14F00000ED01E0010315C06EEB03800101EC07006E13 0E5E01005C6E5B5EEC7E0191387F03C0023F5B4BC7FC158EEC1F9C15B815F05D6E5A81A2 14074A7E141F143B1473ECE3F8EB01C1EB038101077FEB0F00010E7F5B49137E49137F5B 48487F48488048C7121F4881001E1670001CEC0FE05A48913807F0E048913803FFC00040 02001380C9EA3E002F367EA434>I<160C161CA25EA45EA45EA44B5AA44B5AA201F8150E D803FE151F486C903907003F80D80F0F16C0000E1380121C0038020E131F170FD8701F15 07A24B148038E03F001703A2D8007E49130718005BA25D4848150EA25F5D48485DA25F4A 485B4C5A0001150301F84A5A4A4848C7FCD800FC141E017E147C013FEB81F090391FE707 E06DB51280010349C8FC9038003FF0020EC9FCA45CA45CA45CA31460324B7EB936>I<01 40151C01E0153E486C157F491680485A120749153F48C9121F170F121E1800121C003C14 C00038497E1403480107140EA400F049485B5AA24B5BA292C7127817706C16F04A495A91 383F80036C017F495A6C48486C485A3A7F07F7F83FD9FFE3B5C7FC6C01C35B02815B6C01 005B6C48EB7FE0D803F0EB1F8031267FA434>I<EC3FE0903801FFFC010713FF011F1480 90397F801FC09038FC0007D801E0EB0180484890C7FC5B48C9FCA57F3803C0F03801EFFE 6CB5FCA200035BD80780C8FC48C9FC121CA25A5AA35AA26C14180070141C5D00785C003E 495A391F800FC06CB55A6C91C7FC000113FC38003FE022287FA527>I<EE1F800160EC7F E001E0903801FFF048484913F848484913FC48C7EA0FC092381F003E000E023C131E0338 130E484A130F15F05D48495AA24A48130E5A4AC7FCA21406020E141C12F0020C14380070 131C177017F0007849EB01E0EE03C06CED0780003E0178130F6CED3F00260FC07013FED8 07F8EB03F83A03FFF01FF06C90B55A6C1580013F49C7FC010F13F8010113C0D903E0C8FC A25C1307A4130F5CA2131FA349C9FCA2133E131C30377DA436>39 D<16C0ED03E01507151F157FEC01FFEC03F9EC0FE1EC3F81ECFF01EB01FCEB07F0EB1FC0 EB3F80EBFE00EA03F8EA0FF0EA1FC0007FC7FC12FCA2127FEA1FC0EA0FF0EA03F8EA00FE EB3F80EB1FC0EB07F0EB01FCEB00FFEC3F81EC0FE1EC03F9EC01FFEC007F151F15071503 ED00C023287DA82A>47 D<121E123FEA7F80EAFFC0A4EA7F80EA3F00121E0A0A7A8917> 58 D<121E123FEA7F80EAFFC0A213E0A2127F123F121E1200A4EA01C0A3EA0380A2EA07 00A2120E5A123C123812100B1A7A8917>I<EF0180EF07C0171FEF7F80933801FE00EE07 F8EE1FE0EE7F80DB01FEC7FCED07F8ED1FE0ED7F80DA01FEC8FCEC07F8EC1FE0EC7F80D9 01FEC9FCEB07F8EB1FE0EB7F80D801FECAFCEA07F8EA1FE0EA7F8000FECBFCA2EA7F80EA 1FE0EA07F8EA01FE38007F80EB1FE0EB07F8EB01FE9038007F80EC1FE0EC07F8EC01FE91 38007F80ED1FE0ED07F8ED01FE9238007F80EE1FE0EE07F8EE01FE9338007F80EF1FC017 07EF0180323279AD41>I<150C151EA2153CA31578A315F0A3EC01E0A3EC03C0A3EC0780 A3EC0F00A3141EA35CA35CA35CA3495AA3495AA2495AA349C7FCA3131EA35BA35BA35BA3 485AA3485AA3485AA348C8FCA3121EA35AA35AA35AA212601F537BBD2A>I<126012F812 FEEA7F80EA1FE0EA07F8EA01FE38007F80EB1FE0EB07F8EB01FE9038007F80EC1FE0EC07 F8EC01FE9138007F80ED1FE0ED07F8ED01FE9238007F80EE1FE0EE07F8EE01FE9338007F 80EF1FC0A2EF7F80933801FE00EE07F8EE1FE0EE7F80DB01FEC7FCED07F8ED1FE0ED7F80 DA01FEC8FCEC07F8EC1FE0EC7F80D901FEC9FCEB07F8EB1FE0EB7F80D801FECAFCEA07F8 EA1FE0EA7F8000FECBFC12F81260323279AD41>I<EC03FC91381FFF80027F7F9138FC07 F0903901E001F890390380007C49C77E130E82011F15806E130F496C14C0A34A14E091C7 FC131E90C8FCA5EC3FF0903801FFFC4913FE90380FE00790391F80031F90393F00019F01 7C14DF01FC13004848ECFFC0485A1207485A001F16805B123F17005B127F5E150148C75B A215035E485D15075E150F5E007E4A5A93C7FC5D6C147E6C5C6D485A390FC007E03907F0 1FC00001B5C8FC6C13FCEB1FE02B3E7DBB2C>64 D<177017F01601A21603831607A2160F 161FA2163FA2167716F716E7ED01C783ED0387A2ED0703150F150E151CA2153815781570 15E083EC01C0A2913803800114071500140EA25CA2023FB5FC5CA29139E00001FFA24948 7F13035C49C8FCA2130EA25B133C133801781680EA01F8D807FC5C267FFFC0017F13FEB5 16FF18FE383C7DBB3E>I<0103B77E4916F06D16FC903B0007FC0003FE4BEB00FFF07F80 F03FC04A5A19E0A2181F4A5A183FA34A4815C0187F198018FF4A4815004D5A4D5A4D5A4A C7EA1FF0EF7FC04C485A92B500FCC7FC5B17FF4AC7EA7FC0EF1FE049486E7E8417078449 5A1703A349481407A449484A5AA260171F49484A5A4D5A604C48C7FC4948495A4C5A01FF EC3FF0007F90B65AB8C8FC16F83B397DB83F>I<DCFF801380030FEBE001037FEBF80391 3A01FF807C07913A07FC001E0FDA1FE09038071F00DA3F80EB03BF4AC76CB4FCEB01FED9 03F86E5A495A495A4948157E013F167C495A91C9FC5B4848167812035B1207491670120F 5B001F94C7FCA2485AA3485AA45B12FFA41707A3170EA2007F5EA25FA2003F5E5F6C7E4C 5A6C6C4A5A00074BC8FC6C6C140E6D143C6C6C14F8D8007FEB03E090393FE01FC0010FB5 C9FC010313FC9038007FC0393D7CBA3B>I<0103B7FC18E018F8903B0007FC0007FE4BEB 01FF9438007F80F03FC04A48141FF00FE019F018074A5AF003F8A34A4815FCA44A5AA44A C81207A4494816F8180FA3494816F0181FA219E04948153F19C0A2F07F80495AF0FF0060 170149484A5A604D5A4D5A49484A5A4D5A4DC7FCEE01FE4948495AEE0FF001FFECFFC000 7F90B6C8FCB712FC16C03E397DB845>I<0103B812F05B7F90260007FCC7123F4B1407F0 03E018014A5AA44A5A19C0A34A5A1770A219804A4849C7FCA31601DAFF005B1603161F92 B5FC495DA29138FE001F160F49486DC8FCA44948010E130360A2040C130E494890C7FC60 A2183C494815381878187018F0495A4D5A17034D5A4948141F177F01FF913803FF80007F 90B7FCB9C7FCA23C397DB83D>I<0103B812E0A390260007FCC7127F4B140FF007C01803 4A5AA44A5A1980A34A5AA217C005E013004A484848C7FCA31603DAFF005B1607A2163F49 90B5C8FCA39138FE003F4948131EA44948131CA44948131893C9FCA3495AA4495AA4495A A2497E007FEBFFC0B67E5D3B397DB835>I<DCFF801340030F01F013C0037FEBFC01913A 01FF803E03913A07FC000F07DA0FF09038038F80DA3FC0EB01DF4AC713FF02FE80D903FC ED7F00EB07F0495A011F824948153E5C49C9FC5B4848163CA2485A1207491638120F5B00 1F94C7FCA2485AA3485AA45B12FF93B512FE5D8104001380177FA24DC7FC127FA34C5A12 3FA27F001F4B5AA26C6C14076C7E00034B5A6C6C141DD800FE1478903A7F8001F078903A 3FF00FE030010FB5EA8010010349C9FC9038003FE03A3D7DBA41>I<0103B5D8FC07B512 F8495D6D8190260007FCC7380FF8004B5DA34A484A5AA44A484A5AA44A484A5AA44A484A C7FCA44AC7485AA392B6FC495EA24AC71203A249484A5AA449484A5AA449484A5AA44948 4A5AA449484A5AA449484AC8FCA201FF5C007F01FF90B512FEB600818003015C45397DB8 45>I<0103B512FC4914FE6D14FC90390007FC005DA34A5AA44A5AA44A5AA44A5AA44AC7 FCA4495AA4495AA4495AA4495AA4495AA4495AA4495AA213FFB67EA292C7FC27397DB824 >I<4AB512FE5CA2DA000113806F1300A34B5AA44B5AA44B5AA44B5AA44B5AA44B5AA44B 5AA44BC7FCA44A5AA4391F8003FCEA3FC0127F13E04848485AA249485A01005B48495A00 70495A4AC8FC003C13FE381F03FC380FFFF06C13C0D801FEC9FC2F3B7BB82E>I<0103B5 00FC90380FFFFC495E6D7013F890260007FCC7000113804BEDFC0019F04E5A4A48EC0780 4EC7FC181C604A4814F04D5AEF03804DC8FC4A48131E5F17705F91397F8003C04CC9FC16 0E161E9138FF007E16FF5D15074948487FED1C7F03387FEDF03F903803FDE0DAFF807FED 001F4A804948130F5C8316074948801603A283494813018382A249486E7EA284173F4948 81A201FF4B7E007F01FF010FB57EB61280150046397DB847>I<0103B6FC5B7F90260007 FEC8FC15F8A34A5AA44A5AA44A5AA44A5AA44AC9FCA4495AA4495AA4494815C01701A2EF 0380495AA2EF0700A249485C170E171EA249485C177C17FC4C5A49481307EE1FF001FF14 FF007F90B6FCB85AA232397DB839>I<902603FFFE93381FFFE049606D60D90007F0E000 505AF101DF157F020E4C485AF1073FA2190E021C4DC7FC191C19386F7E0238EE70FE19E0 A2F001C00270EEC1FCF00381F007016F7E02E092380E03F8181CA21838D901C04C5A1870 6F6C13E0A2D903804A48485AEF0380A2EF0700D90700020E495AA26F6C5AA2010E4B495A 5FA25F49DAF1C049C8FCA2923803FB80A24902FFC712FE5EA201785C4C495A13F8D807FE 6D481303267FFFE04AB512F8B500F049488002E001C05D53397DB851>I<902603FFFC91 383FFFF8496D5C6D82D90007030313006FEC00F86182DA0E7F5DA282153F021C6D495A15 1FA282DA380F4A5A821507A202706D49C7FC150382150102E0150E6F7EA217804948017F 5BA2EE3FC0A249486E5A161FA2EE0FF049C75C17F8160717FC010E02035BA217FE160149 6F5A82A3496F5AA2173F137895C8FC01F881EA07FE267FFFE080B56C140E4A140645397D B843>I<4BB4FC031F13F092B512FC913903FE01FE913A07F0007F80DA1FC0EB1FC04A48 EB0FE002FEC7EA07F0495A4948EC03F8495A4948EC01FC495A494815FE017F150049C9FC 4916FF1201485AA2485AA2120F5B001F5EA2485AA34848ED03FEA44848ED07FCA3EF0FF8 A2EF1FF0A218E0173F18C0EF7F80007F16FF18004C5A4C5A6C6C5D1607001F4B5A6D4A5A 000FED3F806C6C4AC7FC6DEB01FE6C6CEB03F86C6CEB0FF03A007FC07FC06DB5C8FC0107 13F801001380383D7CBA3F>I<0103B7FC4916E06D16F8903B0007FC0007FC4BEB01FEEF 007F19804A48EC3FC0A319E04A5AA44A48EC7FC0A319804A4814FF19004D5AA24AC7485A 4D5AEF0FE0EF3FC0494849B4C7FC91B612FC17F094C8FCD903FCCAFCA4495AA4495AA449 5AA4495AA4495AA213FF007F13FFB67E92CAFC3B397DB835>I<4BB4FC031F13F092B512 FC913903FE01FE913A07F0007F80DA1FC0EB3FC04A48EB1FE002FEC7EA0FF04948140749 48EC03F8495A494815FC49481401013F16FE495A49C8FCEF00FF485A12035B12075B120F 495D121FA2485AA34848ED03FEA44848ED07FCA3EF0FF8A218F0171F18E090C9123F18C0 6DED7F80127FEFFF00913903E001FEDA0FF05B003F496C485A9039C03C3C07001F903970 0C0FF0903AE0E00E1FE0000FED3F802607F1C049C7FC01F9EB06FED803FDEB07F8D801FF 495A3A007FE07FC06DB5EA000313070100018F5BDA000F1306170EEE801E5FEEC0FCEEFF F8A25F5F5F8194C7FC6F5AED00F8384B7CBA42>I<0103B612F849EDFF806D16E0903B00 07FC001FF84BEB03FC717E717E4A48801980A219C04A5AA44A48ECFF80A34D13004A485C 1703604D5A4AC7485AEF3FC0EFFF80DC07FEC7FC4990B512F817C0A29139FE001FE04948 EB07F0707E707E8349481300A449481301A449481303A44948495A194019E0A24948ED01 C0160301FFEE0380007F01FF903901FE0700B6D88000130E92C7EA7FFCCA6C5AEF07F03B 3B7DB83F>I<92381FE0019238FFFC030203EBFE0791390FE01F0F91391F80079F91393E 0001FE5C4A1300495A4948147C13075C130F4A1478131FA3013F1570A28017008080EB1F FCECFF8015F86DEBFF806D14E0826D800100806E7F020F7F1400031F1380150181167F16 3FA30006151F120EA3001EED3F00A2163E167E003E157CA25E003F4A5A484A5A6D495AD8 7DE0495AD878F0013FC7FCD8F07F13FE39E03FFFF8010F13E0D8C00190C8FC303D7CBA33 >I<48B812FE5A5A903AFC003FE00301E09138C0007ED80F80163C90C7FC000E4A5A121E 121CA2484AC7FC18385AA24A5A5AA2481730C748481400A44A5AA44A5AA44A5AA44A5AA4 4A5AA44AC9FCA4495AA4495AA2EB0FFE003FB67EA337397EB831>I<003FB5D8C003B512 80485D6C8126007FC0C7383FF0004AEC0F8095C7FCA249C8120EA448485DA448485DA448 485DA448485DA448484A5AA448484A5AA448484AC8FCA448C8120EA35EA25EA25E6C5DA2 4B5A6C4A5A6C6C49C9FC151E6C6C5B6C6C13F83903F807E0C6B512806D48CAFCEB0FF039 3B7BB839>I<267FFFFE91387FFFC0B5FC7E0001018091380FFC0091C8EA07E0715A4D5A 6C94C7FC5F170E5FA25F17786E14705F017F14015F4C5AA24CC8FC5E6E130E5EA2013F5C 167816705EA24B5A6E485AA24BC9FC011F5B150E5DA25D1578ECF0705D130FECF1C0ECF3 80A202F7CAFC14FF5C5CA26D5AA25C5CA25CA25C3A3B7CB830>I<277FFFFC01B500F090 B51280B56C486E5A6C496C4A7E0003902780000FFCC7381FF8006C48C749EC07E01B80A2 040793C7FC1A0EA262A2040F5D1A78041F157062163F6D027F4A5A16776C03E74A5A83DB 01C74AC8FC61DB0383140E03075D1603030E5DA2031C5D19F003385D4E5A1570DA80F04A 5A15E090267F81C04AC9FC715ADA8380140E0401131EDA8700141C028F5D148E029C5DA2 02B85D17FF02F05D605C95CAFC5C6D485CA291C75B705A133E5F133C01385D513B7CB84E >I<91B500FC90B512E0495D83D9000301C090381FFC006E90C7EA0FE0198096C7FC6E6D 131E60606F6C5B4D5A6F6C485A604DC8FC92381FF00E5F030F133C705A5F923807FDE0EE FFC05F6F90C9FC5E6F7EA36F7F5D5D4B7F167F92380E3FE0151C15389238781FF015F091 3801E00FDA03C07FEC07804A486C7E140E5C4A6D7E5C02F06D7E495A495A49486D7F49C8 FC49157F017F4B7E2603FF8001037F007F01F0011FEBFF80B56C5B6C497F43397EB845> I<B500FE91383FFFE05F19C0000301C0913807FC006C4915E06C5F95C7FC5F6D6C141E5F 17386D6C5C5F6E1301011F4A5A4C5A6D6C91C8FC160E5E6D6C133C5E5E6D6C5B4B5A4B5A 903801FF074BC9FC150E6D139C15B815F86E5A5D5D6E5A147FA392CAFC5CA35C1301A35C 1303A35C1307A2130F0007B512F048805D3B397DB830>I<027FB612FC91B7FC18F84990 39E0000FF092C7121F02FCEC3FE0D903F015C04AEC7F804AECFF0001074A5A4A130391C7 485A494A5A010E5D4C5A4C5A494A5A16FF01184990C7FC90C75B4B5A4B5A4B5A151F4B5A 4B5A5E4BC8FC4A5A4A5A14074A5A5D4A5A4A5A4A48130C02FF141C4990C7FC49485C5C49 5A4948147849481470133F494814F04A5C49C71201485A48484A5A000715074848140F48 48143F49ECFF804848130748B7FCB8C7FCA236397BB839>I<147E903803FF80010FEBC3 8090391F81E7C090383E00FF49137F5B4848EB3F80485A12074848131FED3F00485AA212 3F90C7127E5AA300FE5CA44A5A48151CA3913803F038A2007C1307140F007E011F137000 3E133B6C01F113E0380F83E13A07FFC0FFC06C9038807F803A00FC001F0026267DA42C> 97 D<133FEA0FFF5AA2EA007EA45BA4485AA4485AA4485A14FCEBE3FF01E713C0390FDF 07E09038FC01F013F801F013F8381FE0004913FC5BA248C7FCA4007E1301A448EB03F8A3 15F014074814E0A2EC0FC0A2EC1F801500007C133E147E003C5B383E01F0381F07E06CB4 5A6C90C7FCEA01FC1E3B7CB924>I<EC3FC0903801FFF0010713FC90380FE03E90383F80 0E90387E001F49133F4848137F484813FF485A15FE4848137C001F1438491300123F90C8 FC5AA312FEA55AA31502007C1407007E140E151C003E14386C14F0390F8003E09038E03F C00003B512006C13FC38003FC020267DA424>I<163FED0FFF5DA2ED007EA416FCA4ED01 F8A4ED03F0A4ED07E0147E903803FF87010F13C790391F81EFC090383E00FF49137F5B48 48EB3F80485A12074848131FED3F00485AA2123F90C7127E5AA300FE5CA44A5A48151CA3 913803F038A2007C1307140F007E011F1370003E133B6C01F113E0380F83E13A07FFC0FF C06C9038807F803A00FC001F00283B7DB92B>I<EC3F80903801FFF0010713F890381FE0 7C90383F003C017C131E5B485A1203485A485A001F143C491378003FEB01F0EC07E048B5 12C0150014F890C8FC127E12FEA6127E15021507150E003E141C003F14386C14F0390F80 03E03907E03FC06CB512006C13FC38003FC020267DA427>I<16F8ED03FE4B7E92380F0F 80ED1E1FED3E3FED7C7FA203FC1300163E161C4A48C7FCA54A5AA54A5AA20103B512F05B 5E9026000FC0C7FCA54A5AA64AC8FCA5147EA55CA5495AA5495AA45C1307A25C121E003F 5BEA7F8FA2D8FF0FC9FC131E127EEA7C3CEA3FF86C5AEA07C0294C7CBA29>I<EC07E0EC 1FF891387FFC389138FC1E7C903903F007FCEB07E090380FC003D91F8013F81400491301 137EED03F05B1201A249EB07E01203A34848EB0FC0A4ED1F805BA3ED3F0000035CA26D5B 0001495A3800F80790387C1F7EEB3FFE90381FF8FCEB07E090C7FCA24A5AA4001E495A12 3F486C485AA24848485A4A5A007E017FC7FC387C01FE383FFFF86C13E0000390C8FC2636 7FA428>I<EB03F013FF5AA2380007E0A4495AA4495AA449C9FCA4137EEC03F8EC1FFE4A 6C7E9039FCFC0FC09038FDE007D9FFC07F148048EB00035B5B49130712035BA34848495A A34B5A485AA24BC7FCA2485A92387E0380A348C738FC0700A3EDF80E127E5E5EED787848 EC7FF0007C6E5A0038EC0F80293B7CB930>I<14E0EB01F0EB03F81307A214F0EB03E0EB 01C090C7FCAB13F8EA01FEEA07FF130F000E1380121C1238A2EA701FA338E03F00A3EA00 7EA25BA3485AA3485AA23807E038A3380FC070A3EB80E0A2EB81C0EB83803807870013FF EA03FCC65A15397EB71D>I<150E151FED3F80157FA21600153E151C92C7FCABEC0F80EC 3FE04A7EECF0F8EB01C049487EEB0700A2130E5BA24A5A1338A213004A5AA44A5AA44A5A A44A5AA44AC7FCA4147EA45CA4381E01F8123F387F83F05CEAFF07495A48485AD87C3FC8 FCEA3FFC6C5AEA0FC0214981B722>I<EB03F013FF5AA2380007E0A4495AA4495AA449C8 FCA4137EED07C0ED1FE0ED3FF049EBF838913801E0F8EC0381EC07033901F80E07141C14 3891387003F03A03F0E001C0D9F1C0C7FCEBF38001FFC8FC485A6D7E14F014FC380FC1FE EBC07FEC1F80A248486C7E1670A33A3F001F80E0A3ED81C0127E020F1380158391380787 004814FE007C6D5A0038EB00F8253B7CB92B>I<EB0FC0EA03FF5A7E38001F80A4EB3F00 A4137EA45BA4485AA4485AA4485AA4485AA4485AA448C7FCA4127E1307A3EAFC0EA35BA3 EA7C381378EA3FF06C5AEA0780123B7DB919>I<D803E0017F14FE3D07F803FFC007FF80 486C48D9F00F13E03D1C3E1F81F83F03F0DA3C00EB780126383F78D9FCF07F4AEBFDE000 704990387DC0004AEB7F804A91C7FC91C700FE1301485A017E5CA21200494948495AA34E 5A4848495AA24E5AA24848495A95381F80E0A34848494890383F01C0A395383E03804848 495AF10700190EF01E1E484849C7EA1FFC000F6E6E5A6CC7000EEC03E043267EA449>I< D803E0137F3A07F803FFC0486C4813F03A1C3E1F81F8EC3C0026383F787F5C007049137C 5C5C91C712FC485A137EA2120049495AA34B5A485AA24B5AA2485A92380FC070A3484890 381F80E0A3EE01C0485AEE0380EE07006F5A484814FE000FEC07F86CC7EA01F02C267EA4 32>I<EC1FC0ECFFF801077F90380FE07E90383F801F90397E000F8049EB07C0484814E0 485A485AED03F0485A121F5B003F140790C7FC5AA300FEEC0FE0A316C0151F481580153F 1600157E127E5D4A5A003E495A6C495A390F801FC02607E07FC7FC3803FFFE6C13F03800 3F8024267DA428>I<90390F8003F090391FE01FFC496C487E903970F87C1F9238F00F80 903AE0FDC007C0DAFF8013E04848EB00035C4A14F0A2380381F8A3D800011407495AA449 48EB0FE0A317C04948131F1780A2EE3F00495A6E137E167C5E496C485A4B5A6E485A9138 781F8090267E3FFFC7FCEC1FFCEC07E091C9FC5BA4485AA4485AA3387FFFC0B5FC7E2C35 83A42A>I<02FC13C0903803FF01010F138390391F83C78090383E01EF9038FC00FF4848 137F491400485A000780485A157E485AA2123F90C75A5AA300FE495AA448495AA44A5AA2 140F007C131F007E495A003E137F14FF381F83EF390FFF9F800003131FEA00FC13004AC7 FCA4147EA45CA2130190387FFFF0A322357DA425>I<3903E001F03907F807FC390FFC1F FE391C3E3E0FEC781F39383FE03F157F0070EBC0FF5CA2EC007ED8E07E13381500A21200 5BA4485AA4485AA4485AA4485AA4485A120F6CC8FC20267EA425>I<14FF010313C0010F 13F090381F80F890383E003849133C49137C4913FC00011301A215F80003EB00F06D1300 7F3801FFC014FC6C13FF15806D13C0011F13E013079038003FF0140F1403121E123F5A48 14E0A215C048130700F814800070EB0F000078133E003F13FC6CB45A000713E0000190C7 FC1E267CA427>I<EB0380EB07C0130FA4EB1F80A4EB3F00A4137EA2B512FE14FF14FE38 00FC00A3485AA4485AA4485AA4485AA4485A141CA21438EA3F00A2147014E0A2EB01C038 1F0380EB0F00EA0FFE6C5AEA01F018357DB31E>I<13F8D803FE1438486C147CD80F0F14 FC000E7F121C12384B5AEA701FA3484848485AA3EA007E4B5A5BA34848495AA44848495A EE81C0A392383F0380A2157FA29238FF070000015B9039F8039F0E3900FC0F1F90397FFE 0FFC90391FFC07F8903907F001F02A267EA430>I<01F8EB0380D801FEEB07C0D807FFEB 0FE0D80F0F14F0000E1380121C003814071503D8701F1301A216E038E03F001500A2D800 7E130116C05BA34848EB0380A3ED0700485A1506150EA25DA25D5D12016D5B0000EB03C0 90387E07806DB4C7FCEB1FFCEB07F024267EA428>I<01F816E0D801FE9138E001F0D807 FF903901F003F8D80F0F010314FC000E1380121C00381601923807E000D8701F167CA218 78484848485A1838A2D8007E16784B4813705BA3484849C712E0A3EF01C04848137EA2EF 0380A3EF0700A203FE130E00015E6D487E0000D9039F5B903A7E0F0FC0F0903A3FFE07FF E06D486C5B902703F8007FC7FC36267EA43B>I<90390FC007E090393FF00FF890397FF8 3FFC9039F07C783C3A01C03EF03E3A03801FE07ED80700EBC0FE000E14C115815AED80FC 4890383F00701600A2C7FC147EA45CA4495A1638A2121C267E03F0137012FF16E0130700 FEEC01C0D8FC0FEB03803AF81EF8070039783C7C1E397FF83FFC393FE01FF8390FC007E0 27267DA42F>I<13F8D801FE1470D807FF14F8D80F0F1301000E1380121C1238ED03F0EA 701FA33AE03F0007E0A3EA007EED0FC05BA34848EB1F80A44848EB3F00A4157EA315FE4A 5A00011303EBF8073800FC1F90387FFFF8EB3FF9EB0FE1EB00014A5AA2000E495A121F48 6C485A127F4A5A010090C7FC007E133E007C5B383801F8383E07F0381FFFC06C90C8FCEA 01FC25367EA429>I<D901E013E0EB07F890391FFC01C090393FFE0380A290397FFF8F00 9038F81FFE9038E007FC3801C0001538495BC85A4A5A4A5A4AC7FC140E5C5C14F0495AEB 038049C8FC130E5B4913075B495B4848130E4848131ED807F05B01FE137C390FFF81F839 1F1FFFF0D83C075BD838035B486C5B486C90C7FC147C23267DA427>I<1504150E151E81 A2ED078016C0007FB612F0B712F8A26C15F0C8EA0FC0ED3F00153E5D5D1570251171BB2A >126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FO cmr10 10 96 /FO 96 128 df<B812E0A30001903880007F6C90C7120FEE03F01601A21600A21770A417 781738A41700B3B04813C0B612E0A32D397DB834>0 D<150FA24B7EA24B7EA24B7EA24B 7EA2913801EFF815CF913803C7FC158791380783FE15034A6C7E140EDA1E007F141C023C 6D7E143802786D7E1470161F4A80160F4948801607494880A249C76C7EA2496E7E130E01 1E6E7F131C013C6F7E133801786F7E137001F06F7E5B0001707E5B0003707E5B0007707E 90C9FC48707E120E001E701380121C001FB9FC4818C0A24818E0A2BA12F0A23C3C7CBB45 >I<15E04A7EA44A7EA34A7EA34A7EA34A7E141CA2023C7F1438157FA202707F153FA202 E07F151FA2D901C07F150FA2010380EC8007A201078014001503A2010E801501A2498081 A2013C810138147FA20178811370163F01F0815B161FA20001820003150F486C141FD81F FCEC3FFCB5D88007B512E0A3333C7DBB3A>3 D<007FB712FEB8FCA2D87FC0C7121F6C6C 14036DEC00FF001F163F6C7E6D810007826C7E6C6C81A26C7F6D6C15806E1403133F6D7E 80010F92C7FC6D7E8013036D7E817F6E7E81143F141F6E5A5D6EC9FC140E5C5C5C02F0EC 0380495A495A5C49C81207010E16005B5B01785D5B48485D495D485A48C9B4FC000E1503 48ED3FFE003FB7FC5AB8FC7E31397BB83C>6 D<011FB512FEA39026001FFEC8FCEC07F8 A8EC3FFE0103B512E0011F14FC90397FE7FBFF903AFF07F87F80D803FCEC1FE0D807F86E 7ED80FF06E7ED81FE06E7ED83FC06E7EA2007F8201808000FF1780A7007F170001C05C00 3F5EA2D81FE04A5AD80FF04A5AD807F84A5AD803FC4A5AD800FFEC7F8090277FE7FBFFC7 FC011FB512FC010314E09026003FFEC8FCEC07F8A8EC1FFE011FB512FEA331397BB83C> 8 D<EC0FFE91387FFFC00103B512F890390FFC07FE90391FE000FFD97F80EB3FC049C76C 7ED801FCEC07F000038248486E7E491401000F8248486E7EA2003F1780A249157F007F17 C0A9003F17806D15FFA2001F1700A26C6C4A5AA200075EA26C6C4A5AA200015E6D140700 005E017C5D017E140F013E5DA26D4AC7FC00E017E06D141E0070EE01C06D6C133CA20103 14380078160300386D01781380D83FFFEC7FFFA46C170082333B7CBA3C>10 D<DA07FC13FC91393FFF07FF49B5009F1380903B03FC07FF8FC0903B0FE003FE0FE0903A 1FC007FC1F90383F800FD97F0013F8017EED0FC001FE903907F007804992C7FC1201ACB8 12F8A32801FC0007F0C7FCB3AB486C497E267FFFE0B512F0A3333B7FBA30>I<EC07F8EC 3FFE49B57E903903FC07C090390FE001E090381FC0034948487E90387F000F137E13FE49 6D5A00016E5A6F5A92C8FCA9ED07F0B7FCA33901FC001F1507B3AA486C497E267FFFE0B5 12C0A32A3B7FBA2E>I<EC07FC91387FFF7049B512F0903803FC0790380FE00FEB1FC0EB 3F005B017E130713FE5B1201ACB7FCA33901FC0007B3AB486C497E267FFFF1B512C0A32A 3B7FBA2E>I<DA07FCEB0FF0DA3FFFEB7FFC49B50083B5FC903C03FC07C7F80F80903C0F E000FFC003C0494848EB800749484849487E49484848131F017E5C13FE494A6D5A000102 036E5A72C7FC95C8FCA9F00FE0BAFCA33C01FC0003F8003F180FB3AA486C496C497E267F FFF0B500E3B51280A3413B7FBA45>I<EA01F812FFA312071203B3AA487EB512E0A31325 7EA418>16 D<00C0131800F0137838FC01F8387F07F0383FDFE0380FFF806C13006C5AEA 00F81370150A76B42A>20 D<00E0EB0380A40070EB0700A200785B6C131EA2001F137C38 0FC1F813FF6C5B000113C06C6CC7FC190F78B92A>I<B612F8A41D047AB02A>I<EA07F0EA 1FFC487EEA780FEA700738E00380A538700700EA780FEA3FFE6C5AEA07F0110F6ABB3E> I<121E123FEA7F80EAFFC0A7EA7F80ABEA3F00AA121EAA120EC7FCA8121E123FEA7F80EA FFC0A4EA7F80EA3F00121E0A3C7ABB17>33 D<001E130F003FEB1F80397F803FC039FFC0 7FE0A201E013F0A2007F133F003F131F001EEB0F7000001300A4484813E0A339038001C0 A23907000380A2000EEB070048130E003C131E0038131C001013081C1A7DB92A>I<121E 123FEA7F80EAFFC0A213E0A2127F123F121E1200A4EA01C0A3EA0380A2EA0700A2120E5A 123C123812100B1A7AB917>39 D<14E01301EB03C0EB0780EB0F00131EA25B5B13F85B12 015B12035B1207A2485AA2121F90C7FCA25AA2123EA2127EA5127C12FCB2127C127EA512 3EA2123FA27EA27F120FA26C7EA212037F12017F12007F13787F7FA27FEB0780EB03C0EB 01E01300135278BD20>I<126012F012787E7E7EA26C7E6C7E7F12017F12007F1378137C A27FA2133F7FA21480A2130FA214C0A5130714E0B214C0130FA51480A2131FA21400A25B 133EA25BA2137813F85B12015B12035B485A48C7FCA2121E5A5A5A126013527CBD20>I< 15301578B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43 D<121E123FEA7F80EAFFC0A213E0A2127F123F121E1200A4EA01C0A3EA0380A2EA0700A2 120E5A123C123812100B1A7A8917>I<B512FCA616067F941C>I<121E123FEA7F80EAFFC0 A4EA7F80EA3F00121E0A0A7A8917>I<150C151EA2153CA31578A315F0A3EC01E0A3EC03 C0A3EC0780A3EC0F00A3141EA35CA35CA35CA3495AA3495AA2495AA349C7FCA3131EA35B A35BA35BA3485AA3485AA3485AA348C8FCA3121EA35AA35AA35AA212601F537BBD2A>I< EB03F8EB1FFF017F13C09038FE0FE03901F803F048486C7E48486C7E4848137EA248487F A2003F1580A290C7121F4815C0A64815E0B36C15C0A56C6CEB3F80A3001F1500A26C6C13 7EA26C6C5B6C6C485A6C6C485A3900FE0FE090387FFFC0011F90C7FCEB03F8233A7DB72A >I<EB01C013031307131F13FFB5FCA2131F1200B3B3A8497E007FB512F0A31C3879B72A> I<EB0FF0EB7FFE48B57E3903E03FC03907800FE0390E0007F0486D7E48806E7E5A6E7E12 7CB4FC16807F157FA26CC7FCA2001C14FFC8FC1600A25C5D5D14035D4A5A4A5A5D4A5A4A C7FC143E5C5C495A495A495A5C49C8FC011EEB03805B5B491307484814005B485A48C75A 48B6FC5A5A485CB6FCA321387CB72A>I<EB07F8EB3FFF4913C03901F80FE03903E003F0 D807807F496C7E488013C0486C6C7EA4120F495AD803805BC7FCA25D14035D4A5A5D4A5A EC7F80D91FFEC7FC5CECFF809038000FE0EC03F06E7E6E7E81157F1680A2ED3FC0A216E0 A2123E127F487EA316C090C7FC48147F007C158012706CECFF006C5C000F495A3907C003 F83903F80FF06CB512C06C6C90C7FCEB07F8233A7DB72A>I<157815F8A214011403A214 07A2140F141FA2143F147B147314F3EB01E314C31303EB07831403130F131E131C133C13 78137013F0EA01E013C012031380EA07005A120E121E5A123812785AB712F8A3C73803F8 00AB4A7E0103B512F8A325397EB82A>I<0004140C000F143C9038F003FC90B55A5D5D15 8092C7FC14FC000E13F090C9FCABEB07F8EB1FFE90387FFF809038F80FC0390FE007E090 388003F0496C7E000E6D7E1206C87EA2157FA31680A31218123E127F5AA316005A00785C 00705CA26C495A5D6C495A6C495A6C6C485A3903E03F806CB5C7FC38007FFCEB1FE0213A 7CB72A>I<EC3FC0903801FFF0010713FC90380FE03E90381F000F013E5B49EB3F804913 7F485A485A120749EB3F00000F141E001F91C7FC5BA2123FA3387F0020EB03FF4913C049 7F39FF1C03F090383000F849137C157E497FA2491480151F16C0A290C7FC16E0A47EA57E 6D14C0A2121FED3F80120F6D14000007147E6C6C137C6D5B6C6C485A3900FE07E090383F FFC06D90C7FCEB03FC233A7DB72A>I<1238123C123F90B612E0A316C0481580A2160000 78C7120E00705CA25D00F05C5A5D4A5AC7FC4A5A4AC7FC140EA25C143C14381478147014 F0A2495AA21303A2495AA2130FA3131F5CA3133FA5137FA96DC8FC131E233B7BB82A>I< EB03F8EB1FFF4913C09038FC07E03901F001F03903C000F8157C48487F120F90C7FC815A A37FA26D133EA2D80FF0133C6D137CD807FE5B6D5B6CEBC3E06CEBE7C06CEBFF806D48C7 FC6D7E010F7F013F13E0497FD801F813FC3903E03FFE3807C01F390F8007FF48486C1380 001E1300003EEC7FC048141FED0FE0A2481407A21503A416C0127CED07807EED0F006C14 1E6C6C5BD807E013F83903FC03F0C6B55A013F1380D907FCC7FC233A7DB72A>I<EB03F8 EB1FFF017F7F9038FC07E03901F001F048486C7E48487F000F147C4848137E153E003F14 3F48C7FC1680A248141F16C0A516E0A47E153FA27E7F001F147FA26C6C13DF12073903E0 019F3901F8071F2600FFFE13C0EB7FFCEB1FF8EB00801400ED3F80A31600A2000F147E48 7E486C5B5D14015D49485A6C48485A001EEB1F80260FC07FC7FC3807FFFC000113F03800 3FC0233A7DB72A>I<121E123FEA7F80EAFFC0A4EA7F80EA3F00121EC7FCB0121E123FEA 7F80EAFFC0A4EA7F80EA3F00121E0A247AA317>I<121E123FEA7F80EAFFC0A4EA7F80EA 3F00121EC7FCB0121E123FEA7F8012FF13C0A3127F123F121F1201A4EA0380A312071300 A2120E121E121C5AA212100A347AA317>I<007FB812F8B912FCA26C17F8CCFCAE007FB8 12F8B912FCA26C17F836167B9F41>61 D<1538157CA315FEA34A7EA34A7FA34A7F153FA2 020F7FEC0E1FA2021E7FEC1C0FA2023C7FEC3807A202787FEC7003A202F07FECE001A201 01804A7EA20103814A137FA201078191C7123F91B6FC4981A2010EC7121F011E81011C14 0FA2013C8101381407A201788101701403A201F08116011201486C81D80FFE02071380B5 00C090B512FEA3373C7DBB3E>65 D<B712E016FC16FF0001903980007FC06C90C7EA1FE0 707E707E707EA2707EA283A65FA216035F4C5A160FEE1FE0EE7FC04B485A91B548C7FCA2 707E91C7EA3FE0EE0FF0707E707E707E707EA21880177F18C0A7188017FFA24C13005F16 034C5AEE1FF8486DEB7FF0B812C094C7FC16F832397DB83B>I<4AB4EB0180020FEBE003 027F13F8903A01FF807E07903A03FC000F0FD90FF0EB079FD91FC0EB01DF4948EB00FF49 C8127F13FE4848153F4848151FA24848150F120F5B001F1607A2485AA21703127FA25B94 C7FC12FFAB127FA26DED0380A2123FA36C7EEF0700120F7F0007160E6C7E5F6C7E6C6C5D 017F5D6D6C14F06D6C495AD90FF0495AD903FC010FC7FC903901FF807E6D6CB45A020F13 F002011380313D7BBA3C>I<B712C016F816FE000190398001FF806C90C7EA3FE0EE0FF0 EE03F8707E707E177FA2EF3F8018C0171F18E0170F18F0A3EF07F8A418FCAC18F8A4EF0F F0A218E0A2171F18C0EF3F80A2EF7F0017FE4C5A4C5AEE0FF0EE3FE0486DEBFF80B8C7FC 16F816C036397DB83F>I<B812FCA30001903880001F6C90C71201707E177E173E171EA2 170EA4170F83ED01C0A394C7FCA31503A21507151F91B5FCA3EC001F15071503A21501A2 18E0A3170192C713C0A41703A3EF0780A2170FA2171F173F17FF486D010F1300B9FCA333 397DB839>I<B812F8A30001903880001F6C90C71203EE01FC1600177C173CA2171CA417 1E170EA2ED0380A21700A41507A2150F153F91B5FCA3EC003F150F1507A21503A692C8FC AD4813C0B612C0A32F397DB836>I<DBFF8013C0020FEBF001023F13FC9139FF803E0390 3A03FC000F87D907F0EB03CFD91FC0EB01EF4948EB007F49C8FC01FE153F4848151FA248 48150F485A000F16075B001F1603A2485AA21701127FA25B94C7FC12FFAA93B6FC127FA2 6D9138007FE0EF3FC0123FA36C7EA26C7EA212076C7E6C7EA26C7E017F157F6D7ED91FE0 14EFD907F0EB01C7D903FCEB0783903A00FFC03F0191393FFFFE00020F01F81300020013 80383D7CBA41>I<B648B512FEA30001902680000313006C90C76C5AB3A491B6FCA391C7 1201B3A6486D497EB648B512FEA337397DB83E>I<B612C0A3C6EBC0006D5AB3B3AD497E B612C0A31A397EB81E>I<017FB512C0A39039003FF800EC0FF0B3B3A3121C127FA2EAFF 80A25DEB001FA2007C5C0078495A1238001E49C7FC380F81FC6CB45A000113E038007F80 223B7CB82B>I<B60107B5FCA300010180010013F06C90C8EA7F80053EC7FC5F17705F4C 5A4C5A4CC8FC160E5E5E5E5E4B5A4B5A4BC9FC150E5D153C157E15FE4A7E4A7FEC077F91 380E3FC0021C7FEC381F4A6C7E02E07FECC0074A6C7E02007F15016F7E83167F707E8316 1F707E831607831603707E83828484486D4913F0B6011FEBFF80A339397DB841>I<B612 E0A3000101C0C8FC6C90C9FCB3AD1738A517781770A417F0A21601A216031607160FEE3F E04890388001FFB8FCA32D397DB834>I<B56C92380FFFF8A300016D92381FFC006C60D9 EFE0153BA3D9E7F01573A3D9E3F815E3A2D9E1FCEC01C3A3D9E0FEEC0383A3027FEC0703 A26E6C130EA36E6C131CA36E6C1338A26E6C1370A36E6C13E0A2913901FC01C0A3913900 FE0380A392387F0700A2ED3F8EA3ED1FDCA3ED0FF8A26F5A487E487ED80FFE6D48497EB5 00E00203B512F8A2ED01C045397DB84C>I<B591380FFFFE80A2C66D010013E06EEC3F80 EF1F00D9EFF0140E8013E7EBE3FC8013E1EBE0FF81147F81143F6E7E81140F6E7E811403 6E7E8180ED7F8016C0153FED1FE016F0150FED07F816FC1503ED01FE16FF81EE7F8E17CE 163FEE1FEE17FE160FA216071603A216011600A2486C157E486C153EEA0FFEB500E0141E 170EA237397DB83E>I<EC03FF021F13E091B512FC903901FE01FE903A07F8007F80D90F E0EB1FC0D93F80EB07F049C76C7E01FE6E7E0001824914004848157F0007178049153F00 0F17C049151F001F17E0A24848ED0FF0A3007F17F8491507A300FF17FCAC6C6CED0FF8A4 003F17F06D151F001F17E0A26D153F000F17C0000717806D157F6C6CEDFF0000015E6D14 016C6C4A5A6D6C495A6D6C495A6D6C495AD907F8EB7F80902703FE01FFC7FC0100B512FC 021F13E0020390C8FC363D7BBA41>I<B712C016F816FE000190398001FF806C90C7EA3F C0EE0FE0EE07F0EE03F817FC17FE1601A217FFA717FEA2160317FC17F8EE07F0EE0FE0EE 3FC0923801FF8091B5EAFE0016F816C091C9FCB3A4487FB6FCA330397DB839>I<EC03FF 021F13E091B512FC903901FE01FE903A07F8007F80D90FE0EB1FC0D93FC0EB0FF049486D 7E49C76C7E48486E7E4914004848157F0007178049153F000F17C049151F001F17E0A248 48ED0FF0A3007F17F8A2491507A200FF17FCAC007F17F86D150FA3003F17F0A26C6CED1F E0A36C6C017CEB3FC00007D901FE14806D486C137F000390260783801300D801FC903800 C0FED9FE0E13E1D800FFEC63FC017FEC73F8D93FCEEB7FF0D91FEE6D5AD907FFEB7F806D D981FFC7FC0100D9FFFC130C141F0203131C91C7121E181C161FEF803CEFC0F8EE0FFFA2 18F08218E0827013C0701380EF7E00364B7BBA41>I<B612FEEDFFE016F8000190388007 FE6C90C76C7EEE3FC0707E707E707EA2707EA283A65FA24C5AA24C5A4C5A4C5AEEFF80DB 07FEC8FC91B512F816E0A291380007F8ED01FC6F7E167F707E83161F83A683A560F00380 EE0FF8A3486D0107EB0700B6EB03FC933801FE0E933800FFFCCAEA3FF8EF07F0393B7DB8 3D>I<D90FF813C090383FFE0190B512813901F80FE33907E001F7390F80007F90C7123F 48141F003E140FA2481407A200FC1403A415017EA27E6C91C7FC7F13E0EA3FF8EBFF806C 13F86CEBFF806C14E06C14F86C806C80013F7F01071480D9007F13C0020713E0EC007FED 1FF0150F150716F81503126012E01501A47E16F0A26C14036C15E0A26CEC07C06CEC0F80 D8FBC0EB1F00D8F9F0133ED8F0FF13FC39E03FFFF8010F13E0D8C00190C7FC253D7CBA2E >I<003FB812E0A3D9E003EB003F90260001FE1307007EEE03F0007C160100781600A300 701770A400F01778481738A4C71600B3B0913807FF80011FB612E0A335397DB83C>I<B6 90380FFFFEA300010180010013E06C90C8EA3F80EF1F00170EB3B27F5FA280013F5DA26D 6C5C130F6E5C01074A5A6D6C13036D6C495AD900FE011FC7FC91383F807C6EB45A020713 E002001380373B7DB83E>I<B500FC91B51280A30003018091381FF8006C90C8EA07E060 6C705AA26D6C4AC7FCA280013F150EA26E141E011F151CA26E143C010F1538A26D6C5CA2 8001035DA26E130101015DA26E13036D5DA26E6C48C8FCA215C0023F130EA2EDE01E021F 131CA2EDF03C020F1338A26E6C5AA215FC02035BA215FF6E5BA36E5BA26FC9FCA3153EA3 151C393B7EB83E>I<B5D8FE01B5D8FC01B512C0A300039026C0000790C7381FFC006C90 C76C48EC07F06C735A04015E1A016D6C6E4A5AA36D6C4DC7FC4C7FA26E5F011F9126073F C0130EA26E171E010F020F6D131CEE0E1FA26D6C011E6D5BEE1C0FA26D6C5F4C6C7EA26E 17F001014A6C6C5BA36D6C01F06D485AEEE001A2DA7F804B5A923981C000FFA203C11507 91263FC380D97F87C8FCA203E3158FDA1FE715CE93C7123FA26EB415FC4B141FA202075E 4B140FA36E486E5AA302015E4B1403A202005E4B1401523B7FB855>I<B500FE91387FFF E0A3000101E091380FFE006C49EC07F0017F6F5A606D6C5D6D6C140795C7FC6D6C140E17 1E6D6C141C6D6C143C17386D6C14785F6D6D5B91387FC0015F91383FE0035F91381FF007 6E6C48C8FC160E913807FC1E161C913803FE3C913801FF385E6E13F05E157F6F5AB3A24B 7E023FB512C0A33B397FB83E>89 D<003FB7FCA39039FC0001FE01E0130301805C90C748 5A003E140F5E003C141F007C5D00784A5A157F5E007014FF93C7FC4A5A14035DC712075D 4A5A141F5D143F5D4A5A14FF92C8FC5B5C495A13075C130F4AEB0380495A133F5C137F5C 49C7FC4815075B12035B4848EC0F00120F495C121F495C484814FF007F140349131FB8FC A329397BB833>I<EAFFF8A4EAF000B3B3B3B3A3EAFFF8A40D5378BD17>I<481480390380 01C00007130301001380000EEB070048130EA2485BA2485BA3485BA400EFEB778039FF80 7FC001C013E001E013F0A2007F133FA2393FC01FE0391F800FC0390F0007801C1A76B92A >I<EAFFF8A4EA0078B3B3B3B3A3EAFFF8A40D537FBD17>I<13101338137C13FE487E3803 EF803807C7C0380F83E0381F01F0383E00F848137C00F0131E0060130C170D77B92A>I< EB3FE0EBFFFC000313FF3907C03F80390F800FC0486C6C7E01E07F6E7EA2380FC001D807 807FEA0300C7FCA414FF130FEB7FF13801FF01EA07F8EA0FF0EA1FE0EA3FC0EA7F80A2D8 FF00141CA41403A2387F800714063A3FC01C7E383A1FF0787FF83A0FFFF03FF0000301E0 13E03A007F000F8026277DA52A>97 D<EA03F812FFA3120F1203B0EC0FE0EC7FFC9038F9 FFFE9039FBE03F809039FF800FC09039FE0007E049EB03F0A249EB01F816FCA216FE1500 A216FFA916FEA3150116FCA2ED03F86D14F0ED07E06DEB0FC09039E7801F809039E3E07F 009038C1FFFE9038807FF8C7EA1FC0283B7EB92E>I<EB03FC90381FFF80017F13E09038 FE01F03901F800F83903F001FC3807E003EA0FC0121F90388001F8003FEB00F01560007F 140090C8FCA25AA97EA27FA2003F140E7F001F141C6C7E000714386C7ED801FC13F03900 FF03E090387FFFC0011F1300EB07F81F277DA525>I<ED1FC0EC07FFA3EC007F151FB0EB 07F8EB1FFE90387FFF9F9038FE07DF3901F800FF4848137F4848133F4848131F121F485A A2127F90C7FCA35AA97EA27F123FA2121F6C6C133F157F6C6C13FF3A03F001DFF03A01FC 079FFF39007FFF1FEB3FFED907F013C0283B7DB92E>I<EB07F8EB1FFF017F13809038FC 0FC03901F003E03903E001F0000714F8390FC000FC121F4913FE003F147EA248C7127FA3 5A90B6FCA390C8FCA57EA27F123F15076C7E150E6C7E0007141C6C6C133CD801FC137839 00FF01F090383FFFC06D1380903803FC0020277EA525>I<147E903803FF80010F13C090 381FC7E090383F07F0EB7E0F13FE13FC0001EB07E09038F803C0000390C7FCADB512FCA3 D803F8C7FCB3AB487EB512F8A31C3B7FBA19>I<ED03E090390FF00FF090393FFC1FF890 B5EA3C7C3A01F81FF0FC3A03E007C07C3A07C003E038000FECF000001F80EB8001003F80 A7001F5CEBC003000F5C00075C6C6C485A9038F81F800006B5C7FCEB3FFC380E0FF090C9 FC121EA3121F6C7E90B512C015F86C14FE6CECFF8016C04815E0391F80007F48C7EA0FF0 007E140316F8481401A5007EEC03F0A26CEC07E06C6CEB0FC0D80FE0EB3F803A07FC01FF 000001B512FC6C6C13F0010790C7FC26387EA52A>I<EA03F812FFA3120F1203B0EC07F0 EC1FFCEC7FFF9138F07F809038F9C01FD9FB807F9038FF000F49805BA35BB3A4486C497E B500E1B51280A3293A7EB92E>I<EA03C0487E487E487EA46C5A6C5A6C5AC8FCA9EA01F8 12FFA312071203B3AA487EB512E0A313387EB718>I<EB01E0EB03F0EB07F8EB0FFCA4EB 07F8EB03F0EB01E090C7FCA9EB01FC13FFA313071301B3B3A2123C127E00FF13F8130314 F0A2387E07E0387C0FC0383FFF00EA0FFEEA03F8164984B719>I<EA03F812FFA3120F12 03B1913803FFFCA36E13C0913800FE005D15F04A5A4A5A4A5A4AC7FC141E5C5C14FCEBF9 FE13FBEBFF7F496C7EEBFC1F01F87F6E7E6E7EA26E7E6E7EA26E7E157FA2ED3F8016C048 6CEB7FF0B500E1B5FCA3283A7EB92C>I<EA03F812FFA3120F1203B3B3AD487EB512E0A3 133A7EB918>I<2703F807F8EB0FF000FFD91FFEEB3FFCDA7FFFEBFFFE913AF03F81E07F 3D0FF9C00FC3801F802603FB80D9E7007F020013E601FED907FC6D7EA2495CA2495CB3A4 486C496C497EB500E1B500C3B51280A341257EA446>I<3903F807F000FFEB1FFCEC7FFF 9138F07F80390FF9C01F2603FB807F9038FF000F49805BA35BB3A4486C497EB500E1B512 80A329257EA42E>I<EB03FE90380FFF80013F13E09038FE03F83901F800FC4848137E48 487F4848EB1F80001F15C049130F003F15E0A248C7EA07F0A44815F8A96C15F0A26D130F 003F15E0A26C6CEB1FC0000F15806D133F6C6CEB7F006C6C13FE3900FE03F890387FFFF0 011F13C0D903FEC7FC25277EA52A>I<3903F80FE000FFEB7FFC9038F9FFFE9039FBE07F 803A0FFF801FC03A03FE000FE049EB07F0A249EB03F816FC150116FEA3ED00FFA916FE15 01A316FC150316F86DEB07F0ED0FE06D14C09039FF803F809039FBE07F009038F9FFFE90 38F87FF8EC1FC091C8FCAB487EB512E0A328357EA42E>I<903903F801C090381FFE03EB 7FFF9038FE07873901FC01C73903F000EF0007147F4848133F485A003F141F5B127FA390 C7FC5AA96C7EA3123F7F001F143F6C7E157F6C6C13FF3903F801DF3901FE0F9F39007FFF 1FEB3FFCEB07F090C7FCABED3FE00207B5FCA328357DA42C>I<3803F03F00FFEB7F8090 38F1FFE014C7390FF38FF03803F70F13F69038FE07E09038FC03C0EC018091C7FCA25BB3 A3487EB512F8A31C257EA421>I<EBFF83000313E7000F13FFEA1F80383E003F487F0078 7FA200F87FA37E6C90C7FC6C7EEA7FF8EBFF806C13E06C13F86C7F6C7F00017F6C7E0103 1380EB003F0060EB1FC000E0130FA26C1307A37E15806C130F1500B4131EEBC07C38F3FF F800E15B38C07F801A277DA521>I<131CA5133CA3137CA213FC120112031207121FB6FC A3D801FCC7FCB2EC01C0A93900FE0380A2017E13006D5AEB1FFE6D5AEB03F01A347FB220 >I<D803F8EB0FE000FFEB03FFA3000FEB003F0003140FB3A5151FA2153F1201156F6C6C EBEFF8903A7E03CFFF8090383FFF8F6D130FD903FCEBE00029267EA42E>I<B538C07FFE A33A0FFC001FF0D803F8EB07C016807F00011500A26D5B0000140EA2017F5BA2EC803C01 3F1338A26D6C5AA214E0010F5BA214F101075BA2903803FB80A214FF6D90C7FCA36D5AA2 147CA3143827257EA32C>I<B53A8FFFF07FFEA3260FF8009038800FF8000791397F0007 E0EF03C000031780816DEC80070001027F1400A26D6E5A000002FF130E15EF6DECE01ED9 7F01141C15C70281EBF03CD93F8314381583D91FC36D5AECC701A2D90FE76D5AECEE00A2 D907FEEBFFC04A137FA201035D4A133FA2010192C7FC4A7FA20100141E4A130E37257EA3 3C>I<B500C0B5FCA300039038007FF06C48EB3F806C6C013EC7FC6D133CEB3F806E5A01 1F5B6D6C5A14F1903807FBC06DB45A6D90C8FCA26D5A147F818114FF497FECCFF0EB03C7 90380783F890380F03FCEB0E0190381E00FE497F496D7E01F8133F000181D80FFC497EB5 48B51280A329247FA32C>I<B538C07FFEA33A07FC001FF06C48EB07C016807F00011500 A26C6C130EA26D131E6D131CA26D6C5AA2ECC078011F1370A2ECE0F0010F5B14F0903807 F1C0A214FB01035BA26DB4C7FCA36D5AA2147CA31438A214781470A25CA21301007C5BEA FE035C130749C8FCEAFC1EEA783CEA3FF86C5AEA0FC027357EA32C>I<003FB512FCA390 38C007F8D83E0013F0003C130FEC1FE0003814C00078EB3F80147F0070EBFF005C130149 5A5CEA0007495A5C495A133F90387F800E14005B485A5B0003141E485A5B4848131C001F 143C4848137C4913FC007F1303B6FCA31F247EA325>I<B81280A3290380972A>I<D803E0 1380390FFE01C0391FFFEFE048EBFFC048148000FE140038700FFE382000F81B0879B62A >126 D<001C131C007F137F39FF80FF80A6397F007F00001C131C190A78B72A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FP cmsy10 10 42 /FP 42 113 df<007FB81280B912C0A26C17803204799641>0 D<121E123FEA7F80EAFF C0A4EA7F80EA3F00121E0A0A7A9917>I<0060150600F0150F6C151F007C153E6C157C6C 15F86C6CEB01F06C6CEB03E06C6CEB07C06C6CEB0F806C6CEB1F00017C133E6D5B6D5B90 380F81F0903807C3E0903803E7C06DB45A6D90C7FC147EA214FF497F903803E7C0903807 C3E090380F81F049C67E013E137C497F497F4848EB0F804848EB07C04848EB03E04848EB 01F048C812F8003E157C48153E48151F48150F00601506282874A841>I<EB0380497EA7 397803803C00FC147E00FE14FE397F8383FC393FC387F8390FE38FE03903FBBF803900FF FE00EB3FF8EB0FE0A2EB3FF8EBFFFE3903FBBF80390FE38FE0393FC387F8397F8383FC39 FE0380FE00FC147E0078143C390007C000A76D5A1F247BA62A>I<15301578B3A6007FB8 12F8B912FCA26C17F8C80078C8FCB3A3007FB812F8B912FCA26C17F836367BB641>6 D<EC03FF023F13F091B512FC903903FC38FFD90FC0EB0FC0D93F00EB03F0017CEC00F801 F0153C4848814848814848ED078090C71403000EEE01C0A248EE00E0A2481770A2481738 A448171CA3B912FCA300E0C70038C7121CA400701738A46C1770A26C17E0A26CEE01C0A2 6CEE03806D15076C6CED0F006C6C151E6C6C5D017C15F8013FEC03F0D90FC0EB0FC0D903 FC01FFC7FC0100B512FC023F13F0020390C8FC36367BAF41>8 D<EC03FF023F13F091B5 12FC903903FC00FFD90FC0EB0FC0013FC7EA03F0017CEC00F801F0153C48488148488148 48ED078090C91203000EEE01C0A248EE00E0A2481770A2481738A448171CA3B912FCA300 E0CA121CA400701738A46C1770A26C17E0A26CEE01C0A26CEE03806D15076C6CED0F006C 6C151E6C6C5D017C15F8013FEC03F0D90FC0EB0FC0D903FC01FFC7FC0100B512FC023F13 F0020390C8FC36367BAF41>I<EC03FF023F13F091B512FC903903FC00FFD90FC0EB0FC0 013FC7EA03F0017CEC00F801F0153C484881486C151F486CED3F80013C157BD80E1EEDF1 C06DEC01E1486C6C903803C0E06D6CEB0780486C6C90380F00706D6C131E480178491338 6E5B6E5B6E485A4890260783C0131C913803C780DA01EFC7FCEC00FE157CA215FEEC01EF 913803C78091380783C0007090260F01E0133891381E00F04A13784A7F6C496D13704948 7F6C484890380780E04948EB03C06C48C73801E1C0011EEC00F16C48ED7B8001F8153F6C 48ED1F006C48151E6C6C5D017C15F8013FEC03F0D90FC0EB0FC0D903FC01FFC7FC0100B5 12FC023F13F0020390C8FC36367BAF41>I<EB0FE0EB7FFC497E0003EBFF803907F01FC0 390FC007E0391F0001F0001E1300481478007C147C0078143CA248141EA70078143CA200 7C147C003C14786C14F0001F1301390FC007E03907F01FC06CB51280C6EBFE006D5AEB0F E01F1F7BA42A>14 D<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F8 CCFCAE007FB812F8B912FCA26C17F836287BA841>17 D<020FB6128091B712C01303010F 1680D91FF0C9FC017FCAFC13FC485AEA03E0485A5B48CBFC5A121E5AA25AA45AAA1278A4 7EA27E121F7E6C7E7F6C7EEA01F86C7E137FEB1FF06DB71280010316C01300020F158091 CAFCAE001FB812804817C0A26C1780324479B441>I<007FB512FCB712C016F06C15FCC8 EA03FE9238003F80EE0FC0707EEE01F0707E177883173E171E83A2EF0780A4EF03C0AAEF 0780A4EF0F00A2171E173E173C5F17F84C5AEE07E04C5AEE3F80DB03FEC7FC007FB65AB7 12F016C06C02FCC8FCCCFCAE007FB712FEB9FCA26C5E324479B441>I<EF0180EF07C017 1FEF7F80933801FE00EE07F8EE1FE0EE7F80DB01FEC7FCED07F8ED1FE0ED7F80DA01FEC8 FCEC07F8EC1FE0EC7F80D901FEC9FCEB07F8EB1FE0EB7F80D801FECAFCEA07F8EA1FE0EA 7F8000FECBFCA2EA7F80EA1FE0EA07F8EA01FE38007F80EB1FE0EB07F8EB01FE9038007F 80EC1FE0EC07F8EC01FE9138007F80ED1FE0ED07F8ED01FE9238007F80EE1FE0EE07F8EE 01FE9338007F80EF1FC01707EF018094C7FCAE007FB81280B912C0A26C1780324479B441 >I<126012F812FEEA7F80EA1FE0EA07F8EA01FE38007F80EB1FE0EB07F8EB01FE903800 7F80EC1FE0EC07F8EC01FE9138007F80ED1FE0ED07F8ED01FE9238007F80EE1FE0EE07F8 EE01FE9338007F80EF1FC0A2EF7F80933801FE00EE07F8EE1FE0EE7F80DB01FEC7FCED07 F8ED1FE0ED7F80DA01FEC8FCEC07F8EC1FE0EC7F80D901FEC9FCEB07F8EB1FE0EB7F80D8 01FECAFCEA07F8EA1FE0EA7F8000FECBFC12F81260CCFCAE007FB81280B912C0A26C1780 324479B441>I<D93F801508D9FFF0150C00036D151C4813FE487F4814C09038C07FE027 3F000FF01438003CEB07FC48D901FE147800706D6C6C13F092383FC0034891391FF80FE0 6FB5FC030314C06F14806F6C13000060ED3FFC0040ED07F036137B9D41>24 D<D93F801508D9FFF0150C000301FC151C487F486D7E48809026C07FF01438393F000FF8 003CD903FE1478486D6C14F000709039007FC00392393FF80FE048020FB5FC6F14C00301 14806F1400EE3FFC0040ED07F0CCFCA2D93F801508EBFFF0000301FC151C487F486D7E48 809026C07FF01438393F000FF8003CD903FE1478486D6C14F000709039007FC00392393F F80FE048020FB5FC6F14C0030114806F1400EE3FFC0040ED07F036267BA741>I<020FB6 128091B712C01303010F1680D91FF0C9FC017FCAFC13FC485AEA03E0485A5B48CBFC5A12 1E5AA25AA45AAA1278A47EA27E121F7E6C7E7F6C7EEA01F86C7E137FEB1FF06DB7128001 0316C01300020F1580323279AD41>I<007FB512FCB712C016F06C15FCC8EA03FE923800 3F80EE0FC0707EEE01F0707E177883173E171E83A2EF0780A4EF03C0AAEF0780A4EF0F00 A2171E173E173C5F17F84C5AEE07E04C5AEE3F80DB03FEC7FC007FB65AB712F016C06C02 FCC8FC323279AD41>I<126012F0A41278A37EA2123E121E7E7F6C7EEA03F06C7EEA00FE EB3F80EB1FF0EB07FE903801FFF09039007FFFF0020F90B51280020115C0A2020F158002 7F01F0C7FC902601FFF0C8FCD907FEC9FCEB1FF0EB3F8001FECAFCEA01F8485AEA07C048 5A90CBFC121E123E123CA25AA35AA41260323279AC41>31 D<147014F0A3495AA3495AA2 495AA249CCFC5B131E5B5B5B1203EA07C0EA1F80007FBA12FCBB12FEA26C19FCD81F80CC FCEA07C0EA03F0120013787F7F131F7F6D7EA26D7EA26D7EA36D7EA31470472C7AAA53> I<181C181EA384A3727EA2727EA2727E85180019788585F11F80F107C0F103F0007FBA12 FCBCFCA26C19FCCCEA03F0F107C0F11F80F11E006161611801614E5AA24E5AA24EC7FCA3 181EA3181C482C7BAA53>I<0270153802F0153CA3494881A3494881A249486F7EA249C9 6C7E4983011E160149707E49177849830003183FD807C0EF0F80D81F80EF07E0007FBA12 F8BB12FEA26C19F8D81F80CAEA07E0D807C0EF0F80D803F0EF3F000000183C01785F6D5F 6D4C5A011F16036D5F6D6C4B5AA26D6C4BC7FCA26D6C151EA36D6C5DA302701538472C7A AA53>36 D<D93F801508D9FFF0150C000301FC151C487F486D7E48809026C07FF0143839 3F000FF8003CD903FE1478486D6C14F000709039007FC00392393FF80FE048020FB5FC6F 14C0030114806F14000060ED3FFC0040ED07F0CCFCB0007FB812F8B912FCA26C17F83626 7BA641>39 D<1738173CA283A283717EA2717E717E8417001878007FB812FCB97E727E6C 84CBEA03E0F001F8F0007EF13F80F10FE0F107F8F101FFA2F107F8F10FE0F13F80F17E00 F001F8F003E0007FB95ABA5A4EC7FC6C5FCB1278601701604D5A4D5AA24DC8FC171EA25F A2173848307BAC53>41 D<DA0380141C6F143C4A4880A24AC87EA2021E6F7EA24A6F7E02 7C82027815014A6F7E4948167849B812FC498349834984013ECAEA07C049717E4848EF01 F84848717ED80FC0183FD83F80F01FC0B4CCEA0FF0A2D83F80F01FC0D80FC0F03F00D803 F018FC6C6C4D5AD8007CEF03E06D4D5A6DB95A6D95C7FC6D5F6D5FD901E0C912786D6C5E 02784B5A027C1503023C5E6E4B5AA26E4BC8FCA26E6C141EA26E6C5C4B141C4C307DAC53 >44 D<D90FF0ED0FF0D93FFEED7FFC90B56C49B5FC486E4914804802F0903907F003C027 07C01FF890390FC001E0270F8007FC49C71270271E0003FE013E1478001C6D6C01781438 003C6DD980F8141C003891387FC1F0007891263FC3E0140E007091381FE3C092380FF780 00F003FF1507486E90C8FC816F5A826F7F167F4C7E834B6D140F007003EF150E923803C7 F8922607C3FC141E6C91260F83FE141C92261F01FF143C6CDA1E006D1338001E027C6D6C 1378000E4A90393FE001F027078003F090391FF803E02703C00FE06DB512C06CB5480103 14806C4A6D1400D93FFEC8EA7FFCD90FF0ED0FF048267BA453>49 D<91380FFFFE91B6FC1303010F14FED93FF0C7FC49C8FC13FCEA01F0485A485A5B48C9FC 5A121E5AA25AA45AA3B712FE16FFA216FE00F0C9FCA31278A47EA27E121F7E6C7E7F6C7E 6C7EEA00FC137FEB3FF0010FB512FE010314FF1300020F13FE283279AD37>I<387FFFF0 B6FC15C06C14F0C7EA0FFCEC00FE153FED0F80ED07C0ED03E01501ED00F016F81678163C A2161EA4160FA3007FB7FCB8FCA27EC9120FA3161EA4163CA2167816F816F0ED01E01503 ED07C0ED0F80ED3F0015FEEC0FFC007FB512F0B612C092C7FC6C13F0283279AD37>I<EE 0180EE03C0A2EE0780A2EE0F00A2161EA25EA25EA25EA24B5AA24B5AA24B5A150F93C7FC 151EA25DA25DA25DA24A5AA24A5AA24A5AA24AC8FCA2141EA25CA25CA25CA2495AA2495A A2495AA249C9FCA2131EA25B137C13785BA2485AA2485AA2485AA248CAFCA2121EA25AA2 5AA25AA212602A4E75BB00>54 D<126012F0AD12F812FCA212F812F0AD126006207BA400 >I<0060161800F0163CA200781678A36C16F0A36CED01E0A26CED03C0A36C6CEC0780A3 6C6CEC0F00A26C6C141E90B612FEA26C5DA201F0C7123C01785CA26D5CA36D495AA36D49 5AA26D6C485AA36D6C48C7FCA3903801E01EA26D6C5AA3EC7878A36E5AA2EC1FE0A36E5A A36E5AA26EC8FC2E3C80B92F>I<007FB612F0B712F8A27EC91278B3A5003FB612F85AA2 7EC91278B3A5007FB612F8B7FCA26C15F0253A7CB92E>I<0060161800F0163CB3B30078 1678A2007C16F8003C16F06CED01E0001F15036C6CEC07C0D807E0EC1F80D803FCECFF00 3A01FF8007FE6C6CB512F8011F14E0010391C7FC9038007FF82E347CB137>91 D<EC7FF80103B5FC011F14E0017F14F83A01FF8007FED803FCC77ED807E0EC1F80D80F80 EC07C048C8EA03E0001E150148ED00F0007C16F800781678A248163CB3B3006016182E34 7CB137>I<15FE1407141FEC7FC0ECFE00495AEB03F0A2495AB3A8495AA2495A49C7FC13 FEEA07FCEAFFF0138013F0EA07FCC67E133F6D7E6D7EA26D7EB3A86D7EA2EB01FC6D7EEC 7FC0EC1FFE140714001F537BBD2A>102 D<127EEAFFE013F8EA07FEC67EEB3F806D7E13 0F6D7EB3A86D7EA26D7E6D7E147FEC3FC0EC0FFE1403140FEC3FC0EC7F0014FC495A495A A2495AB3A8495A131F495A01FFC7FCEA07FEEAFFF813E0007EC8FC1F537BBD2A>I<14C0 EB01E0A2EB03C0A3EB0780A3EB0F00A2131EA35BA25BA35BA2485AA3485AA3485AA248C7 FCA3121EA25AA35AA35AA21278A37EA37EA27EA36C7EA26C7EA36C7EA36C7EA21378A37F A27FA37FA2EB0780A3EB03C0A3EB01E0A2EB00C0135278BD20>I<126012F0A21278A37E A37EA27EA36C7EA26C7EA36C7EA26C7EA31378A37FA27FA37FA2EB0780A3EB03C0A3EB01 E0A2EB03C0A3EB0780A3EB0F00A2131EA35BA25BA35BA3485AA2485AA3485AA248C7FCA3 121EA25AA35AA35AA2126013527CBD20>I<126012F0B3B3B3B3A91260045377BD17>I<00 60130C0070131C00F0131EB3B3B3B3A60070131C0060130C175277BD2A>I<126012F0A2 1278A37EA37EA37EA36C7EA36C7EA36C7EA36C7EA31378A37FA37FA37FA36D7EA36D7EA2 6D7EA36D7EA31478A380A380A380A3EC0780A3EC03C0A3EC01E0A3EC00F0A31578A3153C A3151EA2150C1F537BBD2A>110 D<F10180F103C0A2F10780A2F10F00A2191EA261A261 A261A24E5AA24E5AA24E5AA24EC7FCA3181EA260A260A260A24D5AA24D5AA24D5AA24DC8 FCA2171E132001F05D1201486C5D120F486C5D123D00794B5AEAE0FE00404B5AEA007F4C 5A6D7E4CC9FC6D7E161EA26D6C5BA26D6C5BA26D7E5E6D7E4B5AEB00FE4B5AA291387F07 80A2DA3F8FCAFCA2EC1FDEA2EC0FFCA26E5AA25D14035D14015D42547B8345>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FQ cmbx10 10 64 /FQ 64 124 df<913801FFC0023F13F049B512FC010780011FEB00FFEB3FF890267FE001 138049485A481380A2481300A36F13006F5A167C93C7FCA492383FFFC0B8FCA5000390C7 FCB3A9B5D8FC3F13FFA5303A7EB935>12 D<D80F80133ED81FE0EB7F80486CEBFFC0486C 4813E000FF5B01FC14F0A201FE14F8A3007F7F003F7F001F147FD80FDEEB3F78D8001E13 00013E14F8013C14F0A349EB01E0A249EB03C0000114074914804848EB0F0000075C4848 133E48C75A000E143800041410251E7DB932>34 D<EA0F80EA1FE0EA3FF0EA7FF812FF13 FCA213FEA3127F123F121FEA0FDEEA001E133E133CA31378A213F0120113E0EA03C01207 EA0F80EA1F00120E12040F1E7AB91B>39 D<EA0F80EA1FE0EA3FF0EA7FF812FF13FCA213 FEA3127F123F121FEA0FDEEA001E133E133CA31378A213F0120113E0EA03C01207EA0F80 EA1F00120E12040F1E7A8D1B>44 D<B61280A919097F9620>I<EA0780EA1FE0EA3FF0EA 7FF8A2EAFFFCA4EA7FF8A2EA3FF0EA1FE0EA07800E0E7A8D1B>I<49B4FC010F13E0017F 13FC90B57E48EB83FF4848C613804848EB7FC04848EB3FE049131F001F15F0A2003F15F8 49130FA2007F15FCA500FF15FEB3007F15FCA4003F15F8A26D131F001F15F0A2000F15E0 6D133F6C6CEB7FC06C6CEBFF802601FF8313006CEBFFFE6D5B011F13F0010190C7FC2738 7CB630>48 D<143C14FC1301130F137FB5FCA4139FEA001FB3B3A4007FB61280A5213779 B630>I<EB0FFC90387FFFC048B512F0000714FC4880391FC03FFF263F800F1380267FC0 0313C06D6C13E0EAFFF06D6C13F0A2157F16F8A26C5A6C5A6C5A6C5AC8FC16F015FF16E0 A24A13C016804A13005D4A5A4A5A4A5A4A5A5D4AC7FC14FE495AD903F013F8495A495A49 5A90383E00014914F05B4848130348B6FC5A5A5A5A5AB712E0A525377BB630>I<EB03FF 011F13F0017F13FC90B57E3901FE07FF2603F0011380D807E014C0260FF80013E07F486C 14F0A6000F15E06C5A6C485AD801E014C0C714805C16004A5AEC3FF890380FFFF015C015 F015FE90380007FF020113C06E13E0ED7FF016F8A2ED3FFCA2D80FC014FEEA3FF0487EA2 487EA316FCA25B007FEC7FF85B6C48EBFFF06C484813E0260FF80713C06CB612806C1500 C614FC013F13F0010790C7FC27387CB630>I<ED0FC0151F153FA2157F15FF5C5CA25C5C 5CA2143E147E5C495A5C495A1307495A495A5C133E137E5B485A5B1203485A485A5B48C7 FC5A127E5AB81280A5C70001EBC000A90103B61280A529377DB630>I<000C15C0D81F80 130701F8137F90B6FC168016005D5D5D15E05D92C7FC14F8010EC8FC90C9FCA7EB03FE90 381FFFC0017F13F890B57E9038FC07FE9038E003FFD980011380010014C0000E6D13E0C8 FC16F0A316F8A2EA1F80487E487E487EA316F0A25B6C4814E0495A010014C06C4913806C 6C481300390FF01FFE6CB55A6C5C6C14E06C6C1380D91FF8C7FC25387BB630>I<EC0FF8 ECFFFE0103EBFF80010F14C090391FF80FE090397FE003F09038FF800748EB000F4848EB 1FF84848133FA2120F5B001FEC1FF0ED0FE0003FEC07C04990C7FC127FA2EC3FC0ECFFF8 D8FFF113FE01F37FD9F78113809039FF007FC049EB3FE04914F0ED1FF85B16FCA34914FE A4127FA5123F16FCA2121F6D14F8120FED3FF06C6C14E06C6C137F3A01FF81FFC06C90B5 12806DEBFE006D5B010F13F00101138027387CB630>I<123E123F13E090B71280A54816 005E5E5E5E5E007EC7EA07C0007C140F4B5A00FC4AC7FC48147E5D5D4A5AC712034A5A4A 5AA24A5A4AC8FCA25C147E14FE13015C1303A31307A2495AA3131FA4133FA96D5A6D5A6D 5A293A7BB830>I<49B47E010F13F0013F13FC497F9038FE01FF3A01F0003F804848EB1F C00007EC0FE05B000FEC07F0A2121F7FA213F87F01FFEB0FE014C09138F01FC06C13F891 38FE3F806C90B512005D6C14F86C806C806DEBFF806D14C090B612E04815F0EA07FD260F F07F13F848486C13FC383FC00F14034848C613FE157F48C7121F15071503A21501A216FC A26C7EED03F86C7E6DEB07F0D81FF8EB1FE0D80FFEEBFFC06CB612806C1500C614FC013F 13F00103138027387CB630>I<EB03FF011F13E0017F13F890B57E0003EB01FF2607FC00 13804848EB7FC0153F001F15E04848EB1FF0A2007F15F8A300FF15FCA516FEA5007F143F A3123F157F6C7E000F14FF6C6C5A3903FF03DF6CEBFF9F6CEC1FFCEB3FFEEB07F890C7FC A216F8D807C0133F487E486C14F0487E16E0ED7FC0A249EBFF806C484813009038C007FE 390FE01FFC90B55A6C5C000114C06C91C7FCEB1FF027387CB630>I<EA0780EA1FE0EA3F F0EA7FF8A2EAFFFCA4EA7FF8A2EA3FF0EA1FE0EA0780C7FCA9EA0780EA1FE0EA3FF0EA7F F8A2EAFFFCA4EA7FF8A2EA3FF0EA1FE0EA07800E257AA41B>I<ED03E04B7EA24B7EA34B 7EA24B7EA34B7EA292B57EA34A8015F702038015E715E302078015C1020F801580A2021F 80ED007F4A80023E133FA2027E80027C7F02FC814A7FA20101824A7F49B77EA34982A290 270FC000017F4A7FA2011F8291C8127F4982013E153FA2017E82017C81B500F8010FB612 80A5413A7DB948>65 D<B812C017FC17FF18C08428007FF800017F707FEF3FFCA2717EA2 717EA74D5AA24D5AA24D5A4D5A04035B91B712804DC7FCA2EFFFC09126F8000113F0706C 7EEF1FFE717E8319807113C0A319E0A719C05FA24D13804D13005F4CB45AB95A18F018C0 95C7FC17F03B397DB844>I<DB3FFC14C00203B5EAC003021FECF00791B6EAFC0F0103ED FE1F499039FC00FFBF011F01C0EB1FFF4948C71207D97FF8804948804849804849157F48 49153F91C9FC48171F485A180F123F5B1807127FA25B95C7FC12FFAB127FA26DEE07C0A2 123FA27F001FEF0F80A26C7E6CEF1F00806C6D153E6C6D157E6C6D5D6D6C4A5AD93FFEEC 07F090261FFFC0EB1FE0010701FCEBFFC06D90B6C7FC01005D021F14F8020314C0912600 3FFEC8FC3A3B7BB945>I<B87E17F817FF18C018F028007FFC00077F9338007FFEEF1FFF 050713807113C0837113E019F0187F19F8183F19FC181FA219FEA419FFAC19FEA419FC18 3FA219F8187F19F0F0FFE05F4D13C04D1380051F1300EF7FFE933803FFFCB912F018C095 C7FC17FC178040397DB849>I<B912F0A528007FFC000113F8EE003F170F17071703A217 01A21700A2163E18FC187CA4047E1300A216FE150391B5FCA5ECFC031500167E181F163E A2183EA493C7127EA2187C18FCA21701A21703EF07F8170F173FEE03FFB9FCA218F0A338 397DB83F>I<B912C0A528007FFC000713E01600173F171F170F1707A31703A3047C13F0 1701A494C7FC16FCA21501150791B5FCA5ECFC0715011500A2167CA693C8FCABB77EA534 397DB83C>I<B6D8FE07B612F0A526007FFCC70003EBE000B3A291B8FCA502FCC71203B3 A4B6D8FE07B612F0A544397DB84B>72 D<B612FEA539007FFC00B3B3ABB612FEA51F397E B824>I<B600FE010FB512C0A526007FFCC8383F800006FFC7FC4D5A4D5AEF07F04D5A4D 5AEF7F804DC8FC4C5AEE03F84C5A4C5AEE3FC04C5A4CC9FCED01FC15034B7E150FED3FFF 4B7F92B57E14FD91B67E03F37F03E37F15C103007F4A6D7E4A8082707F707FA2707F707F 707FA2707F717E8483717F717F8583B600FE90B612E0A543397DB84B>75 D<B77EA526007FFCC9FCB3AAEF0F80A5171F1800A35FA35F5FA25E16075EEE7FFEB8FCA5 31397DB839>I<B500FC0407B512F06E5E6E5EA3C66C6D4BEBE000A2017D6D157DA2017C 6D15F9A36E6CEC01F1A26E6CEC03E1A26E6CEC07C1A36E6CEC0F81A26E6CEC1F01A26E6D 133EA36E6D137CA26E6D13F8A292397FF001F0A392393FF803E0A292391FFC07C0A29239 0FFE0F80A3923907FF1F00A26F13BEA26F13FCA36F5BA2705AA2705AB500FE0303B612F0 A2EE1FC0A2EE0F8054397DB85B>I<B500FE0207B512F080A28181C66C6D90390003E000 8181A2017D7F017C7F6E7E6E7F6E7FA26E7F6E7F6E7F6E7F6E7FA26F7E6F13806F13C06F 13E06F13F0A26F13F86F13FC6F13FEEE7FFF701383A27013C37013E37013F37013FB7013 FFA28283838383A283838383187FA2183FB500FE151F180F18071803A244397DB84B>I< EDFFF8020FEBFF80027F14F049B612FC01079038C01FFF90271FFE000313C0D93FF80100 7F49486E7E49486E7E48496E7E48496E7E488391C87E48486F1380A2001F18C04981003F 18E0A348486F13F0A400FF18F8AC007F18F06D5DA3003F18E0A26D5D001F18C06D5D6C18 806C6D4A1300A26C6D4A5A6C6D4A5A6C6D4A5AD97FFC49485A6D6C495B90271FFFC01F5B 010790B6C7FC010115FC6D6C14F0020F1480020001F8C8FC3D3B7BB948>I<B8FC17F017 FE8318C028007FFC000713E0040113F09338007FF8EF3FFCA218FE171FA218FFA718FEA2 173F18FCA2EF7FF8933801FFF0040F13E091B71280180017FC17E002FCC9FCB3A2B612FE A538397DB841>I<B712FCEEFFE017FC17FF18C028007FFC000F7F04017F706C7E717EA2 717EA284A760173F604D5A4D5A4C5B040F5B91B7C8FC17FC5F17FE913AFC003FFF80040F 7F707F82707FA2707FAB1AE0F101F0A2EF7FFC1AE0B600FE90383FFE0394381FFF0771EB FFC00503148005001400CBEA0FFC443A7DB848>82 D<D903FF1306013FEBE00E90B5EAF8 3E48ECFE7E48ECFFFE3807FE01390FF0003F4848130F484813031501485A1500167E12FF A2163E7F7F7F6D91C7FCEBFF806C13F8ECFFC015FC6C14FF6C8116E06C816C816C816C81 6C7E011F801307D9003F14801403EC001F030713C0818100788012F8167FA36C1680A27E EEFF007E6D5C01E0130101F8495A9039FF801FF891B55A00FC5DD8F83F1480D8E00749C7 FC39C0007FF02A3B7BB935>I<003FB91280A5267FF801D9F00313C001E015000180163F 0100161FA2007E170FA2007C1707A400FC18E0481703A4C793C7FCB3AC011FB7FCA53B38 7DB742>I<B600FE011FB512C0A526007FFCC8380F8000B3B3133F4EC7FCA26D7E183E01 0F167E6E5D6D6D495A6D6D13036D6DEB0FF06D01FCEB7FE0023FB65A6E92C8FC020314FC 020014F0030F90C9FC423A7DB849>I<B6D8F01FB500FE90387FFFFCA5C601F8C7003F90 C8EA7C00836E19FC017F715C836E1801013F715C806D4B6D495AA26F17076D4B6D5CA26F 170F6D4B6D5C6F137C73131F6D03FC93C7FC6FD9F87F5C6D02016E133EEFF03F03F8177E 6D02036E137C4D7E03FC17FC027F01076F5A03FE496C1381023F010FEDC1F04D7E03FF16 C36E011FEDE3E04D7E049F15E76E01BFEDF7C004FE7F6EEFFF804C7FA26E95C8FC4C80A2 6E5F4C143FA26E496E5AA2037F5E4C140FA2033F5E4C1407A26FC86C5A5E3A7EB863>87 D<01405B01E0EB0380486CEB07C04848EB0F804848EB1F0049131E48C75A48147C001E14 78485CA248495AA300F81303485CD8F7E0EBDF80D8FFF0EBFFC001F814E001FC14F001FE 14F8A3007F7FA2003F7F4914F06C48EB7FE06C48EB3FC0D803E0EB0F80251E77B932>92 D<EB3FFE0003B512E04814F84880391FF007FE393FF801FF6E7F82157F6C4880A26C5AEA 0380C8FCA291B5FC130F90B6FC0003EBF87F481380381FFC00485A5B485A485AA515FF6C 6C5A6C6C487F903AFC0FBFFFC06CB5123F0007497E6CEBF80F39007FC0032A257DA42E> 97 D<13FFB5FCA512077EAEEDFF80020F13F8023F7F91B6FCDAFE031380DAF00013C002 C0EB3FE04A14F091C7121F17F8A2EE0FFCA317FEA917FCA3EE1FF8A217F06EEB3FE06E13 7F02F0EBFFC09026FDFC07138001F8B5EAFE00023F5BD9F00F13F0D9E00190C7FC2F3A7E B935>I<903801FFC0010F13FC017F7F90B6FC48018013802607FE0113C0EA0FFC13F8EA 1FF0003F6D1380A24848EB7F00151C92C7FC12FFA9127F7FA2123F6DEB03E0121F6C6CEB 07C07F6C6CEB0F8000019038E03F006CEBFFFE6D5B010F13F00101138023257DA42A>I< EE7F80ED7FFFA5150381AEEB03FF011F13F1017F13FD48B7FC48EBC07F3907FE000F4848 7F484813015B123FA2485AA312FFA9127FA36C7EA2121F6D5B000F140FD807FE4913C06C 6C6CB512FE6CEBFFFD6C6C13F9011F13C1903803FE012F3A7DB935>I<49B47E011F13F0 017F13FC90B57E0003903881FF803907FE007F4848EB3FC04848EB1FE05B003FEC0FF0A2 485A16F8150712FF90B6FCA401E0C8FCA4127FA27F123F16F86C7E6C6C13016DEB03F06C 6CEB07E00001EBE01F6C90B51280013F1400010F13FC010013C025257DA42C>I<EC0FF8 ECFFFC010713FF491480EB3FF8D97FE113C0EBFFC114815A1401486D1380ED7F00153E92 C7FCA8B6FCA5000390C8FCB3A9B512FEA5223A7DB91D>I<163FD907FEEBFF8090267FFF E113C048B512FB489138FFDFE02607FC03131F380FF000001F159F484890387FCFC0EEC7 80007FEDE000A7003F5DA26C6C495A000F92C7FC3907FC03FE90B55A485C6D13E0261E07 FEC8FC90CAFCA2123FA213C06CB512F8EDFF8016E06C81826C816C81000F815A273FC000 0F13804848130148C8127FA56C6CECFF006C6C495AD81FF0EB07FCD80FFEEB3FF86CB65A 000115C06C6C91C7FC010713F02B377DA530>I<13FFB5FCA512077EAEED7FC0913803FF F84A7F021F7F91383F03FFDA7801138014F04A6C13C05C5CA391C7FCB3A2B5D8FC3F13FF A5303A7DB935>I<13F0EA03FC487E487EA2481380A46C1300A26C5A6C5AEA00F090C7FC A813FF127FA512077EB3A7B512F8A5153B7DBA1B>I<141EEC7F80ECFFC04913E0A24913 F0A46D13E0A26D13C0EC7F80EC1E0091C7FCA8EC0FF0EB0FFFA5EB007F143FB3AF120EEA 3F80EA7FC0A239FFE07FE0A215C0ECFF80D87FC113006CB45A6C5B6C13E0000190C7FC1C 4B86BA1D>I<13FFB5FCA512077EAE92383FFFE0A592380FF0004B5A4B5A4BC7FC15FEEC 03FC4A5A4A5AEC1FC0143F4A7E4A7E81A281ECCFFEEC87FF1407496C7F6E7F6E7FA26F7E 6F7E6F7EA26F7EB539F83FFFF8A52D3A7EB932>I<13FFB5FCA512077EB3B3AAB512FCA5 163A7DB91B>I<01FED97FE0EB0FFC00FF902601FFFC90383FFF8002076D90B57E021FD9 FF0380DA3F03903987E07FF0DA78009039CF001FF800074914DE6C6C48D97FFC6D7E4A5C 4A5CA391C75BB3A2B5D8FC1FB50083B512F0A54C257DA451>I<01FEEB7FC000FF903803 FFF84A7F021F7F91383F03FFDA78011380000713F06C6C486C13C05C5CA391C7FCB3A2B5 D8FC3F13FFA530257DA435>I<903801FFC0010F13F8017F13FF90B67E0003018013E03A 07FE003FF0D80FF8EB0FF8001F81491307003F81491303007F81A300FF1680A9007F1600 A36C6C495AA2001F5D6D130F6C6C495A6C6C495A6C6C6CB45A6C90B55A6C6C91C7FC011F 13FC010113C029257DA430>I<01FFEBFF80B5000F13F8023F7F91B6FCDAFE071380DAF0 0113C000079039C0007FE06C4914F091C7123F17F8161F17FCA2160F17FEA917FC161FA2 17F8163F17F06EEB7FE06E13FFDAF00113C0DAFC07138091B5EAFE00023F5B020F13F002 0190C7FC91C9FCABB512FCA52F357EA435>I<49B4EB0F80011FEBE01F017F13F890B5EA FC3F00039038C07E7F48EB001F4848EB0FFF497F48487F123F497F127FA25B12FFA9127F 7FA2123F7F001F5C6D5B000F5C6CB4133F6C01C0B5FC6CEBFFFD6C6C13F1011F13C19038 03FE0190C7FCAB037F13FEA52F357DA432>I<9038FE07F000FFEB1FFC4A7E4A7E02F813 8002E113C0EA07FF6C13C1A202801380A2ED7F00151C91C8FCB3B512FEA522257EA427> I<90383FF0383903FFFEF84813FF121F383FC00FEB0003007E1301140012FEA27E6D1300 13F8EBFFE06C13FC14FF6C14C06C14E06C14F0000314F8C614FC131F9038007FFE140700 78130112F814007EA26C14FC6C1301018013F89038F00FF090B512E000FD14C000F01400 38E01FF81F257DA426>I<131FA55BA45BA25BA25A5A5A001FEBFFE0B6FCA4000390C7FC B115F8A86CEB01F014816CEBC3E090387FFFC06D13806D1300EB03FC1D357EB425>I<01 FFEC3FC0B5EB3FFFA5000714016C80B3A25DA25D6C5C4B13E06CD9C03E13FF90387FFFFC 6D5B6D13E00103130030257DA435>I<B539F003FFF8A5000390C7EA3E006E137E6C157C 6E13FC6C5DECE001017F5CA2ECF003013F5CECF807011F5CECFC0F010F5C151FD907FE90 C7FCA26E5A6D133E15FE6D5BA26D5BA36E5AA26E5AA26E5AA26E5A2D257EA432>I<B500 F1B538807FFFA50007903B000FF80007E06C6E6C14C0180F6C0180168082DAC00F141F6C 6F14004B5CD97FE0153E17806E48147E013F90393E7FC07C037E14FC90271FF87C3F5B17 E0DAFCFC13E1010F9039F81FF1F014FD9139FFF00FF36DEDFBE017FF6D496C5BA24B7E6D 5EA26D496C90C7FCA36EC75AA2023E147C40257EA445>I<B539F01FFFF0A500019039C0 07F0006C5D6D6C485A6D6C485AECF83F6D6C48C7FC010F13FE6DB45A6D5B5D6D5B7F147F 6E7E814A7E14FF497F903803FBFFD907F37F02E17FEB0FC049486C7E013F6D7ED97F007F 01FE6D7E49130FB539803FFFF8A52D257EA432>I<B539F003FFF8A5000390C7EA3E006E 137E6C157C6E13FC6C5DECE001017F5CA2ECF003013F5CECF807011F5CECFC0F010F5C15 1FD907FE90C7FCA26E5A6D133E15FE6D5BA26D5BA36E5AA26E5AA26E5AA26E5AA2141F92 C8FC5C003F133E387F807E38FFC07C14FC5CEAF80138FC07F0387E1FE0387FFFC06C90C9 FC6C5AEA07F02D357EA432>I<003FB612C0A4D9F0031380D9C007130049485AEB001F00 7E5C4A5A147F007C495A5D495B5B495B000091C7FC495A131F5C90393FF807C0137FEBFF F014E04813C048140F48138014004848EB1F80121F49133F4848137F397FF003FFB7FCA4 22257DA42A>I<B812FEA42F04809830>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FR cmbx12 14.4 33 /FR 33 123 df<B712F0AC240C7FA02D>45 D<153C157EEC01FE1407141F14FF13070003 B5FCB6FCA3147F13F8EAFC00C7FCB3B3B3A4007FB712FEA62F4E76CD43>49 D<EC3FFC49B512E0010F14F8013F14FE90B77E4816E00007D9007F7FD80FF8010F7F01E0 6D7F484801017FD83FF06D7F01FC1680486C806D6E13C0B5FC6E6D13E0A318F082A26C90 C7FCA26C5A6C5A6C5AEA01C0C94813E0A318C05E18805E18004C5AA24B5B5F4B5B4B5B5F 4B90C7FC4B5A4B5A4B5AEDFFE05E4A5B4A90C8FCEC07FC4A48EB03F04A5A4A5A4A5A4AC7 EA07E0495A495A495A5C4948140FEB1F8049C8121F49B7FC90B812C05A5A5A5A5A5A5AA2 B91280A5344E79CD43>I<91380FFF8091B512F8010314FF010F15C0013F15F090267FF8 077F9026FF80017F4848C76C7ED803F881D807FE6E13807F486D15C06E7F486D15E0A76C 5D4A15C06C5B6C90C71480EA00FC90C8481300A24C5A5F4B5B4B5B4B5B4B5B92B5C7FC91 B512FC16F016C016FCEEFF80DA000713E0030113F89238007FFE707E70138018C07013E0 18F0A27013F818FCA3D801E016FEEA07F8EA1FFE487E487FA2B57EA218FCA44A4913F87E 91C714F06C5D01FC16E06C484A13C0494A1380D80FFE91B512002707FFE0075B000190B6 5A6C16F0013F15C0010F92C7FC010114F8D9001F90C8FC374F7ACD43>I<BA12C019FEF1 FFC01AF01AFC1AFFD8000701F8C7000380060080073F7F737F737F85878587A28587A863 61A24F5B6361634F5B077F5B96B55A060349C7FC061F13F892B812E097C8FC861AF81AFE 03F8C86D7E071F13E0737F070313FC737F87851C807413C0A27413E0A31CF0A386A362A3 1CE0A2621CC097B5FC1C80614F14004F5B61073F5B4EB512F0BC5A1B8098C7FC1AF81AC0 07F8C8FC54527CD160>66 D<932601FFF8EC01C0047FD9FF8013070303B600F0130F031F 03FC131F92B8133F0203EFC07F020FDAF000EBF0FF4A49C7EA1FF9027F01F00203B5FC91 B5008080010349C97E4949824949824901E08249498249498290B5CA7EA2484983484918 7FA24849183FA25A4A181F5AA24849180FA45A4A95C7FCA3B5FCAE7EA280A26CF20FC0A4 6C7FA26C1A1F6E19807EA26C6DF03F00A26C6D187E6C7F636D6D16016D6D4C5A6D6D4C5A 6D01F8160F6D6D4C5A6D6D4C5A01006D6CEDFF806E01F0020390C7FC021F01FEEC0FFC6E D9FFF0EBFFF8020391B612E002005F031F93C8FC030315F8DB007F14C0040101FCC9FC52 5479D261>I<B812E0A6D8000701FCC7FCB3B3B3B0B812E0A62B527DD132>73 D<93380FFFC00303B6FC031F15E092B712FC020316FF020FD9FC0014C0023F01C0010F13 F04A90C700037F902601FFFC020013FE49496F7E49496F7F49496F7F49496F7F49496F7F 4990C96C7F4948707FA24849707F48864A83481B804A83481BC0A2481BE04A83A2481BF0 A348497113F8A5B51AFCAF6C1BF86E5FA46C1BF0A26E5F6C1BE0A36C6D4D13C0A26C6D94 B51280A26C1B006C6D4C5B6E5E6C626D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D6D4B5B6D 01FE4AB5C7FC6D6D4A5B6D02E0011F5B023F01FC90B512F0020F90B712C0020394C8FC02 0016FC031F15E0030392C9FCDB001F13E0565479D265>79 D<BAFC19F819FF1AC01AF01A FCD8000701F8C7001F7F0603EBFF80060014C0073F13E07313F0851BF8851BFCA27313FE A31BFFA91BFEA34F13FCA21BF8611BF04F13E04F13C096B5128006031400061F5B92B812 F8621A804FC7FC19E003FCCBFCB3ACB812E0A650527CD15C>I<B912F0F0FF8019F819FF 1AC01AF0D8000701F8C76C7F060F13FE06037F060080737F85737F87A2737FA387A863A3 4F5B6361634F5B96B5C8FC06035B060F5B95B512F092B812C097C9FC19F86119FC9226F8 000313FFDD007F7F727F727F727F727F727FA28684A286A887A61D3C1D7E8784A2736D13 FCA273EBF001B800C06D9038F803F8739038FE07F07390B5FC070115E0736C1480080F14 00CEEA7FF85F537CD164>82 D<EC3FFF0103B512F0011F14FE017F6E7E90B712E048D9C0 077F48D9800113FC486D6C6C7E163F486D6D7E848284A2826C49816C5BA2C648C7FC90C8 FCA44BB5FC4AB6FC143F49B7FC130F013FEBF80790B512800003EBFC004813F0485B485B 485B91C7FC485AA2485AA55EA26C6C5C5E6C6C147D6E01FD13F86C9026C001F9EBFFE06C 9026F00FF014F06C90B5487E0001ED803F6C6C49487E011F01F8010313E0010101C090C8 FC3C387CB641>97 D<EB3FF8B5FCA61203C6FCB3A3EE7FF00303B5FC031F14E0037F14F8 02F9B612FE02FB9038007FFFDAFFF8010F7F03E06D7F4B6D7F92C76C7F4A6E7F4A6F7EA2 727EA285A2841A80A41AC0AB1A80A44E1300A261A24E5AA26E4B5A804D5B03C0495B6F01 0F5BDAE7F0495BDAC3FED9FFFEC7FC0281B65ADA007F14F049011F14C049010749C8FC90 C813E042547CD24B>I<913801FFF0021F13FF91B612E0010315F8010F81499038800FFE 4948486C7ED9FFF8491380485B4A4913C0485B5A485BA25A91C76C1380486F1300A2EE01 FC484891C8FCA412FFAB127FA27FA27EA26C6DEC07E0A27E6EEC0FC06C7FEF1F806C6D14 3F6C6DEC7F006C01FE14FE6D6C495A011F9038E00FF86D90B55A01035D01001580021F01 FCC7FC020113E033387CB63C>I<943801FFC00407B5FCA6EE001F1707B3A3913801FFC0 021F13F891B6FC010315C7010F15F749D9C01FB5FC90397FFE0003494813004801F08048 498048498083485BA24890C8FCA25AA2485AA412FFAB127FA4123F7FA27EA26C7F5F6C6D 5C6C6D5C6C6D91B5FC6C6D010314F06D6C49ECFFC090393FFF803F6D90B512CF0107150F 010114FCD9003F13F00203018049C7FC42547CD24B>I<913803FF80023F13F891B512FE 01036E7E010F15E0013F01017F903A7FFC003FF8D9FFF06D7E48496D7E48496D7E5A4A6D 13805A91C76C13C05AA24817E082485AA218F0A212FFA290B8FCA418E049CAFCA5127FA3 6C7EA36CEE01E0EF03F06C7FEF07E06C7F6C6DEC0FC06C161F6C6DEC3F80D97FFCEC7F00 6DB4EB01FE6D9038E00FFC010790B55A010115E06D6C1480021F49C7FC020013E034387C B63D>I<ED0FFE92B51280020714E0021F14F0027F14F89138FFF83F499038E07FFC0107 1380923800FFFE5B495A5C133FEE7FFC5C017FEC3FF8EE1FF0EE07C093C7FCADB712E0A6 26007FFCC8FCB3B3A5007FB6FCA62F547CD329>I<DA1FFE14FE49B539E007FF800107DA F81F13C0011FDAFE7F13E0017F91B612F09026FFF807143F48D9E001EBF07F4890268000 7F133F4804F813E04890C7383FFC1F19C0484891391FFE070095C7FCA2003F82A8001F5E A36C6C4A5AA26C6D495A6C5E6C9039E001FFE06CD9F8075B4890B65A484BC8FC01E714F8 D807C114E09026C01FFEC9FC91CBFC120FA27FA27F7F7F90B7FC17F06C16FE717E18E06C 836C83846C83488312075AD81FFCC7000114804848EC001F484815077113C0485A83A56C 6C4B1380A26C6C4B13006D5D6C6C4B5A6C6C4B5A6C01E049B45A6C01FE011F5BC690B712 C0013F93C7FC010F15FC010115E0D9000F01FCC8FC3C4F7CB543>I<EB3FF8B5FCA61203 C6FCB3A3EE0FFC93B57E030314E0030F14F84B8092393FC07FFE92387E003F4B6D7EECF9 F0DAFBE0814B7F14FF4B8192C7FCA25CA35CB3ACB6D8FC0FB612C0A642537BD24B>I<13 3FEBFFC0487F487F487FA2487FA66C5BA26C5B6C5B6C5B013FC7FC90C8FCACEB1FF8B5FC A612017EB3B3A4B612F0A61C547BD326>I<153FEDFFC04A13E04A13F04A13F8A24A13FC A66E13F8A26E13F06E13E06E13C0ED3F0092C7FCACED1FFC91B5FCA61401EC007FB3B3AF EA0380EA0FE0EA3FF8487E16F8487E15FF16F0A24A13E0D87FFC14C05C6C484813804948 13006CB512FC6C5C000314E0C691C7FCEB1FF8266C88D329>I<EB1FF8B5FCA612017EB3 B3B3AFB612F8A61D537BD226>108 D<D93FF8D90FFEED7FF0B591267FFFC0903803FFFE 4BB500F0010F6D7E030702FC013F14E0031F6E90B67E922A3FC07FFF01FE037F92267F00 1F903803F800000302FC6DD987E06D7EC6D9F9F8ED8FC0DAFBE003DFC77F4B6D01DE143F 02FF16FE4B4B8192C75CA24A5EA34A5EB3ACB6D8FC07B6D8E03FB6FCA668367BB571>I< D93FF8EB0FFCB591B57E030314E0030F14F84B8092393FC07FFE92387E003F00034A6D7E C6EBF9F0DAFBE0814B7F14FF4B8192C7FCA25CA35CB3ACB6D8FC0FB612C0A642367BB54B >I<913801FFE0021F13FE91B612C0010315F0010F15FC499038807FFE903B7FFC000FFF 8049486D7F4801E001017F48496D7FA248496E7E488391C8123F4883A248834981A2007F 1880A400FF18C0AC007F1880A3003F18006D5DA26C5FA26C6D4A5AA26C6D4A5A6C5F6C6D 495B6C01F801075B6D6C495B90273FFF807F90C7FC010F90B512FC6D5D010015C0023F91 C8FC020113E03A387CB643>I<D93FF8EB7FF0B50103B5FC031F14E0037F14F802F9B612 FE02FB01007FDAFFF8011F7F000302E06D7FC64A6D7F92C76C7F4A6E7F4A8283727EA285 84A21A80A284A21AC0AB1A8060A31A006061A24E5AA26E4A5B6E5C6103C0495B6F011F5B 6F495B9227FE01FFFEC7FC02FDB65ADAFC7F14F0031F14C0030749C8FC030013E093CAFC B0B612FCA6424D7CB54B>I<912601FFC0EB07C0021F01F8130F91B500FE131F0103ECFF 80010FEDC03F499039E01FE07F017F90380007F04948903803F8FF4801F8EB00FD484991 B5FC48824A80484980A2484980A25A91C87E5AA35B12FFAB127FA27FA27EA2806C5EA26C 6D5C6C6D5C5F6C6D91B5FC6C6D5B6C6D5B6D6C130F903A3FFFC07FEF010F90B512CF6D15 0F010114FCD9003F13F00203130091C8FCB0040FB612C0A6424D7CB547>I<90393FF003 FCB5EB1FFF4B13C092B512F002F114F89238FC7FFCECF3F000039039F7E0FFFEC65CECFF 805DA25CEE7FFCEE3FF85CEE0FE093C7FCA35CB3AAB612FEA62F367CB537>I<903901FF E007011FEBFC1F017FEBFF7F48B7FC1207390FFE003FD81FF0130749130148487F5B007F 81A200FF81A37F7F01F891C7FC13FEEBFFF06CEBFF8015FC6C14FF16C06C15F06C816C15 FE6C817E6C6C1580011F15C01303D9003F14E01400030F13F01501007C8000FC157F163F 6C151FA37E17E07F7FEE3FC001F0147F6DECFF806D4913009039FF800FFE91B55A013F5C D8FC1F14E0D8F803148027E0007FF8C7FC2C387CB635>I<147EA614FEA41301A31303A2 1307A2130F131F133F137F13FF1203000F90B6FCB8FCA5C66C48C8FCB3A8EE0FC0AB013F EC1F808017006D5C6DEB807EEDE0FE6DEBFFFC01015C6D5C023F13C0DA03FEC7FC2A4D7E CB34>I<D91FFC913801FFC0B5020FB5FCA60003ED003FC6160FB3AD5FA35FA2017F5DA2 94B5FC6D6CD903F713F0DC07E7EBFFC0903A1FFF801FC76D90B512876D1507010114FC6D 6C13F00207018091C7FC42377BB54B>I<007FB500F890B512FEA6D8003F0180D90FF8C7 FC6D6D5C6D6D495A4D5A6D6D495A6D6D91C8FC6D6D5B6D6D485A4C5A6E6C485A6EEB8FF0 6EEBCFE06E13FF5F6E5C6E91C9FC6E5BA2806F7E6F7F6F7F5D4B7F8392B57E4A80EC03FD DA07F87F4A486C7E4B6C7F021F6D7FEC3FC04A486C7F4A486C7F49486D7F49486D7F5C01 076E7F49486E7E011F6F7FB60103B612C0A642357EB447>120 D<B600F8010FB5FCA6C6 49C8EA7F00A26D6C157E18FE6D5E6F13016D5E6F1303A26D6D5C17076D6D5C170F6D5E6F 131F6D5E6F133F6D93C7FC6F5BA26E6C137E17FE6E5C16816E5C16C36E5C16E7A26EEBF7 E016FF6E5CA26E5CA26E91C8FCA36F5AA26F5AA26F5AA26F5AA35EA25E151F5E153FD81F E091C9FC486C5B486C137E15FE487E4A5A5D14034A5A49485A007F131F9038F07FC0393F A1FF8090B5CAFC6C13FC6C5B000313E0C690CBFC404D7DB447>I<001FB8FC1880A31800 ECC00049C6485B01F8495B495D495B49495B003F5E495B4B5B4B5B94C7FC90C7B5FC4A5B 4A5B5E5C4A5BC75C5C4A5B4A5B93C8FC91B5FC495B4949EB1F805D5B495B5D49153F4949 1400495B92C7FC90B5FC48495C485B4A5C5A48495B4A5B485D4849131F484948B45A91B6 FCB8FCA37E31357CB43C>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop -118 667 a FR(Con)l(ten)l(ts)-118 1104 y FQ(In)m(tro)s(duction) 38 b(to)g(the)f(Theory)i(of)e(Represen)m(tations)g(of)h(Finitely)-118 1204 y(Presen)m(ted)32 b FP(\003)p FQ(-Algebras.)-118 1304 y(I.)f(Represen)m(tations)g(b)m(y)i(b)s(ounded)e(op)s(erators)-118 1403 y FO(V.)d(Ostro)n(vskyi,)c(Y)-7 b(u.)38 b(Samoilenk)n(o)-118 1639 y FQ(Preface)2100 b(3)-118 1834 y(1)76 b(P)m(airs)26 b(of)e(self-adjoin)m(t)g(op)s(erators)h(connected)g(b)m(y)g(quadratic)6 1933 y(relations)31 b(and)i(some)c(generalizations)851 b(7)6 2040 y FO(1.1)84 b(In)n(tro)r(duction)27 b(to)g(represen)n (tations)e(of)i FP(\003)p FO(-algebras)20 b(.)41 b(.)h(.)g(.)f(.)h(.) 111 b(7)197 2147 y(1.1.1)94 b FP(\003)p FO(-Represen)n(tations:)34 b(k)n(ey)27 b(w)n(ords)51 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) 111 b(7)197 2253 y(1.1.2)94 b FN(C)528 2223 y FM(\003)566 2253 y FO(-represen)n(table)25 b FP(\003)p FO(-algebras)48 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(11)197 2360 y(1.1.3)94 b(En)n(v)n(eloping)24 b FP(\003)p FO(-algebras)g(and)j FN(C)1510 2330 y FM(\003)1548 2360 y FO(-algebras)i(.)42 b(.)g(.)f(.)h(.)70 b(15)197 2467 y(1.1.4)94 b FP(\003)p FO(-Represen)n(tations)24 b(of)k(generators)d(and)j(relations)55 b(.)70 b(21)197 2573 y(1.1.5)94 b(P)n(airs)22 b(of)j(self-adjoin)n(t)e (op)r(erators)g(satisfying)g(quad-)463 2673 y(ratic)j(relations)k(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)70 b(24)6 2780 y(1.2)84 b FN(F)250 2792 y FL(n)296 2780 y FO(-algebras)24 b(and)k(their)e(represen)n(tations)e(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(27)197 2887 y(1.2.1)94 b(Ab)r(out)28 b FP(\003)p FO(-represen)n(tations)c(of)k FN(F)1514 2899 y FL(n)1559 2887 y FO(-algebras)18 b(.)42 b(.)g(.)f(.)h(.)70 b(27)197 2993 y(1.2.2)94 b(Examples)34 b(of)i FN(F)1006 3005 y FL(n)1052 2993 y FO(-algebras)d(generated)j(b)n (y)g(idem-)463 3093 y(p)r(oten)n(ts)28 b(and)f(their)g(represen)n (tations)41 b(.)g(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(29)197 3200 y(1.2.3)94 b(Non-comm)n(utativ)n(e)29 b(\\circle",)j(\\pair)f(of)j (in)n(tersect-)463 3299 y(ing)25 b(lines")e(and)j(\\h)n(yp)r(erb)r (ola".)33 b(More)25 b(examples)e(of)463 3399 y FN(F)516 3411 y FK(4)554 3399 y FO(-algebras)55 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(39)6 3505 y(1.3)84 b(Represen)n(tations)17 b(of)i(t)n(w)n (o-dimensional)14 b(Lie)k(algebras,)g(their)197 3605 y(nonlinear)25 b(transformations,)f(and)j(semilinear)c(relations)47 b(.)42 b(.)70 b(42)197 3712 y(1.3.1)94 b(Represen)n(tations)17 b(of)i(t)n(w)n(o-dimensional)13 b(real)18 b(Lie)g(al-)463 3811 y(gebras)25 b(and)h(their)f(nonlinear)f(transformations)e(b)n(y) 463 3911 y(b)r(ounded)28 b(op)r(erators)50 b(.)42 b(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(42)1099 4121 y(i)p eop %%Page: 2 2 2 1 bop -118 -137 a FO(ii)2085 b FJ(Con)n(ten)n(ts)197 96 y FO(1.3.2)94 b(P)n(airs)24 b(of)j(op)r(erators)f(connected)h(b)n(y) f(semilinear)d(re-)463 196 y(lations)36 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(44)197 301 y(1.3.3)94 b(Kleinec)n(k)n(e{Shirok)n(o)m(v)22 b(t)n(yp)r(e)28 b(theorems)68 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)70 b(48)197 407 y(1.3.4)94 b(Irreducible)30 b(represen)n(tations)g(of)j (semilinear)28 b(rela-)463 506 y(tions)38 b(.)j(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(60)197 612 y(1.3.5)94 b(Represen)n(tations)25 b(of)j(semilinear)23 b FN(F)1608 624 y FK(4)1645 612 y FO(-relations)48 b(.)41 b(.)h(.)70 b(66)6 717 y(1.4)84 b(Represen)n(tations)26 b(of)h FN(q)s FO(-relations)46 b(.)c(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)70 b(68)197 822 y(1.4.1)94 b(Finite-dimensional)21 b(represen)n(tations)j(of)j FN(q)s FO(-relations)43 b(68)197 927 y(1.4.2)94 b(Hermitian)25 b FN(q)s FO(-plane)h(and)h FN(q)s FO(-CCR)e(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(70)197 1033 y(1.4.3)94 b(Real)25 b(quan)n(tum)h(plane)g(and)g(real)f (quan)n(tum)h(h)n(yp)r(er-)463 1132 y(b)r(oloid)53 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)70 b(75)6 1238 y(Commen)n(ts)26 b(to)h(Chapter)g(1) 50 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)70 b(77)-118 1430 y FQ(2)76 b(Represen)m(tations)31 b(of)h(dynamical)f FP(\003)p FQ(-algebras)556 b(83)6 1535 y FO(2.1)84 b(Op)r(erator)49 b(relations)f(and)i(one-dimensional)c (dynamical)197 1634 y(systems)69 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) 70 b(83)197 1740 y(2.1.1)94 b(Op)r(erator)24 b(relations)e(connected)j (with)g(one-dimen-)463 1839 y(sional)g(dynamical)f(systems)72 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)70 b(83)197 1945 y(2.1.2)94 b(Finite-dimensional)21 b(represen)n(tations) 61 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)70 b(91)197 2050 y(2.1.3)94 b(In\014nite-dimensional)22 b(represen)n(tations)81 b(.)41 b(.)h(.)g(.)f(.)h(.)70 b(96)6 2155 y(2.2)84 b(Some)27 b(classes)e(of)j FP(\003)p FO(-algebras)23 b(with)k(3)h(and)f(4)g (generators)62 b(.)42 b(.)28 b(106)197 2261 y(2.2.1)94 b(Represen)n(tations)24 b(of)i(graded)f FN(so)p FO(\(3\))h(and)g (four-tup-)463 2360 y(les)g(of)i(pro)5 b(jections)25 b(satisfying)g(a)i(linear)e(relation)43 b(.)f(.)28 b(106)197 2465 y(2.2.2)94 b(Represen)n(tations)36 b(of)i(a)g(class)f(of)h (quadratic)e(alge-)463 2565 y(bras)27 b(with)g(three)g(generators)j(.) 42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(113)197 2670 y(2.2.3)94 b(Op)r(erator)31 b(relations)e(connected)k(with)f(a)g (dynami-)463 2770 y(cal)26 b(system)h(on)g(a)g(plane)76 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(116)197 2875 y(2.2.4)94 b(Represen)n(tation)30 b(of)j(real)d(forms)h (of)i(Witten's)f(\014rst)463 2975 y(deformation)i(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)28 b(119)197 3080 y(2.2.5)94 b(Represen)n(tations)37 b(of)i(the)h(Skly)n(anin)d(algebra)f(and)463 3180 y FN(U)520 3192 y FL(q)556 3180 y FO(\()p FN(sl)r FO(\(2\)\))79 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)28 b(123)6 3285 y(2.3)84 b(Represen)n(tations)26 b(of)h FN(q)s FO(-deformed)f FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))43 b(.)f(.)f(.)h(.)g(.)f(.)h(.)28 b(133)197 3390 y(2.3.1)94 b(Real)26 b(forms)g(of)i FN(U)1034 3402 y FL(q)1070 3390 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) 28 b(133)197 3496 y(2.3.2)94 b(Represen)n(tations)25 b(of)j FN(U)1219 3508 y FL(q)1255 3496 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))50 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)28 b(135)6 3601 y(2.4)84 b(Man)n(y-dimensional)22 b(dynamical)i(systems)78 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(151)197 3706 y(2.4.1)94 b(\\Direct)28 b(pro)r(ducts")g(of)h (one-dimensional)23 b(dynam-)463 3806 y(ical)i(systems)39 b(.)i(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)28 b(152)197 3911 y(2.4.2)94 b(\\T)-7 b(riangular")23 b(dynamical)h(systems.)52 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(155)p eop %%Page: 3 3 3 2 bop -118 -137 a FJ(Con)n(ten)n(ts)2063 b FO(iii)197 96 y(2.4.3)94 b(Op)r(erator)17 b(relations)e(connected)j(with)g(man)n (y-dimen-)463 196 y(sional)25 b(dynamical)f(systems)72 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(161)197 296 y(2.4.4)94 b(Represen)n(tations)16 b(of)i(the)h (non-standard)e(real)f(quan-)463 395 y(tum)27 b(sphere)66 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)28 b(166)197 495 y(2.4.5)94 b(Heisen)n(b)r(erg)48 b(relations)f(for)i(the)h(quan)n(tum)f FN(E)5 b FO(\(2\))463 595 y(group)70 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(169)197 694 y(2.4.6)94 b(Wic)n(k)27 b(algebras)d(related)i(to)i(dynamical)23 b(systems)38 b(.)k(.)28 b(173)6 794 y(2.5)84 b(On)28 b(represen)n(tations)c(of)k(some)e(n)n(uclear)f(algebras)39 b(.)i(.)h(.)g(.)f(.)h(.)28 b(180)197 893 y(2.5.1)94 b(Comm)n(utativ)n (e)24 b(mo)r(dels)f(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)28 b(180)197 993 y(2.5.2)94 b(Cen)n(tered)27 b(op)r(erators)36 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)28 b(185)197 1093 y(2.5.3)94 b(Represen)n(tations)25 b(of)j(Cun)n(tz)g(algebras)67 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)28 b(189)6 1192 y(Commen)n(ts)e(to)h(Chapter)g(2)50 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)28 b(198)-118 1375 y FQ(3)76 b(On)40 b(the)f(complexit)m(y)f(of) h(the)h(description)e(of)i(represen)m(ta-)6 1475 y(tions)31 b(of)h FP(\003)p FQ(-)o(algebras)1432 b(203)6 1574 y FO(3.1)84 b FP(\003)p FO(-Wild)26 b(algebras)e(and)k(relations)53 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(203)197 1674 y(3.1.1)94 b(Ma)5 b(jorization)23 b(of)j FP(\003)p FO(-algebras)c(with)k(resp)r(ect)g(to)h(the)463 1773 y(complexit)n(y)d(of)k(their)e(represen)n(tations)46 b(.)c(.)f(.)h(.)g(.)f(.)h(.)28 b(203)197 1873 y(3.1.2)94 b FP(\003)p FO(-Wildness)25 b(of)j FP(\003)p FO(-algebras)38 b(.)j(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(212)197 1973 y(3.1.3)94 b FP(\003)p FO(-Wild)27 b(algebras)g (generated)h(b)n(y)h(orthogonal)d(pro-)463 2072 y(jections)h(and)g (idemp)r(oten)n(ts)80 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)28 b(214)197 2172 y(3.1.4)94 b FP(\003)p FO(-Wild)25 b(semilinear)e(relations)74 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)28 b(221)197 2272 y(3.1.5)94 b FP(\003)p FO(-Wild)25 b(quadratic)h(and)h(cubic)g(relations)40 b(.)h(.)h(.)g(.)f(.)h(.)28 b(222)197 2371 y(3.1.6)94 b FP(\003)p FO(-Wild)25 b(groups.)36 b(P)n(erio)r(dic)24 b(groups)i(are)h(not)h FP(\003)p FO(-wild)22 b(.)28 b(227)6 2471 y(3.2)84 b(On)22 b(the)g(complexit)n(y) d(of)j(the)g(description)d(of)j(classes)e(of)h(non-)197 2570 y(self-adjoin)n(t)k(op)r(erators)34 b(.)42 b(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(229)197 2670 y(3.2.1)94 b(Classes)44 b(of)j(non-self-adjoin)n(t)d(op) r(erators)g(singled)463 2770 y(out)28 b(b)n(y)f(a)g(quadratic)f(or)g(a) i(cubic)e(relation)70 b(.)42 b(.)g(.)f(.)h(.)28 b(230)197 2869 y(3.2.2)94 b(P)n(artial)31 b(isometries,)i(w)n(eakly)g(cen)n (tered)i(op)r(erators)463 2969 y(and)27 b(algebraic)d(op)r(erators)71 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(234)197 3069 y(3.2.3)94 b(Hyp)r(onormal)16 b(op)r(erators)g(and)j (pairs)e(of)h(comm)n(uting)463 3168 y(completely)24 b(non-unitary)i (isometries)75 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)28 b(236)6 3268 y(Commen)n(ts)e(to)h(Chapter)g(3)50 b(.)41 b(.)h(.)g(.)f(.)h(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)28 b(238)-118 3451 y FQ(Bibliograph)m(y)1777 b(243)-118 3633 y(Index)2077 b(259)p eop %%Page: 4 4 4 3 bop -118 -137 a FO(iv)p eop %%Page: 1 5 1 4 bop -118 -137 a FH(Rev)21 b(Math.)28 b(&)20 b(Math.)27 b(Phys.,)22 b(1999,)1137 -139 y(c)1117 -137 y FM(\015)p FH(1999)f(OP)-5 b(A)20 b(\(Overseas)h(Publishers)f(Asso)r(ciation\)) -118 -70 y(vol.)28 b(11,)20 b(pp.)27 b(1{261)855 b(Amsterdam)21 b(B.V.)g(Published)f(under)h(license)g(in)-118 -4 y(Rep)n(rints)g (available)e(directly)j(from)e(the)i(publisher)87 b(The)21 b(Netherlands)f(under)h(the)g(Ha)n(rw)n(o)r(o)r(d)f(Academic)-118 63 y(Photo)r(cop)n(ying)h(p)r(ermitted)g(b)n(y)h(license)f(only)322 b(Publishers)20 b(imp)n(rint,)g(pa)n(rt)h(of)f(The)i(Go)n(rdon)e(and) 1679 129 y(Breach)h(Publishing)f(Group.)1839 195 y(Printed)i(in)e(Mala) n(ysia)-97 794 y FG(In)m(tro)s(duction)37 b(to)g(the)h(Theory)f(of)h (Represen)m(tations)g(of)366 910 y(Finitely)h(Presen)m(ted)e FF(\003)p FG(-Algebras.)106 1026 y(I.)g(Represen)m(tations)h(b)m(y)g(b) s(ounded)h(op)s(erators)262 1362 y FE(V)-10 b(asyl)30 b(Ostr)n(o)n(vsky)972 1355 y(\025)982 1362 y(\020)i(and)f(Yuri)1408 1355 y(\025)1418 1362 y(\020)g(Samo)1681 1355 y(\025)1691 1362 y(\020lenk)n(o)-106 1487 y FO(Institute)d(of)f(Mathematics,)e (Ukrainian)g(National)g(Academ)n(y)h(of)i(Sciences)928 1906 y FQ(Abstract)195 2059 y FD(This)36 b(review)g(giv)n(es)h (fundamen)n(tals)g(of)e(represen)n(tations)i(of)e(\014nitely)89 2138 y(presen)n(ted)e FC(\003)p FD(-algebras)g(b)n(y)f(b)r(ounded)h(op) r(erators.)55 b(The)32 b(theory)h(is)f(illus-)89 2217 y(trated)h(with)f(n)n(umerous)f(examples)i(of)d FC(\003)p FD(-algebras.)55 b(The)31 b(examples,)j(in)89 2296 y(particular,)25 b(include)f FC(\003)p FD(-algebras)g(with)f(t)n(w)n(o)g(self-adjoin)n (t)h(generators)g(that)89 2375 y(satisfy)c(a)f(quadratic)i(or)d(a)h (more)g(general)h(relation,)h FC(\003)p FD(-algebras)f(with)g(three)89 2453 y(and)35 b(four)e(generators,)k FC(\003)p FD(-algebras)f(that)f (arise)f(from)f(one-)h(and)h(man)n(y-)89 2532 y(dimensional)30 b(discrete)e(dynamical)h(systems,)f(Wic)n(k)g FC(\003)p FD(-algebras,)g(v)l(arious)89 2611 y FC(\003)p FD(-wild)d(algebras.)195 2690 y(This)19 b(b)r(o)r(ok)f(is)g(in)n(tended)j(for)c(graduate)i (studen)n(ts)g(as)f(w)n(ell)h(as)f(for)f(the)i(re-)89 2769 y(searc)n(hers)h(who)g(sp)r(ecialize)i(in)f(the)f(theory)g(of)f (represen)n(tations)j(of)d FC(\003)p FD(-algeb-)89 2848 y(ras)24 b(and)g(related)h(areas.)p eop %%Page: 2 6 2 5 bop -118 -137 a FO(2)p eop %%Page: 3 7 3 6 bop -118 670 a FR(Preface)-118 1110 y FQ(1.)78 b FO(The)42 b(w)n(ord)e(algebra)f(in)i(this)g(b)r(o)r(ok)g(means)f(an)i (asso)r(ciativ)n(e)37 b(algebra)-118 1210 y(o)n(v)n(er)28 b(the)i(\014eld)f(of)h(complex)e(n)n(um)n(b)r(ers)g FI(C)15 b FO(.)50 b(The)29 b(terms)g FP(\003)p FO(-algebra,)e(Banac)n(h)-118 1309 y FP(\003)p FO(-algebra,)33 b FN(C)339 1279 y FM(\003)377 1309 y FO(-algebra,)h FN(W)818 1279 y FM(\003)856 1309 y FO(-algebra,)f(as)h(w)n(ell)e(as)i(their)g(prop)r(erties,)h(are)-118 1409 y(used)27 b(in)g(this)g(b)r(o)r(ok,)h(as)e(a)i(rule,)e(without)h (sp)r(ecial)e(references.)6 1517 y(This)d(b)r(o)r(ok)h(is)e(dev)n(oted) h(to)h FB(r)l(epr)l(esentations)30 b FO(of)23 b(\014nitely)e(presen)n (ted)h FP(\003)p FO(-al-)-118 1617 y(gebras)32 b(\(de\014ned)j(b)n(y)f (a)g(\014nite)f(n)n(um)n(b)r(er)g(of)h(generators)e(and)i(relations\))d (b)n(y)-118 1717 y(b)r(ounded)d(op)r(erators.)-118 1892 y FQ(2.)42 b FO(There)29 b(is)g(a)g(whole)g(domain)e(of)j(Algebra)d (called)h(\\Represen)n(tation)f(the-)-118 1992 y(ory)38 b(of)i(algebras".)68 b(If)40 b(one)f(in)n(tro)r(duces)f(an)h(in)n(v)n (olution)d FP(\003)j FO(in)n(to)f(an)h(alge-)-118 2092 y(bra)e FA(A)h FO(and)g(considers)d(only)i(those)g(represen)n(tations)e (b)n(y)j(op)r(erators)e(in)h(a)-118 2191 y(Hilb)r(ert)c(space)g FN(H)41 b FO(whic)n(h)33 b(preserv)n(e)g(the)h(in)n(v)n(olution)d(\()p FP(\003)p FO(-represen)n(tations\),)-118 2291 y(then)k(these)g (represen)n(tations)d(mak)n(e)i(just)h(an)g(island)d(among)h(all)g(the) i(rep-)-118 2390 y(resen)n(tations)c(of)i FA(A)p FO(.)55 b(Moreo)n(v)n(er,)32 b(indecomp)r(osable)e FP(\003)p FO(-represen)n(tations,)h(so)-118 2490 y(dear)25 b(to)g(the)h (algebraist's)c(heart,)j(coincide,)f(in)h(this)g(case,)g(with)g (irreducible)-118 2590 y(represen)n(tations,)c(and)j(t)n(w)n(o)f FP(\003)p FO(-represen)n(tations)d(are)j(equiv)-5 b(alen)n(t)21 b(if)i(and)h(only)-118 2689 y(if)19 b(they)h(are)f(unitarily)d(equiv)-5 b(alen)n(t.)32 b(And)20 b(hence,)i(the)e(problem)d(of)j(describing)-118 2789 y FP(\003)p FO(-represen)n(tations)i(of)k FA(A)p FO(,)g(up)h(to)f(a)f(unitary)g(equiv)-5 b(alence,)24 b(is)h(a)g(subproblem)-118 2889 y(\(a)33 b(particular)d(case\))j(of)h (the)f(problem)e(of)j(describing)c(all)i(represen)n(tations)-118 2988 y(up)c(to)f(an)h(equiv)-5 b(alence.)6 3097 y(But:)131 3205 y(1\))28 b(Mathematical)d(problems)h(related)h(to)h FP(\003)p FO(-represen)n(tations)d(could)6 3305 y(turn)j(out)g(to)f(b)r (e)h(pith)n(y)f(and)g(in)n(teresting.)131 3413 y(2\))e(Considering)d FP(\003)p FO(-represen)n(tations)f(allo)n(ws)h(one)i(to)h(c)n(hange)f (the)h(ac-)6 3513 y(cen)n(t)g(sharply)-7 b(,)23 b(from)g(algebra)f(to)i (functional)e(analysis,)g(and)i(to)h(consider)6 3612 y(not)42 b(only)e(represen)n(tations)f(b)n(y)i(b)r(ounded)h(op)r (erators)d(in)i(an)g(in\014nite-)6 3712 y(dimensional)20 b(space)j FN(H)7 b FO(,)25 b(but)g(also)d(represen)n(tations)f(b)n(y)j (un)n(b)r(ounded)g(op-)6 3811 y(erators.)57 b(Represen)n(tations)33 b(of)i(Lie)f(algebras)e(and)j(their)f(applications)6 3911 y(sho)n(w)27 b(ho)n(w)g(imp)r(ortan)n(t)e(and)j(useful)f(suc)n(h)g (represen)n(tations)e(are.)1089 4121 y(3)p eop %%Page: 4 8 4 7 bop -118 -137 a FO(4)2148 b FJ(Preface)131 96 y FO(3\))33 b(Moreo)n(v)n(er,)g(kno)n(wing)e(only)h FP(\003)p FO(-represen)n (tations)e(can)j(sometimes)6 196 y(b)r(e)24 b(satisfactory)c(to)i (consumers)f(of)i(represen)n(tation)d(theory)-7 b(.)34 b(The)23 b(repre-)6 296 y(sen)n(tation)29 b(theory)h(of)g FP(\003)p FO(-algebras)d(suggests)i(some)f(applications)f(of)j(the)6 395 y(theory)d(to:)255 502 y(a\))51 b(the)f(construction)f(and)h(study) g(of)g(mo)r(dels)f(of)h(quan)n(tum)131 602 y(ph)n(ysics,)29 b(in)g(particular,)e(b)n(y)i(using)g(Wic)n(k)f(algebras)f(and)j(their)e (rep-)131 701 y(resen)n(tations;)255 808 y(b\))g(a)e(study)h(of)g (represen)n(tations)d(of)j FP(\003)p FO(-algebras)c(whic)n(h)j(are)f (gen-)131 908 y(erated)38 b(b)n(y)h(idemp)r(oten)n(ts)e(and)i(the)g (corresp)r(onding)d(resolution)g(of)131 1007 y(the)28 b(iden)n(tit)n(y;)255 1114 y(c\))d(a)e(study)h(of)g(op)r(erator)f (Banac)n(h)f(algebras)f(con)n(taining)g(a)j(dense)131 1214 y FP(\003)p FO(-subalgebra,)19 b(and)i(construction)f(of)i(in)n(v) n(ertibilit)n(y)15 b(sym)n(b)r(ols)k(for)i(op-)131 1313 y(erators)26 b(in)g(the)i(algebra,)d(etc.;)255 1420 y(d\))34 b(structure)f(theorems)e(for)i(algebraicall)o(y)27 b(de\014ned)34 b(classes)d(of)131 1520 y(not)d(self-adjoin)n(t)d(op)r(erators.;)255 1627 y(e\))32 b(the)g(theory)e(of)h(algebras)d(and)j(their)g(represen)n (tations,)e(since)131 1726 y(the)f(island)c(of)k FP(\003)p FO(-represen)n(tations)23 b(could)j(turn)h(out)h(to)f(b)r(e)g(an)g(arc) n(hi-)131 1826 y(p)r(elago,)37 b(and)f(the)h(facts)f(ab)r(out)h(it)f (could)f(b)r(e)i(useful)f(for)g(studying)131 1925 y(b)r(oth)30 b(the)f(algebra)e FA(A)i FO(itself)f(and)h(its)f(represen)n(tations)e (but)k(already)131 2025 y(without)20 b(the)h(in)n(v)n(olution)c(in)j (the)h(algebra.)32 b(In)21 b(particular,)e(ev)n(en)h(suc)n(h)131 2125 y(a)f(traditional)c(part)k(of)g(algebra)d(as)i(the)i(theory)e(of)h (groups)f(\(esp)r(ecially)131 2224 y(coun)n(table)j(groups\))g(has)h (long)e(ago)h(included,)h(in)g(its)f(sto)r(c)n(k-in-trade,)131 2324 y(the)28 b(metho)r(ds)f(of)g(theory)g(of)h FP(\003)p FO(-represen)n(tations;)255 2431 y(f)6 b(\))29 b(other)e(applications.) -118 2602 y FQ(3.)35 b FO(In)23 b(the)h(course)e(of)i(some)e(time,)h (the)h(authors)e(ha)n(v)n(e)g(accum)n(ulated)f(a)i(fairly)-118 2701 y(large)j(n)n(um)n(b)r(er)g(of)i(examples,)e(and)i(dev)n(elop)r (ed)e(tec)n(hniques)h(for)h(calculating)-118 2801 y FP(\003)p FO(-represen)n(tations)j(of)36 b(classes)d(of)i(\014nitely)f(presen)n (ted)h FP(\003)p FO(-algebras.)56 b(These)-118 2900 y(classes)24 b(include)g(certain)g(curv)n(es)h(in)g(the)i(real)d(plane,)h FP(\003)p FO(-algebras)d(generated)-118 3000 y(b)n(y)27 b(idemp)r(oten)n(ts,)f(Wic)n(k)h(algebras,)d(and)k(others.)6 3107 y(There)i(came)f(an)h(idea)f(to)h(presen)n(t)g(these)g(examples,)e (classes)g(of)i(exam-)-118 3207 y(ples,)h(and)g(metho)r(ds)f(used)h(to) g(describ)r(e)f(their)g(represen)n(tations)e(gradually)-7 b(,)-118 3306 y(with)28 b(an)h(increase)d(of)j(complexit)n(y)c(of)j (the)h(problem.)38 b(Actually)-7 b(,)27 b(the)i(c)n(hoice)-118 3406 y(of)21 b(examples)e(and)j(metho)r(ds)f(w)n(as)f(determined)g(b)n (y)h(authors')g(taste)g(and)h(their)-118 3505 y(exp)r(erience)k(in)h (the)h(sub)5 b(ject.)6 3612 y(T)-7 b(rying)22 b(to)h(carry)e(out)i (this)g(idea)f(systematically)-7 b(,)19 b(w)n(e)j(split)g(it)g(in)n(to) g(infor-)-118 3712 y(mation)k(ab)r(out)j FP(\003)p FO(-represen)n (tations)c(of)k(algebras)d(considered)g(in)i(the)h(exam-)-118 3811 y(ples)j(b)n(y)h(b)r(ounded)g(op)r(erators)e(\(I\))j(and)e(un)n(b) r(ounded)i(op)r(erators)d(\(I)r(I\).)j(This)-118 3911 y(b)r(o)r(ok)i(is)e(based)i(on)g(a)f(su\016cien)n(tly)f(large)g(\\zo)r (o")g(of)i(examples)e(that)i(illus-)p eop %%Page: 5 9 5 8 bop -118 -137 a FJ(Preface)2147 b FO(5)-118 96 y(trate)29 b(the)i(notions)d(and)i(metho)r(ds)f(that)i(app)r(ear)e(in)g(studying)g (b)r(ounded)h FP(\003)p FO(-)-118 196 y(represen)n(tations.)h(A)21 b(more)e(accurate)h(title)f(of)i(this)f(b)r(o)r(ok)g(w)n(ould)f(p)r (ossibly)f(b)r(e)-118 296 y(\\Represen)n(tations)23 b(of)k FP(\003)p FO(-algebras)22 b(b)n(y)k(b)r(ounded)h(op)r(erators)e(b)n(y)h (examples",)-118 395 y(but,)i(a)f(similar)c(title)j(has)h(already)f(b)r (een)i(tak)n(en)f(\(see)g([64)o(]\).)-118 553 y FQ(4.)50 b FO(A)32 b(starting)e(p)r(oin)n(t)i(for)f(the)i(exp)r(osition)c(in)j (this)f(review)g(is)g(represen)n(ta-)-118 652 y(tions)f(of)h FP(\003)p FO(-algebras)c(generated)i(b)n(y)i(t)n(w)n(o)f(self-adjoin)n (t)f(generators)f(satisfy-)-118 752 y(ing)e(a)h(quadratic)f(relation)e (\(a)k(\\noncomm)n(utativ)n(e)23 b(curv)n(e)k(of)g(degree)g(t)n(w)n(o)f (in)-118 851 y(the)37 b(real)e(plane"\).)64 b(But)37 b(w)n(e)g(also)e(giv)n(e)g(far)h(reac)n(hing)f(generalizations)d(of) -118 951 y(suc)n(h)i(\\noncomm)n(utativ)n(e)c(curv)n(es":)48 b(a)34 b(theory)g(of)g(represen)n(tations)d(of)j(op-)-118 1051 y(erators)29 b(satisfying)g(a)i(semilinear)26 b(relation)j (\(Sections)h(1.3.2{1.3.5,)g(3.1.4\),)-118 1150 y(an)d(accoun)n(t)f(of) i(noncomm)n(utativ)n(e)23 b(dynamical)h(systems,)i(one-dimensional)-118 1250 y(\(Section)c(2.1\))h(and)f(man)n(y-dimensional)17 b(\(Section)22 b(2.4\),)i(represen)n(tations)c(of)-118 1350 y(algebras)30 b(with)j(three)g(and)g(four)g(generators,)g(whic)n (h)f(app)r(ear)h(in)f(theoreti-)-118 1449 y(cal)22 b(ph)n(ysics)g (\(Sections)g(2.2,)i(2.3\),)g(v)-5 b(arious)21 b FP(\003)p FO(-wild)g(problems)g(\(Sections)h(3.1,)-118 1549 y(3.2\).)-118 1706 y FQ(5.)63 b FO(In)36 b(order)f(to)i(read)e(this)h(b)r(o)r(ok,)i (it)e(is)f(enough)h(to)h(b)r(e)f(familiar)c(with)k(a)-118 1806 y(basic)g(univ)n(ersit)n(y)f(course)i(of)h(op)r(erator)e(theory)h (and)h(in)n(v)n(olutiv)n(e)c(algebras)-118 1905 y(\()p FP(\003)p FO(-algebras\).)e(Of)23 b(course,)g(a)f(part)h(dev)n(oted)f (to)h(a)g(description)e(of)i FP(\003)p FO(-algebras)-118 2005 y(and)33 b(their)f(prop)r(erties)f(w)n(ould)h(b)r(e)h(useful)g(in) f(an)h(enlarged)e(edition,)h(where)-118 2105 y(\014nitely)42 b(generated)h(and)h(\014nitely)e(presen)n(ted)h(algebras)e(and)i FP(\003)p FO(-algebras,)-118 2204 y(prop)r(erties)26 b(of)h(suc)n(h)g(algebras)e(and)i(examples)e(could)i(b)r(e)h(presen)n (ted.)6 2307 y(W)-7 b(e)32 b(w)n(ould)d(lik)n(e)f(to)j(giv)n(e)e(a)h (list)f(of)i(some)e(related)g(monographs,)g(whic)n(h)-118 2406 y(are)d(close,)g(in)h(con)n(ten)n(ts,)g(to)h(this)f(b)r(o)r(ok.)6 2508 y(1\))36 b(Asso)r(ciativ)n(e)d(algebras)g(\(see,)k(e.g.,)h([116)o (,)e(111)n(,)g(200)o(],)i(and)d(the)i(bib-)-118 2608 y(liograph)n(y)32 b(therein\),)37 b(coun)n(table)d(groups)g(\(see,)k (e.g.,)g([41)o(,)e(180)n(,)g(106)o(],)i(and)-118 2708 y(the)d(bibliograph)n(y)29 b(therein\),)36 b(and)e(represen)n(tations)d (of)j(coun)n(table)f(groups)-118 2807 y(and)26 b(asso)r(ciativ)n(e)d (algebras)g(\(see,)k(e.g.,)f([59)o(,)h(13)o(,)f(86)o(],)h(and)g(the)f (bibliograph)n(y)-118 2907 y(therein\).)6 3009 y(2\))c(Dynamical)d (systems,)j(esp)r(ecially)c(one-dimensional)f(\(see,)23 b(e.g.,)g([245)o(,)-118 3109 y(246)n(,)28 b(253)o(],)f(etc.\).)6 3211 y(3\))40 b(F)-7 b(unctional)38 b(analysis)f(and)j(op)r(erator)e (theory)-7 b(,)42 b(including)37 b(sp)r(ectral)-118 3311 y(theory)k(\(see,)k(e.g.,)f([4,)e(102)n(,)g(220)n(,)g(226)o(,)f(37)o(,) h(29)o(],)j(and)d(others\),)i(unitary)-118 3410 y(represen)n(tations)35 b(of)j(groups)e(\(see,)41 b(e.g.,)f([132)o(,)e(296)n(,)g(20],)i (etc.\),)h(op)r(erator)-118 3510 y FP(\003)p FO(-algebras)29 b(and)k(their)f(represen)n(tations)e(\(see,)35 b(e.g.,)f([68)o(,)f(264) o(,)g(72)o(,)g(9,)g(259)o(,)-118 3610 y(197)n(,)25 b(127)n(,)f(130)o(,) g(169)n(,)g(64)o(],)h(and)f(others\),)g(in)g(particular,)d(represen)n (tations)g(b)n(y)-118 3709 y(un)n(b)r(ounded)28 b(op)r(erators)d(\(see) j(e.g.,)f([122)o(,)h(228)n(,)g(121)n(,)g(234)o(],)f(etc.\))6 3811 y(4\))19 b(Quan)n(tum)e(groups)h(and)g(homgeneous)e(spaces,)k (their)e(represen)n(tations)-118 3911 y(\(esp)r(ecially)h FP(\003)p FO(-represen)n(tations\),)g(and)j(their)f(applications)d(to)k (the)g(theory)g(of)p eop %%Page: 6 10 6 9 bop -118 -137 a FO(6)2148 b FJ(Preface)-118 96 y FO(in)n(tegrable)24 b(mo)r(dels)i(\(see,)h(e.g.,)h([126)n(,)g(157)n(,)g (222)o(,)f(54)o(,)h(161)o(,)f(117)o(,)h(137)n(,)g(140)o(]\).)6 196 y(5\))35 b(Applications)c(of)k(the)g(theory)f(of)g FP(\003)p FO(-represen)n(tations)d(to)j(mo)r(dels)f(of)-118 296 y(mathematical)18 b(ph)n(ysics)j(\(see,)i(e.g.,)g([78)o(,)g(48)o(,) g(227)n(,)g(284)n(],)h(etc.\),)g(non-comm)n(u-)-118 395 y(tativ)n(e)30 b(geometry)f(\(see,)j(e.g.,)g([57)o(,)g(162)o(],)g (etc.\),)h(non-comm)n(utativ)n(e)27 b(proba-)-118 495 y(bilit)n(y)21 b(theory)j(\([112)o(,)g(194)n(,)h(42)o(],)g(etc.\),)g (to)f(the)h(construction)d(of)i(in)n(v)n(ertibilit)n(y)-118 595 y(sym)n(b)r(ols)i(\([147)o(,)j(40)o(],)g(etc.\),)g(to)g(the)g (theory)f(of)h(non)f(self-adjoin)n(t)f(op)r(erators)-118 694 y(\(see,)g(e.g.,)h([102)n(,)g(79)o(,)g(169)n(],)g(etc.\).)-118 844 y FQ(6.)74 b FO(References)39 b(to)h(the)h(literature)c(often)k (con)n(tained)d(in)i(the)g(commen)n(ts)-118 943 y(to)c(c)n(hapters)e (do)i(not)f(claim)e(to)j(b)r(e)g(complete)d(and,)38 b(presumably)-7 b(,)35 b(do)h(not)-118 1043 y(con)n(tain)j(a)h(full)f(bibliograph)n(y)d (on)k(b)r(o)r(oks)g(and)g(articles)e(directly)g(related)-118 1143 y(to)32 b(the)h(questions)e(touc)n(hed)h(up)r(on)g(in)g(this)g(b)r (o)r(ok.)50 b(Sometimes,)31 b(the)i(refer-)-118 1242 y(ences)23 b(to)g(original)18 b(sources)k(are)g(replaced)g(with)g(the)i (references)e(to)h(a)n(v)-5 b(ailable)-118 1342 y(monographs)29 b(or)j(reviews)e(con)n(taining)f(additional)g(bibliographical)d(mate-) -118 1441 y(rial;)f(probably)-7 b(,)25 b(the)j(authors)f(to)r(o)g (often)h(refer)f(to)h(sources)e(in)h(Russian)f(and)-118 1541 y(their)g(translations.)6 1641 y(W)-7 b(e)27 b(also)d(included)h (some)g(references)g(related)g(to)h FP(\003)p FO(-represen)n(tations)d (b)n(y)-118 1740 y(un)n(b)r(ounded)41 b(op)r(erators,)i(k)n(eeping)c (in)i(mind)f(the)h(future)h(second)e(v)n(olume)-118 1840 y(of)32 b(this)f(b)r(o)r(ok)g(that)i(will)c(b)r(e)j(dev)n(oted)f(to)h (represen)n(tations)d(b)n(y)i(un)n(b)r(ounded)-118 1940 y(op)r(erators.)-118 2089 y FQ(7.)40 b FO(The)29 b(authors)f(are)f (sincerely)f(grateful)i(to)g(man)n(y)g(mathematicians)c(who)-118 2189 y(con)n(tributed)32 b(to)h(this)f(w)n(ork:)47 b(their)32 b(teac)n(her,)h(professor)e(Y)-7 b(u.)34 b(M.)f(Berezan-)-118 2288 y(sky)-7 b(,)30 b(for)g(his)f(kind)g(atten)n(tion)g(and)h(useful)f (advice,)g(all)f(participan)n(ts)f(of)j(the)-118 2388 y(seminars)h(on)k(algebraic)c(problems)h(of)i(functional)f(analysis)f (in)i(the)h(Insti-)-118 2487 y(tute)43 b(of)g(Mathematics)d(of)j(the)g (Ukrainian)d(National)g(Academ)n(y)h(of)i(Sci-)-118 2587 y(ences,)33 b(colleagues)c(Stanisla)n(v)g(Krugly)n(ak,)h(Konrad)h(Sc)n (hm)r(\177)-44 b(udgen)31 b(and)i(Vic-)-118 2687 y(tor)41 b(Sh)n(ul'man,)i(studen)n(ts)f(Lyudm)n(yla)c(T)-7 b(uro)n(wsk)i(a,)44 b(Alexandra)39 b(Piry)n(atin-)-118 2786 y(sk)-5 b(a)n(y)n(a,)21 b(Eduard)f(V)-7 b(a)n(ysleb,)21 b(Y)-7 b(ury)21 b(Chap)r(o)n(vsky)-7 b(,)21 b(Stanisla)n(v)d(P)n(op)r(o)n(vyc)n(h,)i(Daniil)-118 2886 y(Proskurin,)g(Sla)n(vik)g(Rabanovic)n(h)g(for)i(their)f(v)-5 b(aluable)20 b(con)n(tributions)f(to)j(this)-118 2986 y(b)r(o)r(ok.)6 3085 y(W)-7 b(e)38 b(also)c(gratefully)g(ac)n(kno)n (wlege)f(\014nancial)i(supp)r(ort)h(from)f(the)j(join)n(t)-118 3185 y(gran)n(t)26 b(from)g(the)i(CRDF)h(and)e(Ukrainian)e(Go)n(v)n (ernmen)n(t)g(no.)37 b(UM1-311.)p eop %%Page: 7 11 7 10 bop -118 664 a FR(Chapter)45 b(1)-118 982 y(P)l(airs)g(of)g (self-adjoin)l(t)f(op)t(erators)i(connected)-118 1131 y(b)l(y)f(quadratic)f(relations)h(and)f(some)-118 1281 y(generalizations)-118 1732 y FG(1.1)112 b(In)m(tro)s(duction)37 b(to)g(represen)m(tations)i(of)e FF(\003)p FG(-algebras)-118 1919 y FQ(1.1.1)94 b FP(\003)p FQ(-Represen)m(tations:)40 b(k)m(ey)33 b(w)m(ords)-118 2077 y(1.)38 b FO(A)29 b(represen)n(tation) c(of)j(an)g(algebra)e FA(A)i FO(on)g(a)g(\014nite-dimensional)23 b(Hilb)r(ert)-118 2177 y(\(unitary\))h(space)h FN(H)32 b FO(is)24 b(a)h(homomorphism)20 b FN(\031)28 b FO(of)d FA(A)h FO(in)n(to)e(the)h(algebra)e FN(L)p FO(\()p FN(H)7 b FO(\))-118 2276 y(of)27 b(linear)d(transformations)g(on)j FN(H)7 b FO(.)36 b(A)28 b FP(\003)p FO(-represen)n(tation)23 b(of)28 b(a)e FP(\003)p FO(-algebra)e Fz(A)-118 2376 y FO(is)g(a)h FP(\003)p FO(-homomorphism)20 b FN(\031)29 b FO(from)24 b(the)i(algebra)d(in)n(to)h(the)i FP(\003)p FO(-algebra)c FN(L)p FO(\()p FN(H)7 b FO(\))26 b(of)-118 2475 y(b)r(ounded)k(op)r(erators)d(on)j(a)f(separable)e(Hilb)r(ert)h (space)h FN(H)7 b FO(.)42 b(The)30 b(dimension)-118 2575 y(of)d(the)h(represen)n(tation)d(is)i(the)h(dimension)c(of)k FN(H)7 b FO(.)6 2677 y(W)-7 b(e)22 b(emphasize)d(that)j(in)f(this)g(c)n (hapter)f(w)n(e)i(restrict)d(ourselv)n(es)g(to)i(consid-)-118 2777 y(ering)j(only)g(\014nite-dimensional)d(represen)n(tations)i(of)j FA(A)p FO(,)g(if)f FA(A)h FO(is)f(an)g(algebra)-118 2877 y(without)30 b(in)n(v)n(olution,)d(and)j FP(\003)p FO(-represen)n (tations)c(b)n(y)k(b)r(ounded)h(op)r(erators)d(on)-118 2976 y(a)f(separable)e(Hilb)r(ert)h(space)h(\(dim)13 b FN(H)30 b FP(\024)22 b(1)p FO(\))28 b(if)f Fz(A)g FO(is)g(a)g FP(\003)p FO(-algebra.)-73 3134 y FQ(2.)72 b FO(In)40 b(the)g(represen)n(tation)d(theory)h(of)i(algebras,)f(represen)n (tations)e(are)-118 3233 y(studied)h(up)h(to)f(some)g(equiv)-5 b(alence.)67 b(W)-7 b(e)39 b(call)d(represen)n(tations)g(of)j FA(A)p FO(,)i FN(\031)-118 3333 y FO(on)32 b FN(H)39 b FO(and)d(~)-46 b FN(\031)35 b FO(on)500 3312 y(~)478 3333 y FN(H)7 b FO(,)33 b(equiv)-5 b(alen)n(t)30 b(if)i(there)f(exists) g(an)h(in)n(v)n(ertible)c(op)r(erator)-118 3433 y FN(C)15 b FO(:)28 b FN(H)i FP(7!)234 3412 y FO(~)212 3433 y FN(H)k FO(that)28 b(in)n(tert)n(wines)d(the)j(represen)n(tations)d FN(\031)31 b FO(and)h(~)-47 b FN(\031)t FO(,)27 b(i.e.,)587 3621 y FN(C)6 b(\031)s FO(\()p FN(x)p FO(\))25 b(=)i(~)-47 b FN(\031)t FO(\()p FN(x)p FO(\))p FN(C)q(;)181 b FP(8)p FN(x)23 b FP(2)g FA(A)p FN(:)6 3811 y FO(In)40 b(the)g(represen)n (tation)c(theory)j(of)g FP(\003)p FO(-algebras,)f(represen)n(tations)f (are)-118 3911 y(studied)h(up)h(to)g(a)f(unitary)f(equiv)-5 b(alence.)68 b(Represen)n(tations)37 b(of)i Fz(A)p FO(,)i FN(\031)h FO(on)1089 4121 y(7)p eop %%Page: 8 12 8 11 bop -118 -137 a FO(8)917 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FN(H)38 b FO(and)e(~)-47 b FN(\031)35 b FO(on)377 75 y(~)355 96 y FN(H)7 b FO(,)33 b(are)d(said)g(to)i(b)r(e)f(unitarily) e(equiv)-5 b(alen)n(t)29 b(if)i(there)g(exists)f(a)-118 196 y(unitary)c(op)r(erator)g FN(U)18 b FO(:)28 b FN(H)h FP(7!)862 175 y FO(~)840 196 y FN(H)35 b FO(suc)n(h)27 b(that)591 355 y FN(U)9 b(\031)s FO(\()p FN(x)p FO(\))24 b(=)j(~)-46 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(U;)181 b FP(8)p FN(x)22 b FP(2)i Fz(A)p FN(:)6 514 y FO(T)-7 b(o)44 b(ev)n(ery)f FP(\003)p FO(-algebra)e Fz(A)p FO(,)48 b(one)43 b(can)h(asso)r(ciate)e(a)i(category)e FP(\003)p FO(-Rep)13 b Fz(A)p FO(,)-118 614 y(whose)27 b(ob)5 b(jects)27 b(are)f FP(\003)p FO(-represen)n(tations)e(of)k Fz(A)f FO(considered)f(up)i(to)f(a)g(unitary)-118 713 y(equiv)-5 b(alence)25 b(and)i(its)g(morphisms)d(are)j(in)n(tert)n(wining)d(op)r (erators.)6 813 y(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h(follo)n(wing)23 b(simple)i(prop)r(osition.)-118 960 y FQ(Prop)s(osition)30 b(1.)41 b FB(A)n(ny)34 b(two)g(\014nite-dimensional)44 b FP(\003)p FB(-r)l(epr)l(esentations)33 b(of)i(a)-118 1060 y FP(\003)p FB(-algebr)l(a)25 b Fz(A)g FB(ar)l(e)g(e)l(quivalent)g (if)h(and)f(only)g(if)h(they)f(ar)l(e)g(unitarily)h(e)l(quivalent.)-118 1207 y(Pr)l(o)l(of.)43 b FO(One)31 b(has)g(to)g(pro)n(v)n(e)f(only)g (that)i(the)g(equiv)-5 b(alence)29 b(of)38 b FP(\003)p FO(-represen)n(ta-)-118 1306 y(tions)26 b FN(\031)31 b FO(on)d FN(H)34 b FO(and)e(~)-47 b FN(\031)31 b FO(on)758 1285 y(~)737 1306 y FN(H)j FO(implies)24 b(their)j(unitary)f(equiv)-5 b(alence.)6 1406 y(Let)28 b FN(C)16 b FO(:)27 b FN(H)j FP(7!)507 1385 y FO(~)485 1406 y FN(H)35 b FO(b)r(e)28 b(an)f(in)n(v)n(ertible)d(op)r(erator)i(suc)n(h)h(that)589 1565 y FN(C)6 b(\031)s FO(\()p FN(x)p FO(\))25 b(=)i(~)-46 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(C)q(;)182 b FP(8)p FN(x)22 b FP(2)i Fz(A)p FN(:)536 b FO(\(1.1\))6 1724 y(Let)29 b(us)f(consider)e(the)j(p)r(olar)d(decomp)r(osition)f(of)j (the)g(op)r(erator)f FN(C)6 b FO(,)29 b FN(C)h FO(=)-118 1823 y FN(U)9 b(A)p FO(,)23 b(where)e FN(A)i FO(=)g(\()p FN(C)6 b(C)625 1793 y FM(\003)664 1823 y FO(\))696 1793 y FK(1)p FL(=)p FK(2)822 1823 y FO(is)20 b(an)h(in)n(v)n(ertible)d(p)r (ositiv)n(e)h(op)r(erator)g(on)j FN(H)28 b FO(and)-118 1923 y FN(U)39 b FO(is)28 b(a)i(unitary)e(op)r(erator)g(from)h FN(H)37 b FO(to)1197 1902 y(~)1175 1923 y FN(H)g FO(\()p FN(U)1379 1893 y FM(\000)p FK(1)1495 1923 y FO(=)26 b FN(U)1652 1893 y FM(\003)1690 1923 y FO(\).)44 b(Then)30 b(it)g(follo)n(ws)-118 2023 y(from)c(\(1.1\))h(that)435 2182 y FN(\031)s FO(\()p FN(x)p FO(\))e(=)d FN(A)770 2147 y FM(\000)p FK(1)860 2182 y FN(U)926 2147 y FM(\000)p FK(1)1019 2182 y FO(~)-47 b FN(\031)t FO(\()p FN(x)p FO(\))p FN(U)9 b(A;)181 b FP(8)p FN(x)22 b FP(2)i Fz(A)p FN(:)382 b FO(\(1.2\))-118 2341 y(T)-7 b(aking)26 b(adjoin)n(ts)g(of)h (b)r(oth)h(sides,)f(w)n(e)g(obtain)440 2500 y(~)-47 b FN(\031)s FO(\()p FN(x)p FO(\))25 b(=)d FN(U)9 b(A)836 2465 y FM(\000)p FK(1)925 2500 y FN(\031)s FO(\()p FN(x)p FO(\))p FN(AU)1214 2465 y FM(\000)p FK(1)1305 2500 y FN(;)180 b FP(8)p FN(x)22 b FP(2)i Fz(A)p FN(:)382 b FO(\(1.3\))-118 2658 y(Consequen)n(tly)-7 b(,)-54 2817 y FN(A)8 2783 y FK(2)46 2817 y FN(\031)s FO(\()p FN(x)p FO(\))24 b(=)f FN(AU)447 2783 y FM(\000)p FK(1)540 2817 y FO(~)-46 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(U)9 b(A)24 b FO(=)f FN(AU)1065 2783 y FM(\000)p FK(1)1154 2817 y FN(U)9 b(A)1282 2783 y FM(\000)p FK(1)1371 2817 y FN(\031)s FO(\()p FN(x)p FO(\))p FN(AU)1660 2783 y FM(\000)p FK(1)1751 2817 y FN(U)g(A)23 b FO(=)f FN(\031)s FO(\()p FN(x)p FO(\))p FN(A)2212 2783 y FK(2)2251 2817 y FN(:)-118 2976 y FO(Since)33 b FN(A)h FO(is)e(a)h(p)r(ositiv)n(e)f (op)r(erator,)h(w)n(e)g(ha)n(v)n(e)g FN(A)1432 2946 y FK(2)1469 2976 y FN(\031)s FO(\()p FN(x)p FO(\))i(=)d FN(\031)s FO(\()p FN(x)p FO(\))p FN(A)1985 2946 y FK(2)2024 2976 y FO(,)k FN(x)d FP(2)h FA(A)p FO(,)-118 3076 y(whic)n(h)h(implies) c(that)36 b FN(A\031)s FO(\()p FN(x)p FO(\))j(=)d FN(\031)s FO(\()p FN(x)p FO(\))p FN(A)h FO(for)e(an)n(y)g FN(x)i FP(2)g Fz(A)p FO(.)61 b(F)-7 b(rom)34 b(this)h(w)n(e)-118 3176 y(obtain)553 3335 y FN(U)9 b(\031)s FO(\()p FN(x)p FO(\))p FN(A)24 b FO(=)f FN(U)9 b(A\031)s FO(\()p FN(x)p FO(\))24 b(=)j(~)-46 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(U)9 b(A;)-118 3494 y FO(and)27 b(since)g FN(A)h FO(is)e(in)n(v)n (ertible,)591 3652 y FN(U)9 b(\031)s FO(\()p FN(x)p FO(\))24 b(=)j(~)-46 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(U;)181 b FP(8)p FN(x)22 b FP(2)i Fz(A)p FN(:)-118 3811 y FO(Hence)29 b(w)n(e)f(ha)n(v)n(e)g(a)g(unitary)g(equiv)-5 b(alence)26 b(of)j(the)g(represen)n(tations)d FN(\031)32 b FO(on)d FN(H)-118 3911 y FO(and)j(~)-47 b FN(\031)31 b FO(on)258 3890 y(~)237 3911 y FN(H)6 b FO(.)p 2278 3911 4 57 v 2282 3858 50 4 v 2282 3911 V 2331 3911 4 57 v eop %%Page: 9 13 9 12 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)626 b FO(9)-118 96 y FQ(3.)35 b FO(In)26 b(the)f(general)e(represen)n (tation)g(theory)h(one)h(distinguishes)d(irreducible)-118 196 y(and)37 b(indecomp)r(osable)d(represen)n(tations)g(in)j(the)g(set) h(of)f(all)e FB(\014nite-dimen-)-118 296 y(sional)42 b FO(represen)n(tations.)75 b(A)41 b(represen)n(tation)d FN(\031)12 b FO(:)33 b FA(A)46 b FP(7!)f FN(L)p FO(\()p FN(H)7 b FO(\))41 b(is)f(called)-118 395 y(irreducible)19 b(if)j(there)g(exists)f(no)i(non-trivial)18 b(subspace)k(of)g FN(H)30 b FO(in)n(v)-5 b(arian)n(t)19 b(with)-118 495 y(resp)r(ect)k(to)g(all)e(op)r(erators)g FN(\031)s FO(\()p FN(x)p FO(\),)26 b FN(x)d FP(2)h FA(A)p FO(.)35 b(A)24 b(represen)n(tation)c FN(\031)12 b FO(:)29 b FA(A)23 b FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\))-118 595 y(is)25 b(called)e(indecomp)r(osable)g(if)i(there)h(exists)f(no)h(decomp)r (osition)c FN(H)30 b FO(=)22 b FN(H)2221 607 y FK(1)2274 595 y FO(+)-118 694 y FN(H)-49 706 y FK(2)24 694 y FO(in)n(to)34 b(a)h(sum)g(of)h(the)g(t)n(w)n(o)e(non-trivial)e(subspaces)i(that)i (are)f(in)n(v)-5 b(arian)n(t)-118 794 y(with)25 b(resp)r(ect)h(to)g (all)e(the)i(op)r(erators)e FN(\031)s FO(\()p FN(x)p FO(\))j FN(x)d FP(2)f FA(A)p FO(,)k(and)e FN(H)1752 806 y FK(1)1805 794 y FP(\\)15 b FN(H)1944 806 y FK(2)2005 794 y FO(=)22 b FP(f)p FO(0)p FP(g)p FO(.)35 b(It)-118 893 y(is)25 b(clear)f(that)i(an)n(y)f(irreducible)d(represen)n(tation)i (is)g(indecomp)r(osable.)33 b(Th)n(us)-118 993 y(the)40 b(set)g(of)g(irreducible)c(represen)n(tations)h(is)h(a)i(subset)f(of)h (the)g(set)g(of)g(all)-118 1093 y(indecomp)r(osable)19 b(represen)n(tations.)33 b(The)23 b(size)f(of)i(this)e(subset)i(in)e (the)i(whole)-118 1192 y(set)d(of)g(indecomp)r(osable)d(represen)n (tations)g(dep)r(ends)j(on)g(the)h(structure)e(of)i FA(A)p FO(.)6 1292 y(A)e(description)c(of)k(all)c(indecomp)r(osable)g (\(particularly)f(irreducible\))g(rep-)-118 1392 y(resen)n(tations)21 b(is)h(one)h(of)g(the)h(most)e(imp)r(ortan)n(t)f(problems)g(of)i (represen)n(tation)-118 1491 y(theory)-7 b(.)6 1591 y(In)22 b(the)g(case)e(where)h Fz(A)g FO(is)f(an)h(algebra)d(with)j(an)g(in)n (v)n(olution,)e(one)i(can)f(con-)-118 1690 y(sider)g(b)r(oth)i (irreducible)c FP(\003)p FO(-represen)n(tations)g(and)k(indecomp)r (osable)c FP(\003)p FO(-repre-)-118 1790 y(sentations.)33 b(Ho)n(w)n(ev)n(er,)20 b(in)g(this)g(case)f(these)i(notions)e (coincide.)32 b(Namely)-7 b(,)20 b(the)-118 1890 y(follo)n(wing)j (simple)i(prop)r(osition)g(holds.)-118 2052 y FQ(Prop)s(osition)30 b(2.)41 b FB(A)27 b FP(\003)p FB(-r)l(epr)l(esentation)f FN(\031)31 b FB(is)d(inde)l(c)l(omp)l(osable)h(if)f(and)g(only)-118 2151 y(if)j(it)e(is)h(irr)l(e)l(ducible.)-118 2313 y(Pr)l(o)l(of.)43 b FO(It)i(is)e(su\016cien)n(t)g(to)h(pro)n(v)n(e)f(that)h(an)n(y)f (indecomp)r(osable)e FP(\003)p FO(-repre-)-118 2413 y(sen)n(tation)d (is)h(irreducible.)70 b(Assume)39 b(the)i(con)n(trary)-7 b(,)41 b(that)f(is,)i(let)d(an)h(in-)-118 2513 y(decomp)r(osable)23 b(represen)n(tation)g(b)r(e)k(reducible,)d(i.e.,)h(there)h(exists)f(a)h (prop)r(er)-118 2612 y(subspace)h FN(H)298 2624 y FK(1)363 2612 y FO(in)g FN(H)34 b FO(in)n(v)-5 b(arian)n(t)24 b(with)k(resp)r(ect)f(to)g(all)f FN(\031)s FO(\()p FN(x)p FO(\),)j FN(x)23 b FP(2)h Fz(A)p FO(.)-118 2762 y FQ(Lemma)29 b(1.)41 b FB(The)36 b(subsp)l(ac)l(e)e FN(H)924 2732 y FM(?)917 2782 y FK(1)1011 2762 y FO(=)d FP(f)p FN(y)j FP(2)e FN(H)16 b FO(:)29 b(\()p FN(y)s(;)14 b(f)9 b FO(\))31 b(=)g(0)p FN(;)48 b FP(8)p FN(f)39 b FP(2)32 b FN(H)2166 2774 y FK(1)2203 2762 y FP(g)i FB(is)-118 2861 y(non-trivial)c(and)h (invariant)f(with)h(r)l(esp)l(e)l(ct)e(to)h(al)t(l)h FN(\031)s FO(\()p FN(x)p FO(\))p FB(,)g FN(x)24 b FP(2)f Fz(A)p FB(.)-118 3026 y(Pr)l(o)l(of.)43 b FO(If)28 b FN(y)e FP(2)d FN(H)446 2996 y FM(?)439 3047 y FK(1)502 3026 y FO(,)28 b(then)603 3204 y(\()p FN(\031)s FO(\()p FN(x)p FO(\))p FN(y)s(;)14 b(f)9 b FO(\))24 b(=)e(\()p FN(y)s(;)14 b(\031)s FO(\()p FN(x)1312 3170 y FM(\003)1351 3204 y FO(\))p FN(f)9 b FO(\))24 b(=)e(0)-118 3382 y(for)27 b(all)e FN(f)37 b FO(from)26 b(the)i(in)n(v)-5 b(arian)n(t)24 b(subspace)j FN(H)1305 3394 y FK(1)1342 3382 y FO(,)h(i.e.)36 b FN(\031)s FO(\()p FN(x)p FO(\))p FN(y)27 b FP(2)d FN(H)1920 3352 y FM(?)1913 3402 y FK(1)1975 3382 y FO(.)p 2278 3382 4 57 v 2282 3329 50 4 v 2282 3382 V 2331 3382 4 57 v 6 3547 a(The)k(con)n(tradiction)c(immediately)f(follo)n(ws)i(from) h(the)i(lemma.)p 2278 3547 V 2282 3494 50 4 v 2282 3547 V 2331 3547 4 57 v -118 3712 a FQ(4.)52 b FO(Let)32 b FN(H)40 b FO(b)r(e)33 b(a)g(separable)d(\(generally)f(sp)r(eaking,)k (in\014nite-dimensional)o(\))-118 3811 y(Hilb)r(ert)27 b(space.)40 b(F)-7 b(ollo)n(wing)25 b(the)k(general)e(strategy)h(of)g (represen)n(tation)f(the-)-118 3911 y(ory)-7 b(,)27 b(w)n(e)g(tak)n(e)g (irreducible)e(represen)n(tations)g(to)i(b)r(e)h(the)h(\\simplest")24 b(among)p eop %%Page: 10 14 10 13 bop -118 -137 a FO(10)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(all)j FP(\003)p FO(-represen)n(tations.)42 b(A)31 b FP(\003)p FO(-represen)n(tation)c FN(\031)12 b FO(:)29 b Fz(A)e FP(7!)h FN(L)p FO(\()p FN(H)7 b FO(\))31 b(is)e(called)f(ir-) -118 196 y(reducible,)c(if)h(there)g(exists)f(no)h(non-trivial)d (subspace)j(in)f FN(H)33 b FO(in)n(v)-5 b(arian)n(t)22 b(with)-118 296 y(resp)r(ect)36 b(to)h(all)d(the)j(op)r(erators)d FN(\031)s FO(\()p FN(x)p FO(\))k(\()p FN(x)h FP(2)g Fz(A)p FO(\).)63 b(The)37 b(follo)n(wing)32 b(form)k(of)-118 395 y(Sc)n(h)n(ur's)27 b(lemma,)d(giv)n(es)i(an)h(equiv)-5 b(alen)n(t)25 b(condition)h(of)h(irreducibilit)n(y)-7 b(.)-118 564 y FQ(Prop)s(osition)30 b(3.)41 b FB(A)28 b FP(\003)p FB(-r)l(epr)l(esentation)g FN(\031)s FO(\()p FP(\001)p FO(\))h FB(is)f(irr)l(e)l(ducible)i(if)f(and)g(only)g(if)-118 664 y(any)h(b)l(ounde)l(d)g(op)l(er)l(ator)h FN(C)e FP(2)24 b FN(L)p FO(\()p FN(H)7 b FO(\))29 b FB(such)h(that)587 848 y FN(C)6 b(\031)s FO(\()p FN(x)p FO(\))25 b(=)e FN(\031)s FO(\()p FN(x)p FO(\))p FN(C)q(;)185 b FP(8)p FN(x)23 b FP(2)g Fz(A)p FN(;)-118 1032 y FB(is)30 b(a)g(multiple)g(of)h(the)f (identity,)h(i.e.,)g FN(C)f FO(=)22 b FN(cI)7 b FB(,)31 b(with)f FN(c)23 b FP(2)g FI(C)15 b FB(.)-118 1201 y(Pr)l(o)l(of.)43 b FO(If)30 b FN(A)c FO(=)g FN(A)468 1171 y FM(\003)536 1201 y FO(comm)n(utes)h(with)i FN(\031)s FO(\()p FP(\001)p FO(\),)i(i.e.)42 b FN(A\031)s FO(\()p FN(x)p FO(\))27 b(=)f FN(\031)s FO(\()p FN(x)p FO(\))p FN(A)p FO(,)32 b FP(8)p FN(x)25 b FP(2)i Fz(A)-118 1301 y FO(then)630 1485 y FN(E)691 1497 y FL(A)745 1485 y FO(\(\001\))14 b FN(\031)s FO(\()p FN(x)p FO(\))25 b(=)e FN(\031)s FO(\()p FN(x)p FO(\))14 b FN(E)1402 1497 y FL(A)1457 1485 y FO(\(\001\))-118 1669 y(for)38 b(all)f FN(x)43 b FP(2)f Fz(A)d FO(and)g(Borel)e(sets)i (\001)j FP(\032)g FI(R)1284 1639 y FK(1)1366 1669 y FO(\(here)d FN(E)1651 1681 y FL(A)1706 1669 y FO(\(\001\))h(is)e(a)g(sp)r(ectral) -118 1769 y(pro)5 b(jector)35 b(of)i(the)g(op)r(erator)f FN(A)p FO(\).)65 b(In)37 b(this)f(case,)j FN(H)1596 1781 y FK(\001)1693 1769 y FO(=)f FN(E)1857 1781 y FL(A)1912 1769 y FO(\(\001\))p FN(H)44 b FO(is)36 b(an)-118 1869 y(in)n(v)-5 b(arian)n(t)24 b(subspace)j(in)g FN(H)7 b FO(.)6 1969 y(If)26 b(the)g(represen)n(tation)c FN(\031)29 b FO(is)24 b(irreducible,)e(then)k(all)d(suc)n(h)i FN(H)1911 1981 y FK(\001)1995 1969 y FO(are)f(either)-118 2069 y FP(f)p FO(0)p FP(g)36 b FO(or)g FN(H)7 b FO(,)39 b(i.e.,)g(the)e(sp)r (ectral)f(measure)f FN(E)1330 2081 y FL(A)1384 2069 y FO(\()p FP(\001)p FO(\))j(is)e(concen)n(trated)g(at)h(one)-118 2168 y(p)r(oin)n(t)27 b FN(a)c FP(2)g FI(R)298 2138 y FK(1)341 2168 y FO(,)28 b(and)f FN(A)d FO(=)e FN(aI)7 b FO(.)6 2269 y(If)34 b FN(C)39 b FO(=)33 b FN(A)22 b FO(+)g FN(iB)37 b FO(\()p FN(A)c FO(=)g FN(A)878 2239 y FM(\003)916 2269 y FO(,)i FN(B)i FO(=)32 b FN(B)1238 2239 y FM(\003)1309 2269 y FP(2)h FN(L)p FO(\()p FN(H)7 b FO(\)\))34 b(comm)n(utes)d(with)i(an)-118 2369 y(irreducible)19 b(represen)n(tation)h FN(\031)s FO(\()p FP(\001)p FO(\))k(of)f(the)g FP(\003)p FO(-algebra)c Fz(A)p FO(,)k(then)h(the)f(op)r(erators)-118 2468 y FN(A)p FO(,)36 b FN(B)i FO(also)31 b(comm)n(ute)h(with)i FN(\031)s FO(\()p FP(\001)p FO(\),)i(and,)f(consequen)n(tly)-7 b(,)34 b FN(C)40 b FO(=)33 b FN(aI)c FO(+)22 b FN(ibI)40 b FO(=)-118 2568 y(\()p FN(a)19 b FO(+)f FN(ib)p FO(\))p FN(I)7 b FO(;)27 b FN(a)p FO(,)g FN(b)c FP(2)h FI(R)p FO(.)6 2668 y(Con)n(v)n(ersely)-7 b(,)25 b(if)j(a)f(represen)n(tation)e FN(\031)s FO(\()p FP(\001)p FO(\))k(is)d(reducible)g(and)i FN(H)1968 2680 y FK(1)2033 2668 y FO(is)f(a)g(sub-)-118 2768 y(space)34 b(in)n(v)-5 b(arian)n(t)32 b(with)j(resp)r(ect)g(to)g FN(\031)s FO(\()p FN(x)p FO(\),)j FN(x)e FP(2)g Fz(A)p FO(,)h(then,)g(b)n(y)e(the)h(lemma,)-118 2868 y FN(H)-42 2837 y FM(?)-49 2888 y FK(1)41 2868 y FO(is)27 b(also)e(in)n(v)-5 b(arian)n(t.)34 b(Then)28 b(the)g(op)r(erator)145 3106 y FN(C)h FO(=)321 2989 y Fy(\022)423 3051 y FN(c)459 3063 y FK(1)496 3051 y FN(I)532 3063 y FL(H)586 3071 y Fx(1)797 3051 y FO(0)503 3151 y(0)162 b FN(c)743 3163 y FK(2)780 3151 y FN(I)816 3171 y FL(H)874 3151 y Fw(?)870 3189 y Fx(1)970 2989 y Fy(\023)1045 3106 y FN(;)180 b(c)1284 3118 y FK(1)1344 3106 y FP(6)p FO(=)23 b FN(c)1468 3118 y FK(2)1505 3106 y FN(;)180 b(c)1744 3118 y FK(1)1781 3106 y FN(;)14 b(c)1854 3118 y FK(2)1914 3106 y FP(2)24 b FI(C)15 b FN(;)-118 3340 y FO(comm)n(utes)24 b(with)i(the)h(represen) n(tation)c(and)k(is)e(not)h(a)g(m)n(ultiple)d(of)k(the)g(iden-)-118 3439 y(tit)n(y)-7 b(.)p 2278 3439 4 57 v 2282 3386 50 4 v 2282 3439 V 2331 3439 4 57 v -118 3612 a FB(R)l(emark)30 b(1.)42 b FO(It)31 b(is)f(p)r(ossible)f(to)i(de\014ne)g(the)h(notion)e (of)h(an)f(indecomp)r(osable)-118 3712 y(represen)n(tation)40 b(in)j(the)h(case)e(where)h FN(H)50 b FO(is)42 b(a)h(separable)e(Hilb)r (ert)g(space)-118 3811 y(\(dim)12 b FN(H)36 b FO(=)29 b FP(1)p FO(\))j(and)f(to)g(pro)n(v)n(e)e(an)i(analog)e(of)i(the)h (previous)d(prop)r(ositions.)-118 3911 y(Ho)n(w)n(ev)n(er)d(w)n(e)h (are)f(not)i(going)d(to)j(do)f(it)g(here.)p eop %%Page: 11 15 11 14 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(11)6 96 y(Irreducible)21 b(represen)n(tations)g(and)j(their)e(in)n (tert)n(wining)e(op)r(erators)i(form)-118 196 y(a)d(full)g (sub-category)-7 b(,)19 b FP(\003)p FO(-Irrep)12 b Fz(A)p FO(,)21 b(in)f(the)g(category)e FP(\003)p FO(-Rep)13 b Fz(A)p FO(.)34 b(The)20 b(condition)-118 296 y(for)33 b(a)g(sub-category)e(to)i(b)r(e)h(full)e(means)g(that)h(the)h(em)n(b)r (edding)d(functor)j FN(F)-118 395 y FO(from)24 b FP(\003)p FO(-Irrep)12 b Fz(A)25 b FO(in)n(to)f FP(\003)p FO(-Rep)14 b Fz(A)25 b FO(is)f(an)h(isomorphism)c(on)k(the)h(corresp)r(onding)-118 495 y(morphisms.)31 b(In)23 b(what)f(follo)n(ws,)f(w)n(e)h(will)d (mainly)g(deal)i(with)h FP(\003)p FO(-algebras)d(and)-118 595 y(their)30 b FP(\003)p FO(-represen)n(tations;)e(th)n(us)j(w)n(e)g (will)d(sometimes)f(omit)i(the)j(in)n(v)n(olution)-118 694 y(sign)18 b(with)h(the)h(w)n(ords)e(algebra,)g(morphism,)g (category)-7 b(,)19 b(and)g(represen)n(tation,)-118 794 y(if)27 b(no)g(am)n(biguit)n(y)d(can)j(arise.)-118 1007 y FQ(1.1.2)94 b FN(C)239 977 y FM(\003)277 1007 y FQ(-represen)m(table) 32 b FP(\003)p FQ(-algebras)-118 1160 y(1.)j FO(An)24 b(imp)r(ortan)n(t)e(class)g(of)i FP(\003)p FO(-algebras)c(is)j(the)h (class)f(of)h(all)d FP(\003)p FO(-algebras)f(that)-118 1260 y(ha)n(v)n(e)33 b(\\su\016cien)n(tly)f(man)n(y")g FP(\003)p FO(-represen)n(tations.)53 b(The)35 b(latter)e(means)f(that) -118 1359 y(there)27 b(exists)g(a)g(residual)e(family)g(\(r.f.\))38 b(of)27 b FP(\003)p FO(-represen)n(tations,)d(i.e.,)j(for)g(an)n(y)-118 1459 y FN(x)f FP(2)g Fz(A)p FO(,)j FN(x)d FP(6)p FO(=)f(0,)k(there)g (exists)e(a)i FP(\003)p FO(-represen)n(tation)c FN(\031)33 b FO(\(it)c(can)f(b)r(e)i(c)n(hosen)e(to)-118 1559 y(b)r(e)e (irreducible\))c(suc)n(h)i(that)i FN(\031)s FO(\()p FN(x)p FO(\))e FP(6)p FO(=)f(0.)36 b(F)-7 b(or)24 b(an)n(y)h FN(C)1536 1528 y FM(\003)1574 1559 y FO(-algebra)e(there)i(alw)n(a)n (ys)-118 1658 y(exists)h(a)h(r.f.)37 b(of)28 b FP(\003)p FO(-represen)n(tations.)6 1758 y(If)41 b(a)e FP(\003)p FO(-algebra)d Fz(A)j FO(is)g FN(C)815 1728 y FM(\003)853 1758 y FO(-represen)n(table,)h(i.e.,)i(there)e(exists)e(a)h FP(\003)p FO(-iso-)-118 1857 y(morphism)31 b(of)j Fz(A)g FO(on)g(a)g FP(\003)p FO(-subalgebra)c(of)35 b(a)f FN(C)1404 1827 y FM(\003)1442 1857 y FO(-algebra,)f(then)i(it)f(is)f(clear)-118 1957 y(that)28 b Fz(A)f FO(has)g(a)g(r.f.)-118 2084 y FB(R)l(emark)j(2.)42 b FO(1\).)37 b(A)28 b FN(C)588 2053 y FM(\003)626 2084 y FO(-algebra)d(con)n(taining)f(a)j(dense)h FP(\003)p FO(-subalgebra)c(whic)n(h)-118 2183 y(is)f FP(\003)p FO(-isomorphic)c(to)k(a)h(giv)n(en)e(one)h(is)g(not)h(unique) f(in)g(general.)34 b(F)-7 b(or)23 b(example,)-118 2283 y(the)31 b FP(\003)p FO(-algebra)c FI(C)15 b FO([)p FN(a)34 b FO(=)28 b FN(a)683 2253 y FM(\003)721 2283 y FO(])i(is)g(isomorphic)c (to)31 b(a)f(dense)h(subalgebra)d(in)h(an)n(y)-118 2382 y FN(C)-53 2352 y FM(\003)-15 2382 y FO(-algebra)23 b FN(C)6 b FO(\()p FN(K)g FO(\))27 b(of)f(con)n(tin)n(uous)e(functions)h (on)h(an)g(in\014nite)f(compact)g(set)-118 2482 y FN(K)30 b FP(\032)24 b FI(R)126 2452 y FK(1)169 2482 y FO(.)40 b(Ho)n(w)n(ev)n(er,)27 b FN(C)6 b FO(\()p FN(K)759 2494 y FK(1)796 2482 y FO(\))29 b(is)e(isomorphic)e(to)j FN(C)6 b FO(\()p FN(K)1633 2494 y FK(2)1670 2482 y FO(\))29 b(if)f(and)h(only)e(if)h FN(K)2302 2494 y FK(1)-118 2582 y FO(and)f FN(K)114 2594 y FK(2)179 2582 y FO(are)f(homeomorphic.)6 2681 y(2\).)62 b(Not)36 b(an)n(y)f FP(\003)p FO(-algebra)d(is)j FN(C)1027 2651 y FM(\003)1065 2681 y FO(-represen)n(table.)59 b(Moreo)n(v)n(er,)35 b(not)g(an)n(y)-118 2781 y FP(\003)p FO(-algebra)29 b(p)r(ossesses)j(a)g(r.f.)53 b(of)33 b FP(\003)p FO(-represen)n(tations.)49 b(F)-7 b(or)33 b(example,)f(if)g (the)-118 2881 y(in)n(v)n(olution)h(in)j FA(A)h FO(is)f(not)h(prop)r (er)f(\(an)h(in)n(v)n(olution)c(is)j(prop)r(er)g(if)g FN(xx)2116 2850 y FM(\003)2194 2881 y FO(=)i(0)-118 2980 y(implies)30 b FN(x)j FO(=)g(0\),)i(then)f(suc)n(h)f(a)g FP(\003)p FO(-algebra)d(cannot)j(b)r(e)h FN(C)1771 2950 y FM(\003)1810 2980 y FO(-represen)n(table;)-118 3080 y(moreo)n(v)n(er)24 b(it)j(do)r(es)g(not)h(p)r(ossess)e(r.f.)37 b(of)28 b FP(\003)p FO(-represen)n(tations.)6 3206 y(Nev)n(ertheless,) 33 b(if)g Fz(A)e FO(=)h FI(C)15 b FO([)p FN(G)q FO(])38 b(=)32 b FI(C)15 b FP(h)p FN(g)41 b FP(j)32 b FN(g)j FP(2)e FN(G)p FP(i)h FO(is)e(a)g(group)g FP(\003)p FO(-algebra)-118 3306 y(of)c(a)g(coun)n(table)f(discrete)g(group)g FN(G)h FO(with)g(a)g(natural)f(in)n(v)n(olution)d(\(giv)n(en)j(on)-118 3405 y(basis)34 b(v)n(ectors)g FN(g)39 b FP(2)e FN(G)f FO(b)n(y)f FN(g)822 3375 y FM(\003)897 3405 y FO(=)h FN(g)1041 3375 y FM(\000)p FK(1)1130 3405 y FO(\),)i(then)e(the)g (follo)n(wing)c(prop)r(osition)-118 3505 y(holds.)-118 3658 y FQ(Prop)s(osition)e(4.)41 b FB(The)31 b(involutive)f(algebr)l(a) h FI(C)15 b FO([)p FN(G)q FO(])35 b FB(is)c FN(C)1698 3628 y FM(\003)1736 3658 y FB(-r)l(epr)l(esentable.)-118 3811 y(Pr)l(o)l(of.)43 b FO(Indeed,)23 b(an)n(y)e(op)r(erator)f FN(\031)957 3823 y FL(r)994 3811 y FO(\()p FN(x)p FO(\))i(of)g(the)g (righ)n(t)e(regular)e(represen)n(tation)-118 3911 y FN(\031)-71 3923 y FL(r)3 3911 y FO(on)36 b FN(L)184 3923 y FK(2)221 3911 y FO(\()p FN(G)p FO(\))i(is)e(non-zero)f(for)h(an)n(y)g(non-zero)f (elemen)n(t)g FN(x)k FP(2)g FI(C)15 b FO([)p FN(G)p FO(])q(,)45 b(and)p eop %%Page: 12 16 12 15 bop -118 -137 a FO(12)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(hence)h(the)g FP(\003)p FO(-algebra)d FI(C)15 b FO([)p FN(G)p FO(])32 b(is)25 b(isomorphic)d(to)j FN(\031)1455 108 y FL(r)1493 96 y FO(\()p FI(C)15 b FO([)p FN(G)p FO(]\))32 b(whic)n(h)25 b(is)g(a)h(dense)-118 196 y FP(\003)p FO(-subalgebra)33 b(of)k(the)g FN(C)697 166 y FM(\003)736 196 y FO(-algebra)d FN(C)1129 166 y FM(\003)1123 217 y FL(r)1167 196 y FO(\()p FN(G)p FO(\))k(generated)e(b)n(y)g(the)i(op)r (erators)-118 296 y FN(\031)s FO(\()p FN(x)p FO(\),)29 b FN(x)23 b FP(2)h FI(C)15 b FO([)p FN(G)p FO(].)p 2278 296 4 57 v 2282 243 50 4 v 2282 296 V 2331 296 4 57 v -118 461 a FQ(2.)36 b FO(The)28 b(follo)n(wing)23 b(prop)r(osition)i (holds.)-118 624 y FQ(Prop)s(osition)30 b(5.)41 b FB(Consider)31 b(the)f(fol)t(lowing)i(pr)l(op)l(erties)f(of)g(a)f FP(\003)p FB(-algebr)l(a)6 b FO(:)-45 802 y(\()p FN(i)p FO(\))41 b FB(the)30 b(algebr)l(a)h(is)g FN(C)663 772 y FM(\003)701 802 y FB(-r)l(epr)l(esentable)6 b FO(;)-74 967 y(\()p FN(ii)p FO(\))41 b FB(ther)l(e)30 b(exists)g(a)g(r)l(esidual)g(family)i (of)f(its)e(r)l(epr)l(esentations)7 b FO(;)-102 1131 y(\()p FN(iii)p FO(\))40 b FB(the)k(involution)g(on)f(the)g(algebr)l(a) i(is)e(c)l(ompletely)i(pr)l(op)l(er)f FO(\()p FB(i.e.,)k(the)89 1231 y(e)l(quality)389 1168 y Fy(P)477 1189 y FL(n)477 1256 y(k)q FK(=1)615 1231 y FN(x)662 1243 y FL(k)704 1231 y FN(x)751 1200 y FM(\003)751 1254 y FL(k)815 1231 y FO(=)23 b(0)i FB(implies)j FN(x)1298 1243 y FL(k)1362 1231 y FO(=)23 b(0)p FB(,)j FN(k)g FO(=)d(1)p FB(,)j FN(:)14 b(:)g(:)27 b FB(,)g FN(n)p FB(;)h FN(n)23 b FP(2)g FI(N)t FO(\))q(;)-88 1395 y(\()p FN(iv)s FO(\))41 b FB(the)c (involution)g(in)f(the)g(algebr)l(a)i(is)e(pr)l(op)l(er)h FO(\()p FB(i.e.,)j FN(xx)1833 1365 y FM(\003)1907 1395 y FO(=)34 b(0)i FB(implies)89 1495 y FN(x)24 b FO(=)f(0\))p FB(.)6 1673 y(Then)31 b FO(\()p FN(i)p FO(\))23 b FP(\))g FO(\()p FN(ii)p FO(\))g FP(\))g FO(\()p FN(iii)p FO(\))g FP(\))g FO(\()p FN(iv)s FO(\))p FB(.)6 1773 y(Neither)31 b(of)f(the)g(inverse)h(implic)l(ations)g(holds.)-118 1936 y(Pr)l(o)l(of.)43 b FO(The)35 b(direct)e(implications)c(are)34 b(easily)d(v)n(eri\014ed.)56 b(T)-7 b(o)34 b(see)g(that)h(\()p FN(ii)p FO(\))-118 2035 y(do)r(es)24 b(not)g(imply)e(\()p FN(i)p FO(\),)j(one)f(has)f(to)h(consider)f(the)h FP(\003)p FO(-algebra)d FN(C)6 b FO(\([0)p FN(;)14 b FP(1)p FO(\)\))25 b(of)f(all)-118 2135 y(con)n(tin)n(uous)j(functions)h FN(f)9 b FO(\()p FP(\001)p FO(\))29 b(on)g([0)p FN(;)14 b FP(1)p FO(\))29 b(with)f(the)i(natural)d(in)n(v)n(olution)e(and)-118 2234 y(p)r(oin)n(t)n(wise)g(m)n(ultiplication.)32 b(It)c(is)f(ob)n (vious)e(that)j(this)f(algebra)e(has)i(a)h(r.f.)g(of)-118 2334 y(one-dimensional)19 b(represen)n(tations,)k(ho)n(w)n(ev)n(er)f (it)j(is)e(not)i FN(C)1771 2304 y FM(\003)1810 2334 y FO(-represen)n(table.)-118 2434 y(Indeed,)34 b(assume)d(that)i FN(C)6 b FO(\([0)p FN(;)14 b FP(1)p FO(\)\))34 b(is)d(em)n(b)r(edded)i (as)f(a)g FP(\003)p FO(-subalgebra)d(in)j(a)-118 2533 y FN(C)-53 2503 y FM(\003)-15 2533 y FO(-algebra)g Fz(A)p FO(,)j(then)g(there)f(exists)f(an)h(in)n(teger)e FN(N)44 b FO(suc)n(h)34 b(that)g FP(k)p FN(f)9 b FP(k)33 b FN(<)h(N)9 b FO(,)-118 2633 y(where)229 2865 y FN(f)g FO(\()p FN(x)p FO(\))24 b(=)502 2723 y Fy(\()568 2809 y FN(x)19 b FP(\000)f FN(n;)83 b FO(if)27 b(2)p FN(n)c(<)f(x)i(<)f FO(2)p FN(n)17 b FO(+)h(1)p FN(;)568 2928 y(n;)232 b FO(if)27 b(2)p FN(n)18 b FP(\000)g FO(1)23 b FP(\024)f FN(x)i FP(\024)f FO(2)p FN(n)o(;)1757 2865 y(n)g FP(2)g FI(N)t FN(:)6 3114 y FO(This)j(implies)d(that)k(the)g(elemen)n(t)e FN(N)9 b(I)23 b FP(\000)16 b FN(f)35 b FO(is)26 b(in)n(v)n(ertible)d (in)j Fz(A)p FO(,)g(but)i(it)e(is)-118 3214 y(a)h(zero)g(divisor)d(in)j FN(C)6 b FO(\([0)p FN(;)14 b FP(1)p FO(\)\),)28 b(whic)n(h)f(giv)n(es)e (a)i(con)n(tradiction.)6 3313 y(In)34 b(order)e(to)h(sho)n(w)f(that)i (\()p FN(iii)p FO(\))f(do)r(es)g(not)g(imply)e(\()p FN(ii)p FO(\),)j(consider)e(the)h FP(\003)p FO(-)-118 3413 y(algebra)k(with)j (unit)f FN(e)p FO(,)k Fz(A)g FO(=)g FI(C)15 b FP(h)q FN(a;)f(x)50 b FP(j)43 b FN(a)1265 3383 y FM(\003)1303 3413 y FN(a)h FO(=)f FN(q)s(aa)1627 3383 y FM(\003)1665 3413 y FN(;)28 b(xx)1810 3383 y FM(\003)1876 3413 y FO(+)e FN(aa)2055 3383 y FM(\003)2136 3413 y FO(=)44 b FN(e)p FP(i)p FO(,)-118 3513 y(where)d(0)47 b FN(<)g(q)j(<)d FO(1.)80 b(The)42 b(set)g(of)g(w)n(ords)f(whic)n(h)g(do)h(not)g(con)n (tain)e(the)-118 3612 y(sub-w)n(ords)31 b FN(a)322 3582 y FM(\003)360 3612 y FN(a)i FO(and)g FN(xx)698 3582 y FM(\003)771 3612 y FO(forms)e(a)i(linear)d(basis)i(for)g(this)h (algebra.)51 b(T)-7 b(ak)n(e)-118 3712 y(an)35 b(arbitrary)d FN(z)39 b FO(=)545 3650 y Fy(P)646 3712 y FN(\013)699 3724 y FL(i)727 3712 y FN(u)775 3724 y FL(i)816 3712 y FN(x)863 3682 y FL(k)898 3690 y Fv(i)965 3712 y FP(2)d Fz(A)p FO(,)h(where)e(the)h(w)n(ords)e FN(u)1869 3724 y FL(i)1931 3712 y FO(do)h(not)g(end)-118 3811 y(with)40 b FN(x)p FO(,)45 b(and)40 b FN(k)416 3823 y FL(i)489 3811 y FP(\025)45 b FO(0.)76 b(Denote)41 b(b)n(y)f FN(F)12 b FO(\()p FN(z)t FO(\))41 b(the)g(sum)f(of)g(co)r(e\016cien)n(ts)g(at) -118 3911 y(those)27 b(w)n(ords)f FN(u)385 3923 y FL(i)412 3911 y FN(x)459 3881 y FL(k)494 3889 y Fv(i)553 3911 y FO(of)i(minimal)23 b(length)j(that)i(can)f(also)f(b)r(e)i(written)e (in)h(the)p eop %%Page: 13 17 13 16 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(13)-118 96 y(form)34 b FN(w)r(w)208 66 y FM(\003)248 96 y FO(.)60 b(De\014ne)36 b FN(J)44 b FO(=)36 b FP(f)p FN(j)14 b FO(:)30 b FN(l)r FO(\()p FN(u)1037 108 y FL(j)1071 96 y FO(\))37 b FP(\024)f FN(l)r FO(\()p FN(u)1348 108 y FL(i)1374 96 y FO(\))p FN(;)28 b FP(8)p FN(i)p FP(g)p FO(.)59 b(W)-7 b(e)36 b(will)d(sho)n(w)h(that)-118 196 y FN(F)12 b FO(\()p FN(z)t(z)65 166 y FM(\003)102 196 y FO(\))28 b(=)255 134 y Fy(P)343 221 y FL(j)s FM(2)p FL(J)478 196 y FP(j)p FN(\013)554 208 y FL(j)590 196 y FP(j)613 166 y FK(2)650 196 y FO(.)46 b(Indeed,)32 b(the)f(canonical)c(form)i(of)i FN(u)1874 208 y FL(i)1901 196 y FN(x)1948 166 y FL(k)1983 174 y Fv(i)2014 196 y FN(x)2061 166 y FM(\003)p FL(k)2130 174 y Fv(j)2166 196 y FN(u)2214 166 y FM(\003)2214 218 y FL(j)2283 196 y FO(is)-118 311 y FP(\000)p FN(u)-5 323 y FL(i)22 244 y Fy(\000)60 249 y(P)147 336 y FK(1)p FM(\024)p FL(s)p FM(\024)p FK(min\()p FL(k)487 344 y Fv(i)513 336 y FL(;k)568 344 y Fv(j)599 336 y FK(\))643 311 y FN(x)690 281 y FL(k)725 289 y Fv(i)752 281 y FM(\000)p FL(s)839 311 y FO(\()p FN(x)918 281 y FM(\003)957 311 y FO(\))989 281 y FL(k)1024 289 y Fv(j)1056 281 y FM(\000)p FL(s)1143 244 y Fy(\001)1195 311 y FN(u)1243 281 y FM(\003)1243 333 y FL(j)1295 311 y FO(+)14 b(CF)o(\()p FN(u)1567 323 y FL(i)1594 311 y FN(u)1642 281 y FM(\003)1642 333 y FL(j)1680 311 y FO(\),)26 b(where)f(CF\()p FN(u)2193 323 y FL(i)2221 311 y FN(u)2269 281 y FM(\003)2269 333 y FL(i)2306 311 y FO(\))-118 411 y(denotes)f(the)h(canonical)d(form)h(of)i FN(u)1014 423 y FL(i)1041 411 y FN(u)1089 381 y FM(\003)1089 432 y FL(i)1127 411 y FO(.)36 b(Since)24 b FN(u)1448 423 y FL(i)1500 411 y FO(and)h FN(u)1707 423 y FL(j)1766 411 y FO(do)f(not)h(end)g(with)-118 510 y FN(x)p FO(,)33 b(CF\()p FN(u)179 522 y FL(i)206 510 y FN(u)254 480 y FM(\003)254 532 y FL(j)292 510 y FO(\))e(is)g(a)f(w)n(ord)g(of)i (length)e FP(j)p FN(u)1151 522 y FL(i)1178 510 y FP(j)21 b FO(+)g FP(j)p FN(u)1379 522 y FL(j)1413 510 y FP(j)p FO(.)48 b(So)31 b(if)g(the)g(unique)g(short-)-118 610 y(est)g(w)n(ord,)h(CF\()p FN(u)443 622 y FL(i)471 610 y FN(u)519 580 y FM(\003)519 632 y FL(j)556 610 y FO(\),)h(in)e(CF\()p FN(z)t(z)977 580 y FM(\003)1014 610 y FO(\))h(has)f(minimal)c(length)k (in)f FN(z)t(z)2001 580 y FM(\003)2038 610 y FO(,)j(then)f FN(i)p FO(,)-118 710 y FN(j)40 b FP(2)c FN(J)43 b FO(\(hence)35 b FP(j)p FN(u)477 722 y FL(i)504 710 y FP(j)h FO(=)f FP(j)p FN(u)734 722 y FL(j)768 710 y FP(j)p FO(\).)60 b(Let)35 b(us)g(sho)n(w)f(that)h(if)g FN(u)1708 722 y FL(i)1735 710 y FN(u)1783 680 y FM(\003)1783 731 y FL(j)1856 710 y FO(=)g FN(w)r(w)2078 680 y FM(\003)2117 710 y FO(,)i(then)-118 809 y FN(u)-70 821 y FL(i)-11 809 y FO(=)31 b FN(u)133 821 y FL(j)167 809 y FO(.)53 b(Let)33 b FN(u)445 821 y FL(i)505 809 y FO(b)r(e)g(a)f(w)n(ord)g(in)g FN(a)p FO(,)i FN(a)1155 779 y FM(\003)1193 809 y FO(.)52 b(If)33 b FN(u)1404 821 y FL(i)1464 809 y FO(ends)g(with)f FN(a)p FO(,)j(or)c FN(u)2109 821 y FL(j)2177 809 y FO(ends)-118 909 y(with)k FN(a)123 879 y FM(\003)161 909 y FO(,)j(then)e(CF\()p FN(u)613 921 y FL(i)641 909 y FN(u)689 879 y FM(\003)689 931 y FL(j)727 909 y FO(\))g(is)e FN(u)934 921 y FL(i)962 909 y FN(u)1010 879 y FM(\003)1010 931 y FL(j)1083 909 y FO(\(as)h(in)g(the)h(free)g FP(\003)p FO(-algebra\),)e(and)i(w)n(e) -118 1009 y(conclude)i(from)h FN(u)492 1021 y FL(i)519 1009 y FN(u)567 978 y FM(\003)567 1030 y FL(j)648 1009 y FO(=)k FN(w)r(w)878 978 y FM(\003)957 1009 y FO(that)d FN(u)1197 1021 y FL(i)1267 1009 y FO(=)j FN(u)1423 1021 y FL(j)1458 1009 y FO(.)73 b(In)40 b(the)g(opp)r(osite)e(case,)-118 1120 y(write)g FN(u)154 1132 y FL(i)223 1120 y FO(=)k FN(v)370 1132 y FL(i)398 1120 y FN(a)442 1090 y FM(\003)p FL(k)556 1120 y FO(and)d FN(u)777 1132 y FL(j)854 1120 y FO(=)j FN(v)1001 1132 y FL(j)1036 1120 y FN(a)1080 1090 y FL(m)1143 1120 y FO(,)g(where)d FN(v)1500 1132 y FL(i)1567 1120 y FO(ends)g(with)g FN(a)2013 1090 y FM(\003)2090 1120 y FO(and)g FN(v)2303 1132 y FL(j)-118 1220 y FO(ends)f(with)f FN(a)p FO(.)68 b(Then)38 b(CF\()p FN(u)837 1232 y FL(i)865 1220 y FN(u)913 1190 y FM(\003)913 1241 y FL(j)950 1220 y FO(\))j(=)f FN(q)1168 1190 y FL(k)q(m)1267 1220 y FN(v)1307 1232 y FL(i)1335 1220 y FN(a)1379 1190 y FL(m)1442 1220 y FN(a)1486 1190 y FM(\003)p FL(k)1561 1220 y FN(v)1604 1190 y FM(\003)1601 1241 y FL(j)1642 1220 y FO(.)68 b(If)39 b FN(m)h(>)g(k)s FO(,)g(then,)-118 1332 y(since)c FN(u)143 1344 y FL(i)170 1332 y FN(u)218 1301 y FM(\003)218 1353 y FL(j)295 1332 y FO(=)j FN(w)r(w)521 1301 y FM(\003)561 1332 y FO(,)h(w)n(e)d(ha)n(v)n(e)f(that)i FN(v)1187 1344 y FL(i)1215 1332 y FN(a)1259 1301 y FL(m)1318 1309 y Fx(1)1393 1332 y FO(=)h FN(w)r FO(,)i FN(a)1666 1301 y FL(m)p FM(\000)p FL(m)1836 1309 y Fx(1)1872 1332 y FN(a)1916 1301 y FM(\003)p FL(k)1991 1332 y FN(v)2034 1301 y FM(\003)2031 1353 y FL(j)2112 1332 y FO(=)e FN(w)2277 1301 y FM(\003)2316 1332 y FO(,)-118 1431 y(whic)n(h)28 b(is)g(imp)r(ossible,)d(since)i FN(w)32 b FO(ends)d(with)f FN(a)h FO(and)g FN(a)1591 1401 y FM(\003)1658 1431 y FO(sim)n(ultaneously)-7 b(.)36 b(So)-118 1531 y FN(m)e FO(=)f FN(k)k FO(and)d FN(w)i FO(=)e FN(v)569 1543 y FL(i)597 1531 y FN(a)641 1501 y FL(k)715 1531 y FO(=)g FN(v)854 1543 y FL(j)889 1531 y FN(a)933 1501 y FL(k)973 1531 y FO(.)57 b(Hence,)36 b FN(v)1371 1543 y FL(i)1432 1531 y FO(=)e FN(v)1571 1543 y FL(j)1640 1531 y FO(and)g FN(u)1856 1543 y FL(i)1917 1531 y FO(=)f FN(u)2063 1543 y FL(j)2098 1531 y FO(.)56 b(No)n(w)-118 1630 y(let)30 b FN(u)53 1642 y FL(k)123 1630 y FO(=)f FN(u)265 1642 y FL(k)q(;)p FK(1)358 1630 y FN(x)405 1600 y FK(#)464 1630 y FN(u)512 1642 y FL(k)q(;)p FK(2)605 1630 y FO(,)k FN(k)f FO(=)c FN(i)p FO(,)k FN(j)5 b FO(,)33 b(where)d FN(u)1299 1642 y FL(k)q(;)p FK(1)1424 1630 y FO(do)r(es)h(not)g(con)n (tain)f FN(x)2110 1600 y FK(#)2200 1630 y FO(\()p FN(x)2279 1600 y FK(#)-118 1730 y FO(stands)j(for)g(either)g FN(x)h FO(or)f FN(x)758 1700 y FM(\003)796 1730 y FO(\).)56 b(Then)33 b(it)h(follo)n(ws)c(from)i FN(u)1747 1742 y FL(i)1775 1730 y FN(u)1823 1700 y FM(\003)1823 1752 y FL(j)1893 1730 y FO(=)h FN(w)r(w)2113 1700 y FM(\003)2186 1730 y FO(that)-118 1830 y FN(u)-70 1842 y FK(1)p FL(;)p FK(1)48 1830 y FO(=)28 b FN(u)189 1842 y FK(2)p FL(;)p FK(1)310 1830 y FO(and)i FN(u)522 1842 y FK(1)p FL(;)p FK(2)612 1830 y FN(u)660 1800 y FM(\003)660 1850 y FK(2)p FL(;)p FK(2)778 1830 y FO(=)e FN(w)930 1842 y FK(1)968 1830 y FN(w)1029 1800 y FM(\003)1027 1850 y FK(1)1067 1830 y FO(.)47 b(By)31 b(induction)e(on)i FN(l)r FO(\()p FN(u)1869 1842 y FL(i)1895 1830 y FO(\),)h(w)n(e)f(obtain)-118 1929 y(the)h(desired)f(result.)48 b(This)31 b(pro)n(v)n(es)f(that,)j (if)f FN(u)1384 1941 y FL(i)1411 1929 y FN(u)1459 1899 y FM(\003)1459 1951 y FL(j)1527 1929 y FO(=)e FN(w)r(w)1744 1899 y FM(\003)1783 1929 y FO(,)j(then)g FN(u)2081 1941 y FL(i)2138 1929 y FO(=)d FN(u)2281 1941 y FL(j)2316 1929 y FO(.)-118 2038 y(F)-7 b(rom)28 b(this)g(it)h(follo)n(ws)d(that)j FN(F)12 b FO(\()p FN(z)t(z)988 2008 y FM(\003)1025 2038 y FO(\))26 b(=)1174 1976 y Fy(P)1261 2063 y FL(j)s FM(2)p FL(J)1397 2038 y FP(j)p FN(\013)1473 2050 y FL(j)1508 2038 y FP(j)1531 2008 y FK(2)1569 2038 y FO(.)41 b(Using)28 b(the)i(existence)-118 2138 y(of)i FN(F)12 b FO(\()p FP(\001)p FO(\),)34 b(it)e(is)f(easy)h(to)g(sho)n(w)f(that)i(the)g FP(\003)p FO(-algebra)28 b(is)j(completely)f(prop)r(er.)-118 2237 y(But)c(in)g(ev)n(ery)f(represen)n(tation,)f(w)n(e)i(ha)n(v)n(e)f (that)h FP(k)p FN(aa)1543 2207 y FM(\003)1581 2237 y FP(k)c FO(=)h FP(k)p FN(a)1819 2207 y FM(\003)1857 2237 y FN(a)p FP(k)f FO(=)h FN(q)17 b FP(k)p FN(aa)2237 2207 y FM(\003)2274 2237 y FP(k)p FO(,)-118 2337 y(hence)28 b FP(k)p FN(aa)243 2307 y FM(\003)280 2337 y FP(k)23 b FO(=)f(0,)27 b(and)h(so)f FN(A)h FO(is)e(not)i FN(C)1174 2307 y FM(\003)1212 2337 y FO(-represen)n(table.)6 2442 y(T)-7 b(o)26 b(see)g(that)g(\()p FN(iv)s FO(\))g(do)r(es)g(not)g (imply)d(\()p FN(iii)p FO(\),)j(w)n(e)g(refer)f(the)i(reader)d(to)i ([283)o(])-118 2542 y(\(see)h(also)f([71)o(])i(and)f(the)h(references)f (therein\).)p 2278 2542 4 57 v 2282 2489 50 4 v 2282 2542 V 2331 2542 4 57 v 6 2750 a FQ(3.)34 b FO(It)19 b(is)g(natural)e(to)i(consider,)g(among)f(all)f FP(\003)p FO(-algebras)e(whic)n(h)j(ha)n(v)n(e)h(a)g(r.f.,)-118 2850 y(those)30 b(whic)n(h)g(p)r(osses)g(a)g(residual)e(family)f(of)k (\014nite-dimensional)25 b(represen-)-118 2949 y(tations;)h(w)n(e)h (call)f(these)h(algebras)e(residually)e(\014nite)28 b(dimensional)23 b(\(r.f.d.\).)-118 3131 y FQ(Prop)s(osition)30 b(6.)41 b FB(If)32 b FN(G)h FB(is)f(a)h(r)l(esidual)t(ly)h(\014nite)d(gr)l(oup) i FO(\()p FB(i.e.)47 b FP(8)p FN(g)29 b FP(6)p FO(=)e FN(e)k FB(ther)l(e)-118 3231 y(exists)f(a)i(normal)f(sub)l(gr)l(oup)g FN(G)877 3243 y FL(g)941 3231 y FP(63)26 b FN(g)33 b FB(such)e(that)g FN(G=G)1628 3243 y FL(g)1698 3231 y FB(is)g(a)h(\014nite)e(gr)l(oup)5 b FO(\))p FB(,)-118 3331 y(then)29 b FI(C)15 b FO([)p FN(G)q FO(])36 b FB(is)30 b(r)l(esidual)t(ly)h(\014nite)e(dimensional.)-118 3513 y(Pr)l(o)l(of.)43 b FO(Let)19 b(us)f(\014rst)h(recall)c(an)k(equiv)-5 b(alen)n(t)16 b(de\014nition)h(of)i(the)g(residual)c(\014nite-)-118 3612 y(ness)37 b(of)h(a)g(group)f FN(G)p FO(:)58 b(for)37 b(an)n(y)h(\014nite)f(set)h FP(f)p FN(g)1394 3624 y FK(1)1430 3612 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(g)1669 3624 y FL(n)1714 3612 y FP(g)37 b FO(of)h(non-iden)n(tit)n(y)-118 3712 y(elemen)n(ts)25 b(of)i FN(G)h FO(there)f(exists)f(a)h(normal)d (subgroup)i(that)i(do)r(es)f(not)g(con)n(tain)-118 3811 y(an)n(y)e(of)h(these)h(elemen)n(ts)d(and)i(suc)n(h)g(that)g(the)h (quotien)n(t)e(group)g(of)h FN(G)g FO(b)n(y)g(this)-118 3911 y(subgroup)g(is)h(\014nite.)p eop %%Page: 14 18 14 17 bop -118 -137 a FO(14)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(Let)34 b FN(\013)f FO(=)344 34 y Fy(P)432 121 y FL(k)486 96 y FN(c)522 108 y FL(k)563 96 y FN(g)603 108 y FL(k)644 96 y FO(,)h FN(c)737 108 y FL(k)811 96 y FP(6)p FO(=)e(0,)i FN(k)i FO(=)c(1,)h FN(:)14 b(:)g(:)27 b FO(,)35 b FN(n)p FO(,)g(b)r(e)f(an)f(elemen)n(t)e(of)i(the)-118 196 y(group)20 b(algebra.)32 b(Cho)r(osing)19 b(a)i(normal)d(subgroup)i FN(N)31 b FO(that)21 b(do)r(es)g(not)g(con)n(tain)-118 296 y(the)e(elemen)n(ts)d FN(g)386 308 y FL(i)413 296 y FN(g)456 260 y FM(\000)p FK(1)453 319 y FL(j)545 296 y FO(,)k FN(i)j FP(6)p FO(=)g FN(j)5 b FO(;)21 b FN(i)p FO(,)f FN(j)28 b FO(=)23 b(1,)18 b FN(:)c(:)g(:)27 b FO(,)21 b FN(n)p FO(,)f(w)n(e)e(conclude)f(that)i(the)f(image)-118 395 y(of)27 b FN(\013)d FP(2)f FI(C)15 b FO([)p FN(G)p FO(])34 b(under)27 b(the)g(homomorphism)22 b FI(C)15 b FO([)p FN(G)q FO(])29 b FP(7!)23 b FI(C)15 b FO([)p FN(G=)-5 b(N)9 b FO(])33 b(of)27 b FI(C)15 b FO([)p FN(G)q FO(])33 b(in)n(to)-118 495 y(the)24 b(group)f(algebra)e(of)j(\014nite)g (group)f(is)g(non-zero)f(and)i(hence)g(it)f(is)g(non-zero)-118 595 y(in)k(the)h(\014nite-dimensional)22 b(regular)j(represen)n(tation) g(of)i FI(C)15 b FO([)p FN(G)q(=)-5 b(N)9 b FO(].)p 2278 595 4 57 v 2282 542 50 4 v 2282 595 V 2331 595 4 57 v -118 758 a FQ(4.)55 b FO(The)34 b(follo)n(wing)c(list)j(of)h FN(C)858 728 y FM(\003)896 758 y FO(-algebras)d(is)i(naturally)e (connected)j(with)f(a)-118 857 y(group)26 b FN(G)p FO(:)-17 1013 y(1.)41 b FN(C)154 982 y FM(\003)148 1036 y FL(f)193 1013 y FO(\()p FN(G)p FO(\).)c(This)23 b(algebra)e(is)i(the)i (completion)c(of)j FI(C)15 b FO([)p FN(G)p FO(])31 b(in)23 b(the)i(follo)n(wing)89 1112 y(norm:)335 1281 y FP(k)p FN(\013)p FP(k)472 1296 y FL(C)524 1276 y Fw(\003)520 1316 y Fv(f)557 1296 y FK(\()p FL(G)p FK(\))688 1281 y FO(=)210 b(sup)776 1355 y FL(\031)r FM(2\003)p FK(-f)5 b FL(:)p FK(d)p FL(:)p FK(Rep)k Fu(C)t FK([)p FL(G)p FK(])1290 1281 y FP(k)p FN(\031)s FO(\()p FN(\013)p FO(\))p FP(k)p FN(;)180 b(\013)24 b FP(2)f FI(C)15 b FO([)p FN(G)p FO(])p FN(;)89 1513 y FO(where)23 b(sup)g(is)f(tak)n(en)g(o)n(v)n(er)f (all)g(\014nite-dimensional)c FP(\003)p FO(-represen)n(tations)89 1612 y(of)28 b FI(C)15 b FO([)p FN(G)p FO(].)-17 1773 y(2.)41 b FN(C)154 1743 y FM(\003)148 1793 y FL(r)193 1773 y FO(\()p FN(G)p FO(\).)67 b(This)37 b(algebra)d(is)j(the)g (completion)e(of)i FI(C)15 b FO([)p FN(G)p FO(])44 b(in)37 b(the)g(righ)n(t)89 1872 y(regular)25 b(norm)577 2041 y FP(k)p FN(\013)p FP(k)714 2056 y FL(C)762 2064 y Fv(r)795 2056 y FK(\()p FL(G)p FK(\))926 2041 y FO(=)d FP(k)p FN(\031)1102 2053 y FL(r)1139 2041 y FO(\()p FN(\013)p FO(\))p FP(k)p FN(;)180 b(\013)24 b FP(2)f FI(C)15 b FO([)p FN(G)q FO(])p FN(;)89 2210 y FO(where)26 b FN(\031)375 2222 y FL(r)412 2210 y FO(\()p FN(\013)p FO(\))g(is)f(the)h(op)r (erator)e(of)i(the)g(righ)n(t)e(regular)e(represen)n(tation)89 2310 y(of)28 b FN(G)g FO(on)f FN(L)449 2322 y FK(2)486 2310 y FO(\()p FN(G)p FO(\).)-17 2470 y(3.)41 b FN(C)154 2440 y FM(\003)193 2470 y FO(\()p FN(G)p FO(\).)c(This)23 b(algebra)e(is)i(the)i(completion)c(of)j FI(C)15 b FO([)p FN(G)p FO(])31 b(in)23 b(the)i(follo)n(wing)89 2570 y(norm:)385 2739 y FP(k)p FN(\013)p FP(k)522 2754 y FL(C)574 2737 y Fw(\003)608 2754 y FK(\()p FL(G)p FK(\))739 2739 y FO(=)159 b(sup)826 2812 y FL(\031)r FM(2\003)p FK(-Rep)11 b Fu(C)t FK([)p FL(G)p FK(])1239 2739 y FP(k)p FN(\031)s FO(\()p FN(\013)p FO(\))p FP(k)p FN(;)181 b(\013)23 b FP(2)g FI(C)15 b FO([)p FN(G)q FO(])p FN(:)-118 3006 y FO(The)35 b FP(\003)p FO(-algebra)c FI(C)15 b FO([)p FN(G)p FO(])41 b(is)34 b FP(\003)p FO(-b)r(ounded,)i(i.e.,)g FP(k)p FN(\031)s FO(\()p FN(\013)p FO(\))p FP(k)f(\024)g FN(C)1775 3018 y FL(\013)1858 3006 y FN(<)f FP(1)h FO(for)f(an)n(y)-118 3105 y FN(\013)f FP(2)g FI(C)15 b FO([)p FN(G)p FO(],)41 b(and)34 b(an)n(y)e FN(\031)k FP(2)d(\003)p FO(-Rep)13 b FI(C)i FO([)p FN(G)q FO(])39 b(\(for)33 b(more)f(detailed)f (information)-118 3205 y(ab)r(out)e FP(\003)p FO(-b)r(oundedness)g(see) g(1.1.3\).)42 b(Hence)30 b(all)d(norms)h(de\014ned)h(ab)r(o)n(v)n(e)g (are)-118 3304 y(\014nite)h(for)h(an)n(y)f FN(\013)e FP(2)h FI(C)15 b FO([)p FN(G)q FO(].)52 b(The)31 b(follo)n(wing)c(prop) r(osition)h(can)i(b)r(e)h(found)g(in)-118 3404 y([56)o(].)-118 3559 y FQ(Prop)s(osition)f(7.)41 b FB(If)30 b FN(G)g FB(is)g(a)g(r)l(esidual)t(ly)i(\014nite)d(gr)l(oup,)i(then)71 3728 y FP(k)p FN(\013)p FP(k)208 3743 y FL(L)254 3751 y Fx(1)285 3743 y FK(\()p FL(G)p FK(\))416 3728 y FP(\025)23 b(k)p FN(\013)p FP(k)641 3743 y FL(C)693 3726 y Fw(\003)727 3743 y FK(\()p FL(G)p FK(\))857 3728 y FP(\025)g(k)p FN(\013)p FP(k)1082 3743 y FL(C)1134 3723 y Fw(\003)1130 3763 y Fv(f)1168 3743 y FK(\()p FL(G)p FK(\))1299 3728 y FP(\025)f(k)p FN(\013)p FP(k)1523 3743 y FL(C)1575 3726 y Fw(\003)1571 3760 y Fv(r)1609 3743 y FK(\()p FL(G)p FK(\))1740 3728 y FP(\025)g(k)p FN(\013)p FP(k)1964 3743 y FL(L)2010 3751 y Fx(2)2042 3743 y FK(\()p FL(G)p FK(\))-118 3911 y FB(for)30 b(any)h FN(\013)23 b FP(2)h FI(C)14 b FO([)q FN(G)p FO(])p FB(.)p eop %%Page: 15 19 15 18 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(15)-118 96 y FB(Pr)l(o)l(of.)43 b FO(The)28 b(only)e(non-trivial)d (part)k(is)g(the)h(pro)r(of)f(of)g(the)h(inequalit)n(y)425 276 y FP(k)p FN(\013)p FP(k)562 291 y FL(C)614 271 y Fw(\003)610 312 y Fv(f)648 291 y FK(\()p FL(G)p FK(\))779 276 y FP(\025)22 b(k)p FN(\013)p FP(k)1003 291 y FL(C)1055 275 y Fw(\003)1051 308 y Fv(r)1089 291 y FK(\()p FL(G)p FK(\))1197 276 y FN(;)180 b FP(8)p FN(\013)23 b FP(2)g FI(C)15 b FO([)p FN(G)p FO(])p FN(:)-118 475 y FO(Let)20 b FN(\013)k FO(=)187 413 y Fy(P)275 433 y FL(m)275 500 y(k)q FK(=1)414 475 y FN(c)450 487 y FL(k)491 475 y FN(g)531 487 y FL(k)594 475 y FP(6)p FO(=)f(0,)e(then)g FP(k)p FN(\013)p FP(k)1087 490 y FL(C)1139 474 y Fw(\003)1135 507 y Fv(r)1173 490 y FK(\()p FL(G)p FK(\))1304 475 y FO(=)h FN(d)h(>)g FO(0,)e(and)g(there)f(exists)f FN(\015)27 b FP(2)-118 575 y FN(L)-61 587 y FK(2)-24 575 y FO(\()p FN(G)p FO(\))20 b(suc)n(h)f(that)h FP(k)p FN(\015)5 b FP(k)608 590 y FL(L)654 598 y Fx(2)685 590 y FK(\()p FL(G)p FK(\))816 575 y FO(=)22 b(1)e(and)f FP(k)p FN(\013\015)5 b FP(k)1303 590 y FL(L)1349 598 y Fx(2)1380 590 y FK(\()p FL(G)p FK(\))1511 575 y FP(\025)23 b FN(d)s FP(\000)s FN("=)p FO(2.)31 b(Hence,)22 b(there)-118 674 y(exists)g FN(\016)k FO(=)258 612 y Fy(P)346 633 y FL(n)346 699 y FK(1)405 674 y FN(\016)442 686 y FL(k)482 674 y FN(h)530 686 y FL(k)595 674 y FO(suc)n(h)e(that)g FP(k)p FN(\016)s FP(k)1079 689 y FL(L)1125 697 y Fx(2)1156 689 y FK(\()p FL(G)p FK(\))1286 674 y FO(=)f(1)h(and)f FP(k)p FN(\015)15 b FP(\000)c FN(\016)s FP(k)1855 689 y FL(L)1901 697 y Fx(2)1932 689 y FK(\()p FL(G)p FK(\))2063 674 y FP(\024)23 b FN("=)p FO(2)p FN(d)o FO(.)-118 774 y(This)g(implies)e FP(k)p FN(\013\015)c FP(\000)12 b FN(\013\016)s FP(k)714 789 y FL(L)760 797 y Fx(2)792 789 y FK(\()p FL(G)p FK(\))923 774 y FP(\024)22 b FN("=)p FO(2,)i(and)h FP(k)p FN(\013\016)s FP(k)1516 789 y FL(L)1562 797 y Fx(2)1593 789 y FK(\()p FL(G)p FK(\))1724 774 y FP(\025)e FN(d)12 b FP(\000)g FN(")p FO(.)36 b(F)-7 b(urther,)-118 874 y(let)30 b(us)h(c)n(ho)r(ose)e (a)i(normal)c(subgroup)j FN(N)40 b FO(of)30 b(the)i(group)d FN(G)i FO(whic)n(h)f(do)r(es)h(not)-118 973 y(con)n(tain)k(non-trivial) e(elemen)n(ts)i(among)g FN(g)1268 985 y FL(k)1308 973 y FN(g)1351 938 y FM(\000)p FK(1)1348 998 y FL(l)1440 973 y FO(,)k FN(h)1550 985 y FL(t)1579 973 y FN(h)1627 943 y FM(\000)p FK(1)1627 994 y FL(s)1716 973 y FO(,)h FN(g)1819 985 y FL(k)1859 973 y FN(h)1907 985 y FL(t)1936 973 y FN(h)1984 943 y FM(\000)p FK(1)1984 994 y FL(s)2073 973 y FN(g)2116 938 y FM(\000)p FK(1)2113 998 y FL(l)2205 973 y FO(;)i FN(k)s FO(,)-118 1073 y FN(l)27 b FO(=)e(1,)j FN(:)14 b(:)g(:)28 b FO(,)h FN(m)p FO(,)g FN(t)p FO(,)h FN(s)25 b FO(=)g(1,)k FN(:)14 b(:)g(:)27 b FO(,)j FN(n)p FO(,)f(and)g(suc)n(h)f(that)i(the)f(quotien)n(t)f(group)g(of)-118 1173 y FN(G)i FO(b)n(y)f(this)g(subgroup)g(is)f(\014nite.)43 b(Then,)30 b(in)f(the)h(regular)d(represen)n(tation)g(of)-118 1272 y FN(G=)-5 b(N)9 b FO(,)28 b(w)n(e)f(ha)n(v)n(e)490 1452 y FP(k)p FN(\013\016)s FP(k)667 1467 y FL(L)713 1475 y Fx(2)744 1467 y FK(\()p FL(G=)l(N)6 b FK(\))964 1452 y FO(=)23 b FP(k)p FN(\013\016)s FP(k)1229 1467 y FL(L)1275 1475 y Fx(2)1306 1467 y FK(\()p FL(G)p FK(\))1437 1452 y FP(\025)f FN(d)d FP(\000)f FN(":)-118 1632 y FO(Hence,)31 b(for)f(an)n(y)f FN(")h FO(w)n(e)g(ha)n(v)n(e)f FP(k)p FN(\013)p FP(k)969 1647 y FL(C)1021 1627 y Fw(\003)1017 1667 y Fv(f)1055 1647 y FK(\()p FL(G)p FK(\))1190 1632 y FP(\025)e FN(d)21 b FP(\000)e FN(")p FO(.)45 b(Therefore)29 b FP(k)p FN(\013)p FP(k)2053 1647 y FL(C)2105 1627 y Fw(\003)2101 1667 y Fv(f)2139 1647 y FK(\()p FL(G)p FK(\))2274 1632 y FP(\025)-118 1746 y(k)p FN(\013)p FP(k)19 1761 y FL(C)71 1744 y Fw(\003)67 1778 y Fv(r)105 1761 y FK(\()p FL(G)p FK(\))212 1746 y FO(.)p 2278 1746 4 57 v 2282 1693 50 4 v 2282 1746 V 2331 1746 4 57 v -118 1912 a FB(R)l(emark)h(3.)42 b FO(If)26 b FN(G)g FO(is)e(a)i(residually)21 b(\014nite)k(group,)g(then)h(w)n(e)f(ha)n(v)n(e)g(the)h(follo)n(w-)-118 2011 y(ing)g(sequence)h(of)h FP(\003)p FO(-homomorphisms)620 2191 y FN(C)685 2157 y FM(\003)723 2191 y FO(\()p FN(G)p FO(\))c FP(7!)f FN(C)1047 2157 y FM(\003)1041 2212 y FL(f)1086 2191 y FO(\()p FN(G)p FO(\))g FP(7!)g FN(C)1409 2157 y FM(\003)1403 2212 y FL(r)1448 2191 y FO(\()p FN(G)p FO(\))p FN(;)-118 2371 y FO(whic)n(h)f(are)g(iden)n(tical)e(on)j(the)g (dense)g FP(\003)p FO(-subalgebra)c FI(C)c FO([)p FN(G)q FO(].)41 b(Let)23 b(us)g(note)g(that)-118 2471 y(these)28 b(homomorphisms)22 b(are)k(epimorphisms.)6 2571 y(Ho)n(w)n(ev)n(er,)j (in)g(general,)f(neither)h FN(C)1127 2540 y FM(\003)1165 2571 y FO(\()p FN(G)p FO(\))i(nor)e FN(C)1540 2540 y FM(\003)1534 2591 y FL(r)1579 2571 y FO(\()p FN(G)p FO(\))h(is)f(a)h (r.f.d.)g FP(\003)p FO(-alge-)-118 2670 y(bra.)k(Indeed,)24 b FN(C)414 2640 y FM(\003)453 2670 y FO(\()p FA(F)541 2682 y FK(2)576 2670 y FO(\),)g(where)e FA(F)946 2682 y FK(2)1004 2670 y FO(is)f(the)i(free)g(group)e(with)h(t)n(w)n(o)g (generators,)-118 2770 y(is)33 b(a)h(r.f.d.)g FP(\003)p FO(-algebra)d([55)o(];)38 b(ho)n(w)n(ev)n(er,)c FN(C)1227 2740 y FM(\003)1221 2790 y FL(r)1265 2770 y FO(\()p FA(F)1353 2782 y FK(2)1389 2770 y FO(\))g(is)g(not)g(a)g(r.f.d.)g FP(\003)p FO(-algebra,)-118 2869 y(since)27 b(it)g(is)g(simple)f([207)n (].)39 b(The)28 b(residually)c(\014nite)k(group)f FN(S)5 b(L)p FO(\(2)p FN(;)14 b FI(Z)n FO([)2059 2837 y FK(1)p 2053 2851 35 4 v 2053 2898 a FL(p)2097 2869 y FO(]\))29 b(\()p FN(p)f FO(is)-118 2980 y(a)e(prime)f(n)n(um)n(b)r(er\))h(is)f (an)i(example)d(of)j(a)f(r.f.)h(group)f FN(G)h FO(for)f(whic)n(h)g FN(C)2088 2950 y FM(\003)2126 2980 y FO(\()p FN(G)p FO(\))i(is)-118 3080 y(not)f(r.f.d.)h(see)g([105)n(].)-118 3295 y FQ(1.1.3)94 b(En)m(v)m(eloping)31 b FP(\003)p FQ(-algebras)g(and)h FN(C)1360 3265 y FM(\003)1398 3295 y FQ(-algebras)-118 3448 y(1.)54 b FO(Sometimes)31 b(it)i(is)f(p)r(ossible)g(to)h(reduce)g (the)h(study)g(of)g(represen)n(tations)-118 3548 y(of)d(a)g FP(\003)p FO(-algebra)d Fz(A)j FO(to)h(the)f(study)h(of)f FP(\003)p FO(-represen)n(tations)d(of)j(its)g(en)n(v)n(eloping)-118 3647 y FP(\003)p FO(-algebra,)24 b FN(\033)s FO(-)p FN(C)408 3617 y FM(\003)446 3647 y FO(-algebra,)h(or)i FN(C)955 3617 y FM(\003)993 3647 y FO(-algebra.)34 b(Let)28 b(us)f(recall)e(the) j(de\014nition.)-118 3811 y FQ(De\014nition)j(1.)40 b FB(L)l(et)34 b Fz(A)26 b FB(b)l(e)g(a)h FP(\003)p FB(-algebr)l(a.)38 b(The)27 b(p)l(air)h FO(\()1604 3790 y(~)1595 3811 y Fz(A)p FO(;)14 b FN(\036)9 b FO(:)28 b Fz(A)23 b FP(7!)1999 3790 y FO(~)1990 3811 y Fz(A)o FO(\))p FB(,)28 b(wher)l(e)-109 3889 y FO(~)-118 3911 y Fz(A)e FB(is)g(a)h FP(\003)p FB(-algebr)l(a)g(and)f FN(\036)h FB(is)g(a)f FP(\003)p FB(-homomorphism,)j(is)e(c)l(al)t(le)l(d)h(an)e(enveloping)p eop %%Page: 16 20 16 19 bop -118 -137 a FO(16)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FP(\003)p FB(-algebr)l(a)32 b(of)g(the)f(algebr)l(a)h Fz(A)f FB(if)h(for)g(any)g FP(\003)p FB(-r)l(epr)l(esentation)e FN(\031)12 b FO(:)29 b Fz(A)c FP(7!)h FN(L)p FO(\()p FN(H)7 b FO(\))-118 196 y FB(of)27 b(the)g(algebr)l(a)g Fz(A)f FB(ther)l(e)g(exists)g(a)h(unique)f FP(\003)p FB(-r)l(epr)l(esentation)k FO(~)-46 b FN(\031)12 b FO(:)1962 174 y(~)1953 196 y Fz(A)23 b FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\))-118 296 y FB(such)30 b(that)f(the)h(fol)t(lowing)j(diagr)l(am) e(is)f(c)l(ommutative.)874 829 y FO(~)865 851 y Fz(A)286 b FN(L)p FO(\()p FN(H)7 b FO(\))p 949 824 239 4 v 1104 822 a Ft(-)1047 894 y FO(~)-47 b FN(\031)1154 633 y(\031)953 549 y Ft(@)1036 632 y(@)1119 715 y(@)1169 764 y(@)-83 b(R)865 436 y Fz(A)p 893 764 4 299 v 895 764 a Ft(?)812 636 y FN(\036)-118 1019 y FB(Example)31 b(1.)42 b FO(Let)27 b(\006)h(b)r(e)f(an)n(y)g(family)d(of)j(elemen)n(ts)e(of)i(a)g FP(\003)p FO(-algebra)d Fz(A)j FO(whic)n(h)-118 1118 y(are)i(in)n(v)n(ertible)d(in)k(an)n(y)f FP(\003)p FO(-represen)n (tation)e FN(\031)12 b FO(:)29 b Fz(A)e FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\).)44 b(Let)31 b(us)f(denote)-118 1218 y(the)f(algebra)e(of)i(quotien)n(ts)e(of)i Fz(A)g FO(with)g(resp)r(ect)g(to)g(\006)g(b)n(y)1753 1196 y(~)1747 1218 y Fz(U)c FO(=)g Fz(A)p FO([\006)2059 1188 y FM(\000)p FK(1)2148 1218 y FO(])k(\(see)-118 1318 y([87)o(]\).)36 b(Let)24 b FN(\036)9 b FO(:)29 b Fz(A)22 b FP(7!)551 1296 y FO(~)545 1318 y Fz(U)i FO(b)r(e)g(the)h(natural)d(homomorphism.) 30 b(Then)24 b(w)n(e)g(obtain)-118 1417 y(an)i(en)n(v)n(eloping)e FP(\003)p FO(-algebra)f(for)k(the)g(the)g FP(\003)p FO(-algebra)c Fz(A)p FO(.)37 b(Let)27 b(us)f(note)h(that)g(in)-118 1517 y(the)h(case)f(where)g Fz(A)g FO(is)f FN(C)676 1487 y FM(\003)715 1517 y FO(-represen)n(table,)f FN(\036)j FO(is)e(an)h(injection.)-118 1646 y FQ(2.)36 b FO(If)85 1624 y(~)76 1646 y Fz(A)27 b FO(carries)d(the)k(structure)e(of)i(a)f FN(C)1153 1616 y FM(\003)1191 1646 y FO(-algebra,)d(then)32 b(~)-46 b FN(\031)31 b FO(is)26 b(a)h(con)n(tin)n(uous)-118 1746 y FP(\003)p FO(-homomorphism)j(from)k(the)j FN(C)967 1716 y FM(\003)1005 1746 y FO(-algebra)1341 1724 y(~)1332 1746 y Fz(A)e FO(to)h(the)g FN(C)1753 1716 y FM(\003)1792 1746 y FO(-algebra)d FN(L)p FO(\()p FN(H)7 b FO(\).)-118 1845 y(In)40 b(this)f(case)f(the)i(pair)e(\()738 1824 y(~)729 1845 y Fz(A)p FN(;)14 b(\036)p FO(\))40 b(is)f(called)e(an)j (en)n(v)n(eloping)c FN(C)1903 1815 y FM(\003)1941 1845 y FO(-algebra)h(of)-118 1945 y(the)32 b(algebra)c Fz(A)p FO(.)48 b(The)31 b(en)n(v)n(eloping)d FN(C)1105 1915 y FM(\003)1143 1945 y FO(-algebra)g(is)j(unique)f(in)h(the)g(class)f (of)-118 2045 y FN(C)-53 2015 y FM(\003)-15 2045 y FO(-algebras,)h(if)g (it)h(exists)f(\(see)i(Theorem)d(1\).)51 b(Indeed,)34 b(in)e(this)f(case)2200 2023 y(~)2191 2045 y Fz(A)h FO(is)-118 2144 y(an)27 b(en)n(v)n(eloping)e FN(\033)s FO(-)p FN(C)549 2114 y FM(\003)587 2144 y FO(-algebra)g(whic)n(h)i(is)g(unique)g(b)n(y) g(Theorem)f(1.)37 b(W)-7 b(e)28 b(will)-118 2244 y(denote)f(the)h(en)n (v)n(eloping)d FN(C)766 2214 y FM(\003)804 2244 y FO(-algebra)g(of)i (algebra)e Fz(A)i FO(b)n(y)g FN(C)1775 2214 y FM(\003)1814 2244 y FO(\()p Fz(A)p FO(\).)6 2344 y(Let)35 b Fz(A)f FO(=)g FI(C)15 b FP(h)p FN(x)489 2356 y FK(1)532 2344 y FN(;)f(:)g(:)g(:)g(;)g(x)764 2356 y FL(n)809 2344 y FN(;)g(x)893 2313 y FM(\003)893 2364 y FK(1)932 2344 y FN(;)g(:)g(:)g(:)f(;)h(x)1163 2313 y FM(\003)1163 2364 y FL(n)1243 2344 y FP(j)35 b FN(P)1354 2356 y FL(k)1395 2344 y FO(\()p FP(\001)p FO(\))g(=)f(0)p FN(;)48 b(k)37 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)p FP(i)35 b FO(=)-118 2443 y FI(C)15 b FP(h)p FN(x)16 2455 y FK(1)59 2443 y FN(;)f(:)g(:)g(:)27 b(;)14 b(x)304 2455 y FL(n)350 2443 y FN(;)g(x)434 2413 y FM(\003)434 2464 y FK(1)472 2443 y FN(;)g(:)g(:)g(:)28 b(;)14 b(x)718 2413 y FM(\003)718 2464 y FL(n)798 2443 y FP(j)35 b FN(J)8 b FP(i)35 b FO(denote)f(the)h FP(\003)p FO(-algebra)c(with)j(generators)-118 2543 y FN(x)-71 2555 y FL(j)-36 2543 y FO(,)41 b FN(x)75 2513 y FM(\003)75 2564 y FL(j)114 2543 y FO(,)g FN(j)46 b FO(=)41 b(1,)d FN(:)14 b(:)g(:)27 b FO(,)41 b FN(m)p FO(,)h(and)c(relations)d FN(P)1366 2555 y FL(k)1407 2543 y FO(\()p FN(x)1486 2555 y FK(1)1524 2543 y FN(;)14 b(:)g(:)g(:)g(;)g (x)1756 2555 y FL(n)1801 2543 y FN(;)g(x)1885 2513 y FM(\003)1885 2563 y FK(1)1924 2543 y FN(;)g(:)g(:)g(:)f(;)h(x)2155 2513 y FM(\003)2155 2563 y FL(n)2201 2543 y FO(\))41 b(=)-118 2642 y(0,)i(\(where)e FN(P)329 2654 y FL(k)370 2642 y FO(\()p FP(\001)p FO(\))g(are)f(non-comm)n(utativ)n(e)c(p)r (olynomials\),)j(i.e.,)44 b Fz(A)c FO(is)f(the)-118 2742 y(quotien)n(t)28 b(of)i(the)f(free)h FP(\003)p FO(-algebra)25 b FI(C)15 b FP(h)q FN(x)1109 2754 y FK(1)1152 2742 y FN(;)f(:)g(:)g(:)g(;)g(x)1384 2754 y FL(n)1430 2742 y FN(;)g(x)1514 2712 y FM(\003)1514 2763 y FK(1)1552 2742 y FN(;)g(:)g(:)g(:)g(;)g(x)1784 2712 y FM(\003)1784 2763 y FL(n)1829 2742 y FP(i)30 b FO(with)f(resp)r(ect)-118 2842 y(to)e(the)h(t)n(w)n(o-sided)d FP(\003)p FO(-ideal)f FN(J)36 b FO(generated)26 b(b)n(y)h(the)h(relations.)34 b(In)27 b(the)h(sequel,)-118 2941 y(w)n(e)22 b(will)e(sometimes)f(omit) i(the)i(sym)n(b)r(ols)d FN(x)1234 2911 y FM(\003)1234 2962 y FK(1)1273 2941 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(x)1518 2911 y FM(\003)1518 2962 y FL(n)1564 2941 y FO(,)23 b(if)g(it)f(is)f(clear)g(from)g(the)-118 3041 y(con)n(text,)27 b(that)h(the)g(considered)e(algebra)e(is)j(a)g FP(\003)p FO(-algebra.)6 3141 y(If)32 b(a)e FP(\003)p FO(-algebra)d Fz(A)p FO(,)32 b(generated)d(b)n(y)i(generators)d FN(x)1595 3153 y FK(1)1633 3141 y FO(,)j FN(:)14 b(:)g(:)27 b FO(,)32 b FN(x)1913 3153 y FL(n)1989 3141 y FO(and)f(p)r(oly-)-118 3240 y(nomial)d(relations)h FN(P)553 3252 y FL(k)594 3240 y FO(\()p FN(x)673 3252 y FK(1)711 3240 y FN(;)14 b(:)g(:)g(:)f(;)h(x)942 3252 y FL(n)988 3240 y FN(;)g(x)1072 3210 y FM(\003)1072 3261 y FK(1)1111 3240 y FN(;)g(:)g(:)g(:)f(;)h(x) 1342 3210 y FM(\003)1342 3261 y FL(n)1388 3240 y FO(\))30 b(=)g(0,)i FN(k)h FO(=)c(1,)j FN(:)14 b(:)g(:)27 b FO(,)33 b FN(m)p FO(,)g(has)-118 3340 y(the)28 b(en)n(v)n(eloping)c FN(C)498 3310 y FM(\003)537 3340 y FO(-algebra)864 3318 y(~)855 3340 y Fz(A)p FO(,)j(then)h(w)n(e)g(will)c(denote)k(it)f(b)n(y) 245 3513 y FN(C)310 3479 y FM(\003)348 3513 y FO(\()p Fz(A)p FO(\))c(=)g FN(C)648 3479 y FM(\003)686 3513 y FO(\()p FN(x)765 3525 y FK(1)803 3513 y FN(;)14 b(:)g(:)g(:)g(;)g(x) 1035 3525 y FL(n)1080 3513 y FO(;)g FN(P)1170 3525 y FL(k)1211 3513 y FO(\()p FP(\001)p FO(\))24 b(=)f(0)p FN(;)14 b(k)25 b FO(=)e(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)p FO(\))495 3638 y(=)23 b FN(C)648 3604 y FM(\003)686 3638 y FO(\()p FN(x)765 3650 y FK(1)803 3638 y FN(;)14 b(:)g(:)g(:)g(;)g(x) 1035 3650 y FL(n)1080 3638 y FO(;)g FN(J)8 b FO(\))p FN(:)6 3811 y FO(A)27 b FP(\003)p FO(-algebra)c Fz(A)k FO(is)e(called)g FP(\003)p FO(-b)r(ounded)h(if)g(for)g(an)n(y)g FN(x)e FP(2)f Fz(A)j FO(there)h(exists)e(a)-118 3911 y(n)n(um)n(b)r(er)i FN(C)244 3923 y FL(x)311 3911 y FN(<)e FP(1)j FO(suc)n(h)h(that)g FP(k)p FN(\031)s FO(\()p FN(x)p FO(\))p FP(k)c(\024)f FN(C)1300 3923 y FL(x)1371 3911 y FO(for)k(an)n(y)g(of)h(its)f FP(\003)p FO(-represen)n(ta-)p eop %%Page: 17 21 17 20 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(17)-118 96 y(tions)26 b FN(\031)12 b FO(:)29 b Fz(A)22 b FP(7!)h FN(L)p FO(\()p FN(H)7 b FO(\).)37 b(F)-7 b(or)27 b(a)g(\014nitely-generated)e FP(\003)p FO(-algebra)720 271 y Fz(A)e FO(=)g FI(C)15 b FP(h)p FN(x)1024 283 y FK(1)1068 271 y FN(;)f(:)g(:)g(:)f(;)h(x)1299 283 y FL(n)1368 271 y FP(j)23 b FN(J)8 b FP(i)-118 446 y FO(to)30 b(b)r(e)g FP(\003)p FO(-b)r(ounded)g(it)g(is)f(su\016cien)n(t)g(that)h(for)g(an)n (y)f(of)h(its)f FP(\003)p FO(-represen)n(tations)-118 545 y FN(\031)12 b FO(:)32 b Fz(A)40 b FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\))38 b(and)g(an)n(y)f FN(k)43 b FO(=)d(1,)d FN(:)14 b(:)g(:)28 b FO(,)41 b FN(n)p FO(,)f(there)e (exists)f FN(C)1910 557 y FL(k)1989 545 y FO(suc)n(h)g(that)-118 645 y FP(k)p FN(\031)s FO(\()p FN(x)53 657 y FL(k)94 645 y FO(\))p FP(k)23 b(\024)g FN(C)338 657 y FL(k)379 645 y FO(.)6 744 y(A)31 b FP(\003)p FO(-algebra)c Fz(A)j FO(has)f(the)i(en)n(v)n(eloping)c FN(C)1324 714 y FM(\003)1362 744 y FO(-algebra)g FN(C)1748 714 y FM(\003)1787 744 y FO(\()p Fz(A)p FO(\))j(if)g(and)g(only)-118 844 y(if)23 b(it)g(is)g FP(\003)p FO(-b)r(ounded.)35 b(In)24 b(this)g(case)f(the)h (en)n(v)n(eloping)c FN(C)1591 814 y FM(\003)1630 844 y FO(-algebra)g(is)j(the)h(com-)-118 944 y(pletion)g(of)j Fz(A)e FO(with)h(resp)r(ect)g(to)g(the)h(norm)e FP(k)p FN(x)p FP(k)d FO(=)h(sup)1631 964 y FL(\031)r FM(2\003)p FK(-Rep)o(\()p Fs(A)p FK(\))2006 944 y FP(k)p FN(\031)s FO(\()p FN(x)p FO(\))p FP(k)g(\024)-118 1043 y FN(C)-59 1055 y FL(x)6 1043 y FN(<)g FP(1)k FO(\(sup)h(is)f(tak)n(en)g(o)n(v)n (er)f(all)f FP(\003)p FO(-represen)n(tations)f FN(\031)s FO(\).)-118 1173 y FB(Example)31 b(2.)42 b FO(The)36 b(group)f FP(\003)p FO(-algebra)e FI(C)15 b FO([)p FN(G)p FO(])43 b(is)35 b FP(\003)p FO(-b)r(ounded,)j(since)d(for)g(an)n(y)-118 1273 y(generator)29 b FN(g)k FP(2)d FN(G)i FO(and)g(an)n(y)f FN(\031)i FP(2)d(\003)p FO(-Rep)13 b FI(C)i FO([)q FN(G)p FO(])38 b(the)32 b(op)r(erator)e FN(\031)s FO(\()p FN(g)s FO(\))i(is)f(uni-)-118 1372 y(tary)e(and)h FP(k)p FN(\031)s FO(\()p FN(g)s FO(\))p FP(k)c FO(=)h(1.)43 b(Hence)30 b(there)g(exists)f(the)h(en)n(v)n(eloping)c FN(C)2009 1342 y FM(\003)2048 1372 y FO(-algebra)-118 1472 y FN(C)-53 1442 y FM(\003)-15 1472 y FO(\()p FN(G)p FO(\))e(=)f FN(C)291 1442 y FM(\003)329 1472 y FO(\()p FI(C)16 b FO([)p FN(G)p FO(]\).)-118 1601 y FB(Example)31 b(3.)42 b FO(The)28 b FP(\003)p FO(-algebra)243 1776 y FA(P)298 1788 y FL(n)366 1776 y FO(=)22 b FI(C)507 1709 y Fy(\012)553 1776 y FN(p)595 1788 y FK(1)632 1776 y FN(;)14 b(:)g(:)g(:)f(;)h(p)858 1788 y FL(n)926 1776 y FP(j)23 b FN(p)1014 1742 y FK(2)1014 1796 y FL(k)1078 1776 y FO(=)g FN(p)1208 1788 y FL(k)1271 1776 y FO(=)g FN(p)1401 1742 y FM(\003)1401 1796 y FL(k)1442 1776 y FN(;)41 b(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)1939 1709 y Fy(\013)-118 1951 y FO(is)33 b FP(\003)p FO(-b)r(ounded,)i (since)e FP(k)p FN(P)716 1963 y FL(k)757 1951 y FP(k)g(\024)g FO(1,)i FN(k)i FO(=)c(1,)h FN(:)14 b(:)g(:)28 b FO(,)35 b FN(n)p FO(.)56 b(Then)35 b(there)e(exists)g(a)-118 2050 y(unique)d FN(C)220 2020 y FM(\003)259 2050 y FO(\()p FA(P)346 2062 y FL(n)391 2050 y FO(\))f(=)f FN(C)610 2020 y FM(\003)649 2050 y FO(\()p FN(p)723 2062 y FK(1)760 2050 y FN(;)14 b(:)g(:)g(:)g(;)g(p)987 2062 y FL(n)1032 2050 y FO(;)g FN(p)1111 2020 y FK(2)1111 2074 y FL(k)1180 2050 y FO(=)28 b FN(p)1315 2062 y FL(k)1385 2050 y FO(=)g FN(p)1520 2020 y FM(\003)1520 2074 y FL(k)1561 2050 y FN(;)45 b(k)32 b FO(=)c(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)p FO(\).)47 b(It)32 b(is)-118 2150 y(kno)n(wn)26 b(that)h(the)h FN(C)531 2120 y FM(\003)569 2150 y FO(-algebra)c FN(C)952 2120 y FM(\003)991 2150 y FO(\()p FA(P)1078 2162 y FK(2)1115 2150 y FO(\))j(can)g(also)e(b)r(e)i(de\014ned)h(as)e FN(C)2056 2120 y FM(\003)2094 2150 y FO(\()p FA(P)2181 2162 y FK(2)2219 2150 y FO(\))d(=)-118 2249 y FP(f)p FN(f)39 b FP(2)32 b FN(C)6 b FO(\([0)p FN(;)14 b FO(1])p FN(;)g(M)473 2261 y FK(2)509 2249 y FO(\()p FI(C)i FO(\)\))37 b FP(j)32 b FN(f)9 b FO(\(0\))p FN(;)14 b(f)9 b FO(\(1\))32 b(are)26 b(diagonal)m FP(g)32 b FO(\(see)g([271)o(,)h(219)n(])g(and) -118 2349 y(others\).)-118 2479 y FQ(3.)j FO(If)28 b(a)g FP(\003)p FO(-algebra)c Fz(A)e FO(=)h FI(C)15 b FP(h)p FN(x)810 2491 y FK(1)853 2479 y FN(;)f(:)g(:)g(:)g(;)g(x)1085 2491 y FL(n)1130 2479 y FP(i)p FN(=J)36 b FO(is)26 b(not)i FP(\003)p FO(-b)r(ounded,)f(it)g(do)r(es)h(not)-118 2578 y(ha)n(v)n(e)g(an)g(en)n(v)n(eloping)d FN(C)665 2548 y FM(\003)704 2578 y FO(-algebra,)h FN(C)1112 2548 y FM(\003)1150 2578 y FO(\()p FN(x)1229 2590 y FK(1)1267 2578 y FN(;)14 b(:)g(:)g(:)g(;)g(x)1499 2590 y FL(n)1545 2578 y FO(;)g FN(J)8 b FO(\).)40 b(Ho)n(w)n(ev)n(er,)28 b(w)n(e)g(can)-118 2678 y(alw)n(a)n(ys)21 b(\014nd)j(an)f(en)n(v)n (eloping)d FN(\033)s FO(-)p FN(C)964 2648 y FM(\003)1003 2678 y FO(-algebra.)32 b(Let)24 b(us)g(sk)n(etc)n(h)f(this)g(construc-) -118 2778 y(tion.)44 b(If)30 b(in)g(the)h(de\014nition)d(of)i(the)h(en) n(v)n(eloping)c FN(C)1514 2747 y FM(\003)1552 2778 y FO(-algebra,)h(one)i(replaces)-118 2877 y(the)e(condition)d(that)j(the) g(diagram)872 3428 y Fy(e)865 3450 y Fz(A)286 b FN(L)p FO(\()p FN(H)7 b FO(\))p 949 3422 239 4 v 1104 3420 a Ft(-)1044 3496 y Fy(e)-48 b FN(\031)1154 3230 y(\031)953 3146 y Ft(@)1036 3229 y(@)1119 3312 y(@)1169 3362 y(@)-83 b(R)865 3033 y Fz(A)p 893 3362 4 299 v 895 3362 a Ft(?)812 3233 y FN(\036)-118 3637 y FO(is)22 b(comm)n(utativ)n(e)d(for)k(all)e FP(\003)p FO(-represen)n(tations)e(of)k Fz(A)g FO(b)n(y)f(the)i (condition)d(that)i(it)-118 3736 y(is)d(comm)n(utativ)n(e)e(only)j(for) g(the)h(represen)n(tations)c(sub)5 b(ject)22 b(to)g(the)g(restriction) 321 3911 y FP(k)p FN(\031)s FO(\()p FN(x)492 3923 y FL(k)533 3911 y FO(\))p FP(k)h(\024)g FN(d)761 3923 y FL(k)802 3911 y FN(;)180 b(d)1048 3923 y FL(k)1112 3911 y FN(>)23 b FO(0)p FN(;)179 b(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)p eop %%Page: 18 22 18 21 bop -118 -137 a FO(18)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(then)h(there)f(exists)e(a)i FN(C)637 66 y FM(\003)676 96 y FO(-algebra)1001 75 y(~)992 96 y Fz(A)g FO(making)d(this)j (diagram)c(comm)n(utativ)n(e.)-118 196 y(W)-7 b(e)28 b(will)d(denote)i(this)g FN(C)676 166 y FM(\003)715 196 y FO(-algebra)d(b)n(y)331 383 y FN(C)396 348 y FM(\003)435 383 y FO(\()p FN(x)514 395 y FK(1)552 383 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(x)797 395 y FL(n)843 383 y FO(;)28 b FP(k)p FN(x)983 395 y FL(k)1023 383 y FP(k)23 b(\024)g FN(d)1219 395 y FL(k)1260 383 y FN(;)k(k)f FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h (n)p FO(;)g FN(J)8 b FO(\))p FN(:)-118 571 y FB(Example)31 b(4.)42 b FO(The)35 b(algebra)e FI(C)15 b FO([)p FN(a)41 b FO(=)35 b FN(a)1111 541 y FM(\003)1149 571 y FO(])h(of)f(complex)e(p) r(olynomials)d(of)35 b(one)-118 671 y(real)g(v)-5 b(ariable)33 b FN(t)39 b FP(2)g FI(R)590 641 y FK(1)670 671 y FO(is)c(not)i FP(\003)p FO(-b)r(ounded,)i(since)c(for)i(an)n(y)f FN(\025)i FP(2)h FI(R)k FO(there)-118 771 y(exists)26 b(a)h(represen)n(tation)e FN(\031)770 783 y FL(\025)814 771 y FO(\()p FN(a)p FO(\))f(=)e FN(\025I)35 b FO(with)27 b FP(k)p FN(\031)1430 783 y FL(\025)1474 771 y FO(\()p FN(a)p FO(\))p FP(k)c FO(=)p FP(j)f FN(\025)i FP(j)p FO(.)6 872 y(Ho)n(w)n(ev)n(er,)k(there)h (exists)f(a)h FN(C)947 842 y FM(\003)985 872 y FO(-algebra)d Fz(A)1365 884 y FL(d)1429 872 y FO(=)g FN(C)6 b FO([)p FP(\000)p FN(d;)14 b(d)p FO(])29 b(with)g(one)g(self-)-118 972 y(adjoin)n(t)24 b(generator)f FN(a)p FO(\()p FN(t)p FO(\))h(=)e FN(t)k FO(and)f(a)g(homomorphism)20 b FN(\022)12 b FO(:)27 b FI(C)15 b FO([)p FN(a)29 b FO(=)23 b FN(a)2026 942 y FM(\003)2064 972 y FO(])g FP(3)h FN(a)f FP(7!)-118 1072 y FN(a)p FO(\()p FP(\001)p FO(\))g FP(2)h FN(C)6 b FO([)p FP(\000)p FN(d;)14 b(d)p FO(],)28 b(whic)n(h)e(satis\014es)g (the)i(follo)n(wing)c(conditions:)-17 1244 y(1.)41 b FP(k)p FN(a)p FO(\()p FP(\001)p FO(\))p FP(k)23 b(\024)g FN(d)p FO(;)-17 1418 y(2.)41 b(for)k(an)n(y)f FP(\003)p FO(-represen)n(tation)d FN(\031)12 b FO(:)34 b FI(C)15 b FO([)p FN(a)58 b FO(=)51 b FN(a)1492 1388 y FM(\003)1530 1418 y FO(])h FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\))45 b(suc)n(h)f(that)89 1518 y FP(k)p FN(\031)s FO(\()p FN(a)p FO(\))p FP(k)e(\024)f FN(d)e FO(there)f(exists)f(a)h(unique)g(represen) n(tation)i(~)-46 b FN(\031)13 b FO(:)31 b FN(C)2118 1487 y FM(\003)2156 1518 y FO(\()p FN(a)42 b FO(=)89 1617 y FN(a)133 1587 y FM(\003)171 1617 y FO(;)f FP(k)p FN(a)p FP(k)22 b(\024)h FN(d)p FO(\))g FP(7!)g FN(L)p FO(\()p FN(H)7 b FO(\))27 b(suc)n(h)g(that)g(the)g(follo)n(wing)22 b(diagram)i(is)h(com-)89 1717 y(m)n(utativ)n(e)994 2310 y Fz(A)1054 2322 y FL(d)1360 2310 y FN(L)p FO(\()p FN(H)7 b FO(\))p 1117 2291 219 4 v 1252 2289 a Ft(-)1205 2361 y FO(~)-47 b FN(\031)1302 2100 y(\031)1101 2015 y Ft(@)1184 2098 y(@)1267 2181 y(@)1317 2231 y(@)-83 b(R)872 1895 y FI(C)15 b FO([)p FN(a)29 b FO(=)22 b FN(a)1153 1865 y FM(\003)1191 1895 y FO(])p 1041 2231 4 299 v 1043 2231 a Ft(?)969 2111 y FN(\022)-118 2544 y FB(Example)31 b(5.)42 b FO(The)31 b FP(\003)p FO(-algebra)c FA(Q)920 2556 y FK(1)985 2544 y FO(=)h FP(h)p FN(q)s(;)14 b(q)1227 2514 y FM(\003)1294 2544 y FP(j)28 b FN(q)1385 2514 y FK(2)1450 2544 y FO(=)g FN(q)s(;)45 b FO(\()p FN(q)1723 2514 y FM(\003)1761 2544 y FO(\))1793 2514 y FK(2)1859 2544 y FO(=)28 b FN(q)1992 2514 y FM(\003)2030 2544 y FP(i)p FO(,)k(gener-)-118 2644 y(ated)27 b(b)n(y)g(one)f(idemp)r(oten)n(t)g (is)g(not)h FP(\003)p FO(-b)r(ounded,)g(since)f(for)g(an)n(y)h FN(\025)c FP(2)h FI(R)33 b FO(there)-118 2743 y(exists)26 b(the)i(represen)n(tation)768 2975 y FN(\031)815 2987 y FL(\025)859 2975 y FO(\()p FN(q)s FO(\))c(=)1074 2858 y Fy(\022)1177 2925 y FO(1)82 b FN(\025)1177 3024 y FO(0)k(0)1391 2858 y Fy(\023)-118 3212 y FO(with)27 b FP(k)p FN(\031)160 3224 y FL(\025)203 3212 y FO(\()p FN(q)s FO(\))p FP(k)d(!)f(1)k FN(;)42 b(\025)23 b FP(!)g(1)p FO(.)37 b(Ho)n(w)n(ev)n(er)26 b(there)h(exists)640 3399 y FN(C)705 3364 y FM(\003)743 3331 y Fy(\000)781 3399 y FN(q)s(;)14 b(q)898 3364 y FM(\003)936 3399 y FO(;)g FP(k)p FN(q)s FP(k)22 b(\024)h FN(d)p FO(;)42 b FN(q)1355 3364 y FK(2)1415 3399 y FO(=)23 b FN(q)1543 3331 y Fy(\001)334 3523 y FO(=)f FP(f)p FN(f)32 b FP(2)23 b FN(C)6 b FO(\([0)p FN(;)14 b(d)p FO(])p FN(;)g(M)997 3535 y FK(2)1034 3523 y FO(\()p FI(C)h FO(\))q(\))9 b(:)34 b FN(f)9 b FO(\(0\))27 b(is)g(diagonal)l FP(g)p FN(:)6 3712 y FO(F)-7 b(or)39 b(ev)n(ery)e FP(\003)p FO(-algebra)e Fz(A)42 b FO(=)f FI(C)15 b FP(h)p FN(x)1110 3724 y FK(1)1153 3712 y FN(;)f(:)g(:)g(:)27 b(;)14 b(x)1398 3724 y FL(m)1462 3712 y FO(;)g FN(J)8 b FP(i)39 b FO(w)n(e)f(can)h(construct)f(a)-118 3811 y(top)r(ological)27 b FP(\003)p FO(-algebra)681 3790 y(~)672 3811 y Fz(A)k FO(whic)n(h)f(is)h(also)e(an)i(en)n(v)n (eloping)e FP(\003)p FO(-algebra)e(of)32 b Fz(A)p FO(.)-118 3911 y(Indeed,)43 b(in)d(the)g(previous)e(section)h(w)n(e)h(ha)n(v)n(e) f(constructed)g(the)i(algebras)p eop %%Page: 19 23 19 22 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(19)-118 96 y Fz(A)-58 108 y FL(n)30 96 y FO(=)43 b FN(C)203 66 y FM(\003)241 96 y FO(\()p FN(x)320 108 y FK(1)358 96 y FN(;)14 b(:)g(:)g(:)g(;)g(x)590 108 y FL(m)653 96 y FO(;)g FP(k)p FN(x)779 108 y FL(j)814 96 y FP(k)43 b(\024)g FN(n;)28 b FO(1)42 b FP(\024)h FN(j)49 b FP(\024)43 b FN(m)p FO(;)14 b FN(J)8 b FO(\))40 b(and)f FP(\003)p FO(-homomor-)-118 196 y(phisms)33 b FN(\036)216 208 y FL(n)271 196 y FO(:)d Fz(A)35 b FP(\000)-48 b(!)35 b Fz(A)614 208 y FL(n)694 196 y FO(with)f(appropriate)f(univ)n(ersal)e (prop)r(erties.)57 b(Since)-118 296 y(k)n(er)13 b FN(\036)56 308 y FL(n)132 296 y FP(\023)30 b FO(k)n(er)13 b FN(\036)401 308 y FL(n)p FK(+1)531 296 y FO(,)33 b(there)f(exists)f(a)h FP(\003)p FO(-homomorphism)26 b FN( )1829 266 y FL(n)p FK(+1)1826 316 y FL(n)1967 296 y FO(:)j Fz(A)2079 308 y FL(n)p FK(+1)2239 296 y FP(\000)-48 b(!)-118 395 y Fz(A)-58 407 y FL(n)14 395 y FO(suc)n(h)28 b(that)f(the)h(follo)n(wing) c(diagrams)g(comm)n(ute)622 968 y Fz(A)682 980 y FL(n)p FK(+1)1493 968 y Fz(A)1553 980 y FL(n)p 835 951 634 4 v 1386 949 a Ft(-)1059 1034 y FN( )1116 1003 y FL(n)p FK(+1)1113 1054 y FL(n)1101 563 y Fz(A)693 761 y FN(\036)742 773 y FL(n)p FK(+1)990 676 y Ft(\000)907 759 y(\000)824 842 y(\000)774 891 y(\000)-83 b(\011)1390 763 y FN(\036)1439 775 y FL(n)1189 676 y Ft(@)1272 759 y(@)1355 842 y(@)1405 891 y(@)g(R)6 1225 y FO(Consider)24 b(a)h(subalgebra)840 1204 y(~)831 1225 y Fz(A)g FO(in)f(the)i(Cartesian)d(pro)r(duct)1832 1163 y Fy(Q)1911 1250 y FL(n)p FM(2)p Fu(N)2056 1225 y Fz(A)2116 1237 y FL(n)2186 1225 y FO(con-)-118 1325 y(sisting)34 b(of)j(elemen)n(ts)e FN(f)17 b FO(:)31 b FI(N)48 b FP(\000)-48 b(!)38 b([)1006 1337 y FL(n)p FM(2)p Fu(N)1139 1325 y Fz(A)1199 1337 y FL(n)1280 1325 y FO(suc)n(h)f(that)g FN( )1723 1295 y FL(n)p FK(+1)1720 1345 y FL(n)1852 1325 y FO(\()p FN(f)9 b FO(\()p FN(n)24 b FO(+)h(1\)\))38 b(=)-118 1425 y FN(f)9 b FO(\()p FN(n)p FO(\),)35 b FN(n)e FP(2)g FI(N)t FO(.)60 b(Then)644 1403 y(~)635 1425 y Fz(A)33 b FO(is)f(a)h(top)r(ological)c FP(\003)p FO(-algebra)h(endo)n (w)n(ed)i(with)h(the)-118 1524 y(w)n(eak)n(est)f(top)r(ology)f(suc)n(h) i(that)g(the)h(maps)e FN(\031)1339 1536 y FL(n)1393 1524 y FO(:)1455 1503 y(~)1446 1524 y Fz(A)g FP(\000)-48 b(!)32 b Fz(A)1730 1536 y FL(n)1775 1524 y FO(,)j FN(f)41 b FP(7!)32 b FN(f)9 b FO(\()p FN(n)p FO(\))34 b(are)-118 1624 y(con)n(tin)n(uous.)65 b(W)-7 b(e)38 b(will)d(denote)967 1602 y(~)958 1624 y Fz(A)j FO(b)n(y)f(lim)1181 1663 y FP( )-32 b(\000)1310 1624 y Fz(A)1370 1636 y FL(n)1415 1624 y FO(.)67 b(W)-7 b(e)39 b(need)f(the)g(follo)n(wing)-118 1732 y(statemen)n(t)26 b([150)o(,)i(Lemma)d(3.9].)-118 1900 y FQ(Lemma)k(2.)41 b FB(L)l(et)33 b FN(A)g FB(b)l(e)g(a)g FN(C)823 1870 y FM(\003)862 1900 y FB(-algebr)l(a)h(and)f FN(B)k FB(b)l(e)c(a)g FN(C)1685 1870 y FM(\003)1724 1900 y FB(-sub)l(algebr)l(a)g(of)h FN(A)p FB(.)-118 2000 y(Assume)c(that)h FN(\031)e Fr(\026)c FN(B)k FO(=)c FN(\031)732 1969 y FM(0)781 2000 y Fr(\026)g FN(B)35 b FB(for)d(any)f(two)h(r)l(epr)l (esentations)f FN(\031)s FB(,)h FN(\031)2123 1969 y FM(0)2177 2000 y FB(of)g FN(A)-118 2099 y FB(implies)f(that)f FN(\031)c FO(=)d FN(\031)547 2069 y FM(0)571 2099 y FB(.)38 b(In)30 b(this)g(c)l(ase)g FN(B)d FO(=)c FN(A)p FB(.)6 2267 y FO(The)28 b(follo)n(wing)c(theorem)i(holds.)-118 2436 y FQ(Theorem)k(1.)41 b FB(The)e(p)l(air)f FO(\()800 2414 y(~)791 2436 y Fz(A)p FN(;)14 b(\036)p FO(\))38 b FB(is)g(a)g(unique)f (enveloping)i FN(\033)s FB(-)p FN(C)2019 2406 y FM(\003)2057 2436 y FB(-algebr)l(a)-118 2535 y(for)c Fz(A)p FB(.)53 b(The)35 b(homomorphism)j FN(\036)d FB(has)g(dense)g(r)l(ange.)53 b(Mor)l(e)l(over)36 b FO(\()2075 2514 y(~)2066 2535 y Fz(A)p FN(;)14 b(\036)p FO(\))35 b FB(is)-118 2635 y(also)c(an)e (enveloping)j FP(\003)p FB(-algebr)l(a.)-118 2803 y(Pr)l(o)l(of.)43 b FO(Let)37 b(us)g(notice)f(that)h(the)h(homomorphisms)32 b FN(\031)1688 2815 y FL(n)1742 2803 y FO(:)1805 2781 y(~)1796 2803 y Fz(A)39 b FP(\000)-49 b(!)39 b Fz(A)2093 2815 y FL(n)2175 2803 y FO(ha)n(v)n(e)-118 2903 y(dense)24 b(ranges.)34 b(The)24 b(top)r(ology)e(of)i(the)g FN(\033)s FO(-)p FN(C)1273 2873 y FM(\003)1312 2903 y FO(-algebra)1636 2881 y(~)1627 2903 y Fz(A)f FO(can)h(b)r(e)h(de\014ned)f(b)n(y)-118 3002 y(a)32 b(coun)n(table)e(increasing)f(family)h FN(p)1030 3014 y FL(n)1075 3002 y FO(\()p FP(\001)p FO(\))j(of)f FN(C)1359 2972 y FM(\003)1398 3002 y FO(-semi-norms.)46 b(Using)31 b(argu-)-118 3102 y(men)n(ts)e(similar)d(to)k(Can)n(tor's)f (diagonal)d(metho)r(d,)31 b(w)n(e)f(can)g(pro)n(v)n(e)e(that)j(it)f(is) -118 3202 y(also)25 b(dense)i(in)f(the)h(top)r(ology)e(de\014ned)i(b)n (y)g(the)g(family)d FN(p)1690 3214 y FL(n)1735 3202 y FO(\()p FP(\001)p FO(\),)k(whic)n(h)e(pro)n(v)n(es)-118 3301 y(that)35 b(the)h(homomorphism)30 b FN(\036)36 b FO(has)f(dense)g(range.)59 b(Let)35 b FN(\031)k FO(b)r(e)c(a)g (represen-)-118 3401 y(tation)f(of)h Fz(A)g FO(in)f FN(L)p FO(\()p FN(H)7 b FO(\).)59 b(If)36 b FN(\031)i FP(2)e FA(R)1041 3413 y FL(n)1086 3401 y FO(,)i(de\014ne)h(~)-46 b FN(\031)39 b FO(=)c FN(F)12 b FO(\()p FN(\031)s FO(\)\()p FN(\031)1838 3413 y FL(n)1884 3401 y FO(\).)60 b(Since)34 b(the)-118 3501 y(algebra)24 b FN(\036)p FO(\()p FN(A)p FO(\))k(is)e(dense)h(in)f FA(A)p FO(,)32 b(~)-46 b FN(\031)30 b FO(is)c(uniquely)f(de\014ned.)37 b(Let)27 b(us)g(pro)n(v)n(e)e(that) -118 3600 y(the)j(en)n(v)n(eloping)c FN(\033)s FO(-)p FN(C)576 3570 y FM(\003)615 3600 y FO(-algebra)g(is)j(unique.)6 3700 y(1.)45 b(W)-7 b(e)31 b(sa)n(y)f(that)g FN(\031)638 3712 y FK(1)704 3700 y FP(\024)d FN(\031)843 3712 y FK(2)911 3700 y FO(for)j FN(\031)1088 3712 y FK(1)1126 3700 y FO(,)h FN(\031)1227 3712 y FK(2)1293 3700 y FP(2)d FO(Rep\()p Fz(A)p FO(\))j(i\013)f(k)n(er)12 b FN(\031)1948 3712 y FK(2)2014 3700 y FP(\022)27 b FO(k)n(er)13 b FN(\031)2278 3712 y FK(1)2316 3700 y FO(.)-118 3800 y(Then)25 b(the)h(set)f(Rep)14 b Fz(A)24 b FO(is)g(a)h(net)g(since)f FN(\031)1142 3812 y FL(j)1200 3800 y FP(\024)f FN(\031)1335 3812 y FK(1)1386 3800 y FP(\010)13 b FN(\031)1511 3812 y FK(2)1548 3800 y FO(.)37 b(Cho)r(ose)24 b(an)n(y)g(co-\014nite)-118 3911 y(subnet)k FN(\031)197 3923 y FL(n)242 3911 y FO(,)g(and)g (de\014ne)703 3889 y(~)694 3911 y Fz(A)754 3923 y FL(p)815 3911 y FO(=)23 b(lim)903 3950 y FP( )-32 b(\000)p 1032 3839 217 4 v 1032 3911 a FN(\031)1079 3923 y FL(n)1125 3911 y FO(\()p Fz(A)p FO(\).)p eop %%Page: 20 24 20 23 bop -118 -137 a FO(20)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(2.)37 b(Let)27 b(\()297 75 y(~)288 96 y Fz(A)p FN(;)14 b(\036)p FO(\))28 b(b)r(e)g(an)f(en)n(v)n(eloping)d FN(\033)s FO(-)p FN(C)1273 66 y FM(\003)1311 96 y FO(-algebra)g(of)k Fz(A)p FO(.)36 b(W)-7 b(e)28 b(will)c(pro)n(v)n(e)-118 196 y(that)j FN(\036)p FO(\()p Fz(A)p FO(\))h(is)e(dense)h(in)677 174 y(~)668 196 y Fz(A)o FO(.)37 b(Let)28 b FA(B)f FO(b)r(e)h(a)e FN(C)1272 166 y FM(\003)1311 196 y FO(-algebra)e(and)j FN(j)14 b FO(:)28 b Fz(A)22 b FP(\000)-48 b(!)23 b FA(B)k FO(b)r(e)h(a)-118 296 y(con)n(tin)n(uous)19 b FP(\003)p FO(-surjection.)32 b(If)21 b FN(\031)s FO(,)i FN(\031)993 266 y FM(0)1040 296 y FP(2)g FO(Rep)14 b FA(B)21 b FO(and)g FN(\031)8 b FP(\016)d FN(j)10 b FP(\016)5 b FN(\036)22 b FO(=)h FN(\031)1917 266 y FM(0)1946 296 y FP(\016)5 b FN(j)10 b FP(\016)5 b FN(\036)p FO(,)21 b(then)-118 395 y(b)n(y)h(the)i(uniqueness)d(of)28 b(~)-47 b FN(\031)27 b FO(in)22 b(the)h(de\014nition)e(of)i(the)g(en)n(v)n(eloping)d (algebra,)h(w)n(e)-118 495 y(ha)n(v)n(e)i FN(\031)15 b FP(\016)d FN(j)27 b FO(=)c FN(\031)385 465 y FM(0)420 495 y FP(\016)12 b FN(j)5 b FO(.)36 b(Then)24 b FN(\031)i FO(=)d FN(\031)996 465 y FM(0)1044 495 y FO(since)g FN(j)29 b FO(is)23 b(surjectiv)n(e.)35 b(Then)24 b(Lemma)e(2)-118 595 y(implies)g(that)j FN(j)20 b FP(\016)13 b FN(\036)p FO(\()p Fz(A)p FO(\))27 b(is)d(dense)h(in)g FA(B)p FO(.)37 b(Represen)n(t)1564 573 y(~)1555 595 y Fz(A)26 b FO(as)e(lim)1740 633 y FP( )-32 b(\000)1870 595 y FA(B)1933 607 y FL(n)2004 595 y FO(\(see)25 b([199)o(,)-118 703 y(Prop)r(osition)g(1.2])j(or)g ([10)o(,)h(Prop)r(osition)c(5.6]\),)k(where)f FA(B)1715 715 y FL(n)1789 703 y FO(is)g(a)g FN(C)2009 672 y FM(\003)2048 703 y FO(-algebra)-118 802 y(and)33 b FN(j)83 814 y FL(n)160 802 y FO(:)224 780 y(~)215 802 y Fz(A)f FP(\000)-49 b(!)32 b FA(B)501 814 y FL(n)579 802 y FO(is)g(the)i(canonical)29 b(surjection.)52 b(Previous)30 b(argumen)n(ts)-118 902 y(sho)n(w)d(that)p 268 830 318 4 v 28 w FN(j)302 914 y FL(n)348 902 y FO(\()p FN(\036)p FO(\()p Fz(A)p FO(\)\))e(=)e FA(B)761 914 y FL(n)806 902 y FO(.)38 b(F)-7 b(rom)27 b(this)g(it)g(follo)n(ws)e(that)k FN(\036)p FO(\()p Fz(A)p FO(\))f(is)f(a)h(dense)-118 1001 y FP(\003)p FO(-algebra)c(in)348 980 y(~)339 1001 y Fz(A)o FO(,)k(since)e FN(z)691 1013 y FL(k)755 1001 y FP(\000)-48 b(!)23 b FN(z)31 b FO(in)1053 980 y(~)1044 1001 y Fz(A)d FO(i\013)f FN(j)1265 1013 y FL(n)1310 1001 y FO(\()p FN(z)1381 1013 y FL(k)1422 1001 y FO(\))c FP(\000)-48 b(!)23 b FN(j)1634 1013 y FL(n)1679 1001 y FO(\()p FN(z)t FO(\))28 b(for)f(all)e FN(n)p FO(.)6 1107 y(3.)78 b(W)-7 b(e)42 b(will)d(pro)n(v)n(e)g(that) 917 1086 y(~)908 1107 y Fz(A)45 b FP(')1133 1086 y FO(~)1124 1107 y Fz(A)1184 1119 y FL(p)1222 1107 y FO(.)78 b(P)n(assing)39 b(if)h(necessary)g(to)h(the)-118 1207 y(quotien)n(t)c(with)g(resp)r (ect)h(to)g(the)g(radical,)f(w)n(e)h(can)f(assume)f(that)j FN(\036)f FO(is)f(an)-118 1306 y(injection,)27 b(and)i(so)f FN(\036)p FO(\()p Fz(A)p FO(\))d FP(')g Fz(A)p FO(.)39 b(Since)p 1139 1234 174 4 v 28 w FN(\036)p FO(\()p Fz(A)p FO(\))26 b(=)e Fz(A)p FO(,)29 b(w)n(e)f(only)f(need)i(to)f(pro)n(v)n(e) -118 1406 y(that)39 b(the)h(top)r(ology)d FN(\034)617 1418 y FK(1)693 1406 y FO(on)i FN(\036)p FO(\()p Fz(A)p FO(\))h(induced)f(from)1570 1384 y(~)1561 1406 y Fz(A)g FO(coincides)e(with)h(the)-118 1506 y(top)r(ology)28 b FN(\034)262 1518 y FK(2)331 1506 y FO(on)i FN(\036)p FO(\()p Fz(A)p FO(\))i(de\014ned)f(b)n(y)f(the)h(semi-norms)c FP(k)t FO(~)-46 b FN(\031)1739 1518 y FL(n)1784 1506 y FO(\()p FP(\001)p FO(\))p FP(k)p FO(.)47 b(Note)30 b(that)-118 1605 y(the)21 b(top)r(ology)e FN(\034)389 1617 y FK(1)448 1605 y FO(is)g(de\014ned)j(b)n(y)f(the)g(semi-norms)c FP(k)5 b(\001)g(k)1599 1617 y FL(n)1665 1605 y FO(induced)20 b(b)n(y)h(faithful)-118 1705 y(represen)n(tations)i(of)k FA(B)613 1717 y FL(n)658 1705 y FO(,)g(and)g(since)e(the)i(subnet)g FN(\031)1527 1717 y FL(n)1599 1705 y FO(is)f(co-\014nite)f(in)h(the)h (set)-118 1805 y(of)g(all)d(represen)n(tations,)g(the)j(top)r(ology)e FN(\034)1207 1817 y FK(2)1271 1805 y FO(is)h(stronger)f(then)i FN(\034)1901 1817 y FK(1)1939 1805 y FO(.)37 b(But)27 b(since)-118 1904 y(ev)n(ery)k(represen)n(tations)e FN(\031)733 1916 y FL(n)811 1904 y FO(can)i(b)r(e)i(lifted)e(to)1419 1883 y(~)1410 1904 y Fz(A)p FO(,)i FN(\034)1562 1916 y FK(1)1632 1904 y FO(is)e(stronger)f(then)j FN(\034)2278 1916 y FK(2)2316 1904 y FO(.)-118 2004 y(This)26 b(pro)n(v)n(es)g(that) i FN(\034)544 2016 y FK(1)605 2004 y FO(=)22 b FN(\034)728 2016 y FK(2)766 2004 y FO(.)6 2110 y(Since)35 b(the)h(algebra)690 2088 y(~)681 2110 y Fz(A)g FO(is)e(a)i(metrizable)c(lo)r(cally)g FN(C)1696 2080 y FM(\003)1734 2110 y FO(-algebra,)j(b)n(y)g([83,)-118 2209 y(Corollary)18 b(4.7],)k(ev)n(ery)f FP(\003)p FO(-represen)n (tation)d(of)1338 2188 y(~)1329 2209 y Fz(A)j FO(is)g(con)n(tin)n (uous.)33 b(And)23 b(th)n(us)k(~)-47 b FN(\031)-118 2309 y FO(is)17 b(uniquely)g(de\014ned)i(ev)n(en)f(without)g(the)h (requiremen)n(t)c(of)k(b)r(eing)f(con)n(tin)n(uous.)-118 2409 y(It)41 b(pro)n(v)n(es)d(that)i(the)h FP(\003)p FO(-algebra)984 2387 y(~)975 2409 y Fz(A)f FO(is)f(also)f(an)i(en)n(v)n (eloping)d FP(\003)p FO(-algebra)g(of)-118 2508 y Fz(A)p FO(.)p 2278 2508 4 57 v 2282 2455 50 4 v 2282 2508 V 2331 2508 4 57 v 6 2724 a(Note)k(that)f(the)h(homomorphism)35 b FN(\036)41 b FO(is)e(an)h(injection)e(if)i(and)g(only)f(if)-118 2824 y Fz(A)34 b FO(has)f(a)h(r.f.)h(of)f(represen)n(tations.)54 b(In)34 b(this)f(case)h(it)g(is)f(natural)f(to)i(call)e Fz(A)-118 2923 y FN(\033)s FO(-)p FN(C)25 2893 y FM(\003)63 2923 y FO(-represen)n(table.)-118 3068 y FB(R)l(emark)e(4.)42 b FO(In)25 b(the)f(case)g(where)g Fz(A)g FO(is)f(not)h(\014nitely)f (generated,)h(w)n(e)g(can)g(also)-118 3168 y(construct)35 b(an)h(en)n(v)n(eloping)c(pro-)p FN(C)1009 3138 y FM(\003)1047 3168 y FO(-algebra)g(\(the)37 b(index)e FN(n)h FO(in)f(the)h(ab)r(o)n (v)n(e)-118 3268 y(construction)c(should)g(b)r(e)i(replaced)e(b)n(y)i (the)g(m)n(ulti-index)c FN(\013)j FP(2)h FI(N)2035 3238 y Fu(I)2068 3268 y FO(,)h(where)-118 3367 y FP(f)p FN(x)-29 3379 y FL(\013)18 3367 y FO(;)14 b FN(\013)24 b FP(2)f FI(I)-7 b FP(g)14 b FO(is)19 b(the)i(set)f(of)g(generators)e(of)i Fz(A)p FO(\).)35 b(In)20 b(suc)n(h)g(a)g(case,)g(all)e(the)j(ab)r(o)n (v)n(e)-118 3467 y(statemen)n(ts)39 b(hold)h(true)g(except)h(that)g (the)g FP(\003)p FO(-algebra)c(is)j(not)g(necessarily)-118 3567 y(en)n(v)n(eloping,)20 b(and)i(the)g(range)f(of)h(the)g (homomorphism)17 b FN(\036)23 b FO(need)f(not)g(b)r(e)h(dense)-118 3666 y(but)42 b(only)d(quasi-dense,)k(i.e.,)h(suc)n(h)d(that,)k(for)40 b(an)n(y)h(represen)n(tation)d FN(\031)49 b FP(2)-118 3766 y FO(Rep\()p Fz(A)p FO(\),)28 b(the)g(set)f FN(\031)s FO(\()p FN(\036)p FO(\()645 3744 y(~)636 3766 y Fz(A)q FO(\)\))i(is)d(dense)i(in)e(Im)13 b FN(\031)s FO(.)-118 3911 y FQ(4.)36 b FO(It)28 b(is)e(con)n(v)n(enien)n(t)g(to)h(adopt)h (the)g(follo)n(wing)23 b(de\014nition:)p eop %%Page: 21 25 21 24 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(21)-118 96 y FQ(De\014nition)31 b(2.)40 b FB(We)28 b(wil)t(l)g(say)f(that)g(a)h FP(\003)p FB(-algebr)l(a)f(is)h(of)f(typ)l (e)h(I)40 b FO(\()p FB(nucle)l(ar)9 b FO(\))27 b FB(i\013)-118 196 y Fz(A)-58 208 y FL(n)16 196 y FB(is)j(of)h(typ)l(e)f(I)43 b FO(\()p FB(nucle)l(ar)9 b FO(\))30 b FB(for)h(al)t(l)g FN(n)22 b FP(2)i FI(N)t FB(.)-118 410 y FQ(1.1.4)94 b FP(\003)p FQ(-Represen)m(tations)30 b(of)h(generators)h(and)g (relations)-118 563 y(1.)k FO(T)-7 b(o)27 b(an)n(y)g FP(\003)p FO(-represen)n(tation)d(of)k(a)f(\014nitely)f(generated)g FP(\003)p FO(-algebra)140 733 y Fz(B)d FO(=)g FI(C)378 666 y Fy(\012)423 733 y FN(x)470 745 y FK(1)508 733 y FN(;)14 b(:)g(:)g(:)f(;)h(x)739 745 y FL(n)785 733 y FN(;)g(x)869 699 y FM(\003)869 754 y FK(1)907 733 y FN(;)g(:)g(:)g(:)g (;)g(x)1139 699 y FM(\003)1139 754 y FL(n)1207 733 y FP(j)462 867 y FN(P)515 879 y FL(j)550 867 y FO(\()p FN(x)629 879 y FK(1)667 867 y FN(;)g(:)g(:)g(:)g(;)g(x)899 879 y FL(n)944 867 y FN(;)g(x)1028 833 y FM(\003)1028 888 y FK(1)1067 867 y FN(;)g(:)g(:)g(:)g(;)g(x)1299 833 y FM(\003)1299 888 y FL(n)1344 867 y FO(\))23 b(=)g(0)p FN(;)41 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)2042 800 y Fy(\013)-118 1038 y FO(b)n(y)25 b(b)r(ounded)h(op)r(erators)e (there)h(corresp)r(onds)f(a)h(family)e(of)i(b)r(ounded)h(op)r(era-)-118 1137 y(tors)h FP(f)p FN(X)160 1149 y FL(i)210 1137 y FO(=)22 b FN(\031)s FO(\()p FN(x)426 1149 y FL(i)455 1137 y FO(\))p FN(;)14 b(X)600 1107 y FM(\003)593 1159 y FL(i)661 1137 y FO(=)22 b FN(\031)s FO(\()p FN(x)877 1149 y FL(i)906 1137 y FO(\))938 1107 y FM(\003)999 1137 y FO(=)h FN(\031)s FO(\()p FN(x)1216 1107 y FM(\003)1216 1159 y FL(i)1255 1137 y FO(\))p FP(g)1329 1107 y FL(n)1329 1159 y(i)p FK(=1)1468 1137 y FO(suc)n(h)28 b(that)108 1308 y FN(P)161 1320 y FL(j)197 1308 y FO(\()p FN(X)298 1320 y FK(1)335 1308 y FN(;)14 b(:)g(:)g(:)g(;)g(X)589 1320 y FL(n)634 1308 y FN(;)g(X)747 1273 y FM(\003)740 1328 y FK(1)784 1308 y FN(;)g(:)g(:)g(:)f(;)h(X)1044 1273 y FM(\003)1037 1328 y FL(n)1082 1308 y FO(\))24 b(=)e(0)p FN(;)180 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)g(;)g(m:) 226 b FO(\(1.4\))-118 1478 y(Con)n(v)n(ersely)-7 b(,)35 b(a)g(family)e(of)j(b)r(ounded)g(op)r(erators)e FP(f)p FN(X)1605 1490 y FL(i)1632 1478 y FN(;)14 b(X)1745 1448 y FM(\003)1738 1500 y FL(i)1782 1478 y FP(g)1824 1448 y FL(n)1824 1500 y(i)p FK(=1)1935 1478 y FO(,)38 b(satisfying)-118 1578 y(\(1.4\),)k(can)d(b)r(e)h(uniquely)e(extended)h(to)h(a)f (represen)n(tation)e(of)i(the)h(whole)-118 1677 y FP(\003)p FO(-algebra)31 b Fz(B)p FO(.)60 b(F)-7 b(or)34 b(an)n(y)g(\014nitely)g (presen)n(ted)g FP(\003)p FO(-algebra,)g(one)g(can)h(c)n(ho)r(ose)-118 1777 y(self-adjoin)n(t)28 b(generators)h FN(a)762 1789 y FL(i)818 1777 y FO(=)f FN(a)955 1747 y FM(\003)955 1799 y FL(i)993 1777 y FO(,)k FN(i)c FO(=)g(1,)j FN(:)14 b(:)g(:)27 b FO(,)32 b FN(l)g FO(\(their)e(n)n(um)n(b)r(er)g(ma)n(y)f (b)r(e)-118 1877 y(larger)e(than)k FN(n)p FO(\),)g(connected)g(b)n(y)f (self-adjoin)n(t)e(relations)f FN(Q)1798 1889 y FL(j)1833 1877 y FO(\()p FN(a)1909 1889 y FK(1)1946 1877 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(a)2189 1889 y FL(l)2214 1877 y FO(\))28 b(=)-118 1976 y FN(Q)-52 1946 y FM(\003)-52 1998 y FL(j)-14 1976 y FO(\()p FN(a)62 1988 y FK(1)99 1976 y FN(;)14 b(:)g(:)g(:)g(;)g(a)328 1988 y FL(l)353 1976 y FO(\),)25 b FN(l)f FO(=)f(1,)h FN(:)14 b(:)g(:)27 b FO(,)e FN(r)h FO(\(their)e(n)n(um)n(b)r(er)e(ma)n(y)g(also)g (increase\);)h(there-)-118 2076 y(fore,)37 b(an)n(y)e(represen)n (tation)d FN(\031)39 b FO(of)d(the)g(algebra)d Fz(B)j FO(=)g FI(C)15 b FP(h)p FN(a)1773 2088 y FK(1)1817 2076 y FN(;)f(:)g(:)g(:)f(;)h(a)2045 2088 y FL(l)2107 2076 y FP(j)36 b FN(a)2210 2088 y FL(i)2274 2076 y FO(=)-118 2175 y FN(a)-74 2145 y FM(\003)-74 2197 y FL(i)-36 2175 y FN(;)14 b(i)33 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(l)r FO(;)27 b FN(Q)531 2187 y FL(j)565 2175 y FO(\()p FN(a)641 2187 y FK(1)679 2175 y FN(;)14 b(:)g(:)g(:)g(;)g(a)908 2187 y FL(l)933 2175 y FO(\))34 b(=)f(0)p FN(;)28 b(j)38 b FO(=)c(1)p FN(;)14 b(:)g(:)g(:)f(;)h(r)r FP(i)35 b FO(is)e(uniquely)f(deter-)-118 2275 y(mined)27 b(b)n(y)i(a)g(family)d (of)j(self-adjoin)n(t)e(op)r(erators)g FN(A)1531 2287 y FL(i)1584 2275 y FO(=)f FN(A)1737 2245 y FM(\003)1737 2297 y FL(i)1801 2275 y FO(=)f FN(\031)s FO(\()p FN(a)2017 2287 y FL(i)2045 2275 y FO(\),)30 b FN(i)25 b FO(=)g(1,)-118 2375 y FN(:)14 b(:)g(:)27 b FO(,)h FN(l)r FO(,)f(suc)n(h)h(that)447 2545 y FN(Q)513 2557 y FL(j)548 2545 y FO(\()p FN(A)642 2557 y FK(1)680 2545 y FN(;)14 b(:)g(:)g(:)f(;)h(A)926 2557 y FL(l)952 2545 y FO(\))23 b(=)g(0)p FN(;)179 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(r)n(:)395 b FO(\(1.5\))-78 2716 y FQ(2.)59 b FO(Since)34 b(the)h(prop)r(erties)e (of)i(the)h(represen)n(tation)c(of)j(an)f(algebra)f(\(irre-)-118 2815 y(ducibilit)n(y)27 b(etc.\))47 b(are)30 b(completely)d(determined) i(b)n(y)i(the)g(represen)n(tation)d(of)-118 2915 y(its)c(generators,)f (in)h(what)h(follo)n(ws,)d(w)n(e)j(will)d(use)i(an)h(equiv)-5 b(alen)n(t)23 b(language)f(of)-118 3014 y(represen)n(tations)j(of)i (relation)e(\(1.5\))i(b)n(y)g(b)r(ounded)h(self-adjoin)n(t)d(op)r (erators.)6 3114 y(In)43 b(studying)e(families)d(of)k(self-adjoin)n(t)e (op)r(erators)g FN(A)1791 3126 y FK(1)1829 3114 y FO(,)i FN(:)14 b(:)g(:)28 b FO(,)46 b FN(A)2150 3126 y FL(n)2195 3114 y FO(,)g(as)-118 3214 y(usual,)24 b(the)i(role)e(of)h(the)h (simplest)c(families)g(of)j(op)r(erators)e(is)h(pla)n(y)n(ed)g(b)n(y)h (irre-)-118 3313 y(ducible)e(ones.)35 b(A)25 b(family)d(of)j (self-adjoin)n(t)d(op)r(erators)h FN(A)1662 3325 y FL(k)1726 3313 y FO(=)1814 3246 y Fy(R)1853 3343 y Fu(R)1913 3313 y FN(\025)1961 3325 y FL(k)2016 3313 y FN(dE)2120 3325 y FL(k)2162 3313 y FO(\()p FN(\025)2242 3325 y FL(k)2283 3313 y FO(\),)-118 3413 y FN(k)j FO(=)c(1,)27 b FN(:)14 b(:)g(:)27 b FO(,)g FN(n)p FO(,)g(is)f(irreducible,)d(if)j(there)h(is)e (no)h(non-trivial)d(\(di\013eren)n(t)j(from)-118 3513 y FN(H)42 b FO(and)35 b FP(f)p FO(0)p FP(g)p FO(\))e(subspace)i(in)f FN(H)42 b FO(in)n(v)-5 b(arian)n(t)32 b(with)i(resp)r(ect)h(to)g(all)e (op)r(erators)-118 3612 y FN(E)-57 3624 y FL(k)-16 3612 y FO(\(\001\),)j FN(k)h FO(=)c(1,)g FN(:)14 b(:)g(:)28 b FO(,)35 b FN(n)p FO(;)i(\001)d FP(2)f Fz(B)p FO(\()p FI(R)1096 3582 y FK(1)1139 3612 y FO(\).)56 b(If)35 b(the)f(op)r (erators)e(of)h(the)i(family)-118 3712 y(are)27 b(b)r(ounded,)j(the)f (irreducibilit)n(y)22 b(of)29 b(the)g(family)c(means)i(that)i(there)g (is)e(no)-118 3811 y(non-trivial)d(subspace)j(in)g FN(H)7 b FO(,)29 b(in)n(v)-5 b(arian)n(t)24 b(with)k(resp)r(ect)g(to)g(all)e (op)r(erators)g(of)-118 3911 y(the)i(family)d(\()p FN(A)373 3923 y FL(k)414 3911 y FO(\))446 3881 y FL(n)446 3935 y(k)q FK(=1)571 3911 y FO(.)p eop %%Page: 22 26 22 25 bop -118 -137 a FO(22)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(The)33 b(follo)n(wing)c(condition)h(is)i(equiv)-5 b(alen)n(t)30 b(to)j(irreducibilit)n(y:)41 b(a)33 b(collec-)-118 196 y(tion)e(of)h(self-adjoin)n(t)d(op)r(erators)h(\()p FN(A)1053 208 y FL(k)1094 196 y FO(\))1126 166 y FL(n)1126 220 y(k)q FK(=1)1283 196 y FO(is)h(irreducible)d(if)j(an)n(y)g(b)r (ounded)-118 296 y(op)r(erator)23 b FN(C)32 b FO(comm)n(uting)22 b(with)j(all)e FN(A)1096 308 y FL(k)1137 296 y FO(,)j FN(k)g FO(=)c(1,)j FN(:)14 b(:)g(:)28 b FO(,)e FN(n)f FO(\(i.e.,)g(with)g(all)e(their)-118 395 y(sp)r(ectral)j(pro)5 b(jections\),)26 b(is)g(a)h(m)n(ultiple)d(of)k(the)g(iden)n(tit)n(y)e (op)r(erator.)-95 532 y FQ(3.)33 b FO(F)-7 b(or)19 b(a)g(single)f(b)r (ounded)i(self-adjoin)n(t)d(op)r(erator)h FN(A)23 b FO(=)g FN(A)1751 501 y FM(\003)1789 532 y FO(,)f(its)c(irreducibil-)-118 631 y(it)n(y)29 b(means)g(that)h(dim)13 b FN(H)34 b FO(=)27 b(1,)j(and)h(this)e(op)r(erator)g(is)g(a)h(m)n(ultiplicatio)o(n)25 b(b)n(y)-118 731 y(a)30 b(constan)n(t,)g FN(A)f FO(=)e FN(\025)p FO(,)32 b FN(\025)c FP(2)g FI(R)p FO(,)37 b(and)31 b(the)f(sp)r(ectral)f(theorem)g(for)h(a)g(b)r(ounded)-118 830 y(self-adjoin)n(t)35 b(op)r(erator)h(giv)n(es)f(its)h(decomp)r (osition)e(in)n(to)i(irreducible)e(ones:)-118 945 y FN(A)42 b FO(=)92 878 y Fy(R)147 899 y FM(k)p FL(A)p FM(k)131 974 y(\000k)p FL(A)p FM(k)319 945 y FN(\025)14 b(dE)485 957 y FL(A)540 945 y FO(\()p FN(\025)p FO(\),)43 b(where)38 b FN(E)1030 957 y FL(A)1084 945 y FO(\()p FP(\001)p FO(\))i(is)d(the)j (sp)r(ectral)d(measure)g(of)h(the)-118 1054 y(op)r(erator)26 b FN(A)p FO(.)-71 1190 y FQ(4.)78 b FO(Irreducible)38 b(represen)n(tations)h(of)i(a)g(pair)e(of)i(b)r(ounded)h(self-adjoin)n (t)-118 1290 y(op)r(erators)25 b(exist)i(in)g(a)g(Hilb)r(ert)f(space)h (of)h(arbitrary)c(dimension.)6 1389 y(Let)k(dim)12 b FN(H)30 b FO(=)22 b FN(n)p FO(,)27 b FN(e)632 1401 y FK(1)669 1389 y FO(,)g FN(:)14 b(:)g(:)28 b FO(,)f FN(e)933 1401 y FL(n)978 1389 y FO(,)g(b)r(e)h(an)f(orthonormal)c(basis)i(in)h FN(H)7 b FO(.)37 b(T)-7 b(ak)n(e)-118 1489 y(op)r(erators)22 b FN(A)j FO(and)f FN(B)29 b FO(suc)n(h)24 b(that,)h(in)f(the)h(basis)d (\()p FN(e)1472 1501 y FL(k)1513 1489 y FO(\))1545 1459 y FL(n)1545 1513 y(k)q FK(=1)1670 1489 y FO(,)j(they)g(are)e(giv)n(en)g (b)n(y)-118 1589 y(the)28 b(matrices)78 1845 y FN(A)23 b FO(=)251 1653 y Fy(0)251 1799 y(B)251 1853 y(@)323 1717 y FN(\025)371 1729 y FK(1)698 1717 y FO(0)497 1814 y(.)529 1839 y(.)561 1864 y(.)345 1971 y(0)285 b FN(\025)720 1983 y FL(n)766 1653 y Fy(1)766 1799 y(C)766 1853 y(A)852 1845 y FN(;)97 b(B)27 b FO(=)c(\()p FN(b)1218 1857 y FL(ij)1276 1845 y FO(\))p FN(;)180 b(b)1547 1857 y FL(ij)1629 1845 y FO(=)1713 1823 y(\026)1716 1845 y FN(b)1752 1857 y FL(j)s(i)1810 1845 y FN(;)42 b(\025)1923 1857 y FL(j)1981 1845 y FP(2)24 b FI(R)p FN(;)-118 2117 y FO(where)j FN(\025)170 2129 y FL(i)221 2117 y FP(6)p FO(=)c FN(\025)357 2129 y FL(j)392 2117 y FO(,)28 b FN(i)23 b FP(6)p FO(=)f FN(j)5 b FO(,)28 b(and)f FP(8)p FN(i)g FO(there)g(exists)f FN(j)5 b FO(,)28 b FN(i)23 b FP(6)p FO(=)f FN(j)5 b FO(,)28 b(suc)n(h)f(that)h FN(b)2099 2129 y FL(ij)2180 2117 y FP(6)p FO(=)23 b(0.)6 2217 y(The)28 b(pair)e(of)i(self-adjoin)n(t)d(op) r(erators)h FN(A)p FO(,)i FN(B)t FO(,)h(is)d(irreducible.)34 b(Indeed,)28 b(if)-118 2317 y FN(C)34 b FO(=)27 b(\()p FN(c)135 2329 y FL(ij)194 2317 y FO(\))226 2287 y FL(n)226 2338 y(i;j)s FK(=1)419 2317 y FO(is)i(a)h(matrix)e(comm)n(uting)f(with) i(the)i(op)r(erators)e FN(A)h FO(and)g FN(B)t FO(,)-118 2416 y(then)e(the)g(condition)d([)p FN(A;)14 b(C)6 b FO(])24 b(=)f(0)k(giv)n(es)685 2693 y FN(C)j FO(=)861 2502 y Fy(0)861 2648 y(B)861 2701 y(@)934 2565 y FN(c)970 2577 y FK(11)1344 2565 y FO(0)1128 2662 y(.)1160 2687 y(.)1193 2712 y(.)966 2820 y(0)295 b FN(c)1339 2832 y FL(nn)1425 2502 y Fy(1)1425 2648 y(C)1425 2701 y(A)1512 2693 y FN(;)-118 2970 y FO(and)39 b([)p FN(C)q(;)14 b(B)t FO(])44 b(=)f(0)c(implies)d FN(c)828 2982 y FK(11)942 2970 y FO(=)42 b FP(\001)14 b(\001)g(\001)43 b FO(=)g FN(c)1333 2982 y FL(nn)1463 2970 y FO(=)f FN(c)p FO(,)h(i.e.,)f FN(C)49 b FO(=)43 b FN(cI)7 b FO(,)43 b(and)-118 3070 y(therefore,)27 b(the)h(pair)e FN(A)p FO(,)i FN(B)j FO(is)c (irreducible.)6 3170 y(No)n(w)d(let)f FN(H)31 b FO(b)r(e)25 b(a)f(separable)d(in\014nite-dimensional)d(Hilb)r(ert)23 b(space,)h(and)-118 3269 y(let)31 b(\()p FN(e)77 3281 y FL(k)118 3269 y FO(\))150 3239 y FM(1)150 3293 y FL(k)q FK(=1)307 3269 y FO(b)r(e)h(an)g(orthonormal)c(basis)i(in)h FN(H)7 b FO(.)49 b(A)33 b(pair)d(of)i(b)r(ounded)g(self-)-118 3369 y(adjoin)n(t)26 b(op)r(erators)g(ha)n(ving)f(the)j(follo)n(wing)c (matrix)h(represen)n(tation)393 3719 y FN(A)f FO(=)566 3452 y Fy(0)566 3598 y(B)566 3648 y(B)566 3698 y(B)566 3748 y(B)566 3801 y(@)639 3513 y FN(\025)687 3525 y FK(1)1192 3513 y FO(0)812 3610 y(.)845 3635 y(.)877 3660 y(.)988 3768 y FN(\025)1036 3780 y FL(n)661 3923 y FO(0)1169 3865 y(.)1201 3889 y(.)1233 3915 y(.)1261 3452 y Fy(1)1261 3598 y(C)1261 3648 y(C)1261 3698 y(C)1261 3748 y(C)1261 3801 y(A)1348 3719 y FN(;)96 b(B)28 b FO(=)22 b(\()p FN(b)1713 3731 y FL(ij)1772 3719 y FO(\))p FN(;)p eop %%Page: 23 27 23 26 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(23)-118 96 y(where)33 b FN(\025)176 108 y FL(i)238 96 y FP(6)p FO(=)h FN(\025)385 108 y FL(j)420 96 y FO(,)i FN(i)d FP(6)p FO(=)h FN(j)5 b FO(;)37 b FP(j)p FN(\025)810 108 y FL(k)851 96 y FP(j)d(\024)g FN(C)40 b(<)33 b FP(1)p FO(,)j FN(k)g FO(=)e(1,)h(2,)f FN(:)14 b(:)g(:)27 b FO(;)38 b FN(b)1944 108 y FL(ij)2036 96 y FO(=)2131 75 y(\026)2134 96 y FN(b)2170 108 y FL(j)s(i)2228 96 y FO(,)e FN(i)p FO(,)-118 196 y FN(j)29 b FO(=)c(1,)j(2,)g FN(:)14 b(:)g(:)28 b FO(;)h FP(8)p FN(i)23 b FP(6)p FO(=)i FN(j)33 b FP(9)p FN(b)736 208 y FL(ij)819 196 y FP(6)p FO(=)24 b(0;)1002 134 y Fy(P)1090 154 y FM(1)1090 221 y FL(j)s FK(=1)1222 196 y FP(j)p FN(b)1281 208 y FL(ij)1340 196 y FP(j)1363 166 y FK(2)1425 196 y FP(\024)g FN(K)30 b(<)24 b FP(1)29 b(8)p FN(i)23 b FO(=)h(1,)29 b(2,)f FN(:)14 b(:)g(:)28 b FO(,)-118 296 y(is)e(irreducible.)-118 433 y FQ(5.)35 b FO(In)26 b(general,)e(it)h(is)g(not)h(necessary)e(that)i(irreducible) c(pairs)h(connected)j(b)n(y)-118 533 y(the)i(relation)d(\(1.5\))i (exist)f(in)h(ev)n(ery)g(dimension.)6 632 y(P)n(airs)22 b(of)j(comm)n(uting)d(b)r(ounded)j(self-adjoin)n(t)e(op)r(erators)g FN(A)g FO(=)g FN(A)2097 602 y FM(\003)2135 632 y FO(,)j FN(B)h FO(=)-118 732 y FN(B)-51 702 y FM(\003)-13 732 y FO(,)39 b FN(AB)j FO(=)37 b FN(B)t(A)p FO(,)j(ha)n(v)n(e)35 b(only)g(one-dimensional)c(irreducible)i(represen)n(ta-)-118 831 y(tions,)44 b(dim)12 b FN(H)53 b FO(=)45 b(1,)g FN(A)h FO(=)f FN(\025)886 843 y FK(1)924 831 y FO(,)g FN(B)50 b FO(=)c FN(\025)1264 843 y FK(2)1301 831 y FO(,)f(\()p FN(\025)1449 843 y FK(1)1487 831 y FN(;)14 b(\025)1572 843 y FK(2)1610 831 y FO(\))46 b FP(2)h FI(R)1844 843 y FK(2)1887 831 y FO(.)78 b(The)42 b(join)n(t)-118 931 y(sp)r(ectral)34 b(measure)f FN(E)594 946 y FK(\()p FL(A)670 954 y Fx(1)703 946 y FL(;A)773 954 y Fx(2)805 946 y FK(\))835 931 y FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))37 b(=)f FN(E)1181 943 y FL(A)1231 951 y Fx(1)1267 931 y FO(\()p FP(\001)p FO(\))25 b FP(\012)e FN(E)1528 943 y FL(A)1578 951 y Fx(2)1615 931 y FO(\()p FP(\001)p FO(\))36 b(on)f(the)h(plane)e FI(R)2295 901 y FK(2)-118 1031 y FO(giv)n(es)41 b(a)i(decomp)r(osition)d(of)j(the)h(pair)d FN(A)1266 1043 y FK(1)1353 1031 y FO(=)1467 964 y Fy(R)1506 1060 y Fu(R)1553 1044 y Fx(2)1598 1031 y FN(\025)1646 1043 y FK(1)1698 1031 y FN(dE)1802 1046 y FK(\()p FL(A)1878 1054 y Fx(1)1911 1046 y FL(;A)1981 1054 y Fx(2)2013 1046 y FK(\))2043 1031 y FO(\()p FN(\025)2123 1043 y FK(1)2161 1031 y FN(;)14 b(\025)2246 1043 y FK(2)2283 1031 y FO(\),)-118 1130 y FN(A)-56 1142 y FK(2)4 1130 y FO(=)92 1063 y Fy(R)131 1160 y Fu(R)178 1143 y Fx(2)224 1130 y FN(\025)272 1142 y FK(2)323 1130 y FN(dE)427 1145 y FK(\()p FL(A)503 1153 y Fx(1)536 1145 y FL(;A)606 1153 y Fx(2)638 1145 y FK(\))668 1130 y FO(\()p FN(\025)748 1142 y FK(1)786 1130 y FN(;)g(\025)871 1142 y FK(2)909 1130 y FO(\))28 b(in)n(to)e(irreducible)e(ones.)-118 1268 y FQ(6.)59 b FO(It)36 b(ma)n(y)d(happ)r(en)j(that)f(there)h(are)e (no)h(pairs)e(of)i(b)r(ounded)h(self-adjoin)n(t)-118 1367 y(op)r(erators)25 b FN(A)p FO(,)j FN(B)t FO(,)g(connected)g(b)n(y) f(relation)e(\(1.5\))i(at)h(all.)6 1467 y(F)-7 b(or)23 b(example,)f(there)i(are)e(no)i(b)r(ounded)f(pairs)f(of)i(self-adjoin)n (t)d(op)r(erators)-118 1566 y FN(A)p FO(,)33 b FN(B)j FO(\(in)31 b(particular,)e(no)i(irreducible)d(pairs\),)j(connected)h(b) n(y)f(the)h(canon-)-118 1666 y(ical)e(comm)n(utation)e(relations)h (\(CCR\),)k([)p FN(A;)14 b(B)t FO(])31 b(=)g FN(iI)7 b FO(.)50 b(Indeed,)34 b(otherwise,)-118 1766 y(follo)n(wing,)23 b(e.g.,)28 b([220)n(],)g(w)n(e)f(w)n(ould)g(ha)n(v)n(e)676 1918 y FN(A)738 1884 y FL(n)783 1918 y FN(B)c FP(\000)18 b FN(B)t(A)1081 1884 y FL(n)1149 1918 y FO(=)23 b FN(i)14 b(nA)1392 1884 y FL(n)p FM(\000)p FK(1)1522 1918 y FN(;)-118 2070 y FO(and)451 2222 y FN(n)g FP(k)p FN(A)619 2188 y FL(n)p FM(\000)p FK(1)749 2222 y FP(k)22 b FO(=)h FN(n)14 b FP(k)p FN(A)p FP(k)1111 2188 y FL(n)p FM(\000)p FK(1)1263 2222 y FP(\024)23 b FO(2)14 b FP(k)p FN(A)p FP(k)1553 2188 y FL(n)1596 2222 y FP(k)p FN(B)t FP(k)p FN(:)-118 2374 y FO(Since)33 b FP(k)p FN(A)p FP(k)h(6)p FO(=)f(0,)j(the)f(latter) d(implies)f FP(k)p FN(A)p FP(k)14 b(k)p FN(B)t FP(k)32 b(\025)i FN(n=)p FO(2)f(for)h(all)e FN(n)p FO(,)k(whic)n(h)-118 2474 y(con)n(tradicts)25 b(the)j(assumption)d(that)j FN(A)g FO(and)g FN(B)j FO(are)c(b)r(ounded.)6 2573 y(The)c(fact)f(that) h(pairs)d(of)i(op)r(erators)f(satisfying)e(the)k(CCR)f(pla)n(y)f(a)h (crucial)-118 2673 y(role)34 b(in)g(mo)r(dels)g(of)h(mathematical)c(ph) n(ysics)j(stresses)g(the)i(need)g(to)f(study)-118 2773 y(b)r(oth)j(b)r(ounded)g(and)g(un)n(b)r(ounded)g(families)c(of)j(op)r (erators)f(satisfying)f(the)-118 2872 y(relations.)-118 3010 y FQ(7.)68 b FO(As)38 b(is)f(commonly)e(accepted)i(in)h(represen)n (tation)d(theory)-7 b(,)41 b(collections)-118 3109 y(of)27 b(op)r(erators)e(are)h(studied)h(up)g(to)g(unitary)f(equiv)-5 b(alence.)34 b(Tw)n(o)26 b(collections,)-118 3209 y(\()p FN(A)-24 3221 y FL(k)17 3209 y FO(\))49 3179 y FL(n)49 3232 y(k)q FK(=1)195 3209 y FO(on)20 b(a)f(Hilb)r(ert)g(space)g FN(H)7 b FO(,)22 b(and)e(\()1186 3188 y(~)1164 3209 y FN(A)1226 3221 y FL(k)1267 3209 y FO(\))1299 3179 y FL(n)1299 3232 y(k)q FK(=1)1445 3209 y FO(on)f(a)h(Hilb)r(ert)f(space)2129 3188 y(~)2107 3209 y FN(H)7 b FO(,)22 b(are)-118 3317 y(unitarily)29 b(equiv)-5 b(alen)n(t,)32 b(if)h(there)f(exists)g(a)g (unitary)f(op)r(erator)g FN(U)18 b FO(:)30 b FN(H)38 b FP(\000)-48 b(!)2285 3296 y FO(~)2263 3317 y FN(H)-118 3417 y FO(suc)n(h)27 b(that)h(the)g(diagrams)868 3590 y FN(H)1067 3542 y FL(A)1117 3551 y Fv(k)986 3590 y FP(\000)-28 b(\000)-19 b(\000)g(\000)-28 b(!)41 b FN(H)827 3755 y FL(U)879 3661 y Fy(?)879 3711 y(?)879 3760 y(y)1286 3661 y(?)1286 3711 y(?)1286 3760 y(y)1342 3755 y FL(U)890 3925 y FO(~)868 3946 y FN(H)1084 3882 y FK(~)1067 3897 y FL(A)1117 3906 y Fv(k)986 3946 y FP(\000)-28 b(\000)-19 b(\000)g(\000)-28 b(!)1298 3925 y FO(~)1276 3946 y FN(H)p eop %%Page: 24 28 24 27 bop -118 -137 a FO(24)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(are)h(comm)n(utativ)n(e)e(for)j(all)f FN(k)g FO(=)c(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(,)f(i.e.,)g FN(U)9 b(A)1568 108 y FL(k)1632 96 y FO(=)1742 75 y(~)1720 96 y FN(A)1782 108 y FL(k)1823 96 y FN(U)g FO(.)6 196 y(The)29 b(description)d(of)j(b)r(ounded)g(represen)n(tations)d(of)j(a) f FP(\003)p FO(-algebra)d Fz(B)k FO(up)-118 296 y(to)39 b(unitary)e(equiv)-5 b(alence)37 b(is)h(the)h(same)e(as)i(the)g (description)d(of)j(b)r(ounded)-118 395 y(represen)n(tations)25 b(of)i(the)h(generators)d(up)j(to)g(unitary)e(equiv)-5 b(alence.)6 495 y(The)36 b(basic)d(problem)g(of)i FP(\003)p FO(-represen)n(tation)d(theory)j(is)f(to)h(describ)r(e)f(all)-118 595 y(irreducible)d(families)g(of)j(b)r(ounded)i(self-adjoin)n(t)c(op)r (erators)g FN(A)1928 607 y FK(1)1966 595 y FO(,)j FN(:)14 b(:)g(:)27 b FO(,)37 b FN(A)2270 607 y FL(n)2316 595 y FO(,)-118 694 y(satisfying)31 b(the)j(giv)n(en)e(relations)e (\(1.4\),)35 b(up)f(to)f(unitary)g(equiv)-5 b(alence;)34 b(and)-118 794 y(this)27 b(is)f(the)i(task)f(w)n(e)h(will)c(b)r(e)k (dealing)d(with)i(in)g(the)h(sequel.)-118 1009 y FQ(1.1.5)94 b(P)m(airs)47 b(of)g(self-adjoin)m(t)f(op)s(erators)h(satisfying)f (quadratic)174 1109 y(relations)-118 1262 y FO(In)41 b(this)g(c)n(hapter,)i(w)n(e)e(will)d(study)-7 b(,)45 b(in)c(particular,)g(pairs)e(of)i(self-adjoin)n(t)-118 1362 y(b)r(ounded)28 b(op)r(erators)e FN(A)p FO(,)i FN(B)t FO(,)f(whic)n(h)g(satisfy)f(the)i(follo)n(wing)c(relation)-100 1541 y FN(P)-47 1553 y FK(2)-9 1541 y FO(\()p FN(A;)14 b(B)t FO(\))24 b(=)f FN(\013A)448 1507 y FK(2)504 1541 y FO(+)18 b FN(\014)t FP(f)p FN(A;)c(B)t FP(g)k FO(+)g FN(i)p FI(~)p FO([)p FN(B)t(;)c(A)p FO(])k(+)g FN(\015)5 b(B)1491 1507 y FK(2)1547 1541 y FO(+)18 b FN(\016)s(A)h FO(+)f FN(\017B)k FO(+)c FN(\037I)30 b FO(=)23 b(0)p FN(;)748 1666 y(\013;)14 b(\014)t(;)g FI(~)p FN(;)g(\015)5 b(;)14 b(\016)o(;)g(\017;)g(\037)22 b FP(2)h FI(R)p FN(:)702 b FO(\(1.6\))-118 1845 y FQ(1.)36 b FO(Let)28 b(us)f(start)g(with)g (the)h(homogeneous)d(quadratic)h(relation)94 1985 y FN(q)p 94 2022 41 4 v 100 2098 a(i)144 2041 y FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(\013A)582 2007 y FK(2)638 2041 y FO(+)18 b FN(\014)t FP(f)p FN(A;)c(B)t FP(g)k FO(+)g FN(\015)5 b(B)1238 2007 y FK(2)1275 2041 y FN(;)180 b(\013;)14 b(\014)t(;)g(\015)5 b(;)14 b(q)27 b FP(2)c FI(R)p FN(:)208 b FO(\(1.7\))-118 2244 y FQ(Prop)s(osition)30 b(8.)41 b FB(By)d(using)f(a)g(non-de)l(gener)l(ate)g(line)l(ar)h(tr)l (ansformation,)-118 2343 y(r)l(elation)f FO(\(1.7\))29 b FB(c)l(an)h(b)l(e)g(r)l(e)l(duc)l(e)l(d)f(to)h(one)g(of)h(the)f(fol)t (lowing)i(forms)7 b FO(:)p 405 2456 1410 4 v 403 2555 4 100 v 447 2525 a(\(0)521 2537 y FK(0)558 2525 y FO(\))p 944 2555 V 397 w(\()p FN(I)g(V)1110 2537 y FK(0)1148 2525 y FO(\))p 1813 2555 V 403 2655 V 579 2625 a(0)22 b(=)h(0)p 944 2655 V 425 w([)p FN(A;)14 b(B)t FO(])24 b(=)e(0)p 1813 2655 V 405 2658 1410 4 v 403 2758 4 100 v 447 2728 a(\()p FN(I)515 2740 y FK(0)553 2728 y FO(\))p 944 2758 V 402 w(\()p FN(V)1067 2740 y FK(0)1105 2728 y FO(\))p 1813 2758 V 403 2858 4 101 v 550 2828 a FN(A)612 2798 y FK(2)672 2828 y FO(=)h(0)p 944 2858 V 1153 2795 a FK(1)p 1153 2809 34 4 v 1158 2857 a FL(i)1196 2828 y FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(A)1581 2798 y FK(2)p 1813 2858 4 101 v 405 2861 1410 4 v 403 2961 4 100 v 447 2931 a FO(\()p FN(I)7 b(I)558 2943 y FK(0)596 2931 y FO(\))p 944 2961 V 359 w(\()p FN(V)20 b(I)1123 2943 y FK(0)1160 2931 y FO(\))p 1813 2961 V 403 3061 4 101 v 447 3031 a FN(A)509 3001 y FK(2)565 3031 y FO(+)e FN(B)715 3001 y FK(2)775 3031 y FO(=)23 b(0)p 944 3061 V 997 2998 a FK(1)p 997 3012 34 4 v 1002 3060 a FL(i)1040 3031 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)f FN(q)s FO(\()p FN(A)1498 3001 y FK(2)1554 3031 y FO(+)18 b FN(B)1704 3001 y FK(2)1741 3031 y FO(\))p 1813 3061 4 101 v 403 3160 4 100 v 944 3160 V 1252 3131 a(\()p FN(q)27 b(>)22 b FO(0\))p 1813 3160 V 405 3164 1410 4 v 403 3263 4 100 v 447 3234 a(\()p FN(I)7 b(I)g(I)601 3246 y FK(0)639 3234 y FO(\))p 944 3263 V 316 w(\()p FN(V)20 b(I)7 b(I)1166 3246 y FK(0)1203 3234 y FO(\))p 1813 3263 V 403 3363 4 101 v 447 3334 a FN(A)509 3303 y FK(2)565 3334 y FP(\000)18 b FN(B)715 3303 y FK(2)775 3334 y FO(=)23 b(0)p 944 3363 V 997 3301 a FK(1)p 997 3315 34 4 v 1002 3362 a FL(i)1040 3334 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)f FN(q)s FO(\()p FN(A)1498 3303 y FK(2)1554 3334 y FP(\000)18 b FN(B)1704 3303 y FK(2)1741 3334 y FO(\))p 1813 3363 4 101 v 403 3463 4 100 v 944 3463 V 1252 3433 a(\()p FN(q)27 b(>)22 b FO(0\))p 1813 3463 V 405 3466 1410 4 v -118 3612 a FB(Pr)l(o)l(of.)43 b FO(By)29 b(using)e(a)i(non-degenerate)e(linear)f (transformation,)g(w)n(e)j(can)f(re-)-118 3712 y(duce)23 b(the)g(symmetric)c(quadratic)h(form)i FN(\013A)1281 3682 y FK(2)1327 3712 y FO(+)8 b FN(\014)t FP(f)p FN(A;)14 b(B)t FP(g)8 b FO(+)g FN(\015)d(B)1897 3682 y FK(2)1957 3712 y FO(to)22 b(a)g(diago-)-118 3811 y(nal)i(form,)h(i.e.,)g(w)n(e)g (can)g(assume)f FN(\014)j FO(=)c(0,)i FN(\013)p FO(,)i FN(\015)g FP(2)d(f\000)p FO(1)p FN(;)14 b FO(0)p FN(;)g FO(1)p FP(g)p FO(.)33 b(If)26 b FN(q)g FO(=)d(0,)i(then)-118 3911 y(equation)h(\(1.7\))i(will)d(tak)n(e)j(one)f(of)h(the)h(forms)d (\(0)1452 3923 y FK(0)1489 3911 y FO(\){\()p FN(I)7 b(I)g(I)1717 3923 y FK(0)1755 3911 y FO(\).)39 b(If)28 b FN(q)f FP(6)p FO(=)c(0,)28 b(then)p eop %%Page: 25 29 25 28 bop -118 -137 a FJ(1.1.)36 b(In)n(tro)r(duction)26 b(to)i(represen)n(tations)c(of)k FP(\003)p FJ(-algebras)585 b FO(25)-118 96 y(b)n(y)29 b(using)e(the)j(same)d(transformation,)f(w)n (e)i(reduce)h(the)g(righ)n(t-hand)e(side)h(of)-118 196 y(equalit)n(y)i(\(1.7\))h(to)g(the)h(corresp)r(onding)d(form)h(and)h (then:)45 b(if)31 b FN(q)s(i)p FO([)p FN(A;)14 b(B)t FO(])30 b(=)f(0,)-118 296 y(w)n(e)36 b(get)h(\()p FN(I)7 b(V)284 308 y FK(0)322 296 y FO(\);)42 b(if)36 b FP(\000)p FN(q)s(i)p FO([)p FN(A;)14 b(B)t FO(])38 b(=)g FN(A)1053 266 y FK(2)1091 296 y FO(,)h(replace)c FN(B)41 b FO(b)n(y)c FN(q)s(B)k FO(to)c(get)f(\()p FN(V)2155 308 y FK(0)2193 296 y FO(\);)42 b(if)-118 395 y FP(\000)p FN(q)s(i)p FO([)p FN(A;)14 b(B)t FO(])23 b(=)f FN(A)400 365 y FK(2)451 395 y FP(\006)13 b FN(B)596 365 y FK(2)633 395 y FO(,)26 b(w)n(e)f(get)g(\()p FN(V)19 b(I)1073 407 y FK(0)1111 395 y FO(\))25 b(and)g(\()p FN(V)19 b(I)7 b(I)1505 407 y FK(0)1543 395 y FO(\))25 b(with)g FN(q)j FO(replaced)23 b(with)-118 495 y FN(q)-78 465 y FM(\000)p FK(1)11 495 y FO(;)28 b(b)n(y)f(substituting)f FN(A)i FO(with)f(sign)13 b FN(q)s(A)28 b FO(w)n(e)f(get)g FN(q)f(>)d FO(0.)p 2278 495 4 57 v 2282 442 50 4 v 2282 495 V 2331 495 4 57 v -118 658 a FQ(2.)34 b FO(By)22 b(applying)e(a)h(similar)d(argumen)n(t,) j(w)n(e)g(can)h(pro)n(v)n(e)f(the)h(follo)n(wing)c(state-)-118 757 y(men)n(t.)-118 910 y FQ(Prop)s(osition)30 b(9.)41 b FB(By)h(using)f(an)g(a\016ne)h(change)g(of)h(variables,)j(e)l (quation)-118 1009 y FO(\(1.1.5\))29 b FB(c)l(an)h(b)l(e)f(r)l(e)l(duc) l(e)l(d)h(to)g(one)g(of)g(the)g(fol)t(lowing)j(forms:)p -103 1108 2427 4 v -105 1232 4 125 v -62 1191 a FO(\(0)12 1203 y FK(0)49 1191 y FO(\))d(0)23 b(=)g(0)p 690 1232 V 427 w(\(0)807 1203 y FK(1)844 1191 y FO(\))30 b FN(\037I)g FO(=)23 b(0)p FN(;)43 b(\037)23 b FP(6)p FO(=)g(0)p 1577 1232 V 196 w(\(0)1695 1203 y FK(2)1732 1191 y FO(\))30 b FN(A)23 b FO(=)g(0)p 2322 1232 V -103 1236 2427 4 v -105 1360 4 125 v -62 1319 a(\()p FN(I)6 1331 y FK(0)44 1319 y FO(\))30 b FN(A)168 1288 y FK(2)229 1319 y FO(=)23 b(0)p 690 1360 V 374 w(\()p FN(I)801 1331 y FK(1)839 1319 y FO(\))30 b FN(A)963 1288 y FK(2)1024 1319 y FO(=)22 b FN(I)p 1577 1360 V 474 w FO(\()p FN(I)1689 1331 y FK(2)1727 1319 y FO(\))30 b FN(A)1851 1288 y FK(2)1911 1319 y FO(=)23 b FN(B)p 2322 1360 V -105 1485 V 690 1485 V 733 1443 a FO(\()p FN(I)808 1413 y FM(0)801 1464 y FK(1)839 1443 y FO(\))30 b FN(A)963 1413 y FK(2)1024 1443 y FO(=)22 b FP(\000)p FN(I)p 1577 1485 V 2322 1485 V -103 1488 2427 4 v -105 1613 4 125 v -62 1571 a FO(\()p FN(I)7 b(I)49 1583 y FK(0)87 1571 y FO(\))30 b FN(A)211 1541 y FK(2)267 1571 y FO(+)18 b FN(B)417 1541 y FK(2)478 1571 y FO(=)23 b(0)p 690 1613 V 125 w(\()p FN(I)7 b(I)844 1583 y FK(1)882 1571 y FO(\))30 b FN(A)1006 1541 y FK(2)1062 1571 y FO(+)18 b FN(B)1212 1541 y FK(2)1273 1571 y FO(=)k FN(I)p 1577 1613 V 2322 1613 V -105 1737 V 690 1737 V 733 1696 a FO(\()p FN(I)7 b(I)851 1665 y FM(0)844 1716 y FK(1)882 1696 y FO(\))30 b FN(A)1006 1665 y FK(2)1062 1696 y FO(+)18 b FN(B)1212 1665 y FK(2)1273 1696 y FO(=)k FP(\000)p FN(I)p 1577 1737 V 2322 1737 V -103 1740 2427 4 v -105 1865 4 125 v -62 1823 a FO(\()p FN(I)7 b(I)g(I)92 1835 y FK(0)130 1823 y FO(\))30 b FN(A)254 1793 y FK(2)310 1823 y FP(\000)18 b FN(B)460 1793 y FK(2)521 1823 y FO(=)23 b(0)p 690 1865 V 82 w(\()p FN(I)7 b(I)g(I)887 1835 y FK(1)925 1823 y FO(\))30 b FN(A)1049 1793 y FK(2)1105 1823 y FP(\000)18 b FN(B)1255 1793 y FK(2)1316 1823 y FO(=)k FN(I)p 1577 1865 V 2322 1865 V -105 1989 V -62 1948 a FB(or)31 b FP(f)109 1927 y FO(~)88 1948 y FN(A)o(;)206 1927 y FO(~)186 1948 y FN(B)t FP(g)23 b FO(=)g(0)p 690 1989 V 285 w FB(or)30 b FP(f)904 1927 y FO(~)882 1948 y FN(A;)1001 1927 y FO(~)981 1948 y FN(B)t FP(g)23 b FO(=)f FN(I)p 1577 1989 V 2322 1989 V -103 1993 2427 4 v -105 2117 4 125 v -62 2076 a FO(\()p FN(I)7 b(V)61 2088 y FK(0)99 2076 y FO(\))30 b([)p FN(A;)14 b(B)t FO(])24 b(=)f(0)p 690 2117 V 206 w(\()p FN(I)7 b(V)856 2088 y FK(1)894 2076 y FO(\))966 2043 y FK(1)p 966 2057 34 4 v 971 2104 a FL(i)1009 2076 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(I)p 1577 2117 4 125 v 253 w FO(\()p FN(I)7 b(V)1744 2088 y FK(2)1782 2076 y FO(\))1854 2043 y FK(1)p 1854 2057 34 4 v 1859 2104 a FL(i)1897 2076 y FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(A)p 2322 2117 4 125 v -103 2121 2427 4 v -105 2245 4 125 v -62 2204 a FO(\()p FN(V)18 2216 y FK(0)56 2204 y FO(\))128 2171 y FK(1)p 128 2185 34 4 v 133 2232 a FL(i)171 2204 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)f FN(A)557 2174 y FK(2)p 690 2245 4 125 v 733 2204 a FO(\()p FN(V)813 2216 y FK(1)851 2204 y FO(\))923 2171 y FK(1)p 923 2185 34 4 v 928 2232 a FL(i)966 2204 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(A)1351 2174 y FK(2)1407 2204 y FO(+)c FN(I)p 1577 2245 4 125 v 95 w FO(\()p FN(V)1701 2216 y FK(2)1739 2204 y FO(\))1811 2171 y FK(1)p 1811 2185 34 4 v 1816 2232 a FL(i)1854 2204 y FO([)p FN(A;)c(B)t FO(])p 2322 2245 4 125 v -105 2370 V 690 2370 V 733 2328 a(\()p FN(V)832 2298 y FM(0)813 2349 y FK(1)856 2328 y FO(\))927 2295 y FK(1)p 927 2309 34 4 v 932 2357 a FL(i)971 2328 y FO([)p FN(A;)g(B)t FO(])23 b(=)g FN(A)1356 2298 y FK(2)1412 2328 y FP(\000)18 b FN(I)p 1577 2370 4 125 v 395 w FO(=)23 b FN(A)2076 2298 y FK(2)2132 2328 y FO(+)18 b FN(B)p 2322 2370 V -103 2373 2427 4 v -105 2498 4 125 v -62 2456 a FO(\()p FN(V)i(I)74 2468 y FK(0)111 2456 y FO(\))183 2423 y FK(1)p 183 2437 34 4 v 188 2485 a FL(i)226 2456 y FO([)p FN(A;)14 b(B)t FO(])p 690 2498 4 125 v 295 w(\()p FN(V)19 b(I)868 2468 y FK(1)906 2456 y FO(\))978 2423 y FK(1)p 978 2437 34 4 v 983 2485 a FL(i)1021 2456 y FO([)p FN(A;)14 b(B)t FO(])p 1577 2498 4 125 v 2322 2498 V -105 2597 4 100 v 129 2567 a(=)23 b FN(q)s FO(\()p FN(A)351 2537 y FK(2)407 2567 y FO(+)18 b FN(B)557 2537 y FK(2)595 2567 y FO(\))p FN(;)p 690 2597 V 222 w FO(=)23 b FN(q)s FO(\()p FN(A)1094 2537 y FK(2)1150 2567 y FO(+)18 b FN(B)1300 2537 y FK(2)1338 2567 y FO(\))h(+)f FN(I)7 b(;)p 1577 2597 V 2322 2597 V -105 2722 4 125 v 458 2680 a(q)26 b(>)d FO(0)p 690 2722 V 694 w FN(q)k FP(6)p FO(=)22 b(0)p 1577 2722 V 2322 2722 V -103 2725 2427 4 v -105 2850 4 125 v -62 2808 a(\()p FN(V)e(I)7 b(I)117 2820 y FK(0)154 2808 y FO(\))226 2775 y FK(1)p 226 2789 34 4 v 231 2837 a FL(i)269 2808 y FO([)p FN(A;)14 b(B)t FO(])p 690 2850 4 125 v 252 w(\()p FN(V)19 b(I)7 b(I)911 2820 y FK(1)949 2808 y FO(\))1021 2775 y FK(1)p 1021 2789 34 4 v 1026 2837 a FL(i)1064 2808 y FO([)p FN(A;)14 b(B)t FO(])p 1577 2850 4 125 v 2322 2850 V -105 2949 4 100 v 129 2919 a(=)23 b FN(q)s FO(\()p FN(A)351 2889 y FK(2)407 2919 y FP(\000)18 b FN(B)557 2889 y FK(2)595 2919 y FO(\))p FN(;)p 690 2949 V 222 w FO(=)23 b FN(q)s FO(\()p FN(A)1094 2889 y FK(2)1150 2919 y FP(\000)18 b FN(B)1300 2889 y FK(2)1338 2919 y FO(\))h(+)f FN(I)7 b(;)p 1577 2949 V 2322 2949 V -105 3074 4 125 v 458 3032 a(q)26 b(>)d FO(0)p 690 3074 V 694 w FN(q)k FP(6)p FO(=)22 b(0)p 1577 3074 V 2322 3074 V -103 3077 2427 4 v 6 3214 a(Our)j(next)g(goal)e(is)g(to)i (describ)r(e)f(for)g(eac)n(h)g(of)h(the)h(relations)c(\(0)1980 3226 y FK(0)2017 3214 y FO(\){\()p FN(V)d(I)7 b(I)2269 3226 y FK(0)2306 3214 y FO(\))-118 3313 y(pairs)26 b(of)h(b)r(ounded)h (self-adjoin)n(t)d(op)r(erators,)h(whic)n(h)g(satisfy)h(this)f (relation.)6 3413 y(Considering)f(the)j(solutions)d(of)i(the)h (equations)e(w)n(e)h(ha)n(v)n(e:)6 3513 y(\(0)80 3525 y FK(1)117 3513 y FO(\))h FN(\037I)i FO(=)23 b(0,)k FN(\037)c FP(6)p FO(=)g(0.)36 b(There)28 b(are)e(no)h(pairs)f FN(A)p FO(,)i FN(B)k FO(whic)n(h)26 b(satisfy)h(\(0)2216 3525 y FK(1)2253 3513 y FO(\);)6 3612 y(\(0)80 3624 y FK(2)117 3612 y FO(\))d FN(A)f FO(=)g(0.)35 b(Since)22 b FN(B)27 b FO(=)c FN(B)903 3582 y FM(\003)941 3612 y FO(,)h(it)e(is)g(an)h (arbitrary)d(b)r(ounded)j(self-adjoin)n(t)-118 3712 y(op)r(erator;)36 b(the)f(only)e(irreducible)d(represen)n(tations)i(are)h (one-dimensional,)-118 3811 y FN(A)23 b FO(=)g(0,)k FN(B)h FO(=)22 b FN(b)p FO(,)28 b(and)f(their)g(structure)g(is)g(giv)n(en)f(b) n(y)h(the)h(structure)f(theorem)-118 3911 y(for)g(a)g(single)e(op)r (erator)h FN(B)t FO(;)p eop %%Page: 26 30 26 29 bop -118 -137 a FO(26)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(\()p FN(I)74 108 y FK(0)112 96 y FO(\))35 b FN(A)241 66 y FK(2)311 96 y FO(=)e(0.)55 b(Because)33 b FN(A)h FO(=)f FN(A)1109 66 y FM(\003)1147 96 y FO(,)i(w)n(e)f(ha)n(v)n(e)e FN(A)1593 66 y FK(2)1664 96 y FO(=)h FN(A)h FO(=)f(0,)i(and)e(the)-118 196 y(case)23 b(\()p FN(I)122 208 y FK(0)160 196 y FO(\))h(is)e (similar)d(to)24 b(\(0)736 208 y FK(2)773 196 y FO(\).)36 b(The)24 b(structure)f(of)g(an)n(y)g(solution)f(of)h(equation)-118 296 y(\()p FN(I)-50 308 y FK(0)-12 296 y FO(\))40 b(is)e(the)i(follo)n (wing:)56 b FN(A)43 b FO(=)f(0,)g FN(B)k FO(=)1252 229 y Fy(R)1291 325 y Fu(R)1338 309 y Fx(1)1383 296 y FN(\025)14 b(dE)1549 308 y FL(B)1607 296 y FO(\()p FN(\025)p FO(\),)44 b(where)38 b FN(E)2098 308 y FL(B)2156 296 y FO(\()p FP(\001)p FO(\))i(is)-118 395 y(the)31 b(resolution)d(of)j(the)h(iden)n (tit)n(y)d(for)h(the)i(op)r(erator)d FN(B)35 b FO(concen)n(trated)30 b(on)h(a)-118 495 y(compact)26 b(set)i FN(K)g FP(\032)23 b FI(R)p FO(;)6 595 y(\()p FN(I)81 564 y FM(0)74 615 y FK(1)112 595 y FO(\))28 b FN(A)234 564 y FK(2)295 595 y FO(=)23 b FP(\000)p FN(I)7 b FO(.)36 b(This)27 b(equation)f(do)r(es)h (not)h(ha)n(v)n(e)e(solutions.)6 694 y(\()p FN(I)74 706 y FK(2)112 694 y FO(\))h FN(A)233 664 y FK(2)294 694 y FO(=)c FN(B)t FO(.)36 b(The)27 b(structure)f(of)h(an)n(y)f(b)r (ounded)h(solution)d(of)i(equation)-118 794 y(\()p FN(I)-50 806 y FK(2)-12 794 y FO(\))19 b(has)f(the)h(form)e FN(A)23 b FO(=)672 727 y Fy(R)711 823 y Fu(R)758 807 y Fx(1)803 794 y FN(\025)14 b(dE)969 806 y FL(A)1024 794 y FO(\()p FN(\025)p FO(\),)22 b FN(B)27 b FO(=)1358 727 y Fy(R)1398 823 y Fu(R)1445 807 y Fx(1)1490 794 y FN(\025)1538 764 y FK(2)1590 794 y FN(dE)1694 806 y FL(A)1748 794 y FO(\()p FN(\025)p FO(\),)22 b(where)c FN(E)2197 806 y FL(A)2251 794 y FO(\()p FP(\001)p FO(\))-118 893 y(is)27 b(the)h(resolution)d(of) j(the)h(iden)n(tit)n(y)d(for)i(the)g(op)r(erator)e FN(A)p FO(,)j(concen)n(trated)e(on)-118 993 y(a)g(compact)f(set)i FN(K)g FP(\032)23 b FI(R)p FO(;)6 1093 y(\()p FN(I)7 b(I)117 1105 y FK(0)155 1093 y FO(\))28 b FN(A)277 1063 y FK(2)333 1093 y FO(+)18 b FN(B)483 1063 y FK(2)544 1093 y FO(=)k(0.)37 b(Here,)27 b FN(A)d FO(=)e FN(B)27 b FO(=)c(0;)6 1192 y(\()p FN(I)7 b(I)124 1162 y FM(0)117 1213 y FK(1)155 1192 y FO(\))28 b FN(A)277 1162 y FK(2)333 1192 y FO(+)18 b FN(B)483 1162 y FK(2)544 1192 y FO(=)k FP(\000)p FN(I)7 b FO(.)37 b(There)27 b(are)g(no)g(solutions.)6 1292 y(The)h(rest)g(of)f(the)i(relations)24 b(can)k(b)r(e)g(divided)e (in)n(to)h(four)g(groups:)36 b FP(\003)p FO(-wild)-118 1392 y(relations,)25 b FN(F)296 1404 y FK(4)334 1392 y FO(-relations,)g(Lie)j(algebras)d(and)k(their)e(nonlinear)f (transforma-)-118 1491 y(tions,)g(and)i FN(q)s FO(-relations)c(\(see)j (the)h(table)f(b)r(elo)n(w\))p -72 1595 2365 4 v -74 1720 4 125 v 831 1678 a FP(\003)p FO(-wild)e(relations)p 2291 1720 V -72 1723 2365 4 v -74 1847 4 125 v -31 1806 a(\(0)43 1818 y FK(0)81 1806 y FO(\))i(0)c(=)g(0)p 663 1847 V 371 w(\()p FN(I)774 1818 y FK(1)812 1806 y FO(\))28 b FN(A)934 1776 y FK(2)995 1806 y FO(=)22 b FN(I)p 1548 1847 V 2291 1847 V -72 1851 2365 4 v -74 1975 4 125 v 896 1934 a(F)949 1946 y FK(4)987 1934 y FO(-relations)p 2291 1975 V -72 1979 2365 4 v -74 2103 4 125 v 663 2103 V 706 2062 a(\()p FN(I)7 b(I)817 2074 y FK(1)855 2062 y FO(\))28 b FN(A)977 2032 y FK(2)1033 2062 y FO(+)18 b FN(B)1183 2032 y FK(2)1244 2062 y FO(=)k FN(I)p 1548 2103 V 2291 2103 V -72 2106 2365 4 v -74 2231 4 125 v -31 2190 a FO(\()p FN(I)7 b(I)g(I)123 2202 y FK(0)162 2190 y FO(\))27 b FP(f)p FN(A;)14 b(B)t FP(g)23 b FO(=)g(0)p 663 2231 V 82 w(\()p FN(I)7 b(I)g(I)860 2202 y FK(1)898 2190 y FO(\))28 b FP(f)p FN(A;)14 b(B)t FP(g)22 b FO(=)h FN(I)p 1548 2231 V 2291 2231 V -72 2234 2365 4 v -74 2359 4 125 v 231 2317 a FO(Lie)j(algebras)f(and)i(their)g(nonlinear)e (transformations)p 2291 2359 V -72 2362 2365 4 v -74 2487 4 125 v -31 2445 a(\()p FN(I)7 b(V)92 2457 y FK(0)130 2445 y FO(\))28 b([)p FN(A;)14 b(B)t FO(])24 b(=)f(0)p 663 2487 V 150 w(\()p FN(I)7 b(V)829 2457 y FK(1)867 2445 y FO(\))937 2413 y FK(1)p 937 2427 34 4 v 942 2474 a FL(i)980 2445 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(I)p 1548 2487 4 125 v 253 w FO(\()p FN(I)7 b(V)1715 2457 y FK(2)1753 2445 y FO(\))1823 2413 y FK(1)p 1823 2427 34 4 v 1828 2474 a FL(i)1866 2445 y FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(A)p 2291 2487 4 125 v -72 2490 2365 4 v -74 2615 4 125 v -31 2573 a FO(\()p FN(V)49 2585 y FK(0)87 2573 y FO(\))157 2540 y FK(1)p 157 2554 34 4 v 162 2602 a FL(i)200 2573 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)f FN(A)586 2543 y FK(2)p 663 2615 4 125 v 706 2573 a FO(\()p FN(V)786 2585 y FK(1)824 2573 y FO(\))894 2540 y FK(1)p 894 2554 34 4 v 899 2602 a FL(i)937 2573 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(A)1322 2543 y FK(2)1378 2573 y FO(+)c FN(I)p 1548 2615 4 125 v 95 w FO(\()p FN(V)1672 2585 y FK(2)1710 2573 y FO(\))1752 2540 y FK(1)p 1752 2554 34 4 v 1757 2602 a FL(i)1795 2573 y FO([)p FN(A;)c(B)t FO(])p 2291 2615 4 125 v -74 2739 V 663 2739 V 706 2698 a(\()p FN(V)805 2667 y FM(0)786 2718 y FK(1)829 2698 y FO(\))898 2665 y FK(1)p 898 2679 34 4 v 903 2726 a FL(i)942 2698 y FO([)p FN(A;)g(B)t FO(])23 b(=)g FN(A)1327 2667 y FK(2)1383 2698 y FP(\000)18 b FN(I)p 1548 2739 4 125 v 393 w FO(=)23 b FN(A)2045 2667 y FK(2)2101 2698 y FO(+)18 b FN(B)p 2291 2739 V -72 2742 2365 4 v -74 2867 4 125 v 922 2825 a(q)s FO(-relations)p 2291 2867 V -72 2870 2365 4 v -74 2995 4 125 v -31 2953 a(\()p FN(V)i(I)105 2965 y FK(0)142 2953 y FO(\))212 2921 y FK(1)p 212 2935 34 4 v 217 2982 a FL(i)255 2953 y FO([)p FN(A;)14 b(B)t FO(])p 663 2995 4 125 v 239 w(\()p FN(V)19 b(I)841 2965 y FK(1)879 2953 y FO(\))949 2921 y FK(1)p 949 2935 34 4 v 954 2982 a FL(i)992 2953 y FO([)p FN(A;)14 b(B)t FO(])p 1548 2995 4 125 v 2291 2995 V -74 3094 4 100 v 102 3065 a(=)23 b FN(q)s FO(\()p FN(A)324 3034 y FK(2)380 3065 y FO(+)18 b FN(B)530 3034 y FK(2)568 3065 y FO(\))p FN(;)p 663 3094 V 220 w FO(=)23 b FN(q)s FO(\()p FN(A)1065 3034 y FK(2)1121 3065 y FO(+)18 b FN(B)1271 3034 y FK(2)1309 3065 y FO(\))h(+)f FN(I)7 b(;)p 1548 3094 V 2291 3094 V -74 3219 4 125 v 431 3177 a(q)26 b(>)d FO(0)p 663 3219 V 692 w FN(q)j FP(6)p FO(=)d(0)p 1548 3219 V 2291 3219 V -72 3222 2365 4 v -74 3347 4 125 v -31 3305 a(\()p FN(V)d(I)7 b(I)148 3317 y FK(0)185 3305 y FO(\))255 3273 y FK(1)p 255 3287 34 4 v 260 3334 a FL(i)298 3305 y FO([)p FN(A;)14 b(B)t FO(])p 663 3347 4 125 v 196 w(\()p FN(V)19 b(I)7 b(I)884 3317 y FK(1)922 3305 y FO(\))992 3273 y FK(1)p 992 3287 34 4 v 997 3334 a FL(i)1035 3305 y FO([)p FN(A;)14 b(B)t FO(])p 1548 3347 4 125 v 2291 3347 V -74 3446 4 100 v 102 3417 a(=)23 b FN(q)s FO(\()p FN(A)324 3386 y FK(2)380 3417 y FP(\000)18 b FN(B)530 3386 y FK(2)568 3417 y FO(\))p FN(;)p 663 3446 V 220 w FO(=)23 b FN(q)s FO(\()p FN(A)1065 3386 y FK(2)1121 3417 y FP(\000)18 b FN(B)1271 3386 y FK(2)1309 3417 y FO(\))h(+)f FN(I)7 b(;)p 1548 3446 V 2291 3446 V -74 3571 4 125 v 431 3529 a(q)26 b(>)d FO(0)p 663 3571 V 692 w FN(q)j FP(6)p FO(=)d(0)p 1548 3571 V 2291 3571 V -72 3574 2365 4 v 6 3712 a(In)37 b(what)f(follo)n(ws,)f(our)h(aim)e (is,)j(in)f(particular,)f(to)h(study)g(irreducible)-118 3811 y(represen)n(tations)e(for)j(eac)n(h)g(of)h(these)f(groups)f(of)i (relations.)63 b(Represen)n(ta-)-118 3911 y(tions)31 b(of)i FN(F)243 3923 y FK(4)280 3911 y FO(-relations)c(are)j(describ)r (ed)f(in)h(Section)f(1.2.3,)i(represen)n(tations)p eop %%Page: 27 31 27 30 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(27)-118 96 y(of)28 b(Lie)f(algebras)f(and)i(their)f(nonlinear)e (transformations)g(b)n(y)j(b)r(ounded)g(op-)-118 196 y(erators)g(are)g(describ)r(ed)h(in)f(Section)h(1.3.1,)g(represen)n (tations)e(of)i FN(q)s FO(-relations)-118 296 y(are)22 b(describ)r(ed)g(in)g(Sections)f(1.4.1{1.4.3.)33 b(Ab)r(out)23 b FP(\003)p FO(-wild)e(relations)e(see)k(Sec-)-118 395 y(tion)k(3.1.5.)-118 639 y FG(1.2)112 b Fq(F)200 654 y Fp(n)247 639 y FG(-algebras)40 b(and)e(their)f(represen)m(tations) -118 822 y FQ(1.2.1)94 b(Ab)s(out)31 b FP(\003)p FQ(-represen)m (tations)g(of)h FN(F)1375 834 y FL(n)1420 822 y FQ(-algebras)-118 976 y(1.)61 b FO(Let)37 b FN(F)229 988 y FL(n)310 976 y FO(denote)f(the)h(standard)e(p)r(olynomial)c(of)36 b(degree)f FN(n)h FO(in)f FN(n)h FO(non-)-118 1075 y(comm)n(uting)24 b(v)-5 b(ariables:)243 1271 y FN(F)296 1283 y FL(n)341 1271 y FO(\()p FN(x)420 1283 y FK(1)458 1271 y FN(;)14 b(x)542 1283 y FK(2)580 1271 y FN(;)g(:)g(:)g(:)f(;)h(x)811 1283 y FL(n)857 1271 y FO(\))23 b(=)1023 1192 y Fy(X)1000 1370 y FL(\033)r FM(2)p FL(S)1126 1378 y Fv(n)1167 1271 y FO(\()p FP(\000)p FO(1\))1338 1237 y FL(p)p FK(\()p FL(\033)r FK(\))1468 1271 y FN(x)1515 1286 y FL(\033)r FK(\(1\))1659 1271 y FN(:)14 b(:)g(:)g(x)1817 1286 y FL(\033)r FK(\()p FL(n)p FK(\))1955 1271 y FN(;)-118 1541 y FO(where)29 b FN(p)p FO(\()p FN(\033)s FO(\))h(is)e(the)h(parit) n(y)f(of)h(the)h(p)r(erm)n(utation)d FN(\033)s FO(,)j FN(S)1658 1553 y FL(n)1732 1541 y FO(is)e(the)i(symmetric)-118 1641 y(group.)35 b(In)25 b(the)h(sequel,)e(w)n(e)h(sa)n(y)f(that)i FN(A)g FO(is)e(a)h FN(F)1387 1653 y FL(n)1432 1641 y FO(-algebra,)e(if)31 b FP(8)p FN(x)1946 1653 y FK(1)1983 1641 y FN(;)14 b(:)g(:)g(:)g(;)g(x)2215 1653 y FL(n)2283 1641 y FP(2)-118 1741 y FN(A)p FO(,)28 b(w)n(e)f(ha)n(v)n(e)g(that)g FN(F)541 1753 y FL(n)587 1741 y FO(\()p FN(x)666 1753 y FK(1)704 1741 y FN(;)14 b(:)g(:)g(:)g(;)g(x)936 1753 y FL(n)981 1741 y FO(\))23 b(=)g(0.)6 1841 y(The)31 b(follo)n(wing)c (Amitsur{Levitski)f(theorem)k(tak)n(es)g(place)f([5]:)43 b FB(the)33 b(al-)-118 1940 y(gebr)l(a)d FN(M)180 1952 y FL(n)225 1940 y FO(\()p FI(C)15 b FO(\))36 b FB(is)30 b(a)h FN(F)594 1952 y FK(2)p FL(n)672 1940 y FB(-algebr)l(a,)h(but)d (not)g(a)h FN(F)1420 1952 y FK(2)p FL(n)p FM(\000)p FK(1)1584 1940 y FB(-algebr)l(a)p FO(.)6 2040 y FN(F)59 2052 y FL(n)105 2040 y FO(-algebras)21 b(form)j(one)g(of)h(the)g(most)f (simple)e(class)h(of)i(algebras,)d(if)i(con-)-118 2140 y(sidered)h(from)g(the)j(standp)r(oin)n(t)d(of)i(the)g(structure)f(of)h (irreducible)c(represen-)-118 2240 y(tations.)-118 2407 y FQ(Theorem)30 b(2.)41 b FB(Consider)31 b(the)f(fol)t(lowing)i (statements:)-40 2574 y FO(\(i\))41 b FB(ther)l(e)36 b(exists)f(a)h(r)l(esidual)h(family)g FA(L)f FB(of)g(irr)l(e)l(ducible) h(r)l(epr)l(esentations)89 2674 y FO(Irrep)13 b FN(A)24 b FP(\033)e FA(L)i FP(3)f FN(\031)33 b FB(such)d(that)g FO(dim)12 b FN(H)1275 2686 y FL(\031)1343 2674 y FP(\024)23 b FN(n)29 b FB(for)i(al)t(l)g FN(\031)26 b FP(2)d FA(L)p FO(;)-63 2842 y(\(ii\))40 b FB(ther)l(e)24 b(exists)f(a)h(r)l(esidual)h (family)g FA(L)f FB(of)h(r)l(epr)l(esentations)f FO(Rep)14 b FN(A)23 b FP(\033)g FA(L)g FP(3)89 2942 y FN(\031)33 b FB(such)d(that)g FO(dim)12 b FN(H)749 2954 y FL(\031)817 2942 y FP(\024)23 b FN(n)30 b FB(for)g(al)t(l)h FN(\031)26 b FP(2)e FA(L)p FO(;)-86 3109 y(\(iii\))39 b FN(A)30 b FB(is)h(a)f FN(F)396 3121 y FK(2)p FL(n)474 3109 y FB(-algebr)l(a)6 b FO(;)-84 3277 y(\(iv\))41 b FB(for)31 b(any)f FN(\031)d FP(2)c FO(Irrep)13 b FN(A)p FB(,)30 b FO(dim)12 b FN(H)1063 3289 y FL(\031)1132 3277 y FP(\024)22 b FN(n)p FB(.)-118 3445 y(We)31 b(have)g(the)g(fol)t(lowing)i(implic)l (ations:)47 b FO(\(i\))30 b FP(\))h FO(\(ii\))e FP(\))h FO(\(iii\))f FP(\))h FO(\(iv\))p FB(.)41 b(Nei-)-118 3544 y(ther)30 b(of)g(the)g(inverse)h(implic)l(ations)g(hold.)-118 3712 y(Pr)l(o)l(of.)43 b FO(Here)25 b(w)n(e)f(will)e(only)i(pro)n(v)n (e)f(that)j(\(ii\))d FP(\))i FO(\(iii\),)f(since)g(it)g(is)g(this)g (state-)-118 3811 y(men)n(t)35 b(that)i(will)c(b)r(e)k(used)f(later)e (in)i(examples)d(to)j(pro)n(v)n(e)f(that)h(the)h(corre-)-118 3911 y(sp)r(onding)26 b(algebra)f(is)h(a)i FN(F)732 3923 y FK(2)p FL(n)810 3911 y FO(-algebra.)p eop %%Page: 28 32 28 31 bop -118 -137 a FO(28)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(Assume)30 b(that)h(\(ii\))e(holds,)h(but)i FN(A)e FO(is)g(not)h(a)f FN(F)1496 108 y FK(2)p FL(n)1574 96 y FO(-algebra.)43 b(Then)31 b(there)-118 196 y(exist)f FN(x)129 208 y FK(1)167 196 y FO(,)h FN(:)14 b(:)g(:)28 b FO(,)k FN(x)448 208 y FK(2)p FL(n)556 196 y FP(2)e FN(A)i FO(suc)n(h)f(that)g FN(F)1162 208 y FK(2)p FL(n)1241 196 y FO(\()p FN(x)1320 208 y FK(1)1358 196 y FN(;)14 b(:)g(:)g(:)g(;)g(x)1590 208 y FK(2)p FL(n)1668 196 y FO(\))30 b(=)f FN(x)h FP(6)p FO(=)e(0.)48 b(Let)32 b(us)-118 296 y(c)n(ho)r(ose)26 b FN(\031)h FP(2)c FA(L)28 b FO(suc)n(h)f(that)h FN(\031)s FO(\()p FN(x)p FO(\))d FP(6)p FO(=)d(0.)37 b(Then)28 b(w)n(e)f(get)67 477 y FN(\031)s FO(\()p FN(F)202 489 y FK(2)p FL(n)282 477 y FO(\()p FN(x)361 489 y FK(1)399 477 y FN(;)14 b(:)g(:)g(:)f(;)h(x)630 489 y FK(2)p FL(n)709 477 y FO(\)\))23 b(=)g FN(F)937 489 y FK(2)p FL(n)1016 477 y FO(\()p FN(\031)s FO(\()p FN(x)1177 489 y FK(1)1215 477 y FO(\))p FN(;)14 b(:)g(:)g(:)g(;)g(\031)s FO(\()p FN(x)1561 489 y FK(2)p FL(n)1640 477 y FO(\)\))24 b(=)f FN(\031)s FO(\()p FN(x)p FO(\))h FP(6)p FO(=)f(0)p FN(:)-118 658 y FO(But)31 b(since)g(dim)12 b FN(H)479 670 y FL(\031)553 658 y FP(\024)29 b FN(n)p FO(,)j(this)f(con)n(tradicts)e(the)i (Amitsur{Levitski)c(theo-)-118 757 y(rem.)6 857 y(Neither)j(of)f(the)i (in)n(v)n(erse)c(implications)e(of)30 b(Theorem)e(1)i(holds.)42 b(Indeed,)-118 957 y(to)d(see)f(that)i(\(ii\))e(do)r(es)g(not)h(imply)e (\(i\),)k(and)e(\(iv\))g(do)r(es)f(not)h(imply)e(\(iii\),)-118 1056 y(consider)22 b(the)j(nilp)r(oten)n(t)e(algebra)f(of)i(complex)f FN(n)12 b FP(\002)g FN(n)24 b FO(matrices)e(of)i(the)h(form)753 1360 y FN(X)30 b FO(=)939 1168 y Fy(0)939 1314 y(B)939 1367 y(@)1012 1232 y FO(0)262 b FP(\003)1141 1329 y FO(.)1173 1354 y(.)1206 1379 y(.)1012 1486 y(0)g(0)1358 1168 y Fy(1)1358 1314 y(C)1358 1367 y(A)1444 1360 y FN(;)-118 1664 y FO(whic)n(h)26 b(only)h(has)g(the)h(trivial)c(irreducible)g (represen)n(tation)g(for)k(an)n(y)e FN(n)d FP(2)h FI(N)t FO(.)6 1763 y(Condition)30 b(\(iii\))f(do)r(es)i(not)g(imply)e(\(ii\).) 47 b(F)-7 b(or)31 b(example,)f(the)i(algebra)d(of)-118 1863 y(matrices)c FN(X)k FP(2)23 b FN(M)472 1875 y FL(n)517 1863 y FO(\()p FI(C)15 b FO(\))34 b(of)28 b(the)g(form)595 2249 y FN(X)i FO(=)781 1982 y Fy(0)781 2128 y(B)781 2178 y(B)781 2228 y(B)781 2278 y(B)781 2331 y(@)854 2043 y FN(a)898 2055 y FK(11)1086 2043 y FP(\003)117 b FN(:)14 b(:)g(:)133 b FP(\003)890 2198 y FO(0)1095 2132 y(.)1095 2165 y(.)1095 2198 y(.)1484 2132 y(.)1484 2165 y(.)1484 2198 y(.)899 2287 y(.)899 2320 y(.)899 2353 y(.)164 b FP(\003)117 b FN(:)14 b(:)g(:)133 b FP(\003)890 2453 y FO(0)119 b FN(:)14 b(:)g(:)131 b FO(0)117 b FN(a)1482 2465 y FK(11)1553 1982 y Fy(1)1553 2128 y(C)1553 2178 y(C)1553 2228 y(C)1553 2278 y(C)1553 2331 y(A)-118 2630 y FO(is)30 b(a)g FN(F)94 2642 y FK(2)p FL(n)p FM(\000)p FK(2)258 2630 y FO(-algebra.)44 b(But)31 b(for)g(an)n(y)f(represen)n (tation)e FN(\031)s FO(,)33 b(dim)12 b FN(H)1951 2642 y FL(\031)2025 2630 y FP(\024)28 b FN(n)21 b FP(\000)f FO(1,)-118 2729 y(and)27 b(the)h(nilp)r(oten)n(t)e(elemen)n(t)673 3083 y FN(S)i FO(=)839 2841 y Fy(0)839 2988 y(B)839 3037 y(B)839 3087 y(B)839 3140 y(@)912 2905 y FO(0)110 b(1)290 b(0)1041 3002 y(.)1074 3027 y(.)1106 3052 y(.)1221 3002 y(.)1253 3027 y(.)1286 3052 y(.)1244 3160 y(0)110 b(1)912 3259 y(0)442 b(0)1438 2841 y Fy(1)1438 2988 y(C)1438 3037 y(C)1438 3087 y(C)1438 3140 y(A)1524 3083 y FN(;)-118 3446 y FO(w)n(e)34 b(ha)n(v)n(e)f(that)i FN(\031)s FO(\()p FN(S)534 3416 y FL(n)p FM(\000)p FK(1)664 3446 y FO(\))g(=)f(\()p FN(\031)s FO(\()p FN(S)5 b FO(\)\))1064 3416 y FL(n)p FM(\000)p FK(1)1230 3446 y FO(=)34 b(0,)i(since)d FN(\031)s FO(\()p FN(S)5 b FO(\))35 b(is)e(a)h(nilp)r(oten)n(t)-118 3546 y(elemen)n(t)g(in)h FN(M)383 3558 y FL(n)p FM(\000)p FK(1)513 3546 y FO(\()p FI(C)15 b FO(\))q(.)68 b(Hence)36 b(suc)n(h)g(represen)n(tations)d(do)j(not)g(separate)-118 3646 y FN(S)-62 3616 y FL(n)p FM(\000)p FK(1)95 3646 y FO(and)28 b(the)g(zero)e(elemen)n(t)g(of)i(the)g(algebra.)p 2278 3646 4 57 v 2282 3593 50 4 v 2282 3646 V 2331 3646 4 57 v -118 3811 a FQ(2.)43 b FO(If)30 b Fz(A)f FO(is)g(a)g FP(\003)p FO(-algebra,)e(and)j(one)f(only)g(considers)e(its)i FP(\003)p FO(-representations,)-118 3911 y(then,)f(eviden)n(tly)-7 b(,)26 b(\(i\))h FP(,)h FO(\(ii\).)p eop %%Page: 29 33 29 32 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(29)6 96 y(\(iii\))31 b(do)r(es)h(not)h(imply)d(\(ii\),)j(since,)g (for)f(example,)g(the)h(algebra)d FN(M)2169 108 y FL(n)2214 96 y FO(\()p FI(C)15 b FO(\))-118 196 y(with)23 b(a)g(non-standard)f (in)n(v)n(olution)e(do)r(es)j(not)h(ha)n(v)n(e)f(non-zero)f FP(\003)p FO(-represen)n(ta-)-118 296 y(tions.)6 395 y(Condition)d(\(iv\))h(do)r(es)g(not)g(imply)e(\(iii\).)32 b(F)-7 b(or)20 b(example,)f(the)i(W)-7 b(eyl)20 b FP(\003)p FO(-alge-)-118 495 y(bra)30 b FI(C)15 b FP(h)q FN(P)47 b FO(=)28 b FN(P)378 465 y FM(\003)416 495 y FN(;)14 b(Q)29 b FO(=)f FN(Q)707 465 y FM(\003)774 495 y FP(j)i FO([)p FN(P)r(;)14 b(Q)p FO(])29 b(=)g FN(iI)7 b FP(i)31 b FO(of)g(di\013eren)n(tial)d(op)r(erators)h(with)-118 595 y(the)k(co)r(e\016cien)n(ts)e(b)r(eing)h(p)r(olynomials)c(in)k(one) h(v)-5 b(ariable)30 b(do)r(es)i(not)h(ha)n(v)n(e)e FP(\003)p FO(-)-118 694 y(represen)n(tations)25 b(in)i(b)r(ounded)h(op)r (erators,)e(but)j(it)e(is)g(not)g(a)h FN(F)1875 706 y FL(n)1920 694 y FO(-algebra)d(for)-118 794 y(an)n(y)i FN(n)c FP(2)g FI(N)t FO(.)-118 942 y FQ(3.)43 b FO(If)30 b Fz(A)g FO(is)e(a)i FN(C)397 912 y FM(\003)435 942 y FO(-algebra)d(then)j(all)e(conditions)f(of)j(the)h(lemma)26 b(are)j(equiv-)-118 1042 y(alen)n(t,)k(since)f(for)h(a)g(Banac)n(h)f (semi-simple)27 b(algebra)k Fz(A)p FO(,)j(the)g(set)f(Irrep)13 b Fz(A)33 b FO(of)-118 1142 y(its)j(irreducible)d(represen)n(tations)h (is)i(a)h(residual)d(family)g(\(see)j([147)n(]\))h(and,)-118 1241 y(therefore,)27 b(\()p FN(iv)s FO(\))c FP(\))g FO(\()p FN(i)p FO(\).)-118 1457 y FQ(1.2.2)94 b(Examples)37 b(of)i FN(F)782 1469 y FL(n)827 1457 y FQ(-algebras)g(generated)g(b)m(y)h (idemp)s(oten)m(ts)174 1556 y(and)32 b(their)g(represen)m(tations)-118 1709 y FO(Here,)27 b(w)n(e)g(giv)n(e)e(a)i(n)n(um)n(b)r(er)f(of)h (examples)e(of)i(algebras)d(and)j FP(\003)p FO(-algebras)c(gen-)-118 1809 y(erated)d(b)n(y)g(idemp)r(oten)n(ts,)h(and)g(construct)f(a)g (residual)e(family)g(of)j(represen)n(ta-)-118 1909 y(tions)k(or)h FP(\003)p FO(-represen)n(tations)d FN(\031)30 b FO(for)c(eac)n(h)g(of)h (them)f(suc)n(h)h(that)g(dim)12 b FN(H)2110 1921 y FL(\031)2178 1909 y FP(\024)23 b FN(n)p FO(,)-118 2008 y(and,)k(therefore,)g(sho)n (w)g(that)h(these)f(algebras)e(are)i FN(F)1552 2020 y FK(2)p FL(n)1630 2008 y FO(-algebras.)-118 2157 y FQ(1.)36 b FO(Represen)n(tations)26 b(of)h(the)h FN(F)888 2169 y FK(4)926 2157 y FO(-algebra)d(generated)h(b)n(y)h(t)n(w)n(o)g(idemp)r (oten)n(ts)-118 2256 y FN(q)-81 2268 y FK(1)-44 2256 y FO(,)h FN(q)44 2268 y FK(2)81 2256 y FO(,)g(and)f(the)h(unit)g (elemen)n(t,)212 2437 y FN(Q)278 2449 y FK(2)338 2437 y FO(=)22 b FI(C)15 b FP(h)q FN(q)549 2449 y FK(1)592 2437 y FN(;)f(q)666 2449 y FK(2)726 2437 y FP(j)23 b FN(q)809 2449 y FK(1)847 2402 y(2)907 2437 y FO(=)g FN(q)1032 2449 y FK(1)1069 2437 y FN(;)28 b(q)1157 2449 y FK(2)1194 2402 y(2)1254 2437 y FO(=)23 b FN(q)1379 2449 y FK(2)1416 2437 y FP(i)338 2571 y FO(=)f FI(C)15 b FP(h)q FN(u)28 b FO(=)23 b(2)p FN(q)755 2583 y FK(1)810 2571 y FP(\000)18 b FN(e;)28 b(v)e FO(=)d(2)p FN(q)1216 2583 y FK(2)1271 2571 y FP(\000)18 b FN(e)23 b FP(j)g FN(u)1510 2537 y FK(2)1569 2571 y FO(=)g FN(e;)k(v)1789 2537 y FK(2)1850 2571 y FO(=)c FN(e)p FP(i)-118 2752 y FO(are)37 b(w)n(ell)e(kno)n(wn;) 42 b(nev)n(ertheless)36 b(w)n(e)h(presen)n(t)g(a)g(description)e(of)j (the)g(irre-)-118 2851 y(ducible)27 b(represen)n(tation)f(of)i(the)h (algebra)d(to)i(sho)n(w)g(the)h(sc)n(heme)e(of)i(in)n(v)n(esti-)-118 2951 y(gations)c(w)n(e)j(will)c(follo)n(w)h(in)i(more)f(complicated)e (examples.)6 3051 y(All)38 b(\014nite)i(dimensional)35 b(irreducible)g(represen)n(tations)i(of)i FN(Q)2030 3063 y FK(2)2067 3051 y FO(,)k(up)d(to)-118 3150 y(equiv)-5 b(alence,)25 b(are:)6 3250 y(a\))i(four)g(one-dimensional)21 b(represen)n(tations:)34 b FN(\031)1543 3262 y FK(0)p FL(;)p FK(0)1633 3250 y FO(\()p FN(q)1702 3262 y FK(1)1740 3250 y FO(\))23 b(=)g(0,)k FN(\031)2022 3262 y FK(0)p FL(;)p FK(0)2112 3250 y FO(\()p FN(q)2181 3262 y FK(2)2219 3250 y FO(\))c(=)-118 3349 y(0;)j FN(\031)20 3361 y FK(1)p FL(;)p FK(0)111 3349 y FO(\()p FN(q)180 3361 y FK(1)217 3349 y FO(\))e(=)e(1,)27 b FN(\031)499 3361 y FK(1)p FL(;)p FK(0)589 3349 y FO(\()p FN(q)658 3361 y FK(2)696 3349 y FO(\))c(=)g(0;)j FN(\031)977 3361 y FK(0)p FL(;)p FK(1)1068 3349 y FO(\()p FN(q)1137 3361 y FK(1)1174 3349 y FO(\))e(=)e(0,)27 b FN(\031)1456 3361 y FK(0)p FL(;)p FK(1)1546 3349 y FO(\()p FN(q)1615 3361 y FK(2)1653 3349 y FO(\))c(=)g(1;)j FN(\031)1934 3361 y FK(1)p FL(;)p FK(1)2024 3349 y FO(\()p FN(q)2093 3361 y FK(1)2131 3349 y FO(\))d(=)g(1,)-118 3449 y FN(\031)-71 3461 y FK(1)p FL(;)p FK(1)19 3449 y FO(\()p FN(q)88 3461 y FK(2)126 3449 y FO(\))g(=)g(1;)6 3549 y(b\))i(the)g(family)-7 b(,)22 b(parameterized)f(b)n(y)j FN(z)i FP(2)e FI(C)15 b FP(nf)p FO(0)p FN(;)f FO(1)p FP(g)p FO(,)28 b(of)c(t)n(w)n (o-dimensional)-118 3648 y(represen)n(tations:)239 3874 y FN(\031)286 3886 y FL(z)324 3874 y FO(\()p FN(q)393 3886 y FK(1)431 3874 y FO(\))f(=)574 3757 y Fy(\022)635 3823 y FO(1)83 b(0)635 3923 y(0)g(0)801 3757 y Fy(\023)876 3874 y FN(;)97 b(\031)1043 3886 y FL(z)1082 3874 y FO(\()p FN(q)1151 3886 y FK(2)1188 3874 y FO(\))23 b(=)1331 3757 y Fy(\022)1483 3823 y FN(z)248 b FO(1)1392 3923 y FN(z)22 b FP(\000)c FN(z)1579 3893 y FK(2)1698 3923 y FO(1)g FP(\000)g FN(z)1884 3757 y Fy(\023)1959 3874 y FN(:)p eop %%Page: 30 34 30 33 bop -118 -137 a FO(30)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(Ev)n(ery)e FN(\031)k FP(2)c FO(Irrep)13 b FN(Q)652 108 y FK(2)713 96 y FO(is)24 b(one-)f(or)h(t)n(w)n(o-dimensional.)30 b(Indeed,)c(the)e(space)-118 196 y FN(H)43 b FO(=)36 b FI(C)15 b FP(h)p FN(e)220 208 y FL(\025)269 196 y FN(;)f(\031)s FO(\()p FN(v)s FO(\))p FN(e)502 208 y FL(\025)546 196 y FP(i)36 b FO(\()p FN(e)685 208 y FL(\025)764 196 y FO(is)e(an)i(eigen)n(v)n(ector)c(of)j FN(\031)s FO(\()p FN(u)p FO(\))24 b FP(\001)g FN(\031)s FO(\()p FN(v)s FO(\),)39 b FP(k)p FN(e)2053 208 y FL(\025)2096 196 y FP(k)c FO(=)h(1,)-118 296 y FN(\025)23 b FP(6)p FO(=)g(0\))k(is)g(in)n (v)-5 b(arian)n(t)24 b(for)k(the)f(represen)n(tation)e FN(\031)s FO(.)-118 436 y FQ(2.)35 b FO(In)24 b(the)h(algebra)c FN(Q)585 448 y FK(2)622 436 y FO(,)k(one)f(can)g(in)n(tro)r(duce)f(t)n (w)n(o)g(natural)f(structures)i(of)g(an)-118 535 y(algebra)h(with)i(in) n(v)n(olution:)6 635 y(1\))h(the)g FP(\003)293 647 y FK(1)330 635 y FO(-algebra)304 794 y FA(P)359 806 y FK(2)419 794 y FO(=)23 b FI(C)15 b FP(h)p FN(p)635 757 y FM(\003)669 765 y Fx(1)635 817 y FK(1)735 794 y FO(=)22 b FN(p)864 806 y FK(1)901 794 y FN(;)14 b(p)980 757 y FM(\003)1014 765 y Fx(1)980 817 y FK(2)1074 794 y FO(=)22 b FN(p)1203 806 y FK(2)1263 794 y FP(j)i FN(p)1352 760 y FK(2)1352 815 y(1)1412 794 y FO(=)e FN(p)1541 806 y FK(1)1578 794 y FN(;)14 b(p)1657 760 y FK(2)1657 815 y(2)1717 794 y FO(=)23 b FN(p)1847 806 y FK(2)1884 794 y FP(i)419 929 y FO(=)g FI(C)15 b FP(h)p FN(u)641 895 y FM(\003)675 903 y Fx(1)741 929 y FO(=)22 b FN(u;)14 b(v)956 895 y FM(\003)990 903 y Fx(1)1049 929 y FO(=)23 b FN(v)j FP(j)d FN(u)1297 895 y FK(2)1357 929 y FO(=)g FN(v)1488 895 y FK(2)1548 929 y FO(=)g FN(e)p FP(i)419 1054 y FO(=)g FI(C)29 b FO([)6 b FI(Z)665 1066 y FK(2)715 1054 y FP(\003)18 b FI(Z)836 1066 y FK(2)868 1054 y FO(])-118 1213 y(is)31 b(a)g(group)f FP(\003)324 1225 y FK(1)361 1213 y FO(-algebra)f (generated)h(b)n(y)i(t)n(w)n(o)f(unitary)f(self-adjoin)n(t)f(genera-) -118 1313 y(tors.)57 b(Irreducible)32 b(t)n(w)n(o-dimensional)d FP(\003)p FO(-represen)n(tations)i(of)k FA(P)1957 1325 y FK(2)2029 1313 y FO(\(up)g(to)g(a)-118 1412 y(unitary)26 b(equiv)-5 b(alence\))25 b(are:)79 1623 y FN(\031)126 1635 y FL(\036)170 1623 y FO(\()p FN(p)244 1635 y FK(1)282 1623 y FO(\))e(=)425 1506 y Fy(\022)486 1572 y FO(1)82 b(0)486 1672 y(0)g(0)652 1506 y Fy(\023)727 1623 y FN(;)97 b(\031)894 1635 y FL(\036)938 1623 y FO(\()p FN(p)1012 1635 y FK(2)1049 1623 y FO(\))24 b(=)1192 1506 y Fy(\022)1324 1571 y FO(cos)1436 1541 y FK(2)1487 1571 y FN(\036)154 b FO(cos)13 b FN(\036)h FO(sin)f FN(\036)1253 1673 y FO(cos)g FN(\036)h FO(sin)g FN(\036)159 b FO(sin)1867 1638 y FK(2)1919 1673 y FN(\036)2044 1506 y Fy(\023)2118 1623 y FN(;)-118 1833 y(\036)39 b FP(2)h FO(\(0)p FN(;)14 b(\031)s(=)p FO(2\).)64 b(They)37 b(are)f(equiv)-5 b(alen)n(t)35 b(to)i(the)h(represen)n(tations)c FN(\031)2101 1845 y FL(z)2139 1833 y FO(,)40 b FN(z)i FP(2)-118 1933 y FO(\(0)p FN(;)14 b FO(1\))22 b FP(\032)h FI(R)p FO(.)6 2032 y(2\))28 b(the)g FP(\003)293 2044 y FK(2)330 2032 y FO(-algebra)299 2192 y FA(Q)354 2204 y FK(1)415 2192 y FO(=)22 b FI(C)15 b FP(h)q FN(q)626 2204 y FK(1)669 2192 y FN(;)f(q)746 2155 y FM(\003)780 2163 y Fx(2)743 2214 y FK(1)839 2192 y FP(j)24 b FN(q)926 2158 y FK(2)923 2212 y(1)986 2192 y FO(=)e FN(q)1110 2204 y FK(1)1148 2192 y FP(i)h FO(=)g FI(C)15 b FP(h)p FN(u;)f(u)1510 2158 y FM(\003)1544 2166 y Fx(2)1609 2192 y FP(j)23 b FN(u)1703 2158 y FK(2)1763 2192 y FO(=)f FN(e)p FP(i)-118 2351 y FO(is)35 b(a)i FP(\003)95 2363 y FK(2)132 2351 y FO(-algebra)c(generated)j(b)n(y)g(an) g(idemp)r(oten)n(t)g(and)g(its)g(adjoin)n(t.)63 b(Irre-)-118 2451 y(ducible)26 b(t)n(w)n(o-dimensional)c FP(\003)p FO(-represen)n(tations)i(of)j FA(Q)1576 2463 y FK(1)1641 2451 y FO(are:)565 2655 y FN(\031)612 2667 y FL(\013)660 2655 y FO(\()p FN(q)729 2667 y FK(1)767 2655 y FO(\))c(=)910 2538 y Fy(\022)971 2605 y FO(1)82 b FN(\013)971 2704 y FO(0)88 b(0)1149 2538 y Fy(\023)1224 2655 y FN(;)179 b(\013)24 b(>)f FO(0)p FN(:)-118 2864 y FO(They)k(are)g(equiv)-5 b(alen)n(t)25 b(to)j(the)g(represen)n(tations)c FN(\031)1496 2876 y FL(z)1535 2864 y FO(,)k FN(z)e FP(2)d FO(\(1)p FN(;)14 b FP(1)p FO(\))24 b FP(\032)e FI(R)p FO(.)-118 3004 y FQ(3.)34 b FO(The)23 b(algebra)c FN(Q)508 3016 y FK(2)567 3004 y FO(and)j(b)r(oth)h(the)g FP(\003)p FO(-algebras)18 b(constructed)k(from)f(it,)i(ha)n(v)n(e)-118 3104 y(a)k(residual)e(family)f(of)k(\014nite-dimensional)22 b(represen)n(tations.)-118 3252 y FQ(Prop)s(osition)30 b(10.)41 b FB(The)32 b(two-dimensional)g(r)l(epr)l(esentations)f FN(\031)1974 3264 y FL(z)2013 3252 y FB(,)h FN(z)c FP(2)e FI(C)40 b FP(n)-118 3351 y(f)p FO(0)p FN(;)14 b FO(1)p FP(g)p FB(,)30 b FO(\()p FB(as)h(wel)t(l)h(as)g FN(\031)607 3363 y FL(\036)651 3351 y FB(,)g FN(\036)26 b FP(2)g FO(\(0)p FN(;)14 b(\031)s(=)p FO(2\))p FB(,)31 b(and)g FN(\031)1406 3363 y FL(\013)1454 3351 y FB(,)h FN(\013)26 b(>)f FO(0\))30 b FB(form)i(a)g(r)l(esidual)-118 3451 y(family.)40 b FN(Q)234 3463 y FK(2)301 3451 y FB(is)30 b(a)g FN(F)515 3463 y FK(4)553 3451 y FB(-algebr)l(a.)-118 3598 y(Pr)l(o)l(of.)43 b FO(Let)28 b(us)f(consider)f(an)n(y)h FN(x)c FP(2)h FN(Q)1093 3610 y FK(2)1129 3598 y FO(.)345 3835 y FN(x)g FO(=)f FN(\013)557 3847 y FK(0)612 3835 y FO(+)713 3730 y FL(N)766 3738 y Fx(1)695 3756 y Fy(X)702 3933 y FL(i)p FK(=1)829 3835 y FN(a)873 3847 y FL(i)901 3835 y FO(\()p FN(q)970 3847 y FK(1)1007 3835 y FN(q)1044 3847 y FK(2)1082 3835 y FO(\))1114 3801 y FL(i)1160 3835 y FO(+)1260 3730 y FL(N)1313 3738 y Fx(2)1243 3756 y Fy(X)1245 3933 y FL(j)s FK(=1)1377 3835 y FN(b)1413 3847 y FL(j)1447 3835 y FO(\()p FN(q)1516 3847 y FK(2)1554 3835 y FN(q)1591 3847 y FK(1)1628 3835 y FO(\))1660 3801 y FL(j)p eop %%Page: 31 35 31 34 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(31)612 166 y(+)713 61 y FL(N)766 69 y Fx(3)696 87 y Fy(X)695 266 y FL(k)q FK(=0)830 166 y FN(c)866 178 y FL(k)907 166 y FO(\()p FN(q)976 178 y FK(1)1013 166 y FN(q)1050 178 y FK(2)1088 166 y FO(\))1120 132 y FL(k)1161 166 y FN(q)1198 178 y FK(1)1254 166 y FO(+)1354 61 y FL(N)1407 69 y Fx(4)1337 87 y Fy(X)1344 266 y FL(l)p FK(=0)1470 166 y FN(d)1513 178 y FL(l)1539 166 y FO(\()p FN(q)1608 178 y FK(2)1646 166 y FN(q)1683 178 y FK(1)1720 166 y FO(\))1752 132 y FL(l)1778 166 y FN(q)1815 178 y FK(2)1852 166 y FN(;)-118 432 y(\013)-65 444 y FK(0)-28 432 y FO(,)28 b FN(a)67 444 y FL(i)95 432 y FO(,)f FN(b)181 444 y FL(j)216 432 y FO(,)h FN(c)303 444 y FL(k)343 432 y FO(,)g FN(d)437 444 y FL(l)486 432 y FP(2)23 b FI(C)15 b FO(.)43 b(Then)28 b(one)f(has)145 700 y FN(\031)192 712 y FL(z)230 700 y FO(\()p FN(x)p FO(\))d(=)453 558 y Fy( )519 637 y FN(\013)572 649 y FK(0)717 637 y FO(0)543 762 y(0)107 b FN(\013)745 774 y FK(0)783 558 y Fy(!)867 700 y FO(+)950 558 y Fy( )1016 581 y(P)1103 601 y FL(N)1156 609 y Fx(1)1103 668 y FL(i)p FK(=1)1229 643 y FN(a)1273 655 y FL(i)1300 643 y FN(z)1343 613 y FL(i)1453 581 y Fy(P)1541 601 y FL(N)1594 609 y Fx(1)1541 668 y FL(i)p FK(=1)1666 643 y FN(a)1710 655 y FL(i)1738 643 y FN(z)1781 613 y FL(i)p FM(\000)p FK(1)1172 768 y FO(0)438 b(0)1893 558 y Fy(!)361 985 y FO(+)444 843 y Fy( )635 856 y(P)722 877 y FL(N)775 885 y Fx(2)722 944 y FL(j)s FK(=1)855 919 y FN(b)891 931 y FL(j)926 919 y FN(z)969 889 y FL(j)1211 919 y FO(0)510 993 y Fy(P)597 1013 y FL(N)650 1021 y Fx(2)597 1080 y FL(j)s FK(=1)730 1055 y FN(b)766 1067 y FL(j)801 1055 y FN(z)844 1025 y FL(j)878 1055 y FO(\(1)18 b FP(\000)g FN(z)t FO(\))83 b(0)1252 843 y Fy(!)1337 985 y FO(+)1420 843 y Fy( )1485 865 y(P)1573 886 y FL(N)1626 894 y Fx(3)1573 953 y FL(k)q FK(=0)1712 928 y FN(c)1748 940 y FL(k)1788 928 y FN(z)1831 898 y FL(k)1955 928 y FO(0)1658 1052 y(0)255 b(0)1996 843 y Fy(!)361 1270 y FO(+)444 1128 y Fy( )635 1144 y(P)722 1165 y FL(N)775 1173 y Fx(4)722 1231 y FL(l)p FK(=0)846 1207 y FN(d)889 1219 y FL(l)914 1207 y FN(z)957 1176 y FL(l)p FK(+1)1399 1144 y Fy(P)1487 1165 y FL(N)1540 1173 y Fx(4)1487 1231 y FL(l)p FK(=0)1610 1207 y FN(d)1653 1219 y FL(l)1679 1207 y FN(z)1722 1176 y FL(l)510 1281 y Fy(P)597 1301 y FL(N)650 1309 y Fx(4)597 1368 y FL(l)p FK(=0)721 1343 y FN(d)764 1355 y FL(l)789 1343 y FN(z)832 1313 y FL(l)p FK(+1)941 1343 y FO(\(1)18 b FP(\000)g FN(z)t FO(\))1274 1281 y Fy(P)1362 1301 y FL(N)1415 1309 y Fx(4)1362 1368 y FL(l)p FK(=0)1485 1343 y FN(d)1528 1355 y FL(l)1554 1343 y FN(z)1597 1313 y FL(l)1621 1343 y FO(\(1)h FP(\000)f FN(z)t FO(\))1871 1128 y Fy(!)1951 1270 y FN(:)-118 1535 y FO(It)h(easily)e(follo)n(ws)f(from)i(the)h(structure)g(of)g(the)h (matrix)c FN(\031)1647 1547 y FL(z)1686 1535 y FO(\()p FN(x)p FO(\))k(that)g FN(\031)2036 1547 y FL(z)2074 1535 y FO(\()p FN(x)p FO(\))k(=)f(0)-118 1635 y(for)k(an)n(y)g FN(z)f FP(2)e FI(C)39 b FP(n)18 b(f)p FO(0)p FN(;)c FO(1)p FP(g)25 b FO(if)j(and)f(only)f(if)h FN(x)d FO(=)e(0.)p 2278 1635 4 57 v 2282 1582 50 4 v 2282 1635 V 2331 1635 4 57 v -118 1834 a FQ(4.)51 b FO(The)33 b(structure)f(of)g(indecomp)r (osable)d(represen)n(tations)h(of)i(the)h(algebra)-118 1934 y FN(Q)-52 1946 y FK(2)6 1934 y FO(is)19 b(more)h(complicated)d (than)k(the)g(structure)g(of)f(irreducible)e(ones.)34 b(Let)21 b(us)-118 2034 y(presen)n(t)f(the)h(list)e(of)h(all)e (indecomp)r(osable)f(represen)n(tations)h(of)i FN(Q)1940 2046 y FK(2)2018 2034 y FO([172)o(,)g(92)o(].)6 2137 y(F)-7 b(or)27 b(dim)13 b FN(H)29 b FO(=)23 b(2)p FN(k)s FO(,)291 2374 y FN(\031)338 2386 y FL(\025)382 2374 y FO(\()p FN(u)p FO(\))g(=)605 2256 y Fy(\022)666 2323 y FN(I)702 2335 y FL(k)876 2323 y FO(0)684 2422 y(0)100 b FP(\000)p FN(I)927 2434 y FL(k)968 2256 y Fy(\023)1043 2374 y FN(;)d(\031)1210 2386 y FL(\025)1254 2374 y FO(\()p FN(v)s FO(\))24 b(=)e FP(\006)1551 2256 y Fy(\022)1613 2323 y FN(A)87 b(B)1612 2422 y(C)i(D)1831 2256 y Fy(\023)1906 2374 y FN(;)-118 2610 y FO(where)27 b FN(I)158 2622 y FL(k)227 2610 y FO(is)f(the)i(iden)n(tit)n(y)e(matrix)f(of)j(order)e FN(k)s FO(,)i(and)65 3001 y FN(B)f FO(=)243 2735 y Fy(0)243 2881 y(B)243 2931 y(B)243 2980 y(B)243 3030 y(B)243 3083 y(@)315 2796 y FP(\000)p FO(2\(1)18 b FP(\000)g FN(\025)p FO(\))677 2766 y FM(\000)p FK(1)850 2796 y FP(\000)p FO(2\(1)f FP(\000)h FN(\025)p FO(\))1211 2766 y FM(\000)p FK(2)1384 2796 y FP(\001)c(\001)g(\001)96 b(\000)p FO(2\(1)18 b FP(\000)g FN(\025)p FO(\))1939 2766 y FM(\000)p FL(k)850 2951 y FP(\000)p FO(2\(1)f FP(\000)h FN(\025)p FO(\))1211 2921 y FM(\000)p FK(1)1395 2893 y FO(.)1428 2918 y(.)1460 2943 y(.)1793 2884 y(.)1793 2918 y(.)1793 2951 y(.)708 3106 y Fo(0)1395 3048 y FO(.)1428 3073 y(.)1460 3098 y(.)1579 3106 y FP(\000)p FO(2\(1)g FP(\000)g FN(\025)p FO(\))1941 3076 y FM(\000)p FK(2)1579 3205 y FP(\000)p FO(2\(1)g FP(\000)g FN(\025)p FO(\))1941 3175 y FM(\000)p FK(1)2032 2735 y Fy(1)2032 2881 y(C)2032 2931 y(C)2032 2980 y(C)2032 3030 y(C)2032 3083 y(A)2119 3001 y FN(;)92 3544 y(C)29 b FO(=)268 3278 y Fy(0)268 3424 y(B)268 3474 y(B)268 3523 y(B)268 3573 y(B)268 3626 y(@)340 3339 y FO(2)p FN(\025)p FO(\(1)19 b FP(\000)f FN(\025)p FO(\))686 3309 y FM(\000)p FK(1)883 3339 y FO(2\(1)f FP(\000)h FN(\025)p FO(\))1179 3309 y FM(\000)p FK(2)1377 3339 y FP(\001)c(\001)g(\001)119 b FO(2\(1)17 b FP(\000)h FN(\025)p FO(\))1889 3309 y FM(\000)p FL(k)858 3494 y FO(2)p FN(\025)p FO(\(1)h FP(\000)f FN(\025)p FO(\))1204 3464 y FM(\000)p FK(1)1388 3436 y FO(.)1420 3461 y(.)1453 3486 y(.)1776 3427 y(.)1776 3461 y(.)1776 3494 y(.)717 3649 y Fo(0)1388 3591 y FO(.)1420 3615 y(.)1453 3641 y(.)1594 3649 y(2\(1)g FP(\000)g FN(\025)p FO(\))1891 3619 y FM(\000)p FK(2)1570 3748 y FO(2)p FN(\025)p FO(\(1)g FP(\000)g FN(\025)p FO(\))1915 3718 y FM(\000)p FK(1)2005 3278 y Fy(1)2005 3424 y(C)2005 3474 y(C)2005 3523 y(C)2005 3573 y(C)2005 3626 y(A)2092 3544 y FN(;)244 3881 y(A)23 b FO(=)g FP(\000)p FN(B)f FP(\000)c FN(I)686 3893 y FL(k)727 3881 y FN(;)97 b(D)25 b FO(=)e FP(\000)p FN(C)h FP(\000)18 b FN(I)1296 3893 y FL(k)1338 3881 y FN(;)180 b(\025)23 b FP(2)h FI(C)39 b FP(n)18 b(f)p FO(1)p FP(g)p FN(:)p eop %%Page: 32 36 32 35 bop -118 -137 a FO(32)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(F)-7 b(or)27 b(dim)12 b FN(H)30 b FO(=)23 b(2)p FN(k)d FO(+)e(1,)249 320 y FN(\031)296 332 y FL(\025)340 320 y FO(\()p FN(u)p FO(\))23 b(=)563 203 y Fy(\022)624 270 y FN(I)660 282 y FL(k)q FK(+1)919 270 y FO(0)684 369 y(0)142 b FP(\000)p FN(I)969 381 y FL(k)1010 203 y Fy(\023)1085 320 y FN(;)97 b(\031)1252 332 y FL(\025)1296 320 y FO(\()p FN(v)s FO(\))24 b(=)e FP(\006)1593 203 y Fy(\022)1655 270 y FN(A)87 b(B)1654 369 y(C)i(D)1873 203 y Fy(\023)1948 320 y FN(;)-118 544 y FO(where)202 907 y FN(A)24 b FO(=)375 641 y Fy(0)375 787 y(B)375 837 y(B)375 886 y(B)375 936 y(B)375 989 y(@)456 702 y FO(1)91 b(2)83 b FP(\001)14 b(\001)g(\001)97 b FO(2)589 856 y(1)725 798 y(.)758 823 y(.)790 848 y(.)917 790 y(.)917 823 y(.)917 856 y(.)725 953 y(.)758 978 y(.)790 1003 y(.)908 1011 y(2)448 1111 y Fo(0)401 b FO(1)949 641 y Fy(1)949 787 y(C)949 837 y(C)949 886 y(C)949 936 y(C)949 989 y(A)1036 907 y FN(;)97 b(B)27 b FO(=)1333 641 y Fy(0)1333 787 y(B)1333 837 y(B)1333 886 y(B)1333 936 y(B)1333 989 y(@)1406 702 y FP(\000)p FO(2)82 b FP(\001)14 b(\001)g(\001)97 b(\000)p FO(2)1438 856 y(0)1607 798 y(.)1639 823 y(.)1671 848 y(.)1830 790 y(.)1830 823 y(.)1830 856 y(.)1607 953 y(.)1639 978 y(.)1671 1003 y(.)1789 1011 y FP(\000)p FO(2)1430 1111 y Fo(0)332 b FO(0)1895 641 y Fy(1)1895 787 y(C)1895 837 y(C)1895 886 y(C)1895 936 y(C)1895 989 y(A)1981 907 y FN(;)49 1474 y(C)29 b FO(=)225 1208 y Fy(0)225 1354 y(B)225 1404 y(B)225 1453 y(B)225 1503 y(B)225 1556 y(@)306 1269 y FO(0)91 b(2)83 b FP(\001)14 b(\001)g(\001)96 b FO(2)83 b(2)439 1424 y(0)575 1365 y(.)607 1390 y(.)640 1416 y(.)766 1357 y(.)766 1390 y(.)766 1424 y(.)891 1357 y(.)891 1390 y(.)891 1424 y(.)575 1520 y(.)607 1545 y(.)640 1570 y(.)757 1578 y(2)g(2)297 1678 y Fo(0)401 b FO(0)83 b(2)923 1208 y Fy(1)923 1354 y(C)923 1404 y(C)923 1453 y(C)923 1503 y(C)923 1556 y(A)1010 1474 y FN(;)97 b(D)25 b FO(=)1311 1208 y Fy(0)1311 1354 y(B)1311 1404 y(B)1311 1453 y(B)1311 1503 y(B)1311 1556 y(@)1384 1269 y FP(\000)p FO(1)82 b FP(\000)p FO(2)g FP(\001)14 b(\001)g(\001)97 b(\000)p FO(2)1573 1424 y FP(\000)p FO(1)1774 1365 y(.)1806 1390 y(.)1838 1416 y(.)1998 1357 y(.)1998 1390 y(.)1998 1424 y(.)1774 1520 y(.)1806 1545 y(.)1838 1570 y(.)1956 1578 y FP(\000)p FO(2)1408 1678 y Fo(0)489 b FP(\000)p FO(1)2062 1208 y Fy(1)2062 1354 y(C)2062 1404 y(C)2062 1453 y(C)2062 1503 y(C)2062 1556 y(A)2149 1474 y FN(:)6 1855 y FO(The)26 b(algebra)d FN(Q)530 1867 y FK(2)593 1855 y FO(is)i(the)h(group)f (algebra)e(of)j(the)g(Co)n(xeter)f(group)2156 1837 y Fn(r)p 2156 1839 125 4 v 99 w(r)2185 1820 y FM(1)-118 1954 y FO(generated)h(b)n(y)i(t)n(w)n(o)e(\015ips)h(without)g(an)n(y)g (relations.)-118 2102 y FQ(5.)43 b FO(Consider)28 b(the)j(Co)n(xeter)d (group)h FN(G)1109 2114 y FL(M)1213 2102 y FO(with)h(a)g(matrix)d FN(M)36 b FO(=)26 b(\()p FN(m)2062 2114 y FL(ij)2121 2102 y FO(\))2153 2072 y FL(m)2153 2124 y(i;j)s FK(=1)2316 2102 y FO(,)-118 2202 y FN(m)-45 2214 y FL(ij)52 2202 y FP(2)40 b FI(N)35 b FP([)25 b(f1g)p FO(;)41 b FN(m)616 2214 y FL(ii)706 2202 y FO(=)e(1)d FN(m)961 2214 y FL(ij)1059 2202 y FO(=)i FN(m)1235 2214 y FL(j)s(i)1333 2202 y FN(>)h FO(1,)g FN(i)f FP(6)p FO(=)h FN(j)5 b FO(;)42 b FN(i)p FO(,)e FN(j)k FO(=)38 b(1,)f FN(:)14 b(:)g(:)28 b FO(,)-118 2302 y FN(m)p FO(,)f(whic)n(h)f(is)h(de\014ned)g(in)g(terms)f(of)h (generators)e(\()p FN(w)1526 2314 y FL(i)1554 2302 y FO(\))1586 2271 y FL(m)1586 2323 y(i)p FK(=1)1725 2302 y FO(and)i(the)h(relations)-118 2401 y(\()p FN(w)-27 2413 y FL(i)1 2401 y FN(w)60 2413 y FL(j)96 2401 y FO(\))128 2371 y FL(m)187 2379 y Fv(ij)287 2401 y FO(=)42 b FN(e)p FO(,)f FN(i)p FO(,)h FN(j)48 b FO(=)42 b(1,)d FN(:)14 b(:)g(:)27 b FO(,)42 b FN(m)p FO(;)j(if)39 b FN(m)1375 2413 y FL(ij)1476 2401 y FO(=)j FP(1)p FO(,)g(then)e(there)f(is)f(no) -118 2501 y(relation)33 b(b)r(et)n(w)n(een)k(the)g(generators)d FN(w)1147 2513 y FL(i)1211 2501 y FO(and)i FN(w)1440 2513 y FL(j)1476 2501 y FO(.)63 b(If)37 b(the)g(Cartan)e(matrix)-118 2600 y FN(K)28 b FO(=)69 2533 y Fy(\000)114 2600 y FP(\000)7 b FO(cos)12 b FN(\031)s(=m)475 2612 y FL(ij)533 2533 y Fy(\001)571 2550 y FL(m)571 2634 y(i;j)s FK(=1)733 2600 y FO(,)24 b(whic)n(h)c(corresp)r(onds)g(to)i FN(M)9 b FO(,)22 b(is)f(p)r(ositiv)n(e)e(de\014nite)-118 2716 y(\(all)30 b(its)h(principal)d(minors)h(are)i(p)r(ositiv)n(e\),)g(then) h(the)h(group)d FN(G)1937 2728 y FL(M)2043 2716 y FO(is)h(\014nite;) -118 2815 y(if)h(det)14 b FN(K)37 b FO(=)30 b(0,)k(but)f(the)g(other)f (principal)d(minors)g(are)j(p)r(ositiv)n(e,)f(then)i(the)-118 2915 y(group)26 b FN(G)182 2927 y FL(M)283 2915 y FO(is)g(in\014nite,)g (but)i FN(G)887 2927 y FL(M)988 2915 y FO(is)e(a)h(semi-direct)d(pro)r (duct)j(of)g(the)h(lattice)-118 3014 y FI(Z)-57 2984 y FL(m)p FM(\000)p FK(1)108 3014 y FO(and)23 b(a)g(\014nite)f(group)g FN(G)833 3026 y FL(f)877 3014 y FO(\()p FN(M)9 b FO(\),)24 b FN(G)1143 3026 y FL(M)1240 3014 y FO(=)f FI(Z)1389 2984 y FL(m)p FM(\000)p FK(1)1540 3014 y FI(o)9 b FN(G)1679 3026 y FL(f)1723 3014 y FO(\()p FN(M)g FO(\))23 b(see)46 b([41)o(])23 b(and)-118 3114 y(others.)6 3214 y(Since)28 b(the)g(Co)n(xeter)e(group)h FN(G)977 3226 y FL(M)1079 3214 y FO(is)g(generated)g(b)n(y)g(\015ips)g FN(w)1894 3184 y FK(2)1892 3235 y FL(i)1956 3214 y FO(=)c FN(e)p FO(,)28 b FN(i)23 b FO(=)g(1,)-118 3313 y FN(:)14 b(:)g(:)27 b FO(,)i FN(m)p FO(,)f FI(C)15 b FO([)p FN(G)325 3325 y FL(M)404 3313 y FO(])29 b(also)d(giv)n(es)g(an)i(example)d(of)j(an)g (algebra)e(generated)h(b)n(y)h FN(m)-118 3413 y FO(pro)5 b(jections.)36 b(There)27 b(is)g(a)g(natural)g(in)n(v)n(olution)d(in)j FI(C)15 b FO([)p FN(G)1657 3425 y FL(M)1737 3413 y FO(])28 b(suc)n(h)g(that)g(all)e(of)-118 3513 y(the)i(group)e(elemen)n(ts)g (are)g(unitary)-7 b(,)26 b FN(g)1090 3482 y FM(\003)1151 3513 y FO(=)d FN(g)1282 3482 y FM(\000)p FK(1)1371 3513 y FO(.)37 b(\(Generally)25 b(sp)r(eaking,)g(this)-118 3612 y(is)h(not)i(the)g(unique)f(in)n(v)n(olution)d(that)j(can)h(b)r(e) g(de\014ned)g(on)f FI(C)15 b FO([)p FN(G)1904 3624 y FL(M)1984 3612 y FO(].\))6 3712 y(The)42 b(dimensions)c(of)j(the)h (irreducible)c FP(\003)p FO(-represen)n(tations)g FN(\031)2026 3724 y FL(\013)2115 3712 y FO(of)j(the)-118 3811 y(group)17 b FP(\003)p FO(-al)n(gebra)g(of)i(the)g(Co)n(xeter)e(group)h FN(G)1271 3823 y FL(M)1368 3811 y FO(=)k FI(Z)1517 3781 y FL(m)p FM(\000)p FK(1)1659 3811 y FI(o)p FN(G)1789 3823 y FL(f)1851 3811 y FO(are)c(ma)5 b(jorized)-118 3911 y(b)n(y)32 b(the)g(n)n(um)n(b)r(er)f FP(j)p FN(G)544 3923 y FL(f)587 3911 y FP(j)p FO(.)50 b(These)32 b(represen)n(tations)d (form)i(a)g(residual)e(family)-7 b(,)p eop %%Page: 33 37 33 36 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(33)-118 96 y(b)r(ecause)28 b(irreducible)e FP(\003)p FO(-represen)n(tations)f(of)j FI(C)15 b FO([)q FN(G)1487 108 y FL(M)1566 96 y FO(],)30 b(with)e(the)i(in)n(v)n(olution)-118 196 y FN(g)-75 166 y FM(\003)-11 196 y FO(=)25 b FN(g)122 166 y FM(\000)p FK(1)211 196 y FO(,)30 b(mak)n(e)d(a)i(residual)e (family)-7 b(.)39 b(Hence)30 b FI(C)15 b FO([)p FN(G)1540 208 y FL(M)1620 196 y FO(])29 b(is)g(a)g FN(F)1882 211 y FK(2)p FM(j)p FL(G)1987 220 y Fv(f)2024 211 y FM(j)2048 196 y FO(-algebra)-118 296 y(whic)n(h)d(is)h(generated)f(b)n(y)i (\015ips.)-118 422 y FB(R)l(emark)i(5.)42 b FO(It)27 b(is)f(a)g(v)n(ery)g(di\016cult)g(problem)e(to)j(describ)r(e)e (indecomp)r(osable)-118 521 y(represen)n(tations)37 b(of)j FI(C)15 b FO([)p FN(G)719 533 y FL(M)799 521 y FO(])40 b(\(except)g(for)f(the)h(case)f(where)h(the)g(Co)n(xeter)-118 621 y(group)26 b FN(G)182 633 y FL(M)284 621 y FO(is)g(a)i(\014nite)f (group)f(or)h(is)f FI(Z)12 b(o)18 b(Z)1288 633 y FK(2)1319 621 y FO(\))28 b([39)o(].)-118 763 y FQ(6.)43 b FO(No)n(w)29 b(let)g Fz(A)373 775 y FL(k)440 763 y FO(=)e FI(C)15 b FP(h)p FN(u)666 720 y FK(\()p FL(k)q FK(\))666 785 y(1)764 763 y FN(;)f(:)g(:)g(:)g(;)g(u)997 720 y FK(\()p FL(k)q FK(\))997 773 y FL(n)1038 782 y Fv(k)1116 763 y FP(j)27 b FO(\(\))1230 775 y FL(k)1271 763 y FP(i)j FO(b)r(e)h FN(F)1502 775 y FK(2)p FL(m)1594 784 y Fv(k)1634 763 y FO(-algebras)c(generated)-118 880 y(b)n(y)40 b(the)g(\015ips)g FN(u)402 837 y FK(\()p FL(k)q FK(\))402 902 y(1)494 880 y FO(,)g FN(:)14 b(:)g(:)28 b FO(,)44 b FN(u)797 837 y FK(\()p FL(k)q FK(\))797 890 y FL(n)838 899 y Fv(k)929 880 y FO(and)c(relations)d(\(\))1517 892 y FL(k)1559 880 y FO(,)43 b(suc)n(h)d(that)h FN(\031)2068 850 y FK(\()p FL(k)q FK(\))2201 880 y FO(is)e(a)-118 980 y(residual)25 b(family)g(with)i(dim)12 b FN(H)856 999 y FL(\031)897 982 y Fx(\()p Fv(k)q Fx(\))1006 980 y FP(\024)23 b FN(m)1167 992 y FL(k)1208 980 y FO(,)28 b FN(k)f FO(=)c(1,)k FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(\).)38 b(Of)28 b(course,)e(the)-118 1079 y(algebra)190 1256 y FI(C)15 b FP(h)p FN(u)324 1213 y FK(\(1\))324 1278 y(1)419 1256 y FN(;)f(:)g(:)g(:)g(;)g(u)652 1222 y FK(\()p FL(n)p FK(\))652 1277 y FL(n)693 1285 y Fv(n)771 1256 y FP(j)23 b FO(\(\))881 1268 y FK(1)919 1256 y FN(;)14 b(:)g(:)g(:)g(;)g FO(\(\))1168 1268 y FL(n)1213 1256 y FN(;)28 b FO([)p FN(u)1335 1213 y FK(\()p FL(k)q FK(\))1335 1279 y FL(i)1427 1256 y FN(;)14 b(u)1512 1213 y FK(\()p FL(l)p FK(\))1512 1279 y FL(j)1589 1256 y FO(])23 b(=)g(0)p FN(;)k(k)f FP(6)p FO(=)d FN(l)r FP(i)-118 1438 y FO(is)j(a)i FN(F)88 1450 y FK(2)p FL(m)180 1458 y Fx(1)212 1450 y FM(\001\001\001\001)o(\001)p FL(m)370 1458 y Fv(n)415 1438 y FO(-algebra)c(ha)n(ving)i(the)i(residual)c (family)h FN(\031)1757 1408 y FK(\(1\))1865 1438 y FP(\012)18 b(\001)c(\001)g(\001)k(\012)g FN(\031)2196 1408 y FK(\()p FL(n)p FK(\))2294 1438 y FO(.)-118 1581 y FQ(7.)37 b FO(Examples)24 b(of)k(algebras)d(that)j(w)n(e)f(will)e(consider)h(in)h (the)i(sequel)d(are)h(also)-118 1692 y(de\014ned)d(b)n(y)f(generators)f FN(u)722 1649 y FK(\(1\))722 1714 y(1)810 1692 y FO(,)i FN(:)14 b(:)g(:)28 b FO(,)c FN(u)1077 1649 y FK(\()p FL(n)p FK(\))1077 1701 y FL(n)1118 1709 y Fv(n)1174 1692 y FO(,)h(but)f(if)f(the)h(upp)r(er)g(indices)e(are)h(not)-118 1791 y(equal,)30 b(the)h(generators)d(pairwise)f(comm)n(ute)i(or)h(an)n (ti-comm)n(ute.)42 b(In)30 b(items)-118 1902 y(7)g(and)h(8,)g(these)g (relations)c(are)j(as)g(follo)n(ws:)40 b FN(u)1377 1859 y FK(\()p FL(k)q FK(\))1377 1926 y FL(i)1470 1902 y FN(u)1518 1859 y FK(\()p FL(l)p FK(\))1518 1926 y FL(j)1623 1902 y FO(=)28 b FN(\017)1750 1914 y FL(k)q(l)1825 1902 y FN(u)1873 1859 y FK(\()p FL(l)p FK(\))1873 1926 y FL(j)1950 1902 y FN(u)1998 1859 y FK(\()p FL(k)q FK(\))1998 1926 y FL(i)2090 1902 y FO(,)k FN(k)f FP(6)p FO(=)d FN(l)-118 2002 y FO(\()p FN(\017)-52 2014 y FL(k)q(l)36 2002 y FO(=)d(+1)k(or)f FP(\000)p FO(1,)h FN(\017)558 2014 y FL(k)q(l)645 2002 y FO(=)d FN(\017)770 2014 y FL(lk)832 2002 y FO(\),)k FN(k)s FO(,)f FN(l)f FO(=)d(1,)k FN(:)14 b(:)g(:)27 b FO(,)j FN(n)p FO(,)g(and)f(do)g(not)g(dep)r(end)h(on)-118 2102 y FN(i)23 b FO(=)f(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(n)339 2114 y FL(k)407 2102 y FO(and)f FN(j)h FO(=)23 b(1,)k FN(:)14 b(:)g(:)28 b FO(,)g FN(n)1036 2114 y FL(l)1061 2102 y FO(.)6 2201 y(Let)g Fz(A)215 2213 y FL(n;\017)335 2201 y FO(b)r(e)g(an)f(algebra)e(generated)i(b)n (y)g FN(s)1387 2213 y FK(1)1424 2201 y FO(,)h FN(:)14 b(:)g(:)f FO(,)28 b FN(s)1675 2213 y FL(n)1720 2201 y FO(,)32 2367 y Fz(A)92 2379 y FL(n;\017)207 2367 y FO(=)23 b FI(C)349 2299 y Fy(\012)394 2367 y FN(s)433 2379 y FK(1)470 2367 y FN(;)14 b(:)g(:)g(:)g(;)g(s)694 2379 y FL(n)762 2367 y FP(j)23 b FN(s)847 2332 y FK(2)847 2387 y FL(i)907 2367 y FO(=)g(1)p FN(;)k(s)1126 2379 y FL(i)1154 2367 y FN(s)1193 2379 y FL(j)1250 2367 y FO(=)c FN(\017)1372 2379 y FL(ij)1430 2367 y FN(s)1469 2379 y FL(j)1504 2367 y FN(s)1543 2379 y FL(i)1571 2367 y FO(;)k FN(i;)14 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n) 2113 2299 y Fy(\013)2152 2367 y FN(;)728 2491 y(\017)23 b FO(=)g(\()p FN(\017)939 2503 y FL(ij)997 2491 y FO(\))p FN(;)180 b(\017)1266 2503 y FL(ii)1340 2491 y FO(=)23 b(1)p FN(:)6 2656 y FO(The)31 b(algebra)d Fz(A)534 2668 y FL(n;\017)657 2656 y FO(is)i(\014nite)g(dimensional)d(and)j (semi-simple,)d(it)j(has)g(a)-118 2756 y(\014nite)21 b(residual)e(family)f(of)j(irreducible)d FP(\003)p FO(-represen)n (tations)g FN(\031)1818 2768 y FL(p)1878 2756 y FO(with)j FN(s)2100 2726 y FM(\003)2100 2778 y FL(i)2161 2756 y FO(=)i FN(s)2288 2768 y FL(i)2316 2756 y FO(,)-118 2856 y FN(i)g FO(=)f(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(,)f(and)h(is)e(an)h FN(F)802 2868 y FL(m)866 2856 y FO(-algebra,)d(where)k FN(m)23 b FP(\025)f FO(2)1673 2825 y FL(n=)p FK(2+1)1869 2856 y FO(.)-118 3010 y FQ(8.)34 b FO(Let)20 b Fz(B)205 3025 y FK(\()p Fs(A)279 3034 y Fv(k)316 3025 y FK(\))p FL(;\017)417 3010 y FO(=)j FI(C)15 b FP(h)p FN(u)639 2966 y FK(\(1\))639 3032 y(1)734 3010 y FN(;)f(:)g(:)g(:)f(;)h(u)966 2966 y FK(\()p FL(n)p FK(\))966 3019 y FL(n)1007 3027 y Fv(n)1086 3010 y FP(j)23 b FO(\(\))1196 3022 y FK(1)1234 3010 y FN(;)14 b(:)g(:)g(:)f(;)h FO(\(\))1482 3022 y FL(n)1528 3010 y FO(;)28 b FN(u)1627 2966 y FK(\()p FL(k)q FK(\))1627 3033 y FL(i)1719 3010 y FN(u)1767 2966 y FK(\()p FL(l)p FK(\))1767 3033 y FL(j)1867 3010 y FO(=)22 b FN(\017)1988 3022 y FL(k)q(l)2050 3010 y FN(u)2098 2966 y FK(\()p FL(l)p FK(\))2098 3033 y FL(j)2175 3010 y FN(u)2223 2966 y FK(\()p FL(k)q FK(\))2223 3033 y FL(i)2316 3010 y FN(;)-118 3109 y(k)k FP(6)p FO(=)c FN(l)r(;)28 b(k)s(;)14 b(l)24 b FO(=)e(1)14 b FN(:)g(:)g(:)f(;)h(n)p FO(;)28 b FN(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)1054 3121 y FL(k)1095 3109 y FN(;)28 b(j)g FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)1571 3121 y FL(l)1597 3109 y FP(i)p FO(.)6 3209 y(This)38 b(is)e(a)i FN(F)433 3225 y FK(2)466 3208 y Fv(n)p Fx(+1)578 3225 y FL(m)637 3233 y Fx(1)670 3225 y FM(\001\001\001\001)n(\001)p FL(m)827 3233 y Fv(n)872 3209 y FO(-algebra)d(whic)n(h)i(has)h(a)f(residual)f(family)f(of)-118 3308 y FP(\003)p FO(-rep)o(resentations)23 b FN(\031)577 3278 y FK(\(1\))681 3308 y FP(\012)15 b(\001)f(\001)g(\001)g(\012)h FN(\031)1002 3278 y FK(\()p FL(n)p FK(\))1114 3308 y FP(\012)f FN(\031)1240 3320 y FL(p)1305 3308 y FO(with)25 b(dim)12 b FN(H)1713 3328 y FL(\031)1754 3311 y Fx(\(1\))1832 3328 y FM(\012\001\001\001)o(\012)p FL(\031)2036 3311 y Fx(\()p Fv(n)p Fx(\))2121 3328 y FM(\012)p FL(\031)2212 3336 y Fv(p)2274 3308 y FP(\024)-118 3439 y FO(2)-76 3409 y FL(n)-27 3439 y FP(\001)t FN(m)73 3451 y FK(1)114 3439 y FP(\001)t(\001)i(\001)g(\001)s(\001)t FN(m)341 3451 y FL(n)407 3439 y FO(\(here,)22 b FN(\031)687 3409 y FK(\(1\))780 3439 y FP(\012)t(\001)14 b(\001)g(\001)s(\012)t FN(\031)1068 3409 y FK(\()p FL(n)p FK(\))1169 3439 y FP(\012)t FN(\031)1285 3451 y FL(p)1323 3439 y FO(\()p FN(u)1403 3396 y FK(\()p FL(k)q FK(\))1403 3462 y FL(i)1496 3439 y FO(\))23 b(=)g(1)t FP(\012)t(\001)14 b(\001)g(\001)s(\012)t FN(\031)1973 3409 y FK(\()p FL(k)q FK(\))2065 3439 y FO(\()p FN(u)2145 3396 y FK(\()p FL(k)q FK(\))2145 3462 y FL(i)2238 3439 y FO(\))t FP(\012)-118 3539 y(\001)g(\001)g(\001)k (\012)g FO(1)g FP(\012)g FN(\031)270 3551 y FL(p)309 3539 y FO(\()p FN(s)380 3551 y FL(k)421 3539 y FO(\)\).)-118 3700 y FQ(9.)58 b FO(In)35 b(items)e(7)i(and)g(8)f(ab)r(o)n(v)n(e,)i (the)f(generators)e FN(u)1553 3657 y FK(\()p FL(k)q FK(\))1553 3723 y FL(i)1680 3700 y FO(and)i FN(u)1897 3657 y FK(\()p FL(l)p FK(\))1897 3723 y FL(j)2009 3700 y FO(comm)n(ute)-118 3800 y(or)26 b(an)n(ticomm)n(ute)e(indep)r(enden)n(tly)i(of)h FN(i)g FO(and)g FN(j)5 b FO(.)37 b(In)28 b(this)e(item,)g(whether)h (the)-118 3911 y(generators)e FN(u)332 3868 y FK(\()p FL(k)q FK(\))332 3934 y FL(i)452 3911 y FO(and)i FN(u)661 3868 y FK(\()p FL(l)p FK(\))661 3934 y FL(j)766 3911 y FO(comm)n(ute)e(or)i(not)h(dep)r(ends)g(on)f FN(i)p FO(,)g FN(j)5 b FO(.)p eop %%Page: 34 38 34 37 bop -118 -137 a FO(34)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(Let)38 b FN(A)227 108 y FL(k)308 96 y FO(=)h FI(C)14 b FP(h)q FN(u;)g(v)s(;)g(s)702 108 y FK(1)744 96 y FN(;)g(:)g(:)g(:)g(;)g(s)968 108 y FL(k)1048 96 y FP(j)40 b FN(u)1159 66 y FK(2)1235 96 y FO(=)f FN(v)1382 66 y FK(2)1459 96 y FO(=)g FN(s)1602 66 y FK(2)1602 118 y FL(i)1678 96 y FO(=)g FN(e;)28 b(i)39 b FO(=)g(1)p FN(;)14 b(:)g(:)g(:)f(;)h(k)s(;)-118 196 y(us)-31 208 y FL(i)21 196 y FO(=)24 b FN(\013)163 208 y FL(i)191 196 y FN(s)230 208 y FL(i)257 196 y FN(u;)k(v)s(s)438 208 y FL(i)490 196 y FO(=)d FN(\014)627 208 y FL(i)654 196 y FN(s)693 208 y FL(i)721 196 y FN(v)s(;)j(s)854 208 y FL(i)881 196 y FN(s)920 208 y FL(j)980 196 y FO(=)c FN(\017)1103 208 y FL(ij)1161 196 y FN(s)1200 208 y FL(j)1235 196 y FN(s)1274 208 y FL(i)1302 196 y FO(;)k FN(i;)14 b(j)29 b FO(=)24 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(k)s FP(i)p FO(,)29 b(where)f FN(\013)2221 208 y FL(i)2274 196 y FO(=)-118 296 y FP(\006)p FO(1,)f FN(\014)86 308 y FL(i)136 296 y FO(=)c FP(\006)p FO(1,)k FN(\017)415 308 y FL(ij)496 296 y FO(=)c FN(\017)618 308 y FL(j)s(i)699 296 y FO(=)g FP(\006)p FO(1;)j FN(i)d FP(6)p FO(=)g FN(j)5 b FO(,)27 b FN(\017)1206 308 y FL(ii)1280 296 y FO(=)c(1.)6 400 y(Of)g(course,)g(this)f(algebra)f(should)g(ha)n(v)n(e)h(b)r(een)i (denoted)f(b)n(y)f FN(A)1955 412 y FL(k)q(;\013;\014)s(;\017)2167 400 y FO(,)i(but)-118 500 y(w)n(e)j(lea)n(v)n(e)e(out)j(the)g(sym)n(b)r (ols)d FN(\013)p FO(,)j FN(\014)t FO(,)g(and)f FN(\017)h FO(for)f(brevit)n(y)-7 b(.)6 604 y(Let)35 b(us)g(notice)f(that)h(the)g (algebras)d FI(C)15 b FP(h)p FN(u;)f(v)s(;)g(s)1486 616 y FK(1)1564 604 y FP(j)35 b FN(us)1709 616 y FK(1)1781 604 y FO(=)g FP(\000)p FN(s)1985 616 y FK(1)2021 604 y FN(u;)28 b(v)s(s)2202 616 y FK(1)2274 604 y FO(=)-118 704 y FP(\000)p FN(s)-14 716 y FK(1)23 704 y FN(v)s FP(i)34 b FO(and)g FI(C)15 b FP(h)q FN(u;)f(v)s(;)g(s)591 716 y FK(1)667 704 y FP(j)35 b FN(us)812 716 y FK(1)882 704 y FO(=)f FP(\000)p FN(s)1085 716 y FK(1)1122 704 y FN(u;)27 b(v)s(s)1302 716 y FK(1)1373 704 y FO(=)34 b FN(s)1511 716 y FK(1)1548 704 y FN(v)s FP(i)h FO(w)n(ere)e(considered)f(in)-118 803 y([224)n(,)c(149)o(].)-118 984 y FQ(Lemma)h(3.)41 b FB(The)d(algebr)l(a)g FN(A)863 996 y FL(k)940 984 y FB(is)f(a)g FN(F)1168 1001 y FK(2)1201 984 y Fv(k)q Fx(+2)9 b FB(-algebr)l(a,)39 b(and)e(has)h(a)f(r)l(esidual)-118 1083 y(family)31 b FA(L)193 1095 y FL(k)264 1083 y FB(subje)l(ct)f(to)g (the)g(c)l(ondition)6 b FO(:)39 b FP(8)p FN(\031)26 b FP(2)d FA(L)1431 1095 y FL(k)1472 1083 y FB(,)31 b FO(dim)12 b FN(H)1749 1095 y FL(\031)1817 1083 y FP(\024)23 b FO(2)1947 1053 y FL(k)q FK(+1)2071 1083 y FB(.)-118 1264 y(Pr)l(o)l(of.)43 b FO(F)-7 b(or)29 b FN(k)f FO(=)e(0,)j FN(A)611 1276 y FK(0)678 1264 y FO(is)f(a)h FN(F)887 1276 y FK(4)925 1264 y FO(-algebra,)d(and)j(has)g(a)g(residual)e(family)f FA(L)2278 1276 y FK(0)2316 1264 y FO(,)-118 1363 y(since)36 b(the)i(algebra)d FN(A)611 1375 y FK(0)688 1363 y FO(=)40 b FN(Q)859 1375 y FK(2)896 1363 y FO(.)67 b(Let)37 b FN(k)43 b FO(=)c FN(n)p FO(,)i(and)c(assume)f(that)i(all)d FN(A)2293 1375 y FL(n)-118 1463 y FO(are)h FN(F)83 1479 y FK(2)116 1462 y Fv(n)p Fx(+2)232 1463 y FO(-algebras)e(and)k(there)f (exists)f FA(L)1285 1475 y FL(n)1368 1463 y FO(with)h(dim)12 b FN(H)1788 1475 y FL(\031)1872 1463 y FP(\024)39 b FO(2)2018 1433 y FL(n)p FK(+1)2147 1463 y FO(.)66 b(By)-118 1563 y(induction,)39 b(consider)d(the)j(algebra)d FN(A)1139 1575 y FL(n)p FK(+1)1268 1563 y FO(.)69 b(It)39 b(con)n(tains)d(the)i (subalgebra)-118 1662 y FN(B)28 b FO(=)23 b FN(C)6 b FP(h)p FN(u;)14 b(v)s(;)g(s)362 1674 y FK(1)399 1662 y FN(;)g(:)g(:)g(:)f(;)h(s)622 1674 y FL(n)691 1662 y FP(j)24 b FN(\013)791 1674 y FL(i)818 1662 y FN(;)14 b(\014)902 1674 y FL(i)930 1662 y FN(;)g(\017)1001 1674 y FL(ij)1059 1662 y FP(i)p FO(.)38 b(Clearly)-7 b(,)25 b FN(B)32 b FO(is)27 b(isomorphic)c(to)28 b(some)-118 1762 y FN(A)-56 1774 y FL(n)-11 1762 y FO(.)34 b(Therefore,)19 b(the)g(claim)d(is)h(true)h(for)g FN(B)t FO(.)34 b(Let)19 b FA(L)1466 1774 y FL(n)1530 1762 y FO(b)r(e)g(a)f(residual)e(family)f (for)-118 1861 y FN(B)31 b FO(with)26 b(dim)12 b FN(H)385 1873 y FL(\031)453 1861 y FP(\024)23 b FO(2)583 1831 y FL(n)p FK(+1)712 1861 y FO(.)36 b(W)-7 b(e)27 b(construct)f(a)g (residual)e(family)g(for)i FN(A)2095 1873 y FL(n)p FK(+1)2251 1861 y FO(b)n(y)-118 1961 y(applying)c(the)j(follo)n(wing)20 b(pro)r(cedure:)34 b FP(8)p FN(\031)26 b FP(2)d FA(L)1378 1973 y FL(n)1424 1961 y FO(,)i FN(\031)h FO(:)e FN(H)1661 1973 y FL(\031)1729 1961 y FP(!)f FN(H)1904 1973 y FL(\031)1949 1961 y FO(,)i(in)n(tro)r(duce)-114 2061 y(^)-46 b FN(\031)26 b FP(2)e FA(L)93 2073 y FL(n)p FK(+1)222 2061 y FO(,)32 b(^)-46 b FN(\031)26 b FO(:)e FN(H)462 2073 y FL(\031)525 2061 y FP(\010)18 b FN(H)677 2073 y FL(\031)745 2061 y FP(!)23 b FN(H)920 2073 y FL(\031)984 2061 y FP(\010)18 b FN(H)1136 2073 y FL(\031)1181 2061 y FO(,)28 b(b)n(y)46 2301 y(^)-46 b FN(\031)s FO(\()p FN(u)p FO(\))23 b(=)315 2184 y Fy(\022)376 2251 y FN(\031)s FO(\()p FN(u)p FO(\))235 b(0)437 2350 y(0)143 b FN(\013)675 2362 y FL(n)p FK(+1)804 2350 y FN(\031)s FO(\()p FN(u)p FO(\))966 2184 y Fy(\023)1041 2301 y FN(;)102 b FO(^)-47 b FN(\031)t FO(\()p FN(v)s FO(\))23 b(=)1430 2184 y Fy(\022)1491 2251 y FN(\031)s FO(\()p FN(v)s FO(\))231 b(0)1549 2350 y(0)141 b FN(\014)1779 2362 y FL(n)p FK(+1)1909 2350 y FN(\031)s FO(\()p FN(v)s FO(\))2067 2184 y Fy(\023)2142 2301 y FN(;)257 2534 y FO(^)-47 b FN(\031)t FO(\()p FN(s)374 2546 y FL(i)402 2534 y FO(\))23 b(=)545 2417 y Fy(\022)606 2483 y FN(\031)s FO(\()p FN(s)727 2495 y FL(i)755 2483 y FO(\))246 b(0)676 2583 y(0)152 b FN(\017)904 2595 y FL(n)p FK(+1)p FL(i)1057 2583 y FN(\031)s FO(\()p FN(s)1178 2595 y FL(i)1206 2583 y FO(\))1238 2417 y Fy(\023)1313 2534 y FN(;)180 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)747 2766 y FO(^)-47 b FN(\031)t FO(\()p FN(s)864 2778 y FL(n)p FK(+1)993 2766 y FO(\))23 b(=)1136 2649 y Fy(\022)1198 2716 y FO(0)83 b FN(I)1197 2815 y(I)91 b FO(0)1366 2649 y Fy(\023)1441 2766 y FN(;)-118 3004 y(I)43 b FO(:)36 b FN(H)89 3016 y FL(\031)170 3004 y FP(!)h FN(H)359 3016 y FL(\031)439 3004 y FO(is)d(the)i(iden)n(tit)n(y)e(op)r(erator.)58 b(Let)36 b(us)f(sho)n(w)g(that)h FA(L)2118 3016 y FL(n)p FK(+1)2283 3004 y FO(is)-118 3104 y(indeed)31 b(a)g(residual)e(family) -7 b(.)46 b(F)-7 b(or)31 b FP(8)p FN(x)e FP(2)h FN(A)1250 3116 y FL(n)p FK(+1)1412 3104 y FO(there)h(exists)f(an)i(expansion)-118 3204 y FN(x)23 b FO(=)g FN(b)76 3216 y FK(1)131 3204 y FO(+)c FN(s)254 3216 y FL(n)p FK(+1)383 3204 y FN(b)419 3216 y FK(2)456 3204 y FO(,)27 b(with)g FN(b)731 3216 y FK(1)768 3204 y FO(,)h FN(b)855 3216 y FK(2)915 3204 y FP(2)c FN(B)t FO(.)37 b(If)28 b FN(b)1240 3216 y FK(2)1300 3204 y FP(6)p FO(=)22 b(0,)28 b(then)g FP(9)p FN(\031)e FP(2)e FA(L)1926 3216 y FL(n)1999 3204 y FO(suc)n(h)j(that)-118 3303 y FN(\031)s FO(\()p FN(b)0 3315 y FK(2)37 3303 y FO(\))d FP(6)p FO(=)e(0)28 b FP(\))402 3545 y FO(^)-47 b FN(\031)s FO(\()p FN(x)p FO(\))25 b(=)670 3428 y Fy(\022)731 3495 y FN(\031)s FO(\()p FN(b)849 3507 y FK(1)887 3495 y FO(\))83 b FP(\003)731 3594 y FN(\031)s FO(\()p FN(b)849 3606 y FK(2)887 3594 y FO(\))g FP(\003)1044 3428 y Fy(\023)1128 3545 y FP(6)p FO(=)22 b(0)p FN(;)184 b FO(^)-46 b FN(\031)26 b FP(2)e FA(L)1671 3557 y FL(n)p FK(+1)1800 3545 y FN(:)6 3792 y FO(If)32 b FN(b)129 3804 y FK(2)194 3792 y FO(=)d(0,)i FN(b)420 3804 y FK(1)486 3792 y FP(6)p FO(=)d(0,)j(then)h FP(9)p FN(\031)g FP(2)d FA(L)1136 3804 y FL(n)1181 3792 y FO(:)44 b FN(\031)s FO(\()p FN(b)1366 3804 y FK(1)1404 3792 y FO(\))29 b FP(6)p FO(=)f(0)i FP(\))36 b FO(^)-47 b FN(\031)t FO(\()p FN(b)1863 3804 y FK(1)1900 3792 y FO(\))29 b FP(6)p FO(=)f(0.)47 b(Note)-118 3892 y(that)28 b FP(8)t FO(^)-46 b FN(\031)25 b FP(2)f FA(L)319 3904 y FL(n)p FK(+1)448 3892 y FO(,)k(dim)17 b(^)-47 b FN(\031)27 b FP(\024)22 b FO(2)854 3862 y FL(n)p FK(+2)983 3892 y FO(.)p 2278 3892 4 57 v 2282 3839 50 4 v 2282 3892 V 2331 3892 4 57 v eop %%Page: 35 39 35 38 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(35)-118 96 y FB(R)l(emark)30 b(6.)42 b FO(There)c(is)g(a)h (natural)e(in)n(v)n(olution)e(in)j FN(A)1599 108 y FL(k)1679 96 y FO(giv)n(en)f(b)n(y)i FN(u)2082 66 y FM(\003)2162 96 y FO(=)i FN(u)p FO(,)-118 196 y FN(v)-75 166 y FM(\003)-14 196 y FO(=)23 b FN(v)s FO(,)i FN(s)204 166 y FM(\003)204 218 y FL(i)266 196 y FO(=)d FN(s)392 208 y FL(i)420 196 y FO(.)36 b(Since)23 b(there)i(exists)e(a)h(residual)d(family)h(for)i FN(Q)1941 208 y FK(2)2002 196 y FO(suc)n(h)g(that)-118 296 y FP(8)p FN(\031)h FP(2)f FA(L)139 308 y FK(0)203 296 y FO(the)i(op)r(erators)e FN(\031)s FO(\()p FN(u)p FO(\),)j FN(\031)s FO(\()p FN(v)s FO(\))g(are)e(self-adjoin)n(t)f (\(see)i(1.2.2,)f(item)g(2\),)-118 395 y(there)19 b(exists)e(a)h (residual)e(family)g(for)j FN(A)1093 407 y FL(k)1153 395 y FO(satisfying)d(the)j(condition:)30 b FP(8)p FN(\031)c FP(2)d FA(L)2297 407 y FL(k)-118 495 y FO(the)32 b(op)r(erators)e FN(\031)s FO(\()p FN(u)p FO(\),)j FN(\031)s FO(\()p FN(v)s FO(\),)h FN(\031)s FO(\()p FN(s)954 507 y FL(i)983 495 y FO(\))e(are)e(self-adjoin)n(t.)47 b(Therefore,)32 b(there)g(is)-118 595 y(a)38 b(residual)d(family)h(for)i FN(A)747 607 y FL(k)826 595 y FO(consisting)e(only)h(of)h(irreducible)d FP(\003)p FO(-rep)o(resen-)-118 694 y(tations.)6 794 y(F)-7 b(or)29 b(a)g(description)d(of)k(irreducible)25 b FP(\003)p FO(-rep)o(resentations)i(of)i FN(A)1970 806 y FL(k)2011 794 y FO(,)h(see)f(also)-118 893 y([232)n(].)-118 1013 y FB(R)l(emark)h(7.)42 b FP(8)27 b FN(k)s FO(,)g FN(\013)522 1025 y FL(i)550 1013 y FO(,)h FN(\014)648 1025 y FL(i)675 1013 y FO(,)g FN(\017)760 1025 y FL(ij)818 1013 y FO(,)g(the)g(algebra)d FN(A)1365 1025 y FL(k)1434 1013 y FO(is)h(semi-simple.)6 1133 y(Moreo)n(v)n(er,)f(the)j(follo)n (wing)c(theorem)i(holds.)-118 1273 y FQ(Theorem)k(3.)41 b FB(L)l(et)d FN(Q)621 1285 y FK(2)p FL(;m)776 1273 y FO(=)h FN(A)942 1285 y FL(m)1044 1273 y FB(with)h FN(\013)1287 1285 y FL(i)1354 1273 y FO(=)f(1)p FB(,)i FN(\014)1613 1285 y FL(i)1680 1273 y FO(=)e(1)p FB(,)i FN(\017)1926 1285 y FL(ij)2024 1273 y FO(=)e(1)p FB(,)i(for)-118 1372 y(al)t(l)36 b FN(i)p FB(,)g FN(j)5 b FB(.)55 b(Then)36 b(every)g(algebr)l(a)g FN(A)1010 1384 y FL(k)1086 1372 y FB(is)g(isomorphic)h(to)e FN(M)1797 1384 y FK(2)1830 1368 y Fv(n)1875 1372 y FO(\()p FN(Q)1973 1384 y FK(2)p FL(;m)2089 1372 y FO(\))g FB(or)h(to)-118 1472 y FN(M)-37 1484 y FK(2)-4 1468 y Fv(n)40 1472 y FO(\()p FN(Z)6 b FO(\()p FN(A)229 1484 y FL(k)271 1472 y FO(\)\))p FB(,)30 b(wher)l(e)h FN(Z)6 b FO(\()p FN(A)782 1484 y FL(k)823 1472 y FO(\))30 b FB(is)g(the)g(c)l(enter)f(of)i FN(A)1521 1484 y FL(k)1562 1472 y FB(.)-118 1612 y(Pr)l(o)l(of.)43 b FO(Let)28 b(us)f(split)f(the)i(pro)r(of)f(in)n(to)g(four)g(steps.)6 1712 y(1\).)40 b(Let)28 b(us)h(de\014ne)f(the)h(algebra)d FN(A)1138 1724 y FL(k)1204 1712 y FO(=)e FI(C)15 b FP(h)p FN(u;)f(v)s(;)g(s)1583 1724 y FK(1)1626 1712 y FN(;)g(:)g(:)g(:)f(;)h (s)1849 1724 y FL(k)1914 1712 y FP(j)25 b FN(\013)2015 1724 y FL(i)2043 1712 y FN(;)14 b(\014)2127 1724 y FL(i)2154 1712 y FN(;)g(\017)2225 1724 y FL(ij)2283 1712 y FP(i)p FO(.)-118 1811 y(Supp)r(ose)23 b(that)h(there)f(exist)f FN(i)p FO(,)i FN(j)k FO(suc)n(h)23 b(that)g FN(\017)1309 1823 y FL(ij)1390 1811 y FO(=)g FP(\000)p FO(1,)g(for)g(example,)f FN(s)2138 1823 y FK(1)2175 1811 y FN(s)2214 1823 y FK(2)2274 1811 y FO(=)-118 1911 y FP(\000)p FN(s)-14 1923 y FK(2)23 1911 y FN(s)62 1923 y FK(1)99 1911 y FO(.)64 b(Then,)39 b(using)c(the)i(follo)n(wing)32 b(substitution)j(of)i(generators)d FN(s)2199 1881 y FM(0)2199 1931 y FK(1)2274 1911 y FO(=)-118 2010 y FN(s)-79 2022 y FK(1)-42 2010 y FN(;)14 b(s)34 1980 y FM(0)34 2031 y FK(2)94 2010 y FO(=)23 b FN(s)221 2022 y FK(2)258 2010 y FN(;)360 2350 y(s)399 2315 y FM(0)399 2370 y FL(j)457 2350 y FO(=)544 2105 y Fy(8)544 2179 y(>)544 2204 y(>)544 2229 y(>)544 2254 y(<)544 2404 y(>)544 2428 y(>)544 2453 y(>)544 2478 y(:)618 2173 y FN(s)657 2185 y FL(j)692 2173 y FN(;)503 b(\017)1252 2185 y FK(1)p FL(j)1343 2173 y FO(=)23 b FN(\017)1465 2185 y FK(2)p FL(j)1555 2173 y FO(=)g(1)p FN(;)618 2293 y(s)657 2305 y FK(1)694 2293 y FN(s)733 2305 y FL(j)768 2293 y FN(;)427 b(\017)1252 2305 y FK(1)p FL(j)1343 2293 y FO(=)23 b FP(\000)p FN(\017)1530 2305 y FK(2)p FL(j)1620 2293 y FO(=)g(1)p FN(;)618 2413 y(s)657 2425 y FK(2)694 2413 y FN(s)733 2425 y FL(j)768 2413 y FN(;)427 b(\017)1252 2425 y FK(1)p FL(j)1343 2413 y FO(=)23 b FP(\000)p FN(\017)1530 2425 y FK(2)p FL(j)1620 2413 y FO(=)g FP(\000)p FO(1)p FN(;)618 2461 y Fy(p)p 701 2461 171 4 v 71 x FO(\()p FP(\000)p FO(1\))14 b FN(s)925 2544 y FK(1)962 2532 y FN(s)1001 2544 y FK(2)1038 2532 y FN(s)1077 2544 y FL(j)1112 2532 y FN(;)83 b(\017)1252 2544 y FK(1)p FL(j)1343 2532 y FO(=)23 b FN(\017)1465 2544 y FK(2)p FL(j)1555 2532 y FO(=)g FP(\000)p FO(1)p FN(;)363 2861 y(u)411 2827 y FM(0)457 2861 y FO(=)544 2616 y Fy(8)544 2691 y(>)544 2716 y(>)544 2741 y(>)544 2766 y(<)544 2915 y(>)544 2940 y(>)544 2965 y(>)544 2990 y(:)618 2685 y FN(u;)503 b(\013)1245 2697 y FK(1)1305 2685 y FO(=)23 b FN(\013)1446 2697 y FK(2)1506 2685 y FO(=)g(1)p FN(;)618 2804 y(s)657 2816 y FK(1)694 2804 y FN(u;)427 b(\013)1245 2816 y FK(1)1305 2804 y FO(=)23 b FP(\000)p FN(\013)1511 2816 y FK(2)1571 2804 y FO(=)f(1)p FN(;)618 2924 y(s)657 2936 y FK(2)694 2924 y FN(u;)427 b(\013)1245 2936 y FK(1)1305 2924 y FO(=)23 b FP(\000)p FN(\013)1511 2936 y FK(2)1571 2924 y FO(=)f FP(\000)p FO(1)p FN(;)618 2972 y Fy(p)p 701 2972 V 71 x FO(\()p FP(\000)p FO(1\))14 b FN(s)925 3055 y FK(1)962 3043 y FN(s)1001 3055 y FK(2)1038 3043 y FN(u;)83 b(\013)1245 3055 y FK(1)1305 3043 y FO(=)23 b FN(\013)1446 3055 y FK(2)1506 3043 y FO(=)g FP(\000)p FO(1)p FN(;)367 3372 y(v)410 3338 y FM(0)457 3372 y FO(=)544 3127 y Fy(8)544 3202 y(>)544 3227 y(>)544 3252 y(>)544 3277 y(<)544 3426 y(>)544 3451 y(>)544 3476 y(>)544 3501 y(:)618 3196 y FN(v)s(;)503 b(\014)1234 3208 y FK(1)1295 3196 y FO(=)22 b FN(\014)1429 3208 y FK(2)1490 3196 y FO(=)g(1)p FN(;)618 3316 y(s)657 3328 y FK(1)694 3316 y FN(v)s(;)427 b(\014)1234 3328 y FK(1)1295 3316 y FO(=)22 b FP(\000)p FN(\014)1494 3328 y FK(2)1554 3316 y FO(=)h(1)p FN(;)618 3435 y(s)657 3447 y FK(2)694 3435 y FN(v)s(;)427 b(\014)1234 3447 y FK(1)1295 3435 y FO(=)22 b FP(\000)p FN(\014)1494 3447 y FK(2)1554 3435 y FO(=)h FP(\000)p FO(1)p FN(;)618 3484 y Fy(p)p 701 3484 V 71 x FO(\()p FP(\000)p FO(1\))14 b FN(s)925 3567 y FK(1)962 3555 y FN(s)1001 3567 y FK(2)1038 3555 y FN(v)s(;)83 b(\014)1234 3567 y FK(1)1295 3555 y FO(=)22 b FN(\014)1429 3567 y FK(2)1490 3555 y FO(=)g FP(\000)p FO(1)p FN(;)-118 3712 y FO(w)n(e)41 b(obtain)e(that)j FN(A)545 3724 y FL(k)631 3712 y FO(=)k FI(C)14 b FP(h)q FN(u)876 3682 y FM(0)905 3712 y FN(;)g(v)985 3682 y FM(0)1008 3712 y FN(;)g(s)1084 3682 y FM(0)1084 3732 y FK(1)1121 3712 y FN(;)g(:)g(:)g(:)g(;)g(s)1345 3682 y FM(0)1345 3735 y FL(k)1431 3712 y FP(j)45 b FN(\013)1552 3682 y FM(0)1552 3732 y FK(1)1635 3712 y FO(=)g FN(\013)1798 3682 y FM(0)1798 3732 y FK(2)1881 3712 y FO(=)g FN(\014)2042 3682 y FM(0)2038 3732 y FK(1)2121 3712 y FO(=)g FN(\014)2282 3682 y FM(0)2278 3732 y FK(2)2316 3712 y FN(;)-118 3811 y(\017)-84 3781 y FM(0)-84 3832 y FK(12)19 3811 y FO(=)33 b FP(\000)p FO(1)p FN(;)26 b(\017)307 3781 y FM(0)307 3833 y FK(1)p FL(j)408 3811 y FO(=)33 b FN(\017)540 3781 y FM(0)540 3833 y FK(2)p FL(j)641 3811 y FO(=)f(1)p FN(;)27 b(j)38 b(>)33 b FO(2)p FP(i)p FO(.)55 b(Let)33 b FN(A)1368 3823 y FL(k)q FM(\000)p FK(2)1528 3811 y FO(denote)h(the)f(subalgebra) -118 3911 y FI(C)15 b FP(h)p FN(s)7 3881 y FM(0)7 3932 y FK(3)50 3911 y FN(;)f(:)g(:)g(:)g(;)g(s)274 3881 y FM(0)274 3935 y FL(k)315 3911 y FN(;)g(u)400 3881 y FM(0)422 3911 y FN(;)g(v)502 3881 y FM(0)526 3911 y FP(i)p FO(.)p eop %%Page: 36 40 36 39 bop -118 -137 a FO(36)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FQ(Lemma)k(4.)41 b FB(The)31 b(algebr)l(a)g FN(A)849 108 y FL(k)920 96 y FB(is)f(isomorphic)i(to)e FN(M)1615 108 y FK(2)1652 96 y FO(\()p FN(A)1746 108 y FL(k)q FM(\000)p FK(2)1872 96 y FO(\))p FB(.)-118 260 y(Pr)l(o)l(of.)43 b FO(Let)35 b FN(A)360 272 y FL(k)401 260 y FO(,)h FN(A)522 272 y FL(k)q FM(\000)p FK(2)682 260 y FO(b)r(e)f(algebras)c(describ)r (ed)j(ab)r(o)n(v)n(e.)55 b(Then)35 b FP(8)p FN(x)f FP(2)g FN(A)2297 272 y FL(k)-118 360 y FO(there)28 b(exists)g(a)g(unique)g (decomp)r(osition)d FN(x)h FO(=)e FN(s)1416 330 y FM(0)1416 380 y FK(1)1453 360 y FN(a)1497 372 y FK(1)1554 360 y FO(+)19 b FN(s)1677 330 y FM(0)1677 380 y FK(2)1714 360 y FN(a)1758 372 y FK(2)1814 360 y FO(+)g FN(s)1937 330 y FM(0)1937 380 y FK(1)1974 360 y FN(s)2013 330 y FM(0)2013 380 y FK(2)2050 360 y FN(a)2094 372 y FK(3)2151 360 y FO(+)f FN(a)2278 372 y FK(4)2316 360 y FO(,)-118 459 y(where)31 b FN(a)170 471 y FL(i)228 459 y FP(2)f FN(A)375 471 y FL(k)q FM(\000)p FK(2)501 459 y FO(,)j FN(i)d FO(=)f(1)j FN(:)14 b(:)g(:)27 b FO(,)33 b(4.)49 b(This)31 b(implies)d(the)k(follo) n(wing)c(iden)n(tit)n(y)-118 559 y FN(x)g FO(=)50 492 y Fy(\000)88 559 y FO(\(1)20 b(+)g FN(s)306 529 y FM(0)306 580 y FK(1)343 559 y FO(\))14 b FN(s)428 571 y FK(2)465 559 y FN(a)509 529 y FM(0)509 580 y FK(1)567 559 y FO(+)20 b(\(1)g FP(\000)g FN(s)870 529 y FM(0)870 580 y FK(1)907 559 y FO(\))14 b FN(s)992 571 y FK(2)1030 559 y FN(a)1074 529 y FM(0)1074 580 y FK(2)1131 559 y FO(+)20 b(\(1)g(+)g FN(s)1434 529 y FM(0)1434 580 y FK(1)1471 559 y FO(\))14 b FN(a)1561 529 y FM(0)1561 580 y FK(3)1619 559 y FO(+)20 b(\(1)g FP(\000)g FN(s)1922 529 y FM(0)1922 580 y FK(1)1959 559 y FO(\))14 b FN(a)2049 529 y FM(0)2049 580 y FK(4)2087 492 y Fy(\001)2125 559 y FN(=)p FO(2.)44 b(It)-118 659 y(can)27 b(b)r(e)h(easily)d(v)n(eri\014ed)h(that)i FN( )12 b FO(:)28 b FN(A)1026 671 y FL(k)1090 659 y FP(!)23 b FN(M)1277 671 y FK(2)1314 659 y FO(\()p FN(A)1408 671 y FL(k)q FM(\000)p FK(2)1534 659 y FO(\),)-6 882 y FN( )s FO(\()p FN(s)122 848 y FM(0)122 902 y FK(1)159 882 y FO(\))h(=)302 765 y Fy(\022)365 831 y FN(e)115 b FO(0)363 931 y(0)83 b FP(\000)p FN(e)591 765 y Fy(\023)666 882 y FN(;)97 b( )s FO(\()p FN(s)914 848 y FM(0)914 902 y FK(2)952 882 y FO(\))23 b(=)1095 765 y Fy(\022)1156 831 y FO(0)84 b FN(e)1157 931 y(e)g FO(0)1322 765 y Fy(\023)1397 882 y FN(;)97 b( )s FO(\()p FN(s)1645 848 y FM(0)1645 902 y FL(j)1680 882 y FO(\))23 b(=)1823 765 y Fy(\022)1884 828 y FN(s)1923 798 y FM(0)1923 850 y FL(j)2057 828 y FO(0)1900 931 y(0)99 b FN(s)2080 901 y FM(0)2080 953 y FL(j)2115 765 y Fy(\023)2190 882 y FN(;)374 1117 y( )s FO(\()p FN(u)511 1083 y FM(0)535 1117 y FO(\))23 b(=)678 1000 y Fy(\022)739 1067 y FN(u)787 1037 y FM(0)907 1067 y FO(0)753 1166 y(0)98 b FN(u)941 1136 y FM(0)963 1000 y Fy(\023)1038 1117 y FN(;)f( )s FO(\()p FN(v)1290 1083 y FM(0)1314 1117 y FO(\))23 b(=)1457 1000 y Fy(\022)1518 1067 y FN(v)1561 1037 y FM(0)1680 1067 y FO(0)1531 1166 y(0)95 b FN(v)1711 1136 y FM(0)1734 1000 y Fy(\023)1809 1117 y FN(;)-118 1343 y FO(is)30 b(an)g(isomorphism.)42 b(The)31 b(in)n(v)n(erse)d(mapping)h FN( )1469 1313 y FM(\000)p FK(1)1587 1343 y FO(:)f FN(M)1719 1355 y FK(2)1756 1343 y FO(\()p FN(A)1850 1355 y FL(k)q FM(\000)p FK(2)1976 1343 y FO(\))h FP(!)g FN(A)2211 1355 y FL(k)2283 1343 y FO(is)-118 1443 y(de\014ned)f(b)n(y)f(the)h(form)n(ula:)-118 1658 y FN( )-61 1624 y FM(\000)p FK(1)42 1541 y Fy(\022)103 1608 y FN(a)147 1578 y FM(0)147 1628 y FK(3)267 1608 y FN(a)311 1578 y FM(0)311 1628 y FK(1)103 1707 y FN(a)147 1677 y FM(0)147 1728 y FK(2)267 1707 y FN(a)311 1677 y FM(0)311 1728 y FK(4)348 1541 y Fy(\023)432 1658 y FO(=)530 1602 y(1)p 530 1639 42 4 v 530 1715 a(2)582 1591 y Fy(\000)620 1658 y FO(\(1+)p FN(s)798 1624 y FM(0)798 1679 y FK(1)834 1658 y FO(\))p FN(s)905 1670 y FK(2)943 1658 y FN(a)987 1624 y FM(0)987 1679 y FK(1)1024 1658 y FO(+\(1)p FP(\000)p FN(s)1267 1624 y FM(0)1267 1679 y FK(1)1303 1658 y FO(\))p FN(s)1374 1670 y FK(2)1411 1658 y FN(a)1455 1624 y FM(0)1455 1679 y FK(2)1492 1658 y FO(+\(1+)p FN(s)1735 1624 y FM(0)1735 1679 y FK(1)1772 1658 y FO(\))p FN(a)1848 1624 y FM(0)1848 1679 y FK(3)1885 1658 y FO(+\(1)p FP(\000)p FN(s)2128 1624 y FM(0)2128 1679 y FK(1)2164 1658 y FO(\))p FN(a)2240 1624 y FM(0)2240 1679 y FK(4)2277 1591 y Fy(\001)2316 1658 y FN(:)-118 1874 y FO(whic)n(h)e(completes)g(the)i(pro)r(of)f(of)g(the)h(lemma.)p 2278 1874 4 57 v 2282 1821 50 4 v 2282 1874 V 2331 1874 4 57 v 6 2038 a(Using)f(this)g(lemma)d(one)j(can)h(obtain)e(the)i (follo)n(wing)23 b(prop)r(osition:)-118 2187 y FQ(Prop)s(osition)30 b(11.)41 b FB(The)31 b(algebr)l(a)g FN(A)1075 2199 y FL(k)1140 2187 y FO(=)23 b FI(C)15 b FP(h)p FN(u;)f(v)s(;)g(s)1518 2199 y FK(1)1561 2187 y FN(;)g(:)g(:)g(:)g(;)g(s)1785 2199 y FL(k)1849 2187 y FP(j)24 b FN(\013)1949 2199 y FL(i)1976 2187 y FN(;)14 b(\014)2060 2199 y FL(i)2088 2187 y FN(;)g(\017)2159 2199 y FL(ij)2217 2187 y FP(i)30 b FB(is)-118 2287 y(isomorphic)i(to)e FN(M)488 2299 y FK(2)521 2283 y Fv(m)580 2287 y FO(\()p FN(A)674 2299 y FL(k)q FM(\000)p FK(2)p FL(m)859 2287 y FO(\))g FB(for)h(some)f FN(m)p FB(,)g(wher)l(e)239 2457 y FN(A)301 2469 y FL(k)q FM(\000)p FK(2)p FL(m)509 2457 y FO(=)23 b FI(C)15 b FP(h)p FN(u)731 2423 y FM(0)760 2457 y FN(;)f(v)840 2423 y FM(0)863 2457 y FN(;)g(s)939 2423 y FM(0)939 2478 y FK(2)p FL(m)p FK(+1)1119 2457 y FN(;)g(:)g(:)g(:)g(;)g(s)1343 2423 y FM(0)1343 2478 y FL(k)1406 2457 y FP(j)24 b FN(\013)1506 2423 y FM(0)1506 2478 y FL(i)1533 2457 y FN(;)14 b(\014)1621 2423 y FM(0)1617 2478 y FL(i)1645 2457 y FN(;)g(\017)1716 2423 y FM(0)1716 2478 y FL(ij)1797 2457 y FO(=)23 b(1)p FP(i)p FN(:)6 2628 y FO(So,)k(w)n(e)g(m)n(ust)f(study)i(the)f (structure)g(of)g(the)h(algebra)c FN(A)1787 2640 y FL(k)1855 2628 y FO(with)j(the)g(con-)-118 2728 y(dition)f(that)i FN(\017)336 2740 y FL(ij)417 2728 y FO(=)22 b(1.)6 2827 y(2\))33 b(W)-7 b(e)32 b(further)h(assume,)f(without)g(loss)e(of)i (generalit)n(y)-7 b(,)31 b(that)h(for)g(some)-118 2927 y FN(m)25 b FP(2)h FI(N)t FO(,)35 b(the)30 b(relations)25 b FN(\013)709 2939 y FL(i)762 2927 y FO(=)g FN(\014)899 2939 y FL(i)927 2927 y FO(,)k(1)c FP(\024)g FN(i)g(<)g(m)p FO(,)k FN(\013)1458 2939 y FL(i)1511 2927 y FP(6)p FO(=)c FN(\014)1648 2939 y FL(i)1676 2927 y FO(,)k FN(m)d FP(\024)f FN(i)g FP(\024)f FN(k)s FO(,)30 b(hold.)-118 3026 y(Let)e(us)f(in)n (tro)r(duce)f(the)i(new)g(generators)d FN(u)1270 2996 y FM(0)1316 3026 y FO(=)e FN(u)p FO(,)k FN(v)1545 2996 y FM(0)1592 3026 y FO(=)22 b FN(v)s FO(,)649 3267 y FN(s)688 3233 y FM(0)688 3288 y FL(j)746 3267 y FO(=)833 3125 y Fy(\()900 3211 y FN(s)939 3223 y FL(j)974 3211 y FN(;)163 b FO(1)22 b FP(\024)h FN(j)28 b(<)23 b(m;)900 3330 y(s)939 3342 y FL(j)974 3330 y FN(s)1013 3342 y FL(k)1054 3330 y FN(;)83 b(m)23 b FP(\024)f FN(j)28 b(<)23 b(k)s(:)-118 3513 y FO(Then)30 b FN(A)163 3525 y FL(k)230 3513 y FO(=)c FI(C)15 b FP(h)p FN(u)455 3482 y FM(0)484 3513 y FN(;)f(v)564 3482 y FM(0)587 3513 y FN(;)g(s)663 3482 y FM(0)663 3533 y FK(1)700 3513 y FN(;)g(:)g(:)g(:)g(;)g(s)924 3482 y FM(0)924 3536 y FL(k)990 3513 y FP(j)27 b FN(\013)1093 3525 y FL(i)1147 3513 y FO(=)e FN(\014)1284 3525 y FL(i)1312 3513 y FN(;)j(i)d(<)h(k)s(;)i(\013)1658 3525 y FL(k)1725 3513 y FO(=)e FP(\006)p FO(1)p FN(;)g(\014)2019 3525 y FL(k)2086 3513 y FO(=)g FP(\006)p FO(1)p FP(i)p FO(,)-118 3612 y(and)k(there)f(are)g(t)n(w)n(o)g(p)r(ossibilities:)36 b FN(\013)1096 3624 y FL(k)1164 3612 y FO(=)27 b FN(\014)1303 3624 y FL(k)1373 3612 y FO(or)i FN(\013)1530 3624 y FL(k)1598 3612 y FP(6)p FO(=)d FN(\014)1736 3624 y FL(k)1777 3612 y FO(.)44 b(The)30 b(\014rst)g(case)-118 3712 y(is)c(considered)g(in)h (3\),)g(and)h(the)g(second)f(in)g(4\).)6 3811 y(3\))j(Let)h(us)f (arrange)e(the)i(family)e FP(f)p FN(s)1157 3823 y FL(j)1191 3811 y FP(g)i FO(so)f(that,)j(for)d(some)g FN(m)p FO(,)i(the)f(con-) -118 3911 y(ditions)i FN(\013)214 3923 y FL(i)276 3911 y FO(=)h FN(\014)421 3923 y FL(i)483 3911 y FO(=)h(1,)h(1)f FP(\024)f FN(i)h(<)f(m)p FO(,)j(and)e FN(\013)1370 3923 y FL(i)1432 3911 y FO(=)g FN(\014)1578 3923 y FL(i)1639 3911 y FO(=)g FP(\000)p FO(1,)h FN(m)f FP(\024)f FN(i)h FP(\024)g FN(k)s FO(,)p eop %%Page: 37 41 37 40 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(37)-118 96 y(hold.)66 b(Using)37 b(the)h(new)g(generators)e FN(s)1159 66 y FM(0)1159 118 y FL(i)1226 96 y FO(=)k FN(s)1370 108 y FL(i)1398 96 y FO(,)g(1)g FP(\024)g FN(i)g(<)f(m)p FO(,)i FN(s)1997 66 y FM(0)1997 118 y FL(i)2064 96 y FO(=)f FN(s)2208 108 y FL(i)2236 96 y FN(s)2275 108 y FL(k)2316 96 y FO(,)-118 196 y FN(m)29 b FP(\024)g FN(i)g(<)g(k)24 b FP(\000)d FO(1,)32 b FN(u)528 166 y FM(0)580 196 y FO(=)d FN(u)p FO(,)j FN(v)820 166 y FM(0)873 196 y FO(=)d FN(v)s FO(,)k(w)n(e)e(obtain)f(the)i(algebra)d FN(A)1958 208 y FL(k)2030 196 y FO(with)i(the)-118 296 y(co)r(e\016cien)n(ts)c FN(\013)356 266 y FM(0)356 317 y FL(i)408 296 y FO(=)c FN(\014)547 266 y FM(0)543 317 y FL(i)595 296 y FO(=)h(1,)k FN(i)c(<)f(k)s FO(.)39 b(If)29 b FN(\013)1163 266 y FM(0)1163 319 y FL(k)1228 296 y FO(=)24 b FN(\014)1368 266 y FM(0)1364 319 y FL(k)1429 296 y FO(=)g(1,)k(then)g(the)h(algebra)c FN(A)2297 308 y FL(k)-118 395 y FO(is)f(isomorphic)e(to)k FN(Q)548 407 y FK(2)p FL(;k)641 395 y FO(.)36 b(In)26 b(the)g(case)f(where)g FN(\013)1408 365 y FM(0)1408 419 y FL(k)1472 395 y FO(=)e FN(\014)1611 365 y FM(0)1607 419 y FL(k)1671 395 y FO(=)f FP(\000)p FO(1,)j(w)n(e)h(ha)n(v)n(e)e (the)-118 495 y(follo)n(wing)f(prop)r(osition:)-118 646 y FQ(Prop)s(osition)30 b(12.)41 b FN(A)624 658 y FL(k)710 646 y FO(=)j FI(C)15 b FP(h)q FN(u;)f(v)s(;)g(s)1110 658 y FK(1)1152 646 y FN(;)g(:)g(:)g(:)g(;)g(s)1376 658 y FL(k)1462 646 y FP(j)45 b FN(\013)1583 658 y FL(i)1655 646 y FO(=)g FN(\014)1812 658 y FL(i)1885 646 y FO(=)f(1)p FN(;)27 b(i)45 b(<)g(k)s(;)-118 745 y(\013)-65 757 y FL(k)-1 745 y FO(=)23 b FN(\014)134 757 y FL(k)197 745 y FO(=)g FP(\000)p FO(1)p FP(i)446 723 y(\030)446 749 y FO(=)534 745 y FN(M)615 757 y FK(2)652 745 y FO(\()p FN(Z)6 b FO(\()p FN(A)841 757 y FL(k)882 745 y FO(\)\))p FB(.)-118 913 y(Pr)l(o)l(of.)43 b FO(Let)29 b(us)g(denote)g FN(B)g FO(=)c FI(C)15 b FP(h)p FN(s)976 925 y FK(1)1019 913 y FN(;)f(:)g(:)g(:)g(;)g(s)1243 925 y FL(k)q FM(\000)p FK(1)1368 913 y FN(;)g(f)t(;)g(f)1537 883 y FM(\000)p FK(1)1626 913 y FP(i)p FO(,)29 b FN(f)34 b FO(=)25 b(\(1)19 b(+)g FN(s)2091 925 y FL(k)2132 913 y FO(\))p FN(uv)j FO(+)-118 1012 y(\(1)c FP(\000)g FN(s)96 1024 y FL(k)137 1012 y FO(\))p FN(v)s(u)p FO(,)28 b(and)f(write)g(an)n(y)f FN(x)e FP(2)f FN(A)1052 1024 y FL(k)1121 1012 y FO(in)k(the)h(form)-2 1234 y FN(x)c FO(=)166 1178 y(1)p 166 1215 42 4 v 166 1291 a(2)218 1167 y Fy(\000)256 1234 y FO(\(1)18 b(+)g FN(s)470 1246 y FL(k)511 1234 y FO(\))c FN(a)601 1246 y FK(1)656 1234 y FO(+)k(\(1)h FP(\000)f FN(s)954 1246 y FL(k)995 1234 y FO(\))c FN(a)1085 1246 y FK(2)1140 1234 y FO(+)k(\(1)g(+)h FN(s)1438 1246 y FL(k)1478 1234 y FO(\))14 b FN(ua)1616 1246 y FK(3)1671 1234 y FO(+)19 b(\(1)f FP(\000)g FN(s)1969 1246 y FL(k)2010 1234 y FO(\))c FN(ua)2148 1246 y FK(4)2184 1167 y Fy(\001)-118 1441 y FN(a)-74 1453 y FL(i)-23 1441 y FP(2)24 b FN(B)t FO(.)37 b(Note,)28 b(that)g(this)f(decomp)r(osition)e(is)h(unique.)37 b(Indeed,)28 b(if)f(w)n(e)h(ha)n(v)n(e)-118 1540 y(another)f(one:)3 1742 y FN(x)d FO(=)171 1686 y(1)p 171 1723 V 171 1799 a(2)223 1674 y Fy(\000)261 1742 y FO(\(1)18 b(+)g FN(s)475 1754 y FL(k)516 1742 y FO(\))c FN(b)598 1754 y FK(1)653 1742 y FO(+)k(\(1)g FP(\000)g FN(s)950 1754 y FL(k)991 1742 y FO(\))c FN(b)1073 1754 y FK(2)1129 1742 y FO(+)k(\(1)g(+)g FN(s)1426 1754 y FL(k)1467 1742 y FO(\))c FN(ub)1597 1754 y FK(3)1652 1742 y FO(+)k(\(1)g FP(\000)g FN(s)1949 1754 y FL(k)1990 1742 y FO(\))c FN(ub)2120 1754 y FK(4)2156 1674 y Fy(\001)2194 1742 y FN(;)-118 1961 y(b)-82 1973 y FL(i)-32 1961 y FP(2)24 b FN(B)t FO(,)i(then)h(w)n(e)f(obtain)f(the)i (iden)n(tit)n(y)d(0)f(=)g(1)p FN(=)p FO(2)1455 1894 y Fy(\000)1491 1961 y FO(\(1)16 b(+)f FN(s)1700 1973 y FL(k)1741 1961 y FO(\)\()p FN(b)1841 1973 y FK(1)1894 1961 y FP(\000)h FN(a)2019 1973 y FK(1)2056 1961 y FO(\))g(+)f(\(1)h FP(\000)-118 2069 y FN(s)-79 2081 y FL(k)-38 2069 y FO(\)\()p FN(b)62 2081 y FK(2)117 2069 y FP(\000)h FN(a)243 2081 y FK(2)280 2069 y FO(\))h(+)f(\(1)h(+)f FN(s)625 2081 y FL(k)666 2069 y FO(\))p FN(u)p FO(\()p FN(b)814 2081 y FK(3)869 2069 y FP(\000)g FN(a)995 2081 y FK(3)1032 2069 y FO(\))h(+)f(\(1)h FP(\000)f FN(s)1377 2081 y FL(k)1418 2069 y FO(\))p FN(u)p FO(\()p FN(b)1566 2081 y FK(4)1621 2069 y FP(\000)g FN(a)1747 2081 y FK(4)1784 2069 y FO(\))1816 2002 y Fy(\001)1854 2069 y FO(.)37 b(Multiplying)-118 2168 y(this)27 b(form)n(ula)d(b)n(y)k(1)18 b(+)g FN(s)646 2180 y FL(k)686 2168 y FO(,)28 b(w)n(e)f(conclude)g(that)698 2351 y(0)c(=)f(2\(1)c(+)g FN(s)1106 2363 y FL(k)1147 2351 y FO(\)\()p FN(b)1247 2363 y FK(1)1303 2351 y FP(\000)g FN(a)1430 2363 y FK(1)1467 2351 y FO(\))p FN(:)-118 2534 y FO(Com)n(bining)24 b(it)j(with)g(the)h(iden)n(tit)n(y)312 2717 y FN(u)p FO(\(1)18 b(+)g FN(s)574 2729 y FL(k)615 2717 y FO(\)\()p FN(b)715 2729 y FK(1)771 2717 y FP(\000)g FN(a)898 2729 y FK(1)935 2717 y FO(\))p FN(u)23 b FO(=)f(\(1)d FP(\000)f FN(s)1340 2729 y FL(k)1381 2717 y FO(\)\()p FN(b)1481 2729 y FK(1)1536 2717 y FP(\000)g FN(a)1663 2729 y FK(1)1701 2717 y FO(\))23 b(=)g(0)p FN(;)-118 2900 y FO(w)n(e)g(obtain)g(that)h(\()p FN(b)499 2912 y FK(1)547 2900 y FP(\000)11 b FN(a)667 2912 y FK(1)703 2900 y FO(\))24 b(=)e(0.)36 b(In)24 b(the)g(same)e(w)n(a)n(y)g(w)n(e)i (sho)n(w)f(that)h FN(b)2087 2912 y FK(2)2147 2900 y FO(=)e FN(a)2278 2912 y FK(2)2316 2900 y FO(,)-118 3000 y FN(b)-82 3012 y FK(3)-22 3000 y FO(=)g FN(a)109 3012 y FK(3)147 3000 y FO(,)i FN(b)230 3012 y FK(4)290 3000 y FO(=)f FN(a)422 3012 y FK(4)459 3000 y FO(.)36 b(No)n(w)23 b(w)n(e)g(can)h (write)e(the)j(form)n(ula)20 b(for)k(the)g(isomorphism)-118 3100 y FN( )12 b FO(:)28 b FN(A)61 3112 y FL(k)125 3100 y FP(!)23 b FN(M)312 3112 y FK(2)349 3100 y FO(\()p FN(B)t FO(\),)768 3250 y FN( )s FO(\()p FN(x)p FO(\))h(=)1048 3133 y Fy(\022)1109 3199 y FN(a)1153 3211 y FK(1)1273 3199 y FN(a)1317 3211 y FK(3)1109 3299 y FN(a)1153 3311 y FK(4)1273 3299 y FN(a)1317 3311 y FK(2)1354 3133 y Fy(\023)1429 3250 y FN(:)-118 3445 y FO(A)e(direct)f(v)n(eri\014cation) e(sho)n(ws)i(that)h FN( )j FO(is)c(an)h(epimorphism)17 b(and)22 b(the)g(algebra)-118 3544 y FN(B)32 b FO(is)26 b(isomorphic)e(to)j(the)h FN(Z)6 b FO(\()p FN(A)882 3556 y FL(k)923 3544 y FO(\).)p 2278 3544 4 57 v 2282 3492 50 4 v 2282 3544 V 2331 3544 4 57 v 6 3712 a(4\))34 b(Consider)e(the)i (second)f(p)r(ossibilit)n(y)-7 b(.)50 b(One)34 b(can)f(assume)f(that)i FN(\013)2200 3724 y FL(k)2274 3712 y FO(=)-118 3811 y FP(\000)p FN(\014)-6 3823 y FL(k)75 3811 y FO(=)41 b FP(\000)p FO(1)d(\(otherwise)e(w)n(e)i(replace)f FN(u)h FO(with)g FN(v)j FO(or)d(vice)f(v)n(ersa\).)68 b(Using)-118 3911 y(the)32 b(previous)d(results)g(w)n(e)i(ha)n(v)n(e)f FN(\013)1004 3923 y FL(j)1069 3911 y FO(=)e FN(\014)1209 3923 y FL(j)1244 3911 y FO(,)33 b FN(j)h(<)28 b(k)s FO(.)48 b(Then)32 b(b)n(y)f(the)g(metho)r(d)p eop %%Page: 38 42 38 41 bop -118 -137 a FO(38)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(describ)r(ed)33 b(in)h(3\),)j(w)n(e)d(obtain)g(the)h(iden)n (tities)d FN(\013)1457 108 y FL(j)1527 96 y FO(=)i FN(\014)1673 108 y FL(j)1743 96 y FO(=)h(1,)h FN(j)k(<)34 b(k)26 b FP(\000)d FO(1,)-118 196 y FN(\013)-65 208 y FL(k)q FM(\000)p FK(1)102 196 y FO(=)41 b FN(\014)255 208 y FL(k)q FM(\000)p FK(1)381 196 y FO(.)70 b(Th)n(us,)41 b FN(\013)774 208 y FL(j)851 196 y FO(=)g FN(\014)1004 208 y FL(j)1080 196 y FO(=)g(1,)g FN(j)47 b(<)41 b(k)28 b FP(\000)e FO(1,)41 b FN(\013)1800 208 y FL(k)1882 196 y FO(=)g FP(\000)p FN(\014)2100 208 y FL(k)2140 196 y FO(,)h(and)-118 296 y FN(\013)-65 308 y FL(k)q FM(\000)p FK(1)92 296 y FO(equals)29 b(+1)g(or)h FP(\000)p FO(1.)45 b(These)31 b(cases)e(are)h(considered)f (in)h(the)h(follo)n(wing)-118 395 y(prop)r(ositions.)-118 545 y FQ(Prop)s(osition)f(13.)41 b FN(A)624 557 y FL(k)689 545 y FO(=)23 b FI(C)15 b FP(h)q FN(u;)f(v)s(;)g(s)1068 557 y FK(1)1110 545 y FN(;)g(:)g(:)g(:)g(;)g(s)1334 557 y FL(k)1398 545 y FP(j)25 b FN(\013)1499 557 y FL(i)1550 545 y FO(=)f FN(\014)1686 557 y FL(i)1737 545 y FO(=)g(1)p FN(;)j(i)d(<)f(k)s(;)28 b(\013)2209 557 y FL(k)2274 545 y FO(=)-118 644 y FP(\000)p FN(\014)-6 656 y FL(k)57 644 y FO(=)23 b FP(\000)p FO(1)p FP(i)306 622 y(\030)306 649 y FO(=)394 644 y FN(M)475 656 y FK(2)512 644 y FO(\()p FN(Q)610 656 y FK(2)p FL(;k)q FM(\000)p FK(1)788 644 y FO(\))p FB(.)-118 804 y(Pr)l(o)l(of.)43 b FO(If)24 b(w)n(e)g(denote)g FN(v)644 816 y FK(1)705 804 y FO(=)e(1)p FN(=)p FO(2\(\(1)11 b(+)g FN(s)1150 816 y FL(k)1190 804 y FO(\))p FN(v)k FO(+)c(\(1)g FP(\000)g FN(s)1553 816 y FL(k)1593 804 y FO(\))p FN(uv)s(u)p FO(\),)25 b FN(v)1884 816 y FK(2)1944 804 y FO(=)e(1)p FN(=)p FO(2\(\(1)11 b FP(\000)-118 904 y FN(s)-79 916 y FL(k)-38 904 y FO(\))p FN(v)22 b FO(+)c(\(1)g(+)g FN(s)353 916 y FL(k)394 904 y FO(\))p FN(uv)s(u)p FO(\),)27 b(then)93 1057 y FN(A)155 1069 y FL(k)220 1035 y FP(\030)220 1061 y FO(=)307 1057 y FI(C)15 b FP(h)q FN(u;)f(v)519 1069 y FK(1)561 1057 y FN(;)g(v)638 1069 y FK(2)676 1057 y FN(;)g(s)752 1069 y FK(1)789 1057 y FN(;)g(:)g(:)g(:)f(;)h(s)1012 1069 y FL(k)1053 1057 y FP(i)1109 1035 y(\030)1109 1061 y FO(=)1196 1057 y FN(M)1277 1069 y FK(2)1314 1057 y FO(\()p FN(A)1408 1069 y FL(k)q FM(\000)p FK(1)1534 1057 y FO(\))1590 1035 y FP(\030)1590 1061 y FO(=)1677 1057 y FN(M)1758 1069 y FK(2)1795 1057 y FO(\()p FN(Q)1893 1069 y FK(2)p FL(;k)q FM(\000)p FK(1)2072 1057 y FO(\))p FN(;)-118 1210 y FO(where)28 b FN(A)185 1222 y FL(k)q FM(\000)p FK(1)337 1210 y FO(=)d FI(C)15 b FP(h)p FN(v)553 1222 y FK(1)596 1210 y FN(;)f(v)673 1222 y FK(2)711 1210 y FN(;)g(s)787 1222 y FK(1)824 1210 y FN(;)g(:)g(:)g(:)g(;)g(s)1048 1222 y FL(k)q FM(\000)p FK(1)1173 1210 y FP(i)1231 1188 y(\030)1231 1214 y FO(=)1321 1210 y FN(Q)1387 1222 y FK(2)p FL(;k)q FM(\000)p FK(1)1565 1210 y FO(.)41 b(Indeed,)30 b(an)n(y)e FN(x)e FP(2)f FN(A)2297 1222 y FL(k)-118 1310 y FO(can)30 b(b)r(e)h(represen)n(ted)f(in)g(the)h(form)f FN(x)e FO(=)g(1)p FN(=)p FO(2)1338 1242 y Fy(\000)1375 1310 y FO(\(1)20 b(+)g FN(s)1593 1322 y FL(k)1634 1310 y FO(\))14 b FN(a)1724 1322 y FK(1)1782 1310 y FO(+)20 b(\(1)g FP(\000)g FN(s)2085 1322 y FL(k)2126 1310 y FO(\))14 b FN(a)2216 1322 y FK(2)2274 1310 y FO(+)-118 1418 y(\(1)k(+)g FN(s)96 1430 y FL(k)137 1418 y FO(\))c FN(ua)275 1430 y FK(3)330 1418 y FO(+)k(\(1)g FP(\000)g FN(s)627 1430 y FL(k)668 1418 y FO(\))c FN(ua)806 1430 y FK(4)843 1350 y Fy(\001)881 1418 y FO(,)28 b FN(a)976 1430 y FL(i)1026 1418 y FP(2)c FN(A)1167 1430 y FL(k)q FM(\000)p FK(1)1293 1418 y FO(,)k(and)f(the)h(form)n(ula)768 1629 y FN( )s FO(\()p FN(x)p FO(\))c(=)1048 1512 y Fy(\022)1109 1578 y FN(a)1153 1590 y FK(1)1273 1578 y FN(a)1317 1590 y FK(3)1109 1678 y FN(a)1153 1690 y FK(4)1273 1678 y FN(a)1317 1690 y FK(2)1354 1512 y Fy(\023)1429 1629 y FN(:)-118 1831 y FO(giv)n(es)h(the)j(needed)g(isomorphism)22 b FN( )12 b FO(:)28 b FN(A)1169 1843 y FL(k)1234 1831 y FP(!)23 b FN(M)1421 1843 y FK(2)1458 1831 y FO(\()p FN(A)1552 1843 y FL(k)q FM(\000)p FK(1)1678 1831 y FO(\).)p 2278 1831 4 57 v 2282 1779 50 4 v 2282 1831 V 2331 1831 4 57 v -118 1991 a FQ(Prop)s(osition)30 b(14.)41 b FN(A)624 2003 y FL(k)710 1991 y FO(=)j FI(C)15 b FP(h)q FN(u;)f(v)s(;)g(s)1110 2003 y FK(1)1152 1991 y FN(;)g(:)g(:)g(:)g(;)g(s)1376 2003 y FL(k)1462 1991 y FP(j)45 b FN(\013)1583 2003 y FL(i)1655 1991 y FO(=)g FN(\014)1812 2003 y FL(i)1885 1991 y FO(=)f(1)p FN(;)27 b(i)45 b(<)g(k)s(;)-118 2091 y(\013)-65 2103 y FL(k)q FM(\000)p FK(1)84 2091 y FO(=)23 b FN(\014)219 2103 y FL(k)q FM(\000)p FK(1)367 2091 y FO(=)g FP(\000)p FO(1)p FN(;)k(\013)665 2103 y FL(k)729 2091 y FO(=)22 b FP(\000)p FN(\014)928 2103 y FL(k)992 2091 y FO(=)h FP(\000)p FO(1)p FP(i)1241 2069 y(\030)1241 2095 y FO(=)1329 2091 y FN(M)1410 2103 y FK(4)1446 2091 y FO(\()p FN(Z)6 b FO(\()p FN(A)1635 2103 y FL(k)1677 2091 y FO(\)\))p FN(:)-118 2251 y FB(Pr)l(o)l(of.)43 b FO(It)27 b(is)e(easy)g(to)i(v)n(erify)d(that)j(the)g(elemen)n(t)d(1)p FN(=)p FO(2)1576 2184 y Fy(\000)1613 2251 y FO(\(1)16 b(+)f FN(s)1822 2263 y FL(k)1863 2251 y FO(\)\()p FN(uv)s FO(\))2050 2221 y FK(2)2104 2251 y FO(+)g(\(1)h FP(\000)-118 2359 y FN(s)-79 2371 y FL(k)-38 2359 y FO(\)\()p FN(v)s(u)p FO(\))149 2329 y FK(2)187 2292 y Fy(\001)259 2359 y FO(lies)31 b(in)i(the)i(cen)n(ter)e(of)h(the)g(algebra)d FN(A)1523 2371 y FL(k)1564 2359 y FO(,)36 b(and,)f(moreo)n(v)n(er,)d(this)-118 2459 y(elemen)n(t)21 b(and)j(the)g(family)c FP(f)p FN(s)811 2471 y FL(i)838 2459 y FN(;)28 b(i)23 b(<)f(k)13 b FP(\000)d FO(1)p FP(g)23 b FO(generate)f(the)i(cen)n(ter.)35 b(Then)23 b(the)-118 2558 y(form)n(ulas)h FN( )12 b FO(:)28 b FN(A)398 2570 y FL(k)462 2558 y FP(!)c FN(M)650 2570 y FK(4)686 2558 y FO(\()p FN(Z)6 b FO(\()p FN(A)875 2570 y FL(k)917 2558 y FO(\)\))362 2861 y FN( )s FO(\()p FN(x)p FO(\))24 b(=)642 2644 y Fy(0)642 2790 y(B)642 2840 y(B)642 2893 y(@)714 2711 y FN(x)87 b FO(0)h(0)h(0)717 2810 y(0)d FN(x)g FO(0)j(0)717 2910 y(0)g(0)c FN(x)i FO(0)717 3009 y(0)i(0)f(0)e FN(x)1153 2644 y Fy(1)1153 2790 y(C)1153 2840 y(C)1153 2893 y(A)1240 2861 y FN(;)180 b(x)23 b FP(2)g FN(Z)6 b FO(\()p FN(A)1748 2873 y FL(k)1789 2861 y FO(\))p FN(;)-69 3317 y( )s FO(\()p FN(u)p FO(\))24 b(=)211 3101 y Fy(0)211 3247 y(B)211 3297 y(B)211 3350 y(@)284 3167 y FO(0)83 b(0)f(1)h(0)284 3267 y(0)g(0)f(0)h(1)284 3366 y(1)g(0)f(0)h(0)284 3466 y(0)g(1)f(0)h(0)699 3101 y Fy(1)699 3247 y(C)699 3297 y(C)699 3350 y(A)786 3317 y FN(;)96 b( )s FO(\(1)p FN(=)p FO(2\(1)18 b(+)g FN(s)1334 3329 y FL(k)q FM(\000)p FK(1)1459 3317 y FO(\))p FN(v)s FO(\))24 b(=)1678 3101 y Fy(0)1678 3247 y(B)1678 3297 y(B)1678 3350 y(@)1751 3167 y FO(0)82 b(1)h(0)f(0)1751 3267 y(1)g(0)h(0)f(0)1751 3366 y(0)g(0)h(0)f(0)1751 3466 y(0)g(0)h(0)f(0)2166 3101 y Fy(1)2166 3247 y(C)2166 3297 y(C)2166 3350 y(A)2252 3317 y FN(;)-26 3774 y( )s FO(\()p FN(s)102 3786 y FL(k)q FM(\000)p FK(1)228 3774 y FO(\))23 b(=)371 3557 y Fy(0)371 3703 y(B)371 3753 y(B)371 3806 y(@)444 3624 y FO(1)82 b(0)115 b(0)147 b(0)444 3723 y(0)82 b(1)115 b(0)147 b(0)444 3823 y(0)82 b(0)h FP(\000)p FO(1)114 b(0)444 3923 y(0)82 b(0)115 b(0)g FP(\000)p FO(1)988 3557 y Fy(1)988 3703 y(C)988 3753 y(C)988 3806 y(A)1074 3774 y FN(;)97 b( )s FO(\()p FN(s)1322 3786 y FL(k)1363 3774 y FO(\))24 b(=)1506 3557 y Fy(0)1506 3703 y(B)1506 3753 y(B)1506 3806 y(@)1579 3624 y FO(1)115 b(0)g(0)f(0)1579 3723 y(0)83 b FP(\000)p FO(1)f(0)114 b(0)1579 3823 y(0)h(0)g(1)f(0)1579 3923 y(0)h(0)g(0)82 b FP(\000)p FO(1)2123 3557 y Fy(1)2123 3703 y(C)2123 3753 y(C)2123 3806 y(A)2210 3774 y FN(;)p eop %%Page: 39 43 39 42 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(39)-118 96 y(determine)26 b(the)i(needed)f(isomorphism.)p 2278 96 4 57 v 2282 44 50 4 v 2282 96 V 2331 96 4 57 v -118 271 a(The)h(pro)r(of)f(of)g(the)h(theorem)e(is)h(completed.)p 2278 271 V 2282 218 50 4 v 2282 271 V 2331 271 4 57 v 6 445 a(By)e(using)e(Theorem)f(2.1,)i(it)g(is)f(p)r(ossible)f(to)j (obtain)e(a)h(description)e(of)i(all)-118 545 y(irreducible)g(represen) n(tations)g(of)k(the)g(algebras)d FN(A)1492 557 y FL(k)1533 545 y FO(.)-118 767 y FQ(1.2.3)94 b(Non-comm)m(utativ)m(e)39 b(\\circle",)k(\\pair)f(of)f(in)m(tersecting)g(li-)174 866 y(nes")h(and)h(\\h)m(yp)s(erb)s(ola".)74 b(More)42 b(examples)e(of)j FN(F)2073 878 y FK(4)2111 866 y FQ(-alge-)174 966 y(bras)-118 1121 y(1.)55 b FO(No)n(w)33 b(w)n(e)h(describ)r(e)e(b)r (ounded)i(irreducible)d(solutions)g(of)j(the)g(relations)-118 1221 y(\()p FN(I)7 b(I)-7 1233 y FK(1)31 1221 y FO(\))28 b FN(A)153 1190 y FK(2)210 1221 y FO(+)18 b FN(B)360 1190 y FK(2)421 1221 y FO(=)23 b FN(I)36 b FO(\(\\non-comm)n(utativ)n (e)23 b(circle"\),)i(\()p FN(I)7 b(I)g(I)1780 1233 y FK(0)1819 1221 y FO(\))28 b FN(A)1941 1190 y FK(2)1997 1221 y FP(\000)19 b FN(B)2148 1190 y FK(2)2209 1221 y FO(=)k(0)-118 1320 y(\(\\non-comm)n(utativ)n(e)e(pair)j(of)h(in)n (tersecting)e(lines"\),)h(whic)n(h)g(is)g(the)i(same)e(as)-118 1430 y(the)c(relations)c FP(f)407 1409 y Fy(e)388 1430 y FN(A;)504 1409 y Fy(e)487 1430 y FN(B)t FP(g)22 b FO(=)h(0,)e(and)e (\()p FN(I)7 b(I)g(I)1099 1442 y FK(1)1137 1430 y FO(\))20 b FN(A)1251 1400 y FK(2)1291 1430 y FP(\000)r FN(B)1425 1400 y FK(2)1485 1430 y FO(=)j FN(I)j FO(\(\\non-comm)n(utativ)n(e)-118 1540 y(h)n(yp)r(erb)r(ola"\),)c(whic)n(h)h(is)g(the)i(same)d(as)h(the)h (relation)d FP(f)1621 1519 y Fy(e)1601 1540 y FN(A;)1718 1519 y Fy(e)1700 1540 y FN(B)t FP(g)i FO(=)f FN(I)7 b FO(.)36 b(W)-7 b(e)25 b(sho)n(w)-118 1640 y(that)g(these)g(relations)d (are)i FN(F)793 1652 y FK(4)831 1640 y FO(-relations,)e(i.e.,)i(the)i (corresp)r(onding)c(algebras)-118 1740 y(are)k FN(F)73 1752 y FK(4)111 1740 y FO(-algebras.)-118 1909 y FQ(Prop)s(osition)k (15.)41 b FB(Irr)l(e)l(ducible)27 b(self-adjoint)h(solutions)f FN(A)p FB(,)g FN(B)k FB(of)c(the)f(r)l(ela-)-118 2008 y(tions)k FO(\()p FN(I)7 b(I)199 2020 y FK(1)237 2008 y FO(\))p FN(;)44 b FO(\()p FN(I)7 b(I)g(I)490 2020 y FK(0)528 2008 y FO(\))p FN(;)44 b FO(\()p FN(I)7 b(I)g(I)781 2020 y FK(1)819 2008 y FO(\))30 b FB(ar)l(e)g(the)g(fol)t(lowing)7 b FO(:)-26 2178 y(1\))41 b FB(one-dimensional)53 b FO(\(dim)12 b FN(H)52 b FO(=)44 b(1\))p FB(,)h FN(A)g FO(=)g FN(\025)1553 2190 y FK(1)1591 2178 y FQ(1)p FB(,)g FN(B)k FO(=)44 b FN(\025)1978 2190 y FK(2)2016 2178 y FQ(1)p FB(,)h(wher)l(e)89 2293 y(the)30 b(p)l(air)h FO(\()p FN(\025)478 2305 y FK(1)516 2293 y FN(;)14 b(\025)601 2305 y FK(2)639 2293 y FO(\))30 b FB(b)l(elongs)h(to)e(the)h(cir)l(cle)h FN(K)1522 2250 y FK(\(1\))1516 2322 y(\()p FL(I)5 b(I)1605 2330 y Fx(1)1637 2322 y FK(\))1691 2293 y FO(=)23 b FP(f)p FO(\()p FN(\025)1901 2305 y FK(1)1938 2293 y FN(;)14 b(\025)2023 2305 y FK(2)2061 2293 y FO(\))23 b FP(2)h FI(R)2249 2263 y FK(2)2316 2293 y FP(j)89 2431 y FN(\025)137 2401 y FK(2)137 2452 y(1)176 2431 y FO(+)q FN(\025)290 2401 y FK(2)290 2452 y(2)351 2431 y FO(=)f(1)p FP(g)p FB(,)f(the)g(p)l(air)h(of)g(interse)l(cting)f(lines)g FN(K)1656 2388 y FK(\(1\))1650 2459 y(\()p FL(I)5 b(I)g(I)1773 2467 y Fx(0)1805 2459 y FK(\))1858 2431 y FO(=)23 b FP(f)p FO(\()p FN(\025)2068 2443 y FK(1)2105 2431 y FN(;)14 b(\025)2190 2443 y FK(2)2228 2431 y FO(\))23 b FP(2)89 2569 y FI(R)143 2539 y FK(2)221 2569 y FP(j)71 b FN(\025)363 2539 y FK(2)363 2590 y(1)436 2569 y FO(=)34 b FN(\025)583 2539 y FK(2)583 2590 y(2)621 2569 y FP(g)p FB(,)k(or)e(the)h(hyp)l(erb) l(ola)h FN(K)1431 2526 y FK(\(1\))1425 2597 y(\()p FL(I)5 b(I)g(I)1548 2605 y Fx(1)1580 2597 y FK(\))1645 2569 y FO(=)34 b FP(f)p FO(\()p FN(\025)1866 2581 y FK(1)1904 2569 y FN(;)14 b(\025)1989 2581 y FK(2)2026 2569 y FO(\))35 b FP(2)h FI(R)2238 2539 y FK(2)2316 2569 y FP(j)89 2688 y FN(\025)137 2658 y FK(2)137 2708 y(1)194 2688 y FP(\000)18 b FN(\025)325 2658 y FK(2)325 2708 y(2)385 2688 y FO(=)23 b(1)p FP(g)p FB(,)29 b(r)l(esp)l(e)l(ctively)7 b FO(;)-26 2858 y(2\))41 b FB(two-dimensional)f FO(\(dim)13 b FN(H)30 b FO(=)22 b(2\))p FB(,)360 3093 y FN(A)h FO(=)g FN(\025)581 3105 y FK(1)633 2975 y Fy(\022)694 3042 y FO(1)115 b(0)694 3142 y(0)82 b FP(\000)p FO(1)924 2975 y Fy(\023)999 3093 y FN(;)99 b(B)27 b FO(=)c FN(\025)1347 3105 y FK(2)1398 2975 y Fy(\022)1460 3042 y FO(cos)13 b FN(\036)127 b FO(sin)13 b FN(\036)1464 3142 y FO(sin)g FN(\036)88 b FP(\000)14 b FO(cos)e FN(\036)1970 2975 y Fy(\023)2045 3093 y FN(;)89 3327 y FO(0)30 b FN(<)g(\036)g(<)g(\031)s FB(,)35 b(wher)l(e)f(the)g(p)l(air)h FO(\()p FN(\025)1175 3339 y FK(1)1213 3327 y FN(;)14 b(\025)1298 3339 y FK(2)1335 3327 y FO(\))34 b FB(b)l(elongs)g(r)l(esp)l(e)l(ctively)h(to)e(the)89 3427 y(set)8 b FO(:)332 3611 y FN(K)403 3626 y FK(\()p FL(I)d(I)492 3634 y Fx(1)525 3626 y FK(\))578 3611 y FO(=)22 b FP(f)p FO(\()p FN(\025)787 3623 y FK(1)825 3611 y FN(;)14 b(\025)910 3623 y FK(2)947 3611 y FO(\))24 b FP(j)f FN(\025)1097 3623 y FK(1)1158 3611 y FN(>)f FO(0)p FN(;)28 b(\025)1386 3623 y FK(2)1446 3611 y FN(>)23 b FO(0)p FN(;)k(\025)1674 3577 y FK(2)1674 3632 y(1)1730 3611 y FO(+)18 b FN(\025)1861 3577 y FK(2)1861 3632 y(2)1922 3611 y FO(=)23 b(1)p FP(g)p FN(;)298 3746 y(K)369 3761 y FK(\()p FL(I)5 b(I)g(I)492 3769 y Fx(0)525 3761 y FK(\))578 3746 y FO(=)22 b FP(f)p FO(\()p FN(\025)787 3758 y FK(1)825 3746 y FN(;)14 b(\025)910 3758 y FK(2)947 3746 y FO(\))24 b FP(j)f FN(\025)1097 3758 y FK(1)1158 3746 y FN(>)f FO(0)p FN(;)28 b(\025)1386 3758 y FK(2)1446 3746 y FN(>)23 b FO(0)p FN(;)k(\025)1674 3712 y FK(2)1674 3767 y(1)1730 3746 y FP(\000)18 b FN(\025)1861 3712 y FK(2)1861 3767 y(2)1922 3746 y FO(=)23 b(0)p FP(g)p FN(;)298 3881 y(K)369 3896 y FK(\()p FL(I)5 b(I)g(I)492 3904 y Fx(1)525 3896 y FK(\))578 3881 y FO(=)22 b FP(f)p FO(\()p FN(\025)787 3893 y FK(1)825 3881 y FN(;)14 b(\025)910 3893 y FK(2)947 3881 y FO(\))24 b FP(j)f FN(\025)1097 3893 y FK(1)1158 3881 y FN(>)f FO(0)p FN(;)28 b(\025)1386 3893 y FK(2)1446 3881 y FN(>)23 b FO(0)p FN(;)k(\025)1674 3847 y FK(2)1674 3902 y(1)1730 3881 y FP(\000)18 b FN(\025)1861 3847 y FK(2)1861 3902 y(2)1922 3881 y FO(=)23 b(1)p FP(g)p FN(:)p eop %%Page: 40 44 40 43 bop -118 -137 a FO(40)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Since)34 b(the)i(op)r(erators)d FN(A)954 66 y FK(2)1026 96 y FO(and)i FN(B)1262 66 y FK(2)1335 96 y FO(b)r(elong)e(to)i(the)h(cen)n(ter)e(of)h(the)-118 196 y(algebra,)28 b(they)i(are)g(scalar)d(in)i(an)h(irreducible)d (represen)n(tation,)h FN(A)2030 166 y FK(2)2095 196 y FO(=)f FN(\025)2235 166 y FK(2)2235 217 y(1)2273 196 y FN(I)7 b FO(,)-118 296 y FN(B)-51 266 y FK(2)9 296 y FO(=)23 b FN(\025)145 266 y FK(2)145 316 y(2)183 296 y FN(I)7 b FO(.)35 b(If)24 b(the)g(represen)n(tation)d(is)h(not)i (one-dimensional,)19 b(then)24 b FN(\025)2126 308 y FK(1)2186 296 y FN(>)f FO(0,)-118 395 y FN(\025)-70 407 y FK(2)7 395 y FN(>)39 b FO(0,)h(and)d(\()p FN(A=\025)571 407 y FK(1)609 395 y FO(\))641 365 y FK(2)718 395 y FO(=)i(\()p FN(B)t(=\025)1011 407 y FK(2)1049 395 y FO(\))1081 365 y FK(2)1158 395 y FO(=)g FN(I)7 b FO(.)67 b(Then)38 b(the)f(prop)r (osition)e(fol-)-118 495 y(lo)n(ws)27 b(from)h(the)h(description)e(of)i (irreducible)c(pairs)i(of)j(unitary)d(self-adjoin)n(t)-118 595 y(op)r(erators)e FN(U)32 b FO(=)23 b FN(A=\025)578 607 y FK(1)615 595 y FO(,)28 b FN(V)42 b FO(=)23 b FN(B)t(=\025)1001 607 y FK(2)1038 595 y FO(.)p 2278 595 4 57 v 2282 542 50 4 v 2282 595 V 2331 595 4 57 v -118 757 a FB(R)l(emark)30 b(8.)42 b FO(F)-7 b(or)38 b(the)h(\\circle")c(\()p FN(I)7 b(I)1035 769 y FK(1)1073 757 y FO(\))39 b FN(A)1206 726 y FK(2)1269 757 y FO(+)26 b FN(B)1427 726 y FK(2)1506 757 y FO(=)41 b FN(I)k FO(only)38 b(b)r(ounded)h(so-)-118 856 y(lutions)34 b(exist,)j(since)d FP(k)p FN(A)p FP(k)i(\024)h FO(1,)g FP(k)p FN(B)t FP(k)f(\024)h FO(1.)61 b(In)36 b(con)n(trast)f(with)g(this,)i(for)-118 956 y(relations)31 b(\()p FN(I)7 b(I)g(I)380 968 y FK(0)418 956 y FO(\))35 b(and)f(\()p FN(I)7 b(I)g(I)807 968 y FK(1)845 956 y FO(\),)37 b(a)d(class)e(of)i(\\in)n(tegrable")d(represen)n(tations)-118 1055 y(b)n(y)25 b(un)n(b)r(ounded)h(op)r(erators)d(w)n(as)h(de\014ned)i (and)f(in)n(v)n(estigated)d(\(see)k([184)n(])g(and)-118 1155 y(others\).)6 1255 y(There)35 b(is)f(no)h(need)h(to)f(use)g(un)n (b)r(ounded)g(op)r(erators)e(for)i(studying)f(the)-118 1354 y(irreducible)24 b(represen)n(tations)g(of)k(the)f(relations)e(\() p FN(I)7 b(I)g(I)1597 1366 y FK(0)1635 1354 y FO(\))28 b(and)f(\()p FN(I)7 b(I)g(I)2010 1366 y FK(1)2048 1354 y FO(\))28 b(either.)-118 1454 y(Irreducibilit)n(y)h(here)k(implies)c (that)34 b(the)g(op)r(erators)d FN(A)1632 1424 y FK(2)1704 1454 y FO(and)i FN(B)1938 1424 y FK(2)2009 1454 y FO(comm)n(ute)-118 1554 y(with)27 b(b)r(oth)h FN(A)g FO(and)g FN(B)t FO(,)g(and)g(hence)f (are)g(scalar.)35 b(Th)n(us)27 b(the)h(op)r(erators)e FN(A)p FO(,)i FN(B)-118 1653 y FO(are)23 b(b)r(ounded)i(in)e(an)n(y)h (irreducible)c(represen)n(tation)i(whic)n(h)h(are)g(giv)n(en)g(in)g (the)-118 1753 y(prop)r(osition)h(ab)r(o)n(v)n(e.)6 1852 y(If)31 b(w)n(e)f(consider)e(reducible)h(represen)n(tations)e(of)j(\()p FN(I)7 b(I)g(I)1731 1864 y FK(0)1769 1852 y FO(\))31 b(and)f(\()p FN(I)7 b(I)g(I)2150 1864 y FK(1)2188 1852 y FO(\))31 b(b)n(y)-118 1952 y(un)n(b)r(ounded)20 b(op)r(erators,)g (new)g(represen)n(tations)e(app)r(ear)h(\(see)h([184)o(])g(and)g(oth-) -118 2052 y(ers\).)-118 2176 y FQ(2.)36 b FO(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(follo)n(wing)c(prop)r(osition.)-118 2326 y FQ(Prop)s(osition)30 b(16.)41 b FB(The)30 b(fol)t(lowing)j (algebr)l(as)k FO(\()p FB(without)31 b(involution)6 b FO(\))303 2488 y FI(C)357 2421 y Fy(\012)402 2488 y FN(x;)14 b(y)26 b FP(j)d FN(x)646 2454 y FK(2)702 2488 y FO(+)18 b FN(y)829 2454 y FK(2)889 2488 y FO(=)23 b FN(e)1016 2421 y Fy(\013)1078 2488 y FO(=)g FI(C)1220 2421 y Fy(\012)1265 2488 y FN(a;)14 b(b)22 b FP(j)h FN(a)1494 2454 y FK(2)1550 2488 y FP(\000)18 b FN(b)1669 2454 y FK(2)1729 2488 y FO(=)23 b FN(e)1856 2421 y Fy(\013)1894 2488 y FN(;)-118 2651 y FB(and)400 2813 y FI(C)454 2746 y Fy(\012)499 2813 y FN(x;)14 b(y)26 b FP(j)d FN(x)743 2779 y FK(2)804 2813 y FO(=)f FN(y)935 2779 y FK(2)972 2746 y Fy(\013)1035 2813 y FO(=)g FI(C)1176 2746 y Fy(\012)1221 2813 y FN(a;)14 b(b)23 b FP(j)g(f)p FN(a;)14 b(b)p FP(g)21 b FO(=)i(0)1759 2746 y Fy(\013)1798 2813 y FN(;)-118 2975 y FB(ar)l(e)30 b FN(F)76 2987 y FK(4)114 2975 y FB(-algebr)l(as.)-118 3125 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)28 b(giv)n(e)e(the)i(pro)r(of)f (for)g(the)h(algebra)d FI(C)1431 3058 y Fy(\012)1476 3125 y FN(a;)14 b(b)22 b FP(j)i(f)p FN(a;)14 b(b)p FP(g)21 b FO(=)i(0)2014 3058 y Fy(\013)2080 3125 y FO(de\014ned)-118 3225 y(b)n(y)34 b(t)n(w)n(o)g(generators)f FN(a)p FO(,)j FN(b)f FO(and)g(the)g(relation)d FN(ab)22 b FO(+)h FN(ba)35 b FO(=)g(0.)58 b(As)35 b(a)f(v)n(ector)-118 3324 y(space,)39 b(it)e(is)f(the)i(same)e(as)h(the)g(space)g(of)g(complex)e(p)r (olynomials)e(in)j(t)n(w)n(o)-118 3424 y(v)-5 b(ariables)24 b(but)k(with)g(the)f(follo)n(wing)d(m)n(ultiplication)e(of)27 b(terms:)414 3586 y FN(a)458 3552 y FL(k)493 3560 y Fx(1)530 3586 y FN(b)566 3552 y FL(k)601 3560 y Fx(2)656 3586 y FP(\001)19 b FN(a)742 3552 y FL(j)769 3560 y Fx(1)806 3586 y FN(b)842 3552 y FL(j)869 3560 y Fx(2)928 3586 y FO(=)k(\()p FP(\000)p FO(1\))1187 3552 y FL(k)1222 3560 y Fx(2)1254 3552 y FM(\001)p FL(j)1301 3560 y Fx(1)1338 3586 y FN(a)1382 3552 y FL(k)1417 3560 y Fx(1)1450 3552 y FK(+)p FL(k)1536 3560 y Fx(2)1572 3586 y FN(b)1608 3552 y FL(j)1635 3560 y Fx(1)1668 3552 y FK(+)p FL(j)1746 3560 y Fx(2)1783 3586 y FN(:)-118 3749 y FO(Let)28 b(us)f(supply)g (this)g(algebra)e(b)n(y)i(in)n(v)n(olution)d(de\014ned)k(as)f(follo)n (ws:)256 3911 y FN(a)22 b FO(=)h FN(a)454 3877 y FM(\003)492 3911 y FN(;)97 b(b)23 b FO(=)f FN(b)794 3877 y FM(\003)832 3911 y FN(;)97 b FO(\()p FN(a)1028 3877 y FL(k)1063 3885 y Fx(1)1100 3911 y FN(b)1136 3877 y FL(k)1171 3885 y Fx(2)1208 3911 y FO(\))1240 3877 y FM(\003)1301 3911 y FO(=)23 b(\()p FP(\000)p FO(1\))1560 3877 y FL(k)1595 3885 y Fx(1)1627 3877 y FM(\001)p FL(k)1682 3885 y Fx(2)1719 3911 y FN(a)1763 3877 y FL(k)1798 3885 y Fx(1)1834 3911 y FN(b)1870 3877 y FL(k)1905 3885 y Fx(2)1942 3911 y FN(:)p eop %%Page: 41 45 41 44 bop -118 -137 a FJ(1.2.)36 b FN(F)101 -125 y FL(n)147 -137 y FJ(-algebras)24 b(and)j(their)g(represen)n(tations)847 b FO(41)6 96 y(Irreducible)25 b FP(\003)p FO(-represen)n(tations)f(of)k (the)g FP(\003)p FO(-algebra)493 285 y FI(C)547 217 y Fy(\012)592 285 y FN(a;)14 b(b)23 b FP(j)g FN(a)g FO(=)g FN(a)977 250 y FM(\003)1015 285 y FN(;)k(b)c FO(=)g FN(b)1248 250 y FM(\003)1286 285 y FN(;)k FP(f)p FN(a;)14 b(b)p FP(g)22 b FO(=)g(0)1688 217 y Fy(\013)-118 473 y FO(can)27 b(b)r(e)h(obtained)e(as)h(follo)n(ws:)-26 647 y(1\))41 b(one-dimensional)28 b(\(dim)12 b FN(H)38 b FO(=)32 b(1\),)h FN(A)f FO(=)f FN(\025)1460 659 y FK(1)1498 647 y FN(I)7 b FO(,)34 b FN(B)h FO(=)c FN(\025)1840 659 y FK(2)1878 647 y FN(I)7 b FO(,)34 b(where)e(the)89 747 y(pair)26 b(\()p FN(\025)340 759 y FK(1)378 747 y FN(;)14 b(\025)463 759 y FK(2)501 747 y FO(\))28 b(b)r(elongs)e(to)h(the)h(set)591 935 y FN(K)668 900 y FK(\(1\))779 935 y FO(=)23 b FP(f)p FO(\()p FN(\025)989 947 y FK(1)1026 935 y FN(;)14 b(\025)1111 947 y FK(2)1149 935 y FO(\))23 b FP(2)h FI(R)1337 900 y FK(2)1403 935 y FP(j)f FN(\025)1497 947 y FK(1)1535 935 y FN(\025)1583 947 y FK(2)1644 935 y FO(=)f(0)p FP(g)p FO(;)-26 1161 y(2\))41 b(t)n(w)n(o-dimensional)22 b(\(dim)13 b FN(H)30 b FO(=)22 b(2\),)533 1399 y FN(A)i FO(=)e FN(\025)754 1411 y FK(1)806 1282 y Fy(\022)867 1349 y FO(1)115 b(0)867 1448 y(0)82 b FP(\000)p FO(1)1097 1282 y Fy(\023)1172 1399 y FN(;)97 b(B)27 b FO(=)c FN(\025)1518 1411 y FK(2)1570 1282 y Fy(\022)1631 1349 y FO(0)82 b(1)1631 1448 y(1)g(0)1797 1282 y Fy(\023)1872 1399 y FN(;)89 1637 y FO(where)28 b(the)f(pair)f(\()p FN(\025)723 1649 y FK(1)761 1637 y FN(;)14 b(\025)846 1649 y FK(2)884 1637 y FO(\))28 b(b)r(elongs)e(to)489 1825 y FN(K)566 1791 y FK(\(2\))678 1825 y FO(=)d FP(f)p FO(\()p FN(\025)888 1837 y FK(1)925 1825 y FN(;)14 b(\025)1010 1837 y FK(2)1048 1825 y FO(\))23 b FP(2)g FI(R)1235 1791 y FK(2)1302 1825 y FP(j)g FN(\025)1396 1837 y FK(1)1456 1825 y FN(>)g FO(0)p FN(;)k(\025)1684 1837 y FK(2)1745 1825 y FN(>)c FO(0)p FP(g)p FN(:)6 2052 y FO(Let)31 b(us)f(sho)n(w)g(that)h(these)f(represen)n(tations)e (separate)h(elemen)n(ts)f(of)i(the)-118 2152 y(algebra.)k(Let)547 2352 y FN(x)24 b FO(=)e FN(\013e)d FO(+)f FN(\014)t(a)g FO(+)g FN(\015)5 b(b)18 b FO(+)1280 2273 y Fy(X)1303 2450 y FL(i;j)1414 2352 y FN(c)1450 2364 y FL(ij)1508 2352 y FN(a)1552 2318 y FL(i)1580 2352 y FN(b)1616 2318 y FL(j)1650 2352 y FN(:)-118 2628 y FO(If)33 b FN(\031)s FO(\()p FN(x)p FO(\))g(=)e(0)i(for)f(an)n(y)g(one-dimensional)27 b(represen)n(tation,)32 b(then)h FN(\013)f FO(=)f FN(\014)36 b FO(=)-118 2728 y FN(\015)29 b FO(=)23 b(0.)39 b(If,)28 b(further,)h FN(\031)s FO(\()p FN(x)p FO(\))c(=)f(0)k(for)f(an)n(y)h(t) n(w)n(o-dimensional)22 b(represen)n(tation,)-118 2828 y(then)28 b(w)n(e)f(ha)n(v)n(e:)254 3016 y FN(\031)s FO(\()p FN(x)p FO(\))d(=)f FN(\031)577 2924 y Fy(\020)627 2937 y(X)650 3114 y FL(i;j)761 3016 y FN(c)797 3028 y FL(ij)855 3016 y FN(a)899 2982 y FL(i)926 3016 y FN(b)962 2982 y FL(j)997 2924 y Fy(\021)439 3287 y FO(=)527 3195 y Fy(\020)667 3208 y(X)576 3387 y FL(i)p FK(=2)p FL(k)q(;j)s FK(=2)p FL(l)890 3287 y FN(c)926 3299 y FL(ij)985 3287 y FN(\025)1033 3253 y FL(i)1033 3308 y FK(1)1071 3287 y FN(\025)1119 3247 y FL(j)1119 3309 y FK(2)1156 3195 y Fy(\021)1220 3170 y(\022)1281 3236 y FO(1)82 b(0)1281 3336 y(0)g(1)1447 3170 y Fy(\023)518 3560 y FO(+)601 3468 y Fy(\020)782 3481 y(X)650 3660 y FL(i)p FK(=2)p FL(k)q FK(+1)p FL(;j)s FK(=2)p FL(l)1048 3560 y FN(c)1084 3572 y FL(ij)1143 3560 y FN(\025)1191 3526 y FL(i)1191 3581 y FK(1)1228 3560 y FN(\025)1276 3520 y FL(j)1276 3582 y FK(2)1314 3468 y Fy(\021)1378 3443 y(\022)1439 3509 y FO(1)114 b(0)1439 3609 y(0)82 b FP(\000)p FO(1)1669 3443 y Fy(\023)518 3833 y FO(+)601 3741 y Fy(\020)782 3754 y(X)650 3933 y FL(i)p FK(=2)p FL(k)q(;j)s FK(=2)p FL(l)p FK(+1)1048 3833 y FN(c)1084 3845 y FL(ij)1143 3833 y FN(\025)1191 3799 y FL(i)1191 3854 y FK(1)1228 3833 y FN(\025)1276 3793 y FL(j)1276 3855 y FK(2)1314 3741 y Fy(\021)1378 3716 y(\022)1439 3782 y FO(0)g(1)1439 3882 y(1)g(0)1605 3716 y Fy(\023)p eop %%Page: 42 46 42 45 bop -118 -137 a FO(42)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)518 134 y FO(+)601 42 y Fy(\020)824 55 y(X)650 234 y FL(i)p FK(=2)p FL(k)q FK(+1)p FL(;j)s FK(=2)p FL(l)p FK(+1)1132 134 y FN(c)1168 146 y FL(ij)1227 134 y FN(\025)1275 100 y FL(i)1275 154 y FK(1)1312 134 y FN(\025)1360 94 y FL(j)1360 156 y FK(2)1398 42 y Fy(\021)1462 17 y(\022)1555 83 y FO(0)115 b(1)1523 183 y FP(\000)p FO(1)82 b(0)1753 17 y Fy(\023)1837 134 y FO(=)23 b(0)-118 373 y(whic)n(h)j(implies)e FN(c)437 385 y FL(ij)519 373 y FO(=)e(0,)28 b FP(8)p FN(i)p FO(,)e FN(j)5 b FO(,)28 b(i.e.,)f FN(x)c FO(=)g(0.)6 473 y(Then)36 b(b)n(y)g(Theorem)e(2,)j(the)f(algebra)d FI(C)1320 405 y Fy(\012)1365 473 y FN(x;)14 b(y)40 b FP(j)d FN(x)1637 442 y FK(2)1711 473 y FO(=)f FN(y)1856 442 y FK(2)1893 405 y Fy(\013)1969 473 y FO(=)g FI(C)15 b FP(h)p FN(a;)f(b)43 b FP(j)-118 572 y(f)p FN(a;)14 b(b)p FP(g)21 b FO(=)i(0)p FP(i)k FO(is)g(an)g FN(F)545 584 y FK(4)583 572 y FO(-algebra.)p 2278 572 4 57 v 2282 519 50 4 v 2282 572 V 2331 572 4 57 v -118 732 a FQ(3.)66 b FO(Let)38 b(us)f(observ)n(e)f(that)i(the)g(algebra)d FN(Q)1314 744 y FK(4)p FL(;)p FK(2)1443 732 y FO(=)k FI(C)1601 665 y Fy(\012)1647 732 y FN(q)1684 744 y FK(1)1721 732 y FN(;)14 b(q)1795 744 y FK(2)1832 732 y FN(;)g(q)1906 744 y FK(3)1943 732 y FN(;)g(q)2017 744 y FK(4)2094 732 y FP(j)40 b FN(q)2197 702 y FK(2)2194 754 y FL(i)2274 732 y FO(=)-118 849 y FN(q)-81 861 y FL(i)-53 849 y FN(;)27 b(i)i FO(=)g(1)p FN(;)14 b FO(2)p FN(;)g FO(3)p FN(;)g FO(4;)477 787 y Fy(P)564 807 y FK(4)564 874 y FL(i)p FK(=1)690 849 y FN(q)727 861 y FL(i)784 849 y FO(=)29 b(2)p FN(e)959 782 y Fy(\013)1029 849 y FO(is)h(also)f(a)i FN(F)1412 861 y FK(4)1450 849 y FO(-algebra.)45 b(Its)32 b(represen)n(ta-)-118 949 y(tion)27 b(will)d(b)r(e)k(describ)r(ed)f(b)r (elo)n(w)f(in)h(Section)f(2.2.1.)-118 1086 y FQ(4.)68 b FO(The)38 b(algebra)e FI(C)562 1019 y Fy(\012)607 1086 y FN(q)644 1098 y FK(1)681 1086 y FN(;)14 b(q)755 1098 y FK(2)792 1086 y FN(;)g(q)866 1098 y FK(3)904 1086 y FN(;)g(q)978 1098 y FK(4)1056 1086 y FP(j)41 b FN(q)1160 1056 y FK(2)1157 1110 y FL(k)1238 1086 y FO(=)g FN(q)1381 1098 y FL(k)1422 1086 y FN(;)1472 1019 y Fy(\002)1507 1086 y FN(q)1544 1098 y FL(k)1585 1086 y FN(;)1622 1024 y Fy(P)1709 1045 y FK(4)1709 1111 y FL(j)s FK(=1)1842 1086 y FN(q)1879 1098 y FL(j)1914 1019 y Fy(\003)1990 1086 y FO(=)f(0)p FN(;)27 b(k)44 b FO(=)-118 1201 y(1)p FN(;)14 b FO(2)p FN(;)g FO(3)p FN(;)g FO(4)161 1134 y Fy(\013)227 1201 y FO(is)29 b(not)g(a)g FN(F)586 1213 y FL(n)632 1201 y FO(-algebra)d(for)j(all)e(\014nite)i FN(n)p FO(,)h(since)e(it)h(has)g(irreducible)-118 1301 y(represen)n(tations)c(in)h(all)g(\014nite)h(dimensions)d(\(see)k (Section)e(2.2.1\).)-118 1439 y FQ(5.)36 b FO(The)28 b(algebra)94 1667 y FN(Q)160 1679 y FK(5)p FL(;\025)279 1667 y FO(=)23 b FI(C)421 1574 y Fy(D)478 1667 y FN(q)515 1679 y FK(1)552 1667 y FN(;)14 b(:)g(:)g(:)f(;)h(q)773 1679 y FK(5)834 1667 y FP(j)23 b FN(q)920 1632 y FK(2)917 1687 y FL(k)981 1667 y FO(=)f FN(q)1105 1679 y FL(k)1146 1667 y FN(;)28 b(k)e FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)1715 1563 y FK(5)1672 1588 y Fy(X)1674 1764 y FL(j)s FK(=1)1806 1667 y FN(q)1843 1679 y FL(j)1901 1667 y FO(=)22 b FN(\025e)2075 1574 y Fy(E)-118 1904 y FO(is)36 b(not)i(a)f FN(F)265 1916 y FL(n)310 1904 y FO(-algebra)e(for)i(all)e FN(\025)40 b FP(2)g FI(C)58 b FO(and)37 b(all)e(\014nite)j FN(n)f FO(since)f(it)h(has)g(an)-118 2003 y(irreducible)21 b(in\014nite-dimensional)e(represen)n(tation)k(\(see)50 b([218)o(]\).)36 b(The)25 b(con-)-118 2103 y(struction)f(of)i(this)f (represen)n(tation,)e(whic)n(h)i(can)g(b)r(e)h(found)g(in)51 b([218)o(],)26 b(gener-)-118 2203 y(alizes)f(the)j(one)f(for)g(\014v)n (e)g(idemp)r(oten)n(ts)f(with)h(the)h(zero)f(sum)f(giv)n(en)g(in)55 b([19)o(].)-118 2439 y FG(1.3)112 b(Represen)m(tations)40 b(of)f(t)m(w)m(o-dimensional)j(Lie)d(alge-)137 2555 y(bras,)78 b(their)69 b(nonlinear)h(transformations,)79 b(and)137 2671 y(semilinear)39 b(relations)-118 2853 y FQ(1.3.1)94 b(Represen)m(tations)23 b(of)h(t)m(w)m(o-dimensional)d(real)j(Lie)g (algebras)174 2953 y(and)29 b(their)f(nonlinear)h(transformations)e(b)m (y)j(b)s(ounded)e(op-)174 3053 y(erators)-118 3206 y(1.)49 b FO(Lie)31 b(algebras)e(\()p FN(I)7 b(V)601 3218 y FK(0)639 3206 y FO(\),)33 b(\()p FN(I)7 b(V)850 3218 y FK(1)888 3206 y FO(\),)34 b(\()p FN(I)7 b(V)1100 3218 y FK(2)1138 3206 y FO(\),)33 b(similarly)26 b(to)32 b(the)h(relations)28 b(\()p FN(V)2245 3218 y FK(0)2283 3206 y FO(\),)-118 3305 y(\()p FN(V)-38 3317 y FK(1)0 3305 y FO(\),)g(\()p FN(V)182 3275 y FM(0)163 3326 y FK(1)206 3305 y FO(\),)g(can)g(b)r(e)g (treated)f(from)g(a)g(general)f(p)r(oin)n(t)h(of)h(view.)36 b(Namely)-7 b(,)26 b(all)-118 3405 y(these)i(relations)c(are)j (particular)d(cases)i(of)i(the)g(relation)807 3558 y([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(iP)1212 3570 y FK(2)1250 3558 y FO(\()p FN(A)p FO(\))p FN(;)769 b FO(\(1.8\))-118 3712 y(where)27 b FN(P)175 3724 y FK(2)213 3712 y FO(\()p FN(A)p FO(\))h(is)f(a)g(real)e(quadratic)h(p)r(olynomial.)6 3811 y(If)31 b FN(A)c FO(=)f FN(A)334 3781 y FM(\003)373 3811 y FO(,)k FN(B)h FO(=)c FN(B)679 3781 y FM(\003)744 3811 y FP(2)g FN(L)p FO(\()p FN(H)7 b FO(\),)30 b(then)h(their)d(study) i(can)g(b)r(e)g(p)r(erformed)-118 3911 y(b)n(y)d(using)f(the)i(follo)n (wing)c(theorem.)p eop %%Page: 43 47 43 46 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(43)-118 96 y FQ(Theorem)30 b(4.)41 b FO(\(D.)20 b(Kleinec)n(k)n(e,)d(F.V.)i(Shirok)n(o)n(v\).)33 b FB(If)22 b FN(P)33 b FB(and)22 b FN(Q)f FB(ar)l(e)h(b)l(ounde)l(d) -118 196 y(op)l(er)l(ators,)49 b(and)44 b FO([)p FN(P)r(;)14 b FO([)p FN(P)r(;)g(Q)p FO(]])50 b(=)e(0)p FB(,)g(then)c(the)g(op)l(er) l(ator)h FO([)p FN(P)r(;)14 b(Q)p FO(])44 b FB(is)g(quasi-)-118 296 y(nilp)l(otent,)30 b(i.e.,)744 488 y FO(lim)697 538 y FL(n)p FP(\000)-46 b(!)p FM(1)943 452 y Fv(n)933 413 y Fy(p)p 1016 413 333 4 v 75 x FP(k)p FO([)p FN(P)r(;)14 b(Q)p FO(])1262 464 y FL(n)1307 488 y FP(k)23 b FO(=)f(0)p FN(:)-118 709 y FB(Pr)l(o)l(of.)43 b FO(The)33 b(Kleinec)n(k)n (e{Shirok)n(o)m(v)28 b(theorem)j(follo)n(ws,)h(for)h(example,)f(from) -118 809 y(the)f(form)n(ula)d(ad)423 772 y FL(n)423 830 y(P)492 809 y FN(Q)558 779 y FL(n)631 809 y FO(=)g FN(n)p FO(!)14 b(\(ad)931 821 y FL(P)1000 809 y FN(Q)p FO(\))1098 779 y FL(n)1143 809 y FO(,)32 b(where)e(ad)1529 821 y FL(P)1598 809 y FN(X)k FO(=)28 b FN(P)12 b(X)27 b FP(\000)20 b FN(X)7 b(P)42 b FO(is)29 b(a)-118 909 y(b)r(ounded)f(op)r(erator)e (in)h FN(L)p FO(\()p FN(H)7 b FO(\),)27 b(and)h FP(k)14 b FO(ad)1205 921 y FL(P)1274 909 y FP(k)22 b(\024)h FO(2)p FP(k)p FN(P)12 b FP(k)p FO(.)35 b(Then)157 1108 y Fv(n)147 1070 y Fy(p)p 230 1070 V 75 x FP(k)p FO([)p FN(P)r(;)14 b(Q)p FO(])476 1121 y FL(n)521 1145 y FP(k)23 b(\024)741 1089 y FO(2)p 684 1126 156 4 v 707 1172 a Fv(n)697 1142 y FP(p)p 766 1142 73 4 v 71 x FN(n)p FO(!)863 1145 y FP(k)p FN(P)12 b FP(k)17 b(\001)i(k)p FN(Q)p FP(k)i(\000)-48 b(!)23 b FO(0)p FN(;)180 b(n)23 b FP(\000)-49 b(!)23 b(1)p FN(:)p 2278 1145 4 57 v 2282 1092 50 4 v 2282 1145 V 2331 1145 4 57 v 6 1393 a FO(In)31 b(Section)f(1.3.3)f(b)r(elo)n(w)g (w)n(e)h(will)e(giv)n(e)g(man)n(y)h(analogies)e(of)j(this)g(theo-)-118 1492 y(rem.)-118 1656 y FQ(2.)36 b FO(No)n(w)27 b(w)n(e)g(consider)f(b) r(ounded)i(self-adjoin)n(t)d(op)r(erators)h(satisfying)g(\(1.8\))o(.) -118 1836 y FQ(Prop)s(osition)k(17.)41 b FB(Irr)l(e)l(ducible)25 b(p)l(airs,)i FN(A)p FB(,)f FN(B)i FB(of)d(b)l(ounde)l(d)g (self-adjoint)h(op-)-118 1936 y(er)l(ators)d(which)i(satisfy)f(the)f(r) l(elation)30 b FO(\(1.8\))22 b FB(ar)l(e)h(one-dimensional,)k(and)c (they)-118 2036 y(ar)l(e)36 b(given)h(by)7 b FO(:)51 b FN(A)35 b FO(=)f FN(\025)p FB(,)k FN(B)h FO(=)33 b FN(\026)p FB(,)38 b(wher)l(e)f FO(\()p FN(\025;)14 b(\026)p FO(\))35 b FP(2)g FN(M)43 b FO(=)34 b FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))34 b FP(2)h FI(R)2238 2006 y FK(2)2316 2036 y FP(j)-118 2135 y FN(P)-65 2147 y FK(2)-28 2135 y FO(\()p FN(\025)p FO(\))24 b(=)f(0)p FP(g)p FB(.)6 2240 y(A)n(n)29 b(arbitr)l(ary)i(b)l(ounde)l(d)f(p)l(air)h(has)g(the)f (form)356 2470 y FN(A)24 b FO(=)529 2357 y Fy(Z)576 2545 y FL(M)663 2470 y FN(\025)14 b(dE)5 b FO(\()p FN(\025;)14 b(\026)p FO(\))p FN(;)100 b(B)28 b FO(=)1334 2357 y Fy(Z)1380 2545 y FL(M)1468 2470 y FN(\026)14 b(dE)5 b FO(\()p FN(\025;)14 b(\026)p FO(\))p FN(;)-118 2709 y FB(wher)l(e)30 b FN(E)5 b FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))31 b FB(is)f(the)g(r)l (esolution)g(of)g(the)g(identity)h(on)e FN(M)9 b FB(.)-118 2889 y(Pr)l(o)l(of.)43 b FO(Indeed,)35 b(\(1.8\))d(implies)d([)p FN(A;)14 b FO([)p FN(A;)g(B)t FO(]])33 b(=)f(0,)i(and)e(b)n(y)h(the)g (Kleinec)n(k)n(e{)-118 2989 y(Shirok)n(o)n(v)27 b(theorem,)i(the)h(op)r (erator)f([)p FN(A;)14 b(B)t FO(])30 b(is)f(quasi-nilp)r(oten)n(t.)41 b(But)30 b(since)-118 3088 y([)p FN(A;)14 b(B)t FO(])25 b(is)e(sk)n(ew-adjoin)n(t,)g(it)h(yields)e([)p FN(A;)14 b(B)t FO(])23 b(=)g(0.)36 b(Then)24 b FN(P)1700 3100 y FK(2)1738 3088 y FO(\()p FN(A)p FO(\))g(=)e(0,)j(and)f(the)-118 3188 y(statemen)n(t)e(follo)n(ws)f(from)h(the)i(sp)r(ectral)e(theorem)g (for)h(a)g(pair)f(of)h(comm)n(uting)-118 3287 y(self-adjoin)n(t)i(op)r (erators.)p 2278 3287 V 2282 3235 50 4 v 2282 3287 V 2331 3287 4 57 v -118 3492 a FB(R)l(emark)30 b(9.)42 b FO(The)i(prop)r(osition)d(ab)r(o)n(v)n(e)i(implies)d(that)k(if)g(the) g(p)r(olynomial)-118 3591 y FN(P)-65 3603 y FK(2)-28 3591 y FO(\()p FP(\001)p FO(\))28 b(has)f(no)g(real)e(ro)r(ots,)h(then) i(there)f(are)f(no)g(b)r(ounded)i(self-adjoin)n(t)d(pairs)-118 3691 y(that)37 b(satisfy)g(\(1.8\))o(.)66 b(In)37 b(particular,)f (there)g(are)g(no)h(b)r(ounded)g(pairs)e(that)-118 3791 y(satisfy)i(the)j(CCR)e(\(relation)f(\()p FN(I)7 b(V)998 3803 y FK(1)1036 3791 y FO(\)\),)42 b(or)c([)p FN(A;)14 b(B)t FO(])42 b(=)f FN(i)p FO(\()p FN(A)1761 3761 y FK(2)1825 3791 y FO(+)25 b FN(I)7 b FO(\))39 b(\(relation)-118 3890 y(\()p FN(V)-38 3902 y FK(1)0 3890 y FO(\)\).)p eop %%Page: 44 48 44 47 bop -118 -137 a FO(44)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FB(R)l(emark)30 b(10.)42 b FO(Relation)18 b(\()p FN(V)747 108 y FK(2)785 96 y FO(\))i([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(i)p FO(\()p FN(A)1283 66 y FK(2)1323 96 y FO(+)s FN(B)t FO(\))c(can)g(also)f(b)r(e)i(rewritten)e(in)-118 196 y(the)28 b(equiv)-5 b(alen)n(t)26 b(form)g([)p FN(A;)14 b(A)800 166 y FK(2)856 196 y FO(+)19 b FN(B)t FO(])k(=)g FN(i)p FO(\()p FN(A)1264 166 y FK(2)1320 196 y FO(+)18 b FN(B)t FO(\),)29 b(whic)n(h)d(can)i(b)r(e)g(reduced)-118 296 y(b)n(y)38 b(a)g(non-degenerate)f(nonlinear)f(c)n(hange)h(of)h(v)-5 b(ariables,)1845 275 y(~)1823 296 y FN(A)42 b FO(=)f FN(A)2095 266 y FK(2)2158 296 y FO(+)25 b FN(B)t FO(,)-98 374 y(~)-118 395 y FN(B)33 b FO(=)28 b FN(B)t FO(,)k(to)f(the)g(form)f ([)689 374 y(~)667 395 y FN(A;)786 374 y FO(~)766 395 y FN(B)t FO(])f(=)f FN(i)1029 374 y FO(~)1007 395 y FN(A)p FO(.)47 b(Since)1381 374 y(~)1359 395 y FN(A)31 b FO(and)1637 374 y(~)1617 395 y FN(B)k FO(are)30 b(also)f(b)r(ounded)-118 495 y(self-adjoin)n(t)f(op)r(erators,)h(b)n(y)h(the)h(Kleinec)n(k)n (e{Shirok)n(o)n(v)24 b(theorem,)30 b(w)n(e)g(ha)n(v)n(e)-118 595 y([)-73 574 y(~)-95 595 y FN(A;)24 574 y FO(~)4 595 y FN(B)t FO(])23 b(=)g(0,)k(and)h([)p FN(A;)14 b(B)t FO(])23 b(=)g(0,)k(whic)n(h)g(yields)e FN(A)1407 564 y FK(2)1468 595 y FO(=)d FP(\000)p FN(B)t FO(.)6 695 y(Irreducible)34 b(represen)n(tations)g(of)j(relation\()p FN(V)1482 707 y FK(2)1517 695 y FO(\))g(are)f(one-dimensional,)-118 794 y FN(A)23 b FO(=)g FN(\025)p FO(,)28 b FN(B)f FO(=)c FN(\026)p FO(,)28 b(\()p FN(\025;)14 b(\026)p FO(\))24 b FP(2)f FN(M)815 809 y FK(\()p FL(V)880 817 y Fx(2)912 809 y FK(\))965 794 y FO(=)g FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))23 b FP(2)h FI(R)1450 764 y FK(2)1516 794 y FP(j)f FN(\025)1610 764 y FK(2)1671 794 y FO(=)g FP(\000)p FN(\026)p FP(g)p FO(.)6 895 y(An)j(arbitrary)21 b(pair)i(of)i(b)r(ounded)g(self-adjoin)n(t)d(op)r(erators)h(satisfying) e(re-)-118 994 y(lation)k(\()p FN(V)197 1006 y FK(2)235 994 y FO(\))j(has)f(the)h(form)251 1220 y FN(A)23 b FO(=)424 1107 y Fy(Z)470 1296 y FL(M)533 1307 y Fx(\()p Fv(V)589 1319 y Fx(2)623 1307 y(\))667 1220 y FN(\025)14 b(dE)5 b FO(\()p FN(\025;)14 b(\026)p FO(\))p FN(;)99 b(A)23 b FO(=)1332 1107 y Fy(Z)1378 1296 y FL(M)1441 1307 y Fx(\()p Fv(V)1497 1319 y Fx(2)1530 1307 y(\))1575 1220 y FN(\025)14 b(dE)5 b FO(\()p FN(\025;)14 b(\026)p FO(\))p FN(;)-118 1474 y FO(where)27 b FN(E)5 b FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))28 b(is)f(the)h(resolution)c(of)k(the)g(iden) n(tit)n(y)e(on)h FN(M)1714 1489 y FK(\()p FL(V)1779 1497 y Fx(2)1811 1489 y FK(\))1841 1474 y FO(.)-118 1608 y FQ(3.)34 b FO(If)23 b(in)f(the)h(study)f(of)h(represen)n(tations)c(of)k (relations)c(giv)n(en)i(b)n(y)h(the)h(families)-118 1708 y(\()p FN(I)7 b(V)19 b FO(\))37 b(and)f(\()p FN(V)19 b FO(\),)39 b(w)n(e)e(restrict)d(ourselv)n(es)g(to)i(only)f(the)i (\014nite-dimensional)-118 1807 y(lev)n(el,)h(then)h(instead)e(of)h (the)g(Kleinec)n(k)n(e{Shirok)n(o)m(v)32 b(theorem,)40 b(whic)n(h)d(w)n(as)-118 1907 y(pro)n(v)n(ed)26 b(in)h(1956,)f(w)n(e)h (could)g(apply)g(its)f(\014nite-dimensional)d(v)-5 b(arian)n(t,)26 b(whic)n(h)-118 2007 y(is)j(the)i(follo)n(wing)26 b(theorem)j(pro)n(v)n (ed)g(m)n(uc)n(h)h(earlier)c(\(1935\).)45 b(W)-7 b(e)30 b(form)n(ulate)-118 2106 y(it)d(in)g(the)h(simplest)d(case.)-118 2274 y FQ(Theorem)30 b(5.)41 b FO(\(N.)f(Jacobson\).)67 b FB(If)40 b FO(dim)13 b FN(H)48 b(<)41 b FP(1)f FB(and)h FO([)p FN(P)r(;)14 b FO([)p FN(P)r(;)g(Q)p FO(]])42 b(=)g(0)p FB(,)-118 2374 y(then)29 b(the)h(op)l(er)l(ator)h FO([)p FN(P)r(;)14 b(Q)p FO(])30 b FB(is)g(nilp)l(otent.)-118 2541 y(R)l(emark)g(11.)42 b FO(In)34 b(the)f(relations)d(\()p FN(I)7 b(V)20 b FO(\))33 b(and)g(\()p FN(V)19 b FO(\))34 b(no)f(new)g(irreducible)c(rep-)-118 2641 y(resen)n(tations)34 b(arise)g(when)i(passing)e(from)h(the)h(\014nite-dimensional)31 b(to)36 b(the)-118 2741 y(b)r(ounded)28 b(case.)6 2841 y(Of)g(course,)f(considering)d(un)n(b)r(ounded)k(op)r(erators,)e(for)h (whic)n(h)f(the)i(Klei-)-118 2940 y(nec)n(k)n(e{Shirok)n(o)n(v)i (theorem)j(fails,)h(one)h(obtains)e(a)h(m)n(uc)n(h)f(more)g(consisten)n (t)-118 3040 y(represen)n(tation)25 b(theory)i(of)g(the)h(relations)d (\()p FN(I)7 b(V)19 b FO(\))28 b(and)f(\()p FN(V)19 b FO(\).)-118 3259 y FQ(1.3.2)94 b(P)m(airs)31 b(of)f(op)s(erators)g (connected)h(b)m(y)g(semilinear)c(relations)-118 3413 y FO(In)i(Sections)f(1.3.2{1.3.5)e(w)n(e)i(study)h(a)g(wide)f(class)f (of)i(so-called)d(semilinear)-118 3513 y(relations)32 b(whic)n(h)i(connect)i(pairs)d(of)i(b)r(ounded)h(op)r(erators)d FN(A)k FO(=)e FN(A)2081 3482 y FM(\003)2120 3513 y FO(,)i FN(B)j FP(2)-118 3612 y FN(L)p FO(\()p FN(H)7 b FO(\).)61 b(They)36 b(app)r(ear)f(as)g(a)g(generalization)c(of)36 b(the)g(relations)d(\()p FN(I)7 b(V)19 b FO(\))36 b(and)-118 3712 y(\()p FN(V)19 b FO(\))i(studied)e(in)g(previous)f(sections.)33 b(W)-7 b(e)21 b(pro)n(v)n(e)d(man)n(y)h(Kleinec)n(k)n(e{Shirok)n(o)m(v) -118 3811 y(t)n(yp)r(e)32 b(theorems)e(\(Section)i(1.3.3\).)49 b(F)-7 b(or)31 b(general)f(semilinear)d(relations,)j(w)n(e)-118 3911 y(describ)r(e)21 b(prop)r(erties)g(of)i(irreducible)c(represen)n (tations)h(\(Section)i(1.3.4\).)34 b(All)p eop %%Page: 45 49 45 48 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(45)-118 96 y(irreducible)15 b(represen)n(tations)i(are)h(classi\014ed)e(up)k(to)f(unitary)e(equiv) -5 b(alence)17 b(for)-118 196 y(semilinear)23 b FN(F)328 208 y FK(4)365 196 y FO(-relations)h(\(Section)j(1.3.5\).)-118 367 y FQ(1.)38 b FO(Consider)27 b(b)r(ounded)i(op)r(erators)d FN(A)f FO(=)f FN(A)1289 337 y FM(\003)1327 367 y FO(,)29 b FN(B)f FP(2)d FN(L)p FO(\()p FN(H)7 b FO(\),)29 b(whic)n(h)e(satisfy) g(a)-118 466 y(relation)e(of)i(the)h(form:)693 605 y FL(n)654 630 y Fy(X)660 807 y FL(i)p FK(=1)788 709 y FN(f)829 721 y FL(i)856 709 y FO(\()p FN(A)p FO(\))p FN(B)t(g)1089 721 y FL(i)1117 709 y FO(\()p FN(A)p FO(\))c(=)f FN(h)p FO(\()p FN(A)p FO(\))p FN(;)616 b FO(\(1.9\))-118 983 y(where)35 b FN(f)171 995 y FL(i)198 983 y FO(\()p FP(\001)p FO(\),)j FN(g)386 995 y FL(i)414 983 y FO(\()p FP(\001)p FO(\),)g FN(h)p FO(\()p FP(\001)p FO(\),)g FN(i)e FO(=)g(1,)f FN(:)14 b(:)g(:)27 b FO(,)38 b FN(n)p FO(,)f(are)e(complex)e(b)r(ounded)j(Borel)-118 1082 y(mappings)27 b(de\014ned)k(on)e FI(R)37 b FO(or)29 b(a)g(subset)h FN(D)r FO(,)h FN(\033)s FO(\()p FN(A)p FO(\))d FP(\032)f FN(D)r FO(.)44 b(Relation)28 b(\(1.9\))i(is)-118 1182 y(called)24 b(semilinear)e(and)k(the)h(op)r(erators)e FN(A)p FO(,)i FN(B)k FO(are)26 b(called)e(a)i(represen)n(tation)-118 1282 y(of)h(\(1.9\).)6 1388 y(If)h FN(f)130 1400 y FL(i)158 1388 y FO(\()p FP(\001)p FO(\),)g FN(g)336 1400 y FL(i)363 1388 y FO(\()p FP(\001)p FO(\),)h FN(h)p FO(\()p FP(\001)p FO(\),)f FN(i)23 b FO(=)g(1,)k FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(,)g(are)f(p)r(olynomials,)22 b(then)29 b(the)f(op)r(er-)-118 1488 y(ators)c FN(A)p FO(,)i FN(B)t FO(,)g FN(B)382 1458 y FM(\003)446 1488 y FO(de\014ne)g(a)f(represen)n(tation)d(of)j(the)h FP(\003)p FO(-algebra)c Fz(A)j FO(with)g(three)-118 1588 y(generators)k FN(a)g FO(=)f FN(a)498 1557 y FM(\003)536 1588 y FO(,)33 b FN(b)p FO(,)f FN(b)719 1557 y FM(\003)788 1588 y FO(satisfying)c(the)k(relation)1616 1525 y Fy(P)1704 1546 y FL(n)1704 1613 y(i)p FK(=1)1829 1588 y FN(f)1870 1600 y FL(i)1897 1588 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)2109 1600 y FL(i)2136 1588 y FO(\()p FN(a)p FO(\))30 b(=)-118 1687 y FN(h)p FO(\()p FN(a)p FO(\).)37 b(This)26 b(algebra)f(is)h(the)i (quotien)n(t)e(algebra)e(of)k(the)f(free)g FP(\003)p FO(-algebra)d(with)-118 1787 y(three)c(generators)e FI(C)d FP(h)p FN(a)29 b FO(=)23 b FN(a)773 1757 y FM(\003)811 1787 y FN(;)14 b(b;)g(b)957 1757 y FM(\003)994 1787 y FP(i)21 b FO(with)f(resp)r(ect)g(to)h(the)f(t)n(w)n(o-sided)f FP(\003)p FO(-ideal)-118 1886 y(generated)26 b(b)n(y)i(the)g(elemen)n (t)825 1824 y Fy(P)913 1845 y FL(n)913 1911 y(i)p FK(=1)1039 1886 y FN(f)1080 1898 y FL(i)1107 1886 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)1319 1898 y FL(i)1346 1886 y FO(\()p FN(a)p FO(\))19 b FP(\000)f FN(h)p FO(\()p FN(a)p FO(\).)6 1993 y(Unless)32 b(otherwise)e(stated,)j(w)n(e)f(assume)e FN(f)1384 2005 y FL(i)1412 1993 y FO(,)j FN(g)1508 2005 y FL(i)1535 1993 y FO(,)h FN(h)e FO(to)g(b)r(e)h(de\014ned)f(here)-118 2093 y(on)i(the)h(whole)e FI(R)p FO(.)63 b(The)35 b(general)d(case)h (can)h(b)r(e)h(easily)d(deriv)n(ed)g(from)h(this)-118 2192 y(one.)-118 2339 y FB(R)l(emark)d(12.)42 b FO(As)31 b(b)r(efore,)g(if)f(the)h(op)r(erators)d FN(A)g FO(=)f FN(A)1582 2309 y FM(\003)1621 2339 y FO(,)k FN(B)k FO(are)29 b(un)n(b)r(ounded,)-118 2439 y(then)k(it)f(is)g(necessary)f(to)h (de\014ne)h(the)g(meaning)e(of)h(the)h(op)r(erator)e(equalit)n(y)-118 2539 y(\(1.9\).)46 b(W)-7 b(e)31 b(in)n(v)n(estigate)d(the)j(question)e (of)i(the)g(\\correct")e(de\014nition)g(of)h(re-)-118 2638 y(lation)e(\(1.9\))i(with)g(un)n(b)r(ounded)h(op)r(erators)d(for)i (some)f(sp)r(ecial)f(relations)f(in)-118 2738 y([233)n(].)-118 2885 y FQ(2.)60 b FO(The)36 b(study)g(of)g(b)r(ounded)g(represen)n (tations)c(of)k(\(1.9\))g(can)f(b)r(e)h(reduced)-118 2984 y(to)27 b(the)g(study)g(of)g(op)r(erators)e(satisfying)f(the)j (corresp)r(onding)d(homogeneous)-118 3084 y(relation:)767 3223 y FL(n)728 3248 y Fy(X)734 3425 y FL(i)p FK(=1)861 3327 y FN(f)902 3339 y FL(i)930 3327 y FO(\()p FN(A)p FO(\))p FN(B)t(g)1163 3339 y FL(i)1191 3327 y FO(\()p FN(A)p FO(\))g(=)e(0)p FN(:)633 b FO(\(1.10\))6 3612 y(If)39 b(the)g(function)f FN(\036)p FO(\()p FN(t)p FO(\))j(=)g FN(h)p FO(\()p FN(t)p FO(\))p FN(=)1078 3550 y Fy(P)1165 3570 y FL(n)1165 3637 y(i)p FK(=1)1291 3612 y FN(f)1332 3624 y FL(i)1359 3612 y FO(\()p FN(t)p FO(\))p FN(g)1493 3624 y FL(i)1521 3612 y FO(\()p FN(t)p FO(\))e(is)e(b)r(ounded)h(on)g (the)-118 3712 y(sp)r(ectrum)f(of)i(the)g(op)r(erator)d FN(A)42 b FO(=)e FN(A)1131 3682 y FM(\003)1170 3712 y FO(,)h(then)e(the)g(pair)d(of)j(op)r(erators)d FN(A)p FO(,)-118 3811 y FN(B)47 b FO(=)42 b FN(\036)p FO(\()p FN(A)p FO(\))f(is)d(a)h(particular)d(solution)h(of)i(the)h (inhomogeneous)c(relation)-118 3911 y(\(1.9\),)e(and)f(the)g(general)e (solution)f(of)j(\(1.9\),)h(with)f(the)g(op)r(erator)e FN(A)j FO(\014xed,)p eop %%Page: 46 50 46 49 bop -118 -137 a FO(46)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(consists)f(of)i(all)d(pairs)h(\()p FN(A)p FO(,)j FN(B)19 b FO(+)c FN(\036)p FO(\()p FN(A)p FO(\)\))27 b(where)f(\()p FN(A)p FO(,)h FN(B)t FO(\))f(satis\014es)e(\(1.10\).)36 b(Let)119 301 y(~)108 323 y FN(\036)q FO(\()p FN(t)p FO(\))23 b(=)363 181 y Fy(\()430 266 y FN(h)p FO(\()p FN(t)p FO(\))p FN(=)628 204 y Fy(P)715 225 y FL(n)715 291 y(i)p FK(=1)841 266 y FN(f)882 278 y FL(i)909 266 y FO(\()p FN(t)p FO(\))14 b FN(g)1057 278 y FL(i)1085 266 y FO(\()p FN(t)p FO(\))p FN(;)83 b FO(if)1375 204 y Fy(P)1463 225 y FL(n)1463 291 y(i)p FK(=1)1588 266 y FN(f)1629 278 y FL(i)1657 266 y FO(\()p FN(t)p FO(\))14 b FN(g)1805 278 y FL(i)1832 266 y FO(\()p FN(t)p FO(\))24 b FP(6)p FO(=)f(0)p FN(;)430 386 y FO(0)p FN(;)790 b FO(otherwise)n FN(:)-118 549 y FQ(Prop)s(osition)30 b(18.)41 b FB(If)61 b FO(\(1.9\))43 b FB(has)i(a)50 b FO(\()p FB(b)l(ounde)l(d)9 b FO(\))44 b FB(solution)g FO(\()p FN(A;)14 b(B)t FO(\))p FB(,)49 b(then)-118 649 y(ther)l(e)30 b(exists)f FN(C)g(>)23 b FO(0)29 b FB(such)h(that)402 850 y FP(j)p FN(h)p FO(\()p FN(t)p FO(\))p FP(j)23 b(\024)g FN(C)766 754 y Fy(\014)766 804 y(\014)766 854 y(\014)834 746 y FL(n)794 771 y Fy(X)800 948 y FL(i)p FK(=1)928 850 y FN(f)969 862 y FL(i)996 850 y FO(\()p FN(t)p FO(\))p FN(g)1130 862 y FL(i)1158 850 y FO(\()p FN(t)p FO(\))1252 754 y Fy(\014)1252 804 y(\014)1252 854 y(\014)1280 850 y FN(;)184 b(t)23 b FP(2)g FN(\033)s FO(\()p FN(A)p FO(\))p FN(:)-118 1094 y FB(A)n(ny)40 b(solution)h FN(A)p FB(,)j FN(B)h FB(of)60 b FO(\(1.9\))40 b FB(has)i(the)f(form)g FN(B)47 b FO(=)c FN(B)1776 1064 y FM(0)1826 1094 y FO(+)1928 1072 y(~)1917 1094 y FN(\036)p FO(\()p FN(A)p FO(\))f FB(wher)l(e)-118 1194 y FO(\()p FN(A;)14 b(B)80 1163 y FM(0)104 1194 y FO(\))30 b FB(is)g(a)g(r)l(epr)l(esentation)g(of)48 b FO(\(1.10\))o FB(.)-118 1335 y(Pr)l(o)l(of.)43 b FO(Let)24 b FN(A)p FO(,)h FN(B)j FO(satisfy)23 b(\(1.9\),)i(and)e(let)g FN( )12 b FO(:)28 b FN(L)p FO(\()p FN(H)7 b FO(\))23 b FP(!)g Fz(A)h FO(b)r(e)g(a)f(conditional)-118 1435 y(exp)r(ectation)34 b(on)i(the)g FN(W)702 1404 y FM(\003)740 1435 y FO(-algebra)c Fz(A)k FO(of)f(all)f(op)r(erators)f(whic)n(h)i (comm)n(ute)-118 1534 y(with)25 b(the)h(op)r(erator)e FN(A)p FO(.)36 b(Applying)24 b FN( )k FO(to)e(b)r(oth)g(sides)e(of)h (equation)f(\(1.9\),)i(w)n(e)-118 1634 y(obtain)626 1675 y FL(n)586 1700 y Fy(X)593 1877 y FL(i)p FK(=1)720 1779 y FN(f)761 1791 y FL(i)788 1779 y FO(\()p FN(A)p FO(\))14 b FN( )s FO(\()p FN(B)t FO(\))g FN(g)1170 1791 y FL(i)1199 1779 y FO(\()p FN(A)p FO(\))24 b(=)e FN(h)p FO(\()p FN(A)p FO(\))p FN(:)-118 1987 y FO(F)-7 b(rom)26 b(this)h(w)n(e)g(conclude)f (that)i(\()p FN(A;)14 b( )s FO(\()p FN(B)t FO(\)\))30 b(is)c(a)h(solution)e(of)34 b(\(1.9\))28 b(and)586 2206 y FN( )s FO(\()p FN(B)t FO(\))828 2102 y FL(n)789 2127 y Fy(X)795 2304 y FL(i)p FK(=1)923 2206 y FN(f)964 2218 y FL(i)991 2206 y FO(\()p FN(A)p FO(\))14 b FN(g)1171 2218 y FL(i)1199 2206 y FO(\()p FN(A)p FO(\))24 b(=)e FN(h)p FO(\()p FN(A)p FO(\))p FN(:)-118 2450 y FO(Th)n(us,)28 b FN( )s FO(\()p FN(B)t FO(\))e(=)430 2428 y(~)419 2450 y FN(\036)p FO(\()p FN(A)p FO(\))k(on)e(the)h(image)c(of)k(the)g(op)r (erator)1699 2388 y Fy(P)1787 2408 y FL(n)1787 2475 y(i)p FK(=1)1912 2450 y FN(f)1953 2462 y FL(i)1981 2450 y FO(\()p FN(A)p FO(\))14 b FN(g)2161 2462 y FL(i)2189 2450 y FO(\()p FN(A)p FO(\).)-118 2549 y(Since)28 b FN( )s FO(\()p FN(B)t FO(\))i(is)e(a)g(b)r(ounded)i(op)r(erator,)e(there)g(exists)g FN(C)k(>)25 b FO(0)j(suc)n(h)h(that)g(for)-118 2649 y(ev)n(ery)d FN(t)d FP(2)h FN(\033)s FO(\()p FN(A)p FO(\))29 b(and)599 2587 y Fy(P)687 2607 y FL(n)687 2674 y(i)p FK(=1)812 2649 y FN(f)853 2661 y FL(i)880 2649 y FO(\()p FN(t)p FO(\))14 b FN(g)1028 2661 y FL(i)1056 2649 y FO(\()p FN(t)p FO(\))24 b FP(6)p FO(=)e(0)28 b(w)n(e)f(ha)n(v)n(e)660 2872 y FP(j)p FN(h)p FO(\()p FN(t)p FO(\))p FP(j)c(\024)g FN(C)1024 2776 y Fy(\014)1024 2826 y(\014)1024 2876 y(\014)1091 2768 y FL(n)1052 2793 y Fy(X)1058 2969 y FL(i)p FK(=1)1186 2872 y FN(f)1227 2884 y FL(i)1254 2872 y FO(\()p FN(t)p FO(\))p FN(g)1388 2884 y FL(i)1416 2872 y FO(\()p FN(t)p FO(\))1510 2776 y Fy(\014)1510 2826 y(\014)1510 2876 y(\014)1538 2872 y FN(:)-118 3105 y FO(On)31 b(the)g(other)g(hand,)h(k) n(er)750 3043 y Fy(P)838 3063 y FL(n)838 3130 y(i)p FK(=1)964 3105 y FN(f)1005 3117 y FL(i)1032 3105 y FO(\()p FN(A)p FO(\))14 b FN(g)1212 3117 y FL(i)1240 3105 y FO(\()p FN(A)p FO(\))29 b FP(\032)g FO(k)n(er)13 b FN(h)p FO(\()p FN(A)p FO(\),)32 b(whic)n(h)e(implies)-118 3216 y(that)e(the)g (inequalit)n(y)c(holds)j(for)g(ev)n(ery)g FN(t)c FP(2)h FN(\033)s FO(\()p FN(A)p FO(\),)29 b(and)f(\()p FN(A;)14 b(B)23 b FP(\000)1988 3194 y FO(~)1977 3216 y FN(\036)p FO(\()p FN(A)p FO(\)\))29 b(is)e(a)-118 3315 y(represen)n(tation)e(of)i (relation)e(\(1.10\).)p 2278 3315 4 57 v 2282 3262 50 4 v 2282 3315 V 2331 3315 4 57 v 6 3475 a(It)33 b(is)f(easy)f(to)i(pro) n(v)n(e)d(that)j(the)g(corresp)r(ondence)e(b)r(et)n(w)n(een)h (irreducible)-118 3575 y(represen)n(tations)19 b(of)i(\(1.9\))g(and)h (\(1.10\))f(preserv)n(es)e(unitary)h(equiv)-5 b(alence.)33 b(So,)-118 3674 y(in)c(studying)f(b)r(ounded)i(represen)n(tations,)d(w) n(e)i(can)g(restrict)f(ourselv)n(es)e(only)-118 3774 y(to)h(the)h(homogeneous)d(relations)g(\(1.10\).)-118 3911 y FQ(3.)36 b FO(T)-7 b(o)27 b(the)h(semilinear)23 b(relation)i(\(1.10\))o(,)j(w)n(e)f(asso)r(ciate:)p eop %%Page: 47 51 47 50 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(47)-26 96 y(a\))41 b(the)28 b(c)n(haracteristic)c(function:)524 338 y(\010\()p FN(t;)14 b(s)p FO(\))23 b(=)904 234 y FL(n)865 259 y Fy(X)871 436 y FL(i)p FK(=1)999 338 y FN(f)1040 350 y FL(i)1067 338 y FO(\()p FN(t)p FO(\))14 b FN(g)1215 350 y FL(i)1243 338 y FO(\()p FN(s)p FO(\))p FN(;)180 b FO(\()p FN(t;)14 b(s)23 b FP(2)h FI(R)p FO(\);)-30 622 y(b\))41 b(the)28 b(c)n(haracteristic)c(binary)i(relation:)665 797 y(\000)d(=)f FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FO(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0)p FP(g)g(\032)h FI(R)1697 763 y FK(2)1740 797 y FO(;)-21 1004 y(c\))41 b(an)g(orien)n(ted)f (graph)g(\()p FI(R)p FN(;)14 b FO(\000\),)51 b(where)41 b(the)h(edge)1738 986 y Fn(r)p 1738 988 117 4 v 8 w Ft(-)8 b Fn(r)1725 1069 y FL(t)97 b(s)1904 1004 y FO(,)45 b FN(t)p FO(,)g FN(s)g FP(2)i FI(R)p FO(,)89 1141 y(b)r(elongs)26 b(to)i(the)g(graph)e(if)h(and)h(only)e(if)h(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0.)6 1301 y(If)32 b(no)g(confusion)e (arises,)g(w)n(e)h(will)e(simply)f(write)i(\000)i(for)f(the)h(graph)e (\()p FN(D)r FO(,)-118 1400 y(\000\))40 b(and)g(call)e(it)i(the)g (graph)g(of)g(relation)d(\(1.10\).)74 b(An)n(y)40 b(subset)g FN(M)53 b FP(\032)44 b FI(R)-118 1500 y FO(determines)28 b(a)j(subgraph)e(\000)g Fr(\026)857 1512 y FL(M)961 1500 y FO(suc)n(h)h(that)h(its)f(v)n(ertices)f(are)h(p)r(oin)n(ts)f(of)i FN(M)-118 1600 y FO(and)c(its)g(edges)g(are)g(edges)f(of)i(\000)g(whic) n(h)e(connect)h(p)r(oin)n(ts)g(of)h FN(M)9 b FO(.)6 1699 y(If)33 b(\010\()p FN(t;)14 b(s)p FO(\))30 b(=)g(0)i(is)e(equiv)-5 b(alen)n(t)30 b(to)i(\010\()p FN(s;)14 b(t)p FO(\))30 b(=)g(0,)j(the)f(graph)f(\000)h(together)-118 1799 y(with)d(the)i(edge) 452 1781 y Fn(r)p 452 1783 V 8 w Ft(-)9 b Fn(r)648 1799 y FO(also)28 b(con)n(tains)g(the)j(edge)1524 1781 y Fn(r)p 1532 1783 V -17 w Ft(\033)j Fn(r)1704 1799 y FO(.)44 b(In)31 b(this)e(case,)h(w)n(e)-118 1899 y(will)25 b(consider)g(the)j (graph)f(as)g(non-orien)n(ted.)6 1998 y(In)32 b(what)g(follo)n(ws)c (\000)654 2010 y FL(s)721 1998 y FO(denote)k(the)g(set)f FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))30 b FP(j)g FO(\010\()p FN(t;)14 b(s)p FO(\))30 b(=)f(0)p FN(;)e FO(\010\()p FN(s;)14 b(t)p FO(\))30 b(=)-118 2098 y(0)p FP(g)22 b(\032)h FI(R)130 2068 y FK(2)200 2098 y FO(or)k(the)h(corresp)r(onding)d (non-orien)n(ted)g(graph.)6 2197 y(Consider)h(some)g(examples)f(of)j (semilinear)22 b(relations.)6 2297 y(1\))k(Relations)d(ad)558 2309 y FL(A)612 2297 y FO(\()p FN(B)t FO(\))h(=)e([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(AB)d FP(\000)c FN(B)t(A)23 b FO(=)g(0)i(and)h(\(ad)1988 2309 y FL(A)2042 2297 y FO(\))2074 2267 y FL(n)2119 2297 y FO(\()p FN(B)t FO(\))e(=)-118 2397 y([)p FN(A;)14 b(:)g(:)g(:)g FO([)p FN(A;)g(B)t FO(])g FN(:)g(:)g(:)g FO(])35 b(=)g(0)f(ha)n(v)n(e)g(the)h(c)n (haracteristic)c(functions)j(of)h(the)g(form)-118 2496 y(\010\()p FN(t;)14 b(s)p FO(\))36 b(=)g FN(t)24 b FP(\000)f FN(s)35 b FO(and)h(\010\()p FN(t;)14 b(s)p FO(\))36 b(=)g(\()p FN(t)24 b FP(\000)f FN(s)p FO(\))1247 2466 y FL(n)1328 2496 y FO(resp)r(ectiv)n(ely)-7 b(,)34 b(and)i(de\014ne)f(the)-118 2596 y(same)f(graph,)i(all)d(connected)i(comp)r(onen)n(ts)e(of)j(whic)n (h)e(are)g(the)i(follo)n(wing:)-77 2712 y Fn(r)p -77 2713 4 4 v -81 2709 V -86 2704 V -90 2699 V -94 2695 V -97 2690 V -100 2686 V -102 2682 V -104 2678 V -106 2674 V -108 2670 V -109 2666 V -109 2663 V -110 2659 V -110 2656 V -109 2653 V -108 2650 V -107 2647 V -106 2644 V -104 2641 V -102 2639 V -102 2639 V -99 2636 V -97 2634 V -94 2632 V -92 2631 V -89 2629 V -87 2628 V -84 2627 V -82 2627 V -79 2626 V -77 2626 V -74 2626 V -72 2627 V -69 2627 V -67 2628 V -64 2629 V -62 2631 V -59 2632 V -57 2634 V -54 2636 V -52 2639 V -77 2713 V -72 2709 V -67 2704 V -63 2699 V -60 2695 V -56 2690 V -53 2686 V -51 2682 V -49 2678 V -47 2674 V -45 2670 V -44 2666 V -44 2663 V -43 2659 V -43 2656 V -44 2653 V -45 2650 V -46 2647 V -47 2644 V -49 2641 V -52 2639 V -96 2795 a FL(\025)-35 2730 y FO(,)28 b FN(\025)23 b FP(2)h FI(R)p FO(.)6 2867 y(2\))k(Characteristic)c(functions)j (corresp)r(onding)d(to)k(the)g(relations)430 3042 y(ad)517 3054 y FL(A;)p FM(\000)p FK(1)676 3042 y FO(\()p FN(B)t FO(\))c(=)e FP(f)p FN(A;)14 b(B)t FP(g)23 b FO(=)f FN(AB)h FO(+)18 b FN(B)t(A)24 b FO(=)e(0)-118 3217 y(and)30 b(\(ad)166 3229 y FL(A;)p FM(\000)p FK(1)325 3217 y FO(\))357 3187 y FL(n)402 3217 y FO(\()p FN(B)t FO(\))e(=)f FP(f)p FN(A;)14 b(:)g(:)g(:)f FP(f)p FN(A;)h(B)t FP(g)g FN(:)g(:)g(:)f FP(g)27 b FO(=)g(0)i(are)g(\010\()p FN(t;)14 b(s)p FO(\))28 b(=)f FN(t)20 b FO(+)g FN(s)30 b FO(and)-118 3316 y(\010\()p FN(t;)14 b(s)p FO(\))23 b(=)g(\()p FN(t)5 b FO(+)g FN(s)p FO(\))431 3286 y FL(n)476 3316 y FO(,)22 b(resp)r(ectiv)n(ely)-7 b(.)32 b(As)21 b(b)r(efore,)h(they)f(de\014ne)g(the)h(same)d(graph)-118 3416 y(with)27 b(connected)g(comp)r(onen)n(ts)g(of)g(the)h(form:)701 3607 y Fn(r)p 701 3609 V 696 3604 V 692 3599 V 688 3595 V 684 3590 V 681 3586 V 678 3581 V 675 3577 V 673 3573 V 671 3569 V 670 3565 V 669 3562 V 668 3558 V 668 3555 V 668 3551 V 668 3548 V 669 3545 V 670 3542 V 672 3539 V 674 3537 V 676 3534 V 676 3534 V 678 3532 V 681 3530 V 683 3528 V 686 3526 V 688 3525 V 691 3524 V 693 3523 V 696 3522 V 698 3522 V 701 3522 V 703 3522 V 706 3522 V 708 3523 V 711 3524 V 713 3525 V 716 3526 V 718 3528 V 721 3530 V 723 3532 V 726 3534 V 701 3609 V 706 3604 V 710 3599 V 714 3595 V 718 3590 V 721 3586 V 724 3581 V 727 3577 V 729 3573 V 731 3569 V 732 3565 V 733 3562 V 734 3558 V 734 3555 V 734 3551 V 734 3548 V 733 3545 V 732 3542 V 730 3539 V 728 3537 V 726 3534 V 684 3690 a FK(0)742 3625 y FN(;)904 3607 y Fn(r)p 904 3609 125 4 v 99 w(r)884 3690 y FL(\025)39 b FM(\000)p FL(\025)1070 3625 y FN(;)180 b FO(\()p FN(\025)24 b(>)e FO(0\))p FN(:)6 3811 y FO(3\))40 b(Let)g FN(AB)t(A)k FO(=)f FN(\013B)t FO(,)g FN(\013)h FP(2)f FI(R)p FO(.)79 b(Then)40 b(\010\()p FN(t;)14 b(s)p FO(\))44 b(=)f FN(ts)26 b FP(\000)g FN(\013)p FO(,)43 b(and)d(all)-118 3911 y(connected)27 b(comp)r(onen)n(ts)f(of)i (the)g(corresp)r(onding)d(graph)h(are)h(of)g(the)h(form:)p eop %%Page: 48 52 48 51 bop -118 -137 a FO(48)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(a\))j FN(\013)23 b FP(6)p FO(=)g(0)207 303 y Fn(r)p 207 305 125 4 v 99 w(r)127 386 y FL(\013\025)209 361 y Fw(\000)p Fx(1)312 386 y FL(\025)373 321 y FN(;)180 b FO(\()p FN(\025)656 287 y FK(2)717 321 y FP(6)p FO(=)22 b FN(\013;)14 b FO(0\))1176 303 y Fn(r)1160 386 y FK(0)1218 321 y FN(;)1462 303 y Fn(r)p 1462 305 4 4 v 1457 300 V 1453 295 V 1449 290 V 1445 286 V 1442 281 V 1439 277 V 1436 273 V 1434 269 V 1432 265 V 1431 261 V 1430 257 V 1429 254 V 1429 250 V 1429 247 V 1429 244 V 1430 241 V 1431 238 V 1433 235 V 1435 232 V 1437 230 V 1437 230 V 1440 228 V 1442 225 V 1445 224 V 1447 222 V 1450 221 V 1452 219 V 1455 219 V 1457 218 V 1460 218 V 1462 217 V 1465 218 V 1467 218 V 1470 219 V 1472 219 V 1475 221 V 1477 222 V 1480 224 V 1482 225 V 1485 228 V 1487 230 V 1462 305 V 1467 300 V 1471 295 V 1475 290 V 1479 286 V 1482 281 V 1485 277 V 1488 273 V 1490 269 V 1492 265 V 1493 261 V 1494 257 V 1495 254 V 1495 250 V 1495 247 V 1495 244 V 1494 241 V 1493 238 V 1491 235 V 1489 232 V 1487 230 V 1387 386 a FM(\006)1439 344 y(p)p 1494 344 44 3 v 42 x FL(\013)1559 321 y FN(;)180 b FO(\()p FN(\013)24 b(>)e FO(0\);)6 519 y(b\))31 b(if)e FN(\013)d FO(=)g(0,)k(then)g(an)n (y)f(v)n(ertex)g FN(\025)e FP(2)f FI(R)g FP(n)19 b(f)p FO(0)p FP(g)29 b FO(is)f(connected)i(with)f(0)g(b)n(y)-118 619 y(the)f(edge)255 601 y Fn(r)p 255 603 125 4 v 100 w(r)236 684 y FL(\025)88 b FK(0)422 619 y FO(.)6 757 y(4\))30 b(An)n(y)f(connected)h(comp)r(onen)n(t)e(of)i(the)g(graph)e (corresp)r(onding)f(to)i(the)-118 857 y(relation)77 1043 y FN(A)139 1009 y FK(2)176 1043 y FN(B)23 b FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)559 1009 y FM(\000)p FK(1)648 1043 y FO(\))p FN(AB)t(A)h FO(+)f FN(B)t(A)1102 1009 y FK(2)1163 1043 y FO(=)23 b(0)p FN(;)179 b(q)26 b FP(2)d FI(R)i FP(n)18 b(f\000)p FO(1)p FN(;)c FO(0)p FN(;)g FO(1)p FP(g)p FN(;)-118 1229 y FO(is)26 b(either)h(an)g(in\014nite)f(c) n(hain:)865 1396 y Fn(r)p 865 1398 V 99 w(r)p 989 1398 V 100 w(r)797 1479 y FL(q)829 1454 y Fw(\000)p Fx(1)907 1479 y FL(t)45 b(t)83 b(q)r(t)726 1400 y FN(:)14 b(:)g(:)327 b(:)14 b(:)g(:)1280 1414 y(;)180 b(t)23 b FP(6)p FO(=)f(0)p FN(;)-118 1658 y FO(or)25 1640 y Fn(r)p 25 1642 4 4 v 20 1637 V 16 1632 V 12 1628 V 8 1623 V 5 1619 V 2 1614 V -1 1610 V -3 1606 V -5 1602 V -6 1598 V -7 1595 V -8 1591 V -8 1588 V -8 1584 V -8 1581 V -7 1578 V -6 1575 V -4 1572 V -2 1570 V 0 1567 V 0 1567 V 3 1565 V 5 1563 V 8 1561 V 10 1559 V 13 1558 V 15 1557 V 18 1556 V 20 1555 V 23 1555 V 25 1555 V 28 1555 V 30 1555 V 33 1556 V 35 1557 V 38 1558 V 40 1559 V 43 1561 V 45 1563 V 48 1565 V 50 1567 V 25 1642 V 30 1637 V 34 1632 V 38 1628 V 42 1623 V 45 1619 V 48 1614 V 51 1610 V 53 1606 V 55 1602 V 56 1598 V 57 1595 V 58 1591 V 58 1588 V 58 1584 V 58 1581 V 57 1578 V 56 1575 V 54 1572 V 52 1570 V 50 1567 V 9 1723 a FK(0)67 1658 y FN(:)-118 1915 y FQ(1.3.3)94 b(Kleinec)m(k)m(e{Shirok)m(o)m(v)32 b(t)m(yp)s(e)g(theorems)-118 2071 y FO(W)-7 b(e)41 b(b)r(egin)f(with)g(the)i(study)f(of)g(the)g (structure)f(of)h(op)r(erators)e FN(A)45 b FO(=)g FN(A)2277 2041 y FM(\003)2316 2071 y FO(,)-118 2171 y FN(B)f FO(satisfying)39 b(\(1.10\))o(;)46 b(in)40 b(particular,)f(w)n(e)h(will)d(lo)r(ok)i(at)h (the)g(connection)-118 2270 y(b)r(et)n(w)n(een)27 b(the)g(sp)r(ectral)f (prop)r(erties)f(of)i(the)h(op)r(erator)d FN(A)j FO(and)f(the)g (structure)-118 2370 y(of)j(the)g(op)r(erator)f FN(B)t FO(.)44 b(It)30 b(turns)g(out)g(that)g(for)g(a)g(broad)e(class)h(of)h (semilinear)-118 2470 y(relations)36 b(the)j(c)n(haracteristic)c (binary)j(relation)e(corresp)r(onding)g(to)j(them)-118 2569 y(completely)26 b(de\014nes)k(the)g(solutions)d(of)37 b(\(1.10\))o(.)43 b(This)29 b(fact)h(pro)n(vides)d(man)n(y)-118 2669 y(Kleinec)n(k)n(e{Shirok)n(o)m(v)22 b(t)n(yp)r(e)27 b(theorems.)-118 2823 y FQ(1.)42 b FO(W)-7 b(e)30 b(start)f(b)n(y)g (studying)g(\014nite-dimensional)24 b(represen)n(tations)i(of)k(semi-) -118 2923 y(linear)j(relations.)58 b(Note)36 b(that)g(the)g(same)e (argumen)n(ts)f(w)n(ork)h(in)h(the)h(more)-118 3022 y(general)30 b(case)i(where)h(the)g(op)r(erator)e FN(A)i FO(in)f(the)h(represen)n (tation)d(has)j(a)f(dis-)-118 3122 y(crete)23 b(sp)r(ectrum.)35 b(So)24 b(let)f FN(\033)s FO(\()p FN(A)p FO(\))h(=)f FP(f)p FN(\025)1081 3134 y FK(1)1118 3122 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)1364 3134 y FL(m)1428 3122 y FP(g)23 b FO(and)h FN(H)1720 3134 y FL(\025)1759 3142 y Fv(j)1818 3122 y FO(b)r(e)g(eigenspaces)-118 3221 y(of)30 b FN(A)h FO(corresp)r(onding)d(to)i FN(\025)761 3233 y FL(j)797 3221 y FO(.)45 b(With)31 b(resp)r(ect)f(to)g(the)h(decomp)r(osition)c FN(H)35 b FO(=)-118 3321 y FN(H)-49 3333 y FL(\025)-10 3341 y Fx(1)46 3321 y FP(\010)19 b FN(:)14 b(:)g(:)19 b FP(\010)g FN(H)399 3333 y FL(\025)438 3341 y Fv(m)498 3321 y FO(,)29 b(the)h(op)r(erator)d FN(B)33 b FO(can)c(b)r(e)g (written)f(in)g(the)i(form)d(of)i(the)-118 3421 y(blo)r(c)n(k)d (matrix:)34 b FN(B)28 b FO(=)22 b(\()p FN(B)676 3433 y FL(ln)743 3421 y FO(\))775 3391 y FL(m)775 3444 y(l;n)p FK(=1)946 3421 y FN(:)-118 3591 y FQ(Prop)s(osition)30 b(19.)41 b FB(F)-6 b(or)33 b(op)l(er)l(ators)g FN(A)p FB(,)h FN(B)j FB(to)c(de\014ne)g(a)g(r)l(epr)l(esentation)g(of)-118 3691 y(the)24 b(r)l(elation)30 b FO(\(1.10\))p FB(,)25 b(it)e(is)h(ne)l(c)l(essary)g(and)g(su\016cient)g(that)f(the)h(blo)l (ck)h(matrix)-118 3791 y FN(B)49 b FO(=)c(\()p FN(B)199 3803 y FL(sn)276 3791 y FO(\))308 3761 y FL(m)308 3811 y(s;n)p FK(=1)531 3791 y FB(b)l(e)d(supp)l(orte)l(d)g(by)h FO(\000)i Fr(\026)1280 3806 y FL(\033)r FK(\()p FL(A)p FK(\))1468 3791 y FO(\()p FB(i.e.,)j FN(B)1749 3803 y FL(sn)1871 3791 y FO(=)d(0)c FB(for)i(any)-118 3890 y FO(\()p FN(\025)-38 3902 y FL(s)-2 3890 y FN(;)14 b(\025)83 3902 y FL(n)129 3890 y FO(\))32 b FN(=)-51 b FP(2)23 b FO(\000\))p FB(.)p eop %%Page: 49 53 49 52 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(49)-118 96 y FB(Pr)l(o)l(of.)43 b FO(The)28 b(statemen)n(t)e(follo)n(ws)f(immediately)e(from)j(the)i (equalit)n(y)22 184 y Fy(\020)72 197 y(X)78 374 y FL(i)p FK(=1)206 276 y FN(f)247 288 y FL(i)274 276 y FO(\()p FN(A)p FO(\))14 b FN(B)k(g)535 288 y FL(i)563 276 y FO(\()p FN(A)p FO(\))689 184 y Fy(\021)739 334 y FL(k)q(j)834 276 y FO(=)23 b(\010\()p FN(\025)1062 288 y FL(k)1103 276 y FN(;)14 b(\025)1188 288 y FL(j)1223 276 y FO(\))g FN(B)1332 288 y FL(k)q(j)1404 276 y FN(;)180 b(k)s(;)14 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(m:)p 2278 276 4 57 v 2282 223 50 4 v 2282 276 V 2331 276 4 57 v -118 514 a FQ(2.)43 b FO(A)30 b(pair)e FN(A)f FO(=)f FN(A)507 484 y FM(\003)545 514 y FO(,)31 b FN(B)g FO(=)26 b FN(B)851 484 y FM(\003)889 514 y FO(,)31 b(is)d(a)i(represen) n(tation)d(of)i(relation)e(\(1.10\))i(if)-118 614 y(and)e(only)g(if)g (the)h(blo)r(c)n(k)e(matrix)f FN(B)i FO(=)c(\()p FN(B)1207 626 y FL(ij)1265 614 y FO(\))1297 584 y FL(n)1297 636 y(i;j)s FK(=1)1488 614 y FO(is)j(supp)r(orted)h(b)n(y)113 773 y(\000)165 785 y FL(s)200 773 y FP(j)223 788 y FL(\033)r FK(\()p FL(A)p FK(\))393 773 y FO(=)c FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(2)g FN(\033)s FO(\()p FN(A)p FO(\))d FP(\002)e FN(\033)s FO(\()p FN(A)p FO(\))24 b FP(j)f FO(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)e(\010\()p FN(s;)14 b(t)p FO(\))24 b(=)e(0)p FP(g)p FN(:)-118 932 y FO(In)29 b(fact,)h(if)f FN(A)d FO(=)g FN(A)495 902 y FM(\003)533 932 y FO(,)k FN(B)g FO(=)25 b FN(B)836 902 y FM(\003)904 932 y FO(is)j(a)h(represen)n(tation)d(of)k(\(1.10\),) f(then)g FN(A)p FO(,)h FN(B)-118 1032 y FO(also)25 b(satisfy)i(the)h (relation)753 1149 y FL(n)714 1174 y Fy(X)720 1351 y FL(i)p FK(=1)850 1253 y FO(\026)-45 b FN(g)887 1265 y FL(i)915 1253 y FO(\()p FN(A)p FO(\))14 b FN(B)1154 1231 y FO(\026)1136 1253 y FN(f)1177 1265 y FL(i)1205 1253 y FO(\()p FN(A)p FO(\))24 b(=)e(0)p FN(:)-118 1489 y FO(Hence)h(the)g(blo)r(c)n(k)f(matrix)e FN(B)27 b FO(=)c(\()p FN(B)1015 1501 y FL(ij)1073 1489 y FO(\))1105 1459 y FL(m)1105 1511 y(i;j)s FK(=1)1291 1489 y FO(is)f(supp)r(orted)g(b)n(y)h (\(\000)9 b FP(\\)g FO(\000)2075 1459 y FM(\003)2114 1489 y FO(\))p FP(j)2169 1504 y FL(\033)r FK(\()p FL(A)p FK(\))2316 1489 y FO(,)-118 1597 y(where)30 b(\000)177 1567 y FM(\003)242 1597 y FO(=)e FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))27 b FP(2)h FI(R)f FP(\002)19 b FI(R)34 b FP(j)28 b FO(\010\()p FN(s;)14 b(t)p FO(\))28 b(=)1311 1535 y Fy(P)1399 1555 y FL(n)1399 1622 y(i)p FK(=1)1524 1597 y FN(g)1564 1609 y FL(i)1591 1597 y FO(\()p FN(t)p FO(\))14 b FN(f)1740 1609 y FL(i)1768 1597 y FO(\()p FN(s)p FO(\))28 b(=)g(0)p FP(g)p FO(,)i(whic)n(h)-118 1697 y(giv)n(es)25 b(the)j(required)e(statemen)n(t.)-118 1837 y FQ(3.)45 b FO(No)n(w)30 b(w)n(e)g(will)e(try)i(to)h(remo)n(v)n (e)d(the)j(condition)d(in)i(Prop)r(osition)d(19)i(that)-118 1936 y FN(\033)s FO(\()p FN(A)p FO(\))j(is)e(discrete.)46 b(F)-7 b(or)31 b(this)f(purp)r(ose,)i(w)n(e)f(will)d(consider)h(a)i (more)e(general)-118 2036 y(situation.)6 2136 y(Let)39 b FN(M)9 b FO(,)40 b FN(N)47 b FO(b)r(e)38 b(normal)d(op)r(erators)h (acting)h(on)g(Hilb)r(ert)g(spaces)g FN(H)2242 2148 y FL(M)2316 2136 y FO(,)-118 2235 y FN(H)-49 2247 y FL(N)14 2235 y FO(,)28 b(resp)r(ectiv)n(ely)-7 b(,)24 b(and)k(let)f FN(E)876 2247 y FL(M)950 2235 y FO(\()p FP(\001)p FO(\),)h FN(E)1149 2247 y FL(N)1212 2235 y FO(\()p FP(\001)p FO(\))h(b)r(e)f (their)e(sp)r(ectral)g(measures.)-118 2382 y FQ(De\014nition)31 b(3.)40 b FB(We)28 b(say)g(that)g(a)g(subset)e FN(F)35 b FP(\032)23 b FI(C)29 b FP(\002)14 b FI(C)53 b FO(\()p FN(M)t(;)14 b(N)9 b FO(\))p FB(-supp)l(orts)27 b(an)-118 2482 y(op)l(er)l(ator)k FN(B)13 b FO(:)28 b FN(H)405 2494 y FL(N)491 2482 y FP(!)23 b FN(H)666 2494 y FL(M)769 2482 y FB(if)740 2641 y FN(E)801 2653 y FL(M)875 2641 y FO(\()p FN(\013)p FO(\))14 b FN(B)19 b(E)1149 2653 y FL(N)1212 2641 y FO(\()p FN(\014)t FO(\))24 b(=)f(0)-118 2800 y FB(for)30 b(any)h(p)l(air)f FO(\()p FN(\013;)14 b(\014)t FO(\))31 b FB(of)g(Bor)l(el)g(sets)e(such)h(that)g FO(\()p FN(\013)19 b FP(\002)f FN(\014)t FO(\))h FP(\\)g FN(F)35 b FO(=)22 b FI(?)p FB(.)6 2948 y FO(It)30 b(is)f(not)h (di\016cult)e(to)i(pro)n(v)n(e)e(that)h(there)h(exists)e(a)h(smallest)e (closed)h(set)-118 3047 y FN(F)38 b FO(supp)r(orting)25 b FN(B)31 b FO(\(tak)n(e)26 b(the)h(complemen)n(t)d(to)i(the)h(union)f (of)g(all)e(suc)n(h)j(op)r(en)-118 3147 y(sets)37 b FN(\013)26 b FP(\002)f FN(\014)t FO(\).)68 b(W)-7 b(e)38 b(will)d(denote)j(this)f (set)h(b)n(y)f(supp)1603 3167 y FL(M)s(;N)1752 3147 y FO(\()p FN(B)t FO(\).)68 b(It)38 b(is)f(clear)-118 3246 y(that)f(supp)241 3267 y FL(M)s(;N)389 3246 y FO(\()p FN(B)t FO(\))i FP(\032)d FN(\033)s FO(\()p FN(M)9 b FO(\))25 b FP(\002)e FN(\033)s FO(\()p FN(N)9 b FO(\);)40 b(in)35 b(particular,)g(it)g(is)f(a)h(subset)h(of)-118 3346 y FI(R)7 b FP(\002)g FI(R)33 b FO(when)22 b FN(M)31 b FO(and)22 b FN(N)30 b FO(are)21 b(self-adjoin)n(t.)33 b(W)-7 b(e)22 b(shall)e(also)f(write)i(supp)2133 3366 y FL(M)2207 3346 y FO(\()p FN(B)t FO(\))-118 3446 y(instead)26 b(of)i(supp)434 3466 y FL(M)s(;M)593 3446 y FO(\()p FN(B)t FO(\).)-118 3593 y FQ(Theorem)i(6.)41 b FB(If)30 b FN(A)p FB(,)h FN(B)j FB(is)c(a)g(r)l(epr)l(esentation)g(of)g(r)l(elation)37 b FO(\(1.10\))o FB(,)31 b(then)839 3752 y FO(supp)1010 3772 y FL(A)1064 3752 y FO(\()p FN(B)t FO(\))24 b FP(\032)f FO(\000)p FN(;)-118 3911 y FB(wher)l(e)30 b FO(\000)g FB(is)g(the)g(binary)h(r)l(elation)f(c)l(orr)l(esp)l(onding)h(to)k FO(\(1.10\))o FB(.)p eop %%Page: 50 54 50 53 bop -118 -137 a FO(50)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Let)34 b FN(\013)22 b FP(\002)g FN(\014)38 b FO(not)c(in)n(tersect)e(\000.)54 b(Denote)34 b(b)n(y)f FN(\026)h FO(the)f(scalar)e(sp)r(ectral)-118 196 y(measure)i(of)i(the)g(op)r(erator)e FN(A)p FO(.)59 b(By)35 b(Luzin's)f(theorem,)h(for)g(ev)n(ery)e FN(")i(>)g FO(0,)-118 296 y(there)d(exist)e(compact)h(sets)h FN(\013)856 266 y FM(0)910 296 y FP(\032)e FN(\013)i FO(and)g FN(\014)1307 266 y FM(0)1361 296 y FP(\032)e FN(\014)36 b FO(suc)n(h)c(that)g FN(\026)p FO(\()p FN(\013)22 b FP(n)f FN(\013)2188 266 y FM(0)2211 296 y FO(\))31 b FN(<)-118 395 y(")p FO(,)41 b FN(\026)p FO(\()p FN(\014)31 b FP(n)25 b FN(\014)263 365 y FM(0)287 395 y FO(\))41 b FN(<)g(")p FO(,)g(and)e(the)g (functions)f FN(f)1315 365 y FM(0)1306 417 y FL(i)1379 395 y FO(=)j FN(f)1526 407 y FL(i)1595 395 y Fr(\026)1630 407 y FL(\013)1673 391 y Fw(0)1699 395 y FO(,)h FN(g)1807 365 y FM(0)1804 417 y FL(i)1872 395 y FO(=)f FN(g)2018 407 y FL(i)2087 395 y Fr(\026)2122 407 y FL(\014)2163 391 y Fw(0)2228 395 y FO(are)-118 495 y(con)n(tin)n(uous.)33 b(Put)24 b FN(P)35 b FO(=)23 b FN(E)724 507 y FL(A)778 495 y FO(\()p FN(\013)863 465 y FM(0)887 495 y FO(\),)i FN(Q)e FO(=)f FN(E)1204 507 y FL(A)1259 495 y FO(\()p FN(\014)1342 465 y FM(0)1366 495 y FO(\),)j FN(B)1513 465 y FM(0)1559 495 y FO(=)e FN(P)12 b(B)t(Q)d FO(:)27 b FN(QH)j FP(!)23 b FN(P)12 b(H)7 b FO(.)-118 595 y(The)28 b(op)r(erator)d(of)j(m)n(ultiplication,)617 838 y(\001)9 b(:)28 b FN(X)h FP(7!)990 734 y FL(n)950 759 y Fy(X)956 936 y FL(i)p FK(=1)1084 838 y FN(f)1134 804 y FM(0)1125 858 y FL(i)1157 838 y FO(\()p FN(A)p FO(\))14 b FN(X)21 b(g)1430 804 y FM(0)1427 858 y FL(i)1454 838 y FO(\()p FN(A)p FO(\))p FN(;)-118 1097 y FO(acts)38 b(on)h(the)g(space)f FN(L)p FO(\()p FN(QH)r(;)14 b(P)e(H)7 b FO(\).)70 b(The)39 b(sp)r(ectrum)f(of)h(the)g(op)r(erator)e(\001)-118 1196 y(b)r(elongs)26 b(to)h(the)h(set)258 1348 y Fy(n)352 1336 y FL(n)313 1361 y Fy(X)319 1538 y FL(i)p FK(=1)447 1440 y FN(f)488 1452 y FL(i)515 1440 y FO(\()p FN(\025)p FO(\))14 b FN(g)681 1452 y FL(i)709 1440 y FO(\()p FN(\026)p FO(\))24 b FP(j)f FN(\025)g FP(2)h FN(\013)1096 1405 y FM(0)1120 1440 y FN(;)j(\026)c FP(2)h FN(\014)1373 1405 y FM(0)1396 1348 y Fy(o)1475 1440 y FO(=)e(\010\()p FN(\013)1707 1405 y FM(0)1750 1440 y FP(\002)c FN(\014)1884 1405 y FM(0)1907 1440 y FO(\))p FN(;)-118 1698 y FO(see,)29 b(for)f(example)f([226)n(].)41 b(By)29 b(the)g(condition)e(of)h(the)i (theorem,)e(\010\()p FN(\013)2105 1668 y FM(0)2148 1698 y FP(\002)19 b FN(\014)2283 1668 y FM(0)2306 1698 y FO(\))-118 1798 y(do)r(es)40 b(not)g(con)n(tain)e(zero,)k(hence)f(the)f(op)r (erator)e(\001)j(is)e(in)n(v)n(ertible.)70 b(Since)-118 1898 y(\001\()p FN(B)50 1868 y FM(0)74 1898 y FO(\))25 b(=)f(0,)k(w)n(e)g(ha)n(v)n(e)f(that)i FN(B)876 1868 y FM(0)924 1898 y FO(=)24 b(0,)29 b(i.e.,)f FN(E)1326 1910 y FL(A)1380 1898 y FO(\()p FN(\013)1465 1868 y FM(0)1489 1898 y FO(\))14 b FN(B)t(E)1663 1910 y FL(A)1718 1898 y FO(\()p FN(\014)1801 1868 y FM(0)1824 1898 y FO(\))25 b(=)f(0.)40 b(Letting)-118 1997 y FN(")23 b FP(!)g FO(0,)k(w)n(e)g (obtain)g(that)g FN(E)763 2009 y FL(A)818 1997 y FO(\()p FN(\013)p FO(\))14 b FN(B)t(E)1077 2009 y FL(A)1132 1997 y FO(\()p FN(\014)t FO(\))24 b(=)f(0.)p 2278 1997 4 57 v 2282 1945 50 4 v 2282 1997 V 2331 1997 4 57 v 6 2163 a(In)28 b(a)f(similar)c(w)n(a)n(y)-7 b(,)27 b(more)e(general)h(results) g(can)h(b)r(e)h(pro)n(v)n(ed.)-118 2328 y FQ(Theorem)i(7.)41 b FB(If)30 b FN(M)9 b FB(,)30 b FN(N)38 b FB(ar)l(e)31 b(normal)f(op)l(er)l(ators)h(and)726 2468 y FL(n)686 2493 y Fy(X)693 2670 y FL(i)p FK(=1)820 2572 y FN(f)861 2584 y FL(i)888 2572 y FO(\()p FN(M)9 b FO(\))14 b FN(B)k(g)1177 2584 y FL(i)1205 2572 y FO(\()p FN(N)9 b FO(\))23 b(=)g(0)p FN(;)605 b FO(\(1.11\))-118 2826 y FB(then)792 2925 y FO(supp)963 2946 y FL(M)s(;N)1111 2925 y FO(\()p FN(B)t FO(\))24 b FP(\032)f FO(\000)p FN(;)-118 3085 y FB(wher)l(e)30 b FO(\000)23 b(=)279 3018 y Fy(\010)327 3085 y FO(\()p FN(t;)14 b(s)p FO(\))24 b FP(2)f FI(C)40 b FP(\002)18 b FI(C)44 b FP(j)23 b FO(\010\()p FN(t;)14 b(s)p FO(\))23 b(=)1231 3023 y Fy(P)1318 3043 y FL(n)1318 3110 y(i)p FK(=1)1444 3085 y FN(f)1485 3097 y FL(i)1512 3085 y FO(\()p FN(t)p FO(\))14 b FN(g)1660 3097 y FL(i)1688 3085 y FO(\()p FN(s)p FO(\))23 b(=)g(0)1944 3018 y Fy(\011)1992 3085 y FB(.)-118 3299 y FQ(4.)35 b FO(F)-7 b(or)23 b(an)n(y)g FN(F)35 b FP(\032)23 b FI(C)j FP(\002)11 b FI(C)50 b FO(let)23 b(us)h(denote)g(b)n(y)g Fz(M)p FO(\()p FN(F)12 b FO(\))24 b(the)g(set)g(of)g(all)e(op)r(erators)-118 3399 y(supp)r(orted)27 b(b)n(y)g FN(F)12 b FO(.)37 b(If)27 b(necessary)-7 b(,)26 b(w)n(e)h(will)d(write)j(more)e(explicitly:)33 b Fz(M)2155 3411 y FL(A)2209 3399 y FO(\()p FN(F)12 b FO(\))-118 3499 y(or)32 b Fz(M)76 3511 y FL(M)s(;N)225 3499 y FO(\()p FN(F)12 b FO(\).)53 b(Similarly)-7 b(,)28 b(w)n(e)33 b(will)d(denote)j(b)n(y)f(\001)h(\(instead)f(of)h(\001)2083 3511 y FL(M)s(;N)2265 3499 y FO(or)-118 3598 y(\001)-49 3610 y FL(A)5 3598 y FO(\))28 b(the)g(m)n(ultiplication)22 b(op)r(erator)675 3846 y FN(X)29 b FP(7!)919 3743 y FL(n)879 3767 y Fy(X)885 3944 y FL(i)p FK(=1)1013 3846 y FN(f)1054 3858 y FL(i)1081 3846 y FO(\()p FN(M)9 b FO(\))14 b FN(X)20 b(g)1378 3858 y FL(i)1406 3846 y FO(\()p FN(N)9 b FO(\))p eop %%Page: 51 55 51 54 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(51)-118 96 y(on)27 b FN(L)p FO(\()p FN(H)155 108 y FL(N)218 96 y FN(;)14 b(H)324 108 y FL(M)398 96 y FO(\).)6 196 y(As)29 b(an)f(analogue)d(of)j(Prop)r (osition)d(19,)j(it)f(w)n(ould)g(b)r(e)i(natural)d(to)i(exp)r(ect)-118 296 y(that)g(the)g(equalit)n(y)856 473 y Fz(M)p FO(\(\000\))c(=)e(k)n (er)13 b(\001)762 b(\(1.12\))-118 651 y(holds.)41 b(Theorem)28 b(6)h(sho)n(ws)g(that)g Fz(M)p FO(\(\000\))e FP(\033)f FO(k)n(er)13 b(\001.)43 b(W)-7 b(e)30 b(shall)d(see)i(that)h(the)-118 751 y(in)n(v)n(erse)e(inclusion)g(is)i(true)h(under)f(some)g (restrictions)d(on)k(the)g(smo)r(othness)-118 850 y(of)c(the)h (functions)f FN(f)518 862 y FL(i)546 850 y FO(,)g FN(g)636 862 y FL(i)664 850 y FO(,)g(and)h(it)f(is)f(not)i(true)f(in)g(the)h (general)d(case.)6 950 y(Let)42 b(us)f(establish)e(some)h(additional)e (results.)77 b(Denote)42 b(b)n(y)f(pr)2117 970 y FK(1)2155 950 y FO(,)k(pr)2301 970 y FK(2)-118 1050 y FO(the)40 b(pro)5 b(jections)37 b(on)n(to)i(the)g(comp)r(onen)n(ts)f(in)h FN(\033)s FO(\()p FN(M)9 b FO(\))27 b FP(\002)f FN(\033)s FO(\()p FN(N)9 b FO(\).)73 b(Let)40 b FN(S)47 b FO(=)-118 1149 y(\()p FN(\033)s FO(\()p FN(M)9 b FO(\))19 b FP(\002)f FN(\033)s FO(\()p FN(N)9 b FO(\)\))20 b FP(\\)f FO(\000.)-118 1311 y FQ(Lemma)29 b(5.)41 b FB(L)l(et)c FN(g)527 1323 y FL(i)592 1311 y FP(2)g FO(Lip)13 b FN(\033)s FO(\()p FN(N)c FO(\))p FB(,)40 b FN(k)g FO(=)d(1)p FB(,)g FN(:)14 b(:)g(:)28 b FB(,)40 b FN(n)p FB(.)61 b(If)38 b FO(pr)1863 1332 y FK(1)1914 1311 y FN(S)k FO(=)37 b FN(\033)s FO(\()p FN(M)9 b FO(\))p FB(,)-118 1411 y(then)477 1556 y FP(k)p FO(\001)p FP(k)22 b(\024)h FO(2)835 1453 y FL(n)795 1477 y Fy(X)801 1654 y FL(i)p FK(=1)929 1556 y FP(k)p FN(f)1012 1568 y FL(i)1039 1556 y FP(k)14 b(k)p FN(g)1177 1568 y FL(i)1203 1556 y FP(k)1245 1568 y FK(Lip)1359 1556 y FO(diam)e FN(\033)s FO(\()p FN(N)d FO(\))-118 1780 y FB(wher)l(e)30 b FP(k)p FN(f)199 1792 y FL(i)226 1780 y FP(k)23 b FO(=)f(sup)p FP(f)p FN(f)586 1792 y FL(i)613 1780 y FO(\()p FN(t)p FO(\))i FP(j)f FN(t)g FP(2)g FN(\033)s FO(\()p FN(M)9 b FO(\))p FP(g)p FB(.)-118 1942 y(Pr)l(o)l(of.)43 b FO(Set)37 b FN(C)44 b FO(=)38 b(2)556 1880 y Fy(P)643 1900 y FL(n)643 1967 y(i)p FK(=1)769 1942 y FP(k)p FN(f)852 1954 y FL(i)879 1942 y FP(k)14 b(k)p FN(g)1017 1954 y FL(i)1043 1942 y FP(k)1085 1954 y FK(Lip)1199 1942 y FO(diam)d FN(\033)s FO(\()p FN(N)e FO(\))38 b(and)e(\014x)h FN(s)1956 1954 y FK(0)2031 1942 y FP(2)i FN(\033)s FO(\()p FN(N)9 b FO(\).)-118 2042 y(Then)25 2265 y(\001\()p FN(B)t FO(\))24 b(=)376 2161 y FL(n)336 2186 y Fy(X)343 2363 y FL(i)p FK(=1)470 2265 y FN(f)511 2277 y FL(i)539 2265 y FO(\()p FN(M)9 b FO(\))14 b FN(B)j(g)827 2277 y FL(i)855 2265 y FO(\()p FN(s)926 2277 y FK(0)963 2265 y FO(\))i(+)1136 2161 y FL(n)1097 2186 y Fy(X)1103 2363 y FL(i)p FK(=1)1231 2265 y FN(f)1272 2277 y FL(i)1299 2265 y FO(\()p FN(M)9 b FO(\))14 b FN(B)k FO(\()p FN(g)1620 2277 y FL(i)1647 2265 y FO(\()p FN(N)9 b FO(\))19 b FP(\000)f FN(g)1929 2277 y FL(i)1957 2265 y FO(\()p FN(s)2028 2277 y FK(0)2065 2265 y FO(\))p FN(I)7 b FO(\))p FN(:)-118 2520 y FO(Since)26 b FN(\033)s FO(\()p FN(g)220 2532 y FL(i)248 2520 y FO(\()p FN(N)9 b FO(\))17 b FP(\000)g FN(g)527 2532 y FL(i)554 2520 y FO(\()p FN(s)625 2532 y FK(0)663 2520 y FO(\))p FN(I)7 b FO(\))23 b(=)g FP(f)p FN(g)963 2532 y FL(i)990 2520 y FO(\()p FN(t)p FO(\))17 b FP(\000)g FN(g)1223 2532 y FL(i)1250 2520 y FO(\()p FN(s)1321 2532 y FK(0)1359 2520 y FO(\))23 b FP(j)g FN(t)g FP(2)h FN(\033)s FO(\()p FN(N)9 b FO(\))p FP(g)p FO(,)27 b(w)n(e)g(ha)n(v)n(e)e(that)397 2698 y FP(k)p FN(g)479 2710 y FL(i)506 2698 y FO(\()p FN(N)9 b FO(\))19 b FP(\000)f FN(g)788 2710 y FL(i)815 2698 y FO(\()p FN(s)886 2710 y FK(0)923 2698 y FO(\))p FN(I)7 b FP(k)23 b(\024)g(k)p FN(g)1233 2710 y FL(i)1260 2698 y FP(k)1302 2710 y FK(Lip)1416 2698 y FO(diam)11 b FN(\033)s FO(\()p FN(N)e FO(\))p FN(:)-118 2876 y FO(Th)n(us,)314 2942 y Fy(\015)314 2991 y(\015)314 3041 y(\015)400 2933 y FL(n)360 2958 y Fy(X)366 3135 y FL(i)p FK(=1)494 3037 y FN(f)535 3049 y FL(i)562 3037 y FO(\()p FN(M)g FO(\))14 b FN(B)k FO(\()p FN(g)883 3049 y FL(i)911 3037 y FO(\()p FN(N)9 b FO(\))19 b FP(\000)f FN(g)1193 3049 y FL(i)1220 3037 y FO(\()p FN(s)1291 3049 y FK(0)1328 3037 y FO(\))p FN(I)7 b FO(\))1435 2942 y Fy(\015)1435 2991 y(\015)1435 3041 y(\015)1505 3037 y FP(\024)1603 2981 y FO(1)p 1603 3018 42 4 v 1603 3094 a(2)1668 3037 y FN(C)f FP(k)p FN(B)t FP(k)p FN(:)-118 3269 y FO(F)-7 b(urthermore,)24 b FN(\033)435 3202 y Fy(\000)473 3207 y(P)561 3227 y FL(n)561 3294 y(i)p FK(=1)687 3269 y FN(g)727 3281 y FL(i)754 3269 y FO(\()p FN(s)825 3281 y FK(0)862 3269 y FO(\))14 b FN(f)949 3281 y FL(i)977 3269 y FO(\()p FN(M)9 b FO(\))1131 3202 y Fy(\001)1192 3269 y FO(=)23 b FP(f)p FO(\010\()p FN(t;)14 b(s)1520 3281 y FK(0)1556 3269 y FO(\))24 b FP(j)f FN(t)g FP(2)g FN(\033)s FO(\()p FN(M)9 b FO(\))p FP(g)p FO(.)37 b(By)25 b(the)-118 3369 y(condition)k(of)h(the)i (theorem,)e(for)g(an)n(y)g FN(t)f FP(2)g FN(\033)s FO(\()p FN(M)9 b FO(\))31 b(there)g(exists)f FN(s)e FO(=)g FN(s)p FO(\()p FN(t)p FO(\))h FP(2)-118 3468 y FN(\033)s FO(\()p FN(N)9 b FO(\))28 b(suc)n(h)g(that)f(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0.)37 b(Therefore,)528 3646 y FP(j)p FO(\010\()p FN(t;)14 b(s)749 3658 y FK(0)786 3646 y FO(\))p FP(j)24 b FO(=)e FP(j)p FO(\010\()p FN(t;)14 b(s)1173 3658 y FK(0)1211 3646 y FO(\))k FP(\000)h FO(\010\()p FN(t;)14 b(s)p FO(\()p FN(t)p FO(\)\))p FP(j)475 3751 y Fy(\014)475 3801 y(\014)475 3850 y(\014)542 3743 y FL(n)503 3767 y Fy(X)509 3944 y FL(i)p FK(=1)623 3846 y FO(\()p FN(g)695 3858 y FL(i)722 3846 y FO(\()p FN(s)793 3858 y FK(0)831 3846 y FO(\))19 b FP(\000)f FN(g)1005 3858 y FL(i)1032 3846 y FO(\()p FN(s)p FO(\()p FN(t)p FO(\)\)\))c FN(f)1316 3858 y FL(i)1344 3846 y FO(\()p FN(s)p FO(\))1447 3751 y Fy(\014)1447 3801 y(\014)1447 3850 y(\014)1499 3846 y FP(\024)1596 3790 y FO(1)p 1596 3827 V 1596 3903 a(2)1661 3846 y FN(C)q(:)p eop %%Page: 52 56 52 55 bop -118 -137 a FO(52)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(Hence,)152 26 y Fy(\015)152 76 y(\015)198 34 y(P)285 55 y FL(n)285 121 y(i)p FK(=1)411 96 y FN(g)451 108 y FL(i)478 96 y FO(\()p FN(s)549 108 y FK(0)587 96 y FO(\))14 b FN(f)674 108 y FL(i)701 96 y FO(\()p FN(M)9 b FO(\))855 26 y Fy(\015)855 76 y(\015)924 96 y FP(\024)23 b FN(C)t(=)p FO(2.)36 b(Th)n(us,)522 264 y Fy(\015)522 314 y(\015)522 364 y(\015)608 256 y FL(n)569 281 y Fy(X)575 458 y FL(i)p FK(=1)702 360 y FN(f)743 372 y FL(i)771 360 y FO(\()p FN(M)9 b FO(\))14 b FN(B)k(g)1060 372 y FL(i)1087 360 y FO(\()p FN(s)1158 372 y FK(0)1195 360 y FO(\))1227 264 y Fy(\015)1227 314 y(\015)1227 364 y(\015)1297 360 y FP(\024)1394 304 y FO(1)p 1394 341 42 4 v 1394 417 a(2)1460 360 y FN(C)6 b FP(k)p FN(B)t FP(k)p FN(;)-118 625 y FO(and)27 b FP(k)p FO(\001\()p FN(B)t FO(\))p FP(k)c(\024)g FN(C)6 b FP(k)p FN(B)t FP(k)p FO(.)p 2278 625 4 57 v 2282 573 50 4 v 2282 625 V 2331 625 4 57 v -118 815 a FQ(Lemma)29 b(6.)41 b FB(L)l(et)22 b FN(E)5 b FO(\()p FP(\001)p FO(\))23 b FB(b)l(e)f(a)h(pr)l(oje)l(ction-value)l(d)g(me)l (asur)l(e)f(on)h(a)f(c)l(omp)l(act)h(set)-118 914 y FN(K)6 b FB(,)31 b(let)f FN(\013)185 926 y FK(1)223 914 y FB(,)h FN(:)14 b(:)g(:)27 b FB(,)32 b FN(\013)513 926 y FL(N)607 914 y FB(b)l(e)e(Bor)l(el)i(subsets)e(of)i FN(K)k FB(such)31 b(that)g(the)f(interse)l(ction)-118 1014 y(of)g(any)h FN(m)18 b FO(+)g(1)29 b FB(subsets)g FN(\013)717 1026 y FL(i)775 1014 y FB(is)h(empty)g(.)39 b(Then)712 1175 y FL(N)682 1200 y Fy(X)684 1377 y FL(j)s FK(=1)816 1279 y FP(k)p FN(E)5 b FO(\()p FN(\013)1009 1291 y FL(j)1044 1279 y FO(\))p FN(\030)t FP(k)1158 1245 y FK(2)1218 1279 y FP(\024)22 b FN(m)p FP(k)p FN(\030)t FP(k)1502 1245 y FK(2)-118 1551 y FB(for)30 b(any)h(ve)l(ctor)f FN(\030)t FB(.)-118 1726 y(Pr)l(o)l(of.)43 b FO(Since)26 b FN(E)5 b FO(\()p FN(\013)509 1738 y FL(j)545 1726 y FO(\))23 b(=)688 1659 y Fy(R)727 1756 y FL(K)805 1726 y FN(\037)857 1738 y FL(\013)900 1746 y Fv(j)935 1726 y FO(\()p FN(t)p FO(\))14 b FN(dE)5 b FO(\()p FN(t)p FO(\),)29 b(where)d FN(\037)1589 1738 y FL(\013)1632 1746 y Fv(j)1695 1726 y FO(is)g(the)h(c)n(haracteris-)-118 1826 y(tic)g(function)g(of)h FN(\013)475 1838 y FL(j)510 1826 y FO(,)f(w)n(e)h(ha)n(v)n(e)e(that)129 1995 y FL(N)98 2020 y Fy(X)101 2197 y FL(j)s FK(=1)232 2099 y FN(E)5 b FO(\()p FN(\013)383 2111 y FL(j)419 2099 y FO(\))23 b(=)562 1986 y Fy(Z)608 2174 y FL(K)672 2006 y Fy(\020)721 2020 y(X)766 2197 y FL(j)855 2099 y FN(\037)907 2111 y FL(\013)950 2119 y Fv(j)985 2099 y FO(\()p FN(t)p FO(\))1079 2006 y Fy(\021)1143 2099 y FN(dE)5 b FO(\()p FN(t)p FO(\))24 b FP(\024)f FN(m)1545 1986 y Fy(Z)1591 2174 y FL(K)1669 2099 y FN(dE)5 b FO(\()p FN(t)p FO(\))23 b(=)g FN(mI)7 b(:)-118 2395 y FO(Therefore,)281 2332 y Fy(P)369 2353 y FL(N)369 2420 y(j)s FK(=1)502 2395 y FP(k)p FN(E)e FO(\()p FN(\013)695 2407 y FL(j)730 2395 y FO(\))p FN(\030)t FP(k)844 2365 y FK(2)904 2395 y FO(=)992 2332 y Fy(P)1079 2353 y FL(N)1079 2420 y(j)s FK(=1)1198 2395 y FO(\()p FN(E)g FO(\()p FN(\013)1381 2407 y FL(j)1417 2395 y FO(\))p FN(\030)t(;)14 b(\030)t FO(\))24 b FP(\024)f FN(m)14 b FP(k)p FN(\030)t FP(k)1921 2365 y FK(2)1957 2395 y FO(.)p 2278 2395 V 2282 2342 50 4 v 2282 2395 V 2331 2395 4 57 v -118 2584 a FQ(Theorem)30 b(8.)41 b FB(L)l(et)29 b(one)h(of)h(the)f(fol)t(lowing)i(c)l(onditions)f(hold)9 b FO(:)-58 2759 y(\(a\))41 b FN(g)129 2771 y FL(i)157 2759 y FB(,)30 b FN(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)p FB(,)30 b(ar)l(e)g(p)l(olynomials)7 b FO(;)-63 2937 y(\(b\))42 b FN(g)129 2949 y FL(i)183 2937 y FP(2)27 b FO(Lip)13 b FN(\033)s FO(\()p FN(N)c FO(\))p FB(,)33 b FN(i)26 b FO(=)g(1)p FB(,)32 b FN(:)14 b(:)g(:)27 b FB(,)33 b FN(n)p FB(,)f(and)g(the)g(Hausdor\013)g(dimension)h(of)89 3036 y FN(\033)s FO(\()p FN(M)9 b FO(\))31 b FB(is)f(less)g(then)f FO(2)p FB(.)6 3211 y(Then)i Fz(M)p FO(\(\000\))23 b(=)g(k)n(er)13 b(\001)p FB(.)-118 3386 y(Pr)l(o)l(of.)43 b FO(Let)29 b FN(B)h FP(2)c Fz(M)p FO(\(\000\),)k(then)f(supp)1083 3406 y FL(M)s(;N)1246 3386 y FN(B)h FP(\032)25 b FN(S)5 b FO(.)41 b(F)-7 b(or)28 b(an)n(y)g FN(\014)i FP(\032)25 b FN(\033)s FO(\()p FN(N)9 b FO(\))30 b(w)n(e)-118 3503 y(will)f(denote)j(the)g(set)h(pr)675 3523 y FK(1)712 3503 y FO(\(pr)823 3467 y FM(\000)p FK(1)823 3525 y(2)912 3503 y FO(\()p FN(\014)t FO(\)\))h(b)n(y)1224 3481 y(~)1212 3503 y FN(\014)5 b FO(.)50 b(It)32 b(is)f(easy)g(to)h(sho)n(w)f(that)i (if)e FN(\014)-118 3603 y FO(is)j(closed)g(then)i(so)f(is)636 3581 y(~)624 3603 y FN(\014)t FO(,)j(and)d(that)h(pr)1172 3623 y FK(1)1209 3603 y FO(\(pr)1320 3567 y FM(\000)p FK(1)1320 3625 y(2)1409 3603 y FO(\()p FP([)1496 3572 y FM(1)1496 3626 y FL(k)q FK(=1)1621 3603 y FN(\014)1668 3615 y FL(k)1709 3603 y FO(\)\))h(=)f FP([)1966 3572 y FM(1)1966 3626 y FL(k)q FK(=1)2114 3581 y FO(~)2091 3603 y FN(\014)2138 3615 y FL(k)2179 3603 y FO(,)h(for)-118 3712 y(an)n(y)25 b(sequence)f FP(f)p FN(\014)467 3724 y FL(k)508 3712 y FP(g)p FO(.)36 b(Hence)865 3690 y(~)853 3712 y FN(\014)30 b FO(is)24 b(a)h(Borel)f(set)h(for)g(an)n(y)g FN(F)1757 3724 y FL(\033)1802 3712 y FO(-set)g FN(\014)30 b FO(\(here)25 b(w)n(e)-118 3811 y(consider)k(only)g(suc)n(h)i FN(\014)t FO(\).)47 b(Since)31 b(\()p FN(\033)s FO(\()p FN(M)9 b FO(\))21 b FP(n)1290 3790 y FO(~)1278 3811 y FN(\014)5 b FO(\))21 b FP(\002)f FN(\033)s FO(\()p FN(N)9 b FO(\))32 b(do)r(es)e(not)h(in)n(tersect)-118 3911 y FN(S)5 b FO(,)27 b(one)h(has)f(\()p FN(I)e FP(\000)18 b FN(E)526 3923 y FL(M)600 3911 y FO(\()644 3889 y(~)632 3911 y FN(\014)5 b FO(\)\))14 b FN(B)t(E)890 3923 y FL(N)954 3911 y FO(\()p FN(\014)t FO(\))24 b(=)e(0.)p eop %%Page: 53 57 53 56 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(53)6 96 y(Let)26 b FN(")c(>)h FO(0)i(and)f(let)h FN(\033)s FO(\()p FN(N)9 b FO(\))23 b(=)g FP([)1001 62 y FL(K)1001 117 y(j)s FK(=1)1120 96 y FN(\013)1173 108 y FL(j)1209 96 y FO(,)i(where)g FN(\013)1548 108 y FL(i)1589 96 y FP(\\)13 b FN(\013)1710 108 y FL(j)1768 96 y FO(=)23 b FI(?)p FO(,)i(diam)12 b FN(\013)2216 108 y FL(j)2274 96 y FN(<)-118 196 y(")p FO(,)34 b(and)f FN(K)k FO(=)32 b FN(K)6 b FO(\()p FN(")p FO(\))33 b(is)f(the)h(least)f (p)r(ossible)f(for)h(all)f(suc)n(h)h(decomp)r(ositions.)-118 296 y(Then)19 530 y(\001\()p FN(B)t FO(\))24 b(=)361 426 y FL(K)331 451 y Fy(X)333 628 y FL(j)s FK(=1)465 530 y FO(\001\()p FN(B)t FO(\))p FN(E)726 542 y FL(N)790 530 y FO(\()p FN(\013)875 542 y FL(j)910 530 y FO(\))f(=)1083 426 y FL(K)1053 451 y Fy(X)1056 628 y FL(j)s FK(=1)1187 530 y FN(E)1248 542 y FL(M)1322 530 y FO(\()8 b(~)-50 b FN(\013)1407 542 y FL(j)1442 530 y FO(\))14 b(\001\()p FN(B)t FO(\))p FN(E)1749 542 y FL(N)1813 530 y FO(\()p FN(\013)1898 542 y FL(j)1934 530 y FO(\))p FN(:)137 b FO(\(1.13\))-118 801 y(Set)28 b FN(\015)g FO(=)183 734 y Fy(\010)232 801 y FN(t)23 b FP(2)g FN(\033)s FO(\()p FN(M)9 b FO(\))g(:)641 739 y Fy(P)729 759 y FL(n)729 826 y(i)p FK(=1)855 801 y FN(f)896 813 y FL(i)923 801 y FO(\()p FN(t)p FO(\))14 b FN(g)1071 813 y FL(i)1099 801 y FO(\()p FN(N)9 b FO(\))23 b(=)g(0)1392 734 y Fy(\011)1439 801 y FO(.)37 b(Then)864 975 y FN(E)925 987 y FL(M)999 975 y FO(\()p FN(\015)5 b FO(\)\001)24 b(=)e(0)p FN(:)770 b FO(\(1.14\))-118 1149 y(In)28 b(fact,)f(for)h(an)n(y)e FN(Y)42 b FP(2)23 b FN(L)p FO(\()p FN(H)782 1161 y FL(N)845 1149 y FN(;)14 b(H)951 1161 y FL(M)1025 1149 y FO(\),)28 b(w)n(e)f(ha)n(v)n(e)367 1390 y FN(E)428 1402 y FL(M)502 1390 y FO(\()p FN(\015)5 b FO(\)\001\()p FN(Y)19 b FO(\))24 b(=)965 1286 y FL(n)925 1311 y Fy(X)931 1488 y FL(i)p FK(=1)1059 1390 y FN(E)1120 1402 y FL(M)1194 1390 y FO(\()p FN(\015)5 b FO(\))14 b FN(f)1361 1402 y FL(i)1388 1390 y FO(\()p FN(M)9 b FO(\))14 b FN(Y)19 b(g)1663 1402 y FL(i)1690 1390 y FO(\()p FN(N)9 b FO(\))p FN(;)-118 1637 y FO(and)153 1857 y(\()p FN(E)246 1869 y FL(M)320 1857 y FO(\()p FN(\015)c FO(\)\001\()p FN(Y)20 b FO(\))p FN(\030)t(;)14 b(\027)5 b FO(\))24 b(=)939 1753 y FL(n)899 1778 y Fy(X)905 1955 y FL(i)p FK(=1)1019 1857 y FO(\()p FN(Y)19 b(g)1158 1869 y FL(i)1185 1857 y FO(\()p FN(N)9 b FO(\))p FN(\030)t(;)14 b(E)1463 1869 y FL(M)1538 1857 y FO(\()p FN(\015)5 b FO(\))14 b(\()p FN(f)1737 1869 y FL(i)1764 1857 y FO(\()p FN(M)9 b FO(\)\))1950 1822 y FM(\003)1989 1857 y FN(\027)c FO(\))342 2133 y(=)469 2029 y FL(n)430 2054 y Fy(X)436 2231 y FL(i)p FK(=1)564 2020 y Fy(Z)610 2209 y FL(\015)652 2066 y Fy(\000)690 2133 y FN(Y)19 b(g)797 2145 y FL(i)824 2133 y FO(\()p FN(N)9 b FO(\))p FN(\030)t(;)p 1041 2061 163 4 v 14 w(f)1082 2145 y FL(i)1110 2133 y FO(\()p FN(t)p FO(\))14 b FN(dE)1322 2145 y FL(M)1396 2133 y FO(\()p FN(t)p FO(\))p FN(\027)1536 2066 y Fy(\001)342 2410 y FO(=)430 2297 y Fy(Z)476 2485 y FL(\015)519 2318 y Fy(\020)608 2306 y FL(n)568 2331 y Fy(X)574 2508 y FL(i)p FK(=1)702 2410 y FN(Y)k(g)808 2422 y FL(i)836 2410 y FO(\()p FN(N)9 b FO(\))14 b FN(f)1031 2422 y FL(i)1058 2410 y FO(\()p FN(t)p FO(\))p FN(\030)t(;)g(dE)1333 2422 y FL(M)1408 2410 y FO(\()p FN(t)p FO(\))p FN(\027)1548 2318 y Fy(\021)1621 2410 y FO(=)23 b(0)p FN(;)-118 2657 y FO(for)k(an)n(y)g FN(\030)g FP(2)c FN(H)376 2669 y FL(N)439 2657 y FO(,)28 b FN(\027)g FP(2)c FN(H)707 2669 y FL(M)781 2657 y FO(.)6 2756 y(Relation)i(\(1.14\))h(implies)479 3011 y(\001\()p FN(B)t FO(\))d(=)821 2908 y FL(K)791 2933 y Fy(X)794 3109 y FL(j)s FK(=1)925 3011 y FN(E)986 3023 y FL(M)1060 3011 y FO(\()8 b(^)-50 b FN(\013)1145 3023 y FL(j)1180 3011 y FO(\))14 b(\001\()p FN(B)t FO(\))g FN(E)1501 3023 y FL(N)1565 3011 y FO(\()p FN(\013)1650 3023 y FL(j)1686 3011 y FO(\))p FN(;)-118 3274 y FO(where)35 b(^)-50 b FN(\013)175 3286 y FL(j)233 3274 y FO(=)31 b(~)-50 b FN(\013)374 3286 y FL(j)427 3274 y FP(n)18 b FN(\015)5 b FO(.)37 b(Hence,)124 3532 y FP(j)p FO(\(\001\()p FN(B)t FO(\))p FN(\030)t(;)14 b(\027)5 b FO(\))p FP(j)25 b FO(=)669 3437 y Fy(\014)669 3487 y(\014)669 3537 y(\014)727 3429 y FL(K)697 3454 y Fy(X)699 3630 y FL(j)s FK(=1)817 3465 y Fy(\000)855 3532 y FN(E)916 3544 y FL(M)990 3532 y FO(\()8 b(^)-50 b FN(\013)1075 3544 y FL(j)1110 3532 y FO(\)\001\()p FN(B)t FO(\))p FN(E)1403 3544 y FL(N)1468 3532 y FO(\()p FN(\013)1553 3544 y FL(j)1588 3532 y FO(\))p FN(\030)t(;)14 b(E)1758 3544 y FL(M)1832 3532 y FO(\()8 b(^)-50 b FN(\013)1917 3544 y FL(j)1953 3532 y FO(\))p FN(\027)2031 3465 y Fy(\001)2069 3437 y(\014)2069 3487 y(\014)2069 3537 y(\014)-76 3835 y FP(\024)11 3743 y Fy(\020)91 3731 y FL(K)61 3756 y Fy(X)64 3933 y FL(j)s FK(=1)181 3764 y Fy(\015)181 3814 y(\015)227 3835 y FN(E)288 3847 y FL(M)362 3835 y FO(\()8 b(^)-50 b FN(\013)447 3847 y FL(j)482 3835 y FO(\)\001\()p FN(B)t FO(\))p FN(E)775 3847 y FL(N)840 3835 y FO(\()p FN(\013)925 3847 y FL(j)960 3835 y FO(\))p FN(\030)1032 3764 y Fy(\015)1032 3814 y(\015)1079 3785 y FK(2)1116 3743 y Fy(\021)1166 3760 y FK(1)p FL(=)p FK(2)1270 3743 y Fy(\020)1350 3731 y FL(K)1320 3756 y Fy(X)1322 3933 y FL(j)s FK(=1)1453 3835 y FP(k)p FN(E)1556 3847 y FL(M)1630 3835 y FO(\()8 b(^)-50 b FN(\013)1715 3847 y FL(j)1750 3835 y FO(\))p FN(\027)5 b FP(k)1870 3801 y FK(2)1908 3743 y Fy(\021)1957 3760 y FK(1)p FL(=)p FK(2)2061 3835 y FN(:)42 b FO(\(1.15\))p eop %%Page: 54 58 54 57 bop -118 -137 a FO(54)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(Since)i(^)-50 b FN(\013)144 108 y FL(j)202 96 y FP(\032)23 b FO(pr)368 117 y FK(1)405 96 y FO(\(\()8 b(^)-50 b FN(\013)522 108 y FL(j)561 96 y FP(\002)s FN(\013)682 108 y FL(j)716 96 y FO(\))s FP(\\)s FN(S)5 b FO(\),)21 b(w)n(e)f(can)f(apply)f(Lemma)g(5)h(to)h(the)g(op)r(erators)-118 196 y FN(N)9 b(E)19 208 y FL(N)82 196 y FO(\()p FN(\013)167 208 y FL(j)202 196 y FO(\),)28 b FN(M)9 b(E)436 208 y FL(M)510 196 y FO(\()f(^)-50 b FN(\013)595 208 y FL(j)630 196 y FO(\).)38 b(W)-7 b(e)27 b(obtain)167 371 y FP(k)p FN(E)270 383 y FL(M)344 371 y FO(\()8 b(^)-50 b FN(\013)429 383 y FL(j)464 371 y FO(\)\001\()p FN(B)t FO(\))p FN(E)757 383 y FL(N)822 371 y FO(\()p FN(\013)907 383 y FL(j)942 371 y FO(\))p FP(k)23 b(\024)f FN(C)6 b FP(k)p FN(B)t FP(k)14 b FO(diam)d FN(\013)1602 383 y FL(j)1661 371 y FP(\024)22 b FN(C)6 b FP(k)p FN(B)t FP(k)14 b FN(";)258 520 y FP(k)p FN(E)361 532 y FL(M)435 520 y FO(\()8 b(^)-50 b FN(\013)520 532 y FL(j)555 520 y FO(\)\001\()p FN(B)t FO(\))p FN(E)848 532 y FL(N)912 520 y FO(\()p FN(\013)997 532 y FL(j)1033 520 y FO(\))p FN(\030)t FP(k)23 b(\024)f FN(C)6 b FP(k)p FN(B)t FP(k)14 b FN(")p FP(k)p FN(E)1629 532 y FL(N)1691 520 y FO(\()p FN(\013)1776 532 y FL(j)1812 520 y FO(\))p FN(\030)t FP(k)p FN(;)274 631 y FL(K)243 656 y Fy(X)246 833 y FL(j)s FK(=1)377 735 y FP(k)p FN(E)480 747 y FL(M)554 735 y FO(\()8 b(^)-50 b FN(\013)639 747 y FL(j)674 735 y FO(\)\001\()p FN(B)t FO(\))p FN(E)967 747 y FL(N)1032 735 y FO(\()p FN(\013)1117 747 y FL(j)1152 735 y FO(\))p FN(\030)t FP(k)1266 701 y FK(2)1326 735 y FP(\024)23 b FN(C)1479 701 y FK(2)1516 735 y FP(k)p FN(B)t FP(k)1667 701 y FK(2)1704 735 y FN(")1743 701 y FK(2)1780 735 y FP(k)p FN(\030)t FP(k)1904 701 y FK(2)1940 735 y FN(;)-118 994 y FO(since)j(the)i(sets)g FN(\013)444 1006 y FL(j)506 994 y FO(are)f(m)n(utually)e(disjoin)n(t.)6 1093 y(First)e(let)g(all)f FN(g)473 1105 y FL(i)500 1093 y FO(\()p FP(\001)p FO(\))i(b)r(e)h(p)r(olynomials)18 b(and)24 b(let)f FN(m)h FO(b)r(e)g(the)g(greatest)f(degree)-118 1193 y(of)31 b(the)g(p)r(olynomials)26 b FN(g)631 1205 y FL(i)658 1193 y FO(\()p FP(\001)p FO(\).)47 b(Then,)32 b(for)e(an)n(y)g FN(t)f FP(2)f FN(\033)s FO(\()p FN(M)9 b FO(\))22 b FP(n)e FN(\015)5 b FO(,)31 b(the)g(equation)-118 1293 y(\010\()p FN(t;)14 b(s)p FO(\))30 b(=)g(0)i(do)r(es)f(not)h(ha)n (v)n(e)f(more)f(then)j FN(m)f FO(ro)r(ots.)48 b(Th)n(us,)33 b FN(t)f FO(can)g(not)g(b)r(e-)-118 1392 y(long)h(to)j(more)d(than)j FN(m)f FO(sets)43 b(^)-50 b FN(\013)927 1404 y FL(j)962 1392 y FO(,)37 b(b)r(ecause)e(for)g(an)n(y)f FN(t)i FP(2)45 b FO(^)-51 b FN(\013)1846 1404 y FL(j)1917 1392 y FO(there)35 b(exists)-118 1492 y FN(s)p FO(\()p FN(t)p FO(\))26 b FP(2)f FN(\013)174 1504 y FL(j)239 1492 y FO(suc)n(h)j(that)h(\010\()p FN(t;)14 b(s)p FO(\()p FN(t)p FO(\)\))27 b(=)e(0,)j(and)h(the)h(sets)e FN(\013)1666 1504 y FL(j)1731 1492 y FO(are)f(disjoin)n(t.)40 b(Ap-)-118 1605 y(plying)21 b(Lemma)h(6,)i(w)n(e)f(can)g(conclude)g (that)1291 1543 y Fy(P)1378 1564 y FL(K)1378 1630 y(j)s FK(=1)1511 1605 y FP(k)p FN(E)1614 1617 y FL(M)1688 1605 y FO(\()8 b(^)-50 b FN(\013)1773 1617 y FL(j)1808 1605 y FO(\))p FN(\027)5 b FP(k)1928 1575 y FK(2)1988 1605 y FP(\024)23 b FN(m)p FP(k)p FN(\027)5 b FP(k)2279 1575 y FK(2)2316 1605 y FO(.)-118 1724 y(Hence,)32 b FP(j)p FO(\(\001\()p FN(B)t FO(\))p FN(\030)t(;)14 b(\027)5 b FO(\))p FP(j)30 b(\024)d FN(C)6 b FP(k)p FN(B)t FP(k)14 b FN(")p FP(k)p FN(\030)t FP(k)g FN(m)1191 1694 y FK(1)p FL(=)p FK(2)1294 1724 y FP(k)p FN(\027)5 b FP(k)1424 1694 y FK(2)1460 1724 y FO(.)46 b(Letting)30 b FN(")e FP(!)g FO(0,)j(w)n(e)f(ob-)-118 1823 y(tain)d(\001\()p FN(B)t FO(\))d(=)e(0.)6 1923 y(No)n(w)32 b(assume)f(that)h(the)h (second)f(condition)e(holds.)49 b(W)-7 b(e)33 b(can)f(estimate)-118 2023 y(the)c(second)f(factor)g(in)f(the)i(righ)n(t-hand)e(side)g(of)i (\(1.15\))f(as)g(follo)n(ws:)231 2186 y Fy(\020)311 2175 y FL(K)281 2200 y Fy(X)283 2377 y FL(j)s FK(=1)415 2279 y FP(k)p FN(E)518 2291 y FL(M)591 2279 y FO(\()8 b(^)-50 b FN(\013)676 2291 y FL(j)711 2279 y FO(\))p FN(\027)5 b FP(k)831 2244 y FK(2)869 2186 y Fy(\021)918 2204 y FK(1)p FL(=)p FK(2)1046 2279 y FP(\024)22 b FO(\()p FN(K)6 b FP(k)p FN(\027)f FP(k)1372 2244 y FK(2)1409 2279 y FO(\))1441 2244 y FK(1)p FL(=)p FK(2)1568 2279 y FO(=)23 b FN(K)1733 2244 y FK(1)p FL(=)p FK(2)1837 2279 y FP(k)p FN(\027)5 b FP(k)p FN(:)-118 2553 y FO(Th)n(us)36 b FP(k)p FO(\001\()p FN(B)t FO(\))p FP(k)h(\024)f FN(C)6 b FP(k)p FN(B)t FP(k)14 b FN("K)869 2523 y FK(1)p FL(=)p FK(2)1008 2553 y FO(\()p FN(K)43 b FO(=)37 b FN(K)6 b FO(\()p FN(")p FO(\)\).)62 b(Since)35 b(dim)13 b FN(\033)s FO(\()p FN(M)c FO(\))37 b FN(<)g FO(2,)-118 2653 y FN(K)6 b FO(\()p FN(")p FO(\))23 b(=)f FN(o)p FO(\()p FN(")283 2623 y FM(\000)p FK(2)373 2653 y FO(\),)28 b(this)f(implies)c(\001\()p FN(B)t FO(\))h(=)f(0.)p 2278 2653 4 57 v 2282 2600 50 4 v 2282 2653 V 2331 2653 4 57 v -118 2817 a FB(R)l(emark)30 b(13.)42 b FO(The)34 b(restriction)c(on)j(dim)12 b FN(\033)s FO(\()p FN(M)d FO(\))34 b(in)e(Theorem)f(8)i(can)g(b)r(e)g(re-)-118 2917 y(mo)n(v)n(ed)e(\(see)h([248)o(]\),)i(but)g(the)f(pro)r(of)f(b)r (ecomes)f(m)n(uc)n(h)h(more)f(complicated,)-118 3017 y(so)c(w)n(e)g(omit)f(this)h(generalization)22 b(here.)-118 3193 y FQ(5.)34 b FO(As)21 b(a)g(corollary)c(from)j(Theorem)f(8)i(w)n (e)g(obtain)f(the)h(follo)n(wing)c(Kleinec)n(k)n(e{)-118 3293 y(Shirok)n(o)n(v)24 b(t)n(yp)r(e)k(theorems.)-118 3452 y FQ(Theorem)i(9.)41 b FB(If)34 b(two)g(p)l(olynomial)h(semiline)l (ar)g(r)l(elations)f(have)h(the)e(same)-118 3552 y(gr)l(aph,)e(then)f (their)g(r)l(epr)l(esentations)g(c)l(oincide.)-118 3712 y(Example)h(6.)42 b FO(By)32 b(Theorem)e(9,)i(the)h(semilinear)27 b(relations)h(\(ad)1921 3724 y FL(A;q)2027 3712 y FO(\))2059 3682 y FL(n)2105 3712 y FN(B)34 b FO(=)c(0)-118 3811 y(and)36 b(ad)140 3823 y FL(A;q)260 3811 y FN(B)42 b FO(=)37 b FN(AB)29 b FP(\000)24 b FN(q)s(B)t(A)38 b FO(=)g(0)e(ha)n(v)n (e)f(the)i(same)d(represen)n(tations)g(b)n(y)-118 3911 y(the)25 b(b)r(ounded)g(op)r(erators)e FN(A)g FO(=)g FN(A)958 3881 y FM(\003)996 3911 y FO(,)j FN(B)t FO(,)f(since)f(the)h (c)n(haracteristic)20 b(functions)p eop %%Page: 55 59 55 58 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(55)-118 96 y(\010\()p FN(t;)14 b(s)p FO(\))32 b(=)g FN(t)22 b FP(\000)f FN(q)s(s)33 b FO(and)g(\010\()p FN(t;)14 b(s)p FO(\))32 b(=)f(\()p FN(t)23 b FP(\000)e FN(q)s(s)p FO(\))1298 66 y FL(n)1344 96 y FO(,)34 b(corresp)r(onding)c(to)j(the)g(\014rst)-118 196 y(and)40 b(the)h(second)f(relations)d(resp)r(ectiv)n(ely)-7 b(,)40 b(de\014ne)h(the)g(same)d(graph.)74 b(In)-118 296 y(particular,)28 b(for)i FN(q)g FO(=)e(1)i(w)n(e)g(obtain)f(that)h (b)r(ounded)h(op)r(erators)e FN(A)f FO(=)f FN(A)2179 266 y FM(\003)2217 296 y FO(,)k FN(B)-118 395 y FO(satisfy)22 b(the)i(relation)c(\(ad)696 407 y FL(A)750 395 y FO(\))782 365 y FL(n)828 395 y FN(B)27 b FO(=)c([)p FN(A;)14 b(:)g(:)g(:)f FO([)p FN(A;)h(B)t FO(])g FN(:)g(:)g(:)g FO(])24 b(=)e(0)h(if)g(and)g (only)f(if)h FN(A)p FO(,)-118 495 y FN(B)k FO(comm)n(ute,)22 b(and)h(for)f FN(q)27 b FO(=)22 b FP(\000)p FO(1,)h(a)g(pair)e(of)i(b)r (ounded)h(op)r(erators)d FN(A)i FO(=)g FN(A)2186 465 y FM(\003)2224 495 y FO(,)h FN(B)-118 595 y FO(is)31 b(a)h(solution)e(of)i(the)h(equation)e(\(ad)1074 607 y FL(A;)p FM(\000)p FK(1)1233 595 y FO(\))1265 564 y FL(n)1310 595 y FN(B)k FO(=)c FP(f)p FN(A;)14 b(:)g(:)g(:)f FP(f)p FN(A;)h(B)t FP(g)g FN(:)g(:)g(:)f FP(g)30 b FO(=)h(0)-118 694 y(i\013)c FN(AB)h FO(=)22 b FP(\000)p FN(B)t(A)p FO(.)-118 819 y FQ(6.)66 b FO(As)38 b(corollaries)32 b(from)k(Theorem)g(8)h(w)n(e)g(will)e(also)g(giv)n(e)h(some)g(results,) -118 919 y(where)e(the)h(form)f(of)h(the)g(c)n(haracteristic)c(binary)i (relation)f(corresp)r(onding)-118 1018 y(to)27 b(the)h(semilinear)22 b(relation)i(\(1.10\))j(is)f(more)f(imp)r(ortan)n(t)g(then)j(the)g (form)e(of)-118 1118 y(the)i(functions)f FN(f)424 1130 y FL(i)451 1118 y FO(,)h FN(g)542 1130 y FL(i)569 1118 y FO(.)-118 1268 y FQ(Theorem)i(10.)41 b FB(L)l(et)24 b FN(f)9 b FB(,)25 b FN(g)s FB(,)g(b)l(e)g(b)l(ounde)l(d)g(Bor)l(el)g (functions.)37 b(A)24 b(p)l(air)h(of)g(b)l(oun-)-118 1367 y(de)l(d)30 b(op)l(er)l(ators)h FN(A)23 b FO(=)g FN(A)626 1337 y FM(\003)664 1367 y FB(,)31 b FN(B)i FB(satis\014es)d (the)g(e)l(quation)815 1530 y FN(f)9 b FO(\()p FN(A)p FO(\))p FN(B)27 b FO(=)c FN(B)t(g)s FO(\()p FN(A)p FO(\))-118 1693 y FB(if)31 b(and)f(only)g(if)h FO(supp)556 1713 y FL(A)610 1693 y FO(\()p FN(B)t FO(\))24 b FP(\032)e FO(\000)p FB(,)30 b(wher)l(e)h FO(\000)23 b(=)f FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FN(f)9 b FO(\()p FN(t)p FO(\))24 b(=)e FN(g)s FO(\()p FN(s)p FO(\))p FP(g)p FB(.)-118 1842 y(Pr)l(o)l(of.)43 b FO(Set)32 b FN(M)38 b FO(=)29 b FN(f)9 b FO(\()p FN(A)p FO(\),)33 b FN(N)38 b FO(=)29 b FN(g)s FO(\()p FN(A)p FO(\).)49 b(Since)31 b FN(E)1456 1854 y FL(M)1530 1842 y FO(\()p FN(\013)p FO(\))f(=)f FN(E)1832 1854 y FL(A)1886 1842 y FO(\()p FN(f)1968 1812 y FM(\000)p FK(1)2057 1842 y FO(\()p FN(\013)p FO(\)\))k(for)-118 1942 y(an)n(y)h(Borel)e(set)j FN(\013)p FO(,)i(the)e(condition)e(supp)1216 1962 y FL(A)1270 1942 y FO(\()p FN(B)t FO(\))j FP(\032)e FO(\000)h(is)e(equiv)-5 b(alen)n(t)33 b(to)h(the)-118 2042 y(follo)n(wing)23 b(one)567 2141 y(supp)738 2162 y FL(M)s(;N)886 2141 y FO(\()p FN(B)t FO(\))h FP(\032)f(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FN(t)g FO(=)f FN(s)p FP(g)p FN(:)-118 2279 y FO(Theorem)k(8)h(giv)n(es)e(the)j(required)e(statemen)n(t.)p 2278 2279 4 57 v 2282 2226 50 4 v 2282 2279 V 2331 2279 4 57 v 6 2441 a(In)h(a)e(similar)d(w)n(a)n(y)i(one)i(sho)n(ws)f(that)h (Theorem)f(8)g(implies)e(the)j(F)-7 b(uglede{)-118 2540 y(Putnam{Rozen)n(blum)34 b(theorem)i(see,)k(for)e(example)d([226)o(])j (and)f(also)f(Sec-)-118 2640 y(tion)23 b(1.4.3.)35 b(In)24 b(particular,)d(if)j(b)r(ounded)h(op)r(erators)d FN(A)h FO(=)g FN(A)1790 2610 y FM(\003)1828 2640 y FO(,)i FN(B)k FO(satisfy)22 b(the)-118 2740 y(relation)k FN(AB)31 b FO(=)25 b FN(q)s(B)t(A)p FO(,)30 b(where)f FP(j)p FN(q)s FP(j)d FO(=)g(1,)j(then)h FN(AB)g FO(=)i(\026)-48 b FN(q)s(B)t(A)p FO(,)30 b(since)e(the)i(sets)-118 2839 y FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(2)g FI(R)i FP(\002)18 b FI(R)29 b FP(j)23 b FN(t)g FO(=)f FN(q)s(s)p FP(g)28 b FO(and)f FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(2)g FI(R)i FP(\002)18 b FI(R)29 b FP(j)23 b FN(t)g FO(=)p 1681 2794 41 4 v 23 w FN(q)s(s)p FP(g)k FO(coincide.)-118 2964 y FB(R)l(emark)j(14.)42 b FO(Sets)34 b(of)g(the)g(form)e FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))33 b FP(j)g FN(f)9 b FO(\()p FN(t)p FO(\))34 b(=)e FN(g)s FO(\()p FN(s)p FO(\))p FP(g)p FO(,)j(where)e FN(f)9 b FO(,)35 b FN(g)i FO(are)-118 3064 y(b)r(ounded)g(Borel)e(functions)i(are)f(called)e (rectangular)h(sets.)65 b(F)-7 b(or)36 b(them,)j(a)-118 3163 y(m)n(uc)n(h)28 b(stronger)g(result)g(is)h(true:)41 b(an)29 b(op)r(erator)f(supp)r(orted)h(b)n(y)h(a)f(rectangu-)-118 3263 y(lar)i(set)i FN(F)45 b FO(satis\014es)32 b(an)n(y)g(relation)e (whose)i(c)n(haracteristic)d(binary)j(relation)-118 3363 y(con)n(tains)23 b FN(F)38 b FO([248)o(].)e(The)26 b(pro)r(of)f(is)f (complicated)e(and)k(w)n(e)f(restrict)f(ourselv)n(es)-118 3462 y(to)j(a)h(more)d(particular)g(result.)-118 3612 y FQ(Theorem)30 b(11.)41 b FB(L)l(et)23 b FN(F)36 b FB(b)l(e)24 b(the)g(gr)l(aph)h(of)g(a)g(function,)g(that)f(is,)i FN(F)35 b FO(=)23 b FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)-118 3712 y FN(s)k FO(=)f FN(')p FO(\()p FN(t)p FO(\))p FP(g)p FB(,)33 b(wher)l(e)g FN(')f FB(is)g(a)h(b)l(ounde)l(d)f (Bor)l(el)h(function.)45 b(If)32 b FO(supp)1944 3732 y FL(A)1998 3712 y FO(\()p FN(B)t FO(\))c FP(\032)e FN(F)12 b FB(,)-118 3811 y(then)21 b(the)h(p)l(air)g FN(A)p FB(,)i FN(B)h FB(gives)e(a)e(r)l(epr)l(esentation)h(of)g(any)g(r)l(elation)g (whose)h(binary)-118 3911 y(r)l(elation)30 b(c)l(ontains)g FN(F)12 b FB(.)p eop %%Page: 56 60 56 59 bop -118 -137 a FO(56)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Let)28 b FN(F)35 b FP(\032)22 b FO(\000)h(=)629 29 y Fy(\010)677 96 y FO(\()p FN(t;)14 b(s)p FO(\))24 b FP(j)917 34 y Fy(P)1005 55 y FL(n)1005 121 y(i)p FK(=1)1130 96 y FN(f)1171 108 y FL(i)1198 96 y FO(\()p FN(t)p FO(\))p FN(g)1332 108 y FL(i)1360 96 y FO(\()p FN(s)p FO(\))g(=)e(0)1616 29 y Fy(\011)1664 96 y FO(.)37 b(Since)57 266 y(supp)228 286 y FL(A)282 266 y FO(\()p FN(B)t FO(\))24 b FP(\032)e(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FN(s)g FO(=)g FN(')p FO(\()p FN(t)p FO(\))p FP(g)g(\032)g(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FN(g)1577 278 y FL(i)1604 266 y FO(\()p FN(s)p FO(\))h(=)f FN(g)1859 278 y FL(i)1886 266 y FO(\()p FN(')p FO(\()p FN(t)p FO(\)\))p FP(g)p FN(;)-118 435 y FO(w)n(e)h(ha)n(v)n(e)f FN(B)t(g)296 447 y FL(i)323 435 y FO(\()p FN(A)p FO(\))h(=)f FN(g)601 447 y FL(i)628 435 y FO(\()p FN(')p FO(\()p FN(A)p FO(\)\))p FN(B)t FO(,)k FN(i)22 b FO(=)h(1,)h FN(:)14 b(:)g(:)27 b FO(,)e FN(n)p FO(,)g(b)n(y)f(Theorem)e(10.)35 b(On)24 b(the)-118 535 y(other)j(hand,)778 592 y FL(n)739 617 y Fy(X)745 794 y FL(i)p FK(=1)872 696 y FN(f)913 708 y FL(i)941 696 y FO(\()p FN(t)p FO(\))14 b FN(g)1089 708 y FL(i)1116 696 y FO(\()p FN(')p FO(\()p FN(t)p FO(\)\))25 b(=)d(0)-118 919 y(for)27 b(an)n(y)g FN(t)c FP(2)g FN(\033)s FO(\()p FN(A)p FO(\),)29 b(hence)756 857 y Fy(P)843 877 y FL(n)843 944 y(i)p FK(=1)969 919 y FN(f)1010 931 y FL(i)1037 919 y FO(\()p FN(A)p FO(\))14 b FN(g)1217 931 y FL(i)1245 919 y FO(\()p FN(')p FO(\()p FN(A)p FO(\)\))25 b(=)e(0.)36 b(It)28 b(follo)n(ws)d(that)132 1055 y FL(n)93 1080 y Fy(X)99 1257 y FL(i)p FK(=1)226 1159 y FN(f)267 1171 y FL(i)295 1159 y FO(\()p FN(A)p FO(\))p FN(B)t(g)528 1171 y FL(i)556 1159 y FO(\()p FN(A)p FO(\))f(=)833 1055 y FL(n)793 1080 y Fy(X)800 1257 y FL(i)p FK(=1)927 1159 y FN(f)968 1171 y FL(i)995 1159 y FO(\()p FN(A)p FO(\)\()p FN(B)t(g)1260 1171 y FL(i)1289 1159 y FO(\()p FN(A)p FO(\))19 b FP(\000)f FN(g)1557 1171 y FL(i)1584 1159 y FO(\()p FN(')p FO(\()p FN(A)p FO(\)\))p FN(B)t FO(\))26 b(=)c(0)p FN(:)p 2278 1159 4 57 v 2282 1106 50 4 v 2282 1159 V 2331 1159 4 57 v -118 1395 a FQ(Corollary)32 b(1.)40 b FB(L)l(et)e FN(A)f FO(=)g FN(A)837 1365 y FM(\003)876 1395 y FB(,)j(and)e(let)f(ther)l(e)h(exist)f(a)h(de)l(c)l(omp)l (osition)i(of)-118 1495 y FN(\033)s FO(\()p FN(A)p FO(\))30 b FB(into)f(Bor)l(el)g(sets)g FN(P)694 1507 y FL(i)722 1495 y FB(,)g FN(\033)s FO(\()p FN(A)p FO(\))24 b(=)f FP([)1119 1507 y FL(i)1147 1495 y FN(P)1200 1507 y FL(i)1228 1495 y FB(,)29 b(such)g(that)g(e)l(ach)g FN(P)1877 1507 y FL(i)1921 1495 y FP(\002)16 b FN(P)2055 1507 y FL(j)2107 1495 y FP(\\)g FO(\000)p FB(,)30 b FN(i)p FB(,)-118 1594 y FN(j)e FO(=)23 b(1)p FB(,)29 b FO(2)p FB(,)h FN(:)14 b(:)g(:)27 b FB(,)k(is)f(the)g(gr)l(aph)h(of)f(a)g(function.)39 b(Then)30 b FO(k)n(er)13 b(\001)1798 1606 y FL(A)1875 1594 y FO(=)23 b Fz(M)2050 1606 y FL(A)2104 1594 y FO(\(\000\))p FB(.)-118 1750 y(Pr)l(o)l(of.)43 b FO(Theorem)25 b(11)h(implies)d FN(E)944 1762 y FL(A)998 1750 y FO(\()p FN(P)1083 1762 y FL(i)1111 1750 y FO(\)\001\()p FN(B)t FO(\))p FN(E)1404 1762 y FL(A)1460 1750 y FO(\()p FN(P)1545 1762 y FL(j)1580 1750 y FO(\))h(=)e(0)27 b(for)f(an)n(y)g FN(i;)14 b(j)31 b FO(and)-118 1849 y(hence)d(\001\()p FN(B)t FO(\))23 b(=)g(0.)p 2278 1849 V 2282 1796 50 4 v 2282 1849 V 2331 1849 4 57 v -118 2013 a FB(R)l(emark)30 b(15.)42 b FO(Let)e(us)g(note)f (that)h(if)46 b(\000)40 b(is)e(suc)n(h)i(that)f(the)i(set)e FP(f)p FN(\026)k FP(2)g FI(R)50 b FP(j)-118 2112 y FO(\()p FN(\025;)14 b(\026)p FO(\))29 b FP(2)f FO(\000)p FP(g)i FO(is)f(\014nite)h(or)f(coun)n(table)g(for)h(an)n(y)g FN(\025)e FP(2)g FI(R)p FO(,)37 b(then)31 b(the)g(condition)-118 2212 y(of)c(Corollary)d(1)j(is)f(satis\014ed.)-118 2384 y FQ(7.)36 b FO(As)26 b(w)n(as)g(noticed,)f(the)i(con)n(v)n(erse)e(of) 32 b(Theorem)25 b(7)h(is)f(false)h(in)f(the)i(general)-118 2483 y(case.)35 b(Here)24 b(w)n(e)g(construct)f(an)h(example)e(whic)n (h)h(sim)n(ultaneously)c(solv)n(es)j(the)-118 2583 y(F)-7 b(uglede{W)g(eiss)25 b(problem.)36 b(Namely)-7 b(,)26 b(w)n(e)i(sho)n(w)f(that)i(there)f(exist)f(a)g(pair)g(of)-118 2683 y(b)r(ounded)j(op)r(erators)d FN(A)f FO(=)g FN(A)833 2652 y FM(\003)871 2683 y FO(,)k FN(B)k FO(and)29 b(con)n(tin)n(uous)e (functions)i FN(f)2002 2695 y FL(i)2029 2683 y FO(,)h FN(g)2122 2695 y FL(i)2179 2683 y FO(suc)n(h)-118 2782 y(that)209 2893 y FL(m)179 2918 y Fy(X)185 3095 y FL(i)p FK(=1)313 2997 y FN(f)354 3009 y FL(i)381 2997 y FO(\()p FN(A)p FO(\))p FN(B)t(g)614 3009 y FL(i)642 2997 y FO(\()p FN(A)p FO(\))24 b(=)f(0)p FN(;)96 b FO(but)1293 2893 y FL(m)1262 2918 y Fy(X)1269 3095 y FL(i)p FK(=1)1414 2975 y FO(\026)1396 2997 y FN(f)1437 3009 y FL(i)1464 2997 y FO(\()p FN(A)p FO(\))p FN(B)22 b FO(\026)-45 b FN(g)1712 3009 y FL(i)1740 2997 y FO(\()p FN(A)p FO(\))23 b FP(6)p FO(=)g(0)p FN(:)-118 3244 y FB(Example)31 b(7.)42 b FO(Let)j FN(D)r FO(\()p FI(R)651 3214 y FL(n)702 3244 y FO(\))g(b)r(e)g(the)g(space)e(of)i(compactly)c(supp)r(orted)k(in-) -118 3343 y(\014nitely)40 b(di\013eren)n(tiable)e(functions)j(on)g FI(R)1242 3313 y FL(n)1293 3343 y FO(,)k FN(D)1432 3313 y FM(0)1455 3343 y FO(\()p FI(R)1542 3313 y FL(n)1593 3343 y FO(\))d(b)r(e)f(its)g(dual)f(space,)-118 3443 y FN(F)12 b(L)4 3413 y FK(1)41 3443 y FO(\()p FI(R)127 3413 y FL(n)178 3443 y FO(\))31 b(the)f(F)-7 b(ourier)29 b(algebra,)f FN(P)12 b(M)d FO(\()p FI(R)1235 3413 y FL(n)1286 3443 y FO(\))31 b(the)f(space)g(dual)f(to)h FN(F)12 b(L)2132 3413 y FK(1)2169 3443 y FO(\()p FI(R)2255 3413 y FL(n)2306 3443 y FO(\))-118 3543 y(\(the)30 b(space)e(of)h(pseudo-measures\),)e FN(F)41 b FO(the)29 b(F)-7 b(ourier)27 b(transform,)g FN(')20 b FP(\003)f FN( )32 b FO(the)-118 3642 y(con)n(v)n(olution)24 b(of)k(t)n(w)n(o)e(functions)h(in)g FN(D)r FO(\()p FI(R)1190 3612 y FL(n)1242 3642 y FO(\),)41 b(~)-55 b FN(')p FO(\()p FN(x)p FO(\))24 b(=)p 1602 3570 231 4 v 23 w FN(')p FO(\()p FP(\000)p FN(x)p FO(\))q(.)6 3742 y(Consider)i(the)i(follo)n(wing)c(p)r (olynomial)f(in)j(six)h(v)-5 b(ariables)109 3911 y FN(p)p FO(\()p FN(x)230 3923 y FK(1)268 3911 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(x)513 3923 y FK(6)551 3911 y FO(\))23 b(=)g FN(x)741 3877 y FK(2)741 3932 y(1)797 3911 y FO(+)18 b FN(x)927 3877 y FK(2)927 3932 y(2)983 3911 y FO(+)g FN(x)1113 3877 y FK(2)1113 3932 y(3)1169 3911 y FP(\000)g FO(1)g(+)g FN(i)p FO(\()p FN(x)1503 3877 y FK(2)1503 3932 y(4)1559 3911 y FO(+)g FN(x)1689 3877 y FK(2)1689 3932 y(5)1745 3911 y FO(+)g FN(x)1875 3877 y FK(2)1875 3932 y(6)1932 3911 y FP(\000)g FO(1\))p FN(:)p eop %%Page: 57 61 57 60 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(57)6 96 y(Let)28 b FN(s)194 108 y FL(i)222 96 y FO(,)f FN(r)309 108 y FL(i)338 96 y FO(,)g FN(i)c FO(=)g(1,)k FN(:)14 b(:)g(:)27 b FO(,)h FN(m)p FO(,)g(b)r(e)g(p)r(olynomials)22 b(satisfying)j(the)j(relation) 656 328 y FN(p)p FO(\()p FN(x)19 b FP(\000)f FN(y)s FO(\))23 b(=)1096 224 y FL(m)1065 249 y Fy(X)1071 426 y FL(i)p FK(=1)1199 328 y FN(s)1238 340 y FL(i)1266 328 y FO(\()p FN(x)p FO(\))14 b FN(r)1428 340 y FL(i)1457 328 y FO(\()p FN(y)s FO(\))-118 579 y(for)30 b(an)n(y)h FN(x;)14 b(y)31 b FP(2)e FI(R)467 549 y FK(6)510 579 y FO(.)48 b(Let)31 b FN(u;)14 b(v)31 b FP(2)e FN(D)r FO(\()p FI(R)1130 549 y FL(n)1181 579 y FO(\))j(and)f FN(a)1454 591 y FL(i)1510 579 y FO(=)d FN(us)1690 591 y FL(i)1717 579 y FO(,)k FN(b)1808 591 y FL(i)1864 579 y FO(=)c FN(v)s(r)2037 591 y FL(i)2066 579 y FO(.)47 b(Ob)n(vi-)-118 679 y(ously)-7 b(,)28 b FN(a)159 691 y FL(i)186 679 y FO(,)h FN(b)274 691 y FL(i)327 679 y FP(2)d FN(D)r FO(\()p FI(R)565 649 y FK(6)608 679 y FO(\).)41 b(Consider)27 b(the)i(op)r(erators)e FN(A)1627 691 y FL(i)1681 679 y FO(=)d FN(M)1851 691 y FL(a)1887 699 y Fv(i)1917 679 y FO(,)30 b FN(B)2033 691 y FL(i)2086 679 y FO(=)24 b FN(M)2256 691 y FL(b)2285 699 y Fv(i)2316 679 y FO(,)-118 779 y FN(i)45 b FO(=)h(1,)41 b FN(:)14 b(:)g(:)27 b FO(,)45 b FN(m)c FO(in)g(the)g(space)g FN(L)1039 748 y FK(2)1075 779 y FO(\()p FI(R)1162 748 y FK(6)1205 779 y FO(\))h(\(here)f FN(M)1586 791 y FL(f)1669 779 y FO(is)g(the)g(op)r(erator)f(of)-118 878 y(m)n(ultiplication)24 b(b)n(y)31 b(the)f(function)g FN(f)9 b FO(\).)46 b(Since)29 b(the)i(F)-7 b(ourier)28 b(transform)h(of)h(a)-118 978 y(pseudo-measure)22 b(\010)i(b)r(elongs)f(to)h FN(L)1014 948 y FM(1)1084 978 y FO(\()p FI(R)1170 948 y FK(6)1214 978 y FO(\),)h(the)g(op)r(erator)e FN(T)34 b FO(=)22 b FN(F)2001 948 y FM(\000)p FK(1)2091 978 y FN(M)2172 990 y FL(F)9 b FK(\010)2274 978 y FN(F)-118 1077 y FO(is)28 b(w)n(ell)e(de\014ned)j(in)f(the)h(space)f FN(L)945 1047 y FK(2)982 1077 y FO(\()p FI(R)1068 1047 y FK(6)1112 1077 y FO(\).)41 b(F)-7 b(urthermore)26 b(a)j(direct)e(computa-)-118 1177 y(tion)g(sho)n(ws)f(that)442 1300 y Fy(\020)522 1288 y FL(m)491 1313 y Fy(X)498 1490 y FL(i)p FK(=1)625 1392 y FN(M)706 1404 y FL(b)735 1412 y Fv(i)765 1392 y FN(T)12 b(M)907 1404 y FL(a)943 1412 y Fv(i)972 1392 y FN(';)i( )1120 1300 y Fy(\021)1193 1392 y FO(=)23 b FP(h)p FN(p)14 b FO(\010)p FN(;)g(u')k FP(\003)1655 1370 y Fy(f)1649 1392 y FO(\026)-45 b FN(v)s( )s FP(i)348 b FO(\(1.16\))-118 1644 y(for)30 b(an)n(y)f FN(u)p FO(,)i FN(v)s FO(,)g FN(')p FO(,)g FN( )g FP(2)c FN(D)r FO(\()p FI(R)803 1614 y FK(6)846 1644 y FO(\),)k(where)f FP(h\001)p FN(;)14 b FP(\001i)31 b FO(is)e(the)i(pairing)c(of)j(the)h(spaces)-118 1744 y FN(D)r FO(\()p FI(R)39 1713 y FK(6)82 1744 y FO(\),)44 b FN(D)252 1713 y FM(0)275 1744 y FO(\()p FI(R)361 1713 y FK(6)405 1744 y FO(\))c(\(since)f FN(P)12 b(M)d FO(\()p FI(R)966 1713 y FK(6)1009 1744 y FO(\))44 b FP(\032)f FN(D)r FO(\()p FI(R)1351 1713 y FK(6)1394 1744 y FO(\))1426 1713 y FM(0)1449 1744 y FO(,)h FN(p)14 b FO(\010)43 b FP(2)h FN(D)1845 1713 y FM(0)1868 1744 y FO(\()p FI(R)1954 1713 y FK(6)1998 1744 y FO(\))c(for)f(an)n(y)-118 1843 y(p)r(olynomial)23 b FN(p)k FO(in)g(six)f(v)-5 b(ariables\).)34 b(Since)27 b(the)h(set)487 2012 y FN(L)22 b FO(=)h FP(f)p FN(u')18 b FP(\003)905 1990 y FO(~)879 2012 y(\026)-45 b FN(v)s( )26 b FP(j)d FN(';)14 b( )s(;)g(u;)g(v)26 b FP(2)d FN(D)r FO(\()p FI(R)1617 1978 y FK(6)1660 2012 y FO(\))p FP(g)-118 2182 y FO(is)33 b(dense)i(in)f FN(D)r FO(\()p FI(R)467 2152 y FK(6)510 2182 y FO(\),)j(the)e(existence)e(of)i (a)f(pseudo-measure)e(\010)i(suc)n(h)g(that)-118 2281 y FN(p)14 b FO(\010)23 b(=)f(0)h(and)30 b(\026)-49 b FN(p)13 b FO(\010)23 b FP(6)p FO(=)g(0)g(w)n(ould)f(imply)e(the)j (existence)f(of)h(functions)g FN(u)p FO(,)h FN(v)i FO(suc)n(h)-118 2381 y(that)146 2504 y Fy(\020)226 2492 y FL(m)195 2517 y Fy(X)202 2694 y FL(i)p FK(=1)329 2596 y FN(M)419 2562 y FM(\003)410 2616 y FL(b)439 2624 y Fv(i)469 2596 y FN(T)12 b(M)620 2562 y FM(\003)611 2616 y FL(a)647 2624 y Fv(i)676 2596 y FN(';)i( )824 2504 y Fy(\021)897 2596 y FO(=)23 b FP(h)7 b FO(\026)-49 b FN(p)14 b FO(\010)p FN(;)19 b FO(\026)-47 b FN(u)o(')19 b FP(\003)1359 2574 y Fy(f)1350 2596 y FN(v)s( )s FP(i)24 b(6)p FO(=)e(0)p FN(;)1206 2741 y FO(for)27 b(some)f FN(')p FO(,)i FN( )f FP(2)c FN(D)r FO(\()p FI(R)1962 2706 y FK(6)2006 2741 y FO(\))p FN(;)146 2849 y Fy(\020)235 2837 y FL(n)195 2862 y Fy(X)202 3039 y FL(i)p FK(=1)329 2941 y FN(M)410 2953 y FL(b)439 2961 y Fv(i)469 2941 y FN(T)12 b(M)611 2953 y FL(a)647 2961 y Fv(i)676 2941 y FN(';)i( )824 2849 y Fy(\021)897 2941 y FO(=)23 b FP(h)p FN(p)14 b FO(\010)p FN(;)g(u')k FP(\003)1359 2919 y Fy(f)1353 2941 y FO(\026)-45 b FN(v)s( )s FP(i)24 b FO(=)e(0)p FN(;)1206 3086 y FO(for)27 b(all)f FN(')p FO(,)i FN( )e FP(2)d FN(D)r FO(\()p FI(R)1870 3051 y FK(6)1913 3086 y FO(\))p FN(:)-118 3255 y FO(Hence,)30 b(for)f(the)h(op)r(erator)e FN(A)827 3267 y FL(i)855 3255 y FO(,)i FN(B)971 3267 y FL(i)998 3255 y FO(,)g FN(T)41 b FO(constructed)29 b(relativ)n(ely)c(to)30 b(\010,)g FN(u)p FO(,)f FN(v)s FO(,)-118 3355 y(w)n(e)e(obtain)433 3466 y FL(m)403 3491 y Fy(X)409 3668 y FL(i)p FK(=1)537 3570 y FN(B)600 3582 y FL(i)627 3570 y FN(T)12 b(A)750 3582 y FL(i)800 3570 y FO(=)23 b(0)82 b(and)1273 3466 y FL(m)1243 3491 y Fy(X)1249 3668 y FL(i)p FK(=1)1377 3570 y FN(B)1444 3535 y FM(\003)1440 3590 y FL(i)1482 3570 y FN(T)12 b(A)1605 3535 y FM(\003)1605 3590 y FL(i)1666 3570 y FP(6)p FO(=)22 b(0)p FN(:)308 b FO(\(1.17\))-118 3811 y(Since)22 b(an)n(y)g(\014nite)g(comm)n(utativ) n(e)d(family)h(of)i(normal)e(op)r(erators)h(can)h(b)r(e)h(real-)-118 3911 y(ized)i(as)h(a)g(family)e(of)i(con)n(tin)n(uous)f(functions)h(of) g(one)g(self-adjoin)n(t)e(op)r(erator,)p eop %%Page: 58 62 58 61 bop -118 -137 a FO(58)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(one)31 b(can)g(\014nd)h(a)f(self-adjoin)n(t)e(op)r(erator)h FN(A)i FO(and)f(con)n(tin)n(uous)e(functions)i FN(f)2288 108 y FL(i)2316 96 y FO(,)-118 196 y FN(g)-78 208 y FL(i)-18 196 y FO(suc)n(h)h(that)g FN(B)421 208 y FL(i)480 196 y FO(=)e FN(f)616 208 y FL(i)643 196 y FO(\()p FN(A)p FO(\),)35 b FN(A)889 208 y FL(i)948 196 y FO(=)30 b FN(g)1083 208 y FL(i)1110 196 y FO(\()p FN(A)p FO(\).)52 b(Then)33 b(\(1.17\))f(is)f(equiv)-5 b(alen)n(t)30 b(to)-118 296 y(the)e(follo)n(wing)210 450 y FL(m)179 475 y Fy(X)185 652 y FL(i)p FK(=1)313 554 y FN(f)354 566 y FL(i)381 554 y FO(\()p FN(A)p FO(\))14 b FN(T)26 b(g)636 566 y FL(i)663 554 y FO(\()p FN(A)p FO(\))e(=)e(0)83 b(and)1286 450 y FL(m)1256 475 y Fy(X)1262 652 y FL(i)p FK(=1)1407 532 y FO(\026)1389 554 y FN(f)1430 566 y FL(i)1458 554 y FO(\()p FN(A)p FO(\))14 b FN(T)28 b FO(\026)-45 b FN(g)1712 566 y FL(i)1739 554 y FO(\()p FN(A)p FO(\))24 b FP(6)p FO(=)f(0)p FN(:)-118 823 y FO(Th)n(us,)k(it)g(remains)d(to)j(pro)n(v)n (e)f(the)i(existence)e(of)h(a)g(pseudo-measure)e(\010)i(suc)n(h)-118 922 y(that)38 b FN(p)14 b FO(\010)40 b(=)g(0)e(and)45 b(\026)-49 b FN(p)14 b FO(\010)40 b FP(6)p FO(=)g(0.)68 b(In)38 b(order)f(to)h(do)g(this,)i(w)n(e)d(consider)f(the)-118 1022 y FN(\016)s FO(-measure)24 b FN(\026)322 1034 y FK(0)386 1022 y FO(of)i(the)h(surface)f FN(S)956 992 y FK(2)1020 1022 y FO(in)f FI(R)1169 992 y FK(3)1213 1022 y FO(.)36 b(Let)27 b FN(\026)c FO(=)g FN(\026)1631 1034 y FK(0)1684 1022 y FP(\002)16 b FN(\026)1815 1034 y FK(0)1852 1022 y FO(.)37 b(It)27 b(is)e(easy)h(to)-118 1122 y(sho)n(w)h(that)380 1340 y FP(j)p FN(F)12 b(\026)p FO(\()p FN(\025)p FO(\))p FP(j)24 b FO(=)765 1244 y Fy(\014)765 1294 y(\014)765 1344 y(\014)793 1227 y(Z)839 1415 y Fu(R)886 1399 y Fx(6)931 1340 y FN(e)970 1305 y FL(i)p FK(\()p FL(\025;x)p FK(\))1160 1340 y FN(d\026)p FO(\()p FN(x)p FO(\))1364 1244 y Fy(\014)1364 1294 y(\014)1364 1344 y(\014)1416 1340 y FO(=)e FN(O)r FO(\()p FP(j)p FN(\025)p FP(j)1694 1305 y FM(\000)p FK(1)1785 1340 y FO(\))p FN(;)-118 1588 y FO(as)27 b FP(j)p FN(\025)p FP(j)c(!)h FO(+)p FP(1)j FO(in)g FI(R)534 1558 y FK(6)577 1588 y FO(.)6 1694 y(Since)38 b FN(F)12 b FO(\()373 1661 y FL(@)p 341 1675 103 4 v 341 1723 a(@)t(x)418 1731 y Fv(i)454 1694 y FN(f)d FO(\))40 b(=)g FN(\025)729 1706 y FL(i)757 1694 y FN(F)12 b(f)d FO(,)40 b(the)e(F)-7 b(ourier)36 b(transform)g FN(F)12 b FO(\()1897 1657 y FL(@)t(\026)p 1885 1675 V 1885 1723 a(@)t(x)1962 1731 y Fv(i)1998 1694 y FO(\))38 b(b)r(elongs)-118 1817 y(to)29 b FN(L)42 1787 y FM(1)112 1817 y FO(\()p FI(R)198 1787 y FK(6)241 1817 y FO(\),)h(so)f(that)633 1780 y FL(@)t(\026)p 621 1798 V 621 1845 a(@)t(x)698 1853 y Fv(i)759 1817 y FP(2)d FN(P)12 b(M)d FO(\()p FI(R)1081 1787 y FK(6)1124 1817 y FO(\).)42 b(The)29 b(pseudo-measure)2017 1780 y FL(@)t(\026)p 2005 1798 V 2005 1845 a(@)t(x)2082 1853 y Fv(i)2148 1817 y FO(has)f(a)-118 1930 y(compact)20 b(supp)r(ort.)35 b(Th)n(us)21 b FN(L\026)i FP(2)g FN(P)12 b(M)d FO(\()p FI(R)1199 1899 y FK(6)1242 1930 y FO(\))22 b(for)f(an)n(y)g(\014rst-order)f(di\013eren)n(tial)-118 2029 y(op)r(erator)26 b FN(L)h FO(with)g(smo)r(oth)f(co)r(e\016cien)n (ts.)36 b(Let)169 2264 y FN(L)23 b FO(=)g(\(1)18 b(+)g FN(i)p FO(\))c FN(x)634 2276 y FK(4)724 2208 y FN(@)p 681 2245 134 4 v 681 2321 a(@)5 b(x)777 2333 y FK(1)843 2264 y FP(\000)18 b FO(\(1)g FP(\000)g FN(i)p FO(\))c FN(x)1223 2276 y FK(1)1313 2208 y FN(@)p 1271 2245 V 1271 2321 a(@)5 b(x)1367 2333 y FK(4)1414 2264 y FN(;)97 b FO(and)83 b(\010)23 b(=)f FN(L\026:)-118 2501 y FO(It)30 b(is)e(easy)g(to)i(see)f(that)g FN(Lp)d FO(=)f(0)k(and)h FN(L)7 b FO(\026)-49 b FN(p)25 b FO(=)h(4\(1)19 b(+)g FN(i)p FO(\))14 b FN(x)1670 2513 y FK(1)1707 2501 y FN(x)1754 2513 y FK(4)1792 2501 y FO(,)30 b(whic)n(h)e(implies)-118 2600 y(that)g(for)f(an)n(y)g(function)g FN(')h FO(in)f FN(D)r FO(\()p FI(R)1007 2570 y FK(6)1050 2600 y FO(\),)132 2797 y FP(h)p FN(p)14 b FO(\010)p FN(;)g(')p FP(i)23 b FO(=)g FP(h)p FN(L\026;)14 b(p')p FP(i)23 b FO(=)g FP(h)p FN(\026;)14 b(L)p FO(\()p FN(p')p FO(\))p FP(i)23 b FO(=)g FP(h)p FN(\026;)14 b(pL)p FO(\()p FN(')p FO(\))p FP(i)24 b FO(=)e(0)p FN(;)132 2921 y FP(h)7 b FO(\026)-49 b FN(p)14 b FO(\010)p FN(;)g(')p FP(i)23 b FO(=)g FP(h)p FN(\026;)14 b(L)p FO(\()7 b(\026)-49 b FN(p)p FO(\))p FN(')p FP(i)19 b FO(+)f FP(h)p FN(\026;)j FO(\026)-49 b FN(pL')p FP(i)23 b FO(=)g(4\(1)18 b(+)g FN(i)p FO(\))p FP(h)p FN(x)1756 2933 y FK(1)1793 2921 y FN(x)1840 2933 y FK(4)1892 2921 y FN(\026;)c(')p FP(i)p FN(:)-118 3117 y FO(Th)n(us,)27 b FN(p)14 b FO(\010)23 b(=)g(0)k(and)34 b(\026)-49 b FN(p)14 b FO(\010)23 b(=)f(4\(1)c(+)g FN(i)p FO(\))c FN(x)1137 3129 y FK(1)1175 3117 y FN(x)1222 3129 y FK(4)1273 3117 y FN(\026)23 b FP(6)p FO(=)g(0)p FN(:)-118 3264 y FB(R)l(emark)30 b(16.)42 b FO(Instead)f(of)g(one)g(op)r(erator)e FN(A)46 b FO(=)e FN(A)1549 3234 y FM(\003)1629 3264 y FO(one)d(can)f(consider)f(a)-118 3363 y(family)28 b(of)i(comm)n(uting)d (self-adjoin)n(t)h(op)r(erators)h FQ(A)f FO(=)g(\()p FN(A)1758 3375 y FL(k)1799 3363 y FO(\))1831 3333 y FL(m)1831 3387 y(k)q FK(=1)1987 3363 y FO(whic)n(h)i(are)-118 3463 y(connected)23 b(with)f(a)h(b)r(ounded)h(op)r(erator)d FN(B)27 b FO(b)n(y)c(a)g(semilinear)18 b(relation)i(of)j(the)-118 3563 y(form)736 3701 y FL(n)697 3726 y Fy(X)703 3903 y FL(i)p FK(=1)831 3805 y FN(f)872 3817 y FL(i)899 3805 y FO(\()p FQ(A)p FO(\))14 b FN(B)k(g)1170 3817 y FL(i)1198 3805 y FO(\()p FQ(A)p FO(\))24 b(=)e(0)p FN(;)616 b FO(\(1.18\))p eop %%Page: 59 63 59 62 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(59)-118 96 y(where)23 b FN(f)159 108 y FL(i)186 96 y FO(\()p FQ(A)p FO(\),)i FN(g)410 108 y FL(i)437 96 y FO(\()p FQ(A)p FO(\))f(are)e(Borel)f(b)r (ounded)j(functions)e(of)h(the)h(family)c FQ(A)p FO(.)36 b(As)-118 196 y(b)r(efore,)27 b(one)g(can)h(asso)r(ciate)d(the)j(set) 333 445 y(\000)23 b(=)g FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(2)g FI(R)863 411 y FL(m)951 445 y FP(\002)18 b FI(R)1088 411 y FL(m)1180 445 y FP(j)1265 341 y FL(n)1226 366 y Fy(X)1232 543 y FL(i)p FK(=1)1360 445 y FN(f)1401 457 y FL(i)1428 445 y FO(\()p FN(t)p FO(\))p FN(g)1562 457 y FL(i)1590 445 y FO(\()p FN(s)p FO(\))23 b(=)g(0)p FP(g)-118 709 y FO(to)32 b(relation)d(\(1.18\))o(.)51 b(Supp)r(ort)33 b(of)f(the)g(op)r(erator)f FN(B)36 b FO(\(supp)1798 730 y Fm(A)1872 709 y FN(B)f FP(\032)30 b FN(\033)s FO(\()p FQ(A)p FO(\))23 b FP(\002)-118 809 y FN(\033)s FO(\()p FQ(A)p FO(\)\))35 b(with)e(resp)r(ect)h(to)f(the)h (family)d FQ(A)j FO(is)e(de\014ned)i(in)f(the)h(same)e(w)n(a)n(y)g(as) -118 909 y(in)j(De\014nition)f(3)i(but)g(with)f(the)h(join)n(t)f (resolution)e(of)i(the)h(iden)n(tit)n(y)e FN(E)2190 921 y Fm(A)2251 909 y FO(\()p FP(\001)p FO(\))-118 1008 y(for)e(the)i (family)c FQ(A)j FO(instead)e(of)i FN(E)979 1020 y FL(M)1053 1008 y FO(\()p FP(\001)p FO(\))h(and)f FN(E)1402 1020 y FL(N)1465 1008 y FO(\()p FP(\001)p FO(\).)54 b(If)33 b Fz(M)1804 1020 y Fu(A)1850 1008 y FO(\(\000\))h(is)d(the)j(set)-118 1108 y(of)k(all)e(b)r(ounded)j(op)r(erators)d(whose)i(supp)r(ort)g(b)r (elongs)f(to)h(\000)g(and)h(k)n(er)12 b(\001)2292 1120 y Fu(A)-118 1208 y FO(is)38 b(the)h(op)r(erator)e(of)i(m)n (ultiplication,)d FN(X)48 b FP(7!)1395 1145 y Fy(P)1483 1166 y FL(n)1483 1232 y(i)p FK(=1)1608 1208 y FN(f)1649 1220 y FL(i)1677 1208 y FO(\()p FQ(A)p FO(\))14 b FN(B)k(g)1948 1220 y FL(i)1975 1208 y FO(\()p FQ(A)p FO(\),)43 b(then)-118 1307 y(using)27 b(similar)d(argumen)n(ts)i(to)j(those)f(giv)n(en)f(in)h (Section)g(1.3.3,)f(item)g(4,)i(one)-118 1407 y(can)40 b(pro)n(v)n(e)f(that)h(k)n(er)13 b(\001)670 1419 y Fm(A)775 1407 y FP(\032)44 b Fz(M)971 1419 y Fm(A)1032 1407 y FO(\(\000\).)76 b(But)40 b(the)h(con)n(v)n(erse)e(inclusion)e(is)-118 1506 y(not)30 b(true)f(already)e(for)i FN(m)h FO(greater)e(than)i(2)f (and)g(p)r(olynomials)c FN(f)1955 1518 y FL(k)1996 1506 y FO(,)30 b FN(g)2089 1518 y FL(k)2129 1506 y FO(.)44 b(The)-118 1606 y(corresp)r(onding)34 b(example)g(can)j(b)r(e)g(deriv)n (ed)e(using)g(similar)d(argumen)n(ts)j(to)-118 1706 y(those)27 b(giv)n(en)f(in)h(Example)e(7.)-118 1843 y FB(R)l(emark)30 b(17.)42 b FO(As)28 b(has)f(b)r(een)h(sho)n(wn)f(ab)r(o)n(v)n(e,)f(an)h (op)r(erator)f FN(B)t FO(,)i FN(A)p FO(-supp)r(orted)-118 1943 y(b)n(y)40 b(the)g(set)g(\000)k(=)g FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))43 b FP(j)834 1880 y Fy(P)922 1901 y FL(n)922 1968 y(i)p FK(=1)1048 1943 y FN(f)1089 1955 y FL(i)1116 1943 y FO(\()p FN(t)p FO(\))14 b FN(g)1264 1955 y FL(i)1292 1943 y FO(\()p FN(s)p FO(\))44 b(=)f(0)p FP(g)p FO(,)g(migh)n(t)38 b(not)i(satisfy)-118 2042 y(the)31 b(corresp)r(onding)d(semilinear)d(relation)1268 1980 y Fy(P)1356 2001 y FL(n)1356 2067 y(i)p FK(=1)1481 2042 y FN(f)1522 2054 y FL(i)1549 2042 y FO(\()p FN(A)p FO(\))p FN(B)19 b(g)1797 2054 y FL(i)1824 2042 y FO(\()p FN(A)p FO(\))29 b(=)f(0)i(if)g(the)-118 2142 y(functions)c FN(f)280 2154 y FL(i)307 2142 y FO(\()p FP(\001)p FO(\),)h FN(g)484 2154 y FL(i)511 2142 y FO(\()p FP(\001)p FO(\))g(are)f(not)g(smo)r (oth.)35 b(But)27 b(this)e(is)h(true)g(if)g(the)g(op)r(erator)-118 2242 y FN(B)i FO(is)22 b(pseudo-in)n(tegral.)32 b(Let)23 b(us)h(recall)c(the)k(de\014nition)e(of)i(a)f(pseudo-in)n(tegral)-118 2341 y(op)r(erator.)6 2443 y(Let)36 b FN(\026)g FO(b)r(e)g(the)g (scalar)d(sp)r(ectral)h(measure)f(of)j(a)f(self-adjoin)n(t)e(op)r (erator)-118 2543 y FN(A)p FO(,)28 b(let)g FN(H)i FO(=)303 2476 y Fy(R)359 2496 y FM(\010)342 2572 y FL(\033)r FK(\()p FL(A)p FK(\))503 2543 y FN(l)528 2555 y FK(2)565 2543 y FO(\()p FN(N)9 b FO(\()p FN(\025)p FO(\)\))14 b FN(d\026)p FO(\()p FN(\025)p FO(\),)30 b FN(N)9 b FO(\()p FN(\025)p FO(\))25 b FP(2)f FQ(N)18 b FP([)h(f1g)p FO(\),)28 b(b)r(e)g(the)g(sp)r (ectral)-118 2653 y(resolution)33 b(of)j FN(A)p FO(,)i(and)e(let)g FN(m)g FO(b)r(e)g(a)g(b)r(ounded)g(measure)e(on)i FI(R)30 b FP(\002)24 b FI(R)42 b FO(suc)n(h)-118 2752 y(that)32 b(the)g(pro)5 b(jections)30 b(of)i FN(m)f FO(on)h(b)r(oth)g(co)r (ordinates)e(are)g(ma)5 b(jorized)29 b(b)n(y)i FN(\026)p FO(.)-118 2852 y(The)d(measure)d FN(m)j FO(is)f(said)f(to)i(b)r(e)g (regular.)34 b(Since)27 b FN(l)1503 2864 y FK(2)1540 2852 y FO(\()p FN(N)9 b FO(\))28 b(for)f FN(N)33 b FP(2)23 b FI(N)38 b FO(can)27 b(b)r(e)-118 2952 y(em)n(b)r(edded)g(in)n(to)f FN(l)467 2964 y FK(2)504 2952 y FO(\()p FP(1)p FO(\),)j(w)n(e)e(can)g (de\014ne)h(the)g(bilinear)c(form)523 3181 y(\()o FN(~)-41 b(x)q(;)13 b(~)-41 b(y)r FO(\))24 b FP(7!)845 3068 y Fy(Z)9 b(Z)983 3181 y FO(\()o FN(~)-41 b(x)q FO(\()p FN(s)p FO(\))p FN(;)13 b(~)-41 b(y)s FO(\()p FN(t)p FO(\)\))14 b FN(dm)p FO(\()p FN(t;)g(s)p FO(\))p FN(;)-118 3411 y FO(where)20 b(\()o FN(~)-41 b(x)q FO(\()p FN(s)p FO(\))p FN(;)13 b(~)-41 b(y)s FO(\()p FN(t)p FO(\)\))22 b(is)d(the)j(scalar)c (pro)r(duct)j(in)f FN(l)1389 3423 y FK(2)1426 3411 y FO(\()p FP(1)p FO(\).)35 b(If)21 b FN(m)g FO(is)f(regular,)f(then)-118 3510 y(the)29 b(bilinear)c(form)j(de\014nes)h(some)e(b)r(ounded)i(op)r (erator)e FN(T)1733 3522 y FL(m)1796 3510 y FO(.)41 b(The)28 b(op)r(erator)-118 3610 y FN(T)-69 3622 y FL(m)32 3610 y FO(is)38 b(called)e(a)j(\\pseudo-in)n(tegral)34 b(op)r(erator)j (constructed)h(relativ)n(ely)d(to)-118 3710 y FN(m)p FO(".)6 3811 y(It)e(w)n(as)f(sho)n(wn)g(b)n(y)g(Arv)n(eson)g(that)g (the)i(pseudo-in)n(tegral)28 b(op)r(erator)j FN(T)2276 3823 y FL(m)-118 3911 y FO(is)j(supp)r(orted)h(b)n(y)g(an)n(y)f (pseudo-closed)f(set)i(on)g(whic)n(h)f(the)i(measure)d FN(m)i FO(is)p eop %%Page: 60 64 60 63 bop -118 -137 a FO(60)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(concen)n(trated)j(\(the)i(set)g FN(E)h FP(\032)26 b FI(R)g FP(\002)19 b FI(R)36 b FO(is)28 b(called)f(pseudo-closed)h(if) h(its)f(com-)-118 196 y(plemen)n(t)j(is)g(a)h(union)f(of)h(coun)n (tably)f(man)n(y)f(measurable)f(sets)j(of)g(the)h(form)-118 296 y FN(X)25 b FP(\002)18 b FN(Y)41 b FP(\032)23 b FI(R)h FP(\002)18 b FI(R)p FO(\))q(.)6 395 y(Let)31 b(\000)e(b)r(e)i(the)f(c)n (haracteristic)c(binary)i(relation)f(corresp)r(onding)g(to)j(the)-118 495 y(semilinear)14 b(relation)j(\(1.10\),)j(and)f FN(m)g FO(a)g(regular)e(measure)g(with)i(supp)14 b FN(m)23 b FP(\032)g FO(\000.)-118 595 y(Then,)28 b FN(A)p FO(,)g FN(T)284 607 y FL(m)374 595 y FO(is)e(a)i(represen)n(tation)d(of)i (relation)e(\(1.10\).)36 b(Indeed,)204 767 y Fy(\000)281 730 y FL(n)242 755 y Fy(X)248 932 y FL(i)p FK(=1)375 834 y FN(f)416 846 y FL(i)444 834 y FO(\()p FN(A)p FO(\))14 b FN(T)633 846 y FL(m)710 834 y FN(g)750 846 y FL(i)777 834 y FO(\()p FN(A)p FO(\))o FN(~)-41 b(x)q(;)13 b(~)-41 b(y)1032 742 y Fy(\021)1104 834 y FO(=)1232 730 y FL(n)1192 755 y Fy(X)1198 932 y FL(i)p FK(=1)1312 767 y Fy(\000)1350 834 y FN(T)1399 846 y FL(m)1462 834 y FN(g)1502 846 y FL(i)1529 834 y FO(\()p FN(A)p FO(\))o FN(~)g(x)q(;)p 1740 767 69 4 v 14 w(f)1781 846 y FL(i)1808 834 y FO(\()p FN(A)p FO(\))o FN(~)g(y)1979 767 y Fy(\001)393 1111 y FO(=)520 1007 y FL(n)480 1032 y Fy(X)486 1209 y FL(i)p FK(=1)614 998 y Fy(Z)9 b(Z)752 1186 y FK(\000)811 1111 y FN(g)851 1123 y FL(i)879 1111 y FO(\()p FN(s)p FO(\))14 b FN(f)1037 1123 y FL(i)1064 1111 y FO(\()p FN(t)p FO(\))g(\()o FN(~)-41 b(x)q FO(\()p FN(s)p FO(\))p FN(;)13 b(~)-41 b(y)s FO(\()p FN(t)p FO(\)\))14 b FN(dm)p FO(\()p FN(t;)g(s)p FO(\))393 1363 y(=)480 1250 y Fy(Z)c(Z)619 1439 y FK(\000)678 1363 y FO(\010\()p FN(t;)k(s)p FO(\))g(\()o FN(~)-41 b(x)q FO(\()p FN(s)p FO(\))p FN(;)13 b(~)-41 b(y)s FO(\()p FN(t)p FO(\)\))14 b FN(dm)p FO(\()p FN(t;)g(s)p FO(\))24 b(=)e(0)p FN(:)-118 1619 y FQ(1.3.4)94 b(Irreducible)31 b(represen)m(tations)h(of)g(semilinear)d(relations)-118 1773 y FO(No)n(w)21 b(w)n(e)g(study)h(irreducible)c(represen)n(tations) h(of)i(semilinear)c(relations,)i(i.e.,)-118 1872 y(an)27 b(irreducible)d(families)g(of)j(op)r(erators)f FN(A)d FO(=)g FN(A)1408 1842 y FM(\003)1446 1872 y FO(,)28 b FN(B)t FO(,)g FN(B)1682 1842 y FM(\003)1748 1872 y FO(satisfying)753 2003 y FL(n)714 2028 y Fy(X)720 2205 y FL(i)p FK(=1)847 2107 y FN(f)888 2119 y FL(i)916 2107 y FO(\()p FN(A)p FO(\))14 b FN(B)k(g)1177 2119 y FL(i)1205 2107 y FO(\()p FN(A)p FO(\))24 b(=)e(0)p FN(:)-118 2353 y FO(It)k(is)e(clear)f(that)j (if)f FN(f)540 2365 y FL(i)567 2353 y FO(,)h FN(g)656 2365 y FL(i)708 2353 y FO(are)f(p)r(olynomials,)c(then)k(suc)n(h)g(a)g (family)e(of)i(op)r(era-)-118 2452 y(tors)i(de\014nes)g(an)g (irreducible)d(represen)n(tation)h(of)j(the)g FP(\003)p FO(-algebra)c(generated)-118 2552 y(b)n(y)34 b FN(a)i FO(=)e FN(a)227 2522 y FM(\003)265 2552 y FO(,)j FN(b)p FO(,)g FN(b)457 2522 y FM(\003)529 2552 y FO(and)e(the)g(relation)1160 2490 y Fy(P)1248 2510 y FL(n)1248 2577 y(i)p FK(=1)1374 2552 y FN(f)1415 2564 y FL(i)1442 2552 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)1654 2564 y FL(i)1681 2552 y FO(\()p FN(a)p FO(\))36 b(=)e(0.)59 b(In)35 b(what)-118 2652 y(follo)n(ws)26 b(w)n(e)i(shall)e(mean)i(b)n(y)g(a)h(represen)n(tation) d(of)j(the)g(semilinear)24 b(relation)-118 2751 y(a)i(triple)e(\()p FN(A)g FO(=)f FN(A)438 2721 y FM(\003)476 2751 y FN(;)14 b(B)t(;)g(B)684 2721 y FM(\003)722 2751 y FO(\))27 b(satisfying)d(the)i (semilinear)c(relation)h(instead)j(of)-118 2851 y(the)i(pair)e(\()p FN(A)d FO(=)g FN(A)463 2821 y FM(\003)501 2851 y FN(;)14 b(B)t FO(\),)29 b(if)e(it)g(do)r(es)g(not)g(lead)g(to)g(an)n(y)g (confusion.)-118 2997 y FQ(1.)58 b FO(W)-7 b(e)36 b(b)r(egin)e(with)g (\014nite-dimensional)c(representations)i(and)j(establish)-118 3096 y(a)e(connection)f(b)r(et)n(w)n(een)i(irreducible)c(represen)n (tations)g(of)40 b(\(1.10\))33 b(and)g(the)-118 3196 y(corresp)r(onding)25 b(graph.)-118 3354 y FQ(Prop)s(osition)30 b(20.)41 b FB(If)24 b(a)h(family)h FO(\()p FN(A;)14 b(B)t(;)g(B)1259 3324 y FM(\003)1298 3354 y FO(\))24 b FB(de\014nes)h(a)f (\014nite-dimensional)-118 3454 y(irr)l(e)l(ducible)43 b(r)l(epr)l(esentation)f(of)60 b FO(\(1.10\))o FB(,)46 b(then)41 b(the)h(c)l(orr)l(esp)l(onding)h(gr)l(aph)-118 3553 y FO(\000)23 b Fr(\026)-8 3568 y FL(\033)r FK(\()p FL(A)p FK(\))168 3553 y FB(is)30 b(c)l(onne)l(cte)l(d.)39 b(F)-6 b(or)30 b(every)h(\014nite)e(c)l(onne)l(cte)l(d)h(sub)l(gr)l (aph)h FO(\()p FN(D)r(;)14 b FO(\000)23 b Fr(\026)2221 3565 y FL(D)2281 3553 y FO(\))p FB(,)-118 3653 y(ther)l(e)29 b(exists)f(an)g(irr)l(e)l(ducible)i(r)l(epr)l(esentation)f FO(\()p FN(A;)14 b(B)t(;)g(B)1675 3623 y FM(\003)1714 3653 y FO(\))29 b FB(with)g FN(\033)s FO(\()p FN(A)p FO(\))24 b(=)f FN(D)r FB(.)-118 3811 y(Pr)l(o)l(of.)43 b FO(In)23 b(fact,)i(if)d(\000)h Fr(\026)606 3826 y FL(\033)r FK(\()p FL(A)p FK(\))752 3811 y FO(=)g(\000)892 3823 y FK(1)939 3811 y FP([)10 b FO(\000)1056 3823 y FK(2)1116 3811 y FO(is)22 b(the)i(union)e(of)h(t)n(w)n(o)g(connected)g(sub-)-118 3911 y(graphs)i(then,)i(b)n(y)e(Theorem)g(6,)h(the)g(sp)r(ectral)f (subspaces)g FN(A)i FO(corresp)r(onding)p eop %%Page: 61 65 61 64 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(61)-118 96 y(to)25 b(the)g(v)n (ertices)e(of)38 b(\000)576 108 y FK(1)638 96 y FO(and)25 b(\000)849 108 y FK(2)911 96 y FO(are)f(in)n(v)-5 b(arian)n(t)22 b(with)i(resp)r(ect)h(to)f FN(B)30 b FO(and)24 b FN(B)2277 66 y FM(\003)2316 96 y FO(.)-118 196 y(This)i(sho)n(ws)h(that)h(\()p FN(A;)14 b(B)t(;)g(B)792 166 y FM(\003)831 196 y FO(\))28 b(is)e(reducible.)6 296 y(Let)38 b(\000)i Fr(\026)292 311 y FL(\033)r FK(\()p FL(A)p FK(\))476 296 y FO(b)r(e)e(connected.)68 b(The)38 b(family)c(of)k(op)r(erators)e FN(A)p FO(,)41 b FN(B)t FO(,)f FN(B)2300 266 y FM(\003)-118 395 y FO(de\014ned)28 b(b)n(y)242 680 y FN(A)23 b FO(=)415 488 y Fy(0)415 634 y(B)415 687 y(@)487 552 y FN(\025)535 564 y FK(1)871 552 y FO(0)660 648 y(.)693 673 y(.)725 698 y(.)509 806 y(0)285 b FN(\025)884 818 y FL(m)947 488 y Fy(1)947 634 y(C)947 687 y(A)1034 680 y FN(;)180 b FP(f)p FN(\025)1327 692 y FK(1)1364 680 y FN(;)14 b(:)g(:)g(:)f(;)h(\025)1596 692 y FL(m)1660 680 y FP(g)22 b FO(=)h FN(D)r(;)1300 897 y(\025)1348 909 y FL(i)1399 897 y FP(6)p FO(=)g FN(\025)1535 909 y FL(j)1570 897 y FN(;)180 b(i)23 b FP(6)p FO(=)g FN(j;)237 1131 y(B)k FO(=)c(\()p FN(b)483 1143 y FL(ij)541 1131 y FO(\))573 1096 y FL(m)573 1151 y(i;j)s FK(=1)735 1131 y FN(;)180 b(b)974 1143 y FL(ij)1056 1131 y FO(=)1143 989 y Fy(\()1210 1074 y FO(0)p FN(;)83 b FO(\()p FN(\025)1438 1086 y FL(i)1466 1074 y FN(;)14 b(\025)1551 1086 y FL(j)1586 1074 y FO(\))33 b FN(=)-51 b FP(2)23 b FO(\000)p FP(j)1795 1086 y FL(D)1855 1074 y FN(;)1210 1194 y FO(1)p FN(;)83 b FO(\()p FN(\025)1438 1206 y FL(i)1466 1194 y FN(;)14 b(\025)1551 1206 y FL(j)1586 1194 y FO(\))24 b FP(2)f FO(\000)p FP(j)1795 1206 y FL(D)1855 1194 y FN(;)-118 1367 y FO(is)40 b(irreducible.)73 b(Indeed,)44 b(the)d(relation)d([)p FN(C)q(;)14 b(A)p FO(])46 b(=)f(0)40 b(implies)d(that)42 b FN(C)47 b FO(is)-118 1467 y(diagonal,)37 b FN(C)48 b FO(=)40 b(diag\()p FN(c)684 1479 y FK(1)721 1467 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(c)956 1479 y FK(2)993 1467 y FO(\).)69 b(F)-7 b(rom)37 b([)p FN(C)q(;)14 b(B)t FO(])42 b(=)f(0,)f(it)e(follo)n(ws)d(that)-118 1566 y FN(c)-82 1578 y FL(k)-41 1566 y FN(b)-5 1578 y FL(k)q(l)85 1566 y FO(=)27 b FN(b)213 1578 y FL(k)q(l)275 1566 y FN(c)311 1578 y FL(l)337 1566 y FO(.)46 b(Since)30 b(\000)e Fr(\026)741 1581 y FL(\033)r FK(\()p FL(A)p FK(\))918 1566 y FO(is)h(connected,)j(w)n(e)e(ha)n(v)n(e)g(that)h(there)f(exists) -118 1666 y(a)36 b(p)r(erm)n(utation)e(\()p FN(l)501 1678 y FK(1)538 1666 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(l)762 1678 y FL(m)824 1666 y FO(\))38 b FP(2)g FN(S)1038 1678 y FL(m)1138 1666 y FO(suc)n(h)e(that)g(\()p FN(\025)1602 1678 y FL(l)1623 1687 y Fv(k)1664 1666 y FN(;)14 b(\025)1749 1678 y FL(l)1770 1687 y Fv(k)q Fx(+1)1882 1666 y FO(\))38 b FP(2)g FO(\000)f Fr(\026)2169 1681 y FL(\033)r FK(\()p FL(A)p FK(\))2316 1666 y FO(,)-118 1766 y(hence)32 b FN(b)153 1778 y FL(l)174 1787 y Fv(k)210 1778 y FL(;l)251 1787 y Fv(k)q Fx(+1)392 1766 y FO(=)e(1.)50 b(This)31 b(implies)d(that)33 b FN(c)1303 1778 y FK(1)1370 1766 y FO(=)d FN(:)14 b(:)g(:)31 b FO(=)f FN(c)1724 1778 y FL(m)1787 1766 y FO(,)j(and)f FN(C)k FO(=)30 b FN(c)2235 1778 y FK(1)2273 1766 y FN(I)7 b FO(.)-118 1865 y(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n(e)f FN(\033)s FO(\()p FN(A)p FO(\))f(=)d FN(D)r FO(.)p 2278 1865 4 57 v 2282 1812 50 4 v 2282 1865 V 2331 1865 4 57 v 6 2027 a(W)-7 b(e)20 b(also)c(giv)n(e)h(a)i (reform)n(ulation)14 b(of)19 b(this)f(statemen)n(t)g(for)g(the)h (symmetrical)-118 2127 y(case.)-118 2276 y FQ(Prop)s(osition)30 b(21.)41 b FB(If)f FN(A)23 b FO(=)g FN(A)894 2246 y FM(\003)932 2276 y FB(,)h FN(B)j FO(=)c FN(B)1226 2246 y FM(\003)1286 2276 y FB(is)f(an)h(irr)l(e)l(ducible)g(\014nite-dimen-)-118 2376 y(sional)j(r)l(epr)l(esentation)g(of)44 b FO(\(1.10\))o FB(,)27 b(then)e(the)g(gr)l(aph)i FO(\000)1618 2388 y FL(s)1676 2376 y Fr(\026)1711 2391 y FL(\033)r FK(\()p FL(A)p FK(\))1883 2376 y FB(is)e(c)l(onne)l(cte)l(d.)-118 2475 y(F)-6 b(or)41 b(every)h(\014nite)e(c)l(onne)l(cte)l(d)h(sub)l(gr) l(aph)g FO(\()p FN(D)r(;)14 b FO(\000)1432 2487 y FL(s)1511 2475 y Fr(\026)1546 2490 y FL(\033)r FK(\()p FL(A)p FK(\))1692 2475 y FO(\))p FB(,)44 b(ther)l(e)d(exists)f(an)-118 2575 y(irr)l(e)l(ducible)d(p)l(air)g FN(A)d FO(=)g FN(A)725 2545 y FM(\003)764 2575 y FB(,)j FN(B)i FO(=)33 b FN(B)1093 2545 y FM(\003)1167 2575 y FB(satisfying)45 b FO(\(1.10\))35 b FB(such)h(that)g FN(D)g FO(=)-118 2674 y FN(\033)s FO(\()p FN(A)p FO(\))p FB(.)6 2824 y FO(The)23 b(pro)r(of)f(is)g(the)h (same)e(as)h(the)h(one)f(giv)n(en)f(ab)r(o)n(v)n(e,)h(but)i(with)e (\000)2036 2836 y FL(s)2094 2824 y FO(replac-)-118 2923 y(ing)28 b(\000.)40 b(Note)29 b(that)g(the)g(constructed)g(op)r(erator) e FN(B)33 b FO(is)28 b(self-adjoin)n(t)e(b)r(ecause)-118 3023 y(the)i(graph)e(\000)312 3035 y FL(s)375 3023 y FO(is)h(symmetrical.)-118 3164 y FQ(2.)80 b FO(In)42 b(what)g(follo)n(ws,)h(w)n(e)f(in)n(v)n(estigate)d(irreducible)f (represen)n(tations)i(of)-118 3264 y(\(1.10\))26 b(on)h(a)g(Hilb)r(ert) e(space)i FN(H)7 b FO(,)27 b(where)f FN(H)34 b FO(is)26 b(not)h(necessary)e(\014nite-dimen-)-118 3363 y(sional.)33 b(First,)24 b(w)n(e)g(pro)n(v)n(e)f(an)h(analogue)e(of)i(the)h(theorem) e(on)h(connectedness)-118 3463 y(of)j(the)h(graph)d(\(Prop)r(osition)f (20\).)36 b(F)-7 b(or)27 b(this)f(purp)r(ose,)h(w)n(e)g(recall)d(some)i (def-)-118 3563 y(initions.)-118 3712 y FQ(De\014nition)31 b(4.)40 b FB(A)d(subset)f FN(F)47 b FP(\032)36 b FI(R)29 b FP(\002)23 b FI(R)43 b FB(is)37 b(c)l(al)t(le)l(d)h(mar)l(ginal)t(ly) g(nul)t(l)f(with)-118 3811 y(r)l(esp)l(e)l(ct)g(to)h(a)f(me)l(asur)l(e) h FN(\026)f FO(\()p FB(or)h(a)g(class)g(of)h(me)l(asur)l(es)e(e)l (quivalent)g(to)h FN(\026)p FO(\))g FB(if)-118 3911 y FN(F)d FP(\032)23 b FO(\()p FN(\013)c FP(\002)f FI(R)p FO(\))24 b FP([)19 b FO(\()p FI(R)25 b FP(\002)18 b FN(\014)t FO(\))p FB(,)31 b(wher)l(e)f FN(\026)p FO(\()p FN(\013)p FO(\))24 b(=)f FN(\026)p FO(\()p FN(\014)t FO(\))h(=)e(0)p FB(.)p eop %%Page: 62 66 62 65 bop -118 -137 a FO(62)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)6 96 y FO(Let,)j(further,)535 276 y(\000\()p FN(M)9 b FO(\))23 b(=)g FP(f)p FN(y)i FP(j)e(9)p FN(x)h FP(2)f FN(M)18 b FO(:)27 b(\()p FN(x;)14 b(y)s FO(\))24 b FP(2)g FO(\000)p FP(g)p FN(;)446 411 y FO(\000)498 377 y FM(\000)p FK(1)587 411 y FO(\()p FN(M)9 b FO(\))23 b(=)g FP(f)p FN(x)g FP(j)g(9)p FN(y)j FP(2)d FN(M)18 b FO(:)27 b(\()p FN(x;)14 b(y)s FO(\))24 b FP(2)g FO(\000)p FP(g)p FN(;)-118 590 y FO(and)35 b(let)f FN(M)268 560 y FL(c)336 590 y FO(b)r(e)i(the)f(complemen)n(t)d (of)j(the)g(set)g FN(M)9 b FO(.)58 b(In)35 b(what)g(follo)n(ws)d(w)n(e) -118 690 y(shall)23 b(assume)g(that)j(\000\()p FN(M)9 b FO(\))25 b(and)g(\000)978 660 y FM(\000)p FK(1)1067 690 y FO(\()p FN(M)9 b FO(\))26 b(are)e(Borel)f(for)i(an)n(y)g(Borel)e (set)i FN(M)9 b FO(.)6 790 y(The)32 b(concepts)f(de\014ned)i(b)r(elo)n (w)d(generalize)e(those)j(of)h(quasi-in)n(v)-5 b(ariance)-118 889 y(and)32 b(ergo)r(dicit)n(y)d(of)k(a)f(measure.)50 b(F)-7 b(or)31 b(graphs)h(related)e(to)j(dynamical)c(sys-)-118 989 y(tems,)e(w)n(e)g(will)e(also)g(discuss)h(these)i(notions)e(in)h (Section)g(2.1.1.)-118 1152 y FQ(De\014nition)k(5.)40 b FB(We)29 b(c)l(al)t(l)g(a)f(set)g FN(M)j FP(\032)23 b FI(R)34 b FB(right-invariant)j FO(\()p FB(left-invariant)8 b FO(\))-118 1252 y FB(with)25 b(r)l(esp)l(e)l(ct)f(to)h FO(\000)f FB(and)h(a)g(me)l(asur)l(e)f FN(\026)p FB(,)i(if)f(the)g(set) f FN(M)1555 1222 y FL(c)1595 1252 y FP(\002)7 b FN(M)15 b FP(\\)7 b FO(\000)25 b(\()p FB(r)l(esp)l(e)l(ctively)-118 1352 y FN(M)15 b FP(\002)6 b FN(M)139 1322 y FL(c)178 1352 y FP(\\)g FO(\000\))25 b FB(is)f(mar)l(ginal)t(ly)i(nul)t(l)9 b FO(;)26 b FN(M)32 b FB(is)25 b(invariant)g(if)g(it)f(is)g(left-)h (and)f(right-)-118 1451 y(invariant.)47 b(A)32 b(me)l(asur)l(e)g FN(\026)h FB(is)f(c)l(al)t(le)l(d)43 b FO(\()p FB(left-,)33 b(right-)p FO(\))g FB(quasi-invariant)h(with)-118 1551 y(r)l(esp)l(e)l(ct)28 b(to)g FO(\000)g FB(if)h(the)f(set)g FO(\000\()p FN(M)9 b FO(\))15 b FP([)g FO(\000)1015 1521 y FM(\000)p FK(1)1104 1551 y FO(\()p FN(M)9 b FO(\))29 b(\()p FB(r)l(esp)l(e)l(ctively)g FO(\000)1803 1521 y FM(\000)p FK(1)1892 1551 y FO(\()p FN(M)9 b FO(\))p FB(,)29 b FO(\000\()p FN(M)9 b FO(\)\))-118 1651 y FB(is)33 b(of)h(non-zer)l(o) f(me)l(asur)l(e)f(for)i(every)g(Bor)l(el)f(set)g FN(M)41 b FB(such)33 b(that)g FN(\026)p FO(\()p FN(M)9 b FO(\))29 b FP(6)p FO(=)g(0)p FB(,)-118 1750 y FO(\000\()p FN(M)9 b FO(\))p FB(,)39 b FO(\000)204 1720 y FM(\000)p FK(1)293 1750 y FO(\()p FN(M)9 b FO(\))36 b FP(2)g Fz(B)p FO(\()p FI(R)q FO(\))p FB(.)66 b(A)36 b(sp)l(e)l(ctr)l(al)h(typ)l(e)g FN(\026)g FB(is)g(c)l(al)t(le)l(d)47 b FO(\()p FB(left-,)40 b(right-)r FO(\))-118 1850 y FB(er)l(go)l(dic)26 b(with)g(r)l(esp)l(e)l (ct)e(to)h FO(\000)g FB(if)h(any)33 b FO(\()p FB(left-,)26 b(right-)p FO(\))g FB(invariant)g(set)e(is)i(a)f FN(\026)p FB(-nul)t(l)-118 1949 y(set)k(or)h(has)h(the)f FN(\026)p FB(-nul)t(l)f(c)l(omplement.)6 2113 y FO(Note)e(that)f(for)g(an)n(y)g (op)r(erator)e FN(A)g FO(=)e FN(A)1234 2083 y FM(\003)1299 2113 y FO(there)k(exists)f(the)i(trivial)c(repre-)-118 2213 y(sen)n(tation)29 b(\()p FN(A;)14 b FO(0)p FN(;)g FO(0\))31 b(of)g(the)h(relation)c(\(1.10\).)46 b(Therefore,)31 b(it)g(is)e(natural)h(to)-118 2312 y(consider)25 b(represen)n(tations)g (without)i(trivial)d(parts.)-118 2476 y FQ(De\014nition)31 b(6.)40 b FB(We)31 b(c)l(al)t(l)h(a)f(r)l(epr)l(esentation)f FN(A)p FB(,)i FN(B)t FB(,)f FN(B)1672 2446 y FM(\003)1741 2476 y FP(\003)p FB(-c)l(omplete)f(if)i(for)-118 2575 y(any)c(non-zer)l(o)g(sp)l(e)l(ctr)l(al)g(subsp)l(ac)l(e)g FN(W)40 b FB(of)29 b(the)f(op)l(er)l(ator)h FN(A)f FB(one)g(of)h(the)f (op)l(er-)-118 2675 y(ators)i FN(B)k FB(and)c FN(B)418 2645 y FM(\003)486 2675 y FB(is)g(not)g(e)l(qual)g(to)f(zer)l(o)h(on)g FN(W)12 b FB(.)6 2839 y FO(It)20 b(is)f(easy)f(to)i(sho)n(w)e(that)i (an)n(y)f(irreducible)d(represen)n(tation)g(of)k(dimension)-118 2938 y(greater)26 b(than)h(one)h(is)e FP(\003)p FO(-complete.)-118 3102 y FQ(Prop)s(osition)k(22.)41 b FB(If)21 b(a)h(r)l(epr)l (esentation)f FO(\()p FN(A;)14 b(B)t(;)g(B)1538 3072 y FM(\003)1577 3102 y FO(\))22 b FB(of)39 b FO(\(1.10\))21 b FB(is)g FP(\003)p FB(-c)l(omp-)-118 3201 y(lete,)36 b(then)f(the)f(sp)l(e)l(ctr)l(al)h(me)l(asur)l(e)f(of)i(the)e(op)l(er)l (ator)i FN(A)e FB(is)h(quasi-invariant)-118 3301 y(with)30 b(r)l(esp)l(e)l(ct)g(to)f FO(\000)p FB(.)-118 3465 y(Pr)l(o)l(of.)43 b FO(If)37 b FN(E)295 3477 y FL(A)349 3465 y FO(\()p FN(M)9 b FO(\))39 b FP(6)p FO(=)e(0)f(but)h FN(E)944 3477 y FL(A)999 3465 y FO(\(\000)1083 3434 y FM(\000)p FK(1)1172 3465 y FO(\()p FN(M)9 b FO(\)\))38 b(=)g(0)e(and)g FN(E)1808 3477 y FL(A)1863 3465 y FO(\(\000\()p FN(M)9 b FO(\)\))38 b(=)g(0,)-118 3564 y(then)405 3664 y FN(B)t(E)533 3676 y FL(A)587 3664 y FO(\()p FN(M)9 b FO(\))23 b(=)g FN(E)913 3676 y FL(A)967 3664 y FO(\(\000)1051 3630 y FM(\000)p FK(1)1141 3664 y FO(\()p FN(M)9 b FO(\)\))p FN(B)t(E)1455 3676 y FL(A)1510 3664 y FO(\()p FN(M)g FO(\))23 b(=)f(0)-118 3811 y(and)399 3911 y FN(B)466 3877 y FM(\003)505 3911 y FN(E)566 3923 y FL(A)620 3911 y FO(\()p FN(M)9 b FO(\))23 b(=)g FN(E)946 3923 y FL(A)1000 3911 y FO(\(\000\()p FN(M)9 b FO(\)\))p FN(B)1337 3877 y FM(\003)1376 3911 y FN(E)1437 3923 y FL(A)1492 3911 y FO(\()p FN(M)g FO(\))23 b(=)f(0)p FN(;)p eop %%Page: 63 67 63 66 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(63)-118 96 y(hence)30 b FN(B)i Fr(\026)27 b FN(E)333 108 y FL(A)388 96 y FO(\()p FN(M)9 b FO(\))p FN(H)34 b FO(=)27 b(0)j(and)g FN(B)1040 66 y FM(\003)1106 96 y Fr(\026)e FN(E)1230 108 y FL(A)1284 96 y FO(\()p FN(M)9 b FO(\))p FN(H)35 b FO(=)27 b(0.)44 b(Th)n(us,)31 b(the)g(repre-)-118 196 y(sen)n(tation)26 b(\()p FN(A;)14 b(B)t(;)g(B)542 166 y FM(\003)581 196 y FO(\))28 b(is)e(not)i FP(\003)p FO(-complete.)p 2278 196 4 57 v 2282 143 50 4 v 2282 196 V 2331 196 4 57 v 6 375 a(One)37 b(can)g(also)e(sho)n(w)h(that)i(if)f(the)g(sp)r(ectral)f (measure)f(of)i FN(A)h FO(is)e(quasi-)-118 474 y(in)n(v)-5 b(arian)n(t,)41 b(then)g(there)f(exists)g(a)g(b)r(ounded)h(op)r(erator) e FN(B)45 b FO(suc)n(h)40 b(that)h(the)-118 574 y(triple)33 b(\()p FN(A;)14 b(B)t(;)g(B)413 544 y FM(\003)452 574 y FO(\))36 b(is)e(a)h FP(\003)p FO(-complete)e(represen)n(tation)g(of) 42 b(\(1.10\))o(.)60 b(But)36 b(w)n(e)-118 674 y(omit)26 b(the)i(pro)r(of)f(here.)-118 844 y FQ(Prop)s(osition)j(23.)41 b FB(If)f(a)f(r)l(epr)l(esentation)h FO(\()p FN(A;)14 b(B)t(;)g(B)1593 814 y FM(\003)1632 844 y FO(\))40 b FB(of)58 b FO(\(1.10\))39 b FB(is)g(irr)l(e-)-118 944 y(ducible,)31 b(then)f(the)g(sp)l(e)l(ctr)l(al)g(me)l(asur)l(e)f(of)i FN(A)f FB(is)g(er)l(go)l(dic)h(with)f(r)l(esp)l(e)l(ct)g(to)g FO(\000)p FB(.)-118 1115 y(Pr)l(o)l(of.)43 b FO(Let)22 b(\()p FN(A;)14 b(B)t(;)g(B)587 1084 y FM(\003)625 1115 y FO(\))22 b(b)r(e)g(an)f(irreducible)d(represen)n(tation)g(of)k(the)g (relation)-118 1214 y(\(1.10\))o(.)35 b(If)22 b(the)g(sp)r(ectral)d(t)n (yp)r(e)j(of)f FN(A)h FO(is)e(not)i(ergo)r(dic,)e(then)i(there)f (exists)f FN(M)32 b FP(\032)-118 1314 y FI(R)k FO(suc)n(h)30 b(that)g FN(E)405 1326 y FL(A)459 1314 y FO(\()p FN(M)9 b FO(\))27 b FP(6)p FO(=)g(0,)j FN(E)888 1326 y FL(A)943 1314 y FO(\()p FN(M)9 b FO(\))27 b FP(6)p FO(=)f FN(I)7 b FO(,)31 b(and)f(the)g(sets)g(\000)20 b FP(\\)g FO(\()p FN(M)2055 1284 y FL(c)2109 1314 y FP(\002)g FN(M)9 b FO(\),)-118 1413 y(\000)23 b FP(\\)h FO(\()p FN(M)32 b FP(\002)24 b FN(M)360 1383 y FL(c)393 1413 y FO(\))35 b(are)g(marginally)29 b(n)n(ull)k(sets.)60 b(Hence,)37 b(\000)23 b FP(\\)h FO(\()p FN(M)1971 1383 y FL(c)2028 1413 y FP(\002)f FN(M)9 b FO(\))36 b FP(\032)-118 1513 y FO(\()p FN(M)-5 1525 y FK(1)56 1513 y FP(\002)25 b FI(R)p FO(\))30 b FP([)25 b FO(\()p FI(R)31 b FP(\002)24 b FN(M)629 1525 y FK(2)666 1513 y FO(\),)39 b(where)d FN(\026)p FO(\()p FN(M)1172 1525 y FK(1)1209 1513 y FO(\))j(=)f FN(\026)p FO(\()p FN(M)1546 1525 y FK(2)1583 1513 y FO(\))g(=)g(0.)64 b(F)-7 b(rom)35 b(this)h(it)-118 1613 y(follo)n(ws)29 b(that)i FN(E)403 1625 y FL(A)458 1613 y FO(\()p FN(M)580 1583 y FL(c)614 1613 y FO(\))p FN(B)t(E)774 1625 y FL(A)828 1613 y FO(\()p FN(M)9 b FO(\))30 b(=)g FN(E)1168 1625 y FL(A)1222 1613 y FO(\()p FN(M)1344 1583 y FL(c)1399 1613 y FP(n)21 b FN(M)1543 1625 y FK(1)1579 1613 y FO(\))p FN(B)t(E)1739 1625 y FL(A)1794 1613 y FO(\()p FN(M)30 b FP(n)21 b FN(M)2081 1625 y FK(2)2118 1613 y FO(\))30 b(=)f(0,)-118 1712 y(i.e.,)i(the)g(sp)r(ectral)e(subspace)h FN(E)915 1724 y FL(A)970 1712 y FO(\()p FN(M)9 b FO(\))p FN(H)37 b FO(is)30 b(in)n(v)-5 b(arian)n(t)28 b(with)i(resp)r(ect)h(to) f FN(A)p FO(,)-118 1812 y FN(B)t FO(.)48 b(In)31 b(the)h(same)d(w)n(a)n (y)h(one)h(can)g(sho)n(w)f(that)i FN(E)1423 1824 y FL(A)1477 1812 y FO(\()p FN(M)9 b FO(\))p FN(H)38 b FO(is)30 b(in)n(v)-5 b(arian)n(t)28 b(with)-118 1912 y(resp)r(ect)21 b(to)g FN(A)p FO(,)i FN(B)430 1882 y FM(\003)468 1912 y FO(.)35 b(This)20 b(con)n(tradicts)f(the)i(irreducibilit)n(y)16 b(of)21 b FN(A)p FO(,)i FN(B)t FO(,)g FN(B)2130 1882 y FM(\003)2168 1912 y FO(.)p 2278 1912 V 2282 1859 50 4 v 2282 1912 V 2331 1912 4 57 v 6 2090 a(Note)33 b(that)f(the)h(con)n (v)n(erse)d(statemen)n(t)i(is)f(also)f(true.)51 b(Namely)-7 b(,)32 b(for)f(an)n(y)-118 2190 y(ergo)r(dic)g(measure)g FN(\026)j FO(there)f(exists)f(an)h(irreducible)d(represen)n(tation)g FN(A)p FO(,)35 b FN(B)t FO(,)-118 2289 y FN(B)-51 2259 y FM(\003)18 2289 y FO(of)30 b(relation)d(\(1.10\))j(suc)n(h)g(that)h FN(\026)f FO(is)f(a)h(sp)r(ectral)f(scalar)f(measure)g(of)i FN(A)p FO(.)-118 2389 y(Ho)n(w)n(ev)n(er)c(w)n(e)h(will)e(not)i (discuss)f(the)i(pro)r(of)f(here.)-118 2543 y FQ(3.)36 b FO(Let)28 b(\()p FN(A;)14 b(B)t FO(\))28 b(satisfy)f(the)h(follo)n (wing)23 b(semilinear)g(relation:)842 2729 y FN(AB)28 b FO(=)22 b FN(B)t(F)12 b FO(\()p FN(A)p FO(\))p FN(;)763 b FO(\(1.19\))-118 2915 y(where)29 b FN(F)12 b FO(\()p FP(\001)p FO(\))30 b(is)f(a)g(b)r(ounded)h(Borel)e(mapping)g(de\014ned) i(on)f FI(R)p FO(.)49 b(It)30 b(is)f(easy)g(to)-118 3014 y(sho)n(w)24 b(that)h(in)f(this)g(case)g(the)h(in)n(v)-5 b(ariance)21 b(of)k(a)f(set)h(\001)e FP(2)h Fz(B)p FO(\()p FI(R)q FO(\))31 b(with)24 b(resp)r(ect)-118 3114 y(to)i(\000)g(and)g(a) f(measure)f FN(\026)i FO(means)f(its)g(in)n(v)-5 b(ariance)23 b(with)i(resp)r(ect)h(to)g(the)g(map-)-118 3214 y(ping)c FN(F)12 b FO(\()p FP(\001)p FO(\),)24 b(i.e.,)f FN(\026)p FO(\()p FN(F)12 b FO(\(\001\)\))25 b(=)d FN(\026)p FO(\(\001\))i(and)f FN(\026)p FO(\()p FN(F)1349 3184 y FM(\000)p FK(1)1438 3214 y FO(\(\001\)\))i(=)d FN(\026)p FO(\(\001\).)36 b(A)24 b(measure)-118 3313 y FN(\026)34 b FO(is)f(quasi-in)n(v)-5 b(arian)n(t)30 b(with)j(resp)r(ect)h(to)h(\000)f(if)g(the)g(measures)e FN(\026)p FO(\()p FN(F)12 b FO(\()p FP(\001)p FO(\)\))36 b(and)-118 3413 y FN(\026)p FO(\()p FN(F)29 3383 y FM(\000)p FK(1)118 3413 y FO(\()p FP(\001)p FO(\)\))26 b(are)f(absolutely)d(con)n (tin)n(uous)h(with)i(resp)r(ect)g(to)g FN(\026)p FO(.)36 b(The)25 b(ergo)r(dic-)-118 3513 y(it)n(y)e(of)h(the)g(sp)r(ectral)e (measure)g(with)h(resp)r(ect)h(to)g(\000)f(means)g(ergo)r(dicit)n(y)d (of)k(the)-118 3612 y(measure)d FN(E)262 3624 y FL(A)316 3612 y FO(\()p FP(\001)p FO(\))i(with)g(resp)r(ect)f(to)h FN(F)12 b FO(\()p FP(\001)p FO(\))23 b(\(i.e.,)h(for)e(an)n(y)g FN(F)12 b FO(\()p FP(\001)p FO(\)-in)n(v)-5 b(arian)n(t)20 b(Borel)-118 3712 y(set)28 b(\001)c FP(\032)g FI(R)p FO(,)34 b(either)28 b FN(E)603 3724 y FL(A)657 3712 y FO(\(\001\))d(=)e(0)28 b(or)f FN(E)1136 3724 y FL(A)1191 3712 y FO(\(\001\))e(=)e FN(I)7 b FO(\).)39 b(Therefore,)28 b(b)n(y)f(Prop)r(o-)-118 3811 y(sition)c(23,)i(w)n(e)g(ha)n(v)n(e)f (that,)i(if)f(\()p FN(A;)14 b(B)t(;)g(B)1124 3781 y FM(\003)1163 3811 y FO(\))25 b(is)g(an)g(irreducible)c(represen)n(tation)-118 3911 y(of)34 b(\(1.19\))o(,)28 b(then)g FN(E)496 3923 y FL(A)550 3911 y FO(\()p FP(\001)p FO(\))h(is)d(ergo)r(dic)g(with)h (resp)r(ect)g(to)g FN(F)12 b FO(\()p FP(\001)p FO(\).)p eop %%Page: 64 68 64 67 bop -118 -137 a FO(64)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FQ(4.)34 b FO(In)20 b(the)g(analysis)d(of)j(the)h(represen)n (tations)c(of)27 b(\(1.19\))o(,)22 b(the)e(b)r(eha)n(vior)e(of)i(the) -118 196 y(dynamical)25 b(system)j FN(F)21 b FO(:)28 b FI(R)k FP(\000)-49 b(!)26 b FI(R)35 b FO(pla)n(ys)27 b(a)h(cen)n(tral)f(role.)39 b(The)29 b(structure)g(of)-118 296 y(represen)n(tations)17 b(of)27 b(\(1.19\))20 b(dep)r(ends)h(on)f (the)h(structure)f(of)g(the)h(orbits)e(of)h(the)-118 395 y(corresp)r(onding)g(dynamical)g(system)i(\(see)i(Section)e(2.1.1)g (for)h(details\).)34 b(Here)-118 495 y(w)n(e)k(giv)n(e)e(the)i(corresp) r(onding)e(results)g(for)i(general)e(semilinear)d(relations.)-118 595 y(W)-7 b(e)28 b(b)r(egin)f(with)g(the)h(follo)n(wing)23 b(de\014nition.)-118 763 y FQ(De\014nition)31 b(7.)40 b FB(A)29 b(set)f FN(E)34 b FB(is)29 b(c)l(al)t(le)l(d)38 b FO(\000)p FB(-invariant)30 b(if)f FN(E)21 b FP(\002)16 b FN(E)1830 733 y FL(c)1880 763 y FP(\\)g FO(\000)23 b(=)g FI(?)28 b FB(and)-118 862 y FN(E)-52 832 y FL(c)1 862 y FP(\002)18 b FN(E)25 b FP(\\)19 b FO(\000)24 b(=)g FI(?)p FB(.)41 b(A)30 b(minimal)i FO(\000)24 b Fr(\026)1071 874 y FL(M)1145 862 y FB(-invariant)31 b(set)f FN(O)1728 874 y FL(M)1802 862 y FO(\()p FN(E)5 b FO(\))31 b FB(c)l(ontaining)-118 962 y(the)22 b(set)g FN(E)28 b FB(is)23 b(said)g(to)g(b)l(e)f(a)h(tr)l (aje)l(ctory)30 b FO(\()p FB(semi-tr)l(aje)l(ctory)7 b FO(\))24 b FB(of)g(the)e(set)g FN(E)28 b FP(\032)23 b FN(M)-118 1062 y FB(with)30 b(r)l(esp)l(e)l(ct)g(to)f FO(\000)23 b Fr(\026)542 1074 y FL(M)616 1062 y FB(.)6 1230 y FO(The)33 b(concept)f(of)g(a)g(tra)5 b(jectory)31 b(generalizes)d(that)33 b(of)f(an)g(orbit)f(for)h(dy-)-118 1329 y(namical)21 b(systems.)35 b(The)25 b(follo)n(wing)c(result)j(is)g (an)h(analogue)e(of)i(the)g(theorem)-118 1429 y(on)31 b(the)g(connectedness)g(of)g(the)h(graph)e(supp)r(orting)g(an)h (irreducible)c(\014nite-)-118 1529 y(dimensional)c(represen)n(tation.) -118 1697 y FQ(Theorem)30 b(12.)41 b FB(L)l(et)29 b FN(M)38 b FB(b)l(e)30 b(a)g(c)l(omp)l(act)h(set.)6 1797 y FO(\()p FB(a)6 b FO(\))33 b FB(If)e(ther)l(e)g(is)g FN(x)26 b FP(2)g FN(M)39 b FB(such)31 b(that)g(the)g(tr)l(aje)l(ctory)h FN(O)1749 1809 y FL(M)1823 1797 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))f FB(is)g(dense)-118 1897 y(in)f FN(M)9 b FB(,)30 b(then)g FN(M)39 b FB(is)30 b(the)h(sp)l(e)l(ctrum)e(of)i(an) f(irr)l(e)l(ducible)i(r)l(epr)l(esentation,)f(i.e.,)-118 1997 y(ther)l(e)21 b(exists)g(an)h(irr)l(e)l(ducible)h(r)l(epr)l (esentation)e FO(\()p FN(A;)14 b(B)t(;)g(B)1639 1966 y FM(\003)1678 1997 y FO(\))22 b FB(such)f(that)h FN(\033)s FO(\()p FN(A)p FO(\))i(=)-118 2096 y FN(M)9 b FB(.)6 2196 y FO(\()p FB(b)c FO(\))32 b FB(If)g(the)f(set)g FN(M)39 b FB(is)32 b(the)f(sp)l(e)l(ctrum)f(of)i(an)f(irr)l(e)l (ducible)h(r)l(epr)l(esentation)-118 2296 y(of)56 b FO(\(1.10\))o FB(,)40 b(then)d(the)h(tr)l(aje)l(ctory)g FN(O)1064 2308 y FL(M)1138 2296 y FO(\()p FN(G)25 b FP(\\)f FN(M)9 b FO(\))37 b FB(is)h(dense)g(in)g FN(M)46 b FB(for)38 b(any)-118 2396 y(op)l(en)30 b(set)g FN(G)p FB(,)g(wher)l(e)g FN(G)19 b FP(\\)g FN(M)32 b FP(6)p FO(=)22 b FI(?)p FB(.)-118 2564 y(Pr)l(o)l(of.)43 b FO(Let)i FN(O)371 2576 y FL(M)445 2564 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))g(b)r(e)f(dense)h(in)f FN(M)9 b FO(.)87 b(Then)45 b(there)g(is)e(a)h(sequence)-118 2664 y FP(f)p FN(\025)-28 2676 y FL(n)17 2664 y FP(g)59 2633 y FM(1)59 2684 y FL(n)p FK(=1)225 2664 y FP(\032)38 b FN(O)391 2676 y FL(M)465 2664 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))e(whic)n(h)f(is)g(dense)i(in)e FN(M)9 b FO(.)63 b(No)n(w)36 b(let)f FN(\026)i FO(b)r(e)f(a)g(mea-)-118 2763 y(sure)25 b(concen)n(trated)f(on)i FP(f)p FN(\025)749 2775 y FL(n)794 2763 y FP(g)836 2733 y FM(1)836 2784 y FL(n)p FK(=1)965 2763 y FO(,)g(and)f FN(\026)p FO(\()p FN(\025)1303 2775 y FL(n)1349 2763 y FO(\))f FP(6)p FO(=)e(0)j(for)g (an)n(y)g FN(\025)1887 2775 y FL(n)1933 2763 y FO(.)36 b(Then)26 b FN(\026)g FO(is)-118 2863 y(ergo)r(dic.)35 b(In)27 b(fact,)g(let)g FN(S)32 b FO(b)r(e)c(in)n(v)-5 b(arian)n(t)24 b(with)j(resp)r(ect)g(to)g(\000,)g(and)g FN(\026)p FO(\()p FN(S)5 b FO(\))23 b FP(6)p FO(=)g(0,)-118 2962 y FN(\026)p FO(\()p FN(S)20 2932 y FL(c)54 2962 y FO(\))45 b FP(6)p FO(=)f(0.)75 b(Then)41 b FN(S)32 b FP(\\)27 b(f)p FN(\025)865 2974 y FL(n)911 2962 y FP(g)953 2932 y FM(1)953 2983 y FL(n)p FK(=1)1126 2962 y FP(6)p FO(=)44 b FI(?)p FO(.)76 b(F)-7 b(rom)39 b(this)h(it)g(follo)n(ws)d (that)-118 3062 y FN(S)25 b FP(\\)20 b FN(O)96 3074 y FL(M)170 3062 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))27 b FP(6)p FO(=)g FI(?)p FO(.)43 b(If)31 b FN(x)c FP(2)g FN(S)5 b FO(,)31 b(then)f FN(O)1221 3074 y FL(M)1295 3062 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))d FP(\032)g FN(S)5 b FO(,)30 b(hence)g FN(\026)p FO(\()p FN(S)2089 3032 y FL(c)2123 3062 y FO(\))d(=)g(0,)-118 3162 y(whic)n(h)18 b(con)n(tradicts)g(the)i(assumption.)31 b(Assume)19 b(that)h FN(x)j FP(62)h FN(S)5 b FO(,)21 b(hence)e FN(x)24 b FP(2)f FN(S)2282 3132 y FL(c)2316 3162 y FO(.)-118 3261 y(Since)g FN(S)28 b FO(is)22 b(in)n(v)-5 b(arian)n(t)21 b(with)i(resp)r(ect)g(to) g(\000,)i(so)d(is)h(the)h(complemen)n(t)c(of)k FN(S)k FO(and)-118 3361 y FN(O)-55 3373 y FL(M)19 3361 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))9 b FP(\\)g FN(S)343 3331 y FL(c)400 3361 y FP(6)p FO(=)23 b FI(?)p FO(.)35 b(F)-7 b(rom)22 b(the)h(condition)e FN(x)i FP(2)h FN(O)1535 3373 y FL(M)1609 3361 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))9 b FP(\\)g FN(S)1933 3331 y FL(c)1990 3361 y FO(w)n(e)23 b(obtain)-118 3461 y FN(O)-55 3473 y FL(M)19 3461 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))f FP(\\)h FN(S)370 3430 y FL(c)436 3461 y FO(=)33 b FN(O)597 3473 y FL(M)671 3461 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\).)55 b(Hence)33 b FN(O)1259 3473 y FL(M)1333 3461 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))g FP(\032)g FN(S)1715 3430 y FL(c)1748 3461 y FO(,)i(and)e FN(\026)p FO(\()p FN(S)5 b FO(\))34 b(=)e(0,)-118 3560 y(whic)n(h)26 b(giv)n(es)g(a)h(con)n(tradiction.)-118 3712 y FQ(5.)36 b FO(Let)26 b FN(G)h FO(b)r(e)g(an)f(op)r(en)g(set,)h FN(G)16 b FP(\\)h FN(M)32 b FP(6)p FO(=)22 b FI(?)p FO(,)p 1277 3640 449 4 v 27 w FN(O)1340 3724 y FL(M)1414 3712 y FO(\()p FN(G)d FP(\\)f FN(M)9 b FO(\))23 b FP(6)p FO(=)g FN(M)9 b FO(,)26 b(and)h FN(E)2197 3724 y FL(A)2251 3712 y FO(\()p FP(\001)p FO(\))-118 3811 y(b)r(e)i(the)g(sp)r(ectral)e (measure)f(of)j(the)g(op)r(erator)d FN(A)j FO(with)f(sp)r(ectrum)g FN(M)9 b FO(.)39 b(Then)-118 3911 y FN(E)-57 3923 y FL(A)-3 3911 y FO(\()p FN(G)p FO(\))25 b FP(6)p FO(=)d(0)28 b(and)g(there)f (exists)g(a)g(compact)f(set)i FN(F)35 b FP(\032)23 b FN(G)c FP(\\)g FN(M)9 b FO(,)28 b FN(E)1979 3923 y FL(A)2033 3911 y FO(\()p FN(F)12 b FO(\))24 b FP(6)p FO(=)f(0.)p eop %%Page: 65 69 65 68 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(65)-118 96 y(The)24 b(set)g FN(O)238 108 y FL(M)312 96 y FO(\()p FN(F)12 b FO(\))24 b(is)f(a)g(Borel)f(set.)36 b(Since)p 1201 24 267 4 v 23 w FN(O)1264 108 y FL(M)1338 96 y FO(\()p FN(F)12 b FO(\))23 b FP(6)p FO(=)g FN(M)9 b FO(,)24 b FN(E)1776 108 y FL(A)1831 96 y FO(\()p FN(O)1926 108 y FL(M)2000 96 y FO(\()p FN(F)12 b FO(\)\))24 b FP(6)p FO(=)f FN(I)7 b FO(.)-118 196 y(Th)n(us,)28 b FN(E)177 208 y FL(A)231 196 y FO(\()p FP(\001)p FO(\))h(is)e(not)h(ergo)r(dic)e (with)i(resp)r(ect)g(to)g(\000)c Fr(\026)1556 208 y FL(M)1629 196 y FO(,)k(b)r(ecause)g FN(O)2051 208 y FL(M)2125 196 y FO(\()p FN(F)12 b FO(\))29 b(is)-118 296 y(an)e(in)n(v)-5 b(arian)n(t)25 b(set.)p 2278 296 4 57 v 2282 243 50 4 v 2282 296 V 2331 296 4 57 v -118 561 a FQ(De\014nition)31 b(8.)40 b FB(A)33 b(set)g FN(\034)43 b FB(is)34 b(said)g(to)g(b)l(e)f (a)h(me)l(asur)l(able)g(se)l(ction)f(of)h FO(\()p FN(D)r(;)14 b FO(\000\))-118 661 y FB(if)29 b FN(\034)k FP(\032)23 b FN(D)30 b FB(is)f(a)g(Bor)l(el)g(set)f(and)h(every)g(tr)l(aje)l (ctory)g FN(O)1536 673 y FL(D)1597 661 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))f FB(with)h(r)l(esp)l(e)l(ct)g(to)-118 761 y FO(\000)23 b Fr(\026)-8 773 y FL(D)81 761 y FB(interse)l(cts)30 b FN(\034)39 b FB(at)30 b(exactly)g(one)g(p)l(oint.)-118 964 y FQ(Prop)s(osition)g(24.)41 b FB(L)l(et)33 b FO(\()p FN(D)r(;)14 b FO(\000\))34 b FB(have)h(a)f(me)l(asur)l(able)g(se)l (ction.)51 b(Then,)36 b(for)-118 1064 y(any)30 b(irr)l(e)l(ducible)g(r) l(epr)l(esentation)g(of)48 b FO(\(1.10\))29 b FB(with)h FN(\033)s FO(\()p FN(A)p FO(\))24 b FP(\032)e FN(D)r FB(,)30 b(ther)l(e)g(exists)-118 1163 y(a)g(unique)f(tr)l(aje)l(ctory)i FN(O)658 1175 y FL(D)718 1163 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))f FB(for)h(which)g FN(E)1370 1175 y FL(A)1424 1163 y FO(\()p FN(O)1519 1175 y FL(D)1580 1163 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\)\))24 b(=)e FN(I)7 b FB(.)-118 1367 y(Pr)l(o)l(of.)43 b FO(Let)34 b FN(\034)44 b FO(b)r(e)34 b(a)f(measurable)e(section)h(of)i(\()p FN(D)r(;)14 b FO(\000\).)56 b(Assume)33 b(that)h FN(\026)g FO(is)-118 1466 y(not)28 b(concen)n(trated)e(on)i(an)n(y)f(orbit)g FN(O)1061 1478 y FL(D)1121 1466 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\).)38 b(W)-7 b(e)28 b(pro)n(v)n(e)e(that)i(there)g(exists) -118 1566 y(a)34 b(partition)e(of)i FN(\034)44 b FO(in)n(to)33 b(t)n(w)n(o)h(sets)g FN(\034)44 b FO(=)34 b FN(\034)1215 1578 y FK(1)1276 1566 y FP([)23 b FN(\034)1390 1578 y FK(2)1428 1566 y FO(,)36 b FN(\034)1523 1578 y FK(1)1583 1566 y FP(\\)24 b FN(\034)1698 1578 y FK(2)1770 1566 y FO(=)34 b FI(?)p FO(,)h(suc)n(h)f(that)-118 1666 y FN(\026)p FO(\()p FN(O)27 1678 y FL(D)88 1666 y FO(\()p FN(\034)156 1678 y FK(1)194 1666 y FO(\)\))44 b FN(>)g FO(0,)g FN(\026)p FO(\()p FN(O)665 1678 y FL(D)725 1666 y FO(\()p FN(\034)793 1678 y FK(2)831 1666 y FO(\)\))h FN(>)f FO(0.)75 b(Consider)38 b(the)j(tra)5 b(jectories)38 b FN(T)2203 1678 y FL(i)2274 1666 y FO(=)-118 1765 y FN(O)-53 1735 y FL(i)-55 1788 y(D)5 1765 y FO(\()p FN(\034)73 1777 y FL(i)102 1765 y FO(\),)d FN(i)c FO(=)h(1,)i(2,)g(for)f(an)n(y)f (t)n(w)n(o)h(sets)f FN(\034)1208 1777 y FL(i)1270 1765 y FO(satisfying)e(the)j(conditions)e FN(\034)42 b FO(=)-118 1865 y FN(\034)-82 1877 y FK(1)-27 1865 y FP([)18 b FN(\034)82 1877 y FK(2)119 1865 y FO(,)28 b FN(\034)206 1877 y FK(1)261 1865 y FP(\\)18 b FN(\034)370 1877 y FK(2)430 1865 y FO(=)23 b FI(?)p FO(.)36 b(By)27 b(the)h(de\014nition)d(of)i(a)g (measurable)d(section,)i FN(T)2229 1877 y FK(1)2283 1865 y FP(\\)-118 1965 y FN(T)-69 1977 y FK(2)-9 1965 y FO(=)c FI(?)h FO(and)g FN(T)372 1977 y FK(1)418 1965 y FP([)9 b FN(T)531 1977 y FK(2)592 1965 y FO(=)23 b FN(D)r FO(.)35 b(Assuming)21 b(that)j(for)e(an)n(y)g(suc)n(h)h(decomp)r(osition)-118 2064 y FN(D)28 b FO(=)e FN(T)119 2076 y FK(1)175 2064 y FP([)20 b FN(T)299 2076 y FK(2)366 2064 y FO(one)29 b(of)g(the)h(v)-5 b(alues)28 b FN(\026)p FO(\()p FN(T)1141 2076 y FK(1)1178 2064 y FO(\))i(and)f FN(\026)p FO(\()p FN(T)1534 2076 y FK(2)1572 2064 y FO(\))g(is)g(equal)f(to)h(zero,)g(w)n (e)-118 2164 y(can)23 b(\014nd)h(a)e(decreasing)f(sequence)i FP(f)p FN(T)1097 2134 y FL(k)1136 2164 y FP(g)g FO(suc)n(h)g(that)h FN(E)1621 2176 y FL(A)1675 2164 y FO(\()p FN(T)1768 2134 y FL(k)1808 2164 y FO(\))f(=)g FN(I)30 b FO(for)23 b(an)n(y)g FN(k)-118 2263 y FO(and)44 2201 y Fy(T)127 2263 y FN(T)188 2233 y FL(k)252 2263 y FO(=)h FN(O)404 2275 y FL(D)464 2263 y FO(\()p FP(f)p FN(x)p FP(g)p FO(\))29 b(for)e(some)g FN(x)e FP(2)f FN(\034)9 b FO(.)40 b(F)-7 b(rom)26 b(the)j(latter)e (argumen)n(t)f(w)n(e)-118 2363 y(can)33 b(conclude)f(that)i FN(\026)g FO(is)e(concen)n(trated)h(on)g(an)g(orbit)f(whic)n(h)h(con)n (tradicts)-118 2463 y(the)28 b(assumption.)6 2575 y(Let)21 b FN(\034)32 b FO(=)23 b FN(\034)340 2587 y FK(1)381 2575 y FP([)s FN(\034)475 2587 y FK(2)533 2575 y FO(b)r(e)e(the)f (required)e(decomp)r(osition,)g(i.e.)34 b FN(\026)p FO(\()p FN(O)1942 2587 y FL(D)2003 2575 y FO(\()p FN(\034)2071 2587 y FL(i)2099 2575 y FO(\)\))23 b FN(>)g FO(0,)-118 2674 y FN(i)28 b FO(=)g(1)p FN(;)14 b FO(2.)45 b(Since)30 b(b)r(oth)i(sets)e FN(O)869 2686 y FL(D)930 2674 y FO(\()p FN(\034)998 2686 y FK(1)1035 2674 y FO(\))i(and)e FN(O)1326 2686 y FL(D)1387 2674 y FO(\()p FN(\034)1455 2686 y FK(2)1493 2674 y FO(\))h(are)f(in)n(v)-5 b(arian)n(t)27 b(with)j(re-)-118 2774 y(sp)r(ect)24 b(to)f(\000)p FP(j)265 2786 y FL(D)325 2774 y FO(,)i(the)f(existence)e(of)i(the)f(decomp)r(osition)d(implies)g (a)j(con)n(tradic-)-118 2874 y(tion)k(to)i(ergo)r(dicit)n(y)c(of)j(the) h(measure)d FN(\026)p FO(.)40 b(Th)n(us,)28 b FN(\026)h FO(is)e(concen)n(trated)g(on)h(the)-118 2973 y(tra)5 b(jectory)26 b(of)h(some)f(p)r(oin)n(t)h FN(x)p FO(.)p 2278 2973 V 2282 2921 50 4 v 2282 2973 V 2331 2973 4 57 v 6 3239 a(If)40 b(there)g(is)e(no)h(measurable)d(section)i(for)h (the)h(graph)e(\()p FN(D)r FO(,)43 b(\000\))d(of)f(the)-118 3339 y(semilinear)20 b(relation)h(\(1.10\))o(,)26 b(then)f(the)g (structure)f(of)h(represen)n(tations)c(with)-118 3438 y(b)r(ounded)g(op)r(erators)d(\()p FN(A;)c(B)t(;)g(B)876 3408 y FM(\003)915 3438 y FO(\))21 b(is)e(more)f(complicated:)30 b(there)20 b(migh)n(t)f(exist)-118 3538 y(irreducible)h(represen)n (tations)g(suc)n(h)j(that)h(the)g(sp)r(ectrum)e(of)i(the)g(op)r(erator) d FN(A)-118 3638 y FO(is)26 b(not)i(discrete.)-118 3795 y FB(R)l(emark)i(18.)42 b FO(The)22 b(same)d(theorems)h(are)g(v)-5 b(alid)19 b(for)i(the)h(represen)n(tation)c(with)-118 3895 y FN(B)27 b FO(=)c FN(B)127 3865 y FM(\003)165 3895 y FO(,)28 b(but)g(with)f(\000)609 3907 y FL(s)672 3895 y FO(instead)g(of)g(\000.)p eop %%Page: 66 70 66 69 bop -118 -137 a FO(66)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FQ(1.3.5)94 b(Represen)m(tations)30 b(of)i(semilinear)d FN(F)1491 108 y FK(4)1529 96 y FQ(-relations)-118 250 y FO(The)e(problem)e(of)i(describing)e(all)g(irreducible)e(represen)n (tations)i(of)i(semilin-)-118 349 y(ear)h(relations)d(up)30 b(to)e(unitary)g(equiv)-5 b(alence)26 b(migh)n(t)h(b)r(e)i(v)n(ery)f (di\016cult.)40 b(The)-118 449 y(complexit)n(y)23 b(of)k(the)g (description)d(dep)r(ends)j(on)g(the)g(structure)f(of)g(the)i(corre-) -118 549 y(sp)r(onding)e(graph.)-118 715 y FQ(Theorem)k(13.)41 b FO(1)p FB(.)d(If)30 b(al)t(l)h(c)l(onne)l(cte)l(d)f(c)l(omp)l(onents) f(of)i(the)f(gr)l(aph)h FO(\000)f FB(c)l(orr)l(e-)-118 815 y(sp)l(onding)36 b(to)g(a)f(semiline)l(ar)i(r)l(elation)f(ar)l(e)f (of)i(the)e(form:)1810 797 y Fn(r)1852 815 y FB(,)1956 797 y Fn(r)p 1956 798 117 4 v 1989 796 a Ft(-)2080 797 y Fn(r)2122 815 y FB(,)i(then)-118 925 y(any)f(irr)l(e)l(ducible)i(r)l (epr)l(esentation)e FO(\()p FN(A;)14 b(B)t(;)g(B)1306 895 y FM(\003)1345 925 y FO(\))36 b FB(of)h(the)g(r)l(elation)g(is)f (one-)g(or)-118 1025 y(two-dimensional)9 b FO(:)-51 1191 y(\()p FB(i)f FO(\))43 b FN(A)24 b FO(=)e FN(\025)p FB(,)31 b FN(B)c FO(=)c(0)p FB(,)30 b FN(\025)23 b FP(2)g FI(R)p FO(;)-76 1358 y(\()p FB(ii)8 b FO(\))43 b FN(A)35 b FO(=)285 1290 y Fy(\000)337 1322 y FL(\025)376 1330 y Fx(1)452 1322 y FK(0)356 1383 y(0)43 b FL(\025)471 1391 y Fx(2)518 1290 y Fy(\001)556 1358 y FB(,)38 b FN(B)h FO(=)820 1290 y Fy(\000)871 1329 y FK(0)24 b(0)873 1382 y FL(b)i FK(0)975 1290 y Fy(\001)1013 1358 y FB(,)38 b(wher)l(e)f FO(\()p FN(\025)1397 1370 y FK(1)1435 1358 y FN(;)14 b(\025)1520 1370 y FK(2)1557 1358 y FN(;)g(b)p FO(\))36 b FB(b)l(elongs)h(to)f(the) g(set)89 1471 y FN(K)160 1483 y FK(1)220 1471 y FO(=)308 1403 y Fy(\010)356 1471 y FO(\()p FN(\025)436 1483 y FK(1)474 1471 y FN(;)14 b(\025)559 1483 y FK(2)597 1471 y FN(;)g(b)p FO(\))23 b FP(2)g FI(R)857 1441 y FK(3)923 1471 y FP(j)h FO(\010\()p FN(\025)1110 1483 y FK(1)1147 1471 y FN(;)14 b(\025)1232 1483 y FK(2)1270 1471 y FO(\))23 b(=)g(0;)k FN(\025)1553 1483 y FK(1)1614 1471 y FP(6)p FO(=)c FN(\025)1750 1483 y FK(2)1787 1471 y FN(;)28 b(b)23 b(>)f FO(0)2026 1403 y Fy(\011)2074 1471 y FB(.)6 1637 y FO(2)p FB(.)54 b(If)35 b(al)t(l)h(c)l(onne)l(cte)l(d)f(c)l(omp)l (onents)g(of)g(the)g(gr)l(aph)i FO(\000)1706 1649 y FL(s)1776 1637 y FB(ar)l(e)e(of)h(the)f(form)-77 1753 y Fn(r)-35 1771 y FB(,)70 1753 y Fn(r)p 70 1754 4 4 v 65 1749 V 61 1745 V 57 1740 V 53 1735 V 50 1731 V 47 1727 V 44 1723 V 42 1718 V 41 1715 V 39 1711 V 38 1707 V 37 1704 V 37 1700 V 37 1697 V 38 1694 V 38 1691 V 40 1688 V 41 1685 V 43 1682 V 45 1680 V 45 1680 V 48 1677 V 50 1675 V 53 1673 V 55 1672 V 58 1670 V 60 1669 V 63 1668 V 65 1668 V 68 1667 V 70 1667 V 73 1667 V 75 1668 V 78 1668 V 80 1669 V 83 1670 V 85 1672 V 88 1673 V 90 1675 V 93 1677 V 95 1680 V 70 1754 V 75 1749 V 79 1745 V 83 1740 V 87 1735 V 90 1731 V 93 1727 V 96 1723 V 98 1718 V 100 1715 V 101 1711 V 102 1707 V 103 1704 V 103 1700 V 103 1697 V 103 1694 V 102 1691 V 101 1688 V 99 1685 V 97 1682 V 95 1680 V 112 1771 a FB(,)217 1753 y Fn(r)p 217 1754 125 4 v 100 w(r)383 1771 y FB(,)k(then)d(any)h(irr)l(e)l(ducible)h (r)l(epr)l(esentation)f FO(\()p FN(A;)14 b(B)40 b FO(=)c FN(B)2164 1740 y FM(\003)2202 1771 y FO(\))h FB(of)-118 1870 y(the)30 b(r)l(elation)g(is)g(one-)g(or)g(two-dimensional)9 b FO(:)-51 2036 y(\()p FB(i)f FO(\))43 b FN(A)24 b FO(=)e FN(\025)p FB(,)31 b FN(B)c FO(=)c(0)p FB(,)30 b FN(\025)23 b FP(2)g FI(R)p FO(;)-76 2203 y(\()p FB(ii)8 b FO(\))43 b FN(A)31 b FO(=)e FN(\025)p FB(,)35 b FN(B)f FO(=)c FN(b)p FB(,)k(wher)l(e)g FO(\()p FN(\025;)14 b(b)p FO(\))34 b FB(b)l(elongs)g(to)g(the)f(set)g FN(K)1865 2215 y FK(2)1932 2203 y FO(=)c FP(f)p FO(\()p FN(\025;)14 b(b)p FO(\))30 b FP(2)89 2302 y FI(R)143 2272 y FK(2)210 2302 y FP(j)23 b FO(\010\()p FN(\025;)14 b(\025)p FO(\))24 b(=)f(0)p FP(g)p FO(;)-101 2469 y(\()p FB(iii)8 b FO(\))43 b FN(A)i FO(=)304 2402 y Fy(\000)356 2433 y FL(\025)395 2441 y Fx(1)471 2433 y FK(0)375 2495 y(0)e FL(\025)490 2503 y Fx(2)537 2402 y Fy(\001)575 2469 y FB(,)h FN(B)k FO(=)820 2402 y Fy(\000)872 2442 y FK(0)25 b FL(b)874 2495 y(b)h FK(0)975 2402 y Fy(\001)1013 2469 y FB(,)45 b(wher)l(e)d FO(\()p FN(\025)1409 2481 y FK(1)1447 2469 y FN(;)14 b(\025)1532 2481 y FK(2)1569 2469 y FN(;)g(b)p FO(\))41 b FB(b)l(elongs)h(to)f FN(K)2193 2481 y FK(3)2274 2469 y FO(=)89 2515 y Fy(\010)138 2582 y FO(\()p FN(\025)218 2594 y FK(1)256 2582 y FN(;)14 b(\025)341 2594 y FK(2)378 2582 y FN(;)g(b)p FO(\))23 b FP(2)h FI(R)639 2552 y FK(3)705 2582 y FP(j)f FO(\010\()p FN(\025)891 2594 y FK(1)929 2582 y FN(;)14 b(\025)1014 2594 y FK(2)1052 2582 y FO(\))23 b(=)g(\010\()p FN(\025)1335 2594 y FK(2)1372 2582 y FN(;)14 b(\025)1457 2594 y FK(1)1495 2582 y FO(\))23 b(=)g(0;)k FN(\025)1778 2594 y FK(1)1839 2582 y FP(6)p FO(=)c FN(\025)1975 2594 y FK(2)2012 2582 y FN(;)28 b(b)23 b(>)f FO(0)2251 2515 y Fy(\011)2299 2582 y FB(.)6 2748 y FO(Note)41 b(that)g(if)g FN(f)544 2760 y FL(i)571 2748 y FO(,)j FN(g)678 2760 y FL(i)746 2748 y FO(are)c(p)r(olynomials,)f(then)j(the)f(algebra)d FI(C)15 b FP(h)p FN(a;)f(b)51 b FP(j)-118 2785 y Fy(P)-30 2806 y FL(n)-30 2873 y(i)p FK(=1)95 2848 y FN(f)136 2860 y FL(i)163 2848 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)375 2860 y FL(i)402 2848 y FO(\()p FN(a)p FO(\))24 b(=)f(0)p FP(i)k FO(is)f(an)i FN(F)975 2860 y FK(4)1012 2848 y FO(-algebra.)-118 3014 y FB(Pr)l(o)l(of.)43 b FO(Under)37 b(the)g(ab)r(o)n(v)n(e)e(condition)g(w)n(e)h(ha)n(v)n(e)g(that)h(\000)f (is)g(the)h(graph)f(of)-118 3114 y(a)f(bijectiv)n(e)g(mapping)e FN(\036)9 b FO(:)32 b FN(N)827 3126 y FK(1)901 3114 y FP(!)37 b FN(N)1088 3126 y FK(2)1125 3114 y FO(,)h(where)e FN(N)1502 3126 y FK(1)1563 3114 y FP(\\)24 b FN(N)1709 3126 y FK(2)1783 3114 y FO(=)37 b FI(?)e FO(\(i.e.,)j(\000)f(=)-118 3214 y FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))35 b FP(2)g FN(N)286 3226 y FK(2)347 3214 y FP(\002)23 b FN(N)502 3226 y FK(1)574 3214 y FP(j)35 b FN(t)g FO(=)g FN(\036)p FO(\()p FN(s)p FO(\))p FP(g)p FO(\),)i(and)e(\000)1304 3226 y FL(s)1374 3214 y FO(is)f(the)h(graph)f(of)h(a)f(bijectiv)n(e) -118 3313 y(mapping)28 b FN(\036)9 b FO(:)29 b FN(N)37 b FP(!)27 b FN(N)39 b FO(\(i.e.,)31 b(\000)c(=)g FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))28 b FP(2)g FN(N)1419 3283 y FK(2)1483 3313 y FP(j)g FN(t)f FO(=)h FN(\036)p FO(\()p FN(s)p FO(\))p FP(g)p FO(\).)45 b(Moreo)n(v)n(er,)-118 3413 y FN(\036)32 b FO(is)e(measurable)d(under)k(the)h(assumption)d (that)i(\000\()p FN(M)9 b FO(\))31 b(and)g(\000)1953 3383 y FM(\000)p FK(1)2042 3413 y FO(\()p FN(M)9 b FO(\))32 b(are)-118 3513 y(Borel)d(sets)j(for)f(an)n(y)g(Borel)e FN(M)9 b FO(.)49 b(Therefore,)31 b(b)n(y)g(Corollary)c(8,)33 b(if)e(\()p FN(A;)14 b(B)t FO(\))30 b FP(2)-118 3612 y FN(L)p FO(\()p FN(H)7 b FO(\))22 b(is)f(a)g(represen)n(tation)f(of)i (a)g(relation)d(with)i(suc)n(h)h(a)g(graph,)g(then)g(\()p FN(A)p FO(,)i FN(B)t FO(\))-118 3712 y(satisfy)i(the)j(relation)c FN(AB)j FO(=)23 b FN(B)t(\036)p FO(\()p FN(A)p FO(\),)29 b(and)f(the)g(sp)r(ectrum)f(of)h(the)g(op)r(erator)-118 3811 y FN(A)g FO(b)r(elongs)f(to)h FN(N)440 3823 y FK(1)495 3811 y FP([)20 b FN(N)637 3823 y FK(2)702 3811 y FO(in)27 b(the)i(\014rst)e(case,)h(and)g(to)g FN(N)37 b FO(in)27 b(the)i(second)e(one.)-118 3911 y(Clearly)-7 b(,)25 b(the)j(op)r (erators)d FN(A)p FO(,)j FN(B)t(B)944 3881 y FM(\003)1010 3911 y FO(comm)n(ute.)p eop %%Page: 67 71 67 70 bop -118 -137 a FJ(1.3.)36 b(Lie)26 b(algebras)f(and)i (semilinear)c(relations)878 b FO(67)6 96 y(Let)34 b(us)e(consider)f (the)j(\014rst)e(case.)53 b(W)-7 b(e)33 b(ha)n(v)n(e)f FN(B)1563 66 y FK(2)1632 96 y FO(=)g(0)g(and,)j(therefore,)-118 196 y(k)n(er)13 b FN(B)39 b FP(6)p FO(=)34 b(0.)58 b(Denote)35 b(the)g(subspace)f(k)n(er)13 b FN(B)27 b FP(\\)d FO(k)n(er)13 b FN(B)1613 166 y FM(\003)1686 196 y FO(b)n(y)34 b FN(H)1877 208 y FK(1)1915 196 y FO(.)58 b(It)35 b(is)f(easy)-118 296 y(to)41 b(sho)n(w)g(that)h FN(H)480 308 y FK(1)559 296 y FO(is)e(in)n(v)-5 b(arian)n(t)39 b(with)i(resp)r(ect)g(to)h FN(A)p FO(,)k FN(B)t FO(,)f FN(B)1969 266 y FM(\003)2007 296 y FO(,)g(and)d(all)-118 395 y(irreducible)31 b(represen)n(tations)i (de\014ned)i(on)g FN(H)1369 407 y FK(1)1442 395 y FO(are)f (one-dimensional)c(and)-118 495 y(giv)n(en)38 b(b)n(y)i(\()p FN(i)p FO(\).)75 b(No)n(w)39 b(assume)g(that)h(k)n(er)13 b FN(B)31 b FP(\\)c FO(k)n(er)13 b FN(B)1616 465 y FM(\003)1698 495 y FO(=)43 b FI(?)p FO(.)74 b(Set)41 b FN(H)2193 507 y FK(0)2274 495 y FO(=)-118 595 y(k)n(er)13 b FN(B)27 b FP(\\)d FO(\(k)n(er)13 b FN(B)400 564 y FM(\003)438 595 y FO(\))470 564 y FM(?)527 595 y FO(.)59 b(Since)34 b FN(H)902 607 y FK(0)975 595 y FP(\032)h FN(E)1136 607 y FL(A)1190 595 y FO(\()p FN(\036)p FO(\()p FN(N)1370 607 y FK(1)1408 595 y FO(\)\))p FN(H)7 b FO(,)38 b FN(B)1676 564 y FM(\003)1714 595 y FN(H)1783 607 y FK(0)1856 595 y FP(\032)d FN(E)2017 607 y FL(A)2071 595 y FO(\()p FN(N)2170 607 y FK(1)2207 595 y FO(\))p FN(H)7 b FO(,)-118 694 y(and)21 b FN(N)104 706 y FK(1)148 694 y FP(\\)7 b FN(\036)p FO(\()p FN(N)358 706 y FK(1)396 694 y FO(\))23 b(=)g FI(?)p FO(,)f(the)g(subspaces)f FN(H)1229 706 y FK(0)1266 694 y FO(,)i FN(B)1379 664 y FM(\003)1418 694 y FN(H)1487 706 y FK(0)1546 694 y FO(are)d(orthogonal.)32 b(More-)-118 794 y(o)n(v)n(er,)c FN(H)154 806 y FK(0)191 794 y FO(,)i FN(B)311 764 y FM(\003)349 794 y FN(H)418 806 y FK(0)484 794 y FO(are)f(in)n(v)-5 b(arian)n(t)26 b(with)i(resp)r(ect)h(to)g FN(A)p FO(,)h FN(B)t(B)1803 764 y FM(\003)1842 794 y FO(,)f(whic)n(h)f(imply)-118 893 y(that)23 b(giv)n(en)f(\001)h FP(2)g Fz(B)p FO(\()p FI(R)600 863 y FK(2)643 893 y FO(\),)i(the)e (subspace)g FN(E)5 b FO(\(\001\))p FN(H)1472 905 y FK(0)1520 893 y FP(\010)10 b FN(B)1662 863 y FM(\003)1699 893 y FN(E)5 b FO(\(\001\))p FN(H)1967 905 y FK(0)2029 893 y FO(is)22 b(in)n(v)-5 b(ari-)-118 993 y(an)n(t)21 b(with)g(resp)r(ect) g(to)g FN(A)p FO(,)h FN(B)t FO(,)h FN(B)864 963 y FM(\003)903 993 y FO(,)f(where)f FN(E)5 b FO(\()p FP(\001)p FO(\))22 b(is)e(the)i(join)n(t)e(resolution)e(of)j(the)-118 1093 y(iden)n(tit)n(y)31 b(for)i(the)h(comm)n(uting)c(pair)h(of)j(op)r (erators)d FN(A)p FO(,)k FN(B)t(B)1817 1063 y FM(\003)1889 1093 y FO(restricted)d(to)-118 1192 y FN(H)-49 1204 y FK(0)-12 1192 y FO(.)k(F)-7 b(rom)23 b(this)h(it)g(follo)n(ws)d(that)k (\001)f(is)g(concen)n(trated)f(in)g(one)h(p)r(oin)n(t)g(if)g FN(A)p FO(,)h FN(B)t FO(,)-118 1292 y FN(B)-51 1262 y FM(\003)10 1292 y FO(is)c(an)h(irreducible)d(family)-7 b(.)32 b(Th)n(us)22 b(for)g(suc)n(h)g(a)g(family)d(of)k(op)r(erators)d (there)-118 1392 y(exists)33 b(a)h(join)n(t)f(eigen)n(v)n(ector)e FN(e)j FP(2)g FN(H)1063 1404 y FK(0)1135 1392 y FO(for)g FN(A)p FO(,)i FN(B)t(B)1524 1361 y FM(\003)1562 1392 y FO(,)g(and)f FP(f)p FN(e;)14 b(B)1975 1361 y FM(\003)2012 1392 y FN(e)p FP(g)33 b FO(de\014ne)-118 1491 y(an)28 b(orthogonal)e(basis)h(of)h(the)h(represen)n(tation)d(space.)40 b(The)29 b(corresp)r(onding)-118 1591 y(irreducible)24 b(represen)n(tation)h(is)h(giv)n(en)g(b)n(y)h(\()p FN(ii)p FO(\).)6 1697 y(In)32 b(the)f(case)f FN(A)f FO(=)f FN(A)685 1667 y FM(\003)724 1697 y FO(,)j FN(B)36 b FO(w)n(e)30 b(ha)n(v)n(e)g(that)h(the)h(op)r(erators)d FN(A\036)p FO(\()p FN(A)p FO(\),)k FN(A)21 b FO(+)-118 1797 y FN(\036)p FO(\()p FN(A)p FO(\))30 b(comm)n(ute)c(with)j FN(A)p FO(,)g FN(B)t FO(,)g(and)g(hence)f(due)h(to)g(the)g(irreducibilit)n(y) -7 b(,)23 b(they)-118 1896 y(are)i(m)n(ultiples)d(of)k(the)h(iden)n (tit)n(y)-7 b(,)25 b(i.e.,)g FN(A\036)p FO(\()p FN(A)p FO(\))g(=)d FN(a)1480 1908 y FK(1)1518 1896 y FN(I)33 b FO(and)26 b FN(A)15 b FO(+)g FN(\036)p FO(\()p FN(A)p FO(\))25 b(=)d FN(a)2235 1908 y FK(2)2273 1896 y FN(I)7 b FO(.)-118 1996 y(Then)37 b FN(A)170 1966 y FK(2)233 1996 y FP(\000)24 b FN(a)366 2008 y FK(2)404 1996 y FN(A)h FO(+)f FN(a)624 2008 y FK(1)661 1996 y FN(I)47 b FO(=)39 b(0,)g(and)e(so)g(the)h(sp)r(ectrum)e(of)i FN(A)f FO(is)f FN(\033)s FO(\()p FN(A)p FO(\))41 b(=)-118 2095 y FP(f)p FN(\025)-28 2107 y FK(1)9 2095 y FN(;)14 b(\025)94 2107 y FK(2)132 2095 y FP(g)p FO(,)22 b(where)e FN(\025)500 2107 y FK(1)538 2095 y FO(,)i FN(\025)631 2107 y FK(2)690 2095 y FO(are)d(the)j(ro)r(ots)d(of)i(the)h(equation)d FN(\025)1766 2065 y FK(2)1808 2095 y FP(\000)5 b FN(a)1922 2107 y FK(2)1959 2095 y FN(\025)g FO(+)g FN(a)2126 2107 y FK(1)2186 2095 y FO(=)23 b(0.)-118 2195 y(Hence)j(the)f(sp)r(ectrum)g (of)h FN(A)f FO(is)g(discrete)f(as)h(so)r(on)f(as)h(\()p FN(A)p FO(,)i FN(B)t FO(\))f(is)e(irreducible.)-118 2295 y(In)d(addition,)f FN(B)391 2265 y FK(2)449 2295 y FO(comm)n(utes)e (with)i FN(A)p FO(,)i FN(B)j FO(and)20 b(is)g(a)g(m)n(ultiple)d(of)j (the)h(iden)n(tit)n(y)-7 b(,)-118 2394 y FN(B)-51 2364 y FK(2)16 2394 y FO(=)30 b FN(b)147 2364 y FK(2)184 2394 y FN(I)7 b FO(.)50 b(If)32 b FN(b)e FP(6)p FO(=)f(0)j(and)f FN(e)825 2406 y FL(\025)864 2414 y Fx(1)933 2394 y FO(is)g(an)g(eigen)n (v)n(ector)e(of)j FN(A)p FO(,)h(then)f FN(e)2025 2406 y FL(\025)2064 2414 y Fx(1)2101 2394 y FO(,)h FN(B)t(e)2263 2406 y FL(\025)2302 2414 y Fx(1)-118 2494 y FO(de\014ne)d(an)g(in)n(v) -5 b(arian)n(t)27 b(subspace;)k(moreo)n(v)n(er)26 b FN(B)t(e)1433 2506 y FL(\025)1472 2514 y Fx(1)1539 2494 y FO(is)j(an)h(eigen)n(v)n (ector)d(with)-118 2594 y(the)40 b(eigen)n(v)-5 b(alue)37 b FN(\036)p FO(\()p FN(\025)575 2606 y FK(1)613 2594 y FO(\))44 b(=:)e FN(\025)867 2606 y FK(2)905 2594 y FO(.)73 b(Therefore,)42 b(b)n(y)d(the)h(irreducibilit)n(y)-7 b(,)37 b(the)-118 2693 y(op)r(erators)25 b FN(A)p FO(,)j FN(B)j FO(can)c(b)r(e)h(at)f(most)f(t)n(w)n(o-dimensional.)31 b(If)d FN(\036)p FO(\()p FN(\025)1883 2705 y FK(1)1921 2693 y FO(\))c FP(6)p FO(=)e FN(\025)2112 2705 y FK(1)2177 2693 y FO(\(i.e.,)-118 2793 y(\010\()p FN(\025)22 2805 y FK(1)60 2793 y FN(;)14 b(\025)145 2805 y FK(1)182 2793 y FO(\))26 b FP(6)p FO(=)f(0\),)k(then)g(normalizing)24 b(the)29 b(orthogonal)d(basis)h FN(e)1905 2805 y FL(\025)1944 2813 y Fx(1)1981 2793 y FO(,)i FN(B)t(e)2139 2805 y FL(\025)2178 2813 y Fx(1)2244 2793 y FO(w)n(e)-118 2892 y(get)22 b(an)f(orthogonal)e (basis)h(in)i(whic)n(h)e(op)r(erators)g FN(A)p FO(,)k FN(B)i FO(are)21 b(of)h(the)g(form)f(\()p FN(iii)p FO(\).)-118 2992 y(F)-7 b(or)20 b FN(\036)p FO(\()p FN(\025)p FO(\))25 b(=)e FN(\025)e FO(one)g(has)g(that)g FN(A)p FO(,)i FN(B)i FO(comm)n(ute,)c(and)f(hence)i(w)n(e)e(can)h(c)n(ho)r(ose)f(a)-118 3092 y(join)n(t)k(eigen)n(v)n(ector)d FN(e)540 3104 y FL(\025;b)633 3092 y FO(,)k(whic)n(h)f(de\014ne)h(an)f(in)n(v)-5 b(arian)n(t)22 b(subspace.)35 b(F)-7 b(rom)23 b(this)-118 3191 y(one)31 b(can)g(conclude)g(that)h(the)g(corresp)r(onding)c (irreducible)g(represen)n(tation)-118 3291 y(is)e(one)i(dimensional)23 b(and)k(giv)n(en)f(b)n(y)h(\()p FN(i)p FO(\))h(or)f(\()p FN(ii)p FO(\).)p 2278 3291 4 57 v 2282 3238 50 4 v 2282 3291 V 2331 3291 4 57 v 6 3509 a(If)39 b(the)f(graph)e(\000)i(corresp)r (onding)d(to)j(a)f(semilinear)c(relation)i(con)n(tains)-118 3642 y(the)i(subgraphs:)520 3625 y Fn(r)p 520 3626 4 4 v 515 3621 V 511 3617 V 506 3612 V 503 3607 V 499 3603 V 497 3599 V 494 3594 V 492 3590 V 490 3586 V 489 3583 V 488 3579 V 487 3575 V 487 3572 V 487 3569 V 487 3566 V 488 3563 V 489 3560 V 491 3557 V 493 3554 V 495 3552 V 495 3552 V 497 3549 V 500 3547 V 502 3545 V 505 3544 V 507 3542 V 510 3541 V 512 3540 V 515 3540 V 517 3539 V 520 3539 V 522 3539 V 525 3540 V 527 3540 V 530 3541 V 532 3542 V 535 3544 V 537 3545 V 540 3547 V 542 3549 V 545 3552 V 520 3626 V 525 3621 V 529 3617 V 533 3612 V 537 3607 V 540 3603 V 543 3599 V 545 3594 V 548 3590 V 549 3586 V 551 3583 V 552 3579 V 553 3575 V 553 3572 V 553 3569 V 552 3566 V 552 3563 V 550 3560 V 549 3557 V 547 3554 V 545 3552 V 598 3642 a FO(or)751 3625 y Fn(r)p 751 3626 117 4 v 784 3624 a Ft(-)875 3625 y Fn(r)p 875 3626 V 908 3624 a Ft(-)1000 3625 y Fn(r)1095 3642 y FO(\(and)g(with)f (an)n(y)g(other)h(orien)n(tation\),)-118 3800 y(and)26 b(the)g(graph)f(\000)469 3812 y FL(s)531 3800 y FO(con)n(tains)f(the)j (subgraphs:)1463 3782 y Fn(r)p 1463 3784 4 4 v 1458 3779 V 1454 3774 V 1450 3770 V 1446 3765 V 1443 3761 V 1440 3756 V 1437 3752 V 1435 3748 V 1433 3744 V 1432 3740 V 1431 3737 V 1430 3733 V 1430 3730 V 1430 3726 V 1430 3723 V 1431 3720 V 1432 3717 V 1434 3715 V 1436 3712 V 1438 3709 V 1438 3709 V 1441 3707 V 1443 3705 V 1446 3703 V 1448 3701 V 1451 3700 V 1453 3699 V 1456 3698 V 1458 3697 V 1461 3697 V 1463 3697 V 1465 3697 V 1468 3697 V 1470 3698 V 1473 3699 V 1475 3700 V 1478 3701 V 1480 3703 V 1483 3705 V 1485 3707 V 1488 3709 V 1463 3784 V 1468 3779 V 1472 3774 V 1476 3770 V 1480 3765 V 1483 3761 V 1486 3756 V 1489 3752 V 1491 3748 V 1493 3744 V 1494 3740 V 1495 3737 V 1496 3733 V 1496 3730 V 1496 3726 V 1496 3723 V 1495 3720 V 1494 3717 V 1492 3715 V 1490 3712 V 1488 3709 V 1463 3784 125 4 v 100 w(r)1646 3800 y FO(,)f(or)1837 3782 y Fn(r)p 1837 3784 V 100 w(r)p 1962 3784 V 99 w(r)2128 3800 y FO(,)g(then)-118 3911 y(the)32 b(problem)d(of)j(describing)c(all)i(irreducible)e (represen)n(tations)h(\()p FN(A;)14 b(B)t(;)g(B)2268 3881 y FM(\003)2306 3911 y FO(\))p eop %%Page: 68 72 68 71 bop -118 -137 a FO(68)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(and)j(\()p FN(A;)14 b(B)30 b FO(=)24 b FN(B)424 66 y FM(\003)462 96 y FO(\))29 b(resp)r(ectiv)n(ely)d(b)r(ecomes)i(v)n (ery)f(complicated)e(\(the)30 b(corre-)-118 196 y(sp)r(onding)19 b FP(\003)p FO(-algebra)e(is)i(wild\).)33 b(W)-7 b(e)21 b(refer)f(the)h(reader)e(to)h(Sections)g(3.1.1)f(and)-118 296 y(3.1.2)27 b(for)g(a)h(precise)e(de\014nition)h(of)h FP(\003)p FO(-wild)e(algebras,)f(and)j(to)g(Section)f(3.1.4)-118 395 y(for)g(the)h(pro)r(of)f(of)h(the)f(ab)r(o)n(v)n(e)g(fact.)-118 634 y FG(1.4)112 b(Represen)m(tations)38 b(of)g Fq(q)t FG(-relations)-118 816 y FQ(1.4.1)94 b(Finite-dimensional)27 b(represen)m(tations)32 b(of)f FN(q)s FQ(-relations)-118 969 y(1.)36 b FO(When)28 b(studying)f(represen)n(tations)d(of)k FN(q)s FO(-relations,)25 1170 y(\()p FN(V)20 b(I)161 1182 y FK(0)198 1170 y FO(\))p FN(;)14 b FO(\()p FN(V)20 b(I)7 b(I)446 1182 y FK(0)483 1170 y FO(\))609 1114 y(1)p 609 1151 42 4 v 615 1227 a FN(i)660 1170 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(\013)p FO(\()p FN(A)1130 1135 y FK(2)1187 1170 y FP(\006)c FN(B)1337 1135 y FK(2)1374 1170 y FO(\))p FN(;)181 b(\013)23 b(>)g FO(0)p FN(;)25 1370 y FO(\()p FN(V)d(I)161 1382 y FK(1)198 1370 y FO(\))p FN(;)14 b FO(\()p FN(V)20 b(I)7 b(I)446 1382 y FK(1)483 1370 y FO(\))609 1313 y(1)p 609 1350 V 615 1427 a FN(i)660 1370 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(\013)p FO(\()p FN(A)1130 1335 y FK(2)1187 1370 y FP(\006)c FN(B)1337 1335 y FK(2)1374 1370 y FO(\))h(+)f FN(I)7 b(;)180 b(\013)24 b FP(2)f FI(R)h FP(n)18 b(f)p FO(0)p FP(g)p FN(;)-118 1559 y FO(it)k(w)n(ould)g(b)r(e)h(nice)f(to)h(ha)n(v)n(e)f(a)h(kind)f (of)h(theorem)e(of)i(the)h(Kleinec)n(k)n(e{Shirok)n(o)m(v)-118 1659 y(t)n(yp)r(e)k(for)f(general)e(relations)f(of)k(the)g(form)687 1803 y(1)p 687 1841 V 693 1917 a FN(i)738 1860 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(f)9 b FO(\()p FN(A)p FO(\))19 b(+)f FN(\036)p FO(\()p FN(B)t FO(\))p FN(;)-118 2059 y FO(where)i FN(f)9 b FO(\()p FP(\001)p FO(\),)22 b FN(\036)p FO(\()p FP(\001)p FO(\))g(are)d(real)f(functions)i(of)g FN(t)j FP(2)h FI(R)1364 2029 y FK(1)1407 2059 y FO(.)35 b(In)20 b(the)h(\014nite-dimensional)-118 2159 y(case)27 b(\(dim)12 b FN(H)30 b(<)23 b FP(1)p FO(\))28 b(w)n(e)f(ha)n(v)n(e)f (the)i(follo)n(wing)c(statemen)n(t.)-118 2312 y FQ(Prop)s(osition)30 b(25.)41 b FB(If)48 b FN(A)30 b FB(and)g FN(B)k FB(ar)l(e)c (self-adjoint)i(op)l(er)l(ators)f(in)f(a)g(\014nite-)-118 2411 y(dimensional)h(sp)l(ac)l(e)g FN(H)7 b FB(,)30 b(and)831 2578 y FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(T)29 b FO(+)18 b FN(S;)-118 2744 y FB(wher)l(e)762 2910 y FO([)p FN(A;)c(T)e FO(])22 b(=)h([)p FN(B)t(;)14 b(S)5 b FO(])23 b(=)f(0)p FN(;)-118 3077 y FB(then)29 b FO([)p FN(A;)14 b(B)t FO(])24 b(=)f(0)p FB(.)-118 3230 y(Pr)l(o)l(of.)43 b FO(Indeed,)28 b(since)94 3396 y([)p FN(A;)14 b(B)t FO(])306 3362 y FK(2)367 3396 y FO(=)23 b(\()p FN(T)29 b FO(+)18 b FN(S)5 b FO(\)[)p FN(A;)14 b(B)t FO(])24 b(=)e FN(T)12 b FO(\()p FN(AB)23 b FP(\000)18 b FN(B)t(A)p FO(\))h(+)f FN(S)5 b FO(\()p FN(AB)23 b FP(\000)18 b FN(B)t(A)p FO(\))367 3521 y(=)23 b FN(A)p FO(\()p FN(T)12 b(B)t FO(\))18 b FP(\000)g FO(\()p FN(T)12 b(B)t FO(\))p FN(A)18 b FO(+)g(\()p FN(S)5 b(A)p FO(\))p FN(B)24 b FP(\000)18 b FN(B)t FO(\()p FN(S)5 b(A)p FO(\))367 3645 y(=)23 b([)p FN(A;)14 b(T)e(B)t FO(])17 b(+)h([)p FN(S)5 b(A;)14 b(B)t FO(])p FN(;)-118 3811 y FO(w)n(e)27 b(conclude)f(that)i (T)-7 b(r[)p FN(A;)14 b(B)t FO(])823 3781 y FK(2)883 3811 y FO(=)23 b(0.)37 b(But)27 b([)p FN(A;)14 b(B)t FO(])28 b(is)f(a)g(sk)n(ew-adjoin)n(t)e(op)r(era-)-118 3911 y(tor,)i(therefore,)g(T)-7 b(r)o([)p FN(A;)14 b(B)t FO(])708 3881 y FK(2)769 3911 y FO(=)23 b(0)k(implies)d([)p FN(A;)14 b(B)t FO(])23 b(=)g(0.)p 2278 3911 4 57 v 2282 3858 50 4 v 2282 3911 V 2331 3911 4 57 v eop %%Page: 69 73 69 72 bop -118 -137 a FJ(1.4.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-relations)1128 b FO(69)-118 96 y FQ(Corollary)32 b(2.)40 b FB(Irr)l(e)l(ducible)30 b(\014nite-dimensional)f(r)l(epr)l (esentations)g(of)g(r)l(ela-)-118 196 y(tions)36 b FO(\()p FN(V)19 b(I)229 208 y FK(0)267 196 y FO(\))p FB({)p FO(\()p FN(V)h(I)7 b(I)520 208 y FK(1)558 196 y FO(\))36 b FB(ar)l(e)h (one-dimensional,)j FN(A)34 b FO(=)h FN(\025)p FB(,)j FN(B)h FO(=)34 b FN(\026)p FB(,)39 b FO(\()p FN(\025;)14 b(\026)p FO(\))35 b FP(2)-118 296 y FN(M)-37 311 y FK(\()p FM(\001)p FK(\))38 296 y FO(\()p FN(\013)p FO(\))p FB(,)c(wher)l(e)5 496 y FN(M)86 511 y FK(\()p FL(V)14 b(I)194 519 y Fx(0)227 511 y FK(\))257 496 y FO(\()p FN(\013)p FO(\))24 b(=)f FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))24 b FP(2)f FI(R)883 462 y FK(2)949 496 y FP(j)g FN(\025)h FO(=)e(0)p FN(;)14 b(\026)23 b FO(=)g(0)p FP(g)p FN(;)182 b FB(for)31 b(al)t(l)g FN(\013)23 b(>)g FO(0)p FB(,)5 621 y FN(M)86 636 y FK(\()p FL(V)14 b(I)194 644 y Fx(1)227 636 y FK(\))257 621 y FO(\()p FN(\013)p FO(\))24 b(=)f FI(?)p FN(;)183 b(\013)24 b(>)e FO(0)p FN(;)5 793 y(M)86 808 y FK(\()p FL(V)14 b(I)194 816 y Fx(1)227 808 y FK(\))257 793 y FO(\()p FN(\013)p FO(\))24 b(=)f FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))24 b FP(2)f FI(R)883 759 y FK(2)949 793 y FP(j)g FN(\025)1043 759 y FK(2)1099 793 y FO(+)18 b FN(\026)1232 759 y FK(2)1293 793 y FO(=)k FP(\000)1461 737 y FO(1)p 1455 774 54 4 v 1455 850 a FN(\013)1518 793 y FP(g)p FN(;)183 b(\013)24 b(<)f FO(0)p FN(;)-29 955 y(M)52 970 y FK(\()p FL(V)14 b(I)5 b(I)194 978 y Fx(0)227 970 y FK(\))257 955 y FO(\()p FN(\013)p FO(\))24 b(=)f FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))24 b FP(2)f FI(R)883 921 y FK(2)949 955 y FP(j)g FN(\025)1043 921 y FK(2)1104 955 y FO(=)g FN(\026)1242 921 y FK(2)1279 955 y FP(g)p FN(;)183 b FB(for)31 b(al)t(l)g FN(\013)23 b(>)g FO(0)p FB(,)-29 1128 y FN(M)52 1143 y FK(\()p FL(V)14 b(I)5 b(I)194 1151 y Fx(1)227 1143 y FK(\))257 1128 y FO(\()p FN(\013)p FO(\))24 b(=)f FP(f)p FO(\()p FN(\025;)14 b(\026)p FO(\))24 b FP(2)f FI(R)883 1094 y FK(2)949 1128 y FP(j)g FN(\025)1043 1094 y FK(2)1099 1128 y FP(\000)18 b FN(\026)1232 1094 y FK(2)1293 1128 y FO(=)k FP(\000)1461 1072 y FO(1)p 1455 1109 V 1455 1185 a FN(\013)1518 1128 y FP(g)p FN(;)183 b FB(for)31 b(al)t(l)g FN(\013)23 b FP(6)p FO(=)g(0)p FB(.)-118 1346 y(R)l(emark)30 b(19.)42 b FO(In)37 b(the)f(\014nite-dimensional)31 b(case,)38 b(the)e(corresp)r(onding)d(gen-)-118 1445 y(eralization)f(of)37 b(the)g(Jacobson)e(theorem)g(is)h(not)h(true.)64 b(Indeed,)40 b(ev)n(en)c(for)-118 1545 y(dim)12 b FN(H)32 b FO(=)24 b(3,)29 b(there)f(exist)g(matrices)d FN(A)p FO(,)30 b FN(B)t FO(,)f FN(T)12 b FO(,)28 b FN(S)5 b FO(,)29 b(suc)n(h)f(that)h (the)g(relations)-118 1644 y([)p FN(P)r(;)14 b(T)e FO(])23 b(=)f([)p FN(Q;)14 b(S)5 b FO(])23 b(=)f(0)27 b(hold,)g(but)h(the)g (matrix)d([)p FN(P)r(;)14 b(Q)p FO(])28 b(is)f(not)g(nilp)r(oten)n(t.)6 1748 y(The)e(follo)n(wing)20 b(example)h(is)j(due)g(to)g(V.S.)h(Guba)f (and)g(is)f(giv)n(en)f(in)48 b([229)n(].)-118 1889 y FB(Example)31 b(8.)42 b FO(Let)342 2174 y FN(A)23 b FO(=)515 2007 y Fy(0)515 2157 y(@)588 2074 y FO(2)82 b(0)h(0)588 2173 y(0)f(1)h(0)588 2273 y(0)f(0)h(0)878 2007 y Fy(1)878 2157 y(A)965 2174 y FN(;)97 b(B)27 b FO(=)1263 2007 y Fy(0)1263 2157 y(@)1368 2074 y FO(2)147 b(4)114 b(4)1368 2173 y(6)g FP(\000)p FO(4)82 b(4)1335 2273 y FP(\000)p FO(3)115 b(6)f(2)1755 2007 y Fy(1)1755 2157 y(A)1841 2174 y FN(;)186 2506 y(T)34 b FO(=)357 2339 y Fy(0)357 2489 y(@)429 2406 y FO(4)104 b(0)f(0)429 2505 y(0)83 b(16)f(0)429 2605 y(0)104 b(0)f(4)762 2339 y Fy(1)762 2489 y(A)848 2506 y FN(;)97 b(S)28 b FO(=)1134 2339 y Fy(0)1134 2489 y(@)1239 2406 y FO(4)147 b(4)115 b(8)1207 2505 y FP(\000)p FO(6)94 b(16)f(4)1239 2605 y(6)115 b FP(\000)p FO(6)82 b(4)1627 2339 y Fy(1)1627 2489 y(A)1722 2506 y FO(=)23 b FN(B)1877 2472 y FK(2)1914 2506 y FN(=)p FO(4;)-118 2796 y(then)28 b([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(T)30 b FO(+)18 b FN(S)5 b FO(,)27 b([)p FN(T)7 b(;)14 b(A)p FO(])23 b(=)f([)p FN(S;)14 b(B)t FO(])24 b(=)e(0,)28 b(but)g FN(\033)s FO(\([)p FN(A;)14 b(B)t FO(]\))24 b FP(6)p FO(=)f FP(f)p FO(0)p FP(g)p FO(.)-118 2937 y FQ(2.)34 b FO(A)22 b(similar)c(fact)k(to)g(Prop)r(osition)c(25)j (holds)g(for)g(op)r(erators)f(on)i(an)f(in\014nite-)-118 3036 y(dimensional)i FN(H)7 b FO(,)28 b(but)g(under)g(the)g(additional) c(assumption)h(that)j(the)h(op)r(er-)-118 3136 y(ators)d FN(T)39 b FO(and)27 b FN(S)33 b FO(are)26 b(compact.)-118 3314 y FQ(Prop)s(osition)k(26.)41 b FB(If)28 b FN(A)c FO(=)e FN(A)882 3284 y FM(\003)921 3314 y FB(,)29 b FN(B)e FO(=)c FN(B)1220 3284 y FM(\003)1281 3314 y FP(2)g FN(L)p FO(\()p FN(H)7 b FO(\))p FB(,)29 b(and)38 b FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(T)j FO(+)15 b FN(S)5 b FB(,)-118 3413 y(wher)l(e)28 b FN(T)12 b FB(,)28 b FN(S)5 b FB(,)28 b(ar)l(e)h(c)l(omp)l(act)f(op)l(er)l(ators)h(on)f FN(H)34 b FB(such)28 b(that)g FO([)p FN(A;)14 b(T)e FO(])22 b(=)h([)p FN(B)t(;)14 b(S)5 b FO(])23 b(=)-118 3513 y(0)p FB(,)30 b(then)f FO([)p FN(A;)14 b(B)t FO(])24 b(=)e(0)p FB(.)-118 3691 y(Pr)l(o)l(of.)43 b FO(Since)27 b([)p FN(A;)14 b(B)t FO(])28 b(is)e(sk)n(ew-adjoin)n(t,)760 3881 y([)p FN(A;)14 b(B)t FO(])24 b(=)e(\()p FN(T)1164 3893 y FK(1)1220 3881 y FO(+)c FN(S)1354 3893 y FK(1)1391 3881 y FO(\))p FN(;)680 b FO(\(1.20\))p eop %%Page: 70 74 70 73 bop -118 -137 a FO(70)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(where)f(the)h(compact)e(op)r(erators)g FN(T)1002 108 y FK(1)1062 96 y FO(=)g(\()p FN(T)h FP(\000)13 b FN(T)1394 66 y FM(\003)1430 96 y FO(\))p FN(=)p FO(2,)25 b(and)g FN(S)1804 108 y FK(1)1864 96 y FO(=)d(\()p FN(S)c FP(\000)13 b FN(S)2186 66 y FM(\003)2223 96 y FO(\))p FN(=)p FO(2)-118 196 y(are)26 b(sk)n(ew-adjoin)n(t.)34 b(By)27 b(the)g(F)-7 b(uglede{Putnam{Rosen)n(blum)21 b(theorem)26 b(\(for)-118 296 y(its)h(form)n(ulation)c(and)28 b(pro)r(of)f(see)g(Section)g(1.4.3)f(b)r(elo)n(w\),)g(w)n(e)i(ha)n(v)n (e)733 477 y([)p FN(T)805 489 y FK(1)841 477 y FN(;)14 b(A)p FO(])24 b(=)e([)p FN(S)1148 489 y FK(1)1186 477 y FN(;)14 b(B)t FO(])23 b(=)f(0)p FN(:)-118 659 y FO(F)-7 b(urther,)20 b(let)e FP(k)p FN(T)401 671 y FK(1)437 659 y FP(k)23 b(\025)f(k)p FN(S)682 671 y FK(1)719 659 y FP(k)h FN(>)f FO(0.)34 b(Cho)r(ose)17 b(an)i(eigen)n(v)-5 b(alue)15 b FN(\026)k FO(of)f(the)h(op)r(erator)-118 759 y FN(T)-69 771 y FK(1)-3 759 y FO(suc)n(h)28 b(that)i FP(j)p FN(\026)p FP(j)25 b FO(=)h FP(k)p FN(T)670 771 y FK(1)706 759 y FP(k)p FO(,)j(and)g(write)f FN(H)1246 771 y FL(\026)1316 759 y FO(=)d FP(f)p FN(f)33 b FP(2)26 b FN(H)16 b FO(:)28 b FN(T)1788 771 y FK(1)1825 759 y FN(f)34 b FO(=)25 b FN(\026f)9 b FP(g)p FO(.)41 b(The)-118 858 y(space)33 b FN(H)179 870 y FL(\026)258 858 y FO(is)g (\014nite-dimensional)c(and)34 b(in)n(v)-5 b(arian)n(t)31 b(with)i(resp)r(ect)h(to)g FN(A)p FO(.)57 b(If)-118 958 y FN(e)23 b FP(2)g FN(H)91 970 y FL(\026)136 958 y FO(,)k FP(k)p FN(e)p FP(k)22 b FO(=)h(1,)k(and)g FN(Ae)c FO(=)g FN(\025e)p FO(,)28 b FN(\025)23 b FP(2)h FI(R)p FO(,)33 b(then,)c(due)e(to)h(\(1.20\))o(,)g(w)n(e)f(ha)n(v)n(e)672 1140 y(\()p FN(A)19 b FP(\000)f FN(\025I)7 b FO(\))p FN(B)t(e)24 b FO(=)e FN(\026e)c FO(+)g FN(S)1449 1152 y FK(1)1487 1140 y FN(e;)-118 1321 y FO(whic)n(h)32 b(implies)c(\()p FN(S)494 1333 y FK(1)532 1321 y FN(e;)14 b(e)p FO(\))31 b(=)g FP(\000)p FN(\026)p FO(,)j(and)e(therefore,)i FP(k)p FN(S)1616 1333 y FK(1)1652 1321 y FP(k)d(\025)g(j)p FN(\026)p FP(j)p FO(.)53 b(But)33 b(since)-118 1421 y FP(k)p FN(S)-25 1433 y FK(1)12 1421 y FP(k)f(\024)h(k)p FN(T)275 1433 y FK(1)311 1421 y FP(k)g FO(=)g FP(j)p FN(\026)p FP(j)p FO(,)j(w)n(e)d(ha)n(v)n(e)f FP(k)p FN(S)1057 1433 y FK(1)1094 1421 y FP(k)h FO(=)g FP(j)p FN(\026)p FP(j)p FO(,)i(whic)n(h)e(yields)e FN(S)1955 1433 y FK(1)1993 1421 y FN(e)h FO(=)h FP(\000)p FN(\026e)p FO(.)-118 1521 y(Then)k FN(S)159 1533 y FK(1)235 1521 y FO(=)i FP(\000)p FN(T)453 1533 y FK(1)527 1521 y FO(on)e(the)g(whole)f(subspace)g FN(H)1474 1533 y FL(\026)1519 1521 y FO(.)66 b(Since)36 b(the)i(subspaces)-118 1620 y FN(H)-49 1632 y FL(\026)27 1620 y FO(and)31 b FP(f)p FN(f)37 b FP(2)29 b FN(H)36 b FP(j)29 b FN(S)604 1632 y FK(1)641 1620 y FN(f)37 b FO(=)29 b FP(\000)p FN(\026f)9 b FP(g)30 b FO(coincide,)g(the)h(op)r(erator)f FN(B)35 b FO(also)29 b(maps)-118 1720 y FN(H)-49 1732 y FL(\026)32 1720 y FO(in)n(to)34 b FN(H)277 1732 y FL(\026)322 1720 y FO(.)63 b(Then)36 b FN(T)682 1732 y FK(1)755 1720 y FO(is)f(compact,)i(and)f(w)n(e)f(ha)n(v)n(e)g FN(S)1764 1732 y FK(1)1839 1720 y FO(=)i FP(\000)p FN(T)2055 1732 y FK(1)2127 1720 y FO(on)f(all)-118 1819 y(eigenspaces)18 b(of)i(the)h(op)r(erator)d FN(T)913 1831 y FK(1)950 1819 y FO(,)k(and)e(therefore,)h(on)f(the)g(whole)f FN(H)7 b FO(.)34 b(Hence)-118 1919 y([)p FN(A;)14 b(B)t FO(])23 b(=)g(0.)p 2278 1919 4 57 v 2282 1866 50 4 v 2282 1919 V 2331 1919 4 57 v 6 2085 a(In)d(Section)e(1.4.2,)h(when)h(in)n(v)n (estigating)14 b(the)20 b(relation)c(\()p FN(V)j FO(1)1857 2097 y FK(1)1894 2085 y FO(\),)i(w)n(e)e(will)d(see)-118 2185 y(that)36 b(the)f(corresp)r(onding)e(statemen)n(t)h(for)h (arbitrary)e(b)r(ounded)j(op)r(erators)-118 2284 y(do)r(es)27 b(not)h(hold.)-118 2500 y FQ(1.4.2)94 b(Hermitian)29 b FN(q)s FQ(-plane)j(and)g FN(q)s FQ(-CCR)-118 2653 y FO(W)-7 b(e)19 b(will)d(consecutiv)n(ely)g(study)j(represen)n(tations)c (of)k(relations)c(\()p FN(V)20 b(I)1997 2665 y FK(0)2034 2653 y FO(\),)h(\()p FN(V)e(I)2245 2665 y FK(1)2283 2653 y FO(\),)-118 2753 y(\()p FN(V)g(I)7 b(I)60 2765 y FK(0)98 2753 y FO(\),)28 b(\()p FN(V)19 b(I)7 b(I)359 2765 y FK(1)397 2753 y FO(\))28 b(b)n(y)f(b)r(ounded)h(self-adjoin)n(t)d(op)r (erators.)-118 2902 y FQ(1.)42 b FO(\()p FN(V)19 b(I)134 2914 y FK(0)172 2902 y FO(\).)43 b(Consider)27 b(the)j(pairs)e(of)h(b)r (ounded)h(self-adjoin)n(t)d(op)r(erators)g(sat-)-118 3001 y(isfying)e(the)j(relation)507 3183 y([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(i\013)p FO(\()p FN(A)1006 3149 y FK(2)1062 3183 y FO(+)18 b FN(B)1212 3149 y FK(2)1250 3183 y FO(\))p FN(;)180 b(\013)23 b(>)g FO(0)p FN(:)-118 3365 y FQ(Prop)s(osition)30 b(27.)41 b FB(If)34 b(a)g(p)l(air)g(of)h(b) l(ounde)l(d)f(self-adjoint)h(op)l(er)l(ators)g FN(A)p FB(,)g FN(B)t FB(,)-118 3464 y(satis\014es)30 b FO(\()p FN(V)19 b(I)329 3476 y FK(0)367 3464 y FO(\))p FB(,)30 b(then)g FN(A)23 b FO(=)g FN(B)k FO(=)c(0)p FB(.)-118 3630 y(Pr)l(o)l(of.)43 b FO(In)n(tro)r(duce)32 b(the)h(op)r(erators)d FN(X)38 b FO(=)31 b FN(A)22 b FO(+)f FN(iB)t FO(,)33 b FN(X)1644 3600 y FM(\003)1713 3630 y FO(=)e FN(A)22 b FP(\000)f FN(iB)t FO(.)51 b(Then)-118 3729 y(the)28 b(op)r(erators)d FN(X)34 b FO(and)28 b FN(X)733 3699 y FM(\003)798 3729 y FO(satisfy)e(the)i(Hermitian)d FN(q)s FO(-plane)h(relation:)589 3911 y(\(1)18 b FP(\000)g FN(\013)p FO(\))c FN(X)7 b(X)1015 3877 y FM(\003)1076 3911 y FO(=)22 b(\(1)d(+)f FN(\013)p FO(\))p FN(X)1500 3877 y FM(\003)1538 3911 y FN(X)r(;)p eop %%Page: 71 75 71 74 bop -118 -137 a FJ(1.4.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-relations)1128 b FO(71)-118 96 y(but)28 b(since)f FN(\013)c(>)g FO(0,)k(putting)g FN(q)f FO(=)d(\(1)18 b FP(\000)g FN(\013)p FO(\))p FN(=)p FO(\(1)h(+)f FN(\013)p FO(\),)28 b(w)n(e)f(get)835 257 y FN(X)911 223 y FM(\003)948 257 y FN(X)i FO(=)23 b FN(q)s(X)7 b(X)1326 223 y FM(\003)1363 257 y FN(:)740 b FO(\(1.21\))-118 417 y(F)-7 b(or)35 b FN(q)k FP(\024)c FO(0)g(\()p FN(\013)i FP(\025)f FO(1\),)h(this)d (equation)g(p)r(ossesses)g(only)g(the)i(zero)e(solution)-118 517 y FN(X)29 b FO(=)23 b FN(X)144 487 y FM(\003)204 517 y FO(=)g(0,)g(since)f(for)g FN(q)k FP(\024)c FO(0,)i(the)f (non-negativ)n(e)d(op)r(erator)g FN(X)1967 487 y FM(\003)2005 517 y FN(X)29 b FO(should)-118 616 y(b)r(e)f(equal)e(to)h(the)h(non-p)r (ositiv)n(e)d(one)i FN(q)s(X)7 b(X)1271 586 y FM(\003)1308 616 y FO(.)37 b(But)28 b(then)g FN(A)23 b FO(=)g FN(B)k FO(=)c(0.)6 716 y(W)-7 b(e)32 b(will)d(carry)h(out)h(a)h(more)d (detailed)h(in)n(v)n(estigation)d(of)k(the)h(case)f(1)e FN(>)-118 816 y(q)d(>)d FO(0)k(\(0)c FN(<)f(\013)i(<)e FO(1\).)-118 965 y FQ(Lemma)29 b(7.)41 b FB(If)k FN(X)7 b FB(,)48 b FN(X)663 935 y FM(\003)751 965 y FP(2)i FN(L)p FO(\()p FN(H)7 b FO(\))p FB(,)49 b(and)k FO(\(1.21\))44 b FB(holds,)50 b(then)45 b FO(k)n(er)13 b FN(X)56 b FO(=)-118 1065 y(k)n(er)13 b FN(X)83 1035 y FM(\003)120 1065 y FB(.)-118 1226 y(Pr)l(o)l(of.)43 b FO(Indeed,)28 b(w)n(e)f(ha)n(v)n(e)g (k)n(er)12 b FN(X)30 b FO(=)22 b(k)n(er)13 b FN(X)1260 1196 y FM(\003)1298 1226 y FN(X)29 b FO(=)23 b(k)n(er)12 b FN(X)7 b(X)1760 1196 y FM(\003)1820 1226 y FO(=)23 b(k)n(er)13 b FN(X)2109 1196 y FM(\003)2146 1226 y FO(.)p 2278 1226 4 57 v 2282 1174 50 4 v 2282 1226 V 2331 1226 4 57 v 6 1388 a(The)26 b(represen)n(tation)d(space)i(of)h(the)g (relation)d(\(1.21\))i(no)n(w)h(has)f(the)h(form)-118 1488 y FN(H)k FO(=)22 b FN(H)137 1500 y FK(0)181 1488 y FP(\010)7 b FN(H)322 1500 y FK(1)358 1488 y FO(,)23 b(where)f FN(H)708 1500 y FK(0)767 1488 y FO(and)f FN(H)991 1500 y FK(1)1050 1488 y FO(are)g(subspaces)f(in)n(v)-5 b(arian)n(t)19 b(with)i(resp)r(ect)-118 1587 y(to)37 b FN(X)7 b FO(,)38 b FN(X)206 1557 y FM(\003)243 1587 y FO(.)65 b(On)36 b FN(H)547 1599 y FK(0)623 1587 y FO(=)i(k)n(er)12 b FN(X)45 b FO(=)38 b(k)n(er)13 b FN(X)1268 1557 y FM(\003)1342 1587 y FO(w)n(e)36 b(ha)n(v)n(e)g FN(X)44 b FO(=)38 b FN(X)1966 1557 y FM(\003)2042 1587 y FO(=)g(0;)j(on)-118 1687 y FN(H)-49 1699 y FK(1)11 1687 y FO(=)23 b FN(H)175 1657 y FM(?)168 1707 y FK(0)258 1687 y FO(these)28 b(op)r(erators)e (are)g(non-degenerate.)6 1786 y(Consider)i(the)j(represen)n(tations)26 b(of)37 b(\(1.21\))29 b(on)g FN(H)1610 1798 y FK(1)1648 1786 y FO(.)43 b(F)-7 b(or)30 b(the)g(p)r(olar)e(de-)-118 1886 y(comp)r(osition)h(of)j(the)h(op)r(erator)d FN(X)37 b FO(=)31 b FN(U)9 b(C)d FO(,)33 b(with)f(unitary)f FN(U)41 b FO(and)32 b FN(C)38 b(>)30 b FO(0,)-118 1986 y(w)n(e)d(ha)n(v)n(e)190 2146 y FN(C)255 2112 y FK(2)292 2146 y FN(U)32 b FO(=)23 b FN(U)9 b FO(\()p FN(q)s(C)672 2112 y FK(2)709 2146 y FO(\))p FN(;)97 b FO(and)83 b FN(C)1143 2112 y FK(2)1181 2146 y FN(U)1247 2112 y FM(\003)1308 2146 y FO(=)22 b FN(U)1461 2112 y FM(\003)1499 2146 y FO(\()p FN(q)1571 2112 y FM(\000)p FK(1)1661 2146 y FN(C)1726 2112 y FK(2)1763 2146 y FO(\))p FN(:)308 b FO(\(1.22\))-118 2306 y(But)24 b(then)g(if)e FN(\025)i(>)f FO(0)g(b)r(elongs)e(to)j(the)g(sp)r(ectrum) e FN(\033)s FO(\()p FN(C)1559 2276 y FK(2)1597 2306 y FO(\))i(of)g(the)f(op)r(erator)f FN(C)2278 2276 y FK(2)2316 2306 y FO(,)-118 2406 y(then)35 b FN(\033)s FO(\()p FN(C)225 2376 y FK(2)263 2406 y FO(\))f FP(\033)428 2344 y Fy(S)497 2431 y FL(k)q FM(2)p Fu(Z)635 2406 y FN(q)675 2376 y FL(k)716 2406 y FN(\025)p FO(.)57 b(F)-7 b(or)34 b FN(\025)g FO(whic)n(h)g(is)f(an)g(eigen)n(v)-5 b(alue)32 b(of)i FN(C)2108 2376 y FK(2)2145 2406 y FO(,)i(this)-118 2506 y(fact)23 b(directly)e(follo)n(ws)g(from)h(\(1.22\))o(.)35 b(F)-7 b(or)23 b(the)h(case)e(where)h FN(\025)g FO(b)r(elongs)f(to)h (the)-118 2605 y(con)n(tin)n(uous)i(sp)r(ectrum)i(of)h(the)g(op)r (erator)e FN(C)1298 2575 y FK(2)1335 2605 y FO(,)i(see)f(Section)g (2.1.1)f(b)r(elo)n(w.)6 2705 y(Since)18 b(the)h(set)468 2643 y Fy(S)538 2730 y FL(k)q FM(2)p Fu(Z)676 2705 y FN(q)716 2675 y FL(k)757 2705 y FN(\025)g FO(is)e(un)n(b)r(ounded,)k (for)d(b)r(ounded)h(represen)n(tations)-118 2805 y(of)27 b(\()p FN(V)20 b(I)112 2817 y FK(0)149 2805 y FO(\))28 b(w)n(e)g(ha)n(v)n(e)e FN(A)d FO(=)g FN(B)k FO(=)c(0.)p 2278 2805 V 2282 2752 50 4 v 2282 2805 V 2331 2805 4 57 v -118 2966 a FB(R)l(emark)30 b(20.)42 b FO(The)25 b(argumen)n(ts)d(ab)r(o)n(v)n(e)i(enable)f(us)h(to)h(write)e(an)i (explicit)d(for-)-118 3066 y(m)n(ula)h(for)j(a)f(family)e(of)j (irreducible)c FB(unb)l(ounde)l(d)34 b FO(represen)n(tations)23 b(of)i(\()p FN(V)19 b(I)2245 3078 y FK(0)2283 3066 y FO(\),)16 3575 y FN(X)30 b FO(=)202 3159 y Fy(0)202 3305 y(B)202 3355 y(B)202 3405 y(B)202 3454 y(B)202 3504 y(B)202 3554 y(B)202 3604 y(B)202 3654 y(B)202 3703 y(B)202 3753 y(B)202 3806 y(@)280 3226 y FO(.)312 3250 y(.)344 3276 y(.)280 3380 y(.)312 3405 y(.)344 3431 y(.)565 3439 y(0)703 b Fo(0)455 3475 y Fy(p)p 538 3475 178 4 v 73 x FN(q)578 3524 y FM(\000)p FK(1)667 3548 y FN(\025)122 b FO(0)799 3584 y FP(p)p 868 3584 49 4 v 71 x FN(\025)167 b FO(0)999 3692 y Fy(p)p 1082 3692 126 4 v 73 x FN(q)1122 3741 y FK(2)1160 3765 y FN(\025)111 b FO(0)556 3920 y Fo(0)1296 3861 y FO(.)1328 3886 y(.)1360 3912 y(.)1476 3861 y(.)1508 3886 y(.)1540 3912 y(.)1568 3159 y Fy(1)1568 3305 y(C)1568 3355 y(C)1568 3405 y(C)1568 3454 y(C)1568 3504 y(C)1568 3554 y(C)1568 3604 y(C)1568 3654 y(C)1568 3703 y(C)1568 3753 y(C)1568 3806 y(A)1654 3575 y FN(;)180 b(\025)24 b FP(2)f FO([1)p FN(;)14 b(q)s FO(\))p FN(;)p eop %%Page: 72 76 72 75 bop -118 -137 a FO(72)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(if)g(w)n(e)h(de\014ne)g(what)f(is)g(a)h(correct)e(meaning)f(of) j(the)g(relation)d(for)i(un)n(b)r(ounded)-118 196 y(op)r(erators)g (\(see)56 b([187)n(])28 b(etc.\).)6 296 y(Note)41 b(that)g(the)g (closure)d(of)i(the)h(symmetric)c(matrix)h FN(X)33 b FO(+)27 b FN(X)2092 266 y FM(\003)2170 296 y FO(from)-118 395 y(the)i(set)h(of)f(\014nite)f(v)n(ectors,)h(is)f(not)h(a)f (self-adjoin)n(t)f(op)r(erator;)h(its)h(de\014ciency)-118 495 y(indices)c(are)i(\(1,1\))g(\(see)56 b([24)o(]\).)-118 619 y FQ(2.)34 b FO(\()p FN(V)19 b(I)126 631 y FK(1)164 619 y FO(\).)35 b(Consider)20 b(pairs)f(of)j(b)r(ounded)g(self-adjoin)n (t)d(op)r(erators)g(satisfying)-118 719 y(the)28 b(relation)271 881 y FP(\000)p FN(i)14 b FO([)p FN(A;)g(B)t FO(])22 b(=)h FN(\013)p FO(\()p FN(A)848 847 y FK(2)905 881 y FO(+)18 b FN(B)1055 847 y FK(2)1092 881 y FO(\))h(+)f FN(I)7 b(;)180 b(\013)23 b FP(2)h FI(R)1681 847 y FK(1)1742 881 y FP(n)18 b(f)p FO(0)p FP(g)p FN(:)-118 1043 y FO(Making)24 b(a)h(c)n(hange)g(of)g(v)-5 b(ariables)23 b FN(X)29 b FO(=)23 b FN(A)14 b FO(+)h FN(iB)t FO(,)25 b FN(X)1517 1013 y FM(\003)1578 1043 y FO(=)e FN(A)14 b FP(\000)h FN(iB)t FO(,)26 b(w)n(e)f(get)g(the)-118 1142 y(follo)n(wing)487 1304 y(\(1)18 b FP(\000)g FN(\013)p FO(\))c FN(X)7 b(X)913 1270 y FM(\003)973 1304 y FO(=)23 b(\(1)18 b(+)g FN(\013)p FO(\))c FN(X)1411 1270 y FM(\003)1449 1304 y FN(X)25 b FO(+)18 b(2)p FN(I)7 b(:)392 b FO(\(1.23\))6 1466 y(F)-7 b(or)32 b FN(\013)f FP(\025)f FO(1)i(this)f(equation)g(do)r(es)g(not)h (ha)n(v)n(e)f(solutions,)g(since)g(the)h(left-)-118 1566 y(hand)27 b(side)e(con)n(tains)g(a)h(non-p)r(ositiv)n(e)e(op)r(erator,) h(and)i(the)g(op)r(erator)e(on)h(the)-118 1665 y(righ)n(t-hand)f(side)i (is)f(strictly)g(p)r(ositiv)n(e.)6 1765 y(Assuming)g FN(\013)d(<)g FO(1,)k(w)n(e)g(ha)n(v)n(e)87 1961 y FN(X)7 b(X)239 1927 y FM(\003)299 1961 y FO(=)397 1905 y(1)18 b(+)g FN(\013)p 397 1942 197 4 v 397 2018 a FO(1)g FP(\000)g FN(\013)603 1961 y(X)679 1927 y FM(\003)716 1961 y FN(X)25 b FO(+)981 1905 y(2)p 903 1942 V 903 2018 a(1)18 b FP(\000)g FN(\013)1110 1961 y(I)30 b FO(=)22 b FN(q)s(X)1379 1927 y FM(\003)1417 1961 y FN(X)j FO(+)18 b(\()p FN(q)j FO(+)d(1\))p FN(I)7 b(;)219 b FO(\(1.24\))-118 2158 y(where)30 b FN(q)g FO(=)d(\(1)18 b(+)g FN(\013)q FO(\))p FN(=)p FO(\(1)g FP(\000)g FN(\013)p FO(\))28 b FN(>)f FP(\000)p FO(1.)44 b(T)-7 b(o)30 b(rewrite)e(the)j(latter)e(in)g(the)i(form)-118 2258 y(used)i(in)g(the)g(literature,)f(in)n(tro)r(duce)g(the)i(op)r (erator)d FN(a)h FO(=)1760 2197 y FP(p)p 1830 2197 184 4 v 1830 2258 a FN(q)21 b FO(+)d(1)13 b FN(X)7 b FO(,)35 b FN(a)2204 2228 y FM(\003)2274 2258 y FO(=)-118 2297 y FP(p)p -49 2297 V 60 x FN(q)22 b FO(+)c(1)13 b FN(X)224 2327 y FM(\003)261 2357 y FO(.)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)825 2519 y FN(aa)913 2485 y FM(\003)974 2519 y FO(=)c FN(q)s(a)1146 2485 y FM(\003)1184 2519 y FN(a)18 b FO(+)g FN(I)7 b(:)731 b FO(\(1.25\))-118 2681 y FB(R)l(emark)30 b(21.)42 b FO(F)-7 b(or)40 b FN(q)48 b FO(=)c FP(\000)p FO(1,)f(equation)c (\(1.25\))o(,)44 b FP(f)p FN(a;)14 b(a)1668 2651 y FM(\003)1706 2681 y FP(g)44 b FO(=)g FN(I)7 b FO(,)44 b(p)r(ossesses)-118 2781 y(irreducible)24 b(one-dimensional)e(and)28 b(t)n(w)n (o-dimensional)21 b(solutions.)6 2905 y(F)-7 b(urther,)27 b(in)e(the)h(in)n(terv)-5 b(al)24 b FN(q)i(>)d FP(\000)p FO(1)i(select)g(the)h(p)r(oin)n(ts)f FN(q)h FO(=)d(0)j(\()p FN(\013)e FO(=)e FP(\000)p FO(1\),)-118 3005 y(and)32 b FN(q)i FO(=)d(1)h(\()p FN(\013)g FO(=)f(0\),)j(corresp)r(onding)29 b(to)k(the)g(co-isometry)28 b FN(aa)1964 2975 y FM(\003)2033 3005 y FO(=)j FN(I)40 b FO(and)-118 3105 y(CCR)28 b([)p FN(a;)14 b(a)239 3074 y FM(\003)276 3105 y FO(])24 b(=)e FN(I)7 b FO(:)p -48 3187 2317 4 v -50 3286 4 100 v 9 3256 a FN(q)p 103 3286 V 192 w FO(1)22 b FN(<)h(q)p 561 3286 V 192 w(q)j FO(=)d(1)p 866 3286 V 145 w(0)f FN(<)h(q)j(<)d FO(1)p 1388 3286 V 177 w FN(q)j FO(=)c(0)p 1758 3286 V 139 w FP(\000)p FO(1)g FN(<)g(q)k(<)d FO(0)p 2267 3286 V -48 3290 2317 4 v -50 3389 4 100 v 2 3359 a FN(\013)p 103 3389 V 100 w FO(0)g FN(<)f(\013)i(<)e FO(1)p 561 3389 V 99 w FN(\013)i FO(=)f(0)p 866 3389 V 99 w FP(\000)p FO(1)f FN(<)g(\013)i(<)f FO(0)p 1388 3389 V 99 w FN(\013)g FO(=)g FP(\000)p FO(1)p 1758 3389 V 168 w FN(\013)g(<)g FP(\000)p FO(1)p 2267 3389 V -48 3393 2317 4 v -118 3550 a FQ(3.)37 b FO(F)-7 b(or)28 b FN(q)f(>)c FO(1,)28 b(the)h(op)r(erator) d FN(a)912 3520 y FM(\003)978 3550 y FO(is)h(non-degenerate.)36 b(Let)29 b FN(a)23 b FO(=)h FN(U)9 b(C)34 b FO(b)r(e)28 b(its)-118 3650 y(p)r(olar)18 b(decomp)r(osition)f(suc)n(h)j(that)h(k)n (er)13 b FN(U)32 b FO(=)22 b(k)n(er)13 b FN(C)6 b FO(,)22 b(and)f FN(U)29 b FO(is)19 b(a)h(co-isometry;)-118 3749 y FN(C)29 b FP(\025)23 b FO(0.)36 b(Then)466 3911 y FN(U)9 b(C)597 3877 y FK(2)634 3911 y FN(U)700 3877 y FM(\003)761 3911 y FO(=)23 b FN(q)s(C)6 b(U)1020 3877 y FM(\003)1058 3911 y FN(U)j(C)25 b FO(+)18 b FN(I)30 b FO(=)22 b FN(q)s(C)1549 3877 y FK(2)1605 3911 y FO(+)c FN(I)7 b(;)p eop %%Page: 73 77 73 76 bop -118 -137 a FJ(1.4.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-relations)1128 b FO(73)-118 96 y(whic)n(h)26 b(giv)n(es)479 263 y FN(C)544 229 y FK(2)581 263 y FN(U)647 229 y FM(\003)708 263 y FO(=)d FN(U)862 229 y FM(\003)900 263 y FO(\()p FN(q)s(C)1037 229 y FK(2)1093 263 y FO(+)18 b FN(I)7 b FO(\))23 b(=)g FN(U)1428 229 y FM(\003)1466 263 y FN(f)9 b FO(\()p FN(C)1613 229 y FK(2)1650 263 y FO(\))p FN(;)517 398 y(C)582 363 y FK(2)619 398 y FN(U)32 b FO(=)23 b FN(U)9 b(q)902 363 y FM(\000)p FK(1)991 398 y FO(\()p FN(C)1088 363 y FK(2)1144 398 y FP(\000)18 b FN(I)7 b FO(\))23 b(=)g FN(f)1463 363 y FM(\000)p FK(1)1552 398 y FO(\()p FN(C)1649 363 y FK(2)1686 398 y FO(\))p FN(:)385 b FO(\(1.26\))-118 564 y(If)28 b FN(\025)23 b FP(2)h FN(\033)s FO(\()p FN(C)262 534 y FK(2)300 564 y FO(\),)k(then)g(the)g(p)r(oin)n(ts)e FN(f)9 b FO(\()p FN(\025)p FO(\))24 b(=)e FN(q)s(\025)d FO(+)f(1,)27 b FN(f)9 b FO(\()p FN(f)g FO(\()p FN(\025)p FO(\)\))24 b(=)e FN(q)1946 534 y FK(2)1984 564 y FN(\025)c FO(+)g FN(q)j FO(+)d(1,)-118 664 y FN(:)c(:)g(:)27 b FO(,)i(also)d(b)r(elong)g (to)i FN(\033)s FO(\()p FN(C)739 634 y FK(2)777 664 y FO(\).)39 b(But)28 b(this)f(set)h(of)g(p)r(oin)n(ts)f(is)g(un)n(b)r (ounded,)h(i.e.,)-118 763 y FB(for)i FN(q)d(>)22 b FO(1)29 b FB(ther)l(e)h(ar)l(e)g(no)g(r)l(epr)l(esentations)g(by)g(b)l(ounde)l (d)h(op)l(er)l(ators)p FO(.)-118 890 y FB(R)l(emark)f(22.)42 b FO(F)-7 b(or)22 b FN(q)k(>)d FO(1)e(one)h(can)f(consider)f(a)i (formal)d(un)n(b)r(ounded)j(solution)-118 989 y(of)34 b(\(1.25\))27 b(giv)n(en)f(b)n(y)h(the)h(follo)n(wing)23 b(Jacobi)j(matrix)311 1513 y FN(a)d FO(=)466 1097 y Fy(0)466 1243 y(B)466 1293 y(B)466 1343 y(B)466 1393 y(B)466 1443 y(B)466 1493 y(B)466 1542 y(B)466 1592 y(B)466 1642 y(B)466 1692 y(B)466 1745 y(@)538 1155 y FO(0)189 b(0)538 1255 y(1)g(0)731 b Fo(0)663 1291 y Fy(q)p 746 1291 170 4 v 756 1349 a FK(1)p FM(\000)p FL(q)873 1333 y Fx(2)p 756 1367 150 4 v 772 1415 a FK(1)p FM(\000)p FL(q)1027 1387 y FO(0)1003 1504 y(.)1036 1529 y(.)1068 1554 y(.)1266 1504 y(.)1298 1529 y(.)1330 1554 y(.)1179 1599 y Fy(q)p 1262 1599 179 4 v 1272 1654 a FK(1)p FM(\000)p FL(q)1389 1638 y Fv(n)p 1272 1672 159 4 v 1292 1720 a FK(1)p FM(\000)p FL(q)1551 1691 y FO(0)760 1870 y Fo(0)1528 1812 y FO(.)1560 1837 y(.)1592 1862 y(.)1707 1812 y(.)1740 1837 y(.)1772 1862 y(.)1800 1097 y Fy(1)1800 1243 y(C)1800 1293 y(C)1800 1343 y(C)1800 1393 y(C)1800 1443 y(C)1800 1493 y(C)1800 1542 y(C)1800 1592 y(C)1800 1642 y(C)1800 1692 y(C)1800 1745 y(A)1886 1513 y FN(:)-118 2033 y FO(In)32 b(this)g(case,)g(the)g (problem)e(with)i(in)n(tro)r(ducing)d(un)n(b)r(ounded)k(op)r(erators)d (is)-118 2132 y(not)k(quite)g(trivial.)53 b(F)-7 b(or)34 b(example,)g(the)g(closure)f(of)h(the)h(follo)n(wing)30 b(Jacobi)-118 2232 y(matrix)78 2801 y FN(a)18 b FO(+)g FN(a)267 2767 y FM(\003)328 2801 y FO(=)416 2310 y Fy(0)416 2456 y(B)416 2506 y(B)416 2556 y(B)416 2606 y(B)416 2656 y(B)416 2705 y(B)416 2755 y(B)416 2805 y(B)416 2855 y(B)416 2905 y(B)416 2955 y(B)416 3004 y(B)416 3054 y(B)416 3107 y(@)489 2380 y FO(0)188 b(1)489 2512 y(1)g(0)949 2416 y Fy(q)p 1032 2416 170 4 v 1042 2475 a FK(1)p FM(\000)p FL(q)1159 2458 y Fx(2)p 1042 2493 150 4 v 1058 2540 a FK(1)p FM(\000)p FL(q)1730 2512 y Fo(0)613 2592 y Fy(q)p 696 2592 170 4 v 706 2650 a FK(1)p FM(\000)p FL(q)823 2634 y Fx(2)p 706 2668 150 4 v 722 2716 a FK(1)p FM(\000)p FL(q)1055 2687 y FO(0)1372 2629 y(.)1404 2654 y(.)1436 2679 y(.)1032 2805 y(.)1064 2830 y(.)1096 2855 y(.)1372 2805 y(.)1404 2830 y(.)1436 2855 y(.)1629 2771 y Fy(q)p 1712 2771 179 4 v 1722 2826 a FK(1)p FM(\000)p FL(q)1839 2809 y Fv(n)p 1722 2844 159 4 v 1742 2892 a FK(1)p FM(\000)p FL(q)1285 2949 y Fy(q)p 1368 2949 179 4 v 1378 3005 a FK(1)p FM(\000)p FL(q)1495 2988 y Fv(n)p 1378 3023 159 4 v 1398 3070 a FK(1)p FM(\000)p FL(q)1739 3042 y FO(0)1978 2984 y(.)2010 3009 y(.)2042 3034 y(.)710 3220 y Fo(0)1716 3162 y FO(.)1748 3187 y(.)1780 3212 y(.)1978 3162 y(.)2010 3187 y(.)2042 3212 y(.)2070 2310 y Fy(1)2070 2456 y(C)2070 2506 y(C)2070 2556 y(C)2070 2606 y(C)2070 2656 y(C)2070 2705 y(C)2070 2755 y(C)2070 2805 y(C)2070 2855 y(C)2070 2905 y(C)2070 2955 y(C)2070 3004 y(C)2070 3054 y(C)2070 3107 y(A)-118 3386 y FO(de\014ned)30 b(on)g(the)h(set)f(of)g(\014nite)g (v)n(ectors,)g(is)f(not)h(self-adjoin)n(t,)e(but)j(has)f(de\014-)-118 3486 y(ciency)c(indices)g(\(1,1\))h(see)55 b([24)o(,)28 b(52)o(].)-118 3612 y FQ(4.)34 b FO(F)-7 b(or)21 b(0)h FN(<)h(q)j(<)d FO(1,)f(the)f(op)r(erator)f FN(a)1033 3582 y FM(\003)1092 3612 y FO(is)g(also)g(non-degenerate,)g(in)h(the)h (p)r(olar)-118 3712 y(decomp)r(osition)16 b FN(a)23 b FO(=)g FN(U)9 b(C)d FO(,)21 b FN(U)29 b FO(is)19 b(a)h(co-isometry)-7 b(,)17 b(and)j(equalit)n(y)f(\(1.26\))g(holds.)6 3811 y(If)i FN(\025)j FP(2)f FN(\033)s FO(\()p FN(C)379 3781 y FK(2)417 3811 y FO(\),)g(then)d(all)e(p)r(oin)n(ts)i FN(f)9 b FO(\()p FN(\025)p FO(\),)22 b FN(f)9 b FO(\()p FN(f)g FO(\()p FN(\025)p FO(\)\),)22 b FN(:)14 b(:)g(:)34 b FO(should)19 b(also)f(b)r(elong)-118 3911 y(to)29 b(the)g(sp)r (ectrum)g FN(\033)s FO(\()p FN(C)641 3881 y FK(2)679 3911 y FO(\).)42 b(T)-7 b(o)28 b(the)i(p)r(oin)n(t)e FN(\025)e FO(=)f(\(1)20 b FP(\000)f FN(q)s FO(\))1676 3881 y FM(\000)p FK(1)1794 3911 y FO(\(the)30 b(stationary)p eop %%Page: 74 78 74 77 bop -118 -137 a FO(74)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(p)r(oin)n(t)f(of)i(this)e(mapping\))g(there)h(corresp)r(onds)e (a)i(circle)e(of)i(one-dimensional)-118 196 y(op)r(erators)30 b FN(a)g FO(=)f(\(1)21 b FP(\000)g FN(q)s FO(\))675 166 y FM(\000)p FK(1)p FL(=)p FK(2)832 196 y FN(e)871 166 y FL(i\036)938 196 y FO(,)33 b FN(\036)e FP(2)f FN(S)1215 166 y FK(1)1252 196 y FO(.)50 b(If)32 b FN(\025)e(>)g FO(\(1)21 b FP(\000)g FN(q)s FO(\))1838 166 y FM(\000)p FK(1)1927 196 y FO(,)33 b(the)g(corre-)-118 296 y(sp)r(onding)26 b(solutions)f(are)i(un)n(b)r(ounded.)-118 427 y FB(R)l(emark)j(23.)42 b FO(There)e(exists)f(a)h(series)f(of)h(suc)n(h)g(inequiv)-5 b(alen)n(t)37 b(un)n(b)r(ounded)-118 527 y(represen)n(tations)19 b(of)29 b(\(1.25\))21 b(dep)r(ending)h(on)f(a)h(parameter)e FN(\025)j FP(2)h FO(\()p FN(\025)1971 539 y FK(0)2009 527 y FN(;)14 b(q)s(\025)2134 539 y FK(0)2179 527 y FO(+)8 b(1],)-118 627 y FN(\025)-70 639 y FK(0)-9 627 y FN(>)22 b FO(\(1)d FP(\000)f FN(q)s FO(\))326 597 y FM(\000)p FK(1)415 627 y FO(.)6 758 y(If)34 b(\(1)22 b FP(\000)g FN(q)s FO(\))350 728 y FM(\000)p FK(1)472 758 y FN(>)33 b(\025)g(>)f FO(0,)i(then)g FN(\025)f FP(2)g FN(\033)s FO(\()p FN(C)1358 728 y FK(2)1396 758 y FO(\))h(implies)c(that)j FN(\033)s FO(\()p FN(C)2082 728 y FK(2)2121 758 y FO(\))g(con-)-118 858 y(tains)26 b(all)g(p)r(oin)n(ts)g FN(f)9 b FO(\()p FN(\025)p FO(\),)28 b FN(f)9 b FO(\()p FN(f)g FO(\()p FN(\025)p FO(\)\),)29 b FN(:)14 b(:)g(:)27 b FO(,)h(tending)f(to)g(\(1) 18 b FP(\000)g FN(q)s FO(\))1814 828 y FM(\000)p FK(1)1904 858 y FO(;)28 b(also)d FN(\033)s FO(\()p FN(C)2268 828 y FK(2)2306 858 y FO(\))-118 958 y(con)n(tains)32 b(all)h(p)r(oin)n(ts) g FN(f)643 927 y FM(\000)p FK(1)732 958 y FO(\()p FN(\025)p FO(\),)k FN(f)954 927 y FM(\000)p FK(1)1043 958 y FO(\()p FN(f)1125 927 y FM(\000)p FK(1)1214 958 y FO(\()p FN(\025)p FO(\)\),)f FN(:)14 b(:)g(:)28 b FO(,)36 b(unless)d(this)h(sequence)-118 1057 y(con)n(tains)23 b(the)j(zero)e(p)r(oin)n(t.)35 b(Otherwise)23 b(k)n(er)13 b FN(U)32 b FO(=)22 b(k)n(er)13 b FN(C)29 b FO(=)23 b(0,)i(whic)n(h)f(implies)-118 1157 y FN(f)-68 1127 y FK(\()p FM(\000)p FL(n)p FK(\))80 1157 y FO(\()p FN(\025)p FO(\))36 b FP(2)f FN(\033)s FO(\()p FN(C)465 1127 y FK(2)503 1157 y FO(\))g(for)f(an)n(y)g FN(n)p FO(.)57 b(But)35 b FN(f)1220 1127 y FK(\()p FM(\000)p FL(m)p FK(\))1386 1157 y FO(\()p FN(\025)p FO(\))h FN(<)e FO(0)g(for)g(some)f FN(m)i FP(2)g FI(N)t FO(.)-118 1256 y(This)27 b(means)g(that)h(equation)f(\(1.25\))g(has)h(the)h FB(unique)f FO(irreducible)c(in\014nite-)-118 1356 y(dimensional)f (represen)n(tation)i(b)n(y)i(b)r(ounded)h(op)r(erators,)311 1889 y FN(a)23 b FO(=)466 1473 y Fy(0)466 1619 y(B)466 1669 y(B)466 1719 y(B)466 1768 y(B)466 1818 y(B)466 1868 y(B)466 1918 y(B)466 1968 y(B)466 2017 y(B)466 2067 y(B)466 2120 y(@)538 1531 y FO(0)189 b(0)538 1630 y(1)g(0)731 b Fo(0)663 1667 y Fy(q)p 746 1667 170 4 v 756 1725 a FK(1)p FM(\000)p FL(q)873 1708 y Fx(2)p 756 1743 150 4 v 772 1791 a FK(1)p FM(\000)p FL(q)1027 1762 y FO(0)1003 1880 y(.)1036 1905 y(.)1068 1930 y(.)1266 1880 y(.)1298 1905 y(.)1330 1930 y(.)1179 1974 y Fy(q)p 1262 1974 179 4 v 1272 2030 a FK(1)p FM(\000)p FL(q)1389 2013 y Fv(n)p 1272 2048 159 4 v 1292 2095 a FK(1)p FM(\000)p FL(q)1551 2067 y FO(0)760 2245 y Fo(0)1528 2187 y FO(.)1560 2212 y(.)1592 2237 y(.)1707 2187 y(.)1740 2212 y(.)1772 2237 y(.)1800 1473 y Fy(1)1800 1619 y(C)1800 1669 y(C)1800 1719 y(C)1800 1768 y(C)1800 1818 y(C)1800 1868 y(C)1800 1918 y(C)1800 1968 y(C)1800 2017 y(C)1800 2067 y(C)1800 2120 y(A)1886 1889 y FN(;)-118 2426 y FO(with)20 b(a)g(v)-5 b(acuum)20 b(v)n(ector)f FN(e)707 2438 y FK(0)764 2426 y FO(suc)n(h)h(that)h FN(ae)1200 2438 y FK(0)1260 2426 y FO(=)i(0)d(\(the)h(F)-7 b(o)r(c)n(k)20 b(represen)n(tation\).)6 2526 y(Notice)27 b(that)h(the)g(sp)r(ectrum)f(of)g(the)h(b)r(ounded)g (self-adjoin)n(t)d(op)r(erator)422 2943 y FN(a)18 b FO(+)g FN(a)611 2908 y FM(\003)672 2943 y FO(=)760 2651 y Fy(0)760 2797 y(B)760 2847 y(B)760 2897 y(B)760 2947 y(B)760 2997 y(B)760 3050 y(@)833 2700 y FO(0)188 b(1)833 2832 y(1)g(0)1293 2737 y Fy(q)p 1376 2737 170 4 v 1386 2795 a FK(1)p FM(\000)p FL(q)1503 2778 y Fx(2)p 1386 2813 150 4 v 1402 2861 a FK(1)p FM(\000)p FL(q)957 2912 y Fy(q)p 1040 2912 170 4 v 1050 2971 a FK(1)p FM(\000)p FL(q)1167 2954 y Fx(2)p 1050 2989 150 4 v 1066 3036 a FK(1)p FM(\000)p FL(q)1399 3008 y FO(0)1634 2950 y(.)1666 2975 y(.)1698 3000 y(.)1376 3125 y(.)1408 3150 y(.)1440 3176 y(.)1634 3125 y(.)1666 3150 y(.)1698 3176 y(.)1726 2651 y Fy(1)1726 2797 y(C)1726 2847 y(C)1726 2897 y(C)1726 2947 y(C)1726 2997 y(C)1726 3050 y(A)-118 3364 y FO(is)26 b(concen)n(trated)h(on)g(the)h(in)n(terv) -5 b(al)25 b([)p FP(\000)p FO(2)p FN(=)p FO(\(1)17 b FP(\000)h FN(q)s FO(\))p FN(;)c FO(2)p FN(=)p FO(\(1)k FP(\000)g FN(q)s FO(\)].)-118 3513 y FQ(5.)45 b FO(F)-7 b(or)30 b FN(q)h FO(=)d(0,)j(w)n(e)f(ha)n(v)n(e)f FN(aa)818 3482 y FM(\003)884 3513 y FO(=)f FN(I)7 b FO(,)31 b(i.e.,)g FN(a)g FO(is)e(a)h(co-isometry)-7 b(.)42 b(Irreducible)-118 3612 y(represen)n(tations)19 b(are)j(the)h(follo)n(wing:)30 b(a)22 b(circle)e(of)i(one-dimensional)17 b(unitary)-118 3712 y(op)r(erators)37 b FN(a)42 b FO(=)h FN(e)494 3682 y FL(i\036)561 3712 y FO(,)f FN(\036)h FP(2)g FN(S)872 3682 y FK(1)909 3712 y FO(,)f(and)d(a)g(single)e(op)r(erator)g(adjoin)n (t)h(to)h(the)-118 3811 y(unilateral)26 b(shift)k(op)r(erator)e(in)i FN(l)910 3823 y FK(2)947 3811 y FO(\()p FI(Z)1040 3823 y FK(+)1090 3811 y FO(\).)44 b(In)31 b(this)e(case,)h(an)n(y)f (isometry)e(in)j FN(H)-118 3911 y FO(giv)n(es)21 b(rise)h(to)h(a)g (unique)f(decomp)r(osition)e(of)j FN(H)30 b FO(in)n(to)22 b(in)n(v)-5 b(arian)n(t)20 b(with)j(resp)r(ect)p eop %%Page: 75 79 75 78 bop -118 -137 a FJ(1.4.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-relations)1128 b FO(75)-118 96 y(to)19 b FN(a)g FO(and)f FN(a)234 66 y FM(\003)291 96 y FO(subspaces,)i FN(H)30 b FO(=)22 b FN(H)942 108 y FK(uni)1040 96 y FP(\010)q FN(H)1175 108 y FK(shift)1326 96 y FO(suc)n(h)c(that)i FN(a)j Fr(\026)1778 108 y FL(H)1832 116 y Fx(uni)1938 96 y FO(is)18 b(a)g(unitary)-118 196 y(op)r(erator,)37 b(and)g FN(a)i Fr(\026)540 208 y FL(H)594 217 y Fx(shift)750 196 y FO(is)d(unitarily)e(equiv)-5 b(alen)n(t)34 b(to)j(a)g(m)n (ultiple)c(of)k(the)-118 296 y(op)r(erator)31 b(adjoin)n(t)h(to)g(the)i (unilateral)29 b(shift)j(\(H.)i(W)-7 b(old's)32 b(decomp)r(osition\).) -118 395 y(One)27 b(can)g(easily)e(obtain)h(this)h(decomp)r(osition)d (putting)589 648 y FN(H)658 660 y FK(shift)813 648 y FO(=)928 544 y FM(1)915 569 y Fy([)901 748 y FL(k)q FK(=0)1022 648 y FO(\()p FN(a)1098 614 y FM(\003)1136 648 y FO(\))1168 614 y FL(k)1209 648 y FO(\()p FN(H)i FP(\011)18 b FN(a)1463 614 y FM(\003)1501 648 y FN(H)7 b FO(\))p FN(:)-118 917 y FQ(6.)36 b FO(F)-7 b(or)27 b FP(\000)p FO(1)22 b FN(<)h(q)j(<)d FO(0,)k(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing:)6 1021 y(a\))e(a)f(circle)d(of)k(one-dimensional)20 b(represen)n(tations)i FN(a)h FO(=)g(\(1)13 b FP(\000)h FN(q)s FO(\))2053 991 y FM(\000)p FK(1)p FL(=)p FK(2)2209 1021 y FN(e)2248 991 y FL(i\036)2316 1021 y FO(,)-118 1121 y FN(\036)23 b FP(2)h FN(S)89 1090 y FK(1)126 1121 y FO(;)6 1224 y(b\))29 b(the)e(F)-7 b(o)r(c)n(k)28 b(represen)n(tation)d(b)n(y)i(b)r(ounded)h (op)r(erators)311 1773 y FN(a)23 b FO(=)466 1357 y Fy(0)466 1503 y(B)466 1553 y(B)466 1602 y(B)466 1652 y(B)466 1702 y(B)466 1752 y(B)466 1802 y(B)466 1851 y(B)466 1901 y(B)466 1951 y(B)466 2004 y(@)538 1414 y FO(0)189 b(0)538 1514 y(1)g(0)740 b(0)663 1550 y Fy(q)p 746 1550 170 4 v 756 1609 a FK(1)p FM(\000)p FL(q)873 1592 y Fx(2)p 756 1627 150 4 v 772 1674 a FK(1)p FM(\000)p FL(q)1027 1646 y FO(0)1003 1763 y(.)1036 1788 y(.)1068 1814 y(.)1266 1763 y(.)1298 1788 y(.)1330 1814 y(.)769 1950 y(0)1179 1858 y Fy(q)p 1262 1858 179 4 v 1272 1913 a FK(1)p FM(\000)p FL(q)1389 1897 y Fv(n)p 1272 1931 159 4 v 1292 1979 a FK(1)p FM(\000)p FL(q)1551 1950 y FO(0)1528 2071 y(.)1560 2096 y(.)1592 2121 y(.)1707 2071 y(.)1740 2096 y(.)1772 2121 y(.)1800 1357 y Fy(1)1800 1503 y(C)1800 1553 y(C)1800 1602 y(C)1800 1652 y(C)1800 1702 y(C)1800 1752 y(C)1800 1802 y(C)1800 1851 y(C)1800 1901 y(C)1800 1951 y(C)1800 2004 y(A)1886 1773 y FN(:)-118 2316 y FO(There)23 b(are)f(no)h (represen)n(tations)d(b)n(y)k(un)n(b)r(ounded)f(op)r(erators,)f(since)h (for)f(eac)n(h)-118 2416 y FP(\000)p FN(q)-13 2386 y FM(\000)p FK(1)99 2416 y FN(<)g(\025)i FP(2)f FN(\033)s FO(\()p FN(C)483 2386 y FK(2)521 2416 y FO(\))28 b(the)g(p)r(oin)n(t)f FN(f)991 2386 y FM(\000)p FK(1)1080 2416 y FO(\()p FN(\025)p FO(\))d FN(<)f FO(0)k(also)e(b)r(elongs)h(to)i FN(\033)s FO(\()p FN(C)2086 2386 y FK(2)2124 2416 y FO(\).)-118 2653 y FQ(1.4.3)94 b(Real)31 b(quan)m(tum)h(plane)f(and)h(real)g(quan)m (tum)g(h)m(yp)s(erb)s(oloid)-118 2814 y(1.)37 b FO(Consider,)26 b(\014nally)-7 b(,)26 b(pairs)g(of)i(self-adjoin)n(t)d(op)r(erators)h (whic)n(h)h(satisfy)f(the)-118 2914 y(relations)e(\()p FN(V)19 b(I)7 b(I)397 2926 y FK(0)435 2914 y FO(\))28 b(and)g(\()p FN(V)19 b(I)7 b(I)835 2926 y FK(1)873 2914 y FO(\),)314 3104 y([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(i\013)p FO(\()p FN(A)813 3070 y FK(2)870 3104 y FP(\000)c FN(B)1020 3070 y FK(2)1057 3104 y FO(\))p FN(;)180 b(\013)24 b(>)e FO(0)p FN(;)314 3239 y FO([)p FN(A;)14 b(B)t FO(])24 b(=)e FN(i\013)p FO(\()p FN(A)813 3205 y FK(2)870 3239 y FP(\000)c FN(B)1020 3205 y FK(2)1057 3239 y FO(\))h(+)f FN(iI)7 b(;)179 b(\013)24 b FP(2)f FI(R)i FP(n)18 b(f)p FO(0)p FP(g)p FN(:)6 3434 y FO(Represen)n(tations)23 b(of)j(relations)c(\()p FN(V)d(I)7 b(I)1214 3446 y FK(0)1252 3434 y FO(\))26 b(and)f(\()p FN(V)19 b(I)7 b(I)1647 3446 y FK(1)1685 3434 y FO(\))26 b(b)n(y)f(b)r(ounded)h(self-)-118 3534 y(adjoin)n(t)g(op)r(erators)g(can)h(b)r(e)h(obtained)e(from)h(the) h(follo)n(wing)23 b(theorem.)-118 3712 y FQ(Theorem)30 b(14.)41 b FO(\(B.)27 b(F)-7 b(uglede,)27 b(C.R.)g(Putnam,)g(M.)g (Rosen)n(blum\))p FB(.)37 b(L)l(et)29 b FN(M)9 b FB(,)-118 3811 y FN(N)g FB(,)25 b FN(T)34 b FP(2)23 b FN(L)p FO(\()p FN(H)7 b FO(\))p FB(,)25 b(and)f(the)f(op)l(er)l(ators)i FN(M)32 b FB(and)24 b FN(N)32 b FB(b)l(e)23 b(normal.)37 b(If)24 b FN(M)9 b(T)34 b FO(=)22 b FN(T)12 b(N)d FB(,)-118 3911 y(then)29 b FN(M)156 3881 y FM(\003)194 3911 y FN(T)34 b FO(=)23 b FN(T)12 b(N)502 3881 y FM(\003)569 3911 y FB(as)30 b(wel)t(l.)p eop %%Page: 76 80 76 79 bop -118 -137 a FO(76)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FB(Pr)l(o)l(of.)43 b FO(The)37 b(pro)r(of)f(is)g(based)h(on)g(the) g(unitarit)n(y)d(of)j(the)h(op)r(erator-v)-5 b(alued)-118 196 y(functions)21 b(exp\()p FN(z)t(M)526 166 y FM(\003)570 196 y FP(\000)12 b FO(\026)-47 b FN(z)s(M)9 b FO(\))22 b(and)f(exp\()p FP(\000)p FN(z)t(N)1326 166 y FM(\003)1370 196 y FO(+)12 b(\026)-47 b FN(z)s(N)9 b FO(\).)35 b(Since)21 b FN(M)1951 166 y FL(k)1991 196 y FN(T)34 b FO(=)23 b FN(T)12 b(N)2299 166 y FL(k)-118 296 y FO(for)38 b(all)f FN(k)44 b FO(=)e(1,)f(2,)d FN(:)14 b(:)g(:)28 b FO(,)42 b(and)c(exp\()5 b(\026)-47 b FN(z)t(M)9 b FO(\))p FN(T)53 b FO(=)41 b FN(T)25 b FO(exp\()5 b(\026)-47 b FN(z)s(N)9 b FO(\),)42 b FP(8)p FN(z)i FP(2)e FI(C)15 b FO(,)48 b(w)n(e)-118 395 y(conclude)26 b(that)i(for)f(the)h(b)r(ounded)g(en)n (tire)e(op)r(erator-v)-5 b(alued)25 b(function)i FN(F)12 b FO(\()p FP(\001)p FO(\))141 580 y FN(F)g FO(\()p FN(z)t FO(\))23 b(=)g(exp\()p FN(z)t(M)716 545 y FM(\003)771 580 y FP(\000)g FO(\026)-47 b FN(z)t(M)9 b FO(\))p FN(T)24 b FO(exp\()5 b(\026)-47 b FN(z)t(N)27 b FP(\000)18 b FN(z)t(N)1590 545 y FM(\003)1627 580 y FO(\))336 704 y(=)23 b(exp\()p FN(z)t(M)716 670 y FM(\003)753 704 y FO(\)\(exp\()p FP(\000)5 b FO(\026)-47 b FN(z)s(M)9 b FO(\)\))p FN(T)25 b FO(exp\()5 b(\026)-47 b FN(z)t(N)9 b FO(\)\))14 b(exp\()p FP(\000)p FN(z)t(N)2010 670 y FM(\003)2047 704 y FO(\))336 829 y(=)23 b FN(F)12 b FO(\(0\))23 b(=)f FN(T)12 b FO(;)-118 1013 y(therefore,)33 b(exp\()p FN(z)t(M)552 983 y FM(\003)589 1013 y FO(\))p FN(T)44 b FO(=)31 b FN(T)25 b FO(exp\()p FN(z)t(N)1162 983 y FM(\003)1199 1013 y FO(\))33 b(for)g(all)d FN(z)35 b FP(2)e FI(C)15 b FO(.)58 b(Th)n(us,)34 b FN(M)2144 983 y FM(\003)2182 1013 y FN(T)43 b FO(=)-118 1113 y FN(T)12 b(N)19 1083 y FM(\003)56 1113 y FO(.)p 2278 1113 4 57 v 2282 1060 50 4 v 2282 1113 V 2331 1113 4 57 v -118 1286 a FQ(2.)36 b FO(Relation)25 b(\()p FN(V)19 b(I)7 b(I)504 1298 y FK(0)542 1286 y FO(\),)507 1470 y([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(i\013)p FO(\()p FN(A)1006 1436 y FK(2)1062 1470 y FP(\000)18 b FN(B)1212 1436 y FK(2)1250 1470 y FO(\))p FN(;)180 b(\013)23 b(>)g FO(0)p FN(;)-118 1655 y FO(can)k(b)r(e)h(rewritten)e(in)h(the)h(form)341 1859 y FN(X)7 b(Y)41 b FO(=)22 b FN(q)s(Y)d(X)r(;)180 b(q)26 b FO(=)1134 1802 y(1)18 b FP(\000)g FN(i\013)p 1134 1839 226 4 v 1134 1916 a FO(1)g(+)g FN(i\013)1392 1859 y FP(6)p FO(=)23 b(1)p FN(;)96 b FP(j)p FN(q)s FP(j)24 b FO(=)e(1)-118 2083 y(\(real)37 b(quan)n(tum)g(sphere)h FI(R)772 2053 y FK(2)772 2104 y FL(q)816 2083 y FO(\);)44 b(here,)d FN(X)48 b FO(=)41 b FN(A)26 b FO(+)f FN(B)t FO(,)42 b FN(Y)60 b FO(=)41 b FN(A)26 b FP(\000)f FN(B)t FO(.)70 b(F)-7 b(or)-118 2183 y(b)r(ounded)39 b(self-adjoin)n(t)c(op)r (erators,)k(w)n(e)f(also)f(ha)n(v)n(e)g(due)h(to)g(the)h(F)-7 b(uglede{)-118 2282 y(Putnam{Rosen)n(blum)23 b(theorem:)877 2467 y FN(X)7 b(Y)41 b FO(=)29 b(\026)-49 b FN(q)17 b(Y)i(X)r(;)-118 2651 y FO(whic)n(h)27 b(implies,)d(since)j FN(q)g FP(6)p FO(=)i(\026)-48 b FN(q)s FO(,)28 b(that)h FN(X)7 b(Y)41 b FO(=)24 b FN(Y)18 b(X)30 b FO(=)24 b(0,)j(i.e.,)h([)p FN(A;)14 b(B)t FO(])24 b(=)f(0)28 b(and)-118 2751 y FN(A)-56 2721 y FK(2)4 2751 y FO(=)23 b FN(B)159 2721 y FK(2)196 2751 y FO(.)6 2851 y(Therefore,)33 b(all)d(irreducible)e(represen)n (tations)i(of)i(\()p FN(V)19 b(I)7 b(I)1805 2863 y FK(0)1843 2851 y FO(\))32 b(b)n(y)g(b)r(ounded)-118 2951 y(op)r(erators)25 b(are)i(one-dimensional.)-118 3103 y FQ(3.)36 b FO(No)n(w)27 b(consider)f(b)r(ounded)i(pairs)d(satisfying)g(\()p FN(V)19 b(I)7 b(I)1596 3115 y FK(1)1634 3103 y FO(\),)296 3287 y FP(\000)p FN(i)p FO([)p FN(A;)14 b(B)t FO(])23 b(=)g FN(\013)p FO(\()p FN(A)860 3253 y FK(2)916 3287 y FP(\000)18 b FN(B)1066 3253 y FK(2)1104 3287 y FO(\))g(+)g FN(I)7 b(;)180 b(\013)24 b FP(2)f FI(R)i FP(n)18 b(f)p FO(0)p FP(g)p FN(:)-118 3472 y FO(F)-7 b(or)27 b(the)h(self-adjoin)n(t)d(op)r (erators)h FN(X)j FO(=)23 b FN(A)18 b FO(+)g FN(B)t FO(,)28 b FN(Y)42 b FO(=)23 b FN(A)18 b FP(\000)g FN(B)t FO(,)28 b(w)n(e)f(ha)n(v)n(e)10 3692 y FN(X)7 b(Y)41 b FO(=)23 b FN(q)s(Y)c(X)24 b FP(\000)19 b FN(i)p FO(\()p FN(q)i FO(+)d(1\))p FN(I)7 b(;)180 b(q)26 b FO(=)1229 3636 y(1)18 b FP(\000)g FN(\013i)p 1229 3673 V 1229 3749 a FO(1)g(+)g FN(\013i)1487 3692 y FP(6)p FO(=)23 b(1)p FN(;)96 b FP(j)p FN(q)s FP(j)24 b FO(=)e(1)p FN(:)128 b FO(\(1.27\))-118 3911 y(A)20 b FP(\003)p FO(-algebra)c(generated)i(b)n(y)j(\(1.27\))d (is)h(called)e(a)i(real)f(quan)n(tum)g(h)n(yp)r(erb)r(oloid.)p eop %%Page: 77 81 77 80 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(1)1494 b FO(77)6 96 y(F)-7 b(or)27 b(b)r(ounded)h(op)r(erators,)e(\(1.27\))h (implies)450 282 y FN(X)7 b FO(\()p FN(X)g(Y)36 b FO(+)18 b FN(\013)854 248 y FM(\000)p FK(1)943 282 y FN(I)7 b FO(\))24 b(=)e FN(q)17 b FO(\()p FN(X)7 b(Y)37 b FO(+)18 b FN(\013)1512 248 y FM(\000)p FK(1)1602 282 y FN(I)7 b FO(\))p FN(X)r(;)-118 468 y FO(and,)26 b(due)h(to)g(the)f(F)-7 b(uglede{Putnam{Rosen)n(blum)21 b(theorem,)k(w)n(e)h(also)f(ha)n(v)n(e) 450 654 y FN(X)7 b FO(\()p FN(X)g(Y)36 b FO(+)18 b FN(\013)854 620 y FM(\000)p FK(1)943 654 y FN(I)7 b FO(\))24 b(=)29 b(\026)-49 b FN(q)17 b FO(\()p FN(X)7 b(Y)37 b FO(+)18 b FN(\013)1512 620 y FM(\000)p FK(1)1602 654 y FN(I)7 b FO(\))p FN(X)r(;)-118 840 y FO(whic)n(h)26 b(is)h(p)r(ossible)e(only) h(if)651 1026 y FN(X)7 b(Y)18 b(X)29 b FO(=)23 b FN(X)1055 992 y FK(2)1091 1026 y FN(Y)42 b FO(=)23 b FP(\000)p FN(\013)1387 992 y FM(\000)p FK(1)1476 1026 y FN(X)r(:)556 b FO(\(1.28\))-118 1212 y(By)30 b(\(1.28\))o(,)g FN(H)349 1224 y FK(0)412 1212 y FO(=)c(k)n(er)13 b FN(X)35 b FO(is)28 b(in)n(v)-5 b(arian)n(t)27 b(under)i FN(A)p FO(,)h FN(B)t FO(;)g(but)g(then)g(b)n(y)g(\(1.27\),)-118 1312 y(w)n(e)23 b(get)g FN(H)203 1324 y FK(0)263 1312 y FO(=)g FP(f)p FO(0)p FP(g)p FO(.)34 b(On)23 b(the)h(subspace)f FN(H)1226 1282 y FM(?)1219 1333 y FK(0)1281 1312 y FO(,)i(the)f(op)r(erator)d FN(X)30 b FO(is)22 b(in)n(v)n(ertible,)-118 1412 y(and)27 b FN(X)7 b(Y)41 b FO(=)23 b FN(Y)18 b(X)7 b FO(.)37 b(Th)n(us)27 b(w)n(e)g(ha)n(v)n(e)f([)p FN(A;)14 b(B)t FO(])23 b(=)g(0,)k(and)g FN(\013)p FO(\()p FN(A)1744 1382 y FK(2)1800 1412 y FP(\000)18 b FN(B)1950 1382 y FK(2)1987 1412 y FO(\))h(+)e FN(I)30 b FO(=)23 b(0.)6 1513 y(Therefore,)28 b(irreducible)d(represen)n (tations)g(of)k(\()p FN(V)19 b(I)7 b(I)1669 1525 y FK(1)1706 1513 y FO(\))29 b(b)n(y)f(b)r(ounded)h(op-)-118 1613 y(erators)d(are)g(one-dimensional.)-118 1749 y FB(R)l(emark)k(24.)42 b FO(F)-7 b(or)41 b(a)h(p)r(ossible)d(de\014nition)h(of)h(in)n (tegrable)e(pairs)g(of)j(un)n(b)r(o-)-118 1848 y(unded)32 b(pairs)f(of)h(self-adjoin)n(t)d(op)r(erators)i(satisfying)e(\()p FN(V)19 b(I)7 b(I)1799 1860 y FK(0)1837 1848 y FO(\))32 b(and)g(\()p FN(V)19 b(I)7 b(I)2245 1860 y FK(1)2283 1848 y FO(\),)-118 1948 y(and)27 b(the)h(structure)f(of)h(suc)n(h)f (pairs,)f(see)h([235)o(,)g(236)o(].)-118 2198 y FG(Commen)m(ts)38 b(to)f(Chapter)h(1)-118 2383 y FQ(Section)31 b(1.1.)6 2485 y FO(1.1.1.)55 b(W)-7 b(e)34 b(giv)n(e)e(basic)g(de\014nitions)g (and)i(prop)r(erties)e(of)i FP(\003)p FO(-represen)n(ta-)-118 2584 y(tions)27 b(of)i FP(\003)p FO(-algebras)c(b)n(y)j(b)r(ounded)h (op)r(erators,)e(i.e.,)h FP(\003)p FO(-homomorphisms)23 b(of)-118 2684 y FP(\003)p FO(-algebras)h(in)n(to)i(the)i FP(\003)p FO(-algebra)c(of)k(op)r(erators)d(on)i(a)h(Hilb)r(ert)e (space)h FN(H)7 b FO(.)6 2785 y(Notice)36 b(that)g(in)f(the)i(b)r(o)r (ok)f(w)n(e)g(do)g(not)g(consider)e FP(\003)p FO(-homomorphisms)-118 2885 y(in)n(to)25 b(the)i FP(\003)p FO(-algebra)22 b(of)27 b(op)r(erators)d(on)i(spaces)f(with)h(inde\014nite)f(metric.)34 b(F)-7 b(or)-118 2984 y(suc)n(h)22 b(represen)n(tations,)f(see,)j (e.g.,)f([95)o(,)g(14)o(,)g(96)o(,)g(241)o(,)g(242)o(,)g(165)n(,)g(133) o(],)h(as)e(w)n(ell)-118 3084 y(as)27 b(the)h(bibliograph)n(y)23 b(cited)k(there.)6 3212 y(1.1.2.)81 b(Since)42 b(in)f(this)h(b)r(o)r (ok,)k(our)c(main)e(concern)i(is)f(the)i(represen-)-118 3312 y(tation)34 b(theory)g(of)h FP(\003)p FO(-algebras,)e(the)i (problems)d(of)j FN(C)1609 3281 y FM(\003)1648 3312 y FO(-represen)n(tabilit)n(y)30 b(of)-118 3411 y FP(\003)p FO(-algebras)37 b(is)j(v)n(ery)g(imp)r(ortan)n(t)e(in)i(what)h(follo)n (ws.)74 b(But)42 b(it)e(seems)f(that)-118 3511 y(there)23 b(has)g(not)g(b)r(een)h(m)n(uc)n(h)e(w)n(ork)g(on)h(the)g(dev)n (elopmen)n(t)e(of)i(these)g(problems)-118 3610 y(\(see)k([155)o(])h (and)f(the)h(bibliograph)n(y)23 b(therein\).)6 3712 y(The)39 b(fact)g(that)g(in)f(Prop)r(osition)d(5)k(the)g(implications)33 b(\()p FN(i)p FO(\))42 b FP(\))g FO(\()p FN(ii)p FO(\))g FP(\))-118 3811 y FO(\()p FN(iii)p FO(\))22 b FP(\))h FO(\()p FN(iv)s FO(\))e(hold)f(is)f(easy)-7 b(.)34 b(Coun)n (terexamples)16 b(to)21 b(the)g(implications)15 b(\()p FN(ii)p FO(\))23 b FP(\))-118 3911 y FO(\()p FN(i)p FO(\))34 b(and)g(\()p FN(iii)p FO(\))e FP(\))i FO(\()p FN(ii)p FO(\))g(ha)n(v)n(e)e(b)r(een)j(constructed)e(b)n(y)g(S.)i(P)n(op)r(o)n (vyc)n(h.)53 b(F)-7 b(or)33 b(a)p eop %%Page: 78 82 78 81 bop -118 -137 a FO(78)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(coun)n(terexample)30 b(to)k(\()p FN(iv)s FO(\))f FP(\))h FO(\()p FN(iii)p FO(\),)g(see)g([283)n(,)g(71)o(])g(and)f(the)h (bibliograph)n(y)-118 196 y(therein.)6 298 y(The)22 b(exp)r(osition)d (of)i(questions)f(ab)r(out)h(residual)e(family)f(of)j(\014nite-dimen-) -118 398 y(sional)k(represen)n(tations)f(of)k(a)f(group)f FN(C)1157 367 y FM(\003)1196 398 y FO(-algebra)e(follo)n(ws)h([56)o(].) 6 499 y(F)-7 b(or)34 b(Remark)e(3,)j(see)e(the)i(related)d(bibliograph) n(y:)45 b([55)o(,)34 b(207)n(,)g(105)o(])g(and)-118 599 y(the)28 b(bibliograph)n(y)23 b(giv)n(en)j(there.)6 728 y(1.1.3.)35 b(In)23 b(some)f(cases,)i(the)g(represen)n(tation)d(theory) h(of)i(a)f FP(\003)p FO(-algebra)d(can)-118 828 y(b)r(e)29 b(reduced)f(to)g(that)h(of)g(one)f(of)g(its)g(en)n(v)n(eloping)d FP(\003)p FO(-algebra,)g FN(\033)s FO(-)p FN(C)1986 797 y FM(\003)2025 828 y FO(-algebra,)-118 927 y(or)34 b FN(C)56 897 y FM(\003)95 927 y FO(-algebra)e(\(and)k(vice)e(v)n (ersa\).)58 b(F)-7 b(or)35 b(a)g(\014nitely)f(generated)g FP(\003)p FO(-algebra)-118 1027 y Fz(A)p FO(,)k(its)d(en)n(v)n(eloping) d FN(C)608 997 y FM(\003)646 1027 y FO(-algebra)h(exists)i(and)g(is)g (unique)g(if)h(and)f(only)g(if)g Fz(A)-118 1126 y FO(is)i FP(\003)p FO(-b)r(ounded)i(\(see,)i(e.g.,)g([68)o(,)e(130)n(]\).)71 b(The)38 b FP(\003)p FO(-algebra)d(generated)j(b)n(y)g(a)-118 1226 y(pair)27 b(of)j(orthogonal)c(pro)5 b(jections,)27 b(whic)n(h)h(w)n(as)h(considered)e(in)i(Example)d(3,)-118 1326 y(is)h FP(\003)p FO(-b)r(ounded.)40 b(F)-7 b(or)28 b(an)g(in)n(v)n(estigation)c(of)29 b(its)f(en)n(v)n(eloping)d FN(C)1851 1296 y FM(\003)1889 1326 y FO(-algebra,)h(see)-118 1425 y([271)n(,)i(219)o(])f(and)h(others.)6 1527 y(If)k(a)e(\014nitely) g(generated)g FP(\003)p FO(-algebra)d(is)j(not)h FP(\003)p FO(-b)r(ounded,)g(then)g(one)g(can)-118 1627 y(construct)25 b(an)g(en)n(v)n(eloping)e(pro-)p FN(C)979 1597 y FM(\003)1016 1627 y FO(-algebra)f(\(see)k([8,)f(10)o(,)h(199)o(])g(and)f(the)h(ref-) -118 1726 y(erences)33 b(therein\))h(whic)n(h)f(p)r(ossesses)g(man)n(y) g(useful)g(prop)r(erties)g(of)h(the)h(en-)-118 1826 y(v)n(eloping)24 b FN(C)274 1796 y FM(\003)313 1826 y FO(-algebra.)34 b(The)27 b(exp)r(osition)e(of)j(Theorem)e(1)h(follo)n(ws)d([204)o(].)6 1955 y(1.1.4.)34 b(F)-7 b(or)21 b(\014nitely)e(generated)i FP(\003)p FO(-algebras,)d(w)n(e)j(set)h(a)f(natural)e(language)-118 2055 y(of)27 b(represen)n(tation)e(of)j(generators)d(and)i(relations.)6 2156 y(The)e(sp)r(ectral)e(theorem)g(for)h(a)g(single)f(self-adjoin)n (t)f(op)r(erator)h FN(A)g FO(=)g FN(A)2213 2126 y FM(\003)2274 2156 y FO(=)-118 2204 y Fy(R)-63 2224 y FM(k)p FL(A)p FM(k)-79 2300 y(\000k)p FL(A)p FM(k)109 2271 y FN(\025)14 b(dE)5 b FO(\()p FN(\025)p FO(\))42 b(giv)n(es)c(a)i(decomp)r(osition)d (of)j(the)h(op)r(erator)d FN(A)j FO(in)n(to)e(irre-)-118 2381 y(ducible)29 b(op)r(erators)g(of)i(m)n(ultiplication)25 b(b)n(y)30 b FN(\025)f FP(2)g FI(R)p FO(.)52 b(It)31 b(is)f(a)g(standard)g(part)-118 2480 y(of)36 b(a)g(univ)n(ersit)n(y)d (course)i(and)h(is)f(exp)r(ounded)i(in)e(detail)f(in)i(textb)r(o)r(oks) g(on)-118 2580 y(sp)r(ectral)26 b(theory)h([4)o(,)h(226)o(,)f(220)o(,)h (37)o(,)f(29)o(],)h(etc.)6 2682 y(P)n(airs)19 b(of)i(self-adjoin)n(t)d (op)r(erators)h FN(A)i FO(and)g FN(B)26 b FO(ha)n(v)n(e)19 b(a)i(m)n(uc)n(h)f(more)f(compli-)-118 2781 y(cated)26 b(structure.)36 b(W)-7 b(e)28 b(discuss)d(the)i(complexit)n(y)c(of)k (their)e(unitary)h(descrip-)-118 2881 y(tion)g(in)g(Section)g(3.1.2.)35 b(F)-7 b(or)26 b(a)h(unitary)e(reduction)h(of)g(a)h(pair)e(of)i (Hermitian)-118 2981 y(matrices,)e(see)i([239)o(,)g(244)o(])h(and)f (the)h(bibliograph)n(y)23 b(therein.)6 3083 y(The)j(sp)r(ectral)e (theorem)f(for)i(a)g(pair)f(of)h FB(c)l(ommuting)32 b FO(self-adjoin)n(t)23 b(op)r(era-)-118 3182 y(tors)i(\(see,)h(e.g.,)f ([37,)g(29)o(],)i(etc.\))36 b(giv)n(es)24 b(a)h(decomp)r(osition)d(of)k (a)f(pair)f(of)i(op)r(er-)-118 3282 y(ators,)31 b FN(A)180 3294 y FK(1)246 3282 y FO(=)340 3215 y Fy(R)379 3311 y Fu(R)426 3295 y Fx(2)472 3282 y FN(\025)520 3294 y FK(1)571 3282 y FN(dE)675 3297 y FK(\()p FL(A)751 3305 y Fx(1)784 3297 y FL(;A)854 3305 y Fx(2)886 3297 y FK(\))916 3282 y FO(\()p FN(\025)996 3294 y FK(1)1034 3282 y FN(;)14 b(\025)1119 3294 y FK(2)1157 3282 y FO(\),)32 b FN(A)1306 3294 y FK(2)1373 3282 y FO(=)1467 3215 y Fy(R)1506 3311 y Fu(R)1553 3295 y Fx(2)1598 3282 y FN(\025)1646 3294 y FK(2)1698 3282 y FN(dE)1802 3297 y FK(\()p FL(A)1878 3305 y Fx(1)1911 3297 y FL(;A)1981 3305 y Fx(2)2013 3297 y FK(\))2043 3282 y FO(\()p FN(\025)2123 3294 y FK(1)2161 3282 y FN(;)14 b(\025)2246 3294 y FK(2)2283 3282 y FO(\),)-118 3381 y(in)n(to)27 b(irreducible)e(pairs)i(of)i(op)r(erators)d(of)j(m)n (ultiplication)23 b(b)n(y)28 b FN(\025)1926 3393 y FK(1)1992 3381 y FO(and)h FN(\025)2203 3393 y FK(2)2269 3381 y FO(in)-118 3481 y(a)e(one-dimensional)22 b(space.)6 3583 y(The)31 b(pro)r(of)g(that)g(there)g(are)f(no)h(b)r(ounded)g (self-adjoin)n(t)d(op)r(erators)h(sat-)-118 3682 y(isfying)c(CCR)j (follo)n(ws)d([284)n(])j(\(see)f(also)f([102)n(,)i(220)o(],)g(etc.\).)6 3811 y(1.1.5.)34 b(F)-7 b(or)22 b(a)h(reduction)e(of)i(algebras)c (\(without)k(an)f(in)n(v)n(olution\))d(de\014ned)-118 3911 y(b)n(y)33 b(a)h(pair)e(of)i(generators)d(that)j(are)f(related)f (b)n(y)i(a)f(quadratic)f(relation)f(to)p eop %%Page: 79 83 79 82 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(1)1494 b FO(79)-118 96 y(a)31 b(canonical)d(form,)j(see,)h(e.g.,)h([277)n(].) 49 b(Canonical)28 b(forms)i(for)h(pairs)e(of)j(self-)-118 196 y(adjoin)n(t)d(op)r(erators)g(\()p FP(\003)p FO(-algebras)e(with)k (a)f(pair)f(of)i(self-adjoin)n(t)d(generators\))-118 296 y(whic)n(h)j(satisfy)h(a)g(quadratic)e(relation)g(\(\\non-comm)n (utativ)n(e)e(conics")j(on)h(a)-118 395 y(real)e(plane\))h(can)g(b)r(e) i(found)f(in,)g(e.g.,)h([187)n(,)f(191)o(],)h(etc.)50 b(Newton's)32 b(classi\014-)-118 495 y(cation)e([174)n(])h(\(see)h (also)d([256)n(])j(etc.\))48 b(of)31 b(third-degree)e(curv)n(es)h (discouraged)-118 595 y(the)d(authors)f(to)h(in)n(v)n(estigate)c(the)k (problem)e(of)h(classi\014cation)d(of)k FP(\003)p FO(-algebras)-118 694 y(with)g(t)n(w)n(o)g(generators)e(and)i(one)h(cubic)e(relation.)6 845 y(The)k(remaining)c(part)j(of)g(this)g(section)g(is)f(dev)n(oted)h (to)h(represen)n(tations)-118 944 y(of)c(\\non-comm)n(utativ)n(e)c (curv)n(es)k(of)g(the)h(second)f(degree)f(on)i(the)g(real)d(plane")-118 1044 y(b)n(y)32 b(b)r(ounded)h(op)r(erators,)e(and)h(represen)n (tations)e(of)i FP(\003)p FO(-algebras)c(whic)n(h)k(are)-118 1143 y(more)26 b(general)f(than)j(these)f(\\curv)n(es".)-118 1294 y FQ(Section)k(1.2)6 1394 y FO(1.2.1.)k(W)-7 b(e)25 b(exp)r(ose)g(some)e(kno)n(wn)h(facts)h(ab)r(out)f(algebras)e (satisfying)g(the)-118 1493 y(standard)j(p)r(olynomial)c(iden)n(tit)n (y)j(\(see,)i(e.g.,)g([111)o(,)g(200)n(])g(and)g(the)g(bibliogra-)-118 1593 y(ph)n(y)h(therein\),)g(and)h(their)e(represen)n(tations)f(\(see)i ([147)o(])h(and)f(others\).)6 1693 y(The)h(exp)r(osition)d(of)j (Theorem)e(2)h(follo)n(ws)d([216)o(,)k(217)n(].)6 1818 y(1.2.2.)37 b(F)-7 b(or)27 b(a)h(n)n(um)n(b)r(er)f(of)g(examples)f(of)i (algebras)d(and)i FP(\003)p FO(-algebras)d(gen-)-118 1918 y(erated)h(b)n(y)h(idemp)r(oten)n(ts)f(and)h(orthogonal)d(pro)5 b(jections,)24 b(their)i(represen)n(ta-)-118 2017 y(tions)g(are)h (studied.)6 2117 y(Represen)n(tations)j(of)j(the)f(algebra)e FN(Q)1224 2129 y FK(2)1293 2117 y FO(\(without)i(an)h(in)n(v)n (olution\))28 b(gen-)-118 2217 y(erated)39 b(b)n(y)h(a)f(pair)f(of)i (idemp)r(oten)n(ts)e(w)n(ere)h(studied,)j(e.g.,)h(in)c([281)n(].)74 b(All)-118 2317 y(irreducible)21 b(represen)n(tations)h(of)k FN(Q)1022 2329 y FK(2)1084 2317 y FO(are)e(either)g(one-)g(or)g(t)n(w)n (o-dimensional.)-118 2416 y(The)37 b(description)d(of)j(indecomp)r (osable)c(represen)n(tations)h(can)i(b)r(e)h(deriv)n(ed)-118 2516 y(from)e([172)o(,)h(92)o(].)64 b(The)36 b(problem)e(of)j(the)f (description)e(of)j(represen)n(tations)-118 2616 y(of)30 b(the)h FP(\003)p FO(-algebra)c(generated)i(b)n(y)h(a)g(pair)f(of)i (orthogonal)c(pro)5 b(jections)28 b(on)i(a)-118 2715 y(\014nite-dimensional)c(space)k(is)h(reduced)g(to)g(a)g(description)d (of)k(Jordan's)d(an-)-118 2815 y(gles)19 b(b)r(et)n(w)n(een)h (subspaces)g([120)o(];)j(in)d(the)h(case)e(of)i(a)f(separable)e(Hilb)r (ert)h(space,)-118 2914 y(see)35 b([196)n(,)g(103)o(],)i(etc.)e(\(see,) i(e.g.,)f(a)f(detailed)e(bibliograph)n(y)d(in)35 b([40)o(]\).)59 b(F)-7 b(or)-118 3014 y(\014nite-dimensional)27 b(represen)n(tations)i (of)j(the)h FP(\003)p FO(-algebra)c(generated)i(b)n(y)h(an)-118 3114 y(idemp)r(oten)n(t)18 b(and)h(its)f(adjoin)n(t,)h(see)g([69)o(,)g (113)o(],)i(etc.;)h(for)d(in\014nite-dimensional)-118 3213 y(\(and)28 b(ev)n(en)f(un)n(b)r(ounded\))h(represen)n(tations,)d (see)i([204)n(].)6 3313 y(The)i(algebra)c FN(Q)535 3325 y FK(2)600 3313 y FO(generated)i(b)n(y)h(a)g(pair)e(of)j(idemp)r(oten)n (ts)d(is)h(the)i(group)-118 3413 y(algebra)k(of)k(the)f(simplest)e (in\014nite)h(group)g FI(Z)1365 3425 y FK(2)1420 3413 y FP(\003)24 b FI(Z)1547 3425 y FK(2)1616 3413 y FO(=)37 b FI(Z)18 b(o)24 b(Z)1948 3425 y FK(2)1979 3413 y FO(.)64 b(F)-7 b(or)35 b(the)-118 3513 y(group)30 b FN(G)f FO(=)g FI(Z)370 3482 y FL(k)426 3513 y FI(o)20 b FN(G)576 3525 y FL(f)619 3513 y FO(,)33 b FN(k)e(>)e FO(1,)j(where)e FN(G)1248 3525 y FL(f)1323 3513 y FO(is)g(a)h(\014nite)f(group,)h FI(C)15 b FO([)p FN(G)q FO(])37 b(is)30 b(an)-118 3612 y FN(F)-65 3627 y FK(2)p FM(j)p FL(G)40 3636 y Fv(f)77 3627 y FM(j)101 3612 y FO(-algebra,)23 b(and)i(its)f(irreducible)e FP(\003)p FO(-represen)n(tations)f(can)k(b)r(e)h(obtained)-118 3712 y(using)19 b(Mac)n(k)n(ey's)g(formalism)d(of)21 b(induced)f(represen)n(tations)d([160)o(].)35 b(Ho)n(w)n(ev)n(er,)-118 3811 y(the)29 b(description)d(of)j(all)d(indecomp)r(osable)f(represen)n (tations)g(of)k FN(G)g FO(is)f(a)g(v)n(ery)-118 3911 y(complicated)35 b(problem)g([39)o(].)69 b(Examples)35 b(of)k FN(F)1455 3923 y FL(n)1500 3911 y FO(-algebras)c(generated)i(b)n (y)p eop %%Page: 80 84 80 83 bop -118 -137 a FO(80)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)-118 96 y FO(idemp)r(oten)n(ts,)20 b(whic)n(h)g(are)g(not)h(group)f (algebras,)f(and)i(their)f(represen)n(tations)-118 196 y(are)26 b(discussed)h(in)f([224)o(,)i(81)o(,)g(149)n(].)6 297 y(The)19 b(exp)r(osition)d(of)j(Section)e(1.2.2)g(and)i(the)f(pro)r (of)g(of)h(Theorem)e(3)h(follo)n(ws)-118 397 y([216)n(,)28 b(217)o(].)6 524 y(1.2.3.)74 b(Non-comm)n(utativ)n(e)36 b(\\circle",)k(\\h)n(yp)r(erb)r(ola",)h(\\pair)d(of)i(in)n(ter-)-118 624 y(secting)33 b(lines",)i(are)f(also)f(natural)g(examples)f(of)j FN(F)1568 636 y FK(4)1606 624 y FO(-algebras.)55 b(F)-7 b(or)35 b(their)-118 724 y(represen)n(tations,)25 b(see,)i(e.g.,)g ([191)o(,)g(187)o(].)6 825 y(F)-7 b(or)28 b(algebras)d(that)j(are)f (generated)g(b)n(y)h(idemp)r(oten)n(ts)e(satisfying)f(linear)-118 924 y(relations,)i(see)i([32)o(].)43 b(The)30 b(fact)f(that)h FN(Q)1162 936 y FK(4)p FL(;)p FK(2)1282 924 y FO(is)e(an)h FN(F)1537 936 y FK(4)1575 924 y FO(-algebra)d(follo)n(ws)h(from)-118 1024 y(the)37 b(description)d(of)j(its)f FP(\003)p FO(-represen)n (tations)d([88)o(])k(\(see)g(also)d(Section)i(2.2.1)-118 1124 y(b)r(elo)n(w\).)6 1225 y(Giv)n(en)31 b(in)h(Section)f(1.2.3,)h (items)e(4)i(and)f(5,)i(are)e(examples)f(of)i(algebras)-118 1324 y(generated)26 b(b)n(y)i(idemp)r(oten)n(ts)e(whic)n(h)g(are)h(not) g FN(F)1423 1336 y FL(n)1469 1324 y FO(-algebras)d(for)j(an)n(y)g FN(n)c FP(2)h FI(N)t FO(.)-118 1424 y(This)32 b(follo)n(ws)f(from)h (the)i(structure)f(of)g(their)g(irreducible)d(represen)n(tations)-118 1524 y([32)o(,)e(88)o(,)f(89)o(])h(\(see)g(also)d(Section)54 b(2.2.1\).)-118 1678 y FQ(Section)31 b(1.3)6 1779 y FO(1.3.1.)61 b(Bounded)36 b(represen)n(tations)d(of)i(t)n(w)n(o-dimensional)c(real)j (Lie)g(al-)-118 1878 y(gebras)d(and)h(their)g(nonlinear)e (transformations)f(can)j(easily)e(b)r(e)j(describ)r(ed)-118 1978 y(using)19 b(the)h(Kleinec)n(k)n(e{Shirok)n(o)m(v)14 b(theorem)19 b([247)o(,)h(134)n(])h(\(see)f(also)e([102)n(],)k(etc.\).) 6 2106 y(1.3.2{1.3.5.)62 b(In)38 b(these)f(sections,)g(a)g(more)e (general)g(class)g(of)i(semilin-)-118 2205 y(ear)32 b(relations)e(is)j (studied.)53 b(The)33 b(pro)r(of)g(of)g(the)h(Kleinec)n(k)n(e{Shirok)n (o)m(v)27 b(t)n(yp)r(e)-118 2305 y(theorems)35 b(and)i(the)h(study)f (of)g(their)f(irreducible)d(represen)n(tations)h(follo)n(w)-118 2405 y([35)o(,)28 b(233)n(].)-118 2559 y FQ(Section)j(1.4)6 2660 y FO(1.4.1.)82 b(Irreducible)40 b(represen)n(tations)g(of)j(the)g FN(q)s FO(-relations)d(b)n(y)i(\014nite-)-118 2759 y(dimensional)25 b(and)30 b(compact)f(op)r(erators)f(are)h(one-dimensional.)38 b(Their)29 b(de-)-118 2859 y(scription)c(using)h(Prop)r(ositions)e(25)j (and)g(26)g(follo)n(ws)d([229)o(].)6 2986 y(1.4.2.)56 b(It)35 b(w)n(as)e(noticed)h(in)f([277)o(],)j(etc.,)g(that)f(the)g (Hermitian)c FN(q)s FO(-plane)-118 3086 y(do)r(es)24 b(not)g(ha)n(v)n(e)g(non-trivial)c(represen)n(tations)h(b)n(y)k(b)r (ounded)f(op)r(erators.)34 b(W)-7 b(e)-118 3186 y(follo)n(w)35 b([187)o(].)69 b(Bounded)38 b(represen)n(tations)e(of)i (one-dimensional)33 b FN(q)s FO(-CCR,)-118 3285 y FN(q)45 b FP(2)d FI(R)p FO(,)47 b(w)n(ere)38 b(studied)g(in)g(n)n(umerous)f (pap)r(ers)h(\(see,)k(e.g.,)f([36)o(,)e(158)o(];)44 b(for)-118 3385 y(detailed)17 b(references,)j(see,)g(e.g.,)g([137)o(]\).)34 b(The)20 b(exp)r(osition)c(follo)n(ws)g([185)o(,)j(187)n(].)-118 3485 y(F)-7 b(or)30 b(un)n(b)r(ounded)i(represen)n(tations)c(of)j FN(q)s FO(-CCR,)g FN(q)g FP(2)e FI(R)p FO(,)38 b(see)31 b([139)o(,)g(187)n(,)g(52,)-118 3584 y(109)n(,)d(191)o(],)f(etc.)6 3712 y(1.4.3.)35 b(Real)23 b(quan)n(tum)g(plane)h(and)g(real)e(quan)n (tum)i(h)n(yp)r(erb)r(oloid)e(do)i(not)-118 3811 y(ha)n(v)n(e)c (non-trivial)d(represen)n(tations)i(b)n(y)i(b)r(ounded)g(op)r(erators.) 33 b(In)22 b([185)n(,)g(187)n(],)-118 3911 y(this)40 b(is)f(pro)n(v)n(ed)g(b)n(y)h(using)f(the)h(F)-7 b(uglede{Putnam{Rosen) n(blum)35 b(theorem)p eop %%Page: 81 85 81 84 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(1)1494 b FO(81)-118 96 y(\(see,)41 b(e.g.,)h([102)n(,)d(226)o(]\).)71 b(F)-7 b(or)38 b(un)n(b)r(ounded)h(represen)n(tations)d(of)i(the)i (real)-118 196 y(quan)n(tum)26 b(h)n(yp)r(erb)r(oloid,)f(see)j([235)n (,)g(236)n(].)p eop %%Page: 82 86 82 85 bop -118 -137 a FO(82)875 b FJ(Chapter)27 b(1.)36 b(P)n(airs)25 b(of)j(self-adjoin)n(t)d(op)r(erators)p eop %%Page: 83 87 83 86 bop -118 664 a FR(Chapter)45 b(2)-118 982 y(Represen)l(tations)h (of)f(dynamical)e Fl(\003)p FR(-algebras)-118 1433 y FG(2.1)112 b(Op)s(erator)53 b(relations)h(and)g(one-dimensional)i(dy-) 137 1549 y(namical)39 b(systems)-118 1736 y FQ(2.1.1)94 b(Op)s(erator)38 b(relations)g(connected)g(with)g(one-dimensional)174 1836 y(dynamical)31 b(systems)-118 1994 y(1.)37 b FO(Consider)25 b(an)j(op)r(erator)e FN(X)34 b FO(satisfying,)25 b(together)i(with)h (its)f(adjoin)n(t)f FN(X)2278 1964 y FM(\003)2316 1994 y FO(,)-118 2093 y(an)h(algebraic)d(relation)h(of)i(the)h(form)783 2282 y FN(X)7 b(X)935 2247 y FM(\003)995 2282 y FO(=)22 b FN(F)12 b FO(\()p FN(X)1255 2247 y FM(\003)1293 2282 y FN(X)7 b FO(\))p FN(;)744 b FO(\(2.1\))-118 2470 y(where)34 b FN(F)12 b FO(\()p FP(\001)p FO(\))d(:)31 b FI(R)41 b FP(\000)-48 b(!)35 b FI(R)41 b FO(is)34 b(a)g(mapping)f(measurable)e (with)k(resp)r(ect)g(to)f(the)-118 2569 y(Borel)25 b FN(\033)s FO(-algebra.)6 2672 y(If)33 b FN(F)12 b FO(\()p FP(\001)p FO(\))32 b(=)f FN(P)427 2684 y FL(n)473 2672 y FO(\()p FP(\001)p FO(\))i(is)f(a)g(real)f(p)r(olynomial,)e(then)k (the)g(pair)e(of)i(op)r(erators)-118 2771 y FN(X)7 b FO(,)31 b FN(X)88 2741 y FM(\003)156 2771 y FO(satisfying)d(relation)f (\(2.1\))k(is)e(a)h(represen)n(tation)e(of)i(a)h FP(\003)p FO(-algebra)c FA(A)-118 2871 y FO(generated)i(b)n(y)h(elemen)n(ts)e FN(x)p FO(,)k FN(x)872 2841 y FM(\003)941 2871 y FO(satisfying)27 b(the)k(relation)d FN(xx)1861 2841 y FM(\003)1927 2871 y FO(=)f FN(P)2072 2883 y FL(n)2118 2871 y FO(\()p FN(x)2197 2841 y FM(\003)2236 2871 y FN(x)p FO(\).)-118 2970 y(In)f(some)e(non-p) r(olynomial)c(cases,)25 b(pairs)f(of)h(op)r(erators)f FN(X)7 b FO(,)25 b FN(X)1868 2940 y FM(\003)1932 2970 y FO(form)f(repre-)-118 3070 y(sen)n(tations)k(of)j FN(C)438 3040 y FM(\003)476 3070 y FO(-algebras)c(or)j(other)f(top)r(ological)d (algebras,)i(but)j(w)n(e)f(will)-118 3170 y(not)35 b(concen)n(trate)e (our)h(atten)n(tion)g(on)g(the)h(underlying)d(algebraic)f(ob)5 b(jects,)-118 3269 y(restricting)24 b(ourselv)n(es)h(to)j(the)g(study)f (of)h(represen)n(tations)d(of)i(the)h(relation.)6 3372 y(W)-7 b(e)36 b(ha)n(v)n(e)e(already)e(considered)i(examples)e(of)j (represen)n(tations)d(of)j(the)-118 3471 y(form)24 b(\(2.1\))h(ab)r(o)n (v)n(e.)34 b(F)-7 b(or)25 b(the)h(Hermitian)c FN(q)s FO(-plane,)i FN(xx)1622 3441 y FM(\003)1684 3471 y FO(=)f FN(q)s(x)1859 3441 y FM(\003)1897 3471 y FN(x)p FO(,)k FN(q)f FP(2)d FI(R)p FO(,)32 b(w)n(e)-118 3571 y(ha)n(v)n(e)f FN(F)12 b FO(\()p FN(\025)p FO(\))33 b(=)e FN(q)s(\025)p FO(,)j(and)f(for)f FN(q)s FO(-CCR,)g FN(xx)1226 3541 y FM(\003)1297 3571 y FO(=)f FN(q)s(x)1480 3541 y FM(\003)1518 3571 y FN(x)23 b FO(+)e(1,)33 b FN(q)i FP(2)d FI(R)p FO(,)40 b(w)n(e)32 b(ha)n(v)n(e)-118 3670 y FN(F)12 b FO(\()p FN(\025)p FO(\))24 b(=)f FN(q)s(\025)18 b FO(+)g(1.)6 3773 y(Belo)n(w)26 b(w)n(e)h(consider)f(other)h(examples)e(of)i (relations)e(of)i(the)h(form)e(\(2.1\).)-118 3911 y FB(Example)31 b(9.)42 b FO(\(Second)27 b(degree)f(mappings\).)35 b(Consider)25 b(the)j(follo)n(wing)23 b(rela-)1069 4121 y(83)p eop %%Page: 84 88 84 87 bop -118 -137 a FO(84)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(tion)513 280 y FN(xx)607 246 y FM(\003)669 280 y FO(=)f FN(\013x)857 246 y FM(\003)896 280 y FN(x)p FO(\()p FN(I)j FP(\000)18 b FN(x)1167 246 y FM(\003)1206 280 y FN(x)p FO(\))p FN(;)180 b(a)23 b(>)g FO(0)p FN(:)460 b FO(\(2.2\))-118 463 y(The)34 b(corresp)r(onding)e(p)r (olynomial)e(is)j FN(F)12 b FO(\()p FN(\025)p FO(\))36 b(=)e FN(\013\025)p FO(\(1)24 b FP(\000)f FN(\025)p FO(\).)58 b(An)n(y)35 b(second-)-118 563 y(degree)f(mapping)e(can)j(b)r(e)g (reduced)f(to)h(suc)n(h)f(a)h(form)e(b)n(y)i(using)e(an)i(a\016ne)-118 663 y(c)n(hange)d(of)h(v)-5 b(ariables;)32 b(ho)n(w)n(ev)n(er,)g(p)r (ositivit)n(y)e(considerations)f(that)k(will)d(b)r(e)-118 762 y(discussed)22 b(b)r(elo)n(w)g(mak)n(e)f(the)j(represen)n(tation)c (problem)h(di\013eren)n(t)h(for)h(di\013er-)-118 862 y(en)n(t)30 b(forms)f(of)h(the)h(p)r(olynomial.)40 b(Here,)31 b(w)n(e)f(consider)e(only)h(relations)e(\(2.2\))-118 961 y(and)779 1145 y FN(xx)873 1111 y FM(\003)935 1145 y FO(=)22 b(\()p FN(x)1101 1111 y FM(\003)1140 1145 y FN(x)d FP(\000)f FN(q)s(I)7 b FO(\))1404 1111 y FK(2)2168 1145 y FO(\(2.3\))-118 1343 y(whic)n(h)40 b(corresp)r(onds)f(to)j(the)f (p)r(olynomial)1335 1322 y(~)1317 1343 y FN(F)12 b FO(\()p FN(t)p FO(\))46 b(=)g(\()p FN(t)27 b FP(\000)h FN(q)s FO(\))1887 1313 y FK(2)1924 1343 y FO(.)78 b(The)41 b(t)n(w)n(o)-118 1443 y(latter)29 b(p)r(olynomials)d(can)k(b)r(e)h(transformed)d(to)i (eac)n(h)g(other)g(b)n(y)g(the)h(c)n(hange)-118 1543 y(of)37 b(v)-5 b(ariables)35 b FN(t)k FO(=)g FP(\000)p FN(\013\025)26 b FO(+)e FN(b)p FO(,)40 b(with)d FN(\013)j FO(=)f(1)25 b(+)1446 1482 y FP(p)p 1515 1482 225 4 v 61 x FO(4)p FN(q)c FO(+)d(1.)67 b(The)37 b(structure)-118 1642 y(of)27 b(b)r(ounded)h(represen)n(tations)c(of)j(suc)n(h)g (relations)e(greatly)g(dep)r(ends)i(on)g(the)-118 1742 y(v)-5 b(alue)26 b(of)i FN(\013)g FO(\(see)g(b)r(elo)n(w\).)-118 1875 y FB(Example)j(10.)42 b FO(\(Con)n(tin)n(ued)28 b(fractions\).)39 b(Another)29 b(in)n(teresting)d(class)h(of)h(re-)-118 1975 y(lations)23 b(related)i(to)h(con)n(tin)n(uous)e(fractions)g(is)h (giv)n(en)f(b)n(y)i(the)g(follo)n(wing)c(rela-)-118 2075 y(tions)547 2258 y FN(xx)641 2224 y FM(\003)703 2258 y FO(=)h(\()p FN(ax)914 2224 y FM(\003)952 2258 y FN(x)c FO(+)f FN(c)p FO(\)\()p FN(bx)1284 2224 y FM(\003)1323 2258 y FN(x)h FO(+)f FN(d)p FO(\))1547 2224 y FM(\000)p FK(1)1637 2258 y FN(;)463 2383 y(a;)c(b;)g(c;)g(d)23 b FP(2)g FI(R)p FN(;)103 b(b)23 b FP(6)p FO(=)f(0)p FN(;)97 b(ad)18 b FP(\000)g FN(bc)23 b FP(6)p FO(=)g(0)p FN(:)-118 2566 y FO(Belo)n(w,)30 b(w)n(e)h(will)d(sho)n(w)i(ho)n(w)h(represen)n (tations)d(of)j(the)h(relation)c(dep)r(end)k(on)-118 2666 y(the)c(v)-5 b(alues)26 b(of)i(the)g(parameters.)6 2766 y(In)c(particular,)c(the)k(t)n(w)n(o-parameter)19 b(unit)k(quan)n(tum)f(disk)g(algebra)e([135)o(])-118 2865 y(is)26 b(generated)h(b)n(y)g(the)h(relation)345 3049 y FN(q)s(z)t(z)471 3015 y FM(\003)527 3049 y FP(\000)18 b FN(z)653 3015 y FM(\003)690 3049 y FN(z)26 b FO(=)d FN(q)f FP(\000)c FO(1)g(+)g FN(\026)p FO(\(1)g FP(\000)g FN(z)t(z)1439 3015 y FM(\003)1476 3049 y FO(\)\(1)g FP(\000)g FN(z)1726 3015 y FM(\003)1764 3049 y FN(z)t FO(\))p FN(;)379 3173 y FO(0)23 b FP(\024)f FN(\026)h FP(\024)g FO(1)p FN(;)96 b FO(0)23 b FP(\024)g FN(q)j FP(\024)c FO(1)p FN(;)97 b FO(\()p FN(\026;)14 b(q)s FO(\))23 b FP(6)p FO(=)g(\(0)p FN(;)14 b FO(1\))p FN(;)-118 3357 y FO(whic)n(h)26 b(can)i(b)r(e)g(rewritten)e(in)h(the)h(form)e(\(2.1\))h(with)574 3588 y FN(F)12 b FO(\()p FN(\025)p FO(\))24 b(=)873 3532 y(\()p FN(q)e FO(+)c FN(\026)p FO(\))p FN(\025)h FO(+)f(1)g FP(\000)g FN(q)j FP(\000)d FN(\026)p 873 3569 741 4 v 1047 3645 a(\026\025)g FO(+)g(1)g FP(\000)g FN(\026)1623 3588 y(:)-118 3811 y FB(R)l(emark)30 b(25.)42 b FO(As)27 b(w)n(e)e(will)f(see,)i(in)f(man)n(y)g(examples)e(un)n(b)r(ounded)k(op) r(erators)-118 3911 y FN(X)32 b FO(naturally)22 b(arise)i(as)h (represen)n(tations)d(of)k(suc)n(h)f(a)g(relation.)33 b(Therefore,)25 b(it)p eop %%Page: 85 89 85 88 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(85)-118 96 y(is)26 b(necessary)g(to)i (accurately)d(de\014ne)j(the)g(meaning)d(of)j(the)g(relation)d(for)i (un-)-118 196 y(b)r(ounded)k(op)r(erators.)44 b(Occasionally)-7 b(,)26 b(deriv)n(ed)j(form)n(ulae)f(will)f(mak)n(e)i(sense)-118 296 y(for)38 b(un)n(b)r(ounded)g(op)r(erators,)h(to)r(o,)i(or)c(will)f (yield)g(un)n(b)r(ounded)j(op)r(erators;)-118 395 y(ho)n(w)n(ev)n(er,) 28 b(w)n(e)h(will)e(not)j(discuss)e(problems)f(related)h(to)h(un)n(b)r (ounded)h(op)r(era-)-118 495 y(tors)d(here.)-118 657 y FQ(2.)35 b FO(Let)24 b FN(X)30 b FO(b)r(e)24 b(b)r(ounded.)36 b(F)-7 b(or)23 b(a)g(p)r(olar)f(decomp)r(osition)e(of)k(the)g(op)r (erator)e FN(X)7 b FO(,)-118 756 y(w)n(e)29 b(ha)n(v)n(e)e FN(X)32 b FO(=)25 b FN(U)9 b(C)d FO(,)30 b(where)f FN(C)i FO(=)25 b FN(C)1060 726 y FM(\003)1124 756 y FO(=)g(\()p FN(X)1322 726 y FM(\003)1360 756 y FN(X)7 b FO(\))1468 726 y FK(1)p FL(=)p FK(2)1572 756 y FO(,)30 b FN(U)38 b FO(is)28 b(a)g(partial)e(isom-)-118 856 y(etry)-7 b(,)34 b(and)f(k)n(er)12 b FN(U)41 b FO(=)31 b(k)n(er)13 b FN(C)38 b FO(=)32 b(k)n(er)12 b FN(X)7 b FO(,)34 b FN(U)1205 826 y FM(\003)1243 856 y FN(U)42 b FO(is)31 b(an)i(orthogonal)d(pro)5 b(jection)-118 956 y(on)n(to)27 b(\(k)n(er)13 b FN(C)6 b FO(\))323 925 y FM(?)379 956 y FO(.)37 b(T)-7 b(aking)26 b(in)n(to)g(accoun)n(t)h(relation)e(\(2.1\))o(,)j(w)n(e)f(get)784 1112 y FN(U)9 b(C)915 1078 y FK(2)953 1112 y FN(U)1019 1078 y FM(\003)1080 1112 y FO(=)22 b FN(F)12 b FO(\()p FN(C)1329 1078 y FK(2)1367 1112 y FO(\))p FN(;)746 b FO(\(2.4\))-118 1268 y(whic)n(h)26 b(implies)e(that)428 1425 y FN(U)9 b(C)559 1391 y FK(2)620 1425 y FO(=)22 b FN(F)12 b FO(\()p FN(C)869 1391 y FK(2)907 1425 y FO(\))p FN(U;)97 b(C)1181 1391 y FK(2)1218 1425 y FN(U)1284 1391 y FM(\003)1345 1425 y FO(=)23 b FN(U)1499 1391 y FM(\003)1537 1425 y FN(F)12 b FO(\()p FN(C)1699 1391 y FK(2)1737 1425 y FO(\))p FN(:)376 b FO(\(2.5\))-118 1581 y(Notice)26 b(that)i(\(2.4\))g(implies)c(that)j(k)n(er)13 b FN(U)1173 1551 y FM(\003)1234 1581 y FO(=)23 b(k)n(er)13 b FN(X)1523 1551 y FM(\003)1583 1581 y FO(=)23 b(k)n(er)12 b FN(F)g FO(\()p FN(C)1957 1551 y FK(2)1995 1581 y FO(\))28 b(as)f(w)n(ell.)-118 1720 y FQ(3.)51 b FO(No)n(w)31 b(w)n(e)h(establish)f(some)f(relations)g (whic)n(h)h(follo)n(w)f(from)h(\(2.1\))o(.)52 b(First)-118 1820 y(notice)34 b(that)h FN(X)7 b FO(\()p FN(X)505 1790 y FM(\003)542 1820 y FN(X)g FO(\))35 b(=)g FN(F)12 b FO(\()p FN(X)958 1790 y FM(\003)995 1820 y FN(X)7 b FO(\))p FN(X)41 b FO(and)35 b(\()p FN(X)1490 1790 y FM(\003)1528 1820 y FN(X)7 b FO(\))p FN(X)1712 1790 y FM(\003)1784 1820 y FO(=)35 b FN(X)1960 1790 y FM(\003)1997 1820 y FN(F)12 b FO(\()p FN(X)2170 1790 y FM(\003)2208 1820 y FN(X)7 b FO(\).)-118 1920 y(F)-7 b(or)28 b(an)n(y)h FN(k)f FO(=)d(1,)k(2,)g FN(:)14 b(:)g(:)27 b FO(,)j(w)n(e)f(recursiv)n (ely)c(get)k(the)g(relations)d FN(X)7 b FO(\()p FN(X)2063 1890 y FM(\003)2100 1920 y FN(X)g FO(\))2208 1890 y FL(k)2274 1920 y FO(=)-118 2019 y(\()p FN(F)12 b FO(\()p FN(X)87 1989 y FM(\003)125 2019 y FN(X)7 b FO(\)\))265 1989 y FL(k)306 2019 y FN(X)g FO(,)43 b(and)e(\()p FN(X)731 1989 y FM(\003)769 2019 y FN(X)7 b FO(\))877 1989 y FL(k)917 2019 y FN(X)993 1989 y FM(\003)1075 2019 y FO(=)45 b FN(X)1261 1989 y FM(\003)1299 2019 y FO(\()p FN(F)12 b FO(\()p FN(X)1504 1989 y FM(\003)1542 2019 y FN(X)7 b FO(\)\))1682 1989 y FL(k)1722 2019 y FO(.)77 b(F)-7 b(rom)39 b(the)i(t)n(w)n(o)-118 2119 y(latter)31 b(relations)e(w)n(e)k (get)f FN(X)7 b(P)12 b FO(\()p FN(X)975 2089 y FM(\003)1012 2119 y FN(X)7 b FO(\))31 b(=)g FN(P)12 b FO(\()p FN(X)1420 2089 y FM(\003)1457 2119 y FN(X)7 b FO(\))p FN(X)39 b FO(for)32 b(p)r(olynomials)c(of)-118 2219 y FN(X)-42 2188 y FM(\003)-5 2219 y FN(X)7 b FO(,)27 b(and)f(in)g(the)g(case)g(of) h(b)r(ounded)f(op)r(erators,)f(a)h(standard)g(appro)n(xima-)-118 2318 y(tion)32 b(pro)r(cedure)g(using)g(the)h(sp)r(ectral)f(decomp)r (osition)e(of)j(the)g(self-adjoin)n(t)-118 2418 y(op)r(erator)26 b FN(X)293 2388 y FM(\003)330 2418 y FN(X)34 b FO(yields)-67 2574 y FN(X)7 b(\036)p FO(\()p FN(X)166 2540 y FM(\003)203 2574 y FN(X)g FO(\))23 b(=)f FN(\036)p FO(\()p FN(F)12 b FO(\()p FN(X)675 2540 y FM(\003)714 2574 y FN(X)7 b FO(\)\))p FN(X)r(;)96 b(\036)p FO(\()p FN(X)1201 2540 y FM(\003)1240 2574 y FN(X)7 b FO(\))p FN(X)1424 2540 y FM(\003)1484 2574 y FO(=)22 b FN(X)1647 2540 y FM(\003)1685 2574 y FN(\036)p FO(\()p FN(F)12 b FO(\()p FN(X)1939 2540 y FM(\003)1977 2574 y FN(X)7 b FO(\)\))51 b(\(2.6\))-118 2731 y(for)31 b(an)n(y)g(b)r(ounded)i(measurable)28 b(function)k FN(\036)p FO(\()p FP(\001)p FO(\).)51 b(In)32 b(particular,)e(\(2.6\))i (im-)-118 2830 y(plies)25 b(that)j(for)f(all)f FN(n)d FO(=)f(1,)27 b(2,)h FN(:)14 b(:)g(:)27 b FO(,)533 2987 y FN(X)609 2953 y FL(n)653 2987 y FN(\036)p FO(\()p FN(X)810 2953 y FM(\003)848 2987 y FN(X)7 b FO(\))23 b(=)g FN(\036)p FO(\()p FN(F)1213 2953 y FL(n)1259 2987 y FO(\()p FN(X)1367 2953 y FM(\003)1404 2987 y FN(X)7 b FO(\))p FN(X)1588 2953 y FL(n)1632 2987 y FN(;)430 3111 y(\036)p FO(\()p FN(X)587 3077 y FM(\003)625 3111 y FN(X)g FO(\)\()p FN(X)841 3077 y FM(\003)879 3111 y FO(\))911 3077 y FL(n)979 3111 y FO(=)23 b(\()p FN(X)1175 3077 y FM(\003)1212 3111 y FO(\))1244 3077 y FL(n)1290 3111 y FN(\036)p FO(\()p FN(F)1436 3077 y FL(n)1482 3111 y FO(\()p FN(X)1590 3077 y FM(\003)1628 3111 y FN(X)7 b FO(\)\))p FN(;)391 b FO(\(2.7\))-118 3268 y(where)27 b FN(F)187 3238 y FL(n)232 3268 y FO(\()p FP(\001)p FO(\))h(is)f(the)h FN(n)p FO(-th)f(iteration)e(of)j FN(F)12 b FO(\()p FP(\001)p FO(\).)6 3367 y(W)-7 b(e)28 b(ha)n(v)n(e)f(the)h(follo)n(wing)23 b(statemen)n(t.)-118 3513 y FQ(Prop)s(osition)30 b(28.)41 b FB(L)l(et)27 b FN(X)34 b FB(b)l(e)28 b(a)g(b)l(ounde)l(d)g(op)l(er)l(ator)h (satisfying)37 b FO(\(2.1\))27 b FB(with)-118 3612 y(a)39 b(b)l(ounde)l(d)f(me)l(asur)l(able)h(r)l(e)l(al)f(function)h FN(F)12 b FO(\()p FP(\001)p FO(\))p FB(,)41 b(and)e(let)f FN(X)45 b FO(=)38 b FN(U)9 b(C)44 b FB(b)l(e)38 b(the)-118 3712 y(p)l(olar)47 b(de)l(c)l(omp)l(osition)g(intr)l(o)l(duc)l(e)l(d)f (ab)l(ove,)51 b(such)46 b(that)f FO(k)n(er)13 b FN(C)1917 3682 y FK(2)2006 3712 y FO(=)52 b(k)n(er)12 b FN(U)d FB(,)-118 3811 y FO(k)n(er)k FN(F)f FO(\()p FN(C)169 3781 y FK(2)206 3811 y FO(\))33 b(=)f(k)n(er)13 b FN(U)559 3781 y FM(\003)597 3811 y FB(.)54 b(The)36 b(r)l(elation)42 b FO(\(2.1\))35 b FB(is)g(e)l(quivalent)g(to)g(any)h(of)f(the)-118 3911 y(fol)t(lowing)7 b FO(:)p eop %%Page: 86 90 86 89 bop -118 -137 a FO(86)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-33 96 y FB(a)6 b FO(\))42 b FN(C)154 66 y FK(2)192 96 y FN(U)258 66 y FM(\003)319 96 y FO(=)23 b FN(U)473 66 y FM(\003)510 96 y FN(F)12 b FO(\()p FN(C)672 66 y FK(2)710 96 y FO(\);)-28 260 y FB(b)5 b FO(\))42 b FN(X)7 b(\036)p FO(\()p FN(X)322 229 y FM(\003)360 260 y FN(X)g FO(\))26 b(=)h FN(\036)p FO(\()p FN(F)12 b FO(\()p FN(X)840 229 y FM(\003)878 260 y FN(X)7 b FO(\)\))p FN(X)g FB(,)32 b(for)h(any)f(me)l(asur)l(able)h(function)e FN(\036)p FO(\()p FP(\001)p FO(\))89 359 y FB(b)l(ounde)l(d)g(on)e(the) h(sp)l(e)l(ctrum)f(of)i FN(X)1185 329 y FM(\003)1222 359 y FN(X)7 b FO(;)-27 522 y FB(c)e FO(\))41 b FN(\036)p FO(\()p FN(C)235 492 y FK(2)274 522 y FO(\))p FN(U)372 492 y FM(\003)460 522 y FO(=)50 b FN(U)641 492 y FM(\003)679 522 y FN(\036)p FO(\()p FN(F)12 b FO(\()p FN(C)922 492 y FK(2)961 522 y FO(\)\))p FB(,)49 b(for)d(any)f(me)l(asur)l(able)h (function)e FN(\036)p FO(\()p FP(\001)p FO(\))89 622 y FB(b)l(ounde)l(d)31 b(on)e(the)h(sp)l(e)l(ctrum)f(of)i FN(X)1185 592 y FM(\003)1222 622 y FN(X)7 b FO(;)-35 785 y FB(d)i FO(\))41 b FN(E)150 800 y FL(C)202 783 y Fx(2)239 785 y FO(\(\001\))p FN(U)438 755 y FM(\003)500 785 y FO(=)22 b FN(U)653 755 y FM(\003)691 785 y FN(E)752 800 y FL(C)804 783 y Fx(2)841 785 y FO(\()p FN(F)938 755 y FM(\000)p FK(1)1027 785 y FO(\(\001\)\))31 b FB(for)f(al)t(l)h (me)l(asur)l(able)g FO(\001)23 b FP(\032)f FI(R)p FO(.)-118 960 y FB(Her)l(e)34 b FN(E)146 975 y FL(C)198 958 y Fx(2)235 960 y FO(\()p FP(\001)p FO(\))h FB(is)f(a)h(r)l(esolution)g(of)g(the)f (identity)h(of)h(the)e(op)l(er)l(ator)i FN(C)2109 930 y FK(2)2146 960 y FB(,)g(and)-118 1060 y FN(F)-53 1030 y FM(\000)p FK(1)36 1060 y FO(\(\001\))42 b FB(denotes)g(the)g(ful)t(l) g(pr)l(e-image)h(of)f FO(\001)g(\()p FB(mor)l(e)g(pr)l(e)l(cisely,)k (the)c(ful)t(l)-118 1160 y(pr)l(e-image)31 b(of)f FO(\001)19 b FP(\\)g FN(F)12 b FO(\()p FI(R)p FO(\)\))q FB(.)-118 1320 y(Pr)l(o)l(of.)43 b FO(It)29 b(w)n(as)e(already)e(sho)n(wn)j(that) g(\(2.1\))g(is)f(equiv)-5 b(alen)n(t)26 b(to)i(a\),)g(and)g(that)-118 1419 y(\(2.1\))e(implies)c(b\).)37 b(T)-7 b(o)26 b(sho)n(w)f(that)i (b\))f(implies)d(\(2.1\))o(,)k(tak)n(e)f FN(\036)p FO(\()p FN(\025)p FO(\))e(=)f FN(\025)p FO(.)37 b(Mul-)-118 1519 y(tiplying)24 b(b)n(y)k FN(X)380 1489 y FM(\003)444 1519 y FO(from)f(the)g(righ)n(t,)f(w)n(e)h(ha)n(v)n(e)f FN(X)7 b(X)1472 1489 y FM(\003)1509 1519 y FN(X)g(X)1661 1489 y FM(\003)1721 1519 y FO(=)23 b FN(F)12 b FO(\()p FN(X)1982 1489 y FM(\003)2019 1519 y FN(X)7 b FO(\))p FN(X)g(X)2279 1489 y FM(\003)2316 1519 y FO(,)-118 1619 y(whic)n(h)43 b(implies)e(that)j(relation)e(\(2.1\))i(holds)f(on)h(v)n(ectors)f (orthogonal)e(to)-118 1718 y(k)n(er)13 b FN(X)7 b(X)159 1688 y FM(\003)195 1718 y FO(.)37 b(On)28 b(the)g(other)f(hand,)g(k)n (er)13 b FN(X)1185 1688 y FM(\003)1246 1718 y FO(=)22 b(k)n(er)13 b FN(U)1524 1688 y FM(\003)1585 1718 y FO(=)23 b(k)n(er)12 b FN(F)g FO(\()p FN(X)1970 1688 y FM(\003)2008 1718 y FN(X)7 b FO(\).)6 1818 y(The)36 b(pro)r(of)f(that)h(c\))g(is)e (equiv)-5 b(alen)n(t)33 b(to)j(a\))f(can)g(b)r(e)h(done)g(in)e(the)i (same)-118 1918 y(w)n(a)n(y)c(as)i(b\))g(w)n(as)f(deriv)n(ed.)54 b(Relation)31 b(d\))k(is)d(a)i(particular)c(case)j(of)h(c\))g(with)-118 2017 y FN(\036)p FO(\()p FN(\025)p FO(\))43 b(=)e FN(\037)244 2029 y FK(\001)303 2017 y FO(\()p FN(\025)p FO(\))e(\(notice)f(that)h FN(E)994 2032 y FL(F)9 b FK(\()p FL(C)1123 2016 y Fx(2)1156 2032 y FK(\))1186 2017 y FO(\(\001\))42 b(=)f FN(E)1528 2032 y FL(C)1580 2015 y Fx(2)1617 2017 y FO(\()p FN(F)1714 1987 y FM(\000)p FK(1)1803 2017 y FO(\(\001\)\)\);)46 b(using)37 b(a)-118 2117 y(sp)r(ectral)25 b(represen)n(tation)f(for)i (measurable)d(functions)j(of)h(the)g(p)r(ositiv)n(e)d(self-)-118 2216 y(adjoin)n(t)37 b(op)r(erator)g FN(C)585 2186 y FK(2)622 2216 y FO(,)42 b FN(\036)p FO(\()p FN(C)833 2186 y FK(2)871 2216 y FO(\))g(=)1050 2150 y Fy(R)1120 2216 y FN(\036)p FO(\()p FN(\025)p FO(\))14 b FN(dE)1399 2231 y FL(C)1451 2215 y Fx(2)1489 2216 y FO(\()p FN(\025)p FO(\),)42 b(one)c(can)g(easily)e(see)-118 2316 y(that)28 b(c\))g(follo)n(ws)c(from)i(d\))j(as)d(w)n(ell.)p 2278 2316 4 57 v 2282 2263 50 4 v 2282 2316 V 2331 2316 4 57 v -118 2481 a FQ(4.)33 b FO(Op)r(erators)18 b(of)i(the)g(form)f (\(2.1\))g(form)g(a)g(sub)r(class)f(in)h(the)i(class)c(of)j(cen)n (tered)-118 2580 y(op)r(erators.)72 b(Recall)38 b(that)i(a)g(b)r (ounded)g(op)r(erator)f FN(X)46 b FO(is)39 b(cen)n(tered,)k(if)c(the) -118 2680 y(op)r(erators)34 b FN(X)334 2650 y FL(l)358 2680 y FO(\()p FN(X)466 2650 y FM(\003)504 2680 y FO(\))536 2650 y FL(l)562 2680 y FO(,)k(\()p FN(X)731 2650 y FM(\003)768 2680 y FO(\))800 2650 y FL(k)842 2680 y FN(X)918 2650 y FL(k)958 2680 y FO(,)g FN(k)s(;)14 b(l)37 b FO(=)f(1,)i(2,)d FN(:)14 b(:)g(:)28 b FO(,)37 b(form)e(a)g(comm)n(uting)-118 2779 y(family)-7 b(.)-118 2940 y FQ(Prop)s(osition)30 b(29.)41 b FB(A)23 b(b)l(ounde)l(d)g(op)l(er)l(ator)h FN(X)30 b FB(satisfying)i FO(\(2.1\))23 b FB(is)g(c)l(enter)l(e)l(d.) -118 3039 y(If)h(a)h(p)l(air)g(of)g(op)l(er)l(ators,)i(a)d (self-adjoint)i FN(C)31 b FB(and)24 b(a)h(p)l(artial)g(isometry)g FN(U)9 b FB(,)25 b(such)-118 3139 y(that)30 b FO(k)n(er)12 b FN(U)32 b FO(=)23 b(k)n(er)13 b FN(C)543 3109 y FK(2)580 3139 y FB(,)31 b FO(k)n(er)12 b FN(U)826 3109 y FM(\003)887 3139 y FO(=)23 b(k)n(er)13 b FN(F)f FO(\()p FN(C)1262 3109 y FM(\003)1301 3139 y FO(\))p FB(,)30 b(satisfy)h(r)l(elation)37 b FO(\(2.5\))o FB(,)31 b(then)-118 3239 y FN(U)38 b FB(is)30 b(a)h(c)l(enter)l(e)l(d)e(p)l(artial)i(isometry.)-118 3399 y(Pr)l(o)l(of.)43 b FO(Let)28 b(us)f(sho)n(w)g(that)h FN(X)34 b FO(is)27 b(cen)n(tered.)36 b(W)-7 b(rite)27 b FN(A)1625 3411 y FL(k)1689 3399 y FO(=)c FN(X)1853 3369 y FL(k)1893 3399 y FO(\()p FN(X)2001 3369 y FM(\003)2039 3399 y FO(\))2071 3369 y FL(k)2112 3399 y FO(,)k FN(B)2225 3411 y FL(l)2274 3399 y FO(=)-118 3498 y(\()p FN(X)-10 3468 y FM(\003)28 3498 y FO(\))60 3468 y FL(l)85 3498 y FN(X)161 3468 y FL(l)186 3498 y FO(,)32 b FN(k)s(;)14 b(l)29 b FP(\025)f FO(1.)46 b(First)30 b(consider)e(the)k(op)r(erators) c FN(A)1693 3510 y FL(k)1734 3498 y FO(.)47 b(Applying)30 b(\(2.7\))-118 3598 y(w)n(e)d(ha)n(v)n(e)98 3773 y FN(A)160 3785 y FL(k)224 3773 y FO(=)c FN(X)388 3739 y FL(k)428 3773 y FO(\()p FN(X)536 3739 y FM(\003)574 3773 y FO(\))606 3739 y FL(k)670 3773 y FO(=)f FN(X)833 3739 y FL(k)q FM(\000)p FK(1)958 3773 y FO(\()p FN(X)7 b(X)1142 3739 y FM(\003)1180 3773 y FO(\)\()p FN(X)1320 3739 y FM(\003)1358 3773 y FO(\))1390 3739 y FL(k)q FM(\000)p FK(1)224 3911 y FO(=)23 b FN(X)388 3877 y FL(k)q FM(\000)p FK(1)513 3911 y FN(F)12 b FO(\()p FN(X)686 3877 y FM(\003)723 3911 y FN(X)7 b FO(\)\()p FN(X)939 3877 y FM(\003)977 3911 y FO(\))1009 3877 y FL(k)q FM(\000)p FK(1)1158 3911 y FO(=)23 b FN(F)1311 3877 y FL(n)1356 3911 y FO(\()p FN(X)1464 3877 y FM(\003)1501 3911 y FN(X)7 b FO(\))p FN(X)1685 3877 y FL(n)p FM(\000)p FK(1)1814 3911 y FO(\()p FN(X)1922 3877 y FM(\003)1960 3911 y FO(\))1992 3877 y FL(n)p FM(\000)p FK(1)p eop %%Page: 87 91 87 90 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(87)224 96 y(=)23 b FN(F)377 62 y FL(n)422 96 y FO(\()p FN(X)530 62 y FM(\003)567 96 y FN(X)7 b FO(\))p FN(X)751 62 y FL(n)p FM(\000)p FK(2)881 96 y FN(F)12 b FO(\()p FN(X)1054 62 y FM(\003)1091 96 y FN(X)7 b FO(\)\()p FN(X)1307 62 y FM(\003)1345 96 y FO(\))1377 62 y FL(n)p FM(\000)p FK(2)224 231 y FO(=)23 b FN(F)377 197 y FL(n)422 231 y FO(\()p FN(X)530 197 y FM(\003)567 231 y FN(X)7 b FO(\))p FN(F)740 197 y FL(n)p FM(\000)p FK(1)870 231 y FO(\()p FN(X)978 197 y FM(\003)1016 231 y FN(X)g FO(\))14 b FN(:)g(:)g(:)f(F)f FO(\()p FN(X)1421 197 y FM(\003)1459 231 y FN(X)7 b FO(\))p FN(:)-118 401 y FO(Since)21 b(all)e(the)k(op)r(erators)d FN(A)764 413 y FL(k)805 401 y FO(,)j FN(k)j FP(\025)c FO(1,)h(are)e(functions)g(of)h (the)g(single)d(op)r(erator)-118 500 y FN(X)-42 470 y FM(\003)-5 500 y FN(X)7 b FO(,)27 b(they)h(comm)n(ute)e(with)h(eac)n(h) g(other.)6 600 y(No)n(w)32 b(consider)d(a)i(pair)f FN(A)837 612 y FL(k)878 600 y FO(,)j FN(B)997 612 y FL(l)1022 600 y FO(.)49 b(Again,)31 b(applying)g(\(2.7\))g(and)g(the)h(ob-)-118 700 y(tained)27 b(represen)n(tation)d(for)k FN(A)868 712 y FL(k)909 700 y FO(,)f(w)n(e)h(get)21 869 y FN(B)84 881 y FL(l)110 869 y FN(A)172 881 y FL(k)236 869 y FO(=)23 b(\()p FN(X)432 835 y FM(\003)469 869 y FO(\))501 835 y FL(l)527 869 y FN(X)603 835 y FL(l)628 869 y FN(X)704 835 y FL(k)744 869 y FO(\()p FN(X)852 835 y FM(\003)890 869 y FO(\))922 835 y FL(k)236 1007 y FO(=)g(\()p FN(X)432 973 y FM(\003)469 1007 y FO(\))501 973 y FL(l)527 1007 y FN(X)603 973 y FL(l)628 1007 y FN(F)693 973 y FL(k)734 1007 y FO(\()p FN(X)842 973 y FM(\003)879 1007 y FN(X)7 b FO(\))p FN(F)1052 973 y FL(k)q FM(\000)p FK(1)1178 1007 y FO(\()p FN(X)1286 973 y FM(\003)1323 1007 y FN(X)g FO(\))14 b FN(:)g(:)g(:)g(F)e FO(\()p FN(X)1729 973 y FM(\003)1766 1007 y FN(X)7 b FO(\))236 1145 y(=)23 b(\()p FN(X)432 1110 y FM(\003)469 1145 y FO(\))501 1110 y FL(l)527 1145 y FN(F)592 1110 y FL(k)q FK(+)p FL(l)705 1145 y FO(\()p FN(X)813 1110 y FM(\003)851 1145 y FN(X)7 b FO(\))p FN(F)1024 1110 y FL(k)q FK(+)p FL(l)p FM(\000)p FK(1)1222 1145 y FO(\()p FN(X)1330 1110 y FM(\003)1367 1145 y FN(X)g FO(\))14 b FN(:)g(:)g(:)f(F)1664 1110 y FL(l)p FK(+1)1774 1145 y FO(\()p FN(X)1882 1110 y FM(\003)1920 1145 y FN(X)7 b FO(\))p FN(X)2104 1110 y FL(l)236 1282 y FO(=)23 b FN(F)389 1248 y FL(k)429 1282 y FO(\()p FN(X)537 1248 y FM(\003)575 1282 y FN(X)7 b FO(\))p FN(F)748 1248 y FL(k)q FM(\000)p FK(1)873 1282 y FO(\()p FN(X)981 1248 y FM(\003)1019 1282 y FN(X)g FO(\))14 b FN(:)g(:)g(:)f(F)f FO(\()p FN(X)1424 1248 y FM(\003)1462 1282 y FN(X)7 b FO(\)\()p FN(X)1678 1248 y FM(\003)1715 1282 y FO(\))1747 1248 y FL(l)1773 1282 y FN(X)1849 1248 y FL(l)1897 1282 y FO(=)23 b FN(A)2047 1294 y FL(k)2088 1282 y FN(B)2151 1294 y FL(l)2176 1282 y FN(;)-118 1452 y FO(and)j(therefore,)f(the)h (op)r(erators)e FN(A)981 1464 y FL(k)1048 1452 y FO(and)i FN(B)1271 1464 y FL(l)1296 1452 y FO(,)h FN(k)s(;)14 b(l)24 b FP(\025)f FO(1,)j(also)d(comm)n(ute)h(with)-118 1551 y(eac)n(h)j(other.)6 1651 y(No)n(w)33 b(w)n(e)f(sho)n(w)g(that)h (the)g(op)r(erators)e FN(B)1308 1663 y FL(l)1333 1651 y FO(,)j FN(l)f FP(\025)e FO(1,)j(also)d(comm)n(ute)f(with)-118 1751 y(eac)n(h)d(other.)36 b(Indeed,)28 b(for)f FN(k)f(>)c(l)r FO(,)28 b(w)n(e)f(ha)n(v)n(e)39 1920 y FN(B)102 1932 y FL(k)143 1920 y FN(B)206 1932 y FL(l)254 1920 y FO(=)c(\()p FN(X)450 1886 y FM(\003)487 1920 y FO(\))519 1886 y FL(k)561 1920 y FN(X)637 1886 y FL(k)677 1920 y FO(\()p FN(X)785 1886 y FM(\003)822 1920 y FO(\))854 1886 y FL(l)880 1920 y FN(X)956 1886 y FL(l)1004 1920 y FO(=)g(\()p FN(X)1200 1886 y FM(\003)1237 1920 y FO(\))1269 1886 y FL(l)1295 1920 y FO(\()p FN(X)1403 1886 y FM(\003)1441 1920 y FO(\))1473 1886 y FL(k)q FM(\000)p FL(l)1587 1920 y FN(X)1663 1886 y FL(k)q FM(\000)p FL(l)1777 1920 y FN(X)1853 1886 y FL(l)1877 1920 y FO(\()p FN(X)1985 1886 y FM(\003)2023 1920 y FO(\))2055 1886 y FL(l)2081 1920 y FN(X)2157 1886 y FL(l)254 2058 y FO(=)g(\()p FN(X)450 2024 y FM(\003)487 2058 y FO(\))519 2024 y FL(l)545 2058 y FN(B)608 2070 y FL(k)q FM(\000)p FL(l)722 2058 y FN(A)784 2070 y FL(l)810 2058 y FN(X)886 2024 y FL(l)934 2058 y FO(=)f(\()p FN(X)1129 2024 y FM(\003)1167 2058 y FO(\))1199 2024 y FL(l)1225 2058 y FN(A)1287 2070 y FL(l)1313 2058 y FN(B)1376 2070 y FL(k)q FM(\000)p FL(l)1490 2058 y FN(X)1566 2024 y FL(l)254 2196 y FO(=)h(\()p FN(X)450 2161 y FM(\003)487 2196 y FO(\))519 2161 y FL(l)545 2196 y FN(X)621 2161 y FL(l)646 2196 y FO(\()p FN(X)754 2161 y FM(\003)792 2196 y FO(\))824 2161 y FL(l)849 2196 y FO(\()p FN(X)957 2161 y FM(\003)995 2196 y FO(\))1027 2161 y FL(k)q FM(\000)p FL(l)1142 2196 y FN(X)1218 2161 y FL(k)q FM(\000)p FL(l)1331 2196 y FN(X)1407 2161 y FL(l)1455 2196 y FO(=)f FN(B)1605 2208 y FL(l)1631 2196 y FN(B)1694 2208 y FL(k)1735 2196 y FN(:)-118 2365 y FO(Th)n(us,)27 b(the)h(op)r(erator)e FN(X)34 b FO(is)27 b(cen)n(tered.)6 2465 y(It)g(remains)22 b(to)k(pro)n(v)n(e)e(the)i(second)g(statemen)n(t)e(of)i(the)g(prop)r (osition.)34 b(Let)-118 2564 y(\001)23 b(=)g FN(\033)s FO(\()p FN(C)209 2534 y FK(2)247 2564 y FO(\))15 b FP(n)e(f)p FO(0)p FP(g)p FO(.)35 b(Then)26 b FN(E)809 2579 y FL(C)861 2563 y Fx(2)897 2564 y FO(\(\001\))h(is)d(a)h(pro)5 b(jection)23 b(on)n(to)i(the)h(co-k)n(ernel)d(of)-118 2664 y FN(C)-53 2634 y FK(2)-16 2664 y FO(,)32 b(and,)f(since)e(k)n(er)13 b FN(U)36 b FO(=)27 b(k)n(er)13 b FN(C)6 b FO(,)32 b(w)n(e)e(ha)n(v)n (e)f FN(E)1368 2679 y FL(C)1420 2662 y Fx(2)1456 2664 y FO(\(\001\))g(=)e FN(U)1776 2634 y FM(\003)1814 2664 y FN(U)9 b FO(.)45 b(Then)31 b(\(2.5\))-118 2764 y(implies)24 b(that)k FN(U)9 b(U)476 2734 y FM(\003)536 2764 y FO(=)23 b FN(U)9 b(E)c FO(\(\001\))p FN(U)955 2734 y FM(\003)1016 2764 y FO(=)23 b FN(E)1165 2779 y FL(C)1217 2762 y Fx(2)1254 2764 y FO(\()p FN(F)1351 2734 y FM(\000)p FK(1)1440 2764 y FO(\(\001\)\).)38 b(Similarly)-7 b(,)101 2933 y FN(U)167 2899 y FL(k)208 2933 y FO(\()p FN(U)306 2899 y FM(\003)344 2933 y FO(\))376 2899 y FL(k)440 2933 y FO(=)23 b FN(U)594 2899 y FL(k)q FM(\000)p FK(1)719 2933 y FN(U)9 b(U)851 2899 y FM(\003)889 2933 y FO(\()p FN(U)987 2899 y FM(\003)1025 2933 y FO(\))1057 2899 y FL(k)q FM(\000)p FK(1)1206 2933 y FO(=)23 b FN(U)1360 2899 y FL(k)q FM(\000)p FK(1)1486 2933 y FN(U)9 b(U)1618 2899 y FM(\003)1655 2933 y FN(U)g(U)1787 2899 y FM(\003)1825 2933 y FO(\()p FN(U)1923 2899 y FM(\003)1961 2933 y FO(\))1993 2899 y FL(k)q FM(\000)p FK(1)440 3071 y FO(=)23 b FN(U)594 3037 y FL(k)634 3071 y FN(E)695 3086 y FL(C)747 3069 y Fx(2)784 3071 y FO(\(\001\)\()p FN(U)1015 3037 y FM(\003)1054 3071 y FO(\))1086 3037 y FL(k)1150 3071 y FO(=)g FN(E)1299 3086 y FL(C)1351 3069 y Fx(2)1387 3071 y FO(\()p FN(F)1484 3037 y FM(\000)p FL(k)1577 3071 y FO(\(\001\)\))p FN(:)-118 3249 y FO(Therefore,)f(the)g (pro)5 b(jections)20 b FN(U)902 3219 y FL(k)943 3249 y FO(\()p FN(U)1041 3219 y FM(\003)1079 3249 y FO(\))1111 3219 y FL(k)1152 3249 y FO(,)j FN(k)j FP(\025)d FO(1,)g(comm)n(ute)d(b) r(oth)i(with)g FN(U)2235 3219 y FM(\003)2273 3249 y FN(U)-118 3349 y FO(and)g(eac)n(h)f(other.)34 b(The)22 b(rest)g(of)f(the)i(comm)n (uting)18 b(relations)h(can)j(b)r(e)g(obtained)-118 3449 y(in)29 b(the)h(same)f(w)n(a)n(y)g(as)g(w)n(as)g(done)h(for)f FN(X)36 b FO(in)30 b(the)g(pro)r(of)f(of)h(the)h(\014rst)e(part)h(of) -118 3548 y(the)e(prop)r(osition.)p 2278 3548 4 57 v 2282 3496 50 4 v 2282 3548 V 2331 3548 4 57 v -118 3712 a FQ(5.)33 b FO(No)n(w)19 b(w)n(e)h(will)c(sho)n(w)j(that)h(prop)r (erties)d(of)j(represen)n(tations)c(of)k(the)g(relation)-118 3811 y(\(2.1\))33 b(dep)r(end)h(on)f(prop)r(erties)f(of)h(the)h (corresp)r(onding)c(dynamical)g(system)-118 3911 y FN(\025)23 b FP(7!)g FN(F)12 b FO(\()p FN(\025)p FO(\),)29 b(and)f(study)f(this)g (dep)r(endency)h(in)f(detail.)p eop %%Page: 88 92 88 91 bop -118 -137 a FO(88)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(Prop)s(osition)30 b(30.)41 b FB(L)l(et)31 b(the)h(op)l(er)l(ators)h FN(C)38 b FB(and)33 b FN(U)40 b FB(satisfy)33 b(r)l(elation)39 b FO(\(2.5\))p FB(.)-118 196 y(If)25 b FN(e)3 208 y FL(\025)71 196 y FB(is)g(an)f(eigenve)l(ctor)i(of)f(the)g(op)l(er)l(ator)h FN(C)1307 166 y FK(2)1344 196 y FB(,)g(c)l(orr)l(esp)l(onding)g(to)f (an)f(eigen-)-118 296 y(value)33 b FN(\025)p FB(,)i(then)d FN(U)460 266 y FM(\003)498 296 y FN(e)537 308 y FL(\025)613 296 y FB(is)h(either)h(zer)l(o)f(or)g(is)g(again)h(an)f(eigenve)l(ctor) h(of)g FN(C)2276 266 y FK(2)2313 296 y FB(,)-118 395 y(c)l(orr)l(esp)l(onding)d(to)f(the)g(eigenvalue)h FN(F)12 b FO(\()p FN(\025)p FO(\))p FB(.)-118 559 y(Pr)l(o)l(of.)43 b FO(Indeed,)f(from)c(\(2.5\))o(,)k(w)n(e)d(see)g(that)g FN(C)1428 529 y FK(2)1465 559 y FN(U)1531 529 y FM(\003)1569 559 y FN(e)1608 571 y FL(\025)1693 559 y FO(=)j FN(U)1866 529 y FM(\003)1904 559 y FN(F)12 b FO(\()p FN(C)2066 529 y FK(2)2104 559 y FO(\))i FN(e)2189 571 y FL(\025)2274 559 y FO(=)-118 658 y FN(F)e FO(\()p FN(\025)p FO(\))i FN(U)139 628 y FM(\003)178 658 y FN(e)217 670 y FL(\025)260 658 y FO(.)p 2278 658 4 57 v 2282 606 50 4 v 2282 658 V 2331 658 4 57 v 6 824 a(In)26 b(particular,)d(if)i(the)h(op)r(erator) e FN(U)34 b FO(is)25 b(unitary)-7 b(,)25 b(then)h FN(U)1795 794 y FM(\003)1833 824 y FN(e)1872 836 y FL(\025)1941 824 y FO(is)e(not)i(zero,)-118 923 y(whic)n(h)h(implies)e(that,)j(in)g (this)f(case,)h FN(F)12 b FO(\()p FP(\001)p FO(\))29 b(maps)e(the)h(sp)r(ectrum)g(of)g FN(C)2132 893 y FK(2)2198 923 y FO(in)n(to)-118 1023 y(itself.)6 1123 y(A)c(similar)19 b(fact)24 b(holds)e(for)h(all)e(p)r(oin)n(ts)i(of)g(the)h(sp)r(ectrum)f (of)g(the)h(op)r(erator)-118 1222 y FN(C)-53 1192 y FK(2)-16 1222 y FO(.)-118 1386 y FQ(Prop)s(osition)30 b(31.)41 b FB(If)28 b(the)g(op)l(er)l(ator)h FN(X)34 b FB(in)h FO(\(2.1\))27 b FB(is)i(invertible)35 b FO(\()p FB(i.e.,)c FN(U)36 b FB(in)-118 1485 y FO(\(2.5\))29 b FB(is)h(unitary)7 b FO(\))p FB(,)31 b(then)f FN(F)12 b FO(\()p FP(\001)p FO(\))30 b FB(maps)g(the)g(sp)l(e)l(ctrum)f FN(\033)s FO(\()p FN(C)1745 1455 y FK(2)1784 1485 y FO(\))h FB(into)f(itself.) -118 1649 y(Pr)l(o)l(of.)43 b FO(Let)29 b FN(\025)e FP(2)f FN(\033)s FO(\()p FN(C)595 1619 y FK(2)633 1649 y FO(\).)43 b(Then)29 b(there)g(exists)f(a)h(sequence)g(of)g(unit)g(v)n(ectors)-118 1749 y FN(e)-79 1761 y FL(n)-11 1749 y FP(2)23 b FN(H)7 b FO(,)28 b FN(n)23 b FP(\025)f FO(1,)27 b(suc)n(h)h(that)g FP(k)p FN(C)921 1718 y FK(2)958 1749 y FN(e)997 1761 y FL(n)1060 1749 y FP(\000)18 b FN(\025e)1230 1761 y FL(n)1275 1749 y FP(k)23 b(\000)-49 b(!)24 b(1)p FO(,)j FN(n)c FP(\000)-48 b(!)23 b(1)p FO(.)37 b(Then)169 1928 y FP(k)p FN(C)276 1894 y FK(2)313 1928 y FN(U)379 1894 y FM(\003)417 1928 y FN(e)456 1940 y FL(n)520 1928 y FP(\000)18 b FN(F)12 b FO(\()p FN(\025)p FO(\))p FN(U)846 1894 y FM(\003)884 1928 y FN(e)923 1940 y FL(n)968 1928 y FP(k)23 b FO(=)g FP(k)p FN(U)1229 1894 y FM(\003)1266 1928 y FO(\()p FN(F)12 b FO(\()p FN(C)1460 1894 y FK(2)1498 1928 y FO(\))p FN(e)1569 1940 y FL(n)1633 1928 y FP(\000)18 b FN(F)12 b FO(\()p FN(\025)p FO(\))p FN(e)1932 1940 y FL(n)1977 1928 y FO(\))p FP(k)333 2063 y FO(=)22 b FP(k)p FN(F)12 b FO(\()p FN(C)624 2029 y FK(2)661 2063 y FO(\))p FN(e)732 2075 y FL(n)796 2063 y FP(\000)18 b FN(F)12 b FO(\()p FN(\025)p FO(\))p FN(e)1095 2075 y FL(n)1141 2063 y FP(k)22 b(\000)-48 b(!)23 b FO(0)p FN(;)179 b(n)23 b FP(\000)-48 b(!)23 b(1)p FN(;)-118 2242 y FO(and)k FP(k)p FN(U)151 2212 y FM(\003)189 2242 y FN(e)228 2254 y FL(n)273 2242 y FP(k)22 b FO(=)h(1,)k FN(n)c FP(\025)g FO(1.)36 b(This)27 b(implies)c(that)28 b FN(F)12 b FO(\()p FN(\025)p FO(\))24 b FP(2)g FN(\033)s FO(\()p FN(C)1857 2212 y FK(2)1895 2242 y FO(\).)p 2278 2242 V 2282 2190 50 4 v 2282 2242 V 2331 2242 4 57 v 6 2408 a(Prop)r(osition)33 b(30)i(pro)n(vides)e(a)j(w)n(a)n(y)e(for)i (constructing)e(represen)n(tations)-118 2507 y(of)28 b(relation)d(\(2.1\).)39 b(Indeed,)29 b(c)n(ho)r(ose)e(a)h(sequence)f (of)i(p)r(ositiv)n(e)c(n)n(um)n(b)r(ers)i FN(\025)2274 2519 y FL(k)2316 2507 y FO(,)-118 2607 y FN(k)f FP(2)d FI(Z)p FO(,)d(\(if)26 b(it)f(exists\))g(suc)n(h)h(that)g FN(F)12 b FO(\()p FN(\025)1091 2619 y FL(k)1133 2607 y FO(\))23 b(=)g FN(\025)1324 2619 y FL(k)q FK(+1)1475 2607 y FO(for)j(all)d FN(k)j FP(2)e FI(Z)o FO(,)d(and)k(de\014ne)-118 2707 y(the)j(op)r(erators)d FN(C)6 b FO(,)28 b FN(U)37 b FO(in)27 b FN(l)724 2719 y FK(2)761 2707 y FO(\()p FI(Z)o FO(\))22 b(as)27 b(follo)n(ws:)435 2886 y FN(C)500 2852 y FK(2)537 2886 y FN(e)576 2898 y FL(k)640 2886 y FO(=)22 b FN(\025)775 2898 y FL(k)817 2886 y FN(;)97 b(U)9 b(e)1042 2898 y FL(k)1105 2886 y FO(=)23 b FN(e)1232 2898 y FL(k)q FM(\000)p FK(1)1357 2886 y FN(;)180 b(k)26 b FP(2)d FI(Z)p FN(:)376 b FO(\(2.8\))-118 3065 y(Then)30 b(the)h(op)r(erators)d FN(U)9 b FO(,)30 b FN(C)6 b FO(,)31 b(satisfy)f(\(2.5\),)h(and)e(therefore,)h(giv)n(e)e(a)i(repre-)-118 3165 y(sen)n(tation)i(of)41 b(\(2.1\))o(.)56 b(Belo)n(w,)34 b(w)n(e)g(will)d(study)j(whether)g(other)f(irreducible)-118 3265 y(represen)n(tations)28 b(exist)h(and)i(what)g(is)e(their)h (structure.)46 b(In)31 b(particular,)d(for)-118 3364 y(\\simple")21 b(mappings)h FN(F)12 b FO(\()p FP(\001)p FO(\),)26 b(all)d(irreducible)e(represen)n(tations)h(with)i(unitary) -118 3464 y FN(U)29 b FO(ha)n(v)n(e)19 b(suc)n(h)h(a)g(form,)g(and)g (can)g(b)r(e)g(classi\014ed)e(up)i(to)h(a)e(unitary)g(equiv)-5 b(alence.)-118 3612 y FQ(6.)80 b FO(Let)43 b(a)f(Borel)e(set)j(\001)f (b)r(e)h(suc)n(h)f(that)h(\001)48 b FP(\032)f FN(F)12 b FO(\()p FI(R)p FO(\))49 b(and)42 b FN(F)2003 3582 y FM(\000)p FK(1)2093 3612 y FO(\(\001\))48 b(=)-118 3712 y(\001.)66 b(Prop)r(osition)34 b(28)i(implies)e(that)k FN(E)1160 3727 y FL(C)1212 3710 y Fx(2)1248 3712 y FO(\(\001\))p FN(U)1447 3682 y FM(\003)1525 3712 y FO(=)h FN(U)1695 3682 y FM(\003)1733 3712 y FN(E)1794 3727 y FL(C)1846 3710 y Fx(2)1882 3712 y FO(\()p FN(F)1979 3682 y FM(\000)p FK(1)2069 3712 y FO(\(\001\)\))h(=)-118 3811 y FN(U)-52 3781 y FM(\003)-14 3811 y FN(E)47 3826 y FL(C)99 3810 y Fx(2)135 3811 y FO(\(\001\),)33 b(i.e.,)f FN(E)547 3826 y FL(C)599 3810 y Fx(2)635 3811 y FO(\(\001\))g(is)e(a)h(pro)5 b(jection)29 b(on)n(to)h(an)h(in)n(v)-5 b(arian)n(t)28 b(subspace)-118 3911 y(in)35 b FN(H)7 b FO(.)62 b(Therefore,)36 b(to)g(study)g(irreducible)c(represen)n(tations,)j(w)n(e)h(need)g(to)p eop %%Page: 89 93 89 92 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(89)-118 96 y(study)23 b(the)h(\\smallest")19 b(in)n(v)-5 b(arian)n(t)20 b(subsets)k(\(in)f (the)g(sense)g(indicated)f(ab)r(o)n(v)n(e\))-118 196 y(that)d(con)n(tains)f(the)h(sp)r(ectrum)g(of)g FN(C)1011 166 y FK(2)1049 196 y FO(.)34 b(Belo)n(w)17 b(w)n(e)i(will)e(use)i (some)f(facts)h(ab)r(out)-118 296 y(dynamical)26 b(systems)j(and)h (their)f(in)n(v)-5 b(arian)n(t)27 b(sets,)k(as)e(w)n(ell)f(as)i(prop)r (erties)e(of)-118 395 y(the)g(corresp)r(onding)d(sp)r(ectral)h (measures.)-118 546 y FQ(7.)68 b FO(Recall)36 b(some)g(basic)h(notions) f(and)i(facts)g(ab)r(out)h(discrete)d(dynamical)-118 646 y(systems.)44 b(A)31 b(discrete)e(time)g(one-dimensional)c (dynamical)i(system)i(is)g(just)-118 746 y(a)e(\(con)n(tin)n(uous)f(or) h(measurable\))d(mapping)h FI(R)1362 716 y FK(1)1429 746 y FP(3)e FN(\025)h FP(7!)f FN(F)12 b FO(\()p FN(\025)p FO(\))24 b FP(2)f FI(R)2018 716 y FK(1)2061 746 y FO(.)6 846 y(First)35 b(of)i(all,)f(w)n(e)f(recall)e(the)k(notion)e(of)h(a)g (tra)5 b(jectory)34 b(or)h(an)h(orbit)f(of)-118 945 y(a)f(dynamical)c (system.)55 b(T)-7 b(raditionally)g(,)31 b(a)i(tra)5 b(jectory)33 b(or)g(an)h(orbit)e(of)i(the)-118 1045 y(dynamical)24 b(system)i FN(F)12 b FO(\()p FP(\001)p FO(\))24 b(:)f FI(R)832 1015 y FK(1)898 1045 y FP(7!)g FI(R)1058 1015 y FK(1)1129 1045 y FO(is)k(the)h(set)306 1295 y(Orb)o(\()p FN(\025)p FO(\))c(=)f FP(f)p FN(\025;)14 b(F)e FO(\()p FN(\025)p FO(\))p FN(;)i(F)1079 1261 y FK(2)1117 1295 y FO(\()p FN(\025)p FO(\))p FN(;)g(:)g(:)g(:)g FP(g)23 b FO(=)1559 1192 y FM(1)1546 1217 y Fy([)1529 1392 y FL(n)p FK(=0)1668 1295 y FN(F)1733 1261 y FL(n)1778 1295 y FO(\()p FN(\025)p FO(\))q FN(:)-118 1561 y FO(Here)32 b FN(F)148 1531 y FL(n)193 1561 y FO(\()p FP(\001)p FO(\))h(=)e FN(F)12 b FO(\()p FN(F)571 1531 y FL(n)p FM(\000)p FK(1)701 1561 y FO(\()p FP(\001)p FO(\)\),)35 b FN(n)d FO(=)f(1,)j(2,)e FN(:)14 b(:)g(:)27 b FO(,)35 b(and)d FN(F)1665 1531 y FK(0)1702 1561 y FO(\()p FP(\001)p FO(\))i(is)d(the)j(iden)n(tit)n(y) -118 1660 y(transformation.)e(Since)24 b(w)n(e)g(are)g(in)n(terested)f (in)h(the)h(action)e(of)h FN(F)1962 1630 y FM(\000)p FK(1)2076 1660 y FO(as)g(w)n(ell,)-118 1760 y(will)k(sa)n(y)i(that)h(t) n(w)n(o)f(p)r(oin)n(ts,)h FN(\025)857 1772 y FK(1)926 1760 y FO(and)g FN(\025)1139 1772 y FK(2)1176 1760 y FO(,)h(b)r(elong)e(to)h(the)g(same)e(tra)5 b(jectory)-7 b(,)-118 1860 y(if)32 b FN(F)28 1829 y FL(k)69 1860 y FO(\()p FN(\025)149 1872 y FK(1)187 1860 y FO(\))f(=)g FN(\025)394 1872 y FK(2)464 1860 y FO(or)h FN(F)636 1829 y FL(k)676 1860 y FO(\()p FN(\025)756 1872 y FK(2)794 1860 y FO(\))g(=)f FN(\025)1002 1872 y FK(1)1072 1860 y FO(for)h(some)f FN(k)j FP(\025)c FO(0.)51 b(W)-7 b(e)33 b(also)e(need)h(the)-118 1959 y(notion)d(of)i(the)g(tra)5 b(jectory)28 b(decomp)r(osition:)40 b(w)n(e)30 b(sa)n(y)f(that)i(t)n(w) n(o)f(p)r(oin)n(ts,)g FN(\025)2301 1971 y FK(1)-118 2059 y FO(and)24 b FN(\025)88 2071 y FK(2)126 2059 y FO(,)h(b)r(elong)d(to)i (the)h(same)e(elemen)n(t)f(of)i(the)h(tra)5 b(jectory)22 b(decomp)r(osition,)-118 2158 y(if)35 b(these)i(p)r(oin)n(ts)e(\\meet)g (in)g(the)h(future",)j(i.e.,)e(if)f FN(F)1595 2128 y FL(k)1635 2158 y FO(\()p FN(\025)1715 2170 y FK(1)1753 2158 y FO(\))i(=)f FN(F)1990 2128 y FL(m)2053 2158 y FO(\()p FN(\025)2133 2170 y FK(2)2171 2158 y FO(\))f(for)-118 2258 y(some)g FN(k)s FO(,)41 b FN(m)f(>)g FO(0.)67 b(A)38 b(p)r(oin)n(t)g FN(\025)936 2270 y FK(0)1014 2258 y FP(2)i FI(R)1163 2228 y FK(1)1206 2258 y FO(,)h(suc)n(h)d(that)g FN(F)1723 2228 y FL(m)1786 2258 y FO(\()p FN(\025)1866 2270 y FK(0)1904 2258 y FO(\))i(=)g FN(\025)2129 2270 y FK(0)2205 2258 y FO(and)-118 2358 y FN(F)-53 2328 y FL(n)-8 2358 y FO(\()p FN(\025)72 2370 y FK(0)110 2358 y FO(\))d FP(6)p FO(=)g FN(\025)329 2370 y FK(0)402 2358 y FO(for)f(0)g FN(<)h(n)g(<)f(m)p FO(,)i(is)d(called)e(a)j(p)r(erio)r (dic)e(p)r(oin)n(t)h(of)h(p)r(erio)r(d)-118 2457 y FN(m)p FO(.)54 b(The)34 b(p)r(erio)r(dic)d(p)r(oin)n(ts)i FN(\025)837 2469 y FK(0)874 2457 y FO(,)i FN(F)12 b FO(\()p FN(\025)1077 2469 y FK(0)1115 2457 y FO(\),)34 b FN(:)14 b(:)g(:)28 b FO(,)35 b FN(F)1452 2427 y FL(m)p FM(\000)p FK(1)1600 2457 y FO(\()p FN(\025)1680 2469 y FK(0)1718 2457 y FO(\))e(form)g(a)g (cycle)f(of)-118 2557 y(p)r(erio)r(d)54 b FN(m)p FO(.)6 2657 y(Bearing)39 b(in)i(mind)e(the)j(represen)n(tations)c(of)j (relation)d(\(2.5\))j(and)g(the)-118 2757 y(fact)34 b(that)h(these)f (in)n(v)-5 b(arian)n(t)31 b(sets)j(should)f(carry)f(the)j(sp)r(ectral)e (measure)f(of)-118 2856 y(the)37 b(op)r(erator)e FN(C)443 2826 y FK(2)481 2856 y FO(,)k(w)n(e)d(need)h(to)g(consider)e(a)h (measurable)e(mapping)g(of)j(a)-118 2956 y(measurable)e(space.)68 b(In)38 b(this)f(case,)j(the)f(mapping)d FN(F)12 b FO(\()p FP(\001)p FO(\))39 b(giv)n(es)d(rise)g(to)i(a)-118 3056 y(mapping)25 b(of)j(Borel)d(measures)h(on)h(the)h(line)e(b)n(y)h(the)h (form)n(ula)726 3239 y FN(d\032)p FO(\()p FN(\025)p FO(\))c FP(7!)f FN(d\032)p FO(\()p FN(F)1237 3205 y FM(\000)p FK(1)1326 3239 y FO(\()p FN(\025)p FO(\)\))p FN(:)6 3424 y FO(A)41 b(Borel)c(measure)h FN(\032)p FO(\()p FP(\001)p FO(\))j(is)e(called)e(quasi-in)n(v)-5 b(arian)n(t)35 b(with)k(resp)r(ect)h(to)-118 3523 y FN(F)12 b FO(\()p FP(\001)p FO(\),)28 b(if)f FN(\032)p FO(\()p FN(F)301 3493 y FM(\000)p FK(1)391 3523 y FO(\()p FP(\001)p FO(\)\))h(is)f (absolutely)d(con)n(tin)n(uous)i(with)h(resp)r(ect)g(to)h FN(\032)p FO(\()p FP(\001)p FO(\).)-118 3691 y FQ(Prop)s(osition)i(32.) 41 b FB(If)31 b(the)g(op)l(er)l(ator)h FN(U)40 b FB(in)d FO(\(2.5\))30 b FB(is)i(unitary,)f(and)h FN(F)12 b FO(\()p FP(\001)p FO(\))31 b FB(is)-118 3791 y(one-to-one,)h(then)g(the)f(sp)l (e)l(ctr)l(al)h(me)l(asur)l(e)f(of)50 b FN(C)1452 3761 y FK(2)1521 3791 y FB(is)32 b(quasi-invariant)h(with)-118 3890 y(r)l(esp)l(e)l(ct)c(to)h FN(F)12 b FO(\()p FP(\001)p FO(\))30 b FB(and)h FN(F)661 3860 y FM(\000)p FK(1)750 3890 y FO(\()p FP(\001)p FO(\))p FB(.)p eop %%Page: 90 94 90 93 bop -118 -137 a FO(90)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Indeed,)33 b(w)n(e)f(can)f(tak)n(e)g(the)i(sp)r(ectral)d(measure)g(to)h(b)r(e)i (in)e(the)h(form)-118 196 y FN(\032)p FO(\()p FP(\001)p FO(\))g(=)f(\()p FN(E)233 211 y FL(C)285 194 y Fx(2)322 196 y FO(\()p FP(\001)p FO(\))p FN(!)s(;)14 b(!)s FO(\),)34 b(where)f FN(!)i FO(is)d(a)g(v)n(ector)f(of)i(maximal)28 b(sp)r(ectral)j(t)n(yp)r(e.)-118 296 y(Then)d(the)g(statemen)n(t)e (follo)n(ws)f(from)h(the)i(equalit)n(y)60 485 y FN(\032)p FO(\()p FN(F)200 450 y FM(\000)p FK(1)289 485 y FO(\()p FP(\001)p FO(\)\))c(=)f(\()p FN(E)613 500 y FL(C)665 483 y Fx(2)702 485 y FO(\()p FN(F)799 450 y FM(\000)p FK(1)888 485 y FO(\()p FP(\001)p FO(\)\))p FN(!)s(;)14 b(!)s FO(\))432 619 y(=)23 b(\()p FN(U)618 585 y FM(\003)656 619 y FN(E)717 634 y FL(C)769 618 y Fx(2)805 619 y FO(\()p FN(F)902 585 y FM(\000)p FK(1)992 619 y FO(\()p FP(\001)p FO(\)\))p FN(!)s(;)14 b(U)1269 585 y FM(\003)1307 619 y FN(!)s FO(\))23 b(=)g(\()p FN(E)1598 634 y FL(C)1650 618 y Fx(2)1687 619 y FO(\()p FP(\001)p FO(\))p FN(U)1840 585 y FM(\003)1878 619 y FN(!)s(;)14 b(U)2036 585 y FM(\003)2074 619 y FN(!)s FO(\))-118 808 y(and)27 b(a)h(similar)22 b(relation)j(for)i FN(U)9 b FO(.)p 2278 808 4 57 v 2282 756 50 4 v 2282 808 V 2331 808 4 57 v 6 999 a(W)-7 b(e)23 b(also)e(need)h(the)h(notion)f(of)g(ergo)r(dicit)n(y)d(of)k(a)f (measure.)33 b(A)23 b(Borel)d(mea-)-118 1099 y(sure)28 b FN(\032)p FO(\()p FP(\001)p FO(\))h(is)f(called)e(ergo)r(dic)h(with)h (resp)r(ect)g(to)h(the)g(action)e(of)i(a)f(dynamical)-118 1198 y(system)k FN(F)12 b FO(\()p FP(\001)p FO(\),)35 b(if)e(for)g(an)n(y)g(measurable)d FN(F)12 b FO(\()p FP(\001)p FO(\)-in)n(v)-5 b(arian)n(t)30 b(set)j(\001)g FP(\032)f FI(R)p FO(,)41 b(\(i.e.,)-118 1298 y(the)28 b(set)h(\001)f(suc)n(h)g(that)g FN(F)686 1268 y FM(\000)p FK(1)775 1298 y FO(\(\001\))d(=)f(\001\),)k(w)n(e)g(ha)n(v)n(e)f(that)i (either)e FN(\032)p FO(\(\001\))d(=)g(0)k(or)-118 1398 y FN(\032)p FO(\()p FI(R)d FP(n)17 b FO(\001\))24 b(=)f(0.)-118 1573 y FQ(Prop)s(osition)30 b(33.)41 b FB(If)36 b(a)h(p)l(air)g FN(X)7 b FB(,)38 b FN(X)1126 1543 y FM(\003)1164 1573 y FB(,)g(satisfying)45 b FO(\(2.1\))p FB(,)38 b(is)f(irr)l(e)l (ducible,)-118 1673 y(then)31 b(the)h(sp)l(e)l(ctr)l(al)g(me)l(asur)l (e)g(of)g(the)g(op)l(er)l(ator)42 b FN(C)1472 1643 y FK(2)1541 1673 y FB(is)32 b(er)l(go)l(dic)h(with)g(r)l(esp)l(e)l(ct) -118 1772 y(to)d FN(F)12 b FO(\()p FP(\001)p FO(\))p FB(.)-118 1948 y(Pr)l(o)l(of.)43 b FO(Indeed,)f(otherwise)c(there)h(w)n (ould)f(exist)g(a)h(non-trivial)c(in)n(v)-5 b(arian)n(t)-118 2047 y(subset)35 b(\001)g FP(\032)g FN(\033)s FO(\()p FN(C)495 2017 y FK(2)533 2047 y FO(\).)59 b(As)35 b(w)n(as)e(noticed)h (ab)r(o)n(v)n(e,)h(in)g(this)f(case)g FN(E)2026 2062 y FL(C)2078 2045 y Fx(2)2114 2047 y FO(\(\001\))i(is)-118 2147 y(a)d(pro)5 b(jection)31 b(on)n(to)h(an)h(in)n(v)-5 b(arian)n(t)30 b(subspace)j(whic)n(h)f(is)g(non-trivial)d(if)k(and)-118 2247 y(only)26 b(if)h(the)h(sp)r(ectral)e(measure)g(of)h(\001)h(is)f (neither)f(zero)h(nor)g(one.)p 2278 2247 V 2282 2194 50 4 v 2282 2247 V 2331 2247 4 57 v 6 2437 a(The)d(simplest)d(in)n(v)-5 b(arian)n(t)21 b(sets)i(are)g(elemen)n(ts)e(of)j(the)g(tra)5 b(jectory)22 b(decom-)-118 2537 y(p)r(osition)31 b(of)h(the)i (dynamical)29 b(system)i(\(in)i(the)g(one-to-one)e(case,)j(they)f(are) -118 2637 y(just)e(orbits\).)44 b(The)30 b(simplest)e(class)g(of)i (quasi-in)n(v)-5 b(arian)n(t)26 b(ergo)r(dic)i(measures)-118 2736 y(is)36 b(formed)f(b)n(y)i(atomic)d(measures)h(concen)n(trated)h (on)g(an)h(elemen)n(t)e(of)i(tra-)-118 2836 y(jectory)23 b(decomp)r(osition;)f(ho)n(w)n(ev)n(er,)h(an)h(atomic)e(measure)g (concen)n(trated)h(on)-118 2936 y(an)28 b(orbit)f(is)g(also)f(quasi-in) n(v)-5 b(arian)n(t)23 b(and)28 b(ergo)r(dic.)37 b(Belo)n(w,)26 b(w)n(e)i(will)e(see)h(that)-118 3035 y(only)c(suc)n(h)h(measures)e (corresp)r(onding)g(to)i(an)g(orbit)g(giv)n(e)e(rise)h(to)h (irreducible)-118 3135 y(represen)n(tations)h(of)i(the)h(relation.)6 3238 y(The)23 b(existence)d(of)i(non-atomic)d(quasi-in)n(v)-5 b(arian)n(t)17 b(ergo)r(dic)j(measures)g(de-)-118 3337 y(p)r(ends)39 b(on)g(top)r(ological)c(prop)r(erties)i(of)i(the)h (dynamical)c(system.)70 b(In)39 b(the)-118 3437 y(one-to-one)26 b(case,)h(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing)c(fact.)-118 3612 y FQ(Prop)s(osition)30 b(34.)41 b FB(If)30 b(a)h(dynamic)l(al)h (system)e FN(\025)24 b FP(7!)g FN(F)12 b FO(\()p FN(\025)p FO(\))31 b FB(with)g(one-to-one)-118 3712 y FN(F)12 b FO(\()p FP(\001)p FO(\))35 b FB(p)l(ossesses)h(a)f(me)l(asur)l(able)g (se)l(ction,)i(i.e.,)h(a)d(me)l(asur)l(able)g(set)g(that)g(in-)-118 3811 y(terse)l(cts)26 b(any)i(orbit)g(in)f(a)h(single)g(p)l(oint,)g (then)f(any)h(er)l(go)l(dic)g(me)l(asur)l(e)f(is)h(c)l(on-)-118 3911 y(c)l(entr)l(ate)l(d)h(on)h(a)g(single)g(orbit)h(of)f(the)g (dynamic)l(al)i(system.)p eop %%Page: 91 95 91 94 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(91)6 96 y(In)38 b(the)g(non-bijectiv)n (e)d(case,)k(the)f(condition)d(of)j(existence)e(of)h(a)g(mea-)-118 196 y(surable)25 b(section)g(is)h(replaced)f(b)n(y)i(the)g(follo)n (wing)c(condition)i(of)i(existence)f(of)-118 296 y FN(M)9 b FO(-partition:)6 395 y FB(the)30 b(dynamic)l(al)h(system)e(p)l (ossesses)g(an)h FN(M)9 b FB(-p)l(artition,)29 b(i.e.,)j(ther)l(e)d (exists)-118 495 y(a)h(p)l(artition)h FI(R)e FO(=)465 433 y Fy(S)534 520 y FL(n)p FM(2)p Fu(N)680 495 y FO(\001)749 507 y FL(n)795 495 y FB(,)h FO(\001)919 507 y FL(k)983 495 y FP(2)23 b Fz(B)p FO(\()p FI(R)q FO(\))p FB(,)36 b(such)30 b(that)6 595 y FO(1\))37 b FB(for)h(any)f FN(k)j FB(ther)l(e)d(exists)g FN(j)42 b FB(such)37 b(that)g FN(F)12 b FO(\(\001)1570 607 y FL(k)1611 595 y FO(\))36 b(=)g(\001)1849 607 y FL(j)1921 595 y FB(and)i FN(F)12 b FO(\()p FP(\001)p FO(\))37 b FB(is)-118 694 y(one-to-one)30 b(on)f FO(\001)483 706 y FL(k)524 694 y FO(;)6 794 y(2\))g FB(if)h(for)g(some)g FN(n)p FO(,)f FN(k)s FB(,)g(the)g(mapping)i FN(F)1272 764 y FL(n)1317 794 y FO(\()p FP(\001)p FO(\))f FB(maps)f FO(\001)1718 806 y FL(k)1788 794 y FB(into)h(itself)p FO(,)g FB(then)-118 893 y FN(F)-53 863 y FL(n)-8 893 y FO(\()p FP(\001)p FO(\))g FB(is)g(the)g(identity)h(on)f FO(\001)828 905 y FL(k)869 893 y FB(.)6 993 y FO(W)-7 b(e)28 b(also)e(men)n(tion)f(the)j(follo)n(wing)c(statemen)n(t)j(from)f ([274)n(].)-118 1158 y FQ(Theorem)k(15.)41 b FB(L)l(et)32 b FN(I)39 b FB(b)l(e)32 b(some)g(\014nite)g(interval,)i(and)f FN(F)12 b FO(\()p FP(\001)p FO(\))d(:)29 b FN(I)35 b FP(\000)-47 b(!)28 b FN(I)39 b FB(b)l(e)32 b(a)-118 1258 y(c)l(ontinuous)k(p)l(artial)t(ly)j(monotone)e(mapping)45 b FO(\()p FB(i.e.,)c FN(I)j FB(de)l(c)l(omp)l(oses)38 b(into)f(a)-118 1357 y(\014nite)29 b(union)f(of)i(sub-intervals,)g(on)f (which)i FN(F)12 b FO(\()p FP(\001)p FO(\))30 b FB(is)f(monotone)6 b FO(\))p FB(.)40 b(Then)29 b(the)-118 1457 y(fol)t(lowing)j(c)l (onditions)f(ar)l(e)f(e)l(quivalent)8 b FO(:)6 1557 y FB(i)g FO(\))33 b FB(ther)l(e)f(exists)f(an)h FN(M)9 b FB(-p)l(artition)32 b(of)h FN(I)7 b FB(,)32 b(e)l(ach)h(element)e(of) i(which)g(is)f(an)-118 1656 y(interval)39 b FO(\()p FB(p)l(ossibly,)32 b(a)e(single)h(p)l(oint)8 b FO(\);)6 1756 y FB(ii)g FO(\))30 b FB(any)e(quasi-invariant)h(er)l(go)l(dic)g(me)l(asur)l(e)f(is)g(c)l (onc)l(entr)l(ate)l(d)g(on)g(a)g(sin-)-118 1855 y(gle)i(element)g(of)g (the)g(tr)l(aje)l(ctory)h(de)l(c)l(omp)l(osition)6 b FO(;)6 1955 y FB(iii)i FO(\))32 b FB(the)e(set)g(of)g(p)l(erio)l(dic)i (p)l(oints,)e FO(P)n(er)13 b FN(F)f FB(,)30 b(is)g(close)l(d)9 b FO(;)6 2067 y FB(iv)g FO(\))34 b FB(for)h(some)e FN(m)c FP(\025)g FO(0)p FB(,)34 b(the)g(r)l(elation)g FO(Fix)o(\()p FN(F)1461 2037 y FK(2)1494 2012 y Fv(m)p Fx(+1)1624 2067 y FO(\))c(=)f(Fix)o(\()p FN(F)1998 2037 y FK(2)2031 2012 y Fv(m)2091 2067 y FO(\))k FB(holds)-118 2167 y FO(\(Fix)o(\()p FN(F)12 b FO(\))30 b FB(denotes)h(the)f(set)f(of)i(\014xe)l(d)e(p)l (oints)h(of)g FN(F)12 b FO(\))p FB(.)6 2332 y FO(Belo)n(w)26 b(w)n(e)h(study)h(the)g(corresp)r(ondence)d(b)r(et)n(w)n(een)j(the)g (orbits)e(and)h(irre-)-118 2431 y(ducible)f(represen)n(tations)e(of)k (the)g(relation.)-118 2647 y FQ(2.1.2)94 b(Finite-dimensional)27 b(represen)m(tations)-118 2800 y FO(If)g(the)g(sequence)f FN(\025)497 2812 y FL(k)565 2800 y FO(in)f(\(2.8\))i(is)e(p)r(erio)r (dic,)g(the)i(corresp)r(onding)c(irreducible)-118 2900 y(represen)n(tation)17 b(is)i(\014nite-dimensional.)29 b(W)-7 b(e)20 b(will)d(sho)n(w)i(here)h(that)g(all)d(\014nite-)-118 2999 y(dimensional)23 b(represen)n(tations)j(of)i(relation)d(\(2.1\))j (are)f(related)f(to)i(cycles)f(of)-118 3099 y(the)21 b(corresp)r(onding)d(dynamical)g(system.)33 b(Then)22 b(w)n(e)e(apply)g(the)h(Shark)n(o)n(vsky)-118 3199 y(theorem)29 b(on)i(existence)e(of)i(cycles)e(to)i(to)f(study)h(irreducible)c (\014nite-dimen-)-118 3298 y(sional)e(represen)n(tations;)f(these)k (results)e(are)h(illustrated)d(with)j(examples.)-118 3447 y FQ(1.)49 b FO(Let)32 b(us)g(classify)e(irreducible)e(pairs)i FN(X)7 b FO(,)33 b FN(X)1393 3417 y FM(\003)1462 3447 y FO(of)f(op)r(erators)e(on)i(a)f(\014nite-)-118 3547 y(dimensional)e(space,)35 b(ob)r(eying)d(relation)f(\(2.1\))i (\(irreducible)d(\014nite-dimen-)-118 3646 y(sional)25 b(represen)n(tations)f(of)34 b(\(2.1\)\))28 b(up)g(to)f(unitary)f (equiv)-5 b(alence.)-118 3811 y FQ(Theorem)30 b(16.)41 b FB(A)n(ny)29 b(cycle)i FN(O)896 3823 y FL(\025)963 3811 y FO(=)22 b FP(f)p FN(\025;)14 b(F)e FO(\()p FN(\025)p FO(\))p FN(;)i(:)g(:)g(:)h(;)f(F)1605 3781 y FL(n)p FM(\000)p FK(1)1735 3811 y FO(\()p FN(\025)p FO(\))p FP(g)30 b FB(of)g(p)l(erio)l(d)h FN(n)p FB(,)-118 3911 y(such)38 b(that)f FN(\025)h FP(\025)e FO(0)h FB(and)h FN(t)g(>)e FO(0)h FB(for)i(al)t(l)f(other)g(p)l(oints)g FN(t)f FP(2)h FN(O)1904 3923 y FL(\025)1948 3911 y FB(,)i(de\014nes)d(a)p eop %%Page: 92 96 92 95 bop -118 -137 a FO(92)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(family)31 b(of)g FN(n)p FB(-dimensional)g (irr)l(e)l(ducible)g(r)l(epr)l(esentations)f(of)48 b FO(\(2.1\):)336 434 y FN(X)29 b FO(=)522 193 y Fy(0)522 339 y(B)522 388 y(B)522 438 y(B)522 491 y(@)651 256 y FO(0)832 185 y Fy(p)p 915 185 178 4 v 71 x FN(F)12 b FO(\()p FN(\025)p FO(\))453 b(0)659 344 y FB(.)659 378 y(.)659 411 y(.)915 353 y(.)950 378 y(.)985 403 y(.)1184 353 y(.)1219 378 y(.)1254 403 y(.)651 518 y FO(0)518 b(0)1370 447 y Fy(p)p 1453 447 309 4 v 71 x FN(F)1518 494 y FL(n)p FM(\000)p FK(1)1648 518 y FO(\()p FN(\025)p FO(\))595 619 y FN(e)634 588 y FL(i\036)701 619 y FN(\025)193 b FO(0)f FN(:)14 b(:)g(:)272 b FO(0)1761 193 y Fy(1)1761 339 y(C)1761 388 y(C)1761 438 y(C)1761 491 y(A)1848 434 y FN(;)297 b FO(\(2.9\))-118 772 y FB(wher)l(e)30 b FN(\036)24 b FP(2)f FO([0)p FN(;)14 b FO(2)p FN(\031)s FO(\))p FB(,)30 b(if)h FN(\025)23 b(>)g FO(0)p FB(,)29 b(or)i(a)f(single)g(r)l(epr)l (esentation,)h(if)f FN(\025)24 b FO(=)e(0)p FB(.)6 871 y(These)31 b(ar)l(e)f(al)t(l,)h(up)f(to)g(unitary)f(e)l(quivalenc)l(e,) i(distinct)f(irr)l(e)l(ducible)h(r)l(ep-)-118 971 y(r)l(esentations)e (of)49 b FO(\(2.1\))29 b FB(in)h(a)g(\014nite-dimensional)h(sp)l(ac)l (e.)-118 1113 y(Pr)l(o)l(of.)43 b FO(Indeed,)28 b(b)n(y)f(a)h(direct)e (calculation)e(one)j(can)g(c)n(hec)n(k)g(that)h(the)g(repre-)-118 1212 y(sen)n(tation)23 b(\(2.9\))h(satis\014es)e(the)j(necessary)e (relation.)33 b(Let)24 b(us)h(sho)n(w)e(that)i(it)f(is)-118 1312 y(irreducible.)41 b(An)n(y)30 b(b)r(ounded)h(self-adjoin)n(t)c(op) r(erator)i FN(T)12 b FO(,)30 b(whic)n(h)f(comm)n(utes)-118 1412 y(with)23 b FN(X)31 b FO(and)24 b FN(X)401 1382 y FM(\003)438 1412 y FO(,)h(comm)n(utes)d(with)h FN(C)1123 1382 y FK(2)1185 1412 y FO(and)h FN(U)33 b FO(in)n(tro)r(duced)23 b(ab)r(o)n(v)n(e)f(as)i(w)n(ell.)-118 1511 y(But,)37 b(in)e(the)h(selected)e(basis)f FN(C)928 1481 y FK(2)1001 1511 y FO(is)h(diagonal)e(with)j(distinct)f(eigen)n(v)-5 b(alues)-118 1611 y FN(\025)p FO(,)29 b FN(:)14 b(:)g(:)27 b FO(,)h FN(F)222 1581 y FL(n)p FM(\000)p FK(1)353 1611 y FO(\()p FN(\025)p FO(\).)39 b(Then)28 b FN(T)40 b FO(is)26 b(diagonal,)f(and)j(the)h(comm)n(utation)24 b(with)k FN(U)-118 1711 y FO(implies)c(that)k FN(T)34 b FO(=)22 b FN(cI)7 b FO(.)6 1810 y(W)-7 b(e)20 b(sho)n(w)e(that)h(these)g(are)f (all)e(the)j(irreducible)d(represen)n(tations)g(of)25 b(\(2.1\).)6 1910 y(Assume)33 b(that)g FN(U)42 b FO(is)32 b(unitary)-7 b(.)52 b(Since)32 b(dim)12 b FN(H)39 b(<)32 b FP(1)p FO(,)i(the)g(op)r(erator)d FN(C)2301 1880 y FK(2)-118 2009 y FO(can)41 b(b)r(e)h(diagonalized,)f(and)h(has)f (non-negativ)n(e)e(eigen)n(v)-5 b(alues.)76 b(T)-7 b(ak)n(e)41 b(an)-118 2109 y(eigen)n(v)-5 b(alue)35 b FN(t)j FO(and)g(a)g(unit)g (eigen)n(v)n(ector)d FN(e)1276 2121 y FL(t)1343 2109 y FO(of)j FN(C)1513 2079 y FK(2)1551 2109 y FO(,)j FN(C)1680 2079 y FK(2)1717 2109 y FN(e)1756 2121 y FL(t)1826 2109 y FO(=)f FN(te)2000 2121 y FL(t)2029 2109 y FO(.)68 b(It)39 b(fol-)-118 2209 y(lo)n(ws)27 b(from)g(\(2.5\))i(that)g FN(U)710 2179 y FM(\003)748 2209 y FN(e)787 2221 y FL(t)841 2209 y FO(=)c FN(e)970 2224 y FL(F)9 b FK(\()p FL(t)p FK(\))1130 2209 y FO(is)28 b(again)f(a)h(unit)h(eigen)n(v)n(ector)c(of) k FN(C)2278 2179 y FK(2)2316 2209 y FO(,)-118 2317 y FN(C)-53 2287 y FK(2)-16 2317 y FN(U)50 2287 y FM(\003)88 2317 y FN(e)127 2329 y FL(t)179 2317 y FO(=)23 b FN(F)12 b FO(\()p FN(t)p FO(\))p FN(e)465 2332 y FL(F)d FK(\()p FL(t)p FK(\))597 2317 y FO(.)37 b(Consider)25 b(the)i(sequence)g(of)f (unit)h(v)n(ectors)e(\()p FN(U)2136 2287 y FM(\003)2174 2317 y FO(\))2206 2287 y FL(k)2248 2317 y FN(e)2287 2329 y FL(t)2316 2317 y FO(,)-118 2417 y FN(k)31 b FP(\025)d FO(0.)45 b(Since)30 b(there)h(is)e(only)g(a)i(\014nite)f(n)n(um)n(b)r (er)f(of)i(eigen)n(v)-5 b(alues)27 b(of)k FN(C)2152 2387 y FK(2)2189 2417 y FO(,)h(w)n(e)-118 2516 y(conclude)27 b(that,)j(for)e(some)g FN(n)d FP(\025)f FO(1)29 b(and)g(some)e(eigen)n (v)-5 b(alue)26 b FN(t)1802 2528 y FK(0)1839 2516 y FO(,)j FN(F)1956 2486 y FL(n)2002 2516 y FO(\()p FN(t)2064 2528 y FK(0)2101 2516 y FO(\))d(=)e FN(t)2278 2528 y FK(0)2316 2516 y FO(.)-118 2616 y(The)j(linear)e(span)i FN(H)546 2628 y FK(0)610 2616 y FO(of)g(eigenspaces)e(of)i FN(C)1302 2586 y FK(2)1340 2616 y FO(,)g(corresp)r(onding)e(to)i(eigen)n(v)-5 b(al-)-118 2716 y(ues)27 b FN(t)55 2728 y FK(0)93 2716 y FO(,)g FN(:)14 b(:)g(:)28 b FO(,)g FN(F)384 2685 y FL(n)p FM(\000)p FK(1)514 2716 y FO(\()p FN(t)576 2728 y FK(0)613 2716 y FO(\),)g(is)f(in)n(v)-5 b(arian)n(t)24 b(with)j(resp)r(ect)h(to)f FN(C)1768 2685 y FK(2)1833 2716 y FO(and)h FN(U)2061 2685 y FM(\003)2099 2716 y FO(.)6 2815 y(W)-7 b(e)32 b(sho)n(w)e(that)i(the)f(space)g FN(H)987 2827 y FK(0)1055 2815 y FO(is)f(also)f(in)n(v)-5 b(arian)n(t)28 b(with)j(resp)r(ect)g(to)g FN(U)9 b FO(.)-118 2915 y(Assume)20 b(that)h FN(U)9 b(e)461 2927 y FL(t)486 2935 y Fx(0)543 2915 y FO(do)r(es)20 b(not)h(b)r(elong)e(to)i FN(H)1286 2927 y FK(0)1323 2915 y FO(.)35 b(Denote)21 b(b)n(y)f FN(u)h FO(the)g(pro)5 b(jection)-118 3014 y(of)38 b FN(U)9 b(e)92 3026 y FL(t)117 3034 y Fx(0)192 3014 y FO(on)39 b(the)g(orthogonal)c(complemen)n(t)h(of)i FN(H)1548 3026 y FK(0)1585 3014 y FO(.)70 b(F)-7 b(rom)38 b(the)h(relation)-118 3114 y FN(E)-57 3129 y FL(C)-5 3112 y Fx(2)31 3114 y FO(\(\001\))p FN(U)230 3084 y FM(\003)296 3114 y FO(=)26 b FN(U)453 3084 y FM(\003)490 3114 y FN(E)551 3129 y FL(C)603 3112 y Fx(2)640 3114 y FO(\()p FN(F)737 3084 y FM(\000)p FK(1)826 3114 y FO(\(\001\)\))31 b(for)e(an)n(y)g (measurable)d(\001,)31 b(w)n(e)e(ha)n(v)n(e)f(that)-118 3214 y FN(u)h FO(b)r(elongs)f(to)i(the)g(space)f FN(H)801 3226 y FK(1)868 3214 y FO(generated)g(b)n(y)g(eigen)n(v)n(ectors)e (corresp)r(onding)-118 3313 y(to)g(all)f(eigen)n(v)-5 b(alues)24 b FN(\025)g FP(2)f FO(\003)g(=)f FP(f)p FN(F)12 b FO(\()p FN(\025)p FO(\))24 b(=)e FN(t)1206 3325 y FK(0)1244 3313 y FN(;)14 b(\025)23 b FP(6)p FO(=)g FN(F)1505 3283 y FL(n)p FM(\000)p FK(1)1635 3313 y FO(\()p FN(t)1697 3325 y FK(0)1734 3313 y FO(\))p FP(g)p FO(.)6 3413 y(Since)30 b(an)n(y)f(p)r(oin)n(t)h(has)g(a)g(single)e(image,)g(for)i(all)e FN(k)i(>)d FO(0,)k(the)g(sets)f(\003)2206 3425 y FL(k)2274 3413 y FO(=)-118 3513 y FN(F)-53 3482 y FM(\000)p FL(k)40 3513 y FO(\(\003\))20 b(are)f(disjoin)n(t.)32 b(W)-7 b(e)20 b(ha)n(v)n(e)f(that)h FN(U)1195 3482 y FL(k)1236 3513 y FN(H)1305 3525 y FK(1)1365 3513 y FP(\032)i FN(E)1513 3528 y FL(C)1565 3511 y Fx(2)1602 3513 y FO(\()p FN(F)1699 3482 y FM(\000)p FL(k)1792 3513 y FO(\(\003\)\))p FN(H)2015 3525 y FK(1)2053 3513 y FO(,)f(and)f(all)-118 3612 y(these)29 b(spaces)f(are)g(non-zero)f(and)i(orthogonal)c(to)k(eac)n(h)f(other.)40 b(But)30 b(this)e(is)-118 3712 y(imp)r(ossible,)23 b(since)k FN(H)34 b FO(is)27 b(\014nite-dimensional.)k(Therefore,)26 b FN(H)1869 3724 y FK(0)1934 3712 y FO(is)h(in)n(v)-5 b(arian)n(t)-118 3811 y(with)30 b(resp)r(ect)g(to)g FN(X)37 b FO(and)31 b FN(X)812 3781 y FM(\003)849 3811 y FO(,)g(and,)h(due)e (to)h(the)g(irreducibilit)n(y)-7 b(,)25 b(coincides)-118 3911 y(with)i(the)h(whole)e(of)i FN(H)7 b FO(.)p eop %%Page: 93 97 93 96 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(93)6 96 y(The)31 b(op)r(erator)e FN(U)584 66 y FL(n)659 96 y FO(comm)n(utes)g(with)h FN(C)1310 66 y FK(2)1347 96 y FO(,)i(since)d FN(F)1673 66 y FL(n)1718 96 y FO(\()p FN(t)1780 108 y FK(0)1818 96 y FO(\))f(=)g FN(t)2001 108 y FK(0)2038 96 y FO(.)46 b(It)31 b(also)-118 196 y(ob)n(viously)h(comm)n(utes)h(with)j FN(U)44 b FO(and)36 b FN(U)1190 166 y FM(\003)1227 196 y FO(.)62 b(Therefore,)36 b(the)g(irreducibilit)n(y)-118 296 y(implies)16 b FN(U)222 266 y FL(n)290 296 y FO(=)22 b FN(\013I)7 b FO(,)22 b FP(j)p FN(\013)p FP(j)i FO(=)f(1.)33 b(In)20 b(the)h(basis)d FN(e)1294 308 y FL(t)1322 296 y FO(,)i FN(:)14 b(:)g(:)28 b FO(,)21 b FN(e)1573 311 y FL(F)1624 295 y Fv(n)p Fw(\000)p Fx(1)1738 311 y FK(\()p FL(t)p FK(\))1819 296 y FO(,)h(the)e(op)r (erators)-118 395 y(act)27 b(as)g(needed.)6 504 y(Consider)e(the)i (case)e(of)i(non-unitary)d FN(U)9 b FO(.)37 b(No)n(w,)26 b(k)n(er)13 b FN(C)1742 473 y FK(2)1802 504 y FO(=)23 b(k)n(er)13 b FN(U)31 b FP(6)p FO(=)23 b FP(f)p FO(0)p FP(g)p FO(,)-118 603 y(and)28 b(there)g(exists)e(a)i(unit)g(v)n(ector)f FN(e)1021 615 y FK(0)1086 603 y FO(suc)n(h)g(that)i FN(U)9 b(e)1559 615 y FK(0)1619 603 y FO(=)24 b FN(C)1773 573 y FK(2)1810 603 y FN(e)1849 615 y FK(0)1910 603 y FO(=)f(0.)38 b(Again,)-118 703 y(consider)30 b(the)i(v)n(ectors)e(\()p FN(U)741 673 y FM(\003)779 703 y FO(\))811 673 y FL(k)852 703 y FN(e)891 715 y FK(0)928 703 y FO(,)j FN(k)g FO(=)c(0,)j(1,)g FN(:)14 b(:)g(:)27 b FO(.)50 b(Relation)29 b(\(2.5\))i(implies)-118 802 y(that)38 b FN(e)111 814 y FL(k)190 802 y FO(is)f(either)g(an)h (eigen)n(v)n(ector)d(of)j FN(C)1268 772 y FK(2)1343 802 y FO(with)g(the)h(eigen)n(v)-5 b(alue)35 b FN(F)2169 772 y FL(k)2209 802 y FO(\(0\),)-118 902 y(or)f(the)h(zero)f(v)n (ector.)57 b(Consider)33 b(t)n(w)n(o)h(p)r(ossibilities:)46 b(there)35 b(exists)e FN(n)i FO(suc)n(h)-118 1002 y(that)f FN(e)107 1014 y FL(k)180 1002 y FO(=)f(0,)i(or)e(else)f(there)h(are)g (n)n(um)n(b)r(ers)f FN(k)37 b FO(and)c FN(n)p FO(,)i FN(k)h(<)d(n)p FO(,)i(suc)n(h)e(that)-118 1101 y FN(F)-53 1071 y FL(n)-8 1101 y FO(\(0\))23 b(=)g FN(F)274 1071 y FL(k)315 1101 y FO(\(0\).)6 1209 y(W)-7 b(e)27 b(sho)n(w)e(that)i (the)f(second)g(alternativ)n(e)d(is)i(imp)r(ossible.)32 b(Indeed,)27 b(since)-118 1309 y(k)n(er)13 b FN(U)73 1279 y FM(\003)138 1309 y FO(=)26 b(k)n(er)13 b FN(F)f FO(\()p FN(C)516 1279 y FK(2)554 1309 y FO(\),)31 b(w)n(e)f(conclude)f (that)h(the)g(m)n(ultiplicit)n(y)25 b(of)30 b(the)g(eigen-)-118 1409 y(v)-5 b(alue)30 b FN(F)165 1379 y FL(k)206 1409 y FO(\(0\))h(is)g(the)g(same)f(as)h(that)h(of)f FN(F)1242 1379 y FL(n)1287 1409 y FO(\(0\).)48 b(On)32 b(the)f(other)g(hand,)h(t) n(w)n(o)-118 1508 y(orthogonal)f(eigenspaces,)i FN(H)843 1520 y FL(k)q FM(\000)p FK(1)1002 1508 y FO(and)h FN(H)1239 1520 y FL(n)1318 1508 y FO(that)h(corresp)r(ond)d(to)i FN(F)2107 1478 y FL(k)q FM(\000)p FK(1)2232 1508 y FO(\(0\))-118 1608 y(and)e FN(F)113 1578 y FL(n)158 1608 y FO(\(0\),)i(are)d(mapp)r (ed)h(b)n(y)g FN(U)971 1578 y FM(\003)1041 1608 y FO(in)n(to)g(the)g (eigenspace)e FN(H)1842 1620 y FL(k)1916 1608 y FO(corresp)r(ond-)-118 1708 y(ing)35 b(to)h FN(F)204 1678 y FL(k)245 1708 y FO(\(0\).)62 b(Since)35 b(the)i(space)e(is)g(\014nite-dimensional,)e (there)j(exists)f(a)-118 1807 y(non-zero)25 b(v)n(ector)g FN(w)k FO(in)d(the)h(direct)e(sum)g FN(H)1269 1819 y FL(k)q FM(\000)p FK(1)1411 1807 y FP(\010)16 b FN(H)1561 1819 y FL(n)1633 1807 y FO(suc)n(h)26 b(that)h FN(U)2064 1777 y FM(\003)2102 1807 y FN(w)e FO(=)e(0,)-118 1907 y(whic)n(h)j(con)n(tradicts)g(the)i(condition)d(k)n(er)13 b FN(U)1245 1877 y FM(\003)1306 1907 y FO(=)22 b(k)n(er)13 b FN(F)f FO(\()p FN(C)1680 1877 y FK(2)1718 1907 y FO(\).)6 2015 y(Then,)46 b(there)c(exists)e FN(n)i FO(suc)n(h)g(that)g FN(e)1261 2027 y FL(k)1348 2015 y FO(=)47 b(0,)e FN(k)50 b FP(\025)c FN(n)p FO(,)g(i.e.,)e FN(e)2106 2027 y FL(n)p FM(\000)p FK(1)2283 2015 y FP(2)-118 2115 y FO(k)n(er)13 b FN(U)73 2085 y FM(\003)146 2115 y FO(=)35 b(k)n(er)13 b FN(F)f FO(\()p FN(C)533 2085 y FK(2)571 2115 y FO(\),)37 b(and)e FN(F)897 2085 y FL(n)942 2115 y FO(\(0\))h(=)f(0;)j(th)n(us)d (0)g(is)f(a)h(p)r(erio)r(dic)d(p)r(oin)n(t)j(of)-118 2214 y(p)r(erio)r(d)26 b FN(n)p FO(,)i(and)f(the)h(form)n(ula)d(follo)n (ws.)p 2278 2214 4 57 v 2282 2162 50 4 v 2282 2214 V 2331 2214 4 57 v -118 2449 a FQ(2.)55 b FO(The)35 b(presen)n(ted)e (theorem)g(reduces)g(the)h(problem)e(of)i(classi\014cation)c(of)-118 2549 y(\014nite-dimensional)18 b(irreducible)i(represen)n(tations)h(\() p FN(C)1623 2519 y FK(2)1661 2549 y FN(;)14 b(U)9 b FO(\))24 b(to)f(the)i(descrip-)-118 2684 y(tion)33 b(of)g(cycles)f(of)i(the)g (dynamical)c(system)j FI(R)1394 2654 y FK(1)1470 2630 y FL(F)9 b FK(\()p FM(\001)p FK(\))1482 2684 y FP(\000)-49 b(!)45 b FI(R)1680 2654 y FK(1)1723 2684 y FO(.)56 b(So,)35 b(let)e(us)g(lo)r(ok)-118 2784 y(ho)n(w)27 b(the)h(facts)f(ab)r(out)h (cycles)e(of)h(dynamical)d(systems)j(can)g(b)r(e)h(used)f(in)g(the)-118 2883 y(con)n(text)g(of)h(represen)n(tation)d(theory)-7 b(.)6 2991 y(Shark)n(o)n(vsky's)31 b(theorem)i(establishes)e(the)j (follo)n(wing)c(order)i(in)h(the)i(set)-118 3091 y(of)27 b(natural)f(n)n(um)n(b)r(ers.)-118 3283 y FQ(Theorem)k(17.)41 b FB(L)l(et)29 b FN(F)21 b FO(:)28 b FN(I)i FP(7!)23 b FN(I)37 b FB(b)l(e)30 b(a)g(c)l(ontinuous)f(mapping)i(of)g(the)e (close)l(d)-118 3382 y(interval)37 b FN(I)42 b FB(into)37 b(itself.)58 b(If)36 b(the)g(dynamic)l(al)i(system)e(p)l(ossesses)g(a)h (cycle)g(of)-118 3482 y(p)l(erio)l(d)j FN(m)p FB(,)h(then)e(for)g(any)g FN(m)853 3452 y FM(0)901 3482 y FN(/)25 b(m)p FB(,)41 b(ther)l(e)d(exists)h(a)g(cycle)g(of)h(p)l(erio)l(d)g FN(m)2290 3452 y FM(0)2313 3482 y FB(,)-118 3582 y(wher)l(e)27 b FN(/)e FB(denotes)h(the)g(fol)t(lowing)j(or)l(der)e(on)f(the)g(set)f FI(N)36 b FB(of)27 b(natur)l(al)e(numb)l(ers)7 b FO(:)-15 3782 y(1)18 b FN(/)g FO(2)g FN(/)g FP(\001)c(\001)g(\001)k FN(/)g FO(2)442 3747 y FL(n)505 3782 y FN(/)g FP(\001)c(\001)g(\001)k FN(/)g FO(2)782 3747 y FK(2)838 3782 y FP(\001)g FO(5)g FN(/)g FO(2)1041 3747 y FK(2)1096 3782 y FP(\001)h FO(3)f FN(/)g FP(\001)c(\001)g(\001)k FN(/)g FO(2)g FP(\001)g FO(5)g FN(/)g FO(2)g FP(\001)h FO(3)f FN(/)g FP(\001)c(\001)g(\001)k FN(/)g FO(5)g FN(/)g FO(3)p FN(:)2126 3881 y FO(\(2.10\))p eop %%Page: 94 98 94 97 bop -118 -137 a FO(94)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(F)-6 b(or)34 b(any)g FN(m)p FB(,)h(ther)l(e)f(exists)f(a)h(c)l(ontinuous)f(mapping)1635 75 y FO(~)1616 96 y FN(F)21 b FO(:)30 b FN(I)37 b FP(7!)30 b FN(I)41 b FB(such)34 b(that)-118 196 y(the)k(dynamic)l(al)i(system)e (has)h(a)g(cycle)g(of)g(p)l(erio)l(d)h FN(m)p FB(,)h(and)d(do)l(es)h (not)f(have)-118 296 y(cycles)31 b(of)f(p)l(erio)l(ds)h FN(m)573 266 y FM(0)626 296 y FB(for)g FN(m)18 b(/)g(m)983 266 y FM(0)1006 296 y FB(.)6 469 y FO(This)k(theorem)f(giv)n(es)f(the)j (follo)n(wing)18 b(statemen)n(t)k(ab)r(out)g(the)h(dimensions)-118 569 y(of)k(irreducible)d(represen)n(tations)h(of)34 b(\(2.1\).)-118 743 y FQ(Prop)s(osition)c(35.)41 b FB(L)l(et)g FN(F)21 b FO(:)32 b FN(I)52 b FP(7!)44 b FN(I)49 b FB(b)l(e)42 b(a)g(c)l(ontinuous)f(mapping)i(of)f(the)-118 843 y(interval)31 b FN(I)37 b FB(into)31 b(itself.)41 b(If)31 b(ther)l(e)f(exist)g(irr)l (e)l(ducible)i FN(m)p FB(-dimensional)g(r)l(epr)l(e-)-118 942 y(sentations)e(of)48 b FO(\(2.1\))30 b FB(such)g(that)g FN(\033)s FO(\()p FN(C)1099 912 y FK(2)1138 942 y FO(\))24 b FP(\032)f FN(I)7 b FB(,)30 b(then)g(for)h(any)g FN(m)1931 912 y FM(0)1973 942 y FN(/)18 b(m)p FB(,)30 b(ther)l(e)-118 1042 y(exist)f FN(m)148 1012 y FM(0)171 1042 y FB(-dimensional)j(irr)l (e)l(ducible)f(r)l(epr)l(esentations)f(of)48 b FO(\(2.1\))p FB(.)6 1144 y(F)-6 b(or)23 b(any)h FN(m)p FB(,)g(ther)l(e)f(exists)f(a) i(c)l(ontinuous)e(mapping)1673 1123 y FO(~)1654 1144 y FN(F)f FO(:)28 b FN(I)i FP(7!)23 b FN(I)30 b FB(such)23 b(that)-118 1244 y(the)38 b(r)l(elation)46 b FO(\(2.1\))38 b FB(has)h(an)g FN(m)p FB(-dimensional)g(irr)l(e)l(ducible)h(r)l(epr)l (esentation)-118 1343 y(and)e(do)l(es)h(not)e(have)i(irr)l(e)l(ducible) g(r)l(epr)l(esentations)f(of)h(dimension)g FN(m)2175 1313 y FM(0)2236 1343 y FB(for)-118 1443 y FN(m)18 b(/)g(m)106 1413 y FM(0)129 1443 y FB(.)6 1617 y FO(F)-7 b(or)27 b(con)n(tin)n(uous)f(mappings)f FN(F)12 b FO(\()p FP(\001)p FO(\),)28 b(the)g(follo)n(wing)c(corollaries)e(hold.)-118 1790 y FQ(Corollary)32 b(3.)40 b FB(If)33 b(r)l(elation)39 b FO(\(2.1\))31 b FB(with)i(c)l(ontinuous)e FN(F)12 b FO(\()p FP(\001)p FO(\))32 b FB(has)h(irr)l(e)l(ducible)-118 1890 y(r)l(epr)l(esentations)j(with)h(a)g(dimension)h(which)g(is)f(not) f(e)l(qual)g(to)h(a)f(p)l(ower)i(of)-118 1990 y FO(2)p FB(,)45 b(then)d(ther)l(e)h(ar)l(e)f(in\014nitely)h(many)f(dimensions)i (for)f(which)50 b FO(\(2.1\))42 b FB(has)-118 2089 y(irr)l(e)l(ducible) 31 b(r)l(epr)l(esentations.)-118 2263 y FQ(Corollary)h(4.)40 b FB(The)34 b(existenc)l(e)e(of)i(a)f(thr)l(e)l(e-dimensional)h(irr)l (e)l(ducible)f(r)l(ep-)-118 2363 y(r)l(esentation)24 b(of)42 b FO(\(2.1\))23 b FB(implies)j(that)31 b FO(\(2.1\))24 b FB(has)g(irr)l(e)l(ducible)h(r)l(epr)l(esentations)-118 2462 y(of)30 b(any)h(dimension)f FN(n)23 b FP(2)h FI(N)t FB(.)-118 2636 y(Example)31 b(11.)42 b FO(\(Second-degree)24 b(mapping.)34 b(Finite-dimensional)18 b(represen-)-118 2736 y(tations\).)36 b(Consider)25 b(the)j(follo)n(wing)c(relation)789 2924 y FN(xx)883 2889 y FM(\003)945 2924 y FO(=)e(\()p FN(x)1111 2889 y FM(\003)1150 2924 y FN(x)d FP(\000)f FN(q)s FO(\))1371 2889 y FK(2)1409 2924 y FN(:)-118 3111 y FO(The)k(corresp)r(onding)d(dynamical)g(system)i(is)g(generated)g(b)n (y)h(the)g(p)r(olynomial)-118 3211 y FN(P)-65 3223 y FL(q)-28 3211 y FO(\()p FN(\025)p FO(\))43 b(=)f(\()p FN(\025)27 b FP(\000)f FN(q)s FO(\))504 3181 y FK(2)542 3211 y FO(.)72 b(According)37 b(to)i(the)h(argumen)n(ts)d(ab)r(o)n(v)n (e,)j(all)d(\014nite-)-118 3311 y(dimensional)26 b(represen)n(tations)i (are)i(describ)r(ed)f(in)h(terms)g(of)g(cycles)g(of)g(this)-118 3410 y(mapping.)41 b(Let)30 b(us)g(lo)r(ok)e(at)i(ho)n(w)f(the)i(v)-5 b(alue)28 b(of)i FN(q)j FO(a\013ects)d(the)g(existence)f(of)-118 3510 y(cycles)d(and)h(their)g(order)f(\(see,)i(e.g.,)f([245)o(]\))6 3612 y(F)-7 b(or)20 b FN(q)26 b(<)d FP(\000)p FO(1)p FN(=)p FO(4,)d(there)h(are)e(no)h(stationary)e(p)r(oin)n(ts,)j(and)g (therefore,)g(there)-118 3712 y(are)28 b(no)h(cycles)f(at)h(all.)40 b(F)-7 b(or)28 b FN(q)h FO(=)d FP(\000)p FO(1)p FN(=)p FO(4,)i(there)h(exists)f(a)h(unique)f(stationary)-118 3811 y(p)r(oin)n(t,)23 b FN(\025)g FO(=)g(1)p FN(=)p FO(4,)f(and)g(no)g(other)g(cycles.)33 b(T)-7 b(o)22 b(this)g(p)r(oin)n (t,)g(there)h(corresp)r(onds)-118 3911 y(a)k(circle)e(of)j (one-dimensional)22 b(represen)n(tations,)j FN(X)k FO(=)23 b FN(e)1696 3881 y FL(i\036)1763 3911 y FN(=)p FO(2,)k FN(\036)c FP(2)h FO([0)p FN(;)14 b FO(2)p FN(\031)s FO(\).)p eop %%Page: 95 99 95 98 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(95)6 96 y(F)-7 b(or)37 b FP(\000)p FO(1)p FN(=)p FO(4)h FN(<)h(q)j(<)d FO(3)p FN(=)p FO(4,)g(there)e(are)g(t)n(w)n(o)f(stationary)f(p)r(oin)n(ts,)k FN(\025)2144 108 y FK(0)p FL(;)p FK(1)2274 96 y FO(=)-118 196 y(\(2)p FN(q)28 b FO(+)d(1)g FP(\006)269 135 y(p)p 338 135 225 4 v 61 x FO(4)p FN(q)c FO(+)d(1)o(\))p FN(=)p FO(2,)41 b(whic)n(h)c(giv)n(e)f(t)n(w)n(o)i(one-dimensional)32 b(families)j(of)-118 296 y(irreducible)26 b(represen)n(tations.)40 b(There)29 b(are)g(no)g(other)g(cycles)f(and)i(no)f(other)-118 395 y(irreducible)24 b(\014nite-dimensional)e(represen)n(tations.)6 498 y(As)33 b FN(q)j FO(increases)30 b(from)h(3/4)h(to)g FN(q)1064 467 y FM(\003)1134 498 y FP(\031)f FO(1)p FN(:)p FO(4,)i(cycles)e(of)i(order)e(2,)i(2)2098 467 y FK(2)2135 498 y FO(,)g FN(:)14 b(:)g(:)28 b FO(,)-118 597 y(2)-76 567 y FL(n)-31 597 y FO(,)k(and)f(the)h(corresp)r(onding)c(families)f (of)32 b(irreducible)27 b(represen)n(tations)i(of)-118 697 y(the)f(corresp)r(onding)d(dimensions)f(arise.)6 799 y(F)-7 b(or)36 b FN(q)k FO(=)d FN(q)383 769 y FM(\003)421 799 y FO(,)i(there)d(exist)f(cycles)f(of)i(an)n(y)g(order)e(2)1691 769 y FL(k)1732 799 y FO(,)k FN(k)i FP(\025)d FO(1,)h(and)e(no)-118 899 y(other)j(cycles;)44 b(an)n(y)39 b(irreducible)d (\014nite-dimensional)e(represen)n(tation)j(has)-118 998 y(dimension)24 b(2)316 968 y FL(k)384 998 y FO(for)k(some)e FN(k)f FP(\025)e FO(0.)6 1101 y(As)30 b FN(q)e(>)e(q)327 1071 y FM(\003)394 1101 y FO(increases,)h(other)i(cycles)e(arise)g(in)i (the)g(order)f(describ)r(ed)g(b)n(y)-118 1200 y(the)37 b(Shark)n(o)n(vsky)c(theorem.)62 b(Starting)35 b(from)g(some)g FN(q)41 b FP(\031)c FO(1)p FN(:)p FO(75,)g(there)g(are)-118 1300 y(cycles)22 b(of)i(order)e(3,)j(and)e(therefore,)h(cycles)e(of)i (all)e(other)h(orders.)34 b(Therefore,)-118 1400 y(for)29 b(suc)n(h)h FN(q)j FO(there)d(are)f(irreducible)e(represen)n(tations)g (of)j(an)n(y)f(\014nite)h(dimen-)-118 1499 y(sion.)58 b(Notice)35 b(that)g(for)g(some)e(v)-5 b(alues)34 b(of)i FN(q)s FO(,)h(zero)d(ma)n(y)g(b)r(ecome)g(p)r(erio)r(dic)-118 1599 y(p)r(oin)n(t)23 b(\(e.g.,)h(for)f FN(q)j FO(=)d(1,)h(zero)f(is)f (a)h(p)r(erio)r(dic)f(p)r(oin)n(t)h(of)g(the)h(second)f(order\);)h(in) -118 1698 y(this)f(case,)h(the)g(corresp)r(onding)d(one-parameter)f (family)h(of)j(represen)n(tations)-118 1798 y(degenerates)i(in)n(to)g (a)i(single)d(irreducible)f(represen)n(tation.)6 1900 y(Relation)i(\(2.2\))724 2088 y FN(xx)818 2054 y FM(\003)880 2088 y FO(=)d FN(\013x)1068 2054 y FM(\003)1107 2088 y FN(x)p FO(\()p FN(I)j FP(\000)18 b FN(x)1378 2054 y FM(\003)1417 2088 y FN(x)p FO(\))-118 2276 y(has)k(a)g(similar)17 b(set)22 b(of)h(\014nite-dimensional)17 b(represen)n(tations,)j(but)j (the)g(family)-118 2376 y(corresp)r(onding)k(to)j(the)h(\014xed)f(p)r (oin)n(t)f FN(\025)f FO(=)f(0)i(degenerates)g(in)n(to)g(the)h(unique) -118 2475 y(trivial)16 b(represen)n(tation,)j(and)h(there)f(is)g(no)g (degeneration)f(of)i(represen)n(tations)-118 2575 y(corresp)r(onding)f (to)k(other)f(cycles.)33 b(The)23 b(corresp)r(onding)d(critical)f(v)-5 b(alues)21 b(of)h FN(\013)-118 2675 y FO(are:)39 b FN(\013)26 b FO(=)f(3)k(\(t)n(w)n(o-dimensional)24 b(represen)n(tations)i (arise\),)i FN(\013)e FO(=)f FN(\013)1995 2645 y FM(\003)2059 2675 y FP(\031)g FO(3)p FN(:)p FO(569)-118 2774 y(\(there)34 b(are)g(represen)n(tations)d(with)j(dimensions)d(of)j(an)n(y)g(p)r(o)n (w)n(er)f(of)h(2,)i(and)-118 2874 y(no)23 b(others\),)i FN(\013)e FP(\031)g FO(3)p FN(:)p FO(8)g(\(there)h(is)f(a)g (three-dimensional)c(represen)n(tation,)j(and)-118 2974 y(th)n(us)28 b(represen)n(tations)c(with)j(an)n(y)g(dimensions\).)-118 3112 y FB(Example)k(12.)42 b FO(\(Con)n(tin)n(uous)31 b(fractions.)50 b(Finite-dimensional)27 b(represen)n(ta-)-118 3211 y(tions\).)76 b(Consider)39 b(op)r(erator)g(relations)f(whic)n(h)h (arise)g(from)h(the)h(M\177)-42 b(obius)-118 3311 y(mapping)547 3499 y FN(xx)641 3465 y FM(\003)703 3499 y FO(=)23 b(\()p FN(ax)914 3465 y FM(\003)952 3499 y FN(x)c FO(+)f FN(c)p FO(\)\()p FN(bx)1284 3465 y FM(\003)1323 3499 y FN(x)h FO(+)f FN(d)p FO(\))1547 3465 y FM(\000)p FK(1)1637 3499 y FN(;)463 3623 y(a;)c(b;)g(c;)g(d)23 b FP(2)g FI(R)p FN(;)103 b(b)23 b(>)f FO(0)p FN(;)97 b(ad)18 b FP(\000)g FN(bc)23 b FP(6)p FO(=)g(0)p FN(:)368 b FO(\(2.11\))-118 3811 y(According)37 b(to)j(Theorem)d(16,)42 b(in)d(order)f(to)h (describ)r(e)f(\014nite-dimensional)-118 3911 y(represen)n(tations)28 b(of)k(relation)c(\(12\),)33 b(one)e(needs)g(to)h(\014nd)g(cycles)e(of) h(the)h(dy-)p eop %%Page: 96 100 96 99 bop -118 -137 a FO(96)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(namical)g(system)i(generated)h(b)n(y)g(the)h (mapping)836 301 y FN(F)12 b FO(\()p FN(z)t FO(\))23 b(=)1128 245 y FN(az)f FO(+)c FN(c)p 1128 282 224 4 v 1128 358 a(bz)k FO(+)c FN(d)1362 301 y(:)741 b FO(\(2.12\))-118 511 y(T)-7 b(o)27 b(do)h(that,)g(w)n(e)f(follo)n(w)e([252)o(].)37 b(First,)27 b(consider)f(\014xed)i(p)r(oin)n(ts)e(of)i(the)g(map-)-118 610 y(ping.)58 b(If)35 b(\()p FN(d)24 b FP(\000)f FN(a)p FO(\))474 580 y FK(2)535 610 y FO(+)g(4)p FN(bc)35 b FO(=)g(0,)h(then)g(there)f(exists)e(a)i(single)d(\014xed)j(p)r(oin)n(t) -118 710 y FN(\030)-82 722 y FK(1)-21 710 y FO(=)22 b(\()p FN(a)d FP(\000)f FN(d)p FO(\))p FN(=)p FO(2)p FN(b)p FO(;)27 b(otherwise,)e(there)j(are)e(t)n(w)n(o)h(\014xed)h(p)r(oin)n (ts,)e FN(\030)1898 722 y FK(1)1936 710 y FO(,)i FN(\030)2023 722 y FK(2)2060 710 y FO(.)6 809 y(If)34 b(there)e(exists)f(a)i(single) d(stationary)g(p)r(oin)n(t)i FN(\030)1515 821 y FK(1)1553 809 y FO(,)i(then)f FN(F)45 b FO(is)31 b(conjugate)-118 909 y(to)i(the)h(shift)f(in)g(the)h(complex)d(plane,)i FN(F)45 b FO(=)32 b FN(\036)1409 879 y FM(\000)p FK(1)1521 909 y FP(\016)22 b FN(T)34 b FP(\016)22 b FN(\036)p FO(,)35 b(where)e FN(\036)p FO(\()p FN(z)t FO(\))g(=)-118 1009 y(1)p FN(=)p FO(\()p FN(z)22 b FP(\000)c FN(\030)178 1021 y FK(1)216 1009 y FO(\),)30 b FN(T)12 b FO(\()p FN(w)r FO(\))25 b(=)f FN(w)e FO(+)d FN(l)r FO(,)29 b FN(l)d FO(=)f(2)p FN(b=)p FO(\()p FN(a)18 b FO(+)h FN(d)p FO(\).)40 b(W)-7 b(e)30 b(see)e(that,)h(in)f(this)g(case,)-118 1108 y FN(\030)-82 1120 y FK(1)-13 1108 y FO(is)i(a)h(unique)g(p)r (erio)r(dic)e(p)r(oin)n(t.)47 b(Represen)n(tations)29 b(exist)h(only)g(if)h FN(\030)2113 1120 y FK(1)2180 1108 y FP(\025)e FO(0;)-118 1208 y(for)39 b FN(a)44 b FO(=)f FN(d)p FO(,)g FN(X)50 b FO(=)43 b(0)c(is)g(a)g(single)f(solution)f(of) 46 b(\(12\),)d(for)c FN(\030)1883 1220 y FK(1)1964 1208 y FN(>)k FO(0)d(there)-118 1308 y(exists)g(a)g(one-parameter)e(family)g (of)j(one-dimensional)36 b(represen)n(tations,)-118 1407 y FN(X)29 b FO(=)23 b FN(\013)121 1345 y FP(p)p 190 1345 74 4 v 62 x FN(\030)226 1419 y FK(1)264 1407 y FO(,)28 b FP(j)p FN(\013)p FP(j)23 b FO(=)g(1.)6 1507 y(If)29 b(there)e(are)g(t)n(w)n(o)g(stationary)f(p)r(oin)n(ts,)h FN(\030)1303 1519 y FK(1)1340 1507 y FO(,)h FN(\030)1427 1519 y FK(2)1465 1507 y FO(,)g(one)g(can)f(construct)g(one-)-118 1606 y(dimensional)i(represen)n(tations)h(quite)j(similarly)-7 b(,)29 b(if)j(one)h(or)g(b)r(oth)g(of)h(these)-118 1706 y(p)r(oin)n(ts)26 b(are)h(non-negativ)n(e.)6 1806 y(If)37 b(there)g(are)e(no)h(stationary)e(p)r(oin)n(ts)h(\(they)i(are)f (conjugate)f(complex-)-118 1905 y(v)-5 b(alued\),)27 b(write)847 2124 y FN(\036)p FO(\()p FN(z)t FO(\))c(=)1124 2067 y FN(z)e FP(\000)d FN(\030)1303 2079 y FK(1)p 1124 2104 218 4 v 1124 2180 a FN(z)j FP(\000)d FN(\030)1303 2192 y FK(2)1351 2124 y FO(;)-118 2348 y(then)37 b FN(F)49 b FO(=)38 b FN(\036)334 2318 y FM(\000)p FK(1)448 2348 y FP(\016)23 b FN(T)36 b FP(\016)23 b FN(\036)p FO(,)40 b(where)c FN(T)12 b(w)39 b FO(=)e FN(q)s(w)i FO(with)d FN(q)41 b FO(=)c(\()p FN(a)25 b FP(\000)f FN(\030)2027 2360 y FK(1)2064 2348 y FN(b)p FO(\))p FN(=)p FO(\()p FN(a)g FP(\000)-118 2447 y FN(\030)-82 2459 y FK(2)-45 2447 y FN(b)p FO(\))38 b(=)e(\()p FN(d)25 b FP(\000)f FN(\030)387 2459 y FK(2)424 2447 y FN(b)p FO(\))p FN(=)p FO(\()p FN(d)g FP(\000)g FN(\030)758 2459 y FK(1)796 2447 y FN(b)p FO(\).)62 b(No)n(w)36 b(it)f(is)g(easy)g(to)h(see)g(that) h(either)e FN(q)k FO(is)-118 2547 y(the)32 b FN(n)p FO(-th)h(ro)r(ot)e (of)h(unit)n(y)f(for)h(some)f FN(n)p FO(,)i(and)f(all)e(p)r(oin)n(ts)h (are)g(p)r(erio)r(dic)f(with)-118 2647 y(p)r(erio)r(d)24 b FN(n)p FO(,)i(or)f(there)h(are)e(no)i(p)r(erio)r(dic)d(p)r(oin)n(ts)i (at)g(all.)34 b(In)26 b(the)g(p)r(erio)r(dic)d(case,)-118 2746 y(represen)n(tations)36 b(corresp)r(ond)i(to)h(orbits)f(with)h (all)e(non-negativ)n(e)g(p)r(oin)n(ts,)-118 2846 y(and)26 b(these)g(represen)n(tations)d(are)i(constructed)h(according)d(to)j (Theorem)e(16.)6 2946 y(Notice)37 b(that)g(in)g(this)g(example)e(the)j (rule)e(ab)r(out)h(dimensions)d(of)j(rep-)-118 3045 y(resen)n(tations) 30 b(established)f(b)n(y)j(the)h(Shark)n(o)n(vsky)c(theorem)i(do)r(es)h (not)g(hold:)-118 3145 y(there)19 b(can)h(b)r(e)g(only)e (one-dimensional)d(and)k FN(n)p FO(-dimensional)c(irreducible)h(rep-) -118 3244 y(resen)n(tations.)-118 3459 y FQ(2.1.3)94 b(In\014nite-dimensional)28 b(represen)m(tations)-118 3612 y FO(In)22 b(order)e(to)i(describ)r(e)e(the)i(general)d(case,)j (recall)d(that,)k(according)c(to)i(Prop)r(o-)-118 3712 y(sition)26 b(29,)h(the)h(op)r(erator)e FN(U)37 b FO(is)26 b(a)i(cen)n(tered)f(partial)e(isometry)-7 b(.)35 b(W)-7 b(e)28 b(will)d(see)-118 3811 y(that)32 b(in)f(the)h(irreducible)27 b(represen)n(tation)i(with)i(non-unitary)f FN(U)9 b FO(,)32 b(the)g(pair)-118 3911 y FN(U)9 b FO(,)24 b FN(U)61 3881 y FM(\003)122 3911 y FO(is)e(again)f(irreducible;)h(since)g(all)f (irreducible)e(partial)i(isometries)e(can)p eop %%Page: 97 101 97 100 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(97)-118 96 y(easily)23 b(b)r(e)j(describ)r(ed,)f(this)g(enables)g(us)h(to)f(giv)n(e)f(a)i (complete)d(description)h(of)-118 196 y(all)h(irreducible)f(represen)n (tations)h(in)i(the)h(non-unitary)d(case.)6 296 y(In)h(the)g(unitary)e (case,)h(t)n(w)n(o)g(classes)e(of)i(represen)n(tations)e(can)i(arise:) 33 b(rep-)-118 395 y(resen)n(tations)18 b(in)j(whic)n(h)f FN(U)29 b FO(acts)21 b(as)f(a)h(shift)g(in)f FN(l)1362 407 y FK(2)1420 395 y FO(\(in)h(this)f(case)g(the)h(sp)r(ectrum)-118 495 y(of)34 b FN(C)48 465 y FK(2)120 495 y FO(lies)f(on)h(a)g(single)e (orbit\),)j(and)f(represen)n(tations)d(corresp)r(onding)h(to)-118 595 y(non-trivial)d(ergo)r(dic)i(measures.)50 b(The)33 b(latter)f(class)f(of)i(represen)n(tations)d(is)-118 694 y(to)r(o)f(complicated)e(to)j(b)r(e)g(classi\014ed)d(up)j(to)g(a)f (unitary)g(equiv)-5 b(alence)27 b(for)i(the)-118 794 y(momen)n(t;)c(ho)n(w)n(ev)n(er,)h(non-trivial)d(ergo)r(dic)i(measures) g(can)i(arise)e(only)h(if)h(the)-118 894 y(corresp)r(onding)18 b(dynamical)f(system)j(do)r(es)g(not)h(ha)n(v)n(e)f(a)g(measurable)e (section.)-118 1044 y FQ(1.)36 b FO(W)-7 b(e)28 b(start)f(with)g(the)h (description)d(of)j(cen)n(tered)f(partial)d(isometries.)-118 1210 y FQ(Theorem)30 b(18.)41 b FB(A)n(ny)e(irr)l(e)l(ducible)i(c)l (enter)l(e)l(d)f(p)l(artial)h(isometry)f(is)h(one)f(of)-118 1310 y(the)30 b(fol)t(lowing:)-40 1493 y FO(\(i\))41 b FB(a)31 b(one-dimensional)g(unitary)f(op)l(er)l(ator)h FN(U)g FO(=)23 b FN(\013)p FB(,)31 b FP(j)p FN(\013)p FP(j)23 b FO(=)g(1;)-63 1660 y(\(ii\))40 b FB(a)31 b(unilater)l(al)f (shift)g(op)l(er)l(ator)h(in)f FN(l)1170 1672 y FK(2)1207 1660 y FB(,)g FN(U)9 b(e)1367 1672 y FL(k)1430 1660 y FO(=)23 b FN(e)1557 1672 y FL(k)q FK(+1)1682 1660 y FO(;)-86 1827 y(\(iii\))39 b FB(an)c(adjoint)h(to)f(the)g(unilater)l(al)g(shift) h(op)l(er)l(ator)f(in)g FN(l)1776 1839 y FK(2)1813 1827 y FB(,)i FN(U)9 b(e)1980 1839 y FL(k)2052 1827 y FO(=)32 b FN(e)2188 1839 y FL(k)q FM(\000)p FK(1)2313 1827 y FB(,)89 1926 y FN(k)26 b(>)d FO(1)p FB(,)30 b FN(U)9 b(e)448 1938 y FK(1)507 1926 y FO(=)23 b(0;)-84 2093 y(\(iv\))41 b FB(a)35 b(\014nite-dimensional)h(op)l(er)l(ator)g(in)e FI(C)1340 2063 y FL(n)1426 2093 y FB(of)h(the)g(form)g FN(U)9 b(e)1981 2105 y FL(k)2053 2093 y FO(=)32 b FN(e)2189 2105 y FL(k)q FK(+1)2313 2093 y FB(,)89 2193 y FN(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FB(,)30 b FN(U)9 b(e)825 2205 y FL(n)893 2193 y FO(=)22 b(0)p FB(,)30 b(for)h(some)f FN(n)23 b FO(=)f(1)p FB(,)30 b FO(2)p FB(,)g FN(:)14 b(:)g(:)27 b FB(.)-118 2376 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)28 b(start)f(with)g(a)g(simple)e (fact.)-118 2527 y FQ(Prop)s(osition)30 b(36.)41 b FB(The)23 b(op)l(er)l(ators)g FN(U)1143 2497 y FL(k)1184 2527 y FO(\()p FN(U)1282 2497 y FM(\003)1320 2527 y FO(\))1352 2497 y FL(k)1393 2527 y FB(,)h FO(\()p FN(U)1540 2497 y FM(\003)1578 2527 y FO(\))1610 2497 y FL(l)1636 2527 y FN(U)1702 2497 y FL(l)1727 2527 y FB(,)h FN(k)s FB(,)f FN(l)g FO(=)f(1)p FB(,)g FO(2)p FB(,)g FN(:)14 b(:)g(:)27 b FB(,)-118 2626 y(ar)l(e)j(pr)l(oje)l(ctions.)-118 2794 y(Pr)l(o)l(of.)43 b FO(Since)29 b FN(U)9 b(U)493 2764 y FM(\003)531 2794 y FN(U)35 b FO(=)27 b FN(U)38 b FO(and)30 b(the)g(op)r(erators)e FN(U)1555 2764 y FM(\003)1593 2794 y FN(U)39 b FO(and)30 b FN(U)1919 2764 y FL(k)q FM(\000)p FK(1)2044 2794 y FO(\()p FN(U)2142 2764 y FM(\003)2180 2794 y FO(\))2212 2764 y FL(k)q FM(\000)p FK(1)-118 2894 y FO(comm)n(ute,)25 b(w)n(e)j(ha)n(v)n(e)e(b)n(y)h(induction)f(that)83 3077 y FN(U)149 3042 y FL(k)190 3077 y FO(\()p FN(U)288 3042 y FM(\003)326 3077 y FO(\))358 3042 y FL(k)399 3077 y FN(U)465 3042 y FL(k)506 3077 y FO(\()p FN(U)604 3042 y FM(\003)642 3077 y FO(\))674 3042 y FL(k)738 3077 y FO(=)d FN(U)9 b(U)958 3042 y FL(k)q FM(\000)p FK(1)1083 3077 y FO(\()p FN(U)1181 3042 y FM(\003)1219 3077 y FO(\))1251 3042 y FL(k)q FM(\000)p FK(1)1378 3077 y FN(U)1444 3042 y FM(\003)1481 3077 y FN(U)g(U)1613 3042 y FL(k)q FM(\000)p FK(1)1739 3077 y FO(\()p FN(U)1837 3042 y FM(\003)1875 3077 y FO(\))1907 3042 y FL(k)q FM(\000)p FK(1)2033 3077 y FN(U)2099 3042 y FM(\003)738 3215 y FO(=)23 b FN(U)9 b(U)958 3180 y FM(\003)996 3215 y FN(U)g(U)1128 3180 y FL(k)q FM(\000)p FK(1)1253 3215 y FO(\()p FN(U)1351 3180 y FM(\003)1389 3215 y FO(\))1421 3180 y FL(k)q FM(\000)p FK(1)1547 3215 y FN(U)1613 3180 y FL(k)q FM(\000)p FK(1)1739 3215 y FO(\()p FN(U)1837 3180 y FM(\003)1875 3215 y FO(\))1907 3180 y FL(k)q FM(\000)p FK(1)2033 3215 y FN(U)2099 3180 y FM(\003)738 3352 y FO(=)23 b FN(U)9 b(U)958 3318 y FL(k)q FM(\000)p FK(1)1083 3352 y FO(\()p FN(U)1181 3318 y FM(\003)1219 3352 y FO(\))1251 3318 y FL(k)q FM(\000)p FK(1)1378 3352 y FN(U)1444 3318 y FM(\003)1505 3352 y FO(=)22 b FN(U)1658 3318 y FL(k)1699 3352 y FO(\()p FN(U)1797 3318 y FM(\003)1835 3352 y FO(\))1867 3318 y FL(k)1908 3352 y FN(:)-118 3544 y FO(Similarly)-7 b(,)20 b(since)k FN(U)516 3514 y FM(\003)554 3544 y FN(U)9 b(U)686 3514 y FM(\003)746 3544 y FO(=)23 b FN(U)900 3514 y FM(\003)963 3544 y FO(and)i FN(U)9 b(U)1254 3514 y FM(\003)1316 3544 y FO(and)25 b(\()p FN(U)1573 3514 y FM(\003)1611 3544 y FO(\))1643 3514 y FL(k)q FM(\000)p FK(1)1769 3544 y FN(U)1835 3514 y FL(k)q FM(\000)p FK(1)1986 3544 y FO(comm)n(ute,)-118 3644 y(w)n(e)i(get)g(that)h(\()p FN(U)420 3614 y FM(\003)459 3644 y FO(\))491 3614 y FL(k)532 3644 y FN(U)598 3614 y FL(k)666 3644 y FO(is)e(a)i(pro)5 b(jection.)p 2278 3644 4 57 v 2282 3591 50 4 v 2282 3644 V 2331 3644 4 57 v 6 3811 a(Denote)39 b(these)f(pro)5 b(jections)36 b(b)n(y)h FN(P)1141 3823 y FL(k)1223 3811 y FO(=)j(\()p FN(U)1426 3781 y FM(\003)1464 3811 y FO(\))1496 3781 y FL(k)1538 3811 y FN(U)1604 3781 y FL(k)1644 3811 y FO(,)h FN(P)1761 3823 y FM(\000)p FL(k)1895 3811 y FO(=)f FN(U)2066 3781 y FL(k)2106 3811 y FO(\()p FN(U)2204 3781 y FM(\003)2242 3811 y FO(\))2274 3781 y FL(k)2316 3811 y FO(,)-118 3911 y FN(k)26 b FO(=)c(1,)28 b(2,)f FN(:)14 b(:)g(:)27 b FO(;)h FN(P)451 3923 y FK(0)512 3911 y FO(=)22 b FN(I)7 b FO(.)p eop %%Page: 98 102 98 101 bop -118 -137 a FO(98)485 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(Prop)s(osition)30 b(37.)41 b FB(F)-6 b(or)30 b(al)t(l)g FN(k)c FP(2)e FI(Z)o FB(,)g(the)30 b(fol)t(lowing)i(r)l(elations)f(hold)841 282 y FN(P)894 294 y FL(k)936 282 y FN(U)g FO(=)23 b FN(U)9 b(P)1231 294 y FL(k)q FK(+1)1356 282 y FN(:)747 b FO(\(2.13\))-118 469 y FB(Pr)l(o)l(of.)43 b FO(Indeed,)28 b(for)f FN(k)f(>)d FO(0,)k(w)n(e)g(ha)n(v)n(e)106 654 y FN(P)159 666 y FL(k)200 654 y FN(U)32 b FO(=)22 b(\()p FN(U)474 620 y FM(\003)513 654 y FO(\))545 620 y FL(k)586 654 y FN(U)652 620 y FL(k)692 654 y FN(U)32 b FO(=)23 b(\()p FN(U)967 620 y FM(\003)1005 654 y FO(\))1037 620 y FL(k)1078 654 y FN(U)1144 620 y FL(k)1185 654 y FN(U)9 b(U)1317 620 y FM(\003)1354 654 y FN(U)32 b FO(=)23 b FN(U)9 b(U)1663 620 y FM(\003)1700 654 y FO(\()p FN(U)1798 620 y FM(\003)1837 654 y FO(\))1869 620 y FL(k)1910 654 y FN(U)1976 620 y FL(k)2016 654 y FN(U)289 792 y FO(=)22 b FN(U)9 b FO(\()p FN(U)540 758 y FM(\003)578 792 y FO(\))610 758 y FL(k)q FK(+1)736 792 y FN(U)802 758 y FL(k)q FK(+1)949 792 y FO(=)23 b FN(U)9 b(P)1156 804 y FL(k)q FK(+1)1281 792 y FN(;)54 930 y(P)107 942 y FM(\000)p FL(k)200 930 y FN(U)32 b FO(=)22 b FN(U)442 896 y FL(k)483 930 y FO(\()p FN(U)581 896 y FM(\003)619 930 y FO(\))651 896 y FL(k)692 930 y FN(U)32 b FO(=)23 b FN(U)9 b(U)1001 896 y FL(k)q FM(\000)p FK(1)1126 930 y FO(\()p FN(U)1224 896 y FM(\003)1262 930 y FO(\))1294 896 y FL(k)q FM(\000)p FK(1)1420 930 y FN(U)1486 896 y FM(\003)1524 930 y FN(U)289 1068 y FO(=)22 b FN(U)9 b(U)508 1033 y FM(\003)546 1068 y FN(U)g(U)678 1033 y FL(k)q FM(\000)p FK(1)803 1068 y FO(\()p FN(U)901 1033 y FM(\003)940 1068 y FO(\))972 1033 y FL(k)q FM(\000)p FK(1)1121 1068 y FO(=)22 b FN(U)9 b(U)1340 1033 y FL(k)q FM(\000)p FK(1)1466 1068 y FO(\()p FN(U)1564 1033 y FM(\003)1602 1068 y FO(\))1634 1033 y FL(k)q FM(\000)p FK(1)1783 1068 y FO(=)23 b FN(U)9 b(P)1990 1080 y FM(\000)p FL(k)q FK(+1)-118 1253 y FO(and)40 b FN(P)109 1265 y FM(\000)p FK(1)199 1253 y FN(U)53 b FO(=)44 b(\()p FN(U)9 b(U)582 1223 y FM(\003)620 1253 y FO(\))p FN(U)54 b FO(=)44 b FN(U)9 b(I)51 b FO(=)44 b FN(U)9 b(P)1253 1265 y FK(0)1290 1253 y FO(,)44 b FN(P)1410 1265 y FK(0)1448 1253 y FN(U)53 b FO(=)44 b FN(I)7 b(U)53 b FO(=)45 b FN(U)9 b FO(\()p FN(U)2094 1223 y FM(\003)2131 1253 y FN(U)g FO(\))45 b(=)-118 1353 y FN(U)9 b(P)1 1365 y FK(1)38 1353 y FO(.)p 2278 1353 4 57 v 2282 1300 50 4 v 2282 1353 V 2331 1353 4 57 v 6 1530 a(The)24 b(case)f(where)f FN(U)33 b FO(is)22 b(unitary)g(is)g(trivial.)32 b(Supp)r(ose)24 b(that)g(the)f(op)r (erator)-118 1630 y FN(U)-52 1600 y FM(\003)17 1630 y FO(has)30 b(a)h(non-trivial)26 b(k)n(ernel)j(\(the)j(case)e(of)h(a)f (non-trivial)d(k)n(ernel)i(of)i FN(U)40 b FO(is)-118 1730 y(similar\).)31 b(Let)22 b FN(f)32 b FP(2)23 b FO(k)n(er)13 b FN(U)703 1700 y FM(\003)741 1730 y FO(.)35 b(F)-7 b(or)22 b(ev)n(ery)f FN(k)26 b FO(=)d(1,)g(2,)f FN(:)14 b(:)g(:)28 b FO(,)c(consider)c(the)j(v)n(ector)-118 1829 y(\()p FN(U)-20 1799 y FM(\003)18 1829 y FO(\))50 1799 y FL(k)91 1829 y FN(U)157 1799 y FL(k)198 1829 y FN(f)9 b FO(.)36 b(The)28 b(follo)n(wing)c(situations)h(ma)n(y)h(o)r(ccur:)6 1930 y(a\))g(\()p FN(U)204 1900 y FM(\003)242 1930 y FO(\))274 1900 y FL(k)315 1930 y FN(U)381 1900 y FL(k)422 1930 y FN(f)31 b FO(=)23 b FN(f)34 b FO(for)25 b(all)e FN(k)j FO(=)d(1,)i(2,)h FN(:)14 b(:)g(:)27 b FO(.)36 b(Then)26 b(the)g(v)n(ector)e FN(f)2060 1942 y FK(0)2120 1930 y FO(=)f FN(f)34 b FO(is)-118 2030 y(a)27 b(join)n(t)g(eigen)n(v)n (ector)d(of)k(a)f(comm)n(uting)d(family)h(\()p FN(P)1510 2042 y FL(k)1551 2030 y FO(\).)6 2131 y(b\))k(for)e(some)f FN(k)f(>)e FO(0,)k(the)h(follo)n(wing)c(conditions)h(hold:)469 2317 y(\()p FN(U)567 2282 y FM(\003)605 2317 y FO(\))637 2282 y FL(l)663 2317 y FN(U)729 2282 y FL(l)754 2317 y FN(f)32 b FO(=)23 b FN(f)t(;)180 b(l)24 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(k)21 b FP(\000)d FO(1)p FN(;)838 2454 y FO(\()p FN(U)936 2420 y FM(\003)974 2454 y FO(\))1006 2420 y FL(k)1047 2454 y FN(U)1113 2420 y FL(k)1154 2454 y FN(f)32 b FP(6)p FO(=)22 b FN(f)t(:)-118 2649 y FO(Put)31 b FN(f)89 2661 y FK(0)156 2649 y FO(=)e FN(f)g FP(\000)21 b FO(\()p FN(U)504 2619 y FM(\003)542 2649 y FO(\))574 2619 y FL(k)615 2649 y FN(U)681 2619 y FL(k)722 2649 y FN(f)38 b FP(6)p FO(=)29 b(0.)48 b(Then)32 b(\()p FN(U)1327 2619 y FM(\003)1365 2649 y FO(\))1397 2619 y FL(k)1438 2649 y FN(U)1504 2619 y FL(k)1545 2649 y FN(f)1586 2661 y FK(0)1652 2649 y FO(=)d(0,)j(whic)n(h)e(implies)-118 2749 y(that)j FN(U)133 2718 y FL(k)174 2749 y FN(f)215 2761 y FK(0)284 2749 y FO(=)e(0,)67 b FN(U)578 2718 y FL(k)q FK(+1)703 2749 y FN(f)744 2761 y FK(0)813 2749 y FO(=)31 b(0,)j(etc.,)h(and)d FN(f)1402 2761 y FK(0)1472 2749 y FO(is)g(a)h(join)n(t)f(eigen)n(v)n(ector)e(of)-118 2848 y(the)e(comm)n(uting)c(family)h(\()p FN(P)795 2860 y FL(k)836 2848 y FO(\).)6 2949 y(In)h(b)r(oth)h(the)f(cases,)f(the)h (relation)d(\(2.13\))i(implies)d(that)k FN(f)1855 2961 y FK(0)1892 2949 y FO(,)h FN(U)9 b(f)2049 2961 y FK(0)2085 2949 y FO(,)27 b FN(U)2201 2919 y FK(2)2238 2949 y FN(f)2279 2961 y FK(0)2316 2949 y FO(,)-118 3049 y FN(:)14 b(:)g(:)27 b FO(,)j(are)e(orthogonal)e(join)n(t)j(eigenspaces)d(of)k(the)f(family) d(\()p FN(P)1833 3061 y FL(k)1875 3049 y FO(\))j(and)g(can)g(b)r(e)-118 3148 y(c)n(hosen)19 b(to)h(b)r(e)g(a)g(basis)e(of)i(the)g(space.)34 b(The)20 b(rest)f(of)h(the)g(pro)r(of)g(is)e(ob)n(vious.)p 2278 3148 V 2282 3096 50 4 v 2282 3148 V 2331 3148 4 57 v -118 3326 a FQ(2.)40 b FO(T)-7 b(o)29 b(apply)e(this)h(theorem)g (to)h(the)g(description)d(of)j(irreducible)c(solutions)-118 3426 y(of)34 b(\(2.1\),)27 b(w)n(e)h(need)f(the)h(follo)n(wing)c(fact.) -118 3596 y FQ(Theorem)30 b(19.)41 b FB(L)l(et)35 b(the)h(p)l(air)h FO(\()p FN(X)r(;)14 b(X)1137 3566 y FM(\003)1175 3596 y FO(\))36 b FB(satisfying)44 b FO(\(2.1\))36 b FB(b)l(e)g(irr)l(e)l (ducible.)-118 3696 y(If)d(one)h(of)g(the)f(op)l(er)l(ators)h FN(X)7 b FB(,)34 b FN(X)950 3666 y FM(\003)1020 3696 y FB(has)g(a)g(non-zer)l(o)e(kernel,)j(then)e(the)g(p)l(air)-118 3795 y FO(\()p FN(U;)14 b(U)74 3765 y FM(\003)111 3795 y FO(\))31 b FB(is)g(irr)l(e)l(ducible,)h(i.e.,)g(any)f(b)l(ounde)l(d)g (op)l(er)l(ator)g(c)l(ommuting)f(with)h FN(U)-118 3895 y FB(and)f FN(U)109 3865 y FM(\003)177 3895 y FB(is)g(a)g(multiple)g (of)h(the)f(identity.)p eop %%Page: 99 103 99 102 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)895 b FO(99)-118 96 y FB(Pr)l(o)l(of.)43 b FO(First,)28 b(consider)e(the)j(case)f(k)n(er)13 b FN(U)33 b FP(6)p FO(=)24 b(0.)40 b(T)-7 b(ak)n(e)27 b(a)h(v)n(ector)f FN(e)1983 108 y FK(0)2045 96 y FP(2)e FO(k)n(er)13 b FN(U)c FO(,)-118 196 y(and)25 b(consider)e(v)n(ectors)g FN(e)680 208 y FL(k)744 196 y FO(=)g(\()p FN(U)930 166 y FM(\003)968 196 y FO(\))1000 166 y FL(k)1041 196 y FO(,)j FN(k)g FO(=)c(1,)k(2,)e FN(:)14 b(:)g(:)28 b FO(.)36 b(Since)24 b(k)n(er)13 b FN(U)32 b FO(=)22 b(k)n(er)13 b FN(C)6 b FO(,)-118 296 y(eac)n(h)27 b FN(e)108 308 y FL(k)176 296 y FO(is)f(an)i(eigen)n(v)n(ector)c(of)k FN(C)966 266 y FK(2)1031 296 y FO(with)f(the)h(eigen)n(v)-5 b(alue)25 b FN(F)1825 266 y FL(k)1865 296 y FO(\(0\).)6 398 y(W)-7 b(e)35 b(sho)n(w)d(that)j(these)e(v)n(ectors)g(form)f(a)h (basis)f(in)i(the)g(space.)54 b(Indeed,)-118 498 y(the)35 b(linear)d(span)493 477 y(~)471 498 y FN(H)42 b FO(of)34 b(these)h(v)n(ectors)e(is)h(in)n(v)-5 b(arian)n(t)31 b(with)k(resp)r(ect)f(to)h FN(U)2301 468 y FM(\003)-118 598 y FO(and)27 b FN(C)108 567 y FK(2)146 598 y FO(.)6 700 y(T)-7 b(ak)n(e)29 b(a)f(v)n(ector)g FN(e)569 712 y FL(k)635 700 y FP(6)p FO(=)d(0)k(with)f(some)g FN(k)g(>)d FO(0,)k(and)g(assume)f(that)k(~)-45 b FN(e)2123 712 y FL(k)q FM(\000)p FK(1)2274 700 y FO(=)-118 800 y FN(U)9 b(e)-13 812 y FL(k)50 800 y FO(=)23 b FN(U)9 b(U)270 770 y FM(\003)308 800 y FN(e)347 812 y FL(k)q FM(\000)p FK(1)500 800 y FO(is)27 b(not)h(in)850 779 y(~)829 800 y FN(H)6 b FO(.)38 b(Since)27 b FN(U)1248 770 y FM(\003)1286 800 y FN(U)9 b(U)1418 770 y FM(\003)1479 800 y FO(=)23 b FN(U)1633 770 y FM(\003)1670 800 y FO(,)28 b(w)n(e)g(get)f FN(U)2048 770 y FM(\003)2089 800 y FO(~)-45 b FN(e)2125 812 y FL(k)q FM(\000)p FK(1)2274 800 y FO(=)-118 899 y FN(e)-79 911 y FL(k)-39 899 y FO(;)23 b(therefore,)f(\()s(~)-45 b FN(e)445 911 y FL(k)q FM(\000)p FK(1)575 899 y FP(\000)t FN(e)683 911 y FL(k)q FM(\000)p FK(1)808 899 y FO(\))24 b FP(2)f FO(k)n(er)13 b FN(U)1133 869 y FM(\003)1194 899 y FO(=)22 b(k)n(er)13 b FN(F)f FO(\()p FN(C)1568 869 y FK(2)1606 899 y FO(\),)22 b(whic)n(h)e(is)f(the)i(image)-118 999 y(of)33 b(the)h(pro)5 b(jection)31 b FN(E)592 1014 y FL(C)644 997 y Fx(2)681 999 y FO(\()p FN(F)778 969 y FM(\000)p FK(1)867 999 y FO(\(0\)\).)55 b(On)33 b(the)g(other)g (hand,)i(\()s(~)-45 b FN(e)1907 1011 y FL(k)q FM(\000)p FK(1)2055 999 y FP(\000)22 b FN(e)2181 1011 y FL(k)q FM(\000)p FK(1)2306 999 y FO(\))-118 1099 y(b)r(elongs)31 b(to)h FN(E)352 1114 y FL(C)404 1097 y Fx(2)440 1099 y FO(\(\001\))p FN(H)7 b FO(,)34 b(where)e(\001)f(=)f FN(F)1211 1069 y FM(\000)p FK(1)1300 1099 y FO(\()p FN(F)1397 1069 y FL(k)1438 1099 y FO(\(0\)\).)52 b(If)32 b(the)h(latter)e(v)n (ector)-118 1198 y(is)24 b(non-zero,)g(this)h(w)n(ould)f(imply)e(that)k (b)r(oth)f FN(e)1361 1210 y FL(k)q FM(\000)p FK(1)1512 1198 y FO(and)k(~)-46 b FN(e)1710 1210 y FL(k)q FM(\000)p FK(1)1861 1198 y FO(b)r(elong)24 b(to)h(the)-118 1298 y(k)n(ernel)h(of)j FN(U)289 1268 y FM(\003)355 1298 y FO(and)f(that)h FN(e)737 1310 y FL(k)802 1298 y FO(=)24 b(0.)38 b(Therefore,)31 b(~)-45 b FN(e)1434 1310 y FL(k)q FM(\000)p FK(1)1584 1298 y FO(=)23 b FN(e)1711 1310 y FL(k)q FM(\000)p FK(1)1837 1298 y FO(,)29 b(and)2073 1277 y(~)2051 1298 y FN(H)i FO(=)24 b FN(H)7 b FO(.)-118 1398 y(The)28 b(op)r(erator)d FN(U)37 b FO(is)26 b(adjoin)n(t)h(to)g (the)h(unilateral)c(shift)j(in)g FN(l)1773 1410 y FK(2)1810 1398 y FO(.)6 1500 y(No)n(w)f(consider)f(the)i(case)e(k)n(er)13 b FN(U)1024 1470 y FM(\003)1085 1500 y FP(6)p FO(=)23 b(0.)36 b(F)-7 b(or)26 b(eac)n(h)f FN(k)h FP(\025)d FO(0,)j(in)n(tro)r (duce)f(the)-118 1600 y(subspace)i FN(H)298 1612 y FL(k)362 1600 y FO(=)22 b FN(E)510 1615 y FL(C)562 1598 y Fx(2)599 1600 y FO(\()p FN(F)696 1570 y FM(\000)p FL(k)789 1600 y FO(\(0\)\))p FN(H)7 b FO(.)37 b(The)28 b(equalities)597 1788 y FN(U)9 b(E)724 1803 y FL(C)776 1787 y Fx(2)812 1788 y FO(\(\001\))24 b(=)e FN(E)1117 1803 y FL(C)1169 1787 y Fx(2)1206 1788 y FO(\()p FN(F)1303 1754 y FM(\000)p FK(1)1392 1788 y FO(\(\001\)\))p FN(U;)558 1923 y(E)619 1938 y FL(C)671 1921 y Fx(2)708 1923 y FO(\(\001\))p FN(U)907 1889 y FM(\003)969 1923 y FO(=)g FN(U)1122 1889 y FM(\003)1160 1923 y FN(E)1221 1938 y FL(C)1273 1921 y Fx(2)1310 1923 y FO(\()p FN(F)1407 1889 y FM(\000)p FK(1)1496 1923 y FO(\(\001\)\))465 b(\(2.14\))-118 2112 y(for)20 b(all)f(measurable)e(\001)22 b(imply)c(that)j FN(U)9 b(H)1163 2124 y FL(k)1227 2112 y FO(=)22 b FN(H)1383 2124 y FL(k)q FK(+1)1508 2112 y FO(,)h FN(U)1620 2082 y FM(\003)1658 2112 y FN(H)1727 2124 y FL(k)q FK(+1)1875 2112 y FO(=)f FN(H)2031 2124 y FL(k)2072 2112 y FO(,)g FN(k)k FP(\025)d FO(1,)-118 2211 y(and)j(the)g(span)f(of)h(these)g (subspaces)f(is)g(an)h(in)n(v)-5 b(arian)n(t)23 b(subspace;)i(due)h(to) g(the)-118 2311 y(irreducibilit)n(y)-7 b(,)22 b(it)27 b(is)g(the)h(whole)e(of)h FN(H)7 b FO(.)6 2414 y(Note)24 b(that)f(k)n(er)13 b FN(U)569 2384 y FM(\003)630 2414 y FO(=)22 b(k)n(er)13 b FN(F)f FO(\()p FN(C)1004 2384 y FK(2)1042 2414 y FO(\))23 b(=)g FN(H)1254 2426 y FK(1)1291 2414 y FO(.)36 b(First)22 b(w)n(e)h(sho)n(w)f(that)h FN(H)2111 2426 y FK(1)2172 2414 y FO(is)f(an)-118 2513 y(eigenspace)j(of)j FN(C)448 2483 y FK(2)485 2513 y FO(,)g(i.e.,)f (there)h(exists)e(a)h(single)e(p)r(oin)n(t)i FN(\025)1700 2525 y FK(0)1766 2513 y FO(of)g(the)h(sp)r(ectrum)-118 2613 y(of)38 b FN(C)52 2583 y FK(2)127 2613 y FO(suc)n(h)f(that)h FN(F)12 b FO(\()p FN(\025)659 2625 y FK(0)697 2613 y FO(\))40 b(=)f(0.)67 b(Indeed,)41 b(tak)n(e)36 b(an)n(y)h(measurable)e (subset)-118 2713 y FN(\016)27 b FP(\032)e FN(F)101 2682 y FM(\000)p FK(1)190 2713 y FO(\(0\).)39 b(Consider)27 b(the)i(subspaces)e FN(H)1306 2682 y FL(\016)1299 2736 y(k)1367 2713 y FO(=)d FN(E)1517 2728 y FL(C)1569 2711 y Fx(2)1605 2713 y FO(\()p FN(F)1702 2682 y FM(\000)p FL(k)1795 2713 y FO(\()p FN(\016)s FO(\)\))p FN(H)33 b FP(\032)24 b FN(H)2191 2725 y FL(k)q FK(+1)2316 2713 y FO(,)-118 2812 y FN(k)36 b FP(\025)d FO(0.)54 b(Equalities)30 b(\(2.14\))j(imply)e(that)j(the)g(span)f(of)h(the)g(subspaces)e FN(H)2302 2782 y FL(\016)2295 2836 y(k)-118 2912 y FO(is)h(an)i(in)n(v) -5 b(arian)n(t)31 b(subspace;)38 b(the)d(irreducibilit)n(y)29 b(then)35 b(implies)c(that)k(there)-118 3011 y(exists)24 b(a)h(single)e(p)r(oin)n(t)i FN(\025)668 3023 y FK(0)705 3011 y FO(,)h FN(F)12 b FO(\()p FN(\025)899 3023 y FK(0)937 3011 y FO(\))24 b(=)e(0,)k(suc)n(h)f(that)h FN(E)1595 3026 y FL(C)1647 3010 y Fx(2)1683 3011 y FO(\()p FN(F)1780 2981 y FM(\000)p FK(1)1869 3011 y FO(\(0\))15 b FP(n)e FN(\025)2093 3023 y FK(0)2131 3011 y FO(\))23 b(=)g(0.)6 3114 y(Th)n(us,)32 b FN(H)313 3126 y FK(1)381 3114 y FO(is)e(an)g(eigenspace)f(of)i FN(C)1159 3084 y FK(2)1196 3114 y FO(.)47 b(No)n(w)30 b(w)n(e)h(sho)n(w)f(that)h FN(H)2045 3126 y FK(2)2113 3114 y FO(is)f(also)-118 3214 y(an)37 b(eigenspace.)62 b(Since)37 b FN(U)9 b(U)832 3184 y FM(\003)906 3214 y FO(comm)n(utes)35 b(with)h FN(F)12 b FO(\()p FN(C)1666 3184 y FK(2)1704 3214 y FO(\),)40 b FN(U)9 b(U)1931 3184 y FM(\003)2006 3214 y FO(maps)35 b FN(H)2301 3226 y FK(2)-118 3313 y FO(in)n(to)24 b(itself,)g(and)h(is) g(a)g(pro)5 b(jection)23 b(on)i(it.)35 b(Actually)-7 b(,)24 b(since)h(k)n(er)12 b FN(U)1940 3283 y FM(\003)2001 3313 y FO(=)23 b FN(H)2158 3325 y FK(1)2195 3313 y FO(,)j(w)n(e)-118 3413 y(conclude)k(that)i FN(U)9 b(U)543 3383 y FM(\003)612 3413 y FO(is)31 b(the)h(iden)n(tit)n(y)d(on)j FN(H)1346 3425 y FK(2)1383 3413 y FO(.)49 b(No)n(w)31 b(w)n(e)g(aply)f(similar)d (ar-)-118 3513 y(gumen)n(ts)f(as)h(ab)r(o)n(v)n(e.)36 b(T)-7 b(ak)n(e)27 b(an)n(y)g(measurable)e FN(\016)h FP(\032)d FN(F)1591 3482 y FM(\000)p FK(1)1680 3513 y FO(\()p FN(\025)1760 3525 y FK(0)1798 3513 y FO(\),)28 b(and)g(consider)-118 3612 y(the)i(subspaces)f FN(H)485 3582 y FL(\016)478 3636 y(k)549 3612 y FO(=)e FN(E)702 3627 y FL(C)754 3610 y Fx(2)790 3612 y FO(\()p FN(F)887 3582 y FM(\000)p FL(k)980 3612 y FO(\()p FN(\016)s FO(\)\))h FP(\032)f FN(H)1305 3624 y FL(k)q FK(+2)1430 3612 y FO(,)k FN(k)f FP(\025)d FO(0.)44 b(Relations)27 b(\(2.14\))-118 3712 y(and)34 b FN(U)9 b(U)182 3682 y FM(\003)255 3712 y Fr(\026)34 b FN(H)393 3724 y FK(2)465 3712 y FO(=)g FN(I)42 b FO(imply)32 b(that)i(the)h(span)g(of)f FN(H)1597 3682 y FL(\016)1590 3735 y(k)1633 3712 y FO(,)j FN(k)g FP(\025)e FO(0,)g(and)g FN(H)2211 3724 y FK(1)2283 3712 y FO(is)-118 3811 y(an)28 b(in)n(v)-5 b(arian)n(t)26 b(subspace;)i(therefore,)g(b)n(y)h(the)g(irreducibilit)n(y)-7 b(,)23 b FN(H)1933 3823 y FK(2)1999 3811 y FO(is)k(also)g(an)-118 3911 y(eigenspace)e(of)j FN(C)448 3881 y FK(2)485 3911 y FO(.)p eop %%Page: 100 104 100 103 bop -118 -137 a FO(100)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(Con)n(tin)n(uing)37 b(this)h(pro)r(cess,)j(w)n (e)e(conclude)e(that)j(all)c FN(H)1820 108 y FL(k)1861 96 y FO(,)42 b FN(k)j FP(\025)d FO(1,)g(are)-118 196 y(eigenspaces)21 b(of)i FN(C)472 166 y FK(2)509 196 y FO(.)36 b(Also,)23 b(in)f(the)i(irreducible)c(case,)j(all)e(these)i (eigenspaces)-118 296 y(are)j(one-dimensional;)d(therefore,)k FN(U)36 b FO(is)26 b(a)i(unilateral)c(shift)j(in)g FN(l)1954 308 y FK(2)1991 296 y FO(.)p 2278 296 4 57 v 2282 243 50 4 v 2282 296 V 2331 296 4 57 v -118 464 a FQ(3.)55 b FO(No)n(w)33 b(w)n(e)g(apply)g(the)h(results)e(on)i(cen)n(tered)f (partial)e(isometries)f(to)j(the)-118 563 y(description)d(of)j(op)r (erators)e(satisfying)h(\(2.1\))h(\(note)g(that)g(the)h(same)d(result) -118 663 y(can)38 b(b)r(e)h(obtained)f(using)f([273)o(]\).)70 b(Since)38 b(w)n(e)g(are)g(in)n(terested)f(in)h(in\014nite-)-118 763 y(dimensional)32 b(represen)n(tations,)k(only)f(isometries,)g(or)h (co-isometries)31 b(ma)n(y)-118 862 y(arise)25 b(in)i(the)h (in\014nite-dimensional)22 b(non-unitary)j(case.)-118 1029 y FQ(Theorem)30 b(20.)41 b FB(A)n(ny)k(in\014nite-dimensional)i (irr)l(e)l(ducible)g(r)l(epr)l(esentation)-118 1129 y(of)40 b(r)l(elations)46 b FO(\(2.1\))p FB(,)41 b(for)f(which)g FO(k)n(er)13 b FN(X)32 b FP([)25 b FO(k)n(er)13 b FN(X)1471 1098 y FM(\003)1548 1129 y FP(6)p FO(=)39 b FP(f)p FO(0)p FP(g)p FB(,)h(fal)t(ls)g(into)f(one)-118 1228 y(of)30 b(the)g(fol)t(lowing)j(classes)7 b FO(:)-40 1395 y(\(i\))41 b FB(in\014nite-dimensional)31 b(in)f FN(l)965 1407 y FK(2)1002 1395 y FO(:)379 1578 y FN(X)7 b(e)494 1590 y FK(1)553 1578 y FO(=)23 b(0)p FN(;)98 b(X)7 b(e)919 1590 y FL(j)976 1578 y FO(=)1064 1507 y Fy(p)p 1147 1507 84 4 v 71 x FN(\025)1195 1590 y FL(j)1230 1578 y FN(e)1269 1590 y FL(j)s FM(\000)p FK(1)1389 1578 y FN(;)183 b(j)28 b FO(=)23 b(2)p FN(;)14 b FO(3)p FN(;)g(:)g(:)g(:)e(;)793 1714 y(\025)841 1726 y FL(j)899 1714 y FN(>)23 b FO(0)p FN(;)98 b(\025)1198 1726 y FL(j)1257 1714 y FO(=)22 b FN(F)1409 1680 y FL(j)s FM(\000)p FK(1)1529 1714 y FO(\(0\))89 1897 y(\()p FB(F)-6 b(o)l(ck)31 b(r)l(epr)l(esentation)6 b FO(\))p FB(,)-63 2064 y FO(\(ii\))40 b FB(in\014nite)30 b(dimensional)h(in)f FN(l)965 2076 y FK(2)1002 2064 y FO(:)592 2247 y FN(X)7 b(e)707 2259 y FL(j)764 2247 y FO(=)851 2176 y Fy(p)p 934 2176 V 71 x FN(\025)982 2259 y FL(j)1018 2247 y FN(e)1057 2259 y FL(j)s FK(+1)1176 2247 y FN(;)183 b(j)28 b FO(=)23 b(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)e(;)508 2383 y(\025)556 2395 y FL(j)614 2383 y FN(>)23 b FO(0)p FN(;)98 b(\025)913 2395 y FL(j)971 2383 y FP(2)24 b FN(F)1115 2349 y FM(\000)p FL(j)1202 2383 y FO(\(0\))p FN(;)98 b(F)12 b FO(\()p FN(\025)1574 2395 y FL(j)s FK(+1)1694 2383 y FO(\))23 b(=)g FN(\025)1885 2395 y FL(j)89 2566 y FO(\()p FB(anti-F)-6 b(o)l(ck)31 b(r)l(epr)l(esentations)7 b FO(\))p FB(.)-118 2733 y(Pr)l(o)l(of.)43 b FO(Indeed,)31 b(the)g(phase)e FN(U)39 b FO(of)30 b(the)h(op)r(erator) d FN(X)37 b FO(is)29 b(a)g(partial)f(isometry)-118 2832 y(\(non-unitary\),)k(th)n(us)g(b)n(y)f(Theorem)g(19,)h(the)g(represen)n (tation)e(space)h(of)h(an)-118 2932 y(irreducible)18 b FN(X)29 b FO(is)21 b(the)h(same)f(as)g(for)h(an)g(irreducible)c FN(U)9 b FO(.)35 b(Use)22 b(Theorem)f(18)g(to)-118 3032 y(represen)n(t)29 b(the)h(op)r(erator)f FN(U)9 b FO(;)31 b(the)g(rest)e(of)h(the)h(pro)r(of)e(follo)n(ws)e(immediately)-118 3131 y(from)f(\(2.5\).)p 2278 3131 4 57 v 2282 3079 50 4 v 2282 3131 V 2331 3131 4 57 v -118 3299 a FB(Example)31 b(13.)42 b FO(\(Mapping)26 b(of)g(degree)g(t)n(w)n(o.)35 b(F)-7 b(o)r(c)n(k)26 b(and)h(an)n(ti-F)-7 b(o)r(c)n(k)24 b(represen-)-118 3399 y(tations\).)35 b(F)-7 b(or)23 b(relation)e(\(2.2\),)k(zero)e(is)g(a)h(stationary)e(p)r(oin)n(t;)i (therefore,)g(this)-118 3499 y(relation)h(admits)g(no)i(F)-7 b(o)r(c)n(k)28 b(or)e(an)n(ti-F)-7 b(o)r(c)n(k)26 b(represen)n (tations.)6 3598 y(Ho)n(w)n(ev)n(er,)g(this)g(is)g(not)i(true)f(for)f (\(2.3\);)i(b)r(elo)n(w,)e(w)n(e)g(lo)r(ok)g(at)h(suc)n(h)g(repre-)-118 3698 y(sen)n(tations)f(for)h(the)h(relation)786 3881 y FN(xx)880 3847 y FM(\003)942 3881 y FO(=)23 b(\()p FN(x)1109 3847 y FM(\003)1147 3881 y FN(x)c FP(\000)f FN(q)s(I)7 b FO(\))p FN(:)692 b FO(\(2.15\))p eop %%Page: 101 105 101 104 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)853 b FO(101)-118 96 y(As)21 b(follo)n(ws)c(from) j(the)h(previous)d(theorem,)j(w)n(e)f(need)h(to)g(consider)e (half-orbits)-118 196 y(lying)25 b(in)i(the)h(p)r(ositiv)n(e)d(region,) h(and)h(coming)e(from)h(or)h(going)e(to)j(zero.)6 296 y(1.)36 b(F)-7 b(o)r(c)n(k)25 b(represen)n(tation.)34 b(The)25 b(image)e(of)32 b(0)25 b(under)h(the)g(action)e(of)h(p)r(o)n (w-)-118 395 y(ers)j(of)h FN(F)41 b FO(is)28 b(alw)n(a)n(ys)e(p)r (ositiv)n(e)h(except)i(for)g(the)g(case)f FN(q)h FO(=)c(0.)41 b(Therefore,)28 b(for)-118 495 y(an)n(y)20 b FN(q)26 b FP(6)p FO(=)d(0,)f(one)f(can)g(construct)g(the)g(F)-7 b(o)r(c)n(k)21 b(represen)n(tation)e(of)27 b(\(2.15\).)34 b(Ho)n(w-)-118 595 y(ev)n(er,)23 b(for)h FN(q)i(<)d FP(\000)p FO(1)p FN(=)p FO(4,)f(the)j(sequence)e FN(F)1133 564 y FL(n)1178 595 y FO(\(0\))h(is)f(un)n(b)r(ounded,)i(and)f(therefore,) -118 694 y(the)19 b(corresp)r(onding)d(represen)n(tation)g(is)i(un)n(b) r(ounded.)35 b(F)-7 b(or)18 b(all)f FN(q)26 b(>)c FP(\000)p FO(1)p FN(=)p FO(4,)d(for)-118 794 y(whic)n(h)f(zero)f(is)h(not)h(a)f (p)r(erio)r(dic)f(p)r(oin)n(t,)j(there)e(exists)g(a)g(unique)g(b)r (ounded)h(F)-7 b(o)r(c)n(k)-118 893 y(represen)n(tation;)35 b(the)g(sp)r(ectrum)f(of)g FN(C)1144 863 y FK(2)1216 893 y FO(lies)f(b)r(et)n(w)n(een)h(zero)g(and)g(the)h(\014rst)-118 993 y(stationary)28 b(p)r(oin)n(t)i(for)g FN(q)h(<)c FO(0,)k(and)f(on)h(the)f(in)n(terv)-5 b(al)28 b([0)p FN(;)14 b(q)1759 963 y FK(2)1796 993 y FO(\))31 b(for)f FN(q)h FP(2)d FO(\(0)p FN(;)14 b FO(2].)-118 1093 y(Notice)31 b(that)h(for)f FN(q)i FO(=)c(1,)k(zero)d(is)h(a)g(p)r(erio)r(dic)f(p)r (oin)n(t)h(of)h(the)g(second)f(order,)-118 1192 y(and)d(the)g(corresp)r (onding)e(represen)n(tation)f(is)i(t)n(w)n(o-dimensional.)33 b(Similarly)-7 b(,)-118 1292 y(for)26 b(those)g(v)-5 b(alues)25 b(of)33 b FN(q)d FO(for)c(whic)n(h)f FN(q)30 b FO(is)25 b(a)h(p)r(erio)r(dic)f(p)r(oin)n(t)g(of)i(some)e(order)g FN(n)p FO(,)-118 1392 y(the)j(corresp)r(onding)d(F)-7 b(o)r(c)n(k)27 b(represen)n(tation)e(is)h FN(n)p FO(-dimensional.)6 1491 y(F)-7 b(or)27 b FN(q)g(>)22 b FO(2,)27 b(the)h(F)-7 b(o)r(c)n(k)28 b(represen)n(tation)c(is)j(un)n(b)r(ounded.)6 1591 y(2.)73 b(An)n(ti-F)-7 b(o)r(c)n(k)38 b(represen)n(tations.)69 b(No)n(w)39 b(w)n(e)g(consider)f(p)r(ositiv)n(e)f(half-)-118 1690 y(orbits)e(going)g(in)n(to)h(zero.)63 b(It)38 b(is)d(ob)n(vious)g (that)i(this)f(is)g(p)r(ossible)f(only)g(for)-118 1790 y FN(q)k(>)c FO(0.)59 b(F)-7 b(or)35 b(0)g FN(<)g(q)k(<)c FO(1,)i(there)e(exists)f(a)h(unique)f(sequence)h(of)g(p)r(ositiv)n(e) -118 1890 y(pre-images)22 b(of)k(zero,)f FN(F)651 1860 y FM(\000)p FL(k)744 1890 y FO(\(0\))e FP(\000)-48 b(!)23 b FN(\025)1044 1902 y FK(2)1082 1890 y FO(,)j(where)g FN(\025)1418 1902 y FK(2)1481 1890 y FO(is)f(the)i(second)e(stationary) -118 1989 y(p)r(oin)n(t)i(of)g(the)h(mapping.)6 2089 y(F)-7 b(or)38 b FN(q)45 b FO(=)c(1,)g(zero)c(is)h(a)g(p)r(erio)r(dic)e (p)r(oin)n(t)i(of)h(p)r(erio)r(d)e(2,)k(and)e(zero)e(has)-118 2189 y(t)n(w)n(o)29 b(p)r(ositiv)n(e)e(pre-images,)g(0)j(and)f(2.)43 b(Ho)n(w)n(ev)n(er,)29 b(the)h(represen)n(tation)d(that)-118 2288 y(corresp)r(onds)39 b(to)j(0)e(is)h(t)n(w)n(o-dimensional,)e(and)i (w)n(e)g(again)e(ha)n(v)n(e)h(a)h(single)-118 2388 y (in\014nite-dimensional)17 b(an)n(ti-F)-7 b(o)r(c)n(k)22 b(represen)n(tation.)33 b(Indeed,)24 b(one)f(can)g(easily)-118 2487 y(see)k(that)h(there)f(are)g(at)g(least)g(coun)n(tably)e(man)n(y)h (in)n(v)n(erse)g(orbits.)6 2587 y(F)-7 b(or)19 b FN(q)26 b(>)d FO(1,)e(zero)d(has)h(t)n(w)n(o)g(p)r(ositiv)n(e)e(pre-images,)g (0)23 b FN(<)f(t)1753 2599 y FK(1)1814 2587 y FN(<)g(q)k(<)d(t)2082 2599 y FK(2)2142 2587 y FN(<)g(\025)2278 2599 y FK(2)2316 2587 y FO(,)-118 2687 y(whic)n(h)33 b(generate)g(at)h(least)f(coun)n (tably)g(man)n(y)g(sequences)g(of)h(p)r(ositiv)n(e)e(pre-)-118 2786 y(images.)-118 2950 y FQ(Prop)s(osition)e(38.)41 b FB(F)-6 b(or)27 b FO(1)c FN(<)g(q)j FP(\024)c FN(q)1057 2920 y FM(\003)1123 2950 y FB(ther)l(e)28 b(ar)l(e)g(c)l(ountably)g (many)f(ine)l(quiv-)-118 3050 y(alent)37 b(irr)l(e)l(ducible)h(anti-F) -6 b(o)l(ck)37 b(r)l(epr)l(esentations.)60 b(F)-6 b(or)37 b FN(q)i(>)c(q)1884 3020 y FM(\003)1922 3050 y FB(,)k(ther)l(e)e(is)g (a)-118 3149 y(c)l(ontinuum)28 b(of)j(such)f(r)l(epr)l(esentations.) -118 3313 y(Pr)l(o)l(of.)43 b FO(First)28 b(mark)g(some)g(p)r(oin)n(ts) g(on)h(the)h(line.)41 b(Denote)30 b(b)n(y)f FN(\025)1933 3325 y FK(1)1970 3313 y FO(,)h FN(\025)2071 3325 y FK(2)2109 3313 y FO(,)g FN(\025)2210 3325 y FK(1)2274 3313 y FN(<)-118 3413 y(\025)-70 3425 y FK(2)-32 3413 y FO(,)c(the)g(stationary)d(p)r (oin)n(ts)i(of)h FN(F)12 b FO(\()p FP(\001)p FO(\);)27 b FN(\025)1142 3383 y FM(0)1142 3434 y FK(1)1202 3413 y FP(6)p FO(=)c FN(\025)1338 3425 y FK(1)1402 3413 y FO(is)h(the)i(second)f(pre-image)e(of)-118 3513 y FN(\025)-70 3525 y FK(1)-32 3513 y FO(,)28 b FN(F)12 b FO(\()p FN(\025)164 3482 y FM(0)164 3533 y FK(1)202 3513 y FO(\))25 b(=)f FN(\025)396 3525 y FK(1)434 3513 y FO(,)29 b(and)f FN(\025)696 3482 y FM(0)q(0)696 3533 y FK(1)764 3513 y FN(<)c(\025)901 3525 y FK(1)967 3513 y FO(is)k(a)g(pre-image)d(of)j FN(\025)1649 3525 y FK(1)1716 3513 y FO(whic)n(h)f(is)h(di\013eren)n(t)-118 3612 y(from)f FN(\025)127 3624 y FK(1)193 3612 y FO(and)i FN(\025)404 3582 y FM(0)404 3633 y FK(1)470 3612 y FO(with)f(resp)r (ect)g(to)h FN(F)1113 3582 y FK(2)1150 3612 y FO(\()p FP(\001)p FO(\),)g FN(F)1354 3582 y FK(2)1391 3612 y FO(\()p FN(\025)1471 3582 y FM(0)q(0)1471 3633 y FK(1)1515 3612 y FO(\))24 b(=)h FN(\025)1709 3624 y FK(1)1746 3612 y FO(.)40 b(Also)27 b(in)n(tro)r(duce)-118 3712 y FN(\025)-70 3724 y FP(\003)-1 3712 y FO(=)c FN(q)127 3682 y FK(2)187 3712 y FO(=)g FN(F)340 3682 y FK(2)377 3712 y FO(\(0\).)6 3811 y(Let)34 b FN(q)i FP(\024)c FN(q)371 3781 y FM(\003)410 3811 y FO(.)54 b(Then)34 b FN(F)12 b FO(\()p FP(\001)p FO(\))34 b(has)f(cycles)f(only)g(of)h(orders)f(2)1880 3781 y FL(k)1920 3811 y FO(,)j FN(k)h FP(\025)d FO(1.)54 b FN(F)-118 3911 y FO(maps)29 b(the)i(in)n(terv)-5 b(al)27 b([0)p FN(;)14 b(\025)700 3923 y FP(\003)746 3911 y FO(])30 b(in)n(to)g(itself,)f(and)h(one)g(can)g(easily)e(see)i(that)g(the)p eop %%Page: 102 106 102 105 bop -118 -137 a FO(102)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(n)n(um)n(b)r(er)18 b(of)h(in)n(v)n(erse)d (orbits)i(that)h(lea)n(v)n(e)d(this)j(in)n(terv)-5 b(al)16 b(do)r(es)j(not)g(in\015uence)f(the)-118 196 y(total)30 b(cardinalit)n(y)d(of)k(the)g(in)n(v)n(erse)e(orbits,)i(pro)n(vided)e (it)h(is)g(in\014nite.)47 b(Th)n(us,)-118 296 y(w)n(e)30 b(need)h(to)f(in)n(v)n(estigate)d(the)k(n)n(um)n(b)r(er)f(of)g(orbits)f (that)i(go)e(to)i(zero)e(and)i(lie)-118 395 y(completely)24 b(inside)i([0)p FN(;)14 b(\025)685 407 y FP(\003)731 395 y FO(].)6 495 y(Consider)36 b(t)n(w)n(o)g(in)n(terv)-5 b(als,)36 b FN(I)939 465 y FK(1)932 516 y(1)1016 495 y FO(=)i([)p FN(\025)1190 465 y FM(0)q(0)1190 516 y FK(1)1233 495 y FN(;)14 b(\025)1318 507 y FK(1)1355 495 y FO(])38 b(and)f FN(I)1630 465 y FK(1)1623 516 y(2)1706 495 y FO(=)h([)p FN(\025)1880 507 y FK(1)1918 495 y FN(;)14 b(\025)2003 465 y FM(0)2003 516 y FK(1)2040 495 y FO(].)66 b(They)-118 595 y(are)31 b(in)n(v)-5 b(arian)n(t)29 b(with)i(resp)r (ect)h(to)g FN(F)1031 565 y FK(2)1068 595 y FO(\()p FP(\001)p FO(\).)50 b(Indeed,)34 b(it)d(is)g(su\016cien)n(t)g(to)h(sho)n(w)-118 694 y(that)j FN(\025)117 664 y FM(0)117 715 y FK(1)189 694 y FN(>)f(\025)336 706 y FP(\003)382 694 y FO(;)k(if)c(not,)i(the)f (dynamical)c(system)i FN(F)1608 664 y FK(2)1679 694 y FO(on)i(the)f(in)n(terv)-5 b(al)32 b FN(I)2301 664 y FK(1)2294 715 y(2)-118 794 y FO(w)n(ould)22 b(ha)n(v)n(e)h(the)h(same)f (dynamics)e(as)i(the)i(quadratic)c(mapping)h(in)h(the)i(case)-118 894 y(of)36 b FN(q)k FP(\025)d FO(2)f([246)o(],)i(and)e FN(F)709 864 y FK(2)747 894 y FO(\()p FP(\001)p FO(\))g(w)n(ould)f(ha)n (v)n(e)g(cycles)g(of)h(an)n(y)f(p)r(erio)r(d,)j(whic)n(h)-118 993 y(con)n(tradicts)h(the)j(condition)d FN(q)49 b FP(\024)c FN(q)1094 963 y FM(\003)1132 993 y FO(.)78 b(Notice)41 b(also)e(that)i(in)g(this)f(case,)-118 1093 y FN(\025)-70 1063 y FM(00)-70 1114 y FK(1)4 1093 y FN(<)32 b FO(0.)51 b(Therefore,)33 b(there)g(are)f(no)g(orbits)f(of)i FN(F)1511 1063 y FK(2)1548 1093 y FO(\()p FP(\001)p FO(\))g(whic)n(h)f(mo)n(v)n (e)f(inside)-118 1193 y(b)r(oth)d FN(I)121 1162 y FK(1)114 1213 y(1)186 1193 y FO(and)f FN(I)390 1162 y FK(1)383 1213 y(2)428 1193 y FO(.)6 1292 y(No)n(w)22 b(w)n(e)g(in)n(v)n (estigate)c(the)23 b(set)f(of)g(orbits)e(of)i(the)h(mapping)d FN(F)1923 1262 y FK(2)1960 1292 y FO(\()p FP(\001)p FO(\))j(on)e(eac)n (h)-118 1392 y(of)36 b(the)h(in)n(terv)-5 b(als,)36 b FN(I)547 1362 y FK(1)540 1413 y(1)621 1392 y FO(and)g FN(I)834 1362 y FK(1)827 1413 y(2)871 1392 y FO(.)63 b(On)37 b(eac)n(h)e(of)i(these)f(in)n(terv)-5 b(als,)36 b FN(F)2057 1362 y FK(2)2094 1392 y FO(\()p FP(\001)p FO(\))h(has)-118 1492 y(similar)24 b(dynamical)h(prop)r(erties)i(as)i (a)f(quadratic)f(mapping;)h(in)g(particular,)-118 1591 y(for)g FN(F)75 1561 y FK(2)112 1591 y FO(\()p FP(\001)p FO(\))h(there)g(are)e(cycles)g(only)g(of)i(orders)e(2)1392 1561 y FL(k)1432 1591 y FO(,)i FN(k)e FP(\025)e FO(1.)39 b(Again,)27 b(there)i(are)-118 1691 y(four)f(in)n(terv)-5 b(als,)25 b FN(I)455 1661 y FK(2)448 1715 y FL(k)492 1691 y FO(,)k FN(k)e FO(=)c(1,)28 b FN(:)14 b(:)g(:)28 b FO(,)g(4,)g(whic)n(h)f(are)g(in)n(v)-5 b(arian)n(t)25 b(with)j(resp)r(ect)g(to)-118 1791 y FN(F)-53 1760 y FK(4)-16 1791 y FO(\()p FP(\001)p FO(\),)23 b(and)e(whic)n(h)f (determine)e(the)k(cardinalit)n(y)16 b(of)21 b(orbits.)33 b(Con)n(tin)n(uing)18 b(this)-118 1890 y(pro)r(cess,)32 b(w)n(e)g(get)g(either)f(a)h(\014nite)f(n)n(um)n(b)r(er)g(of)h(in)n (terv)-5 b(als)30 b(where)h FN(F)2053 1860 y FL(n)2098 1890 y FO(\()p FP(\001)p FO(\))i(has)-118 1990 y(only)c(one)h(orbit)f (going)f(to)i(zero)f(\(for)h FN(q)h(<)c(q)1296 1960 y FM(\003)1334 1990 y FO(\),)k(or)f(a)f(Can)n(tor)g(set)i(on)f(whic)n(h) -118 2089 y FN(F)12 b FO(\()p FP(\001)p FO(\))32 b(is)f(one-to-one)e (\(for)j FN(q)h FO(=)c FN(q)931 2059 y FM(\003)969 2089 y FO(\);)34 b(in)d(b)r(oth)h(cases,)g(w)n(e)f(ha)n(v)n(e)f(a)h(coun)n (table)-118 2189 y(n)n(um)n(b)r(er)26 b(of)i(orbits)e(going)f(to)j (zero.)6 2289 y(T)-7 b(o)31 b(pro)n(v)n(e)e(that)i(for)f FN(q)h(>)d(q)873 2259 y FM(\003)941 2289 y FO(there)j(is)e(a)i(con)n (tin)n(uum)d(of)j(an)n(ti-F)-7 b(o)r(c)n(k)28 b(rep-)-118 2389 y(resen)n(tations,)41 b(consider)d(\014rst)i(the)h(case)e(of)h FN(q)47 b FO(=)d FN(q)1582 2401 y FK(0)1664 2389 y FP(\031)f FO(1)p FN(:)p FO(54,)f(suc)n(h)e(that)-118 2488 y FN(\025)-70 2500 y FP(\003)4 2488 y FO(=)28 b FN(\025)145 2458 y FM(0)145 2509 y FK(1)183 2488 y FO(,)k(i.e.,)f FN(F)464 2458 y FK(2)501 2488 y FO(\(0\))d(=)g FN(\025)776 2500 y FK(1)814 2488 y FO(.)47 b(In)31 b(this)f(case,)h(the)g(in)n(terv)-5 b(als)28 b FN(I)1884 2458 y FK(1)1877 2509 y(1)1952 2488 y FO(and)j FN(I)2160 2458 y FK(1)2153 2509 y(2)2228 2488 y FO(are)-118 2588 y(in)n(v)-5 b(arian)n(t)22 b(with)j(resp)r(ect)g(to) g FN(F)861 2558 y FK(2)899 2588 y FO(\()p FP(\001)p FO(\),)h(and)f(on)g (eac)n(h)g(of)g(these)h(in)n(terv)-5 b(als,)23 b FN(F)2214 2558 y FK(2)2251 2588 y FO(\()p FP(\001)p FO(\))-118 2688 y(has)30 b(the)h(same)e(dynamics)f(as)h(the)i(quadratic)e(mapping) f(with)i FN(q)h FO(=)c(2.)45 b(One)-118 2787 y(can)32 b(easily)e(see)i(that)h(in)f(this)g(case)g(the)h(set)g(of)g(pre-images) c(of)j(zero)g(under)-118 2887 y FN(F)-53 2857 y FK(2)-16 2887 y FO(\()p FP(\001)p FO(\))26 b(con)n(tains)d(an)i(in\014nite)e (binary)h(tree,)h(and)g(therefore,)f(an)h(uncoun)n(table)-118 2986 y(n)n(um)n(b)r(er)h(of)i(paths)f(in)g(it.)6 3086 y(F)-7 b(or)38 b(an)h(arbitrary)c FN(q)44 b(>)d(q)884 3056 y FM(\003)923 3086 y FO(,)g(one)d(can)g(sho)n(w)g(using)f([246)o (])h(that)h(some)-118 3186 y(iteration)23 b(of)j FN(F)12 b FO(\()p FP(\001)p FO(\))26 b(p)r(ossesses)e(the)i(same)e(prop)r(ert)n (y)-7 b(,)25 b(and)h(therefore,)f(there)h(is)-118 3286 y(an)h(uncoun)n(table)f(n)n(um)n(b)r(er)h(of)g(orbits)f(passing)g (through)h(zero.)p 2278 3286 4 57 v 2282 3233 50 4 v 2282 3286 V 2331 3286 4 57 v -118 3454 a FB(Example)k(14.)42 b FO(\(Tw)n(o-parameter)25 b(unit)j(quan)n(tum)f(disk\).)38 b(F)-7 b(or)28 b(relations)c(re-)-118 3553 y(lated)g(to)i(con)n(tin)n (uous)d(fractions,)h(w)n(e)h(consider)f(only)g(a)h(v)n(ery)f(sp)r (ecial)f(case)i(of)-118 3653 y(the)j(mapping)567 3879 y FN(F)12 b FO(\()p FN(\025)p FO(\))24 b(=)866 3823 y(\()p FN(q)e FO(+)c FN(\026)p FO(\))p FN(\025)h FO(+)f(1)g FP(\000)g FN(q)j FP(\000)d FN(\026)p 866 3860 741 4 v 1040 3936 a(\026\025)h FO(+)f(1)g FP(\000)g FN(\026)1616 3879 y(;)487 b FO(\(2.16\))p eop %%Page: 103 107 103 106 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)853 b FO(103)-118 96 y(with)29 b(0)d FP(\024)h FN(\026)f FP(\024)h FO(1,)j(0)c FP(\024)g FN(q)k FP(\024)c FO(1,)k(\()p FN(\026;)14 b(q)s FO(\))27 b FP(6)p FO(=)f(\(0)p FN(;)14 b FO(1\).)43 b(This)29 b(mapping)f(is)g(related)-118 196 y(to)23 b(the)h(t)n(w)n(o-parameter)c (unit)j(quan)n(tum)f(disk)g(algebra)f(in)n(tro)r(duced)h(in)h([136)n (],)-118 296 y(whic)n(h)j(is)h(generated)f(b)n(y)i(the)g(relation)352 465 y FN(q)s(z)t(z)478 431 y FM(\003)534 465 y FP(\000)18 b FN(z)660 431 y FM(\003)697 465 y FN(z)26 b FO(=)d FN(q)e FP(\000)e FO(1)e(+)h FN(\026)p FO(\(1)h FP(\000)f FN(z)t(z)1446 431 y FM(\003)1482 465 y FO(\)\(1)h FP(\000)f FN(z)1733 431 y FM(\003)1770 465 y FN(z)t FO(\))p FN(:)-118 635 y FO(Putting)34 b FN(x)i FO(=)f FN(z)421 604 y FM(\003)458 635 y FO(,)i(w)n(e)e(come)e(to)i(a)g(relation)d(of)j(the)g(form)f (\(2.1\))h(with)f(the)-118 734 y FN(F)12 b FO(\()p FP(\001)p FO(\))35 b(in)n(tro)r(duced)e(in)h(this)g(w)n(a)n(y)-7 b(.)57 b(If)35 b FN(\026)f FO(=)h(0,)h(w)n(e)e(get)g(the)h FN(q)s FO(-CCR)g(relation)-118 834 y(considered)25 b(ab)r(o)n(v)n(e;)i (th)n(us)g(w)n(e)h(assume)e(that)h FN(\026)c FP(6)p FO(=)g(0.)6 933 y(The)42 b(mapping)d FN(F)12 b FO(\()p FP(\001)p FO(\))42 b(is)e(one-to-one)g(and)h(p)r(ossesses)f(t)n(w)n(o)g (stationary)-118 1033 y(p)r(oin)n(ts,)22 b FN(t)180 1045 y FK(1)240 1033 y FO(=)g(1)6 b FP(\000)g FO(\(1)18 b FP(\000)g FN(q)s FO(\))p FN(=\026)k FO(and)f FN(t)992 1045 y FK(2)1052 1033 y FO(=)i(1,)f(that)g(corresp)r(ond)e(to)h(t)n(w)n (o)g(families)-118 1133 y(of)27 b(one-dimensional)c(represen)n (tations.)34 b(Consider)25 b(the)j(follo)n(wing)c(cases.)6 1232 y(1.)48 b(Let)32 b FN(q)g FO(=)d(1.)47 b(In)32 b(this)f(case,)g FN(t)1054 1244 y FK(1)1121 1232 y FO(=)d FN(t)1244 1244 y FK(2)1282 1232 y FO(,)k(and)f(b)r(esides)g(the)g(one-dimen-)-118 1332 y(sional)18 b(family)-7 b(,)20 b(there)h(exists)f(a)g(unique)h (in\014nite-dimensional)15 b(b)r(ounded)21 b(rep-)-118 1432 y(resen)n(tation)35 b(that)i(corresp)r(onds)e(to)i(the)g(sequence) g(of)g(the)g(pre-images)d(of)-118 1531 y(zero,)26 b FN(\025)128 1543 y FL(k)193 1531 y FO(=)c FN(F)345 1501 y FM(\000)p FL(k)438 1531 y FO(\(0\))h(=)g FN(\026k)s(=)p FO(\(1)17 b(+)h FN(\026k)s FO(\),)28 b FN(k)e FP(\025)d FO(1.)36 b(The)28 b(op)r(erator)e FN(z)31 b FO(is)626 1710 y FN(z)t(e)708 1722 y FL(k)771 1710 y FO(=)859 1637 y Fy(p)p 942 1637 175 4 v 73 x FN(\025)990 1722 y FL(k)q FM(\000)p FK(1)1130 1710 y FN(e)1169 1722 y FL(k)q FM(\000)p FK(1)1294 1710 y FN(;)588 1857 y(z)631 1823 y FM(\003)668 1857 y FN(e)707 1869 y FL(k)771 1857 y FO(=)859 1780 y Fy(p)p 942 1780 90 4 v 77 x FN(\025)990 1869 y FL(k)1045 1857 y FN(e)1084 1869 y FL(k)q FK(+1)1208 1857 y FN(;)180 b(k)26 b FP(\025)d FO(1)p FN(:)6 2026 y FO(2.)76 b(Let)41 b(1)26 b FP(\000)h FN(\026)45 b(<)f(q)k(<)c FO(1.)76 b(No)n(w)40 b(there)g(are)g(t)n(w)n (o)g(one-dimensional)-118 2126 y(families,)e(and)h(the)g(represen)n (tation)e(corresp)r(onding)f(to)j(the)g(sequence)g(of)-118 2226 y(pre-images)21 b(of)k(zero,)f FN(\025)631 2238 y FL(k)695 2226 y FO(=)f FN(F)848 2195 y FM(\000)p FL(k)941 2226 y FO(\(0\))g(=)f(\(1)13 b FP(\000)g FN(q)1362 2195 y FM(\000)p FL(k)1467 2226 y FO(+)g FN(\026q)1635 2195 y FM(\000)p FK(\()p FL(k)q FK(+1\))p FL(=)p FK(2)1930 2226 y FO([)p FN(k)s FO(])2022 2203 y FM(p)p 2077 2203 33 3 v 36 x FL(q)2114 2226 y FO(\))p FN(=)p FO(\(1)g(+)-118 2340 y FN(\026q)-28 2310 y FM(\000)p FK(\()p FL(k)q FK(+1\))p FL(=)p FK(2)268 2340 y FO([)p FP(\000)p FN(k)s FO(])425 2317 y FM(p)p 479 2317 V 479 2353 a FL(q)515 2340 y FO(\),)23 b FN(k)i FP(\025)e FO(1)d(\(w)n(e)g(use)g(the)h(notation)d([)p FN(n)p FO(])1650 2352 y FL(q)1710 2340 y FO(=)23 b(\()p FN(q)1870 2310 y FL(n)1919 2340 y FP(\000)t FN(q)2028 2310 y FM(\000)p FL(n)2124 2340 y FO(\))p FN(=)p FO(\()p FN(q)7 b FP(\000)-118 2448 y FN(q)-78 2418 y FM(\000)p FK(1)11 2448 y FO(\)\).)38 b(The)28 b(represen)n(tation)d(acts)i(b)n(y) g(the)h(same)f(form)n(ula)d(with)k(the)g FN(\025)2173 2460 y FL(k)2242 2448 y FO(in-)-118 2548 y(tro)r(duced)d(in)g(this)f(w) n(a)n(y)-7 b(.)35 b(But)26 b(b)r(esides)e(these)h(represen)n(tations,)e (there)i(exists)-118 2647 y(another)30 b(family)f(of)i(b)r(ounded)h (represen)n(tations)d(suc)n(h)i(that)h(the)f(k)n(ernels)f(of)-118 2747 y FN(z)h FO(and)c FN(z)156 2717 y FM(\003)221 2747 y FO(are)g(zero)f(\(see)i(b)r(elo)n(w\).)6 2847 y(3.)38 b(F)-7 b(or)28 b FN(q)f FO(=)c(1)18 b FP(\000)h FN(\026)p FO(,)28 b(zero)f(is)g(a)h(\014xed)g(p)r(oin)n(t,)f(whic)n(h)g(corresp)r (onds)f(to)i(the)-118 2946 y(trivial)c(represen)n(tation.)34 b(In)28 b(this)f(case,)f(there)i(exists)e(a)h(family)e(of)i(b)r(ounded) -118 3046 y(represen)n(tations)e(with)i(zero)f(k)n(ernels)g(of)h FN(z)k FO(and)c FN(z)1468 3016 y FM(\003)1534 3046 y FO(\(see)g(b)r(elo)n(w\).)6 3145 y(4.)69 b(F)-7 b(or)38 b(0)i FP(\024)g FN(q)k(<)d FO(1)25 b FP(\000)g FN(\026)p FO(,)41 b(the)e(v)-5 b(alue)37 b FN(t)1354 3157 y FK(1)1430 3145 y FO(is)g(negativ)n(e,)i(and)f(there)g(is)-118 3245 y(again)17 b(only)i(one)g(family)e(of)i(one-dimensional)c(represen)n (tations)i(corresp)r(ond-)-118 3345 y(ing)31 b(to)i(the)g(\014xed)g(p)r (oin)n(t)f FN(t)738 3357 y FK(2)807 3345 y FO(=)f(1,)j(and)f(a)f (single)e(b)r(ounded)j(represen)n(tation)-118 3444 y(corresp)r(onding) 38 b(to)i(the)i(sequence)e(of)h(images)c(of)k(zero,)i FN(\025)1821 3456 y FL(k)1907 3444 y FO(=)i FN(F)2082 3414 y FL(k)2123 3444 y FO(\(0\))g(=)-118 3544 y(\(1)18 b FP(\000)g FN(q)97 3514 y FL(k)156 3544 y FP(\000)h FN(\026q)330 3514 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)574 3544 y FO([)p FN(k)s FO(])666 3521 y FM(p)p 721 3521 V 36 x FL(q)758 3544 y FO(\))p FN(=)p FO(\(1)f FP(\000)g FN(\026q)1097 3514 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)1342 3544 y FO([)p FN(k)s FO(])1434 3521 y FM(p)p 1488 3521 V 1488 3557 a FL(q)1525 3544 y FO(\),)28 b FN(k)e FP(\025)c FO(0,)583 3738 y FN(z)t(e)665 3750 y FL(k)728 3738 y FO(=)816 3661 y Fy(p)p 899 3661 90 4 v 77 x FN(\025)947 3750 y FL(k)1002 3738 y FN(e)1041 3750 y FL(k)q FK(+1)1165 3738 y FN(;)545 3881 y(z)588 3847 y FM(\003)626 3881 y FN(e)665 3893 y FL(k)728 3881 y FO(=)816 3808 y Fy(p)p 899 3808 175 4 v 73 x FN(\025)947 3893 y FL(k)q FM(\000)p FK(1)1087 3881 y FN(e)1126 3893 y FL(k)q FM(\000)p FK(1)1251 3881 y FN(;)180 b(k)26 b FP(\025)d FO(1)p FN(:)p eop %%Page: 104 108 104 107 bop -118 -137 a FO(104)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(4.)72 b FO(No)n(w,)42 b(it)d(remains)e(to)i (consider)f(the)i(case)f(of)g(the)h(unitary)e(op)r(erator)-118 196 y FN(U)9 b FO(.)70 b(W)-7 b(e)39 b(distinguish)d(b)r(et)n(w)n(een)j (t)n(w)n(o)f(cases:)58 b(represen)n(tations)36 b(related)h(to)-118 296 y(a)e(single)f(orbit,)i(and)g(represen)n(tations)d(related)h(to)i (a)f(non-trivial)d(ergo)r(dic)-118 395 y(quasi-in)n(v)-5 b(arian)n(t)22 b(measure.)6 495 y(If)f(there)g(exists)e(a)h(measurable) d(section)i(of)i(the)g(dynamical)16 b(system,)21 b(then)-118 595 y(an)n(y)32 b(ergo)r(dic)f(measure)h(is)g(concen)n(trated)g(on)h(a) f(single)f(orbit;)k(in)d(this)h(case)-118 694 y(w)n(e)27 b(will)e(pro)n(vide)g(a)j(complete)d(description)g(of)j(the)g(represen) n(tations.)-118 855 y FQ(Theorem)i(21.)41 b FB(L)l(et)32 b(the)g(op)l(er)l(ator)i FN(X)k FB(b)l(e)33 b(invertible,)h(and)f(the)g (dynamic)l(al)-118 954 y(system)d(p)l(ossess)i(a)f(me)l(asur)l(able)g (se)l(ction.)42 b(A)n(ny)30 b(in\014nite-dimensional)i(irr)l(e-)-118 1054 y(ducible)f(r)l(epr)l(esentation)f(of)48 b FO(\(2.1\))29 b FB(has)i(the)f(form)579 1243 y FN(X)7 b(e)694 1255 y FL(k)757 1243 y FO(=)845 1166 y Fy(p)p 928 1166 90 4 v 77 x FN(\025)976 1255 y FL(k)1031 1243 y FN(e)1070 1255 y FL(k)q FM(\000)p FK(1)1195 1243 y FN(;)184 b(k)26 b FP(2)d FI(Z)o FN(;)493 b FO(\(2.17\))-118 1419 y FB(wher)l(e)43 b FN(\025)177 1431 y FL(k)218 1419 y FB(,)j FN(k)j FP(2)d FI(Z)p FB(,)40 b(is)j(any)f(se)l(quenc)l(e)g(of)i(p)l(ositive)f(numb)l (ers)f(such)h(that)-118 1519 y FN(F)12 b FO(\()p FN(\025)27 1531 y FL(k)68 1519 y FO(\))24 b(=)e FN(\025)259 1531 y FL(k)q FK(+1)385 1519 y FB(,)29 b FN(k)d FP(2)d FI(Z)p FB(.)32 b(Two)d(such)g(r)l(epr)l(esentations)g(ar)l(e)f(unitarily)i(e)l (quiv-)-118 1618 y(alent)g(if)g(and)h(only)f(if)h(the)e(c)l(orr)l(esp)l (onding)i(se)l(quenc)l(es)f(c)l(oincide.)-118 1779 y(Pr)l(o)l(of.)43 b FO(First)27 b(let)g(the)h(mapping)e FN(F)12 b FO(\()p FP(\001)p FO(\))28 b(b)r(e)h(one-to-one)d(on)h(the)i(sp)r(ectrum)e(of) -118 1879 y FN(C)-53 1848 y FK(2)-16 1879 y FO(.)35 b(In)22 b(this)f(case,)h(an)n(y)e(ergo)r(dic)g(quasi-in)n(v)-5 b(arian)n(t)16 b(measure)j(is)i(concen)n(trated)-118 1978 y(on)33 b(a)h(single)d(orbit)h(of)i(the)g(dynamical)c(system,)k (and)f(is)g(equiv)-5 b(alen)n(t)31 b(to)j(an)-118 2078 y(atomic)j(measure)g(concen)n(trated)h(at)h(p)r(oin)n(ts)f(of)h(the)h (orbit.)70 b(Cho)r(ose)38 b(an)n(y)-118 2177 y(p)r(oin)n(t)24 b FN(\025)i FO(of)f(the)h(orbit,)e(and)h(tak)n(e)g(a)f(unit)h(eigen)n (v)n(ector)d FN(e)1671 2189 y FK(0)1734 2177 y FO(corresp)r(onding)g (to)-118 2277 y FN(\025)p FO(.)53 b(W)-7 b(rite)32 b FN(e)281 2289 y FL(k)354 2277 y FO(=)f FN(U)516 2247 y FL(k)557 2277 y FN(e)596 2289 y FK(0)633 2277 y FO(,)j FN(k)h FP(2)d FI(Z)p FO(.)47 b(W)-7 b(e)33 b(claim)d(that)j(the)g (space)f(spanned)h(b)n(y)-118 2377 y(these)28 b(v)n(ectors)e(is)g(in)n (v)-5 b(arian)n(t)25 b(under)i FN(X)34 b FO(and)27 b FN(X)1384 2347 y FM(\003)1422 2377 y FO(.)37 b(F)-7 b(or)27 b FN(k)f(<)c FO(0,)28 b(w)n(e)f(ha)n(v)n(e)107 2562 y FN(C)172 2527 y FK(2)209 2562 y FN(e)248 2574 y FL(k)312 2562 y FO(=)22 b FN(C)464 2527 y FK(2)502 2562 y FN(U)568 2527 y FL(k)609 2562 y FN(e)648 2574 y FK(0)707 2562 y FO(=)h FN(C)860 2527 y FK(2)898 2562 y FN(U)964 2527 y FM(\003)1002 2521 y(j)p FL(k)q FM(j)1082 2562 y FN(e)1121 2574 y FK(0)312 2709 y FO(=)f FN(U)465 2675 y FM(\003)503 2668 y(j)p FL(k)q FM(j)584 2709 y FN(F)649 2675 y FM(j)p FL(k)q FM(j)729 2709 y FO(\()p FN(C)826 2675 y FK(2)864 2709 y FO(\))14 b FN(e)949 2721 y FK(0)1009 2709 y FO(=)22 b FN(U)1162 2675 y FM(\003)1200 2668 y(j)p FL(k)q FM(j)1281 2709 y FN(F)1346 2675 y FM(j)p FL(k)q FM(j)1426 2709 y FO(\()p FN(\025)p FO(\))14 b FN(e)1591 2721 y FK(0)1652 2709 y FO(=)22 b FN(F)1804 2675 y FM(j)p FL(k)q FM(j)1884 2709 y FO(\()p FN(\025)p FO(\))14 b FN(e)2049 2721 y FL(k)2091 2709 y FN(:)-118 2885 y FO(Also,)42 b(since)d FN(F)12 b FO(\()p FP(\001)p FO(\))41 b(is)e(one-to-one,)j(w)n(e)e(ha)n (v)n(e)f(in)g(this)h(case)f(that)i FN(C)2127 2855 y FK(2)2164 2885 y FN(U)53 b FO(=)-118 2985 y FN(U)9 b(F)13 2955 y FM(\000)p FK(1)102 2985 y FO(\()p FN(C)199 2955 y FK(2)236 2985 y FO(\),)29 b(whic)n(h)d(implies)e(for)j(eac)n(h)g FN(k)f(>)c FO(0:)217 3161 y FN(C)282 3126 y FK(2)319 3161 y FN(e)358 3173 y FL(k)422 3161 y FO(=)g FN(C)574 3126 y FK(2)612 3161 y FN(U)9 b(e)717 3173 y FL(k)q FM(\000)p FK(1)865 3161 y FO(=)23 b FN(:)14 b(:)g(:)422 3298 y FO(=)22 b FN(C)574 3264 y FK(2)612 3298 y FN(U)678 3264 y FL(k)718 3298 y FN(e)757 3310 y FK(0)817 3298 y FO(=)h FN(U)971 3264 y FL(k)1012 3298 y FN(F)1077 3264 y FM(\000)p FL(k)1169 3298 y FO(\()p FN(C)1266 3264 y FK(2)1304 3298 y FO(\))14 b FN(e)1389 3310 y FK(0)1449 3298 y FO(=)23 b FN(U)1603 3264 y FL(k)1643 3298 y FN(F)1708 3264 y FM(\000)p FL(k)1801 3298 y FO(\()p FN(\025)p FO(\))14 b FN(e)1966 3310 y FK(0)422 3436 y FO(=)22 b FN(F)574 3402 y FM(\000)p FL(k)667 3436 y FO(\()p FN(\025)p FO(\))14 b FN(e)832 3448 y FL(k)873 3436 y FN(:)-118 3612 y FO(Therefore,)40 b(all)d FN(e)461 3624 y FL(k)540 3612 y FO(are)h(eigen)n(v)-5 b(alues)36 b(of)i FN(C)1301 3582 y FK(2)1339 3612 y FO(,)k(and)c (hence,)k(of)d FN(C)6 b FO(.)70 b(In)39 b(the)-118 3712 y(c)n(hosen)27 b(basis,)e(the)j(op)r(erator)e FN(X)34 b FO(acts)27 b(as)g(stated)h(in)f(the)h(theorem.)6 3811 y(In)j(the)g(general)e(case,)h(consider)f(a)h(comm)n(uting)e(family)f (of)k(self-adjoin)n(t)-118 3911 y(op)r(erators,)39 b FN(C)345 3923 y FL(k)428 3911 y FO(=)i FN(U)600 3881 y FL(k)641 3911 y FN(C)706 3881 y FK(2)744 3911 y FN(U)810 3881 y FM(\000)p FL(k)902 3911 y FO(.)70 b(The)39 b(relations)d(are)h FN(C)1734 3923 y FL(k)1776 3911 y FN(U)1842 3881 y FM(\003)1921 3911 y FO(=)k FN(U)2093 3881 y FM(\003)2131 3911 y FN(C)2190 3923 y FL(k)q FK(+1)2316 3911 y FO(.)p eop %%Page: 105 109 105 108 bop -118 -137 a FJ(2.1.)36 b(One-dimensional)22 b(dynamical)i(systems)853 b FO(105)-118 96 y(No)n(w)35 b(w)n(e)h(ha)n(v)n(e)f(a)g(one-to-one)f(action)h(of)h FI(Z)29 b FO(on)36 b(the)g(in\014nite-dimensional)-118 196 y(space)30 b FI(R)161 166 y Fu(Z)210 196 y FO(.)48 b(Using)30 b(the)i(same)d(argumen)n(ts)g(as)i(ab)r(o)n(v)n(e,)g(one)g (concludes)e(that)-118 296 y(the)40 b(join)n(t)f(sp)r(ectrum)f(of)i (the)g(comm)n(utativ)n(e)35 b(family)i(is)i(concen)n(trated)f(on)-118 395 y(a)32 b(single)e(orbit,)j(the)g(represen)n(tation)d(space)i(is)g (generated)f(b)n(y)i FN(\016)s FO(-functions)-118 495 y(concen)n(trated)20 b(at)i(p)r(oin)n(ts)e(of)h(the)h(orbit,)g FN(U)30 b FO(acts)21 b(as)g(a)g(shift,)h(and)g(the)g(form)n(ula)-118 595 y(follo)n(ws.)p 2278 595 4 57 v 2282 542 50 4 v 2282 595 V 2331 595 4 57 v -118 760 a FB(R)l(emark)30 b(26.)42 b FO(Notice)28 b(that)h(w)n(e)g(do)f(not)h(assume)e(here)h(that)h(the)g (dynamical)-118 860 y(system)19 b(is)f(one-to-one.)33 b(Lo)r(oking)18 b(closely)e(at)k(the)g(sp)r(ectrum)f(of)27 b FN(C)1973 830 y FK(2)2010 860 y FO(,)22 b(one)d(sees)-118 959 y(that)26 b(it)e(can)h(just)h(b)r(e)g(a)f(c)n(hain)f(of)h(p)r(oin)n (ts,)g(or)f(a)h(c)n(hain)f(\\glued")f(to)j(a)f(cycle)f(or)-118 1059 y(stationary)g(p)r(oin)n(t.)36 b(In)27 b(the)h(case)e(of)h(the)g (mapping)e(on)i(the)g(sp)r(ectrum)g(b)r(eing)-118 1159 y(one-to-one,)f(only)g(c)n(hains)g(ma)n(y)g(arise.)-118 1290 y FB(Example)31 b(15.)42 b FO(\(Mapping)22 b(of)h(degree)e(t)n(w)n (o.)35 b(Con)n(tin)n(ued:)e FN(q)26 b(<)d(q)1899 1260 y FM(\003)1937 1290 y FO(\).)35 b(Since)22 b(for)-118 1390 y FN(q)k(<)d(q)73 1360 y FM(\003)111 1390 y FO(,)h(relation)c (\(2.3\))j(p)r(ossesses)f(only)f(a)i(\014nite)g(n)n(um)n(b)r(er)f(of)h (cycles,)f(the)i(set)-118 1490 y(of)30 b(p)r(erio)r(dic)e(p)r(oin)n(ts) h(is)g(closed;)h(therefore,)g(the)h(corresp)r(onding)c(dynamical)-118 1589 y(system)21 b(is)f(\\simple",)g(and)i(therefore,)g(all)d (in\014nite-dimensional)d(irreducible)-118 1689 y(represen)n(tations)h (with)j(unitary)e FN(U)29 b FO(are)20 b(describ)r(ed)f(b)n(y)h (sequences)f(of)h(p)r(ositiv)n(e)-118 1789 y(n)n(um)n(b)r(ers)30 b FN(\025)269 1801 y FL(k)310 1789 y FO(,)j FN(k)f FP(2)e FI(Z)o FO(,)d(suc)n(h)k(that)h FN(F)12 b FO(\()p FN(\025)1157 1801 y FL(k)1199 1789 y FO(\))29 b(=)h FN(\025)1403 1801 y FL(k)q FK(+1)1559 1789 y FO(for)h(all)f FN(k)k FO(according)29 b(to)-118 1888 y(\(2.17\))o(.)-118 2020 y FB(Example)i(16.)42 b FO(\(Tw)n(o-parameter)34 b(unit)i(quan)n(tum)g(disk.)63 b(Non-degenerate)-118 2120 y(case\).)d(F)-7 b(or)35 b(1)23 b FP(\000)g FN(\026)36 b FP(\024)g FN(q)k(<)35 b FO(1,)i(the)f (dynamical)c(system)i(\(2.16\))h(p)r(ossesses)-118 2219 y(a)d(family)d(of)j(b)r(ounded)h(p)r(ositiv)n(e)c(orbits.)49 b(T)-7 b(ak)n(e)31 b FN(\025)1503 2231 y FK(0)1572 2219 y FP(2)g FO(\()p FN(t)1720 2231 y FK(1)1757 2219 y FN(;)14 b FO(1\))32 b(as)g(an)f(initial)-118 2319 y(p)r(oin)n(t;)c(then)h FN(H)i FO(=)22 b FN(l)522 2331 y FK(2)559 2319 y FO(\()p FI(Z)p FO(\),)g(and)666 2509 y FN(z)t(e)748 2521 y FL(k)811 2509 y FO(=)898 2435 y Fy(p)p 981 2435 175 4 v 74 x FN(\025)1029 2521 y FL(k)q FM(\000)p FK(1)1170 2509 y FN(e)1209 2521 y FL(k)q FM(\000)p FK(1)1334 2509 y FN(;)628 2655 y(z)671 2621 y FM(\003)708 2655 y FN(e)747 2667 y FL(k)811 2655 y FO(=)898 2578 y Fy(p)p 981 2578 90 4 v 77 x FN(\025)1029 2667 y FL(k)1085 2655 y FN(e)1124 2667 y FL(k)1164 2655 y FN(;)180 b(k)26 b FP(2)d FI(Z)p FN(:)-118 2835 y FO(where)59 3063 y FN(\025)107 3075 y FL(k)171 3063 y FO(=)269 2998 y FN(\025)317 3010 y FK(0)354 2998 y FO(\()p FN(q)426 2968 y FL(k)486 2998 y FO(+)18 b FN(\026q)659 2968 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)904 2998 y FO([)p FN(k)s FO(])996 2975 y FM(p)p 1050 2975 33 3 v 1050 3011 a FL(q)1087 2998 y FO(\))h(+)f(1)g FP(\000)g FO(\()p FN(q)1436 2968 y FL(k)1495 2998 y FO(+)g FN(\026q)1668 2968 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)1913 2998 y FO([)p FN(k)s FO(])2005 2975 y FM(p)p 2060 2975 V 36 x FL(q)2096 2998 y FO(\))p 269 3043 1861 4 v 516 3121 a FN(\025)564 3133 y FK(0)601 3121 y FN(\026q)691 3097 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)936 3121 y FO([)p FN(k)s FO(])1028 3098 y FM(p)p 1083 3098 33 3 v 36 x FL(q)1138 3121 y FO(+)g(1)g FP(\000)g FN(\026q)1454 3097 y FK(\()p FL(k)q FM(\000)p FK(1\))p FL(=)p FK(2)1699 3121 y FO([)p FN(k)s FO(])1791 3098 y FM(p)p 1845 3098 V 1845 3134 a FL(q)2139 3063 y FN(:)-118 3305 y FO(Tw)n(o)30 b(suc)n(h)h(represen)n(tations)e(corresp)r(onding)f(to)j FN(\025)1533 3317 y FK(0)1602 3305 y FO(and)h FN(\025)1816 3274 y FM(0)1816 3325 y FK(0)1885 3305 y FO(are)e(unitarily)-118 3404 y(equiv)-5 b(alen)n(t)23 b(if)h(and)h(only)e(if)h FN(\025)806 3416 y FK(0)869 3404 y FO(and)h FN(\025)1076 3374 y FM(0)1076 3425 y FK(0)1138 3404 y FO(are)f(on)h(the)g(same)e (orbit.)35 b(Therefore,)-118 3504 y(for)k(the)g(measurable)d(section)i (one)h(can)g(tak)n(e)f(an)n(y)h(in)n(terv)-5 b(al)36 b(of)j(the)h(form)-118 3545 y Fy(\002)-84 3612 y FN(\025)-36 3624 y FK(0)2 3612 y FN(;)49 3572 y FK(\()p FL(q)r FK(+)p FL(\026)p FK(\))p FL(\025)263 3580 y Fx(0)297 3572 y FK(+1)p FM(\000)p FL(q)r FM(\000)p FL(\026)p 49 3593 509 4 v 159 3641 a(\026\025)238 3649 y Fx(0)271 3641 y FK(+1)p FM(\000)p FL(\026)567 3545 y Fy(\001)605 3612 y FO(.)6 3712 y(Also,)i(there)e(exists)e(a)i(family)c(of)k(un)n(b)r (ounded)g(p)r(ositiv)n(e)e(orbits)g(lying)-118 3811 y(to)d(the)h(righ)n (t)e(of)i(1.)60 b(The)36 b(set)f(of)h(suc)n(h)f(orbits)f(also)g(p)r (ossesses)g(a)h(measur-)-118 3911 y(able)h(section,)i(and)f(one)g(can)f (use)h(the)h(form)n(ula)c(ab)r(o)n(v)n(e)i(to)h(construct)f(the)p eop %%Page: 106 110 106 109 bop -118 -137 a FO(106)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(corresp)r(onding)33 b(class)i(of)h (irreducible)d(represen)n(tations.)60 b(Ho)n(w)n(ev)n(er,)37 b(these)-118 196 y(represen)n(tations)25 b(are)h(un)n(b)r(ounded.)6 313 y(If)h(the)g(dynamical)22 b(system)j(do)r(es)h(not)g(ha)n(v)n(e)f (a)h(measurable)d(section,)i(the)-118 413 y(represen)n(tations)g (listed)g(in)i(the)h(theorem)e(ab)r(o)n(v)n(e)h(do)g(not)h(form,)e(in)h (general,)-118 513 y(the)h(complete)d(list)h(of)i(irreducible)c (represen)n(tations.)-118 630 y FB(Example)31 b(17.)42 b FO(\(Mapping)20 b(of)g(degree)g(t)n(w)n(o.)33 b FN(q)27 b FP(\025)22 b FN(q)1460 600 y FM(\003)1498 630 y FO(\).)35 b(F)-7 b(or)20 b(an)n(y)g FN(q)26 b FP(\025)d FN(q)2071 600 y FM(\003)2109 630 y FO(,)f(there)-118 729 y(exists)29 b(an)i(in)n(v)-5 b(arian)n(t)28 b(set)j FN(K)36 b FO(homeomorphic)27 b(to)k(a)f(Can)n(tor)g(set)h(where)f(the)-118 829 y(mapping)23 b(is)i(one-to-one)f(and)h(whic)n(h)g(carries)d(a)k(non-trivial)21 b(ergo)r(dic)j(quasi-)-118 929 y(in)n(v)-5 b(arian)n(t)23 b(probabilit)n(y)g(measure)i FN(\026)p FO(.)37 b(Consider)24 b(the)j(space)f FN(L)1851 941 y FK(2)1888 929 y FO(\()p FN(K)q(;)14 b(d\026)p FO(\),)28 b(and)-118 1028 y(de\014ne)g FN(X)34 b FO(b)n(y)202 1172 y(\()p FN(X)7 b(f)i FO(\)\()p FN(\025)p FO(\))24 b(=)e FN(\025)663 1097 y Fy(p)p 747 1097 584 4 v 747 1172 a FN(d\026)p FO(\()p FN(F)12 b FO(\()p FN(\025)p FO(\)\))p FN(=d\026)p FO(\()p FN(\025)p FO(\))k FN(f)9 b FO(\()p FN(F)j FO(\()p FN(\025)p FO(\)\))p FN(;)164 1317 y FO(\()p FN(X)272 1282 y FM(\003)310 1317 y FN(f)d FO(\)\()p FN(\025)p FO(\))24 b(=)e FN(F)680 1282 y FM(\000)p FK(1)770 1317 y FO(\()p FN(\025)p FO(\))882 1242 y Fy(p)p 966 1242 673 4 v 966 1317 a FN(d\026)p FO(\()p FN(F)1156 1293 y FM(\000)p FK(1)1245 1317 y FO(\()p FN(\025)p FO(\)\))p FN(=d\026)p FO(\()p FN(\025)p FO(\))16 b FN(f)9 b FO(\()p FN(F)1799 1282 y FM(\000)p FK(1)1888 1317 y FO(\()p FN(\025)p FO(\)\))p FN(;)-118 1460 y FO(where)39 b FN(F)12 b FO(\()p FN(\025)p FO(\))45 b(=)e(\()p FN(\025)28 b FP(\000)e FN(q)s FO(\))735 1430 y FK(2)773 1460 y FO(.)74 b(Then)40 b(the)h(op)r(erators)d(satisfy)g(the)j(relation)-118 1560 y FN(X)7 b(X)34 1530 y FM(\003)116 1560 y FO(=)44 b(\()p FN(X)333 1530 y FM(\003)371 1560 y FN(X)33 b FP(\000)27 b FN(q)s(I)7 b FO(\))680 1530 y FK(2)718 1560 y FO(,)44 b(the)d(represen)n(tation)d(is)i(irreducible,)g(but)h(an)n(y)-118 1659 y(orbit)26 b(has)h(sp)r(ectral)f(measure)g(of)h FN(X)1039 1629 y FM(\003)1077 1659 y FN(X)34 b FO(equal)26 b(to)h(zero.)-118 1777 y FB(R)l(emark)j(27.)42 b FO(Notice)30 b(that)h(the)g(description)d(of)j(non-trivial)c(ergo)r(dic)h(mea-)-118 1876 y(sures)f(can)h(b)r(e)g(a)g(v)n(ery)f(complicated)e(task;)i(also,) g(there)h(can)f(b)r(e)i(lots)e(of)h(uni-)-118 1976 y(tarily)i(inequiv) -5 b(alen)n(t)30 b(irreducible)f(represen)n(tations)h(corresp)r(onding) g(to)j(the)-118 2075 y(same)26 b(non-trivial)d(ergo)r(dic)j(measure.) -118 2310 y FG(2.2)112 b(Some)42 b(classes)g(of)f FF(\003)p FG(-algebras)h(with)f(3)g(and)h(4)f(gen-)137 2426 y(erators)-118 2608 y FO(The)34 b(metho)r(d)f(dev)n(elop)r(ed)f(in)h(Section)f(2.1)h (can)h(b)r(e)g(carried)d(o)n(v)n(er)h(to)h(other)-118 2708 y(classes)j(of)i(op)r(erator)f(relations.)66 b(The)39 b(idea)e(b)r(ehind)h(this)g(is)f(to)i(consider)-118 2807 y(some)31 b(comm)n(utativ)n(e)d(family)i(of)i(self-adjoin)n(t)e(op)r (erators,)i(and)g(to)g(see)g(ho)n(w)-118 2907 y(other)27 b(op)r(erators)f(act)h(on)g(their)g(\(generalized\))d(eigen)n(v)n (ectors.)-118 3116 y FQ(2.2.1)94 b(Represen)m(tations)43 b(of)i(graded)g FN(so)p FO(\(3\))g FQ(and)h(four-tuples)e(of)174 3216 y(pro)5 b(jections)31 b(satisfying)g(a)i(linear)e(relation)-118 3369 y FO(In)24 b(this)e(section,)h(w)n(e)h(study)f(irreducible)d (represen)n(tations)h(of)i(a)g(graded)g(ana-)-118 3469 y(logue)35 b(of)j(the)g(Lie)e(algebra)e FN(so)p FO(\(3\),)41 b(and)c(sho)n(w)f(ho)n(w)h(they)h(are)e(related)g(to)-118 3568 y(four-tuples)19 b(of)h(pro)5 b(jections)18 b(satisfying)f(a)j (linear)d(relation)h(of)i(a)g(sp)r(ecial)d(t)n(yp)r(e.)6 3668 y(Consider)27 b(a)h(triple)e(of)j(b)r(ounded)f(self-adjoin)n(t)e (op)r(erators)h FN(A)p FO(,)i FN(B)t FO(,)g FN(C)34 b FO(sat-)-118 3767 y(isfying)25 b(the)j(follo)n(wing)c(relations)339 3911 y FP(f)p FN(A;)14 b(B)t FP(g)22 b FO(=)h FN(C)q(;)97 b FP(f)p FN(B)t(;)14 b(C)6 b FP(g)23 b FO(=)f FN(A;)97 b FP(f)p FN(C)q(;)14 b(A)p FP(g)23 b FO(=)g FN(B)t(;)258 b FO(\(2.18\))p eop %%Page: 107 111 107 110 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(107)-118 96 y(where)26 b FP(f)p FN(A;)14 b(B)t FP(g)23 b FO(=)f FN(AB)g FO(+)16 b FN(B)t(A)27 b FO(denotes)g(the)g(an)n(ticomm)n(utator)c(of)j(the)i(op)r(er-)-118 196 y(ators)d FN(A)i FO(and)g FN(B)t FO(.)36 b(As)27 b(w)n(as)f(men)n(tioned)e(ab)r(o)n(v)n(e,)i(suc)n(h)g(triples)f(of)h (self-adjoin)n(t)-118 296 y(op)r(erators)k(can)j(b)r(e)g(considered)d (as)i(represen)n(tations)e(of)i(the)h(graded)e FN(so)p FO(\(3\))-118 395 y(algebra.)g(W)-7 b(e)19 b(describ)r(e)f(all)e(suc)n (h)i(irreducible)e(families)e(up)19 b(to)g(unitary)e(equiv-)-118 495 y(alence.)6 597 y(On)35 b(the)g(other)e(hand,)k(consider)32 b(a)i(four-tuple)f(of)i(orthogonal)c(pro)5 b(jec-)-118 697 y(tions,)31 b FN(P)166 709 y FK(1)204 697 y FO(,)h FN(P)312 709 y FK(2)350 697 y FO(,)g FN(P)458 709 y FK(3)495 697 y FO(,)h FN(P)604 709 y FK(2)673 697 y FO(that)e(are)g(connected)g (b)n(y)g(a)g(linear)e(relation)f(of)j(the)-118 796 y(form)623 984 y FN(\013)14 b FO(\()p FN(P)775 996 y FK(1)831 984 y FO(+)k FN(P)967 996 y FK(2)1023 984 y FO(+)g FN(P)1159 996 y FK(3)1215 984 y FO(+)g FN(P)1351 996 y FK(4)1389 984 y FO(\))23 b(=)g FN(I)7 b(:)-118 1172 y FO(It)31 b(turns)g(out)f(that)h(the)h(unitary)d(description)f(of)j(suc)n(h)f (collections)d(of)k(pro-)-118 1272 y(jections)26 b(is)h(closely)e (related)h(to)h(represen)n(tations)e(of)34 b(\(2.18\))o(.)-118 1445 y FQ(Theorem)c(22.)41 b FB(A)n(ny)35 b(irr)l(e)l(ducible)h(family) h(of)f(self-adjoint)i(op)l(er)l(ators)e(sat-)-118 1545 y(isfying)43 b FO(\(2.18\))34 b FB(is)h(\014nite-dimensional.)55 b(F)-6 b(or)35 b(any)g FN(l)e FP(\025)f FO(1)i FB(ther)l(e)h(exist)f (four)-118 1644 y(irr)l(e)l(ducible)d(r)l(epr)l(esentations)f(of)h (dimension)g FO(2)p FN(l)r FB(,)e(and)h(\014ve)g(irr)l(e)l(ducible)h(r) l(ep-)-118 1744 y(r)l(esentations)e(of)i(dimension)g FO(2)p FN(l)19 b FP(\000)f FO(1)p FB(,)30 b(which)h(act)f(as)g(fol)t (lows.)6 1846 y FO(1)p FB(.)39 b(F)-6 b(our)29 b(r)l(epr)l(esentations) h(with)g(any)h(\014nite)e(dimension)i FO(dim)12 b FN(H)30 b FO(=)22 b FN(n)p FO(:)75 2069 y FN(Ae)176 2081 y FL(k)240 2069 y FO(=)h FN(\013)381 2081 y FK(1)432 2069 y FO(\()p FP(\000)p FO(1\))603 2034 y FL(k)q FK(+1)727 1976 y Fy(\020)777 2069 y FN(k)e FP(\000)934 2012 y FO(1)p 934 2049 42 4 v 934 2126 a(2)986 1976 y Fy(\021)1049 2069 y FN(e)1088 2081 y FL(k)1129 2069 y FN(;)183 b(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)74 2251 y(B)t(e)180 2263 y FK(1)240 2251 y FO(=)338 2194 y FN(\013)391 2206 y FK(1)428 2194 y FN(\013)481 2206 y FK(2)518 2194 y FN(n)p 338 2232 231 4 v 432 2308 a FO(2)592 2251 y FN(e)631 2263 y FK(1)686 2251 y FP(\000)779 2194 y FN(\013)832 2206 y FK(1)p 779 2232 91 4 v 804 2308 a FO(2)893 2169 y Fy(p)p 977 2169 231 4 v 977 2251 a FN(n)1027 2227 y FK(2)1082 2251 y FP(\000)k FO(1)13 b FN(e)1259 2263 y FK(2)1296 2251 y FN(;)70 2467 y(B)t(e)176 2479 y FL(k)240 2467 y FO(=)338 2411 y FN(\013)391 2423 y FK(1)442 2411 y FO(\()p FP(\000)p FO(1\))613 2381 y FL(k)q FM(\000)p FK(1)p 338 2448 401 4 v 517 2524 a FO(2)762 2392 y Fy(p)p 845 2392 480 4 v 75 x FN(n)895 2443 y FK(2)951 2467 y FP(\000)18 b FO(\()p FN(k)j FP(\000)d FO(1\))1287 2443 y FK(2)1338 2467 y FN(e)1377 2479 y FL(k)q FM(\000)p FK(1)320 2684 y FO(+)413 2628 y FN(\013)466 2640 y FK(1)504 2628 y FO(\()p FP(\000)p FO(1\))675 2598 y FL(k)p 413 2665 302 4 v 543 2741 a FO(2)739 2609 y Fy(p)p 822 2609 337 4 v 75 x FN(n)872 2660 y FK(2)928 2684 y FP(\000)g FO(\()p FN(k)s FO(\))1121 2660 y FK(2)1172 2684 y FN(e)1211 2696 y FL(k)q FK(+1)1336 2684 y FN(;)183 b(k)26 b FO(=)d(2)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FN(;)66 2898 y(B)t(e)172 2910 y FL(n)240 2898 y FO(=)338 2842 y FN(\013)391 2854 y FK(1)428 2842 y FO(\()p FP(\000)p FO(1\))599 2812 y FL(n)p FM(\000)p FK(1)p 338 2879 392 4 v 512 2955 a FO(2)753 2829 y FP(p)p 822 2829 235 4 v 69 x FO(2)p FN(n)g FP(\000)g FO(1)13 b FN(e)1109 2910 y FL(n)p FM(\000)p FK(1)1239 2898 y FN(;)76 3098 y(C)6 b(e)180 3110 y FK(1)240 3098 y FO(=)338 3042 y FN(\013)391 3054 y FK(2)428 3042 y FN(n)p 338 3079 141 4 v 387 3155 a FO(2)502 3098 y FN(e)541 3110 y FK(1)596 3098 y FO(+)689 3042 y(1)p 689 3079 42 4 v 689 3155 a(2)754 3016 y Fy(p)p 837 3016 231 4 v 82 x FN(n)887 3074 y FK(2)943 3098 y FP(\000)18 b FO(1)13 b FN(e)1120 3110 y FK(2)1157 3098 y FN(;)72 3298 y(C)6 b(e)176 3310 y FL(k)240 3298 y FO(=)338 3242 y(1)p 338 3279 42 4 v 338 3355 a(2)403 3223 y Fy(p)p 486 3223 480 4 v 75 x FN(n)536 3274 y FK(2)592 3298 y FP(\000)18 b FO(\()p FN(k)j FP(\000)d FO(1\))928 3274 y FK(2)979 3298 y FN(e)1018 3310 y FL(k)q FM(\000)p FK(1)320 3498 y FO(+)413 3441 y(1)p 413 3479 42 4 v 413 3555 a(2)479 3423 y Fy(p)p 562 3423 337 4 v 75 x FN(n)612 3474 y FK(2)667 3498 y FP(\000)g FO(\()p FN(k)s FO(\))860 3474 y FK(2)912 3498 y FN(e)951 3510 y FL(k)q FK(+1)1075 3498 y FN(;)184 b(k)26 b FO(=)c(2)p FN(;)14 b(:)g(:)g(:)g(;)g(n)k FP(\000)g FO(1)p FN(;)68 3697 y(C)6 b(e)172 3709 y FL(n)240 3697 y FO(=)338 3641 y(1)p 338 3678 42 4 v 338 3754 a(2)403 3628 y FP(p)p 472 3628 235 4 v 69 x FO(2)p FN(n)18 b FP(\000)g FO(1)13 b FN(e)759 3709 y FL(n)p FM(\000)p FK(1)889 3697 y FN(;)184 b FB(wher)l(e)30 b FN(\013)1383 3709 y FK(1)1421 3697 y FB(,)g FN(\013)1529 3709 y FK(2)1589 3697 y FO(=)23 b FP(\006)p FO(1)o(;)6 3911 y(2)p FB(.)38 b(A)n(n)28 b(additional)j(r)l(epr)l(esentation)e(for)h(e)l(ach)f(o)l (dd)h(value)f(of)g(the)g(dimen-)p eop %%Page: 108 112 108 111 bop -118 -137 a FO(108)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(sion,)31 b FO(dim)12 b FN(H)30 b FO(=)22 b FN(n)h FO(=)g(2)p FN(l)c FP(\000)f FO(1:)93 322 y FN(Ae)194 334 y FL(k)258 322 y FO(=)k(\()p FP(\000)p FO(1\))516 287 y FL(k)q FK(+1)665 266 y FN(n)c FO(+)g(1)g FP(\000)g FO(2)p FN(k)p 665 303 382 4 v 835 379 a FO(2)1070 322 y FN(e)1109 334 y FL(k)1149 322 y FN(;)184 b(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)91 522 y(B)t(e)197 534 y FK(1)258 522 y FO(=)355 465 y(1)p 355 502 42 4 v 355 578 a(2)420 452 y FP(p)p 490 452 193 4 v 490 522 a FN(n)k FP(\000)g FO(1)13 b FN(e)735 534 y FK(2)772 522 y FN(;)88 738 y(B)t(e)194 750 y FL(k)258 738 y FO(=)355 682 y(\()p FP(\000)p FO(1\))526 652 y FL(k)p 355 719 212 4 v 440 795 a FO(2)590 663 y Fy(p)p 673 663 659 4 v 75 x FO(\()p FN(k)22 b FP(\000)c FO(1\)\()p FN(n)g FP(\000)g FN(k)k FO(+)c(1\))13 b FN(e)1384 750 y FL(k)q FM(\000)p FK(1)338 955 y FO(+)431 899 y(\()p FP(\000)p FO(1\))602 869 y FL(k)q FK(+1)p 431 936 296 4 v 558 1012 a FO(2)750 880 y Fy(p)p 833 880 308 4 v 75 x FN(k)s FO(\()p FN(n)19 b FP(\000)f FN(k)s FO(\))c FN(e)1194 967 y FL(k)q FK(+1)1318 955 y FN(;)184 b(k)26 b FO(=)c(2)p FN(;)14 b(:)g(:)g(:)f(;)h(n)19 b FP(\000)f FO(1)p FN(;)83 1155 y(B)t(e)189 1167 y FL(n)258 1155 y FO(=)k FP(\000)420 1099 y FO(1)p 420 1136 42 4 v 420 1212 a(2)485 1086 y FP(p)p 554 1086 193 4 v 69 x FN(n)c FP(\000)h FO(1)13 b FN(e)800 1167 y FL(n)p FM(\000)p FK(1)930 1155 y FN(;)93 1355 y(C)6 b(e)197 1367 y FK(1)258 1355 y FO(=)355 1299 y(1)p 355 1336 42 4 v 355 1412 a(2)420 1286 y FP(p)p 490 1286 193 4 v 490 1355 a FN(n)18 b FP(\000)g FO(1)13 b FN(e)735 1367 y FK(2)772 1355 y FN(;)90 1555 y(C)6 b(e)194 1567 y FL(k)258 1555 y FO(=)355 1498 y(1)p 355 1536 42 4 v 355 1612 a(2)420 1480 y Fy(p)p 503 1480 659 4 v 75 x FO(\()p FN(k)22 b FP(\000)c FO(1\)\()p FN(n)g FP(\000)g FN(k)k FO(+)c(1\))13 b FN(e)1214 1567 y FL(k)q FM(\000)p FK(1)338 1754 y FO(+)431 1698 y(1)p 431 1735 42 4 v 431 1811 a(2)496 1679 y Fy(p)p 579 1679 308 4 v 75 x FN(k)s FO(\()p FN(n)19 b FP(\000)f FN(k)s FO(\))13 b FN(e)939 1766 y FL(k)q FK(+1)1064 1754 y FN(;)184 b(k)26 b FO(=)c(2)p FN(;)14 b(:)g(:)g(:)f(;)h(n)19 b FP(\000)f FO(1)p FN(;)85 1954 y(C)6 b(e)189 1966 y FL(n)258 1954 y FO(=)355 1898 y(1)p 355 1935 42 4 v 355 2011 a(2)420 1885 y FP(p)p 490 1885 193 4 v 490 1954 a FN(n)18 b FP(\000)g FO(1)13 b FN(e)735 1966 y FL(n)p FM(\000)p FK(1)865 1954 y FN(:)-118 2166 y FB(Pr)l(o)l(of.)43 b FO(The)37 b(pro)r(of)g(of)g (the)h(theorem)e(is)g(essen)n(tially)e(based)i(on)i(the)f(same)-118 2266 y(ideas)g(as)h(the)h(pro)r(of)f(for)g(non-graded)f FN(so)p FO(\(3\).)70 b(In)n(tro)r(duce)39 b(the)g(op)r(erators)-118 2365 y FN(E)-57 2377 y FK(0)15 2365 y FO(=)33 b FN(A)p FO(,)k FN(E)296 2377 y FK(1)367 2365 y FO(=)d FN(B)27 b FO(+)c FN(C)6 b FO(,)36 b FN(E)829 2377 y FK(2)901 2365 y FO(=)d FN(C)d FP(\000)22 b FN(B)t FO(.)57 b(It)35 b(is)e(easy)g(to)h(c)n(hec)n(k)g(that)g(the)-118 2465 y(relations)24 b(are)j(equiv)-5 b(alen)n(t)25 b(to)28 2652 y FP(f)p FN(E)131 2664 y FK(0)168 2652 y FN(;)14 b(E)266 2664 y FK(1)303 2652 y FP(g)23 b FO(=)g FN(E)517 2664 y FK(1)554 2652 y FN(;)97 b FP(f)p FN(E)777 2664 y FK(0)814 2652 y FN(;)14 b(E)912 2664 y FK(2)949 2652 y FP(g)23 b FO(=)g FP(\000)p FN(E)1228 2664 y FK(2)1265 2652 y FN(;)97 b(E)1451 2617 y FK(2)1446 2672 y(1)1506 2652 y FP(\000)18 b FN(E)1655 2617 y FK(2)1650 2672 y(2)1716 2652 y FO(=)k(2)p FN(E)1906 2664 y FK(0)1944 2652 y FN(;)159 b FO(\(2.19\))-118 2838 y(and)27 b(that)h(the)g(op)r(erator)147 3025 y(\001)23 b(=)g FN(A)389 2991 y FK(2)445 3025 y FO(+)18 b FN(B)595 2991 y FK(2)651 3025 y FO(+)g FN(C)799 2991 y FK(2)859 3025 y FO(=)23 b FN(E)1013 2991 y FK(2)1008 3046 y(0)1069 3025 y FO(+)18 b FN(E)1218 2991 y FK(2)1213 3046 y(1)1273 3025 y FP(\000)g FN(E)1417 3037 y FK(0)1478 3025 y FO(=)23 b FN(E)1632 2991 y FK(2)1627 3046 y(0)1687 3025 y FO(+)18 b FN(E)1836 2991 y FK(2)1831 3046 y(2)1892 3025 y FO(+)g FN(E)2036 3037 y FK(0)-118 3212 y FO(comm)n(utes)32 b(with)j FN(A)p FO(,)i FN(B)t FO(,)g FN(C)6 b FO(.)60 b(This)34 b(implies,)f(in)h(particular,)g(that)h(an)n(y)f(ir-)-118 3311 y(reducible)d(represen)n(tation)g(is)h(b)r(ounded,)j(since)d FN(A)1521 3281 y FK(2)1559 3311 y FO(,)j FN(B)1684 3281 y FK(2)1721 3311 y FO(,)g FN(C)1844 3281 y FK(2)1915 3311 y FO(are)d(p)r(ositiv)n(e)-118 3411 y(op)r(erators,)26 b(and)h(their)g(sum)f(is)h(a)g(m)n(ultiple)d(of)k(the)g(iden)n(tit)n(y) -7 b(.)6 3513 y(One)29 b(can)g(easily)d(see)j(from)f(relations)e (\(2.19\))i(that)i FN(E)1724 3525 y FK(1)1791 3513 y FO(maps)e(an)g(eigen-)-118 3612 y(v)n(ector)d FN(e)169 3624 y FL(\025)239 3612 y FO(of)h FN(E)393 3624 y FK(0)457 3612 y FO(to)h(an)f(eigen)n(v)n(ector)d FN(e)1141 3624 y FK(1)p FM(\000)p FL(\025)1269 3612 y FO(,)k(and)f FN(E)1540 3624 y FK(2)1604 3612 y FO(maps)g FN(e)1860 3624 y FL(\025)1929 3612 y FO(in)n(to)f FN(e)2135 3624 y FM(\000)p FK(1)p FM(\000)p FL(\025)2316 3612 y FO(.)-118 3712 y(Then)i(w)n(e)g(ha)n(v)n (e)e(a)i(dynamical)c(system)j(on)h FI(R)32 b FO(generated)26 b(b)n(y)h(t)n(w)n(o)f(\015ips)g(with)-118 3811 y(resp)r(ect)35 b(to)f(the)h(p)r(oin)n(ts)f(1)p FN(=)p FO(2,)i(and)e FP(\000)p FO(1)p FN(=)p FO(2.)57 b(F)-7 b(or)34 b(an)h(irreducible)c (represen-)-118 3911 y(tation,)f(the)h(sp)r(ectral)f(measure)e(of)j FN(E)1100 3923 y FK(0)1168 3911 y FO(m)n(ust)f(b)r(e)h(ergo)r(dic)e (with)h(resp)r(ect)h(to)p eop %%Page: 109 113 109 112 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(109)-118 96 y(the)35 b(action)f(of)h(this)g (dynamical)c(system,)36 b(since,)g(otherwise,)f(an)n(y)g(ergo)r(dic) -118 196 y(comp)r(onen)n(t)27 b(generates)g(an)h(in)n(v)-5 b(arian)n(t)25 b(subspace.)39 b(The)28 b(dynamical)d(system)-118 296 y(p)r(ossesses)c(a)i(measurable)c(section,)k(i.e.,)g(a)f(set)h (that)g(meets)f(ev)n(ery)g(orbit)f(only)-118 395 y(once.)61 b(F)-7 b(or)36 b(suc)n(h)f(a)h(set)g(one)f(can)h(tak)n(e)f([)p FP(\000)p FO(1)p FN(=)p FO(2)p FN(;)14 b FO(1)p FN(=)p FO(2].)58 b(Then)37 b(an)n(y)e(ergo)r(dic)-118 495 y(measure)25 b(is)i(concen)n(trated)f(at)i(a)f(single)e(orbit)h(of)i(some)e(p)r(oin) n(t.)6 596 y(Th)n(us,)k(the)f(sp)r(ectral)e(measure)g(of)i FN(E)1181 608 y FK(0)1248 596 y FO(is)f(discrete,)g(and)h(w)n(e)f(can)h (c)n(ho)r(ose)-118 696 y(a)e(basis)f(consisting)f(of)i(its)g(eigen)n(v) n(ectors.)34 b(Then)27 b(w)n(e)h(ha)n(v)n(e:)75 882 y FN(E)136 894 y FK(0)174 882 y FN(e)213 894 y FL(\025)279 882 y FO(=)23 b FN(\025e)454 894 y FL(\025)497 882 y FN(;)97 b(E)678 894 y FK(1)716 882 y FN(e)755 894 y FL(\025)821 882 y FO(=)23 b FN(a)953 894 y FK(1)990 882 y FO(\()p FN(\025)p FO(\))14 b FN(e)1155 894 y FK(1)p FM(\000)p FL(\025)1284 882 y FN(;)97 b(E)1465 894 y FK(2)1502 882 y FN(e)1541 894 y FL(\025)1607 882 y FO(=)23 b FN(a)1739 894 y FK(2)1776 882 y FO(\()p FN(\025)p FO(\))14 b FN(e)1941 894 y FM(\000)p FK(1)p FM(\000)p FL(\025)2122 882 y FN(;)-118 1068 y FO(where)31 b FN(\025)i FO(are)e(tak)n(en)h(from)e(some)h (orbit.)49 b(It)32 b(remains)e(to)i(\014nd)g(a)g(condition)-118 1168 y(on)h FN(a)47 1180 y FK(1)85 1168 y FO(,)i FN(a)187 1180 y FK(2)258 1168 y FO(so)e(that)h(the)g(relation)d FN(E)1078 1138 y FK(2)1073 1189 y(1)1138 1168 y FP(\000)22 b FN(E)1291 1138 y FK(2)1286 1189 y(2)1362 1168 y FO(=)33 b(2)p FN(E)1563 1180 y FK(0)1634 1168 y FO(w)n(ould)f(hold,)i(and)g(to) -118 1268 y(c)n(hec)n(k)c(whether)h(the)h(these)f(conditions)e(can)i(b) r(e)g(satis\014ed,)g(and)g(the)h(repre-)-118 1367 y(sen)n(tation)27 b(is)h(irreducible)d(\(the)k(ergo)r(dicit)n(y)d(is)i(a)g(necessary)f (condition,)g(but)-118 1467 y(not)g(su\016cien)n(t)g(in)g(general\).)6 1568 y(One)g(can)g(easily)e(c)n(hec)n(k)h(that)i(necessary)e(and)h (su\016cien)n(t)f(conditions)f(for)-118 1668 y FN(a)-74 1680 y FK(1)-37 1668 y FO(,)j FN(a)58 1680 y FK(2)123 1668 y FO(to)f(form)f(a)h(represen)n(tation)e(are)i(the)h(follo)n (wing:)-29 1854 y FN(a)15 1866 y FK(1)52 1854 y FO(\(1)18 b FP(\000)g FN(\025)p FO(\))24 b(=)p 419 1782 195 4 v 23 w FN(a)463 1866 y FK(1)500 1854 y FO(\()p FN(\025)p FO(\))q FN(;)97 b(a)777 1866 y FK(2)814 1854 y FO(\()p FP(\000)p FO(1)18 b FP(\000)g FN(\025)p FO(\))24 b(=)p 1245 1782 V 22 w FN(a)1289 1866 y FK(2)1326 1854 y FO(\()p FN(\025)p FO(\))q FN(;)97 b(a)1603 1820 y FK(2)1603 1875 y(1)1640 1854 y FO(\()p FN(\025)p FO(\))20 b FP(\000)e FN(a)1899 1820 y FK(2)1899 1875 y(2)1936 1854 y FO(\()p FN(\025)p FO(\))24 b(=)f(2)p FN(\025)-118 2041 y FO(for)h(almost)e(all) g FN(\025)j FO(tak)n(en)f(with)g(resp)r(ect)g(to)g(the)h(sp)r(ectral)e (measure)g(of)h FN(E)2166 2053 y FK(0)2204 2041 y FO(.)35 b(In)-118 2140 y(particular,)d(these)i(relations)d(imply)g(that)j(b)r (oth)h FN(a)1531 2110 y FK(2)1531 2161 y(1)1602 2140 y FO(and)f FN(a)1814 2110 y FK(2)1814 2161 y(2)1885 2140 y FO(are)f(uniquely)-118 2240 y(determined)21 b(b)n(y)i(the)h(v)-5 b(alue)22 b(of)h FN(a)905 2252 y FK(1)966 2240 y FO(at)g(a)g(single)d (p)r(oin)n(t)j(of)g(a)g(non-zero)e(sp)r(ectral)-118 2339 y(measure.)35 b(Actually)-7 b(,)26 b(there)h(exists)f FN(\036)e(>)e FO(0)28 b(suc)n(h)f(that)154 2526 y FN(a)198 2491 y FK(2)198 2546 y(1)235 2526 y FO(\()p FN(\025)p FO(\))d(=)f FP(\000)p FO(\()p FN(\025)18 b FP(\000)g FO(1)p FN(=)p FO(2\))863 2491 y FK(2)918 2526 y FO(+)g FN(\036;)97 b(a)1214 2491 y FK(2)1214 2546 y(2)1251 2526 y FO(\()p FN(\025)p FO(\))24 b(=)f FP(\000)p FO(\()p FN(\025)c FO(+)f(1)p FN(=)p FO(2\))1880 2491 y FK(2)1934 2526 y FO(+)g FN(\036)-118 2712 y FO(on)27 b(the)h(sp)r(ectrum)f(of)g FN(E)658 2724 y FK(0)696 2712 y FO(.)6 2813 y(But)35 b(the)g(latter)e(relations)f(cannot)i(hold)f(for)h(all)e(p)r(oin)n(ts)i (of)g(the)h(orbit,)-118 2913 y(since)d(in)h(the)h(righ)n(t-hand)e (sides)g(w)n(e)h(ha)n(v)n(e)g(functions)f(decreasing)f(to)j FP(\0001)p FO(,)-118 3013 y(while)22 b(the)i(left-hand)f(sides)f(m)n (ust)h(b)r(e)i(non-negativ)n(e.)33 b(T)-7 b(o)23 b(a)n(v)n(oid)e(the)j (con)n(tra-)-118 3112 y(diction,)c(w)n(e)h(need)g(to)g(demand)f(that,)i (on)f(the)g(highest)f(v)n(ector)f(\(the)j(v)n(ector)d FN(e)2295 3124 y FL(\025)-118 3212 y FO(with)28 b(the)g(largest)e FN(\025)p FO(\),)j(the)g(op)r(erator)d FN(E)1156 3224 y FK(2)1222 3212 y FO(acts)i(as)f(zero,)g(and)h(on)g(the)h(lo)n(w)n (est)-118 3311 y(v)n(ector,)e(the)i(op)r(erator)e FN(E)696 3323 y FK(1)762 3311 y FO(is)g(zero.)39 b(This)27 b(means)g(that)i (zero)r(es)e(of)h(b)r(oth)h(the)-118 3411 y(functions)e FN(a)284 3423 y FK(1)349 3411 y FO(and)g FN(a)554 3423 y FK(2)619 3411 y FO(m)n(ust)g(b)r(elong)f(to)h(the)h(same)e(orbit.)6 3513 y(One)k(can)g(easily)d(c)n(hec)n(k)i(that)h(these)g(conditions)e (can)h(b)r(e)i(satis\014ed)d(only)-118 3612 y(for)k(three)g(orbits,)g (corresp)r(onding)d(to)k(the)g(p)r(oin)n(ts)e(0,)i(and)f FP(\006)p FO(1)p FN(=)p FO(2.)50 b(In)32 b(eac)n(h)-118 3712 y(case,)i(only)d(a)i(discrete)f(n)n(um)n(b)r(er)g(of)h(v)-5 b(alues)32 b(of)h FN(\036)h FO(are)e(admissible,)e(one)j(for)-118 3811 y(eac)n(h)26 b(dimension.)34 b(Also,)25 b(the)j(represen)n(tation) c(corresp)r(onding)g(to)j(the)g(orbit)-118 3911 y(that)h(con)n(tains)d (zero)i(has)g(an)g(o)r(dd)h(dimension.)p eop %%Page: 110 114 110 113 bop -118 -137 a FO(110)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(T)-7 b(o)21 b(complete)d(the)j(pro)r(of,)h(it)e (remains)e(to)j(notice)e(that)i(for)f(an)n(y)g(orbit)f(con-)-118 196 y(taining)31 b(zero)i(there)h(corresp)r(onds)e(a)h(unique)g (irreducible)d(represen)n(tation,)-118 296 y(while)e(the)i(orbits)e(of) i FP(\006)p FO(1)p FN(=)p FO(2)e(carry)h(t)n(w)n(o)g(represen)n (tations)e(eac)n(h,)i(dep)r(ending)-118 395 y(on)f(the)g(sign)e(of)i FN(a)451 407 y FK(1)488 395 y FO(\(1)p FN(=)p FO(2\))f(or)h FN(a)852 407 y FK(2)889 395 y FO(\()p FP(\000)p FO(1)p FN(=)p FO(2\).)36 b(Finding)27 b(admissible)c(v)-5 b(alues)27 b(for)g FN(\036)-118 495 y FO(and)33 b(restoring)e(the)j(op)r(erators)d FN(A)p FO(,)k FN(B)t FO(,)h FN(C)j FO(from)33 b FN(E)1532 507 y FK(0)1569 495 y FO(,)i FN(E)1688 507 y FK(1)1726 495 y FO(,)g FN(E)1845 507 y FK(2)1916 495 y FO(is)d(a)h(routine)-118 595 y(calculation.)p 2278 595 4 57 v 2282 542 50 4 v 2282 595 V 2331 595 4 57 v 6 758 a(No)n(w)24 b(w)n(e)g(consider)e (another)i(problem)d(that)k(app)r(eared)e(to)h(b)r(e)h(related)e(to) -118 857 y(the)28 b(one)f(just)h(considered.)6 957 y(The)k(problem)d (is)h(related)g(to)h(general)e(non-orthogonal)f(resolutions)g(of)-118 1056 y(the)h(iden)n(tit)n(y)-7 b(.)40 b(Let)29 b FN(A)575 1068 y FK(1)612 1056 y FO(,)h FN(:)14 b(:)g(:)27 b FO(,)j FN(A)904 1068 y FL(n)978 1056 y FO(b)r(e)g(p)r(ositiv)n(e)c (self-adjoin)n(t)h(op)r(erators)g(in)h(a)-118 1156 y (\014nite-dimensional)22 b(space)27 b FN(H)34 b FO(suc)n(h)28 b(that)943 1282 y FL(n)903 1306 y Fy(X)903 1485 y FL(k)q FK(=1)1038 1385 y FN(A)1100 1397 y FL(k)1164 1385 y FO(=)22 b FN(I)7 b(:)-118 1627 y FO(W)-7 b(riting)24 b(the)i(sp)r(ectral)e (decomp)r(osition)e(for)k(eac)n(h)f(of)h(the)g(op)r(erators)e FN(A)2154 1639 y FL(k)2195 1627 y FO(,)i(w)n(e)-118 1727 y(see)h(that)h(the)g(latter)e(sum)h(can)g(b)r(e)h(rewritten)e(as)607 1836 y FL(m)576 1861 y Fy(X)584 2040 y FL(l)p FK(=1)710 1940 y FN(\013)763 1952 y FL(l)789 1940 y FN(P)842 1952 y FL(l)891 1940 y FO(=)c FN(I)7 b(;)180 b FO(0)23 b FN(<)f(\013)1429 1952 y FL(l)1478 1940 y FP(\024)h FO(1)p FN(;)495 b FO(\(2.20\))-118 2182 y(where)32 b FN(P)180 2194 y FL(l)239 2182 y FO(are)f(orthogonal)f (pro)5 b(jections.)50 b(If)33 b(all)d FN(\013)1539 2194 y FL(l)1596 2182 y FO(=)h(1,)j(the)f(pro)5 b(jections)-118 2281 y(comm)n(ute,)25 b(and)j(w)n(e)f(get)g(an)h(ordinary)d(resolution) f(of)k(the)g(iden)n(tit)n(y)-7 b(.)6 2381 y(The)26 b(complexit)n(y)21 b(of)k(the)h(description)c(problem)h(for)i(families)c(of)k(pro)5 b(jec-)-118 2481 y(tions)27 b(that)i(form)f(a)g(non-orthogonal)d (resolution)h(of)i(the)h(iden)n(tit)n(y)e(dep)r(ends)-118 2580 y(on)g(the)h(n)n(um)n(b)r(er)f(of)g(the)h(pro)5 b(jections)25 b(in)i(\(2.20\))o(.)6 2680 y(Tw)n(o)d(pro)5 b(jections)22 b(satisfying)i(\(2.20\))f(are)h(orthogonal;)e(therefore,) j FN(\013)2214 2692 y FK(1)2274 2680 y FO(=)-118 2779 y FN(\013)-65 2791 y FK(2)-5 2779 y FO(=)e(1.)6 2879 y(A)40 b(triple)e(of)h(pro)5 b(jections)38 b(satisfying)g(\(2.20\))g (has)h(only)f(one-)h(or)g(t)n(w)n(o-)-118 2979 y(dimensional)23 b(irreducible)h(represen)n(tations.)6 3078 y(The)36 b(problem)c(of)j (the)h(unitary)d(description)g(of)i(\014v)n(e)g(or)f(more)f(pro)5 b(jec-)-118 3178 y(tions)24 b(satisfying)f(\(2.20\),)i(is)f(v)n(ery)g (complicated)e(\(\\wild",)h(see)i(Section)f(3.1.3)-118 3278 y(b)r(elo)n(w\).)6 3377 y(Consider)h(four)i(orthogonal)c(pro)5 b(jections,)25 b FN(P)1442 3389 y FK(1)1479 3377 y FO(,)i FN(P)1582 3389 y FK(2)1620 3377 y FO(,)g FN(P)1723 3389 y FK(3)1761 3377 y FO(,)g FN(P)1864 3389 y FK(4)1928 3377 y FO(that)g(satisfy)-118 3477 y(a)g(sp)r(ecial)e(case)i(of)h(the)g (relation)c(\(2.20\))j(with)g(all)f FN(\013)1537 3489 y FL(l)1585 3477 y FO(=)d FN(\013)p FO(,)630 3644 y FN(\013)p FO(\()p FN(P)768 3656 y FK(1)824 3644 y FO(+)18 b FN(P)960 3656 y FK(2)1016 3644 y FO(+)g FN(P)1152 3656 y FK(3)1208 3644 y FO(+)g FN(P)1344 3656 y FK(4)1382 3644 y FO(\))23 b(=)g FN(I)7 b(:)535 b FO(\(2.21\))-118 3811 y(W)-7 b(e)25 b(study)g(for)f(whic)n(h)f FN(\013)i FO(solutions)d(exist,)i(and)g(giv) n(e)f(their)g(unitary)h(descrip-)-118 3911 y(tion.)p eop %%Page: 111 115 111 114 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(111)-118 96 y FQ(Theorem)30 b(23.)41 b FB(Solutions)34 b(of)53 b FO(\(2.21\))34 b FB(exist)g(for)h(the)g (fol)t(lowing)i(values)e(of)-118 196 y FN(\013)p FO(:)6 296 y(1)p FB(.)71 b FN(\013)43 b FO(=)358 263 y FK(1)p 358 277 34 4 v 358 324 a(2)401 296 y FB(.)71 b(Ther)l(e)41 b(ar)l(e)g(six)g(one-dimensional,)k(and)c(a)g(c)l(ontinuous)-118 395 y(family)31 b(of)g(two-dimensional)g(r)l(epr)l(esentations)7 b FO(;)6 495 y(2)p FB(.)53 b FN(\013)32 b FO(=)317 462 y FK(1)p 317 476 V 317 524 a(2)382 495 y FP(\000)496 462 y FK(1)p 478 476 70 4 v 478 524 a(4)p FL(k)558 495 y FB(,)k FN(k)e FP(2)e FI(N)t FB(.)59 b(Ther)l(e)35 b(is)g(one)f(irr)l (e)l(ducible)i(r)l(epr)l(esentation)-118 595 y(with)30 b FO(dim)12 b FN(H)30 b FO(=)23 b(2)p FN(k)e FP(\000)d FO(1;)6 694 y(3)p FB(.)53 b FN(\013)32 b FO(=)317 661 y FK(1)p 317 675 34 4 v 317 723 a(2)382 694 y FO(+)496 661 y FK(1)p 478 675 70 4 v 478 723 a(4)p FL(k)558 694 y FB(,)k FN(k)e FP(2)e FI(N)t FB(.)59 b(Ther)l(e)35 b(is)g(one)f(irr)l (e)l(ducible)i(r)l(epr)l(esentation)-118 794 y(with)30 b FO(dim)12 b FN(H)30 b FO(=)23 b(2)p FN(k)e FO(+)d(1;)6 893 y(4)p FB(.)46 b FN(\013)28 b FO(=)302 861 y FK(1)p 302 875 34 4 v 302 922 a(2)365 893 y FP(\000)521 861 y FK(1)p 460 875 154 4 v 460 922 a(4)p FL(k)q FK(+2)624 893 y FB(,)33 b FN(k)e FP(2)d FI(N)t FB(.)52 b(Ther)l(e)33 b(ar)l(e)f(four)h(irr)l(e)l(ducible)h(r)l(epr)l(esenta-)-118 993 y(tions)c(with)g FO(dim)12 b FN(H)30 b FO(=)23 b FN(k)s FO(;)6 1093 y(5)p FB(.)46 b FN(\013)28 b FO(=)302 1060 y FK(1)p 302 1074 34 4 v 302 1121 a(2)365 1093 y FO(+)521 1060 y FK(1)p 460 1074 155 4 v 460 1121 a(4)p FL(k)q FM(\000)p FK(2)625 1093 y FB(,)33 b FN(k)d FP(2)e FI(N)t FB(.)52 b(Ther)l(e)33 b(ar)l(e)g(four)f(irr)l(e)l(ducible)i(r)l (epr)l(esenta-)-118 1192 y(tions)c(with)g FO(dim)12 b FN(H)30 b FO(=)23 b FN(k)s FB(.)-118 1356 y(Pr)l(o)l(of.)43 b FO(In)n(tro)r(duce)28 b(self-adjoin)n(t)e(unitary)h(op)r(erators)f FN(R)1672 1368 y FL(i)1724 1356 y FO(=)e(2)p FN(P)1908 1368 y FL(i)1954 1356 y FP(\000)19 b FN(I)7 b FO(,)28 b FN(i)c FO(=)g(1,)-118 1456 y FN(:)14 b(:)g(:)27 b FO(,)h(4.)37 b(Then)696 1607 y FK(4)652 1632 y Fy(X)658 1808 y FL(i)p FK(=1)786 1710 y FN(R)849 1722 y FL(i)900 1710 y FO(=)997 1654 y(2)18 b FP(\000)g FO(4)p FN(\013)p 997 1691 238 4 v 1090 1767 a(\013)1259 1710 y(I)30 b FO(=)23 b(2)p FN(hI)7 b(;)-118 1968 y FO(where)27 b(w)n(e)g(put)h FN(h)23 b FO(=)g(\(1)18 b FP(\000)g FO(2)p FN(\013)p FO(\))p FN(=\013)p FO(.)6 2067 y(First)27 b(consider)e(the)j(case)f FN(\013)d FO(=)e(1)p FN(=)p FO(2)k(\()p FN(h)e FO(=)e(0\).)37 b(In)n(tro)r(duce)27 b(the)h(elemen)n(ts)-21 2248 y FN(X)48 2260 y FK(1)108 2248 y FO(=)23 b(\()p FN(R)291 2260 y FK(1)347 2248 y FO(+)18 b FN(R)493 2260 y FK(4)530 2248 y FO(\))p FN(=)p FO(2)p FN(;)97 b(X)835 2260 y FK(2)895 2248 y FO(=)22 b(\()p FN(R)1077 2260 y FK(2)1133 2248 y FO(+)c FN(R)1279 2260 y FK(4)1317 2248 y FO(\))p FN(=)p FO(2)p FN(;)96 b(X)1621 2260 y FK(3)1681 2248 y FO(=)23 b(\()p FN(R)1864 2260 y FK(3)1920 2248 y FO(+)18 b FN(R)2066 2260 y FK(4)2103 2248 y FO(\))p FN(=)p FO(2)p FN(:)-118 2428 y FO(Then)150 2608 y FN(R)213 2620 y FK(1)273 2608 y FO(=)23 b FN(X)430 2620 y FK(1)485 2608 y FP(\000)18 b FN(X)637 2620 y FK(2)693 2608 y FP(\000)g FN(X)845 2620 y FK(3)882 2608 y FN(;)332 b(R)1300 2620 y FK(3)1361 2608 y FO(=)22 b FP(\000)p FN(X)1582 2620 y FK(1)1637 2608 y FP(\000)c FN(X)1789 2620 y FK(2)1845 2608 y FO(+)g FN(X)1997 2620 y FK(3)2034 2608 y FN(;)150 2732 y(R)213 2744 y FK(2)273 2732 y FO(=)23 b FP(\000)p FN(X)495 2744 y FK(1)550 2732 y FO(+)18 b FN(X)702 2744 y FK(2)757 2732 y FP(\000)g FN(X)909 2744 y FK(3)946 2732 y FN(;)268 b(R)1300 2744 y FK(4)1361 2732 y FO(=)22 b FN(X)1517 2744 y FK(1)1573 2732 y FO(+)c FN(X)1725 2744 y FK(2)1780 2732 y FO(+)g FN(X)1932 2744 y FK(3)1969 2732 y FN(;)-118 2912 y FO(and)32 b(the)g(relation)d(holds)h(if)h(and)h(only)e(if)i FN(X)1307 2924 y FK(1)1344 2912 y FO(,)h FN(X)1469 2924 y FK(2)1506 2912 y FO(,)g FN(X)1631 2924 y FK(3)1700 2912 y FO(are)d(pairwise)f(an)n(ti-)-118 3012 y(comm)n(uting)24 b(self-adjoin)n(t)h(op)r(erators)h(suc)n(h)h(that)662 3192 y(\001)c(=)g FN(X)918 3158 y FK(2)911 3213 y(1)973 3192 y FO(+)18 b FN(X)1132 3158 y FK(2)1125 3213 y(2)1187 3192 y FO(+)g FN(X)1346 3158 y FK(2)1339 3213 y(3)1405 3192 y FO(=)23 b FN(I)7 b(:)-118 3372 y FO(Then)28 b(an)f(irreducible)d (represen)n(tation)h(is)i(either)f(one-dimensional,)c(one)28 b(of)-118 3472 y FN(X)-49 3484 y FL(i)1 3472 y FO(=)23 b FP(\006)p FO(1,)k(and)g(the)h(others)f(are)g(zeros,)f(or)h(t)n(w)n (o-dimensional,)57 3697 y FN(X)126 3709 y FK(1)186 3697 y FO(=)c FN(a)332 3580 y Fy(\022)393 3647 y FO(1)114 b(0)393 3746 y(0)82 b FP(\000)p FO(1)623 3580 y Fy(\023)698 3697 y FN(;)97 b(X)887 3709 y FK(2)947 3697 y FO(=)23 b FN(b)1085 3580 y Fy(\022)1145 3647 y FO(0)83 b(1)1145 3746 y(1)g(0)1311 3580 y Fy(\023)1386 3697 y FN(;)97 b(X)1575 3709 y FK(3)1635 3697 y FO(=)23 b FN(c)1773 3580 y Fy(\022)1860 3647 y FO(0)115 b FN(i)1834 3746 y FP(\000)p FN(i)82 b FO(0)2052 3580 y Fy(\023)2127 3697 y FN(;)801 3881 y(a)845 3847 y FK(2)900 3881 y FO(+)18 b FN(b)1019 3847 y FK(2)1075 3881 y FO(+)g FN(c)1194 3847 y FK(2)1254 3881 y FO(=)k(1)p FN(;)p eop %%Page: 112 116 112 115 bop -118 -137 a FO(112)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(and)d(either)f FN(a)j(>)g FO(0,)f FN(b)h(>)g FO(0,)f FN(c)h FP(2)g FI(R)p FO(,)29 b(or)21 b FN(a)i FO(=)f(0,)h FN(b)f(>)h FO(0,)f FN(c)h(>)g FO(0,)f(or)f FN(a)i(>)f FO(0,)h FN(b)f FO(=)h(0,)-118 196 y FN(c)j(>)h FO(0.)43 b(These)29 b(represen)n(tations)e(form)h(the)i(\014rst)g (family)d(in)i(the)h(statemen)n(t)-118 296 y(of)d(the)h(theorem.)36 b(The)27 b(form)n(ulas)e(for)i FN(P)1163 308 y FL(k)1232 296 y FO(are)22 515 y FN(P)75 527 y FK(1)135 515 y FO(=)233 459 y(1)p 233 496 42 4 v 233 572 a(2)298 398 y Fy(\022)399 465 y FO(1)18 b(+)g FN(a)122 b FP(\000)p FN(b)18 b FP(\000)g FN(ic)359 564 y FP(\000)p FN(b)g FO(+)g FN(ic)122 b FO(1)18 b FP(\000)g FN(a)975 398 y Fy(\023)1050 515 y FN(;)139 b(P)1265 527 y FK(2)1326 515 y FO(=)1424 459 y(1)p 1424 496 V 1424 572 a(2)1489 398 y Fy(\022)1557 465 y FO(1)18 b FP(\000)g FN(a)91 b(b)18 b FP(\000)g FN(ic)1550 564 y(b)g FO(+)g FN(ic)90 b FO(1)18 b(+)g FN(a)2036 398 y Fy(\023)2111 515 y FN(;)22 748 y(P)75 760 y FK(3)135 748 y FO(=)233 692 y(1)p 233 729 V 233 805 a(2)298 631 y Fy(\022)399 697 y FO(1)g FP(\000)g FN(a)122 b FP(\000)p FN(b)18 b FO(+)g FN(ic)359 797 y FP(\000)p FN(b)g FP(\000)g FN(ic)122 b FO(1)18 b(+)g FN(a)975 631 y Fy(\023)1050 748 y FN(;)139 b(P)1265 760 y FK(4)1326 748 y FO(=)1424 692 y(1)p 1424 729 V 1424 805 a(2)1489 631 y Fy(\022)1590 697 y FO(1)18 b(+)g FN(a)122 b(b)18 b FO(+)g FN(ic)1550 797 y FP(\000)p FN(b)g FP(\000)g FN(ic)90 b FO(1)18 b FP(\000)g FN(a)2101 631 y Fy(\023)2176 748 y FN(:)6 976 y FO(F)-7 b(or)33 b FN(\013)g FP(6)p FO(=)f(1)p FN(=)p FO(2)g(\()p FN(h)h FP(6)p FO(=)f(0\),)i(in)n(tro)r(duce)e(the)i(elemen) n(ts)d FN(X)1781 988 y FK(1)1818 976 y FO(,)k FN(X)1945 988 y FK(2)1982 976 y FO(,)g FN(X)2109 988 y FK(3)2179 976 y FO(suc)n(h)-118 1076 y(that)642 1272 y FN(X)711 1284 y FK(1)771 1272 y FO(=)893 1216 y(1)p 869 1253 90 4 v 869 1329 a(2)p FN(h)982 1272 y FO(\()p FN(R)1077 1284 y FK(2)1133 1272 y FO(+)18 b FN(R)1279 1284 y FK(3)1335 1272 y FP(\000)g FN(hI)7 b FO(\))p FN(;)642 1472 y(X)711 1484 y FK(2)771 1472 y FO(=)893 1416 y(1)p 869 1453 V 869 1529 a(2)p FN(h)982 1472 y FO(\()p FN(R)1077 1484 y FK(1)1133 1472 y FO(+)18 b FN(R)1279 1484 y FK(3)1335 1472 y FP(\000)g FN(hI)7 b FO(\))p FN(;)642 1672 y(X)711 1684 y FK(3)771 1672 y FO(=)893 1616 y(1)p 869 1653 V 869 1729 a(2)p FN(h)982 1672 y FO(\()p FN(R)1077 1684 y FK(1)1133 1672 y FO(+)18 b FN(R)1279 1684 y FK(2)1335 1672 y FP(\000)g FN(hI)7 b FO(\))p FN(:)-118 1874 y FO(Then)517 2052 y FN(R)580 2064 y FK(1)641 2052 y FO(=)22 b FN(h)14 b FO(\()p FP(\000)p FN(X)956 2064 y FK(1)1011 2052 y FO(+)k FN(X)1163 2064 y FK(2)1219 2052 y FO(+)g FN(X)1371 2064 y FK(3)1426 2052 y FO(+)g(1)p FN(=)p FO(2\))p FN(;)517 2176 y(R)580 2188 y FK(2)641 2176 y FO(=)k FN(h)14 b FO(\()p FN(X)891 2188 y FK(1)947 2176 y FP(\000)k FN(X)1099 2188 y FK(2)1154 2176 y FO(+)g FN(X)1306 2188 y FK(3)1362 2176 y FO(+)g(1)p FN(=)p FO(2\))p FN(;)517 2301 y(R)580 2313 y FK(3)641 2301 y FO(=)k FN(h)14 b FO(\()p FN(X)891 2313 y FK(1)947 2301 y FO(+)k FN(X)1099 2313 y FK(2)1154 2301 y FP(\000)g FN(X)1306 2313 y FK(3)1362 2301 y FO(+)g(1)p FN(=)p FO(2\))p FN(;)517 2425 y(R)580 2437 y FK(4)641 2425 y FO(=)k FN(h)14 b FO(\()p FP(\000)p FN(X)956 2437 y FK(1)1011 2425 y FP(\000)k FN(X)1163 2437 y FK(2)1219 2425 y FP(\000)g FN(X)1371 2437 y FK(3)1426 2425 y FO(+)g(1)p FN(=)p FO(2\))p FN(;)-118 2604 y FO(and)27 b(the)h(relation)d(\(2.21\)) i(is)f(equiv)-5 b(alen)n(t)26 b(to)h(the)h(follo)n(wing)c(set)j(of)h (relations)149 2782 y FP(f)p FN(X)260 2794 y FK(1)296 2782 y FN(;)14 b(X)402 2794 y FK(2)439 2782 y FP(g)23 b FO(=)f FN(X)660 2794 y FK(3)698 2782 y FN(;)96 b FP(f)p FN(X)928 2794 y FK(2)965 2782 y FN(;)14 b(X)1071 2794 y FK(3)1108 2782 y FP(g)22 b FO(=)h FN(X)1329 2794 y FK(1)1366 2782 y FN(;)97 b FP(f)p FN(X)1597 2794 y FK(3)1634 2782 y FN(;)14 b(X)1740 2794 y FK(1)1777 2782 y FP(g)22 b FO(=)h FN(X)1998 2794 y FK(2)2035 2782 y FN(;)786 2917 y(h)834 2883 y FK(2)885 2917 y FO(\(\001)c(+)f(1)p FN(=)p FO(4\))k(=)h(1)p FN(;)-118 3095 y FO(where,)30 b(as)g(ab)r(o)n(v)n(e,)f (\001)f(=)f FN(X)779 3065 y FK(2)772 3116 y(1)836 3095 y FO(+)20 b FN(X)997 3065 y FK(2)990 3116 y(2)1053 3095 y FO(+)g FN(X)1214 3065 y FK(2)1207 3116 y(3)1251 3095 y FO(.)44 b(No)n(w,)31 b(w)n(e)f(can)g(easily)d(describ)r(e)-118 3195 y(all)32 b(irreducible)g(represen)n(tations)f(of)42 b(\(2.21\))34 b(in)g(terms)f(of)i(the)g(irreducible)-118 3294 y(represen)n(tations)20 b(of)30 b(\(2.18\))22 b(with)h(one)g (extra)g(restriction)d FN(h)1770 3264 y FK(2)1821 3294 y FO(\(\001)10 b(+)g(1)p FN(=)p FO(4\))21 b(=)i(1.)6 3394 y(i.)42 b(Represen)n(tations)26 b(of)k(o)r(dd)f(dimension,)e (corresp)r(onding)f(to)j(the)h(orbit)-118 3503 y(con)n(taining)e(zero.) 45 b(Let)31 b(dim)13 b FN(H)35 b FO(=)28 b FN(n)h FO(=)f(2)p FN(k)23 b FO(+)d(1,)31 b(then)h(\001)c(=)1862 3470 y FL(n)1903 3445 y Fx(2)1936 3470 y FM(\000)p FK(1)p 1862 3484 159 4 v 1925 3532 a(4)2031 3503 y FN(I)7 b FO(,)32 b(whic)n(h)-118 3603 y(implies)h(that)k FN(h)i FO(=)f FP(\006)p FO(2)p FN(=n)d FO(and)i FN(\013)i FO(=)1162 3570 y FK(1)p 1162 3584 34 4 v 1162 3631 a(2)1230 3603 y FP(\000)1418 3570 y FK(1)p 1329 3584 211 4 v 1329 3631 a(2\()p FL(n)p FK(+1\))1549 3603 y FO(,)g(or)d FN(\013)k FO(=)1928 3570 y FK(1)p 1928 3584 34 4 v 1928 3631 a(2)1995 3603 y FO(+)2184 3570 y FK(1)p 2095 3584 212 4 v 2095 3631 a(2\()p FL(n)p FM(\000)p FK(1\))2316 3603 y FO(,)-118 3712 y(whic)n(h)26 b(giv)n(e)g(cases)g(2)h(and)g(3)g(of)h(the)g (theorem.)35 b(The)28 b(pro)5 b(jections)25 b(are)h(three-)-118 3811 y(diagonal)19 b(matrices)g(that)k(can)f(easily)e(b)r(e)j(restored) e(from)g(the)i(corresp)r(onding)-118 3911 y(represen)n(tation)i(of)i (the)h(graded)f FN(so)p FO(\(3\).)p eop %%Page: 113 117 113 116 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(113)6 96 y(ii.)34 b(Represen)n(tations)19 b(with)i(an)h(arbitrary)d(dimension)g FN(n)p FO(,)k(corresp)r(onding) -118 196 y(to)j(the)g(orbits)f(con)n(taining)e(1)p FN(=)p FO(2,)i(or)g FP(\000)p FO(1)p FN(=)p FO(2.)35 b(No)n(w)26 b(\001)d(=)f(\()p FN(n)1732 166 y FK(2)1785 196 y FP(\000)15 b FO(1)p FN(=)p FO(4\))f FN(I)7 b FO(,)26 b(whic)n(h)-118 296 y(implies)h FN(h)h FO(=)g FP(\006)p FO(1)p FN(=n)p FO(,)j(and)f FN(\013)f FO(=)938 263 y FK(1)p 938 277 34 4 v 938 324 a(2)1002 296 y FP(\000)1202 263 y FK(1)p 1097 277 244 4 v 1097 324 a(2\(2)p FL(n)p FK(+1\))1350 296 y FO(,)j(or)e FN(\013)f FO(=)1695 263 y FK(1)p 1695 277 34 4 v 1695 324 a(2)1759 296 y FP(\000)1959 263 y FK(1)p 1854 277 244 4 v 1854 324 a(2\(2)p FL(n)p FK(+1\))2107 296 y FO(.)47 b(This)-118 405 y(giv)n(es)25 b(cases)i(4)g(and)g(5.)p 2278 405 4 57 v 2282 352 50 4 v 2282 405 V 2331 405 4 57 v -118 568 a FB(R)l(emark)j(28.)42 b FO(One)28 b(can)f(consider)e (the)j(follo)n(wing)c(relation)674 735 y FN(P)727 747 y FK(1)783 735 y FO(+)18 b FN(P)919 747 y FK(2)975 735 y FO(+)g FN(P)1111 747 y FK(3)1167 735 y FO(+)g FN(P)1303 747 y FK(4)1364 735 y FO(=)k FN(Z)q(;)594 b FO(\(2.22\))-118 902 y(where)34 b FN(P)182 914 y FL(i)245 902 y FO(are)f(orthogonal)f (pro)5 b(jections,)34 b(and)g FN(Z)41 b FO(comm)n(utes)32 b(with)i(them.)-118 1001 y(Since,)43 b(for)e(an)g(irreducible)c (represen)n(tation,)42 b(the)f(cen)n(tral)e(elemen)n(t)g FN(Z)47 b FO(is)-118 1101 y(scalar,)29 b FN(Z)k FO(=)28 b FN(\013I)7 b FO(,)31 b(w)n(e)g(can)f(apply)f(the)i(latter)e(theorem)g (to)h(the)h(description)-118 1201 y(of)22 b(irreducible)d(represen)n (tations)h(of)29 b(\(2.22\))o(.)35 b(Indeed,)24 b(the)f(set)g(of)f (irreducible)-118 1300 y(represen)n(tations)d(of)28 b(\(2.22\))21 b(consists)e(of)j(represen)n(tations)d(of)28 b(\(2.21\))21 b(with)g(all)-118 1400 y FN(\013)p FO(,)26 b(i.e.,)g(all)d(represen)n (tations)g(of)i(the)h(graded)e FN(so)p FO(\(3\))i(algebra)d(\(2.18\))o (,)k(and)e(all)-118 1499 y(represen)n(tations)36 b(of)k(triples)d(of)i (an)n(ti-comm)n(uting)c(self-adjoin)n(t)i(op)r(erators)-118 1599 y(with)27 b(the)h(sum)f(of)g(squares)f(equal)g(to)i(the)g(iden)n (tit)n(y)-7 b(.)-118 1812 y FQ(2.2.2)94 b(Represen)m(tations)25 b(of)h(a)h(class)f(of)g(quadratic)i(algebras)e(with)174 1912 y(three)31 b(generators)-118 2065 y FO(Consider)21 b(an)i(algebra)d(with)j(three)g(generators)d FN(X)7 b FO(,)24 b FN(Y)18 b FO(,)25 b FN(Z)j FO(and)23 b(the)h(relations)748 2232 y FN(X)7 b(Y)36 b FP(\000)18 b FN(q)s(Y)h(X)29 b FO(=)23 b FN(\026Y)5 b(;)756 2357 y(Z)h(X)24 b FP(\000)18 b FN(q)s(X)7 b(Z)28 b FO(=)23 b FN(\026Z)q(;)709 2481 y(\013Y)c(Z)24 b FP(\000)18 b FN(\014)t(Z)6 b(Y)41 b FO(=)23 b FN(P)12 b FO(\()p FN(X)7 b FO(\))p FN(;)-118 2648 y FO(where)27 b FN(q)s FO(,)h FN(\026)p FO(,)f FN(\013)p FO(,)i FN(\014)e FP(2)c FI(C)15 b FO(,)34 b(and)28 b FN(P)12 b FO(\()p FP(\001)p FO(\))28 b(is)e(a)h(quadratic)f(p)r (olynomial)d(in)k FN(X)7 b FO(.)6 2748 y(In)19 b(what)g(follo)n(ws,)e (w)n(e)h(assume)f(that)i FN(\026)p FO(,)i FN(q)26 b FP(2)d FI(R)p FO(,)k FN(q)f FP(6)p FO(=)c FP(\006)p FO(1,)e(and)e(the)h (algebra)-118 2847 y(is)h(equipp)r(ed)h(with)h(an)f(in)n(v)n(olution)c (de\014ned)22 b(on)g(the)g(generators)d(b)n(y)i FN(X)2092 2817 y FM(\003)2153 2847 y FO(=)h FN(X)7 b FO(,)-118 2947 y FN(Y)-51 2917 y FM(\003)10 2947 y FO(=)22 b FN(Z)6 b FO(.)35 b(Notice)22 b(that)h(the)f(ideal)f(generated)g(b)n(y)h(the)h (relations)c(is)j(a)g FP(\003)p FO(-ideal.)-118 3047 y(With)k(suc)n(h)h(an)f(in)n(v)n(olution,)d(the)k(in)n(tro)r(duced)f (family)d(of)k(relations)d(includes)-118 3146 y(the)d(Lie)e(algebra)f FN(su)p FO(\(2\))i(\(for)h FN(q)26 b FO(=)d(1\),)e(and)g(man)n(y)e(of)h (its)g(deformations)e(whic)n(h)-118 3246 y(ha)n(v)n(e)26 b(arisen)g(in)h(recen)n(t)g(pap)r(ers)g(in)f(ph)n(ysics.)6 3346 y(F)-7 b(or)30 b FN(q)h FP(6)p FO(=)c(1,)j(one)g(can)g(rewrite)e (the)j(relation)c(with)j FN(\026)e FO(=)f(0;)k(just)g(replace)-118 3445 y FN(X)38 b FO(with)31 b FN(X)c FO(+)20 b FN(\025I)39 b FO(for)31 b(an)g(appropriate)e(v)-5 b(alue)30 b(of)i FN(\025)p FO(.)49 b(In)31 b(what)h(follo)n(ws,)d(w)n(e)-118 3545 y(assume)d(that)i FN(\026)23 b FO(=)f(0.)6 3644 y(In)n(tro)r(ducing)k(self-adjoin)n(t)g(generators,)f FN(A)e FO(=)g FN(X)7 b FO(,)27 b FN(B)h FO(=)1796 3612 y FK(1)p 1796 3626 34 4 v 1796 3673 a(2)1839 3644 y FO(\()p FN(Y)38 b FO(+)18 b FN(Z)6 b FO(\),)28 b FN(C)h FO(=)-108 3711 y FK(1)p -108 3725 V -108 3773 a(2)-65 3744 y FO(\()p FN(Y)37 b FP(\000)18 b FN(Z)6 b FO(\),)28 b(the)g(relations)c(will)h (tak)n(e)i(the)h(form)385 3911 y(\(1)18 b(+)g FN(q)s FO(\)[)p FN(A;)c(B)t FO(])24 b(=)f FP(\000)p FN(i)14 b FO(\(1)j FP(\000)h FN(q)s FO(\))p FP(f)p FN(A;)c(C)6 b FP(g)p FN(;)p eop %%Page: 114 118 114 117 bop -118 -137 a FO(114)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)387 96 y FO(\(1)18 b(+)g FN(q)s FO(\)[)p FN(A;)c(C)6 b FO(])24 b(=)f FN(i)14 b FO(\(1)j FP(\000)h FN(q)s FO(\))p FP(f)p FN(A;)c(B)t FP(g)p FN(;)252 231 y FP(\000)p FN(i)g FO(\()p FN(\013)k FP(\000)g FN(\014)t FO(\)[)p FN(B)t(;)c(C)6 b FO(])19 b(+)f(\()p FN(\013)i FO(+)e FN(\014)t FO(\)\()p FN(B)1316 197 y FK(2)1372 231 y FP(\000)g FN(C)1520 197 y FK(2)1558 231 y FO(\))23 b(=)1711 199 y FK(1)p 1711 213 34 4 v 1711 260 a(2)1754 231 y FN(P)12 b FO(\()p FN(A)p FO(\))p FN(:)6 421 y FO(No)n(w)35 b(w)n(e)f(turn)h(to)g(the)g(study)g(of)g(irreducible)c(represen)n (tations)h(of)i(the)-118 521 y(in)n(tro)r(duced)17 b(algebras,)h(i.e.,) h(the)g(op)r(erators)e FN(X)7 b FO(,)20 b FN(Y)37 b FO(that)19 b(satisfy)e(the)i(relations)378 708 y FN(X)7 b(Y)41 b FO(=)22 b FN(q)s(Y)d(X)r(;)97 b(\013)14 b(Y)k(Y)1128 674 y FM(\003)1185 708 y FO(+)g FN(\014)g(Y)1400 674 y FM(\003)1438 708 y FN(Y)41 b FO(=)23 b FN(P)12 b FO(\()p FN(X)7 b FO(\))p FN(:)283 b FO(\(2.23\))6 898 y(Using)30 b(the)g(argumen)n(ts)e(that)j(are)e(quite)h(similar)25 b(to)30 b(those)g(used)h(in)e(the)-118 997 y(previous)19 b(section,)i(one)g(sees)g(that)h FN(Y)40 b FO(maps)20 b(eigenspaces)f(of)i(the)h(self-adjoin)n(t)-118 1097 y(op)r(erator)27 b FN(X)36 b FO(in)n(to)28 b(eigenspaces,)g(and)h(the)h (corresp)r(onding)c(dynamical)g(sys-)-118 1197 y(tem)36 b(is)f FN(\025)i FP(7!)h FN(q)s(\025)p FO(;)j(ho)n(w)n(ev)n(er,)36 b(w)n(e)g(need)g(not)g(assume)f(that)h FN(X)43 b FO(is)35 b(p)r(ositiv)n(e,)-118 1296 y(and)i(w)n(e)f(ha)n(v)n(e)g(no)g (restrictions)e(on)i(the)h(k)n(ernels)e(of)i(the)g(op)r(erators.)63 b(The)-118 1396 y(dynamical)35 b(system)i FN(\025)42 b FP(7!)g FN(q)s(\025)d FO(p)r(ossesses)e(a)i(measurable)c(section,)40 b(there-)-118 1496 y(fore,)31 b(the)g(sp)r(ectral)e(measure)f(of)j FN(X)36 b FO(in)30 b(an)n(y)g(irreducible)d(represen)n(tation)h(is)-118 1595 y(concen)n(trated)e(on)i(a)f(single)e(orbit.)6 1697 y(In)n(tro)r(duce)j(the)g(set)349 1939 y FN(M)k FO(=)549 1797 y Fy(\()616 1882 y FO([1)p FN(;)14 b FP(j)p FN(q)s FP(j)p FO(\))k FP([)h FO(\()p FP(\000j)p FN(q)s FP(j)p FN(;)14 b FP(\000)p FO(1])k FP([)h(f)p FO(0)p FP(g)p FN(;)81 b FP(j)p FN(q)s FP(j)23 b FN(>)g FO(1)p FN(;)616 2001 y FO(\()p FP(j)p FN(q)s FP(j)p FN(;)14 b FO(1])k FP([)h FO([)p FP(\000)p FO(1)p FN(;)14 b FP(\000j)p FN(q)s FP(j)p FO(\))k FP(\\)h(f)p FO(0)p FP(g)p FN(;)81 b FP(j)p FN(q)s FP(j)23 b FN(<)g FO(1)p FN(:)-118 2196 y(M)36 b FO(is)27 b(a)g(measurable)d(section)i(of)i(the)g(dynamical)c(system.) 6 2298 y(If)35 b(the)f(orbit)e(consists)g(of)h(the)h(zero)f(p)r(oin)n (t,)i(w)n(e)e(ha)n(v)n(e)f FN(X)40 b FO(=)33 b(0,)i(and)e(w)n(e)-118 2398 y(ha)n(v)n(e)21 b(the)h(relation)d FN(\013Y)g(Y)691 2368 y FM(\003)736 2398 y FO(+)7 b FN(\014)t(Y)926 2368 y FM(\003)964 2398 y FN(Y)42 b FO(=)22 b FN(P)12 b FO(\(0\))p FN(I)29 b FO(whic)n(h)21 b(is)g(either)g(the)h FN(q)s FO(-plane,)-118 2497 y(or)29 b(the)h FN(q)s FO(-CCR)g(relation)d (considered)g(in)j(Section)e(1.4.2.)43 b(In)30 b(what)g(follo)n(ws,) -118 2597 y(w)n(e)d(study)h(the)g(represen)n(tations)c(corresp)r (onding)h(to)j(non-zero)e(orbits.)-118 2770 y FQ(Theorem)k(24.)41 b FB(Unitary)j(r)l(epr)l(esentations)h(c)l(orr)l(esp)l(onding)h(to)e (the)h(orbit)-118 2870 y FN(O)-55 2882 y FL(\025)-11 2870 y FB(,)30 b FN(\025)23 b FP(2)h FN(M)j FP(n)18 b(f)p FO(0)p FP(g)p FB(,)29 b(may)h(b)l(e)g(describ)l(e)l(d)h(in)f(terms)f (of)i(the)f(fol)t(lowing)7 b FO(:)-65 3062 y(\()p FB(a)f FO(\))42 b FB(in\014nite)30 b(se)l(quenc)l(es)f FN(y)788 3074 y FL(k)852 3062 y FN(>)22 b FO(0)p FB(,)30 b FN(k)c FP(2)d FI(Z)p FB(,)h(for)31 b(which)789 3249 y FN(\013)14 b(y)900 3215 y FK(2)897 3270 y FL(k)956 3249 y FO(+)k FN(\014)g(y)1148 3215 y FK(2)1145 3270 y FL(k)q FK(+1)1293 3249 y FO(=)23 b FN(P)12 b FO(\()p FN(q)1518 3215 y FL(k)1559 3249 y FN(\025)p FO(\))487 b(\(2.24\))89 3437 y FB(holds)40 b(for)g(al)t(l)f FN(k)j FP(2)d FI(Z)32 b FO(\()p FB(as)39 b(a)g(rule,)i(these)d(r)l(epr)l(esentations)h(form)g(a)89 3536 y(c)l(ontinuous)29 b(family)j(indexe)l(d)e(by)g FN(\025)p FO(\);)-60 3712 y(\()p FB(b)5 b FO(\))42 b FB(in\014nite)30 b(se)l(quenc)l(es)f FN(y)788 3724 y FL(k)829 3712 y FB(,)h FN(k)c FP(\025)c FN(l)r FB(,)30 b(with)g FN(l)h FB(\014xe)l(d,)f(such)g(that)g FN(y)1975 3724 y FL(l)2023 3712 y FO(=)22 b(0)p FB(,)30 b(and)89 3811 y FN(y)130 3823 y FL(k)199 3811 y FN(>)e FO(0)p FB(,)34 b FN(k)d(>)d(l)r FB(,)33 b(and)42 b FO(\(2.24\))32 b FB(holds)i(for)f(al)t(l)h FN(k)d FP(\025)d FN(l)34 b FO(\()p FB(r)l(epr)l(esentations)89 3911 y(with)d(the)f(lowest)g (weight)8 b FO(\);)p eop %%Page: 115 119 115 118 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(115)-60 96 y(\()p FB(c)5 b FO(\))42 b FB(in\014nite)31 b(se)l(quenc)l(es)g FN(y)791 108 y FL(k)832 96 y FB(,)h FN(k)d FP(\024)c FN(l)r FB(,)32 b(with)g FN(l)g FB(\014xe)l(d,)g(such)g(that)f(for)h FN(y)2130 108 y FL(l)2181 96 y FO(=)26 b(0)p FB(,)89 196 y(and)33 b FN(y)294 208 y FL(k)361 196 y FN(>)26 b FO(0)p FB(,)33 b FN(k)c(<)e(l)r FB(,)32 b(and)41 b FO(\(2.24\))31 b FB(holds)i(for)f(al)t(l)h FN(k)d FP(\024)c FN(l)33 b FO(\()p FB(r)l(epr)l(esenta-)89 296 y(tions)d(with)h(the)e (highest)i(weight)8 b FO(\);)-68 475 y(\()p FB(d)h FO(\))42 b FB(\014nite)33 b(se)l(quenc)l(es)g FN(y)723 487 y FL(k)763 475 y FB(,)h FN(l)d FP(\024)d FN(k)k FP(\024)d FN(m)p FB(,)34 b(with)g FN(l)g FB(and)g FN(m)f FB(\014xe)l(d,)g(such)h(that)89 575 y FN(y)130 587 y FL(l)182 575 y FO(=)27 b FN(y)315 587 y FL(m)404 575 y FO(=)g(0)p FB(,)32 b(and)g FN(y)799 587 y FL(k)867 575 y FN(>)26 b FO(0)32 b FB(for)g FN(l)c(<)f(k)j(<)c(m) p FB(,)33 b(and)40 b FO(\(2.24\))31 b FB(holds)j(for)89 674 y(al)t(l)d FN(l)25 b FP(\024)d FN(k)k FP(\024)d FN(m)29 b FO(\()p FB(\014nite-dimensional)j(r)l(epr)l(esentations)7 b FO(\))p FB(.)-118 870 y(In)31 b(series)40 b FO(\()p FB(a)6 b FO(\))p FB(,)34 b(the)d(r)l(epr)l(esentations)h(ar)l(e)g(unb)l (ounde)l(d.)45 b(In)31 b(series)39 b FO(\()p FB(b)5 b FO(\))33 b FB(and)-118 970 y FO(\()p FB(c)5 b FO(\))p FB(,)33 b(the)g(r)l(epr)l(esentations)f(may)h(b)l(e)f(b)l(ounde)l(d)g (or)h(unb)l(ounde)l(d.)45 b(Besides)33 b(the)-118 1069 y(mentione)l(d)d(r)l(epr)l(esentations,)h(ther)l(e)f(c)l(an)f(stil)t(l) i(b)l(e)f(one-dimensional)i(r)l(epr)l(e-)-118 1169 y(sentations)e FN(Y)43 b FO(=)23 b(0)p FB(,)31 b FN(P)12 b FO(\()p FN(X)7 b FO(\))23 b(=)h(0)p FB(,)30 b(and)h(the)g(r)l(epr)l(esentations)f(c)l (orr)l(esp)l(onding)-118 1268 y(to)g(the)g(zer)l(o)g(orbit.)-118 1444 y(Pr)l(o)l(of.)43 b FO(Using)27 b(the)i(same)d(argumen)n(ts)g(as)i (in)f(the)i(previous)d(section,)h(w)n(e)h(see)-118 1544 y(that)c(the)f(sp)r(ectrum)g(of)h FN(X)29 b FO(lies)22 b(in)g FP(f)p FN(q)1058 1514 y FL(k)1099 1544 y FN(\025;)14 b(k)26 b FP(2)d FI(Z)p FP(g)o FO(,)c(where)k FN(\025)g FP(6)p FO(=)g(0)g(is)f(an)h(initial)-118 1644 y(p)r(oin)n(t)33 b(from)g FN(M)9 b FO(,)36 b(and)e(the)h(space)e FN(H)41 b FO(is)33 b(a)h(direct)f(sum)h(of)g(its)f(eigenspaces)-118 1743 y FN(H)-49 1755 y FL(k)31 1743 y FO(corresp)r(onding)j(to)i(the)i (eigen)n(v)-5 b(alues)35 b FN(\025)1331 1755 y FL(k)1414 1743 y FO(=)42 b FN(q)1561 1713 y FL(k)1602 1743 y FN(\025)p FO(.)71 b(The)39 b(op)r(erator)e FN(Y)-118 1843 y FO(maps)32 b FN(H)174 1855 y FL(k)q FM(\000)p FK(1)334 1843 y FO(in)n(to)g FN(H)577 1855 y FL(k)618 1843 y FO(;)k(write)d FN(Y)944 1855 y FL(k)994 1843 y FO(:)d FN(H)1116 1855 y FL(k)q FM(\000)p FK(1)1275 1843 y FP(\000)-49 b(!)33 b FN(H)1476 1855 y FL(k)1551 1843 y FO(for)g(the)h(corresp)r(onding)-118 1942 y(restrictions.)44 b(F)-7 b(rom)30 b(the)i(second)e(relation)e(in) j(\(2.23\))o(,)h(w)n(e)f(ha)n(v)n(e)f FN(\013)14 b(Y)2107 1954 y FL(k)2148 1942 y FN(Y)2215 1912 y FM(\003)2196 1966 y FL(k)2274 1942 y FO(+)-118 2042 y FN(\014)k(Y)14 2012 y FM(\003)-5 2066 y FL(k)q FK(+1)120 2042 y FN(Y)168 2054 y FL(k)q FK(+1)316 2042 y FO(=)23 b FN(P)12 b FO(\()p FN(q)541 2012 y FL(k)582 2042 y FN(\025)p FO(\))i FN(I)7 b FO(.)6 2145 y(W)-7 b(e)42 b(will)d(sho)n(w)h(that)i(all)d FN(H)945 2157 y FL(k)1027 2145 y FO(are)i(either)f(zero)g(or)h (one-dimensional.)-118 2245 y(Indeed,)25 b FN(Y)220 2257 y FL(k)261 2245 y FN(Y)328 2215 y FM(\003)309 2268 y FL(k)390 2245 y FO(and)f FN(Y)615 2215 y FM(\003)596 2268 y FL(k)q FK(+1)721 2245 y FN(Y)769 2257 y FL(k)q FK(+1)919 2245 y FO(are)f(comm)n(uting)e(self-adjoin)n(t)g(op)r (erators)h(in)-118 2344 y FN(H)-49 2356 y FL(k)-8 2344 y FO(.)58 b(T)-7 b(ak)n(e)34 b(an)h(in)n(v)-5 b(arian)n(t)31 b(subspace)j FN(H)1189 2314 y FK(0)1182 2368 y FL(k)1261 2344 y FP(\032)h FN(H)1430 2356 y FL(k)1471 2344 y FO(,)h(then)g(the)f (image)d(of)j FN(H)2302 2314 y FK(0)2295 2368 y FL(k)-118 2444 y FO(under)29 b(the)g(action)f(of)h FN(Y)19 b FO(,)29 b FN(Y)796 2414 y FM(\003)863 2444 y FO(and)g FN(X)36 b FO(is)28 b(in)n(v)-5 b(arian)n(t)26 b(in)i FN(H)7 b FO(.)42 b(Th)n(us,)29 b FN(H)2109 2456 y FL(k)2179 2444 y FO(do)r(es)-118 2544 y(not)23 b(ha)n(v)n(e)g(prop)r(er)f(subspaces.) 35 b(Th)n(us,)24 b(the)g(op)r(erators)e FN(Y)1666 2556 y FL(k)1707 2544 y FN(Y)1773 2513 y FM(\003)1755 2567 y FL(k)1835 2544 y FO(and)h FN(Y)2059 2513 y FM(\003)2040 2567 y FL(k)q FK(+1)2165 2544 y FN(Y)2213 2556 y FL(k)q FK(+1)-118 2653 y FO(are)28 b(scalar,)f(and)i(w)n(e)g(obtain)f(the)i (relations)c FN(\013)14 b FP(j)p FN(y)1445 2665 y FL(k)1486 2653 y FP(j)1509 2623 y FK(2)1565 2653 y FO(+)20 b FN(\014)e FP(j)p FN(y)1779 2665 y FL(k)q FK(+1)1903 2653 y FP(j)1926 2623 y FK(2)1989 2653 y FO(=)26 b FN(P)12 b FO(\()p FN(q)2217 2623 y FL(k)2258 2653 y FN(\025)p FO(\))-118 2752 y(for)34 b(all)f FN(k)s FO(.)60 b(P)n(assing)32 b(to)j(a)f(unitarily)e(equiv)-5 b(alen)n(t)33 b(represen)n(tation,)h(w)n(e)h(can)-118 2852 y(assume)h(that)j FN(y)411 2864 y FL(k)492 2852 y FP(\025)h FO(0.)69 b(T)-7 b(o)37 b(complete)g(the)h(pro)r(of)g(one)g (can)f(notice)g(that)-118 2952 y(the)h(subspaces)425 2889 y Fy(L)517 2977 y FL(k)q FM(\025)p FL(l)645 2952 y FN(H)714 2964 y FL(k)793 2952 y FO(and)964 2889 y Fy(L)1057 2977 y FL(k)q FM(\024)p FL(l)1185 2952 y FN(H)1254 2964 y FL(k)1332 2952 y FO(are)f(in)n(v)-5 b(arian)n(t)35 b(if)i(and)h(only)e(if)-118 3051 y FN(y)-77 3063 y FL(l)-29 3051 y FO(=)23 b(0.)p 2278 3051 4 57 v 2282 2999 50 4 v 2282 3051 V 2331 3051 4 57 v -118 3244 a FB(R)l(emark)30 b(29.)42 b FO(Consider)21 b(a)i(generalization)18 b(of)23 b(the)h(giv)n(en)d(class)g(of)i(quadratic)-118 3343 y(algebras.)31 b(Replace)18 b(the)i(second)f(relation)e(in)i(\(2.23\))g(with)g(a)g (general)f(second-)-118 3443 y(order)26 b(relation)f(connecting)h FN(X)7 b FO(,)27 b FN(Y)19 b FO(,)27 b FN(Y)1129 3413 y FM(\003)1167 3443 y FO(,)268 3632 y FN(a)312 3644 y FK(11)382 3632 y FN(X)458 3598 y FK(2)513 3632 y FO(+)18 b FN(a)640 3644 y FK(22)711 3632 y FN(Y)777 3598 y FK(2)833 3632 y FO(+)g FN(a)960 3644 y FK(33)1030 3632 y FO(\()p FN(Y)1129 3598 y FM(\003)1167 3632 y FO(\))1199 3598 y FK(2)1255 3632 y FO(+)g FN(a)1382 3644 y FK(12)1453 3632 y FN(X)7 b(Y)36 b FO(+)18 b FN(a)1740 3644 y FK(21)1810 3632 y FN(Y)h(X)383 3757 y FO(+)f FN(a)510 3769 y FK(13)581 3757 y FN(X)7 b(Y)723 3722 y FM(\003)779 3757 y FO(+)18 b FN(a)906 3769 y FK(31)976 3757 y FN(Y)1043 3722 y FM(\003)1100 3757 y FO(+)g FN(a)1227 3769 y FK(23)1297 3757 y FN(Y)h(Y)1430 3722 y FM(\003)1487 3757 y FO(+)f FN(a)1614 3769 y FK(32)1684 3757 y FN(Y)1751 3722 y FM(\003)1789 3757 y FN(Y)550 3881 y FO(+)g FN(a)677 3893 y FK(1)714 3881 y FN(X)24 b FO(+)19 b FN(a)935 3893 y FK(2)972 3881 y FN(Y)37 b FO(+)18 b FN(a)1184 3893 y FK(3)1221 3881 y FN(Y)1288 3847 y FM(\003)1344 3881 y FO(+)g FN(aI)30 b FO(=)23 b(0)p FN(:)p eop %%Page: 116 120 116 119 bop -118 -137 a FO(116)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(One)c(can)g(sho)n(w)f(that,)j(in)e(an)n(y)f (represen)n(tation,)g(the)i(latter)e(relation)e(is)i(equiv-)-118 196 y(alen)n(t)26 b(to)i(the)g(follo)n(wing)23 b(system:)292 377 y FN(a)336 389 y FK(11)406 377 y FN(X)482 343 y FK(2)537 377 y FO(+)18 b FN(a)664 389 y FK(1)701 377 y FN(X)25 b FO(+)18 b FN(aI)25 b FO(+)19 b FN(a)1111 389 y FK(23)1181 377 y FN(Y)f(Y)1314 343 y FM(\003)1371 377 y FO(+)g FN(a)1498 389 y FK(32)1568 377 y FN(Y)1635 343 y FM(\003)1673 377 y FN(Y)41 b FO(=)23 b(0)p FN(;)-41 502 y(a)3 514 y FK(12)74 502 y FN(X)7 b(Y)36 b FO(+)18 b FN(a)361 514 y FK(21)431 502 y FN(Y)h(X)25 b FO(+)18 b FN(a)719 514 y FK(2)756 502 y FN(Y)41 b FO(=)23 b(0)p FN(;)96 b(a)1138 514 y FK(13)1209 502 y FN(X)7 b(Y)1351 467 y FM(\003)1407 502 y FO(+)18 b FN(a)1534 514 y FK(31)1605 502 y FN(Y)1671 467 y FM(\003)1709 502 y FN(X)25 b FO(+)18 b FN(a)1930 514 y FK(3)1967 502 y FN(Y)2034 467 y FM(\003)2095 502 y FO(=)23 b(0)p FN(;)668 636 y(a)712 648 y FK(22)783 636 y FN(Y)849 602 y FK(2)910 636 y FO(=)f(0)p FN(;)97 b(a)1203 648 y FK(22)1273 636 y FN(Y)1340 602 y FK(2)1400 636 y FO(=)22 b(0)p FN(:)-118 852 y FQ(2.2.3)94 b(Op)s(erator)32 b(relations)g(connected)h(with)f(a)h(dynamical)f(sys-)174 951 y(tem)e(on)h(a)i(plane)-118 1105 y FO(Let)f FN(A)d FO(=)g FN(A)282 1075 y FM(\003)321 1105 y FO(,)j FN(X)7 b FO(,)32 b FN(X)583 1075 y FM(\003)652 1105 y FO(b)r(e)g(b)r(ounded)g (op)r(erators)e(that)h(satisfy)f(relations)f(of)-118 1204 y(the)f(form:)779 1385 y FN(AX)h FO(=)23 b FN(X)7 b(F)1156 1397 y FK(1)1193 1385 y FO(\()p FN(A)p FO(\))p FN(;)727 1510 y(X)803 1476 y FM(\003)841 1510 y FN(X)29 b FO(=)23 b FN(F)1080 1522 y FK(2)1117 1510 y FO(\()p FN(A;)14 b(X)7 b(X)1400 1476 y FM(\003)1438 1510 y FO(\))p FN(;)647 b FO(\(2.25\))-118 1691 y(where)29 b FN(F)177 1703 y FK(1)214 1691 y FO(\()p FP(\001)p FO(\))9 b(:)29 b FI(R)j FP(!)26 b FI(R)p FO(,)35 b FN(F)722 1703 y FK(2)760 1691 y FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))9 b(:)29 b FI(R)1022 1661 y FK(2)1091 1691 y FP(!)d FI(R)35 b FO(are)28 b(measurable)e(mappings.)39 b(It)-118 1790 y(follo)n(ws)28 b(from)i(the)h(\014rst)g(relation)d(in)j(\(2.25\))f (that)i(the)f(op)r(erators)e FN(A)p FO(,)k FN(X)7 b(X)2302 1760 y FM(\003)-118 1890 y FO(comm)n(ute)25 b(and,)j(hence,)g FN(F)731 1902 y FK(2)768 1890 y FO(\()p FN(A;)14 b(X)7 b(X)1051 1860 y FM(\003)1088 1890 y FO(\))28 b(is)f(w)n(ell)e (de\014ned.)-118 2022 y FB(R)l(emark)30 b(30.)42 b FO(Instead)27 b(of)g(assuming)d(that)j(the)h(op)r(erator)d FN(A)i FO(is)f (self-adjoin)n(t,)-118 2122 y(one)e(can)f(consider)f(the)j(case)e(of)h (a)g(unitary)e(or)h(normal)e(op)r(erator)i(as)g(w)n(ell.)33 b(In)-118 2221 y(this)25 b(case,)f(w)n(e)h(consider)f(the)h(mapping)e FN(F)1212 2233 y FK(1)1259 2221 y FO(:)28 b FI(T)22 b FP(\000)-48 b(!)23 b FI(T)p FO(,)i(and)h FN(F)1828 2233 y FK(2)1874 2221 y FO(:)i FI(T)13 b FP(\002)h FI(R)29 b FP(\000)-48 b(!)23 b FI(R)-118 2321 y FO(for)32 b(the)g(unitary)f(op) r(erator)f FN(A)p FO(,)k(and)e FN(F)1136 2333 y FK(1)1183 2321 y FO(:)d FI(C)52 b FP(\000)-49 b(!)31 b FI(C)15 b FO(,)39 b(and)32 b FN(F)1791 2333 y FK(2)1838 2321 y FO(:)d FI(C)43 b FP(\002)21 b FI(R)36 b FP(\000)-48 b(!)31 b FI(R)-118 2421 y FO(in)38 b(the)h(normal)c(case.)69 b(The)39 b(mapping)d FN(F)51 b FO(in)n(tro)r(duced)37 b(b)r(elo)n(w,)j(de\014nes)f(a)-118 2520 y(dynamical)27 b(system)j(on)h FI(T)20 b FP(\002)g FI(R)37 b FO(or)30 b FI(C)42 b FP(\002)20 b FI(R)p FO(,)38 b(resp)r(ectiv)n(ely)-7 b(.)44 b(All)30 b(statemen)n(ts)-118 2620 y(ab)r(out)23 b(represen)n(tations)d(of)j(relations)d(\(2.25\))i(also)f(hold)h(true)h (in)g(these)g(cases)-118 2720 y(\(of)28 b(course,)g(no)n(w,)f(the)i(sp) r(ectrum)e(of)h FN(A)h FO(b)r(elongs)e(to)h(the)g(circle)e(or)h (complex)-118 2819 y(plane\).)6 2951 y(As)22 b(b)r(efore,)g(let)f(us)g (consider)e(the)j(p)r(olar)d(decomp)r(osition)f(of)j(the)g(op)r(erator) -118 3051 y FN(X)-42 3021 y FM(\003)18 3051 y FO(=)i FN(U)9 b(C)d FO(,)28 b(where)f FN(C)i FO(=)23 b FN(C)769 3021 y FM(\003)830 3051 y FO(=)g(\()p FN(X)7 b(X)1102 3021 y FM(\003)1139 3051 y FO(\))1171 3021 y FK(1)p FL(=)p FK(2)1275 3051 y FO(,)28 b FN(U)36 b FO(is)27 b(a)g(partial)d(isometry) h(suc)n(h)-118 3151 y(that)j(k)n(er)13 b FN(U)31 b FO(=)23 b(k)n(er)13 b FN(C)6 b FO(.)37 b(Using)26 b(the)i(relations)d(\(2.25\)) i(one)g(can)g(obtain)344 3332 y FN(AU)472 3297 y FM(\003)533 3332 y FO(=)c FN(U)687 3297 y FM(\003)725 3332 y FN(F)778 3344 y FK(1)815 3332 y FO(\()p FN(A)p FO(\))p FN(;)98 b(C)1127 3297 y FK(2)1164 3332 y FN(U)1230 3297 y FM(\003)1291 3332 y FO(=)23 b FN(U)1445 3297 y FM(\003)1483 3332 y FN(F)1536 3344 y FK(2)1574 3332 y FO(\()p FN(A;)14 b(C)1770 3297 y FK(2)1808 3332 y FO(\))p FN(;)263 b FO(\(2.26\))-118 3513 y(and)34 b FN(U)44 b FO(is)33 b(a)h(cen)n(tered)g(partial)e (isometry)f(with)j(k)n(er)13 b FN(U)1662 3482 y FM(\003)1735 3513 y FO(=)34 b(k)n(er)13 b FN(F)2012 3525 y FK(2)2049 3513 y FO(\()p FN(A;)h(C)2245 3482 y FK(2)2283 3513 y FO(\).)-118 3612 y(Con)n(v)n(ersely)-7 b(,)29 b(an)n(y)i(triple)e(of)j (the)f(op)r(erators)f FN(A)f FO(=)g FN(A)1573 3582 y FM(\003)1612 3612 y FO(,)j FN(C)k FP(\025)29 b FO(0,)j(and)f(a)g(cen-) -118 3712 y(tered)39 b(partial)e(isometry)f FN(U)49 b FO(satisfying)37 b(\(2.26\))h(and)i(suc)n(h)f(that)h(k)n(er)12 b FN(U)52 b FO(=)-118 3811 y(k)n(er)13 b FN(C)6 b FO(,)33 b(and)f(k)n(er)13 b FN(U)485 3781 y FM(\003)553 3811 y FO(=)30 b(k)n(er)12 b FN(F)825 3823 y FK(2)863 3811 y FO(\()p FN(A;)i(C)1059 3781 y FK(2)1097 3811 y FO(\),)33 b(de\014ne)f(a)g(represen)n(tation)d FN(A)p FO(,)k FN(X)k FO(=)-118 3911 y FN(C)6 b(U)13 3881 y FM(\003)79 3911 y FO(of)27 b(the)h(relations)d(\(2.25\).)p eop %%Page: 117 121 117 120 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(117)6 96 y(Let)29 b FN(F)37 b FO(=)24 b(\()p FN(F)420 108 y FK(1)458 96 y FN(;)14 b(F)548 108 y FK(2)585 96 y FO(\))9 b(:)29 b FI(R)732 66 y FK(2)800 96 y FP(!)24 b FI(R)961 66 y FK(2)1005 96 y FO(.)39 b(F)-7 b(or)28 b FN(k)g FP(2)d FI(N)t FO(,)35 b(w)n(e)28 b(will)d(denote)k(b)n (y)f FN(F)2210 66 y FL(k)2251 96 y FO(\()p FP(\001)p FO(\))-118 196 y(the)34 b FN(k)s FO(-th)g(iteration)e(of)i FN(F)12 b FO(\()p FP(\001)p FO(\))35 b(and,)g(for)f FN(\025)g FP(2)h FI(R)p FO(,)41 b FN(n)34 b FO(=)g(1,)h(2,)g(b)n(y)f FN(F)2035 166 y FL(k)2023 217 y(n)2076 196 y FO(\()p FN(\025)p FO(\))h(the)-118 296 y FN(n)p FO(-th)27 b(co)r(ordinate)f(of) h FN(F)633 266 y FL(k)674 296 y FO(\()p FN(\025)p FO(\).)6 395 y(Analogously)-7 b(,)18 b(the)i(relations)c(\(2.25\))j(corresp)r (ond)f(to)h(a)g(t)n(w)n(o-dimensional)-118 495 y(dynamical)k(system,)i FN(F)12 b FO(\()p FP(\001)p FO(\))23 b(:)h FI(R)853 465 y FK(2)919 495 y FP(\000)-49 b(!)24 b FI(R)1096 465 y FK(2)1139 495 y FO(.)36 b(The)27 b(p)r(ossibilit)n(y)22 b(of)k(classifying)c(all)-118 595 y(irreducible)c(represen)n(tations)h (of)i(the)h(relations)d(dep)r(ends)j(on)f(the)i(prop)r(erties)-118 694 y(of)k(the)h(dynamical)c(system.)-118 853 y FQ(Prop)s(osition)30 b(39.)41 b FB(L)l(et)35 b FO(\()p FN(A)23 b FO(=)g FN(A)978 823 y FM(\003)1016 853 y FN(;)14 b(X)7 b FO(\))27 b FB(b)l(e)h(a)g(r)l (epr)l(esentation)g(of)46 b FO(\(2.25\))26 b FB(on)-118 952 y(a)g(sp)l(ac)l(e)h FN(H)7 b FB(.)37 b(Then)26 b FN(H)33 b FB(c)l(an)26 b(b)l(e)f(de)l(c)l(omp)l(ose)l(d)j(into)e(ortho) l(gonal)h(subsp)l(ac)l(es)f FN(H)2301 964 y FK(1)-118 1052 y FB(and)33 b FN(H)115 1064 y FK(2)152 1052 y FB(,)h(invariant)g (with)f(r)l(esp)l(e)l(ct)g(to)g FN(A)p FB(,)h FN(X)7 b FB(,)33 b FN(X)1465 1022 y FM(\003)1535 1052 y FB(such)g(that)g(the)g (phase)h FN(U)-118 1152 y FB(of)c FN(X)36 b FB(is)31 b(unitary)e(in)h FN(H)636 1164 y FK(1)703 1152 y FB(and)39 b FO(k)n(er)13 b FN(U)27 b FP([)19 b FO(k)n(er)12 b FN(U)1346 1122 y FM(\003)1407 1152 y FP(6)p FO(=)23 b FP(f)p FO(0)p FP(g)28 b FB(in)i FN(H)1820 1164 y FK(2)1857 1152 y FB(.)6 1310 y FO(Similarly)20 b(to)25 b(the)g(case)f(of)h(relation)d(\(2.1\),) j(irreducible)d(represen)n(tations)-118 1410 y(of)30 b(\(2.25\))g(in)g FN(H)391 1422 y FK(2)459 1410 y FO(can)h(b)r(e)g (completely)c(describ)r(ed.)45 b(There)30 b(is)g(a)g(corresp)r(on-)-118 1510 y(dence)c(b)r(et)n(w)n(een)g(irreducible)c(represen)n(tations)h (and)j(orbits)e(of)i(the)g(dynami-)-118 1609 y(cal)20 b(system)g(going)f(through)i(a)g(p)r(oin)n(t)g(with)g(zero)f(second)h (co)r(ordinate.)33 b(More-)-118 1709 y(o)n(v)n(er,)27 b(since)g FN(C)353 1679 y FK(2)416 1709 y FP(\025)d FO(0,)29 b(the)g(sp)r(ectral)e(measure)f(of)j(the)g(pair)e(\()p FN(A)p FO(,)i FN(C)2003 1679 y FK(2)2041 1709 y FO(\))g(is)e(con-)-118 1808 y(cen)n(trated)38 b(on)g(that)h(part)f(of)g(the)h(orbit)e(where)h (the)h(second)f(co)r(ordinates)-118 1908 y(are)26 b(non-negativ)n(e.)34 b(Namely)-7 b(,)25 b(w)n(e)i(ha)n(v)n(e)f(the)i(follo)n(wing)23 b(description)i(of)i(irre-)-118 2008 y(ducible)f(represen)n(tations.) -118 2166 y FQ(Prop)s(osition)k(40.)41 b FB(A)n(ny)f(irr)l(e)l(ducible) i(r)l(epr)l(esentation)47 b FO(\()p FN(A;)14 b(X)7 b FO(\))40 b FB(of)59 b FO(\(2.25\))-118 2266 y FB(such)32 b(that)f FO(k)n(er)13 b FN(X)26 b FP([)20 b FO(k)n(er)13 b FN(X)740 2236 y FM(\003)804 2266 y FP(6)p FO(=)26 b FP(f)p FO(0)p FP(g)k FB(is)i(unitarily)g(e)l(quivalent)g(to)g(one)g(of) g(the)-118 2366 y(fol)t(lowing)7 b FO(:)6 2465 y(\(i\))p FB(.)39 b FN(H)30 b FO(=)22 b FI(C)398 2435 y FL(n)449 2465 y FB(,)30 b FN(n)23 b FP(2)h FI(N)t FB(,)371 2824 y FN(A)g FO(=)544 2583 y Fy(0)544 2729 y(B)544 2778 y(B)544 2828 y(B)544 2881 y(@)617 2638 y FN(\025)1490 2679 y Fo(0)748 2737 y FN(F)801 2749 y FK(1)839 2737 y FO(\()p FN(\025;)14 b FO(0\))1118 2834 y FB(.)1153 2859 y(.)1187 2884 y(.)860 3009 y Fo(0)381 b FN(F)1365 2966 y FK(\()p FL(n)p FM(\000)p FK(1\))1353 3031 y(1)1547 3009 y FO(\()p FN(\025;)14 b FO(0\))1739 2583 y Fy(1)1739 2729 y(C)1739 2778 y(C)1739 2828 y(C)1739 2881 y(A)1825 2824 y FN(;)358 3356 y(X)30 b FO(=)544 3089 y Fy(0)544 3235 y(B)544 3285 y(B)544 3335 y(B)544 3385 y(B)544 3438 y(@)737 3142 y FO(0)1473 3183 y Fo(0)617 3297 y FN(F)670 3309 y FK(2)708 3297 y FO(\()p FN(\025;)14 b FO(0\))987 3238 y FB(.)1021 3263 y(.)1056 3289 y(.)987 3393 y(.)1021 3418 y(.)1056 3443 y(.)1367 3451 y FO(0)729 3568 y Fo(0)381 b FN(F)1234 3525 y FK(\()p FL(n)p FM(\000)p FK(1\))1222 3590 y(2)1416 3568 y FO(\()p FN(\025;)14 b FO(0\))83 b(0)1732 3089 y Fy(1)1732 3235 y(C)1732 3285 y(C)1732 3335 y(C)1732 3385 y(C)1732 3438 y(A)1818 3356 y FN(;)-118 3738 y FB(wher)l(e)30 b FN(\025)g FB(b)l(elongs)h(to)e(the)h(set)26 3911 y FN(\033)73 3923 y FL(n)142 3911 y FO(=)22 b FP(f)p FN(\025)h FP(2)h FI(R)29 b FP(j)23 b FN(F)615 3877 y FL(k)603 3932 y FK(2)656 3911 y FO(\()p FN(\025;)14 b FO(0\))23 b FN(>)g FO(0)p FN(;)k(k)f FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FN(;)27 b(F)1741 3877 y FL(n)1729 3932 y FK(2)1786 3911 y FO(\()p FN(\025;)14 b FO(0\))24 b(=)e(0)p FP(g)p FO(;)p eop %%Page: 118 122 118 121 bop -118 -137 a FO(118)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(\(ii\))p FB(.)38 b FN(H)30 b FO(=)22 b FN(l)392 108 y FK(2)429 96 y FO(\()p FI(N)t FO(\))q FB(,)278 290 y FN(Ae)379 302 y FL(k)443 290 y FO(=)g FN(F)595 247 y FK(\()p FL(k)q FM(\000)p FK(1\))583 312 y(1)773 290 y FO(\()p FN(\025;)14 b FO(0\))g FN(e)1017 302 y FL(k)1058 290 y FN(;)98 b(X)7 b(e)1294 302 y FL(k)1357 290 y FO(=)23 b FN(F)1510 256 y FL(k)1498 311 y FK(2)1551 290 y FO(\()p FN(\025;)14 b FO(0\))g FN(e)1795 302 y FL(k)q FK(+1)1920 290 y FN(;)-118 468 y FB(wher)l(e)30 b FN(\025)g FB(b)l(elongs)h(to)e(the)h(set)g FN(\033)893 480 y FM(1)987 468 y FO(=)22 b FP(f)p FN(\025)h FP(2)h FI(R)29 b FP(j)23 b FN(F)1460 438 y FL(k)1448 489 y FK(2)1501 468 y FO(\()p FN(\025;)14 b FO(0\))23 b FN(>)g FO(0)p FN(;)k(k)f FP(2)d FI(N)t FP(g)p FO(;)6 568 y(\(iii\))p FB(.)37 b FN(H)30 b FO(=)22 b FN(l)415 580 y FK(2)452 568 y FO(\()p FI(N)5 b FO(\))p FB(,)524 745 y FN(Ae)625 757 y FL(k)689 745 y FO(=)23 b FN(\025)825 757 y FL(k)866 745 y FN(e)905 757 y FL(k)946 745 y FN(;)98 b(X)7 b(e)1182 757 y FL(k)1245 745 y FO(=)23 b FN(\026)1383 757 y FL(k)q FM(\000)p FK(1)1509 745 y FN(e)1548 757 y FL(k)q FM(\000)p FK(1)1673 745 y FN(;)-118 923 y FB(wher)l(e)36 b FN(\025)170 935 y FL(k)245 923 y FO(=)d FN(F)396 935 y FK(1)434 923 y FO(\()p FN(\025)514 935 y FL(k)q FK(+1)639 923 y FO(\))p FB(,)38 b FN(\026)784 935 y FL(k)858 923 y FO(=)33 b FN(F)1009 935 y FK(2)1047 923 y FO(\()p FN(\025)1127 935 y FL(k)q FK(+1)1253 923 y FN(;)14 b(\026)1340 935 y FL(k)q FK(+1)1465 923 y FO(\))p FB(,)37 b FN(\026)1609 935 y FK(1)1680 923 y FO(=)c(0)p FB(,)k(and)f FN(\026)2099 935 y FL(k)2174 923 y FN(>)d FO(0)p FB(,)-118 1023 y FN(k)26 b FO(=)c(2)p FB(,)30 b FN(:)14 b(:)g(:)28 b FB(.)-118 1185 y(R)l(emark)i(31.)42 b FO(Note)28 b(that)f(not)h(all)c (\014nite-dimensional)e(represen)n(tations)j(are)-118 1284 y(necessarily)h(related)i(to)h(cycles)f(of)i(the)g(dynamical)c (system)i(\(see)h(examples)-118 1384 y(b)r(elo)n(w)d(in)h(this)g (section\).)6 1515 y(The)35 b(p)r(ossibilit)n(y)c(of)j(the)i (description)c(of)j(irreducible)c(represen)n(tations)-118 1614 y(in)d FN(H)49 1626 y FK(1)114 1614 y FO(dep)r(ends)h(on)f(top)r (ological)c(prop)r(erties)j(of)h(the)h(t)n(w)n(o-dimensional)23 b(dy-)-118 1714 y(namical)h(system.)6 1814 y(Belo)n(w,)i(w)n(e)h (assume)f(that)i(the)g(mapping)d FN(F)12 b FO(\()p FP(\001)p FO(\))29 b(is)d(one-to-one.)-118 1976 y FQ(Prop)s(osition)k(41.)41 b FB(If)i(the)g(dynamic)l(al)i(system)d FN(F)12 b FO(\()p FP(\001)p FO(\))d(:)34 b FI(R)1775 1946 y FK(2)1865 1976 y FP(!)46 b FI(R)2048 1946 y FK(2)2135 1976 y FB(has)d(a)-118 2075 y(me)l(asur)l(able)36 b(se)l(ction,)i(then)e(any)g(irr)l(e)l (ducible)i(r)l(epr)l(esentation)e(is)g(unitarily)-118 2175 y(e)l(quivalent)30 b(to)g(one)g(of)g(the)g(fol)t(lowing)7 b FO(:)6 2275 y(\(i\))p FB(.)39 b FN(H)30 b FO(=)22 b FI(C)398 2244 y FL(n)449 2275 y FB(,)30 b FN(n)23 b FP(2)h FI(N)t FB(,)314 2638 y FN(A)f FO(=)487 2396 y Fy(0)487 2542 y(B)487 2592 y(B)487 2642 y(B)487 2695 y(@)559 2451 y FN(\025)1446 2493 y Fo(0)691 2551 y FN(F)744 2563 y FK(1)782 2551 y FO(\()p FN(\025;)14 b(\026)p FO(\))1069 2648 y FB(.)1104 2673 y(.)1139 2698 y(.)669 2823 y Fo(0)524 b FN(F)1317 2779 y FK(\()p FL(n)p FM(\000)p FK(1\))1305 2845 y(1)1499 2823 y FO(\()p FN(\025;)14 b(\026)p FO(\))1699 2396 y Fy(1)1699 2542 y(C)1699 2592 y(C)1699 2642 y(C)1699 2695 y(A)1785 2638 y FN(;)301 3169 y(X)29 b FO(=)487 2903 y Fy(0)487 3049 y(B)487 3099 y(B)487 3149 y(B)487 3198 y(B)487 3252 y(@)684 2955 y FO(0)924 b FN(e)1689 2925 y FL(i')1760 2955 y FN(\026)559 3110 y(F)612 3122 y FK(2)650 3110 y FO(\()p FN(\025;)14 b(\026)p FO(\))938 3052 y FB(.)972 3077 y(.)1007 3102 y(.)938 3207 y(.)972 3232 y(.)1007 3257 y(.)1323 3265 y FO(0)676 3382 y Fo(0)385 b FN(F)1185 3339 y FK(\()p FL(n)p FM(\000)p FK(1\))1173 3404 y(2)1367 3382 y FO(\()p FN(\025;)14 b(\026)p FO(\))143 b(0)1810 2903 y Fy(1)1810 3049 y(C)1810 3099 y(C)1810 3149 y(C)1810 3198 y(C)1810 3252 y(A)1897 3169 y FN(;)-118 3556 y FB(wher)l(e)30 b FN(\025)p FB(,)h FN(\026)f FB(b)l(elong)g(to)g (the)g(set)-86 3733 y FN(\033)-39 3745 y FL(n)30 3733 y FO(=)23 b FP(f)p FN(\025)g FP(2)g FI(R)29 b FP(j)23 b FN(F)503 3699 y FL(k)491 3754 y FK(2)544 3733 y FO(\()p FN(\025;)14 b(\026)p FO(\))24 b FN(>)f FO(0)p FN(;)k(k)f FO(=)c(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)19 b FP(\000)f FO(1)p FN(;)27 b(F)1652 3699 y FM(\016)p FL(n)1731 3733 y FO(\()p FN(\025;)14 b(\026)p FO(\))24 b(=)f(\()p FN(\025;)14 b(\026)p FO(\))p FP(g)p FN(;)-118 3911 y(')23 b FP(2)h FO([0)p FN(;)14 b FO(2)p FN(\031)s FO(\);)p eop %%Page: 119 123 119 122 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(119)6 96 y(\(ii\))p FB(.)38 b FN(H)30 b FO(=)22 b FN(l)392 108 y FK(2)429 96 y FO(\()p FI(Z)p FO(\))p FB(,)524 263 y FN(Ae)625 275 y FL(k)689 263 y FO(=)h FN(\025)825 275 y FL(k)866 263 y FN(e)905 275 y FL(k)946 263 y FN(;)98 b(X)7 b(e)1182 275 y FL(k)1245 263 y FO(=)23 b FN(\026)1383 275 y FL(k)q FM(\000)p FK(1)1509 263 y FN(e)1548 275 y FL(k)q FM(\000)p FK(1)1673 263 y FN(;)-118 430 y FB(wher)l(e)30 b FN(\025)164 442 y FL(k)229 430 y FO(=)22 b FN(F)369 442 y FK(1)407 430 y FO(\()p FN(\025)487 442 y FL(k)q FK(+1)613 430 y FO(\))p FB(,)30 b FN(\026)750 442 y FL(k)814 430 y FO(=)23 b FN(F)955 442 y FK(2)992 430 y FO(\()p FN(\025)1072 442 y FL(k)q FK(+1)1198 430 y FN(;)14 b(\026)1285 442 y FL(k)q FK(+1)1410 430 y FO(\))p FB(,)30 b(and)g FN(\026)1708 442 y FL(k)1772 430 y FN(>)23 b FO(0)p FB(,)30 b FN(k)c FP(2)d FI(Z)o FB(.)-118 583 y(R)l(emark)30 b(32.)42 b FO(If)24 b(there)g(exists)e(an)h(ergo)r(dic)f(quasi-in)n(v)-5 b(arian)n(t)18 b(measure)k(whic)n(h)-118 683 y(is)28 b(not)i(concen)n(trated)f(on)g(a)g(single)f(orbit,)g(then)j(one)e(can)g (construct)g(factor)-118 783 y(represen)n(tations)21 b(of)j(the)g(relation)d(whic)n(h)i(are)g(not)h(of)f(t)n(yp)r(e)i(I,)f (pro)n(vided)d(that)-118 882 y(all)k(second)i(co)r(ordinates)e(of)j (the)g(p)r(oin)n(ts)e(of)i(the)g(orbit)e(are)h(p)r(ositiv)n(e.)6 1009 y(In)37 b(the)h(follo)n(wing)32 b(subsections)k(w)n(e)g(consider)f (t)n(w)n(o)h(examples)e(of)j(rela-)-118 1108 y(tions)29 b(from)g(this)h(class:)40 b(represen)n(tations)27 b(of)j(real)f(forms)f (of)j(Witten's)f(\014rst)-118 1208 y(deformation,)d(and)i(represen)n (tations)d(of)j(the)h(Skly)n(anin)c(algebra)h(in)h(the)i(de-)-118 1308 y(generate)35 b(case)g(\(they)i(corresp)r(ond)e(to)h(represen)n (tations)d(of)j(the)h(quan)n(tum)-118 1407 y FN(sl)-54 1419 y FK(2)10 1407 y FO(group\).)-118 1620 y FQ(2.2.4)94 b(Represen)m(tation)30 b(of)h(real)g(forms)f(of)h(Witten's)e(\014rst)j (defor-)174 1720 y(mation)-118 1873 y(1.)49 b FO(Studying)31 b(the)h(Jones)f(p)r(olynomials,)d(their)i(generalizations)d(and)32 b(their)-118 1973 y(connections)23 b(with)i(\\v)n(ertex)e(mo)r(dels")g (in)h(t)n(w)n(o-dimensional)19 b(statistical)j(me-)-118 2072 y(c)n(hanics,)j(Witten)h(\(see)h([285)n(]\))g(in)n(tro)r(duced)f (Hopf)h(algebra)c(deformations)h(of)-118 2172 y(the)k(univ)n(ersal)c (en)n(v)n(eloping)g(algebra)h(of)i FN(su)p FO(\(2\).)37 b(There)27 b(is)f(a)h(family)e(of)i(asso-)-118 2272 y(ciativ)n(e)g (algebras)f(that)j(dep)r(end)h(on)g(a)e(real)g(parameter)f FN(p)p FO(.)42 b(These)29 b(algebras)-118 2371 y(are)19 b(de\014ned)i(b)n(y)f(the)h(generators)c FN(E)991 2383 y FK(0)1029 2371 y FO(,)22 b FN(E)1135 2383 y FK(+)1190 2371 y FO(,)g FN(E)1296 2383 y FM(\000)1373 2371 y FO(and)e(the)h (follo)n(wing)16 b(relations:)612 2571 y FN(p)e(E)729 2583 y FK(0)767 2571 y FN(E)828 2583 y FK(+)901 2571 y FP(\000)k FN(p)1026 2536 y FM(\000)p FK(1)1129 2571 y FN(E)1190 2583 y FK(+)1246 2571 y FN(E)1307 2583 y FK(0)1367 2571 y FO(=)23 b FN(E)1516 2583 y FK(+)1571 2571 y FN(;)550 2753 y FO([)p FN(E)634 2765 y FK(+)690 2753 y FN(;)14 b(E)788 2765 y FM(\000)844 2753 y FO(])23 b(=)g FN(E)1039 2765 y FK(0)1095 2753 y FP(\000)18 b FO(\()p FN(p)g FP(\000)g FN(p)1395 2719 y FM(\000)p FK(1)1484 2753 y FO(\))c FN(E)1596 2719 y FK(2)1591 2774 y(0)1633 2753 y FN(;)618 2936 y(p)g(E)735 2948 y FM(\000)791 2936 y FN(E)852 2948 y FK(0)908 2936 y FP(\000)k FN(p)1033 2902 y FM(\000)p FK(1)1136 2936 y FN(E)1197 2948 y FK(0)1234 2936 y FN(E)1295 2948 y FM(\000)1375 2936 y FO(=)k FN(E)1523 2948 y FM(\000)1579 2936 y FN(:)524 b FO(\(2.27\))6 3114 y(In)25 b(item)e(2)g(w)n(e)h(in)n(tro)r(duce)f(a)h(class)e(of)i FP(\003)p FO(-algebra)d(structures)i(in)h(Witten's)-118 3214 y(\014rst)e(deformation.)33 b(In)23 b(items)e(3)h(an)h(4)f(w)n(e)g (giv)n(e)f(a)h(description)f(of)h(irreducible)-118 3313 y(represen)n(tations)29 b(of)j(these)g FP(\003)p FO(-algebras)d(on)i(a) h(Hilb)r(ert)f(space.)50 b(W)-7 b(e)32 b(use)g(the)-118 3413 y(metho)r(d)38 b(of)h(\\dynamical)34 b(relations")h(dev)n(elop)r (ed)j(ab)r(o)n(v)n(e)f(in)h(this)g(section.)-118 3513 y(Notice,)26 b(that)g(some)f(un)n(b)r(ounded)h(represen)n(tations)e (arise)g(in)h(a)h(natural)f(w)n(a)n(y)-118 3612 y(here;)i(ho)n(w)n(ev)n (er,)e(for)i(un)n(b)r(ounded)g(op)r(erators)f(w)n(e)g(restrict)g (ourselv)n(es)e(with)j(a)-118 3712 y(description)18 b(of)j(a)g(certain) e(class)g(of)i(represen)n(tations,)e(while)h(in)g(the)h(b)r(ounded)-118 3811 y(case,)26 b(w)n(e)h(giv)n(e)d(a)j(complete)e(unitary)g (description)f(of)j(irreducible)c(represen-)-118 3911 y(tations.)p eop %%Page: 120 124 120 123 bop -118 -137 a FO(120)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(2.)43 b FO(Real)29 b(forms.)42 b(Witten's)30 b(\014rst)f(deformation)e(is)i(a)h(family)d(of)j(asso)r (ciativ)n(e)-118 196 y(algebras)i FN(A)275 208 y FL(p)349 196 y FO(giv)n(en)i(b)n(y)h(generators)e(and)i(relations)d(\(2.27\))o (,)38 b FN(p)e FP(2)g FO(\(0)p FN(;)14 b FO(1\))35 b(is)-118 296 y(a)e(parameter.)53 b FN(A)467 308 y FK(1)538 296 y FO(=)33 b FN(su)p FO(\(2\);)k FN(A)951 308 y FK(0)1022 296 y FO(has)c(the)h(follo)n(wing)c(relations)h FN(E)2087 308 y FK(+)2142 296 y FN(E)2203 308 y FK(0)2274 296 y FO(=)-118 395 y FN(E)-57 407 y FK(0)-20 395 y FN(E)41 407 y FK(+)120 395 y FO(=)23 b FN(E)274 365 y FK(2)269 416 y(0)334 395 y FO(=)g(0.)35 b(W)-7 b(e)27 b(consider)d(in)n(v)n (olutions)e(in)j FN(A)1564 407 y FL(p)1603 395 y FO(,)h(whic)n(h)f(are) g(obtained)-118 495 y(from)i(in)n(v)n(olutions)d(in)k(the)g(free)g (algebra,)e(and)i(reduce)g(the)h(linear)c(subspace)-118 595 y(generated)h(b)n(y)i(the)g(relations)c(\(2.27\))o(.)6 694 y(W)-7 b(e)32 b(will)c(sa)n(y)h(that)i(in)n(v)n(olutions)c(are)j (equiv)-5 b(alen)n(t)28 b(if)i(the)i(corresp)r(onding)-118 794 y(real)26 b(forms)g(are)g(isomorphic.)-118 956 y FQ(Lemma)j(8.)41 b FB(Ther)l(e)32 b(ar)l(e)g(two)g(ine)l(quivalent)f (involutions)h(in)g(Witten)-8 b('s)30 b(\014rst)-118 1056 y(deformation)6 b FO(:)702 1264 y FN(E)768 1229 y FM(\003)763 1284 y FK(0)829 1264 y FO(=)23 b FN(E)978 1276 y FK(0)1015 1264 y FN(;)99 b(E)1203 1229 y FM(\003)1198 1284 y FK(+)1277 1264 y FO(=)22 b FN(E)1425 1276 y FM(\000)1482 1264 y FN(;)621 b FO(\(2.28\))677 1446 y FN(E)743 1412 y FM(\003)738 1467 y FK(0)804 1446 y FO(=)23 b FN(E)953 1458 y FK(0)990 1446 y FN(;)99 b(E)1178 1412 y FM(\003)1173 1467 y FK(+)1251 1446 y FO(=)23 b FP(\000)p FN(E)1465 1458 y FM(\000)1521 1446 y FN(:)582 b FO(\(2.29\))-118 1640 y FQ(3.)69 b FO(Relations)36 b(\(2.27\))h(with)h(the)h(in)n(v)n (olution)c(giv)n(en)i(b)n(y)i(\(2.28\))f(or)f(\(2.29\))-118 1740 y(ha)n(v)n(e)24 b(the)i(form)e(\(2.25\))o(.)36 b(The)26 b(corresp)r(onding)c(dynamical)g(system)i(on)h FI(R)2214 1710 y FK(2)2283 1740 y FO(is)-118 1840 y(generated)h(b)n(y)174 2018 y FN(F)12 b FO(\()p FN(x;)i(y)s FO(\))24 b(=)f(\()p FN(p)617 1983 y FM(\000)p FK(1)706 2018 y FO(\(1)18 b(+)g FN(p)923 1983 y FM(\000)p FK(1)1012 2018 y FN(x)p FO(\))p FN(;)c(q)s FO(\()p FN(q)s(y)22 b FP(\000)c FN(x)h FO(+)f(\()p FN(p)h FP(\000)f FN(p)1753 1983 y FM(\000)p FK(1)1841 2018 y FO(\))p FN(x)1920 1983 y FK(2)1958 2018 y FO(\)\))p FN(;)-118 2196 y FO(where)27 b FN(q)f FO(=)d(1)k(\()p FN(q)f FO(=)d FP(\000)p FO(1\))k(for)g(the)h(\014rst)f(\(second\))h (real)e(form.)6 2296 y(It)19 b(is)f(a)g(di\016cult)g(problem)e(to)i (\014nd)h(p)r(ositiv)n(e)e(orbits)g(of)h(a)g(t)n(w)n(o-dimensional)-118 2395 y(nonlinear)30 b(dynamical)f(system.)51 b(W)-7 b(e)33 b(ha)n(v)n(e)f(a)n(v)n(oided)e(these)j(di\016culties)d(b)n(y)-118 2495 y(using)c(the)i(Casimir)c(elemen)n(t,)514 2673 y FN(C)573 2685 y FL(p)635 2673 y FO(=)f FN(p)765 2639 y FM(\000)p FK(1)854 2673 y FN(E)915 2685 y FK(+)970 2673 y FN(E)1031 2685 y FM(\000)1106 2673 y FO(+)18 b FN(p)c(E)1306 2685 y FM(\000)1362 2673 y FN(E)1423 2685 y FK(+)1497 2673 y FO(+)k FN(E)1646 2639 y FK(2)1641 2694 y(0)1683 2673 y FN(:)-118 2851 y FO(F)-7 b(or)29 b(an)n(y)g(irreducible)d(represen)n(tation)g FN(\031)s FO(,)31 b(w)n(e)e(ha)n(v)n(e)f FN(\031)s FO(\()p FN(C)1711 2863 y FL(p)1751 2851 y FO(\))e(=)g FN(\026I)7 b FO(,)31 b(where)e FN(\026)-118 2951 y FO(is)d(a)i(complex)d(n)n(um)n(b)r(er.)6 3051 y(W)-7 b(e)28 b(will)d(w)n(ork)h(with)i(the)f(follo)n(wing)d (system:)398 3258 y FN(E)459 3270 y FK(0)496 3258 y FN(E)557 3270 y FK(+)636 3258 y FO(=)f FN(E)785 3270 y FK(+)840 3258 y FN(f)9 b FO(\()p FN(E)983 3270 y FK(0)1020 3258 y FO(\))p FN(;)98 b(E)1239 3224 y FM(\003)1234 3279 y FK(+)1289 3258 y FN(E)1350 3270 y FK(+)1428 3258 y FO(=)23 b FN(G)1581 3270 y FL(\027)1623 3258 y FO(\()p FN(E)1716 3270 y FK(0)1753 3258 y FO(\))p FN(;)318 b FO(\(2.30\))-118 3448 y(where)27 b FN(\027)h FO(=)23 b FN(\026p)p FO(,)683 3655 y FN(f)9 b FO(\()p FN(X)e FO(\))23 b(=)f FN(p)1025 3621 y FM(\000)p FK(1)1114 3655 y FO(\(1)d(+)f FN(p)1332 3621 y FM(\000)p FK(1)1421 3655 y FN(x)p FO(\))p FN(;)470 3838 y(G)535 3850 y FL(\027)576 3838 y FO(\()p FN(y)s FO(\))24 b(=)896 3782 y FN(q)p 805 3819 222 4 v 805 3895 a FO(1)18 b(+)g FN(p)990 3871 y FK(2)1051 3838 y FO(\()p FP(\000)p FN(y)j FP(\000)d FN(p)1335 3804 y FM(\000)p FK(1)1424 3838 y FN(y)1468 3804 y FK(2)1523 3838 y FO(+)g FN(\027)5 b(I)i FO(\))p FN(:)p eop %%Page: 121 125 121 124 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(121)-118 96 y FQ(Lemma)29 b(9.)41 b FB(F)-6 b(or)27 b(any)g(irr)l(e)l(ducible)g(r)l(epr)l(esentation)g FN(\031)i FB(of)f(the)e(r)l(e)l(al)h(form)g FN(A)2300 108 y FL(p)-118 196 y FB(ther)l(e)32 b(is)h(a)g(unique)f FN(\027)37 b FO(\()p FN(\027)c FP(\025)28 b FO(0)k FB(for)h(the)f (\014rst)g(r)l(e)l(al)g(form)6 b FO(\))p FB(,)35 b(such)e(that)f FN(\031)j FB(is)e(a)-118 296 y(r)l(epr)l(esentation)d(of)48 b FO(\(2.30\))o FB(.)6 395 y(F)-6 b(or)23 b(an)f(arbitr)l(ary)h FN(\027)k FO(\()p FN(\027)i FP(\025)22 b FO(0)g FB(for)g(the)h(\014rst) e(r)l(e)l(al)h(form)6 b FO(\))p FB(,)25 b(every)e(irr)l(e)l(ducible) -118 495 y(r)l(epr)l(esentation)30 b(of)48 b FO(\(2.30\))29 b FB(with)h FO(dim)13 b FN(\031)26 b(>)d FO(1)29 b FB(is)h(a)g(r)l(epr) l(esentation)g(of)h FN(A)2207 507 y FL(p)2245 495 y FB(.)6 648 y FO(The)26 b(dynamical)c(system)i(corresp)r(onding)f(to)i (relations)d(\(2.30\))j(is,)g(actu-)-118 748 y(ally)-7 b(,)25 b(one-dimensional,)e(linear,)h(and)k(dep)r(ends)g(on)f(one)g (real)f(parameter.)6 848 y(Ev)n(ery)33 b(irreducible)f(represen)n (tation)g(of)i(\(2.30\))g(is)g(determined)f(b)n(y)h(the)-118 947 y(subset)27 b(\001)d FP(\032)e FI(R)370 917 y FK(2)414 947 y FO(,)628 1114 y(\001)h(=)g FP(f)p FO(\()p FN(\025)930 1126 y FL(k)971 1114 y FN(;)14 b(\026)1058 1126 y FL(k)1099 1114 y FO(\))p FN(;)g(j)28 b(<)22 b(k)k(<)d(J)8 b FP(g)p FN(;)-118 1281 y FO(where)23 b FN(\025)166 1293 y FL(k)q FK(+1)315 1281 y FO(=)f FN(f)9 b FO(\()p FN(\025)532 1293 y FL(k)573 1281 y FO(\),)25 b FN(\026)703 1293 y FL(k)q FK(+1)851 1281 y FO(=)e FN(G)1004 1293 y FL(\027)1045 1281 y FO(\()p FN(\025)1125 1293 y FL(k)1167 1281 y FO(\),)i FN(\026)1297 1293 y FL(k)1361 1281 y FP(\025)e FO(0,)h FN(\026)1588 1293 y FL(k)1652 1281 y FO(=)e(0)i(for)f FN(k)j FO(=)c FN(j)16 b FO(+)11 b(1)22 b FN(>)-118 1381 y FP(\0001)30 b FO(and)h FN(k)g FO(=)d FN(J)g FP(\000)20 b FO(1)28 b FN(<)g FO(+)p FP(1)p FO(;)k FN(j)5 b FO(,)31 b FN(J)39 b FO(are)30 b(in)n(teger)e(or)i(in\014nities;)g FN(l)2009 1393 y FK(2)2046 1381 y FO(\(\001\))i(is)d(a)-118 1481 y(Hilb)r(ert)d(space)h(with)g(an)g(orthonormal)d(base)j FP(f)p FN(e)1436 1496 y FK(\()p FL(\025)1501 1505 y Fv(k)1537 1496 y FL(;\026)1597 1505 y Fv(k)1633 1496 y FK(\))1672 1481 y FO(:)h(\()p FN(\025)1803 1493 y FL(k)1845 1481 y FN(;)14 b(\026)1932 1493 y FL(k)1972 1481 y FO(\))24 b FP(2)f FO(\001)p FP(g)p FO(,)230 1690 y FN(T)12 b FO(\()p FN(E)384 1702 y FK(0)421 1690 y FO(\))i FN(e)506 1705 y FK(\()p FL(\025)571 1714 y Fv(k)608 1705 y FL(;\026)668 1714 y Fv(k)704 1705 y FK(\))757 1690 y FO(=)23 b FN(\025)893 1702 y FL(k)934 1690 y FN(e)973 1705 y FK(\()p FL(\025)1038 1714 y Fv(k)1074 1705 y FL(;\026)1134 1714 y Fv(k)1171 1705 y FK(\))1201 1690 y FN(;)213 1873 y(T)12 b FO(\()p FN(E)367 1885 y FK(+)421 1873 y FO(\))i FN(e)506 1888 y FK(\()p FL(\025)571 1897 y Fv(k)608 1888 y FL(;\026)668 1897 y Fv(k)704 1888 y FK(\))757 1873 y FO(=)23 b FN(\026)895 1830 y FK(1)p FL(=)p FK(2)895 1898 y FL(k)q FK(+1)1020 1873 y FN(e)1059 1888 y FK(\()p FL(\025)1124 1897 y Fv(k)q Fx(+1)1231 1888 y FL(;\026)1291 1897 y Fv(k)q Fx(+1)1398 1888 y FK(\))1428 1873 y FN(;)180 b(j)28 b(<)23 b(k)e FO(+)d(1)23 b FN(<)f(J)o(;)212 2056 y(T)12 b FO(\()p FN(E)366 2068 y FM(\000)421 2056 y FO(\))i FN(e)506 2071 y FK(\()p FL(\025)571 2080 y Fv(k)608 2071 y FL(;\026)668 2080 y Fv(k)704 2071 y FK(\))757 2056 y FO(=)23 b FN(\026)895 2013 y FK(1)p FL(=)p FK(2)895 2081 y FL(k)999 2056 y FN(e)1038 2071 y FK(\()p FL(\025)1103 2080 y Fv(k)q Fw(\000)p Fx(1)1213 2071 y FL(;\026)1273 2080 y Fv(k)q Fw(\000)p Fx(1)1382 2071 y FK(\))1412 2056 y FN(;)180 b(j)28 b(<)23 b(k)e FP(\000)d FO(1)p FN(;)58 2238 y(T)12 b FO(\()p FN(E)212 2250 y FK(+)267 2238 y FO(\))i FN(e)352 2253 y FK(\()p FL(\025)417 2261 y Fv(J)t Fw(\000)p Fx(1)531 2253 y FL(;\026)591 2261 y Fv(J)t Fw(\000)p Fx(1)704 2253 y FK(\))757 2238 y FO(=)23 b(0)p FN(;)96 b(T)12 b FO(\()p FN(E)1160 2250 y FM(\000)1216 2238 y FO(\))i FN(e)1301 2253 y FK(\()p FL(\025)1366 2261 y Fv(j)r Fx(+1)1468 2253 y FL(;\026)1528 2261 y Fv(j)r Fx(+1)1630 2253 y FK(\))1683 2238 y FO(=)23 b(0)p FN(:)-118 2460 y FQ(4.)36 b FO(Classi\014cation)23 b(of)28 b(represen)n(tations.)-118 2614 y FQ(Theorem)i(25.)41 b FB(Every)26 b(irr)l(e)l(ducible)h(r)l(epr) l(esentation)e(of)h(the)g(\014rst)e(r)l(e)l(al)i(form)-118 2713 y(is)k(b)l(ounde)l(d.)6 2813 y FO(1.)42 b FB(F)-6 b(or)31 b(every)h(non-ne)l(gative)f(inte)l(ger)g FN(m)g FB(ther)l(e)g(is)g(a)g(r)l(epr)l(esentation)h(of)-118 2913 y(dimension)f FN(m)18 b FO(+)g(1)29 b FB(with)492 3119 y FN(\027)g FO(=)659 3063 y FN(p)p 659 3100 42 4 v 659 3176 a FO(4)725 3027 y Fy(\020)o(\020)834 3063 y FO(\(1)18 b FP(\000)g FN(p)1051 3033 y FK(2)p FL(m)1147 3063 y FO(\)\(1)h(+)f FN(p)1397 3033 y FK(2)1434 3063 y FO(\))p 834 3100 633 4 v 834 3176 a(\(1)g(+)g FN(p)1051 3152 y FK(2)p FL(m)1147 3176 y FO(\)\(1)h FP(\000)f FN(p)1397 3152 y FK(2)1434 3176 y FO(\))1476 3027 y Fy(\021)1526 3044 y FK(2)1581 3119 y FP(\000)g FO(1)1706 3027 y Fy(\021)1755 3119 y FN(;)428 3334 y FO(\001)497 3346 y FL(\027)562 3334 y FO(=)k FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)903 3346 y FK(1)940 3334 y FO(\))p FN(;)g FP(\000)p FO(1)22 b FN(<)h(k)j FP(\024)d FN(m)18 b FO(+)g(1)p FP(g)p FO(;)6 3513 y(2.)52 b FB(Ther)l(e)36 b(is)f(a)f(family)j(of)e(one-dimensional) h(r)l(epr)l(esentations)7 b FO(:)48 b FN(E)2205 3525 y FK(0)2274 3513 y FO(=)-118 3612 y FN(p)p FO(\()p FN(p)-2 3582 y FK(2)53 3612 y FP(\000)18 b FO(1\))210 3582 y FM(\000)p FK(1)299 3612 y FB(,)36 b FN(E)421 3624 y FK(+)507 3612 y FO(=)30 b FN(\025)p FB(,)36 b FN(E)772 3624 y FM(\000)859 3612 y FO(=)958 3590 y(\026)954 3612 y FN(\025)q FO(,)f FB(wher)l(e)g FN(\025)f FB(is)g(a)h(c)l(omplex)f(numb)l(er,)h FN(\027)h FO(=)-118 3712 y FN(p)-76 3682 y FK(3)-39 3712 y FO(\(1)18 b FP(\000)g FN(p)178 3682 y FK(2)215 3712 y FO(\))247 3682 y FM(\000)p FK(2)337 3712 y FO(;)6 3811 y(3.)63 b FB(F)-6 b(or)38 b(every)h FN(\027)k FP(2)38 b FO([)p FN(p)767 3781 y FK(3)804 3811 y FO(\(1)18 b FP(\000)g FN(p)1021 3781 y FK(2)1058 3811 y FO(\))1090 3781 y FM(\000)p FK(2)1180 3811 y FO(;)c(+)p FP(1)p FO(\))38 b FB(ther)l(e)g(is)g(a)g(r)l(epr)l(esentation)-118 3911 y(with)30 b(the)g(highest)h(weight,)g FO(\001)829 3923 y FL(\027)894 3911 y FO(=)22 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1235 3923 y FK(2)1272 3911 y FO(\))p FN(;)g(k)26 b(<)d FO(1)p FP(g)p FB(.)p eop %%Page: 122 126 122 125 bop -118 -137 a FO(122)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(4.)63 b FB(F)-6 b(or)38 b(every)h FN(\027)k FP(2)38 b FO([)p FN(p)767 66 y FK(3)804 96 y FO(\(1)18 b FP(\000)g FN(p)1021 66 y FK(2)1058 96 y FO(\))1090 66 y FM(\000)p FK(2)1180 96 y FO(;)c(+)p FP(1)p FO(\))38 b FB(ther)l(e)g(is)g(a)g(r)l(epr)l(esentation)-118 196 y(with)30 b(the)g(lowest)g(weight,)i FO(\001)800 208 y FL(\027)864 196 y FO(=)23 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1206 208 y FK(1)1243 196 y FO(\))p FN(;)g(k)26 b(<)c FO(1)p FP(g)p FB(.)6 360 y FO(In)30 b(the)g(theorem)d(w)n(e)i (used)h(the)f(notation)f FN(f)9 b FO(\()p FN(k)s(;)14 b(x)p FO(\))26 b(=)1778 327 y FK(1)p 1746 341 99 4 v 1746 389 a FL(p)1780 373 y Fx(2)p Fv(k)1854 360 y FO(\()p FN(x)21 b FO(+)2048 323 y FL(p)2082 298 y Fx(2)p Fv(k)2147 323 y FM(\000)p FK(1)p 2048 341 184 4 v 2064 388 a FL(p)2098 372 y Fx(2)2130 388 y FM(\000)p FK(1)2241 360 y FN(p)p FO(\);)-118 477 y FN(g)-78 489 y FL(\027)-37 477 y FO(\()p FN(x)p FO(\))j(=)f FP(\000)p FN(x)8 b FP(\000)g FN(p)421 446 y FM(\000)p FK(1)509 477 y FN(x)556 446 y FK(2)602 477 y FO(+)g FN(\027)d FO(,)24 b FN(x)815 489 y FK(1)876 477 y FN(<)e(x)1010 489 y FK(2)1070 477 y FO(are)g(ro)r(ots)f(of)i(the) g(equation)d FN(g)2010 489 y FL(\027)2051 477 y FO(\()p FN(x)p FO(\))k(=)f(0.)-118 620 y FQ(Theorem)30 b(26.)41 b FB(Ther)l(e)32 b(ar)l(e)g(b)l(ounde)l(d)g(and)g(unb)l(ounde)l(d)f (irr)l(e)l(ducible)i(r)l(epr)l(e-)-118 720 y(sentations)c(of)g(the)g (se)l(c)l(ond)h(r)l(e)l(al)f(form)g(in)g(an)g(in\014nite-dimensional)h (Hilb)l(ert)-118 819 y(sp)l(ac)l(e,)40 b(which)e(ar)l(e)g(given,)h(exc) l(ept)e(for)h(the)f(one-dimensional)i(r)l(epr)l(esenta-)-118 919 y(tion,)30 b(by)g(the)g(fol)t(lowing)7 b FO(:)-17 1062 y(1)p FN(:)41 b FB(b)l(ounde)l(d)31 b(r)l(epr)l(esentations)f (with)g(the)g(highest)h(weight)f(such)g(that)289 1217 y FN(\027)f FP(2)23 b FO([)p FP(\000)p FN(p=)p FO(4;)14 b FN(p)730 1182 y FK(3)765 1217 y FO(\(1)19 b FP(\000)f FN(p)983 1182 y FK(2)1020 1217 y FO(\))1052 1182 y FM(\000)p FK(2)1141 1217 y FO(\))p FN(;)99 b FO(\001)1364 1229 y FL(\027)1429 1217 y FO(=)22 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1770 1229 y FK(1)1807 1217 y FO(\))p FN(;)g(k)26 b(<)d FO(1)p FP(g)p FO(;)-17 1398 y(2)p FN(:)41 b FB(unb)l(ounde)l(d)30 b(r)l(epr)l(esentations)7 b FO(:)122 1553 y(\()p FN(a)p FO(\))42 b FB(a)30 b(family)i(with)e(the)g(highest)h (weight)g(such)e(that)576 1707 y FN(\027)f FP(2)23 b FO(\()p FP(\000)p FN(p=)p FO(4;)k(0\))p FN(;)98 b FO(\001)1260 1719 y FL(\027)1325 1707 y FO(=)22 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1666 1719 y FK(2)1703 1707 y FO(\))p FN(;)g(k)26 b(<)d FO(1)p FP(g)p FO(;)272 1862 y FB(a)30 b(family)i(with)e(the)g(highest)h(weight)g(such)e(that)364 2016 y FN(\027)g FP(2)23 b FO(\()p FN(p)586 1982 y FK(3)624 2016 y FO(\(1)18 b FP(\000)g FN(p)841 1982 y FK(2)878 2016 y FO(\))910 1982 y FM(\000)p FK(2)999 2016 y FO(;)c(+)p FP(1)p FO(\))p FN(;)99 b FO(\001)1407 2028 y FL(\027)1471 2016 y FO(=)23 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1813 2028 y FK(1)1850 2016 y FO(\))p FN(;)g(k)26 b(>)d FP(\000)p FO(1)p FP(g)p FO(;)130 2181 y(\()p FN(b)p FO(\))42 b FB(a)30 b(family)i(with)e(the)g(lowest)g(weight)h(such)f (that)580 2336 y FN(\027)f FP(2)23 b FO([)p FN(p=)p FO(4;)k(0\))p FN(;)98 b FO(\001)1191 2348 y FL(\027)1256 2336 y FO(=)22 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1597 2348 y FK(2)1634 2336 y FO(\))p FN(;)g(k)26 b(>)d FP(\000)p FO(1)p FP(g)p FO(;)272 2501 y FB(a)30 b(family)i(with)e(the)g(lowest)g (weight)h(such)f(that)364 2655 y FN(\027)f FP(2)23 b FO(\()p FN(p)586 2621 y FK(3)624 2655 y FO(\(1)18 b FP(\000)g FN(p)841 2621 y FK(2)878 2655 y FO(\))910 2621 y FM(\000)p FK(2)999 2655 y FO(;)c(+)p FP(1)p FO(\))p FN(;)99 b FO(\001)1407 2667 y FL(\027)1471 2655 y FO(=)23 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(x)1813 2667 y FK(1)1850 2655 y FO(\))p FN(;)g(k)26 b(>)d FP(\000)p FO(1)p FP(g)p FO(;)130 2820 y(\()p FN(c)p FO(\))42 b FB(families)32 b(without)e(highest)h(and) f(lowest)g(weights)7 b FO(:)272 2931 y FB(a)30 b(family)i(indexe)l(d)e (by)g FO(\()p FN(\025;)14 b(\027)5 b FO(\))p FB(,)32 b(wher)l(e)330 3119 y FN(\025)24 b FP(2)f FO(\()p FP(\000)p FN(p)30 b FO(;)14 b FP(\000)p FN(p=)p FO(2\))j FP([)h FO(\()p FP(\000)1106 3063 y FN(p)p 1106 3100 42 4 v 1106 3176 a FO(2)1188 3119 y(;)c(0])p FN(;)98 b(\027)28 b FP(2)c FO(\()p FP(\0001)p FO(;)14 b FN(\025)p FO(\()p FN(\025)19 b FO(+)f FN(p)p FO(\))p FN(p)2122 3085 y FM(\000)p FK(1)2211 3119 y FO(\))p FN(;)837 3314 y FO(\001)906 3329 y FK(\()p FL(\025;\027)t FK(\))1081 3314 y FO(=)23 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(\025)p FO(\))p FN(;)g(k)26 b FP(2)d FI(Z)o FP(g)p FO(;)272 3495 y FB(a)30 b(family)i(indexe)l(d)e(by)g FO(\()p FN(\025;)14 b(\027)5 b FO(\))p FB(,)32 b(wher)l(e)467 3684 y FO(\()p FN(\025;)14 b(\027)5 b FO(\))24 b FP(2)g FO([)p FP(\000)p FN(p)895 3649 y FM(\000)p FK(2)983 3684 y FO(;)14 b FP(\000)p FN(p)k FP(\000)g FO(1\))g FP(\002)g FO(\()p FP(\0001)p FO(;)c FN(p)1662 3649 y FK(3)1699 3684 y FO(\(1)k FP(\000)g FN(p)1916 3649 y FK(2)1953 3684 y FO(\))1985 3649 y FM(\000)p FK(1)2075 3684 y FO(\))p FN(;)837 3866 y FO(\001)906 3881 y FK(\()p FL(\025;\027)t FK(\))1081 3866 y FO(=)23 b FP(f)p FN(f)9 b FO(\()p FN(k)s(;)14 b(\025)p FO(\))p FN(;)28 b(k)d FP(2)f FI(Z)o FP(g)p FN(:)p eop %%Page: 123 127 123 126 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(123)-118 96 y FQ(2.2.5)94 b(Represen)m(tations)22 b(of)i(the)g(Skly)m(anin)h(algebra)f(and)g FN(U)2066 108 y FL(q)2103 96 y FO(\()p FN(sl)r FO(\(2\)\))-118 250 y(In)34 b(this)f(subsection)g(w)n(e)h(apply)e(the)j(tec)n(hnique)e (dev)n(elop)r(ed)f(to)i(a)g(study)g(of)-118 349 y(represen)n(tations)21 b(of)k(t)n(w)n(o)e(algebras)f(that)j(arise)d(naturally)f(in)j(ph)n (ysical)e(mo)r(d-)-118 449 y(els.)-118 594 y FQ(1.)34 b FO(The)22 b(Skly)n(anin)d(algebra)g FA(F)j FO(w)n(as)e(in)n(tro)r (duced)h(in)g([254)n(])h(as)f(an)h(algebra)c(o)n(v)n(er)-118 694 y FI(C)49 b FO(generated)26 b(b)n(y)h(elemen)n(ts)f FN(S)854 706 y FK(0)891 694 y FO(,)i FN(S)993 706 y FK(1)1030 694 y FO(,)g FN(S)1132 706 y FK(2)1169 694 y FO(,)f FN(S)1270 706 y FK(3)1335 694 y FO(that)h(satisfy)e(the)i(relations:)-15 896 y([)p FN(S)59 908 y FK(0)96 896 y FN(;)14 b(S)184 908 y FK(3)221 896 y FO(])23 b(=)g(0)p FN(;)96 b FO([)p FN(S)590 908 y FK(0)628 896 y FN(;)14 b(S)716 908 y FK(1)753 896 y FO(])23 b(=)f FN(iJ)8 b FP(f)p FN(S)1062 908 y FK(2)1099 896 y FN(;)14 b(S)1187 908 y FK(3)1224 896 y FP(g)p FN(;)96 b FO([)p FN(S)1459 908 y FK(0)1496 896 y FN(;)14 b(S)1584 908 y FK(2)1621 896 y FO(])24 b(=)e FP(\000)p FN(iJ)8 b FP(f)p FN(S)1996 908 y FK(3)2032 896 y FN(;)14 b(S)2120 908 y FK(1)2157 896 y FP(g)p FN(;)-31 1079 y FO([)p FN(S)43 1091 y FK(1)80 1079 y FN(;)g(S)168 1091 y FK(2)205 1079 y FO(])23 b(=)g(2)p FN(iS)461 1091 y FK(0)497 1079 y FN(S)548 1091 y FK(3)585 1079 y FN(;)97 b FO([)p FN(S)779 1091 y FK(2)816 1079 y FN(;)14 b(S)904 1091 y FK(3)941 1079 y FO(])24 b(=)e FN(i)p FP(f)p FN(S)1197 1091 y FK(0)1233 1079 y FN(;)14 b(S)1321 1091 y FK(1)1358 1079 y FP(g)p FN(;)97 b FO([)p FN(S)1594 1091 y FK(3)1631 1079 y FN(;)14 b(S)1719 1091 y FK(1)1756 1079 y FO(])23 b(=)g FN(i)p FP(f)p FN(S)2012 1091 y FK(0)2048 1079 y FN(;)14 b(S)2136 1091 y FK(2)2173 1079 y FP(g)p FN(;)2126 1213 y FO(\(2.31\))-118 1397 y(where)33 b FN(J)41 b FO(is)33 b(a)g(real)f(parameter.)52 b(The)33 b(cen)n(ter)g(of)h FA(F)f FO(is)g(generated)f(b)n(y)h(t)n(w)n(o)-118 1497 y(elemen)n(ts,)25 b(\001)313 1509 y FK(1)374 1497 y FO(=)e FN(S)518 1467 y FK(2)513 1518 y(0)573 1497 y FP(\000)18 b FN(J)8 b(S)766 1467 y FK(2)761 1518 y(3)831 1497 y FO(and)27 b(\001)1061 1509 y FK(2)1122 1497 y FO(=)22 b FN(S)1265 1467 y FK(2)1260 1518 y(0)1321 1497 y FO(+)c FN(S)1460 1467 y FK(2)1455 1518 y(1)1515 1497 y FO(+)g FN(S)1654 1467 y FK(2)1649 1518 y(2)1709 1497 y FO(+)g FN(S)1848 1467 y FK(2)1843 1518 y(3)1885 1497 y FO(.)-118 1643 y FQ(2.)34 b FO(The)20 b(quan)n(tum)g(algebra)e FN(U)839 1655 y FL(q)875 1643 y FO(\()p FN(sl)r FO(\(2\)\))j([152)o(,)f (73)o(,)h(118)o(])f(is)g(a)g(complex)e(algebra)-118 1742 y(generated)26 b(b)n(y)i(elemen)n(ts)d FN(k)s FO(,)j FN(k)858 1712 y FM(\000)p FK(1)947 1742 y FO(,)f FN(X)7 b FO(,)27 b FN(Y)47 b FO(satisfying)25 b(the)j(relations)231 1945 y FN(k)s(k)323 1910 y FM(\000)p FK(1)435 1945 y FO(=)22 b FN(k)568 1910 y FM(\000)p FK(1)657 1945 y FN(k)k FO(=)d(1)p FN(;)96 b(k)s(X)29 b FO(=)23 b FN(q)s(X)7 b(k)s(;)96 b(k)s(Y)42 b FO(=)22 b FN(q)1751 1910 y FM(\000)p FK(1)1840 1945 y FN(Y)d(k)s(;)757 2146 y FO([)p FN(X)r(;)14 b(Y)k FO(])23 b(=)1098 2090 y FN(k)1144 2060 y FK(2)1199 2090 y FP(\000)18 b FN(k)1328 2060 y FM(\000)p FK(2)p 1098 2127 320 4 v 1122 2203 a FN(q)j FP(\000)d FN(q)1303 2179 y FM(\000)p FK(1)1427 2146 y FN(;)-118 2359 y FO(where)28 b FN(q)33 b FO(is)28 b(a)g(complex)f(parameter,)g FN(q)i FP(6)p FO(=)c FP(\000)p FO(1,)j(0,)h(1.)41 b(The)30 b(cen)n(ter)e(is)g (gener-)-118 2458 y(ated)f(b)n(y)h FN(C)h FO(=)23 b FN(X)7 b(Y)36 b FO(+)18 b(\()p FN(q)673 2428 y FM(\000)p FK(1)763 2458 y FN(k)809 2428 y FK(2)864 2458 y FO(+)g FN(q)s(k)1033 2428 y FM(\000)p FK(2)1122 2458 y FO(\))p FN(=)p FO(\()p FN(q)k FP(\000)c FN(q)1410 2428 y FM(\000)p FK(1)1499 2458 y FO(\))1531 2428 y FK(2)1568 2458 y FO(.)-118 2587 y FB(R)l(emark)30 b(33.)42 b FO(The)28 b(transp)r(osition)d FN(k)h FP($)d FN(k)1235 2557 y FM(\000)p FK(1)1352 2587 y FO(de\014nes)28 b(an)f(isomorphism)c(b)r(e-)-118 2687 y(t)n(w)n(een)k FN(U)174 2699 y FL(q)210 2687 y FO(\()p FN(sl)r FO(\(2\)\))h(and)g FN(U)693 2703 y FL(q)725 2686 y Fw(\000)p Fx(1)806 2687 y FO(\()p FN(sl)r FO(\(2\)\).)-118 2816 y FQ(3.)33 b FO(The)18 b(t)n(w)n(o)g(algebras)d(in)n(tro)r(duced)i (are)h(closely)d(related.)33 b(Indeed,)20 b(in)n(tro)r(duce)-118 2916 y(the)28 b(corresp)r(ondence)395 3167 y FN(q)s FO(\()p FN(J)8 b FO(\))24 b(=)665 3025 y Fy(\()742 3068 y FK(1+)826 3019 y FM(p)p 880 3019 43 3 v 880 3068 a FL(J)p 741 3082 182 4 v 741 3140 a FK(1)p FM(\000)826 3091 y(p)p 881 3091 43 3 v 49 x FL(J)933 3101 y FN(;)159 b FO(if)27 b FN(J)k(>)22 b FO(0)28 b(and)f FN(J)k FP(6)p FO(=)23 b(1,)742 3198 y FK(1+)p FL(i)849 3153 y FM(p)p 904 3153 95 3 v 45 x(\000)p FL(J)p 741 3215 258 4 v 741 3269 a FK(1)p FM(\000)p FL(i)849 3223 y FM(p)p 904 3223 95 3 v 46 x(\000)p FL(J)1008 3234 y FN(;)84 b FO(if)27 b FN(J)k(<)22 b FO(0.)-118 3423 y(It)32 b(establishes)d(a)i(one-to-one)f(mapping)f(b) r(et)n(w)n(een)j(the)g(sets)f FP(f)p FN(J)38 b FP(2)30 b FI(R)9 b FO(:)35 b FN(J)j FP(6)p FO(=)-118 3522 y(0)p FN(;)14 b FO(1)p FP(g)p FO(,)38 b(and)f FP(f)p FN(q)k FP(2)e FI(R)10 b FO(:)36 b FP(j)p FN(q)s FP(j)k FN(>)e FO(1)p FP(g)24 b([)h(f)p FN(q)41 b FP(2)e FI(C)25 b FO(:)37 b FP(j)p FN(q)s FP(j)i FO(=)f(1)p FN(;)27 b FO(Im)13 b FN(q)42 b(>)d FO(0)p FP(g)p FO(.)64 b(Also,)-118 3622 y(in)n(tro)r(duce)26 b(the)i(elemen)n(ts)419 3849 y FN(A)481 3861 y FK(1)541 3849 y FO(=)629 3707 y Fy(\()696 3792 y FN(S)747 3804 y FK(0)803 3792 y FO(+)886 3722 y FP(p)p 955 3722 55 4 v 70 x FN(J)22 b(S)1074 3804 y FK(3)1111 3792 y FN(;)190 b(J)31 b(>)23 b FO(0)p FN(;)696 3912 y(S)747 3924 y FK(0)803 3912 y FO(+)18 b FN(i)929 3845 y FP(p)p 997 3845 119 4 v 997 3912 a(\000)p FN(J)k(S)1181 3924 y FK(3)1218 3912 y FN(;)83 b(J)31 b(<)23 b FO(0)p FN(;)p eop %%Page: 124 128 124 127 bop -118 -137 a FO(124)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)419 159 y FN(A)481 171 y FK(2)541 159 y FO(=)629 17 y Fy(\()696 102 y FN(S)747 114 y FK(0)803 102 y FP(\000)886 32 y(p)p 955 32 55 4 v 70 x FN(J)e(S)1074 114 y FK(3)1111 102 y FN(;)190 b(J)31 b(>)23 b FO(0)p FN(;)696 222 y(S)747 234 y FK(0)803 222 y FP(\000)18 b FN(i)929 155 y FP(p)p 997 155 119 4 v 997 222 a(\000)p FN(J)k(S)1181 234 y FK(3)1218 222 y FN(;)83 b(J)31 b(<)23 b FO(0)p FN(;)443 441 y(X)29 b FO(=)629 299 y Fy(\()706 352 y FK(1)p 706 366 34 4 v 706 413 a(2)763 318 y FP(p)p 832 318 198 4 v 66 x FO(1)18 b FP(\000)g FN(J)k FO(\()p FN(S)1126 396 y FK(1)1182 384 y FO(+)c FN(iS)1345 396 y FK(2)1382 384 y FO(\))p FN(;)111 b(J)32 b FP(\024)22 b FO(1)p FN(;)696 504 y(i)735 471 y FK(1)p 735 485 34 4 v 735 533 a(2)791 437 y FP(p)p 861 437 198 4 v 861 504 a FN(J)k FP(\000)18 b FO(1)13 b(\()p FN(S)1154 516 y FK(1)1210 504 y FO(+)18 b FN(iS)1373 516 y FK(2)1410 504 y FO(\))p FN(;)83 b(J)32 b(>)22 b FO(1)p FN(;)452 723 y(Y)41 b FO(=)629 581 y Fy(\()706 634 y FK(1)p 706 648 34 4 v 706 695 a(2)763 600 y FP(p)p 832 600 198 4 v 67 x FO(1)18 b FP(\000)g FN(J)k FO(\()p FN(S)1126 679 y FK(1)1182 667 y FP(\000)c FN(iS)1345 679 y FK(2)1382 667 y FO(\))p FN(;)111 b(J)32 b FP(\024)22 b FO(1)p FN(;)696 786 y(i)735 753 y FK(1)p 735 767 34 4 v 735 815 a(2)791 719 y FP(p)p 861 719 198 4 v 861 786 a FN(J)k FP(\000)18 b FO(1)13 b(\()p FN(S)1154 798 y FK(1)1210 786 y FP(\000)18 b FN(iS)1373 798 y FK(2)1410 786 y FO(\))p FN(:)83 b(J)32 b(>)22 b FO(1)p FN(;)-118 983 y FQ(Prop)s(osition)30 b(42.)41 b FB(F)-6 b(or)34 b FN(J)k FP(6)p FO(=)30 b(0)p FB(,)35 b FO(1)p FB(,)f(ther)l(e)g(is)h(an)e(algebr)l(aic)j (isomorphism)-118 1083 y(b)l(etwe)l(en)29 b FA(F)r FN(=)p FO(\(\001)386 1095 y FK(1)440 1083 y FP(\000)18 b FN(I)7 b FO(\))p FB(,)31 b(and)f FN(U)872 1098 y FL(q)r FK(\()p FL(J)5 b FK(\))1002 1083 y FO(\()p FN(sl)r FO(\(2\)\))p FB(.)-118 1256 y(Pr)l(o)l(of.)43 b FO(The)28 b(relations)c(\(2.31\))j (are)g(equiv)-5 b(alen)n(t)25 b(to)i(the)h(follo)n(wing)496 1443 y FN(A)558 1455 y FK(1)595 1443 y FN(X)h FO(=)23 b FN(q)s(X)7 b(A)959 1455 y FK(1)996 1443 y FN(;)97 b(A)1178 1455 y FK(1)1215 1443 y FN(Y)42 b FO(=)23 b FN(q)1433 1409 y FM(\000)p FK(1)1522 1443 y FN(Y)c(A)1651 1455 y FK(1)1688 1443 y FN(;)496 1578 y(A)558 1590 y FK(2)595 1578 y FN(Y)42 b FO(=)22 b FN(q)s(Y)d(A)941 1590 y FK(2)979 1578 y FN(;)97 b(A)1161 1590 y FK(2)1198 1578 y FN(X)29 b FO(=)23 b FN(q)1424 1544 y FM(\000)p FK(1)1513 1578 y FN(X)7 b(A)1651 1590 y FK(2)1688 1578 y FN(;)321 1751 y FO([)p FN(A)406 1763 y FK(1)444 1751 y FN(;)14 b(A)543 1763 y FK(2)580 1751 y FO(])23 b(=)g(0)p FN(;)97 b FO([)p FN(X)r(;)14 b(Y)k FO(])23 b(=)1331 1695 y(1)p 1217 1732 271 4 v 1217 1808 a FN(q)e FP(\000)d FN(q)1398 1784 y FM(\000)p FK(1)1497 1751 y FO(\()p FN(A)1591 1717 y FK(2)1591 1771 y(1)1647 1751 y FP(\000)g FN(A)1792 1717 y FK(2)1792 1771 y(2)1830 1751 y FO(\))p FN(;)-118 1988 y FO(where)33 b FN(q)j FO(=)c FN(q)s FO(\()p FN(J)8 b FO(\).)55 b(Set)34 b FN(k)h FO(=)e FN(A)921 2000 y FK(1)958 1988 y FO(,)i FN(k)1062 1957 y FM(\000)p FK(1)1184 1988 y FO(=)d FN(A)1343 2000 y FK(2)1381 1988 y FO(,)j(and)e(notice)g(that)g FN(A)2104 2000 y FK(1)2142 1988 y FN(A)2204 2000 y FK(2)2274 1988 y FO(=)-118 2087 y(\001)-49 2099 y FK(1)-12 2087 y FO(.)p 2278 2087 4 57 v 2282 2034 50 4 v 2282 2087 V 2331 2087 4 57 v -118 2272 a FQ(4.)40 b FO(No)n(w)29 b(w)n(e)g(describ)r(e)e(the)j(real)d(forms)g(of)i FA(F)r FO(.)40 b(W)-7 b(e)29 b(will)d(assume)i(that)h(in)n(v)n(o-)-118 2372 y(lution)23 b(preserv)n(es)h(natural)f(grading)g(related)g(to)i (the)h(c)n(hosen)e(generators)f FN(S)2288 2384 y FL(i)2316 2372 y FO(,)-118 2471 y FN(i)g FO(=)f(0,)27 b(1,)h(2,)f(3.)-118 2645 y FQ(Prop)s(osition)j(43.)41 b FB(Ther)l(e)30 b(ar)l(e)g(thr)l(e)l (e)g(r)l(e)l(al)g(forms)g(of)h FA(F)r FB(,)d(namely)7 b FO(:)6 2746 y FA(F)62 2758 y FK(1)121 2746 y FO(:)29 b FN(S)229 2716 y FM(\003)224 2768 y FL(i)290 2746 y FO(=)23 b FN(S)429 2758 y FL(i)457 2746 y FB(,)30 b FN(i)22 b FO(=)h(0)p FB(,)30 b FO(1)p FB(,)f FO(2)p FB(,)h FO(3;)6 2848 y FA(F)62 2860 y FK(2)121 2848 y FO(:)f FN(S)229 2818 y FM(\003)224 2869 y FK(0)290 2848 y FO(=)23 b FN(S)429 2860 y FK(0)466 2848 y FB(,)30 b FN(S)577 2818 y FM(\003)572 2869 y FK(3)638 2848 y FO(=)23 b FN(S)777 2860 y FK(3)814 2848 y FB(,)30 b FN(S)925 2818 y FM(\003)920 2869 y FK(1)986 2848 y FO(=)23 b FN(S)1125 2860 y FK(1)1162 2848 y FB(,)30 b FN(S)1273 2818 y FM(\003)1268 2869 y FK(2)1334 2848 y FO(=)23 b FN(S)1473 2860 y FK(2)1510 2848 y FO(;)6 2950 y FA(F)62 2962 y FK(3)132 2950 y FO(:)35 b FN(S)246 2920 y FM(\003)241 2971 y FK(0)318 2950 y FO(=)f FN(S)468 2962 y FK(0)505 2950 y FB(,)k FN(S)624 2920 y FM(\003)619 2971 y FK(3)696 2950 y FO(=)33 b FP(\000)p FN(S)910 2962 y FK(3)947 2950 y FB(,)38 b FN(S)1066 2920 y FM(\003)1061 2971 y FK(1)1138 2950 y FO(=)33 b FN(S)1287 2962 y FK(1)1325 2950 y FB(,)k FN(S)1443 2920 y FM(\003)1438 2971 y FK(2)1515 2950 y FO(=)d FP(\000)p FN(S)1730 2962 y FK(2)1767 2950 y FB(,)j(if)g FN(J)42 b FP(\024)34 b FO(1)p FB(,)j(and)-118 3050 y FN(S)-62 3020 y FM(\003)-67 3071 y FK(1)-1 3050 y FO(=)22 b FP(\000)p FN(S)202 3062 y FK(1)239 3050 y FB(,)30 b FN(S)350 3020 y FM(\003)345 3071 y FK(2)411 3050 y FO(=)23 b FN(S)550 3062 y FK(2)587 3050 y FB(,)30 b(if)h FN(J)g(>)23 b FO(1)p FB(.)6 3152 y(F)-6 b(or)30 b FN(J)h(<)23 b FO(0)p FB(,)30 b(the)g FP(\003)p FB(-algebr)l(as)g FA(F)1003 3164 y FK(1)1068 3152 y FB(and)g FA(F)1285 3164 y FK(2)1350 3152 y FB(ar)l(e)g FP(\003)p FB(-isomorphic.)-118 3325 y(Pr)l(o)l(of.)43 b FO(The)35 b(pro)r(of)f(is)f(a)h(routine)f (calculation.)54 b(F)-7 b(or)34 b FN(J)42 b(<)35 b FO(0,)g(the)g (isomor-)-118 3425 y(phism)26 b FN(j)14 b FO(:)28 b FA(F)282 3437 y FK(1)341 3425 y FP(\000)-48 b(!)23 b FA(F)520 3437 y FK(2)583 3425 y FO(is)k(giv)n(en)f(b)n(y)i FN(j)5 b FO(\()p FN(S)1122 3437 y FK(0)1159 3425 y FO(\))24 b(=)f FP(\000)1368 3358 y(p)p 1437 3358 119 4 v 67 x(\000)p FN(J)8 b(S)1607 3437 y FK(3)1644 3425 y FO(,)28 b FN(j)5 b FO(\()p FN(S)1817 3437 y FK(3)1854 3425 y FO(\))24 b(=)f FN(S)2049 3437 y FL(o)2086 3425 y FN(=)2128 3358 y FP(p)p 2197 3358 V 67 x(\000)p FN(J)8 b FO(,)-118 3524 y FN(j)d FO(\()p FN(S)4 3536 y FK(1)41 3524 y FO(\))24 b(=)e FN(iS)264 3536 y FK(2)301 3524 y FO(,)28 b FN(j)5 b FO(\()p FN(S)474 3536 y FK(2)511 3524 y FO(\))23 b(=)g FP(\000)p FN(S)770 3536 y FK(1)807 3524 y FO(.)p 2278 3524 4 57 v 2282 3472 50 4 v 2282 3524 V 2331 3524 4 57 v 6 3709 a(The)28 b(corresp)r(onding)d(real)g(forms)h(of)i FN(U)1253 3721 y FL(q)1289 3709 y FO(\()p FN(sl)r FO(\(2\)\))g(are)f (the)h(follo)n(wing:)6 3811 y FN(su)93 3823 y FL(q)130 3811 y FO(\(2\))21 b(:)34 b FN(k)360 3781 y FM(\003)421 3811 y FO(=)23 b FN(k)s FO(,)g FN(X)677 3781 y FM(\003)737 3811 y FO(=)g FN(Y)40 b FO(for)21 b FN(q)26 b(>)d FO(0;)g FN(X)1349 3781 y FM(\003)1409 3811 y FO(=)g FP(\000)p FN(Y)40 b FO(for)21 b FN(q)26 b(<)d FO(0,)f FN(k)2055 3781 y FM(\003)2116 3811 y FO(=)h FN(k)2250 3781 y FM(\000)p FK(1)-118 3911 y FO(for)k FN(q)f FP(2)d FI(T)p FO(;)p eop %%Page: 125 129 125 128 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(125)6 96 y FN(su)93 108 y FL(q)130 96 y FO(\(1)p FN(;)14 b FO(1\))35 b(:)55 b FN(k)474 66 y FM(\003)550 96 y FO(=)37 b FN(k)s FO(,)h FN(X)835 66 y FM(\003)911 96 y FO(=)f FP(\000)p FN(Y)54 b FO(for)36 b FN(q)41 b(>)c FO(0;)k FN(X)1678 66 y FM(\003)1753 96 y FO(=)c FN(Y)55 b FO(for)36 b FN(q)41 b(<)c FO(0,)-118 196 y FN(k)-72 166 y FM(\003)-11 196 y FO(=)23 b FN(k)123 166 y FM(\000)p FK(1)239 196 y FO(for)k FN(q)f FP(2)e FI(T)p FO(;)6 296 y FN(sl)70 308 y FL(q)107 296 y FO(\(2)p FN(;)14 b FI(R)p FO(\))37 b(:)44 b FN(X)484 266 y FM(\003)550 296 y FO(=)28 b FN(X)7 b FO(,)32 b FN(Y)840 266 y FM(\003)907 296 y FO(=)d FN(Y)18 b FO(,)32 b FN(k)1168 266 y FM(\003)1235 296 y FO(=)d FN(k)1375 266 y FM(\000)p FK(1)1463 296 y FO(,)k(if)d FN(q)i FP(2)d FI(R)p FO(;)39 b FN(k)1913 266 y FM(\003)1980 296 y FO(=)28 b FN(k)2119 266 y FM(\000)p FK(1)2239 296 y FO(for)-118 395 y FN(q)e FP(2)d FI(R)p FO(.)6 495 y(F)-7 b(or)27 b FN(q)g FP(2)c FI(T)p FO(,)k(the)h FP(\003)p FO(-algebra)c FN(su)993 507 y FL(q)1029 495 y FO(\(1)p FN(;)14 b FO(1\))27 b(is)g(isomorphic)c(to)28 b FN(su)1934 507 y FL(q)1970 495 y FO(\(2\).)-118 627 y FB(R)l(emark)i(34.)42 b FO(In)22 b([163)n(])f FP(\003)p FO(-Hopf)g(algebra)d(structures)i(on)h FN(U)1728 639 y FL(q)1765 627 y FO(\()p FN(sl)r FO(\(2\)\))g(are)f(stud-)-118 726 y(ied.)45 b(In)30 b(our)g(list,)g(the)g(in)n(v)n(olution)d(is)i (compatible)f(with)i(the)h(Hopf)f(algebra)-118 826 y(structure)g(in)f (the)i(cases)e(of)h FN(su)881 838 y FL(q)917 826 y FO(\(2\),)h FN(su)1164 838 y FL(q)1200 826 y FO(\(1)p FN(;)14 b FO(1\))30 b(with)g FN(q)g FP(2)e FI(R)p FO(,)37 b(and)30 b FN(sl)2099 838 y FL(q)2136 826 y FO(\(2)p FN(;)14 b FI(R)p FO(\))-118 926 y(with)27 b FN(q)f FP(2)e FI(T)p FO(.)-118 1057 y FQ(5.)39 b FO(No)n(w)28 b(w)n(e)h(go)e(to)i(the)g(study)g(of)f FP(\003)p FO(-represen)n(tations)d(of)k(real)e(forms)g(of)h(the)-118 1157 y(Skly)n(anin)d(algebra)g(and)i(the)h(corresp)r(onding)d(real)g (forms)h(of)i FN(U)1892 1169 y FL(q)1928 1157 y FO(\()p FN(sl)r FO(\(2\)\).)6 1257 y(First)i(notice)g(that)h(the)h(relations)27 b(in)j FA(F)i FO(are)d(homogeneous,)h(therefore,)-118 1356 y(for)25 b(an)n(y)g FN(\025)f FP(2)f FI(R)p FO(,)32 b FN(\025)24 b FP(6)p FO(=)e(0,)k(the)g(op)r(erators)e(\()p FN(S)1261 1368 y FL(i)1289 1356 y FO(\))i(form)f(represen)n(tations)d (of)k FA(F)g FO(if)-118 1456 y(and)g(only)e(if)i(so)f(do)h(the)g(op)r (erators)e(\()p FN(\025S)1149 1468 y FL(i)1178 1456 y FO(\).)36 b(The)27 b(same)d(holds)h(for)g FN(U)2043 1468 y FL(q)2080 1456 y FO(\()p FN(sl)r FO(\(2\)\),)-118 1555 y(and)d(w)n(e)g(agree)e(to)i(iden)n(tify)f(represen)n(tations)e(\()p FN(\025S)1460 1567 y FL(i)1488 1555 y FO(\))k(with)f(di\013eren)n(t)f FN(\025)i FP(6)p FO(=)g(0)f(of)-118 1655 y FA(F)r FO(,)h(and)h(\()p FN(k)s(;)14 b(X)r(;)g(Y)k FO(\))25 b(and)e(\()p FP(\000)p FN(k)s(;)14 b(X)r(;)g(Y)k FO(\))25 b(of)f FN(U)1206 1667 y FL(q)1242 1655 y FO(\()p FN(sl)r FO(\(2\)\),)h(as)e(w)n(ell)f(as)i (the)g(unitarily)-118 1755 y(equiv)-5 b(alen)n(t)25 b(ones.)-118 1886 y FB(R)l(emark)30 b(35.)42 b FO(In)31 b(the)f(case)f FN(J)35 b FO(=)26 b(0,)k(eac)n(h)f(irreducible)d(represen)n(tation)i (of)h FA(F)-118 1986 y FO(is)e(either)h(one-dimensional)23 b(with)28 b FN(S)1054 1998 y FK(0)1116 1986 y FO(=)c(0,)29 b(or)e(is)h(suc)n(h)g(that)h FN(S)1906 1998 y FK(0)1968 1986 y FO(=)2067 1953 y FK(1)p 2067 1967 34 4 v 2067 2015 a(2)2110 1986 y FN(I)7 b FO(,)29 b(and)-118 2086 y FN(S)-67 2098 y FK(1)-30 2086 y FO(,)f FN(S)72 2098 y FK(2)109 2086 y FO(,)g FN(S)211 2098 y FK(3)275 2086 y FO(form)f(an)g(irreducible)d FP(\003)p FO(-represen)n(tation)g(of)k FN(sl)r FO(\(2\).)6 2217 y(Belo)n(w,)21 b(w)n(e)g(will)e(study)i (irreducible)d FP(\003)p FO(-represen)n(tations)f(of)22 b(the)f(Skly)n(anin)-118 2317 y(algebra,)27 b(and)i(of)h(the)g (deformed)e(en)n(v)n(eloping)e(algebra)h FN(U)1727 2329 y FL(q)1763 2317 y FO(\()p FN(sl)r FO(\(2\)\).)43 b(W)-7 b(e)29 b(will)-118 2417 y(see)e(that)h(in)f(the)h(case)f FA(F)668 2429 y FK(1)703 2417 y FO(,)g(and)h FN(su)1002 2429 y FL(q)1038 2417 y FO(\(2\))g(with)f FN(J)k(<)23 b FO(0,)k(and)g FP(j)p FN(q)s FP(j)c FO(=)g(1)k(whic)n(h)g(is)-118 2516 y(not)f(a)g(ro)r(ot)g(of)g(1,)h(the)g(description)d(problem)f (includes)i(suc)n(h)h(a)g(problem)e(for)-118 2616 y(the)g(irrational)19 b(rotation)i(algebra,)h(and)h(is,)h(therefore,)f(rather)g(complicated.) 6 2716 y(Let)31 b(us)f(b)r(egin)f(with)h FN(J)35 b(>)27 b FO(0;)32 b(in)d(this)g(case,)i FN(q)f FP(2)e FI(R)p FO(,)37 b FP(j)p FN(q)s FP(j)27 b FN(>)g FO(1,)k FN(A)2048 2728 y FK(1)2085 2716 y FO(,)g FN(A)2201 2728 y FK(2)2239 2716 y FO(,)g FN(k)-118 2815 y FO(are)f(self-adjoin)n(t,)f(and)i FN(X)719 2785 y FM(\003)785 2815 y FO(=)d FN(Y)50 b FO(for)30 b FN(J)37 b FP(\024)28 b FO(1)i(\()p FN(q)i(>)c FO(1\),)k(and)f FN(X)1918 2785 y FM(\003)1984 2815 y FO(=)d FP(\000)p FN(Y)49 b FO(for)-118 2915 y FN(J)31 b(>)23 b FO(1)k(\()p FN(q)f(<)d FP(\000)p FO(1\).)-118 3079 y FQ(Theorem)30 b(27.)41 b FB(If)50 b FN(J)37 b FO(=)27 b(1)p FB(,)33 b(then)g FA(F)1077 3091 y FK(1)1145 3079 y FB(has)g(the)g(fol)t(lowing) i(irr)l(e)l(ducible)e(r)l(ep-)-118 3178 y(r)l(esentations)7 b FO(:)-17 3342 y(1)p FN(:)41 b FB(one-dimensional)9 b FO(:)52 b FN(A)823 3354 y FK(1)894 3342 y FO(=)33 b FP(\006)p FO(1)p FB(,)k FN(A)1223 3354 y FK(2)1294 3342 y FO(=)c(1)p FB(,)k FN(X)j FO(=)33 b(0)p FB(;)39 b FN(A)1871 3354 y FK(1)1942 3342 y FO(=)33 b FN(A)2102 3354 y FK(2)2173 3342 y FO(=)h(0)p FB(,)89 3442 y FN(X)c FP(2)23 b FI(T)p FO(;)-17 3607 y(2)p FN(:)41 b(n)p FB(-dimensional)32 b FO(\()p FN(n)23 b FO(=)f(2)p FN(;)14 b FO(3)p FN(;)g(:)g(:)g(:)f FO(\))p FB(:)358 3787 y FN(A)420 3799 y FK(1)458 3787 y FN(f)499 3799 y FK(1)559 3787 y FO(=)22 b(0)p FN(;)99 b(:)14 b(:)g(:)f(;)99 b(A)1104 3799 y FK(1)1142 3787 y FN(f)1183 3799 y FL(n)p FM(\000)p FK(1)1335 3787 y FO(=)23 b(0)p FN(;)98 b(A)1648 3799 y FK(1)1686 3787 y FN(f)1727 3799 y FL(n)1795 3787 y FO(=)22 b FP(\006)p FN(f)1988 3799 y FL(n)2033 3787 y FN(;)358 3911 y(A)420 3923 y FK(2)458 3911 y FN(f)499 3923 y FK(1)559 3911 y FO(=)g(0)p FN(;)99 b(:)14 b(:)g(:)f(;)99 b(A)1104 3923 y FK(2)1142 3911 y FN(f)1183 3923 y FL(n)p FM(\000)p FK(1)1335 3911 y FO(=)23 b(0)p FN(;)98 b(A)1648 3923 y FK(2)1686 3911 y FN(f)1727 3923 y FL(n)1795 3911 y FO(=)22 b(0)p FN(;)p eop %%Page: 126 130 126 129 bop -118 -137 a FO(126)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)382 96 y FN(X)7 b(f)499 108 y FK(1)559 96 y FO(=)22 b FN(f)687 108 y FK(2)724 96 y FN(;)99 b(:)14 b(:)g(:)g(;)98 b(X)7 b(f)1195 108 y FL(n)p FM(\000)p FK(1)1348 96 y FO(=)22 b FN(f)1476 108 y FL(n)1521 96 y FN(;)99 b(X)7 b(f)1760 108 y FL(n)1827 96 y FO(=)23 b(0;)-17 326 y(3)p FN(:)41 b FB(in\014nite-dimensional)9 b FO(:)137 505 y(\()p FN(i)p FO(\))42 b FN(A)334 517 y FK(1)406 505 y FO(=)33 b(0)p FB(,)k FN(A)670 517 y FK(2)708 505 y FN(f)749 517 y FK(1)820 505 y FO(=)c FN(f)959 517 y FK(1)996 505 y FB(,)38 b FN(A)1121 517 y FK(2)1159 505 y FN(f)1200 517 y FL(n)1278 505 y FO(=)c(0)p FB(,)j FN(n)d FO(=)g(2)p FB(,)j FO(3)p FB(,)f FN(:)14 b(:)g(:)49 b FO(;)39 b FN(X)7 b(f)2196 517 y FL(n)2274 505 y FO(=)272 605 y FN(f)313 617 y FL(n)p FK(+1)442 605 y FB(,)30 b FN(n)23 b FO(=)g(1)p FB(,)29 b FO(2)p FB(,)h FN(:)14 b(:)g(:)43 b FO(;)109 745 y(\()p FN(ii)p FO(\))e FN(A)334 757 y FK(2)399 745 y FO(=)28 b(0)p FB(,)k FN(A)653 757 y FK(1)691 745 y FN(f)732 757 y FK(1)796 745 y FO(=)c FN(f)930 757 y FK(1)967 745 y FB(,)33 b FN(A)1087 757 y FK(1)1124 745 y FN(f)1165 757 y FL(n)1238 745 y FO(=)27 b(0)p FB(,)33 b FN(n)28 b FO(=)f(2)p FB(,)33 b FO(3)p FB(,)f FN(:)14 b(:)g(:)46 b FO(;)33 b FN(X)7 b(f)2115 757 y FK(1)2179 745 y FO(=)28 b(0)p FB(,)272 844 y FN(X)7 b(f)389 856 y FL(n)456 844 y FO(=)23 b FN(f)585 856 y FL(n)p FM(\000)p FK(1)715 844 y FB(,)30 b FN(n)23 b FO(=)f(2)p FB(,)30 b FO(3)p FB(,)g FN(:)14 b(:)g(:)27 b FB(.)-118 1024 y(Pr)l(o)l(of.)43 b FO(In)28 b(this)f(case)f(w)n(e)i(ha)n(v)n(e)e (the)i(follo)n(wing)c(relations:)489 1213 y([)p FN(A)574 1225 y FK(1)612 1213 y FN(;)14 b(A)711 1225 y FK(2)748 1213 y FO(])23 b(=)g(0)p FN(;)97 b(X)7 b(A)1182 1225 y FK(1)1241 1213 y FO(=)23 b FN(A)1391 1225 y FK(1)1429 1213 y FN(X)1505 1179 y FM(\003)1565 1213 y FO(=)g(0)p FN(;)386 1348 y FO([)p FN(X)r(;)14 b(X)593 1314 y FM(\003)630 1348 y FO(])23 b(=)g FN(A)826 1314 y FK(2)826 1368 y(1)882 1348 y FP(\000)18 b FN(A)1027 1314 y FK(2)1027 1368 y(2)1064 1348 y FN(;)97 b(A)1246 1360 y FK(2)1284 1348 y FN(X)29 b FO(=)23 b FN(X)1546 1314 y FM(\003)1583 1348 y FN(A)1645 1360 y FK(2)1706 1348 y FO(=)f(0)-118 1537 y(whic)n(h)30 b(imply)f(that)j FN(A)606 1549 y FK(1)643 1537 y FO(,)h FN(A)761 1549 y FK(2)798 1537 y FO(,)g FN(X)7 b(X)1006 1507 y FM(\003)1043 1537 y FO(,)32 b FN(X)1174 1507 y FM(\003)1212 1537 y FN(X)37 b FO(are)31 b(pairwise)d(comm)n(uting)g (op-)-118 1637 y(erators)e(and)633 1826 y FA(H)c FO(=)h FA(H)893 1838 y FK(0)947 1826 y FP(\010)18 b FA(H)1105 1838 y FK(1)1159 1826 y FP(\010)g FA(H)1317 1838 y FK(2)1371 1826 y FP(\010)g FA(H)1529 1838 y FK(3)1565 1826 y FN(;)-118 2016 y FO(where:)6 2119 y FN(A)68 2131 y FK(1)129 2119 y FO(=)23 b FN(A)279 2131 y FK(2)339 2119 y FO(=)g(0,)k FN(X)7 b(X)671 2089 y FM(\003)731 2119 y FO(=)23 b FN(X)895 2089 y FM(\003)932 2119 y FN(X)29 b(>)23 b FO(0)k(in)g FA(H)1359 2131 y FK(0)1395 2119 y FO(,)6 2222 y FN(A)68 2234 y FK(1)129 2222 y FO(=)c FN(X)293 2192 y FM(\003)353 2222 y FO(=)g(0,)k FN(X)609 2192 y FM(\003)646 2222 y FN(X)j FO(=)22 b FN(A)894 2192 y FK(2)894 2242 y(2)955 2222 y FN(>)h FO(0)k(in)g FA(H)1284 2234 y FK(1)1319 2222 y FO(,)6 2325 y FN(A)68 2337 y FK(2)129 2325 y FO(=)c FN(X)29 b FO(=)23 b(0,)k FN(X)7 b(X)647 2295 y FM(\003)707 2325 y FO(=)22 b FN(A)856 2295 y FK(2)856 2345 y(1)917 2325 y FN(>)h FO(0)k(in)g FA(H)1246 2337 y FK(2)1281 2325 y FO(,)6 2428 y FN(X)j FO(=)22 b FN(X)268 2398 y FM(\003)329 2428 y FO(=)h(0,)k FN(A)571 2398 y FK(2)571 2448 y(1)631 2428 y FO(=)c FN(A)781 2398 y FK(2)781 2448 y(2)842 2428 y FN(>)f FO(0)27 b(in)g FA(H)1170 2440 y FK(3)1206 2428 y FO(.)6 2531 y(The)37 b(subspace)f FA(H)617 2543 y FK(3)689 2531 y FO(is)g(in)n(v)-5 b(arian)n(t)33 b(with)k(resp)r(ect)f(to)h FN(A)1804 2543 y FK(1)1841 2531 y FO(,)i FN(A)1965 2543 y FK(2)2003 2531 y FO(,)g FN(X)7 b FO(,)38 b FN(X)2278 2501 y FM(\003)2316 2531 y FO(,)-118 2630 y(th)n(us)28 b(irreducible)23 b(represen)n(tations)i (are)i(one-dimensional)22 b(there.)6 2733 y(Consider)37 b(no)n(w)g(its)g(orthogonal)e(complemen)n(t.)65 b(Note)38 b(that)h FN(X)7 b FA(H)2180 2703 y FM(?)2179 2754 y FK(3)2274 2733 y FP(\032)-118 2849 y FA(H)-43 2861 y FK(0)15 2849 y FP(\010)22 b FA(H)177 2861 y FK(2)213 2849 y FO(,)36 b FN(X)348 2819 y FM(\003)385 2849 y FA(H)461 2819 y FM(?)460 2870 y FK(3)549 2849 y FP(\032)d FA(H)722 2861 y FK(0)781 2849 y FP(\010)22 b FA(H)943 2861 y FK(1)979 2849 y FO(.)55 b(Set)35 b FA(H)1283 2806 y FK(\(1\))1282 2871 y(1)1404 2849 y FO(=)e FP(f)p FN(f)41 b FP(2)34 b FA(H)1790 2861 y FK(1)1859 2849 y FP(j)g FN(X)7 b(f)41 b FP(2)34 b FA(H)2238 2861 y FK(2)2274 2849 y FP(g)p FO(,)-118 2966 y FA(H)-42 2923 y FK(\()p FL(n)p FK(+1\))-43 2988 y(1)174 2966 y FO(=)i FP(f)p FN(f)45 b FP(2)37 b FA(H)570 2978 y FK(1)643 2966 y FP(j)f FN(X)7 b(f)t(;)14 b(:)g(:)g(:)27 b(;)14 b(X)1097 2936 y FL(n)1141 2966 y FN(f)46 b FP(2)37 b FA(H)1395 2978 y FK(0)1430 2966 y FN(;)28 b(X)1557 2936 y FL(n)p FK(+1)1686 2966 y FN(f)45 b FP(2)37 b FA(H)1939 2978 y FK(2)1975 2966 y FP(g)p FO(,)g FN(n)g FP(2)g FI(N)t FO(;)-118 3083 y FA(H)-42 3040 y FK(\()p FM(1)p FK(\))-43 3105 y(1)102 3083 y FO(=)23 b FP(f)p FN(f)32 b FP(2)24 b FA(H)459 3095 y FK(1)518 3083 y FP(j)g FN(X)641 3053 y FL(n)685 3083 y FN(f)33 b FP(2)23 b FA(H)912 3095 y FK(0)948 3083 y FN(;)28 b(n)23 b FP(2)h FI(N)t FP(g)p FO(,)34 b FA(H)1380 3040 y FK(\()p FM(1)p FK(\))1379 3105 y(2)1524 3083 y FO(=)23 b FP(f)p FN(f)32 b FP(2)24 b FA(H)1881 3095 y FK(2)1940 3083 y FP(j)g FO(\()p FN(X)2095 3053 y FM(\003)2133 3083 y FO(\))2165 3053 y FL(n)2210 3083 y FN(f)32 b FP(2)-118 3183 y FA(H)-43 3195 y FK(0)-7 3183 y FN(;)27 b(n)c FP(2)h FI(N)t FP(g)p FO(.)42 b(Then:)-40 3375 y(\(i\))f(the)d(subspace)e FA(H)674 3345 y FK(\()p FL(n)p FK(\))808 3375 y FO(=)i FP(\010)976 3345 y FL(n)976 3397 y(j)s FK(=0)1094 3375 y FN(X)1170 3345 y FL(j)1204 3375 y FA(H)1280 3332 y FK(\()p FL(n)p FK(\))1279 3397 y(1)1413 3375 y FO(is)e(in)n(v)-5 b(arian)n(t)33 b(and)k(the)g(irre-)89 3503 y(ducibilit)n(y)28 b(implies)f(that)k(dim) 12 b FN(X)1158 3473 y FL(n)1203 3503 y FA(H)1279 3460 y FK(\()p FL(n)p FK(\))1278 3525 y(1)1403 3503 y FO(=)28 b(1,)k FN(j)i FO(=)28 b(1,)j(2,)g FN(:)14 b(:)g(:)27 b FO(;)33 b(there-)89 3603 y(fore)27 b(dim)13 b FA(H)482 3573 y FK(\()p FL(n)p FK(\))600 3603 y FO(=)23 b FN(n)18 b FO(+)g(1;)-63 3794 y(\(ii\))40 b(the)f(subspace)617 3771 y(~)601 3794 y FA(H)676 3806 y FK(1)752 3794 y FO(=)h FP(\010)922 3764 y FM(1)922 3814 y FL(n)p FK(=0)1051 3794 y FN(X)1127 3764 y FL(n)1172 3794 y FA(H)1248 3751 y FK(\()p FM(1)p FK(\))1247 3816 y(1)1407 3794 y FO(is)d(in)n(v)-5 b(arian)n(t)35 b(and)j(the)h(irre-)89 3911 y(ducibilit)n(y)24 b(implies)g(that)k(dim)12 b FN(X)1148 3881 y FL(n)1193 3911 y FA(H)1269 3868 y FK(\()p FM(1)p FK(\))1268 3933 y(1)1413 3911 y FO(=)22 b(1,)28 b FN(n)22 b FP(2)i FI(N)t FO(;)p eop %%Page: 127 131 127 130 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(127)-86 100 y(\(iii\))39 b(the)30 b(subspace)598 78 y(~)582 100 y FA(H)657 112 y FK(2)717 100 y FO(=)25 b FP(\010)872 70 y FM(1)872 121 y FL(n)p FK(=0)1001 100 y FN(X)1077 70 y FL(n)1121 100 y FA(H)1197 57 y FK(\()p FM(1)p FK(\))1196 122 y(2)1347 100 y FO(is)i(in)n(v)-5 b(arian)n(t)26 b(and)j(in)f(the)h(irre-)89 217 y(ducible)d(case,)h(dim) 12 b FN(X)802 187 y FL(n)847 217 y FA(H)923 174 y FK(\()p FM(1)p FK(\))922 239 y(2)1067 217 y FO(=)22 b(1,)28 b FN(n)23 b FP(2)g FI(N)t FO(.)-118 394 y(On)33 b(the)g(in)n(v)-5 b(arian)n(t)30 b(subspace)897 371 y(~)880 394 y FA(H)955 406 y FK(0)1023 394 y FO(=)h FP(f)p FN(f)40 b FP(2)33 b FA(H)1405 406 y FK(0)1472 394 y FP(j)g FN(X)1604 364 y FL(n)1648 394 y FN(f)41 b FP(2)32 b FA(H)1892 406 y FK(0)1928 394 y FN(;)27 b FO(\()p FN(X)2086 364 y FM(\003)2124 394 y FO(\))2156 364 y FL(n)2202 394 y FN(f)40 b FP(2)-118 494 y FA(H)-43 506 y FK(0)-7 494 y FO(;)27 b FN(n)j FP(2)h FI(N)t FP(g)p FO(,)38 b(the)33 b(op)r(erator)d FN(X)38 b FO(is)31 b(normal,)f(and)h(irreducible)e(represen)n(ta-)-118 593 y(tions)d(are)h(one-dimensional.)6 693 y(Finally)18 b(w)n(e)j(note)f(that)h(for)g(an)n(y)f(represen)n(tation,)f(w)n(e)i(ha) n(v)n(e)e(the)j(follo)n(wing)-118 793 y(decomp)r(osition:)426 1039 y FA(H)h FO(=)641 935 y FM(1)612 960 y Fy(M)612 1136 y FL(n)p FK(=1)751 1039 y FA(H)827 1005 y FK(\()p FL(n)p FK(\))941 1039 y FP(\010)1040 1016 y FO(~)1024 1039 y FA(H)1099 1051 y FK(0)1153 1039 y FP(\010)1253 1016 y FO(~)1236 1039 y FA(H)1311 1051 y FK(1)1365 1039 y FP(\010)1465 1016 y FO(~)1448 1039 y FA(H)1523 1051 y FK(2)1577 1039 y FP(\010)18 b FA(H)1735 1051 y FK(3)1771 1039 y FN(;)-118 1294 y FO(whic)n(h)26 b(completes)g(the)i(pro)r(of.)p 2278 1294 4 57 v 2282 1242 50 4 v 2282 1294 V 2331 1294 4 57 v -118 1465 a FQ(Theorem)i(28.)41 b FB(F)-6 b(or)42 b FN(q)48 b FP(2)d FI(R)p FB(,)51 b FP(j)p FN(q)s FP(j)45 b FN(>)f FO(1)p FB(,)h(the)d FP(\003)p FB(-algebr)l(a)g FN(su)1886 1477 y FL(q)1922 1465 y FO(\(2\))g FB(has)g(the)-118 1565 y(fol)t(lowing)32 b(irr)l(e)l(ducible)f(r)l(epr)l(esentations)37 b FO(\()p FN(n)23 b FP(2)h FI(N)t FO(\):)107 1749 y FN(k)s(f)194 1761 y FL(j)251 1749 y FO(=)f FP(j)p FN(q)s FP(j)425 1714 y FK(\(1)p FM(\000)p FL(n)p FK(\))p FL(=)p FK(2)674 1749 y FN(q)714 1714 y FL(j)s FM(\000)p FK(1)834 1749 y FN(f)875 1761 y FL(j)910 1749 y FN(;)99 b(X)7 b(f)1149 1761 y FL(j)1206 1749 y FO(=)22 b(\()p FP(j)p FO([)p FN(j)5 b FO(])1433 1761 y FL(q)1484 1749 y FO([)p FN(n)19 b FP(\000)f FN(j)5 b FO(])1721 1761 y FL(q)1757 1749 y FP(j)p FO(\))1812 1714 y FK(1)p FL(=)p FK(2)1917 1749 y FN(f)1958 1761 y FL(j)s FK(+1)2077 1749 y FN(;)851 1873 y(j)28 b FO(=)23 b(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)26 b(;)14 b(n)-118 2057 y FO(\()p FB(we)30 b(use)g(a)g(standar)l(d)g (notation)g FO([)p FN(m)p FO(])1044 2069 y FL(q)1104 2057 y FO(=)23 b(\()p FN(q)1264 2027 y FL(m)1345 2057 y FP(\000)18 b FN(q)1468 2027 y FM(\000)p FL(m)1583 2057 y FO(\))p FN(=)p FO(\()p FN(q)k FP(\000)c FN(q)1871 2027 y FM(\000)p FK(1)1960 2057 y FO(\)\))p FB(.)-118 2225 y(Pr)l(o)l(of.)43 b FO(Apply)d(Prop)r(ositions)d(40)j(and)h(41)f(to)h (the)g(self-adjoin)n(t)d(op)r(erator)-118 2324 y FN(A)27 b FO(=)g FN(k)33 b FO(and)d(the)g(map)f FN(F)12 b FO(\()p FN(l)r(;)i(\025)p FO(\))27 b(=)g(\()p FN(q)s(l)r(;)14 b(\025)20 b FP(\000)g FO(\()p FN(l)1343 2294 y FK(2)1341 2345 y(1)1398 2324 y FP(\000)e FN(l)1508 2289 y FM(\000)p FK(2)1506 2347 y(2)1597 2324 y FO(\))p FN(=)p FP(j)p FN(q)j FP(\000)d FN(q)1875 2294 y FM(\000)p FK(1)1964 2324 y FP(j)p FO(\).)45 b(W)-7 b(e)30 b(see)-118 2424 y(that)28 2640 y FN(F)93 2606 y FL(n)138 2640 y FO(\()p FN(l)r(;)14 b(\025)p FO(\))24 b(=)e FN(\025)d FP(\000)f FO([)p FN(n)p FO(])671 2652 y FL(q)732 2584 y FN(l)759 2554 y FK(2)795 2584 y FN(q)835 2554 y FL(n)p FM(\000)p FK(1)984 2584 y FP(\000)g FN(l)1094 2554 y FM(\000)p FK(2)1183 2584 y FN(q)1223 2554 y FK(1)p FM(\000)p FL(n)p 732 2621 622 4 v 884 2697 a FP(j)p FN(q)j FP(\000)d FN(q)1088 2673 y FM(\000)p FK(1)1178 2697 y FP(j)1386 2640 y(!)23 b(\0001)p FN(;)180 b(n)22 b FP(!)h(\0061)p FN(:)-118 2868 y FO(It)28 b(means)e(that)i(only)e(the)i(\014nite-dimensional)22 b(case)27 b(is)f(realized.)p 2278 2868 4 57 v 2282 2816 50 4 v 2282 2868 V 2331 2868 4 57 v 6 3039 a(No)n(w)i(w)n(e)f(consider) e(the)j(case)f(where)g FN(J)k(>)23 b FO(1.)-118 3207 y FQ(Prop)s(osition)30 b(44.)41 b FB(F)-6 b(or)35 b FN(J)40 b(>)33 b FO(0)p FB(,)j FN(J)k FP(6)p FO(=)33 b(1)p FB(,)j(the)f FP(\003)p FB(-algebr)l(a)h FA(F)1853 3219 y FK(1)1923 3207 y FB(has)g(the)f(fol-)-118 3307 y(lowing)c(irr)l(e)l(ducible)g(r)l (epr)l(esentations)7 b FO(:)-17 3474 y(1)p FN(:)41 b FB(one-dimensional)9 b FO(:)40 b FN(A)811 3486 y FK(1)872 3474 y FO(=)22 b FN(A)1021 3486 y FK(2)1082 3474 y FO(=)h(0)p FN(;)43 b(X)29 b FP(2)23 b FI(T)p FO(;)-17 3643 y(2)p FN(:)41 b(n)p FB(-dimensional)33 b FO(\()p FN(n)23 b FP(2)g FI(N)t FO(\))q(:)40 b FN(A)1028 3655 y FK(1)1089 3643 y FO(=)23 b FN(k)s FB(,)h FN(A)1334 3655 y FK(2)1394 3643 y FO(=)f FP(\006)p FN(k)1593 3613 y FM(\000)p FK(1)1681 3643 y FO(;)i FN(k)s FB(,)g FN(X)7 b FB(,)23 b FN(Y)41 b FB(gener)l(ate)89 3743 y(an)30 b(irr)l(e)l(ducible)h(r)l(epr)l (esentation)f(of)h FN(su)1337 3755 y FL(q)1373 3743 y FO(\(2\);)-17 3911 y(3)p FN(:)41 b FB(in\014nite-dimensional)9 b FO(:)p eop %%Page: 128 132 128 131 bop -118 -137 a FO(128)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)137 96 y FO(\()p FN(i)p FO(\))42 b FN(A)334 108 y FK(1)372 96 y FN(f)413 108 y FL(n)481 96 y FO(=)22 b FN(q)608 66 y FM(\000)p FL(n)705 96 y FN(f)746 108 y FL(n)791 96 y FB(,)30 b FN(A)908 108 y FK(2)969 96 y FO(=)22 b(0)p FB(,)434 344 y FN(X)7 b(f)551 356 y FL(n)618 344 y FO(=)706 227 y Fy(\022)777 288 y FO([)p FN(n)p FO(])873 300 y FL(q)924 288 y FN(q)964 258 y FK(1)p FM(\000)p FL(n)p 777 325 317 4 v 777 401 a FP(j)p FN(q)22 b FP(\000)c FN(q)982 377 y FM(\000)p FK(1)1071 401 y FP(j)1104 227 y Fy(\023)1165 245 y FK(1)p FL(=)p FK(2)1269 344 y FN(f)1310 356 y FL(n)p FM(\000)p FK(1)1440 344 y FN(;)184 b(n)23 b FO(=)f(0)p FN(;)14 b FO(1)p FN(;)g FO(2)p FN(;)g(:)g(:)g(:)e FO(;)109 588 y(\()p FN(ii)p FO(\))41 b FN(A)334 600 y FK(1)395 588 y FO(=)22 b(0)p FB(,)30 b FN(A)641 600 y FK(2)678 588 y FN(f)719 600 y FL(n)787 588 y FO(=)23 b FN(q)915 558 y FM(\000)p FL(n)1012 588 y FN(f)1053 600 y FL(n)1098 588 y FB(,)356 836 y FN(X)7 b(f)473 848 y FL(n)541 836 y FO(=)628 719 y Fy(\022)699 780 y FO([)p FN(n)19 b FO(+)f(1])939 792 y FL(q)989 780 y FN(q)1029 750 y FK(1)p FM(\000)p FL(n)p 699 817 460 4 v 771 893 a FP(j)p FN(q)j FP(\000)d FN(q)975 869 y FM(\000)p FK(1)1064 893 y FP(j)1169 719 y Fy(\023)1230 737 y FK(1)p FL(=)p FK(2)1334 836 y FN(f)1375 848 y FL(n)p FK(+1)1504 836 y FN(;)184 b(n)23 b FO(=)f(0)p FN(;)14 b FO(1)p FN(;)g FO(2)p FN(;)g(:)g(:)g(:)26 b(:)-118 1072 y FB(Pr)l(o)l(of.)43 b FO(Apply)24 b(Prop)r(ositions)d(40)j(and)h(41)f(to)h FN(A)e FO(=)g FN(A)1574 1084 y FK(1)1624 1072 y FO(+)13 b FN(iA)1793 1084 y FK(2)1855 1072 y FO(and)25 b(the)g(map-)-118 1171 y(ping)187 1347 y FN(F)c FO(:)28 b(\()p FN(l)369 1359 y FK(1)406 1347 y FN(;)14 b(l)468 1359 y FK(2)505 1347 y FN(;)g(\025)p FO(\))24 b FP(7!)752 1279 y Fy(\000)790 1347 y FN(q)s(l)855 1359 y FK(1)892 1347 y FN(;)14 b(q)969 1312 y FM(\000)p FK(1)1058 1347 y FN(l)1083 1359 y FK(2)1120 1347 y FN(;)28 b(\025)19 b FP(\000)f FO(\()p FN(l)1380 1312 y FK(2)1378 1367 y(1)1435 1347 y FP(\000)g FN(l)1545 1312 y FK(2)1543 1367 y(2)1582 1347 y FO(\))p FN(=)p FP(j)p FN(q)j FP(\000)d FN(q)1860 1312 y FM(\000)p FK(1)1949 1347 y FP(j)1972 1279 y Fy(\001)2010 1347 y FN(:)-118 1522 y FO(The)31 b(orbits)e(with)h FN(l)512 1534 y FK(1)549 1522 y FN(l)574 1534 y FK(2)639 1522 y FP(6)p FO(=)e(0)i(giv)n(e)f(the) i(case)f(2\);)i(the)f(orbits)e(with)h FN(l)2034 1534 y FK(1)2099 1522 y FO(=)e(0)j(or)-118 1621 y FN(l)-93 1633 y FK(2)-33 1621 y FO(=)23 b(0)k(giv)n(e)e(the)j(cases)f(1\))g(and) h(3\).)p 2278 1621 4 57 v 2282 1569 50 4 v 2282 1621 V 2331 1621 4 57 v 6 1786 a(No)n(w)i(consider)e(the)j(case)e FN(J)36 b(<)27 b FO(0;)k(no)n(w)f FN(q)g FO(=)d FN(e)1492 1756 y FL(i\021)1583 1786 y FP(2)h FI(T)p FO(,)j(0)c FN(<)g(\021)j(<)d(\031)s FO(,)32 b(and)-118 1886 y FN(A)-56 1855 y FM(\003)-56 1906 y FK(1)5 1886 y FO(=)23 b FN(A)155 1898 y FK(2)193 1886 y FN(;)41 b(k)303 1855 y FM(\003)364 1886 y FO(=)23 b FN(k)498 1855 y FM(\000)p FK(1)587 1886 y FO(,)28 b FN(X)714 1855 y FM(\003)774 1886 y FO(=)23 b FN(Y)18 b FO(;)28 b(also)e(note)h(that)h([)p FN(m)p FO(])1629 1898 y FL(q)1689 1886 y FO(=)22 b(sin)o(\()p FN(m\021)s FO(\))q FN(=)p FO(sin)12 b FN(\021)s FO(.)6 1985 y(Decomp)r(ose)25 b FN(\021)s(=\031)j FO(in)n(to)d(a)g(con)n(tin)n (ued)f(fraction)g(\(whic)n(h)h(is)f(\014nite)i(if)f FN(q)k FO(is)24 b(a)-118 2085 y(ro)r(ot)j(of)g(1,)g(i.e.,)g(if)g FN(\021)s(=\031)f FP(2)e FI(Q)5 b FO(,)34 b(and)28 b(in\014nite)e (otherwise\):)699 2243 y FN(\021)p 696 2280 51 4 v 696 2356 a(\031)779 2299 y FO(=)d(0)18 b(+)1240 2243 y(1)p 1020 2280 482 4 v 1020 2400 a FN(a)1064 2412 y FK(1)1119 2400 y FO(+)1331 2343 y(1)p 1212 2381 280 4 v 1212 2457 a FN(a)1256 2469 y FK(2)1312 2457 y FO(+)g FN(:)c(:)g(:)1512 2299 y(:)-118 2611 y FO(The)28 b(denominators)c(of)j(the)h(con)n(v)n (ergen)n(ts)d(of)j(the)g(con)n(tin)n(ued)e(fraction)g(are:)246 2786 y FN(l)271 2798 y FK(0)331 2786 y FO(=)c(1)p FN(;)97 b(l)605 2798 y FK(1)665 2786 y FO(=)23 b FN(a)797 2798 y FK(1)834 2786 y FN(;)97 b(l)979 2798 y FL(p)1040 2786 y FO(=)22 b FN(l)1152 2798 y FL(p)p FM(\000)p FK(2)1294 2786 y FO(+)c FN(a)1421 2798 y FL(p)1459 2786 y FN(l)1484 2798 y FL(p)1518 2806 y Fx(1)1555 2786 y FN(;)180 b(p)23 b FP(\025)f FO(2)p FN(:)-118 2961 y FO(Then)32 b(de\014ne)g(an)g (increasing)d(sequence)i(of)h(in)n(tegers)e(\(\014nite)i(if)38 b FN(q)d FO(is)c(a)g(ro)r(ot)-118 3060 y(of)c(1\):)187 3236 y FN(r)226 3201 y FK(\(1\))224 3256 y FL(m)339 3236 y FO(=)22 b FN(m;)180 b(m)23 b FO(=)g(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)e(;)i(a)1234 3248 y FK(1)1271 3236 y FN(;)186 3377 y(r)225 3342 y FK(\()p FL(p)p FK(\))223 3397 y FL(m)339 3377 y FO(=)22 b FN(l)451 3389 y FL(p)p FM(\000)p FK(2)593 3377 y FO(+)c FN(m)c(l)788 3389 y FL(p)p FM(\000)p FK(1)911 3377 y FN(;)180 b(m)23 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)27 b(;)14 b(a)1660 3389 y FL(p)1698 3377 y FO(;)97 b FN(p)23 b FP(\025)f FO(2)p FN(:)-118 3552 y FO(F)-7 b(or)28 b(example,)e(when)i FN(\021)g FO(=)c FN(\031)s(=)-5 b(N)9 b FO(,)29 b(these)f(n)n(um)n(b)r (ers)f(are)g(1,)i(2,)f FN(:)14 b(:)g(:)27 b FO(,)i FN(N)9 b FO(;)28 b(when)-118 3651 y FN(\021)e FO(=)d FN(\031)s FO(\()119 3583 y FP(p)p 189 3583 42 4 v 189 3651 a FO(5)18 b FP(\000)g FO(1\))o FN(=)p FO(2,)27 b(they)h(are)e(the)i(Fib)r(onacci) e(sequence)h(1,)g(2,)g(3,)h(5,)f FN(:)14 b(:)g(:)27 b FO(.)-118 3811 y FQ(Theorem)j(29.)41 b FB(F)-6 b(or)33 b FN(q)26 b FP(2)d FI(T)g FB(such)h(that)32 b FN(q)27 b FB(is)d(not)f(a)i(r)l(o)l(ot)e(of)i FO(1)p FB(,)g(the)f FP(\003)p FB(-algebr)l(a)-118 3911 y FN(su)-31 3923 y FL(q)5 3911 y FO(\(2\))30 b FB(has)g(the)g(fol)t(lowing)i(irr)l(e)l (ducible)g(r)l(epr)l(esentations)7 b FO(:)p eop %%Page: 129 133 129 132 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(129)-17 100 y(1)p FN(:)41 b FB (\014nite-dimensional)31 b(with)g(dimension)f FN(n)23 b FO(=)g FN(r)1545 57 y FK(\()p FL(p)p FK(\))1543 110 y FL(m)1636 100 y FB(,)30 b FN(p)23 b FP(2)g FI(N)t FO(:)527 290 y FN(k)s(f)614 302 y FL(j)671 290 y FO(=)g FN(i)14 b FO(exp)928 222 y Fy(\000)966 290 y FN(i)1006 255 y FL(c)1040 290 y FO(\(\(1)k FP(\000)g FN(n)p FO(\))p FN(=)p FO(2)g(+)g FN(j)23 b FP(\000)18 b FO(1\))p FN(\021)s FO(\))c FN(f)1859 302 y FL(j)1894 290 y FN(;)497 434 y(X)7 b(f)614 446 y FL(j)671 434 y FO(=)759 359 y Fy(p)p 842 359 634 4 v 75 x FO(\()p FP(\000)p FO(1\))1013 410 y FL(c)1060 434 y FO([)p FN(j)e FO(])p FN(q)17 b FO([)p FN(n)h FP(\000)g FN(j)5 b FO(])p FN(q)17 b(f)1530 446 y FL(j)s FK(+1)1649 434 y FN(;)454 b FO(\(2.32\))89 624 y FB(wher)l(e)26 b FN(c)d FO(=)f(0)p FB(,)k FO(1)e FB(is)g(such)h(that) g(the)f(expr)l(ession)h(under)f(the)h(squar)l(e)g(r)l(o)l(ot)89 724 y(is)31 b(p)l(ositive)6 b FO(;)-17 903 y(2)p FN(:)41 b FB(two)30 b(de)l(gener)l(ate)h(in\014nite-dimensional)9 b FO(:)39 b FB(for)30 b FN(j)e FO(=)23 b(1)p FB(,)30 b FO(2)p FB(,)f FN(:)14 b(:)g(:)28 b FB(,)626 1093 y FN(k)s(f)713 1105 y FL(j)771 1093 y FO(=)22 b FN(i)14 b FO(exp\()p FN(i)p FO(\()p FN(\021)s(=)p FO(2)k(+)g(\()p FN(j)23 b FP(\000)18 b FO(1\))p FN(\021)s FO(\)\))c FN(f)1759 1105 y FL(j)1794 1093 y FN(;)597 1217 y(X)7 b(f)714 1229 y FL(j)771 1217 y FO(=)22 b([)p FN(j)5 b FO(])p FN(q)17 b(f)1038 1229 y FL(j)s FK(+1)1157 1217 y FO(;)89 1407 y FB(and)612 1596 y FN(k)s(f)699 1608 y FL(j)757 1596 y FO(=)23 b FN(i)14 b FO(exp)o(\()p FP(\000)p FN(i)p FO(\()p FN(\021)s(=)p FO(2)j(+)h(\()p FN(j)24 b FP(\000)18 b FO(1\))p FN(\021)s FO(\)\))c FN(f)1810 1608 y FL(j)583 1721 y FN(X)7 b(f)700 1733 y FL(j)757 1721 y FO(=)23 b([)p FN(j)g FP(\000)18 b FO(1])p FN(q)f(f)1168 1733 y FL(j)s FM(\000)p FK(1)1287 1721 y FO(;)-17 1950 y(3)p FN(:)41 b FB(non-de)l(gener)l(ate)26 b(in\014nite-dimensional,)i(which) f(ar)l(e)e(r)l(epr)l(esentations)89 2050 y(of)31 b(the)f(irr)l(ational) h(r)l(otation)f(algebr)l(a)h FA(A)1354 2062 y FL(\021)1395 2050 y FB(.)-118 2226 y(Pr)l(o)l(of.)43 b FO(No)n(w)30 b(w)n(e)g(apply)f(Prop)r(ositions)e(40)i(and)i(41)e(to)h(the)h(unitary) e FN(A)f FO(=)g FN(k)-118 2326 y FO(and)f(the)h(map)f FN(F)21 b FO(:)28 b(\()p FN(e)567 2296 y FL(i\036)634 2326 y FN(;)14 b(\025)p FO(\))24 b FP(7!)f FO(\()p FN(e)952 2296 y FL(i)p FK(\()p FL(\036)p FK(+)p FL(\021)r FK(\))1159 2326 y FN(;)14 b(\025)k FP(\000)g FO(sin)13 b(2)p FN(\036=)p FO(sin)f FN(\021)t FO(\).)6 2429 y(If)28 b FN(\025)137 2441 y FK(0)198 2429 y FO(=)23 b FN(\025)334 2441 y FL(n)402 2429 y FO(=)g(0,)k(then)h(it)g(is)e(easy)h(to)g(see)g(that)530 2619 y FN(\025)578 2631 y FL(j)636 2619 y FO(=)c FN(c)28 b FO(sin)n(\()p FN(n)19 b FP(\000)f FN(j)5 b FO(\))p FN(\021)31 b FO(sin)13 b FN(j)5 b(\021)s(=)14 b FO(sin)1572 2584 y FK(2)1623 2619 y FN(\021)s(;)-118 2808 y FO(where)27 b FN(c)c FO(=)g FP(\006)p FO(1.)36 b(The)28 b(requiremen)n(t)d(sign)n (\(sin\(\()p FN(n)19 b FP(\000)f FN(j)5 b FO(\))p FN(\021)s FO(\))14 b(sin)f FN(j)5 b(\021)s FO(\))24 b(=)e(const)28 b(is)-118 2908 y(equiv)-5 b(alen)n(t)24 b(to)j([\()p FN(n)19 b FP(\000)f FN(j)5 b FO(\))p FN(\021)s(=\031)s FO(])17 b(+)f([)p FN(j)5 b(\021)s(=\031)s FO(])23 b(=)g(const.)36 b(The)27 b(n)n(um)n(b)r(ers)e(satisfying)-118 3024 y(this)i(condition)e (are)i FN(n)c FO(=)f FN(r)746 2980 y FK(\()p FL(p)p FK(\))744 3033 y FL(m)837 3024 y FO(.)6 3127 y(Let)i FN(X)227 3096 y FM(\003)287 3127 y FO(=)f FN(U)441 3056 y FP(p)p 510 3056 68 4 v 71 x FN(B)k FO(and)c FN(U)32 b FO(b)r(e)23 b(unitary)-7 b(.)34 b(If)24 b(the)f(cen)n(tral)e(elemen)n(t)h FN(C)29 b FO(=)23 b FN(cI)7 b FO(,)-118 3226 y(then)28 b FN(c)23 b(>)g FO(1)p FN(=)p FO(2)j(and)i(the)g(op)r(erators)e FN(B)h FO(=)c FN(cI)i FO(+)18 b(\()p FN(e)1472 3196 y FM(\000)p FL(i\021)1588 3226 y FN(k)1634 3196 y FK(2)1689 3226 y FO(+)g FN(e)1811 3196 y FL(i\021)1875 3226 y FN(k)1921 3196 y FM(\000)p FK(2)2010 3226 y FO(\))p FN(=)p FO(4,)27 b FN(k)s FO(,)h FN(U)-118 3326 y FO(form)k(a)i(represen)n(tation)d(of)j (the)g(irrational)29 b(rotation)j(algebra)f FA(A)2016 3338 y FL(\021)2056 3326 y FO(:)50 b FN(U)9 b(k)36 b FO(=)-118 3425 y FN(e)-79 3395 y FK(\()p FL(i\021)r FK(\))36 3425 y FN(k)s(U)9 b FO(.)p 2278 3425 4 57 v 2282 3373 50 4 v 2282 3425 V 2331 3425 4 57 v -118 3619 a FQ(Theorem)30 b(30.)41 b FB(L)l(et)32 b FN(q)f FP(2)d FI(T)k FB(b)l(e)g(a)h(r)l(o)l (ot)g(of)47 b FO(1)p FB(,)33 b FN(\021)e FO(=)d FN(\031)s(M)9 b(=)-5 b(N)9 b FB(.)46 b(Then)33 b FN(su)2196 3631 y FL(q)2232 3619 y FO(\(2\))-118 3719 y FB(has)26 b(the)f(fol)t(lowing)i (irr)l(e)l(ducible)f(r)l(epr)l(esentations)32 b FO(\()p FB(al)t(l)27 b(\014nite-dimensional)9 b FO(\):)-17 3911 y(1)p FN(:)41 b FB(or)l(dinary)7 b FO(:)41 b FN(n)22 b FO(=)h FN(r)663 3868 y FK(\()p FL(p)p FK(\))661 3921 y FL(m)754 3911 y FB(,)30 b FO(1)23 b FP(\024)f FN(n)h FP(\024)g FN(N)9 b FO(;)30 b FN(k)i FB(and)e FN(X)36 b FB(ar)l(e)30 b(as)g(in)60 b FO(\(2.32\))o(;)p eop %%Page: 130 134 130 133 bop -118 -137 a FO(130)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-19 96 y FO(2)p FB(.)41 b(c)l(ontinuous)c(series)7 b FO(:)54 b FN(n)37 b FO(=)f FN(N)46 b FB(for)38 b FN(M)46 b FB(even,)40 b(and)e FN(n)f FO(=)f(2)p FN(N)46 b FB(for)38 b FN(M)89 196 y FB(o)l(dd)9 b FO(:)429 378 y FN(k)s(f)516 390 y FL(j)573 378 y FO(=)23 b(exp\()p FN(\036)c FO(+)f(\()p FN(j)24 b FP(\000)18 b FO(1\))p FN(\021)s FO(\))c FN(f)1349 390 y FL(j)1384 378 y FN(;)400 584 y(X)7 b(f)517 596 y FL(j)573 584 y FO(=)661 467 y Fy(\022)722 584 y FN(h)19 b FP(\000)882 528 y FO(sin)n(\(2)p FN(\036)g FO(+)f(\()p FN(j)24 b FP(\000)18 b FO(1\))p FN(\021)s FO(\))p 882 565 650 4 v 1126 641 a(sin)13 b FN(\021)1555 584 y FO([)p FN(j)5 b FO(])1640 596 y FL(q)1677 467 y Fy(\023)1738 485 y FK(1)p FL(=)p FK(2)1842 584 y FN(f)1883 596 y FL(j)s FK(+1)2002 584 y FN(;)720 766 y(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FN(;)389 911 y(X)7 b(f)506 923 y FL(n)573 911 y FO(=)23 b FN(E)727 876 y FL(i\026)795 836 y FP(p)p 864 836 48 4 v 75 x FN(hf)953 923 y FK(1)990 911 y FN(;)89 1093 y FB(wher)l(e)30 b FN(\036)23 b FP(2)h FN(\034)33 b FO(=)22 b([\()p FN(\031)g FO(+)c FN(\021)s FO(\))q FN(=)p FO(2)d FP(\000)h FN(\031)s(=)p FO(2)p FN(n)o(;)e FO(\()p FN(\031)22 b FO(+)c FN(\021)s FO(\))p FN(=)p FO(2)d(+)h FN(\031)s(=)p FO(2)p FN(N)8 b FO(\))p FB(,)30 b FN(e)2089 1062 y FL(i\026)2179 1093 y FP(2)24 b FI(T)p FB(.)-118 1258 y(Pr)l(o)l(of.)43 b FO(The)32 b(mapping)e FN(F)12 b FO(\()p FP(\001)p FO(\))32 b(is)f(de\014ned)i(for)e(all)f FN(q)j FP(2)e FI(T)h FO(in)f(the)h(same) f(w)n(a)n(y)-7 b(.)-118 1357 y(The)24 b(di\013erence)f(from)g(the)i (previous)d(case)h(is)g(that)h(eac)n(h)g(orbit)e(no)n(w)i(is)f(cyclic) -118 1457 y(with)k(the)h(p)r(erio)r(d)f FN(n)22 b FO(=)h(min)n(\()p FN(l)11 b FO(:)28 b FN(e)929 1427 y FL(il\021)1037 1457 y FO(=)23 b(1\).)6 1557 y(Note)40 b(that)g(the)g(dynamical)c(system)i (in)h(this)f(situation)g(has)h(a)g(Borel)-118 1656 y(section)29 b FN(\034)h FP(\002)20 b FI(R)367 1668 y FK(+)428 1656 y FO(.)45 b(It)31 b(means)e(that)h(eac)n(h)g(irreducible)d(represen)n (tation)g(with)-118 1756 y(unitary)f FN(U)36 b FO(corresp)r(onds)26 b(to)i(a)f(single)e(orbit.)p 2278 1756 4 57 v 2282 1703 50 4 v 2282 1756 V 2331 1756 4 57 v -118 1922 a FQ(Theorem)30 b(31.)41 b FB(F)-6 b(or)35 b FN(J)41 b(<)32 b FO(0)p FB(,)k(the)f FP(\003)p FB(-algebr)l(a)h FA(F)1455 1934 y FK(1)1525 1922 y FB(has)g(the)f(fol)t(lowing)i(irr)l(e-)-118 2021 y(ducible)31 b(r)l(epr)l(esentations)7 b FO(:)-17 2187 y(1)p FN(:)41 b FB(one-dimensional)9 b FO(:)40 b FN(A)811 2199 y FK(1)872 2187 y FO(=)22 b FN(A)1021 2199 y FK(2)1082 2187 y FO(=)h(0)p FB(,)29 b FN(X)h FP(2)23 b FI(T)p FO(;)-17 2353 y(2)p FN(:)41 b(A)151 2365 y FK(1)214 2353 y FO(=)24 b FN(k)s FB(,)31 b FN(A)467 2365 y FK(2)529 2353 y FO(=)24 b FN(k)664 2322 y FM(\000)p FK(1)753 2353 y FO(;)31 b FN(k)s(;)44 b(X)r(;)g(Y)49 b FB(gener)l(ate)31 b(an)g(irr)l(e)l(ducible)h(r)l(epr)l(esen-)89 2452 y(tation)e(of)h FN(su)516 2464 y FL(q)552 2452 y FO(\(2\))p FB(.)-118 2617 y(Pr)l(o)l(of.)43 b FO(Recall)33 b(that)j FN(A)650 2587 y FM(\003)650 2638 y FK(1)724 2617 y FO(=)g FN(A)887 2629 y FK(2)960 2617 y FO(and)f FN(A)1191 2587 y FM(\003)1191 2638 y FK(1)1230 2617 y FN(A)1292 2629 y FK(1)1365 2617 y FO(=)h FN(A)1528 2629 y FK(1)1565 2617 y FN(A)1627 2587 y FM(\003)1627 2638 y FK(1)1701 2617 y FO(comm)n(ute)d(with)i FN(X)-118 2717 y FO(and)h FN(X)128 2687 y FM(\003)165 2717 y FO(,)i(so)d(the)h(irreducible)d(represen)n(tation)g(with)i (degenerate)g FN(A)2141 2729 y FK(1)2214 2717 y FO(can)-118 2817 y(only)26 b(b)r(e)i(trivial.)p 2278 2817 V 2282 2764 50 4 v 2282 2817 V 2331 2817 4 57 v -118 2983 a FQ(6.)38 b FO(No)n(w)28 b(consider)e(represen)n(tations)g FA(F)1142 2995 y FK(2)1177 2983 y FO(.)39 b(Recall)26 b(that)i(in)g(this)f(case)h FN(J)k(>)24 b FO(0,)-118 3082 y FN(q)34 b FP(2)e FI(R)p FO(,)39 b FP(j)p FN(q)s FP(j)32 b FN(>)e FO(1,)k FN(A)530 3094 y FK(1)567 3082 y FO(,)g FN(A)686 3094 y FK(2)724 3082 y FO(,)f FN(k)j FO(are)31 b(self-adjoin)n(t,)g(and)i FN(X)1701 3052 y FM(\003)1769 3082 y FO(=)e FP(\000)p FN(Y)51 b FO(if)38 b FN(J)h FP(\024)31 b FO(1)-118 3182 y(\()p FN(q)26 b(>)d FO(1\),)k FN(X)265 3152 y FM(\003)326 3182 y FO(=)c FN(Y)46 b FO(if)33 b FN(J)f(>)22 b FO(1)27 b(\()p FN(q)g(<)22 b FP(\000)p FO(1\).)-118 3347 y FQ(Prop)s(osition)30 b(45.)41 b FB(F)-6 b(or)43 b FN(J)55 b FO(=)47 b(1)p FB(,)g FA(F)1113 3359 y FK(2)1191 3347 y FB(has)d(only)g (one-dimensional)g(irr)l(e-)-118 3447 y(ducible)31 b(r)l(epr)l (esentations.)-118 3612 y(Pr)l(o)l(of.)43 b FO(As)32 b(in)f(the)i(case)e(of)h FA(F)853 3624 y FK(1)888 3612 y FO(,)h(w)n(e)f(ha)n(v)n(e)e(that)j FN(A)1513 3624 y FK(1)1550 3612 y FO(,)g FN(A)1668 3624 y FK(2)1706 3612 y FO(,)g FN(X)1838 3582 y FM(\003)1875 3612 y FN(X)7 b FO(,)33 b FN(X)7 b(X)2159 3582 y FM(\003)2228 3612 y FO(are)-118 3712 y(comm)n(uting)37 b(self-adjoin)n(t)h(op)r(erators,) k(and)e FN(X)1423 3682 y FM(\003)1461 3712 y FN(X)33 b FP(\000)27 b FN(X)7 b(X)1807 3682 y FM(\003)1888 3712 y FO(=)45 b FN(A)2060 3682 y FK(2)2060 3732 y(1)2124 3712 y FP(\000)27 b FN(A)2278 3682 y FK(2)2278 3732 y(2)2316 3712 y FO(.)-118 3811 y(Hence)h FA(H)22 b FO(=)h FA(H)389 3823 y FK(0)443 3811 y FP(\010)18 b FA(H)601 3823 y FK(1)655 3811 y FP(\010)g FA(H)813 3823 y FK(2)867 3811 y FP(\010)g FA(H)1025 3823 y FK(3)1061 3811 y FO(,)28 b(where)f(all)e FA(H)1542 3823 y FL(j)1603 3811 y FO(are)i(in)n(v)-5 b(arian)n(t)24 b(and:)6 3911 y FN(A)68 3923 y FK(1)129 3911 y FO(=)f FN(A)279 3923 y FK(2)339 3911 y FO(=)g(0,)k FN(X)34 b FO(is)27 b(normal)d(in)j FA(H)1159 3923 y FK(0)1195 3911 y FO(;)p eop %%Page: 131 135 131 134 bop -118 -137 a FJ(2.2.)36 b(Algebras)25 b(with)i(3)h(and)f(4)g (generators)956 b FO(131)6 96 y FN(A)68 108 y FK(1)129 96 y FO(=)23 b FN(X)293 66 y FM(\003)353 96 y FO(=)g(0,)k FN(A)595 66 y FK(2)595 117 y(2)656 96 y FO(=)22 b FP(\000)p FN(X)884 66 y FM(\003)921 96 y FN(X)30 b FP(\))23 b FN(A)1188 108 y FK(2)1248 96 y FO(=)g FN(X)29 b FO(=)23 b(0)k(in)g FA(H)1763 108 y FK(1)1799 96 y FO(;)6 197 y FN(A)68 209 y FK(2)129 197 y FO(=)c FN(X)29 b FO(=)23 b(0,)k FN(A)557 167 y FK(2)557 218 y(1)617 197 y FO(=)c FP(\000)p FN(X)7 b(X)922 167 y FM(\003)981 197 y FP(\))24 b FN(A)1150 209 y FK(1)1210 197 y FO(=)f FN(X)1374 167 y FM(\003)1434 197 y FO(=)g(0)k(in)g FA(H)1763 209 y FK(2)1799 197 y FO(;)6 298 y FN(X)j FO(=)22 b FN(X)268 268 y FM(\003)329 298 y FO(=)h(0,)k FN(A)571 268 y FK(2)571 318 y(1)631 298 y FO(=)c FN(A)781 268 y FK(2)781 318 y(2)846 298 y FO(in)k FA(H)1018 310 y FK(3)1054 298 y FO(.)p 2278 298 4 57 v 2282 245 50 4 v 2282 298 V 2331 298 4 57 v -118 473 a FQ(Theorem)j(32.)41 b FB(F)-6 b(or)36 b FN(q)j FP(2)c FI(R)p FB(,)45 b FP(j)p FN(q)s FP(j)35 b FN(>)g FO(1)p FB(,)j(the)e FP(\003)p FB(-algebr)l(a)h FN(su)1818 485 y FL(q)1854 473 y FO(\(1)p FN(;)14 b FO(1\))36 b FB(has)h(the)-118 572 y(fol)t(lowing)32 b(irr)l(e)l(ducible)f(r)l(epr)l (esentations)7 b FO(:)-17 742 y(1)p FN(:)41 b FB(one-dimensional)9 b FO(:)40 b FN(k)26 b FO(=)d(1)p FB(,)29 b FN(X)h FO(=)22 b(0;)-17 912 y(2)p FN(:)41 b FB(with)31 b(the)f(lowest)g(weight)8 b FO(:)530 1097 y FN(k)s(f)617 1109 y FL(j)674 1097 y FO(=)23 b FN(l)15 b(q)842 1063 y FL(j)s FM(\000)p FK(1)962 1097 y FN(f)1003 1109 y FL(j)1038 1097 y FN(;)500 1304 y(X)7 b(f)617 1316 y FL(j)674 1304 y FO(=)762 1187 y Fy(\022)823 1304 y FO([)p FN(j)e FO(])908 1316 y FL(q)969 1247 y FN(l)996 1217 y FK(2)1032 1247 y FN(q)1072 1217 y FL(j)s FM(\000)p FK(1)1211 1247 y FP(\000)18 b FN(l)1321 1217 y FM(\000)p FK(2)1409 1247 y FN(q)1449 1217 y FK(1)p FM(\000)p FL(j)p 969 1284 601 4 v 1111 1361 a FP(j)p FN(q)j FP(\000)d FN(q)1315 1337 y FM(\000)p FK(1)1404 1361 y FP(j)1579 1187 y Fy(\023)1640 1204 y FK(1)p FL(=)p FK(2)1745 1304 y FN(f)1786 1316 y FL(j)s FK(+1)1905 1304 y FN(;)89 1534 y FB(wher)l(e)31 b FN(l)24 b(>)f FO(1)p FB(,)30 b FN(j)e FO(=)22 b(1)p FB(,)30 b FO(2)p FB(,)g FN(:)14 b(:)g(:)43 b FO(;)-17 1704 y(3)p FN(:)e FB(with)31 b(the)f(highest)h(weight)8 b FO(:)588 1889 y FN(k)s(f)675 1901 y FL(j)733 1889 y FO(=)23 b FN(l)15 b(q)901 1855 y FM(\000)p FL(j)988 1889 y FN(f)1029 1901 y FL(j)1063 1889 y FN(;)559 2095 y(X)7 b(f)676 2107 y FL(j)733 2095 y FO(=)821 1978 y Fy(\022)882 2095 y FO([)p FN(j)e FO(])967 2107 y FL(q)1027 2039 y FN(l)1054 2009 y FM(\000)p FK(2)1143 2039 y FN(q)1183 2009 y FL(j)1236 2039 y FP(\000)18 b FN(l)1346 2009 y FK(2)1383 2039 y FN(q)1423 2009 y FM(\000)p FL(j)p 1027 2076 483 4 v 1110 2152 a FP(j)p FN(q)k FP(\000)c FN(q)1315 2128 y FM(\000)p FK(1)1404 2152 y FP(j)1520 1978 y Fy(\023)1581 1996 y FK(1)p FL(=)p FK(2)1685 2095 y FN(f)1726 2107 y FL(j)s FM(\000)p FK(1)1846 2095 y FN(;)89 2330 y FB(wher)l(e)31 b FO(0)22 b FN(<)h(l)i(<)d FP(j)p FN(q)s FP(j)p FB(,)31 b FN(j)d FO(=)22 b(1)p FB(,)30 b FO(2)p FB(,)g FN(:)14 b(:)g(:)43 b FO(;)-17 2501 y(4)p FN(:)e FB(non-de)l(gener)l(ate)6 b FO(:)462 2685 y FN(k)s(f)549 2697 y FL(j)606 2685 y FO(=)23 b FN(l)15 b(q)774 2651 y FL(j)809 2685 y FN(f)850 2697 y FL(j)885 2685 y FN(;)432 2892 y(X)7 b(f)549 2904 y FL(j)606 2892 y FO(=)694 2775 y Fy(\022)755 2892 y FN(h)18 b FO(+)g([)p FN(j)5 b FO(])989 2904 y FL(q)1036 2836 y FN(l)1063 2806 y FK(2)1100 2836 y FN(q)1140 2806 y FL(j)s FM(\000)p FK(1)1278 2836 y FP(\000)18 b FN(l)1388 2806 y FM(\000)p FK(2)1477 2836 y FN(q)1517 2806 y FK(1)p FM(\000)p FL(j)p 1036 2873 601 4 v 1178 2949 a FP(j)p FN(q)j FP(\000)d FN(q)1382 2925 y FM(\000)p FK(1)1472 2949 y FP(j)1647 2775 y Fy(\023)1708 2792 y FK(1)p FL(=)p FK(2)1812 2892 y FN(f)1853 2904 y FL(j)s FM(\000)p FK(1)1973 2892 y FN(;)89 3127 y FB(wher)l(e)31 b FO(1)22 b FP(\024)h FN(l)i(<)d FP(j)p FN(q)s FP(j)p FB(,)31 b FN(h)22 b(>)h FO(0)p FB(,)30 b FN(j)e FP(2)23 b FI(Z)o FB(.)-118 3296 y(Pr)l(o)l(of.)43 b FO(Apply)30 b(Prop)r(ositions)c(40)k(and)g(41)f(to)h(the)h(self-adjoin)n(t)d FN(A)g FO(=)f FN(k)33 b FO(and)-118 3396 y(the)i(mapping)d FN(F)12 b FO(\()p FN(l)r(;)i(\025)p FO(\))35 b(=)755 3328 y Fy(\000)793 3396 y FN(q)s(l)r(;)14 b(\025)23 b FO(+)g(\()p FN(l)1115 3366 y FK(2)1170 3396 y FP(\000)18 b FN(l)1280 3366 y FM(\000)p FK(2)1369 3396 y FO(\))p FN(=)p FP(j)p FN(q)26 b FP(\000)d FN(q)1657 3366 y FM(\000)p FK(1)1746 3396 y FP(j)1769 3328 y Fy(\001)1807 3396 y FO(.)58 b(It)35 b(is)e(easy)h(to)-118 3495 y(sho)n(w)28 b(that,)h(sa)n(v)n(e)e(for)h(the)h(trivial)24 b(represen)n(tation,)j(w) n(e)h(ha)n(v)n(e)f(only)g(in\014nite-)-118 3595 y(dimensional)c (represen)n(tations)i(with)i(degenerate)f FN(X)34 b FO(or)27 b FN(X)1800 3565 y FM(\003)1837 3595 y FO(.)6 3696 y(The)f(dynamical)21 b(system)j(de\014ned)h(b)n(y)g FN(F)12 b FO(\()p FP(\001)p FO(\))26 b(has)e(a)h(Borel)e(section.)35 b(So)24 b(in)-118 3795 y(the)33 b(case)f(of)h(unitary)e FN(U)41 b FO(\(where)33 b FN(X)1061 3765 y FM(\003)1130 3795 y FO(=)e FN(U)1292 3725 y FP(p)p 1361 3725 68 4 v 70 x FN(B)t FO(\),)k(eac)n(h)d (irreducible)d(repre-)-118 3895 y(sen)n(tation)d(corresp)r(onds)g(to)h (a)g(single)e(orbit.)p 2278 3895 4 57 v 2282 3842 50 4 v 2282 3895 V 2331 3895 4 57 v eop %%Page: 132 136 132 135 bop -118 -137 a FO(132)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(R)l(emark)30 b(36.)42 b FO(1.)53 b(It)33 b(is)f(easy)f(to)i(see)g(that)g(an)f(in\014nite-dimensional)27 b(repre-)-118 196 y(sen)n(tation)f(has)h(a)g(classical)c(limit)h(only)i (if)h FN(l)1259 166 y FK(2)1319 196 y FP(2)c FN(q)1437 166 y Fu(Z)-10 b FL(=)p FK(2)1547 196 y FO(,)28 b(and)f(so)g(the)g(sp)r (ectrum)-118 296 y(of)g FN(k)k FO(b)r(elongs)26 b(to)h FN(q)489 266 y Fu(Z)-10 b FL(=)p FK(2)599 296 y FO(.)6 395 y(2.)36 b(Note)27 b(that)f(all)e(represen)n(tations)g(presen)n(ted) h(ab)r(o)n(v)n(e,)g(except)i(for)e(one-)-118 495 y(dimensional,)e(are)k (un)n(b)r(ounded.)37 b(Since)27 b(w)n(e)g(do)g(not)h(giv)n(e)d(an)i (accurate)g(de\014-)-118 595 y(nition)c(of)i(the)g(represen)n(tation)e (of)i FN(su)1071 607 y FL(q)1107 595 y FO(\(1)p FN(;)14 b FO(1\))24 b(b)n(y)h(un)n(b)r(ounded)g(op)r(erators,)f(w)n(e)-118 694 y(do)33 b(not)h(claim)d(that)j(the)g(list)e(of)i(represen)n (tations)d(presen)n(ted)i(includes)f(all)-118 794 y(irreducible)24 b(represen)n(tations)g(b)n(y)k(un)n(b)r(ounded)g(op)r(erators.)-118 938 y FQ(Prop)s(osition)i(46.)41 b FB(F)-6 b(or)35 b FN(J)40 b(>)33 b FO(1)p FB(,)j FN(J)k FP(6)p FO(=)33 b(0)p FB(,)j(the)f FP(\003)p FB(-algebr)l(a)h FA(F)1853 950 y FK(2)1923 938 y FB(has)g(the)f(fol-)-118 1037 y(lowing)c(irr)l(e) l(ducible)g(r)l(epr)l(esentations)7 b FO(:)-17 1181 y(1)p FN(:)41 b FB(one-dimensional)9 b FO(:)40 b FN(A)811 1193 y FK(1)872 1181 y FO(=)22 b FN(A)1021 1193 y FK(2)1082 1181 y FO(=)h(0)p FB(,)29 b FN(X)h FP(2)23 b FI(T)p FO(;)-17 1336 y(2)p FN(:)41 b FB(in\014nite-dimensional)e(inherite)l(d)g(fr)l (om)f FN(su)1492 1348 y FL(q)1528 1336 y FO(\(1)p FN(;)14 b FO(1\):)53 b FN(A)1851 1348 y FK(1)1925 1336 y FO(=)37 b FN(k)s FB(,)j FN(A)2200 1348 y FK(2)2274 1336 y FO(=)89 1436 y FP(\006)p FN(k)200 1405 y FM(\000)p FK(1)289 1436 y FO(;)30 b FN(k)s FB(,)g FN(X)36 b FB(ar)l(e)30 b(as)g(in)g(The)l(or)l (em)37 b FO(32;)-17 1590 y(3)p FN(:)k FB(with)31 b(the)f(highest)h (weight)8 b FO(:)39 b FB(for)30 b FN(j)e FO(=)23 b(1)p FB(,)30 b FO(2)p FB(,)f FN(:)14 b(:)g(:)28 b FB(,)791 1745 y FN(A)853 1757 y FK(1)913 1745 y FO(=)23 b(0)p FN(;)98 b(A)1226 1757 y FK(2)1264 1745 y FN(f)1305 1757 y FL(j)1362 1745 y FO(=)23 b FN(q)1490 1711 y FL(j)1525 1745 y FN(f)1566 1757 y FL(j)1601 1745 y FN(;)564 1952 y(X)7 b(f)681 1964 y FL(j)738 1952 y FO(=)825 1835 y Fy(\022)886 1952 y FO([)p FN(j)24 b FP(\000)18 b FO(1])1115 1964 y FL(q)1296 1896 y FN(q)1336 1865 y FL(j)p 1175 1933 317 4 v 1175 2009 a FP(j)p FN(q)j FP(\000)d FN(q)1379 1985 y FM(\000)p FK(1)1469 2009 y FP(j)1502 1835 y Fy(\023)1563 1852 y FK(1)p FL(=)p FK(2)1667 1952 y FN(f)1708 1964 y FL(j)s FM(\000)p FK(1)1828 1952 y FO(;)-17 2179 y(4)p FN(:)41 b FB(with)31 b(the)f(lowest)g(weight)8 b FO(:)39 b FB(for)31 b FN(j)d FO(=)22 b(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)f FB(,)791 2334 y FN(A)853 2346 y FK(2)913 2334 y FO(=)23 b(0)p FN(;)98 b(A)1226 2346 y FK(1)1264 2334 y FN(f)1305 2346 y FL(j)1362 2334 y FO(=)23 b FN(q)1490 2300 y FL(j)1525 2334 y FN(f)1566 2346 y FL(j)1601 2334 y FN(;)635 2541 y(X)7 b(f)752 2553 y FL(j)809 2541 y FO(=)897 2424 y Fy(\022)958 2541 y FO([)p FN(j)e FO(])1043 2553 y FL(q)1183 2485 y FN(q)1223 2455 y FL(j)s FK(+1)p 1104 2522 V 1104 2598 a FP(j)p FN(q)21 b FP(\000)d FN(q)1308 2574 y FM(\000)p FK(1)1397 2598 y FP(j)1431 2424 y Fy(\023)1492 2441 y FK(1)p FL(=)p FK(2)1596 2541 y FN(f)1637 2553 y FL(j)s FK(+1)1756 2541 y FO(;)-17 2768 y(5)p FN(:)41 b FB(non-de)l(gener)l (ate)6 b FO(:)39 b FB(for)31 b FN(j)d FP(2)23 b FI(Z)o FB(,)765 2923 y FN(A)827 2935 y FK(1)887 2923 y FO(=)g(0)p FN(;)98 b(A)1200 2935 y FK(2)1238 2923 y FN(f)1279 2935 y FL(j)1336 2923 y FO(=)23 b FN(q)1464 2889 y FM(\000)p FL(j)1551 2923 y FN(f)1592 2935 y FL(j)1626 2923 y FN(;)561 3130 y(X)7 b(f)678 3142 y FL(j)735 3130 y FO(=)822 3013 y Fy(\022)884 3130 y FN(h)18 b FP(\000)g FO([)p FN(j)5 b FO(])1118 3142 y FL(q)1231 3074 y FN(q)1271 3044 y FM(\000)p FL(j)s FM(\000)p FK(1)p 1178 3111 V 1178 3187 a FP(j)p FN(q)22 b FP(\000)c FN(q)1383 3163 y FM(\000)p FK(1)1472 3187 y FP(j)1505 3013 y Fy(\023)1566 3030 y FK(1)p FL(=)p FK(2)1671 3130 y FN(f)1712 3142 y FL(j)s FK(+1)1830 3130 y FN(;)89 3340 y FB(wher)l(e)31 b FN(h)23 b(>)f FP(j)p FN(q)s FP(j)568 3310 y FM(\000)p FK(1)658 3340 y FO(\()p FN(q)f FP(\000)e FN(q)872 3310 y FM(\000)p FK(1)961 3340 y FO(\))993 3310 y FM(\000)p FK(2)1082 3340 y FO(;)619 3495 y FN(A)681 3507 y FK(2)742 3495 y FO(=)j(0)p FN(;)99 b(A)1055 3507 y FK(1)1092 3495 y FN(f)1133 3507 y FL(j)1191 3495 y FO(=)22 b FN(q)1318 3460 y FL(j)s FM(\000)p FK(1)1438 3495 y FN(f)1479 3507 y FL(j)1514 3495 y FN(;)568 3701 y(X)7 b(f)685 3713 y FL(j)742 3701 y FO(=)829 3584 y Fy(\022)890 3701 y FN(h)19 b FO(+)f([)p FN(j)5 b FO(])1125 3713 y FL(q)1264 3645 y FN(q)1304 3615 y FL(j)s FM(\000)p FK(1)p 1185 3682 V 1185 3758 a FP(j)p FN(q)22 b FP(\000)c FN(q)1390 3734 y FM(\000)p FK(1)1479 3758 y FP(j)1512 3584 y Fy(\023)1573 3601 y FK(1)p FL(=)p FK(2)1678 3701 y FN(f)1719 3713 y FL(j)s FK(+1)1837 3701 y FN(;)89 3911 y FB(wher)l(e)31 b FN(h)23 b(>)f FO(\()p FN(q)554 3881 y FK(2)610 3911 y FP(\000)c FO(1\))767 3881 y FM(\000)p FK(1)856 3911 y FB(.)p eop %%Page: 133 137 133 136 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(133)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Apply)24 b(Prop)r(ositions)d(40)j(and)h(41)f (to)h FN(A)e FO(=)g FN(A)1574 108 y FK(1)1624 96 y FO(+)13 b FN(iA)1793 108 y FK(2)1855 96 y FO(and)25 b(the)g(map-)-118 196 y(ping)32 b FN(F)21 b FO(:)30 b(\()p FN(l)256 208 y FK(1)293 196 y FN(;)14 b(l)355 208 y FK(2)392 196 y FN(;)g(\025)p FO(\))32 b FP(7!)h FO(\()p FN(q)s(l)754 208 y FK(1)791 196 y FN(;)28 b(q)882 166 y FM(\000)p FK(1)971 196 y FN(l)996 208 y FK(2)1033 196 y FN(;)g(\025)22 b FO(+)g(\()p FN(l)1300 166 y FK(2)1298 217 y(1)1355 196 y FP(\000)c FN(l)1465 166 y FK(2)1463 217 y(2)1502 196 y FO(\))p FN(=)p FP(j)p FN(q)j FP(\000)d FN(q)1780 166 y FM(\000)p FK(1)1869 196 y FP(j)p FO(\).)54 b(Then)33 b(the)-118 296 y(orbits)25 b(with)h FN(l)330 308 y FK(1)367 296 y FN(l)392 308 y FK(2)452 296 y FP(6)p FO(=)d(0)j(giv)n(e)e(the)j (case)f(2\),)h(the)g(orbits)e(with)h FN(l)1808 308 y FK(1)1868 296 y FO(=)d(0)j(or)f FN(l)2149 308 y FK(2)2209 296 y FO(=)e(0)-118 395 y(giv)n(e)j(the)i(cases)e(1\),)i(3\),)f(4\),)h (5\).)6 496 y(Once)h(again,)f(for)g(the)i(non-degenerate)d(case)h(with) h(unitary)f FN(U)9 b FO(,)29 b(w)n(e)g(use)-118 596 y(the)f(existence)e (of)i(a)f(Borel)e(section)i(for)g(this)g(dynamical)d(system.)p 2278 596 4 57 v 2282 543 50 4 v 2282 596 V 2331 596 4 57 v -118 773 a FB(R)l(emark)30 b(37.)42 b FO(Represen)n(tations)d (corresp)r(onding)f(to)i(the)h(cases)f(2\){5\))g(are)-118 873 y(un)n(b)r(ounded.)-118 1122 y FG(2.3)112 b(Represen)m(tations)38 b(of)g Fq(q)t FG(-deformed)h Fq(U)10 b Fk(\()p Fq(so)p Fk(\(3)p Fq(;)17 b Fj(C)j Fk(\)\))-118 1306 y FQ(2.3.1)94 b(Real)31 b(forms)f(of)i FN(U)825 1318 y FL(q)861 1306 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))-118 1462 y FN(q)s FO(-Deformation)28 b(of)k(the)f(orthogonal)d(Lie)i (algebra)f FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))37 b(w)n(as)31 b(prop)r(osed)-118 1562 y(b)n(y)40 b(D.)g(F)-7 b(airlie)37 b([80)o(].)74 b(This)39 b(non-standard)g FN(q)s FO(-analog)e(of)j FN(U)1860 1574 y FL(q)1896 1562 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))47 b(is)-118 1661 y(constructed)18 b(starting)f(from)h FN(so)p FO(\(3)p FN(;)c FI(C)h FO(\))25 b(de\014ned)19 b(b)n(y)g(the)g (generating)e(elemen)n(ts)-118 1761 y FN(I)-82 1773 y FK(1)-44 1761 y FO(,)31 b FN(I)46 1773 y FK(2)83 1761 y FO(,)h FN(I)174 1773 y FK(3)211 1761 y FO(.)45 b(Namely)-7 b(,)29 b FN(U)660 1773 y FL(q)697 1761 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))37 b(is)29 b(an)h(asso)r(ciativ)n(e)c (algebra)i(generated)-118 1861 y(b)n(y)f FN(I)33 1873 y FK(1)71 1861 y FO(,)h FN(I)158 1873 y FK(2)195 1861 y FO(,)g FN(I)282 1873 y FK(3)348 1861 y FO(satisfying)c(the)k (relations:)631 2046 y FN(q)671 2012 y FK(1)p FL(=)p FK(2)775 2046 y FN(I)811 2058 y FK(1)849 2046 y FN(I)885 2058 y FK(2)941 2046 y FP(\000)18 b FN(q)1064 2012 y FM(\000)p FK(1)p FL(=)p FK(2)1221 2046 y FN(I)1257 2058 y FK(2)1294 2046 y FN(I)1330 2058 y FK(1)1391 2046 y FO(=)23 b FN(I)1515 2058 y FK(3)1553 2046 y FN(;)631 2187 y(q)671 2153 y FK(1)p FL(=)p FK(2)775 2187 y FN(I)811 2199 y FK(2)849 2187 y FN(I)885 2199 y FK(3)941 2187 y FP(\000)18 b FN(q)1064 2153 y FM(\000)p FK(1)p FL(=)p FK(2)1221 2187 y FN(I)1257 2199 y FK(3)1294 2187 y FN(I)1330 2199 y FK(2)1391 2187 y FO(=)23 b FN(I)1515 2199 y FK(1)1553 2187 y FN(;)631 2328 y(q)671 2294 y FK(1)p FL(=)p FK(2)775 2328 y FN(I)811 2340 y FK(3)849 2328 y FN(I)885 2340 y FK(1)941 2328 y FP(\000)18 b FN(q)1064 2294 y FM(\000)p FK(1)p FL(=)p FK(2)1221 2328 y FN(I)1257 2340 y FK(1)1294 2328 y FN(I)1330 2340 y FK(3)1391 2328 y FO(=)23 b FN(I)1515 2340 y FK(2)1553 2328 y FN(:)550 b FO(\(2.33\))-118 2513 y(Note)44 b(that)g(the)h(Lie)e(algebras)e FN(sl)r FO(\(2)p FN(;)14 b FI(C)g FO(\))50 b(and)44 b FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))51 b(are)43 b(isomorphic.)-118 2613 y(Ho)n(w)n(ev)n(er,)27 b(the)j(quan)n(tum)d(algebra)g FN(U)1087 2625 y FL(q)1123 2613 y FO(\()p FN(sl)r FO(\(2)p FN(;)14 b FI(C)g FO(\)\))36 b(di\013ers)28 b(from)f(the)i(algebra)-118 2713 y FN(U)-61 2725 y FL(q)-25 2713 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\).)6 2814 y(Let)41 b(us)f(describ)r(e)f(in)n(v)n(olutions)d(on)k(the)g(algebra)e FN(U)1700 2826 y FL(q)1736 2814 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))q(.)80 b(It)41 b(is)-118 2913 y(clear)26 b(that)j(an)f(in)n(v)n(olution)c(in)k(an)g(algebra)e(with)i (generators)e(and)i(relations)-118 3013 y(is)e(completely)f(de\014ned)j (b)n(y)f(its)g(v)-5 b(alues)26 b(on)i(the)g(generators.)34 b(An)29 b(in)n(v)n(olution)-118 3113 y(ma)n(y)20 b(send)i(linear)d(com) n(binations)f(of)k(generators)e(to)i(linear)d(com)n(binations)f(of)-118 3212 y(generators.)32 b(In)20 b(this)g(case)f(it)h(is)f(said)g(to)h(b)r (e)h(an)f(in)n(v)n(olution)c(of)k(the)h(\014rst)f(order.)-118 3312 y(On)31 b(the)g(other)f(hand,)i(there)e(migh)n(t)f(exist)h(in)n(v) n(olutions)d(whic)n(h)j(map)f(linear)-118 3412 y(com)n(binations)j(of)k (generators)e(to)i(p)r(olynomials)c(of)k(generators)e(of)i(degree)-118 3511 y(higher)23 b(then)i(one.)35 b(If)25 b(linear)d(com)n(binations)e (of)25 b(generators)d(are)h(mapp)r(ed)h(b)n(y)-118 3611 y(the)k(in)n(v)n(olution)c(to)k(p)r(olynomials)23 b(of)28 b(the)h(second)e(degree)g(then)h(w)n(e)g(will)d(call)-118 3710 y(suc)n(h)i(in)n(v)n(olutions)d(quadratic.)6 3811 y(Here)30 b(w)n(e)g(will)d(consider)h(all)g(in)n(v)n(olutions)e(of)k (the)h(\014rst)f(order)e(and)i(some)-118 3911 y(quadratic)25 b(in)n(v)n(olutions)f(for)j(the)h(algebras)d FN(U)1327 3923 y FL(q)1363 3911 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))q(.)p eop %%Page: 134 138 134 137 bop -118 -137 a FO(134)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(Prop)s(osition)30 b(47.)110 b FO(1\))p FN(:)41 b FB(If)f FN(q)k FP(2)e FI(R)p FB(,)48 b FP(j)p FN(q)s FP(j)41 b(6)p FO(=)g(1)p FB(,)h(then)e(al)t(l)g (involutions)h(of)89 196 y(the)30 b(\014rst)e(or)l(der)i(in)f(the)h (algebr)l(a)g FN(U)1193 208 y FL(q)1229 196 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)i FO(\)\))35 b FB(ar)l(e)30 b(e)l(quivalent)f(to)g(the)89 296 y(fol)t(lowing)k(involution)6 b FO(:)462 536 y FN(I)505 501 y FM(\003)498 556 y FK(1)566 536 y FO(=)22 b FN(I)689 548 y FK(2)727 536 y FN(;)99 b(I)892 501 y FM(\003)885 556 y FK(2)953 536 y FO(=)23 b FN(I)1077 548 y FK(1)1114 536 y FN(;)99 b(I)1279 501 y FM(\003)1272 556 y FK(3)1340 536 y FO(=)1428 394 y Fy(\()1495 479 y FN(I)1531 491 y FK(3)1569 479 y FN(;)149 b(q)26 b(>)d FO(0)p FN(;)1495 598 y FP(\000)p FN(I)1596 610 y FK(3)1633 598 y FN(;)85 b(q)26 b(<)d FO(0)p FN(:)-49 811 y FO(2\))p FN(:)41 b FB(If)k FP(j)p FN(q)s FP(j)23 b FO(=)g(1)p FB(,)k FN(q)f FP(6)p FO(=)d FP(\006)p FO(1)p FB(,)k(then)f(any)h(involution)h(of)f(the)g(\014rst)f(or)l(der)i(in)e (the)89 910 y(algebr)l(a)38 b FN(U)434 922 y FL(q)470 910 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))42 b FB(is)36 b(e)l(quivalent)g(to)g(one)g(of)g(the)g(fol)t(lowing)i(two) 89 1010 y(ine)l(quivalent)31 b(involutions)7 b FO(:)154 1171 y FN(a)p FO(\))42 b FN(I)315 1141 y FM(\003)308 1191 y FK(1)376 1171 y FO(=)23 b FN(I)500 1183 y FK(1)538 1171 y FB(,)30 b FN(I)636 1141 y FM(\003)629 1191 y FK(2)697 1171 y FO(=)23 b FP(\000)p FN(I)886 1183 y FK(2)923 1171 y FB(,)30 b FN(I)1021 1141 y FM(\003)1014 1191 y FK(3)1082 1171 y FO(=)23 b FN(I)1206 1183 y FK(3)1244 1171 y FB(,)163 1299 y FN(b)p FO(\))41 b FN(I)315 1268 y FM(\003)308 1319 y FK(1)376 1299 y FO(=)23 b FP(\000)p FN(I)565 1311 y FK(1)602 1299 y FB(,)30 b FN(I)700 1268 y FM(\003)693 1319 y FK(2)762 1299 y FO(=)22 b FP(\000)p FN(I)950 1311 y FK(2)988 1299 y FB(,)30 b FN(I)1086 1268 y FM(\003)1079 1319 y FK(3)1147 1299 y FO(=)23 b FP(\000)p FN(I)1336 1311 y FK(3)1373 1299 y FB(.)-49 1459 y FO(3\))p FN(:)41 b FB(If)j FN(q)26 b FO(=)d FP(\000)p FO(1)p FB(,)i(then)g(e)l(ach)i (involution)f(of)g(the)g(\014rst)e(or)l(der)i(in)g(the)f(algebr)l(a)89 1559 y FN(U)146 1571 y FL(q)183 1559 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))36 b FB(is)30 b(e)l(quivalent)g (either)h(to)k FO(1\))p FB(,)30 b(or)g FO(2)p FN(a)p FO(\))p FB(,)g(or)g FO(2)p FN(b)p FO(\))p FB(.)-118 1715 y(Pr)l(o)l(of.)43 b FO(The)30 b(pro)r(of)f(consists)f(of)h(trivial)e (but)j(rather)e(length)n(y)h(calculations,)-118 1814 y(and)e(w)n(e)h(lea)n(v)n(e)d(them)i(out.)p 2278 1814 4 57 v 2282 1762 50 4 v 2282 1814 V 2331 1814 4 57 v 6 1978 a(The)j(algebras)d FN(U)562 1990 y FL(q)598 1978 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))36 b(also)27 b(ha)n(v)n(e)i(some)f(quadratic)g(in)n(v)n(olutions.)-118 2078 y(Since)38 b(the)g(elemen)n(t)f FN(I)617 2090 y FK(3)693 2078 y FO(is)h(not)g(indep)r(enden)n(t)g(and)h(is)e (determined)g(b)n(y)h FN(I)2301 2090 y FK(1)-118 2177 y FO(and)e FN(I)88 2189 y FK(2)125 2177 y FO(,)i(the)f(algebra)c FN(U)694 2189 y FL(q)730 2177 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))42 b(is)34 b(generated)h(b)n(y)h FN(I)1751 2189 y FK(1)1824 2177 y FO(and)g FN(I)2030 2189 y FK(2)2068 2177 y FO(.)62 b(If)36 b(w)n(e)-118 2277 y(substitute)h FN(I)316 2289 y FK(3)391 2277 y FO(with)g(the)h (left)f(hand)h(side)e(of)i(the)f(\014rst)h(relation)c(in)n(to)i(the) -118 2376 y(second)h(and)h(third)f(relation)e(in)i(\(2.33\),)i(w)n(e)f (will)d(get)i(the)i(follo)n(wing)33 b(t)n(w)n(o)-118 2476 y(relations)24 b(of)k(the)g(third)f(order)f(for)h FN(I)1045 2488 y FK(1)1110 2476 y FO(and)h FN(I)1308 2488 y FK(2)1346 2476 y FO(:)434 2646 y FN(I)477 2612 y FK(2)470 2666 y(2)514 2646 y FN(I)550 2658 y FK(1)606 2646 y FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)903 2612 y FM(\000)p FK(1)992 2646 y FO(\))p FN(I)1060 2658 y FK(2)1098 2646 y FN(I)1134 2658 y FK(1)1172 2646 y FN(I)1208 2658 y FK(2)1264 2646 y FO(+)g FN(I)1383 2658 y FK(1)1421 2646 y FN(I)1464 2612 y FK(2)1457 2666 y(2)1524 2646 y FO(=)23 b FP(\000)p FN(I)1713 2658 y FK(1)1750 2646 y FN(;)434 2781 y(I)477 2746 y FK(2)470 2801 y(1)514 2781 y FN(I)550 2793 y FK(2)606 2781 y FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)903 2746 y FM(\000)p FK(1)992 2781 y FO(\))p FN(I)1060 2793 y FK(1)1098 2781 y FN(I)1134 2793 y FK(2)1172 2781 y FN(I)1208 2793 y FK(1)1264 2781 y FO(+)g FN(I)1383 2793 y FK(2)1421 2781 y FN(I)1464 2746 y FK(2)1457 2801 y(1)1524 2781 y FO(=)23 b FP(\000)p FN(I)1713 2793 y FK(2)1750 2781 y FN(:)353 b FO(\(2.34\))6 2950 y(Let)29 b(us)g(\014nd)g(all)e(in)n(v)n(olutions)d(of)29 b(the)g(\014rst)f(degree)g(in)g(the)h(algebra)d(with)-118 3050 y(generators)f FN(I)320 3062 y FK(1)385 3050 y FO(and)j FN(I)583 3062 y FK(2)648 3050 y FO(and)f(relations)d(\(2.34\).)37 b(The)27 b(elemen)n(t)f FN(I)1932 3062 y FK(3)1997 3050 y FO(is)g(de\014ned)-118 3150 y(b)n(y)h(the)h(relations)d(\(2.33\).)36 b(W)-7 b(e)28 b(ha)n(v)n(e)e(the)i(follo)n(wing)c(statemen)n(t.)-118 3305 y FQ(Prop)s(osition)30 b(48.)41 b FB(A)n(ny)35 b(such)h (involution)h(is)f(isomorphic)j(either)d(to)g(one)-118 3405 y(of)j(the)f(involutions)g(fr)l(om)h(Pr)l(op)l(osition)46 b FO(47)p FB(,)39 b(or)g(to)f(one)g(of)h(the)f(fol)t(lowing)-118 3505 y(ones)7 b FO(:)6 3604 y(1\))30 b FN(q)c FP(2)d FI(R)p FB(,)-6 3849 y FN(I)37 3814 y FM(\003)30 3869 y FK(1)98 3849 y FO(=)g FN(I)222 3861 y FK(1)259 3849 y FN(;)99 b(I)424 3814 y FM(\003)417 3869 y FK(2)485 3849 y FO(=)23 b FN(I)609 3861 y FK(2)647 3849 y FN(;)99 b(I)812 3814 y FM(\003)805 3869 y FK(3)873 3849 y FO(=)960 3707 y Fy(\()1027 3792 y FN(q)1067 3762 y FK(1)p FL(=)p FK(2)1172 3792 y FN(I)1208 3804 y FK(2)1245 3792 y FN(I)1281 3804 y FK(1)1338 3792 y FP(\000)18 b FN(q)1461 3762 y FM(\000)p FK(1)p FL(=)p FK(2)1617 3792 y FN(I)1653 3804 y FK(1)1691 3792 y FN(I)1727 3804 y FK(2)1764 3792 y FN(;)214 b(q)27 b(>)22 b FO(0)p FN(;)1027 3912 y FP(\000)p FO(\()p FN(q)1164 3882 y FK(1)p FL(=)p FK(2)1269 3912 y FN(I)1305 3924 y FK(2)1342 3912 y FN(I)1378 3924 y FK(1)1434 3912 y FP(\000)c FN(q)1557 3882 y FM(\000)p FK(1)p FL(=)p FK(2)1714 3912 y FN(I)1750 3924 y FK(1)1787 3912 y FN(I)1823 3924 y FK(2)1861 3912 y FO(\))p FN(;)85 b(q)27 b(<)22 b FO(0)p FN(;)p eop %%Page: 135 139 135 138 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(135)6 96 y(2\))30 b FN(q)c FP(2)d FI(R)p FB(,)-39 347 y FN(I)4 313 y FM(\003)-3 368 y FK(1)66 347 y FO(=)f FN(I)189 359 y FK(1)227 347 y FN(;)99 b(I)392 313 y FM(\003)385 368 y FK(2)453 347 y FO(=)23 b FP(\000)p FN(I)642 359 y FK(2)679 347 y FN(;)99 b(I)844 313 y FM(\003)837 368 y FK(3)905 347 y FO(=)993 205 y Fy(\()1060 290 y FP(\000)p FO(\()p FN(q)1197 260 y FK(1)p FL(=)p FK(2)1301 290 y FN(I)1337 302 y FK(2)1375 290 y FN(I)1411 302 y FK(1)1467 290 y FP(\000)18 b FN(q)1590 260 y FM(\000)p FK(1)p FL(=)p FK(2)1746 290 y FN(I)1782 302 y FK(1)1820 290 y FN(I)1856 302 y FK(2)1894 290 y FO(\))p FN(;)85 b(q)26 b(>)c FO(0)p FN(;)1060 410 y(q)1100 380 y FK(1)p FL(=)p FK(2)1204 410 y FN(I)1240 422 y FK(2)1278 410 y FN(I)1314 422 y FK(1)1370 410 y FP(\000)c FN(q)1493 380 y FM(\000)p FK(1)p FL(=)p FK(2)1649 410 y FN(I)1685 422 y FK(1)1723 410 y FN(I)1759 422 y FK(2)1797 410 y FN(;)214 b(q)26 b(<)c FO(0)p FN(;)6 598 y FO(3\))30 b FN(q)c FP(2)d FI(R)p FB(,)-28 848 y FN(I)15 814 y FM(\003)8 869 y FK(1)76 848 y FO(=)f FP(\000)p FN(I)264 860 y FK(1)302 848 y FN(;)99 b(I)467 814 y FM(\003)460 869 y FK(2)528 848 y FO(=)22 b FP(\000)p FN(I)716 860 y FK(2)754 848 y FN(;)14 b(I)834 814 y FM(\003)827 869 y FK(3)895 848 y FO(=)983 706 y Fy(\()1049 792 y FO(\()p FN(q)1121 762 y FK(1)p FL(=)p FK(2)1226 792 y FN(I)1262 804 y FK(2)1300 792 y FN(I)1336 804 y FK(1)1392 792 y FP(\000)k FN(q)1515 762 y FM(\000)p FK(1)p FL(=)p FK(2)1671 792 y FN(I)1707 804 y FK(1)1745 792 y FN(I)1781 804 y FK(2)1819 792 y FO(\))p FN(;)150 b(q)26 b(>)c FO(0)p FN(;)1049 911 y FP(\000)p FO(\()p FN(q)1186 881 y FK(1)p FL(=)p FK(2)1291 911 y FN(I)1327 923 y FK(2)1364 911 y FN(I)1400 923 y FK(1)1457 911 y FP(\000)c FN(q)1580 881 y FM(\000)p FK(1)p FL(=)p FK(2)1736 911 y FN(I)1772 923 y FK(1)1810 911 y FN(I)1846 923 y FK(2)1883 911 y FO(\))p FN(;)86 b(q)26 b(<)c FO(0)p FN(;)6 1099 y FO(4\))30 b FP(j)p FN(q)s FP(j)23 b FO(=)g(1)p FB(,)247 1275 y FN(I)290 1240 y FM(\003)283 1295 y FK(1)351 1275 y FO(=)g FN(I)475 1287 y FK(2)512 1275 y FN(;)99 b(I)677 1240 y FM(\003)670 1295 y FK(2)738 1275 y FO(=)23 b FN(I)862 1287 y FK(1)900 1275 y FN(;)99 b(I)1065 1240 y FM(\003)1058 1295 y FK(3)1126 1275 y FO(=)22 b FN(q)1253 1240 y FM(\000)p FK(1)p FL(=)p FK(2)1410 1275 y FN(I)1446 1287 y FK(2)1483 1275 y FN(I)1519 1287 y FK(1)1576 1275 y FP(\000)c FN(q)1699 1240 y FK(1)p FL(=)p FK(2)1803 1275 y FN(I)1839 1287 y FK(1)1877 1275 y FN(I)1913 1287 y FK(2)1951 1275 y FN(:)-118 1489 y FQ(2.3.2)94 b(Represen)m(tations)30 b(of)i FN(U)1040 1501 y FL(q)1076 1489 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)i FO(\)\))-118 1643 y(W)-7 b(e)25 b(will)d(study)i(b)r(ounded)h FP(\003)p FO(-represen)n(tations)c(of)k(the)g(algebra)d FN(U)1956 1655 y FL(q)1992 1643 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))-118 1742 y(on)23 b(a)g(Hilb)r(ert)f(space)h FN(H)7 b FO(.)35 b(W)-7 b(e)24 b(restrict)e(ourselv)n(es)e(to)k(a)f (study)g(of)h(represen)n(ta-)-118 1842 y(tions)j(of)h(the)h FP(\003)p FO(-algebra)c(de\014ned)j(b)n(y)g(the)h(in)n(v)n(olution)24 b FN(I)1665 1812 y FM(\003)1658 1863 y FK(1)1728 1842 y FO(=)f FP(\000)p FN(I)1917 1854 y FK(1)1955 1842 y FO(,)28 b FN(I)2049 1812 y FM(\003)2042 1863 y FK(2)2112 1842 y FO(=)23 b FP(\000)p FN(I)2301 1854 y FK(2)-118 1942 y FO(\(for)h FN(q)j FP(2)c FI(R)31 b FO(and)25 b FP(j)p FN(q)s FP(j)e FO(=)f(1,)j FN(q)i FP(6)p FO(=)22 b FP(\006)p FO(1\),)j(whic)n(h,)g(for)f FN(q)i FO(=)d(1,)i(corresp)r (onds)e(to)h(the)-118 2041 y(compact)i(real)g(form)g(of)h FN(so)p FO(\(3\).)38 b(Henceforth)27 b(w)n(e)g(denote)h(it)f(b)n(y)g FN(U)1981 2053 y FL(q)2018 2041 y FO(\()p FN(so)p FO(\(3\)\).)6 2141 y(W)-7 b(e)28 b(assume)e(the)i(follo)n(wing)c(notations)42 2364 y([)p FN(x)p FO(])135 2376 y FL(q)195 2364 y FO(=)293 2308 y FN(q)333 2277 y FL(x)393 2308 y FP(\000)18 b FN(q)516 2277 y FM(\000)p FL(x)p 293 2345 318 4 v 316 2421 a FN(q)j FP(\000)d FN(q)497 2397 y FM(\000)p FK(1)620 2364 y FN(;)96 b(d)782 2376 y FL(q)819 2364 y FO(\()p FN(m)p FO(\))24 b(=)1591 2308 y(1)p 1077 2345 1069 4 v 1077 2422 a(\()p FN(q)1149 2398 y FL(m)1231 2422 y FO(+)18 b FN(q)1354 2398 y FM(\000)p FL(m)1469 2422 y FO(\)\()p FN(q)1573 2398 y FL(m)p FK(+1)1740 2422 y FO(+)g FN(q)1863 2398 y FM(\000)p FK(\()p FL(m)p FK(+1\))2113 2422 y FO(\))2156 2364 y FN(:)-118 2590 y FO(Let)25 b FN(E)89 2602 y FL(A)143 2590 y FO(\()p FP(\001)p FO(\))g(denote)g(the)g(sp)r(ectral)e(measure)f (of)j(the)f(self-adjoin)n(t)f(op)r(erator)f FN(A)-118 2689 y FO(and)27 b(\()p FQ(F)p FO(\()p FP(\001)p FO(\)\))254 2701 y FL(i)311 2689 y FO(the)h FN(i)p FO(-th)f(co)r(ordinate)f(of)h(a) g(function)h FQ(F)9 b FO(:)28 b FI(R)1688 2659 y FL(n)1762 2689 y FP(!)23 b FI(R)1922 2659 y FL(n)1973 2689 y FO(.)6 2789 y(Let)29 b FN(J)202 2801 y FK(1)239 2789 y FO(,)g FN(J)337 2801 y FK(2)403 2789 y FO(b)r(e)g(de\014ned)g(b)n(y)g FN(I)957 2801 y FK(1)1019 2789 y FO(=)24 b FN(iJ)1183 2801 y FK(1)1249 2789 y FO(and)k FN(I)1447 2801 y FK(2)1510 2789 y FO(=)c FN(iJ)1674 2801 y FK(2)1711 2789 y FO(.)40 b(The)28 b(new)h(gener-)-118 2889 y(ators)d FN(J)136 2901 y FK(1)173 2889 y FO(,)i FN(J)270 2901 y FK(2)335 2889 y FO(of)g FN(U)487 2901 y FL(q)523 2889 y FO(\()p FN(so)p FO(\(3\)\))h(are)d(self-adjoin)n(t)f(and)j(satisfy)e(the)i (relations:)426 3065 y FN(J)480 3030 y FK(2)472 3085 y(2)517 3065 y FN(J)563 3077 y FK(1)619 3065 y FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)916 3030 y FM(\000)p FK(1)1005 3065 y FO(\))p FN(J)1083 3077 y FK(2)1121 3065 y FN(J)1167 3077 y FK(1)1204 3065 y FN(J)1250 3077 y FK(2)1306 3065 y FO(+)g FN(J)1435 3077 y FK(1)1472 3065 y FN(J)1526 3030 y FK(2)1518 3085 y(2)1586 3065 y FO(=)23 b FN(J)1720 3077 y FK(1)1757 3065 y FN(;)346 b FO(\(2.35\))426 3199 y FN(J)480 3165 y FK(2)472 3220 y(1)517 3199 y FN(J)563 3211 y FK(2)619 3199 y FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)916 3165 y FM(\000)p FK(1)1005 3199 y FO(\))p FN(J)1083 3211 y FK(1)1121 3199 y FN(J)1167 3211 y FK(2)1204 3199 y FN(J)1250 3211 y FK(1)1306 3199 y FO(+)g FN(J)1435 3211 y FK(2)1472 3199 y FN(J)1526 3165 y FK(2)1518 3220 y(1)1586 3199 y FO(=)23 b FN(J)1720 3211 y FK(2)1757 3199 y FN(:)346 b FO(\(2.36\))-118 3375 y FQ(1.)36 b FO(Represen)n(tations)25 b(of)j FN(U)749 3387 y FL(q)785 3375 y FO(\()p FN(so)p FO(\(3\)\),)h FN(q)d(>)c FO(0.)-118 3536 y FQ(Theorem)30 b(33.)41 b FB(A)n(ny)35 b(irr)l(e)l(ducible)i(r)l (epr)l(esentation)f(of)h FN(U)1750 3548 y FL(q)1786 3536 y FO(\()p FN(so)p FO(\(3\)\))p FB(,)i FN(q)e(>)d FO(0)p FB(,)-118 3636 y(is)f(\014nite-dimensional.)48 b(F)-6 b(or)32 b(any)h FN(n)28 b FP(\025)g FO(1)p FB(,)33 b(irr)l(e)l(ducible) h(r)l(epr)l(esentations)e(in)-118 3735 y FN(H)e FO(=)22 b FI(C)122 3705 y FL(n)203 3735 y FB(ar)l(e)30 b(unitarily)h(e)l (quivalent)f(to)g(the)f(fol)t(lowing)k(one)6 b FO(:)264 3911 y FN(J)310 3923 y FK(1)347 3911 y FN(e)386 3923 y FL(k)450 3911 y FO(=)23 b([)p FN(k)e FP(\000)d FO(\()p FN(n)h FO(+)f(1\))p FN(=)p FO(2])1073 3923 y FL(q)1108 3911 y FN(e)1147 3923 y FL(k)1188 3911 y FN(;)p eop %%Page: 136 140 136 139 bop -118 -137 a FO(136)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)264 213 y FN(J)310 225 y FK(2)347 213 y FN(e)386 225 y FL(k)450 213 y FO(=)538 18 y Fy(8)538 93 y(>)538 118 y(<)538 267 y(>)538 292 y(:)611 97 y FN(\013)664 109 y FK(1)702 97 y FN(e)741 109 y FK(2)778 97 y FN(;)621 b(k)26 b FO(=)c(1)p FN(;)611 217 y(\013)664 229 y FL(k)705 217 y FN(e)744 229 y FL(k)q FK(+1)887 217 y FO(+)c FN(\013)1023 229 y FL(k)q FM(\000)p FK(1)1149 217 y FN(e)1188 229 y FL(k)q FM(\000)p FK(1)1314 217 y FN(;)85 b FO(2)22 b FP(\024)h FN(k)j FP(\024)c FN(n)d FP(\000)f FO(1)p FN(;)611 336 y(\013)664 348 y FL(n)p FM(\000)p FK(1)795 336 y FN(e)834 348 y FL(n)p FM(\000)p FK(1)964 336 y FN(;)435 b(k)26 b FO(=)c FN(n;)-118 533 y FB(wher)l(e)30 b FN(\013)169 545 y FL(k)233 533 y FO(=)23 b(\()p FN(d)396 545 y FL(q)433 533 y FO(\()p FN(k)f FP(\000)c FO(\()p FN(n)g FO(+)g(1\))p FN(=)p FO(2\)[)p FN(k)s FO(])1078 545 y FL(q)1114 533 y FO([)p FN(n)g FP(\000)g FN(k)s FO(])1357 545 y FL(q)1394 533 y FO(\))1426 503 y FK(1)p FL(=)p FK(2)1531 533 y FB(.)-118 695 y(Pr)l(o)l(of.)43 b FO(The)21 b(pro)r(of)f(of)h(the)g(theorem)e(is)h(based)g(on)h(the)g (tec)n(hnique)f(of)h(semilin-)-118 795 y(ear)f(relations)d(dev)n(elop)r (ed)i(in)h(Section)g(1.3)g(and)g(the)h(tec)n(hnique)f(of)g(dynamical) -118 895 y(systems.)6 994 y(Let)39 b FN(q)k FO(=)e FN(e)391 964 y FL(\033)435 994 y FO(,)g FN(\033)j FP(2)d FI(R)p FO(,)47 b(and)38 b FN(J)1028 1006 y FK(1)1065 994 y FO(,)j FN(J)1175 1006 y FK(2)1250 994 y FO(b)r(e)e(self-adjoin)n(t)d(op)r (erators)g(in)h(a)-118 1094 y(Hilb)r(ert)31 b(space)g FN(H)39 b FO(satisfying)30 b(relations)f(\(2.35\))o({\(2.36\))n(.)51 b(Equation)30 b(\(2.35\))-118 1194 y(is)c(linear)f(with)i(resp)r(ect)h (to)f FN(J)817 1206 y FK(2)882 1194 y FO(with)g(the)h(corresp)r(onding) d(binary)h(relation:)156 1372 y(\000)d(=)f FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)h FO(\010\()p FN(t;)14 b(s)p FO(\))23 b FP(\021)g FN(t)971 1338 y FK(2)1026 1372 y FP(\000)18 b FO(\()p FN(q)k FO(+)c FN(q)1323 1338 y FM(\000)p FK(1)1412 1372 y FO(\))c FN(ts)19 b FO(+)f FN(s)1668 1338 y FK(2)1723 1372 y FP(\000)g FO(1)23 b(=)g(0)p FP(g)p FN(:)-118 1550 y FO(Let)469 1729 y FN(F)522 1741 y FK(1)559 1729 y FO(\()p FN(s)p FO(\))h(=)f FN(s)14 b FO(cosh)e FN(\033)22 b FO(+)1149 1640 y Fy(p)p 1232 1640 483 4 v 89 x FN(s)1271 1705 y FK(2)1322 1729 y FO(sinh)1470 1691 y FK(2)1521 1729 y FN(\033)g FO(+)c(1)p FN(;)469 1886 y(F)522 1898 y FK(2)559 1886 y FO(\()p FN(s)p FO(\))24 b(=)f FN(s)14 b FO(cosh)e FN(\033)22 b FP(\000)1149 1798 y Fy(p)p 1232 1798 V 88 x FN(s)1271 1862 y FK(2)1322 1886 y FO(sinh)1470 1849 y FK(2)1521 1886 y FN(\033)g FO(+)c(1)p FN(;)-118 2065 y FO(and)36 b(sinh)13 b FN(\033)40 b FO(=)d(\()p FN(q)28 b FP(\000)23 b FN(q)628 2035 y FM(\000)p FK(1)717 2065 y FO(\))p FN(=)p FO(2.)62 b(Then)36 b(\010\()p FN(t;)14 b(s)p FO(\))38 b(=)f(\()p FN(t)24 b FP(\000)g FN(F)1741 2077 y FK(1)1778 2065 y FO(\()p FN(s)p FO(\)\)\()p FN(t)h FP(\000)f FN(F)2142 2077 y FK(2)2180 2065 y FO(\()p FN(s)p FO(\)\).)-118 2164 y(Consider)34 b(the)i(parameterization)31 b FN(s)36 b FO(=)h(sinh)12 b FN(\033)s(x)p FO(\()p FN(s)p FO(\)\(sinh)j FN(\033)s FO(\))1854 2134 y FM(\000)p FK(1)1981 2164 y FO(=)36 b([)p FN(x)p FO(\()p FN(s)p FO(\)])2278 2176 y FL(q)2316 2164 y FO(,)-118 2264 y FN(x)p FO(\()p FN(s)p FO(\))24 b FP(2)f FI(R)p FO(,)34 b(whic)n(h)27 b(giv)n(es)e FN(F)739 2276 y FK(1)777 2264 y FO(\()p FN(s)p FO(\))e(=)g([)p FN(x)p FO(\()p FN(s)p FO(\))c(+)f(1])1331 2276 y FL(q)1367 2264 y FO(,)28 b FN(F)1471 2276 y FK(2)1509 2264 y FO(\()p FN(s)p FO(\))23 b(=)g([)p FN(x)p FO(\()p FN(s)p FO(\))c FP(\000)f FO(1])2063 2276 y FL(q)2100 2264 y FO(,)27 b(and)313 2442 y(\000)c(=)g FP(f)p FO(\([)p FN(x)c FO(+)f(1])787 2454 y FL(q)823 2442 y FN(;)c FO([)p FN(x)p FO(])953 2454 y FL(q)990 2442 y FO(\))p FN(;)g FO(\([)p FN(x)20 b FP(\000)e FO(1])1329 2454 y FL(q)1365 2442 y FN(;)c FO([)p FN(x)p FO(])1495 2454 y FL(q)1532 2442 y FO(\))23 b FP(j)h FN(x)f FP(2)g FI(R)p FP(g)p FN(:)-118 2621 y FO(Let)d FN(E)84 2633 y FL(J)121 2641 y Fx(1)158 2621 y FO(\()p FP(\001)p FO(\))g(b)r(e)h(the)f(resolution)d(of)j(the)g(iden) n(tit)n(y)f(for)g(the)h(op)r(erator)f FN(J)2032 2633 y FK(1)2069 2621 y FO(.)34 b(Then,)-118 2720 y(b)n(y)27 b(Theorem)f(8,)h FN(J)486 2732 y FK(1)523 2720 y FO(,)h FN(J)620 2732 y FK(2)685 2720 y FO(satisfy)g(\(2.35\))e(if)h(and)h (only)e(if)694 2899 y FN(E)755 2911 y FL(J)792 2919 y Fx(1)829 2899 y FO(\(\001\))p FN(J)1008 2911 y FK(2)1046 2899 y FN(E)1107 2911 y FL(J)1144 2919 y Fx(1)1180 2899 y FO(\(\001)1281 2864 y FM(0)1305 2899 y FO(\))d(=)g(0)p FN(;)613 b FO(\(2.37\))-118 3077 y(for)39 b(an)n(y)g(\001,)k(\001)394 3047 y FM(0)461 3077 y FP(2)h Fz(B)p FO(\()p FI(R)p FO(\))q(,)49 b(\001)26 b FP(\002)h FO(\001)1080 3047 y FM(0)1130 3077 y FP(\\)g FO(\000)43 b(=)g FI(?)p FO(.)73 b(F)-7 b(or)39 b(the)h(op)r(erator)e FN(A)2301 3089 y FK(1)-118 3177 y FO(de\014ned)e(b)n(y)g FN(J)346 3189 y FK(1)420 3177 y FO(=)h(sinh)13 b FN(\033)s(A)796 3189 y FK(1)834 3177 y FN(=)p FO(sinh)f FN(\033)t FO(,)38 b(condition)c(\(2.37\))h(is)g (equiv)-5 b(alen)n(t)34 b(to)-118 3276 y(the)28 b(follo)n(wing)23 b(one:)-98 3455 y FN(E)-37 3467 y FL(A)13 3475 y Fx(1)49 3455 y FO(\(\001\))p FN(J)228 3467 y FK(2)267 3455 y FN(E)328 3467 y FL(A)378 3475 y Fx(1)414 3455 y FO(\(\001)515 3420 y FM(0)539 3455 y FO(\))g(=)g(0)p FN(;)96 b FO(for)28 b(an)n(y)e(\001)p FN(;)14 b FO(\001)1302 3420 y FM(0)1349 3455 y FP(2)23 b Fz(B)p FO(\()p FI(R)q FO(\))p FN(;)48 b FO(\001)19 b FP(\002)f FO(\001)1930 3420 y FM(0)1971 3455 y FP(\\)h FO(\000)2097 3420 y FM(0)2143 3455 y FO(=)k FI(?)p FN(;)-118 3633 y FO(where)j(\000)173 3603 y FM(0)219 3633 y FO(=)d FP(f)p FO(\()p FN(s)16 b FO(+)g(1)p FN(;)e(s)p FO(\))p FN(;)g FO(\()p FN(s)i FP(\000)g FO(1)p FN(;)e(s)p FO(\))22 b FP(j)h FN(s)g FP(2)h FI(R)p FP(g)p FO(.)42 b(It)27 b(follo)n(ws)c(from)i(Theorem)g(8)-118 3733 y(that)j(the)g(op)r (erators)d FN(A)634 3745 y FK(1)672 3733 y FO(,)j FN(J)769 3745 y FK(2)834 3733 y FO(satisfy)e(the)i(relation)556 3911 y FN(A)618 3877 y FK(2)618 3932 y(1)655 3911 y FN(J)701 3923 y FK(2)757 3911 y FP(\000)18 b FO(2)p FN(A)944 3923 y FK(1)981 3911 y FN(J)1027 3923 y FK(2)1064 3911 y FN(A)1126 3923 y FK(1)1182 3911 y FO(+)g FN(J)1311 3923 y FK(2)1348 3911 y FN(A)1410 3877 y FK(2)1410 3932 y(1)1471 3911 y FO(=)23 b FN(J)1605 3923 y FK(2)1642 3911 y FN(:)p eop %%Page: 137 141 137 140 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(137)-118 96 y(Let)33 b FN(E)97 108 y FK(1)166 96 y FO(=)e(\([)p FN(A)379 108 y FK(1)416 96 y FN(;)14 b(J)499 108 y FK(2)537 96 y FO(])22 b(+)f FN(J)714 108 y FK(2)751 96 y FO(\))p FN(=)p FO(2,)33 b FN(E)989 66 y FM(\003)984 117 y FK(1)1059 96 y FO(=)e(\([)p FN(A)1272 108 y FK(1)1310 96 y FN(;)14 b(J)1393 108 y FK(2)1430 96 y FO(])22 b FP(\000)f FN(J)1607 108 y FK(2)1645 96 y FO(\))p FN(=)p FO(2.)51 b(Then)33 b(one)f(can)-118 196 y(c)n(hec)n(k)27 b(that)310 349 y FN(A)372 361 y FK(1)410 349 y FN(E)471 361 y FK(1)531 349 y FO(=)c FN(E)680 361 y FK(1)718 349 y FO(\()p FN(A)812 361 y FK(1)868 349 y FO(+)18 b FN(I)7 b FO(\))p FN(;)97 b(A)1208 361 y FK(1)1246 349 y FN(E)1312 314 y FM(\003)1307 369 y FK(1)1373 349 y FO(=)22 b FN(E)1526 314 y FM(\003)1521 369 y FK(1)1565 349 y FO(\()p FN(A)1659 361 y FK(1)1715 349 y FP(\000)c FN(I)7 b FO(\))p FN(;)701 473 y(E)767 439 y FM(\003)762 494 y FK(1)805 473 y FN(E)866 485 y FK(1)927 473 y FO(=)22 b FN(F)12 b FO(\()p FN(A)1173 485 y FK(1)1211 473 y FN(;)i(E)1309 485 y FK(1)1346 473 y FN(E)1412 439 y FM(\003)1407 494 y FK(1)1451 473 y FO(\))p FN(;)620 b FO(\(2.38\))-118 626 y(where)50 806 y FN(F)12 b FO(\()p FN(x)194 818 y FK(1)232 806 y FN(;)i(x)316 818 y FK(2)354 806 y FO(\))23 b(=)g FN(x)544 818 y FK(2)605 750 y FO(cosh\(\()p FN(x)874 762 y FK(1)930 750 y FP(\000)18 b FO(1\))p FN(\033)s FO(\))p 605 787 565 4 v 605 863 a(cosh\(\()p FN(x)874 875 y FK(1)930 863 y FO(+)g(1\))p FN(\033)s FO(\))1198 806 y FP(\000)1540 750 y FO(sinh)o(\()p FN(x)1767 762 y FK(1)1805 750 y FN(\033)s FO(\))p 1291 787 847 4 v 1291 863 a(2)c(sinh)e FN(\033)18 b FO(cosh)o(\(\()p FN(x)1841 875 y FK(1)1898 863 y FO(+)g(1\))p FN(\033)s FO(\))2147 806 y FN(:)-118 1007 y FO(Con)n(v)n(ersely)-7 b(,)28 b(an)n(y)h(op)r(erators)f FN(A)913 1019 y FK(1)950 1007 y FO(,)j FN(E)1065 1019 y FK(1)1103 1007 y FO(,)g FN(E)1223 977 y FM(\003)1218 1028 y FK(1)1291 1007 y FO(satisfying)e(\(2.38\))g(determine)f(a)-118 1107 y(represen)n(tation) d FN(J)471 1119 y FK(1)508 1107 y FO(,)j FN(J)605 1119 y FK(2)665 1107 y FO(=)23 b FN(E)814 1119 y FK(1)870 1107 y FO(+)18 b FN(E)1019 1077 y FM(\003)1014 1127 y FK(1)1085 1107 y FO(of)34 b(\(2.35\){\(2.36\))n(.)j(It)28 b(is)f(ob)n(vious)e(that)-118 1206 y(the)c(pair)d(\()p FN(J)259 1218 y FK(1)297 1206 y FO(,)k FN(J)388 1218 y FK(2)425 1206 y FO(\))e(is)g(irreducible)c(if)k(and)g(only)f(if)h (the)g(family)e(\()p FN(A)1900 1218 y FK(1)1938 1206 y FO(,)j FN(E)2043 1218 y FK(1)2081 1206 y FO(,)h FN(E)2192 1176 y FM(\003)2187 1227 y FK(1)2230 1206 y FO(\))f(is)-118 1306 y(irreducible)g(and)k(there)g(is)f(a)g(one-to)g(one)h(corresp)r (ondence)e(b)r(et)n(w)n(een)i(classes)-118 1406 y(of)36 b(unitary)f(equiv)-5 b(alen)n(t)34 b(represen)n(tations)g(of)i(these)h (families.)59 b(Hence,)39 b(in-)-118 1505 y(stead)26 b(of)h(represen)n(tations)c(of)k(relations)c(\(2.35\){\(2.36\))i(w)n(e) h(can)g(talk)g(ab)r(out)-118 1605 y(the)34 b(represen)n(tations)d(of)j (\(2.38\).)56 b(T)-7 b(o)33 b(relation)e(\(2.38\))j(there)g(corresp)r (onds)-118 1704 y(the)c(dynamical)25 b(system)k(\()p FN(x)784 1716 y FK(1)822 1704 y FN(;)14 b(x)906 1716 y FK(2)943 1704 y FO(\))26 b FP(7!)h FQ(F)p FO(\()p FN(x)1250 1716 y FK(1)1288 1704 y FN(;)14 b(x)1372 1716 y FK(2)1409 1704 y FO(\))27 b FP(\021)e FO(\()p FN(x)1637 1716 y FK(1)1695 1704 y FO(+)19 b(1)p FN(;)14 b(F)e FO(\()p FN(x)2002 1716 y FK(1)2059 1704 y FO(+)19 b(1)p FN(;)14 b(x)2269 1716 y FK(2)2306 1704 y FO(\))-118 1804 y(whic)n(h)20 b(has)h(the)h(measurable)c(section)i([0)p FN(;)14 b FO(1\))6 b FP(\002)g FI(R)p FO(.)40 b(Therefore)21 b(the)g(join)n(t)g(sp)r(ec-) -118 1904 y(tral)29 b(measure)f(of)i FN(A)527 1916 y FK(1)564 1904 y FO(,)h FN(E)684 1874 y FM(\003)679 1924 y FK(1)723 1904 y FN(E)784 1916 y FK(1)821 1904 y FO(,)g(is)e(discrete) g(and)h(concen)n(trated)f(on)h(an)g(orbit)-118 2003 y(if)g(\()p FN(A)55 2015 y FK(1)93 2003 y FO(,)h FN(E)208 2015 y FK(1)246 2003 y FO(,)g FN(E)366 1973 y FM(\003)361 2024 y FK(1)405 2003 y FO(\))g(is)e(irreducible,)f(and)i(w)n(e)h(can)f(c)n (ho)r(ose)f(a)i(basis)e(consisting)-118 2103 y(of)e(its)g(eigen)n(v)n (ectors.)34 b(Then)28 b(w)n(e)f(ha)n(v)n(e)266 2256 y FN(A)328 2268 y FK(1)366 2256 y FN(e)405 2268 y FL(x)443 2276 y Fx(1)474 2268 y FL(;x)532 2276 y Fx(2)591 2256 y FO(=)c FN(x)726 2268 y FK(1)778 2256 y FN(e)817 2268 y FL(x)855 2276 y Fx(1)886 2268 y FL(;x)944 2276 y Fx(2)980 2256 y FN(;)97 b(E)1166 2221 y FM(\003)1161 2276 y FK(1)1204 2256 y FN(E)1265 2268 y FK(1)1303 2256 y FN(e)1342 2268 y FL(x)1380 2276 y Fx(1)1412 2268 y FL(;x)1470 2276 y Fx(2)1529 2256 y FO(=)22 b FN(x)1663 2268 y FK(2)1715 2256 y FN(e)1754 2268 y FL(x)1792 2276 y Fx(1)1823 2268 y FL(;x)1881 2276 y Fx(2)1917 2256 y FN(;)-75 2405 y(E)-14 2417 y FK(1)23 2405 y FN(e)62 2417 y FL(x)100 2425 y Fx(1)132 2417 y FL(;x)190 2425 y Fx(2)249 2405 y FO(=)h FN(x)384 2362 y FK(1)p FL(=)p FK(2)384 2428 y(2)489 2405 y FN(e)528 2420 y Fm(F)p FK(\()p FL(x)639 2428 y Fx(1)670 2420 y FL(;x)728 2428 y Fx(2)760 2420 y FK(\))790 2405 y FN(;)97 b(E)976 2371 y FM(\003)971 2426 y FK(1)1014 2405 y FN(e)1053 2417 y FL(x)1091 2425 y Fx(1)1123 2417 y FL(;x)1181 2425 y Fx(2)1240 2405 y FO(=)23 b(\()p FQ(F)1420 2371 y FM(\000)p FK(1)1509 2405 y FO(\()p FN(x)1588 2417 y FK(1)1626 2405 y FN(;)14 b(x)1710 2417 y FK(2)1748 2405 y FO(\)\))1812 2362 y FK(1)p FL(=)p FK(2)1812 2428 y(2)1917 2405 y FN(e)1956 2421 y Fm(F)2003 2404 y Fw(\000)p Fx(1)2080 2421 y FK(\()p FL(x)2144 2429 y Fx(1)2176 2421 y FL(;x)2234 2429 y Fx(2)2265 2421 y FK(\))-118 2558 y FO(where)31 b(\()p FN(x)205 2570 y FK(1)243 2558 y FN(;)14 b(x)327 2570 y FK(2)364 2558 y FO(\))32 b(is)e(tak)n(en)h(from) f(some)g(orbit.)46 b(The)31 b(later)f(cannot)h(hold)f(for)-118 2658 y(all)25 b(p)r(oin)n(ts)i(of)g(the)h(orbit,)e(since)-61 2854 y(\()p FQ(F)31 2819 y FK(\()p FL(k)q FK(\))124 2854 y FO(\()p FN(x)203 2866 y FK(1)241 2854 y FN(;)14 b(x)325 2866 y FK(2)362 2854 y FO(\)\))426 2866 y FK(2)487 2854 y FO(=)23 b FN(x)622 2866 y FK(2)863 2797 y FO(cosh)o(\()p FN(x)1099 2809 y FK(1)1137 2797 y FN(\033)s FO(\))14 b(cosh\(\()p FN(x)1502 2809 y FK(1)1559 2797 y FO(+)k(1\))p FN(\033)s FO(\))p 683 2835 1295 4 v 683 2911 a(cosh\(\()p FN(x)952 2923 y FK(1)1008 2911 y FO(+)g FN(k)s FO(\))p FN(\033)s FO(\))c(cosh\(\()p FN(x)1534 2923 y FK(1)1591 2911 y FO(+)k FN(k)j FO(+)d(1\))p FN(\033)s FO(\))566 3083 y FP(\000)948 3027 y FO(sinh)o(\(2)p FN(x)1217 3039 y FK(1)1273 3027 y FO(+)g FN(k)j FO(+)d(1\))p FN(\033)s FO(\))c(sinh\()p FN(k)s(\033)s FO(\))p 659 3064 1614 4 v 659 3148 a(4)g(sinh)862 3111 y FK(2)913 3148 y FN(\033)j FO(cosh\(\()p FN(x)1246 3160 y FK(1)1302 3148 y FO(+)h FN(k)s FO(\))p FN(\033)s FO(\))c(cosh\(\()p FN(x)1828 3160 y FK(1)1885 3148 y FO(+)k FN(k)j FO(+)d(1\))p FN(\033)s FO(\))487 3312 y FP(\000)-48 b(!)23 b(\000)816 3256 y FO(1)p 685 3293 305 4 v 685 3377 a(4)14 b(sinh)888 3340 y FK(2)939 3377 y FN(\033)999 3312 y(;)180 b(k)26 b FP(!)d(\0061)p FN(;)-118 3513 y FO(while)i(\()p FQ(F)190 3482 y FK(\()p FL(k)q FK(\))283 3513 y FO(\()p FN(x)362 3525 y FK(1)400 3513 y FN(;)14 b(x)484 3525 y FK(2)521 3513 y FO(\)\))585 3525 y FK(2)650 3513 y FO(are)25 b(eigen)n(v)-5 b(alues)24 b(of)i(the)h(self-adjoin)n(t)d(non-negativ)n(e)-118 3612 y(op)r(erator)30 b FN(E)287 3582 y FM(\003)282 3633 y FK(1)325 3612 y FN(E)386 3624 y FK(1)424 3612 y FO(.)49 b(Th)n(us)31 b(there)g(exists)g(the)h(highest)e(v)n(ector)g(\(v)n (ector)h FN(e)2175 3624 y FL(x)2213 3632 y Fx(1)2245 3624 y FL(;x)2303 3632 y Fx(2)-118 3712 y FO(with)26 b(the)h(largest)d FN(x)526 3724 y FK(1)564 3712 y FO(\))i(on)h(whic)n (h)e FN(E)1034 3724 y FK(1)1098 3712 y FO(acts)h(as)g(zero)f(and)i(the) f(lo)n(w)n(est)f(v)n(ector,)-118 3811 y(on)36 b(whic)n(h)g FN(E)319 3781 y FM(\003)314 3832 y FK(1)394 3811 y FO(is)f(zero.)63 b(Using)36 b(this)g(argumen)n(t)e(one)j(can)f(easily)e(get)i(the)-118 3911 y(statemen)n(t.)p 2278 3911 4 57 v 2282 3858 50 4 v 2282 3911 V 2331 3911 4 57 v eop %%Page: 138 142 138 141 bop -118 -137 a FO(138)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(2.)36 b FO(Represen)n(tations)25 b(of)j FN(U)749 108 y FL(q)785 96 y FO(\()p FN(so)p FO(\(3\)\),)h FN(q)d(<)c FO(0.)-118 261 y FQ(Theorem)30 b(34.)41 b FB(A)n(ny)35 b(irr)l(e)l(ducible)i(r)l(epr)l(esentation)f(of)h FN(U)1750 273 y FL(q)1786 261 y FO(\()p FN(so)p FO(\(3\)\))p FB(,)i FN(q)e(<)d FO(0)p FB(,)-118 360 y(is)f(\014nite-dimensional.)47 b(F)-6 b(or)33 b(any)f FN(p)c FP(\025)f FO(1)32 b FB(ther)l(e)g(exist)g (four)h(unitarily)g(non-)-118 460 y(e)l(quivalent)e(irr)l(e)l(ducible)i (r)l(epr)l(esentations)e(of)h(dimension)g FO(2)p FN(p)e FB(and)h(\014ve)g(irr)l(e-)-118 560 y(ducible)g(r)l(epr)l(esentations)f (of)g(dimension)h FO(2)p FN(p)18 b FP(\000)g FO(1)p FB(,)30 b(which)h(act)f(as)g(fol)t(lows)7 b FO(:)6 659 y(1)p FB(.)39 b(F)-6 b(our)29 b(r)l(epr)l(esentations)h(with)g(any)h (\014nite)e(dimension,)i FO(dim)12 b FN(H)30 b FO(=)23 b FN(n)p FB(,)102 840 y FN(J)148 852 y FK(1)185 840 y FN(e)224 852 y FL(k)288 840 y FO(=)f(\()p FP(\000)p FO(1\))546 806 y FL(k)q FM(\000)p FK(1)686 840 y FO([)p FN(k)f FO(+)d(\()p FP(\000)p FO(1\))1027 806 y FL(j)1062 840 y FN(=)p FO(2])1169 852 y FM(\000)p FL(q)1270 840 y FN(e)1309 852 y FL(k)1350 840 y FN(;)102 1103 y(J)148 1115 y FK(2)185 1103 y FN(e)224 1115 y FL(k)288 1103 y FO(=)375 908 y Fy(8)375 983 y(>)375 1008 y(<)375 1157 y(>)375 1182 y(:)449 987 y FN(\013)502 999 y FK(1)553 987 y FN(e)592 999 y FK(2)648 987 y FO(+)g(\()p FP(\000)p FO(1\))902 957 y FL(i)943 987 y FO([)p FN(n)p FO(])1039 999 y FM(\000)p FL(q)1127 987 y FO([1)p FN(=)p FO(2])1299 999 y FM(\000)p FL(q)1400 987 y FN(e)1439 999 y FK(1)1476 987 y FN(;)85 b(k)26 b FO(=)d(1)p FN(;)449 1106 y(\013)502 1118 y FL(k)557 1106 y FN(e)596 1118 y FL(k)q FK(+1)739 1106 y FO(+)18 b FN(\013)875 1118 y FL(k)q FM(\000)p FK(1)1015 1106 y FN(e)1054 1118 y FL(k)q FM(\000)p FK(1)1179 1106 y FN(;)382 b FO(2)23 b FP(\024)f FN(k)k FP(\024)d FN(n)18 b FP(\000)g FO(1)p FN(;)449 1226 y(\013)502 1238 y FL(n)p FM(\000)p FK(1)646 1226 y FN(e)685 1238 y FL(n)p FM(\000)p FK(1)815 1226 y FN(;)746 b(k)26 b FO(=)d FN(n;)-118 1425 y FB(wher)l(e)30 b FN(\013)169 1437 y FL(k)233 1425 y FO(=)23 b(\()p FN(d)396 1437 y FM(\000)p FL(q)485 1425 y FO(\()p FN(k)f FP(\000)c FO(1)p FN(=)p FO(2\))c([)p FN(n)j FP(\000)h FN(k)s FO(])1079 1437 y FM(\000)p FL(q)1181 1425 y FO([)p FN(n)g FO(+)g FN(k)s FO(])1424 1437 y FM(\000)p FL(q)1513 1425 y FO(\))1545 1395 y FK(1)p FL(=)p FK(2)1649 1425 y FB(,)31 b FN(i)p FB(,)e FN(j)f FO(=)23 b(0)p FB(,)30 b FO(1)p FB(.)6 1642 y FO(2.)60 b FB(One)37 b(mor)l(e)g(r)l(epr)l(esentation)h(for)g(e)l (ach)g(o)l(dd)g(dimension,)j FO(dim)12 b FN(H)43 b FO(=)-118 1742 y FN(n)23 b FO(=)f(2)p FN(p)c FP(\000)g FO(1)p FB(,)250 1923 y FN(J)296 1935 y FK(1)334 1923 y FN(e)373 1935 y FL(k)436 1923 y FO(=)23 b(\()p FP(\000)p FO(1\))695 1888 y FL(k)q FM(\000)p FK(1)834 1923 y FO([)p FN(k)e FP(\000)d FO(\()p FN(n)h FO(+)f(1\))p FN(=)p FO(2])1369 1935 y FM(\000)p FL(q)1470 1923 y FN(e)1509 1935 y FL(k)1550 1923 y FN(;)250 2186 y(J)296 2198 y FK(2)334 2186 y FN(e)373 2198 y FL(k)436 2186 y FO(=)524 1991 y Fy(8)524 2065 y(>)524 2090 y(<)524 2240 y(>)524 2265 y(:)598 2069 y FN(\013)651 2081 y FK(1)702 2069 y FN(e)741 2081 y FK(2)778 2069 y FN(;)635 b(k)25 b FO(=)e(1)p FN(;)598 2189 y(\013)651 2201 y FL(k)705 2189 y FN(e)744 2201 y FL(k)q FK(+1)887 2189 y FO(+)18 b FN(\013)1023 2201 y FL(k)q FM(\000)p FK(1)1163 2189 y FN(e)1202 2201 y FL(k)q FM(\000)p FK(1)1328 2189 y FN(;)85 b FO(2)22 b FP(\024)h FN(k)j FP(\024)c FN(n)d FP(\000)f FO(1)p FN(;)598 2309 y(\013)651 2321 y FL(n)p FM(\000)p FK(1)795 2309 y FN(e)834 2321 y FL(n)p FM(\000)p FK(1)964 2309 y FN(;)449 b(k)25 b FO(=)e FN(n;)-118 2507 y FB(wher)l(e)30 b FN(\013)169 2519 y FL(k)233 2507 y FO(=)23 b(\()p FN(d)396 2519 y FM(\000)p FL(q)485 2507 y FO(\()p FN(k)f FP(\000)c FO(\()p FN(n)g FO(+)g(1\))p FN(=)p FO(2\))c([)p FN(k)s FO(])1144 2519 y FM(\000)p FL(q)1246 2507 y FO([)p FN(n)k FP(\000)g FN(k)s FO(])1489 2519 y FM(\000)p FL(q)1577 2507 y FO(\))1609 2477 y FK(1)p FL(=)p FK(2)1714 2507 y FB(.)-118 2672 y(Pr)l(o)l(of.)43 b FO(The)30 b(pro)r(of)f(essen)n(tially)d(go)r(es)j(in)g(the)h(same)e (w)n(a)n(y)h(as)g(that)h(of)f(Theo-)-118 2771 y(rem)i(33.)52 b(Let)33 b FN(q)h FO(=)e FP(\000)p FN(e)638 2741 y FL(\033)682 2771 y FO(,)i FN(\033)h FP(2)d FI(R)p FO(,)40 b(and)33 b FN(J)1238 2783 y FK(1)1275 2771 y FO(,)h FN(J)1378 2783 y FK(2)1448 2771 y FO(b)r(e)f(self-adjoin)n(t)e(op)r(erators)-118 2871 y(satisfying)20 b(\(2.35\){\(2.36\))n(.)35 b(Let)23 b(\000)g(=)f FP(f)p FO(\010\()p FN(t;)14 b(s)p FO(\))23 b FP(\021)g FN(t)1490 2841 y FK(2)1535 2871 y FP(\000)8 b FO(\()p FN(q)j FO(+)d FN(q)1801 2841 y FM(\000)p FK(1)1888 2871 y FO(\))p FN(ts)g FO(+)g FN(s)2109 2841 y FK(2)2168 2871 y FO(=)23 b(1)p FP(g)-118 2971 y FO(b)r(e)32 b(the)g(c)n (haracteristic)c(binary)i(relation)f(corresp)r(onding)g(to)j(\(2.35\).) 48 b(Con-)-118 3070 y(sidering)17 b(the)j(same)e(parameterization)d FN(s)23 b FO(=)f(sinh)13 b FN(\033)s(x)p FO(\()p FN(s)p FO(\))q FN(=)p FO(sinh)g FN(\033)26 b FP(\021)d FO([)p FN(x)p FO(\()p FN(s)p FO(\)])2226 3082 y FM(\000)p FL(q)2316 3070 y FO(,)-118 3170 y FN(x)p FO(\()p FN(s)p FO(\))h FP(2)f FI(R)p FO(,)k(w)n(e)18 b(get)g(\000)23 b(=)g FP(f)p FO(\()p FP(\000)p FO([)p FN(x)p FO(+1])982 3182 y FM(\000)p FL(q)1070 3170 y FN(;)14 b FO([)p FN(x)p FO(])1200 3182 y FM(\000)p FL(q)1289 3170 y FO(\))p FN(;)g FO(\()p FP(\000)p FO([)p FN(x)p FP(\000)p FO(1])1655 3182 y FM(\000)p FL(q)1745 3170 y FN(;)g FO([)p FN(x)p FO(])1875 3182 y FM(\000)p FL(q)1964 3170 y FO(\))23 b FP(j)g FN(x)h FP(2)f FI(R)p FP(g)p FO(.)-118 3270 y(As)i(b)r(efore,)h FN(J)319 3282 y FK(1)356 3270 y FO(,)g FN(J)451 3282 y FK(2)514 3270 y FO(satisfy)e(\(2.35\))g(if)h(and)g(only)f(if)h FN(J)1541 3282 y FK(2)1603 3270 y FO(is)g(concen)n(trated)f(on)h(\000)-118 3369 y(with)i(resp)r(ect)g(to)h FN(J)503 3381 y FK(1)540 3369 y FO(,)g(i.e.,)-73 3550 y FN(E)-12 3562 y FL(J)25 3570 y Fx(1)61 3550 y FO(\(\001\))p FN(J)240 3562 y FK(2)278 3550 y FN(E)339 3562 y FL(J)376 3570 y Fx(1)413 3550 y FO(\(\001)514 3516 y FM(0)537 3550 y FO(\))c(=)e(0)p FN(;)97 b FO(for)27 b(an)n(y)g(\001)p FN(;)14 b FO(\001)1301 3516 y FM(0)1347 3550 y FP(2)24 b Fz(B)p FO(\()p FI(R)p FO(\))q FN(;)47 b FO(\001)19 b FP(\002)f FO(\001)1928 3516 y FM(0)1970 3550 y FP(\\)h FO(\000)k(=)f FI(?)p FN(;)-118 3730 y FO(whic)n(h)k(is)h(equiv)-5 b(alen)n(t)25 b(to)-98 3911 y FN(E)-37 3923 y FL(A)13 3931 y Fx(1)49 3911 y FO(\(\001\))p FN(J)228 3923 y FK(2)267 3911 y FN(E)328 3923 y FL(A)378 3931 y Fx(1)414 3911 y FO(\(\001)515 3877 y FM(0)539 3911 y FO(\))e(=)g(0)p FN(;)96 b FO(for)28 b(an)n(y)e(\001)p FN(;)14 b FO(\001)1302 3877 y FM(0)1349 3911 y FP(2)23 b Fz(B)p FO(\()p FI(R)q FO(\))p FN(;)48 b FO(\001)19 b FP(\002)f FO(\001)1930 3877 y FM(0)1971 3911 y FP(\\)h FO(\000)2097 3877 y FM(0)2143 3911 y FO(=)k FI(?)p FN(;)p eop %%Page: 139 143 139 142 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(139)-118 96 y(where)28 b FN(J)169 108 y FK(1)232 96 y FO(=)d(sinh)13 b FN(\033)s(A)596 108 y FK(1)634 96 y FN(=)p FO(sinh)g FN(\033)32 b FO(and)d(\000)1132 66 y FM(0)1181 96 y FO(=)c FP(f)p FO(\()p FP(\000)p FO(\()p FN(x)19 b FO(+)g(1\))p FN(;)14 b(x)p FO(\))p FN(;)g FO(\()p FP(\000)p FO(\()p FN(x)21 b FP(\000)e FO(1\))p FN(;)14 b(x)p FO(\))26 b FP(j)-118 196 y FN(x)f FP(2)g FI(R)p FP(g)31 b FO(=)24 b FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))25 b FP(j)f FN(t)564 166 y FK(2)621 196 y FO(+)18 b(2)p FN(ts)h FO(+)g FN(s)957 166 y FK(2)1019 196 y FO(=)24 b(1)p FP(g)p FO(.)39 b(F)-7 b(rom)27 b(this)h(and)h(Theorem)d(8,)j(w)n (e)-118 296 y(conclude)d(that)i FN(A)465 308 y FK(1)503 296 y FO(,)f FN(J)599 308 y FK(2)664 296 y FO(satisfy)g(the)g(relation) 556 454 y FN(A)618 420 y FK(2)618 475 y(1)655 454 y FN(J)701 466 y FK(2)757 454 y FO(+)18 b(2)p FN(A)944 466 y FK(1)981 454 y FN(J)1027 466 y FK(2)1064 454 y FN(A)1126 466 y FK(1)1182 454 y FO(+)g FN(J)1311 466 y FK(2)1348 454 y FN(A)1410 420 y FK(2)1410 475 y(1)1471 454 y FO(=)23 b FN(J)1605 466 y FK(2)1642 454 y FN(:)-118 613 y FO(Let)28 b FN(E)92 625 y FK(1)154 613 y FO(=)c(\()p FP(f)p FN(A)379 625 y FK(1)417 613 y FN(;)14 b(J)500 625 y FK(2)537 613 y FP(g)k FO(+)h FN(J)727 625 y FK(2)764 613 y FO(\))p FN(=)p FO(2,)28 b FN(E)992 625 y FK(2)1054 613 y FO(=)c FP(\000)p FO(\()p FP(f)p FN(A)1344 625 y FK(1)1381 613 y FN(;)14 b(J)1464 625 y FK(2)1501 613 y FP(g)19 b(\000)f FN(J)1691 625 y FK(2)1729 613 y FO(\))p FN(=)p FO(2.)38 b(It)29 b(is)e(easy)h(to)-118 712 y(sho)n(w)f(that)g FN(E)328 724 y FK(1)389 712 y FO(=)c FN(E)543 682 y FM(\003)538 733 y FK(1)581 712 y FO(,)28 b FN(E)693 724 y FK(2)753 712 y FO(=)23 b FN(E)907 682 y FM(\003)902 733 y FK(2)945 712 y FO(,)28 b FN(J)1042 724 y FK(2)1102 712 y FO(=)23 b FN(E)1251 724 y FK(1)1307 712 y FO(+)18 b FN(E)1451 724 y FK(2)1516 712 y FO(and)251 871 y FN(A)313 883 y FK(1)351 871 y FN(E)412 883 y FK(1)473 871 y FO(=)k FP(\000)p FN(E)686 883 y FK(1)723 871 y FO(\()p FN(A)817 883 y FK(1)874 871 y FP(\000)c FN(I)7 b FO(\))p FN(;)97 b(A)1214 883 y FK(1)1251 871 y FN(E)1312 883 y FK(1)1373 871 y FO(=)22 b FP(\000)p FN(E)1586 883 y FK(1)1624 871 y FO(\()p FN(A)1718 883 y FK(1)1774 871 y FO(+)c FN(I)7 b FO(\))p FN(;)23 1048 y(E)89 1013 y FK(2)84 1068 y(1)141 1048 y FO(cosh)o(\()p FN(\033)s FO(\()p FN(A)474 1060 y FK(1)531 1048 y FO(+)18 b FN(I)7 b FO(\)\))23 b(=)g FN(E)898 1013 y FK(2)893 1068 y(2)949 1048 y FO(cosh\()p FN(\033)s FO(\()p FN(A)1283 1060 y FK(1)1339 1048 y FP(\000)c FO(1\)\))f FP(\000)1640 991 y FO(sinh)13 b FN(\033)s(A)1914 1003 y FK(1)p 1640 1029 312 4 v 1662 1105 a FO(2)h(sinh)e FN(\033)1962 1048 y(:)141 b FO(\(2.39\))-118 1234 y(Con)n(v)n(ersely)-7 b(,)32 b(an)n(y)i(represen)n(tation)d FN(A)1101 1246 y FK(1)1138 1234 y FO(,)36 b FN(E)1258 1246 y FK(1)1296 1234 y FO(,)f FN(E)1415 1246 y FK(2)1487 1234 y FO(of)41 b(\(2.39\))33 b(de\014nes)h(a)f(rep-)-118 1334 y(resen)n(tation)39 b FN(J)369 1346 y FK(1)453 1334 y FO(=)45 b(sinh)13 b FN(\033)s(A)837 1346 y FK(1)875 1334 y FN(=)h FO(sinh)f FN(\033)s FO(,)45 b FN(J)1257 1346 y FK(2)1341 1334 y FO(=)g FN(E)1512 1346 y FK(1)1578 1334 y FO(+)27 b FN(E)1731 1346 y FK(2)1810 1334 y FO(of)42 b(the)f(algebra)-118 1433 y FN(U)-61 1445 y FL(q)-25 1433 y FO(\()p FN(so)p FO(\(3\)\).)k(Moreo)n(v)n(er,)28 b(there)i(is)f(a)g(one-to-one)g (corresp)r(ondence)f(b)r(et)n(w)n(een)-118 1533 y(irreducible)c(and)j (unitary)f(equiv)-5 b(alen)n(t)26 b(represen)n(tations)e(of)k(b)r(oth)g (ob)5 b(jects.)6 1632 y(W)-7 b(e)30 b(can)f(no)n(w)g(pro)r(ceed)g (analogously)c(as)j(in)h(the)h(pro)r(of)f(of)g(Theorem)f(22)-118 1732 y(on)f(represen)n(tations)d(of)k(graded)e FN(so)p FO(\(3\).)37 b(By)27 b(the)h(same)e(argumen)n(ts,)f(w)n(e)i(can)-118 1832 y(c)n(ho)r(ose)h(a)h(basis)f(consisting)f(of)j(eigen)n(v)n(ectors) c(of)j(the)h(op)r(erator)e FN(A)2013 1844 y FK(1)2080 1832 y FO(for)h(an)n(y)-118 1931 y(irreducible)24 b(represen)n(tation)h FN(A)898 1943 y FK(1)935 1931 y FO(,)j FN(E)1047 1943 y FK(1)1084 1931 y FO(,)g FN(E)1196 1943 y FK(2)1261 1931 y FO(in)f FN(H)7 b FO(.)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)68 2090 y FN(A)130 2102 y FK(1)167 2090 y FN(e)206 2102 y FL(\025)273 2090 y FO(=)22 b FN(\025)14 b(e)461 2102 y FL(\025)505 2090 y FN(;)97 b(E)686 2102 y FK(1)723 2090 y FN(e)762 2102 y FL(\025)828 2090 y FO(=)23 b FN(a)960 2102 y FK(1)997 2090 y FO(\()p FN(\025)p FO(\))14 b FN(e)1162 2102 y FK(1)p FM(\000)p FL(\025)1291 2090 y FN(;)97 b(E)1472 2102 y FK(2)1510 2090 y FN(e)1549 2102 y FL(\025)1615 2090 y FO(=)23 b FN(a)1747 2102 y FK(2)1784 2090 y FO(\()p FN(\025)p FO(\))14 b FN(e)1949 2102 y FM(\000)p FK(1)p FM(\000)p FL(\025)2130 2090 y FN(;)-118 2264 y FO(where)36 b FN(\025)h FO(b)r(elongs)e(to)i(the)g(orbit)e(\012)k(=)f FP(f)p FN(F)1307 2221 y FK(\()p FL(k)q FK(\))1295 2287 y(1)1399 2264 y FN(F)1464 2221 y FK(\()p FL(m)p FK(\))1452 2287 y(2)1579 2264 y FO(\()p FN(\025)p FO(\))p FN(;)14 b(k)s(;)g(m)38 b FP(2)h FI(Z)2077 2276 y FK(+)2127 2264 y FP(g)d FO(and)-118 2364 y FN(F)-65 2376 y FK(1)-27 2364 y FO(\()p FN(\025)p FO(\))30 b(=)f(1)20 b FP(\000)h FN(\025)p FO(,)33 b FN(F)514 2376 y FK(2)551 2364 y FO(\()p FN(\025)p FO(\))e(=)e FP(\000)p FO(1)20 b FP(\000)g FN(\025)p FO(.)49 b(The)31 b(conditions)e(for)i FN(A)1888 2376 y FK(1)1925 2364 y FO(,)i FN(E)2042 2376 y FK(1)2079 2364 y FO(,)g FN(E)2196 2376 y FK(2)2265 2364 y FO(to)-118 2464 y(satisfy)26 b(relation)f(\(2.39\))i(are)f(the)i(follo)n(wing:)358 2622 y FN(a)402 2634 y FK(1)439 2622 y FO(\(1)18 b FP(\000)g FN(\025)p FO(\))24 b(=)p 805 2550 195 4 v 22 w FN(a)849 2634 y FK(1)887 2622 y FO(\()p FN(\025)p FO(\))q FN(;)96 b(a)1163 2634 y FK(2)1201 2622 y FO(\()p FP(\000)p FO(1)17 b FP(\000)h FN(\025)p FO(\))24 b(=)p 1632 2550 V 23 w FN(a)1676 2634 y FK(2)1713 2622 y FO(\()p FN(\025)p FO(\))q FN(;)-59 2757 y FP(j)p FN(a)8 2769 y FK(1)45 2757 y FO(\()p FN(\025)p FO(\))p FP(j)180 2723 y FK(2)232 2757 y FO(cosh)14 b FN(\033)s FO(\()p FN(\025)19 b FO(+)f(1\))23 b(=)g FP(j)p FN(a)888 2769 y FK(2)925 2757 y FO(\()p FN(\025)p FO(\))p FP(j)1060 2723 y FK(2)1112 2757 y FO(cosh)13 b FN(\033)s FO(\()p FN(\025)20 b FP(\000)e FO(1\))g FP(\000)g FO(sinh)13 b FN(\033)s(\025=)p FO(2)h(sinh)e FN(\033)n(:)-118 2915 y FO(Similarly)k(to)22 b(the)g(case)e(of)i(the)g(graded)e FN(so)p FO(\(3\),)k(w)n(e)d(see)g(that)h(the)g(last)f(relation)-118 3015 y(cannot)j(hold)g(for)h(an)n(y)f(p)r(oin)n(t)g(of)h(the)h(orbit)d (\012)i(and)g(there)g(exists)e(the)j(highest)-118 3115 y(v)n(ector)h(on)h(whic)n(h)f FN(E)547 3127 y FK(2)613 3115 y FO(acts)h(as)f(zero)g(and)h(the)h(lo)n(w)n(est)d(v)n(ector)h(on) h(whic)n(h)f(the)-118 3214 y(op)r(erator)38 b FN(E)290 3226 y FK(1)368 3214 y FO(is)h(zero.)73 b(This)39 b(implies)d(that)41 b(the)f(only)f(orbits)f(satisfying)-118 3314 y(these)33 b(conditions)d(are)i(those)g(whic)n(h)g(con)n(tain)f(0)i(and)f FP(\006)p FO(1)p FN(=)p FO(2.)51 b(Using)32 b(these)-118 3414 y(conditions)25 b(one)i(can)g(easily)e(get)j(the)g(statemen)n(t.)p 2278 3414 4 57 v 2282 3361 50 4 v 2282 3414 V 2331 3414 4 57 v -118 3575 a FQ(3.)36 b FO(Represen)n(tations)25 b(of)j FN(U)749 3587 y FL(q)785 3575 y FO(\()p FN(so)p FO(\(3\)\),)h FN(q)h FO(is)d(a)g(ro)r(ot)g(of)g(unit)n(y)-7 b(.)6 3674 y(Let)29 b FN(q)f FO(=)d FN(e)350 3644 y FL(i\033)418 3674 y FO(,)k FN(\033)f FP(2)e FO(\()p FP(\000)p FN(\031)s(;)14 b(\031)s FO(\).)41 b(If)29 b FN(q)j FO(is)27 b(a)i(ro)r(ot)f(of)g(unit) n(y)-7 b(,)29 b(then)g FN(\033)f FO(=)d FN(\031)s(k)s(=n)p FO(,)-118 3849 y(where)i FN(k)s(=n)g FO(is)f(an)i(irreducible)c (fraction.)35 b(Let)28 b FN(s)23 b FO(=)1534 3707 y Fy(\()1601 3792 y FN(n;)124 b(k)86 b FO(is)27 b(ev)n(en)o FN(;)1601 3912 y FO(2)p FN(n;)82 b(k)k FO(is)27 b(o)r(dd)p FN(:)p eop %%Page: 140 144 140 143 bop -118 -137 a FO(140)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(In)k(what)e(follo)n(ws,)e(w)n(e)j(denote)g(b)n (y)f FN(I)1150 108 y FK(1)1215 96 y FO(and)h FN(I)1412 108 y FK(2)1477 96 y FO(the)g(iden)n(tit)n(y)e(1)17 b FP(\002)f FO(1,)27 b(2)16 b FP(\002)h FO(2)-118 196 y(matrices)25 b(resp)r(ectiv)n(ely)-7 b(.)-118 339 y FQ(Theorem)30 b(35.)41 b FB(L)l(et)33 b FN(\033)h FO(=)d FN(\031)s(k)s(=n)p FB(,)k FN(\033)f FP(6)p FO(=)c FN(\031)s(l)r FB(.)51 b(A)n(ny)33 b(irr)l(e)l(ducible)j(r)l(epr)l(esenta-)-118 439 y(tions)30 b(of)g FN(U)242 451 y FL(q)279 439 y FO(\()p FN(so)p FO(\(3\)\))g FB(is)g(unitarily)h(e)l(quivalent)f(to)g(one)g(of) g(the)g(fol)t(lowing)7 b FO(:)6 539 y(1)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 509 y FL(s)393 539 y FB(,)31 b FN(J)495 551 y FK(1)532 539 y FN(e)571 551 y FL(m)657 539 y FO(=)22 b([)p FN(a)d FO(+)f FN(m)p FO(])1009 551 y FL(q)1059 539 y FN(e)1098 551 y FL(m)1161 539 y FB(,)161 825 y FN(J)207 837 y FK(2)245 825 y FN(e)284 837 y FL(m)370 825 y FO(=)457 630 y Fy(8)457 705 y(>)457 730 y(<)457 879 y(>)457 904 y(:)531 709 y FN(\013)584 721 y FK(0)635 709 y FN(e)674 721 y FK(1)729 709 y FO(+)h FN(e)852 679 y FL(i\036)919 709 y FN(\013)972 721 y FL(s)p FM(\000)p FK(1)1106 709 y FN(e)1145 721 y FL(s)p FM(\000)p FK(1)1265 709 y FN(;)220 b(m)23 b FO(=)g(0)p FN(;)531 829 y(\013)584 841 y FL(m)661 829 y FN(e)700 841 y FL(m)p FK(+1)865 829 y FO(+)18 b FN(\013)1001 841 y FL(m)p FM(\000)p FK(1)1163 829 y FN(e)1202 841 y FL(m)p FM(\000)p FK(1)1350 829 y FN(;)135 b FO(1)23 b FP(\024)g FN(m)f FP(\024)h FN(s)18 b FP(\000)g FO(2)p FN(;)531 948 y(\013)584 960 y FL(s)p FM(\000)p FK(2)718 948 y FN(e)757 960 y FL(s)p FM(\000)p FK(2)896 948 y FO(+)g FN(e)1018 918 y FM(\000)p FL(i\036)1137 948 y FN(\013)1190 960 y FL(s)p FM(\000)p FK(1)1324 948 y FN(e)1363 960 y FK(0)1400 948 y FN(;)85 b(m)23 b FO(=)g FN(s)18 b FP(\000)g FO(1)p FN(;)-118 1105 y FB(wher)l(e)153 1288 y FN(\013)206 1300 y FL(m)292 1288 y FO(=)380 1171 y Fy(\022)626 1232 y FN(b)c FO(cos)f FN(a\033)k FO(cos)o(\()p FN(a)i FO(+)f(1\))p FN(\033)p 451 1269 1046 4 v 451 1345 a FO(cos)o(\()p FN(a)h FO(+)f FN(m)g FO(+)g(1\))p FN(\033)f FO(cos\()p FN(a)h FO(+)g FN(m)p FO(\))p FN(\033)458 1544 y FP(\000)773 1487 y FO(sin)13 b FN(m\033)k FO(sin)o(\(2)p FN(a)h FO(+)g FN(m)h FO(+)f(1\))p FN(\033)p 551 1524 1319 4 v 551 1607 a FO(4)c(sin)708 1572 y FK(2)759 1607 y FN(\033)j FO(cos)o(\()p FN(a)i FO(+)f FN(m)g FO(+)g(1\))p FN(\033)f FO(cos\()p FN(a)h FO(+)g FN(m)p FO(\))p FN(\033)1879 1426 y Fy(\023)1940 1444 y FK(1)p FL(=)p FK(2)2044 1544 y FN(;)-118 1757 y FB(the)35 b(p)l(air)g FO(\()p FN(a;)14 b(b)p FO(\))35 b FB(b)l(elongs)g(to)g(the)g(set)f FP(f)p FO(\()p FN(a;)14 b(b)p FO(\))32 b FP(2)j FN(M)30 b FP(\002)22 b FI(R)1684 1726 y FK(+)1777 1757 y FP(j)32 b FN(\013)1885 1769 y FL(m)1980 1757 y FN(>)g FO(0)p FN(;)27 b(m)32 b FO(=)-118 1856 y(0)p FN(;)14 b(:)g(:)g(:)f(;)h(s)24 b FP(\000)g FO(1)p FP(g)p FB(,)39 b FN(\036)f FP(2)f FO([0)p FN(;)14 b FO(2)p FN(\031)s FO(\))p FB(,)40 b(and)47 b FN(\033)s(M)f FO(=)37 b([)p FP(\000)p FN(\031)s(=)p FO(2)p FN(;)14 b(\031)s(=)p FO(2])22 b FP(n)i(f)p FO(\()p FN(\031)s FO(\(2)p FN(l)h FO(+)f(1\))g(+)-118 1956 y FN(m\033)s FO(\))p FN(=)p FO(2)f FP(j)g FN(l)r(;)14 b(m)22 b FP(2)h FI(Z)p FP(g)o FO(;)6 2075 y(2)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 2045 y FL(n)403 2075 y FB(,)30 b FN(k)j FB(is)d(o)l(dd,)h FN(J)847 2087 y FK(1)884 2075 y FN(e)923 2087 y FL(m)1009 2075 y FO(=)23 b([)p FN(a)18 b FO(+)g FN(m)p FO(])1361 2087 y FL(q)1398 2075 y FN(e)1437 2087 y FL(m)1500 2075 y FB(,)181 2362 y FN(J)227 2374 y FK(2)264 2362 y FN(e)303 2374 y FL(m)389 2362 y FO(=)477 2166 y Fy(8)477 2241 y(>)477 2266 y(<)477 2415 y(>)477 2440 y(:)551 2245 y FO(\()p FP(\000)p FO(1\))722 2215 y FL(i)749 2245 y FN(\025)c(e)850 2257 y FK(1)906 2245 y FO(+)k FN(\013)1042 2257 y FK(1)1093 2245 y FN(e)1132 2257 y FK(2)1169 2245 y FN(;)286 b(m)23 b FO(=)f(1)p FN(;)551 2365 y(\013)604 2377 y FL(m)681 2365 y FN(e)720 2377 y FL(m)p FK(+1)885 2365 y FO(+)c FN(\013)1021 2377 y FL(m)p FM(\000)p FK(1)1183 2365 y FN(e)1222 2377 y FL(m)p FM(\000)p FK(1)1370 2365 y FN(;)85 b FO(2)22 b FP(\024)h FN(m)g FP(\024)g FN(n)18 b FP(\000)g FO(1)p FN(;)551 2484 y(\013)604 2496 y FL(n)p FM(\000)p FK(1)748 2484 y FN(e)787 2496 y FL(n)p FM(\000)p FK(1)935 2484 y FO(+)g(\()p FP(\000)p FO(1\))1189 2454 y FL(j)1224 2484 y FN(\025)c(e)1325 2496 y FL(n)1370 2484 y FN(;)85 b(m)23 b FO(=)f FN(n;)-118 2641 y FB(wher)l(e)240 2832 y FN(\013)293 2844 y FL(m)379 2832 y FO(=)467 2715 y Fy(\022)1059 2776 y FO(sin)1161 2741 y FK(2)1212 2776 y FN(m\033)p 538 2813 V 538 2895 a FO(4)14 b(sin)695 2861 y FK(2)746 2895 y FN(\033)j FO(sin)o(\()p FN(m)i FP(\000)f FO(1)p FN(=)p FO(2\))p FN(\033)e FO(sin)o(\()p FN(m)j FO(+)f(1)p FN(=)p FO(2\))p FN(\033)544 3093 y FP(\000)g FN(\025)675 3059 y FK(2)1091 3037 y FO(sin)1193 3002 y FK(2)1230 3037 y FO(\()p FN(\033)s(=)p FO(2\))p 737 3074 1046 4 v 737 3150 a(sin)o(\()p FN(m)g FP(\000)g FO(1)p FN(=)p FO(2\))p FN(\033)e FO(sin)o(\()p FN(m)j FO(+)f(1)p FN(=)p FO(2\))p FN(\033)1792 2976 y Fy(\023)1853 2993 y FK(1)p FL(=)p FK(2)1958 3093 y FN(;)-118 3301 y(a)32 b FO(=)g FP(\000)p FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))22 b FP(\000)f FO(1)p FN(=)p FO(2)p FB(,)35 b FN(\025)g FB(b)l(elongs)h(to)f(the)f(set)h FP(f)p FN(\025)d FP(2)h FI(R)1681 3271 y FK(+)1774 3301 y FP(j)i FN(\013)1885 3313 y FL(m)1980 3301 y FN(>)d FO(0)p FN(;)27 b(m)32 b FO(=)-118 3400 y(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)k FP(\000)g FO(1)p FP(g)p FB(,)29 b FN(i)p FB(,)h FN(j)e FO(=)23 b(0)p FB(,)29 b FO(1;)6 3519 y(3)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 3489 y FK(2)p FL(n)436 3519 y FB(,)30 b FN(k)j FB(is)d(o)l(dd,)617 3796 y FN(J)663 3808 y FK(1)723 3796 y FO(=)811 3604 y Fy(0)811 3751 y(B)811 3804 y(@)883 3668 y FN(\025)931 3680 y FK(1)969 3668 y FN(I)1005 3680 y FK(2)1376 3668 y FO(0)1130 3765 y FB(.)1165 3790 y(.)1200 3815 y(.)942 3923 y FO(0)329 b FN(\025)1361 3935 y FL(n)1407 3923 y FN(I)1443 3935 y FK(2)1480 3604 y Fy(1)1480 3751 y(C)1480 3804 y(A)1567 3796 y FN(;)p eop %%Page: 141 145 141 144 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(141)370 289 y FN(J)416 301 y FK(2)476 289 y FO(=)564 22 y Fy(0)564 168 y(B)564 218 y(B)564 268 y(B)564 318 y(B)564 371 y(@)676 83 y FN(Y)724 95 y FK(1)883 83 y FN(\013)936 95 y FK(1)974 83 y FN(I)1010 95 y FK(2)636 238 y FN(\013)689 250 y FK(1)727 238 y FN(I)763 250 y FK(2)945 238 y FO(0)1212 180 y FB(.)1246 205 y(.)1281 230 y(.)918 335 y(.)953 360 y(.)987 385 y(.)1212 335 y(.)1246 360 y(.)1281 385 y(.)1470 393 y FN(\013)1523 405 y FL(n)p FM(\000)p FK(1)1654 393 y FN(I)1690 405 y FK(2)1130 493 y FN(\013)1183 505 y FL(n)p FM(\000)p FK(1)1314 493 y FN(I)1350 505 y FK(2)1556 493 y FN(Y)1604 505 y FK(2)1728 22 y Fy(1)1728 168 y(C)1728 218 y(C)1728 268 y(C)1728 318 y(C)1728 371 y(A)1814 289 y FN(;)-118 662 y FB(wher)l(e)30 b FN(\025)164 674 y FL(m)251 662 y FO(=)23 b([)p FN(a)18 b FO(+)g FN(m)p FO(])603 674 y FL(q)640 662 y FB(,)30 b FN(a)23 b FO(=)f FP(\000)p FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))d FP(\000)f FO(1)p FN(=)p FO(2)p FB(,)292 884 y FN(Y)340 896 y FK(1)401 884 y FO(=)488 767 y Fy(\022)549 833 y FN(\025)120 b FO(0)553 933 y(0)86 b FP(\000)p FN(\025)794 767 y Fy(\023)869 884 y FN(;)99 b(Y)1039 896 y FK(2)1099 884 y FO(=)23 b FN(\025)1249 767 y Fy(\022)1310 833 y FO(cos)13 b FN(')127 b FO(sin)13 b FN(')1315 933 y FO(sin)g FN(')88 b FP(\000)14 b FO(cos)e FN(')1830 767 y Fy(\023)1905 884 y FN(;)-118 1106 y(\013)-65 1118 y FL(m)26 1106 y FB(is)29 b(the)f(same)h(as)g(in)f FO(2)p FB(,)g FN(\025)h FB(b)l(elongs)g(to)f(the)g(set)g FP(f)p FN(\025)23 b FP(2)h FI(R)1729 1076 y FK(+)1813 1106 y FP(j)f FN(\013)1912 1118 y FL(m)1998 1106 y FN(>)g FO(0)p FN(;)k(m)c FO(=)-118 1206 y(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)k FP(\000)g FO(1)p FP(g)p FB(,)29 b FN(')24 b FP(2)f FO(\(0)p FN(;)14 b(\031)s FO(\);)6 1328 y(4)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 1298 y FL(n)p FK(+1)487 1328 y FB(,)30 b FN(k)j FB(is)d(o)l(dd,)h FN(J)931 1340 y FK(1)969 1328 y FN(e)1008 1340 y FL(m)1093 1328 y FO(=)23 b([)p FN(a)18 b FO(+)g FN(m)p FO(])1445 1340 y FL(q)1482 1328 y FN(e)1521 1340 y FL(m)1584 1328 y FB(,)267 1629 y FN(J)313 1641 y FK(2)350 1629 y FN(e)389 1641 y FL(m)475 1629 y FO(=)562 1434 y Fy(8)562 1509 y(>)562 1534 y(<)562 1683 y(>)562 1708 y(:)636 1513 y FN(\013)689 1525 y FK(1)726 1513 y FN(e)765 1525 y FK(2)802 1513 y FN(;)710 b(m)23 b FO(=)g(1)p FN(;)636 1632 y(\013)689 1644 y FL(m)752 1632 y FN(e)791 1644 y FL(m)p FK(+1)956 1632 y FO(+)18 b FN(\013)1092 1644 y FL(m)p FM(\000)p FK(1)1241 1632 y FN(e)1280 1644 y FL(m)p FM(\000)p FK(1)1427 1632 y FN(;)85 b FO(2)23 b FP(\024)f FN(m)h FP(\024)g FN(n;)636 1752 y(\013)689 1764 y FL(n)734 1752 y FN(e)773 1764 y FL(n)818 1752 y FN(;)694 b(m)23 b FO(=)g FN(n)18 b FO(+)g(1)p FN(;)-118 1950 y FB(wher)l(e)24 b FN(a)f FO(=)g FP(\000)p FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))5 b FP(\000)g FO(1)p FB(,)24 b FN(\013)797 1962 y FK(1)857 1950 y FO(=)f FN(\013)998 1962 y FL(n)1066 1950 y FO(=)1154 1881 y FP(p)p 1223 1881 42 4 v 69 x FO(2)14 b FP(j)p FN(q)8 b FP(\000)d FN(q)1457 1920 y FM(\000)p FK(1)1546 1950 y FP(j)1569 1920 y FM(\000)p FK(1)1644 1950 y FB(,)25 b FN(\013)1747 1962 y FL(m)1833 1950 y FO(=)e FP(j)p FN(q)f FP(\000)c FN(q)2126 1920 y FM(\000)p FK(1)2215 1950 y FP(j)2238 1903 y FM(\000)p FK(1)2313 1950 y FB(,)-118 2050 y FN(m)23 b FO(=)f(2)p FB(,)30 b FN(:)14 b(:)g(:)28 b FB(,)i FN(n)18 b FP(\000)g FO(1;)6 2171 y(5)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 2141 y FK(2)p FL(n)436 2171 y FB(,)30 b FN(k)j FB(is)d(o)l(dd,)-11 2540 y FN(J)35 2552 y FK(1)95 2540 y FO(=)183 2349 y Fy(0)183 2495 y(B)183 2548 y(@)256 2412 y FN(\025)304 2424 y FK(1)341 2412 y FN(I)377 2424 y FK(1)790 2412 y FO(0)503 2509 y FB(.)537 2534 y(.)572 2559 y(.)315 2667 y FO(0)328 b FN(\025)733 2679 y FL(n)p FK(+1)863 2667 y FN(I)899 2679 y FK(1)937 2349 y Fy(1)937 2495 y(C)937 2548 y(A)1023 2540 y FN(;)99 b(J)1191 2552 y FK(2)1251 2540 y FO(=)1339 2274 y Fy(0)1339 2420 y(B)1339 2470 y(B)1339 2520 y(B)1339 2569 y(B)1339 2623 y(@)1444 2335 y FO(0)115 b FN(X)1677 2305 y FM(\003)1670 2355 y FK(1)1412 2490 y FN(X)1481 2502 y FK(1)1637 2490 y FO(0)1807 2432 y FB(.)1841 2456 y(.)1876 2482 y(.)1610 2586 y(.)1645 2611 y(.)1679 2637 y(.)1807 2586 y(.)1841 2611 y(.)1876 2637 y(.)1994 2645 y FN(X)2070 2614 y FM(\003)2063 2665 y FL(n)1797 2744 y FN(X)1866 2756 y FL(n)2030 2744 y FO(0)2108 2274 y Fy(1)2108 2420 y(C)2108 2470 y(C)2108 2520 y(C)2108 2569 y(C)2108 2623 y(A)2194 2540 y FN(;)-118 2914 y FB(wher)l(e)30 b FN(\025)164 2926 y FL(m)251 2914 y FO(=)23 b([)p FN(m)18 b FP(\000)g FO(1)g FP(\000)g FN(\031)s(=)p FO(\(2)p FN(\033)s FO(])918 2926 y FL(q)955 2914 y FB(,)51 3141 y FN(X)120 3153 y FK(1)180 3141 y FO(=)267 3024 y Fy(\022)328 3022 y FP(p)p 398 3022 V 398 3091 a FO(2)13 b FP(j)p FN(q)21 b FP(\000)d FN(q)657 3060 y FM(\000)p FK(1)747 3091 y FP(j)770 3060 y FM(\000)p FK(1)873 3091 y FO(cos)13 b FN(')333 3127 y FP(p)p 402 3127 V 69 x FO(2)h FP(j)p FN(q)21 b FP(\000)d FN(q)662 3166 y FM(\000)p FK(1)751 3196 y FP(j)774 3166 y FM(\000)p FK(1)877 3196 y FO(sin)13 b FN(')1052 3024 y Fy(\023)1127 3141 y FN(;)99 b(X)1318 3153 y FL(n)1386 3141 y FO(=)22 b(\()1505 3068 y FP(p)p 1575 3068 V 1575 3141 a FO(2)13 b FP(j)p FN(q)22 b FP(\000)c FN(q)1835 3107 y FM(\000)p FK(1)1924 3141 y FP(j)1947 3107 y FM(\000)p FK(1)2036 3141 y FN(;)c FO(0\))p FN(;)-118 3371 y(X)-49 3383 y FL(m)37 3371 y FO(=)22 b FP(j)p FN(q)g FP(\000)c FN(q)329 3341 y FM(\000)p FK(1)418 3371 y FP(j)441 3341 y FM(\000)p FK(1)530 3371 y FN(I)566 3383 y FK(2)604 3371 y FB(,)30 b FN(m)23 b FO(=)g(2)p FB(,)30 b FN(:)14 b(:)g(:)27 b FB(,)j FN(n)19 b FP(\000)f FO(1)p FB(,)29 b FN(')24 b FP(2)f FO(\(0)p FN(;)14 b(\031)s(=)p FO(2\);)6 3493 y(6)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 3462 y FK(\()p FL(n)p FK(+1\))p FL(=)p FK(2)606 3493 y FB(,)30 b FN(k)j FB(is)d(even,)g FN(J)1088 3505 y FK(1)1126 3493 y FN(e)1165 3505 y FL(m)1251 3493 y FO(=)22 b(\()p FP(\000)p FO(1\))1509 3462 y FL(j)1544 3493 y FO([)p FN(a)c FO(+)g FN(m)p FO(])1808 3505 y FL(q)1859 3493 y FN(e)1898 3505 y FL(m)1960 3493 y FB(,)36 3794 y FN(J)82 3806 y FK(2)119 3794 y FN(e)158 3806 y FL(m)244 3794 y FO(=)332 3599 y Fy(8)332 3674 y(>)332 3699 y(<)332 3848 y(>)332 3873 y(:)406 3678 y FN(\013)459 3690 y FK(1)510 3678 y FN(e)549 3690 y FK(2)604 3678 y FO(+)g(\()p FP(\000)p FO(1\))858 3647 y FL(i)885 3678 y FP(j)p FN(q)k FP(\000)c FN(q)1090 3647 y FM(\000)p FK(1)1179 3678 y FP(j)1202 3647 y FM(\000)p FK(1)1291 3678 y FN(e)1330 3690 y FK(1)1367 3678 y FN(;)85 b(m)23 b FO(=)g(1)p FN(;)406 3797 y(\013)459 3809 y FL(m)536 3797 y FN(e)575 3809 y FL(m)p FK(+1)740 3797 y FO(+)18 b FN(\013)876 3809 y FL(m)p FM(\000)p FK(1)1038 3797 y FN(e)1077 3809 y FL(m)p FM(\000)p FK(1)1225 3797 y FN(;)227 b FO(2)23 b FP(\024)f FN(m)h FP(\024)g FO(\()p FN(n)c FP(\000)f FO(1\))p FN(=)p FO(2)p FN(;)406 3917 y(\013)459 3932 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)722 3917 y FN(e)761 3932 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)1010 3917 y FN(;)442 b(m)23 b FO(=)g(\()p FN(n)18 b FO(+)g(1\))p FN(=)p FO(2)p FN(;)p eop %%Page: 142 146 142 145 bop -118 -137 a FO(142)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 98 y FB(wher)l(e)32 b FN(a)25 b FO(=)g FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))19 b FP(\000)g FO(1)p FN(=)p FO(2)p FB(,)31 b FN(\013)863 110 y FL(m)951 98 y FO(=)25 b FP(j)p FN(q)c FP(\000)e FN(q)1246 68 y FM(\000)p FK(1)1335 98 y FP(j)1358 51 y FM(\000)p FK(1)1447 98 y FB(,)32 b FN(m)25 b FO(=)g(1)p FB(,)31 b FN(:)14 b(:)g(:)27 b FB(,)32 b FO(\()p FN(n)19 b FP(\000)g FO(3\))p FN(=)p FO(2)p FB(,)-118 198 y FN(\013)-65 213 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)207 198 y FO(=)295 129 y FP(p)p 364 129 42 4 v 69 x FO(2)13 b FP(j)p FN(q)22 b FP(\000)c FN(q)624 167 y FM(\000)p FK(1)713 198 y FP(j)736 167 y FM(\000)p FK(1)825 198 y FB(,)31 b FN(i)22 b FO(=)h(0)p FB(,)29 b FO(1)p FB(,)h FN(j)e FO(=)23 b(0)p FB(,)29 b FO(1;)6 321 y(7)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 291 y FL(n)403 321 y FB(,)30 b FN(k)j FB(is)d(even,)-4 702 y FN(J)42 714 y FK(1)102 702 y FO(=)190 510 y Fy(0)190 656 y(B)190 709 y(@)263 574 y FN(\025)311 586 y FK(1)348 574 y FN(I)384 586 y FK(2)794 574 y FO(0)510 670 y FB(.)545 695 y(.)579 720 y(.)322 828 y FO(0)328 b FN(\025)740 840 y FL(p)p FK(+1)863 828 y FN(I)899 840 y FK(1)937 510 y Fy(1)937 656 y(C)937 709 y(A)1023 702 y FN(;)99 b(J)1191 714 y FK(2)1251 702 y FO(=)1339 435 y Fy(0)1339 581 y(B)1339 631 y(B)1339 681 y(B)1339 731 y(B)1339 784 y(@)1431 495 y FN(Y)122 b(X)1677 465 y FM(\003)1670 516 y FK(1)1412 650 y FN(X)1481 662 y FK(1)1637 650 y FO(0)1803 592 y FB(.)1838 617 y(.)1873 642 y(.)1610 747 y(.)1645 772 y(.)1679 797 y(.)1803 747 y(.)1838 772 y(.)1873 797 y(.)1987 805 y FN(X)2063 775 y FM(\003)2056 825 y FL(p)1797 907 y FN(X)1866 919 y FL(p)2023 907 y FO(0)2101 435 y Fy(1)2101 581 y(C)2101 631 y(C)2101 681 y(C)2101 731 y(C)2101 784 y(A)2187 702 y FN(;)-118 1093 y FB(wher)l(e)30 b FN(p)23 b FO(=)g(\()p FN(n)18 b FP(\000)g FO(1\))p FN(=)p FO(2)p FB(,)29 b FN(\025)712 1105 y FL(m)799 1093 y FO(=)23 b(\()p FP(\000)p FO(1\))1058 1063 y FL(j)1092 1093 y FO([)p FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))c FP(\000)f FO(1)p FN(=)p FO(2)f(+)h FN(m)p FO(])1787 1105 y FL(q)1824 1093 y FB(,)-34 1325 y FN(Y)41 b FO(=)23 b FP(j)p FN(q)e FP(\000)d FN(q)347 1291 y FM(\000)p FK(1)437 1325 y FP(j)460 1291 y FM(\000)p FK(1)563 1208 y Fy(\022)624 1274 y FO(cos)13 b FN(')127 b FO(sin)13 b FN(')628 1374 y FO(sin)g FN(')88 b FP(\000)14 b FO(cos)f FN(')1144 1208 y Fy(\023)1219 1325 y FN(;)99 b(X)1410 1337 y FL(p)1471 1325 y FO(=)22 b(\()1590 1252 y FP(p)p 1660 1252 V 1660 1325 a FO(2)13 b FP(j)p FN(q)22 b FP(\000)c FN(q)1920 1291 y FM(\000)p FK(1)2009 1325 y FP(j)2032 1291 y FM(\000)p FK(1)2121 1325 y FN(;)c FO(0\))p FN(;)-118 1559 y(X)-49 1571 y FL(m)37 1559 y FO(=)22 b FP(j)p FN(q)g FP(\000)c FN(q)329 1529 y FM(\000)p FK(1)418 1559 y FP(j)441 1529 y FM(\000)p FK(1)530 1559 y FN(I)566 1571 y FK(2)604 1559 y FB(,)30 b FN(m)23 b FO(=)g(1)p FB(,)30 b FN(:)14 b(:)g(:)27 b FB(,)j FN(p)18 b FP(\000)g FO(1)p FB(,)30 b FN(')24 b FP(2)f FO(\(0)p FN(;)14 b(\031)s FO(\))p FB(,)30 b FN(j)e FO(=)23 b(0)p FB(,)30 b FO(1;)6 1683 y(8)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 1653 y FL(p)396 1683 y FB(,)31 b FN(p)22 b(<)h(n)30 b FB(if)g FN(k)j FB(is)d(o)l(dd,)h FN(p)23 b(<)f FO(\()p FN(n)d FO(+)f(1\))p FN(=)p FO(2)29 b FB(if)h FN(k)j FB(is)d(even,)81 1862 y FN(J)127 1874 y FK(1)164 1862 y FN(e)203 1874 y FL(m)289 1862 y FO(=)23 b(\()p FP(\000)p FO(1\))548 1828 y FL(i)589 1862 y FO([)p FN(a)18 b FO(+)g FN(m)p FO(])853 1874 y FL(q)904 1862 y FN(e)943 1874 y FL(m)1005 1862 y FN(;)81 2129 y(J)127 2141 y FK(2)164 2129 y FN(e)203 2141 y FL(m)289 2129 y FO(=)377 1934 y Fy(8)377 2009 y(>)377 2034 y(<)377 2183 y(>)377 2208 y(:)450 2009 y FN(\013)503 2021 y FK(1)555 2009 y FN(e)594 2021 y FK(2)649 2009 y FO(+)g(\()p FP(\000)p FO(1\))903 1979 y FL(j)1079 1972 y FK(sin)11 b FL(p\033)p 948 1990 431 4 v 948 2038 a FK(2)g(sin\()p FL(\033)r(=)p FK(2\))h(sin)f FL(\033)1402 2009 y FN(e)1441 2021 y FK(1)1478 2009 y FN(;)85 b(m)23 b FO(=)g(1)p FN(;)450 2136 y(\013)503 2148 y FL(m)580 2136 y FN(e)619 2148 y FL(m)p FK(+1)785 2136 y FO(+)18 b FN(\013)921 2148 y FL(m)p FM(\000)p FK(1)1083 2136 y FN(e)1122 2148 y FL(m)p FM(\000)p FK(1)1269 2136 y FN(;)294 b FO(2)23 b FP(\024)f FN(m)h FP(\024)g FN(p)18 b FP(\000)g FO(1)p FN(;)450 2256 y(\013)503 2268 y FL(p)p FM(\000)p FK(1)641 2256 y FN(e)680 2268 y FL(p)p FM(\000)p FK(1)803 2256 y FN(;)760 b(m)23 b FO(=)g FN(p;)-118 2437 y FB(wher)l(e)203 2668 y FN(\013)256 2680 y FL(m)342 2668 y FO(=)430 2551 y Fy(\022)735 2612 y FO(sin)o(\()p FN(m)c FP(\000)f FN(l)r FO(\))p FN(\033)f FO(sin)o(\()p FN(m)h FO(+)g FN(l)r FO(\))p FN(\033)p 501 2649 1319 4 v 501 2732 a FO(4)c(sin)658 2697 y FK(2)709 2732 y FN(\033)k FO(sin)n(\()p FN(m)h FP(\000)f FO(1)p FN(=)p FO(2\))p FN(\033)e FO(sin)o(\()p FN(m)j FO(+)f(1)p FN(=)p FO(2\))p FN(\033)1829 2551 y Fy(\023)1890 2569 y FK(1)p FL(=)p FK(2)1994 2668 y FN(;)-118 2903 y(a)23 b FO(=)f FN(\031)s(=)p FO(\(2)p FN(\033)s FO(\))d FP(\000)f FO(1)p FN(=)p FO(2)p FB(,)29 b FN(p)23 b FP(2)g(f)p FN(p)g FP(2)g FI(Z)955 2915 y FK(+)1027 2903 y FP(j)h FN(\013)1127 2915 y FL(m)1213 2903 y FN(>)e FO(0)p FN(;)28 b FO(1)22 b FP(\024)h FN(m)g(<)g(p)p FP(g)p FB(,)29 b FN(i)p FB(,)h FN(j)e FO(=)22 b(0)p FB(,)30 b FO(1;)6 3026 y(9)p FB(.)39 b FN(H)29 b FO(=)23 b FI(C)352 2996 y FL(p)396 3026 y FB(,)31 b FN(J)498 3038 y FK(1)535 3026 y FN(e)574 3038 y FL(m)660 3026 y FO(=)22 b([)p FN(a)d FO(+)f FN(m)p FO(])1012 3038 y FL(q)1062 3026 y FN(e)1101 3038 y FL(m)1164 3026 y FB(,)199 3338 y FN(J)245 3350 y FK(2)282 3338 y FN(e)321 3350 y FL(m)407 3338 y FO(=)495 3143 y Fy(8)495 3218 y(>)495 3243 y(<)495 3392 y(>)495 3417 y(:)569 3222 y FN(\013)622 3234 y FK(1)659 3222 y FN(e)698 3234 y FK(2)735 3222 y FN(;)710 b(m)23 b FO(=)f(1)p FN(;)569 3341 y(\013)622 3353 y FL(m)685 3341 y FN(e)724 3353 y FL(m)p FK(+1)889 3341 y FO(+)c FN(\013)1025 3353 y FL(m)p FM(\000)p FK(1)1173 3341 y FN(e)1212 3353 y FL(m)p FM(\000)p FK(1)1360 3341 y FN(;)85 b FO(2)22 b FP(\024)h FN(m)g FP(\024)g FN(p)18 b FP(\000)g FO(1)p FN(;)569 3461 y(\013)622 3473 y FL(l)p FM(\000)p FK(1)732 3461 y FN(e)771 3473 y FL(l)p FM(\000)p FK(1)881 3461 y FN(;)564 b(m)23 b FO(=)f FN(p;)-118 3642 y FB(wher)l(e)196 3874 y FN(\013)249 3886 y FK(1)310 3874 y FO(=)397 3757 y Fy(\022)459 3874 y FP(\000)672 3818 y FO(sin)o(\()p FN(a)c FO(+)g(1\))p FN(\033)p 534 3855 681 4 v 534 3931 a FO(2)c(sin)d FN(\033)18 b FO(cos)o(\()p FN(a)g FO(+)g(2\))p FN(\033)s FO(\))1224 3757 y Fy(\023)1285 3774 y FK(1)p FL(=)p FK(2)1389 3874 y FN(;)p eop %%Page: 143 147 143 146 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(143)171 157 y FN(\013)224 169 y FL(m)310 157 y FO(=)397 40 y Fy(\022)459 157 y FP(\000)756 101 y FO(sin)13 b FN(m\033)k FO(sin)o(\(2)p FN(a)h FO(+)g FN(m)g FO(+)g(1\))p FN(\033)p 534 138 1319 4 v 534 220 a FO(4)c(sin)690 185 y FK(2)741 220 y FN(\033)k FO(cos)o(\()p FN(a)h FO(+)f FN(m)p FO(\))p FN(\033)f FO(cos)o(\()p FN(a)i FO(+)f FN(m)g FO(+)g(1\))p FN(\033)1861 40 y Fy(\023)1922 57 y FK(1)p FL(=)p FK(2)2027 157 y FN(;)-118 394 y(m)35 b FP(6)p FO(=)f(1)p FB(,)k(the)e(p)l(air)i FO(\()p FN(a;)14 b(p)p FO(\))36 b FB(b)l(elongs)h(to)f(the)h(set)f FP(f)p FO(\()p FN(a;)14 b(p)p FO(\))34 b FP(2)i FI(R)29 b FP(\002)22 b FI(Z)2003 406 y FK(+)2087 394 y FP(j)35 b FN(\033)s(a)g FP(6)p FO(=)-108 461 y FL(\031)p -108 475 41 4 v -104 522 a FK(2)-48 493 y FP(\000)9 b FN(l)r(\033)j FO(+)d FN(\031)s(r)r FB(,)26 b FO([)p FN(p)p FO(])414 505 y FL(q)451 493 y FO([2)p FN(a)9 b FO(+)g FN(p)g FO(+)g(1])833 505 y FL(q)891 493 y FO(=)22 b(0)p FB(,)k FO([)p FN(a)9 b FO(+)g FN(m)p FO(])1317 505 y FL(q)1377 493 y FP(6)p FO(=)22 b([)p FN(a)9 b FO(+)g FN(n)p FO(])1687 505 y FL(q)1723 493 y FB(,)27 b FO(1)c FP(\024)f FN(m)h(<)g(n)g FP(\024)f FN(p)p FB(,)-118 593 y FN(\013)-65 605 y FL(m)21 593 y FN(>)h FO(0)p FN(;)14 b(m)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(p)k FP(\000)g FO(1)p FN(;)c(r)25 b FP(2)e FI(Z)p FP(g)o FB(.)6 717 y FO(10)p FB(.)38 b FN(H)30 b FO(=)23 b FI(C)394 687 y FK(1)437 717 y FB(,)30 b FN(J)538 729 y FK(1)598 717 y FO(=)23 b(0)p FB(,)30 b FN(J)829 729 y FK(2)889 717 y FO(=)23 b(0)p FB(.)-118 883 y(Pr)l(o)l(of.)43 b FO(Let)24 b FN(J)333 895 y FK(1)393 883 y FO(=)f FN(J)535 853 y FM(\003)527 904 y FK(1)573 883 y FO(,)h FN(J)666 895 y FK(2)727 883 y FO(=)e FN(J)868 853 y FM(\003)860 904 y FK(2)930 883 y FO(b)r(e)i(op)r(erators)e(on)h (a)g(Hilb)r(ert)g(space)f(satisfy-)-118 983 y(ing)h(\(2.35\){\(2.36\))e (and)j(\000)f(=)g FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(j)g FO(\010\()p FN(t;)14 b(s)p FO(\))23 b FP(\021)g FN(t)1479 952 y FK(2)1527 983 y FP(\000)11 b FO(\()p FN(q)j FO(+)d FN(q)1802 952 y FM(\000)p FK(1)1891 983 y FO(\))j FN(ts)d FO(+)g FN(s)2132 952 y FK(2)2180 983 y FP(\000)g FO(1)p FP(g)-118 1082 y FO(the)26 b(c)n(haracteristic)c (binary)i(relation)f(corresp)r(onding)g(to)j(\(2.35\).)36 b(By)25 b(Theo-)-118 1182 y(rem)h(8,)694 1364 y FN(E)755 1376 y FL(J)792 1384 y Fx(1)829 1364 y FO(\(\001\))p FN(J)1008 1376 y FK(2)1046 1364 y FN(E)1107 1376 y FL(J)1144 1384 y Fx(1)1180 1364 y FO(\(\001)1281 1330 y FM(0)1305 1364 y FO(\))d(=)g(0)p FN(;)613 b FO(\(2.40\))-118 1546 y(for)37 b(an)n(y)g(\001,)j(\001)387 1516 y FM(0)450 1546 y FP(2)g Fz(B)p FO(\()p FI(R)q FO(\),)46 b(\001)25 b FP(\002)g FO(\001)1059 1516 y FM(0)1108 1546 y FP(\\)g FO(\000)39 b(=)h FI(?)p FO(.)66 b(Denote)38 b(b)n(y)f FN(S)2009 1558 y FK(0)2084 1546 y FO(the)h(set)-118 1645 y FP(f)p FN(s)32 b FP(2)g FI(R)39 b FP(j)32 b FO(\()p FN(q)25 b FP(\000)d FN(q)451 1615 y FM(\000)p FK(1)540 1645 y FO(\))572 1615 y FK(2)610 1645 y FN(s)649 1615 y FK(2)708 1645 y FO(+)g(4)32 b FN(<)g FO(0)p FP(g)p FO(,)h(and)g(b)n(y)g FN(S)1445 1657 y FK(1)1516 1645 y FO(its)f(complemen)n(t.)50 b(Then)-118 1745 y FN(R)17 b FP(\002)g FN(S)95 1757 y FK(0)148 1745 y FP(\\)g FO(\000)23 b(=)g FI(?)p FO(,)k(whic)n(h)e(implies)e(that)k FN(J)1240 1757 y FK(2)1278 1745 y FN(E)1339 1757 y FL(J)1376 1765 y Fx(1)1412 1745 y FO(\()p FN(S)1495 1757 y FK(0)1533 1745 y FO(\))p FN(H)j FO(=)22 b(0.)37 b(Hence,)27 b FN(H)2191 1757 y FK(0)2251 1745 y FO(:=)-118 1845 y FN(E)-57 1857 y FL(J)-20 1865 y Fx(1)16 1845 y FO(\()p FN(S)99 1857 y FK(0)137 1845 y FO(\))p FN(H)46 b FO(is)37 b(in)n(v)-5 b(arian)n(t)36 b(with)i(resp)r(ect)g(to)g FN(J)1391 1857 y FK(1)1429 1845 y FO(,)j FN(J)1539 1857 y FK(2)1615 1845 y FO(and)d(an)n(y)g(irreducible)-118 1944 y(represen)n(tation)25 b(in)i FN(H)591 1956 y FK(0)656 1944 y FO(is)f(one-dimensional)c(and)28 b(giv)n(en)e(b)n(y)515 2126 y FN(J)561 2138 y FK(1)621 2126 y FO(=)d(\()p FN(\025)p FO(\))p FN(;)98 b(J)988 2138 y FK(2)1048 2126 y FO(=)23 b(\(0\))p FN(;)180 b(\025)23 b FP(2)h FN(S)1646 2138 y FK(0)1683 2126 y FN(:)-118 2308 y FO(But,)32 b(b)n(y)f(\(2.36\),)g FN(\025)f FO(=)e(0)j(whic)n(h)f (do)r(es)h(not)g(b)r(elong)f(to)h FN(S)1712 2320 y FK(0)1749 2308 y FO(.)47 b(Therefore,)31 b(the)-118 2408 y(sp)r(ectrum)c(of)h FN(J)386 2420 y FK(1)451 2408 y FO(b)r(elongs)e(to)i FN(S)902 2420 y FK(1)939 2408 y FO(.)37 b(Consider)26 b(the)i(follo)n(wing)c(parameteriza-)-118 2508 y(tion)h(of)h FN(S)195 2520 y FK(1)232 2508 y FO(:)37 b FN(\025)23 b FO(=)g(sin)12 b FN(x\033)s(=)i FO(sin)f FN(\033)s FO(,)27 b FN(x)d FP(2)f FI(R)p FO(.)43 b(Let)26 b FA(O)p FO(\()p FP(f)p FN(\025)p FP(g)p FO(\))g(b)r(e)h(the)f(tra)5 b(jectory)25 b(of)-118 2607 y(the)g(p)r(oin)n(t)f FN(\025)i FO(with)e(resp)r(ect)h (to)f(\000,)i(i.e.,)e(the)i(minimal)20 b(subset)25 b FN(M)31 b FP(\032)23 b FI(R)31 b FO(whic)n(h)-118 2707 y(con)n(tains)h FN(\025)i FO(and)g(satis\014es)e(the)j(condition)c(\()p FI(R)e FP(n)23 b FN(M)9 b FO(\))22 b FP(\002)g FN(M)32 b FP(\\)23 b FO(\000)33 b(=)h FI(?)p FO(,)h(and)-118 2806 y FN(M)29 b FP(\002)20 b FO(\()p FI(R)26 b FP(n)20 b FN(M)9 b FO(\))20 b FP(\\)h FO(\000)28 b(=)f FI(?)p FO(.)45 b(In)31 b(our)f(case,)g FA(O)p FO(\()p FP(f)p FN(\025)p FP(g)p FO(\))e(=)f FP(f)p FO(sin)o(\()p FN(x)21 b FO(+)f FN(k)s FO(\))p FN(\033)s(=)14 b FO(sin)e FN(\033)32 b FP(j)-118 2906 y FN(k)26 b FP(2)d FI(Z)p FP(g)o FO(.)6 3006 y(If)28 b(\()p FN(J)167 3018 y FK(1)205 3006 y FN(;)14 b(J)288 3018 y FK(2)325 3006 y FO(\))28 b(is)e(irreducible,)d(then)28 b(the)g(measure)d FN(E)1617 3018 y FL(J)1654 3026 y Fx(1)1691 3006 y FO(\()p FP(\001)p FO(\))j(is)e(ergo)r(dic)f(with)-118 3105 y(resp)r(ect)36 b(to)g(\000,)j(i.e.,)e(either)f FN(E)872 3117 y FL(J)909 3125 y Fx(1)945 3105 y FO(\()p FN(M)9 b FO(\))38 b(=)f(0)f(or)f FN(E)1488 3117 y FL(J)1525 3125 y Fx(1)1562 3105 y FO(\()p FN(M)9 b FO(\))38 b(=)f FN(I)43 b FO(for)36 b(an)n(y)g(set)-118 3205 y FN(M)41 b FO(whic)n(h)31 b(is)h(in)n(v)-5 b(arian)n(t)29 b(with)j(resp)r(ect)g (to)h(\000.)51 b(In)33 b(fact,)g(if)f(it)g(w)n(ere)g(not)g(the)-118 3305 y(case,)c(w)n(e)h(w)n(ould)f(conclude)f(that)j FN(E)1035 3317 y FL(J)1072 3325 y Fx(1)1108 3305 y FO(\()p FN(M)9 b FO(\))p FN(H)36 b FO(or)28 b FN(E)1531 3317 y FL(J)1568 3325 y Fx(1)1605 3305 y FO(\()p FN(M)9 b FO(\))p FN(H)36 b FO(is)28 b(a)g(subspace)-118 3404 y(in)n(v)-5 b(arian)n(t)20 b(with)j(resp)r(ect)g(to)h FN(J)835 3416 y FK(1)872 3404 y FO(,)h FN(J)966 3416 y FK(2)1003 3404 y FO(.)35 b(Moreo)n(v)n(er,)22 b(one)h(can)g(easily)e(c)n(hec)n(k)h(that)-118 3504 y(there)31 b(exists)g(a)g(measurable)d(section)i(of)i(\()p FN(S)1306 3516 y FK(1)1343 3504 y FN(;)14 b FO(\000\),)33 b(i.e.,)f(a)f(set)h (whic)n(h)f(meets)-118 3603 y(ev)n(ery)36 b(tra)5 b(jectory)36 b(only)h(once.)66 b(Th)n(us)38 b(an)n(y)f(ergo)r(dic)e(measure)h(is)g (concen-)-118 3703 y(trated)24 b(on)h(a)f(single)e(tra)5 b(jectory)23 b(of)h(some)f(p)r(oin)n(t,)i(and,)g(hence)g(the)f(sp)r (ectrum)-118 3803 y(of)38 b(the)g(op)r(erator)e FN(J)531 3815 y FK(1)606 3803 y FO(is)g(discrete)g(and)i(concen)n(trated)e(on)i (a)f(tra)5 b(jectory)36 b(if)-118 3911 y(\()p FN(J)-40 3923 y FK(1)-3 3911 y FN(;)14 b(J)80 3923 y FK(2)118 3911 y FO(\))29 b(is)e(an)h(irreducible)d(pair.)38 b(Let)29 b FN(H)1216 3923 y FK(1)1278 3911 y FO(=)24 b FN(E)1428 3923 y FL(A)1483 3911 y FO(\()p FA(O)p FO(\()p FP(f)1658 3871 y FK(sin)o(\()p FL(\031)r(=)p FK(2\))p 1658 3892 242 4 v 1712 3940 a(sin)11 b FL(\033)1909 3911 y FP(g)p FO(\)\))p FN(H)c FO(,)29 b FN(H)2212 3923 y FK(2)2274 3911 y FO(=)p eop %%Page: 144 148 144 147 bop -118 -137 a FO(144)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 97 y FN(E)-57 109 y FL(A)-3 97 y FO(\()p FA(O)p FO(\()p FP(f\000)237 57 y FK(sin)o(\()p FL(\031)r(=)p FK(2\))p 237 78 242 4 v 291 126 a(sin)11 b FL(\033)488 97 y FP(g)p FO(\)\))p FN(H)c FO(,)34 b(if)e FN(k)j FO(is)c(ev)n(en,)j (and)e FN(H)1428 109 y FK(1)1496 97 y FO(=)f FN(E)1653 109 y FL(A)1707 97 y FO(\()p FA(O)p FO(\()p FP(f)1882 57 y FK(sin\()p FL(\031)r(=)p FK(2\))p 1882 78 V 1936 126 a(sin)11 b FL(\033)2134 97 y FP(g)p FO(\)\))p FN(H)c FO(,)-118 218 y FN(H)-49 230 y FK(2)11 218 y FO(=)23 b FN(E)160 230 y FL(A)214 218 y FO(\()p FA(O)p FO(\()p FP(f)389 178 y FK(sin\(\()p FL(\031)r FM(\000)p FL(\033)r FK(\))p FL(=)p FK(2\))p 389 199 387 4 v 515 247 a(sin)11 b FL(\033)785 218 y FP(g)p FO(\)\))p FN(H)c FO(,)28 b(if)f FN(k)j FO(is)d(o)r(dd,)g FN(H)1506 230 y FK(3)1567 218 y FO(=)22 b(\()p FN(H)1755 230 y FK(1)1811 218 y FP(\010)c FN(H)1963 230 y FK(2)2001 218 y FO(\))2033 188 y FM(?)2089 218 y FO(.)6 318 y(An)n(y)36 b(tra)5 b(jectory)33 b(can)i(b)r(e)h (describ)r(ed)e(geometrically)c(in)k(the)i(follo)n(wing)-118 417 y(w)n(a)n(y:)i(if)d FN(t)p FO(,)30 b FN(s)25 b FP(2)h FA(O)p FO(\()p FP(f)p FN(\025)p FP(g)p FO(\))j(and)g(\()p FN(t;)14 b(s)p FO(\))26 b FP(2)g FO(\000,)j(then)h(w)n(e)e(dra)n(w)g (an)h(edge)2095 399 y Fn(r)p 2095 401 125 4 v 100 w(r)2082 482 y FL(t)97 b(s)2290 417 y FO(if)-118 575 y FN(t)29 b FP(6)p FO(=)f FN(s)p FO(,)k(and)f(a)g(lo)r(op)593 557 y Fn(r)p 593 559 4 4 v 588 554 V 584 549 V 580 544 V 576 540 V 573 535 V 570 531 V 567 527 V 565 523 V 563 519 V 562 515 V 561 512 V 560 508 V 560 505 V 560 501 V 560 498 V 561 495 V 562 492 V 564 489 V 566 487 V 568 484 V 568 484 V 571 482 V 573 480 V 576 478 V 578 476 V 581 475 V 583 474 V 586 473 V 588 472 V 591 472 V 593 472 V 596 472 V 598 472 V 601 473 V 603 474 V 606 475 V 608 476 V 611 478 V 613 480 V 616 482 V 618 484 V 593 559 V 598 554 V 602 549 V 606 544 V 610 540 V 613 535 V 616 531 V 619 527 V 621 523 V 623 519 V 624 515 V 625 512 V 626 508 V 626 505 V 626 501 V 626 498 V 625 495 V 624 492 V 622 489 V 620 487 V 618 484 V 581 640 a FL(t)666 575 y FO(if)f FN(t)f FO(=)g FN(s)p FO(.)47 b(Then)31 b(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n(wing)27 b(t)n(yp)r(es)k(of)-118 675 y(tra)5 b(jectories.)6 774 y(An)n(y)32 b(tra)5 b(jectory)29 b(of)j(the)g(p)r(oin)n(t)e FN(t)39 b(=)-51 b FP(2)29 b FA(O)p FO(\()p FP(f)p FO(1)p FN(=)p FO(sin)12 b FN(\033)s FP(g)p FO(\))21 b FP([)g FA(O)p FO(\()p FP(f\000)p FO(1)p FN(=)p FO(sin)11 b FN(\033)s FP(g)p FO(\))21 b FP([)-118 874 y FA(O)p FO(\()p FP(f)p FO(cos)o(\()p FN(\033)s(=)p FO(2\))p FN(=)p FO(sin)12 b FN(\033)s FP(g)p FO(\))p FP([)p FA(O)p FO(\()p FP(f\000)p FO(cos)o(\()p FN(\033)s(=)p FO(2\))p FN(=)p FO(sin)g FN(\033)t FP(g)p FO(\))18 b(is)f(a)h(cycle)f (of)i(length)e FN(s)p FO(,)j(i.e.,)-118 974 y(the)g(follo)n(wing)15 b(graph:)650 956 y Fn(r)p 650 957 125 4 v 99 w(r)266 b(r)p 941 957 V -149 w(r)p 650 957 4 4 v 654 956 V 658 954 V 662 952 V 667 951 V 671 949 V 675 948 V 679 947 V 683 945 V 687 944 V 691 942 V 696 941 V 700 940 V 704 939 V 708 937 V 712 936 V 716 935 V 721 934 V 725 933 V 729 932 V 733 931 V 737 930 V 741 929 V 745 928 V 750 927 V 754 926 V 758 925 V 762 925 V 766 924 V 770 923 V 774 922 V 779 922 V 783 921 V 787 921 V 791 920 V 795 920 V 799 919 V 804 919 V 808 918 V 812 918 V 816 917 V 820 917 V 824 917 V 828 917 V 833 916 V 837 916 V 841 916 V 845 916 V 849 916 V 853 916 V 857 916 V 862 916 V 866 916 V 870 916 V 874 916 V 878 916 V 882 916 V 887 917 V 891 917 V 895 917 V 899 917 V 903 918 V 907 918 V 911 919 V 916 919 V 920 920 V 924 920 V 928 921 V 932 921 V 936 922 V 941 922 V 945 923 V 949 924 V 953 925 V 957 925 V 961 926 V 965 927 V 970 928 V 974 929 V 978 930 V 982 931 V 986 932 V 990 933 V 994 934 V 999 935 V 1003 936 V 1007 937 V 1011 939 V 1015 940 V 1019 941 V 1024 942 V 1028 944 V 1032 945 V 1036 947 V 1040 948 V 1044 949 V 1048 951 V 1053 952 V 1057 954 V 1061 956 V 1065 957 V 614 1030 a FL(\025)653 1038 y Fx(1)739 1030 y FL(\025)778 1038 y Fx(2)1029 1030 y FL(\025)1068 1038 y Fv(s)802 960 y FN(:)14 b(:)g(:)1107 974 y FO(,)21 b(where)d FN(\025)1430 986 y FL(m)1517 974 y FO(=)23 b(sin)n(\(\()p FN(x)d FO(+)e FN(m)p FO(\))p FN(\033)s FO(\))q FN(=)p FO(sin)12 b FN(\033)t FO(,)-118 1119 y FN(x\033)27 b FP(2)c FO([)p FP(\000)p FN(\031)s(=)p FO(2)p FN(;)14 b(\031)s(=)p FO(2])j FP(n)h(f)p FO(\()p FN(\031)s FO(\(2)p FN(l)i FO(+)e(1\))g(+)g FN(m\033)s FO(\))p FN(=)p FO(2)23 b FP(j)g FN(m;)14 b(l)24 b FP(2)f FI(Z)p FP(g)o FO(,)6 1218 y(The)j(tra)5 b(jectories)23 b(of)j(the)g(p)r(oin)n(ts)e FP(f\006)p FO(1)p FN(=)p FO(sin)11 b FN(\033)t FP(g)p FO(,)25 b FP(f\006)p FO(cos)o(\()p FN(\033)s(=)p FO(2\))p FN(=)p FO(sin)12 b FN(\033)s FP(g)26 b FO(are)-118 1318 y(of)h(the)h(form:)6 1418 y(a\))g(if)f FN(k)f FO(=)d(2\(2)p FN(p)17 b FP(\000)h FO(1\),)405 1549 y Fn(r)p 405 1551 V 400 1546 V 395 1541 V 391 1536 V 388 1532 V 384 1527 V 381 1523 V 379 1519 V 377 1515 V 375 1511 V 373 1507 V 372 1503 V 372 1500 V 371 1497 V 371 1493 V 372 1490 V 373 1487 V 374 1484 V 375 1481 V 377 1479 V 380 1476 V 380 1476 V 382 1474 V 385 1472 V 387 1470 V 390 1468 V 392 1467 V 395 1466 V 397 1465 V 400 1464 V 402 1464 V 405 1464 V 407 1464 V 410 1464 V 412 1465 V 415 1466 V 417 1467 V 419 1468 V 422 1470 V 424 1472 V 427 1474 V 429 1476 V 405 1551 V 409 1546 V 414 1541 V 418 1536 V 421 1532 V 425 1527 V 428 1523 V 430 1519 V 432 1515 V 434 1511 V 436 1507 V 437 1503 V 437 1500 V 438 1497 V 438 1493 V 437 1490 V 436 1487 V 435 1484 V 434 1481 V 432 1479 V 429 1476 V 405 1551 125 4 v 99 w(r)141 b(r)p 695 1551 V 100 w(r)280 1621 y FK(cos\()p FL(\033)r(=)p FK(2\))p 280 1642 249 4 v 338 1690 a(sin)11 b FL(\033)711 1662 y FP(\000)835 1629 y FK(1)p 785 1643 134 4 v 785 1690 a(sin)g FL(\033)557 1553 y FN(:)j(:)g(:)1401 1549 y Fn(r)p 1401 1551 4 4 v 1396 1546 V 1392 1541 V 1388 1536 V 1384 1532 V 1381 1527 V 1378 1523 V 1375 1519 V 1373 1515 V 1371 1511 V 1370 1507 V 1369 1503 V 1368 1500 V 1368 1497 V 1368 1493 V 1368 1490 V 1369 1487 V 1370 1484 V 1372 1481 V 1374 1479 V 1376 1476 V 1376 1476 V 1378 1474 V 1381 1472 V 1383 1470 V 1386 1468 V 1388 1467 V 1391 1466 V 1393 1465 V 1396 1464 V 1398 1464 V 1401 1464 V 1403 1464 V 1406 1464 V 1408 1465 V 1411 1466 V 1413 1467 V 1416 1468 V 1418 1470 V 1421 1472 V 1423 1474 V 1426 1476 V 1401 1551 V 1406 1546 V 1410 1541 V 1414 1536 V 1418 1532 V 1421 1527 V 1424 1523 V 1427 1519 V 1429 1515 V 1431 1511 V 1432 1507 V 1433 1503 V 1434 1500 V 1434 1497 V 1434 1493 V 1434 1490 V 1433 1487 V 1432 1484 V 1430 1481 V 1428 1479 V 1426 1476 V 1401 1551 125 4 v 99 w(r)141 b(r)p 1691 1551 V 100 w(r)1234 1662 y FP(\000)1309 1621 y FK(cos\()p FL(\033)r(=)p FK(2\))p 1309 1642 249 4 v 1366 1690 a(sin)11 b FL(\033)1799 1629 y FK(1)p 1749 1643 134 4 v 1749 1690 a(sin)g FL(\033)1553 1553 y FN(:)j(:)g(:)6 1769 y FO(b\))29 b(if)e FN(k)f FO(=)c(4)p FN(p)p FO(,)405 1900 y Fn(r)p 405 1902 4 4 v 400 1897 V 395 1892 V 391 1888 V 388 1883 V 384 1879 V 381 1874 V 379 1870 V 377 1866 V 375 1862 V 373 1858 V 372 1855 V 372 1851 V 371 1848 V 371 1844 V 372 1841 V 373 1838 V 374 1835 V 375 1832 V 377 1830 V 380 1827 V 380 1827 V 382 1825 V 385 1823 V 387 1821 V 390 1819 V 392 1818 V 395 1817 V 397 1816 V 400 1815 V 402 1815 V 405 1815 V 407 1815 V 410 1815 V 412 1816 V 415 1817 V 417 1818 V 419 1819 V 422 1821 V 424 1823 V 427 1825 V 429 1827 V 405 1902 V 409 1897 V 414 1892 V 418 1888 V 421 1883 V 425 1879 V 428 1874 V 430 1870 V 432 1866 V 434 1862 V 436 1858 V 437 1855 V 437 1851 V 438 1848 V 438 1844 V 437 1841 V 436 1838 V 435 1835 V 434 1832 V 432 1830 V 429 1827 V 405 1902 125 4 v 99 w(r)141 b(r)p 695 1902 V 100 w(r)238 2013 y FP(\000)313 1973 y FK(cos\()p FL(\033)r(=)p FK(2\))p 313 1994 249 4 v 370 2041 a(sin)11 b FL(\033)711 2013 y FP(\000)835 1980 y FK(1)p 785 1994 134 4 v 785 2041 a(sin)g FL(\033)557 1904 y FN(:)j(:)g(:)1401 1900 y Fn(r)p 1401 1902 4 4 v 1396 1897 V 1392 1892 V 1388 1888 V 1384 1883 V 1381 1879 V 1378 1874 V 1375 1870 V 1373 1866 V 1371 1862 V 1370 1858 V 1369 1855 V 1368 1851 V 1368 1848 V 1368 1844 V 1368 1841 V 1369 1838 V 1370 1835 V 1372 1832 V 1374 1830 V 1376 1827 V 1376 1827 V 1378 1825 V 1381 1823 V 1383 1821 V 1386 1819 V 1388 1818 V 1391 1817 V 1393 1816 V 1396 1815 V 1398 1815 V 1401 1815 V 1403 1815 V 1406 1815 V 1408 1816 V 1411 1817 V 1413 1818 V 1416 1819 V 1418 1821 V 1421 1823 V 1423 1825 V 1426 1827 V 1401 1902 V 1406 1897 V 1410 1892 V 1414 1888 V 1418 1883 V 1421 1879 V 1424 1874 V 1427 1870 V 1429 1866 V 1431 1862 V 1432 1858 V 1433 1855 V 1434 1851 V 1434 1848 V 1434 1844 V 1434 1841 V 1433 1838 V 1432 1835 V 1430 1832 V 1428 1830 V 1426 1827 V 1401 1902 125 4 v 99 w(r)141 b(r)p 1691 1902 V 100 w(r)1277 1973 y FK(cos)o(\()p FL(\033)r(=)p FK(2\))p 1277 1994 249 4 v 1334 2041 a(sin)11 b FL(\033)1799 1980 y FK(1)p 1749 1994 134 4 v 1749 2041 a(sin)g FL(\033)1553 1904 y FN(:)j(:)g(:)6 2120 y FO(c\))28 b(if)f FN(k)f FO(=)d(2)p FN(p)18 b FP(\000)g FO(1,)405 2251 y Fn(r)p 405 2253 4 4 v 400 2248 V 395 2243 V 391 2239 V 388 2234 V 384 2230 V 381 2225 V 379 2221 V 377 2217 V 375 2213 V 373 2210 V 372 2206 V 372 2202 V 371 2199 V 371 2196 V 372 2192 V 373 2189 V 374 2186 V 375 2184 V 377 2181 V 380 2178 V 380 2178 V 382 2176 V 385 2174 V 387 2172 V 390 2170 V 392 2169 V 395 2168 V 397 2167 V 400 2166 V 402 2166 V 405 2166 V 407 2166 V 410 2166 V 412 2167 V 415 2168 V 417 2169 V 419 2170 V 422 2172 V 424 2174 V 427 2176 V 429 2178 V 405 2253 V 409 2248 V 414 2243 V 418 2239 V 421 2234 V 425 2230 V 428 2225 V 430 2221 V 432 2217 V 434 2213 V 436 2210 V 437 2206 V 437 2202 V 438 2199 V 438 2196 V 437 2192 V 436 2189 V 435 2186 V 434 2184 V 432 2181 V 429 2178 V 405 2253 125 4 v 99 w(r)141 b(r)p 695 2253 V 100 w(r)p 820 2253 4 4 v 815 2248 V 810 2243 V 806 2239 V 803 2234 V 799 2230 V 796 2225 V 794 2221 V 792 2217 V 790 2213 V 789 2210 V 787 2206 V 787 2202 V 786 2199 V 787 2196 V 787 2192 V 788 2189 V 789 2186 V 791 2184 V 792 2181 V 795 2178 V 795 2178 V 797 2176 V 800 2174 V 802 2172 V 805 2170 V 807 2169 V 810 2168 V 812 2167 V 815 2166 V 817 2166 V 820 2166 V 822 2166 V 825 2166 V 827 2167 V 830 2168 V 832 2169 V 835 2170 V 837 2172 V 840 2174 V 842 2176 V 845 2178 V 820 2253 V 824 2248 V 829 2243 V 833 2239 V 837 2234 V 840 2230 V 843 2225 V 845 2221 V 848 2217 V 849 2213 V 851 2210 V 852 2206 V 853 2202 V 853 2199 V 853 2196 V 852 2192 V 852 2189 V 850 2186 V 849 2184 V 847 2181 V 845 2178 V 280 2324 a FK(cos\()p FL(\033)r(=)p FK(2\))p 280 2345 249 4 v 338 2393 a(sin)11 b FL(\033)653 2364 y FP(\000)728 2324 y FK(cos\()p FL(\033)r(=)p FK(2\))p 728 2345 V 785 2393 a(sin)g FL(\033)557 2255 y FN(:)j(:)g(:)1401 2251 y Fn(r)p 1401 2253 125 4 v 99 w(r)141 b(r)p 1691 2253 V 100 w(r)1292 2364 y FP(\000)1417 2331 y FK(1)p 1366 2345 134 4 v 1366 2393 a(sin)11 b FL(\033)1799 2331 y FK(1)p 1749 2345 V 1749 2393 a(sin)g FL(\033)1553 2255 y FN(:)j(:)g(:)6 2467 y FO(In)35 b(what)f(follo)n(ws)d(w)n(e)j(study)h(irreducible)30 b(represen)n(tations)i(in)h(eac)n(h)h(of)-118 2566 y(the)28 b(subspaces)e FN(H)473 2578 y FL(i)501 2566 y FO(,)i FN(i)23 b FO(=)f(1,)27 b(2,)h(3.)6 2666 y(1.)56 b(If)34 b FN(J)262 2678 y FK(1)300 2666 y FO(,)h FN(J)404 2678 y FK(2)476 2666 y FO(is)d(an)i(irreducible)c(represen)n(tation)i (acting)g(in)h FN(H)2081 2678 y FK(3)2118 2666 y FO(,)j(then)-118 2766 y FN(\033)s FO(\()p FN(J)10 2778 y FK(1)48 2766 y FO(\))29 b FP(\032)f FA(O)p FO(\()p FP(f)p FO(sin)12 b FN(x\033)t(=)p FO(sin)g FN(\033)t FP(g)p FO(\))28 b(=)g FP(f)p FO(sin)o(\(\()p FN(x)19 b FO(+)f FN(k)s FO(\))p FN(\033)s FO(\))q FN(=)p FO(sin)12 b FN(\033)32 b FP(j)d FO(0)f FP(\024)g FN(k)k FP(\024)c FN(s)20 b FP(\000)g FO(1)p FP(g)p FO(;)-118 2865 y(moreo)n(v)n(er,)36 b FP(\006)p FO(1)p FN(=)p FO(sin)11 b FN(\033)s FO(,)40 b FP(\006)p FO(cos)o(\()p FN(\033)s(=)p FO(2\))p FN(=)p FO(sin)12 b FN(\033)52 b(=)-52 b FP(2)39 b FA(O)p FO(\()p FP(f)p FO(sin)13 b FN(x\033)t(=)p FO(sin)f FN(\033)s FP(g)p FO(\).)65 b(Denote)-118 2965 y(the)27 b(pro)5 b(jection)25 b(on)n(to)h(the)h(eigenspace)e(of)i(the)g(op)r(erator)e FN(J)1768 2977 y FK(1)1833 2965 y FO(corresp)r(onding)-118 3064 y(to)i FN(\025)31 3076 y FL(k)96 3064 y FO(=)22 b(sin)o(\(\()p FN(x)e FO(+)e FN(k)s FO(\))p FN(\033)s FO(\))q FN(=)p FO(sin)12 b FN(\033)31 b FO(b)n(y)c FN(P)1063 3076 y FL(k)1104 3064 y FO(.)6 3164 y(It)h(follo)n(ws)c(from)i (condition)f(\(2.40\))i(that)h FN(J)1394 3176 y FK(2)1431 3164 y FN(P)1484 3176 y FK(0)1521 3164 y FN(H)1590 3176 y FK(3)1651 3164 y FP(\032)22 b FN(P)1791 3176 y FK(1)1829 3164 y FN(H)1898 3176 y FK(3)1953 3164 y FP(\010)17 b FN(P)2088 3176 y FL(s)p FM(\000)p FK(1)2209 3164 y FN(H)2278 3176 y FK(3)2316 3164 y FO(,)-118 3264 y FN(J)-72 3276 y FK(2)-35 3264 y FN(P)18 3276 y FL(s)p FM(\000)p FK(1)139 3264 y FN(H)208 3276 y FK(3)268 3264 y FP(\032)23 b FN(P)409 3276 y FK(1)446 3264 y FN(H)515 3276 y FK(3)571 3264 y FP(\010)18 b FN(P)707 3276 y FL(s)p FM(\000)p FK(2)828 3264 y FN(H)897 3276 y FK(3)934 3264 y FO(,)28 b(and)f FN(J)1192 3276 y FK(2)1229 3264 y FN(P)1282 3276 y FL(k)1323 3264 y FN(H)1392 3276 y FK(3)1453 3264 y FP(\032)22 b FN(P)1593 3276 y FL(k)q FK(+1)1719 3264 y FN(H)1788 3276 y FK(3)1843 3264 y FP(\010)c FN(P)1979 3276 y FL(k)q FM(\000)p FK(1)2105 3264 y FN(H)2174 3276 y FK(3)2239 3264 y FO(for)-118 3363 y FN(k)32 b FP(6)p FO(=)e(0)p FN(;)14 b(s)20 b FP(\000)h FO(1.)48 b(Th)n(us)31 b(the)h(op)r(erator)e FN(J)1135 3375 y FK(2)1204 3363 y FO(can)h(b)r(e)h(represen)n(ted)e(in) h(the)h(form)-118 3463 y FN(J)-72 3475 y FK(2)-7 3463 y FO(=)c FN(X)f FO(+)20 b FN(X)343 3433 y FM(\003)380 3463 y FO(,)32 b(where)e FN(X)k FO(=)874 3401 y Fy(P)962 3421 y FL(s)p FM(\000)p FK(2)962 3488 y FL(k)q FK(=0)1100 3463 y FN(P)1153 3475 y FL(k)q FK(+1)1279 3463 y FN(J)1325 3475 y FK(2)1362 3463 y FN(P)1415 3475 y FL(k)1477 3463 y FO(+)20 b FN(P)1615 3475 y FK(0)1652 3463 y FN(J)1698 3475 y FK(2)1735 3463 y FN(P)1788 3475 y FL(s)p FM(\000)p FK(1)1909 3463 y FO(.)46 b(Moreo)n(v)n(er,)-118 3563 y(the)28 b(pair)e(\()p FN(J)274 3575 y FK(1)312 3563 y FN(;)14 b(J)395 3575 y FK(2)432 3563 y FO(\))28 b(is)f(irreducible)d (if)j(and)h(only)f(if)g(the)h(triple)e(\()p FN(J)1927 3575 y FK(1)1965 3563 y FN(;)14 b(X)r(;)g(X)2186 3532 y FM(\003)2223 3563 y FO(\))28 b(is)-118 3662 y(irreducible.)33 b(One)26 b(can)g(easily)e(sho)n(w)i(that)h FN(J)1308 3674 y FK(1)1345 3662 y FO(,)g FN(J)1441 3674 y FK(2)1505 3662 y FO(satisfy)e(\(2.36\))h(i\013)g FN(X)7 b FO(,)26 b FN(X)2301 3632 y FM(\003)-118 3762 y FO(are)g(additionally)d (connected)28 b(b)n(y)f(the)h(relations:)548 3911 y FN(\013)601 3923 y FL(k)642 3911 y FN(X)718 3877 y FM(\003)756 3911 y FN(X)7 b(P)885 3923 y FL(k)944 3911 y FO(+)18 b FN(\014)1074 3923 y FL(k)1115 3911 y FN(X)7 b(X)1267 3877 y FM(\003)1303 3911 y FN(P)1356 3923 y FL(k)1421 3911 y FO(=)22 b FN(\015)1551 3923 y FL(k)1592 3911 y FN(I)7 b(;)468 b FO(\(2.41\))p eop %%Page: 145 149 145 148 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(145)-118 96 y(where)33 b FN(\013)181 108 y FL(k)256 96 y FO(=)g FP(\000)p FO(2)14 b(cos)n(\(\()p FN(x)24 b FO(+)e FN(k)j FO(+)e(1\))p FN(\033)s FO(\),)36 b FN(\014)1225 108 y FL(k)1299 96 y FO(=)d(2)14 b(cos)o(\(\()p FN(x)24 b FO(+)e FN(k)j FP(\000)d FO(1\))p FN(\033)s FO(\),)37 b FN(\015)2200 108 y FL(k)2274 96 y FO(=)-118 196 y(sin)o(\(\()p FN(x)19 b FO(+)f FN(k)s FO(\))p FN(\033)s FO(\))q FN(=)p FO(sin)12 b FN(\033)t FO(.)36 b(Since)26 b FP(f\006)p FO(1)p FN(=)p FO(sin)10 b FN(\033)t FP(g)26 b FO(do)r(es)f(not)i(b)r(elong)d(to)i(the)h(tra)5 b(jec-)-118 296 y(tory)-7 b(,)27 b FN(\013)129 308 y FL(k)170 296 y FN(\014)217 308 y FL(k)281 296 y FP(6)p FO(=)22 b(0.)6 395 y(Let)29 b(0)23 b FP(\024)g FN(k)k FP(\024)c FN(s)c FP(\000)f FO(1)28 b(b)r(e)g(the)h(smallest)24 b(n)n(um)n(b)r(er)j(suc)n(h)h(that)g FN(\025)1973 407 y FL(k)2038 395 y FP(2)c FN(\033)s FO(\()p FN(J)2245 407 y FK(1)2283 395 y FO(\).)-118 495 y(Set)31 b FN(C)87 507 y FL(k)156 495 y FO(=)d FN(X)325 465 y FM(\003)363 495 y FN(X)7 b(P)492 507 y FL(k)532 495 y FO(,)32 b(and)e(denote)h(the) g(resolution)d(of)i(the)h(iden)n(tit)n(y)e(for)h FN(C)2297 507 y FL(k)-118 595 y FO(b)n(y)k FN(E)65 607 y FL(C)113 616 y Fv(k)153 595 y FO(\()p FP(\001)p FO(\).)57 b(F)-7 b(rom)33 b(\(2.41\))g(it)g(follo)n(ws)e(that)k([)p FN(A;)14 b(X)1544 564 y FL(s)1579 595 y FO(])34 b(=)f(0,)i([)p FN(X)1933 564 y FM(\003)1971 595 y FN(;)14 b(X)2084 564 y FL(s)2119 595 y FO(])34 b(=)f(0,)-118 694 y(whic)n(h)k(yields)e FN(X)449 664 y FL(s)524 694 y FO(=)40 b FN(cI)45 b FO(if)37 b(\()p FN(J)910 706 y FK(1)948 694 y FN(;)14 b(X)r(;)g(X)1169 664 y FM(\003)1206 694 y FO(\))38 b(is)f(irreducible.)63 b(F)-7 b(rom)37 b(this)g(w)n(e)-118 794 y(conclude)30 b(that)i FP(\010)476 758 y FL(s)p FM(\000)p FK(1)476 819 y FL(l)p FK(=)p FL(k)596 794 y FN(X)672 764 y FL(l)p FM(\000)p FL(k)786 794 y FN(E)847 806 y FL(C)895 815 y Fv(k)935 794 y FO(\(\001\))14 b FN(W)45 b FO(is)30 b(in)n(v)-5 b(arian)n(t)29 b(with)i(resp)r(ect)h(to)f FN(J)2278 806 y FK(1)2316 794 y FO(,)-118 893 y FN(X)7 b FO(,)29 b FN(X)86 863 y FM(\003)153 893 y FO(for)g(an)n(y)g(\001)d FP(2)h Fz(B)p FO(\()p FI(R)p FO(\))36 b(and)29 b(a)h(subspace)e FN(W)42 b FO(suc)n(h)29 b(that)h FN(C)1978 905 y FL(k)2019 893 y FN(W)38 b FP(\032)26 b FN(W)12 b FO(.)-118 993 y(Hence,)37 b(if)d(\()p FN(J)322 1005 y FK(1)360 993 y FN(;)14 b(X)r(;)g(X)581 963 y FM(\003)618 993 y FO(\))35 b(is)f(irreducible,)f(then)j(\001)f(is)f(concen)n(trated)g(in)g(one) -118 1093 y(p)r(oin)n(t,)k(and)e(w)n(e)h(can)f(c)n(ho)r(ose)f(a)h (basis)f(consisting)f(of)i(eigen)n(v)n(ectors)e(of)i FN(J)2278 1105 y FK(1)2316 1093 y FO(,)-118 1192 y(namely)-7 b(,)21 b FP(f)p FN(e)263 1204 y FL(\025)302 1213 y Fv(k)342 1192 y FN(;)14 b(X)7 b(e)494 1204 y FL(\025)533 1213 y Fv(k)573 1192 y FN(=)p FP(jj)p FN(X)g(e)776 1204 y FL(\025)815 1213 y Fv(k)854 1192 y FP(jj)p FN(;)14 b(:)g(:)g(:)28 b(;)14 b(X)1175 1162 y FL(s)p FM(\000)p FL(k)q FM(\000)p FK(1)1383 1192 y FN(e)1422 1204 y FL(\025)1461 1213 y Fv(k)1502 1192 y FN(=)p FP(jj)p FN(X)1666 1162 y FL(s)p FM(\000)p FL(k)q FM(\000)p FK(1)1873 1192 y FN(e)1912 1204 y FL(\025)1951 1213 y Fv(k)1992 1192 y FP(jjg)p FO(,)23 b(where)-118 1292 y FN(e)-79 1304 y FL(\025)-40 1313 y Fv(k)31 1292 y FO(is)29 b(an)h(eigen)n(v)n(ector)d(of)k FN(C)826 1304 y FL(k)897 1292 y FO(whic)n(h)e(exists)g(due)i(to)f(the)h (last)e(argumen)n(ts.)-118 1392 y(Let)e FN(X)106 1361 y FM(\003)144 1392 y FN(X)7 b(e)259 1404 y FL(\025)298 1413 y Fv(k)361 1392 y FP(\021)22 b FN(C)507 1404 y FL(k)549 1392 y FN(e)588 1404 y FL(\025)627 1413 y Fv(k)690 1392 y FO(=)h FN(b)14 b(e)867 1404 y FL(\025)906 1413 y Fv(k)946 1392 y FO(,)27 b FN(b)c FP(2)g FI(R)1187 1404 y FK(+)1248 1392 y FO(.)37 b(Using)26 b(\(2.41\))h(one)g(can)g(get)g(the)-118 1491 y(action)20 b(of)h(the)h(op)r(erators)e FN(J)758 1503 y FK(1)795 1491 y FO(,)j FN(X)7 b FO(,)22 b FN(X)1038 1461 y FM(\003)1076 1491 y FO(,)h(and)e FN(J)1323 1503 y FK(2)1382 1491 y FO(on)g(the)h(basis.)33 b(In)22 b(particular,)-118 1591 y(if)e FN(\033)s FO(\()p FN(J)79 1603 y FK(1)117 1591 y FO(\))k(=)e FP(f)p FN(\025)350 1603 y FK(0)388 1591 y FN(;)14 b(\025)473 1603 y FK(1)510 1591 y FN(;)g(:)g(:)g(:)g(;)g (\025)743 1603 y FL(s)p FM(\000)p FK(1)864 1591 y FP(g)p FO(,)21 b(w)n(e)g(will)d(ha)n(v)n(e)i(represen)n(tations)e(of)j(series) e(1,)-118 1690 y(otherwise,)26 b(represen)n(tations)e(from)j(series)e (9.)6 1790 y(2.)63 b(If)37 b FN(J)272 1802 y FK(1)309 1790 y FO(,)i FN(J)417 1802 y FK(2)490 1790 y FO(is)d(an)g(irreducible) c(represen)n(tation)i(acting)h(in)g FN(H)2110 1802 y FK(1)2148 1790 y FO(,)j FN(H)2278 1802 y FK(2)2316 1790 y FO(,)-118 1890 y(then)28 b(it)f(is)g(not)h(necessary)e(for)h FN(\033)s FO(\()p FN(J)1009 1902 y FK(1)1047 1890 y FO(\))h(to)g(b)r(e) g(simple.)34 b(First)27 b(let)g(us)h(consider)-118 1989 y(the)g(case)f(where)g FN(J)487 2001 y FK(1)552 1989 y FO(is)f(concen)n(trated)h(on)g(the)h(tra)5 b(jectory)26 b(of)h(the)h(form)903 2142 y Fn(r)p 903 2143 4 4 v 898 2138 V 893 2134 V 889 2129 V 886 2124 V 882 2120 V 880 2116 V 877 2112 V 875 2108 V 873 2104 V 872 2100 V 870 2096 V 870 2093 V 869 2089 V 870 2086 V 870 2083 V 871 2080 V 872 2077 V 874 2074 V 875 2071 V 878 2069 V 878 2069 V 880 2066 V 883 2064 V 885 2062 V 888 2061 V 890 2059 V 893 2058 V 895 2057 V 898 2057 V 900 2056 V 903 2056 V 905 2056 V 908 2057 V 910 2057 V 913 2058 V 915 2059 V 918 2061 V 920 2062 V 923 2064 V 925 2066 V 928 2069 V 903 2143 V 907 2138 V 912 2134 V 916 2129 V 920 2124 V 923 2120 V 926 2116 V 928 2112 V 931 2108 V 932 2104 V 934 2100 V 935 2096 V 936 2093 V 936 2089 V 936 2086 V 935 2083 V 935 2080 V 933 2077 V 932 2074 V 930 2071 V 928 2069 V 903 2143 125 4 v 99 w(r)141 b(r)p 1193 2143 V 100 w(r)p 1318 2143 4 4 v 1313 2138 V 1309 2134 V 1305 2129 V 1301 2124 V 1298 2120 V 1295 2116 V 1292 2112 V 1290 2108 V 1288 2104 V 1287 2100 V 1286 2096 V 1285 2093 V 1285 2089 V 1285 2086 V 1285 2083 V 1286 2080 V 1287 2077 V 1289 2074 V 1291 2071 V 1293 2069 V 1293 2069 V 1295 2066 V 1298 2064 V 1300 2062 V 1303 2061 V 1305 2059 V 1308 2058 V 1310 2057 V 1313 2057 V 1315 2056 V 1318 2056 V 1320 2056 V 1323 2057 V 1325 2057 V 1328 2058 V 1330 2059 V 1333 2061 V 1335 2062 V 1338 2064 V 1340 2066 V 1343 2069 V 1318 2143 V 1323 2138 V 1327 2134 V 1331 2129 V 1335 2124 V 1338 2120 V 1341 2116 V 1344 2112 V 1346 2108 V 1347 2104 V 1349 2100 V 1350 2096 V 1351 2093 V 1351 2089 V 1351 2086 V 1350 2083 V 1350 2080 V 1348 2077 V 1347 2074 V 1345 2071 V 1343 2069 V 888 2241 a FL(\025)927 2249 y Fx(1)1299 2241 y FL(\025)1338 2249 y Fv(n)1055 2146 y FN(:)14 b(:)g(:)2126 2160 y FO(\(2.42\))-118 2404 y(where)27 b FN(\025)170 2416 y FK(1)231 2404 y FO(=)22 b FP(\000)393 2364 y FK(cos\()p FL(\033)r(=)p FK(2\))p 393 2385 249 4 v 450 2433 a(sin)11 b FL(\033)651 2404 y FO(,)28 b FN(\025)750 2416 y FL(n)819 2404 y FO(=)916 2364 y FK(cos\()p FL(\033)r(=)p FK(2\))p 916 2385 V 974 2433 a(sin)11 b FL(\033)1175 2404 y FO(,)27 b(and)h FN(\033)s FO(\()p FN(J)1515 2416 y FK(1)1553 2404 y FO(\))23 b(=)g FP(f)p FN(\025)1786 2416 y FK(1)1823 2404 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)2056 2416 y FL(n)2101 2404 y FP(g)p FO(.)6 2504 y(Denote)36 b(the)g(pro)5 b(jection)33 b(on)n(to)i(the)g(eigenspace)f(of)h FN(J)1760 2516 y FK(1)1833 2504 y FO(corresp)r(onding)-118 2603 y(to)h(the)h(eigen)n(v) -5 b(alue)33 b FN(\025)597 2615 y FL(k)676 2603 y FO(=)k FP(\000)p FO(cos)o(\(\(2)p FN(k)21 b FP(\000)d FO(1\))p FN(\033)s(=)p FO(2)o FN(=)p FO(sin)13 b FN(\033)39 b FO(b)n(y)d FN(P)1835 2615 y FL(k)1877 2603 y FO(.)63 b(As)36 b(b)r(efore,)-118 2703 y(the)31 b(op)r(erator)e FN(J)412 2715 y FK(2)481 2703 y FO(can)h(b)r(e)h(represen)n(ted)f(in)g (the)i(form)d FN(J)1689 2715 y FK(2)1755 2703 y FO(=)f FN(X)f FO(+)21 b FN(X)2106 2673 y FM(\003)2164 2703 y FO(+)f FN(Y)f FO(,)-118 2803 y(where)34 b FN(X)42 b FO(=)339 2740 y Fy(P)427 2761 y FL(n)p FM(\000)p FK(1)427 2828 y FL(k)q FK(=1)571 2803 y FN(P)624 2815 y FL(k)q FK(+1)749 2803 y FN(J)795 2815 y FK(2)833 2803 y FN(P)886 2815 y FL(k)927 2803 y FO(,)37 b FN(Y)53 b FO(=)35 b FN(P)1241 2815 y FK(1)1279 2803 y FN(J)1325 2815 y FK(2)1362 2803 y FN(P)1415 2815 y FK(1)1476 2803 y FO(+)23 b FN(P)1617 2815 y FL(n)1662 2803 y FN(J)1708 2815 y FK(2)1745 2803 y FN(P)1798 2815 y FL(n)1844 2803 y FO(,)37 b(and)d FN(Y)54 b FP(6)p FO(=)35 b(0,)-118 2902 y FN(X)7 b(P)11 2914 y FL(k)85 2902 y FP(6)p FO(=)34 b(0,)i(for)d FN(k)k(<)d(n)p FO(.)57 b(Moreo)n(v)n(er,)33 b(\()p FN(J)1201 2914 y FK(1)1239 2902 y FN(;)14 b(J)1322 2914 y FK(2)1359 2902 y FO(\))34 b(is)g(irreducible)c(if)k(and)g(only)-118 3002 y(if)28 b(the)h(family)d(\()p FN(J)436 3014 y FK(1)473 3002 y FN(;)14 b(X)r(;)g(X)694 2972 y FM(\003)731 3002 y FN(;)g(Y)19 b FO(\))29 b(is)e(irreducible.)36 b(Clearly)26 b FN(J)1762 3014 y FK(1)1799 3002 y FO(,)j FN(J)1897 3014 y FK(2)1963 3002 y FO(satisfy)e(the)-118 3102 y(relation)d FN(Q)p FO(\()p FN(J)330 3114 y FK(1)368 3102 y FN(;)14 b(J)451 3114 y FK(2)488 3102 y FO(\))23 b FP(\021)g FN(J)685 3071 y FK(2)677 3122 y(2)722 3102 y FN(J)768 3114 y FK(1)823 3102 y FP(\000)18 b FO(\()p FN(q)j FO(+)d FN(q)1119 3071 y FM(\000)p FK(1)1208 3102 y FO(\))p FN(J)1286 3114 y FK(2)1324 3102 y FN(J)1370 3114 y FK(1)1407 3102 y FN(J)1453 3114 y FK(2)1508 3102 y FO(+)g FN(J)1637 3114 y FK(1)1674 3102 y FN(J)1728 3071 y FK(2)1720 3122 y(2)1783 3102 y FP(\000)g FN(J)1912 3114 y FK(1)1972 3102 y FO(=)23 b(0)k(if)g(and)-118 3201 y(only)f(if)h FN(P)193 3213 y FL(s)229 3201 y FN(Q)p FO(\()p FN(J)373 3213 y FK(1)410 3201 y FN(;)14 b(J)493 3213 y FK(2)530 3201 y FO(\))p FN(P)615 3213 y FL(k)680 3201 y FO(=)22 b(0)28 b(for)f(an)n(y)g(1)22 b FP(\024)h FN(k)s(;)14 b(s)22 b FP(\024)h FN(n)p FO(.)37 b(F)-7 b(rom)26 b(this)h(w)n(e)g(ha)n(v)n(e:)189 3371 y FN(\013)242 3383 y FL(k)283 3371 y FN(X)359 3337 y FM(\003)352 3392 y FL(k)397 3371 y FN(X)466 3383 y FL(k)525 3371 y FO(+)18 b FN(\014)655 3383 y FL(k)696 3371 y FN(X)765 3383 y FL(k)q FM(\000)p FK(1)890 3371 y FN(X)966 3337 y FM(\003)959 3392 y FL(k)q FM(\000)p FK(1)1108 3371 y FO(=)23 b FN(\015)1239 3383 y FL(k)1279 3371 y FN(I)7 b(;)180 b(k)26 b FP(6)p FO(=)d(1)p FN(;)41 b(k)26 b FP(6)p FO(=)c FN(n;)395 3506 y(\013)448 3518 y FK(1)485 3506 y FN(X)561 3472 y FM(\003)554 3527 y FK(1)599 3506 y FN(X)668 3518 y FK(1)723 3506 y FO(+)c FN(\014)853 3518 y FK(1)890 3506 y FN(Y)957 3472 y FK(2)994 3506 y FN(P)1047 3518 y FK(1)1108 3506 y FO(=)23 b FN(\015)1239 3518 y FK(1)1276 3506 y FN(I)7 b(;)193 3641 y(\014)240 3653 y FL(n)285 3641 y FN(X)354 3653 y FL(n)p FM(\000)p FK(1)484 3641 y FN(X)560 3607 y FM(\003)553 3662 y FL(n)p FM(\000)p FK(1)701 3641 y FO(+)18 b FN(\013)837 3653 y FL(n)882 3641 y FN(Y)949 3607 y FK(2)986 3641 y FN(P)1039 3653 y FL(n)1108 3641 y FO(=)23 b FN(\015)1239 3653 y FL(n)1284 3641 y FN(I)7 b(;)776 b FO(\(2.43\))-118 3811 y(where)21 b FN(X)185 3823 y FL(k)248 3811 y FO(=)i FN(X)7 b(P)465 3823 y FL(k)506 3811 y FO(,)22 b FN(\013)604 3823 y FL(k)668 3811 y FO(=)h FP(\000)p FO(2)14 b(sin)n(\(\(2)p FN(k)9 b FO(+)d(1\))p FN(\033)s(=)p FO(2\),)22 b FN(\014)1539 3823 y FL(k)1603 3811 y FO(=)g(2)14 b(sin)n(\(\(2)p FN(k)9 b FP(\000)d FO(3\))p FN(\033)s(=)p FO(2\),)-118 3911 y FN(\015)-75 3923 y FL(k)-11 3911 y FO(=)22 b FP(\000)p FO(cos)o(\(\(2)p FN(k)g FP(\000)c FO(1\))p FN(\033)s(=)p FO(2\))o FN(=)p FO(sin)12 b FN(\033)t FO(.)p eop %%Page: 146 150 146 149 bop -118 -137 a FO(146)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(Set)39 b FN(D)229 108 y FK(0)306 96 y FO(=)h(\()p FN(X)519 66 y FM(\003)556 96 y FO(\))588 66 y FL(n)p FM(\000)p FK(1)719 96 y FN(X)795 66 y FL(n)p FM(\000)p FK(1)924 96 y FN(P)977 108 y FK(1)1015 96 y FO(.)68 b(Then)38 b FN(P)1386 108 y FK(1)1423 96 y FN(H)47 b FP(\032)40 b FO(\(k)n(er)13 b FN(D)1870 108 y FK(0)1907 96 y FO(\))1939 66 y FM(?)1995 96 y FO(,)41 b(b)r(ecause)-118 196 y(if)c(this)f(w)n(ere)h(not)g(the)g(case,)i(w)n(e)e(w)n(ould)f (conclude)g(that)i FN(W)51 b FO(=)38 b(k)n(er)13 b FN(D)2212 208 y FK(0)2274 196 y FP(\010)-118 296 y FN(X)20 b FO(k)n(er)13 b FN(D)165 308 y FK(0)228 296 y FP(\010)25 b FN(:)14 b(:)g(:)26 b FP(\010)g FN(X)608 266 y FL(n)p FM(\000)p FK(2)751 296 y FO(k)n(er)13 b FN(D)945 308 y FK(0)1021 296 y FO(is)38 b(in)n(v)-5 b(arian)n(t)35 b(with)k(resp)r(ect)f(to)h FN(J)2130 308 y FK(1)2167 296 y FO(,)j FN(J)2278 308 y FK(2)2316 296 y FO(,)-118 395 y(whic)n(h)26 b(con)n(tradicts)g(the)i (fact)f(that)h FN(\033)s FO(\()p FN(J)1161 407 y FK(1)1199 395 y FO(\))c(=)e FP(f)p FN(\025)1432 407 y FK(1)1470 395 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)1716 407 y FL(n)1761 395 y FP(g)p FO(.)6 495 y(Let)23 b FN(X)30 b FO(=)22 b FN(U)402 425 y FP(p)p 471 425 189 4 v 70 x FN(X)547 471 y FM(\003)585 495 y FN(X)29 b FO(b)r(e)23 b(the)h(p)r(olar)d(decomp)r(osition)e(of)k(the)g(op)r(erator)e FN(X)7 b FO(.)-118 595 y(Put)22 b FN(Y)87 607 y FK(1)148 595 y FO(=)g FN(Y)d(P)355 607 y FK(1)392 595 y FO(,)24 b FN(Y)487 607 y FK(2)547 595 y FO(=)f(\()p FN(U)733 564 y FM(\003)771 595 y FO(\))803 564 y FL(n)p FM(\000)p FK(1)934 595 y FN(Y)18 b(U)1066 564 y FL(n)p FM(\000)p FK(1)1196 595 y FN(P)1249 607 y FK(1)1287 595 y FO(.)35 b(Let)22 b(us)h(pro)n(v)n(e)d(that)j(\()p FN(J)2061 607 y FK(1)2098 595 y FO(,)h FN(J)2191 607 y FK(2)2228 595 y FO(\))f(is)-118 694 y(irreducible)c(if)j(and)h(only)f(if)g(the)h (pair)e(\()p FN(Y)1149 706 y FK(1)1187 694 y FO(,)j FN(Y)1282 706 y FK(2)1320 694 y FO(\))f(is)f(irreducible.)31 b(In)23 b(fact,)h(if)f FN(W)-118 794 y FO(is)e(in)n(v)-5 b(arian)n(t)19 b(with)j(resp)r(ect)h(to)f FN(Y)911 806 y FK(1)948 794 y FO(,)i FN(Y)1043 806 y FK(2)1080 794 y FO(,)g(then)f FN(W)1401 764 y FM(0)1447 794 y FO(=)g FN(W)d FP(\010)8 b FN(U)h(W)19 b FP(\010)8 b FN(:)14 b(:)g(:)g(U)2119 764 y FL(n)p FM(\000)p FK(1)2249 794 y FN(W)-118 893 y FO(is)24 b(in)n(v)-5 b(arian)n(t)22 b(with)i(resp)r(ect)h(to)g FN(J)922 905 y FK(1)959 893 y FO(,)h FN(X)7 b FO(,)25 b FN(X)1208 863 y FM(\003)1245 893 y FO(,)h FN(Y)19 b FO(,)25 b(and)g(hence)g(with)g(resp)r(ect)g(to)-118 993 y FN(J)-72 1005 y FK(1)-35 993 y FO(,)j FN(J)62 1005 y FK(2)99 993 y FO(:)85 1161 y FN(X)7 b(U)227 1127 y FL(k)267 1161 y FN(W)35 b FO(=)22 b FN(U)533 1086 y FP(p)p 602 1086 V 75 x FN(X)678 1137 y FM(\003)716 1161 y FN(X)6 b(U)857 1127 y FL(k)897 1161 y FN(W)380 1318 y FO(=)22 b FN(U)533 1283 y FL(k)q FK(+1)658 1250 y Fy(\000)696 1318 y FQ(F)756 1283 y FK(\()p FL(k)q FK(\))849 1250 y Fy(\000)887 1318 y FO(\()p FN(\033)g FP(\000)c FN(\031)s FO(\))p FN(=)p FO(2)p FN(;)c(X)1350 1283 y FM(\003)1387 1318 y FN(X)7 b FO(\))1495 1250 y Fy(\001)1533 1351 y FK(2)1570 1250 y Fy(\001)1608 1268 y FK(1)p FL(=)p FK(2)1712 1318 y FN(W)35 b FP(\032)23 b FN(U)1979 1283 y FL(k)q FK(+1)2103 1318 y FN(W)n(;)4 1456 y(Y)c(U)137 1421 y FL(n)p FM(\000)p FK(1)267 1456 y FN(W)35 b FO(=)22 b FN(U)533 1421 y FL(n)p FM(\000)p FK(1)663 1456 y FO(\()p FN(U)761 1421 y FM(\003)799 1456 y FO(\))831 1421 y FL(n)p FM(\000)p FK(1)962 1456 y FN(Y)c(U)1094 1421 y FL(n)p FM(\000)p FK(1)1224 1456 y FN(W)35 b FP(\032)23 b FN(U)1491 1421 y FL(n)p FM(\000)p FK(1)1621 1456 y FN(Y)1669 1468 y FK(1)1706 1456 y FN(W)35 b FP(\032)23 b FN(U)1973 1421 y FL(n)p FM(\000)p FK(1)2103 1456 y FN(W)n(;)-118 1623 y FO(where)73 1791 y FQ(F)p FO(\()p FN(x;)14 b(y)s FO(\))23 b(=)g(\()p FN(F)521 1803 y FK(1)559 1791 y FO(\()p FN(x;)14 b(y)s FO(\))p FN(;)g(F)841 1803 y FK(2)879 1791 y FO(\()p FN(x;)g(y)s FO(\)\))348 1975 y(=)436 1858 y Fy(\022)497 1975 y FN(x)19 b FO(+)f(1)p FN(;)835 1919 y(y)e FO(cos)d FN(x\033)p 735 1956 482 4 v 735 2032 a FO(cos)o(\(\()p FN(x)19 b FO(+)f(2\))p FN(\033)s FO(\))1244 1975 y FP(\000)1459 1919 y FO(sin)o(\(\()p FN(x)i FO(+)e(1\))p FN(\033)s FO(\))p 1337 1956 717 4 v 1337 2032 a(2)c(sin)e FN(\033)17 b FO(cos\(\()p FN(x)i FO(+)f(2\))p FN(\033)s FO(\))2064 1858 y Fy(\023)2125 1975 y FN(:)-118 2188 y FO(Here)31 b(w)n(e)g(use)g(the)g(fact)h(that)f FN(U)9 b(U)984 2158 y FM(\003)1053 2188 y FO(is)30 b(the)i(pro)5 b(jection)29 b(on)i(the)h(orthogonal)-118 2287 y(complemen)n(t)24 b(to)k(k)n(er)o(\()p FN(A)19 b FO(+)f(\(cos)o(\()p FN(\033)s(=)p FO(2\))p FN(=)c FO(sin)e FN(\033)s FO(\))p FN(I)7 b FO(\).)38 b(Moreo)n(v)n(er,)25 b(\(2.43\))i(giv)n(es)99 2455 y FN(Q)165 2467 y FK(2)202 2455 y FO(\()p FN(Y)282 2467 y FK(2)320 2455 y FO(\))c(=)g(\()p FN(U)561 2421 y FM(\003)599 2455 y FO(\))631 2421 y FL(n)p FM(\000)p FK(1)761 2455 y FN(X)7 b(X)913 2421 y FM(\003)950 2455 y FN(U)1016 2421 y FL(n)p FM(\000)p FK(1)1146 2455 y FN(P)1199 2467 y FK(1)1260 2455 y FO(=)22 b(\()p FN(U)1445 2421 y FM(\003)1484 2455 y FO(\))1516 2421 y FL(n)p FM(\000)p FK(2)1646 2455 y FN(X)1722 2421 y FM(\003)1759 2455 y FN(X)7 b(U)1901 2421 y FL(n)p FM(\000)p FK(2)2031 2455 y FN(P)2084 2467 y FK(1)375 2596 y FO(=)463 2529 y Fy(\000)501 2596 y FQ(F)561 2562 y FK(\()p FL(n)p FM(\000)p FK(2\))743 2596 y FO(\(\()p FN(\033)22 b FP(\000)c FN(\031)s FO(\))p FN(=)p FO(2)p FN(;)c(X)1238 2562 y FM(\003)1275 2596 y FN(X)7 b FO(\))1383 2529 y Fy(\001)1421 2629 y FK(2)1458 2596 y FN(P)1511 2608 y FK(1)375 2740 y FO(=)463 2673 y Fy(\000)501 2740 y FQ(F)561 2706 y FK(\()p FL(n)p FM(\000)p FK(2\))743 2740 y FO(\(\()p FN(\033)22 b FP(\000)c FN(\031)s FO(\))p FN(=)p FO(2)p FN(;)c(Q)1228 2752 y FK(1)1265 2740 y FO(\()p FN(Y)1345 2752 y FK(1)1383 2740 y FO(\))1415 2673 y Fy(\001)1476 2740 y FO(=)23 b FN(Q)1630 2752 y FK(1)1666 2740 y FO(\()p FN(Y)1746 2752 y FK(1)1784 2740 y FO(\))c(+)f FN(\013I)7 b(;)-118 2908 y FO(where)33 b FN(Q)194 2920 y FK(1)231 2908 y FO(\()p FN(x)p FO(\))i(=)f(\()p FP(\000)p FN(\014)620 2920 y FK(1)657 2908 y FN(x)704 2878 y FK(2)764 2908 y FO(+)22 b FN(\015)894 2920 y FK(1)932 2908 y FO(\))p FN(=\013)1059 2920 y FK(1)1096 2908 y FO(,)36 b FN(Q)1221 2920 y FK(2)1258 2908 y FO(\()p FN(x)p FO(\))e(=)g(\()p FP(\000)p FN(\013)1652 2920 y FL(n)1697 2908 y FN(x)1744 2878 y FK(2)1805 2908 y FO(+)22 b FN(\015)1935 2920 y FL(n)1980 2908 y FO(\))p FN(=\014)2101 2920 y FL(n)2146 2908 y FO(,)36 b(and)-118 3008 y FN(\013)j FO(=)g(\()p FQ(F)170 2978 y FK(\()p FL(n)p FM(\000)p FK(2\))352 3008 y FO(\(\()p FN(\033)29 b FP(\000)c FN(\031)s FO(\))p FN(=)p FO(2)p FN(;)14 b(x)p FO(\)\))896 3020 y FK(2)958 3008 y FP(\000)25 b FN(x)39 b FO(=)g FN(\015)1281 3020 y FL(n)1326 3008 y FN(=\014)1415 3020 y FL(n)1484 3008 y FP(\000)25 b FN(\015)1617 3020 y FK(1)1654 3008 y FN(=\013)1749 3020 y FK(1)1786 3008 y FO(.)66 b(Hence)37 b FN(Y)2198 2978 y FK(2)2179 3028 y(1)2274 3008 y FO(=)-118 3107 y FN(Y)-51 3077 y FK(2)-70 3128 y(2)-14 3107 y FO(.)77 b(F)-7 b(or)40 b(an)n(y)g(irreducible)d(pair)i(\()p FN(Y)1106 3119 y FK(1)1144 3107 y FN(;)14 b(Y)1229 3119 y FK(2)1266 3107 y FO(\),)45 b(w)n(e)c(ha)n(v)n(e)e FN(Y)1773 3077 y FK(2)1754 3128 y(1)1855 3107 y FO(=)45 b FN(Y)2032 3077 y FK(2)2013 3128 y(2)2114 3107 y FO(=)g FN(\025I)7 b FO(,)-118 3207 y FN(\025)23 b(>)g FO(0.)36 b(Denote)28 b(the)g(corresp)r(onding)d(represen)n(tation)f(space)j(b)n(y)g FN(H)2052 3219 y FK(0)2090 3207 y FO(.)36 b(Then)-118 3307 y(dim)12 b FN(H)103 3319 y FK(0)166 3307 y FP(\024)25 b FO(2.)40 b(Let)29 b FP(f)p FN(e)592 3319 y FL(k)632 3307 y FN(;)14 b(k)28 b FP(2)e FN(K)6 b FP(g)28 b FO(b)r(e)h(an)g (orthonormal)c(basis)i(in)h FN(H)2048 3319 y FK(0)2085 3307 y FO(.)41 b(Then)-118 3406 y FP(f)p FN(U)-10 3376 y FL(m)52 3406 y FN(e)91 3418 y FL(k)165 3406 y FP(j)33 b FN(m)g FO(=)f(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(n)23 b FP(\000)f FO(1)p FN(;)27 b(k)36 b FP(2)d FN(K)6 b FP(g)33 b FO(is)f(an)i(orthonormal)29 b(basis)j(in)h FN(H)7 b FO(,)-118 3506 y(and)27 b(the)h(corresp)r(onding)d(op)r(erators)h FN(J)1134 3518 y FK(1)1171 3506 y FO(,)i FN(J)1268 3518 y FK(2)1333 3506 y FO(act)f(as)g(follo)n(ws:)651 3796 y FN(J)697 3808 y FK(1)757 3796 y FO(=)845 3604 y Fy(0)845 3751 y(B)845 3804 y(@)918 3668 y FN(\025)966 3680 y FK(1)1003 3668 y FN(I)1134 3765 y FO(.)1166 3790 y(.)1199 3815 y(.)1309 3923 y FN(\025)1357 3935 y FL(n)1403 3923 y FN(I)1446 3604 y Fy(1)1446 3751 y(C)1446 3804 y(A)1532 3796 y FN(;)p eop %%Page: 147 151 147 150 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(147)211 289 y FN(J)257 301 y FK(2)318 289 y FO(=)405 22 y Fy(0)405 168 y(B)405 218 y(B)405 268 y(B)405 318 y(B)405 371 y(@)557 83 y FN(Y)605 95 y FK(1)804 83 y FN(\026)854 95 y FK(1)891 83 y FO(\()p FN(\025)p FO(\))p FN(I)478 238 y(\026)528 250 y FK(1)565 238 y FO(\()p FN(\025)p FO(\))p FN(I)192 b FO(0)1255 180 y(.)1287 205 y(.)1319 230 y(.)882 335 y(.)914 360 y(.)947 385 y(.)1255 335 y(.)1287 360 y(.)1319 385 y(.)1550 393 y FN(\026)1600 405 y FL(n)p FM(\000)p FK(1)1730 393 y FO(\()p FN(\025)p FO(\))p FN(I)1130 493 y(\026)1180 505 y FL(n)p FM(\000)p FK(1)1311 493 y FO(\()p FN(\025)p FO(\))p FN(I)216 b(Y)1723 505 y FK(2)1886 22 y Fy(1)1886 168 y(C)1886 218 y(C)1886 268 y(C)1886 318 y(C)1886 371 y(A)1972 289 y FN(;)-118 643 y FO(where)-78 850 y FN(\026)-28 862 y FL(m)35 850 y FO(\()p FN(\025)p FO(\))25 b(=)259 733 y Fy(\022)320 850 y FQ(F)380 815 y FL(m)p FM(\000)p FK(1)528 733 y Fy(\022)599 793 y FN(\033)d FP(\000)c FN(\031)p 599 831 203 4 v 680 907 a FO(2)812 850 y FN(;)c FP(\000)935 793 y FO(sin)o(\()p FN(\033)s(=)p FO(2\))p 924 831 324 4 v 924 907 a(sin)e(3)p FN(\033)s(=)p FO(2\))1270 850 y FN(\025)1318 815 y FK(2)1374 850 y FO(+)1601 793 y(cos)o(\()p FN(\033)s(=)p FO(2\))p 1467 831 577 4 v 1467 907 a(2)i(sin)n(\(3)p FN(\033)s(=)p FO(2\))g(sin)e FN(\033)2054 733 y Fy(\023\023)2176 750 y FK(1)p FL(=)p FK(2)2176 933 y(2)172 1109 y FO(=)259 992 y Fy(\022)533 1053 y FO(sin)635 1018 y FK(2)673 1053 y FO(\()p FN(m\033)s FO(\))19 b FP(\000)f FO(4)p FN(\025)1052 1023 y FK(2)1103 1053 y FO(sin)1205 1018 y FK(2)1242 1053 y FO(\()p FN(\033)s(=)p FO(2\))c(sin)1556 1018 y FK(2)1607 1053 y FN(\033)p 330 1090 1531 4 v 330 1173 a FO(4)g(sin)488 1138 y FK(2)539 1173 y FN(\033)j FO(sin)o(\(\(2)p FN(m)h FP(\000)g FO(1\))p FN(\033)s(=)p FO(2\))c(sin)o(\(\(2)p FN(m)k FO(+)g(1\))p FN(\033)s(=)p FO(2\))1870 992 y Fy(\023)1932 1009 y FK(1)p FL(=)p FK(2)2036 1109 y FN(;)481 1296 y(m)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FN(;)100 1478 y(\025)24 b FP(2)250 1386 y Fy(n)305 1478 y FN(x)g FP(2)f FI(R)537 1382 y Fy(\014)537 1432 y(\014)537 1482 y(\014)588 1386 y(\020)638 1478 y FQ(F)698 1443 y FK(\()p FL(m)p FM(\000)p FK(1\))898 1386 y Fy(\020)957 1422 y FN(\033)f FP(\000)c FN(\031)p 957 1459 203 4 v 1037 1535 a FO(2)1169 1478 y FN(;)c FP(\000)1301 1422 y FO(sin)o(\()p FN(\033)s(=)p FO(2\))p 1281 1459 342 4 v 1281 1535 a(sin)o(\(3)p FN(\033)s(=)p FO(2\))1632 1478 y FN(x)1679 1443 y FK(2)665 1709 y FO(+)892 1653 y(cos)o(\()p FN(\033)s(=)p FO(2\))p 758 1690 577 4 v 758 1766 a(2)g(sin)n(\(3)p FN(\033)s(=)p FO(2\))g(sin)e FN(\033)1345 1617 y Fy(\021)o(\021)1444 1767 y FK(2)1504 1709 y FN(>)23 b FO(0)p FN(;)k FO(1)c FP(\024)f FN(m)h FP(\024)g FN(n)18 b FP(\000)g FO(1)2213 1592 y Fy(\033)2275 1709 y FN(:)6 1913 y FO(An)n(y)27 b(irreducible)c(pair)i(\()p FN(Y)843 1925 y FK(1)881 1913 y FN(;)14 b(Y)966 1925 y FK(2)1004 1913 y FO(\))27 b(is)e(unitarily)f(equiv)-5 b(alen)n(t)24 b(to)j(one)g(of)f(the)-118 2013 y(follo)n(wing:)6 2113 y(a\))e(one-dimensional:)30 b FN(Y)792 2125 y FK(1)852 2113 y FO(=)23 b(\()p FP(\000)p FO(1\))1111 2082 y FL(i)1138 2113 y FN(\025)p FO(,)i FN(Y)1282 2125 y FK(2)1343 2113 y FO(=)d(\()p FP(\000)p FO(1\))1601 2082 y FL(j)1636 2113 y FN(\025)p FO(,)j FN(\025)f(>)e FO(0,)i FN(i;)14 b(j)28 b FO(=)23 b(0)p FN(;)14 b FO(1;)6 2212 y(b\))29 b(t)n(w)n(o-dimensional:)293 2416 y FN(Y)341 2428 y FK(1)402 2416 y FO(=)489 2299 y Fy(\022)550 2366 y FN(\025)120 b FO(0)554 2465 y(0)86 b FP(\000)p FN(\025)795 2299 y Fy(\023)870 2416 y FN(;)97 b(Y)1038 2428 y FK(2)1098 2416 y FO(=)23 b FN(\025)1248 2299 y Fy(\022)1309 2366 y FO(cos)13 b FN(')127 b FO(sin)13 b FN(')1314 2465 y FO(sin)g FN(')88 b FP(\000)14 b FO(cos)e FN(')1829 2299 y Fy(\023)1904 2416 y FN(;)-118 2621 y FO(with)27 b FN(\025)c(>)g FO(0,)k FN(')d FP(2)f FO(\(0)p FN(;)14 b(\031)s FO(\).)6 2720 y(Hence,)23 b(all)d(irreducible)d(represen)n(tations)i FN(J)1401 2732 y FK(1)1438 2720 y FO(,)k FN(J)1530 2732 y FK(2)1589 2720 y FO(ha)n(v)n(e)d(dimensions)e FN(n)k FO(or)-118 2820 y(2)p FN(n)p FO(;)f(moreo)n(v)n(er,)c(t)n(w)n(o)h(suc)n (h)g(pairs)f(\()p FN(J)991 2832 y FK(1)1029 2820 y FN(;)d(J)1112 2832 y FK(2)1149 2820 y FO(\),)21 b(\()p FN(J)1311 2790 y FM(0)1303 2840 y FK(1)1340 2820 y FN(;)14 b(J)1431 2790 y FM(0)1423 2840 y FK(2)1461 2820 y FO(\))19 b(are)f(unitarily)d (equiv)-5 b(alen)n(t)-118 2919 y(if)27 b(and)g(only)f(if)h(the)h (corresp)r(onding)c(pairs)i(\()p FN(Y)1337 2931 y FK(1)1375 2919 y FN(;)14 b(Y)1460 2931 y FK(2)1497 2919 y FO(\),)28 b(\()p FN(Y)1679 2889 y FM(0)1660 2940 y FK(1)1702 2919 y FN(;)14 b(Y)1806 2889 y FM(0)1787 2940 y FK(2)1829 2919 y FO(\))28 b(are)e(unitarily)-118 3019 y(equiv)-5 b(alen)n(t.)35 b(Th)n(us)27 b(w)n(e)g(will)e(get)i(the)h(represen)n (tations)d(of)i(series)f(2)h(and)h(3)f(of)-118 3119 y(the)h(theorem.)6 3218 y(If)k(\000)145 3230 y FK(0)210 3218 y FO(=)c(\000)g Fr(\026)418 3233 y FL(\033)r FK(\()p FL(J)521 3241 y Fx(1)553 3233 y FK(\))614 3218 y FO(is)i(a)g(prop)r(er)g(subgraph)f(of) i(\(2.42\),)g(then)g(argumen)n(ts)-118 3318 y(similar)26 b(to)32 b(those)f(in)f(1\))i(giv)n(e)d(that)j FN(\033)s FO(\()p FN(J)1177 3330 y FK(1)1215 3318 y FO(\))f(is)g(simple)d(if)j (the)h(pair)d FN(J)2075 3330 y FK(1)2113 3318 y FO(,)j FN(J)2214 3330 y FK(2)2283 3318 y FO(is)-118 3418 y(irreducible.)46 b(Using)30 b(\(2.43\),)i(one)g(can)f(easily)e(describ)r(e)h(all)g (unitarily)e(non-)-118 3517 y(equiv)-5 b(alen)n(t)38 b(irreducible)f(represen)n(tations)g(connected)j(with)g(the)h(supp)r (ort)-118 3617 y(\000)-66 3629 y FK(0)-29 3617 y FO(.)6 3716 y(3.)c(No)n(w)27 b(let)g(\000)c Fr(\026)527 3731 y FL(\033)r FK(\()p FL(J)630 3739 y Fx(1)662 3731 y FK(\))720 3716 y FO(b)r(e)28 b(the)g(follo)n(wing)23 b(graph)903 3853 y Fn(r)p 903 3855 125 4 v 99 w(r)141 b(r)p 1193 3855 V 100 w(r)766 3965 y FP(\000)891 3933 y FK(1)p 840 3947 134 4 v 840 3994 a(sin)12 b FL(\033)1301 3933 y FK(1)p 1251 3947 V 1251 3994 a(sin)f FL(\033)1055 3857 y FN(:)j(:)g(:)2126 3871 y FO(\(2.44\))p eop %%Page: 148 152 148 151 bop -118 -137 a FO(148)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(or)d(one)h(of)g(its)f(subgraphs.)34 b(Let)22 b FN(P)944 108 y FL(k)1007 96 y FO(b)r(e)h(the)f(pro)5 b(jection)20 b(on)n(to)h(the)i(eigenspace)-118 196 y(of)h FN(J)19 208 y FK(1)80 196 y FO(corresp)r(onding)d(to)j FN(\025)756 208 y FL(k)820 196 y FO(=)f FP(\000)14 b FO(cos)n(\(\()p FN(k)g FP(\000)d FO(1\))p FN(\033)s FO(\))p FN(=)j FO(sin)f FN(\033)s FO(,)25 b FN(k)h FO(=)d(1,)g FN(:)14 b(:)g(:)28 b FO(,)d FN(n)11 b FO(+)g(1,)-118 296 y FN(X)-49 308 y FL(k)16 296 y FO(=)24 b FN(P)158 308 y FL(k)q FK(+1)283 296 y FN(J)329 308 y FK(2)366 296 y FN(P)419 308 y FL(k)460 296 y FO(,)29 b FN(X)i FO(=)701 233 y Fy(P)788 254 y FL(n)788 321 y(k)q FK(=1)927 296 y FN(X)996 308 y FL(k)1037 296 y FO(.)39 b(Using)27 b(the)i(same)e(argumen)n(ts)f(w)n(e)i(can)-118 395 y(conclude)f(that)i FN(J)451 407 y FK(2)513 395 y FO(=)c FN(X)g FO(+)19 b FN(X)857 365 y FM(\003)895 395 y FO(,)29 b(and)f(that)h(the)g(op)r (erators)e FN(X)1872 407 y FL(k)1913 395 y FO(,)i FN(X)2041 365 y FM(\003)2034 419 y FL(k)2107 395 y FO(satisfy)-118 495 y(the)f(follo)n(wing)23 b(relations:)615 657 y FN(\013)668 669 y FL(k)709 657 y FN(X)785 623 y FM(\003)778 678 y FL(k)823 657 y FN(X)892 669 y FL(k)951 657 y FO(+)18 b FN(\014)1081 669 y FL(k)1122 657 y FN(X)1191 669 y FL(k)q FM(\000)p FK(1)1316 657 y FN(X)1392 623 y FM(\003)1385 678 y FL(k)q FM(\000)p FK(1)1534 657 y FO(=)k FN(\015)1664 669 y FL(k)1705 657 y FN(I)7 b(;)355 b FO(\(2.45\))-118 819 y(where)42 b FN(\013)190 831 y FL(k)278 819 y FO(=)47 b FP(\000)p FO(2)14 b(sin)n(\()p FN(k)s(\033)s FO(\),)46 b FN(\014)888 831 y FL(k)977 819 y FO(=)h(2)14 b(sin)n(\(\()p FN(k)31 b FP(\000)d FO(2\))p FN(\033)s FO(\),)47 b FN(\015)1746 831 y FL(k)1834 819 y FO(=)g FP(\000)14 b FO(cos)n(\(\()p FN(k)32 b FP(\000)-118 919 y FO(1\))p FN(\033)s FO(\))p FN(=)14 b FO(sin)f FN(\033)s FO(;)29 b(in)f(particular,)d FN(\013)870 931 y FL(n)940 919 y FO(=)g(0,)j FN(\014)1170 931 y FK(2)1232 919 y FO(=)c(0,)29 b FN(\013)1468 931 y FL(k)1508 919 y FN(\014)1555 931 y FL(m)1643 919 y FP(6)p FO(=)24 b(0,)29 b FN(k)e FP(6)p FO(=)d FN(n)p FO(,)29 b FN(m)c FP(6)p FO(=)f(2.)-118 1019 y(Moreo)n(v)n(er,)39 b(the)h(pair)e(\()p FN(J)700 1031 y FK(1)737 1019 y FN(;)14 b(J)820 1031 y FK(2)858 1019 y FO(\))39 b(is)f(irreducible)e(if)j(and)g (only)f(if)h(the)h(triple)-118 1118 y(\()p FN(J)-40 1130 y FK(1)-3 1118 y FN(;)14 b(X)r(;)g(X)218 1088 y FM(\003)255 1118 y FO(\))36 b(is)d(irreducible.)55 b(If)35 b FP(f)p FO(1)p FN(=)p FO(sin)12 b FN(\033)s FP(g)35 b FO(or)f FP(f\000)p FO(1)p FN(=)p FO(sin)10 b FN(\033)t FP(g)34 b FO(do)r(es)h(not)g(b)r(e-)-118 1218 y(long)26 b(to)h FN(\033)s FO(\()p FN(J)291 1230 y FK(1)329 1218 y FO(\),)h(then)g FN(X)7 b FO(,)27 b FN(X)803 1188 y FM(\003)869 1218 y FO(satisfy)f(the)i(relation:)238 1380 y FN(X)314 1346 y FM(\003)352 1380 y FN(X)h FO(=)23 b FN(F)591 1392 y FK(1)628 1380 y FO(\()p FN(X)7 b(X)812 1346 y FM(\003)850 1380 y FN(;)14 b(A)p FO(\))83 b(or)f FN(X)7 b(X)1373 1346 y FM(\003)1433 1380 y FO(=)23 b FN(F)1574 1392 y FK(2)1611 1380 y FO(\()p FN(X)1719 1346 y FM(\003)1757 1380 y FN(X)r(;)14 b(A)p FO(\))p FN(;)-118 1542 y FO(resp)r(ectiv)n (ely)-7 b(,)25 b(where)480 1774 y FN(F)533 1786 y FK(1)570 1774 y FO(\()p FN(\026;)14 b(\025)737 1786 y FL(k)779 1774 y FO(\))23 b(=)922 1632 y Fy(\()999 1680 y FL(\015)1034 1689 y Fv(k)1070 1680 y FM(\000)p FL(\014)1160 1689 y Fv(k)1196 1680 y FL(\026)p 999 1698 238 4 v 1078 1746 a(\013)1121 1755 y Fv(k)1246 1717 y FN(;)83 b(k)26 b FP(6)p FO(=)d FN(n;)989 1838 y FO(0)p FN(;)298 b FO(otherwise)n FN(;)480 2057 y(F)533 2069 y FK(2)570 2057 y FO(\()p FN(\026;)14 b(\025)737 2069 y FL(k)779 2057 y FO(\))23 b(=)922 1915 y Fy(\()999 1961 y FL(\015)1034 1970 y Fv(k)1070 1961 y FM(\000)p FL(\013)1165 1970 y Fv(k)1201 1961 y FL(\026)p 999 1979 243 4 v 1083 2027 a(\014)1121 2036 y Fv(k)1251 1998 y FN(;)83 b(k)26 b FP(6)p FO(=)d(2)p FN(;)989 2122 y FO(0)p FN(;)303 b FO(otherwise)o FN(:)-118 2289 y FO(Hence,)37 b(an)n(y)e(irreducible)d(represen)n(tation)h(can)i (b)r(e)g(realized)e(in)i(the)g(space)-118 2389 y FN(H)30 b FO(=)22 b FN(l)93 2401 y FK(2)130 2389 y FO(\(\001\))29 b(b)n(y)e(the)h(form)n(ulae:)558 2551 y FN(J)604 2563 y FK(1)641 2551 y FN(e)680 2563 y FL(k)743 2551 y FO(=)23 b FN(\025)879 2563 y FL(k)920 2551 y FN(e)959 2563 y FL(k)1000 2551 y FN(;)97 b(X)7 b(e)1235 2563 y FL(k)1298 2551 y FO(=)22 b FN(\026)1435 2563 y FL(k)1476 2551 y FN(e)1515 2563 y FL(k)q FK(+1)1640 2551 y FN(;)-118 2713 y FO(where)29 b(\001)e(=)f FP(f)p FN(\025)401 2725 y FL(p)439 2713 y FN(;)14 b(\025)524 2725 y FL(p)p FK(+1)647 2713 y FN(;)g(:)g(:)g(:)27 b(;)14 b(\025)893 2725 y FL(m)983 2713 y FP(j)27 b FO(0)f FP(\024)g FN(p;)14 b(m)26 b FP(\024)h FN(n)p FP(g)p FO(,)i(and)h(either)e(1)p FN(=)p FO(sin)12 b FN(\033)39 b(=)-51 b FP(2)-118 2813 y FO(\001)33 b(or)f FP(\000)p FO(1)p FN(=)p FO(sin)11 b FN(\033)45 b(=)-51 b FP(2)32 b FO(\001,)i(and)f FN(\013)870 2825 y FL(k)911 2813 y FN(\026)961 2825 y FL(k)1024 2813 y FO(+)21 b FN(\014)1157 2825 y FL(k)1198 2813 y FN(\026)1248 2825 y FL(k)q FM(\000)p FK(1)1406 2813 y FO(=)31 b FN(\015)1545 2825 y FL(k)1619 2813 y FO(suc)n(h)h(that)i FN(\026)2047 2825 y FL(k)2119 2813 y FN(>)e FO(0)g(if)-118 2912 y FN(\025)-70 2924 y FL(k)-29 2912 y FN(;)14 b(\025)56 2924 y FL(k)q FK(+1)204 2912 y FP(2)24 b FO(\001)k(and)f FN(\026)591 2924 y FL(k)655 2912 y FO(=)c(0)k(if)g FN(\025)936 2924 y FL(k)1010 2912 y FN(=)-52 b FP(2)24 b FO(\001)k(or)e FN(\025)1325 2924 y FL(k)q FK(+1)1483 2912 y FN(=)-51 b FP(2)23 b FO(\001.)6 3012 y(If)i FP(f\006)p FO(1)p FN(=)p FO(sin)11 b FN(\033)s FP(g)23 b(2)g FN(\033)s FO(\()p FN(J)712 3024 y FK(1)750 3012 y FO(\),)i(then)g(the)f(problem)e (of)i(describing)d(irreducible)-118 3112 y(represen)n(tations)41 b(\()p FN(J)552 3124 y FK(1)589 3112 y FO(,)48 b FN(J)706 3124 y FK(2)743 3112 y FO(\))c(is)f(reduced)g(to)h(that)g(of)f(pairs)f (of)i(op)r(erators)-118 3211 y(satisfying)39 b(some)g(quadratic)g (relations.)75 b(In)42 b(order)e(to)h(sho)n(w)f(that,)45 b(con-)-118 3311 y(sider)36 b(the)i(follo)n(wing)c(op)r(erators)h(in)i (the)h(subspace)f FN(P)1652 3323 y FK(2)1690 3311 y FN(H)7 b FO(:)57 b FN(D)1915 3323 y FK(1)1992 3311 y FO(=)39 b FN(X)2165 3323 y FK(1)2202 3311 y FN(X)2278 3281 y FM(\003)2271 3331 y FK(1)2316 3311 y FO(,)-118 3410 y FN(D)-49 3422 y FK(2)26 3410 y FO(=)f(\()p FN(X)230 3422 y FL(n)289 3410 y FN(:)14 b(:)g(:)g(X)469 3422 y FK(2)506 3410 y FO(\))538 3380 y FM(\003)576 3410 y FN(X)645 3422 y FL(n)704 3410 y FN(:)g(:)g(:)g(X)884 3422 y FK(2)921 3410 y FO(.)64 b(F)-7 b(rom)35 b(\(2.45\))h(w)n(e)h(ha)n(v)n(e)e FN(X)1892 3380 y FM(\003)1885 3431 y FK(1)1930 3410 y FN(X)1999 3422 y FK(1)2074 3410 y FO(=)2191 3373 y FL(\015)2226 3381 y Fx(1)p 2187 3391 76 4 v 2187 3439 a FL(\013)2230 3447 y Fx(1)2273 3410 y FN(I)7 b FO(,)-118 3535 y FN(X)-49 3547 y FL(n)10 3535 y FN(:)14 b(:)g(:)f(X)189 3547 y FK(2)226 3535 y FO(\()p FN(X)327 3547 y FL(n)387 3535 y FN(:)h(:)g(:)f(X)566 3547 y FK(2)603 3535 y FO(\))635 3505 y FM(\003)718 3535 y FO(=)44 b FN(\026I)7 b FO(,)43 b(where)d FN(\026)k FO(=)1442 3473 y Fy(Q)1520 3493 y FL(n)p FM(\000)p FK(3)1520 3560 y FL(k)q FK(=0)1650 3535 y FO(\()p FQ(F)1742 3505 y FL(k)1783 3535 y FO(\()p FN(\025)1863 3547 y FL(n)1909 3535 y FN(;)1958 3498 y FL(\015)1993 3506 y Fv(n)p 1956 3516 79 4 v 1956 3563 a FL(\014)1994 3571 y Fv(n)2045 3535 y FO(\)\))2109 3547 y FK(2)2158 3494 y FL(\015)2193 3502 y Fv(n)p Fx(+1)p 2156 3516 150 4 v 2156 3563 a FL(\014)2194 3571 y Fv(n)p Fx(+1)2316 3535 y FO(,)-118 3649 y FQ(F)p FO(\()p FN(\025)22 3661 y FL(k)q FK(+1)148 3649 y FN(;)14 b(x)p FO(\))43 b(=)g(\()p FN(\025)495 3661 y FL(k)537 3649 y FN(;)14 b FO(\()p FN(\015)649 3661 y FL(k)716 3649 y FP(\000)27 b FN(\013)861 3661 y FL(k)901 3649 y FN(x)p FO(\))p FN(=\014)1069 3661 y FL(k)1111 3649 y FO(\).)73 b(One)40 b(can)f(c)n(hec)n(k)g(that)h FN(\015)2062 3661 y FK(1)2099 3649 y FN(=\013)2194 3661 y FK(1)2274 3649 y FO(=)-118 3749 y(1)p FN(=)p FO(\(2)14 b(sin)154 3714 y FK(2)206 3749 y FN(\033)s FO(\).)37 b(Hence)237 3911 y FN(D)306 3923 y FK(1)343 3911 y FO(\()p FN(D)444 3923 y FK(1)500 3911 y FP(\000)18 b FO(1)p FN(=)p FO(\(2)c(sin)856 3876 y FK(2)907 3911 y FN(\033)s FO(\))p FN(I)7 b FO(\))24 b(=)e(0)p FN(;)97 b(D)1406 3923 y FK(2)1443 3911 y FO(\()p FN(D)1544 3923 y FK(2)1600 3911 y FP(\000)18 b FN(\026I)7 b FO(\))23 b(=)g(0)p FN(:)p eop %%Page: 149 153 149 152 bop -118 -137 a FJ(2.3.)36 b(Represen)n(tations)25 b(of)j FN(q)s FJ(-deforemd)e FN(U)9 b FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(R)p FO(\)\))631 b(149)-118 96 y(Let)33 b FN(H)105 108 y FK(0)176 96 y FO(b)r(e)h(a)e(subspace)h(of)g FN(H)40 b FO(whic)n(h)32 b(is)h(in)n(v)-5 b(arian)n(t)30 b(with)i(resp)r(ect)h(to)h FN(D)2279 108 y FK(1)2316 96 y FO(,)-118 196 y FN(D)-49 208 y FK(2)-12 196 y FO(.)j(Then)154 391 y FN(X)230 357 y FM(\003)223 412 y FK(1)268 391 y FN(H)337 403 y FK(0)393 391 y FP(\010)18 b FN(H)545 403 y FK(0)600 391 y FP(\010)g FN(X)752 403 y FK(2)789 391 y FN(H)858 403 y FK(0)914 391 y FP(\010)g FN(X)1066 403 y FK(3)1103 391 y FN(X)1172 403 y FK(2)1209 391 y FN(H)1278 403 y FK(0)1334 391 y FP(\010)g FN(:)c(:)g(:)k FP(\010)g FN(X)1684 403 y FL(n)1743 391 y FN(:)c(:)g(:)g(X)1923 403 y FK(2)1960 391 y FN(H)2029 403 y FK(0)-118 586 y FO(is)26 b(in)n(v)-5 b(arian)n(t)23 b(with)k(resp)r(ect)g(to)f FN(J)931 598 y FK(1)969 586 y FO(,)h FN(X)7 b FO(,)26 b FN(X)1220 556 y FM(\003)1258 586 y FO(.)36 b(In)28 b(fact,)f(using)e(\(2.45\))i(one)f(can)-118 686 y(easily)33 b(pro)n(v)n(e)h(that)h FN(X)614 656 y FM(\003)607 709 y FL(k)652 686 y FN(X)721 698 y FL(k)775 686 y FN(:)14 b(:)g(:)g(X)955 698 y FK(2)1029 686 y FO(=)35 b FN(\026)1179 698 y FL(k)1220 686 y FN(X)1289 698 y FL(k)q FM(\000)p FK(1)1429 686 y FN(:)14 b(:)g(:)g(X)1609 698 y FK(2)1646 686 y FO(,)37 b(1)f FN(<)g(k)j(<)d(n)g FO(\(here)-118 785 y FN(\026)-68 797 y FL(k)-4 785 y FO(=)22 b(1)p FN(=)p FO(4)14 b(sin)324 750 y FK(2)375 785 y FN(\033)s FO(\))28 b(.)37 b(Moreo)n(v)n(er,)564 980 y FN(X)633 992 y FL(n)p FM(\000)p FK(1)777 980 y FN(:)14 b(:)g(:)g(X)957 992 y FK(2)994 980 y FN(X)1070 946 y FM(\003)1063 1001 y FK(2)1121 980 y FN(:)g(:)g(:)g(X)1308 946 y FM(\003)1301 1001 y FL(n)p FM(\000)p FK(1)1454 980 y FO(=)23 b FN(\025I)7 b(;)-118 1191 y FO(where)31 b FN(\025)e FO(=)297 1129 y Fy(Q)375 1149 y FL(n)p FM(\000)p FK(3)375 1216 y FL(k)q FK(=0)505 1191 y FO(\()p FQ(F)597 1161 y FK(\()p FL(k)q FK(\))690 1191 y FO(\()p FN(\025)770 1203 y FL(n)816 1191 y FN(;)14 b(\015)896 1203 y FL(n)941 1191 y FN(=\014)1030 1203 y FL(n)1075 1191 y FO(\)\))1139 1203 y FK(2)1177 1191 y FO(.)47 b(Assuming)29 b(that)j FN(\025)d FO(=)g(0,)j(w)n(e)e (will)-118 1290 y(ha)n(v)n(e)36 b(that)i FN(X)349 1260 y FM(\003)342 1311 y FK(3)401 1290 y FN(:)14 b(:)g(:)f(X)587 1260 y FM(\003)580 1311 y FL(n)p FM(\000)p FK(1)710 1290 y FN(P)763 1302 y FL(n)809 1290 y FN(H)32 b FP(\010)24 b FN(:)14 b(:)g(:)g(P)1163 1302 y FL(n)1209 1290 y FN(H)31 b FP(\010)25 b FN(X)1468 1302 y FL(n)1513 1290 y FN(P)1566 1302 y FL(n)1612 1290 y FN(H)44 b FO(is)37 b(in)n(v)-5 b(arian)n(t)34 b(with)-118 1390 y(resp)r(ect)k(to)g FN(J)335 1402 y FK(1)372 1390 y FO(,)j FN(J)482 1402 y FK(2)519 1390 y FO(,)g(hence)d FN(\033)s FO(\()p FN(J)952 1402 y FK(1)990 1390 y FO(\))i FP(63)h(f)p FO(1)p FN(=)p FO(sin)11 b FN(\033)t FP(g)38 b FO(if)f(the)h(pair)f(\()p FN(J)2028 1402 y FK(1)2065 1390 y FO(,)k FN(J)2175 1402 y FK(2)2212 1390 y FO(\))e(is)-118 1490 y(irreducible,)24 b(whic)n(h)i(con)n (tradicts)g(the)i(assumption.)34 b(Th)n(us)27 b FN(\025)d FP(6)p FO(=)f(0,)397 1685 y FN(X)473 1650 y FM(\003)466 1705 y FL(n)511 1685 y FN(X)580 1697 y FL(n)639 1685 y FN(:)14 b(:)g(:)g(X)819 1697 y FK(2)856 1685 y FN(H)925 1697 y FK(0)985 1685 y FO(=)23 b FN(\025)1121 1650 y FM(\000)p FK(1)1210 1685 y FN(\025X)1334 1650 y FM(\003)1327 1705 y FL(n)1373 1685 y FN(X)1442 1697 y FL(n)1500 1685 y FN(:)14 b(:)g(:)g(X)1680 1697 y FK(2)1717 1685 y FN(H)1786 1697 y FK(0)250 1820 y FO(=)22 b FN(\025)385 1785 y FM(\000)p FK(1)475 1820 y FO(\()p FN(X)576 1832 y FL(n)p FM(\000)p FK(1)720 1820 y FN(:)14 b(:)g(:)g(X)900 1832 y FK(2)937 1820 y FN(X)1013 1785 y FM(\003)1006 1840 y FK(2)1064 1820 y FN(:)g(:)g(:)g(X)1251 1785 y FM(\003)1244 1840 y FL(n)p FM(\000)p FK(1)1374 1820 y FO(\))p FN(X)1482 1785 y FM(\003)1475 1840 y FL(n)1520 1820 y FN(X)1589 1832 y FL(n)1648 1820 y FN(:)g(:)g(:)f(X)1827 1832 y FK(2)1864 1820 y FN(H)1933 1832 y FK(0)787 1944 y FP(\032)23 b FN(X)944 1956 y FL(n)p FM(\000)p FK(1)1087 1944 y FN(:)14 b(:)g(:)g(X)1267 1956 y FK(2)1304 1944 y FN(H)1373 1956 y FK(0)1410 1944 y FN(:)6 2145 y FO(An)n(y)38 b(irreducible)c(pair)i (\()p FN(D)897 2157 y FK(1)934 2145 y FN(;)14 b(D)1040 2157 y FK(2)1077 2145 y FO(\))38 b(is)e(one-)h(or)g(t)n(w)n (o-dimensional.)61 b(Let)-118 2245 y(us)33 b(describ)r(e)e(the)j (corresp)r(onding)c(irreducible)f(pairs)i FN(J)1679 2257 y FK(1)1716 2245 y FO(,)k FN(J)1820 2257 y FK(2)1857 2245 y FO(.)53 b(Denote)33 b(the)-118 2345 y(phase)h(of)g(the)h(op)r (erator)e FN(X)782 2357 y FL(i)823 2345 y FN(:)14 b(:)g(:)g(X)1003 2357 y FK(2)1074 2345 y FO(b)n(y)34 b FN(U)1253 2357 y FL(i)1281 2345 y FO(,)i(the)f(phase)e(of)i FN(X)1905 2315 y FM(\003)1898 2365 y FK(1)1977 2345 y FO(b)n(y)f FN(U)2165 2315 y FM(\003)2156 2365 y FK(1)2203 2345 y FO(.)57 b(If)-118 2444 y(dim)12 b FN(H)103 2456 y FK(0)163 2444 y FP(\024)23 b FO(2)k(and)g FN(e)520 2456 y FK(1)556 2444 y FO(,)h FN(e)646 2456 y FK(2)710 2444 y FO(is)e(an)g(orthonormal) d(basis)j(in)g(the)h(space)g FN(H)2115 2456 y FK(0)2179 2444 y FO(suc)n(h)-118 2544 y(that)h FN(e)101 2556 y FK(1)161 2544 y FP(2)23 b FO(\(k)n(er)13 b FN(D)465 2556 y FK(2)502 2544 y FO(\))534 2514 y FM(?)590 2544 y FO(,)28 b(then)89 2789 y FN(D)158 2801 y FK(1)218 2789 y FO(=)424 2733 y(1)p 315 2770 259 4 v 315 2852 a(2)14 b(sin)473 2817 y FK(2)524 2852 y FN(\033)598 2672 y Fy(\022)659 2738 y FO(1)k(+)g(cos)13 b FN(')159 b FO(sin)13 b FN(')735 2838 y FO(sin)g FN(')159 b FO(1)18 b FP(\000)g FO(cos)13 b FN(')1387 2672 y Fy(\023)1462 2789 y FN(;)96 b(D)1650 2801 y FK(2)1710 2789 y FO(=)1798 2672 y Fy(\022)1859 2738 y FN(\026)83 b FO(0)1864 2838 y(0)j(0)2034 2672 y Fy(\023)2109 2789 y FN(;)-118 3029 y(U)-61 3041 y FL(i)-34 3029 y FN(e)5 3041 y FK(1)42 3029 y FO(,)47 b FN(U)169 3041 y FL(i)196 3029 y FN(e)235 3041 y FK(2)316 3029 y FO(is)41 b(an)i(orthonormal)c(basis)j(in)g FN(X)1437 3041 y FL(i)1478 3029 y FN(:)14 b(:)g(:)g(X)1658 3041 y FK(2)1695 3029 y FN(H)1764 3041 y FK(0)1801 3029 y FO(,)47 b(2)h FP(\024)h FN(i)f(<)h(n)p FO(,)-118 3129 y(and)27 b(the)h(v)n(ectors)e FN(U)525 3141 y FL(n)570 3129 y FN(e)609 3141 y FK(1)674 3129 y FO(and)h FN(U)892 3141 y FK(1)929 3129 y FO(\(cos)o(\()p FN('=)p FO(2\))p FN(e)1313 3141 y FK(1)1368 3129 y FO(+)18 b(sin)o(\()p FN('=)p FO(2\))p FN(e)1794 3141 y FK(2)1831 3129 y FO(\))28 b(generate)e(the)-118 3228 y(spaces)20 b FN(X)199 3240 y FL(n)258 3228 y FN(:)14 b(:)g(:)g(X)438 3240 y FK(2)475 3228 y FN(H)544 3240 y FK(0)602 3228 y FO(and)21 b FN(X)833 3198 y FM(\003)826 3249 y FK(1)870 3228 y FN(H)939 3240 y FK(0)977 3228 y FO(,)h(resp)r(ectiv)n(ely)-7 b(.)32 b(Note)21 b(that)h(cos)o(\()p FN('=)p FO(2\))14 b FN(e)2232 3240 y FK(1)2274 3228 y FO(+)-118 3328 y(sin)o(\()p FN('=)p FO(2\))g FN(e)239 3340 y FK(2)306 3328 y FP(2)31 b FO(\(k)n(er)13 b FN(D)618 3340 y FK(1)655 3328 y FO(\))687 3298 y FM(?)743 3328 y FO(.)50 b(T)-7 b(o)32 b(describ)r(e)f(the)h(action)f(of)h(the)g (op)r(erator)e FN(J)2301 3340 y FK(2)-118 3428 y FO(on)22 b(this)g(basis,)g(it)g(is)g(su\016cien)n(t)g(to)g(kno)n(w)g(it)g(for)g (the)h(op)r(erators)e FN(X)1951 3440 y FL(i)1978 3428 y FO(.)36 b(W)-7 b(e)23 b(ha)n(v)n(e)109 3698 y FN(X)178 3710 y FL(i)p FK(+1)290 3698 y FN(U)347 3710 y FL(i)374 3698 y FN(e)413 3710 y FL(k)476 3698 y FO(=)564 3606 y Fy(\020)614 3594 y FL(i)p FM(\000)p FK(1)615 3619 y Fy(Y)615 3798 y FL(l)p FK(=2)736 3698 y FN(\026)786 3710 y FL(l)811 3606 y Fy(\021)861 3623 y FM(\000)p FK(1)p FL(=)p FK(2)1017 3698 y FN(X)1086 3710 y FL(i)p FK(+1)1212 3698 y FN(:)14 b(:)g(:)f(X)1391 3710 y FK(2)1428 3698 y FN(e)1467 3710 y FL(k)1531 3698 y FO(=)1619 3642 y FP(p)p 1688 3642 78 4 v 56 x FN(\026)1738 3710 y FL(i)1779 3698 y FN(U)1836 3710 y FL(i)p FK(+1)1948 3698 y FN(e)1987 3710 y FL(k)476 3911 y FO(=)23 b(\(2)14 b(sin)e FN(\033)t FO(\))850 3877 y FM(\000)p FK(1)939 3911 y FN(U)996 3923 y FL(i)p FK(+1)1108 3911 y FN(e)1147 3923 y FL(k)1187 3911 y FN(;)180 b(i)23 b(<)f(n)d FP(\000)f FO(1;)p eop %%Page: 150 154 150 153 bop -118 -137 a FO(150)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)73 163 y FN(X)142 175 y FL(n)187 163 y FN(U)244 175 y FL(n)p FM(\000)p FK(1)374 163 y FN(e)413 175 y FL(k)476 163 y FO(=)564 71 y Fy(\020)614 59 y FL(n)p FM(\000)p FK(1)624 84 y Fy(Y)624 263 y FL(l)p FK(=2)754 163 y FN(\026)804 175 y FL(l)829 71 y Fy(\021)879 88 y FM(\000)p FK(1)p FL(=)p FK(2)1035 163 y FN(X)1104 175 y FL(n)1163 163 y FN(:)14 b(:)g(:)f(X)1342 175 y FK(2)1393 163 y FN(e)1432 175 y FL(k)476 450 y FO(=)564 308 y Fy(\()631 393 y FO(0)p FN(;)678 b(k)26 b FO(=)d(2)p FN(;)631 513 y FO(\()663 444 y FP(p)p 732 444 42 4 v 69 x FO(2)14 b(sin)f FN(\033)s FO(\))986 482 y FK(1)p FL(=)p FK(2)1091 513 y FN(U)1148 525 y FL(n)1192 513 y FN(e)1231 525 y FK(1)1268 513 y FN(;)83 b(k)26 b FO(=)d(1)p FN(;)121 656 y(X)190 668 y FK(1)227 656 y FN(U)293 622 y FM(\003)284 677 y FK(1)331 656 y FO(\(cos\()p FN('=)p FO(2\))14 b FN(e)730 668 y FK(1)785 656 y FO(+)k(sin)o(\()p FN('=)p FO(2\))c FN(e)1225 668 y FK(2)1261 656 y FO(\))476 799 y(=)564 726 y FP(p)p 633 726 V 73 x FO(2)g(sin)e FN(\033)s(X)923 811 y FK(1)961 799 y FN(X)1037 765 y FM(\003)1030 819 y FK(1)1074 799 y FO(\(cos\()p FN('=)p FO(2\))i FN(e)1473 811 y FK(1)1528 799 y FO(+)k(sin)o(\()p FN('=)p FO(2\))c FN(e)1968 811 y FK(2)2004 799 y FO(\))24 b(=)476 941 y(=)f(\()596 869 y FP(p)p 666 869 V 666 941 a FO(2)13 b(sin)g FN(\033)s FO(\))919 907 y FM(\000)p FK(1)1009 941 y FO(\(cos)o(\()p FN('=)p FO(2\))h FN(e)1407 953 y FK(1)1462 941 y FO(+)k(sin)o(\()p FN('=)p FO(2\))c FN(e)1902 953 y FK(2)1939 941 y FO(\))p FN(:)6 1111 y FO(Th)n(us)34 b(w)n(e)g(will)d(get)j(represen)n(tations)d(from)h (series)g(5.)56 b(If)35 b(dim)12 b FN(H)2105 1123 y FK(0)2176 1111 y FO(=)33 b(1,)-118 1211 y(then)f FN(D)144 1223 y FK(1)210 1211 y FP(6)p FO(=)d(0)i(and)g FN(D)611 1223 y FK(2)678 1211 y FP(6)p FO(=)e(0,)j(since)e(otherwise)f FN(\033)s FO(\()p FN(J)1577 1223 y FK(1)1615 1211 y FO(\))h FP(6)p FO(=)f FP(f)p FN(\025)1861 1223 y FK(1)1898 1211 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)2144 1223 y FL(n)p FK(+1)2274 1211 y FP(g)p FO(.)-118 1311 y(If)39 b FI(C)15 b FN(e)69 1323 y FK(1)153 1311 y FO(=)40 b FN(H)327 1323 y FK(0)364 1311 y FO(,)i(then)c(using)f(the)i(same)e(argumen)n(ts)f (one)i(can)g(sho)n(w)f(that)-118 1410 y FP(f)p FN(U)-10 1380 y FM(\003)-19 1431 y FK(1)27 1410 y FN(e)66 1422 y FK(1)103 1410 y FN(;)14 b(e)179 1422 y FK(1)216 1410 y FN(;)g(U)310 1422 y FK(1)347 1410 y FN(e)386 1422 y FK(1)423 1410 y FN(;)g(:)g(:)g(:)f(U)627 1422 y FL(n)672 1410 y FN(e)711 1422 y FK(1)748 1410 y FP(g)31 b FO(is)g(an)g (orthogonal)d(basis)i(in)h FN(H)7 b FO(,)33 b(and)e(the)h(cor-)-118 1510 y(resp)r(onding)26 b(irreducible)e(represen)n(tations)g(\()p FN(J)1369 1522 y FK(1)1407 1510 y FN(;)14 b(J)1490 1522 y FK(2)1527 1510 y FO(\))28 b(form)e(series)g(6.)6 1609 y(4.)37 b(Supp)r(ose)28 b(that)f(the)h(supp)r(ort)g(of)f FN(J)1204 1621 y FK(2)1269 1609 y FO(is)903 1762 y Fn(r)p 903 1763 4 4 v 898 1758 V 893 1754 V 889 1749 V 886 1744 V 882 1740 V 880 1736 V 877 1732 V 875 1728 V 873 1724 V 872 1720 V 870 1716 V 870 1713 V 869 1709 V 870 1706 V 870 1703 V 871 1700 V 872 1697 V 874 1694 V 875 1691 V 878 1689 V 878 1689 V 880 1686 V 883 1684 V 885 1682 V 888 1681 V 890 1679 V 893 1678 V 895 1677 V 898 1677 V 900 1676 V 903 1676 V 905 1676 V 908 1677 V 910 1677 V 913 1678 V 915 1679 V 918 1681 V 920 1682 V 923 1684 V 925 1686 V 928 1689 V 903 1763 V 907 1758 V 912 1754 V 916 1749 V 920 1744 V 923 1740 V 926 1736 V 928 1732 V 931 1728 V 932 1724 V 934 1720 V 935 1716 V 936 1713 V 936 1709 V 936 1706 V 935 1703 V 935 1700 V 933 1697 V 932 1694 V 930 1691 V 928 1689 V 903 1763 125 4 v 99 w(r)141 b(r)p 1193 1763 V 100 w(r)881 1861 y FL(\025)920 1869 y Fx(1)1273 1861 y FL(\025)1312 1872 y Fx(\()p Fv(n)p Fx(+1\))p Fv(=)p Fx(2)1055 1766 y FN(:)14 b(:)g(:)2126 1780 y FO(\(2.46\))-118 2002 y(where)35 b FN(\025)178 2014 y FK(1)251 2002 y FO(=)g FP(\006)p FO(\(cos)13 b FN(\033)s(=)p FO(2\))p FN(=)p FO(sin)f FN(\033)t FO(,)37 b FN(\025)1055 2017 y FK(\()p FL(n)p FK(+1\))p FL(=)p FK(2)1340 2002 y FO(=)e(1)p FN(=)p FO(sin)12 b FN(\033)s FO(.)60 b(Let)36 b(us)f(consider)-118 2102 y(the)f(case)e FN(\025)260 2114 y FK(1)330 2102 y FO(=)g(\(cos)14 b FN(\033)s(=)p FO(2)o(\))p FN(=)p FO(sin)f FN(\033)36 b FO(\(for)e FN(\025)1205 2114 y FK(1)1275 2102 y FO(=)e FP(\000)p FO(\(cos)13 b FN(\033)s(=)p FO(2)o(\))p FN(=)p FO(sin)g FN(\033)37 b FO(the)c(pro)r(of)-118 2202 y(is)28 b(the)i(same\).)41 b(The)29 b(op)r(erator)f FN(J)945 2214 y FK(2)1011 2202 y FO(can)h(b)r(e)h(represen)n(ted)e(in)h(the)h (form)e FN(J)2211 2214 y FK(2)2274 2202 y FO(=)-118 2317 y FN(X)h FO(+)22 b FN(X)143 2287 y FM(\003)203 2317 y FO(+)g FN(Y)d FO(,)35 b(where)f FN(X)40 b FO(=)869 2255 y Fy(P)956 2274 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)956 2342 y FL(k)q FK(=1)1219 2317 y FN(P)1272 2329 y FL(k)q FK(+1)1398 2317 y FN(J)1444 2329 y FK(2)1481 2317 y FN(P)1534 2329 y FL(k)1575 2317 y FO(,)c FN(Y)52 b FO(=)33 b FN(P)1885 2329 y FK(1)1922 2317 y FN(J)1968 2329 y FK(2)2006 2317 y FN(P)2059 2329 y FK(1)2096 2317 y FO(,)j FN(P)2208 2329 y FL(k)2283 2317 y FO(is)-118 2417 y(the)22 b(pro)5 b(jection)20 b(on)h(the)i(eigenspace)c(whic)n(h)i (corresp)r(onds)f(to)h(the)i(eigen)n(v)-5 b(alue)-118 2517 y FN(\025)-70 2529 y FL(k)-6 2517 y FO(=)23 b(\(cos)o(\(\(2)p FN(k)f FP(\000)c FO(1\))p FN(\033)s(=)p FO(2\))o(\))p FN(=)p FO(sin)13 b FN(\033)s FO(.)37 b(Let)28 b FN(X)1236 2529 y FL(k)1300 2517 y FO(=)22 b FN(X)7 b(P)1516 2529 y FL(k)1557 2517 y FO(.)37 b(Then)325 2687 y FN(\013)378 2699 y FL(k)419 2687 y FN(X)495 2652 y FM(\003)488 2707 y FL(k)532 2687 y FN(X)601 2699 y FL(k)660 2687 y FO(+)18 b FN(\014)790 2699 y FL(k)831 2687 y FN(X)900 2699 y FL(k)q FM(\000)p FK(1)1026 2687 y FN(X)1102 2652 y FM(\003)1095 2707 y FL(k)q FM(\000)p FK(1)1243 2687 y FO(=)23 b FN(\015)1374 2699 y FL(k)1415 2687 y FN(I)7 b(;)180 b(k)26 b FP(6)p FO(=)c(1)p FN(;)530 2822 y(\013)583 2834 y FK(1)621 2822 y FN(X)697 2787 y FM(\003)690 2842 y FK(1)734 2822 y FN(X)803 2834 y FK(1)859 2822 y FO(+)c FN(\014)989 2834 y FK(1)1026 2822 y FN(Y)1093 2787 y FK(2)1130 2822 y FN(P)1183 2834 y FK(1)1243 2822 y FO(=)23 b FN(\015)1374 2834 y FK(1)1411 2822 y FN(I)7 b(;)649 b FO(\(2.47\))-118 2992 y(where)39 b FN(\013)187 3004 y FL(k)271 2992 y FO(=)k FP(\000)p FO(2)14 b(sin)n(\(\(2)p FN(k)29 b FO(+)d(1\))p FN(\033)s(=)p FO(2\),)43 b FN(\014)1223 3004 y FL(k)1307 2992 y FO(=)g(2)14 b(sin)n(\(\(2)p FN(k)29 b FP(\000)e FO(3\))p FN(\033)s(=)p FO(2\),)42 b FN(\015)2190 3004 y FL(k)2274 2992 y FO(=)-118 3091 y FP(\000)14 b FO(cos)n(\(\(2)p FN(k)25 b FP(\000)d FO(1\))p FN(\033)s(=)p FO(2\))p FN(=)14 b FO(sin)d FN(\033)s FO(;)36 b(in)c(particular,)f FN(\013)1419 3103 y FL(k)1492 3091 y FP(6)p FO(=)g(0)h(for)h FN(k)h FP(6)p FO(=)d(\()p FN(n)22 b FP(\000)g FO(1\))p FN(=)p FO(2,)-118 3191 y FN(\013)-65 3206 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)214 3191 y FO(=)29 b(0.)48 b(As)32 b(ab)r(o)n(v)n(e,)f(the)h(problem)d(of)j(describing)c(irreducible)g (rep-)-118 3291 y(resen)n(tations)38 b(\()p FN(J)433 3303 y FK(1)470 3291 y FN(;)14 b(J)553 3303 y FK(2)590 3291 y FO(\))41 b(reduces)e(to)i(that)f(of)g(irreducible)d(pairs)h(\()p FN(D)2126 3303 y FK(1)2163 3291 y FN(;)14 b(D)2269 3303 y FK(2)2306 3291 y FO(\))-118 3390 y(satisfying)25 b(some)h(quadratic)f (relation.)34 b(Here)196 3560 y FN(D)265 3572 y FK(1)325 3560 y FO(=)22 b FN(Y)5 b(;)97 b(D)654 3572 y FK(2)714 3560 y FO(=)23 b(\()p FN(X)903 3575 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)1166 3560 y FN(:)14 b(:)g(:)g(X)1346 3572 y FK(1)1383 3560 y FO(\))1415 3526 y FM(\003)1453 3560 y FN(X)1522 3575 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)1785 3560 y FN(:)g(:)g(:)g(X)1965 3572 y FK(1)2002 3560 y FN(;)-118 3741 y FO(and)27 b FN(D)112 3753 y FK(2)149 3741 y FO(\()p FN(D)250 3753 y FK(2)306 3741 y FP(\000)18 b FN(\026I)7 b FO(\))23 b(=)g(0,)k FN(D)788 3711 y FK(2)786 3762 y(1)849 3741 y FO(=)22 b(1)p FN(=)p FO(\(4)14 b(sin)1209 3706 y FK(2)1260 3741 y FN(\033)s FO(\))p FN(I)7 b FO(,)28 b FN(\026)g FO(is)f(de\014ned)h(b) n(y)397 3911 y FN(X)466 3926 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)729 3911 y FN(:)14 b(:)g(:)f(X)908 3923 y FK(1)945 3911 y FO(\()p FN(X)1046 3926 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)1310 3911 y FN(:)h(:)g(:)f(X)1489 3923 y FK(1)1526 3911 y FO(\))1558 3877 y FM(\003)1620 3911 y FO(=)22 b FN(\026I)7 b(;)p eop %%Page: 151 155 151 154 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(151)-118 96 y(whic)n(h)26 b(can)i(easily)c(b)r(e)k(obtained)f(from)f(\(2.47\).)6 204 y(If)39 b FN(H)169 216 y FK(0)245 204 y FO(is)e(in)n(v)-5 b(arian)n(t)35 b(with)j(resp)r(ect)g(to)g FN(D)1374 216 y FK(1)1411 204 y FO(,)j FN(D)1544 216 y FK(2)1581 204 y FO(,)g(then)e FN(H)1914 216 y FK(0)1977 204 y FP(\010)25 b FN(X)7 b(H)2212 216 y FK(0)2274 204 y FP(\010)-118 303 y FN(:)14 b(:)g(:)26 b FP(\010)f FN(X)171 273 y FK(\()p FL(n)p FM(\000)p FK(1\))p FL(=)p FK(2)420 303 y FN(H)489 315 y FK(0)565 303 y FO(is)38 b(in)n(v)-5 b(arian)n(t)35 b(with)k(resp)r(ect)g(to)f FN(J)1674 315 y FK(1)1712 303 y FO(,)j FN(X)7 b FO(,)42 b FN(X)1993 273 y FM(\003)2030 303 y FO(.)71 b(More-)-118 403 y(o)n(v)n(er,)23 b(the)i(dimensions)c (of)j(the)h(irreducible)c(represen)n(tations)g(are)i(\()p FN(n)12 b FO(+)g(1\))p FN(=)p FO(2.)-118 502 y(This)25 b(follo)n(ws)f(b)n(y)i(the)g(same)f(metho)r(d)h(as)g(in)f(the)i (previous)d(case)i(and)g(w)n(e)g(get)-118 602 y(represen)n(tations)f (6,)i(7.)6 709 y(If)45 b(\()p FN(J)184 721 y FK(1)222 709 y FN(;)14 b(J)305 721 y FK(2)342 709 y FO(\))44 b(is)g(irreducible) c(and)k FN(\033)s FO(\()p FN(J)1252 721 y FK(1)1290 709 y FO(\))51 b FP(\032)g(f)p FN(\025)1579 721 y FK(1)1616 709 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(\025)1862 724 y FK(\()p FL(n)p FK(+1\))p FL(=)p FK(2)2111 709 y FP(g)44 b FO(is)f(a)-118 809 y(prop)r(er)f(subset)i(then,)k(as)42 b(b)r(efore,)48 b(w)n(e)43 b(conclude)f(that)h FN(\033)s FO(\()p FN(J)1894 821 y FK(1)1932 809 y FO(\))h(is)f(simple)-118 909 y(and)34 b(the)g(irreducible)d(represen)n(tation)g(is)i(realized)e (in)i FN(l)1695 921 y FK(2)1732 909 y FO(\(\001\),)k(where)c(\001)h(=) -118 1008 y FP(f)p FN(\025)-28 1020 y FL(p)10 1008 y FN(;)14 b(\025)95 1020 y FL(p)p FK(+1)218 1008 y FN(;)g(:)g(:)g(:)g(;)g (\025)451 1020 y FL(m)514 1008 y FP(g)p FO(,)24 b(for)f(some)f(1)h FP(\024)g FN(p;)14 b(m)22 b FP(\024)h FO(\()p FN(n)11 b FO(+)g(1\))p FN(=)p FO(2)21 b(and)j(either)e FN(\025)2130 1020 y FK(1)2200 1008 y FN(=)-51 b FP(2)23 b FO(\001)-118 1108 y(or)k FN(\025)32 1123 y FK(\()p FL(n)p FK(+1\))p FL(=)p FK(2)313 1108 y FN(=)-52 b FP(2)24 b FO(\001.)37 b(W)-7 b(e)28 b(obtain)e(series)g(8)h(if)g FN(\025)1328 1120 y FK(1)1389 1108 y FP(2)c FO(\001)28 b(or)f(9)g(if)g FN(\025)1859 1120 y FK(1)1929 1108 y FN(=)-51 b FP(2)23 b FO(\001.)p 2278 1108 4 57 v 2282 1055 50 4 v 2282 1108 V 2331 1108 4 57 v -118 1334 a FQ(4.)37 b FO(Represen)n(tations)25 b(of)34 b FN(U)756 1346 y FL(q)792 1334 y FO(\()p FN(so)p FO(\(3\)\),)29 b(where)e FN(q)k FO(is)c(not)h(a)f(ro)r(ot)g(of)h(unit)n (y)-7 b(.)37 b(Let)-118 1434 y FN(q)j FO(=)d FN(e)100 1404 y FL(i\033)168 1434 y FO(,)i(where)c FN(\033)51 b(=)-52 b FP(2)38 b FN(\031)s FI(Q)6 b FO(.)69 b(Analysis)34 b(similar)d(to)36 b(that)h(as)e(in)h(the)g(pro)r(of)-118 1533 y(of)e(Theorem)f(35)g(sho)n(ws)g(that)i(if)f(the)h(pair)d(\()p FN(J)1378 1545 y FK(1)1416 1533 y FN(;)14 b(J)1499 1545 y FK(2)1536 1533 y FO(\))35 b(de\014nes)f(a)g(non-trivial)-118 1633 y(irreducible)i(represen)n(tation,)41 b(then)f FN(\033)s FO(\()p FN(J)1216 1645 y FK(1)1254 1633 y FO(\))k FP(2)g FN(S)1480 1645 y FK(1)1561 1633 y FO(=)g([)p FP(\000)p FO(1)o FN(=)p FO(sin)12 b FN(\033)s(;)i FO(1)p FN(=)p FO(sin)e FN(\033)s FO(].)-118 1733 y(Moreo)n(v)n(er,)27 b(the)i(op)r(erator)e FN(J)798 1745 y FK(2)864 1733 y FO(is)h(concen)n(trated)g(on)h(\000)c(=)g FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))25 b FP(j)h FN(t)2042 1703 y FK(2)2098 1733 y FP(\000)19 b FO(\()p FN(q)k FO(+)-118 1832 y FN(q)-78 1802 y FM(\000)p FK(1)11 1832 y FO(\))14 b FN(ts)20 b FO(+)g FN(s)270 1802 y FK(2)335 1832 y FO(=)27 b(0)p FP(g)p FO(.)44 b(An)n(y)30 b(tra)5 b(jectory)29 b(of)h(a)g(p)r(oin)n(t) f FN(\025)f FO(=)f(sin)13 b FN(x\033)s(=)h FO(sin)f FN(\033)31 b FP(2)c FN(S)2301 1844 y FK(1)-118 1932 y FO(with)21 b(resp)r(ect)h(to)g(\000)g(is)e(of)i(the)g(form)f FA(O)p FO(\()p FP(f)p FN(\025)p FP(g)p FO(\))i(=)f FP(f)p FO(sin)o(\()p FN(x)7 b FO(+)g FN(k)s FO(\))p FN(\033)s(=)14 b FO(sin)f FN(\033)26 b FP(j)d FN(k)j FP(2)e FI(Z)o FP(g)-118 2032 y FO(whic)n(h)31 b(is)g(clearly)e(dense)j(in)g FN(S)866 2044 y FK(1)903 2032 y FO(,)i(and)e(there)g(is)f(no)h(measurable)d (section)i(of)-118 2131 y(\()p FN(S)-35 2143 y FK(1)2 2131 y FN(;)14 b FO(\000\).)36 b(In)24 b(this)g(case)f(there)h(exist)e (irreducible)f(represen)n(tations)g(whic)n(h)i(are)-118 2231 y(not)29 b(concen)n(trated)e(on)i(a)g(tra)5 b(jectory)-7 b(.)39 b(The)29 b(description)d(of)j(suc)n(h)f(represen-)-118 2330 y(tations)19 b(is)h(problematic.)31 b(The)21 b(description)d(of)j (irreducible)d(represen)n(tations)-118 2430 y(related)j(to)i(tra)5 b(jectories)20 b(can)j(b)r(e)g(obtained)f(using)g(the)h(same)f(tec)n (hnique)g(\(see)-118 2530 y([231)n(])28 b(for)f(the)h(concrete)f(form)n (ulae\).)-118 2811 y FG(2.4)112 b(Man)m(y-dimensional)41 b(dynamical)f(systems)-118 3007 y FO(In)h(this)f(section)g(w)n(e)h (study)g(represen)n(tations)d(of)j(relations)d(with)i(sev)n(eral)-118 3106 y(generators)22 b(using)i(the)h(man)n(y-dimensional)19 b(dynamical)i(system)j(approac)n(h.)6 3214 y(W)-7 b(e)33 b(start)f(with)g(some)e(examples.)48 b(Firstly)-7 b(,)32 b(in)f(Section)g(2.4.1)g(w)n(e)h(con-)-118 3313 y(sider)18 b(the)j(so-called)16 b(\\direct)j(pro)r(ducts")g(of)h(one-dimensional) 15 b(relations)i(and)-118 3413 y(sho)n(w)26 b(ho)n(w)h(to)g(apply)f (results)f(of)i(Section)f(2.1)h(to)g(classify)d(their)i(irreducible) -118 3513 y(represen)n(tations.)70 b(In)40 b(Section)f(2.4.2)f(w)n(e)h (study)h(the)g(more)e(complicated)-118 3612 y(\\triangular")21 b(case)26 b(and)g(apply)f(the)h(inductiv)n(e)e(algorithm)e(for)k(the)h (descrip-)-118 3712 y(tion)k(of)h(irreducible)d(represen)n(tations)g (to)j(the)g(t)n(wisted)f(CCR)i(and)f(t)n(wisted)-118 3811 y(CAR)e(algebras.)41 b(Then)30 b(\(see)g(Section)e(2.4.3\))h(w)n (e)h(consider)e(families)d(of)30 b(op-)-118 3911 y(erators)g (satisfying)f(a)j(general)d(class)h(of)i(relations)c(whose)k(represen)n (tations)p eop %%Page: 152 156 152 155 bop -118 -137 a FO(152)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(can)33 b(b)r(e)i(describ)r(ed)d(in)h(terms)g (of)h(orbits)e(of)h(some)g(man)n(y-dimensional)27 b(dy-)-118 196 y(namical)33 b(system.)64 b(The)37 b(results)f(obtained)f(are)h (illustrated)e(with)i(sev)n(eral)-118 296 y(examples.)58 b(Namely)-7 b(,)36 b(w)n(e)f(study)h(irreducible)c(represen)n(tations)h (of)j(a)f(non-)-118 395 y(standard)h(quan)n(tum)g(sphere)g(\(see)h (2.4.4\),)i(Heisen)n(b)r(erg)c(relations)f(for)i(the)-118 495 y(quan)n(tum)c FN(E)299 507 y FK(2)371 495 y FO(group)g(\(see)67 b(2.4.5\),)34 b(and)f(a)h(wide)e(class)g(of)h(Wic)n(k)g(algebras)-118 595 y(con)n(taining)24 b FN(q)322 607 y FL(ij)381 595 y FO(-CCR,)j FN(\026)p FO(-CCR)h(and)f(other)g(algebras)e(\(see)i (2.4.6\).)6 694 y(Let)f(us)g(note)g(that)g(all)e(examples)f(presen)n (ted)i(b)r(elo)n(w)g(are)g(Wic)n(k)g(algebras)-118 794 y(with)e(a)g(braided)e(op)r(erator)h FN(T)34 b FO(and)23 b(additional)d(relations)g(that)k(reduce)f(these)-118 893 y(algebras)h(to)j(the)g(form,)f(to)h(whic)n(h)f(dynamical)d (formalism)g(can)j(b)r(e)i(applied.)-118 993 y(F)-7 b(or)36 b(more)e(details)g(w)n(e)i(address)f(the)i(reader)e(to)h(Section)f (2.4.6)g(where)h(w)n(e)-118 1093 y(giv)n(e)26 b(all)f(necessary)h (de\014nitions)f(and)j(facts)f(concerning)f(Wic)n(k)g(algebras.)-118 1307 y FQ(2.4.1)94 b(\\Direct)22 b(pro)s(ducts")f(of)h(one-dimensional) 17 b(dynamical)k(sys-)174 1407 y(tems)-118 1560 y(1.)47 b FO(By)31 b(the)h(\\direct)e(pro)r(duct")h(of)g(one-dimensional)26 b(systems)k(w)n(e)h(mean)f(a)-118 1660 y(dynamical)18 b(system)j(on)g FI(R)708 1630 y FL(d)775 1660 y FO(de\014ned)h(b)n(y)g (the)g(family)d(of)j(mappings)e FN(F)2061 1672 y FL(i)2089 1660 y FO(,)j FN(i)f FO(=)h(1,)-118 1759 y FN(:)14 b(:)g(:)27 b FO(,)d FN(d)g FO(with)e(the)i(prop)r(ert)n(y)d(that)j(the)f FN(i)p FO(-th)g(mapping)d(only)i(c)n(hanges)g(the)h FN(i)p FO(-th)-118 1859 y(co)r(ordinate,)i(i.e.,)115 2034 y FN(F)168 2046 y FL(i)196 2034 y FO(\()225 2012 y FN(~)228 2034 y(\025)q FO(\))f(=)e(\()p FN(\025)500 2046 y FK(1)538 2034 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)771 2046 y FL(i)p FM(\000)p FK(1)884 2034 y FN(;)g(f)962 2046 y FL(i)989 2034 y FO(\()p FN(\025)1069 2046 y FL(i)1097 2034 y FO(\))p FN(;)g(\025)1214 2046 y FL(i)p FK(+1)1327 2034 y FN(;)g(:)g(:)g(:)f(;)h (\025)1559 2046 y FL(d)1598 2034 y FO(\))p FN(;)1830 2012 y(~)1833 2034 y(\025)24 b FP(2)f FI(R)2037 2000 y FL(d)2082 2034 y FN(:)-118 2210 y FQ(2.)55 b FO(Let)34 b(us)g(giv)n(e)e(an)i(example)d(of)j(a)g FP(\003)p FO(-algebra)c(suc)n (h)k(that)g(its)f(irreducible)-118 2309 y(represen)n(tations)k(can)j(b) r(e)h(classi\014ed)d(using)h(the)h(direct)g(pro)r(duct)g(of)g(one-)-118 2409 y(dimensional)29 b(systems.)53 b(Namely)-7 b(,)32 b(w)n(e)h(consider)f(the)h(direct)g(pro)r(duct)g(of)g FN(d)-118 2509 y FO(one-dimensional)28 b FN(q)s FO(-CCR,)33 b(whic)n(h)f(is)h(a)g(particular)d(case)i(of)h(the)h(so-called)-118 2608 y FN(q)-81 2620 y FL(ij)-23 2608 y FO(-CCR)28 b(algebra)d(\(see)i (Section)g(2.4.6\).)36 b(Consider)25 b(the)j FP(\003)p FO(-algebra:)127 2784 y FI(C)181 2716 y Fy(\012)226 2784 y FN(x)273 2796 y FL(i)301 2784 y FN(;)14 b(x)385 2749 y FM(\003)385 2804 y FL(i)447 2784 y FP(j)23 b FN(x)540 2749 y FM(\003)540 2804 y FL(i)578 2784 y FN(x)625 2796 y FL(i)677 2784 y FO(=)f(1)c(+)g FN(q)s(x)994 2796 y FL(i)1022 2784 y FN(x)1069 2749 y FM(\003)1069 2804 y FL(i)1108 2784 y FN(;)507 2917 y(x)554 2883 y FM(\003)554 2938 y FL(i)592 2917 y FN(x)639 2929 y FL(j)698 2917 y FO(=)k FN(x)832 2929 y FL(j)868 2917 y FN(x)915 2883 y FM(\003)915 2938 y FL(i)953 2917 y FN(;)28 b(i;)14 b(j)28 b FO(=)22 b(1)p FN(;)14 b(::;)g(d;)28 b FO(0)22 b FN(<)h(q)j(<)d FO(1)1820 2850 y Fy(\013)1858 2917 y FN(:)245 b FO(\(2.48\))-118 3093 y(The)31 b(classi\014cation)26 b(of)31 b(irreducible)c(represen)n(tations)h(is)i(based)g(on)h(the)g (fol-)-118 3192 y(lo)n(wing)g(prop)r(osition)g(whic)n(h)i(sho)n(ws)f (that)j(there)e(are)g(additional)e(relations)-118 3292 y(b)r(et)n(w)n(een)25 b(generators)e(whic)n(h)h(hold)g(automatically)c (in)k(an)n(y)h(irreducible)c(rep-)-118 3391 y(resen)n(tation)k(b)n(y)j (b)r(ounded)g(op)r(erators.)-118 3552 y FQ(Prop)s(osition)i(49.)41 b FB(L)l(et)g FN(\031)s FO(\()p FP(\001)p FO(\))h FB(b)l(e)g(a)f(b)l (ounde)l(d)h(r)l(epr)l(esentation)g(of)60 b FO(\(2.48\))o FB(.)-118 3651 y(Then)30 b FN(\031)s FO(\()p FN(x)227 3663 y FL(j)263 3651 y FN(x)310 3663 y FL(i)357 3651 y FP(\000)18 b FN(x)487 3663 y FL(i)515 3651 y FN(x)562 3663 y FL(j)597 3651 y FO(\))24 b(=)e(0)p FB(.)-118 3811 y(Pr)l(o)l(of.)43 b FO(A)21 b(simple)e(calculation)e(sho)n(ws)j(that)h FN(X)1365 3823 y FL(ij)1446 3811 y FO(=)i FN(\031)s FO(\()p FN(x)1663 3823 y FL(i)1692 3811 y FN(x)1739 3823 y FL(j)1780 3811 y FP(\000)5 b FN(x)1897 3823 y FL(j)1932 3811 y FN(x)1979 3823 y FL(i)2007 3811 y FO(\))21 b(satis\014es)-118 3911 y(the)31 b(relation)d FN(X)412 3881 y FM(\003)405 3933 y FL(ij)464 3911 y FN(X)533 3923 y FL(ij)619 3911 y FO(=)h FN(q)753 3881 y FK(2)790 3911 y FN(X)859 3923 y FL(ij)917 3911 y FN(X)993 3881 y FM(\003)986 3933 y FL(ij)1045 3911 y FO(,)i(whic)n(h)f(implies)e(that)j FN(X)1878 3923 y FL(ij)1965 3911 y FO(=)d(0)j(\(since)p eop %%Page: 153 157 153 156 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(153)-118 96 y(the)27 b(relation)c FN(x)374 66 y FM(\003)413 96 y FN(x)h FO(=)e FN(q)s(xx)705 66 y FM(\003)771 96 y FO(do)r(es)k(not)h(ha)n(v)n(e)e(an)n(y)h (non-trivial)c(b)r(ounded)27 b(rep-)-118 196 y(resen)n(tation,)e(see)j (Section)e(1.4.2\).)p 2278 196 4 57 v 2282 143 50 4 v 2282 196 V 2331 196 4 57 v 6 361 a(Let)32 b(us)g(note)f(that)h(the)g (elemen)n(ts)d FN(x)1179 373 y FL(j)1214 361 y FN(x)1261 373 y FL(i)1311 361 y FP(\000)20 b FN(x)1443 373 y FL(i)1471 361 y FN(x)1518 373 y FL(j)1585 361 y FO(generate)31 b(a)g(quadratic)-118 461 y(Wic)n(k)23 b(ideal)g(\(see)h(Section)g (2.4.6\).)35 b(So,)25 b(to)f(study)h(the)g(irreducible)c(represen-)-118 561 y(tations,)29 b(one)g(has)g(to)h(consider)d(a)i(family)e(of)j(b)r (ounded)g(op)r(erators)d FN(X)2121 573 y FL(i)2149 561 y FO(,)j FN(X)2278 530 y FM(\003)2271 582 y FL(i)2316 561 y FO(,)-118 660 y FN(i)23 b FO(=)f(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(d)p FO(,)g(satisfying)d(the)j(relations)-12 839 y FN(X)64 805 y FM(\003)57 860 y FL(i)101 839 y FN(X)170 851 y FL(i)221 839 y FO(=)22 b(1)c(+)g FN(q)s(X)560 851 y FL(i)588 839 y FN(X)664 805 y FM(\003)657 860 y FL(i)701 839 y FN(;)97 b(X)897 805 y FM(\003)890 860 y FL(i)935 839 y FN(X)1004 851 y FL(j)1062 839 y FO(=)22 b FN(X)1218 851 y FL(j)1253 839 y FN(X)1329 805 y FM(\003)1322 860 y FL(i)1366 839 y FN(;)97 b(X)1555 851 y FL(i)1583 839 y FN(X)1652 851 y FL(j)1710 839 y FO(=)22 b FN(X)1866 851 y FL(j)1901 839 y FN(X)1970 851 y FL(i)1997 839 y FN(:)106 b FO(\(2.49\))-118 1018 y(It)26 b(is)g(easy)f(to)h(see)g(that) g(this)g(algebra)d(is)i(generated)h(b)n(y)g FN(d)g FO(comm)n(uting)d (alge-)-118 1118 y(bras)416 1297 y Fz(A)476 1309 y FL(i)526 1297 y FO(=)g FI(C)668 1229 y Fy(\012)713 1297 y FN(X)782 1309 y FL(i)809 1297 y FN(;)14 b(X)922 1262 y FM(\003)915 1317 y FL(i)983 1297 y FP(j)23 b FN(X)1105 1262 y FM(\003)1098 1317 y FL(i)1142 1297 y FN(X)1211 1309 y FL(i)1262 1297 y FO(=)f(1)c(+)g FN(q)s(X)1601 1309 y FL(i)1629 1297 y FN(X)1705 1262 y FM(\003)1698 1317 y FL(i)1742 1229 y Fy(\013)1782 1297 y FN(:)6 1476 y FO(Let)29 b(us)g(no)n(w)f(consider) f(the)i(p)r(olar)d(decomp)r(osition)g FN(X)1747 1445 y FM(\003)1740 1497 y FL(i)1809 1476 y FO(=)e FN(U)1955 1488 y FL(i)1983 1476 y FN(C)2042 1488 y FL(i)2070 1476 y FO(.)40 b(Using)-118 1575 y(relations)24 b(\(2.49\))j(one)g(can)h (rewrite)d(the)j(system)f(in)g(an)g(equiv)-5 b(alen)n(t)25 b(form,)384 1754 y FN(C)449 1720 y FK(2)443 1775 y FL(i)487 1754 y FN(U)553 1720 y FM(\003)544 1775 y FL(i)614 1754 y FO(=)e FN(U)768 1720 y FM(\003)759 1775 y FL(i)806 1754 y FO(\(1)18 b(+)g FN(q)s(C)1086 1720 y FK(2)1080 1775 y FL(i)1123 1754 y FO(\))p FN(;)98 b(C)1341 1720 y FK(2)1335 1775 y FL(i)1378 1754 y FN(U)1444 1720 y FM(\003)1435 1775 y FL(j)1505 1754 y FO(=)23 b FN(U)1659 1720 y FM(\003)1650 1775 y FL(j)1697 1754 y FN(C)1762 1720 y FK(2)1756 1775 y FL(i)1799 1754 y FN(;)571 1889 y(C)630 1901 y FL(i)658 1889 y FN(C)717 1901 y FL(j)775 1889 y FO(=)g FN(C)922 1901 y FL(j)957 1889 y FN(C)1016 1901 y FL(i)1044 1889 y FN(;)97 b(U)1221 1901 y FL(j)1256 1889 y FN(U)1313 1901 y FL(i)1363 1889 y FO(=)23 b FN(U)1508 1901 y FL(i)1535 1889 y FN(U)1592 1901 y FL(j)1627 1889 y FN(:)476 b FO(\(2.50\))-118 2068 y(As)28 b(in)g(the)h (one-dimensional)23 b(case,)k(the)i(op)r(erator)d FN(U)1601 2038 y FM(\003)1592 2089 y FL(i)1668 2068 y FO(determines)g(the)i(ac-) -118 2167 y(tion)g(on)g(the)h(sp)r(ectrum)f(of)h FN(C)839 2137 y FK(2)833 2189 y FL(i)876 2167 y FO(,)h FN(x)976 2179 y FL(i)1029 2167 y FP(7!)25 b FO(1)18 b(+)h FN(q)s(x)1368 2179 y FL(i)1396 2167 y FO(.)41 b(The)28 b(action)g(of)g FN(U)2043 2137 y FM(\003)2034 2189 y FL(j)2081 2167 y FO(,)h FN(i)c FP(6)p FO(=)g FN(j)5 b FO(,)-118 2276 y(on)27 b FN(\033)s FO(\()p FN(C)144 2246 y FK(2)138 2298 y FL(i)182 2276 y FO(\))h(is)f(iden)n(tical,)d(b)r(ecause)k(of)f(the)h(relation)d FN(C)1598 2246 y FK(2)1592 2298 y FL(i)1635 2276 y FN(U)1692 2288 y FL(j)1750 2276 y FO(=)e FN(U)1895 2288 y FL(j)1929 2276 y FN(C)1994 2246 y FK(2)1988 2298 y FL(i)2032 2276 y FO(.)-118 2424 y FQ(3.)33 b FO(No)n(w)19 b(w)n(e)g(are)f(able)f(to)i (giv)n(e)f(a)g(classi\014cation)d(of)k(b)r(ounded)h(represen)n(tations) -118 2524 y(up)28 b(to)f(unitary)f(equiv)-5 b(alence)26 b(using)g(the)i(one-dimensional)22 b(tec)n(hnique.)6 2623 y(T)-7 b(o)42 b(classify)e(b)r(ounded)i(irreducible)d(represen)n (tations,)k(one)f(m)n(ust)f(de-)-118 2723 y(scrib)r(e)34 b(b)r(ounded)h(orbits)f(of)h(the)g(dynamical)d(system)h(on)i FI(R)1836 2735 y FK(+)1932 2723 y FO(determined)-118 2823 y(b)n(y)27 b(the)h(mapping)d FN(f)9 b FO(\()p FN(\025)p FO(\))24 b(=)f(1)18 b(+)g FN(q)s(\025)p FO(.)37 b(It)28 b(has)f(only)f(t)n(w)n(o)h(orbits,)-17 2986 y(1.)41 b FP(f)p FN(f)181 2956 y FL(n)226 2986 y FO(\(0\))p FN(;)g(n)23 b FP(\025)g FO(0)p FP(g)p FO(,)k(the)h(\\F)-7 b(o)r(c)n(k")26 b(orbit;)-17 3150 y(2.)41 b(the)28 b(\014xed)g(p)r(oin)n(t)f FP(f)p FO(1)p FN(=)p FO(\(1)17 b FP(\000)h FN(q)s FO(\))p FP(g)p FO(.)-118 3313 y(It)29 b(is)e(easy)h(to)h(see)f(that)h(the)g(sp) r(ectral)e(pro)5 b(jection)27 b FN(E)1574 3331 y FL(C)1626 3311 y Fx(2)1622 3349 y(1)1662 3313 y FO(\()p FA(O)p FO(\),)j(where)e FA(O)g FO(is)g(an)-118 3413 y(orbit)34 b(of)i(the)f(one-dimensional)c(dynamical)h(system)i(\()p FN(f)t(;)14 b FI(R)1822 3425 y FK(+)1883 3413 y FO(\),)38 b(comm)n(utes)-118 3513 y(with)c(all)f(op)r(erators)g(of)i(the)h (represen)n(tation.)56 b(Consequen)n(tly)-7 b(,)35 b(for)g(an)f(the) -118 3612 y(irreducible)28 b(represen)n(tation,)j(the)h(sp)r(ectrum)f (of)h(the)g(op)r(erator)f FN(C)2034 3582 y FK(2)2028 3633 y(1)2103 3612 y FO(lies)f(on)-118 3712 y(a)36 b(single)e(orbit)g (of)j(the)f(dynamical)d(system.)62 b(Moreo)n(v)n(er,)35 b(since)h FN(U)2085 3682 y FM(\003)2076 3732 y FK(1)2159 3712 y FO(is)f(an)-118 3811 y(isometry)-7 b(,)19 b(the)i(sp)r(ectrum)e (coincides)f(with)h(one)h(of)h(the)f(t)n(w)n(o)g(orbits)e(presen)n(ted) -118 3911 y(ab)r(o)n(v)n(e.)p eop %%Page: 154 158 154 157 bop -118 -137 a FO(154)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(Let)38 b(us)f(consider)f(the)i(case)e(where)h FN(\033)s FO(\()p FN(C)1350 66 y FK(2)1344 117 y(1)1388 96 y FO(\))j(=)f FP(f)p FN(f)1656 66 y FL(n)1701 96 y FO(\(0\))p FN(;)27 b(n)40 b FP(\025)f FO(0)p FP(g)p FO(.)65 b(W)-7 b(e)-118 196 y(ha)n(v)n(e:)-34 535 y FN(U)32 501 y FM(\003)23 556 y FK(1)93 535 y FO(=)181 293 y Fy(0)181 440 y(B)181 489 y(B)181 539 y(B)181 592 y(@)254 357 y FO(0)254 457 y(1)82 b(0)378 556 y(1)110 b(0)507 653 y(.)540 678 y(.)572 703 y(.)687 653 y(.)720 678 y(.)752 703 y(.)779 293 y Fy(1)779 440 y(C)779 489 y(C)779 539 y(C)779 592 y(A)871 535 y FP(\012)18 b FN(I)7 b(;)97 b(C)1182 501 y FK(2)1176 556 y(1)1242 535 y FO(=)1330 343 y Fy(0)1330 489 y(B)1330 543 y(@)1402 407 y FN(f)9 b FO(\(0\))1641 507 y FN(f)1691 476 y FK(2)1728 507 y FO(\(0\))1922 603 y(.)1954 628 y(.)1986 653 y(.)2014 343 y Fy(1)2014 489 y(C)2014 543 y(A)2105 535 y FP(\012)18 b FN(I)7 b(:)-118 890 y FO(It)28 b(is)e(easy)h(to)g(deduce)h(from)e(the)i(comm)n(utation) c(relations)h(that)387 1073 y FN(C)452 1039 y FK(2)446 1093 y FL(i)512 1073 y FO(=)e(1)18 b FP(\012)762 1052 y FO(^)743 1073 y FN(C)808 1039 y FK(2)802 1093 y FL(i)845 1073 y FN(;)97 b(U)1031 1039 y FM(\003)1022 1093 y FL(i)1092 1073 y FO(=)23 b(1)18 b FP(\012)1337 1052 y FO(^)1323 1073 y FN(U)1389 1039 y FM(\003)1380 1093 y FL(i)1427 1073 y FN(;)180 b(i)22 b FP(\025)h FO(2)p FN(:)-118 1256 y FO(Hence)37 b(the)f(family)e FP(f)608 1235 y FO(^)594 1256 y FN(U)651 1268 y FL(i)678 1256 y FN(;)734 1235 y FO(^)715 1256 y FN(C)774 1268 y FL(i)802 1256 y FN(;)27 b(i)38 b FP(\025)f FO(2)p FP(g)f FO(satis\014es)f(relations)e (\(2.4.1\).)63 b(More-)-118 1355 y(o)n(v)n(er,)33 b(the)i(family)30 b FP(f)560 1334 y FO(^)541 1355 y FN(C)600 1367 y FL(i)628 1355 y FN(;)679 1334 y FO(^)665 1355 y FN(U)722 1367 y FL(i)749 1355 y FP(g)791 1325 y FL(d)791 1377 y(i)p FK(=2)936 1355 y FO(is)j(irreducible)d(if)j(and)h(only)e(if)h(the)i (family)-118 1455 y FP(f)p FN(C)-17 1467 y FL(i)10 1455 y FN(;)14 b(U)104 1467 y FL(i)132 1455 y FP(g)174 1425 y FL(d)174 1476 y(i)p FK(=1)312 1455 y FO(is)27 b(irreducible.)6 1554 y(Consider)f(no)n(w)h(the)h(case)f(where)g FN(\033)s FO(\()p FN(C)1232 1524 y FK(2)1226 1575 y(1)1270 1554 y FO(\))c(=)g FP(f)p FO(1)p FN(=)p FO(\(1)17 b FP(\000)h FN(q)s FO(\))p FP(g)p FO(.)37 b(In)27 b(this)g(case,)381 1737 y FN(C)446 1703 y FK(2)440 1758 y(1)506 1737 y FO(=)c(\(1)18 b FP(\000)g FN(q)s FO(\))841 1703 y FM(\000)p FK(1)931 1737 y FN(I)7 b(;)97 b(U)1151 1749 y FK(1)1211 1737 y FO(=)22 b FN(\025I)7 b(;)181 b FP(j)p FN(\025)p FP(j)23 b FO(=)g(1)-118 1920 y(\(since)e FN(U)169 1932 y FK(1)228 1920 y FO(comm)n(utes)e(with)j(all)d(op)r(erators)h(of)i(the)g (represen)n(tation\).)33 b(There-)-118 2019 y(fore,)f(in)e(this)h(case) f(w)n(e)h(deal)f(with)h(the)g(\\direct)f(pro)r(duct")h(of)g FN(d)21 b FP(\000)f FO(1)31 b(copies)-118 2119 y(of)c FN(q)s FO(-CCR)g(algebras.)33 b(Let)27 b(us)g(in)n(tro)r(duce,)f(for)g (con)n(v)n(enience,)f(the)j(op)r(erators)-118 2219 y(on)f FN(l)22 2231 y FK(2)59 2219 y FO(\()p FI(N)t FO(\))34 b(giv)n(en)26 b(b)n(y)140 2578 y FN(S)i FO(=)306 2337 y Fy(0)306 2483 y(B)306 2533 y(B)306 2583 y(B)306 2636 y(@)379 2401 y FO(0)379 2500 y(1)82 b(0)503 2600 y(1)111 b(0)633 2697 y(.)665 2721 y(.)697 2747 y(.)812 2697 y(.)845 2721 y(.)877 2747 y(.)905 2337 y Fy(1)905 2483 y(C)905 2533 y(C)905 2583 y(C)905 2636 y(A)991 2578 y FN(;)97 b(C)29 b FO(=)1287 2387 y Fy(0)1287 2533 y(B)1287 2586 y(@)1360 2450 y FN(f)9 b FO(\(0\))1598 2550 y FN(f)1648 2520 y FK(2)1685 2550 y FO(\(0\))1879 2647 y(.)1911 2672 y(.)1943 2697 y(.)1971 2387 y Fy(1)1971 2533 y(C)1971 2586 y(A)2058 2578 y FN(:)6 2934 y FO(Com)n(bining)23 b(the)j(results)f(obtained)g(ab)r(o)n(v)n(e)f(and)i(applying)e (induction)g(in)-118 3033 y FN(d)p FO(,)k(w)n(e)f(can)g(form)n(ulate)e (the)j(follo)n(wing)c(prop)r(osition.)-118 3199 y FQ(Prop)s(osition)30 b(50.)41 b FB(Fix)e(a)i(subset)e FO(\010)i(=)f FP(f)p FO(1)h FP(\024)f FN(i)1523 3211 y FK(1)1601 3199 y FN(<)h(i)1736 3211 y FK(2)1814 3199 y FN(<)g FP(\001)14 b(\001)g(\001)41 b FN(<)f(i)2192 3211 y FL(k)2274 3199 y FP(\024)-118 3299 y FN(d)34 b(;)48 b FO(1)31 b FP(\024)g FN(k)j FP(\024)d FN(d)p FP(g)p FB(.)52 b(T)-6 b(o)35 b(e)l(ach)g(such)f(subset,)i(asso)l (ciate)f(the)g(fol)t(lowing)h(irr)l(e-)-118 3399 y(ducible)31 b(r)l(epr)l(esentation)6 b FO(:)347 3593 y FN(C)412 3559 y FK(2)406 3614 y FL(i)473 3593 y FO(=)622 3514 y Fy(O)560 3693 y FL(j)s(<i;j)s FM(62)p FK(\010)823 3593 y FN(I)25 b FP(\012)18 b FN(C)25 b FP(\012)1196 3514 y Fy(O)1134 3693 y FL(j)s(>i;j)s FM(62)p FK(\010)1397 3593 y FN(I)7 b(;)184 b(i)22 b FP(62)i FO(\010)p FN(;)351 3833 y(U)417 3799 y FM(\003)408 3854 y FL(i)478 3833 y FO(=)627 3754 y Fy(O)566 3933 y FL(j)s(<i;j)s FM(62)p FK(\010)828 3833 y FN(I)i FP(\012)18 b FN(S)23 b FP(\012)1192 3754 y Fy(O)1130 3933 y FL(j)s(>i;j)s FM(62)p FK(\010)1393 3833 y FN(I)7 b(;)183 b(i)23 b FP(62)h FO(\010)p FN(;)p eop %%Page: 155 159 155 158 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(155)386 96 y FN(U)443 108 y FL(i)493 96 y FO(=)23 b FN(\025)629 108 y FL(i)657 96 y FN(I)7 b(;)99 b(C)887 62 y FK(2)881 117 y FL(i)947 96 y FO(=)23 b(\(1)18 b FP(\000)g FN(q)s FO(\))1282 62 y FM(\000)p FK(1)1372 96 y FN(I)7 b(;)183 b(i)23 b FP(2)g FO(\010)p FN(:)-118 247 y FB(Then)32 b(al)t(l)h(irr)l(e)l(ducible)g(b)l (ounde)l(d)e(r)l(epr)l(esentations)h(of)h(the)e(\\dir)l(e)l(ct)h(pr)l (o)l(duct")-118 346 y(of)52 b FN(d)34 b FB(c)l(opies)h(of)f(the)67 b FN(q)s FB(-CCR)34 b(algebr)l(as)h(c)l(an)f(b)l(e)f(obtaine)l(d)i(in)f (such)f(a)h(way.)-118 446 y(Mor)l(e)l(over,)41 b(r)l(epr)l(esentations) d(ar)l(e)g(e)l(quivalent)g(if)h(and)f(only)g(if)g(they)h(c)l(orr)l(e-) -118 546 y(sp)l(ond)30 b(to)g(the)g(same)g FO(\010)p FB(.)-118 686 y FQ(4.)47 b FO(It)31 b(is)f(ob)n(vious)f(that)i(one)g (will)e(classify)f(irreducible)g(represen)n(tations)g(of)-118 785 y(the)g FP(\003)p FO(-algebra)402 936 y Fz(A)22 b FO(=)h FI(C)626 868 y Fy(\012)671 936 y Fz(A)731 948 y FL(i)781 936 y FP(j)h FO([)p Fz(A)911 948 y FL(i)938 936 y FN(;)14 b Fz(A)1035 948 y FL(j)1069 936 y FO(])23 b(=)g(0)p FN(;)k(i;)14 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d)1779 868 y Fy(\013)-118 1086 y FO(in)27 b(suc)n(h)g(a)g(w)n(a)n(y)-7 b(.)36 b(Here)209 1236 y Fz(A)269 1248 y FL(i)319 1236 y FO(=)23 b FI(C)461 1169 y Fy(\012)506 1236 y FN(x)553 1248 y FL(i)581 1236 y FN(;)14 b(x)665 1202 y FM(\003)665 1257 y FL(i)727 1236 y FP(j)23 b FN(x)820 1202 y FM(\003)820 1257 y FL(i)858 1236 y FN(x)905 1248 y FL(i)957 1236 y FO(=)f FN(f)1085 1248 y FL(i)1113 1236 y FO(\()p FN(x)1192 1248 y FL(i)1220 1236 y FN(x)1267 1202 y FM(\003)1267 1257 y FL(i)1306 1236 y FO(\))1338 1169 y Fy(\013)1377 1236 y FN(;)180 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)-118 1386 y FO(and)26 b FN(f)83 1398 y FL(i)110 1386 y FO(\()p FP(\001)p FO(\))9 b(:)29 b FI(R)g FP(7!)23 b FI(R)32 b FO(is)25 b(a)g(one-to-one)g(measurable)d(mapping)i(suc)n(h)i(that)g (the)-118 1486 y(dynamical)e(system)i(\()p FN(f)629 1498 y FL(i)657 1486 y FN(;)14 b FI(R)p FO(\))34 b(has)27 b(a)g(measurable)d(section.)-118 1696 y FQ(2.4.2)94 b(\\T)-8 b(riangular")32 b(dynamical)f(systems.)-118 1849 y(1.)43 b FO(W)-7 b(e)30 b(call)e(a)i(dynamical)c(system)j(de\014ned)h(b)n(y)g (a)f(family)e(of)j(mappings)e FN(F)2311 1861 y FL(i)-118 1949 y FO(triangular)c(if)70 2099 y FN(F)123 2111 y FL(i)151 2099 y FO(\()180 2077 y FN(~)183 2099 y(\025)q FO(\))f(=)g(\()p FN(\025)455 2111 y FK(1)493 2099 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)726 2111 y FL(i)p FM(\000)p FK(1)839 2099 y FN(;)g(f)917 2111 y FL(i)944 2099 y FO(\()p FN(\025)1024 2111 y FK(1)1062 2099 y FN(;)g(:)g(:)g(:)f(;)h(\025)1294 2111 y FL(i)1322 2099 y FO(\))p FN(;)g(q)1428 2111 y FL(ii)p FK(+1)1564 2099 y FN(\025)1612 2111 y FL(i)p FK(+1)1724 2099 y FN(;)g(:)g(:)g(:)f (;)h(q)1945 2111 y FL(id)2008 2099 y FN(\025)2056 2111 y FL(d)2095 2099 y FO(\))p FN(;)-118 2249 y(q)-81 2261 y FL(ij)23 2249 y FP(2)46 b FI(R)p FO(,)51 b FN(j)f FO(=)45 b FN(i)27 b FO(+)g(1,)41 b FN(:)14 b(:)g(:)28 b FO(,)44 b FN(d)p FO(,)h(where)c FN(f)1298 2261 y FL(i)1334 2249 y FO(:)32 b FI(R)1443 2219 y FL(i)1523 2249 y FP(7!)45 b FI(R)1705 2219 y FL(i)1780 2249 y FO(are)40 b(measurable)-118 2349 y(functions.)c(Here)27 b(w)n(e)g(giv)n(e)f(examples)e(of)k FP(\003)p FO(-algebras)23 b(connected)28 b(with)f(\\tri-)-118 2449 y(angular")20 b(dynamical)f(systems.)33 b(These)23 b(are)f FN(\026)p FO(-CCR)g(and)h FN(\026)p FO(-CAR)g(algebras)-118 2548 y(connected)k(with)f(w)n(ell-kno)n(wn)e(t)n(wisted)i(CCR)h(and)f (t)n(wisted)g(CAR)i(algebras)-118 2648 y(constructed)f(b)n(y)g(Pusz)g (and)h(W)-7 b(orono)n(wicz)24 b(\(see)k([211)o(,)f(212)o(,)h(124)n (]\).)6 2748 y(In)g(the)g(examples)d(that)j(follo)n(w,)c(w)n(e)j(sho)n (w)g(that,)h(if)f(the)h(dynamical)c(sys-)-118 2847 y(tem)32 b(connected)g(to)g(the)h(op)r(erator)e(family)f(is)h(\\triangular",)e (then)k(the)g(de-)-118 2947 y(scription)k(of)i(classes)f(of)h (irreducible)d(represen)n(tations)h(can)i(b)r(e)h(obtained)-118 3046 y(inductiv)n(ely)25 b(applying)g(only)h(the)i(one-dimensional)22 b(tec)n(hnique.)-118 3183 y FQ(2.)36 b FN(\026)p FO(-CCR)26 b(algebra.)34 b(Here)26 b(w)n(e)g(study)h(irreducible)c(represen)n (tations)h(of)i(the)-118 3283 y(Wic)n(k)i(algebra)e(connected)j(with)g (the)g(t)n(wisted)g(CCR)g(algebra)d(of)j(Pusz)g(and)-118 3382 y(W)-7 b(orono)n(wicz.)6 3482 y(Let)28 b(us)g(consider)d(the)j (follo)n(wing)c FP(\003)p FO(-algebra:)-7 3652 y FI(C)47 3560 y Fy(D)104 3652 y FN(x)151 3664 y FL(i)179 3652 y FN(;)14 b(x)263 3618 y FM(\003)263 3673 y FL(i)325 3652 y FP(j)23 b FN(x)418 3618 y FM(\003)418 3673 y FL(i)456 3652 y FN(x)503 3664 y FL(i)554 3652 y FO(=)g(1)18 b(+)g FN(\026)835 3618 y FK(2)872 3652 y FN(x)919 3664 y FL(i)947 3652 y FN(x)994 3618 y FM(\003)994 3673 y FL(i)1051 3652 y FP(\000)h FO(\(1)f FP(\000)g FN(\026)1360 3618 y FK(2)1397 3652 y FO(\))1443 3573 y Fy(X)1450 3750 y FL(j)s(<i)1577 3652 y FN(x)1624 3664 y FL(j)1659 3652 y FN(x)1706 3618 y FM(\003)1706 3673 y FL(j)1745 3652 y FN(;)c(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)531 3899 y(x)578 3864 y FM(\003)578 3919 y FL(i)617 3899 y FN(x)664 3911 y FL(j)722 3899 y FO(=)23 b FN(\026x)907 3911 y FL(j)943 3899 y FN(x)990 3864 y FM(\003)990 3919 y FL(i)1028 3899 y FN(;)28 b(\026)23 b FP(2)g FO([0)p FN(;)14 b FO(1])p FN(;)27 b(i)c FP(6)p FO(=)g FN(j)1626 3806 y Fy(E)1676 3899 y FN(:)427 b FO(\(2.51\))p eop %%Page: 156 160 156 159 bop -118 -137 a FO(156)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(W)-7 b(e)41 b(sho)n(w)d(that,)44 b(as)39 b(in)g(the)h(previous)e(example,)i(the)g(additional)c(re-)-118 196 y(lations)e FN(x)206 208 y FL(i)234 196 y FN(x)281 208 y FL(j)355 196 y FO(=)k FN(\026x)555 208 y FL(j)590 196 y FN(x)637 208 y FL(i)665 196 y FO(,)i FN(i)e(>)g(j)5 b FO(,)39 b(hold)c(automatically)d(for)k(an)n(y)g(b)r(ounded)-118 296 y(represen)n(tation)25 b(of)i(the)h FN(\026)p FO(-CCR)g(algebra.) -118 453 y FQ(Prop)s(osition)i(51.)41 b FB(L)l(et)31 b FN(\031)s FO(\()p FP(\001)p FO(\))i FB(b)l(e)g(an)f(irr)l(e)l (ducible)i(b)l(ounde)l(d)e(r)l(epr)l(esentation)-118 553 y(of)e(the)g FN(\026)p FB(-CCR)g(algebr)l(a.)40 b(Then)31 b FN(\031)s FO(\()p FN(x)1067 565 y FL(i)1095 553 y FN(x)1142 565 y FL(j)1196 553 y FP(\000)18 b FN(\026x)1376 565 y FL(j)1412 553 y FN(x)1459 565 y FL(i)1487 553 y FO(\))23 b(=)g(0)p FB(,)29 b FN(i)23 b(>)g(j)5 b FB(.)-118 710 y(Pr)l(o)l(of.)43 b FO(Denote)30 b FN(X)498 722 y FL(ij)582 710 y FO(=)25 b FN(\031)s FO(\()p FN(x)801 722 y FL(j)837 710 y FN(x)884 722 y FL(i)932 710 y FP(\000)19 b FN(\026x)1113 722 y FL(i)1141 710 y FN(x)1188 722 y FL(j)1224 710 y FO(\),)30 b FN(j)h(>)25 b(i)p FO(.)42 b(Then,)30 b(using)e(the)h(com-) -118 810 y(m)n(utation)c(rules)h(\(2.51\),)h(it)g(is)g(easy)f(to)i (obtain)e(that)-95 998 y FN(X)-19 964 y FM(\003)-26 1019 y FL(ij)33 998 y FN(X)102 1010 y FL(ij)183 998 y FO(=)d FN(\026)321 964 y FK(6)358 998 y FN(X)427 1010 y FL(ij)485 998 y FN(X)561 964 y FM(\003)554 1019 y FL(ij)631 998 y FP(\000)18 b FN(\026)p FO(\(1)g FP(\000)g FN(\026)989 964 y FK(2)1026 998 y FO(\))1072 919 y Fy(X)1075 1096 y FL(r)r(<j)1206 998 y FN(X)1275 1010 y FL(ir)1335 998 y FN(X)1411 964 y FM(\003)1404 1019 y FL(ir)178 1236 y FP(\000)g FN(\026)311 1202 y FK(4)349 1236 y FO(\(1)g FP(\000)g FN(\026)574 1202 y FK(2)611 1236 y FO(\))692 1157 y Fy(X)657 1334 y FL(j)s(<s<i)860 1236 y FN(X)929 1248 y FL(sj)996 1236 y FN(X)1072 1202 y FM(\003)1065 1257 y FL(sj)1149 1236 y FO(+)g FN(\026)1282 1202 y FK(2)1319 1236 y FO(\(1)h FP(\000)f FN(\026)1545 1202 y FK(2)1582 1236 y FO(\))1705 1157 y Fy(X)1628 1334 y FL(r)r(<j)s(<s<i)1916 1236 y FN(X)1985 1248 y FL(sr)2053 1236 y FN(X)2129 1202 y FM(\003)2122 1257 y FL(sr)178 1474 y FP(\000)g FN(\026)311 1440 y FK(4)349 1474 y FO(\(1)g FP(\000)g FN(\026)574 1440 y FK(2)611 1474 y FO(\))657 1395 y Fy(X)660 1572 y FL(r)r(<j)791 1474 y FN(X)860 1486 y FL(j)s(r)927 1474 y FN(X)1003 1440 y FM(\003)996 1495 y FL(j)s(r)1082 1474 y FO(+)g(\(1)g FP(\000)g FN(\026)1390 1440 y FK(2)1428 1474 y FO(\))1460 1440 y FK(2)1497 1474 y FO(\(1)h(+)f FN(\026)1723 1440 y FK(2)1760 1474 y FO(\))1845 1395 y Fy(X)1806 1572 y FL(r)r(<s<j)2018 1474 y FN(X)2087 1486 y FL(sr)2155 1474 y FN(X)2231 1440 y FM(\003)2224 1495 y FL(sr)2292 1474 y FN(:)-118 1740 y FO(This)36 b(expression)e(sho)n(ws)i(that)h FN(X)1003 1710 y FM(\003)996 1762 y FL(ij)1054 1740 y FN(X)1123 1752 y FL(ij)1218 1740 y FO(consists)e(of)i(the)g(term)f FN(\026)2047 1710 y FK(6)2084 1740 y FN(X)2153 1752 y FL(ij)2211 1740 y FN(X)2287 1710 y FM(\003)2280 1762 y FL(ij)-118 1840 y FO(and)41 b(the)h(ones)f(that)h(ha)n(v)n(e)e(an)h(index)g(less)f (than)h(\()p FN(i;)14 b(j)5 b FO(\))42 b(with)f(resp)r(ect)g(to)-118 1939 y(the)27 b(lexicographic)22 b(ordering.)33 b(Then)27 b(the)g(induction)e(on)i(the)g(lexicographic)-118 2039 y(ordering)16 b(and)i(the)h(fact)g(that)g(the)g(relation)c FN(x)1289 2009 y FM(\003)1328 2039 y FN(x)24 b FO(=)e FN(q)s(xx)1620 2009 y FM(\003)1678 2039 y FO(do)r(es)c(not)h(ha)n(v)n (e)e(non-)-118 2138 y(trivial)k(b)r(ounded)j(represen)n(tations)e (\014nishes)h(the)i(pro)r(of)f(of)g(the)h(prop)r(osition.)p 2278 2238 4 57 v 2282 2185 50 4 v 2282 2238 V 2331 2238 4 57 v 6 2402 a(This)43 b(means)f(that,)48 b(as)c(far)f(as)g(b)r (ounded)h(represen)n(tations)d(are)h(con-)-118 2502 y(cerned,)37 b(no)f(information)c(is)i(lost)h(if)g(the)h FN(\026)p FO(-CCR)f(relations)e(are)i(replaced)-118 2601 y(with)27 b(the)h(complete)e(t)n(wisted)g(CCR)i(relations:)103 2785 y FN(x)150 2751 y FM(\003)150 2806 y FL(i)188 2785 y FN(x)235 2797 y FL(i)286 2785 y FO(=)23 b(1)18 b(+)g FN(\026)567 2751 y FK(2)604 2785 y FN(x)651 2797 y FL(i)679 2785 y FN(x)726 2751 y FM(\003)726 2806 y FL(i)783 2785 y FP(\000)g FO(\(1)h FP(\000)f FN(\026)1092 2751 y FK(2)1129 2785 y FO(\))1175 2706 y Fy(X)1182 2883 y FL(j)s(<i)1309 2785 y FN(x)1356 2797 y FL(j)1391 2785 y FN(x)1438 2751 y FM(\003)1438 2806 y FL(j)1477 2785 y FN(;)180 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)95 3006 y(x)142 2971 y FM(\003)142 3026 y FL(i)181 3006 y FN(x)228 3018 y FL(j)286 3006 y FO(=)23 b FN(\026x)471 3018 y FL(j)506 3006 y FN(x)553 2971 y FM(\003)553 3026 y FL(i)592 3006 y FN(;)180 b(i)23 b FP(6)p FO(=)f FN(j:)106 3130 y(x)153 3142 y FL(j)188 3130 y FN(x)235 3142 y FL(i)286 3130 y FO(=)h FN(\026x)471 3142 y FL(i)499 3130 y FN(x)546 3142 y FL(j)582 3130 y FN(;)179 b(i)23 b(<)g(j;)97 b(\026)23 b FP(2)h FO([0)p FN(;)14 b FO(1])p FN(:)-118 3302 y FO(Let)19 b(us)h(no)n(w,)g(as)f(usual,)h(consider)d(the)j(p)r(olar)d(decomp)r (osition)f FN(\031)s FO(\()p FN(x)1962 3272 y FM(\003)1962 3324 y FL(i)2001 3302 y FO(\))24 b(=)e FN(U)2201 3314 y FL(i)2229 3302 y FN(C)2288 3314 y FL(i)2316 3302 y FO(,)-118 3402 y FN(i)27 b FO(=)g(1,)j FN(:)14 b(:)g(:)28 b FO(,)j FN(d)p FO(.)45 b(Then)31 b(the)g(relations)c(can)j(b)r(e)g (rewritten)f(in)h(an)g(equiv)-5 b(alen)n(t)-118 3501 y(form,)191 3694 y FN(C)256 3659 y FK(2)250 3714 y FL(i)293 3694 y FN(U)359 3659 y FM(\003)350 3714 y FL(i)420 3694 y FO(=)23 b FN(U)574 3659 y FM(\003)565 3714 y FL(i)612 3601 y Fy(\020)661 3694 y FO(1)18 b(+)g FN(\026)854 3659 y FK(2)892 3694 y FN(C)957 3659 y FK(2)951 3714 y FL(i)1013 3694 y FO(+)1096 3615 y Fy(X)1099 3793 y FL(k)q(<i)1229 3694 y FN(C)1294 3659 y FK(2)1288 3714 y FL(k)1332 3601 y Fy(\021)1381 3694 y FN(;)180 b(i)23 b FO(=)g(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)191 3911 y(C)256 3877 y FK(2)250 3932 y FL(i)293 3911 y FN(U)359 3877 y FM(\003)350 3932 y FL(j)420 3911 y FO(=)23 b FN(U)574 3877 y FM(\003)565 3932 y FL(j)612 3911 y FN(C)677 3877 y FK(2)671 3932 y FL(i)714 3911 y FN(;)180 b(i)23 b(<)f(j;)p eop %%Page: 157 161 157 160 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(157)191 96 y FN(C)256 62 y FK(2)250 117 y FL(i)293 96 y FN(U)359 62 y FM(\003)350 117 y FL(j)420 96 y FO(=)23 b FN(\026)558 62 y FK(2)595 96 y FN(U)661 62 y FM(\003)652 117 y FL(j)699 96 y FN(C)764 62 y FK(2)758 117 y FL(i)802 96 y FN(;)179 b(i)23 b(>)g(j:)925 b FO(\(2.52\))-118 302 y(It)25 b(follo)n(ws)c(from)i(these)h(relations) d(that)k(all)d FN(U)1320 314 y FL(i)1372 302 y FO(are)h(co-isometries.) 31 b(It)25 b(is)e(also)-118 402 y(easy)k(to)h(see)f(that)i(the)f (isometry)d FN(U)1031 372 y FM(\003)1022 424 y FL(i)1097 402 y FO(acts)j(non-trivially)22 b(on)28 b(the)g(sp)r(ectrum)-118 502 y(of)33 b FN(C)47 472 y FK(2)41 523 y FL(j)85 502 y FO(,)27 b FN(i)c(<)f(j)5 b FO(,)27 b(and)g(the)g(corresp)r(onding)d (dynamical)g(system)h(is)h(triangular.)6 613 y(Let)34 b(us)f(classify)e(the)j(irreducible)29 b(represen)n(tations)i(of)i (these)g(op)r(erator)-118 713 y(relations.)j(F)-7 b(or)28 b(an)g(irreducible)d(represen)n(tation,)h(the)j(sp)r(ectrum)e(of)h FN(C)2166 683 y FK(2)2160 733 y(1)2232 713 y FO(co-)-118 812 y(incides)d(with)j(one)f(of)g(the)h(t)n(w)n(o)f(orbits:)-17 1013 y(1.)41 b FN(\033)s FO(\()p FN(C)236 983 y FK(2)230 1034 y(1)275 1013 y FO(\))23 b(=)g FP(f)p FN(f)510 983 y FL(n)554 1013 y FO(\(0\))28 b FN(;)41 b(n)23 b(>)g FO(0)p FP(g)p FO(;)-17 1226 y(2.)41 b FN(\033)s FO(\()p FN(C)236 1196 y FK(2)230 1247 y(1)275 1226 y FO(\))23 b(=)g FP(f)p FO(1)p FN(=)p FO(\(1)16 b FP(\000)i FN(\026)767 1196 y FK(2)805 1226 y FO(\))p FP(g)p FO(.)-118 1427 y(Here)25 b FN(f)9 b FO(\()p FN(x)p FO(\))24 b(=)e(1)14 b(+)g FN(\026)533 1397 y FK(2)571 1427 y FN(x)p FO(,)26 b FN(x)e FP(2)f FI(R)p FO(.)42 b(First,)25 b(consider)f(the)i(case)e (where)h FN(\033)s FO(\()p FN(C)2180 1397 y FK(2)2174 1448 y(1)2219 1427 y FO(\))e(=)-118 1527 y FP(f)p FN(f)-26 1497 y FL(n)18 1527 y FO(\(0\))p FN(;)28 b(n)f(>)f FO(0)p FP(g)p FO(.)43 b(In)30 b(this)f(case)g(the)h(op)r(erators)e FN(C)1521 1497 y FK(2)1515 1547 y(1)1589 1527 y FO(and)h FN(U)1818 1497 y FM(\003)1809 1547 y FK(1)1886 1527 y FO(are)g(unitarily)-118 1627 y(equiv)-5 b(alen)n(t)25 b(to)j(the)g(follo)n(wing)23 b(op)r(erators:)-40 2005 y FN(C)25 1971 y FK(2)19 2026 y(1)85 2005 y FO(=)173 1813 y Fy(0)173 1959 y(B)173 2012 y(@)245 1877 y FN(f)9 b FO(\(0\))484 1977 y FN(f)534 1946 y FK(2)571 1977 y FO(\(0\))765 2073 y(.)797 2098 y(.)829 2123 y(.)857 1813 y Fy(1)857 1959 y(C)857 2012 y(A)948 2005 y FP(\012)18 b FN(I)7 b(;)97 b(U)1260 1971 y FM(\003)1251 2026 y FK(1)1321 2005 y FO(=)1408 1763 y Fy(0)1408 1910 y(B)1408 1959 y(B)1408 2009 y(B)1408 2062 y(@)1481 1827 y FO(0)1481 1927 y(1)83 b(0)1606 2026 y(1)110 b(0)1735 2123 y(.)1767 2148 y(.)1799 2173 y(.)1915 2123 y(.)1947 2148 y(.)1979 2173 y(.)2007 1763 y Fy(1)2007 1910 y(C)2007 1959 y(C)2007 2009 y(C)2007 2062 y(A)2098 2005 y FP(\012)18 b FN(I)7 b(;)-68 2492 y(C)-3 2458 y FK(2)-9 2512 y FL(i)58 2492 y FO(=)146 2250 y Fy(0)146 2396 y(B)146 2446 y(B)146 2496 y(B)146 2549 y(@)218 2314 y FO(1)343 2414 y FN(\026)393 2384 y FK(2)513 2513 y FN(\026)563 2483 y FK(4)688 2610 y FO(.)720 2635 y(.)753 2660 y(.)780 2250 y Fy(1)780 2396 y(C)780 2446 y(C)780 2496 y(C)780 2549 y(A)871 2492 y FP(\012)973 2471 y FO(^)954 2492 y FN(C)1019 2458 y FK(2)1013 2512 y FL(i)1057 2492 y FN(;)97 b(U)1243 2458 y FM(\003)1234 2512 y FL(i)1304 2492 y FO(=)22 b FN(I)k FP(\012)1550 2471 y FO(^)1536 2492 y FN(U)1602 2458 y FM(\003)1593 2512 y FL(i)1640 2492 y FN(;)180 b(i)22 b FO(=)h(2)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)-118 2891 y FO(where)26 b(the)h(families)22 b FP(f)625 2870 y FO(^)606 2891 y FN(C)671 2861 y FK(2)665 2913 y FL(i)708 2891 y FP(g)p FO(,)27 b FP(f)856 2870 y FO(^)842 2891 y FN(U)899 2903 y FL(i)926 2891 y FP(g)f FO(satisfy)f(the)i(relations)c(\(2.52\))j (with)g FN(d)16 b FP(\000)g FO(1)-118 2991 y(generators.)6 3102 y(Consider)28 b(no)n(w)h(the)i(case)d FN(\033)s FO(\()p FN(C)1000 3072 y FK(2)994 3123 y(1)1039 3102 y FO(\))f(=)f FP(f)p FO(1)p FN(=)p FO(\(1)17 b FP(\000)h FN(\026)1539 3072 y FK(2)1576 3102 y FO(\))p FP(g)p FO(.)43 b(Here)30 b FN(U)1972 3114 y FK(1)2039 3102 y FO(is)e(a)i(uni-)-118 3202 y(tary)g(op)r(erator,)g(and)h(it)f(is)g(easy)g(to)h(deduce)g(from) e FN(C)1598 3172 y FK(2)1592 3224 y FL(i)1636 3202 y FN(U)1702 3172 y FM(\003)1693 3224 y FL(i)1768 3202 y FO(=)g FN(\026)1912 3172 y FK(2)1949 3202 y FN(U)2015 3172 y FM(\003)2006 3224 y FL(i)2053 3202 y FN(C)2118 3172 y FK(2)2112 3224 y FL(i)2186 3202 y FO(that)-118 3302 y FN(C)-53 3271 y FK(2)-59 3323 y FL(i)7 3302 y FO(=)23 b(0)f(for)f(a)g(b)r(ounded)i(represen)n(tation.)32 b(This)21 b(follo)n(ws)e(from)h(the)i(fact)g(that)-118 3401 y(if)33 b FN(U)21 3413 y FK(1)86 3401 y FO(is)26 b(unitary)-7 b(,)27 b(then)h FN(\033)s FO(\()p FN(C)815 3371 y FK(2)809 3423 y FL(i)853 3401 y FO(\))g(is)e(in)n(v)-5 b(arian)n(t)25 b(under)i(m)n(ultiplication)22 b(b)n(y)27 b FN(\026)2278 3371 y FK(2)2316 3401 y FO(,)-118 3501 y FN(\026)-68 3471 y FM(\000)p FK(2)21 3501 y FO(.)51 b(Consequen)n(tly)-7 b(,)32 b(the)h(sp)r(ectrum)e(of)i FN(C)1312 3471 y FK(2)1306 3522 y FL(i)1349 3501 y FO(,)h FN(i)c FO(=)h(2,)h FN(:)14 b(:)g(:)27 b FO(,)34 b FN(d)p FO(,)g(is)d(b)r(ounded)-118 3600 y(if)f(and)g(only)e(if)i FN(\033)s FO(\()p FN(C)535 3570 y FK(2)529 3622 y FL(i)573 3600 y FO(\))e(=)f FP(f)p FO(0)p FP(g)i FO(and,)h(since)f(k)n(er)13 b FN(U)1455 3612 y FL(i)1510 3600 y FO(=)27 b(k)n(er)13 b FN(C)1786 3612 y FL(i)1814 3600 y FO(,)31 b(w)n(e)e(ha)n(v)n(e)g (that)-118 3700 y FN(U)-61 3712 y FL(i)-11 3700 y FO(=)23 b(0.)36 b(Then,)25 b(the)h(op)r(erator)d FN(U)945 3712 y FK(1)1007 3700 y FO(is)h(in)h(the)g(cen)n(ter)g(of)g(the)g(represen)n (tation,)-118 3800 y(and)i FN(U)109 3770 y FM(\003)100 3820 y FK(1)170 3800 y FO(=)c FN(\025I)7 b FO(,)28 b FP(j)p FN(\025)p FP(j)c FO(=)e(1.)6 3911 y(T)-7 b(o)28 b(obtain)e(the)i(\014nal)e(result,)h(w)n(e)g(com)n(bine)e(these)j(t)n (w)n(o)f(cases.)p eop %%Page: 158 162 158 161 bop -118 -137 a FO(158)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(Prop)s(osition)30 b(52.)41 b FB(Fix)32 b(a)h(numb)l(er)e FN(i)p FB(,)j FO(1)27 b FP(\024)g FN(i)h FP(\024)f FN(d)p FB(,)34 b(and)f(c)l(onsider)g(the)g (fol-)-118 196 y(lowing)e(r)l(epr)l(esentation)f(of)g(the)g FN(\026)p FB(-CCR)g(algebr)l(a)6 b FO(:)335 431 y FN(C)400 396 y FK(2)394 451 y FL(j)461 431 y FO(=)553 323 y FL(j)s FM(\000)p FK(1)548 352 y Fy(O)551 530 y FL(k)q FK(=1)688 431 y FN(d)p FO(\()p FN(\026)p FO(\))19 b FP(\012)f FN(d)p FO(\()p FN(f)9 b FO(\))19 b FP(\012)1253 327 y FL(i)p FM(\000)p FK(1)1244 352 y Fy(O)1206 530 y FL(k)q FK(=)p FL(j)s FK(+1)1422 431 y FN(I)7 b(;)184 b(j)28 b(<)22 b(i;)334 738 y(U)400 703 y FM(\003)391 758 y FL(j)461 738 y FO(=)553 630 y FL(j)s FM(\000)p FK(1)548 659 y Fy(O)551 838 y FL(k)q FK(=1)688 738 y FN(I)j FP(\012)18 b FN(S)23 b FP(\012)1036 634 y FL(i)p FM(\000)p FK(1)1028 659 y Fy(O)989 838 y FL(k)q FK(=)p FL(j)s FK(+1)1206 738 y FN(I)7 b(;)183 b(j)28 b(<)23 b(i;)335 1041 y(C)400 1007 y FK(2)394 1061 y FL(i)461 1041 y FO(=)653 985 y(1)p 558 1022 231 4 v 558 1098 a(1)18 b FP(\000)g FN(\026)751 1074 y FK(2)821 937 y FL(i)p FM(\000)p FK(1)812 962 y Fy(O)815 1141 y FL(k)q FK(=1)952 1041 y FN(d)p FO(\()p FN(\026)p FO(\))p FN(;)353 1252 y(U)410 1264 y FL(i)461 1252 y FO(=)k FN(\025)596 1264 y FL(i)624 1252 y FN(I)7 b(;)99 b(C)848 1264 y FL(j)906 1252 y FO(=)23 b FN(U)1051 1264 y FL(j)1109 1252 y FO(=)f(0)p FN(;)184 b(j)28 b(>)22 b(i;)-118 1407 y FB(wher)l(e)16 1719 y FN(d)p FO(\()p FN(\026)p FO(\))i(=)285 1477 y Fy(0)285 1623 y(B)285 1673 y(B)285 1723 y(B)285 1776 y(@)358 1541 y FO(1)482 1641 y FN(\026)532 1610 y FK(2)652 1740 y FN(\026)702 1710 y FK(4)827 1837 y FB(.)862 1862 y(.)897 1887 y(.)927 1477 y Fy(1)927 1623 y(C)927 1673 y(C)927 1723 y(C)927 1776 y(A)1013 1719 y FN(;)99 b(d)p FO(\()p FN(f)9 b FO(\))23 b(=)1403 1527 y Fy(0)1403 1673 y(B)1403 1726 y(@)1476 1591 y FN(f)9 b FO(\(0\))1714 1690 y FN(f)1764 1660 y FK(2)1801 1690 y FO(\(0\))1995 1787 y FB(.)2030 1812 y(.)2064 1837 y(.)2094 1527 y Fy(1)2094 1673 y(C)2094 1726 y(A)2181 1719 y FN(;)-118 2051 y(S)34 b FB(is)c(the)g(unilater)l (al)h(shift)f(in)g FN(l)876 2063 y FK(2)913 2051 y FO(\()p FI(N)t FO(\))q FB(,)36 b(and)30 b FN(\025)1302 2063 y FL(i)1353 2051 y FP(2)23 b FI(C)15 b FB(,)37 b FP(j)p FN(\025)1618 2063 y FL(i)1646 2051 y FP(j)23 b FO(=)f(1)p FB(.)6 2151 y(Then)35 b(al)t(l)g(irr)l(e)l(ducible)h(r)l(epr)l (esentations)e(of)h(the)g FN(\026)p FB(-CCR)f(algebr)l(a)h(c)l(oin-) -118 2250 y(cide)e(with)f(the)h(ones)f(c)l(onstructe)l(d)f(for)h(some)h FN(i)p FB(.)44 b(Mor)l(e)l(over,)35 b(two)d(r)l(epr)l(esen-)-118 2350 y(tations)k(ar)l(e)h(e)l(quivalent)g(if)g(and)g(only)g(if)h(they)f (c)l(orr)l(esp)l(ond)g(to)f(the)h(same)g FN(i)-118 2450 y FB(and)30 b FN(\025)91 2462 y FL(i)119 2450 y FB(.)-118 2594 y FQ(3.)35 b FN(\026)p FO(-CAR)26 b(algebra.)34 b(No)n(w)25 b(consider)f(the)i(an)n(ticomm)n(utativ)n(e)21 b(case.)36 b(By)25 b(the)-118 2693 y FN(\026)p FO(-CAR)j(algebra)c(w)n (e)k(call)d(the)j(follo)n(wing)23 b FP(\003)p FO(-algebra:)30 2869 y FI(C)84 2777 y Fy(D)141 2869 y FN(x)188 2881 y FL(i)216 2869 y FN(;)14 b(x)300 2835 y FM(\003)300 2889 y FL(i)361 2869 y FP(j)23 b FN(x)454 2835 y FM(\003)454 2889 y FL(i)493 2869 y FN(x)540 2881 y FL(i)591 2869 y FO(=)g(1)18 b FP(\000)g FN(x)869 2881 y FL(i)897 2869 y FN(x)944 2835 y FM(\003)944 2889 y FL(i)1001 2869 y FP(\000)g FO(\(1)g FP(\000)g FN(\026)1309 2835 y FK(2)1346 2869 y FO(\))1392 2790 y Fy(X)1399 2967 y FL(j)s(<i)1526 2869 y FN(x)1573 2881 y FL(j)1609 2869 y FN(x)1656 2835 y FM(\003)1656 2889 y FL(j)1694 2869 y FN(;)28 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)g(;)g(d;)568 3115 y(x)615 3081 y FM(\003)615 3136 y FL(i)654 3115 y FN(x)701 3127 y FL(j)759 3115 y FO(=)23 b FP(\000)p FN(\026x)1009 3127 y FL(j)1044 3115 y FN(x)1091 3081 y FM(\003)1091 3136 y FL(i)1129 3115 y FN(;)28 b(i)23 b FP(6)p FO(=)g FN(j)1359 3023 y Fy(E)1409 3115 y FN(:)-118 3295 y FB(R)l(emark)30 b(38.)42 b FO(Let)30 b(us)f(note)g(that)g(an)n(y)f(represen)n(tation)f FN(\031)s FO(\()p FP(\001)p FO(\))j(of)f(the)h FN(\026)p FO(-CAR)-118 3395 y(algebra)25 b(is)h(b)r(ounded,)i(since)f FN(\031)s FO(\()p FN(x)951 3365 y FM(\003)951 3417 y FL(i)990 3395 y FN(x)1037 3407 y FL(i)1065 3395 y FO(\))c FP(\025)g FO(0)k(for)g(an)n(y)g FN(i)p FO(,)g(and,)h(therefore,)397 3568 y FQ(1)18 b FP(\000)g FN(\031)s FO(\()p FN(x)675 3580 y FL(i)704 3568 y FN(x)751 3533 y FM(\003)751 3588 y FL(i)789 3568 y FO(\))h FP(\000)f FO(\(1)g FP(\000)g FN(\026)1148 3533 y FK(2)1186 3568 y FO(\))1232 3489 y Fy(X)1239 3666 y FL(j)s(<i)1365 3568 y FN(\031)s FO(\()p FN(x)1494 3580 y FL(j)1531 3568 y FN(x)1578 3533 y FM(\003)1578 3588 y FL(j)1616 3568 y FO(\))23 b FP(\025)g FO(0)p FN(;)-118 3811 y FO(whic)n(h)32 b(implies)d(that)k FP(k)p FN(\031)s FO(\()p FN(x)768 3823 y FL(i)796 3811 y FN(x)843 3781 y FM(\003)843 3833 y FL(i)881 3811 y FO(\))p FP(k)f(\024)f FO(1.)52 b(Then)33 b FN(\031)s FO(\()p FN(x)1551 3823 y FL(i)1579 3811 y FO(\))g(is)f(b)r(ounded)h(for)f(an)n(y)-118 3911 y FN(i)23 b FO(=)f(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(d)p FO(.)p eop %%Page: 159 163 159 162 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(159)6 96 y(T)-7 b(o)32 b(reduce)f(the)i FN(\026)p FO(-CAR)f(algebra)d(to)i(a)h(dynamical)c (form,)k(w)n(e)f(ha)n(v)n(e)g(to)-118 196 y(\014nd)i(additional)d (relations)g(for)i(the)i(generators)c FN(x)1529 208 y FL(i)1558 196 y FO(,)k FN(x)1662 208 y FL(j)1731 196 y FO(so)e(that)h(these)g(re-)-118 296 y(lations)d(w)n(ould)i(b)r(e)h (compatible)c(with)k(the)g(de\014ning)f(relations.)50 b(F)-7 b(or)32 b(Wic)n(k)-118 395 y(algebras)d(suc)n(h)j(relations)d (are)i(describ)r(ed)g(b)n(y)h(Wic)n(k)f(ideals,)g(in)h(particular,)-118 495 y(quadratic)39 b(Wic)n(k)h(ideals)f(\(see)i(Section)g(2.4.6\).)76 b(The)42 b(largest)d(quadratic)-118 595 y(Wic)n(k)26 b(ideal)e(of)j(the)g FN(\026)p FO(-CAR)g(algebra)e(is)g(generated)h(b)n (y)h(the)g(follo)n(wing)c(fam-)-118 694 y(ily)j(of)h(elemen)n(ts:)81 857 y FN(x)128 869 y FL(i)157 857 y FN(x)204 869 y FL(j)257 857 y FO(+)18 b FN(\026x)437 869 y FL(j)473 857 y FN(x)520 869 y FL(i)548 857 y FN(;)97 b FO(1)22 b FP(\024)h FN(j)28 b(<)23 b(i)f FP(\024)h FN(d;)97 b FO(and)83 b FN(x)1536 823 y FK(2)1536 877 y FL(i)1574 857 y FN(;)97 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(d:)-118 1020 y FO(According)23 b(to)i(the)h(previous)d(example,)g(w)n(e)i(ha)n (v)n(e)f(to)h(sho)n(w)f(that)h(the)h(gener-)-118 1119 y(ators)31 b(of)h(this)f(ideal)f(are)i(annihilated)d(in)i(an)n(y)h (irreducible)c(represen)n(tation)-118 1219 y(of)23 b(the)h FN(\026)p FO(-CAR)f(algebra.)33 b(Ho)n(w)n(ev)n(er,)22 b(this)h(is)f(not)h(true)g(for)g(the)h FN(\026)p FO(-CAR)f(alge-)-118 1318 y(bra.)42 b(The)29 b(largest)f(quadratic)f(ideal)g(is)i(v)n(ery)f (large.)40 b(So,)30 b(consider)e(another)-118 1418 y(quadratic)d(Wic)n (k)i(ideal,)81 1558 y(^)80 1581 y FA(I)116 1593 y FK(2)176 1581 y FO(=)263 1514 y Fy(\012)302 1581 y FN(x)349 1593 y FL(i)378 1581 y FN(x)425 1593 y FL(j)478 1581 y FO(+)18 b FN(\026x)658 1593 y FL(j)694 1581 y FN(x)741 1593 y FL(i)769 1581 y FN(;)28 b FO(1)22 b FP(\024)h FN(j)28 b(<)23 b(i)f FP(\024)h FN(d;)14 b FO(;)g FN(x)1425 1547 y FK(2)1425 1601 y FL(i)1463 1581 y FN(;)27 b(i)c FO(=)g(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(d)k FP(\000)g FO(1)2079 1514 y Fy(\013)2118 1581 y FN(:)-118 1744 y FQ(Theorem)30 b(36.)41 b FB(F)-6 b(or)25 b(any)g(irr)l(e)l(ducible)h(r)l(epr)l (esentation)f FN(\031)s FO(\()p FP(\001)p FO(\))h FB(of)f(the)g FN(\026)p FB(-CAR)-118 1853 y(algebr)l(a)31 b FN(\031)s FO(\()247 1830 y(^)245 1853 y FA(I)281 1865 y FK(2)319 1853 y FO(\))23 b(=)g FP(f)p FO(0)p FP(g)28 b FB(holds.)-118 2003 y(Pr)l(o)l(of.)43 b FO(Let)27 b(us)h(denote)f FN(X)733 2015 y FL(i)783 2003 y FO(=)c FN(\031)s FO(\()p FN(x)1000 2015 y FL(i)1028 2003 y FO(\),)28 b FN(A)23 b FO(=)g FN(\031)s FO(\()p FN(x)1413 1973 y FK(2)1413 2024 y(1)1451 2003 y FO(\),)28 b FN(B)f FO(=)c FN(\031)s FO(\()p FN(x)1841 2015 y FK(2)1879 2003 y FN(x)1926 2015 y FK(1)1982 2003 y FO(+)17 b FN(\026x)2161 2015 y FK(1)2199 2003 y FN(x)2246 2015 y FK(2)2283 2003 y FO(\).)-118 2103 y(No)n(w)30 b(w)n(e)h(pro)n(v)n(e)f(that)h FN(A)e FO(=)g FN(B)k FO(=)28 b(0.)47 b(It)32 b(is)e(easy)g(to)h(see)f(from)g(the)i(de\014ning)-118 2202 y(relations)24 b(that)311 2365 y FN(A)373 2331 y FM(\003)412 2365 y FN(A)f FO(=)g FN(AA)709 2331 y FM(\003)747 2365 y FN(;)97 b(A)929 2331 y FM(\003)968 2365 y FN(X)1037 2377 y FL(k)1100 2365 y FO(=)23 b FN(\026)1238 2331 y FK(2)1275 2365 y FN(X)1344 2377 y FL(k)1385 2365 y FN(A)1447 2331 y FM(\003)1485 2365 y FN(;)180 b(k)26 b(>)d FO(1)p FN(:)216 b FO(\(2.53\))-118 2528 y(Since)28 b FN(A)h FO(is)f(a)h(normal)d(op)r(erator,)h(w)n(e)i(can)f(use)h(the)g(F)-7 b(uglede{Putnam)27 b(the-)-118 2627 y(orem)f(and)h(obtain)f(the)i (follo)n(wing)c(relations:)627 2790 y FN(AX)758 2802 y FL(k)822 2790 y FO(=)f FN(\026)960 2756 y FK(2)997 2790 y FN(X)1066 2802 y FL(k)1107 2790 y FN(A;)180 b(k)26 b(>)d FO(1)p FN(:)-118 2953 y FO(\(It)28 b(is)f(ob)n(vious)f(that)i FN(AX)700 2965 y FK(1)761 2953 y FO(=)23 b FN(X)918 2965 y FK(1)955 2953 y FN(A)p FO(\).)39 b(Th)n(us,)28 b(in)f(an)n(y)g (irreducible)d(represen-)-118 3052 y(tation,)33 b(either)f FN(A)g FO(=)g(0)h(or)f(k)n(er)13 b FN(A)32 b FO(=)g FP(f)p FO(0)p FP(g)f FO(\(b)r(ecause)i(k)n(er)13 b FN(A)33 b FO(is)f(an)h(in)n(v)-5 b(arian)n(t)-118 3152 y(subspace\).)36 b(Let)28 b(k)n(er)13 b FN(A)23 b FO(=)g FP(f)p FO(0)p FP(g)p FO(.)35 b(Then)28 b(w)n(e)f(ha)n(v)n(e:)393 3315 y FN(B)460 3280 y FM(\003)499 3315 y FN(B)g FO(=)c FN(\026)727 3280 y FK(2)764 3315 y FN(B)t(B)898 3280 y FM(\003)955 3315 y FO(+)18 b(\(1)g FP(\000)g FN(\026)1263 3280 y FK(4)1300 3315 y FO(\)\(1)h(+)f FN(\026)1558 3280 y FK(2)1595 3315 y FO(\))p FN(AA)1751 3280 y FM(\003)1790 3315 y FN(;)544 3449 y(A)606 3415 y FM(\003)644 3449 y FN(B)27 b FO(=)c FN(\026)872 3415 y FK(2)909 3449 y FN(B)t(A)1038 3415 y FM(\003)1077 3449 y FN(;)97 b(AB)27 b FO(=)c FN(\026)1487 3415 y FK(2)1524 3449 y FN(B)t(A:)450 b FO(\(2.54\))-118 3612 y(It)38 b(follo)n(ws)d(from)h(the)j(condition)c(k)n(er)13 b FN(A)40 b FO(=)g FP(f)p FO(0)p FP(g)c FO(that)i FN(AA)1807 3582 y FM(\003)1886 3612 y FN(>)i FO(0.)67 b(Equa-)-118 3712 y(tion)25 b(\(2.4.2\))g(implies)c(that)26 b FN(B)836 3682 y FM(\003)875 3712 y FN(B)h(>)c FO(0.)35 b(Let)26 b(us)g(consider)d(the)k(p)r(olar)c(decom-)-118 3811 y(p)r(osition)31 b FN(B)271 3781 y FM(\003)341 3811 y FO(=)g FN(W)12 b(T)g FO(,)33 b(k)n(er)13 b FN(W)44 b FO(=)31 b(k)n(er)13 b FN(T)f FO(,)33 b FN(T)43 b FP(\025)32 b FO(0,)i FN(T)1578 3781 y FK(2)1646 3811 y FO(=)d FN(B)t(B)1876 3781 y FM(\003)1914 3811 y FO(,)k(and,)f(since)-118 3911 y FN(B)-51 3881 y FM(\003)-13 3911 y FN(B)27 b(>)c FO(0,)k(w)n(e)h(ha)n(v)n(e)e(that)i FN(W)39 b FO(is)27 b(a)g(co-isometry)-7 b(.)p eop %%Page: 160 164 160 163 bop -118 -137 a FO(160)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(No)n(w)k(w)n(e)f(can)g(rewrite)f(relations)e (\(2.4.2\))j(in)g(an)g(equiv)-5 b(alen)n(t)26 b(form:)302 270 y FN(T)363 235 y FK(2)400 270 y FN(W)490 235 y FM(\003)551 270 y FO(=)d FN(W)729 235 y FM(\003)767 202 y Fy(\000)805 270 y FN(\026)855 235 y FK(2)892 270 y FN(T)953 235 y FK(2)1008 270 y FO(+)18 b(\(1)g FP(\000)g FN(\026)1316 235 y FK(4)1353 270 y FO(\)\(1)h(+)f FN(\026)1611 235 y FK(2)1648 270 y FO(\))p FN(AA)1804 235 y FM(\003)1843 202 y Fy(\001)1881 270 y FN(;)572 404 y(AW)724 370 y FM(\003)785 404 y FO(=)23 b FN(\026)923 370 y FK(2)960 404 y FN(W)1050 370 y FM(\003)1088 404 y FN(A;)97 b(AT)35 b FO(=)22 b FN(T)12 b(A:)477 b FO(\(2.55\))-118 578 y(These)31 b(equalities)c(and)k(relations)d(\(2.53\))j(imply)d(that)j(the)h(sp)r (ectrum)e(of)h FN(A)-118 677 y FO(coincides)19 b(with)i(the)h(set)f FP(f)p FN(\025\026)810 647 y FK(2)p FL(n)888 677 y FN(x)935 689 y FK(1)973 677 y FN(;)28 b(n)23 b FP(2)g FI(N)10 b FP([)d(f)p FO(0)p FP(gg)25 b FO(for)c(some)f FN(\025)k FP(2)f FI(C)15 b FO(,)29 b FP(j)p FN(\025)p FP(j)23 b FO(=)g(1,)-118 777 y FN(x)-71 789 y FK(1)-8 777 y FN(>)i FO(0.)42 b(Since)28 b FN(W)497 747 y FM(\003)564 777 y FO(is)g(an)h(isometry)-7 b(,)27 b(the)i(eigen)n(v)-5 b(alues)26 b(of)j(the)h(op)r(erator)d FN(A)-118 877 y FO(ha)n(v)n(e)g(the)h(same)f(m)n(ultiplicit)n(y)-7 b(,)23 b(and)28 b(c)n(ho)r(osing)d(the)k(corresp)r(onding)c(basis)h(in)-118 976 y(the)f(represen)n(tation)e(space)h(w)n(e)h(can)g(write)f(the)h(op) r(erators)e FN(A)j FO(and)f FN(W)37 b FO(in)24 b(the)-118 1076 y(follo)n(wing)f(form:)78 1372 y FN(A)g FO(=)251 1180 y Fy(0)251 1326 y(B)251 1379 y(@)324 1243 y FN(\025x)419 1255 y FK(1)457 1243 y FN(I)583 1343 y(\025x)678 1355 y FK(1)716 1343 y FN(\026)766 1313 y FK(2)803 1343 y FN(I)934 1440 y FO(.)966 1465 y(.)998 1490 y(.)1026 1180 y Fy(1)1026 1326 y(C)1026 1379 y(A)1112 1372 y FN(;)97 b(W)35 b FO(=)1433 1180 y Fy(0)1433 1326 y(B)1433 1379 y(@)1506 1243 y FO(0)82 b FN(I)1631 1343 y FO(0)110 b FN(I)1761 1440 y FO(.)1793 1465 y(.)1825 1490 y(.)1941 1440 y(.)1973 1465 y(.)2005 1490 y(.)2033 1180 y Fy(1)2033 1326 y(C)2033 1379 y(A)2119 1372 y FN(:)-118 1667 y FO(The)28 b(condition)d FN(T)12 b(A)22 b FO(=)h FN(AT)39 b FO(then)28 b(giv)n(es)d(that)632 2013 y FN(T)34 b FO(=)803 1771 y Fy(0)803 1918 y(B)803 1967 y(B)803 2017 y(B)803 2070 y(@)876 1835 y FN(T)925 1847 y FK(0)1044 1935 y FN(T)1093 1947 y FK(1)1213 2034 y FN(T)1262 2046 y FK(2)1387 2131 y FO(.)1419 2156 y(.)1451 2181 y(.)1479 1771 y Fy(1)1479 1918 y(C)1479 1967 y(C)1479 2017 y(C)1479 2070 y(A)1565 2013 y FN(:)-118 2359 y FO(Since)d(k)n(er)12 b FN(W)42 b FO(=)29 b(k)n(er)13 b FN(T)f FO(,)32 b(w)n(e)f(ha)n(v)n(e)g(that)g FN(T)1236 2371 y FK(0)1303 2359 y FO(=)e(0.)49 b(Moreo)n(v)n(er,)30 b(it)h(is)f(easy)h(to)-118 2458 y(obtain)26 b(from)g(\(2.4.2\))h(that) 280 2631 y FN(T)329 2643 y FL(n)397 2631 y FO(=)c FN(x)532 2643 y FK(1)583 2631 y FO(\(1)18 b(+)g FN(\026)808 2597 y FK(2)846 2631 y FO(\))c FN(\026)942 2597 y FL(n)p FM(\000)p FK(1)1072 2631 y FO(\(1)k FP(\000)g FN(\026)1297 2597 y FK(2)p FL(n)1376 2631 y FO(\))1408 2597 y FK(1)p FL(=)p FK(2)1512 2631 y FN(;)180 b(n)23 b FP(\025)g FO(1)p FN(:)-118 2805 y FO(Since)k FN(X)168 2817 y FK(1)205 2805 y FN(A)c FO(=)g FN(AX)509 2817 y FK(1)546 2805 y FO(,)28 b FN(X)673 2775 y FM(\003)666 2825 y FK(1)710 2805 y FN(A)23 b FO(=)g FN(AX)1021 2775 y FM(\003)1014 2825 y FK(1)1059 2805 y FO(,)28 b(w)n(e)f(get)706 3105 y FN(X)775 3117 y FK(1)836 3105 y FO(=)923 2913 y Fy(0)923 3059 y(B)923 3112 y(@)996 2977 y FN(b)1032 2989 y FK(1)1152 3077 y FN(b)1188 3089 y FK(2)1312 3173 y FO(.)1345 3198 y(.)1377 3223 y(.)1404 2913 y Fy(1)1404 3059 y(C)1404 3112 y(A)1491 3105 y FN(:)-118 3401 y FO(Let)h(us)f(write)g(the)h(op)r(erator)d FN(B)j FO(=)22 b FN(T)12 b(W)1156 3371 y FM(\003)1221 3401 y FO(in)27 b(the)h(matrix)d(form,)623 3746 y FN(B)i FO(=)801 3505 y Fy(0)801 3651 y(B)801 3701 y(B)801 3751 y(B)801 3804 y(@)896 3569 y FO(0)874 3668 y FN(T)923 3680 y FK(1)1064 3668 y FO(0)1042 3768 y FN(T)1091 3780 y FK(2)1239 3768 y FO(0)1216 3865 y(.)1248 3889 y(.)1280 3915 y(.)1396 3865 y(.)1428 3889 y(.)1460 3915 y(.)1488 3505 y Fy(1)1488 3651 y(C)1488 3701 y(C)1488 3751 y(C)1488 3804 y(A)1574 3746 y FN(:)p eop %%Page: 161 165 161 164 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(161)-118 96 y(The)27 b(condition)d FN(X)491 66 y FM(\003)484 117 y FK(1)529 96 y FN(B)j FO(=)c FN(\026B)t(X)900 66 y FM(\003)893 117 y FK(1)964 96 y FO(expressed)j(in)g(terms)g(of)g(the)i(matrix)c(co)r(ef-)-118 196 y(\014cien)n(ts)i(tak)n(es)h(the)h(follo)n(wing)c(form:)795 377 y FN(b)831 343 y FM(\003)831 397 y FL(n)p FK(+1)960 377 y FN(T)1009 389 y FL(n)1077 377 y FO(=)e FN(\026)14 b(T)1277 389 y FL(n)1322 377 y FN(b)1358 343 y FM(\003)1358 397 y FL(n)1403 377 y FN(:)-118 558 y FO(Let)31 b(us)f(note)h(that)g FN(x)561 570 y FK(1)627 558 y FP(6)p FO(=)d(0,)j(hence,)h FN(T)1123 570 y FL(n)1196 558 y FP(6)p FO(=)c(0,)j(and)f(w)n(e)h(ha)n (v)n(e)e FN(b)1905 570 y FL(k)1974 558 y FO(=)f FN(\026)2117 528 y FL(k)q FM(\000)p FK(1)2243 558 y FN(b)2279 570 y FK(1)2316 558 y FO(.)-118 658 y(F)-7 b(rom)26 b(the)i(relation)d FN(x)595 627 y FM(\003)595 678 y FK(1)633 658 y FN(x)680 670 y FK(1)741 658 y FO(=)e(1)18 b FP(\000)g FN(x)1019 670 y FK(1)1056 658 y FN(x)1103 627 y FM(\003)1103 678 y FK(1)1170 658 y FO(it)27 b(follo)n(ws)d(that)404 838 y FN(b)440 804 y FM(\003)440 859 y FK(1)478 838 y FN(b)514 850 y FK(1)574 838 y FO(=)f(1)18 b FP(\000)g FN(b)841 850 y FK(1)878 838 y FN(b)914 804 y FM(\003)914 859 y FK(1)951 838 y FN(;)97 b(\026)1121 804 y FK(2)1159 838 y FN(b)1195 850 y FK(1)1232 838 y FN(b)1268 804 y FM(\003)1268 859 y FK(1)1328 838 y FO(=)23 b(1)18 b FP(\000)g FN(\026)1609 804 y FK(2)1646 838 y FN(b)1682 850 y FK(1)1719 838 y FN(b)1755 804 y FM(\003)1755 859 y FK(1)1793 838 y FN(:)-118 1019 y FO(These)31 b(equalities)d(are)j(compatible)e(if)i(and)g(only)f (if)i FN(\026)1631 989 y FK(2)1698 1019 y FO(=)d(1,)k(whic)n(h)d(is)h (im-)-118 1119 y(p)r(ossible.)65 b(Hence,)40 b FN(x)586 1131 y FK(1)663 1119 y FO(=)g(0,)g FN(A)g FO(=)f(0,)h(and)d(relation)e (\(2.4.2\))i(yields)e(that)-118 1219 y FN(B)-51 1188 y FM(\003)-13 1219 y FN(B)27 b FO(=)c FN(\026)215 1188 y FK(2)252 1219 y FN(B)t(B)386 1188 y FM(\003)425 1219 y FO(.)37 b(Since)26 b FN(B)32 b FO(is)27 b(a)g(b)r(ounded)h(op)r (erator,)e FN(B)h FO(=)c(0.)6 1318 y(Let)38 b(us)f(denote)g FN(B)621 1330 y FL(k)701 1318 y FO(=)h FN(\031)s FO(\()p FN(x)933 1330 y FL(k)975 1318 y FN(x)1022 1330 y FK(1)1084 1318 y FO(+)25 b FN(\026x)1271 1330 y FK(1)1308 1318 y FN(x)1355 1330 y FL(k)1397 1318 y FO(\),)40 b(2)e FN(<)h(k)i FP(\024)e FN(d)p FO(.)66 b(F)-7 b(rom)35 b(the)-118 1418 y(de\014ning)27 b(relations)d(w)n(e)j(obtain:)-88 1611 y FN(B)-21 1576 y FM(\003)-25 1631 y FL(k)17 1611 y FN(B)80 1623 y FL(k)144 1611 y FO(=)22 b FN(\026)281 1576 y FK(2)319 1611 y FN(B)382 1623 y FL(k)422 1611 y FN(B)489 1576 y FM(\003)485 1631 y FL(k)546 1611 y FO(+)c FN(\026)679 1576 y FK(2)717 1611 y FO(\(1)g FP(\000)g FN(\026)942 1576 y FK(2)979 1611 y FO(\))1064 1532 y Fy(X)1025 1711 y FK(1)p FL(<i<k)1236 1611 y FN(B)1299 1623 y FL(i)1327 1611 y FN(B)1394 1576 y FM(\003)1390 1631 y FL(i)1450 1611 y FO(+)g(\(1)h(+)f FN(\026)1759 1576 y FK(2)1796 1611 y FO(\)\(1)h FP(\000)f FN(\026)2054 1576 y FK(4)2091 1611 y FO(\))p FN(AA)2247 1576 y FM(\003)2286 1611 y FN(:)-118 1871 y FO(It)k(is)f(easy)f(to)i(see)f(that)h(the)g(induction) e(in)h FN(d)h FO(giv)n(es)e(that)i FN(B)1709 1883 y FL(k)1773 1871 y FO(=)h(0,)f FN(k)k FO(=)d(3,)e FN(:)14 b(:)g(:)28 b FO(,)-118 1970 y FN(d)p FO(.)37 b(Th)n(us)25 b(the)h(op)r(erators)e FN(X)769 1982 y FL(i)823 1970 y FO(can)h(b)r(e)i(decomp)r(osed)d(in)h (the)i(tensor)e(pro)r(duct,)120 2196 y FN(X)189 2208 y FK(1)249 2196 y FO(=)337 2079 y Fy(\022)398 2146 y FO(0)83 b(0)398 2245 y(1)g(0)564 2079 y Fy(\023)644 2196 y FP(\012)18 b FN(I)7 b(;)97 b(X)959 2208 y FL(k)1022 2196 y FO(=)1110 2079 y Fy(\022)1171 2146 y FO(1)119 b(0)1171 2245 y(0)83 b FP(\000)p FN(\026)1410 2079 y Fy(\023)1490 2196 y FP(\012)1595 2175 y FO(~)1573 2196 y FN(A)1635 2208 y FL(k)1676 2196 y FN(;)180 b(k)26 b(>)d FO(1)p FN(;)-118 2441 y FO(where)d FP(f)180 2420 y FO(~)157 2441 y FN(X)226 2453 y FL(k)266 2441 y FN(;)34 b(k)26 b(>)d FO(1)p FP(g)c FO(satisfy)g(the)h FN(\026)p FO(-CAR)h(relations)c (with)j FN(d)t FP(\000)t FO(1)f(generators.)-118 2541 y(Moreo)n(v)n(er,)30 b(the)h(set)h FP(f)p FN(X)666 2553 y FL(i)693 2541 y FP(g)f FO(is)f(irreducible)e(if)i(and)i(only)e(if)h (the)g(set)h FP(f)2124 2520 y FO(~)2101 2541 y FN(X)2170 2553 y FL(k)2210 2541 y FP(g)f FO(is)-118 2640 y(irreducible,)559 2821 y FN(X)635 2787 y FK(2)628 2842 y FL(j)694 2821 y FO(=)23 b(0)p FN(;)14 b FP(,)990 2800 y FO(~)966 2821 y FN(X)1042 2787 y FK(2)1035 2842 y FL(j)1102 2821 y FO(=)22 b(0)p FN(;)180 b(j)28 b FP(\025)23 b FO(2)p FN(;)-14 2963 y(X)55 2975 y FL(j)90 2963 y FN(X)159 2975 y FL(i)205 2963 y FO(+)18 b FN(\026X)407 2975 y FL(i)435 2963 y FN(X)504 2975 y FL(j)561 2963 y FO(=)23 b(0)g FP(,)843 2942 y FO(~)820 2963 y FN(X)889 2975 y FL(j)947 2942 y FO(~)923 2963 y FN(X)992 2975 y FL(i)1038 2963 y FO(+)18 b FN(\026)1195 2942 y FO(~)1171 2963 y FN(X)1240 2975 y FL(i)1292 2942 y FO(~)1268 2963 y FN(X)1337 2975 y FL(j)1395 2963 y FO(=)k(0)p FN(;)180 b FO(2)22 b FP(\024)h FN(i)g(<)f(j)28 b FP(\024)23 b FN(d:)-118 3144 y FO(The)28 b(pro)r(of)f(is)f(completed)g(b)n(y)h(induction)f(in)h FN(d)p FO(.)p 2278 3144 4 57 v 2282 3091 50 4 v 2282 3144 V 2331 3144 4 57 v -118 3359 a FQ(2.4.3)94 b(Op)s(erator)21 b(relations)f(connected)i(with)f(man)m(y-dimensional)174 3459 y(dynamical)31 b(systems)-118 3612 y FO(In)26 b(this)f(section,)g (w)n(e)g(consider)f(families)e(of)k(op)r(erators)e(satisfying)f(a)i (general)-118 3712 y(class)34 b(of)i(relations)d(whose)i(represen)n (tations)f(can)h(b)r(e)i(describ)r(ed)e(in)g(terms)-118 3811 y(of)40 b(orbits)f(of)i(some)e(dynamical)e(system)i(acting)g(on)i (the)g(sp)r(ectrum)f(of)g(a)-118 3911 y(comm)n(uting)24 b(sub-family)-7 b(.)p eop %%Page: 162 166 162 165 bop -118 -137 a FO(162)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(W)-7 b(e)25 b(consider)e(represen)n(tations)e (of)j(a)h(family)c(of)j(op)r(erator)f(relations)e(sat-)-118 196 y(is\014ed)27 b(b)n(y)g(the)h(op)r(erators)d FN(X)789 208 y FL(j)824 196 y FO(,)j FN(j)g FO(=)23 b(1,)k FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(,)g(of)f(the)h(follo)n(wing)c(form:)538 378 y FN(X)614 344 y FM(\003)607 398 y FL(j)652 378 y FN(X)721 390 y FL(j)779 378 y FO(=)e FN(F)919 390 y FL(j)955 378 y FO(\()p FN(X)1056 390 y FK(1)1093 378 y FN(X)1169 344 y FM(\003)1162 398 y FK(1)1206 378 y FN(;)14 b(:)g(:)g(:)g(;)g(X) 1460 390 y FL(n)1505 378 y FN(X)1581 344 y FM(\003)1574 398 y FL(n)1619 378 y FO(\))p FN(;)532 513 y(X)608 478 y FM(\003)601 533 y FL(j)646 513 y FN(X)715 525 y FL(k)779 513 y FO(=)22 b FN(\026)916 525 y FL(j)s(k)1002 513 y FN(X)1071 525 y FL(k)1111 513 y FN(X)1187 478 y FM(\003)1180 533 y FL(j)1225 513 y FN(;)542 647 y(X)611 659 y FL(j)646 647 y FN(X)715 659 y FL(k)779 647 y FO(=)g FN(\025)914 659 y FL(j)s(k)1000 647 y FN(X)1069 659 y FL(k)1110 647 y FN(X)1179 659 y FL(j)1214 647 y FN(;)889 b FO(\(2.56\))-118 829 y(where)37 b FN(F)185 841 y FK(1)223 829 y FO(\()p FP(\001)p FO(\),)h FN(:)14 b(:)g(:)28 b FO(,)41 b FN(F)613 841 y FL(n)658 829 y FO(\()p FP(\001)p FO(\))9 b(:)32 b FI(R)863 799 y FL(n)954 829 y FP(\000)-48 b(!)40 b FI(R)k FO(are)37 b(measurable)d(mappings,)k FN(\025)2243 841 y FL(j)s(k)2316 829 y FO(,)-118 929 y FN(\026)-68 941 y FL(j)s(k)34 929 y FN(>)31 b FO(0,)i(1)d FP(\024)g FN(j;)14 b(k)34 b FP(\024)d FN(n)p FO(.)50 b(Notice)31 b(that)i(the)g(last)d(t)n(w)n(o)i(relations)d(in)i(\(2.56\))-118 1029 y(imply)24 b(that)k(the)f(op)r(erators)f FN(X)873 1041 y FK(1)910 1029 y FN(X)986 999 y FM(\003)979 1049 y FK(1)1023 1029 y FO(,)h FN(:)14 b(:)g(:)28 b FO(,)f FN(X)1317 1041 y FL(n)1362 1029 y FN(X)1438 999 y FM(\003)1431 1049 y FL(n)1504 1029 y FO(comm)n(ute;)e(this)h(explains)-118 1128 y(the)i(meaning)d(of)j(the)f(functions)g(used)h(in)f(the)h (\014rst)f(relation.)-118 1261 y FB(R)l(emark)j(39.)42 b FO(Assuming)21 b(that)i FN(X)962 1273 y FK(1)999 1261 y FO(,)f FN(:)14 b(:)g(:)28 b FO(,)c FN(X)1285 1273 y FL(n)1352 1261 y FO(are,)f(generally)c(sp)r(eaking,)j(un-)-118 1360 y(b)r(ounded)j(closed)e(densely)g(de\014ned)i(op)r(erators)e(whic) n(h)h(satisfy)f(the)i(relations)-118 1460 y(\(2.56\))o(,)34 b(w)n(e)d(need)i(to)f(tak)n(e)f(some)g(care)g(in)h(writing)d(relations) g(b)r(et)n(w)n(een)k(un-)-118 1560 y(b)r(ounded)i(op)r(erators;)i(our)d (approac)n(h)f(is)h(to)g(rewrite)f(them)i(in)f(a)h(formally)-118 1659 y(equiv)-5 b(alen)n(t)28 b(form)h(that)i(w)n(ould)e(in)n(v)n(olv)n (e)e(only)i(b)r(ounded)i(op)r(erators)e(\(this)h(is)-118 1759 y(similar)25 b(to)30 b(the)h(case)e(of)i(un)n(b)r(ounded)f (represen)n(tations)d(of)k(a)e(real)g(Lie)g(alge-)-118 1859 y(bra,)34 b(whic)n(h)e(can)h(b)r(e)g(describ)r(ed)f(in)h(terms)f (of)h(unitary)e(represen)n(tations)g(of)-118 1958 y(the)d(corresp)r (onding)d(Lie)h(group\).)6 2091 y(W)-7 b(rite)22 b(the)g(p)r(olar)e (decomp)r(ositions,)f FN(X)1245 2103 y FL(j)1303 2091 y FO(=)k FN(C)1450 2103 y FL(j)1485 2091 y FN(U)1542 2103 y FL(j)1577 2091 y FO(,)g FN(j)28 b FO(=)23 b(1,)e FN(:)14 b(:)g(:)28 b FO(,)23 b FN(n)p FO(,)g(where)-118 2190 y FN(C)-59 2202 y FL(j)8 2190 y FO(are)30 b(non-negativ)n(e,)g FN(U)724 2202 y FL(j)791 2190 y FO(are)g(partial)f(isometries,)f(and)k (eac)n(h)e FN(C)2036 2202 y FL(j)2103 2190 y FO(is)h(zero)-118 2290 y(on)c(v)n(ectors)f(orthogonal)f(to)i(the)h(range)e(of)i FN(U)1317 2302 y FL(j)1351 2290 y FO(.)-118 2455 y FQ(Lemma)h(10.)41 b FB(L)l(et)27 b(the)h(op)l(er)l(ators)h FN(X)1089 2467 y FL(j)1123 2455 y FB(,)g FN(j)f FO(=)23 b(1)p FB(,)k FN(:)14 b(:)g(:)28 b FB(,)g FN(n)p FB(,)h(b)l(e)e(b)l(ounde)l(d.)39 b(Then)-118 2555 y(the)30 b(r)l(elations)37 b FO(\(2.56\))29 b FB(ar)l(e)h(e)l(quivalent)g(to)g(the)g(fol)t(lowing)7 b FO(:)321 2737 y FN(C)386 2703 y FK(2)380 2757 y FL(j)423 2737 y FN(U)480 2749 y FL(k)544 2737 y FO(=)22 b FN(q)668 2749 y FL(j)s(k)740 2737 y FN(U)797 2749 y FL(k)838 2737 y FN(C)903 2703 y FK(2)897 2757 y FL(j)940 2737 y FN(;)184 b(j)28 b FP(6)p FO(=)23 b FN(k)s(;)327 2874 y(C)392 2839 y FK(2)386 2894 y FL(j)429 2874 y FN(U)486 2886 y FL(j)544 2874 y FO(=)f FN(U)688 2886 y FL(j)723 2874 y FN(F)776 2886 y FL(j)812 2874 y FO(\()p FN(C)909 2839 y FK(2)903 2894 y(1)946 2874 y FN(;)14 b(:)g(:)g(:)g(;)g(C)1196 2839 y FK(2)1190 2894 y FL(n)1235 2874 y FO(\))p FN(;)184 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(n;)332 3008 y(U)389 3020 y FL(j)423 3008 y FN(U)480 3020 y FL(k)544 3008 y FO(=)22 b FN(U)688 3020 y FL(k)729 3008 y FN(U)786 3020 y FL(j)821 3008 y FN(;)98 b(U)999 3020 y FL(j)1034 3008 y FN(U)1100 2974 y FM(\003)1091 3029 y FL(k)1161 3008 y FO(=)23 b FN(U)1315 2974 y FM(\003)1306 3029 y FL(k)1353 3008 y FN(U)1410 3020 y FL(j)1444 3008 y FN(;)184 b(j)28 b(<)23 b(k)s(;)256 b FO(\(2.57\))-118 3190 y FB(wher)l(e)669 3426 y FN(q)706 3438 y FL(j)s(k)801 3426 y FO(=)889 3284 y Fy(\()956 3369 y FN(\026)1006 3381 y FL(j)s(k)1077 3369 y FN(\025)1125 3381 y FL(j)s(k)1197 3369 y FN(;)103 b(j)28 b(<)22 b(k)s(;)956 3489 y(\026)1006 3501 y FL(j)s(k)1077 3489 y FN(\025)1125 3453 y FM(\000)p FK(1)1125 3514 y FL(j)s(k)1215 3489 y FN(;)85 b(j)28 b(>)22 b(k)s(:)-118 3691 y FB(Mor)l(e)l(over,)40 b(the)d(op)l(er)l(ators)g FO(\()p FN(U)891 3661 y FM(\003)882 3713 y FL(j)929 3691 y FO(\))961 3661 y FL(k)1002 3691 y FN(U)1068 3661 y FL(k)1059 3713 y(j)1109 3691 y FB(,)i FN(U)1239 3661 y FL(m)1230 3715 y(l)1301 3691 y FO(\()p FN(U)1399 3661 y FM(\003)1390 3715 y FL(l)1438 3691 y FO(\))1470 3661 y FL(m)1533 3691 y FB(,)f FN(j)5 b FB(,)39 b FN(l)e FO(=)e(1)p FB(,)h FN(:)14 b(:)g(:)28 b FB(,)38 b FN(n)p FB(;)i FN(k)s FB(,)-118 3791 y FN(m)29 b FO(=)g(1)p FB(,)34 b FO(2)p FB(,)f FN(:)14 b(:)g(:)27 b FB(,)35 b(form)f(a)f(c)l(ommuting)g(family) 42 b FO(\()p FB(in)34 b(p)l(articular,)h(al)t(l)f FN(U)2159 3803 y FL(j)2227 3791 y FB(ar)l(e)-118 3890 y(c)l(enter)l(e)l(d)29 b(p)l(artial)i(isometries)7 b FO(\))p FB(.)p eop %%Page: 163 167 163 166 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(163)-118 96 y FB(Pr)l(o)l(of.)43 b FO(The)38 b(pro)r(of)g(is)f(a)h(rather)f(straigh)n(tforw)n(ard)e (calculation)f(in)n(v)n(olving)-118 196 y(the)28 b(fact)g(that)f FN(U)425 208 y FL(l)451 196 y FN(U)517 166 y FM(\003)508 220 y FL(l)578 196 y FO(=)22 b(sign)13 b FN(C)882 208 y FL(l)907 196 y FO(.)p 2278 196 4 57 v 2282 143 50 4 v 2282 196 V 2331 196 4 57 v 6 362 a(According)19 b(to)i([193)n(],)i (relations)17 b(\(2.57\))j(can)g(b)r(e)h(rewritten)f(in)g(the)h(follo)n (w-)-118 462 y(ing)f(form,)i(in)n(v)n(olving)17 b(only)j(b)r(ounded)i (op)r(erators.)33 b(In)n(tro)r(duce)21 b(the)h(mappings)-118 561 y(of)27 b FI(R)30 531 y FL(n)109 561 y FO(in)n(to)g(itself)f(b)n(y) 0 744 y FQ(F)60 756 y FL(l)85 744 y FO(\()p FN(\025)165 756 y FK(1)203 744 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)436 756 y FL(n)481 744 y FO(\))24 b(=)e(\()p FN(q)693 756 y FK(1)p FL(l)752 744 y FN(\025)800 756 y FK(1)838 744 y FN(;)14 b(:)g(:)g(:)f(;)h(q)1059 756 y FL(l)p FM(\000)p FK(1)e FL(l)1203 744 y FN(\025)1251 756 y FL(l)p FM(\000)p FK(1)1361 744 y FN(;)680 878 y(F)733 890 y FL(l)759 810 y Fy(\000)797 878 y FN(\025)845 890 y FK(1)882 878 y FN(;)i(:)g(:)g(:)g(;)g(x)1114 890 y FL(n)1159 878 y FO(\))p FN(;)g(q)1265 890 y FL(l)p FK(+1)e FL(l)1408 878 y FN(\025)1456 890 y FL(l)p FK(+1)1566 878 y FN(;)i(:)g(:)g(:)f(;)h(q)1787 890 y FL(nl)1854 878 y FN(\025)1902 890 y FL(n)1948 810 y Fy(\001)1986 878 y FN(;)131 b FO(\(2.58\))-118 1060 y FN(l)24 b FO(=)f(1,)k FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(.)36 b(Then)28 b(the)g(relations)c(are)j(equiv)-5 b(alen)n(t)25 b(to)j(the)g(follo)n(wing)654 1243 y FN(E)5 b FO(\(\001\))p FN(U)910 1255 y FL(l)960 1243 y FO(=)22 b FN(U)1104 1255 y FL(l)1129 1243 y FN(E)5 b FO(\()p FQ(F)1287 1208 y FM(\000)p FK(1)1287 1268 y FL(l)1377 1243 y FO(\(\001\)\))p FN(;)-118 1426 y FO(where)38 b FN(E)5 b FO(\()p FP(\001)p FO(\))39 b(is)f(a)g(join)n(t)f(resolution)f (of)i(the)h(iden)n(tit)n(y)e(of)h(the)h(comm)n(uting)-118 1525 y(family)d FN(C)212 1495 y FK(2)206 1546 y(1)249 1525 y FO(,)j FN(:)14 b(:)g(:)27 b FO(,)42 b FN(C)565 1495 y FK(2)559 1546 y FL(n)604 1525 y FO(,)g(\001)c(ranges)f(o)n(v)n (er)g(all)f(measurable)g(subsets)i(of)g FI(R)2264 1495 y FL(n)2316 1525 y FO(,)-118 1625 y FN(l)29 b FO(=)e(1,)i FN(:)14 b(:)g(:)28 b FO(,)j FN(n)p FO(.)45 b(The)30 b(latter)f (relations)e(include)h(only)h(b)r(ounded)i(op)r(erators,)-118 1725 y(and)19 b(will)e(b)r(e)j(used)f(as)g(a)g(precise)e(v)n(ersion)g (of)j(the)g(relations)c(in)i(the)i(un)n(b)r(ounded)-118 1824 y(case.)6 1924 y(According)28 b(to)h([193)o(],)h(it)f(mak)n(es)f (sense)h(to)g(consider)f(suc)n(h)h(relations)d(for)-118 2023 y(whic)n(h)39 b FQ(F)192 2035 y FL(j)227 2023 y FO(\()p FQ(F)319 2035 y FL(k)361 2023 y FO(\()p FP(\001)p FO(\)\))45 b(=)f FQ(F)694 2035 y FL(k)735 2023 y FO(\()p FQ(F)827 2035 y FL(j)863 2023 y FO(\()p FP(\001)p FO(\)\),)g FN(j)50 b FP(6)p FO(=)44 b FN(k)s FO(,)g(whic)n(h)39 b(is)h(equiv)-5 b(alen)n(t)38 b(to)i(the)-118 2123 y(follo)n(wing)23 b(equalities)390 2306 y FN(F)443 2318 y FL(j)478 2306 y FO(\()p FQ(F)570 2318 y FL(k)611 2306 y FO(\()p FN(\025)691 2318 y FK(1)729 2306 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)962 2318 y FL(n)1007 2306 y FO(\)\))24 b(=)e FN(q)1219 2318 y FL(j)s(k)1291 2306 y FN(F)1344 2318 y FL(j)1380 2306 y FO(\()p FN(\025)1460 2318 y FK(1)1498 2306 y FN(;)14 b(:)g(:)g(:)f(;)h(\025)1730 2318 y FL(n)1776 2306 y FO(\))p FN(:)-118 2488 y FO(In)29 b(what)g(follo)n(ws,)d(w)n(e)j(are)f(mostly)e (in)n(terested)i(in)g(the)h(case)f(of)h(the)h(second-)-118 2588 y(order)c(relations,)f(i.e.,)h(linear)f(functions)i FN(F)1258 2600 y FL(j)1294 2588 y FO(\()p FP(\001)p FO(\),)h FN(j)g FO(=)23 b(1,)k FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(.)37 b(If)198 2840 y FN(F)251 2852 y FL(j)286 2840 y FO(\()p FN(\025)366 2852 y FK(1)404 2840 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)637 2852 y FL(n)682 2840 y FO(\))24 b(=)865 2736 y FL(n)825 2761 y Fy(X)832 2940 y FL(l)p FK(=1)959 2840 y FN(\036)1008 2852 y FL(j)s(l)1065 2840 y FN(\025)1113 2852 y FL(l)1157 2840 y FO(+)18 b FN(\013)1293 2852 y FL(j)1328 2840 y FN(I)7 b(;)180 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)-118 3097 y FO(the)28 b(conditions)d(are)298 3280 y FN(\036)347 3292 y FL(j)s(l)403 3280 y FO(\()p FN(q)472 3292 y FL(lk)553 3280 y FP(\000)18 b FN(q)673 3292 y FL(j)s(k)745 3280 y FO(\))h(+)f FN(\036)928 3292 y FL(j)s(k)1000 3280 y FN(\036)1049 3292 y FL(k)q(l)1135 3280 y FO(=)k(0)p FN(;)180 b(l)24 b FP(6)p FO(=)f FN(j;)42 b(l)24 b FP(6)p FO(=)f FN(k)s(;)497 3405 y(\036)546 3417 y FL(j)s(k)618 3405 y FN(\036)667 3417 y FL(k)q(j)762 3405 y FO(=)g(0)p FN(;)96 b(\036)1060 3417 y FL(j)s(k)1133 3405 y FO(\()p FN(\036)1214 3417 y FL(k)q(k)1310 3405 y FP(\000)18 b FN(q)1430 3417 y FL(j)s(k)1502 3405 y FO(\))23 b(=)g(0)p FN(;)655 3529 y(\013)708 3541 y FL(j)743 3529 y FO(\(1)c FP(\000)f FN(q)956 3541 y FL(j)s(k)1027 3529 y FO(\))h(+)f FN(\013)1214 3541 y FL(k)1255 3529 y FN(\036)1304 3541 y FL(j)s(k)1399 3529 y FO(=)23 b(0)p FN(;)574 b FO(\(2.59\))-118 3712 y(for)27 b(all)e FN(j)5 b FO(,)28 b FN(k)e FO(=)c(1)28 b FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(,)f FN(j)h FP(6)p FO(=)23 b FN(k)s FO(.)6 3811 y(In)31 b(what)g(follo)n(ws,)d(w)n(e)i(will)e(assume)h(that)i(the) g FN(n)p FO(-dimensional)25 b(dynam-)-118 3911 y(ical)32 b(system)g(generated)h(b)n(y)h(the)g(mappings)e FQ(F)1419 3923 y FK(1)1456 3911 y FO(\()p FP(\001)p FO(\),)j FN(:)14 b(:)g(:)27 b FO(,)36 b FQ(F)1844 3923 y FL(n)1889 3911 y FO(\()p FP(\001)p FO(\))f(p)r(ossesses)p eop %%Page: 164 168 164 167 bop -118 -137 a FO(164)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(a)k(measurable)e(section,)i(a)g(measurable)d (set)k(whic)n(h)f(meets)g(eac)n(h)g(orbit)f(at)i(a)-118 196 y(single)h(p)r(oin)n(t.)52 b(In)33 b(this)f(case,)i(for)e(an)n(y)g (irreducible)d(represen)n(tation)h(of)j(the)-118 296 y(relations,)k(the)h(sp)r(ectral)f(measure)e(of)j(the)g(comm)n(uting)d (family)g FN(C)2093 266 y FK(2)2087 316 y(1)2130 296 y FO(,)j FN(:)14 b(:)g(:)28 b FO(,)-118 395 y FN(C)-53 365 y FK(2)-59 416 y FL(n)16 395 y FO(is)g(concen)n(trated)g(on)h(\(a)g (subset)g(of)6 b(\))30 b(a)e(single)f(orbit,)h(and)h(w)n(e)g(can)g (clas-)-118 495 y(sify)d(all)f(irreducible)f(represen)n(tations)h(up)i (to)g(unitary)f(equiv)-5 b(alence.)35 b(In)27 b(the)-118 595 y(case)j(of)h(more)e(complicated)e(dynamical)g(systems)i(without)i (a)f(measurable)-118 694 y(section,)c(non-trivial)d(ergo)r(dic)i (measures)g(can)i(arise,)e(whic)n(h)h(giv)n(es)f(rise)h(to)h(a)-118 794 y(m)n(uc)n(h)19 b(more)f(complicated)e(structure)j(of)h(represen)n (tations,)f(including)d(factor)-118 893 y(represen)n(tations)28 b(not)j(of)g(t)n(yp)r(e)h(I)f(etc.)48 b(Notice)30 b(that)h(the)h (linear)c(dynamical)-118 993 y(system)e(of)i(the)g(form)e(\(2.58\))h (alw)n(a)n(ys)e(p)r(ossesses)h(a)h(measurable)d(section.)6 1093 y(W)-7 b(e)28 b(pro)r(ceed)f(with)g(a)g(more)f(detailed)g(study)h (of)h(the)g(irreducible)c(collec-)-118 1193 y(tions)33 b FN(X)161 1205 y FL(j)196 1193 y FO(,)j FN(j)j FO(=)34 b(1,)g FN(:)14 b(:)g(:)28 b FO(,)36 b FN(n)p FO(,)g(satisfying)d (\(2.56\))o(,)k(whic)n(h)c(corresp)r(ond)g(to)h(an)-118 1293 y(orbit)f(\012.)59 b(Denote)36 b(b)n(y)e(\001)i(the)f(supp)r(ort)g (of)g(the)g(sp)r(ectral)e(measure)g(of)i(the)-118 1392 y(comm)n(uting)e(family)g FN(C)649 1362 y FK(2)643 1414 y FL(j)686 1392 y FO(,)38 b FN(j)k FO(=)37 b(1,)f FN(:)14 b(:)g(:)27 b FO(,)39 b FN(n)p FO(.)62 b(It)36 b(is)f(a)g(general)f (fact)i(that)g(in)-118 1492 y(the)28 b(basis)d(of)j(eigen)n(v)n(ectors) c(of)j(the)h(comm)n(uting)c(family)-7 b(,)25 b(the)i(op)r(erators)f FN(X)2304 1504 y FL(j)-118 1591 y FO(act)33 b(as)g(w)n(eigh)n(ted)e (shift)i(op)r(erators)f([193)n(],)j(but)f(w)n(e)f(need)h(to)f(tak)n(e)f (in)n(to)g(ac-)-118 1691 y(coun)n(t)f(that)h FN(C)357 1703 y FL(j)422 1691 y FP(\025)d FO(0,)k(and)e(that)h FN(U)1020 1703 y FL(j)1054 1691 y FN(U)1120 1661 y FM(\003)1111 1713 y FL(j)1190 1691 y FO(is)f(a)g(pro)5 b(jection)29 b(on)j(the)f(co-k)n(ernel)-118 1803 y(\(k)n(er)13 b FN(C)104 1773 y FK(2)98 1824 y FL(j)141 1803 y FO(\))173 1773 y FM(?)230 1803 y FO(,)28 b FN(j)g FO(=)22 b(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(.)-118 1971 y FQ(Lemma)h(11.)41 b FB(F)-6 b(or)30 b(any)g FN(\025)24 b FO(=)e(\()p FN(\025)937 1983 y FK(1)975 1971 y FN(;)14 b(:)g(:)g(:)g(;)g(\025)1208 1983 y FL(n)1253 1971 y FO(\))24 b FP(2)f FO(\001)p FB(,)30 b(we)g(have)6 2071 y FO(i\))f FN(\025)139 2083 y FL(j)198 2071 y FP(\025)23 b FO(0)p FB(,)29 b FN(j)f FO(=)23 b(1)p FB(,)30 b FN(:)14 b(:)g(:)27 b FB(,)j FN(n)p FO(;)6 2171 y(ii\))f FB(either)h FQ(F)411 2183 y FL(j)446 2171 y FO(\()p FN(\025)p FO(\))24 b FP(2)g FO(\001)p FB(,)30 b(or)g FO(\()p FQ(F)984 2183 y FL(j)1019 2171 y FO(\()p FN(\025)p FO(\)\))1163 2183 y FL(j)1223 2171 y FO(=)22 b(0;)6 2272 y(iii\))28 b FB(similarly,)k(either)e FQ(F)802 2236 y FM(\000)p FK(1)802 2295 y FL(j)891 2272 y FO(\()p FN(\025)p FO(\))25 b FP(2)e FO(\001)p FB(,)30 b(or)g FN(\025)1385 2284 y FL(j)1444 2272 y FO(=)23 b(0)p FB(.)-118 2450 y(Pr)l(o)l(of.)43 b FO(i\))27 b(Indeed,)h(since)e FN(C)786 2420 y FK(2)780 2472 y FL(j)847 2450 y FP(\025)d FO(0,)k(w)n(e)g(ha)n(v)n(e)f FN(\025)1388 2462 y FL(j)1447 2450 y FP(\025)d FO(0,)k FN(j)h FO(=)22 b(1,)28 b FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(.)6 2550 y(ii\))23 b(If)h FQ(F)248 2562 y FL(j)283 2550 y FO(\()p FN(\025)p FO(\))33 b FN(=)-51 b FP(2)23 b FO(\001,)i(then)g FN(U)857 2562 y FL(j)891 2550 y FN(e)930 2562 y FL(\025)997 2550 y FO(=)d(0,)i(where)g FN(e)1449 2562 y FL(\025)1516 2550 y FO(is)f(the)h(basis)e(eigen)n(v)n(ector)-118 2650 y(of)38 b(the)g(comm)n(uting)d(family)g(corresp)r(onding)g(to)j(the)h(join)n(t) e(eigen)n(v)-5 b(alue)35 b FN(\025)p FO(.)-118 2750 y(Then)28 b(w)n(e)f(also)e(ha)n(v)n(e)i FN(U)636 2762 y FL(j)670 2750 y FN(U)736 2720 y FM(\003)727 2771 y FL(j)774 2750 y FN(e)813 2765 y Fm(F)860 2773 y Fv(j)891 2765 y FK(\()p FL(\025)p FK(\))1010 2750 y FO(=)22 b(0)27 b(and)543 2947 y FN(C)608 2913 y FK(2)602 2967 y FL(j)646 2947 y FN(e)685 2962 y Fm(F)732 2970 y Fv(j)762 2962 y FK(\()p FL(\025)p FK(\))881 2947 y FO(=)22 b(\()p FQ(F)1060 2959 y FL(j)1096 2947 y FO(\()p FN(\025)p FO(\)\))1240 2959 y FL(j)1290 2947 y FN(e)1329 2962 y Fm(F)1376 2970 y Fv(j)1407 2962 y FK(\()p FL(\025)p FK(\))1525 2947 y FO(=)h(0)p FN(;)-118 3131 y FO(whic)n(h)j(implies)e(that)k FQ(F)641 3143 y FL(j)676 3131 y FO(\()p FN(\025)p FO(\)\))820 3143 y FL(j)880 3131 y FO(=)22 b(0.)6 3231 y(iii\))i(Similarly)-7 b(,)21 b(if)k FQ(F)636 3196 y FM(\000)p FK(1)636 3254 y FL(j)725 3231 y FO(\()p FN(\025)p FO(\))33 b FN(=)-51 b FP(2)23 b FO(\001,)k(then)f FN(U)1311 3201 y FM(\003)1302 3253 y FL(j)1363 3231 y FN(e)1402 3243 y FL(\025)1468 3231 y FO(=)d(0.)36 b(Then)26 b FN(U)1929 3243 y FL(j)1963 3231 y FN(U)2029 3201 y FM(\003)2020 3253 y FL(j)2081 3231 y FN(e)2120 3243 y FL(\025)2186 3231 y FO(=)d(0,)-118 3341 y(and)k FN(C)108 3311 y FK(2)102 3363 y FL(j)160 3341 y FN(e)199 3353 y FL(\025)265 3341 y FO(=)c FN(\025)401 3353 y FL(j)436 3341 y FN(e)475 3353 y FL(\025)541 3341 y FO(=)g(0,)k(whic)n(h)g(giv)n(es)e FN(\025)1210 3353 y FL(j)1269 3341 y FO(=)d(0.)p 2278 3341 4 57 v 2282 3289 50 4 v 2282 3341 V 2331 3341 4 57 v -118 3513 a FQ(Corollary)32 b(5.)40 b FB(If)26 b(for)g(some)g FN(\025)d FO(=)g(\()p FN(\025)1080 3525 y FK(1)1118 3513 y FN(;)14 b(:)g(:)g(:)f(;)h(\025)1350 3525 y FL(n)1396 3513 y FO(\))23 b FP(2)h FO(\012)p FB(,)i(we)g(have)g FN(\025)1994 3525 y FL(j)2053 3513 y FN(>)c FO(0)j FB(and)-118 3612 y FO(\()p FQ(F)-26 3624 y FL(j)9 3612 y FO(\()p FN(\025)p FO(\)\))153 3624 y FL(j)223 3612 y FN(<)33 b FO(0)p FB(,)j(then)g FN(\025)43 b(=)-52 b FP(2)34 b FO(\001)p FB(.)56 b(This)36 b(c)l(ondition)h(implies)g(that)e(irr)l(e)l(ducible)-118 3712 y(r)l(epr)l(esentations)23 b(c)l(orr)l(esp)l(ond)h(only)g(to)f (the)g(orbits)g(such)g(that)g FN(\025)1873 3724 y FL(j)1932 3712 y FN(>)g FO(0)f FB(implies)-118 3811 y FO(\()p FQ(F)-26 3823 y FL(j)9 3811 y FO(\()p FN(\025)p FO(\)\))153 3823 y FL(j)213 3811 y FP(\025)i FO(0)p FB(,)30 b FO(\()p FQ(F)491 3776 y FM(\000)p FK(1)491 3835 y FL(j)581 3811 y FO(\()p FN(\025)p FO(\)\))725 3823 y FL(j)785 3811 y FP(\025)24 b FO(0)p FB(.)39 b(Notic)l(e)31 b(also)g(that)38 b FO(\(2.58\))30 b FB(implies)h(that)g(it)-118 3911 y(fol)t(lows)h(fr)l (om)e FN(\025)402 3923 y FL(j)460 3911 y FN(>)23 b FO(0)29 b FB(that)h FO(\()p FQ(F)881 3923 y FL(k)922 3911 y FO(\()p FN(\025)p FO(\)\))1066 3923 y FL(j)1126 3911 y FN(>)22 b FO(0)29 b FB(for)i FN(k)26 b FP(6)p FO(=)d FN(j)5 b FB(.)p eop %%Page: 165 169 165 168 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(165)6 96 y(Consider)25 b(p)r(ossible)f(t)n(yp)r(es)j(of)f(orbits)f(and)h(describ)r(e)g(the)h (corresp)r(onding)-118 196 y(irreducible)d(represen)n(tations)g(of)34 b(\(2.56\).)-118 347 y FQ(Theorem)c(37.)41 b FB(A)n(ny)30 b(irr)l(e)l(ducible)j(r)l(epr)l(esentation)e(c)l(an)g(b)l(e)g(r)l(e)l (alize)l(d)h(in)f(the)-118 447 y(sp)l(ac)l(e)f FN(l)123 459 y FK(2)160 447 y FO(\(\001\))p FB(.)39 b(F)-6 b(or)30 b(any)37 b FN(l)25 b FO(=)d(1)p FB(,)30 b FN(:)14 b(:)g(:)27 b FB(,)j FN(n)g FB(one)f(of)i(the)e(fol)t(lowing)j(c)l(ases)e(holds)7 b FO(:)6 547 y FN(a)p FO(\))p FB(.)58 b(The)37 b(mapping)45 b FQ(F)750 559 y FL(l)775 547 y FO(\()p FP(\001)p FO(\))37 b FB(p)l(ossesses)g(a)f(stationary)h(p)l(oint)f FN(\025)f FP(2)g FO(\001)h(\()p FB(in)-118 646 y(this)24 b(c)l(ase)h(al)t(l)f (other)h(p)l(oints)f(ar)l(e)g(also)h(stationary)7 b FO(\))p FB(.)38 b(If)25 b FN(\025)1662 658 y FL(l)1711 646 y FO(=)d(0)p FB(,)j(then)f FN(X)2138 658 y FL(l)2186 646 y FO(=)f(0;)-118 746 y FB(otherwise,)31 b(the)f(op)l(er)l(ator)h FN(X)809 758 y FL(l)864 746 y FB(has)g(the)e(form)827 910 y FN(X)896 922 y FL(l)921 910 y FN(e)960 922 y FL(\025)1027 910 y FO(=)22 b FN(\014)1161 922 y FL(l)1200 910 y FN(\025)1248 922 y FL(l)1288 910 y FN(e)1327 922 y FL(\025)1370 910 y FN(;)-118 1075 y FB(wher)l(e)30 b FN(\014)163 1087 y FL(l)218 1075 y FB(is)h(a)f(p)l(ar)l(ameter)g(with)g(the)g(absolute)h (value)f(e)l(qual)g(to)g(one)6 b FO(;)6 1174 y FN(b)p FO(\))p FB(.)53 b(The)35 b(mapping)h FQ(F)726 1186 y FL(l)751 1174 y FO(\()p FP(\001)p FO(\))f FB(do)l(es)g(not)f(have)i (stationary)f(p)l(oints.)52 b(In)34 b(this)-118 1274 y(c)l(ase)c(the)g(op)l(er)l(ator)h FN(X)594 1286 y FL(l)649 1274 y FB(has)f(the)g(form)750 1438 y FN(X)819 1450 y FL(l)844 1438 y FN(e)883 1450 y FL(\025)949 1438 y FO(=)23 b FN(F)1090 1450 y FL(l)1116 1438 y FO(\()p FN(\025)p FO(\))14 b FN(e)1281 1453 y Fm(F)1328 1462 y Fv(l)1352 1453 y FK(\()p FL(\025)p FK(\))1448 1438 y FN(:)655 b FO(\(2.60\))-118 1603 y FB(The)36 b(kernel)g(of)g(the)g(op)l(er)l(ator) h FN(X)959 1615 y FL(l)1019 1603 y FB(is)f(gener)l(ate)l(d)g(by)g(ve)l (ctors)f FN(e)1925 1615 y FL(\025)2004 1603 y FB(such)h(that)-118 1702 y FN(F)-65 1714 y FL(l)-39 1702 y FO(\()p FN(\025)p FO(\))k(=)g(0)p FB(;)j(the)c(kernel)h(of)f FN(X)914 1672 y FM(\003)907 1726 y FL(l)991 1702 y FB(is)g(gener)l(ate)l(d)g(by)h(ve) l(ctors)f FN(e)1911 1714 y FL(\025)1993 1702 y FB(for)h(which)-118 1802 y FN(\025)-70 1814 y FL(l)-21 1802 y FO(=)22 b(0)p FB(.)-118 1953 y(Pr)l(o)l(of.)43 b FO(The)d(pro)r(of)f(is)g(essen)n (tially)c(based)40 b(on)f(the)h(follo)n(wing)c(statemen)n(ts)-118 2053 y(from)26 b([193)o(].)-118 2202 y FQ(Theorem)k(38.)41 b FB(L)l(et)h(the)h(dynamic)l(al)i(system)d(on)h FI(R)1634 2172 y FL(n)1728 2202 y FB(gener)l(ate)l(d)g(by)g(the)-118 2302 y(mappings)c FQ(F)319 2314 y FL(l)345 2302 y FB(,)h FN(l)f FO(=)e(1)p FB(,)h FN(:)14 b(:)g(:)28 b FB(,)40 b FN(n)p FB(,)g(p)l(ossess)f(a)f(me)l(asur)l(able)g(se)l(ction.)63 b(Then,)-118 2402 y(for)38 b(e)l(ach)h(irr)l(e)l(ducible)g(c)l(ol)t(le) l(ction)f(of)h(op)l(er)l(ators)f FN(C)1535 2414 y FL(j)1570 2402 y FB(,)i FN(U)1692 2414 y FL(j)1727 2402 y FB(,)g FN(j)i FO(=)36 b(1)p FB(,)i FN(:)14 b(:)g(:)27 b FB(,)40 b FN(n)p FB(,)-118 2501 y(satisfying)e FO(\(2.57\))p FB(,)30 b(the)g(fol)t(lowing)i(holds.)6 2601 y(i.)47 b(Ther)l(e)33 b(exists)f(a)h(unique)f(orbit)h FO(\012)f FB(of)h(the)g(dynamic)l(al)h(system)e(of)h(ful)t(l)-118 2701 y(sp)l(e)l(ctr)l(al)42 b(me)l(asur)l(e)g(of)h(the)f(c)l(ommuting)g (c)l(ol)t(le)l(ction)h FN(C)1659 2713 y FL(j)1694 2701 y FB(,)j FN(j)k FO(=)45 b(1)p FB(,)d FN(:)14 b(:)g(:)28 b FB(,)45 b FN(n)p FB(,)-118 2800 y FN(E)5 b FO(\(\012\))24 b(=)e(1;)6 2900 y FB(ii.)49 b(If)i FO(k)n(er)13 b FN(U)420 2912 y FL(l)473 2900 y FO(=)29 b FP(f)p FO(0)p FP(g)p FB(,)j(then)h(the)g(sp)l(e)l(ctr)l(al)g(me)l(asur)l(e)f(is)h (quasi-invariant)-118 2999 y(with)42 b(r)l(esp)l(e)l(ct)g(to)g(the)g (mapping)h FQ(F)1026 3011 y FL(l)1052 2999 y FO(\()p FP(\001)p FO(\);)48 b FB(in)42 b(the)g(c)l(ase)g(of)h(unitary)f FN(U)2135 3011 y FL(l)2160 2999 y FB(,)j(the)-118 3099 y(me)l(asur)l(e)29 b(is)h(also)h(quasi-invariant)g(with)f(r)l(esp)l(e)l (ct)g(to)g FQ(F)1647 3064 y FM(\000)p FK(1)1647 3124 y FL(l)1736 3099 y FO(\()p FP(\001)p FO(\);)6 3199 y FB(iii.)44 b(The)32 b(joint)f(sp)l(e)l(ctrum)f(of)i(the)f(c)l(ommuting) g(family)h FN(C)1845 3211 y FL(j)1880 3199 y FB(,)g FN(j)e FO(=)25 b(1)p FB(,)31 b FN(:)14 b(:)g(:)27 b FB(,)-118 3298 y FN(n)p FB(,)j(is)g(simple.)-118 3448 y FQ(Theorem)g(39.)41 b FB(The)29 b(irr)l(e)l(ducible)g(c)l(ol)t(le)l(ction)g FN(C)1444 3460 y FL(j)1480 3448 y FB(,)f FN(U)1590 3460 y FL(j)1625 3448 y FB(,)h FN(j)f FO(=)22 b(1)p FB(,)28 b FN(:)14 b(:)g(:)28 b FB(,)g FN(n)p FB(,)h(sat-)-118 3547 y(isfying)44 b FO(\(2.57\))34 b FB(acts)h(in)g(the)g(sp)l(ac)l(e)h FN(l)1081 3559 y FK(2)1118 3547 y FO(\(\001\))p FB(,)i(wher)l(e)d FO(\001)e FP(\032)f FO(\012)j FB(is)h(a)f(subset)g(of)-118 3647 y(some)25 b(orbit)h FO(\012)f(\()p FB(for)h(unitary)g FN(U)873 3659 y FL(l)898 3647 y FB(,)g FN(l)f FO(=)d(1)p FB(,)j FN(:)14 b(:)g(:)28 b FB(,)e FN(n)p FB(,)h FO(\001)c(=)g(\012\))p FB(,)j(by)g(the)f(fol)t(lowing)-118 3747 y(formulae)474 3911 y FN(C)533 3923 y FL(l)559 3911 y FN(e)598 3923 y FL(\025)664 3911 y FO(=)d FN(x)798 3923 y FL(k)840 3911 y FN(e)879 3923 y FL(\025)922 3911 y FN(;)99 b(U)1101 3923 y FL(l)1126 3911 y FN(e)1165 3923 y FL(\025)1231 3911 y FO(=)23 b FN(u)1367 3923 y FL(l)1392 3911 y FO(\()p FN(\025)p FO(\))14 b FN(e)1557 3926 y Fm(F)1604 3935 y Fv(l)1628 3926 y FK(\()p FL(\025)p FK(\))1724 3911 y FN(;)p eop %%Page: 166 170 166 169 bop -118 -137 a FO(166)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(wher)l(e)38 b FN(u)172 108 y FL(l)197 96 y FO(\()p FN(\025)p FO(\))f FB(ar)l(e)h(c)l(onstants)e (which)i(determine)g(the)f(action)h(of)55 b FN(U)2088 108 y FL(l)2113 96 y FB(.)61 b(The)-118 196 y(subset)29 b FO(\001)h FB(satis\014es)g(the)g(fol)t(lowing)i(\\b)l(oundary)e(c)l (onditions")6 b FO(:)590 386 y FN(u)638 398 y FL(l)663 386 y FO(\()p FN(\025)p FO(\))24 b(=)e(0)p FN(;)99 b FP(8)14 b FN(\025)22 b FP(2)i FO(\001)9 b(:)28 b FQ(F)1449 398 y FL(l)1474 386 y FO(\()p FN(\025)p FO(\))34 b FN(=)-52 b FP(2)24 b FO(\001)p FN(;)376 522 y(u)424 534 y FL(l)449 522 y FO(\()p FQ(F)541 487 y FM(\000)p FK(1)541 547 y FL(l)630 522 y FO(\()p FN(\025)p FO(\)\))h(=)d(0)p FN(;)99 b FP(8)14 b FN(\025)22 b FP(2)i FO(\001)9 b(:)28 b FQ(F)1449 487 y FM(\000)p FK(1)1449 547 y FL(l)1538 522 y FO(\()p FN(\025)p FO(\))33 b FN(=)-51 b FP(2)23 b FO(\001)p FN(;)296 b FO(\(2.61\))-118 712 y FB(and)32 b(is)g(\\c)l(onne)l(cte)l(d")e(in)i (the)f(fol)t(lowing)j(sense)6 b FO(:)42 b FN(u)1505 724 y FL(l)1530 712 y FO(\()p FN(\025)p FO(\))27 b FP(6)p FO(=)f(0)31 b FB(for)h(al)t(l)h FN(\025)26 b FP(2)h FO(\001)p FB(,)-118 812 y(such)j(that)f FQ(F)300 824 y FL(l)326 812 y FO(\()p FN(\025)p FO(\))24 b FP(2)f FO(\001)p FB(,)31 b FN(l)24 b FO(=)f(1)p FB(,)30 b FN(:)14 b(:)g(:)27 b FB(,)j FN(n)p FB(.)6 988 y FO(Let)c FN(\025)h FO(b)r(e)f(a)f (stationary)f(p)r(oin)n(t)h(of)g(the)h(mapping)e FQ(F)1648 1000 y FL(l)1674 988 y FO(\()p FP(\001)p FO(\).)37 b(If)26 b FN(\025)1950 1000 y FL(l)1999 988 y FO(=)c(0,)k(then)-118 1087 y(for)d(all)e(p)r(oin)n(ts)h FN(\025)h FP(2)h FO(\001,)g(the)g (comm)n(utation)c(of)j FQ(F)1417 1099 y FL(l)1442 1087 y FO(\()p FP(\001)p FO(\))h(and)f FQ(F)1770 1099 y FL(k)1811 1087 y FO(\()p FP(\001)p FO(\))h(also)d(implies)-118 1187 y(that)28 b FN(\025)110 1199 y FL(l)160 1187 y FO(=)23 b(0.)38 b(Then)28 b FN(X)637 1199 y FL(l)686 1187 y FO(=)c(0.)37 b(If)29 b FN(\025)1009 1199 y FL(l)1059 1187 y FP(6)p FO(=)23 b(0,)28 b(then)g(also)e FN(\025)1644 1199 y FL(l)1694 1187 y FP(6)p FO(=)d(0)28 b(for)f(all)f FN(\025)e FP(2)g FO(\001.)-118 1286 y(In)34 b(this)f(case,)i(the)g(op)r(erator)d FN(U)915 1298 y FL(l)974 1286 y FO(comm)n(utes)g(with)i(all)d(the)k(op) r(erators)d FN(X)2281 1298 y FL(j)2316 1286 y FO(,)-118 1386 y FN(X)-42 1356 y FM(\003)-49 1408 y FL(j)-5 1386 y FO(,)c(and,)g(therefore,)e(is)h(a)g(m)n(ultiple)d(of)k(the)g(iden)n (tit)n(y)-7 b(.)6 1489 y(In)37 b(the)g(case)f(where)g(the)h(mapping)e FQ(F)1268 1501 y FL(l)1293 1489 y FO(\()p FP(\001)p FO(\))j(do)r(es)e (not)h(ha)n(v)n(e)e(stationary)-118 1589 y(p)r(oin)n(ts,)29 b(the)g(op)r(erator)f FN(U)695 1601 y FL(l)749 1589 y FO(is)h(unitarily)c(equiv)-5 b(alen)n(t)28 b(to)h(the)g(shift)g(op)r (erator;)-118 1689 y(taking)c(in)n(to)g(accoun)n(t)h(that)h FN(X)858 1701 y FL(l)883 1689 y FN(X)959 1659 y FM(\003)952 1712 y FL(l)1019 1689 y FO(=)c FN(C)1172 1659 y FK(2)1166 1712 y FL(l)1210 1689 y FO(,)j(w)n(e)h(get)f(the)h(needed)g(form)n(ula) c(for)-118 1788 y FN(X)-49 1800 y FL(l)-24 1788 y FO(.)p 2278 1788 4 57 v 2282 1736 50 4 v 2282 1788 V 2331 1788 4 57 v -118 2024 a FQ(2.4.4)94 b(Represen)m(tations)40 b(of)h(the)h(non-standard)g(real)g(quan)m(tum)174 2123 y(sphere)-118 2284 y(1.)h FO(The)30 b(algebra)d(of)j(functions)f(on)h (the)g(non-standard)f(three-dimensional)-118 2383 y(real)22 b(quan)n(tum)g(sphere)h(\(see)h([176)o(]\))g(is)e(an)i(asso)r(ciativ)n (e)c FP(\003)p FO(-algebra)g(generated)-118 2483 y(b)n(y)27 b(the)h(elemen)n(ts)e FN(x)p FO(,)i FN(y)s FO(,)f FN(u)p FO(,)h FN(v)s FO(,)f FN(c)p FO(,)h(and)g FN(d)f FO(satisfying)e(the)j (relations:)247 2673 y FN(ux)23 b FO(=)f FN(q)s(xu;)97 b(v)s(x)24 b FO(=)f FN(q)s(xv)s(;)97 b(y)s(u)23 b FO(=)f FN(q)s(uy)s(;)96 b(y)s(v)26 b FO(=)d FN(q)s(v)s(y)s(;)70 2808 y(v)s(u)18 b FP(\000)g FN(uv)26 b FO(=)d(\()p FN(q)e FP(\000)d FN(q)677 2774 y FM(\000)p FK(1)767 2808 y FO(\))c FN(d;)97 b(xy)21 b FP(\000)d FN(q)1208 2774 y FM(\000)p FK(1)1298 2808 y FN(uv)25 b FO(=)e FN(y)s(x)c FP(\000)f FN(q)s(v)s(u)k FO(=)h FN(c)18 b FO(+)g FN(d;)75 2943 y(dx)24 b FO(=)e FN(q)316 2908 y FK(2)354 2943 y FN(xd;)97 b(dv)27 b FO(=)22 b FN(q)801 2908 y FK(2)839 2943 y FN(v)s(d;)97 b(ud)23 b FO(=)g FN(q)1287 2908 y FK(2)1324 2943 y FN(du;)97 b(y)s(d)22 b FO(=)h FN(q)1772 2908 y FK(2)1809 2943 y FN(dy)s(;)207 b FO(\(2.62\))-118 3133 y(with)36 b FN(c)g FO(lying)f(in)g(the)i(cen)n(ter,)i(and)d(the)h(in)n(v)n(olution)c (de\014ned)j(b)n(y)h FN(x)2093 3103 y FM(\003)2169 3133 y FO(=)h FN(y)s FO(,)-118 3232 y FN(u)-70 3202 y FM(\003)-7 3232 y FO(=)24 b FP(\000)p FN(q)187 3202 y FM(\000)p FK(1)276 3232 y FN(v)s FO(,)30 b FN(c)408 3202 y FM(\003)471 3232 y FO(=)24 b FN(c)p FO(,)30 b FN(d)692 3202 y FM(\003)755 3232 y FO(=)25 b FN(d)p FO(.)41 b(F)-7 b(or)28 b(the)h(generators)e FN(x)p FO(,)i FN(u)p FO(,)g FN(c)p FO(,)g FN(d)p FO(,)h(the)f(rela-) -118 3332 y(tions)22 b(\(2.62\))h(ha)n(v)n(e)g(the)h(form)e(\(2.58\),)i (and)g(are)f(equiv)-5 b(alen)n(t)21 b(to)j(the)g(follo)n(wing)-118 3432 y(relations)653 3622 y FN(ux)f FO(=)g FN(q)s(xu;)97 b(u)1162 3588 y FM(\003)1199 3622 y FN(x)24 b FO(=)e FN(q)s(xu)1492 3588 y FM(\003)1531 3622 y FN(;)-46 3757 y(u)2 3722 y FM(\003)40 3757 y FN(u)g FO(=)h FN(q)238 3722 y FM(\000)p FK(2)327 3757 y FN(uu)423 3722 y FM(\003)479 3757 y FP(\000)18 b FO(\(1)g FP(\000)g FN(q)777 3722 y FM(\000)p FK(2)866 3757 y FO(\)\()p FN(xx)1024 3722 y FM(\003)1082 3757 y FP(\000)g FN(c)p FO(\))p FN(;)97 b(x)1400 3722 y FM(\003)1439 3757 y FN(x)23 b FO(=)g FN(q)1637 3722 y FK(2)1674 3757 y FN(xx)1768 3722 y FM(\003)1826 3757 y FO(+)18 b(\(1)g FP(\000)g FN(q)2124 3722 y FK(2)2161 3757 y FO(\))p FN(c;)769 3881 y(d)23 b FO(=)g FN(xx)1017 3847 y FM(\003)1075 3881 y FO(+)18 b FN(uu)1254 3847 y FM(\003)1309 3881 y FP(\000)g FN(c:)675 b FO(\(2.63\))p eop %%Page: 167 171 167 170 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(167)-118 96 y FQ(2.)45 b FO(The)30 b(corresp)r(onding)e(dynamical)e(system)k(on)g FI(R)1564 66 y FK(2)1637 96 y FO(is)g(generated)f(b)n(y)h(the)-118 196 y(mappings)294 384 y FQ(F)354 396 y FK(1)392 384 y FO(\()p FN(\025)472 396 y FK(1)510 384 y FN(;)14 b(\025)595 396 y FK(2)632 384 y FO(\))24 b(=)e(\()p FN(q)847 350 y FK(2)885 384 y FN(\025)933 396 y FK(1)989 384 y FO(+)c(\(1)g FP(\000)g FN(q)1287 350 y FK(2)1324 384 y FO(\))p FN(c;)c(q)1469 350 y FK(2)1507 384 y FN(\025)1555 396 y FK(2)1593 384 y FO(\))p FN(;)294 519 y FQ(F)354 531 y FK(2)392 519 y FO(\()p FN(\025)472 531 y FK(1)510 519 y FN(;)g(\025)595 531 y FK(2)632 519 y FO(\))24 b(=)e(\()p FN(\025)855 531 y FK(1)893 519 y FN(;)14 b(q)970 485 y FM(\000)p FK(2)1059 519 y FN(\025)1107 531 y FK(2)1163 519 y FP(\000)k FO(\(1)h FP(\000)f FN(q)1462 485 y FM(\000)p FK(2)1551 519 y FO(\)\()p FN(\025)1663 531 y FK(1)1720 519 y FP(\000)g FN(c)p FO(\)\))p FN(;)-118 707 y FO(whic)n(h)26 b(satisfy)-7 b(,)27 b(as)g(one)g(can)g(see,)g(the)h(conditions)e(\(2.59\))o(.)6 809 y(An)n(y)i(orbit)e(of)i(the)g(dynamical)c(system)i(consists)g(of)h (the)h(p)r(oin)n(ts)-18 998 y FN(\025)30 963 y FK(\()p FL(k)q(l)p FK(\))168 998 y FO(=)22 b FQ(F)315 963 y FL(k)315 1018 y FK(1)356 998 y FO(\()p FQ(F)448 963 y FL(l)448 1018 y FK(2)486 998 y FO(\()p FN(\025)p FO(\)\))168 1139 y(=)g(\()p FN(q)327 1104 y FK(2)p FL(k)402 1139 y FN(\025)450 1151 y FK(1)506 1139 y FO(+)c(\(1)g FP(\000)g FN(q)804 1104 y FK(2)p FL(k)878 1139 y FO(\))c FN(c;)g(q)1037 1104 y FK(2\()p FL(k)q FM(\000)p FL(l)p FK(\))1236 1139 y FN(\025)1284 1151 y FK(2)1340 1139 y FP(\000)k FN(q)1463 1104 y FK(2)p FL(k)1537 1139 y FO(\(1)g FP(\000)g FN(q)1752 1104 y FM(\000)p FK(2)p FL(l)1863 1139 y FO(\)\()p FN(c)h FP(\000)f FN(\025)2113 1151 y FK(1)2151 1139 y FO(\)\))p FN(;)-118 1336 y FO(where)24 b FN(\025)f FO(=)g(\()p FN(\025)358 1348 y FK(1)396 1336 y FN(;)14 b(\025)481 1348 y FK(2)518 1336 y FO(\),)26 b FQ(F)659 1306 y FL(k)659 1359 y(l)699 1336 y FO(\()p FP(\001)p FO(\))f(is)f(the)g FN(k)s FO(-th)g(iterations)e(of)i(the)g(mapping)e FQ(F)2202 1348 y FL(l)2228 1336 y FO(\()p FP(\001)p FO(\).)6 1438 y(The)f(mapping)d FQ(F)564 1450 y FK(1)601 1438 y FO(\()p FP(\001)p FO(\))k(has)d(a)h(single)e(stationary)g(p)r(oin)n(t)i(\()p FN(c;)14 b FO(0\),)22 b(Stationary)-118 1538 y(p)r(oin)n(ts)29 b(of)h(the)h(mapping)d FQ(F)781 1550 y FK(2)818 1538 y FO(\()p FP(\001)p FO(\))j(ha)n(v)n(e)e(the)h(co)r(ordinates)e(\()p FN(\025;)14 b(c)21 b FP(\000)f FN(\025)p FO(\).)45 b(There)-118 1637 y(are)20 b(no)h(p)r(erio)r(dic)e(p)r(oin)n(ts)h(of)i FQ(F)827 1649 y FK(1)864 1637 y FO(\()p FP(\001)p FO(\),)h FQ(F)1057 1649 y FK(2)1094 1637 y FO(\()p FP(\001)p FO(\),)h(apart)c (from)g(the)i(stationary)c(ones.)-118 1795 y FQ(3.)54 b FO(The)34 b(follo)n(wing)29 b(is)k(a)g(list)f(of)h(orbits,)h(the)f (corresp)r(onding)e(sets)i(\001,)j(and)-118 1895 y(the)28 b(corresp)r(onding)d(irreducible)e(represen)n(tations.)6 1997 y(1\))34 b(A)g(single)e(stationary)f(p)r(oin)n(t)i(\()p FN(c;)14 b FO(0\).)56 b(F)-7 b(or)33 b FN(c)h FO(=)f(0,)i(this)e(orbit) g(corre-)-118 2097 y(sp)r(onds)38 b(to)g(the)g(trivial)d(represen)n (tation)g FN(X)47 b FO(=)41 b FN(U)49 b FO(=)40 b(0,)h(and)d(for)f FN(c)k(>)f FO(0,)-118 2196 y(to)30 b(the)h(family)c(of)j (one-dimensional)25 b(irreducible)i(represen)n(tations)g FN(U)36 b FO(=)27 b(0,)-118 2296 y FN(X)34 b FO(=)27 b FN(\013)14 b(c)p FO(,)31 b(where)f FP(j)p FN(\013)p FP(j)e FO(=)f(1.)44 b(Therefore,)30 b(the)h(set)f(of)g(one-dimensional) 25 b(rep-)-118 2395 y(resen)n(tations)g(is)h(parametrized)f(b)n(y)i(p)r (oin)n(ts)f(of)i(the)g(cone.)6 2498 y(2\))38 b(If)h FN(c)h(>)g FO(0,)g(then)f(there)f(exists)e(a)i(unique)f(orbit)g(that)h(is)f(in)n (v)-5 b(arian)n(t)-118 2597 y(with)38 b(resp)r(ect)g(to)h(the)g (mapping)d FQ(F)1056 2609 y FK(2)1094 2597 y FO(\()p FP(\001)p FO(\))j(and)g(satis\014es)d(the)k(conditions)c(of)-118 2697 y(Lemma)22 b(11.)35 b(Namely)-7 b(,)24 b(this)g(orbit)f(con)n (tains)g(the)i(p)r(oin)n(t)f FN(\025)f FO(=)g(\(0)p FN(;)14 b(c)p FO(\).)36 b(The)25 b(set)-118 2797 y(\001)g(consists)e(of)i(the)g (p)r(oin)n(ts)f FN(\025)806 2766 y FK(\()p FL(k)q FK(\))922 2797 y FO(=)f(\(\(1)13 b FP(\000)g FN(q)1247 2766 y FK(2)p FL(k)1320 2797 y FO(\))p FN(c;)h(q)1465 2766 y FK(2)p FL(k)1540 2797 y FN(c)p FO(\),)25 b(and)g(the)g(irreducible)-118 2896 y(represen)n(tation)e(corresp)r(onding)g(to)j(this)f(orbit)g(is)f (realized)f(on)j(the)g(space)g FN(l)2302 2908 y FK(2)-118 2996 y FO(b)n(y)h(the)h(form)n(ulae:)272 3184 y FN(X)7 b(e)387 3196 y FL(k)449 3184 y FO(=)23 b(\(\(1)c FP(\000)f FN(q)785 3150 y FK(2)p FL(k)859 3184 y FO(\))c FN(c)p FO(\))973 3150 y FK(1)p FL(=)p FK(2)1077 3184 y FN(e)1116 3196 y FL(k)q FK(+1)1241 3184 y FN(;)281 3322 y(U)9 b(e)386 3334 y FL(k)449 3322 y FO(=)23 b FN(\013)14 b(q)644 3287 y FL(k)q FM(\000)p FK(1)770 3258 y FP(p)p 839 3258 36 4 v 64 x FN(c)g(e)928 3334 y FL(k)969 3322 y FN(;)180 b FP(j)p FN(\013)p FP(j)23 b FO(=)g(1)p FN(;)41 b(k)26 b FO(=)c(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)27 b(:)-118 3510 y FO(The)g(represen)n(tations)d(of)j(this)g(series)e(are)h (de\014ned)h(b)n(y)g(the)h(parameters)c FN(c)f(>)-118 3609 y FO(0,)k FN(\013)d FP(2)f FN(S)185 3579 y FK(1)222 3609 y FO(.)6 3712 y(3\))31 b(If)f FN(c)e(>)f FO(0,)j(then)h(the)f (orbits)f(that)i(con)n(tain)d(the)j(p)r(oin)n(ts)e(\()p FN(c;)14 b(y)s FO(\),)31 b FN(y)f(>)d FO(0)-118 3811 y(lie)h(in)h(the)h(\014rst)g(quadran)n(t.)42 b(They)29 b(consist)g(of)g(the)i(p)r(oin)n(ts)d(\()p FN(c;)14 b(q)1937 3781 y FK(2)p FL(n)2016 3811 y FO(\),)31 b FN(n)26 b FP(2)h FI(Z)p FO(,)-118 3911 y(and)j(the)h(set)f(of)h(these)f(orbits)f (is)g(naturally)f(parametrized)f(b)n(y)j(p)r(oin)n(ts)g(of)g(a)p eop %%Page: 168 172 168 171 bop -118 -137 a FO(168)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(circle)e FN(S)152 66 y FK(1)189 96 y FO(.)36 b(The)24 b(corresp)r(onding)e(irreducible)f(represen)n (tations)g(are)j(realized)-118 196 y(on)j FN(l)22 208 y FK(2)59 196 y FO(\()p FI(Z)p FO(\))22 b(b)n(y)27 b(the)h(form)n (ulae:)273 380 y FN(X)7 b(e)388 392 y FL(k)451 380 y FO(=)539 316 y FP(p)p 608 316 36 4 v 64 x FN(c)14 b(e)697 392 y FL(k)q FK(+1)821 380 y FN(;)97 b(U)9 b(e)1046 392 y FL(k)1109 380 y FO(=)23 b FN(\025)14 b(q)1299 346 y FL(k)1354 380 y FN(e)1393 392 y FL(k)q FM(\000)p FK(1)1519 380 y FN(;)180 b(k)25 b FP(2)f FI(Z)o FN(:)-118 564 y FO(The)31 b(parameters)d FN(\025)h FP(2)g FO(\()p FN(q)722 534 y FK(2)760 564 y FN(;)14 b FO(1],)31 b FN(c)e FP(\025)f FO(0,)j(determine)e(the)j(set)f(of)g(represen)n(ta-)-118 664 y(tions)26 b(of)i(this)f(series.)6 764 y(4\))20 b(Consider)e(no)n (w)i(the)g(orbits)e(con)n(taining)f(the)k(p)r(oin)n(ts)e(\(0)p FN(;)14 b(y)s FO(\),)21 b FN(y)26 b(>)c(c)h(>)g FO(0.)-118 864 y(They)30 b(are)f(describ)r(ed)g(b)n(y)g FN(\025)f FP(2)f FO(\()p FN(c)20 b FO(+)g FN(q)1098 834 y FK(2)1135 864 y FN(;)14 b(c)20 b FO(+)g(1])26 b FP(\031)h FN(S)1552 834 y FK(1)1589 864 y FO(.)44 b(The)30 b(set)g(\001)g(consists)-118 963 y(of)d(the)h(p)r(oin)n(ts)-64 1147 y FQ(x)-14 1113 y FK(\()p FL(k)q(;l)p FK(\))143 1147 y FO(=)231 1080 y Fy(\000)269 1147 y FO(\(1)18 b FP(\000)g FN(q)484 1113 y FK(2)p FL(k)558 1147 y FO(\))c FN(c;)g(q)717 1113 y FK(2\()p FL(k)q FM(\000)p FL(l)p FK(\))916 1147 y FN(\025)19 b FO(+)f FN(q)1106 1113 y FK(2)p FL(k)1180 1147 y FO(\(1)h FP(\000)f FN(q)1396 1113 y FM(\000)p FK(2)p FL(l)1506 1147 y FO(\))c FN(c)1588 1080 y Fy(\001)1626 1147 y FN(;)180 b(k)26 b FP(\025)d FO(0)p FN(;)k(l)d FP(2)g FI(Z)o FN(:)-118 1332 y FO(The)g(irreducible)c(represen)n(tation)i(corresp)r(onding)f (to)j(this)f(orbit)g(is)g(realized)-118 1431 y(on)k FN(l)22 1443 y FK(2)59 1431 y FO(\()p FI(N)i FP(\002)18 b FI(Z)o FO(\))k(b)n(y)28 b(the)g(form)n(ulae:)209 1615 y FN(X)7 b(e)324 1627 y FL(k)q(l)408 1615 y FO(=)22 b(\(\(1)d FP(\000)f FN(q)743 1581 y FK(2)p FL(k)817 1615 y FO(\))c FN(c)p FO(\))931 1581 y FK(1)p FL(=)p FK(2)1049 1615 y FN(e)1088 1627 y FL(k)q FK(+1)p FL(;l)1254 1615 y FN(;)218 1772 y(U)9 b(e)323 1784 y FL(k)q(l)408 1772 y FO(=)495 1705 y Fy(\000)533 1772 y FN(q)573 1738 y FK(2\()p FL(k)q FM(\000)p FL(l)p FM(\000)p FK(1\))857 1772 y FN(\025)19 b FO(+)f FN(q)1047 1738 y FK(2)p FL(k)q FM(\000)p FK(2)1206 1772 y FO(\(1)h FP(\000)f FN(q)1422 1738 y FM(\000)p FK(2)p FL(l)1532 1772 y FO(\))c FN(c)1614 1705 y Fy(\001)1652 1722 y FK(1)p FL(=)p FK(2)1770 1772 y FN(e)1809 1784 y FL(k)q(;l)p FK(+1)1975 1772 y FN(;)551 1897 y(k)26 b FO(=)c(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)27 b FO(;)41 b FN(l)25 b FP(2)e FI(Z)o FN(:)-118 2081 y FO(F)-7 b(or)27 b(an)n(y)g FN(c)c FP(\025)g FO(0)k(and)g FN(\025)d FP(2)g FO(\()p FN(c)18 b FO(+)g FN(q)925 2050 y FK(2)963 2081 y FN(;)c(c)k FO(+)g(1])27 b(there)h(exists)e(a)h(unique)g(represen-) -118 2180 y(tation)f(of)i(suc)n(h)f(a)g(form.)6 2281 y(5\))i(The)g(follo)n(wing)c(series)h(of)j(represen)n(tations)d(that)j (dep)r(end)g(on)g(a)f(con-)-118 2380 y(tin)n(uous)18 b(parameter)g(corresp)r(onds)g(\(in)h(the)h(case)f(where)g FN(c)k FP(\025)g FO(0\))d(to)f(the)h(orbits)-118 2480 y(con)n(taining)k(the)j(p)r(oin)n(ts)f(\()p FN(z)t(;)14 b FO(0\),)26 b FN(z)g(>)d(c)p FO(.)36 b(The)27 b(set)g(parameterizing) 22 b(these)27 b(or-)-118 2579 y(bits)e(can)g(b)r(e)h(c)n(hosen)e(to)h (b)r(e)h(the)g(line)e(segmen)n(t)f(\()p FN(c)15 b FO(+)f FN(q)1592 2549 y FK(2)1629 2579 y FN(;)g(c)g FO(+)f(1])23 b FP(3)g FN(\025)p FO(.)37 b(The)26 b(set)-118 2679 y(\001)c(has)f(the) h(form)e(\001)j(=)g FP(f)p FN(\025)712 2649 y FK(\()p FL(k)q(l)p FK(\))849 2679 y FO(=)g(\()p FN(c)6 b FP(\000)g FN(q)1122 2649 y FK(2)p FL(k)1196 2679 y FO(\()p FN(c)g FP(\000)g FN(\025)p FO(\))p FN(;)14 b(q)1498 2649 y FK(2)p FL(k)1573 2679 y FO(\(1)6 b FP(\000)g FN(q)1764 2649 y FM(\000)p FK(2)p FL(l)1875 2679 y FO(\)\()p FN(c)g FP(\000)g FN(\025)p FO(\)\))p FN(;)38 b(l)24 b FP(\025)-118 2779 y FO(0)p FN(;)14 b(k)25 b FP(2)f FI(Z)o FP(g)o FO(,)19 b(and)k(the)h(irreducible)19 b(represen)n(tation)i(that)j(corresp)r (onds)d(to)i FN(c;)14 b(\025)-118 2878 y FO(acts)27 b(in)g FN(l)175 2890 y FK(2)212 2878 y FO(\()p FI(Z)12 b FP(\002)18 b FI(N)t FO(\))34 b(b)n(y:)397 3062 y FN(X)7 b(e)512 3074 y FL(k)q(l)597 3062 y FO(=)22 b(\()p FN(c)d FP(\000)f FN(q)894 3028 y FK(2)p FL(k)q FK(+2)1052 3062 y FO(\()p FN(c)h FP(\000)f FN(\025)p FO(\)\))1334 3028 y FK(1)p FL(=)p FK(2)1453 3062 y FN(e)1492 3074 y FL(k)q FK(+1)p FL(;l)1657 3062 y FN(;)407 3219 y(U)9 b(e)512 3231 y FL(k)q(l)597 3219 y FO(=)22 b FN(q)724 3185 y FL(k)779 3152 y Fy(\000)817 3219 y FO(\(1)c FP(\000)g FN(q)1032 3185 y FM(\000)p FK(2)p FL(l)1143 3219 y FO(\)\()p FN(c)h FP(\000)f FN(\025)p FO(\))1425 3152 y Fy(\001)1463 3169 y FK(1)p FL(=)p FK(2)1582 3219 y FN(e)1621 3231 y FL(k)q(;l)p FK(+1)1786 3219 y FN(;)740 3344 y(k)25 b FP(2)f FI(Z)o FN(;)36 b(l)24 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)26 b(:)-118 3528 y FO(Unlik)n(e)c(the)i(previous)e(series)g(of)i(orbits,)f (in)g(the)h(case)f(where)h FN(c)f(<)f FO(0)i(the)g(orbit)-118 3627 y(con)n(taining)32 b(the)j(p)r(oin)n(t)g(\(0)p FN(;)14 b FO(0\))34 b(determines)f(the)i(follo)n(wing)c(represen)n(tation)-118 3727 y(on)c FN(l)22 3739 y FK(2)59 3727 y FO(\()p FI(N)i FP(\002)18 b FI(N)t FO(\):)132 3911 y FN(X)7 b(e)247 3923 y FL(k)q(l)332 3911 y FO(=)22 b(\(\(1)d FP(\000)f FN(q)667 3877 y FM(\000)p FK(2)p FL(k)q FK(+2)877 3911 y FO(\))c FN(c)p FO(\))991 3877 y FK(1)p FL(=)p FK(2)1109 3911 y FN(e)1148 3923 y FL(k)q FM(\000)p FK(1)p FL(;l)1315 3911 y FN(;)p eop %%Page: 169 173 169 172 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(169)142 96 y FN(U)9 b(e)247 108 y FL(k)q(l)332 96 y FO(=)22 b FN(q)459 62 y FM(\000)p FL(k)566 96 y FO(\(\(1)c FP(\000)g FN(q)813 62 y FM(\000)p FK(2)p FL(l)924 96 y FO(\))c FN(c)p FO(\))1038 62 y FK(1)p FL(=)p FK(2)1156 96 y FN(e)1195 108 y FL(k)q(;l)p FK(+1)1361 96 y FN(;)180 b(k)s(;)14 b(l)24 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)26 b(:)6 291 y FO(6\))f(Finally)-7 b(,)23 b(for)h FN(c)f(>)g FO(0,)i(consider)e(the)i(orbit)e(con)n (taining)f(\(0)p FN(;)14 b FO(0\).)36 b(The)25 b(set)-118 391 y(\001)33 b(has)g(the)g(form)f(\001)h(=)e FP(f)p FN(\025)776 361 y FK(\()p FL(k)q(l)p FK(\))923 391 y FO(=)h(\(\(1)22 b FP(\000)g FN(q)1275 361 y FK(2)p FL(k)1349 391 y FO(\))14 b FN(c;)g(q)1508 361 y FK(2)p FL(k)1581 391 y FO(\(1)22 b FP(\000)g FN(q)1804 361 y FM(\000)p FK(2)p FL(l)1915 391 y FO(\))14 b FN(c)p FO(\))p FN(;)47 b(k)35 b FP(\025)d FO(0)p FN(;)-118 490 y(l)24 b FP(\024)f(\000)p FO(1)p FP(g)p FO(.)36 b(The)27 b(represen)n(tation)e(acts)i(on)h FN(l)1252 502 y FK(2)1289 490 y FO(\()p FI(N)g FP(\002)18 b FI(N)t FO(\):)187 681 y FN(X)7 b(e)302 693 y FL(k)q(l)387 681 y FO(=)22 b(\(\(1)d FP(\000)f FN(q)722 647 y FK(2)p FL(k)796 681 y FO(\))c FN(c)p FO(\))910 647 y FK(1)p FL(=)p FK(2)1028 681 y FN(e)1067 693 y FL(k)q FK(+1)p FL(;l)1233 681 y FN(;)197 822 y(U)9 b(e)302 834 y FL(k)q(l)387 822 y FO(=)22 b FN(q)514 788 y FL(k)q FM(\000)p FK(1)640 822 y FO(\(\(1)d FP(\000)f FN(q)888 788 y FK(2)p FL(l)946 822 y FO(\))c FN(c)p FO(\))1060 788 y FK(1)p FL(=)p FK(2)1179 822 y FN(e)1218 834 y FL(k)q(;l)p FM(\000)p FK(1)1384 822 y FN(;)180 b(k)s(;)14 b(l)24 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)27 b(:)-118 1016 y FB(R)l(emark)j(40.)42 b FO(W)-7 b(e)39 b(note)e(that)i(the)f(represen)n(tations)d(describ)r (ed)h(ab)r(o)n(v)n(e)h(in-)-118 1116 y(clude)29 b(represen)n(tations)e (b)n(y)j(un)n(b)r(ounded)g(op)r(erators.)43 b(Op)r(erators)28 b(of)i(repre-)-118 1216 y(sen)n(tations)c(in)g(the)i(cases)f(1\),)g (2\),)h(and)f(6\))h(are)e(b)r(ounded.)-118 1357 y FQ(4.)43 b FO(It)30 b(is)f(easy)f(to)i(see)g(that,)g(for)g FN(c)c(>)h FO(0,)j(using)e(the)i(follo)n(wing)c(substitution)-118 1456 y(of)h(v)-5 b(ariables,)245 1647 y FN(y)286 1659 y FK(1)346 1647 y FO(=)23 b(\(\(1)18 b FP(\000)g FN(q)681 1612 y FK(2)719 1647 y FO(\))c FN(c)p FO(\))833 1612 y FM(\000)p FK(1)p FL(=)p FK(2)989 1647 y FN(x;)98 b(y)1201 1612 y FM(\003)1198 1667 y FK(2)1262 1647 y FO(=)22 b(\(\(1)d FP(\000)f FN(q)1597 1612 y FK(2)1634 1647 y FO(\))c FN(c)p FO(\))1748 1612 y FM(\000)p FK(1)p FL(=)p FK(2)1905 1647 y FN(u;)-118 1837 y FO(the)34 b(relations)c(\(2.63\))i(can)h(b)r(e)h (reduced)f(to)g(the)g(t)n(wisted)g(CCR)g(algebra)e(of)-118 1937 y(Pusz)c(and)g(W)-7 b(orono)n(wicz:)108 2128 y FN(y)152 2093 y FM(\003)149 2148 y FK(1)190 2128 y FN(y)231 2140 y FK(1)291 2128 y FO(=)23 b(1)18 b(+)g FN(q)562 2093 y FK(2)599 2128 y FN(y)640 2140 y FK(1)677 2128 y FN(y)721 2093 y FM(\003)718 2148 y FK(1)759 2128 y FN(;)97 b(y)923 2093 y FM(\003)920 2148 y FK(2)961 2128 y FN(y)1002 2140 y FK(2)1062 2128 y FO(=)22 b(1)c(+)g FN(q)1332 2093 y FK(2)1369 2128 y FN(y)1410 2140 y FK(2)1447 2128 y FN(y)1491 2093 y FM(\003)1488 2148 y FK(2)1548 2128 y FO(+)g(\(1)g FP(\000)g FN(q)1846 2093 y FK(2)1883 2128 y FO(\))p FN(y)1956 2140 y FK(1)1994 2128 y FN(y)2038 2093 y FM(\003)2035 2148 y FK(1)2075 2128 y FN(;)558 2252 y(y)602 2218 y FM(\003)599 2273 y FK(1)640 2252 y FN(y)681 2264 y FK(2)741 2252 y FO(=)23 b FN(q)17 b(y)924 2264 y FK(2)961 2252 y FN(y)1005 2218 y FM(\003)1002 2273 y FK(1)1043 2252 y FN(;)97 b(y)1204 2264 y FK(2)1241 2252 y FN(y)1282 2264 y FK(1)1342 2252 y FO(=)22 b FN(q)17 b(y)1524 2264 y FK(1)1561 2252 y FN(y)1602 2264 y FK(2)1639 2252 y FN(:)-118 2489 y FQ(2.4.5)94 b(Heisen)m(b)s(erg)30 b(relations)h(for)h (the)f(quan)m(tum)h FN(E)5 b FO(\(2\))32 b FQ(group)-118 2650 y(1.)41 b FO(In)29 b(this)f(section)g(w)n(e)h(study)g FP(\003)p FO(-represen)n(tations)c(of)k(the)h(in)n(v)n(olutiv)n(e)25 b(alge-)-118 2749 y(bra)30 b FA(A)i FO(generated)e(b)n(y)h(the)g (so-called)d(Heisen)n(b)r(erg)h(relations)f([289)o(].)48 b(These)-118 2849 y(relations)23 b(connect)j(generators)e(of)j(a)f (quan)n(tum)f(deformation)f(of)i(the)h(group)-118 2949 y FN(E)5 b FO(\(2\))28 b(and)f(those)g(of)h(its)f(dual.)6 3052 y(First)19 b(w)n(e)h(recall)c(the)21 b(de\014nition)d(of)i(the)g (Heisen)n(b)r(erg)e(relations)e(for)j FN(E)2172 3064 y FL(q)2209 3052 y FO(\(2\).)-118 3152 y(Then)32 b(w)n(e)g(pro)n(v)n(e) f(some)f(auxiliary)e(assertions)h(whic)n(h)i(allo)n(w)e(us)j(to)g (rewrite)-118 3252 y(the)d(relations)c(in)j(a)g(more)f(con)n(v)n(enien) n(t)f(form)h(for)h(us.)40 b(The)28 b(relations)e(whic)n(h)-118 3351 y(are)43 b(obtained)f(can)h(b)r(e)h(considered)e(in)h(the)h (framew)n(ork)c(of)k(the)g(general)-118 3451 y(formalism)32 b(dev)n(elop)r(ed)j(ab)r(o)n(v)n(e,)i(whic)n(h)e(pro)n(vides)g(a)h(w)n (a)n(y)f(of)h(dealing)e(with)-118 3550 y(un)n(b)r(ounded)j(represen)n (tations.)61 b(Finally)-7 b(,)37 b(a)f(complete)f(list)g(of)h (irreducible)-118 3650 y FP(\003)p FO(-represen)n(tations)24 b(of)j(the)h(Heisen)n(b)r(erg)e(relations)e(is)j(giv)n(en.)-118 3811 y FQ(2.)34 b FO(The)20 b(quan)n(tum)f(deformation)f(of)i(the)h (group)e(of)h(motions)e(w)n(as)h(in)n(tro)r(duced)-118 3911 y(and)37 b(in)n(v)n(estigated)d(in)i([267)o(,)h(290)o(,)g(289)o (].)65 b(The)37 b(algebra)e(of)i(\\functions)f(on)p eop %%Page: 170 174 170 173 bop -118 -137 a FO(170)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FN(E)-57 108 y FL(q)-20 96 y FO(\(2\)")37 b(is)f(an)h(algebra)d(generated)i(b)n(y)h(the)h(elemen)n(ts)d FN(v)40 b FO(and)d FN(n)p FO(,)j FN(v)h FO(b)r(eing)-118 196 y(unitary)26 b(and)h FN(n)h FO(b)r(eing)f(normal,)d(whic)n(h)j (satisfy)f(the)i(relation)726 368 y FN(v)s(n)23 b FO(=)g FN(q)s(nv)s(;)180 b(q)26 b(>)c FO(0)p FN(;)645 b FO(\(2.64\))-118 540 y(sub)5 b(ject)39 b(to)f(an)h(additional)c(condition:)57 b(the)40 b(sp)r(ectrum)d(of)i(the)g(op)r(erator)-118 640 y FP(j)p FN(n)p FP(j)34 b FO(lies)e(in)i FP(f)p FN(q)348 610 y FL(n)392 640 y FN(;)28 b(n)34 b FP(2)h FI(Z)o FP(g)o FO(.)51 b(Although)34 b(this)f(condition)f(is)h(rather)g(natural,)-118 740 y(w)n(e)g(will)e(not)i(assume)f(it)h(in)g(our)f(study)i(of)g(the)f (represen)n(tations;)h(one)f(can)-118 839 y(easily)22 b(pic)n(k)i(represen)n(tations)e(satisfying)h(this)h(condition)f(from)h (a)h(wider)e(list)-118 939 y(of)k(represen)n(tations.)6 1039 y(On)32 b(the)g(other)g(hand,)h(using)d(the)i(com)n(ultiplication) 26 b(in)31 b FN(E)1900 1051 y FL(q)1937 1039 y FO(\(2\),)i(the)f(al-) -118 1149 y(gebra)c(of)i(\\con)n(tin)n(uous)e(functions)h(on)1164 1128 y(^)1145 1149 y FN(E)1206 1161 y FL(q)1243 1149 y FO(\(2\)",)h(where)1706 1128 y(^)1687 1149 y FN(E)1748 1161 y FL(q)1785 1149 y FO(\(2\))g(denotes)f(the)-118 1249 y(P)n(on)n(try)n(agin)21 b(dual)j(of)h FN(E)640 1261 y FL(q)677 1249 y FO(\(2\),)h(w)n(as)e(constructed)h(and)g(in)n(v) n(estigated)c(in)k([289)n(].)-118 1349 y(This)d(algebra)e(is)i (generated)g(b)n(y)h(the)h(elemen)n(ts)d FN(N)32 b FO(and)23 b FN(b)g FO(\()p FN(N)32 b FO(is)22 b(self-adjoin)n(t,)-118 1448 y FN(b)27 b FO(is)g(normal\))e(with)i(the)h(relation)821 1620 y FN(N)9 b(b)22 b FO(=)h FN(b)14 b FO(\()p FN(N)27 b FO(+)18 b FN(I)7 b FO(\))p FN(:)726 b FO(\(2.65\))6 1793 y(If)35 b(one)e(considers)e(b)r(oth)j(algebras)d(represen)n(ted)h (on)i(the)g(same)e(Hilb)r(ert)-118 1892 y(space,)i(some)e(natural)g (relations)e(b)r(et)n(w)n(een)j(the)h(generators)d FN(v)s FO(,)36 b FN(n)d FO(and)g(the)-118 1992 y(generators)27 b FN(N)9 b FO(,)30 b FN(b)f FO(\(the)h(Heisen)n(b)r(erg)e(relations,)f (see)i([289)n(]\))h(app)r(ear.)42 b(These)-118 2091 y(relations)24 b(are:)161 2264 y FN(v)s(N)32 b FO(=)23 b(\()p FN(N)28 b FP(\000)18 b FN(I)7 b FO(\))14 b FN(v)s(;)97 b(v)s(b)23 b FO(=)f FN(q)1082 2229 y FM(\000)p FK(1)p FL(=)p FK(2)1239 2264 y FN(bv)s(;)96 b(nN)32 b FO(=)23 b(\()p FN(N)k FO(+)18 b FN(I)7 b FO(\))14 b FN(n;)128 2415 y(bn)214 2381 y FM(\003)275 2415 y FO(=)23 b FN(q)403 2381 y FK(1)p FL(=)p FK(2)507 2415 y FN(n)557 2381 y FM(\003)595 2415 y FN(b;)97 b(nb)17 b FP(\000)h FN(q)977 2381 y FK(1)p FL(=)p FK(2)1082 2415 y FN(bn)k FO(=)h(\(1)18 b FP(\000)g FN(q)1493 2381 y FK(2)1531 2415 y FO(\))c FN(q)1617 2381 y FM(\000)1679 2355 y Fv(N)5 b Fx(+)p Fv(I)p 1679 2368 122 3 v 1725 2401 a Fx(2)1814 2415 y FN(v)s(:)246 b FO(\(2.66\))-118 2587 y FQ(3.)34 b FO(In)22 b(the)h(sequel)d(w)n(e)i(consider)e(a)h FP(\003)p FO(-algebra)e FA(A)j FO(generated)f(b)n(y)g(the)i(elemen)n (ts)-118 2687 y FN(v)s FO(,)30 b FN(n)p FO(,)g FN(N)9 b FO(,)29 b(and)h FN(b)f FO(suc)n(h)g(that)g FN(v)k FO(is)28 b(unitary)-7 b(,)28 b FN(N)39 b FO(is)28 b(self-adjoin)n(t,)f FN(n)i FO(and)h FN(b)f FO(are)-118 2787 y(normal,)c(and)i(the)h (generators)d(satisfy)h(relations)f(\(2.64\))o(,)j(\(2.65\))o(,)g (\(2.66\))o(.)6 2886 y(Instead)k(of)f FN(b)g FO(and)g FN(N)9 b FO(,)32 b(in)n(tro)r(duce)f(the)g(new)h(generators,)e FN(d)f FO(=)g FN(bv)2135 2856 y FM(\003)2205 2886 y FO(and)-118 2986 y FN(M)j FO(=)22 b(\(1)c FP(\000)g FN(q)297 2956 y FK(2)335 2986 y FO(\))c FN(q)421 2956 y FM(\000)p FK(\()p FL(N)6 b FK(+)p FL(I)t FK(\))p FL(=)p FK(2)740 2986 y FO(.)37 b(Then)27 b(the)h(relations)d(are:)48 3158 y FN(v)s(n)e FO(=)f FN(q)s(nv)s(;)97 b(nn)604 3124 y FM(\003)665 3158 y FO(=)23 b FN(n)803 3124 y FM(\003)841 3158 y FN(n;)97 b(M)9 b(d)23 b FO(=)f FN(q)1294 3124 y FK(1)p FL(=)p FK(2)1399 3158 y FN(dM)t(;)97 b(d)1690 3124 y FM(\003)1728 3158 y FN(d)23 b FO(=)g FN(q)1922 3124 y FM(\000)p FK(1)2011 3158 y FN(dd)2097 3124 y FM(\003)2136 3158 y FN(;)179 3299 y(v)s(M)32 b FO(=)22 b FN(q)462 3265 y FK(1)p FL(=)p FK(2)567 3299 y FN(M)9 b(v)s(;)96 b(v)s(d)24 b FO(=)e FN(q)1056 3265 y FM(\000)p FK(1)p FL(=)p FK(2)1213 3299 y FN(dv)s(;)97 b(nM)32 b FO(=)22 b FN(q)1709 3265 y FM(\000)p FK(1)p FL(=)p FK(2)1865 3299 y FN(M)9 b(n;)440 3440 y(nd)533 3406 y FM(\003)595 3440 y FO(=)22 b FN(q)722 3406 y FM(\000)p FK(1)p FL(=)p FK(2)878 3440 y FN(d)921 3406 y FM(\003)960 3440 y FN(n;)97 b(nd)18 b FP(\000)g FN(q)1364 3406 y FK(3)p FL(=)p FK(2)1468 3440 y FN(dn)24 b FO(=)e FN(M)t(:)-118 3612 y FQ(4.)34 b FO(Considering)20 b(the)j(represen)n(tations)d(of)i (the)h(algebra)d FA(A)p FO(,)k(w)n(e)e(ha)n(v)n(e)g(to)g(k)n(eep)-118 3712 y(in)33 b(mind)g(that)h(the)h(op)r(erator)d FN(M)43 b FO(m)n(ust)33 b(b)r(e)h(p)r(ositiv)n(e)e(for)h(0)h FN(<)f(q)k(<)c FO(1)h(and)-118 3811 y(negativ)n(e)c(for)j FN(q)h(>)d FO(1.)52 b(W)-7 b(e)33 b(will)d(consider)h(the)i(case)f(0)f FN(<)g(q)k(<)c FO(1)h(\(the)i(case)-118 3911 y FN(q)26 b(>)d FO(1)k(is)f(quite)h(similar\).)p eop %%Page: 171 175 171 174 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(171)-118 96 y FQ(Lemma)29 b(12.)41 b FB(Supp)l(ose)33 b(we)g(have)g(a)g FP(\003)p FB(-r)l(epr)l(esentation)f(of)i(the)e(Heisenb)l(er)l(g)-118 196 y(r)l(elations)j FO(\()p FB(gener)l(al)t(ly)30 b(sp)l(e)l(aking,)g (by)e(unb)l(ounde)l(d)g(op)l(er)l(ators)7 b FO(\))29 b FB(and)f(ther)l(e)g(ex-)-118 296 y(ists)i(a)g(ve)l(ctor)g FN(f)h FP(2)24 b FN(H)36 b FB(such)30 b(that)g FN(f)h FP(2)24 b FO(k)n(er)13 b FN(n)29 b FB(and)727 479 y FN(nd)-14 b(f)27 b FP(\000)18 b FN(q)997 445 y FK(3)p FL(=)p FK(2)1101 479 y FN(dnf)32 b FO(=)23 b FN(M)9 b(f)-118 662 y FO(\()p FB(it)35 b(is)f(supp)l(ose)l(d)i(that)e(the)h(r)l(e)l(quir)l(e)l(d)f (op)l(er)l(ators)h(ar)l(e)g(de\014ne)l(d)g(on)f FN(f)9 b FO(\))p FB(.)53 b(Then)-118 762 y FN(f)32 b FO(=)22 b(0)p FB(.)-118 928 y(Pr)l(o)l(of.)43 b FO(Indeed,)28 b(since)e FN(n)i FO(is)e(normal,)f FN(f)32 b FP(2)23 b FO(k)n(er)13 b FN(n)1430 898 y FM(\003)1468 928 y FO(.)37 b(But)28 b(this)f(implies)c(that)81 1112 y(\()p FN(M)9 b(f)t(;)14 b(f)9 b FO(\))22 b(=)h(\()p FN(nd)-14 b(f)27 b FP(\000)19 b FN(q)780 1077 y FK(3)p FL(=)p FK(2)884 1112 y FN(dn)14 b(f)t(;)g(f)9 b FO(\))23 b(=)f(\()p FN(nd)-14 b(f)t(;)14 b(f)9 b FO(\))24 b(=)e(\()p FN(d)-14 b(f)t(;)14 b(n)1844 1077 y FM(\003)1883 1112 y FN(f)9 b FO(\))23 b(=)f(0)p FN(;)-118 1295 y FO(whic)n(h)k(is)h(imp)r(ossible)c(b)n(y)28 b(the)g(p)r(ositivit)n(y)c(of)k FN(M)9 b FO(.)p 2278 1295 4 57 v 2282 1242 50 4 v 2282 1295 V 2331 1295 4 57 v 6 1463 a(In)n(tro)r(duce)28 b(the)g(elemen)n(t)689 1685 y FN(y)e FO(=)c FN(nM)9 b(d)18 b FO(+)1149 1628 y FN(q)1189 1598 y FM(\000)p FK(1)p FL(=)p FK(2)p 1137 1665 221 4 v 1137 1741 a FN(q)1177 1718 y FK(2)1233 1741 y FP(\000)g FO(1)1382 1685 y FN(M)1472 1650 y FK(2)1508 1685 y FN(:)6 1907 y FO(By)32 b(the)h(previous)d(lemma)e(w)n(e)k(can)g (supp)r(ose)g(that,)h(for)f(\\go)r(o)r(d")e(repre-)-118 2007 y(sen)n(tations)i(of)i(the)h(Heisen)n(b)r(erg)d(relations,)h(the)h (op)r(erators)f FN(n)h FO(and)g FN(M)43 b FO(are)-118 2107 y(in)n(v)n(ertible)24 b(and)j(that,)h(if)f(w)n(e)h(\014nd)g FN(y)s FO(,)f(w)n(e)g(will)e(b)r(e)j(able)e(to)i(reconstruct)e FN(d)i FO(as)551 2324 y FN(d)c FO(=)e FN(M)795 2290 y FM(\000)p FK(1)884 2324 y FN(n)934 2290 y FM(\000)p FK(1)1023 2324 y FN(y)f FO(+)1267 2268 y(1)p 1178 2305 V 1178 2381 a(1)d FP(\000)g FN(q)1361 2357 y FK(2)1422 2324 y FN(n)1472 2290 y FM(\000)p FK(1)1561 2324 y FN(M)t(:)-118 2547 y FO(Th)n(us,)27 b(replacing)e FN(d)j FO(b)n(y)f FN(y)s FO(,)g(w)n(e)g(get)h(the)g(follo)n(wing)23 b(relations:)181 2730 y FN(v)s(n)g FO(=)g FN(q)s(nv)s(;)97 b(nn)738 2696 y FM(\003)799 2730 y FO(=)23 b FN(n)937 2696 y FM(\003)975 2730 y FN(n;)96 b(y)s(M)32 b FO(=)22 b FN(M)9 b(y)s(;)97 b FO([)p FN(y)s(;)14 b(y)1790 2696 y FM(\003)1827 2730 y FO(])23 b(=)g(0)p FN(;)256 2871 y(v)s(M)32 b FO(=)23 b FN(q)540 2837 y FK(1)p FL(=)p FK(2)644 2871 y FN(M)9 b(v)s(;)97 b(v)s(y)26 b FO(=)d FN(q)s(y)s(v)s(;)96 b(nM)32 b FO(=)23 b FN(q)1632 2837 y FM(\000)p FK(1)p FL(=)p FK(2)1788 2871 y FN(M)9 b(n;)663 2996 y(n)713 2961 y FM(\003)751 2996 y FN(y)26 b FO(=)c FN(q)s(y)s(n)1039 2961 y FM(\003)1077 2996 y FN(;)97 b(ny)25 b FO(=)e FN(q)s(y)s(n:)568 b FO(\(2.67\))6 3179 y(F)-7 b(rom)27 b(no)n(w)g(on,)g(w)n(e)g(deal)f (with)i FP(\003)p FO(-represen)n(tations)23 b(of)28 b(the)g(algebra)d FA(A)p FO(.)-118 3346 y FQ(Prop)s(osition)30 b(53.)41 b FB(Ther)l(e)23 b(ar)l(e)h(no)e(r)l(epr)l(esentations)i(of)41 b FO(\(2.67\))22 b FB(by)h(b)l(ounde)l(d)-118 3445 y(op)l(er)l(ators.) -118 3612 y(Pr)l(o)l(of.)43 b FO(Indeed,)26 b(since)f FN(M)9 b(u)22 b FO(=)g FN(q)922 3582 y FM(\000)p FK(1)p FL(=)p FK(2)1079 3612 y FN(uM)33 b FO(and)26 b FN(u)f FO(is)f(unitary)-7 b(,)25 b(the)g(sp)r(ectrum)-118 3712 y(of)30 b FN(M)39 b FO(is)29 b(in)n(v)-5 b(arian)n(t)27 b(under)j(m)n(ultiplication)25 b(b)n(y)k FN(q)1467 3682 y FM(\000)p FK(1)p FL(=)p FK(2)1624 3712 y FO(.)45 b(But)30 b(since)f FN(M)36 b(>)27 b FO(0,)-118 3811 y(the)k(sp)r(ectrum)f(of)h FN(M)40 b FO(do)r(es)30 b(not)h(con)n(tain)e(zero)h(and,)i(th)n(us,)f (is)f(un)n(b)r(ounded.)p 2278 3911 4 57 v 2282 3858 50 4 v 2282 3911 V 2331 3911 4 57 v eop %%Page: 172 176 172 175 bop -118 -137 a FO(172)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(R)l(emark)30 b(41.)42 b FO(As)36 b(one)e(can)h(see,)i(relations)32 b(\(2.67\))i(ha)n(v)n(e)g(the)i(form) d(\(2.57\).)-118 196 y(This)d(fact)i(enables)e(one)h(to)g(select)f(the) i(class)d(of)j(\\go)r(o)r(d")e(un)n(b)r(ounded)h(rep-)-118 296 y(resen)n(tations,)f(whic)n(h)h(can)h(b)r(e)g(con)n(tin)n(ued)e(to) i(represen)n(tations)d(of)j(the)g(cor-)-118 395 y(resp)r(onding)g(b)r (ounded)j(op)r(erators)d(\(see)j(Section)e(2.4.3\),)i(and)f(to)g(list)f (irre-)-118 495 y(ducible)26 b(represen)n(tations)e(from)j(this)g (class.)6 632 y(In)n(tro)r(duce)h(the)g(follo)n(wing)23 b(op)r(erators)j(on)h(the)h(space)f FN(l)1749 644 y FK(2)1786 632 y FO(\()p FI(Z)o FO(\):)-13 818 y FN(S)5 b(e)82 830 y FL(k)145 818 y FO(=)23 b FN(e)272 830 y FL(k)q FK(+1)396 818 y FN(;)97 b(T)12 b(e)616 830 y FL(k)679 818 y FO(=)23 b FN(k)s(e)852 830 y FL(k)892 818 y FN(;)97 b(Qe)1117 830 y FL(k)1180 818 y FO(=)23 b FN(q)1308 784 y FL(k)q(=)p FK(2)1416 818 y FN(e)1455 830 y FL(k)1518 818 y FO(=)g FN(e)1645 784 y FL(T)7 b(=)p FK(2)1762 818 y FN(e)1801 830 y FL(k)2008 818 y FN(k)26 b FP(2)d FI(Z)o FN(:)-118 1006 y FQ(Theorem)30 b(40.)41 b FB(A)n(l)t(l)27 b(irr)l(e)l(ducible)h FP(\003)p FB(-r)l(epr)l(esentations)e(of)i(the)f(algebr)l(a)h FA(A)p FB(,)g(up)-118 1106 y(to)i(unitary)f(e)l(quivalenc)l(e,)i(ar)l (e:)6 1234 y FN(a)p FO(\))f FB(r)l(epr)l(esentations)g(on)g FN(l)831 1246 y FK(2)868 1234 y FO(\()p FI(Z)p FO(\))12 b FP(\012)18 b FN(l)1114 1246 y FK(2)1152 1234 y FO(\()p FI(Z)o FO(\):)234 1421 y FN(n)23 b FO(=)g FN(\025)14 b(S)23 b FP(\012)18 b FN(Q)680 1387 y FK(2)717 1421 y FN(;)99 b(v)26 b FO(=)d FN(S)1049 1387 y FM(\003)1105 1421 y FP(\012)18 b FN(S)1244 1387 y FM(\003)1282 1421 y FN(;)99 b(N)31 b FO(=)23 b FN(\013)c FP(\000)f FN(T)29 b FP(\012)18 b FN(I)7 b(;)498 1562 y(b)23 b FO(=)g FN(q)685 1527 y FM(\000)p FL(\013=)p FK(2)p FM(\000)p FK(1)936 1562 y FN(\025)984 1527 y FM(\000)p FK(1)1074 1562 y FN(Q)p FO(\()p FN(S)1228 1527 y FM(\003)1265 1562 y FO(\))1297 1527 y FK(2)1353 1562 y FP(\012)18 b FN(Q)1502 1527 y FM(\000)p FK(2)1591 1562 y FN(S)1647 1527 y FM(\003)1685 1562 y FO(;)6 1750 y FN(b)p FO(\))30 b FB(r)l(epr)l(esentations)g(on)g FN(l)823 1762 y FK(2)860 1750 y FO(\()p FI(Z)o FO(\))13 b FP(\012)18 b FN(l)1106 1762 y FK(2)1143 1750 y FO(\()p FI(Z)p FO(\))13 b FP(\012)18 b FN(l)1390 1762 y FK(2)1427 1750 y FO(\()p FI(Z)o FO(\):)-8 1937 y FN(n)23 b FO(=)f FN(\025)14 b(S)24 b FP(\012)18 b FN(Q)438 1902 y FK(2)493 1937 y FP(\012)g FN(I)7 b(;)99 b(v)26 b FO(=)d FN(S)951 1902 y FM(\003)1007 1937 y FP(\012)18 b FN(S)1146 1902 y FM(\003)1203 1937 y FP(\012)g FN(S)1342 1902 y FM(\003)1379 1937 y FN(;)99 b(N)32 b FO(=)23 b FN(\013)c FP(\000)f FN(T)29 b FP(\012)18 b FN(I)26 b FP(\012)18 b FN(I)7 b(;)408 2130 y(b)23 b FO(=)566 2073 y FN(\016)s(q)646 2043 y FK(\()p FL(\013)p FK(+1\))p FL(=)p FK(2)p 564 2110 334 4 v 564 2187 a FN(\025)p FO(\(1)c FP(\000)f FN(q)828 2163 y FK(2)865 2187 y FO(\))908 2130 y FN(Q)974 2095 y FM(\000)p FK(1)1062 2130 y FO(\()p FN(S)1150 2095 y FM(\003)1188 2130 y FO(\))1220 2095 y FK(2)1276 2130 y FP(\012)g FN(Q)1425 2095 y FM(\000)p FK(2)1533 2130 y FP(\012)g FN(Q)1682 2095 y FK(2)1719 2130 y FN(S)1775 2095 y FM(\003)442 2318 y FO(+)g FN(q)565 2284 y FM(\000)p FL(\013=)p FK(2)p FM(\000)p FK(1)816 2318 y FN(\025)864 2284 y FM(\000)p FK(1)954 2318 y FN(Q)p FO(\()p FN(S)1108 2284 y FM(\003)1145 2318 y FO(\))1177 2284 y FK(2)1233 2318 y FP(\012)g FN(Q)1382 2284 y FM(\000)p FK(2)1471 2318 y FN(S)1527 2284 y FM(\003)1583 2318 y FP(\012)g FN(S)1722 2284 y FM(\003)1760 2318 y FN(;)-118 2505 y FB(wher)l(e)30 b FN(\025)p FB(,)h FN(\016)26 b FP(2)d FO(\()p FN(q)s(;)14 b FO(1])p FB(,)30 b FN(\013)24 b FP(2)f FO([0)p FN(;)14 b FO(1\))p FB(.)-118 2677 y(Pr)l(o)l(of.)43 b FO(Applying)27 b(the)i(argumen)n(t)e(as)i(ab)r(o)n(v)n(e,)f(w)n(e)g (obtain)g(that,)h(up)h(to)f(uni-)-118 2776 y(tary)e(equiv)-5 b(alence,)126 2963 y FN(u)174 2975 y FL(n)241 2963 y FO(=)23 b FN(S)g FP(\012)18 b FN(I)26 b FP(\012)18 b FN(I)7 b(;)97 b(u)842 2975 y FL(y)904 2963 y FO(=)23 b FN(I)i FP(\012)18 b FN(S)23 b FP(\012)18 b FN(I)7 b(;)97 b(v)27 b FO(=)22 b FN(S)1666 2928 y FM(\003)1723 2963 y FP(\012)c FN(S)1862 2928 y FM(\003)1918 2963 y FP(\012)g FN(S)2057 2928 y FM(\003)-118 3149 y FO(for)27 b FN(\016)f FP(6)p FO(=)d(0)k(and,)g(otherwise,)436 3336 y FN(u)484 3348 y FL(n)551 3336 y FO(=)c FN(S)g FP(\012)18 b FN(I)7 b(;)97 b(u)1007 3348 y FL(y)1070 3336 y FO(=)22 b(0)p FN(;)97 b(v)26 b FO(=)d FN(S)1529 3301 y FM(\003)1585 3336 y FP(\012)18 b FN(S)1724 3301 y FM(\003)1762 3336 y FN(:)-118 3522 y FO(Then,)28 b(since)450 3708 y FN(b)22 b FO(=)h FN(M)686 3674 y FM(\000)p FK(1)775 3708 y FN(n)825 3674 y FM(\000)p FK(1)914 3708 y FN(y)s(v)e FO(+)d(\(1)g FP(\000)g FN(q)1317 3674 y FK(2)1355 3708 y FO(\))1387 3674 y FM(\000)p FK(1)1476 3708 y FN(n)1526 3674 y FM(\000)p FK(1)1615 3708 y FN(M)9 b(v)s(:)-118 3895 y FO(w)n(e)26 b(get)g(the)h(necessary)e(expressions)f(b)n(y)i(using)f(the)i (expression)d(for)i FN(M)9 b FO(.)p 2278 3895 4 57 v 2282 3842 50 4 v 2282 3895 V 2331 3895 4 57 v eop %%Page: 173 177 173 176 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(173)-118 96 y FQ(2.4.6)94 b(Wic)m(k)32 b(algebras)g(related)f(to)h(dynamical)f(systems)-118 250 y(1.)52 b FO(In)32 b(this)h(section)e(w)n(e)h(consider)f(some)g (Wic)n(k)h(algebras)e(connected)i(with)-118 350 y(dynamical)24 b(systems.)6 450 y(It)h(w)n(as)d(noted)i(ab)r(o)n(v)n(e)e(that)i(the)h (relations)20 b(\(2.59\))j(serv)n(e)g(as)g(some)f(consis-)-118 549 y(tency)32 b(conditions.)47 b(They)32 b(app)r(eared)f(in)g(Section) g(2.4.3)g(as)g(conditions)e(for)-118 649 y(comm)n(utation)24 b(of)j(the)h(mappings)e(obtained)g(from)g(the)i(de\014ning)f (relations.)6 749 y(In)f(this)f(section)e(w)n(e)i(lo)r(ok)f(at)h(this)g (consistency)e(from)h(the)i(p)r(oin)n(t)f(of)g(view)-118 849 y(of)30 b(Wic)n(k)g(algebras.)42 b(W)-7 b(e)31 b(supp)r(ose)f(that) h(the)f(generators)e FN(x)1802 819 y FM(\003)1802 870 y FL(i)1841 849 y FO(,)j FN(x)1942 861 y FL(j)1978 849 y FO(,)g FN(i)p FO(,)g FN(j)i FO(=)27 b(1,)-118 948 y FN(:)14 b(:)g(:)27 b FO(,)h FN(d)p FO(,)g(only)e(satisfy)h(the)g (relations)318 1132 y FN(x)365 1097 y FM(\003)365 1152 y FL(i)403 1132 y FN(x)450 1144 y FL(i)501 1132 y FO(=)c FN(F)642 1144 y FL(i)670 1132 y FO(\()p FN(x)749 1144 y FK(1)787 1132 y FN(x)834 1097 y FM(\003)834 1152 y FK(1)873 1132 y FN(;)14 b(:)g(:)g(:)f(;)h(x)1104 1144 y FL(d)1143 1132 y FN(x)1190 1097 y FM(\003)1190 1152 y FL(d)1230 1132 y FO(\))p FN(;)180 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d;)310 1256 y(x)357 1222 y FM(\003)357 1277 y FL(i)396 1256 y FN(x)443 1268 y FL(j)501 1256 y FO(=)23 b FN(q)626 1268 y FL(ij)685 1256 y FN(x)732 1268 y FL(j)767 1256 y FN(x)814 1222 y FM(\003)814 1277 y FL(i)853 1256 y FN(;)180 b(i)22 b FP(6)p FO(=)h FN(j:)-118 1439 y FO(Then)33 b(additional)d(relations)g(for)j FN(x)1025 1451 y FL(i)1053 1439 y FO(,)i FN(x)1158 1451 y FL(j)1193 1439 y FO(,)g FN(i)d FP(6)p FO(=)g FN(j)38 b FO(de\014ne)c(an)f(ideal)e FA(I)i FO(in)f(the)-118 1539 y(algebra)23 b(generated)i(b)n(y)g FP(f)p FN(x)750 1509 y FM(\003)750 1561 y FL(i)788 1539 y FN(;)14 b(x)872 1551 y FL(i)900 1539 y FN(;)28 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d)p FP(g)p FO(,)26 b(and)g(the)g(consistency)e(con-)-118 1639 y(dition)i(is)g(determined)g (as)h(a)g(sp)r(ecial)e(prop)r(ert)n(y)i(of)g(the)h(ideal)e FA(I)p FO(.)-118 1789 y FQ(2.)41 b FO(T)-7 b(o)28 b(do)h(our)g (reasoning)d(more)h(clearly)f(let)i(us)i(presen)n(t)e(some)g (de\014nitions)-118 1889 y(and)f(prop)r(ositions.)-118 2056 y FQ(De\014nition)k(9.)40 b FB(L)l(et)d FN(I)42 b FO(=)35 b FP(f)p FO(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d)p FP(g)36 b FB(and)h FN(T)1390 2026 y FL(k)q(l)1378 2077 y(ij)1487 2056 y FP(2)f FI(C)15 b FB(,)45 b FN(i)p FB(,)38 b FN(j)5 b FB(,)39 b FN(k)s FB(,)g FN(l)e FP(2)f FN(I)7 b FB(,)38 b(b)l(e)-118 2167 y(such)i(that)f FN(T)321 2137 y FL(k)q(l)309 2189 y(ij)424 2167 y FO(=)546 2146 y(\026)529 2167 y FN(T)590 2137 y FL(lk)578 2189 y(j)s(i)652 2167 y FB(.)68 b(A)39 b(Wick)i(algebr)l(a)g(with)f(the)g(c)l(o)l (e\016cients)g FP(f)p FN(T)2236 2137 y FL(k)q(l)2224 2189 y(ij)2297 2167 y FP(g)-118 2267 y FO(\()p FB(se)l(e)33 b FO([124)n(]\))27 b FB(is)f(a)h FP(\003)p FB(-algebr)l(a)f FA(W)g FB(gener)l(ate)l(d)h(by)g(the)f(elements)g FN(x)1881 2279 y FL(i)1909 2267 y FB(,)h FN(x)2008 2237 y FM(\003)2008 2289 y FL(i)2073 2267 y FB(and)f(the)-118 2367 y(de\014ning)k(r)l (elations)611 2627 y FN(x)658 2593 y FM(\003)658 2648 y FL(i)696 2627 y FN(x)743 2639 y FL(j)802 2627 y FO(=)22 b FN(\016)926 2639 y FL(ij)985 2627 y FO(1)c(+)1191 2523 y FL(d)1149 2548 y Fy(X)1128 2727 y FL(k)q(;l)p FK(=1)1303 2627 y FN(T)1364 2593 y FL(k)q(l)1352 2648 y(ij)1426 2627 y FN(x)1473 2639 y FL(l)1499 2627 y FN(x)1546 2593 y FM(\003)1546 2648 y FL(k)1587 2627 y FN(:)6 2901 y FO(Denote)36 b(b)n(y)f FA(H)h FO(=)g FP(h)p FN(e)705 2913 y FK(1)742 2901 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(e)980 2913 y FL(d)1018 2901 y FP(i)36 b FO(a)f(\014nite-dimensional) 30 b(space)35 b(o)n(v)n(er)f FI(C)15 b FO(,)-118 3001 y(and)27 b(b)n(y)g FA(H)234 2971 y FM(\003)298 3001 y FO(its)g(formal)d(dual.)36 b FA(T)s FO(\()p FA(H)q FN(;)14 b FA(H)1167 2971 y FM(\003)1200 3001 y FO(\))27 b(will)e(denote)i(the)h (tensor)e(algebra)-118 3100 y(o)n(v)n(er)g FA(H)q FO(,)g FA(H)261 3070 y FM(\003)298 3100 y FO(.)37 b(Then)27 b FA(W)h FO(can)f(b)r(e)h(canonically)c(realized)g(as)297 3304 y FA(T)s FO(\()p FA(H)q FN(;)14 b FA(H)573 3270 y FM(\003)606 3304 y FO(\))638 3212 y Fy(.)q(D)756 3304 y FN(e)795 3270 y FM(\003)795 3325 y FL(i)852 3304 y FP(\012)k FN(e)974 3316 y FL(j)1027 3304 y FP(\000)g FN(\016)1147 3316 y FL(ij)1205 3304 y FO(1)g FP(\000)1348 3225 y Fy(X)1482 3304 y FN(T)1543 3270 y FL(k)q(l)1531 3325 y(ij)1604 3304 y FN(e)1643 3316 y FL(l)1687 3304 y FP(\012)g FN(e)1809 3270 y FM(\003)1809 3325 y FL(k)1849 3212 y Fy(E)1900 3304 y FN(:)-118 3512 y FO(In)39 b(this)f (realization,)g(the)h(subalgebra)d(generated)i(b)n(y)h FP(f)p FN(x)1801 3524 y FL(i)1829 3512 y FP(g)f FO(is)g(iden)n (ti\014ed)-118 3612 y(with)27 b FA(T)s FO(\()p FA(H)q FO(\).)6 3712 y(It)h(is)e(ob)n(vious)f(that)i(an)n(y)g(elemen)n(t)e(of) i FA(W)g FO(can)g(b)r(e)h(uniquely)d(represen)n(ted)-118 3811 y(as)38 b(a)g(p)r(olynomial)c(in)k(the)h(non-comm)n(uting)34 b(v)-5 b(ariables)36 b FN(a)1780 3823 y FL(i)1807 3811 y FO(,)42 b FN(a)1916 3781 y FM(\003)1916 3833 y FL(i)1954 3811 y FO(,)f(where)d(in)-118 3911 y(eac)n(h)32 b(monomial,)e(the)k(v) -5 b(ariables)30 b FN(a)1030 3923 y FL(i)1091 3911 y FO(are)i(placed)g(to)i(the)f(left)g(of)g FN(a)2050 3881 y FM(\003)2050 3933 y FL(j)2089 3911 y FO(.)54 b(Suc)n(h)p eop %%Page: 174 178 174 177 bop -118 -137 a FO(174)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(monomials)15 b(are)k(called)e(Wic)n(k)i (ordered)f(monomials,)f(and)i(they)h(form)f(a)g(basis)-118 196 y(in)27 b FA(W)p FO(.)6 296 y(When)34 b(studying)d(the)j(prop)r (erties)c(of)j FA(W)p FO(,)i(one)d(can)g(\014nd)i(the)f(follo)n(wing) -118 395 y(op)r(erators)25 b(useful:)11 561 y FN(T)20 b FO(:)28 b FA(H)18 b FP(\012)g FA(H)k FP(\000)-48 b(!)23 b FA(H)18 b FP(\012)g FA(H)q FN(;)96 b(T)12 b(e)998 573 y FL(k)1056 561 y FP(\012)18 b FN(e)1178 573 y FL(l)1226 561 y FO(=)1313 482 y Fy(X)1336 659 y FL(i;j)1447 561 y FN(T)1508 521 y FL(lj)1496 586 y(ik)1564 561 y FN(e)1603 573 y FL(i)1648 561 y FP(\012)g FN(e)1770 573 y FL(j)1805 561 y FN(;)-5 782 y(T)44 794 y FL(i)80 782 y FO(:)28 b FA(H)207 748 y FM(\012)p FL(n)325 782 y FP(\000)-48 b(!)23 b FA(H)524 748 y FM(\012)p FL(n)620 782 y FN(;)97 b(T)789 794 y FL(i)839 782 y FO(=)23 b(1)17 b FP(\012)i(\001)14 b(\001)g(\001)k(\012)g FO(1)927 824 y Fy(|)p 964 824 117 10 v 117 w({z)p 1155 824 V 117 w(})1064 901 y FL(i)p FM(\000)p FK(1)1342 782 y FP(\012)g FN(T)29 b FP(\012)j FO(1)18 b FP(\012)g(\001)c(\001)g(\001)k(\012)g FO(1)1600 824 y Fy(|)p 1637 824 V 117 w({z)p 1828 824 V 117 w(})1691 901 y FL(n)p FM(\000)p FL(i)p FM(\000)p FK(1)1983 782 y FN(;)-37 1020 y(R)26 1032 y FL(n)80 1020 y FO(:)28 b FA(H)207 985 y FM(\012)p FL(n)325 1020 y FP(\000)-48 b(!)23 b FA(H)524 985 y FM(\012)p FL(n)620 1020 y FN(;)97 b(R)803 1032 y FL(n)871 1020 y FO(=)23 b(1)18 b(+)g FN(T)1151 1032 y FK(1)1206 1020 y FO(+)g FN(T)1338 1032 y FK(1)1375 1020 y FN(T)1424 1032 y FK(2)1479 1020 y FO(+)g FP(\001)c(\001)g(\001)k FO(+)g FN(T)1809 1032 y FK(1)1846 1020 y FN(T)1895 1032 y FK(2)1945 1020 y FN(:)c(:)g(:)g(T)2105 1032 y FL(n)p FM(\000)p FK(1)2235 1020 y FN(:)-118 1173 y FO(These)27 b(op)r(erators)f(determine)f(the)j(Wic)n(k)f(ordering)e(on)i FA(W)p FO(.)-118 1316 y FQ(Prop)s(osition)j(54.)41 b FB(L)l(et)29 b FN(X)g FP(2)24 b FA(H)958 1286 y FM(\012)p FL(n)1053 1316 y FB(.)39 b(Then)-13 1547 y FN(e)26 1513 y FM(\003)26 1568 y FL(i)82 1547 y FP(\012)18 b FN(X)29 b FO(=)23 b FN(\026)p FO(\()p FN(e)472 1513 y FM(\003)472 1568 y FL(i)510 1547 y FO(\))14 b FN(R)619 1559 y FL(n)664 1547 y FN(X)25 b FO(+)18 b FN(\026)p FO(\()p FN(e)962 1513 y FM(\003)962 1568 y FL(i)1000 1547 y FO(\))1089 1444 y FL(d)1047 1468 y Fy(X)1046 1647 y FL(k)q FK(=1)1181 1547 y FN(T)1230 1559 y FK(1)1267 1547 y FN(T)1316 1559 y FK(2)1366 1547 y FN(:)c(:)g(:)g(T)1526 1559 y FL(n)1571 1547 y FO(\()p FN(X)25 b FP(\012)18 b FN(e)1819 1559 y FL(k)1859 1547 y FO(\))c FN(e)1944 1513 y FM(\003)1944 1568 y FL(k)1985 1547 y FN(;)118 b FO(\(2.68\))-118 1780 y FB(wher)l(e)30 b FN(\026)p FO(\()p FN(e)237 1750 y FM(\003)237 1802 y FL(i)275 1780 y FO(\))9 b(:)29 b FA(T)s FO(\()p FA(H)q FO(\))19 b FP(\000)-46 b(!)23 b FA(T)s FO(\()p FA(H)q FO(\))j FB(is)k(de\014ne)l(d)g(by)122 1934 y FN(\026)p FO(\()p FN(e)243 1900 y FM(\003)243 1954 y FL(i)281 1934 y FO(\)1)23 b(=)f(0)p FN(;)98 b(\026)p FO(\()p FN(e)749 1900 y FM(\003)749 1954 y FL(i)788 1934 y FO(\))14 b FN(e)873 1946 y FL(i)896 1954 y Fx(1)951 1934 y FP(\012)k(\001)c(\001)g(\001)k(\012)g FN(e)1271 1946 y FL(i)1294 1954 y Fv(n)1362 1934 y FO(=)23 b FN(\016)1487 1946 y FL(ii)1533 1954 y Fx(1)1570 1934 y FN(e)1609 1946 y FL(i)1632 1954 y Fx(2)1687 1934 y FP(\012)c(\001)14 b(\001)g(\001)k(\012)g FN(e)2008 1946 y FL(i)2031 1954 y Fv(n)2076 1934 y FN(:)6 2088 y FO(Let)31 b(us)g(note)g(that)g(the)g (de\014ning)e(relations)f(of)j(a)f(Wic)n(k)f(algebra)f(giv)n(e)h(a)-118 2187 y(comm)n(utation)24 b(rule)j(only)f(for)h(the)h(generators)e FN(x)1456 2157 y FM(\003)1456 2209 y FL(i)1494 2187 y FO(,)i FN(x)1592 2199 y FL(j)1628 2187 y FO(,)g(and)f(do)h(not)f (induce)-118 2287 y(an)n(y)21 b(relations)e(for)i FN(x)533 2299 y FL(i)561 2287 y FO(,)i FN(x)654 2299 y FL(j)690 2287 y FO(.)35 b(Ho)n(w)n(ev)n(er)20 b(suc)n(h)h(relations)e(are)i(v)n (ery)g(useful)g(in)g(the)-118 2386 y(study)27 b(of)g(represen)n (tations)d(of)j FA(W)p FO(.)37 b(It)27 b(is)f(ob)n(vious)f(that)i(an)n (y)f(relation)e(deter-)-118 2486 y(mines)29 b(a)i(t)n(w)n(o-sided)e (ideal.)45 b(So,)32 b(additional)27 b(relations)h(for)j(the)g (generators)-118 2586 y FN(x)-71 2598 y FL(i)-43 2586 y FO(,)d FN(x)55 2598 y FL(j)118 2586 y FO(can)f(b)r(e)h(describ)r(ed)e (in)h(the)h(terms)f(of)g(sp)r(ecial)e(ideals)g(in)i FA(W)p FO(.)-118 2729 y FQ(De\014nition)k(10.)40 b FB(A)e(Wick)h(ide)l(al)h (is)f(a)g(two-side)l(d)g(ide)l(al)h FA(I)f FP(\032)f FA(T)s FO(\()p FA(H)q FO(\))c FB(such)-118 2828 y(that)c FA(T)s FO(\()p FA(H)215 2798 y FM(\003)249 2828 y FO(\))p FA(I)24 b FP(\032)f FA(IT)s FO(\()p FA(H)628 2798 y FM(\003)662 2828 y FO(\))p FB(.)40 b(If)48 b FA(I)29 b FB(is)i(gener)l(ate)l(d)f (by)h(a)f(set)38 b FA(I)1742 2840 y FK(0)1803 2828 y FP(\032)23 b FA(H)1967 2798 y FM(\012)p FL(n)2062 2828 y FB(,)31 b(then)f FA(I)-118 2928 y FB(is)g(c)l(al)t(le)l(d)h(a)f(homo) l(gene)l(ous)h(Wick)f(ide)l(al)i(of)e(de)l(gr)l(e)l(e)g FN(n)p FB(.)6 3071 y FO(The)22 b(follo)n(wing)17 b(statemen)n(t)j(is)g (a)h(generalization)c(of)k(the)h(fact)f(established)-118 3170 y(in)27 b([124)n(])h(for)f FN(n)c FO(=)g(2.)-118 3313 y FQ(Prop)s(osition)30 b(55.)41 b FB(L)l(et)30 b FN(P)21 b FO(:)28 b FA(H)907 3283 y FM(\012)p FL(n)1027 3313 y FP(\000)-47 b(!)25 b FA(H)1229 3283 y FM(\012)p FL(n)1355 3313 y FB(b)l(e)30 b(a)h(pr)l(oje)l(ction.)43 b(Then)31 b FA(I)2205 3325 y FL(n)2274 3313 y FO(=)-118 3413 y FP(h)p FN(P)12 b FA(H)55 3383 y FM(\012)p FL(n)151 3413 y FP(i)30 b FB(is)g(a)g(Wick)g(ide)l(al)i(if)e(and)g(only)h(if)6 3513 y FO(1)p FN(:)f(R)164 3525 y FL(n)209 3513 y FN(P)35 b FO(=)22 b(0)p FN(;)6 3612 y FO(2)p FN(:)30 b FO([1)18 b FP(\012)g FO(\(1)g FP(\000)g FN(P)12 b FO(\)])i FN(T)625 3624 y FK(1)662 3612 y FN(T)711 3624 y FK(2)761 3612 y FN(:)g(:)g(:)g(T)921 3624 y FL(n)966 3612 y FO([)p FN(P)30 b FP(\012)18 b FO(1])23 b(=)f(0)p FN(:)6 3712 y FB(Mor)l(e)l(over,)44 b(if)58 b FN(T)51 b FB(satis\014es)40 b(the)f(br)l(aid)i(c)l(ondition)g FN(T)1727 3724 y FK(1)1764 3712 y FN(T)1813 3724 y FK(2)1849 3712 y FN(T)1898 3724 y FK(1)1976 3712 y FO(=)f FN(T)2130 3724 y FK(2)2167 3712 y FN(T)2216 3724 y FK(1)2253 3712 y FN(T)2302 3724 y FK(2)-118 3811 y FB(and)d FN(P)48 b FB(is)37 b(a)g(pr)l(oje)l(ction)g (to)g FO(k)n(er)13 b FN(R)1012 3823 y FL(n)1057 3811 y FB(,)38 b(then)f(c)l(ondition)g FO(2)f FB(is)h(automatic)l(al)t(ly) -118 3911 y(ful\014l)t(le)l(d.)p eop %%Page: 175 179 175 178 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(175)6 96 y(In)31 b(the)f(case)g(where)g FN(d)d FO(=)g(2,)k(condition)d(1)i(is)f(called)f(the)i(\\linear",)d (con-)-118 196 y(dition)f(2)h(is)f(called)g(\\quadratic".)-118 349 y FQ(3.)36 b FO(Let)28 b(us)f(consider)f(the)i(follo)n(wing)23 b(class)j(of)i(Wic)n(k)e(algebras:)112 611 y FN(x)159 576 y FM(\003)159 631 y FL(i)198 611 y FN(x)245 623 y FL(i)296 611 y FO(=)c(1)c(+)569 507 y FL(d)526 532 y Fy(X)529 709 y FL(j)s FK(=1)660 611 y FN(\013)713 623 y FL(ij)772 611 y FN(x)819 623 y FL(j)854 611 y FN(x)901 576 y FM(\003)901 631 y FL(j)940 611 y FN(;)97 b(x)1107 576 y FM(\003)1107 631 y FL(i)1145 611 y FN(x)1192 623 y FL(j)1251 611 y FO(=)22 b FN(\025)1386 623 y FL(ij)1445 611 y FN(q)1482 623 y FL(ij)1541 611 y FN(x)1588 623 y FL(j)1623 611 y FN(x)1670 576 y FM(\003)1670 631 y FL(i)1709 611 y FN(;)180 b(i)22 b FP(6)p FO(=)h FN(j;)-118 896 y FO(0)31 b FN(<)f(\013)103 908 y FL(ii)186 896 y FN(<)h FO(1,)i FN(q)417 908 y FL(ij)507 896 y FO(=)d FN(q)639 908 y FL(j)s(i)729 896 y FP(2)i FN(R)879 908 y FK(+)934 896 y FO(,)p 991 828 49 4 v 34 w FN(\025)1039 908 y FL(ij)1129 896 y FO(=)e FN(\025)1272 908 y FL(j)s(i)1331 896 y FO(,)k FP(j)p FN(\025)1459 908 y FL(ij)1518 896 y FP(j)d FO(=)g(1.)51 b(W)-7 b(e)33 b(denote)f(this)-118 996 y(class)26 b(b)n(y)h FA(U)p FO(\()p FN(A;)14 b FO(\003)p FN(;)g(Q)p FO(\))28 b(with)f FN(A)c FO(=)g(\()p FN(\013)1050 1008 y FL(ij)1109 996 y FO(\),)28 b(\003)22 b(=)h(\()p FN(\025)1440 1008 y FL(ij)1499 996 y FO(\),)28 b FN(Q)23 b FO(=)g(\()p FN(q)1828 1008 y FL(ij)1886 996 y FO(\).)-118 1130 y FB(R)l(emark)30 b(42.)42 b FO(1.)54 b(If)34 b(w)n(e)f(c)n(ho)r (ose)f FN(\013)999 1142 y FL(ii)1083 1130 y FO(=)g FN(\026)1230 1100 y FK(2)1267 1130 y FO(,)j FN(\013)1378 1142 y FL(ij)1469 1130 y FO(=)d FN(\026)1616 1100 y FK(2)1676 1130 y FP(\000)22 b FO(1)p FN(;)47 b(j)37 b(<)32 b(i)p FO(,)j FN(\013)2183 1142 y FL(ij)2274 1130 y FO(=)-118 1230 y(0)p FN(;)50 b(j)42 b(>)37 b(i)p FO(,)h FN(\025)313 1242 y FL(ij)410 1230 y FO(=)f(1,)h(and)e FN(q)822 1242 y FL(ij)918 1230 y FO(=)h FN(\026)p FO(,)i(then)e(the)g(obtained)e(algebra)e(is)i(the) -118 1330 y FN(\026)p FO(-CCR)27 b(algebra.)6 1430 y(2.)36 b(F)-7 b(or)25 b FN(\013)307 1442 y FL(ii)381 1430 y FO(=)e FN(q)506 1442 y FL(i)534 1430 y FO(,)j FN(\013)636 1442 y FL(ij)717 1430 y FO(=)d(0)p FN(;)39 b(i)22 b FP(6)p FO(=)h FN(j)5 b FO(,)26 b(the)g(algebra)c(b)r(ecomes)j(the)g FN(q)2071 1442 y FL(ij)2130 1430 y FO(-CCR)-118 1530 y(algebra)g(de\014ned)j(b)n(y)f(the)h(follo)n(wing)23 b(relations:)394 1715 y FN(x)441 1680 y FM(\003)441 1735 y FL(i)480 1715 y FN(x)527 1727 y FL(i)578 1715 y FO(=)g(1)18 b(+)g FN(q)846 1727 y FL(i)873 1715 y FN(x)920 1727 y FL(i)948 1715 y FN(x)995 1680 y FM(\003)995 1735 y FL(i)1034 1715 y FN(;)97 b(x)1201 1680 y FM(\003)1201 1735 y FL(i)1240 1715 y FN(x)1287 1727 y FL(j)1345 1715 y FO(=)23 b FN(\025)1481 1727 y FL(ij)1540 1715 y FN(q)1577 1727 y FL(ij)1635 1715 y FN(x)1682 1727 y FL(j)1718 1715 y FN(x)1765 1680 y FM(\003)1765 1735 y FL(i)1803 1715 y FN(:)6 1901 y FO(The)i(purp)r(ose)e(of)h(this)f(section)g(is)g(to)h(describ)r(e)f (algebras)e(from)h(this)i(class)-118 2000 y(that)g(ha)n(v)n(e)f(a)h (quadratic)e(ideal)g(of)j(maximal)19 b(p)r(ossible)j(rank,)i(and)g(to)g (classify)-118 2100 y(the)k FP(\003)p FO(-represen)n(tations)c(of)j (these)h(algebras)c(b)n(y)k(b)r(ounded)g(op)r(erators.)-118 2252 y FQ(4.)36 b FO(Let)28 b FA(U)23 b FO(=)g FA(U)p FO(\()p FN(A;)14 b FO(\003)p FN(;)g(Q)p FO(\).)37 b(Then)28 b(the)g(op)r(erator)e FN(T)38 b FO(has)27 b(the)h(form:)437 2437 y FN(T)12 b(e)537 2449 y FL(i)582 2437 y FP(\012)18 b FN(e)704 2449 y FL(i)754 2437 y FO(=)23 b FN(\013)895 2449 y FL(ii)960 2437 y FN(e)999 2449 y FL(i)1045 2437 y FP(\012)18 b FN(e)1167 2449 y FL(i)1194 2437 y FN(;)430 2562 y(T)12 b(e)530 2574 y FL(i)574 2562 y FP(\012)18 b FN(e)696 2574 y FL(j)754 2562 y FO(=)23 b FN(\013)895 2574 y FL(ij)967 2562 y FN(e)1006 2574 y FL(i)1052 2562 y FP(\012)18 b FN(e)1174 2574 y FL(j)1227 2562 y FO(+)g FN(\025)1358 2574 y FL(j)s(i)1417 2562 y FN(q)1454 2574 y FL(j)s(i)1526 2562 y FN(e)1565 2574 y FL(j)1618 2562 y FP(\012)g FN(e)1740 2574 y FL(i)1768 2562 y FN(:)-118 2746 y FO(Then)588 2992 y FA(H)g FP(\012)g FA(H)23 b FO(=)995 2888 y FL(d)950 2913 y Fy(M)959 3090 y FL(i)p FK(=1)1089 2992 y FA(H)1164 3004 y FL(i)1208 2992 y FP(\010)1353 2888 y FL(d)1308 2913 y Fy(M)1291 3090 y FL(i;j)s FK(=1)1463 2992 y FA(H)1538 3004 y FL(ij)1595 2992 y FN(;)370 3213 y FA(H)445 3225 y FL(i)494 3213 y FO(=)g FP(h)p FN(e)653 3225 y FL(i)699 3213 y FP(\012)18 b FN(e)821 3225 y FL(i)848 3213 y FP(i)p FN(;)97 b FA(H)1075 3225 y FL(ij)1155 3213 y FO(=)23 b FP(h)p FN(e)1314 3225 y FL(i)1360 3213 y FP(\012)18 b FN(e)1482 3225 y FL(j)1517 3213 y FN(;)c(e)1593 3225 y FL(j)1646 3213 y FP(\012)k FN(e)1768 3225 y FL(i)1795 3213 y FP(i)p FN(:)-118 3398 y FO(The)35 b(\\linear)d(condition")g (means)i(that)h FN(P)47 b FO(m)n(ust)34 b(b)r(e)i(a)e(pro)5 b(jection)33 b(to)i(the)-118 3497 y(subspace)24 b(generated)f(b)n(y)h (eigen)n(v)n(ectors)e(of)31 b FN(T)k FO(corresp)r(onding)22 b(to)i(the)h(eigen-)-118 3597 y(v)-5 b(alue)30 b FP(\000)p FO(1.)49 b(Since)30 b FN(\013)552 3609 y FL(ii)634 3597 y FP(6)p FO(=)f FP(\000)p FO(1,)j(the)g(rank)f(of)g FN(P)44 b FO(is)30 b(maximal)d(if)k(and)h(only)e(if)-118 3696 y(the)e(equalities)640 3881 y(\()p FN(\013)725 3893 y FL(ij)803 3881 y FO(+)18 b(1\)\()p FN(\013)1045 3893 y FL(j)s(i)1122 3881 y FO(+)g(1\))23 b(=)f FN(q)1426 3893 y FL(ij)1485 3881 y FN(q)1522 3893 y FL(j)s(i)2126 3881 y FO(\(2.69\))p eop %%Page: 176 180 176 179 bop -118 -137 a FO(176)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(hold)i(for)h(all)f FN(i)c FP(6)p FO(=)h FN(j)5 b FO(,)28 b(and)277 312 y FN(P)e FA(H)18 b FP(\012)g FA(H)23 b FO(=)717 220 y Fy(D)768 312 y FN(e)807 324 y FL(j)860 312 y FP(\012)18 b FN(e)982 324 y FL(i)1028 312 y FP(\000)1147 256 y FN(\025)1195 268 y FL(ij)1254 256 y FN(q)1291 268 y FL(ij)p 1121 293 255 4 v 1121 369 a FN(\013)1174 381 y FL(ij)1251 369 y FO(+)g(1)1399 312 y FN(e)1438 324 y FL(i)1484 312 y FP(\012)g FN(e)1606 324 y FL(j)1640 312 y FN(;)28 b(i)23 b(<)g(j)1870 220 y Fy(E)1920 312 y FN(:)-118 541 y FO(Denote)36 b(the)h(algebra)d FA(U)p FO(\()p FN(A;)14 b FO(\003)p FN(;)g(Q)p FO(\))36 b(for)g(whic)n(h)g(equations)e(\(2.69\))i(hold)f(b)n(y)-118 641 y FA(U)p FO(\()p FN(A;)14 b FO(\003\).)38 b(The)27 b(\\quadratic)e(condition")g(tak)n(es)i(the)h(form:)679 819 y FN(\013)732 831 y FL(ij)791 819 y FN(\013)844 831 y FL(j)s(i)925 819 y FO(=)23 b(0)p FN(;)179 b(i)23 b FP(6)p FO(=)g FN(j;)251 943 y(\013)304 955 y FL(ij)363 943 y FO(\()p FN(\013)448 955 y FL(ij)525 943 y FO(+)18 b(1)g FP(\000)g FN(\013)804 955 y FL(j)s(j)870 943 y FO(\))23 b(=)g(0)p FN(;)179 b(i)23 b FP(6)p FO(=)g FN(j;)45 1068 y(\013)98 1080 y FL(ik)162 1068 y FO(\()p FN(\013)247 1080 y FL(k)q(j)337 1068 y FP(\000)c FN(\013)474 1080 y FL(ij)532 1068 y FO(\))g(+)f FN(\013)719 1080 y FL(ij)777 1068 y FN(\013)830 1080 y FL(j)s(k)925 1068 y FO(=)23 b(0)p FN(;)179 b(i)23 b FP(6)p FO(=)g FN(j;)42 b(i)22 b FP(6)p FO(=)h FN(k)s(;)41 b(j)28 b FP(6)p FO(=)23 b FN(k)s(:)162 b FO(\(2.70\))-118 1245 y(It)32 b(is)e(con)n(v)n(enien)n (t)g(to)i(consider)d FP(f)p FN(\013)1006 1257 y FL(ij)1064 1245 y FP(g)j FO(as)f(a)g(function)g FN(\013)10 b FO(:)29 b FN(I)f FP(\002)21 b FN(I)37 b FP(\000)-49 b(!)30 b FI(R)38 b FO(and)-118 1345 y(denote)27 b(it)g(b)n(y)h FN(\013)401 1357 y FL(ij)482 1345 y FO(=)23 b FN(\013)p FO(\()p FN(i;)14 b(j)5 b FO(\).)-118 1475 y FB(R)l(emark)30 b(43.)42 b FO(If)31 b FN(\013)g FO(is)e(a)h(solution)d(of)j(system)f (\(2.70\))o(,)i(then)g(for)f(all)e FN(\031)i FP(2)e FN(S)2277 1487 y FL(d)2316 1475 y FO(,)-118 1575 y FN(\013)-65 1545 y FL(\031)-20 1575 y FO(\()p FN(i;)14 b(j)5 b FO(\))44 b(=)g FN(\013)p FO(\()p FN(\031)s FO(\()p FN(i)p FO(\))p FN(;)14 b(\031)s FO(\()p FN(j)5 b FO(\)\))42 b(is)d(also)f(a)i (solution,)h(and)f(if)45 b(^)-48 b FN(x)1813 1587 y FL(i)1885 1575 y FO(=)44 b FN(x)2041 1587 y FL(\031)2080 1595 y Fv(i)2111 1575 y FO(,)f(then)-118 1684 y(the)35 b(\\structure)f (constan)n(ts")f(for)40 b(^)-48 b FN(x)1034 1696 y FL(i)1097 1684 y FO(are)1246 1662 y(^)1243 1684 y FN(\025)1291 1696 y FL(ij)1385 1684 y FO(=)34 b FN(\013)1537 1653 y FL(\031)1583 1684 y FO(\()p FN(i;)14 b(j)5 b FO(\),)1815 1662 y(^)1812 1684 y FN(\025)1860 1696 y FL(ij)1953 1684 y FO(=)35 b FN(\025)2101 1653 y FL(\031)2147 1684 y FO(\()p FN(i;)14 b(j)5 b FO(\).)-118 1783 y(Consequen)n(tly)-7 b(,)25 b(it)h(will)e(su\016ce)i(to)h(describ)r(e)e(solutions)f(of)33 b(\(2.70\))26 b(up)h(to)f(the)-118 1883 y(action)g(of)i FN(S)277 1895 y FL(d)315 1883 y FO(.)-118 2045 y FQ(De\014nition)j(11.) 40 b FB(A)c(solution)h FN(\013)f FB(of)55 b FO(\(2.70\))35 b FB(is)i(c)l(anonic)l(al)g(if)55 b FN(\013)p FO(\()p FN(i;)14 b(j)5 b FO(\))35 b(=)f(0)-118 2144 y FB(for)c(al)t(l)h FN(i)23 b(<)g(j)5 b FB(.)-118 2306 y FQ(Prop)s(osition)30 b(56.)41 b FB(L)l(et)d FN(\013)h FB(b)l(e)g(an)g(arbitr)l(ary)h (solution)f(of)58 b FO(\(2.70\))o FB(.)66 b(Then)-118 2406 y(ther)l(e)30 b(exists)f FN(\031)d FP(2)e FN(S)519 2418 y FL(d)587 2406 y FB(such)30 b(that)g FN(\013)999 2376 y FL(\031)1074 2406 y FB(is)g(a)g(c)l(anonic)l(al)h(solution.)6 2568 y FO(W)-7 b(e)23 b(can)f(supp)r(ose)g(no)n(w)g(that)g FN(\013)993 2580 y FL(ij)1075 2568 y FO(=)g(0,)h FP(8)p FN(i)f(<)h(j)5 b FO(.)35 b(Then)22 b(\(2.70\))g(is)f(reduced)-118 2667 y(to)27 b(the)h(follo)n(wing:)426 2845 y FN(\013)479 2857 y FL(ij)537 2845 y FO(\()p FN(\013)622 2857 y FL(j)s(k)713 2845 y FP(\000)18 b FN(\013)849 2857 y FL(ik)913 2845 y FO(\))23 b(=)g(0)p FN(;)97 b FO(1)22 b FP(\024)h FN(k)j(<)c(j)28 b(<)23 b(i)g FP(\024)f FN(d;)325 2969 y(\013)378 2981 y FL(ij)437 2969 y FO(\(1)c(+)g FN(\013)665 2981 y FL(ij)742 2969 y FP(\000)g FN(\013)878 2981 y FL(j)913 2969 y FO(\))23 b(=)g(0)p FN(;)97 b FO(1)22 b FP(\024)h FN(j)28 b(<)22 b(i)h FP(\024)g FN(d;)857 3094 y(\013)910 3106 y FL(j)968 3094 y FO(=)g FN(\013)1109 3106 y FL(j)s(j)1175 3094 y FN(;)-118 3271 y FO(where)i(the)h(second)f(equation)g(means)f(only)g (the)i(fact)g(that)g(all)e(non-zero)g FN(\013)2280 3283 y FL(ij)-118 3371 y FO(are)i(equal)h(to)g(the)h(same)e(parameter)f FN(\013)1142 3383 y FL(j)1196 3371 y FP(\000)18 b FO(1)27 b(for)g(\014xed)h FN(j)k FO(and)c FN(i)22 b(>)h(j)5 b FO(.)-118 3533 y FQ(De\014nition)31 b(12.)40 b FB(A)30 b(c)l(anonic)l(al)h(solution)f(is)g(c)l(al)t(le)l(d)h(de)l(c)l(omp)l (osable)h(if)634 3710 y FN(I)e FO(=)22 b FN(I)823 3722 y FK(1)880 3710 y FP([)c FN(I)989 3722 y FK(2)1027 3710 y FN(;)99 b(I)1185 3722 y FK(1)1241 3710 y FP(\\)19 b FN(I)1351 3722 y FK(2)1412 3710 y FO(=)j FI(?)p FN(;)-118 3887 y FB(and)30 b(for)h(al)t(l)g FN(i)22 b FP(2)i FN(I)461 3899 y FK(1)498 3887 y FB(,)31 b FN(j)d FP(2)23 b FN(I)730 3899 y FK(2)768 3887 y FB(,)30 b FN(\013)876 3899 y FL(ij)957 3887 y FO(=)23 b FN(\013)1098 3899 y FL(j)s(i)1180 3887 y FO(=)f(0)p FB(.)p eop %%Page: 177 181 177 180 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(177)-118 96 y FB(R)l(emark)30 b(44.)42 b FO(If)19 b(a)f(canonical)d(solution)h(is)i(decomp)r(osable,) f(then)i(there)f(exists)-118 196 y FN(\031)30 b FP(2)d FN(S)92 208 y FL(d)161 196 y FO(suc)n(h)i(that)h FN(\013)585 166 y FL(\031)661 196 y FO(is)e(decomp)r(osable)f(and)j(canonical,)d (and)j(the)g(set)g(of)-118 296 y(indices)25 b(has)j(the)g(form)393 468 y FN(I)429 480 y FK(1)490 468 y FO(=)23 b FP(f)p FO(1)p FN(;)14 b(:)g(:)g(:)e(;)i(m)p FP(g)p FN(;)96 b(I)1115 480 y FK(2)1176 468 y FO(=)23 b FP(f)p FN(m)18 b FO(+)g(1)p FN(;)c(:)g(:)g(:)27 b(;)14 b(d)p FP(g)p FN(:)6 640 y FO(It)31 b(is)f(clear)e(that)j(if)f FN(\013)702 652 y FK(21)801 640 y FO(=)d FP(\001)14 b(\001)g(\001)28 b FO(=)g FN(\013)1164 652 y FL(d)p FK(1)1264 640 y FO(=)g FN(\013)1410 652 y FK(1)1467 640 y FP(\000)20 b FO(1,)31 b(then)h FN(\013)f FO(is)e(indecom-)-118 740 y(p)r(osable.)-118 897 y FQ(Prop)s(osition)h(57.)41 b FB(L)l(et)23 b FN(\013)i FB(b)l(e)f(a)g(c)l(anonic)l(al)i(solution.)37 b(Then)25 b(it)f(is)g(inde)l(c)l(om-)-118 997 y(p)l(osable)31 b(if)g(and)f(only)g (if)h FN(\013)722 1009 y FK(21)816 997 y FO(=)22 b FP(\001)14 b(\001)g(\001)23 b FO(=)g FN(\013)1164 1009 y FL(d)p FK(1)1259 997 y FO(=)f FN(\013)1399 1009 y FK(1)1455 997 y FP(\000)c FO(1)p FB(.)6 1154 y FO(Let)28 b FN(\013)h FO(b)r(e)f(a)f(canonical)e(solution,)g FN(A)f FO(=)f(\()p FN(\013)1380 1166 y FL(ij)1438 1154 y FO(\).)38 b(It)28 b(follo)n(ws)d(from)h(Prop)r(o-)-118 1254 y(sition)32 b(57)i(and)g(Remark)e(44,)k(that)e(w)n(e)h(can)f(supp)r(ose)g(that,)i (for)e(an)n(y)g(\014xed)-118 1354 y FN(j)5 b FO(,)40 b(all)c(non-zero)g FN(\013)510 1366 y FL(ij)568 1354 y FO(,)41 b FN(i)e(>)h(j)5 b FO(,)40 b(are)d(placed)g(b)r(efore)g(all)e (zero)r(es.)67 b(Consider)-122 1445 y FN(~)-118 1467 y(k)45 b FO(=)e(\()p FN(k)153 1479 y FK(1)190 1467 y FN(;)14 b(:)g(:)g(:)28 b(;)14 b(k)432 1479 y FL(d)p FM(\000)p FK(1)556 1467 y FO(\),)42 b(where)d FN(i)j FP(\024)g FN(k)1126 1479 y FL(i)1197 1467 y FP(\024)g FN(d)e FO(are)e(natural)g (n)n(um)n(b)r(ers)f(con-)-118 1567 y(structed)f(as)g(follo)n(ws:)52 b(if,)38 b(for)e(a)g(\014xed)h FN(j)k FO(and)c(all)d FN(i)j(>)h(j)5 b FO(,)39 b FN(\013)1875 1579 y FL(ij)1971 1567 y FO(=)f(0,)g(then)-118 1666 y FN(k)-75 1678 y FL(j)-17 1666 y FO(=)23 b(0;)i(otherwise)e FN(k)571 1678 y FL(j)631 1666 y FO(is)h(the)h(greatest)e(n)n(um)n(b)r(er)h FN(l)i FO(suc)n(h)f(that)g FN(\013)1929 1678 y FL(lj)2008 1666 y FO(=)e FN(\013)2149 1678 y FL(j)2197 1666 y FP(\000)13 b FO(1.)-118 1780 y(The)28 b(c)n(haracteristic)23 b(prop)r(ert)n(y)k (of)993 1758 y FN(~)997 1780 y(k)k FO(is)26 b(the)i(follo)n(wing.)-118 1937 y FQ(Prop)s(osition)i(58.)41 b FB(If)47 b FN(i)23 b(>)g(j)34 b FB(and)d FN(i)22 b FP(\024)h FN(k)1218 1949 y FL(j)1253 1937 y FB(,)30 b(then)g FN(k)1536 1949 y FL(i)1587 1937 y FP(\024)22 b FN(k)1717 1949 y FL(j)1753 1937 y FB(.)6 2108 y FO(Con)n(v)n(ersely)-7 b(,)20 b(let)547 2087 y FN(~)551 2108 y(k)25 b FO(b)r(e)c(a)g(v)n(ector)f(with)h(the)g (c)n(haracteristic)c(prop)r(ert)n(y)-7 b(,)22 b(and)-118 2208 y FN(A)33 b FO(=)g(\()p FN(\013)160 2220 y FL(ij)219 2208 y FO(\))h(b)r(e)g(a)f(matrix)e(suc)n(h)i(that)h FN(\013)1187 2220 y FL(ii)1271 2208 y FO(=)f FN(\013)1422 2220 y FL(i)1450 2208 y FO(,)i FN(\013)1561 2220 y FL(ij)1653 2208 y FO(=)d(0,)j FN(i)e(<)f(j)5 b FO(.)55 b(Then,)-118 2308 y FN(\013)-65 2320 y FL(ij)32 2308 y FO(=)38 b(0)f FN(;)14 b FP(8)p FN(i)37 b(>)h(j)k FO(if)37 b FN(k)672 2320 y FL(j)745 2308 y FO(=)i FN(j)5 b FO(;)41 b(otherwise)35 b FN(\013)1384 2320 y FL(lj)1479 2308 y FO(=)j FN(\013)1635 2320 y FL(j)1695 2308 y FP(\000)24 b FO(1,)39 b FN(j)44 b(<)38 b(l)i FP(\024)e FN(k)2280 2320 y FL(j)2316 2308 y FO(,)-118 2407 y FN(\013)-65 2419 y FL(lj)30 2407 y FO(=)h(0,)g FN(l)i(>)e(k)451 2419 y FL(j)486 2407 y FO(.)66 b(Then)37 b(it)g(is)f(easy)g(to)h(v)n(erify)f(that)h FN(A)h FO(is)e(a)h(matrix)e(of)-118 2521 y(a)e(canonical)e(solution.)53 b(W)-7 b(e)34 b(will)d(denote)i(suc)n(h)h(a)f(matrix)e(b)n(y)j FN(A)p FO(\()2035 2499 y FN(~)2039 2521 y(k)s FO(\).)56 b(The)-118 2620 y(follo)n(wing)23 b(statemen)n(t)k(has)g(b)r(een)h(pro) n(v)n(ed.)-118 2778 y FQ(Theorem)i(41.)41 b FB(L)l(et)22 b FN(\013)h FB(b)l(e)g(a)g(solution)g(of)h(system)29 b FO(\(2.70\))o FB(.)37 b(Then)23 b(ther)l(e)g(exist)-118 2888 y FN(\031)j FP(2)e FN(S)85 2900 y FL(d)146 2888 y FB(and)305 2866 y FN(~)309 2888 y(k)i FB(having)e(the)f(char)l (acteristic)i(pr)l(op)l(erty)f(so)f(that)g FO(\()p FN(\013)1939 2858 y FL(\031)1939 2910 y(ij)1997 2888 y FO(\))h(=)e FN(A)p FO(\()2230 2866 y FN(~)2234 2888 y(k)t FO(\))p FB(.)-118 3011 y(Conversely,)36 b(for)f(any)630 2989 y FN(~)635 3011 y(k)h FB(with)e(char)l(acteristic)h(pr)l(op)l(erty,)h FN(A)30 b FO(=)f FN(A)p FO(\()2048 2989 y FN(~)2052 3011 y(k)t FO(\))34 b FB(gives)-118 3110 y(a)c(solution.)-118 3268 y FQ(5.)50 b FO(No)n(w)31 b(w)n(e)h(describ)r(e)f(irreducible)d (represen)n(tations)i(of)i(the)g(algebras)d(ob-)-118 3368 y(tained.)80 b(Let)43 b FN(A)48 b FO(=)f FN(A)p FO(\()688 3346 y FN(~)692 3368 y(k)s FO(\),)g FA(U)h FO(=)f FA(U)p FO(\()p FN(A;)14 b FO(\003\).)82 b(Then)43 b FA(U)f FO(has)g(the)h(largest)-118 3467 y(quadratic)25 b(ideal)h(generated)g(b)n(y)475 3639 y FN(X)544 3651 y FL(ij)625 3639 y FO(=)d FN(x)760 3651 y FL(j)795 3639 y FN(x)842 3651 y FL(i)889 3639 y FP(\000)18 b FN(\025)1020 3651 y FL(ij)1079 3639 y FN(q)1116 3651 y FL(ij)1188 3639 y FN(x)1235 3651 y FL(i)1263 3639 y FN(x)1310 3651 y FL(j)1346 3639 y FN(;)180 b(i)22 b(<)h(j:)-118 3811 y FQ(Prop)s(osition)30 b(59.)41 b FB(L)l(et)31 b FN(\031)s FO(\()p FP(\001)p FO(\))h FB(b)l(e)g(a)g(b)l(ounde)l(d)g(r)l(epr)l (esentation)g(of)50 b FA(U)p FO(\()p FN(A;)14 b FO(\003\))p FB(.)-118 3911 y(Then)30 b FN(\031)s FO(\()p FN(X)249 3923 y FL(ij)308 3911 y FO(\))23 b(=)g(0)p FB(.)p eop %%Page: 178 182 178 181 bop -118 -137 a FO(178)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(R)l(emark)30 b(45.)42 b FO(The)23 b(pro)r(of)f(basically)c(coincides)h(with)j(the)h(pro)r(of)f(of)g(the)h (anal-)-118 196 y(ogous)g(fact)i(for)g(the)g(t)n(wisted)f(comm)n (utation)d(relations)h(\(see)j(Section)49 b(2.4.2\))-118 296 y(giv)n(en)26 b(b)n(y)h(\003)c(=)f(1,)28 b FN(A)23 b FO(=)g FN(A)p FO(\()p FN(d;)14 b(:)g(:)g(:)g(;)g(d)p FO(\),)28 b(and)f FN(\013)1310 308 y FL(j)1369 296 y FO(=)22 b FN(\026)1506 266 y FK(2)1544 296 y FO(.)6 436 y(This)c(means)f(that)i(in)f(order)f(to)h(describ)r(e)f(irreducible)e (represen)n(tations)h(of)-118 536 y FA(U)p FO(,)27 b(it)e(is)g (necessary)f(to)i(describ)r(e)f(families)d(of)k(op)r(erators)e FP(f)p FN(X)1811 548 y FL(i)1838 536 y FN(;)k(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d)p FP(g)-118 635 y FO(suc)n(h)27 b(that)270 826 y FN(X)346 792 y FM(\003)339 846 y FL(i)384 826 y FN(X)453 838 y FL(i)503 826 y FO(=)c(1)18 b(+)g FN(\013)787 838 y FL(i)815 826 y FN(X)884 838 y FL(i)911 826 y FN(X)987 792 y FM(\003)980 846 y FL(i)1043 826 y FO(+)1200 747 y Fy(X)1126 926 y FL(j)s(<i;k)1286 934 y Fv(j)1318 926 y FM(\025)p FL(i)1394 826 y FO(\()p FN(\013)1479 838 y FL(j)1533 826 y FP(\000)g FO(1\))c FN(X)1773 838 y FL(j)1807 826 y FN(X)1883 792 y FM(\003)1876 846 y FL(j)1920 826 y FN(;)263 1053 y(X)339 1019 y FM(\003)332 1074 y FL(i)377 1053 y FN(X)446 1065 y FL(j)503 1053 y FO(=)23 b FN(\025)639 1065 y FL(ij)698 1053 y FN(q)735 1065 y FL(ij)793 1053 y FN(X)862 1065 y FL(j)897 1053 y FN(X)973 1019 y FM(\003)966 1074 y FL(i)1011 1053 y FN(;)180 b(i)22 b(<)h(j;)280 1178 y(X)349 1190 y FL(j)384 1178 y FN(X)453 1190 y FL(i)503 1178 y FO(=)g FN(\025)639 1190 y FL(ij)698 1178 y FN(q)735 1190 y FL(ij)793 1178 y FN(X)862 1190 y FL(i)890 1178 y FN(X)959 1190 y FL(j)994 1178 y FN(;)180 b(i)22 b(<)h(j;)385 1386 y(q)425 1352 y FK(2)422 1407 y FL(ij)503 1386 y FO(=)591 1244 y Fy(\()658 1330 y FN(\013)711 1342 y FL(i)739 1330 y FN(;)83 b(i)22 b(<)h(j;)28 b(k)1112 1342 y FL(i)1163 1330 y FP(\025)23 b FN(j;)658 1449 y FO(1)p FN(;)122 b FO(otherwise)n FN(:)2126 1386 y FO(\(2.71\))-118 1651 y(Let)25 b FN(X)104 1621 y FM(\003)97 1673 y FL(i)165 1651 y FO(=)d FN(U)309 1663 y FL(i)337 1651 y FN(C)396 1663 y FL(i)449 1651 y FO(b)r(e)j(the)g(p)r (olar)e(decomp)r(osition.)33 b(Then)25 b(system)f(\(2.71\))g(can)-118 1751 y(b)r(e)k(rewritten)e(in)h(an)g(equiv)-5 b(alen)n(t)26 b(form:)404 1941 y FQ(C)p FN(U)539 1907 y FM(\003)530 1962 y FL(i)600 1941 y FO(=)c FN(U)753 1907 y FM(\003)744 1962 y FL(i)791 1941 y FQ(F)851 1953 y FL(i)879 1941 y FO(\()p FQ(C)p FO(\))p FN(;)97 b FQ(C)23 b FO(=)g(\()p FN(C)1409 1907 y FK(2)1403 1962 y(1)1447 1941 y FN(;)14 b(:)g(:)g(:)27 b(;)14 b(C)1710 1907 y FK(2)1704 1962 y FL(d)1748 1941 y FO(\))p FN(;)-7 2079 y FO([)p FN(C)75 2091 y FL(i)104 2079 y FN(;)g(C)200 2091 y FL(j)235 2079 y FO(])23 b(=)g(0)p FN(;)96 b(U)587 2091 y FL(i)614 2079 y FN(U)671 2091 y FL(j)729 2079 y FO(=)p 817 2011 49 4 v 23 w FN(\025)865 2091 y FL(ij)924 2079 y FN(U)981 2091 y FL(j)1015 2079 y FN(U)1072 2091 y FL(i)1100 2079 y FN(;)g(U)1276 2091 y FL(i)1304 2079 y FN(U)1370 2044 y FM(\003)1361 2099 y FL(j)1431 2079 y FO(=)22 b FN(\025)1566 2091 y FL(ij)1625 2079 y FN(U)1691 2044 y FM(\003)1682 2099 y FL(j)1729 2079 y FN(U)1786 2091 y FL(i)1814 2079 y FN(;)179 b(i)23 b(<)g(j;)856 2213 y FQ(F)916 2225 y FL(i)944 2213 y FO(\()p FN(x)1023 2225 y FK(1)1061 2213 y FN(;)14 b(:)g(:)g(:)g(;)g(x)1293 2225 y FL(d)1332 2213 y FO(\))157 2348 y(=)245 2281 y Fy(\000)283 2348 y FN(x)330 2360 y FK(1)368 2348 y FN(;)g(:)g(:)g(:)f(;)h(x)599 2360 y FL(i)p FM(\000)p FK(1)712 2348 y FN(;)g(F)802 2360 y FL(i)830 2348 y FO(\()p FN(x)909 2360 y FK(1)947 2348 y FN(;)g(:)g(:)g(:)g(;)g(x)1179 2360 y FL(d)1218 2348 y FO(\))p FN(;)g(q)1327 2314 y FK(2)1324 2369 y FL(ii)p FK(+1)1459 2348 y FN(x)1506 2360 y FL(i)p FK(+1)1618 2348 y FN(;)g(:)g(:)g(:)g(;)g(q)1843 2314 y FK(2)1840 2369 y FL(id)1902 2348 y FN(x)1949 2360 y FL(d)1988 2281 y Fy(\001)2026 2348 y FN(;)-118 2539 y FO(where)-43 2729 y FN(F)10 2741 y FL(i)38 2729 y FO(\()p FN(x)117 2741 y FK(1)155 2729 y FN(;)g(:)g(:)g(:)g(;)g(x)387 2741 y FL(d)426 2729 y FO(\))23 b(=)g(1)18 b(+)g FN(\013)765 2741 y FL(i)793 2729 y FN(x)840 2741 y FL(i)886 2729 y FO(+)1043 2650 y Fy(X)969 2829 y FL(j)s(<i;k)1129 2837 y Fv(j)1161 2829 y FM(\025)p FL(i)1237 2729 y FO(\()p FN(\013)1322 2741 y FL(j)1376 2729 y FP(\000)g FO(1\))c FN(x)1594 2741 y FL(j)1629 2729 y FN(;)180 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(d:)-118 3010 y FO(Using)27 b(the)i(tec)n(hnique)e(of)h(dynamical)d(systems,)i(w)n(e)h(can)g (reduce)g(the)h(prob-)-118 3110 y(lem)g(of)h(describing)e(irreducible)f (represen)n(tations)g(of)k FA(U)p FO(\()p FN(A;)14 b FO(\003\))31 b(to)f(an)h(anal-)-118 3209 y(ogous)c(problem)g(for)i (\014nite)f(families)d(of)30 b(the)f(unitary)f(op)r(erators)f FP(f)p FN(U)2089 3221 y FL(i)2116 3209 y FP(g)h FO(that)-118 3309 y(satisfy)j(the)i(relations)d FN(U)694 3321 y FL(i)721 3309 y FN(U)778 3321 y FL(j)845 3309 y FO(=)h FN(\025)989 3321 y FL(ij)1048 3309 y FN(U)1105 3321 y FL(j)1139 3309 y FN(U)1196 3321 y FL(i)1224 3309 y FO(,)j FN(i)d(<)g(j)5 b FO(.)53 b(First,)33 b(in)n(tro)r(duce)e(some)-118 3409 y(notations:)k(let)27 b FN(D)r FO(\()p FN(\026)p FO(\))h(denote)g(an)f (op)r(erator)f(in)h FN(l)1452 3421 y FK(2)1489 3409 y FO(\()p FN(N)9 b FO(\))28 b(giv)n(en)e(b)n(y)567 3599 y FN(D)r FO(\()p FN(\026)p FO(\))p FN(e)791 3611 y FL(n)859 3599 y FO(=)d FN(\026)997 3565 y FL(n)p FM(\000)p FK(1)1127 3599 y FN(e)1166 3611 y FL(n)1211 3599 y FN(;)180 b(n)23 b FP(2)g FI(N)t FN(;)558 3807 y(D)r FO(\()p FN(j;)14 b(k)775 3819 y FL(i)804 3807 y FO(\))23 b(=)947 3665 y Fy(\()1014 3751 y FO(1)p FN(;)265 b(j)28 b(>)22 b(k)1536 3763 y FL(i)1564 3751 y FN(;)1014 3870 y(D)r FO(\()p FN(\013)1170 3882 y FL(j)1205 3870 y FO(\))p FN(;)84 b(j)28 b FP(\024)22 b FN(k)1536 3882 y FL(i)1564 3870 y FN(;)p eop %%Page: 179 183 179 182 bop -118 -137 a FJ(2.4.)36 b(Man)n(y-dimensional)22 b(dynamical)i(systems)795 b FO(179)-118 96 y(and)22 b(let)g FN(S)28 b FO(stand)22 b(for)h(the)g(unilateral)c(shift.)35 b(Then)22 b FN(D)r FO(\()p FN(f)1656 108 y FL(j)1691 96 y FO(\))14 b FN(e)1776 108 y FL(n)1844 96 y FO(=)23 b FN(f)1982 61 y FL(n)p FM(\000)p FK(1)1973 120 y FL(j)2112 96 y FO(\(0\))14 b FN(e)2271 108 y FL(n)2316 96 y FO(,)-118 196 y(where)27 b FN(f)163 208 y FL(j)198 196 y FO(\()p FN(x)p FO(\))d(=)e(1)c(+)g FN(\013)616 208 y FL(j)651 196 y FN(x)p FO(,)29 b(and)e FN(f)961 166 y FL(n)1006 196 y FO(\()p FP(\001)p FO(\))h(denotes)f(the)h FN(n)p FO(-th)g(iteration)d(of)i FN(f)9 b FO(.)6 313 y(Let)28 b(1)23 b FP(\024)f FN(i)336 325 y FK(1)396 313 y FN(<)h FP(\001)14 b(\001)g(\001)23 b FN(<)f(i)720 325 y FL(l)769 313 y FP(\024)g FN(d)28 b FO(b)r(e)g(natural)e(n)n(um)n(b)r(ers)g(suc)n (h)h(that)473 530 y FN(k)516 542 y FL(i)539 550 y Fv(j)593 530 y FO(+)18 b(1)23 b FP(\024)g FN(i)858 542 y FL(j)s FK(+1)976 530 y FN(;)180 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(l)20 b FP(\000)e FO(1)p FN(:)-118 767 y FO(Fix)24 b(suc)n(h)h(a)g(family)-7 b(.)33 b(Denote)25 b(\010)e(=)1011 704 y Fy(S)1080 725 y FL(l)1080 792 y(j)s FK(=1)1199 767 y FP(f)p FN(i)1270 779 y FL(j)1317 767 y FO(+)13 b(1)p FN(;)h(:)g(:)g(:)27 b(;)14 b(k)1678 779 y FL(i)1701 787 y Fv(j)1737 767 y FP(g)p FO(.)36 b(Construct)24 b(the)-118 866 y(follo)n(wing)f(irreducible)h(represen)n(tation)h(for)i (a)h(\014xed)f(family)e(of)i FP(f)p FN(i)2002 878 y FK(1)2053 866 y FN(:)14 b(:)g(:)27 b(;)14 b(i)2243 878 y FL(l)2268 866 y FP(g)p FO(:)-20 1083 y FN(C)39 1095 y FL(j)98 1083 y FO(=)22 b FN(U)242 1095 y FL(j)300 1083 y FO(=)h(0)p FN(;)179 b FP(8)p FN(j)27 b FP(2)d FO(\010)p FN(;)-28 1301 y(C)37 1267 y FK(2)31 1322 y FL(j)98 1301 y FO(=)249 1193 y FL(j)s FM(\000)p FK(1)244 1222 y Fy(O)185 1404 y FL(i)p FK(=1)p FL(;i)7 b(=)-41 b FM(2)q FK(\010)442 1301 y FN(D)r FO(\()p FN(j;)14 b(k)659 1313 y FL(i)687 1301 y FO(\))19 b FP(\012)f FN(D)r FO(\()p FN(f)965 1313 y FL(j)1000 1301 y FO(\))h FP(\012)f FN(I)25 b FP(\012)18 b(\001)c(\001)g(\001)19 b(\012)f FN(I)7 b(;)180 b(j)28 b FP(6)p FO(=)22 b FN(i)1901 1313 y FL(k)1942 1301 y FN(;)-29 1611 y(U)37 1577 y FM(\003)28 1632 y FL(j)98 1611 y FO(=)249 1504 y FL(j)s FM(\000)p FK(1)244 1532 y Fy(O)185 1711 y FL(i)p FK(=1)p FL(;i)p FM(62)p FK(\010)442 1611 y FN(D)r FO(\()p FN(\025)593 1623 y FL(ij)652 1611 y FO(\))d FP(\012)f FN(S)23 b FP(\012)18 b FN(I)26 b FP(\012)18 b(\001)c(\001)g(\001)k(\012)g FN(I)7 b(;)180 b(j)28 b FP(6)p FO(=)23 b FN(i)1711 1623 y FL(k)1751 1611 y FN(;)-46 1851 y(U)20 1817 y FM(\003)11 1872 y FL(i)34 1881 y Fv(k)98 1851 y FO(=)258 1772 y Fy(O)185 1951 y FL(i<i)283 1960 y Fv(k)320 1951 y FL(;i)p FM(62)p FK(\010)470 1851 y FN(D)r FO(\()p FN(\025)621 1863 y FL(ii)667 1872 y Fv(k)709 1851 y FO(\))18 b FP(\012)915 1772 y Fy(O)843 1951 y FL(i>i)941 1960 y Fv(k)978 1951 y FL(;i)p FM(62)p FK(\010)1127 1851 y FN(D)r FO(\()p 1230 1784 49 4 v FN(\025)1279 1863 y FL(ii)1325 1872 y Fv(k)1366 1851 y FO(\))h FP(\012)1514 1830 y FO(^)1500 1851 y FN(U)1566 1817 y FM(\003)1557 1872 y FL(i)1580 1881 y Fv(k)1620 1851 y FN(;)180 b(k)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(l)r(;)-49 2114 y(C)16 2079 y FK(2)10 2134 y FL(i)33 2143 y Fv(k)98 2114 y FO(=)304 2057 y(1)p 195 2094 261 4 v 195 2170 a(1)k FP(\000)g FN(\013)391 2182 y FL(i)414 2191 y Fv(k)552 2035 y Fy(O)479 2213 y FL(i<i)577 2222 y Fv(k)614 2213 y FL(;i)p FM(62)p FK(\010)763 2114 y FN(D)r FO(\()p FN(i;)c(k)975 2126 y FL(i)998 2135 y Fv(k)1039 2114 y FO(\))19 b FP(\012)f FN(I)26 b FP(\012)18 b(\001)c(\001)g(\001)k(\012)g FN(I)7 b(;)97 b(k)26 b FO(=)c(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(l)r(;)-118 2437 y FO(where)30 b FP(f)181 2416 y FO(^)167 2437 y FN(U)224 2449 y FL(i)247 2458 y Fv(k)287 2437 y FP(g)g FO(is)f(an)h(irreducible)d(family)g(of)j(unitary)f(op)r (erators)f(satisfying)-118 2546 y(the)g(relations)377 2525 y(^)362 2546 y FN(U)419 2558 y FL(i)461 2525 y FO(^)447 2546 y FN(U)504 2558 y FL(j)561 2546 y FO(=)23 b FN(\025)697 2558 y FL(ij)770 2525 y FO(^)756 2546 y FN(U)813 2558 y FL(j)862 2525 y FO(^)848 2546 y FN(U)905 2558 y FL(j)939 2546 y FO(.)-118 2763 y FQ(Theorem)30 b(42.)41 b FB(A)n(l)t(l)36 b(irr)l(e)l(ducible)i(r)l(epr)l(esentations)e(of)h(the)g(algebr)l(a)g FA(U)g FB(c)l(an)-118 2863 y(b)l(e)g(obtaine)l(d)i(in)e(the)g(way)i (describ)l(e)l(d)f(ab)l(ove)6 b FO(;)43 b FB(mor)l(e)l(over,)d(two)e(r) l(epr)l(esenta-)-118 2963 y(tions)f(ar)l(e)h(unitarily)g(e)l(quivalent) f(if)h(and)g(only)g(if)g(they)g(c)l(orr)l(esp)l(ond)g(to)f(the)-118 3062 y(same)32 b(family)h FP(f)p FN(i)421 3074 y FK(1)457 3062 y FN(;)14 b(:)g(:)g(:)g(;)g(i)671 3074 y FL(l)696 3062 y FP(g)p FB(,)31 b(and)h(the)g(c)l(orr)l(esp)l(onding)g(unitary)g (families)h(ar)l(e)-118 3162 y(unitarily)d(e)l(quivalent.)-118 3380 y(R)l(emark)g(46.)42 b FO(1.)36 b(If)25 b(at)f(least)f(one)h(of)g (the)h FN(\025)1239 3392 y FL(ij)1322 3380 y FO(is)e(not)i(a)f(ro)r(ot) f(of)h(1,)h(then)g(there)-118 3479 y(exists)h(a)h(represen)n(tation)e (that)j(is)f(not)g(of)h(t)n(yp)r(e)f(one.)6 3596 y(2.)49 b(If)32 b(all)d FN(\025)374 3608 y FL(ij)464 3596 y FO(are)i(ro)r(ots)f (of)i(1,)g(then)g(the)g(problem)d(of)j(classi\014cation)27 b(of)-118 3696 y(families)21 b FP(f)p FN(U)281 3708 y FL(i)308 3696 y FP(g)26 b FO(can)f(b)r(e)h(reduced)f(to)h(the)g(case)e (where)h FN(\025)1644 3656 y FL(q)1644 3719 y(ij)1726 3696 y FO(=)e(1;)j(here)f FN(q)h FO(=)d FN(p)2276 3666 y FL(m)-118 3795 y FO(for)h(some)e(prime)g FN(p)p FO(.)36 b(In)24 b(this)g(case,)g(the)h(families)20 b FP(f)p FN(U)1536 3807 y FL(i)1563 3795 y FP(g)j FO(can)h(b)r(e)h(describ)r(ed)e(b)n(y) -118 3895 y(a)k(simple)e(reduction)h(algorithm.)p eop %%Page: 180 184 180 183 bop -118 -137 a FO(180)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FG(2.5)112 b(On)38 b(represen)m(tations)g(of)g (some)g(n)m(uclear)g(algebras)-118 278 y FQ(2.5.1)94 b(Comm)m(utativ)m(e)29 b(mo)s(dels)-118 432 y FO(In)40 b(previous)e(sections,)j(considering)36 b(represen)n(tations)h(of)j(op) r(erator)e(rela-)-118 531 y(tions,)g(w)n(e)f(ha)n(v)n(e)f(seen)h(that)g (the)h(complete)d(unitary)g(description)g(of)i(irre-)-118 631 y(ducible)28 b(represen)n(tations)e(is)i(p)r(ossible)f(only)h(if)h (the)h(underlying)c(dynamical)-118 730 y(system)i(has)h(a)g(measurable) d(section.)41 b(No)n(w)29 b(w)n(e)g(will)e(lo)r(ok)h(at)h(what)g(can)g (b)r(e)-118 830 y(said)23 b(ab)r(out)i(the)g(relation)c(in)j(the)h (case)f(where)g(the)i(dynamical)21 b(system)i(do)r(es)-118 930 y(not)g(necessarily)d(ha)n(v)n(e)i(a)h(measurable)d(section.)34 b(Of)23 b(course,)g(one)g(should)f(not)-118 1029 y(exp)r(ect)i(a)f (complete)f(unitary)g(description;)h(ho)n(w)n(ev)n(er,)f(w)n(e)i(will)d (see)i(that)h(in)f(a)-118 1129 y(certain)i(sense,)i(a)g(uniform)e (realization)e(\(comm)n(utativ)n(e)h(mo)r(del\))h(of)i(all)e(rep-)-118 1229 y(resen)n(tations)31 b(as)i(op)r(erators)e(of)j(m)n(ultiplication) 28 b(and)33 b(w)n(eigh)n(ted)f(shifts)h(can)-118 1328 y(b)r(e)28 b(constructed.)6 1428 y(Instead)h(of)f(considering)d(the)k (relation)d(of)j(the)f(form)g FN(X)7 b(X)1870 1398 y FM(\003)1931 1428 y FO(=)24 b FN(F)12 b FO(\()p FN(X)2193 1398 y FM(\003)2231 1428 y FN(X)7 b FO(\))-118 1527 y(or)33 b(its)g(m)n(ulti-dimensio)o(nal)27 b(v)n(ersion,)33 b(w)n(e)g(will)e (consider)h(a)h(family)e(of)i(com-)-118 1627 y(m)n(uting)k(self-adjoin) n(t)g(or)i(normal)d(op)r(erators)i FQ(A)43 b FO(=)f(\()p FN(A)1719 1639 y FL(k)1760 1627 y FO(\),)h(and)d(a)f(family)-118 1727 y FQ(B)23 b FO(=)g(\()p FN(B)156 1739 y FL(j)191 1727 y FO(\))28 b(whic)n(h)e(satisfy)g(more)g(general)f(relations)774 1907 y FN(A)836 1919 y FL(k)878 1907 y FN(B)941 1919 y FL(j)999 1907 y FO(=)d FN(B)1149 1919 y FL(j)1184 1907 y FN(F)1237 1919 y FL(k)q(j)1309 1907 y FO(\()p FQ(A)p FO(\))-118 2088 y(\(w)n(e)39 b(allo)n(w)c(families)g FQ(A)k FO(and)g FQ(B)g FO(to)g(ha)n(v)n(e)f(di\013eren)n(t)g (cardinalit)n(y)-7 b(,)37 b(and)i(no)-118 2188 y(conditions)f(on)j(the) g(join)n(t)f(sp)r(ectrum)g(of)h FQ(A)g FO(or)f(conditions)e(of)j(the)g (form)-118 2287 y(k)n(er)13 b FN(A)69 2299 y FL(k)133 2287 y FO(=)22 b(k)n(er)13 b FN(B)408 2299 y FL(k)477 2287 y FO(are)26 b(assumed\).)6 2387 y(W)-7 b(e)26 b(start)e(with)g (the)i(case)e(of)g(a)h(unitary)e(op)r(erator)g FN(U)34 b FO(and)25 b(a)f(self-adjoin)n(t)-118 2487 y(op)r(erator)d FN(A)i FO(satisfying)d(a)i(relation)e(of)j(the)g(form)e FN(AU)32 b FO(=)23 b FN(U)9 b(F)j FO(\()p FN(A)p FO(\),)24 b(where)e(the)-118 2586 y(mapping)j FN(F)12 b FO(\()p FP(\001)p FO(\))28 b(is)f(measurable)d(and)k(one-to-one)d(on)j(the)g (sp)r(ectrum)f(of)g FN(A)p FO(.)-118 2751 y FQ(Theorem)j(43.)41 b FB(L)l(et)e(a)g(self-adjoint)i(op)l(er)l(ator)g FN(A)e FB(and)h(a)g(unitary)f(op)l(er)l(a-)-118 2851 y(tor)i FN(U)49 b FB(satisfy)42 b(the)g(r)l(elation)f FN(AU)53 b FO(=)42 b FN(U)9 b(F)j FO(\()p FN(A)p FO(\))42 b FB(with)f(a)h(me)l (asur)l(able)f(map-)-118 2950 y(ping)35 b FN(F)12 b FO(\()p FP(\001)p FO(\))35 b FB(which)h(is)f(one-to-one)f(on)g(the)h(sp)l(e)l (ctrum)e(of)j FN(A)p FB(.)52 b(The)36 b(sp)l(ac)l(e)f FN(H)-118 3050 y FB(c)l(an)42 b(b)l(e)g(uniquely)h(de)l(c)l(omp)l(ose)l (d)g(into)g(a)f(dir)l(e)l(ct)h(sum,)i FN(H)52 b FO(=)46 b FN(H)1984 3062 y FK(1)2048 3050 y FP(\010)27 b FN(H)2209 3062 y FK(2)2274 3050 y FP(\010)-118 3149 y(\001)14 b(\001)g(\001)g (\010)h FN(H)142 3161 y FM(1)212 3149 y FB(,)29 b(of)g(invariant)f (subsp)l(ac)l(es)7 b FO(;)29 b FB(e)l(ach)g FN(H)1369 3161 y FL(m)1460 3149 y FB(is)g(unitarily)f(e)l(quivalent)h(to)-118 3249 y FP(\010)-53 3219 y FL(m)-53 3273 y(k)q FK(=1)71 3249 y FN(L)128 3261 y FK(2)165 3249 y FO(\()p FI(R)251 3219 y FK(1)295 3249 y FN(;)14 b(d\026)425 3261 y FL(m)488 3249 y FO(\))29 b FB(in)f(such)h(a)g(way)g(that)g FN(\026)1293 3261 y FL(m)1356 3249 y FO(\()p FP(\001)p FO(\))g FB(is)g(a)g(pr)l(ob)l (ability)i FN(F)12 b FB(-quasi-)-118 3349 y(invariant)31 b(me)l(asur)l(e,)h FN(m)24 b FO(=)h(1)p FB(,)30 b FN(:)14 b(:)g(:)28 b FB(,)j FP(1)p FB(,)g(and)g(the)g(op)l(er)l(ators)h(in)f FN(H)2030 3361 y FL(m)2123 3349 y FB(act)g(as)-118 3448 y(fol)t(lows)7 b FO(:)225 3629 y(\()p FN(Af)i FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(\025)14 b(f)9 b FO(\()p FN(\025)p FO(\))p FN(;)221 3770 y FO(\()p FN(U)g(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(u)673 3782 y FL(m)735 3770 y FO(\()p FN(\025)p FO(\))14 b FN(\032)904 3782 y FL(m)968 3770 y FO(\()p FN(\025)p FO(\))1080 3736 y FK(1)p FL(=)p FK(2)1199 3770 y FN(f)9 b FO(\()p FN(F)1346 3736 y FM(\000)p FK(1)1435 3770 y FO(\()p FN(\025)p FO(\)\))p FN(;)183 3911 y FO(\()p FN(U)281 3877 y FM(\003)319 3911 y FN(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(u)673 3877 y FM(\003)673 3932 y FL(m)735 3911 y FO(\()p FN(F)12 b FO(\()p FN(\025)p FO(\)\))i FN(\032)1033 3923 y FL(m)1098 3911 y FO(\()p FN(F)e FO(\()p FN(\025)p FO(\)\))1339 3877 y FM(\000)p FK(1)p FL(=)p FK(2)1510 3911 y FN(f)d FO(\()p FN(F)j FO(\()p FN(\025)p FO(\)\))p FN(;)316 b FO(\(2.72\))p eop %%Page: 181 185 181 184 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(181)-118 96 y FB(wher)l(e)37 b FN(f)9 b FO(\()p FP(\001)p FO(\))36 b FB(is)g(a)h(ve)l(ctor)f(function)g(in)g FP(\010)1227 66 y FL(m)1227 120 y(k)q FK(=1)1352 96 y FN(L)1409 108 y FK(2)1445 96 y FO(\()p FI(R)1532 66 y FK(1)1575 96 y FN(;)14 b(d\026)1705 108 y FL(m)1768 96 y FO(\))p FB(,)38 b(the)f(functions)-118 196 y FN(\032)-75 208 y FL(m)-12 196 y FO(\()p FN(\025)p FO(\))g(=)f FN(d\026)331 208 y FL(m)394 196 y FO(\()p FN(F)491 166 y FM(\000)p FK(1)581 196 y FO(\()p FN(\025)p FO(\)\))p FN(=d\026)p FO(\()p FN(\025)p FO(\))j FB(ar)l(e)e(the)g(R)l(adon{Niko)l(dym)i(derivatives,) -118 296 y FN(u)-70 308 y FL(m)-8 296 y FO(\()p FP(\001)p FO(\))h FB(ar)l(e)f(me)l(asur)l(able)h(mappings)g(taking)g(values)f(in) h(unitary)f(op)l(er)l(ators)-118 395 y(on)30 b FI(C)55 365 y FL(m)124 395 y FB(.)-118 575 y(Pr)l(o)l(of.)43 b FO(The)27 b(decomp)r(osition)d(of)k(the)f(space)g FN(H)7 b FO(,)27 b(and)g(form)n(ula)e(for)i FN(A)g FO(follo)n(w)-118 674 y(from)j(the)i(decomp)r(osition)c(of)j(the)h(self-adjoin)n(t)d(op)r (erator)h FN(A)i FO(with)f(resp)r(ect)-118 774 y(to)26 b(m)n(ultiplicit)n(y)21 b(of)26 b(its)f(sp)r(ectrum.)36 b(W)-7 b(e)27 b(need)f(to)g(sho)n(w)f(that)i(the)g(subspaces)-118 874 y FN(H)-49 886 y FL(m)42 874 y FO(are)f(in)n(v)-5 b(arian)n(t)25 b(with)i(resp)r(ect)g(to)h FN(U)9 b FO(,)27 b FN(U)1286 844 y FM(\003)1324 874 y FO(.)6 978 y(W)-7 b(e)41 b(shall)d(pro)n(v)n(e)h(that)h(eac)n(h)g(measure)e FN(\026)1384 990 y FL(m)1487 978 y FO(is)i(quasi-in)n(v)-5 b(arian)n(t.)70 b(Let)-118 1077 y FN(h)-70 1089 y FK(1)-33 1077 y FO(,)36 b FN(:)14 b(:)g(:)27 b FO(,)38 b FN(h)259 1089 y FL(m)358 1077 y FO(b)r(e)e(an)f(orthonormal)d(basis)i(in)h FI(C)1457 1047 y FL(m)1526 1077 y FO(.)61 b(F)-7 b(or)35 b(an)n(y)g(measurable)-118 1177 y(\001)g FP(\032)f FI(R)p FO(,)43 b(consider)32 b(functions)i FN(f)941 1189 y FL(k)982 1177 y FO(\()p FN(\025)p FO(\))i(=)e FN(\037)1281 1189 y FK(\001)1340 1177 y FO(\()p FN(\025)p FO(\))p FN(h)1500 1189 y FL(k)1576 1177 y FP(\032)h FN(H)1745 1189 y FL(m)1808 1177 y FO(,)h(\()p FN(\037)1951 1189 y FK(\001)2010 1177 y FO(\()p FP(\001)p FO(\))g(is)d(the)-118 1277 y(c)n(haracteristic)28 b(function)k(of)h(\001\))f(whic)n(h)g(are)f(ob)n(viously)e(orthogonal)g (in)j FN(H)7 b FO(.)-118 1376 y(F)-7 b(rom)26 b(the)i(relation)d(that)j (connects)f FN(A)h FO(and)f FN(U)9 b FO(,)28 b(w)n(e)f(ha)n(v)n(e)f (that)129 1568 y FN(U)195 1534 y FM(\003)233 1568 y FN(f)274 1580 y FL(k)315 1568 y FO(\()p FN(\025)p FO(\))e(=)e FN(U)604 1534 y FM(\003)642 1568 y FN(E)703 1580 y FL(A)758 1568 y FO(\(\001\))p FN(f)932 1580 y FL(k)973 1568 y FO(\()p FN(\025)p FO(\))451 1703 y(=)g FN(E)5 b FO(\()p FN(F)701 1668 y FM(\000)p FK(1)791 1703 y FO(\(\001\)\))p FN(U)1022 1668 y FM(\003)1061 1703 y FN(f)1102 1715 y FL(k)1142 1703 y FO(\()p FN(\025)p FO(\))24 b(=)f FN(\037)1418 1718 y FL(F)1469 1702 y Fw(\000)p Fx(1)1546 1718 y FK(\(\001\))1657 1703 y FO(\()p FN(\025)p FO(\))p FN(U)1835 1668 y FM(\003)1874 1703 y FN(f)1915 1715 y FL(k)1955 1703 y FO(\()p FN(\025)p FO(\))p FN(;)-118 1894 y FO(and)29 b(all)d FN(U)227 1864 y FM(\003)265 1894 y FN(f)306 1906 y FL(k)347 1894 y FO(\()p FP(\001)p FO(\),)k FN(k)e FO(=)d(1,)k FN(:)14 b(:)g(:)27 b FO(,)j FN(m)f FO(ha)n(v)n(e)e(supp)r(orts)i(in)f FN(F)1718 1864 y FM(\000)p FK(1)1807 1894 y FO(\(\001\).)42 b(Since)28 b(the)-118 1994 y(op)r(erator)17 b FN(U)27 b FO(is)18 b(unitary)-7 b(,)19 b(the)g(latter)e(functions)i(again)d (are)i(orthonormal;)f(this)-118 2093 y(implies)31 b(that)36 b(for)e(almost)f(all)g FN(\025)i FO(with)g(resp)r(ect)g(to)g(the)g(sp)r (ectral)f(measure)-118 2193 y(of)k FN(A)p FO(,)j(the)d(v)n(ectors)f(\() p FN(U)657 2163 y FM(\003)695 2193 y FN(f)736 2205 y FL(k)776 2193 y FO(\)\()p FN(\025)p FO(\),)42 b FN(k)i FO(=)c(1,)d FN(:)14 b(:)g(:)28 b FO(,)41 b FN(m)p FO(,)f(are)d (orthogonal,)h(and)-118 2293 y(therefore,)25 b(the)g(sp)r(ectral)f(m)n (ultiplicit)n(y)c(of)25 b FN(A)g FO(at)h(p)r(oin)n(ts)e(of)h FN(F)12 b FO(\(\001\))26 b(is)e(not)h(less)-118 2392 y(than)k(at)g(p)r(oin)n(ts)g(of)g(\001.)42 b(Applying)27 b(the)j(same)e(argumen)n(ts)f(to)i(the)g(op)r(erator)-118 2492 y FN(U)40 b FO(instead)30 b(of)h FN(U)433 2462 y FM(\003)471 2492 y FO(,)h(w)n(e)f(conclude)f(that)h(the)h(sp)r(ectral)d (m)n(ultiplicit)n(y)d(of)31 b FN(A)h FO(is)-118 2592 y(in)n(v)-5 b(arian)n(t)23 b(with)k(resp)r(ect)f(to)g FN(F)12 b FO(\()p FP(\001)p FO(\).)38 b(This)25 b(implies)e(that)k FN(H)1731 2604 y FL(m)1821 2592 y FO(is)e(an)i(in)n(v)-5 b(arian)n(t)-118 2691 y(subspace,)27 b(and)g(that)h FN(\026)643 2703 y FL(k)712 2691 y FO(is)e(quasi-in)n(v)-5 b(arian)n(t.)6 2795 y(In)28 b(the)g(subspace)f FN(H)669 2807 y FL(m)732 2795 y FO(,)h(in)n(tro)r(duce)e(the)i(unitary)e(op)r(erator)357 3005 y(\()p FN(V)437 3017 y FL(k)479 3005 y FN(f)9 b FO(\)\()p FN(\025)p FO(\))24 b(=)784 2938 y Fy(\000)822 3005 y FN(d\026)p FO(\()p FN(F)12 b FO(\()p FN(\025)p FO(\)\))p FN(=d\026)p FO(\()p FN(\025)p FO(\))1403 2938 y Fy(\001)1444 2955 y FK(1)p FL(=)p FK(2)1548 3005 y FN(f)d FO(\()p FN(F)j FO(\()p FN(\025)p FO(\)\))p FN(:)-118 3211 y FO(Then)45 b(the)g(op)r(erator)643 3190 y(~)628 3211 y FN(U)61 b FO(=)51 b FN(U)9 b(V)64 b FO(comm)n(utes)42 b(with)j FN(A)p FO(,)k(and,)h(therefore,)-118 3311 y(\()-71 3290 y(~)-86 3311 y FN(U)9 b(f)g FO(\)\()p FN(\025)p FO(\))37 b(=)f FN(u)360 3323 y FL(m)423 3311 y FO(\()p FN(\025)p FO(\))14 b FN(f)9 b FO(\()p FN(\025)p FO(\).)63 b(F)-7 b(rom)34 b FN(U)45 b FO(=)1241 3290 y(~)1226 3311 y FN(U)9 b(V)1359 3281 y FM(\003)1397 3311 y FO(,)38 b(w)n(e)d(ha)n(v)n(e)g(the)h(represen)n(ta-)-118 3410 y(tion)27 b(required.)p 2278 3410 4 57 v 2282 3358 50 4 v 2282 3410 V 2331 3410 4 57 v 6 3612 a(Instead)i(of)f(decomp)r (osing)e FN(H)35 b FO(in)n(to)27 b(a)i(direct)e(sum)h(with)g(resp)r (ect)g(to)g(the)-118 3712 y(m)n(ultiplicit)n(y)16 b(of)22 b(the)g(sp)r(ectrum)f(of)h FN(A)p FO(,)i(one)d(can)h(decomp)r(ose)e FN(H)29 b FO(in)n(to)20 b(a)i(direct)-118 3811 y(in)n(tegral)17 b(of)j(generalized)d(eigenspaces)h(of)i FN(A)p FO(.)35 b(Then)21 b(the)f(theorem)f(ab)r(o)n(v)n(e)g(can)-118 3911 y(b)r(e)28 b(reform)n(ulated)c(as)j(follo)n(ws.)p eop %%Page: 182 186 182 185 bop -118 -137 a FO(182)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FQ(Theorem)30 b(44.)41 b FB(L)l(et)28 b FN(A)g FB(and)h FN(U)36 b FB(b)l(e)29 b(as)f(in)g(the)h(pr)l(evious)g (the)l(or)l(em.)38 b(Then)29 b FN(H)-118 196 y FB(c)l(an)h(b)l(e)f(de)l (c)l(omp)l(ose)l(d)j(into)e(a)g(dir)l(e)l(ct)g(inte)l(gr)l(al,)763 433 y FN(H)g FO(=)949 320 y Fy(Z)1032 341 y FM(\010)995 509 y Fu(R)1042 492 y Fx(1)1102 433 y FN(H)1171 445 y FL(\025)1228 433 y FN(d\026)p FO(\()p FN(\025)p FO(\))p FN(;)-118 664 y FB(wher)l(e)38 b(the)f(Bor)l(el)h(pr)l(ob)l(ability)h (me)l(asur)l(e)e FN(\026)g FB(is)g FN(F)12 b FO(\()p FP(\001)p FO(\))p FB(-quasi-invariant,)41 b(and)-118 763 y FN(N)9 b FO(\()p FN(\025)p FO(\))26 b(=)g(dim)12 b FN(H)408 775 y FL(\025)483 763 y FB(is)31 b(invariant)h(with)g(r)l (esp)l(e)l(ct)f(to)g FN(F)12 b FO(\()p FP(\001)p FO(\))p FB(,)33 b(so)e(that)g(the)h(op)l(er)l(a-)-118 863 y(tors)e(take)g(the)g (form)375 1047 y FO(\()p FN(Af)9 b FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(\025)14 b(f)9 b FO(\()p FN(\025)p FO(\))p FN(;)371 1188 y FO(\()p FN(U)g(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(u)p FO(\()p FN(\025)p FO(\))14 b FN(\032)p FO(\()p FN(\025)p FO(\))1104 1153 y FK(1)p FL(=)p FK(2)1223 1188 y FN(f)9 b FO(\()p FN(F)1370 1153 y FM(\000)p FK(1)1459 1188 y FO(\()p FN(\025)p FO(\)\))p FN(;)333 1329 y FO(\()p FN(U)431 1294 y FM(\003)469 1329 y FN(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(u)823 1294 y FM(\003)860 1329 y FO(\()p FN(F)12 b FO(\()p FN(\025)p FO(\)\))i FN(\032)p FO(\()p FN(F)e FO(\()p FN(\025)p FO(\)\))1399 1294 y FM(\000)p FK(1)p FL(=)p FK(2)1572 1329 y FN(f)d FO(\()p FN(F)j FO(\()p FN(\025)p FO(\)\))p FN(;)254 b FO(\(2.73\))-118 1513 y FB(wher)l(e)34 b FN(\032)p FO(\()p FN(\025)p FO(\))d(=)f FN(d\026)p FO(\()p FN(F)591 1482 y FM(\000)p FK(1)680 1513 y FO(\()p FN(\025)p FO(\)\))p FN(=d\026)p FO(\()p FN(\025)p FO(\))36 b FB(is)e(the)g(R)l(adon{Niko)l(dym)h(derivative,) -118 1612 y FN(u)p FO(\()p FP(\001)p FO(\))30 b FB(is)g(a)g(unitary)g (me)l(asur)l(able)g(op)l(er)l(ator-value)l(d)h(function.)6 1780 y FO(The)20 b(follo)n(wing)15 b(statemen)n(t)j(giv)n(es)f (conditions)f(of)j(irreducibilit)n(y)14 b(and)19 b(uni-)-118 1880 y(tary)26 b(equiv)-5 b(alence)24 b(of)j(represen)n(tations)d(in)i (the)h(form)e(pro)n(vided)g(b)n(y)h(the)h(pre-)-118 1979 y(vious)f(theorem.)-118 2147 y FQ(Prop)s(osition)k(60.)41 b FB(If)j(r)l(epr)l(esentation)51 b FO(\(2.73\))43 b FB(is)i(irr)l(e)l(ducible,)k(then)44 b(the)-118 2247 y(me)l(asur)l(e)c FN(\026)p FO(\()p FP(\001)p FO(\))i FB(is)f(er)l(go)l(dic,)k(and)c(the)g(dimension)h(function)f FN(N)9 b FO(\()p FP(\001)p FO(\))41 b FB(is)g(c)l(on-)-118 2347 y(stant)32 b FN(\026)p FB(-a.e.)48 b(If)33 b(the)f(me)l(asur)l(e)h FN(\026)p FO(\()p FP(\001)p FO(\))g FB(is)g(er)l(go)l(dic,)i(and)e FO(dim)12 b FN(H)1863 2359 y FL(\025)1935 2347 y FO(=)28 b(1)k FN(\026)p FB(-a.e.,)-118 2446 y FO(\()p FB(i.e.,)g FN(H)e FO(=)22 b FN(L)327 2458 y FK(2)364 2446 y FO(\()p FI(R)p FN(;)14 b(d\026)p FO(\))q(\))p FB(,)36 b(then)30 b(the)g(r)l(epr)l(esentation)g(is)g(irr)l(e)l(ducible.)6 2546 y(Two)23 b(r)l(epr)l(esentations,)i(c)l(orr)l(esp)l(onding)e(to)f (the)g(me)l(asur)l(e)g FN(\026)p FB(,)i(multiplicity)-118 2646 y(function)i FN(N)9 b FO(\()p FP(\001)p FO(\))p FB(,)27 b(and)g(multiplier)g FN(u)p FO(\()p FP(\001)p FO(\))p FB(,)g(and)f(c)l(orr)l(esp)l(onding)h(to)f(the)g(triple)34 b FO(~)-49 b FN(\026)p FB(,)-94 2725 y FO(~)-118 2746 y FN(N)9 b FO(\()p FP(\001)p FO(\))p FB(,)29 b(and)k FO(~)-47 b FN(u)o FO(\()p FP(\001)p FO(\))p FB(,)29 b(ar)l(e)f (unitarily)h(e)l(quivalent)e(if)i(and)f(only)g(if)h(the)e(me)l(asur)l (es)h FN(\026)-118 2845 y FB(and)i FO(~)-48 b FN(\026)23 b FB(ar)l(e)g(e)l(quivalent)32 b FO(\()p FB(have)25 b(the)e(same)h(zer) l(o)f(sets)7 b FO(\))p FB(,)25 b(multiplicity)g(functions)-118 2945 y(c)l(oincide)39 b FN(\026)p FB(-a.e.,)h(and)d(the)h(multipliers)f (ar)l(e)h(e)l(quivalent)f(in)g(the)g(fol)t(lowing)-118 3045 y(sense)6 b FO(:)63 b FB(ther)l(e)43 b(exists)f(a)h(me)l(asur)l (able)f(unitary)h(op)l(er)l(ator-value)l(d)h(function)-118 3144 y FN(v)s FO(\()p FP(\001)p FO(\))30 b FB(such)g(that)163 3328 y FO(~)-48 b FN(u)p FO(\()p FN(\025)p FO(\))24 b(=)f FN(v)s FO(\()p FN(\025)p FO(\))14 b FN(u)p FO(\()p FN(\025)p FO(\))g FN(v)815 3294 y FM(\003)855 3328 y FO(\()p FN(F)952 3294 y FM(\000)p FK(1)1041 3328 y FO(\()p FN(\025)p FO(\)\))p FN(;)185 b FB(for)30 b FN(\026)p FB(-almost)g(al)t(l)h FN(\025)p FB(.)-118 3513 y(Pr)l(o)l(of.)43 b FO(If)26 b(the)g(measure)d(is)i(not)g(ergo)r(dic,)f(its)h(supp)r(ort)g(can)g(b)r (e)h(decomp)r(osed)-118 3612 y(in)n(to)g(the)h(union)f(of)h(t)n(w)n(o)g (in)n(v)-5 b(arian)n(ts)24 b(subsets)i(of)i(p)r(ositiv)n(e)c(measure.) 35 b(Suc)n(h)27 b(a)-118 3712 y(decomp)r(osition)e(giv)n(es)i(rise)h (to)h(a)f(decomp)r(osition)e(of)j FN(H)36 b FO(in)n(to)28 b(a)h(direct)f(sum)-118 3811 y(of)35 b(in)n(v)-5 b(arian)n(t)32 b(subspaces.)59 b(Since)34 b FN(N)9 b FO(\()p FP(\001)p FO(\))36 b(is)e FN(F)12 b FO(\()p FP(\001)p FO(\)-in)n(v)-5 b(arian)n(t,)34 b(it)g(is)g(constan)n(t)-118 3911 y FN(\026)p FO(-a.e.)p eop %%Page: 183 187 183 186 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(183)6 96 y(Let)27 b FN(\026)g FO(b)r(e)g(ergo)r(dic)d(and)i FN(N)9 b FO(\()p FN(\025)p FO(\))24 b(=)f(1)j(for)g FN(\026)p FO(-almost)e(all)g FN(\025)p FO(.)37 b(An)n(y)26 b(b)r(ounded)-118 196 y(self-adjoin)n(t)19 b(op)r(erator)h FN(C)28 b FO(that)21 b(comm)n(utes)e(with)i FN(A)h FO(is)f(the)h(op)r(erator)d(of)j(m)n(ul-) -118 296 y(tiplication)29 b(b)n(y)j(a)g(b)r(ounded)h(measurable)c (function)k FN(c)p FO(\()p FP(\001)p FO(\).)52 b(If)33 b FN(C)39 b FO(comm)n(utes)-118 395 y(with)g FN(U)9 b FO(,)43 b(then)d(this)f(function)g(is)g(in)n(v)-5 b(arian)n(t,)40 b(and)f(b)n(y)h(the)g(ergo)r(dicit)n(y)-7 b(,)40 b(is)-118 495 y(constan)n(t)27 b FN(\026)p FO(-a.e.,)g(i.e.,)g(the)h(pair)d FN(A)p FO(,)j FN(U)37 b FO(is)26 b(irreducible.)6 595 y(No)n(w)k(consider)f(t)n(w)n(o)g(pairs)g FN(A)p FO(,)i FN(U)40 b FO(on)30 b FN(H)37 b FO(corresp)r(onding)28 b(to)i(a)g(triple)e FN(\026)p FO(,)-118 694 y FN(N)9 b FO(\()p FP(\001)p FO(\),)32 b FN(u)p FO(\()p FP(\001)p FO(\),)h(and)478 673 y(~)456 694 y FN(A)p FO(,)587 673 y(~)573 694 y FN(U)40 b FO(on)810 673 y(~)789 694 y FN(H)e FO(corresp)r(onding)28 b(to)37 b(~)-48 b FN(\026)p FO(,)1667 673 y(~)1643 694 y FN(N)9 b FO(\()p FP(\001)p FO(\),)38 b(~)-48 b FN(u)p FO(\()p FP(\001)p FO(\).)48 b(If)32 b(these)-118 794 y(pairs)23 b(are)h(unitarily)d(equiv)-5 b(alen)n(t,)24 b(then)h(the)h(sp)r(ectral)d(measures)g(of)h(the)i (self-)-118 893 y(adjoin)n(t)31 b(op)r(erators)f FN(A)j FO(and)823 872 y(~)801 893 y FN(A)f FO(are)g(equiv)-5 b(alen)n(t,)31 b(and)h FN(N)9 b FO(\()p FP(\001)p FO(\))31 b(=)1942 872 y(~)1918 893 y FN(N)8 b FO(\()p FP(\001)p FO(\))33 b FN(\026)p FO(-a.e.;)-118 993 y(no)n(w)d(w)n(e)h(can)g (naturally)d(iden)n(tify)i FN(H)37 b FO(and)1300 972 y(~)1278 993 y FN(H)7 b FO(.)48 b(A)31 b(unitary)f(op)r(erator)f FN(V)50 b FO(on)-118 1093 y FN(H)39 b FO(that)33 b(in)n(tert)n(wines)d (these)i(pairs)f(comm)n(utes)f(with)i FN(A)p FO(,)i(and)e(therefore,)h (is)-118 1192 y(a)h(m)n(ultiplication)29 b(b)n(y)35 b(a)f(measurable)d (unitary)i(op)r(erator-v)-5 b(alued)32 b(function)-118 1292 y FN(v)s FO(\()p FP(\001)p FO(\).)45 b(It)31 b(directly)d(follo)n (ws)f(from)i(\(2.73\))g(that)i(the)g(relation)1844 1271 y(~)1830 1292 y FN(U)36 b FO(=)27 b FN(V)19 b(U)9 b(V)2214 1262 y FM(\003)2283 1292 y FO(is)-118 1392 y(equiv)-5 b(alen)n(t)25 b(to)33 b(~)-47 b FN(u)o FO(\()p FN(\025)p FO(\))24 b(=)f FN(v)s FO(\()p FN(\025)p FO(\))14 b FN(u)p FO(\()p FN(\025)p FO(\))g FN(v)1035 1361 y FM(\003)1075 1392 y FO(\()p FN(F)1172 1361 y FM(\000)p FK(1)1261 1392 y FO(\()p FN(\025)p FO(\)\).)p 2278 1392 4 57 v 2282 1339 50 4 v 2282 1392 V 2331 1392 4 57 v -118 1555 a FB(R)l(emark)30 b(47.)42 b FO(One)31 b(can)f(easily)d(see)j(that)h(the) g(same)d(statemen)n(t)i(holds)f(true)-118 1654 y(for)f(a)g(\014nite)g (or)f(coun)n(table)g(comm)n(uting)e(family)h(of)i(self-adjoin)n(t)e(or) i(normal)-118 1754 y(op)r(erators)i FQ(A)i FO(=)e(\()p FN(A)547 1766 y FL(k)589 1754 y FO(\))621 1766 y FL(k)q FM(2)p FL(X)798 1754 y FO(whic)n(h)h(are)h(related)f(with)g(a)h (unitary)f(op)r(erator)-118 1854 y FN(U)45 b FO(b)n(y)37 b(the)g(relations)c FN(A)669 1866 y FL(k)710 1854 y FN(U)47 b FO(=)38 b FN(U)9 b(F)1036 1866 y FL(k)1077 1854 y FO(\()p FQ(A)p FO(\),)39 b FN(k)i FP(2)e FN(X)7 b FO(.)63 b(Indeed,)39 b(w)n(e)d(actually)-118 1953 y(use)27 b(the)h(prop)r(erties)e(of)i(the) g(join)n(t)e(resolution)f(of)j(the)g(iden)n(tit)n(y)d(of)j(this)f(fam-) -118 2053 y(ily)-7 b(,)34 b FN(E)85 2065 y Fm(A)146 2053 y FO(\()p FP(\001)p FO(\).)58 b(T)-7 b(o)34 b(pro)n(v)n(e)f(the)h (theorem,)h(w)n(e)f(use)g(the)h(relation)c FN(E)1979 2065 y Fm(A)2040 2053 y FO(\(\001\))p FN(U)44 b FO(=)-118 2152 y FN(U)9 b(E)9 2164 y FL(A)63 2152 y FO(\()p FQ(F)155 2122 y FM(\000)p FK(1)244 2152 y FO(\()p FQ(A)p FO(\)\),)26 b(where)c FQ(F)9 b FO(:)28 b FI(R)870 2122 y FL(X)962 2152 y FP(\000)-48 b(!)23 b FI(R)1139 2122 y FL(X)1231 2152 y FO(is)f FQ(F)p FO(\()p FP(\001)p FO(\))i(=)f(\()p FN(F)1654 2164 y FL(k)1695 2152 y FO(\()p FP(\001)p FO(\)\))1814 2164 y FL(k)q FM(2)p FL(X)1960 2152 y FO(.)35 b(The)23 b(only)-118 2252 y(di\013erence)f(is)h(that)g(the)h(dynamical)c(system) i(acts)h(on)h FI(R)1642 2222 y FL(X)1711 2252 y FO(,)g(and)f(the)h(sp)r (ectral)-118 2352 y(measure)h FN(\026)j FO(should)f(b)r(e)h(considered) d(on)i(this)g(space.)6 2451 y(Moreo)n(v)n(er,)g(since)g(the)i(decomp)r (osition)c(of)k FN(H)35 b FO(is)28 b(constructed)g(from)f(the)-118 2551 y(comm)n(utativ)n(e)e(family)i FQ(A)p FO(,)i(the)h(comm)n(utativ)n (e)25 b(mo)r(del)j(can)h(b)r(e)g(constructed)-118 2651 y(for)34 b(a)g(\014nite)h(or)f(coun)n(table)f(family)f(of)i(unitary)g (op)r(erators)e FQ(U)k FO(=)e(\()p FN(U)2138 2663 y FL(j)2173 2651 y FO(\))2205 2663 y FL(j)s FM(2)p FL(Y)-118 2750 y FO(whic)n(h)26 b(satisfy)h(the)h(relations)c FN(A)921 2762 y FL(k)962 2750 y FN(B)1025 2762 y FL(j)1083 2750 y FO(=)f FN(B)1234 2762 y FL(j)1269 2750 y FN(F)1322 2762 y FL(k)q(j)1394 2750 y FO(\()p FQ(A)p FO(\),)28 b FN(k)e FP(2)e FN(X)7 b FO(,)27 b FN(j)h FP(2)23 b FN(Y)c FO(.)6 2850 y(T)-7 b(o)24 b(extend)h(this)e(theorem)g(to)h(the)h(case)e (of)h(an)g(arbitrary)e(\(uncoun)n(table\))-118 2949 y(n)n(um)n(b)r(er) 31 b(of)i(op)r(erators,)f(one)h(should)e(require)g(the)i(existence)f (of)g(a)h(n)n(uclear)-118 3049 y(rigging)24 b(for)j(the)h(op)r(erators) e([28)o(,)h(183)o(].)6 3176 y(No)n(w)40 b(consider)e(a)h(family)e(of)j (comm)n(uting)d(b)r(ounded)j(self-adjoin)n(t)d(\(or)-118 3276 y(normal\))26 b(op)r(erators)h(\()p FN(A)660 3288 y FL(k)701 3276 y FO(\),)j(where)f FN(k)f FP(2)d FN(X)36 b FO(ranges)27 b(o)n(v)n(er)g(a)i(\014nite)f(or)g(coun)n(t-)-118 3375 y(able)j(set,)k(and)d(a)h(family)d(of)i(b)r(ounded)i(op)r(erators) c(\()p FN(B)1633 3387 y FL(j)1669 3375 y FO(\),)k FN(j)j FP(2)32 b FN(Y)19 b FO(,)34 b(whic)n(h)e(is)-118 3475 y(connected)27 b(with)g(the)h(op)r(erators)e FN(A)1031 3487 y FL(k)1100 3475 y FO(b)n(y)h(the)h(follo)n(wing)c(relations)756 3643 y FN(A)818 3655 y FL(k)859 3643 y FN(B)922 3655 y FL(j)980 3643 y FO(=)f FN(B)1131 3655 y FL(j)1166 3643 y FN(F)1219 3655 y FL(k)q(j)1291 3643 y FO(\()p FQ(A)p FO(\))p FN(;)676 b FO(\(2.74\))-118 3811 y(where)33 b FN(F)181 3823 y FL(k)q(j)287 3811 y FO(is)f(a)h(measurable)d(function)k (of)f(the)h(comm)n(uting)c(family)h FQ(A)i FO(=)-118 3911 y(\()p FN(A)-24 3923 y FL(k)17 3911 y FO(\),)h FN(k)f FP(2)e FN(X)7 b FO(,)32 b FN(j)k FP(2)30 b FN(Y)19 b FO(,)33 b(suc)n(h)f(that)g FQ(F)1113 3923 y FL(j)1148 3911 y FO(\()p FP(\001)p FO(\))f(=)f(\()p FN(F)1446 3923 y FL(k)q(j)1518 3911 y FO(\()p FP(\001)p FO(\)\))1637 3923 y FL(k)q FM(2)p FL(X)1792 3911 y FO(:)f FI(R)1898 3881 y FL(X)1998 3911 y FP(\000)-49 b(!)31 b FI(R)2182 3881 y FL(X)2283 3911 y FO(is)p eop %%Page: 184 188 184 187 bop -118 -137 a FO(184)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(an)j(injectiv)n(e)f(on)h(the)h(join)n(t)f(sp) r(ectrum)g(of)g FQ(A)h FO(measurable)c(mapping.)-118 261 y FQ(Theorem)30 b(45.)41 b FB(L)l(et)35 b(families)j FO(\()p FN(A)1010 273 y FL(k)1052 261 y FO(\))p FB(,)g FN(A)1209 273 y FL(k)1285 261 y FO(=)c FN(A)1446 231 y FM(\003)1446 285 y FL(k)1487 261 y FB(,)k FN(k)f FP(2)e FN(X)7 b FB(,)37 b FO(\()p FN(B)1953 273 y FL(j)1988 261 y FO(\))p FB(,)h FN(j)i FP(2)34 b FN(Y)19 b FB(,)-118 361 y(satisfy)28 b(r)l(elations)34 b FO(\(2.74\))25 b FB(on)i(a)g(Hilb)l(ert)g(sp)l(ac)l(e)g FN(H)7 b FB(.)37 b(Then)28 b FN(H)33 b FB(c)l(an)27 b(b)l(e)f(de)l(c)l(om-)-118 461 y(p)l(ose)l(d)31 b(into)e(the)h(dir)l(e)l(ct)g(inte)l(gr)l(al)427 695 y FN(H)g FO(=)614 582 y Fy(Z)697 603 y FM(\010)660 771 y Fu(R)707 754 y Fv(X)773 695 y FN(H)842 707 y FL(\025)900 695 y FN(d\026)p FO(\()p FN(\025)p FO(\))p FN(;)185 b(\025)23 b FO(=)g(\()p FN(\025)1552 707 y FL(k)1593 695 y FO(\))1625 707 y FL(k)q FM(2)p FL(X)1770 695 y FN(;)-118 918 y FB(and)30 b(the)g(op)l(er)l(ators)h(act)f(as)g(fol)t(lows)7 b FO(:)-51 1099 y(\()p FN(A)43 1111 y FL(k)84 1099 y FN(f)i FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(\025)438 1111 y FL(k)493 1099 y FN(f)9 b FO(\()p FN(\025)p FO(\))p FN(;)-46 1240 y FO(\()p FN(B)49 1252 y FL(j)84 1240 y FN(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(b)426 1252 y FL(j)460 1240 y FO(\()p FN(\025)p FO(\))14 b FN(\037)638 1267 y FK(\001)693 1239 y Fv(j)693 1285 y Fx(0)730 1240 y FO(\()p FN(\025)p FO(\))g FN(\032)899 1252 y FL(j)935 1240 y FO(\()p FN(\025)p FO(\))1047 1206 y FK(1)p FL(=)p FK(2)1166 1240 y FN(f)9 b FO(\()p FQ(F)1308 1205 y FM(\000)p FK(1)1308 1263 y FL(j)1397 1240 y FO(\()p FN(\025)p FO(\)\))p FN(;)-54 1397 y FO(\()p FN(B)45 1362 y FM(\003)41 1417 y FL(j)84 1397 y FN(f)g FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(b)426 1362 y FM(\003)426 1417 y FL(j)463 1397 y FO(\()p FQ(F)555 1409 y FL(j)591 1397 y FO(\()p FN(\025)p FO(\)\))14 b FN(\037)801 1424 y Fm(F)848 1397 y Fw(\000)p Fx(1)848 1443 y Fv(j)926 1424 y FK(\(\001)1007 1396 y Fv(j)1007 1442 y Fx(0)1039 1424 y FK(\))1069 1397 y FO(\()p FN(\025)p FO(\))g FN(\032)1238 1409 y FL(j)1274 1397 y FO(\()p FQ(F)1366 1409 y FL(j)1401 1397 y FO(\()p FN(\025)p FO(\)\))1545 1362 y FM(\000)p FK(1)p FL(=)p FK(2)1717 1397 y FN(f)9 b FO(\()p FQ(F)1859 1409 y FL(j)1894 1397 y FO(\()p FN(\025)p FO(\)\))p FN(:)65 b FO(\(2.75\))-118 1619 y FB(Her)l(e)34 b FO(\001)154 1579 y FL(j)154 1641 y FK(0)227 1619 y FB(is)h(the)g(set)f(de\014ne)l(d)h(by)g(the)g(pr)l(op)l(erty)h(that)e (for)i(any)f(me)l(asur)l(able)-118 1728 y FN(\016)f FP(\032)c FO(\001)117 1688 y FL(j)117 1750 y FK(0)154 1728 y FB(,)36 b(we)e(have)h FN(B)600 1740 y FL(j)635 1728 y FN(E)696 1740 y Fm(A)757 1728 y FO(\()p FN(\016)s FO(\))c FP(6)p FO(=)f(0;)36 b FB(the)e(me)l(asur)l(e)f FN(\037)1607 1755 y FK(\001)1662 1727 y Fv(j)1662 1773 y Fx(0)1698 1728 y FO(\()p FN(\025)p FO(\))14 b FN(d\026)p FO(\()p FQ(F)2009 1692 y FM(\000)p FK(1)2009 1751 y FL(j)2100 1728 y FO(\()p FN(\025)p FO(\)\))35 b FB(is)-118 1843 y(absolutely)d(c)l(ontinuous)e(with)i(r)l(esp)l(e)l(ct)e(to)h FN(d\026)p FO(\()p FN(\025)p FO(\))p FB(,)i FN(\032)1542 1855 y FL(j)1577 1843 y FO(\()p FP(\001)p FO(\))e FB(is)h(the)f(c)l (orr)l(esp)l(ond-)-118 1943 y(ing)26 b(R)l(adon{Niko)l(dym)h (derivative,)i FN(b)1069 1955 y FL(j)1104 1943 y FO(\()p FP(\001)p FO(\))d FB(is)g(a)g(me)l(asur)l(able)g(op)l(er)l(ator-value)l (d)-118 2053 y(function,)k FN(b)270 2065 y FL(j)305 2053 y FO(\()p FN(\025)p FO(\))24 b FP(6)p FO(=)e(0)30 b FB(for)g FN(\026)p FB(-almost)g(al)t(l)h FN(\025)23 b FP(2)h FO(\001)1415 2014 y FL(j)1415 2076 y FK(0)1452 2053 y FB(.)-118 2218 y(Pr)l(o)l(of.)43 b FO(The)32 b(pro)r(of)f(of)h(the)g(theorem)f(in)g(a) h(general)d(situation)h(is)h(based)g(on)-118 2318 y(the)24 b(theory)f(of)h(decomp)r(osition)d(with)i(resp)r(ect)h(to)g (generalized)c(eigen)n(v)n(ectors)-118 2418 y(of)27 b(families)d(of)j (self-adjoin)n(t)f(op)r(erators,)f(and)j(uses)f(tec)n(hniques)f(b)r(ey) n(ond)h(the)-118 2517 y(scop)r(e)d(of)h(the)g(b)r(o)r(ok;)g(w)n(e)f (consider)f(only)g(the)i(case)f(where)g(the)h(comm)n(utativ)n(e)-118 2617 y(family)h FQ(A)j FO(has)f(a)g(simple)e(join)n(t)i(sp)r(ectrum)g (in)g(order)f(to)i(illustrate)c(the)k(idea)-118 2716 y(of)e(the)h(pro)r(of.)6 2816 y(In)k(this)f(case,)g(the)h(represen)n (tation)d(space)i(is)f FN(L)1547 2828 y FK(2)1584 2816 y FO(\()p FI(R)1670 2786 y FL(X)1739 2816 y FN(;)14 b(d\026)p FO(\),)33 b(where)e FN(\026)h FO(is)-118 2916 y(the)h(sp)r(ectral)d (measure)h(of)h(the)h(comm)n(utativ)n(e)c(family)-7 b(,)30 b(whic)n(h)i(acts)g(as)f(the)-118 3015 y(op)r(erators)g(of)i(m)n (ultiplication,)c(\()p FN(A)1012 3027 y FL(k)1053 3015 y FN(f)9 b FO(\)\()p FN(\025)p FO(\))33 b(=)f FN(\025)1425 3027 y FL(k)1480 3015 y FN(f)9 b FO(\()p FN(\025)p FO(\),)35 b FN(k)g FP(2)e FN(X)7 b FO(.)52 b(T)-7 b(ak)n(e)32 b(the)-118 3115 y(v)n(ector)f FN(e)p FO(\()p FN(\025)p FO(\))g FP(\021)e FO(1;)34 b(then)f(for)e(all)f(c)n(haracteristic)e(functions)j(of)h (measurable)-118 3215 y(subsets)27 b(of)34 b FI(R)324 3184 y FL(X)393 3215 y FO(,)28 b(w)n(e)f(ha)n(v)n(e)154 3396 y(\()p FN(U)243 3408 y FL(j)278 3396 y FN(\037)330 3408 y FL(\016)367 3396 y FO(\)\()p FN(\025)p FO(\))d(=)f(\()p FN(U)712 3408 y FL(j)747 3396 y FN(E)808 3408 y Fm(A)868 3396 y FO(\()p FN(\016)s FO(\))14 b FN(e)p FO(\)\()p FN(\025)p FO(\))535 3531 y(=)23 b FN(E)684 3543 y Fm(A)744 3531 y FO(\()p FQ(F)836 3543 y FL(j)872 3531 y FO(\()p FN(\016)s FO(\)\)\()p FN(U)1097 3543 y FL(j)1133 3531 y FN(e)p FO(\)\()p FN(\025)p FO(\))h(=)e FN(e)1466 3543 y FL(j)1501 3531 y FO(\()p FN(\025)p FO(\))14 b FN(\037)1679 3543 y FL(\016)1716 3531 y FO(\()p FQ(F)1808 3496 y FM(\000)p FK(1)1898 3531 y FO(\()p FN(\025)p FO(\)\))p FN(;)-118 3712 y FO(where)33 b(w)n(e)f(write)h FN(e)513 3724 y FL(j)547 3712 y FO(\()p FN(\025)p FO(\))h(=)e(\()p FN(U)879 3724 y FL(j)914 3712 y FN(e)p FO(\)\()p FN(\025)p FO(\).)55 b(No)n(w)32 b(w)n(e)h(sho)n(w)g(that)g(the)h(measure)-118 3811 y FN(\037)-66 3823 y FK(\001)-11 3831 y Fx(0)25 3811 y FO(\()p FN(\025)p FO(\))14 b FN(d\026)p FO(\()p FN(F)341 3781 y FM(\000)p FK(1)431 3811 y FO(\()p FN(\025)p FO(\)\))34 b(is)e(absolutely)e(con)n(tin)n(uous)h(with)h(resp)r(ect)h (to)f FN(d\026)p FO(\()p FN(\025)p FO(\).)-118 3911 y(Indeed,)37 b(for)e(an)n(y)f(measurable)e FN(\016)39 b FO(w)n(e)c(ha)n(v)n(e)f FN(E)1391 3923 y Fm(A)1451 3911 y FO(\()p FN(\016)s FO(\))p FN(U)1612 3923 y FL(j)1683 3911 y FO(=)i FN(U)1841 3923 y FL(j)1875 3911 y FN(E)1936 3923 y Fm(A)1997 3911 y FO(\()p FQ(F)2089 3876 y FM(\000)p FK(1)2089 3934 y FL(j)2179 3911 y FO(\()p FN(\016)s FO(\)\),)p eop %%Page: 185 189 185 188 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(185)-118 96 y(and)44 b(since)f FN(U)346 66 y FM(\003)337 118 y FL(j)384 96 y FN(U)441 108 y FL(j)526 96 y FO(=)50 b FN(P)694 108 y FL(j)773 96 y FO(comm)n(utes)42 b(with)i FN(E)1447 108 y Fm(A)1507 96 y FO(\()p FQ(F)1599 61 y FM(\000)p FK(1)1599 120 y FL(j)1689 96 y FO(\()p FN(\016)s FO(\)\),)49 b(w)n(e)44 b(see)f(that)-118 196 y(\()p FN(P)-33 208 y FL(j)2 196 y FN(f)9 b FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(\037)360 223 y FK(\001)415 195 y Fv(j)415 241 y Fx(0)451 196 y FO(\()p FN(\025)p FO(\).)38 b(No)n(w)27 b(w)n(e)g(ha)n(v)n(e)568 383 y FN(P)621 395 y FL(j)657 383 y FN(E)718 395 y Fm(A)778 383 y FO(\()p FQ(F)870 348 y FM(\000)p FK(1)960 383 y FO(\()p FN(\016)s FO(\)\))d(=)e FN(U)1273 348 y FM(\003)1264 403 y FL(j)1311 383 y FN(E)1372 395 y Fm(A)1433 383 y FO(\()p FN(\016)s FO(\))p FN(U)1594 395 y FL(j)1629 383 y FN(;)-118 544 y FO(whic)n(h)c(giv)n(es)f(the)i (absolute)e(con)n(tin)n(uit)n(y)g(required,)h(since)g FN(\026)p FO(\()p FP(\001)p FO(\))24 b(=)e(\()p FN(E)2020 556 y Fm(A)2081 544 y FO(\()p FP(\001)p FO(\))p FN(e;)14 b(e)p FO(\).)-118 643 y(Notice)40 b(that)i(supp)14 b FN(e)574 655 y FL(j)608 643 y FO(\()p FP(\001)p FO(\))47 b(=)e(\001)921 603 y FL(j)921 665 y FK(0)958 643 y FO(;)k(then)41 b(w)n(e)g(can)g(rewrite)e(it)i(as)g FN(e)2081 655 y FL(j)2115 643 y FO(\()p FN(\025)p FO(\))47 b(=)-118 743 y FN(u)-70 755 y FL(j)-36 743 y FO(\()p FN(\025)p FO(\))14 b FN(\032)133 755 y FL(j)169 743 y FO(\()p FN(\025)p FO(\))281 713 y FK(1)p FL(=)p FK(2)386 743 y FO(,)27 b(whic)n(h)f(pro)n(v)n(es)e(the) j(form)n(ula)c(for)j(c)n(haracteristic)c(functions)-118 842 y(of)27 b(measurable)e(sets,)i(and)h(therefore,)e(for)h(the)h (whole)f FN(H)7 b FO(.)p 2278 842 4 57 v 2282 790 50 4 v 2282 842 V 2331 842 4 57 v -118 1054 a FQ(2.5.2)94 b(Cen)m(tered)32 b(op)s(erators)-118 1208 y FO(Recall)24 b(that)j(a)f(b)r(ounded)h(op)r(erator)e FN(T)37 b FO(is)26 b(called)e(cen)n(tered)i(if)g(the)h(op)r(erators)-118 1307 y FN(T)-57 1277 y FL(k)-17 1307 y FO(\()p FN(T)76 1277 y FM(\003)113 1307 y FO(\))145 1277 y FL(k)186 1307 y FO(,)h(\()p FN(T)330 1277 y FM(\003)368 1307 y FO(\))400 1277 y FL(k)441 1307 y FN(T)502 1277 y FL(k)569 1307 y FO(form)f(a)g(comm)n(utativ)n(e)d(family)-7 b(,)24 b(i.e.,)j(for)g(all)f FN(k)s(;)14 b(j)27 b FP(2)d FI(N)375 1468 y FO([)p FN(T)459 1434 y FL(k)498 1468 y FO(\()p FN(T)591 1434 y FM(\003)629 1468 y FO(\))661 1434 y FL(k)702 1468 y FN(;)14 b(T)800 1434 y FL(j)834 1468 y FO(\()p FN(T)927 1434 y FM(\003)965 1468 y FO(\))997 1434 y FL(j)1032 1468 y FO(])23 b(=)g([)p FN(T)1250 1434 y FL(k)1290 1468 y FO(\()p FN(T)1383 1434 y FM(\003)1420 1468 y FO(\))1452 1434 y FL(k)1493 1468 y FN(;)14 b FO(\()p FN(T)1623 1434 y FM(\003)1661 1468 y FO(\))1693 1434 y FL(j)1728 1468 y FN(T)1789 1434 y FL(j)1823 1468 y FO(])639 1606 y(=)22 b([\()p FN(T)842 1572 y FM(\003)880 1606 y FO(\))912 1572 y FL(k)953 1606 y FN(T)1014 1572 y FL(k)1054 1606 y FN(;)14 b FO(\()p FN(T)1184 1572 y FM(\003)1221 1606 y FO(\))1253 1572 y FL(j)1289 1606 y FN(T)1350 1572 y FL(j)1384 1606 y FO(])23 b(=)f(0)p FN(:)544 b FO(\(2.76\))6 1767 y(In)25 b(this)e(section)g(w)n(e)h(study)h(b)r(ounded)f(cen)n (tered)g(op)r(erators.)34 b(W)-7 b(e)24 b(rewrite)-118 1867 y(relations)29 b(\(2.76\))j(in)f(the)i(form)e(whic)n(h)g(enables)g (one)h(to)g(construct)f(a)h(com-)-118 1966 y(m)n(utativ)n(e)c(mo)r(del) h(for)h(cen)n(tered)g(op)r(erators,)g(and)g(sho)n(w)g(that)h(the)g (problem)-118 2066 y(of)f(the)g(unitary)f(classi\014cation)c(of)30 b(cen)n(tered)f(op)r(erators)f(is)h(not)h(wild)e(in)h(the)-118 2166 y(sense)c(discussed)e(in)i(Chapter)g(3.)35 b(Similarly)20 b(to)25 b(Section)g(2.1.1,)f(w)n(e)h(describ)r(e)-118 2265 y(cen)n(tered)k(op)r(erators)g(suc)n(h)g(that)i(k)n(er)13 b FN(T)38 b FP(6)p FO(=)27 b FP(f)p FO(0)p FP(g)i FO(or)g(k)n(er)13 b FN(T)1706 2235 y FM(\003)1770 2265 y FP(6)p FO(=)27 b FP(f)p FO(0)p FP(g)p FO(.)43 b(W)-7 b(e)31 b(also)-118 2365 y(describ)r(e,)j(up)h(to)f(the)h(unitary)e(equiv)-5 b(alence,)33 b(all)f(\014nite-dimensional)d(irre-)-118 2464 y(ducible)d(cen)n(tered)h(op)r(erators.)-118 2605 y FQ(1.)46 b FO(W)-7 b(e)31 b(will)d(rewrite)h(relations)f(\(2.76\))i (in)g(a)h(form)e(that)i(will)d(enable)i(us)h(to)-118 2705 y(in)n(v)n(estigate)f(the)j(relations)d(using)h(the)j(formalism)28 b(of)33 b(dynamical)c(systems)-118 2804 y(dev)n(elop)r(ed)h(ab)r(o)n(v) n(e)g(in)g(this)h(c)n(hapter.)47 b(W)-7 b(e)32 b(will)c(sho)n(w)j(that) h(irreducible)27 b(\(or)-118 2904 y(factor\))j(represen)n(tations)f(of) i(the)g(relations)d(fall)h(in)n(to)h(t)n(w)n(o)g(cases:)43 b(the)32 b(case)-118 3004 y(where)i(k)n(er)13 b FN(T)34 b FP([)23 b FO(k)n(er)13 b FN(T)601 2974 y FM(\003)672 3004 y FO(=)34 b FP(f)p FO(0)p FP(g)f FO(and)i(degenerate)e(cases)g (\(similarly)c(to)34 b(the)-118 3103 y(W)-7 b(old)27 b(decomp)r(osition)d(of)j(isometries\),)d(whic)n(h)j(are)f(studied)h (separately)-7 b(.)6 3203 y(Let)30 b FN(T)37 b FO(=)26 b FN(U)9 b(C)36 b FO(b)r(e)29 b(the)h(p)r(olar)e(decomp)r(osition)e(of) j(the)h(op)r(erator)e FN(T)12 b FO(.)41 b(W)-7 b(e)-118 3303 y(also)27 b(in)n(tro)r(duce)g(the)i(comm)n(uting)c(self-adjoin)n (t)i(op)r(erators)f FN(A)1855 3315 y FL(k)1921 3303 y FO(=)f FN(T)2072 3272 y FL(k)2112 3303 y FO(\()p FN(T)2205 3272 y FM(\003)2242 3303 y FO(\))2274 3272 y FL(k)2316 3303 y FO(,)-118 3402 y FN(B)-55 3414 y FL(k)9 3402 y FO(=)d(\()p FN(T)189 3372 y FM(\003)227 3402 y FO(\))259 3372 y FL(k)300 3402 y FN(T)361 3372 y FL(k)401 3402 y FO(,)j FN(k)h FP(\025)d FO(1,)h(and)h(denote)f(the)h(join)n(t)e (resolution)e(of)k(the)g(iden)n(tit)n(y)-118 3502 y(of)i(the)h(comm)n (utativ)n(e)c(self-adjoin)n(t)h(family)g(\()p FN(A)1390 3514 y FL(k)1431 3502 y FN(;)14 b(B)1531 3514 y FL(k)1572 3502 y FO(\))1604 3514 y FL(k)q FM(2)p Fu(N)1760 3502 y FO(b)n(y)27 b FN(E)1936 3514 y Fm(A)p FL(;)p Fm(B)2070 3502 y FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\).)-118 3650 y FQ(Prop)s(osition)30 b(61.)41 b FB(The)32 b(r)l(elations)39 b FO(\(2.76\))31 b FB(ar)l(e)h(e)l(quivalent)g(to)g(the)g(fol)t(low-) -118 3750 y(ing)7 b FO(:)77 3911 y FN(E)138 3923 y Fm(A)p FL(;)p Fm(B)271 3911 y FO(\(\001\))p FN(U)33 b FO(=)23 b FN(U)9 b(E)c FO(\()p FN(F)811 3877 y FM(\000)p FK(1)900 3911 y FO(\(\001\)\))p FN(;)184 b FO(\001)24 b FP(2)f Fz(B)p FO(\()p FI(R)1603 3877 y Fu(N)1673 3911 y FP(\002)18 b FI(R)1810 3877 y Fu(N)1862 3911 y FO(\))p FN(;)209 b FO(\(2.77\))p eop %%Page: 186 190 186 189 bop -118 -137 a FO(186)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(wher)l(e)30 b(the)g(mapping)h FN(F)654 66 y FM(\000)p FK(1)744 96 y FO(\()p FP(\001)p FO(\))f FB(is)g(de\014ne)l(d)g(by)-35 295 y FN(F)30 261 y FM(\000)p FK(1)119 295 y FO(\()p FQ(x)p FN(;)14 b FQ(y)q FO(\))25 b(=)d FN(F)498 261 y FM(\000)p FK(1)587 295 y FO(\(\()p FN(x)698 307 y FK(1)737 295 y FN(;)14 b(x)821 307 y FK(2)858 295 y FN(;)g(:)g(:)g(:)g FO(\))p FN(;)g FO(\()p FN(y)1148 307 y FK(1)1185 295 y FN(;)g(y)1263 307 y FK(2)1300 295 y FN(;)g(:)g(:)g(:)g FO(\)\))241 503 y(=)329 361 y Fy(\()396 447 y FO(\(\()p FN(x)507 459 y FK(2)545 447 y FN(=x)634 459 y FK(1)671 447 y FN(;)g(x)755 459 y FK(3)793 447 y FN(=x)882 459 y FK(1)919 447 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)h FO(\()p FN(x)1214 459 y FK(1)1253 447 y FN(;)g(y)1331 459 y FK(1)1367 447 y FN(x)1414 459 y FK(1)1452 447 y FN(;)g(y)1530 459 y FK(2)1567 447 y FN(x)1614 459 y FK(1)1652 447 y FN(;)g(:)g(:)g(:)f FO(\)\))p FN(;)86 b(x)2019 459 y FK(1)2080 447 y FP(6)p FO(=)22 b(0)p FN(;)396 566 y FO(\(\(0)p FN(;)14 b FO(0)p FN(;)g(:)g(:)g(:)f FO(\))p FN(;)h FO(\(0)p FN(;)g FO(0)p FN(;)g(:)g(:)g(:)f FO(\)\))p FN(;)788 b(x)2019 578 y FK(1)2080 566 y FO(=)22 b(0)p FN(:)6 747 y FB(The)31 b(phase)g FN(U)39 b FB(of)30 b(the)g(op)l(er)l(ator)h FN(T)41 b FB(is)30 b(a)g(c)l(enter)l(e)l(d)f (op)l(er)l(ator.)-118 906 y(Pr)l(o)l(of.)43 b FO(F)-7 b(or)27 b(an)n(y)g FN(k)f FP(2)d FI(N)261 1079 y FN(A)323 1091 y FL(k)364 1079 y FN(U)9 b(C)29 b FO(=)23 b FN(T)667 1045 y FL(k)707 1079 y FO(\()p FN(T)800 1045 y FM(\003)837 1079 y FO(\))869 1045 y FL(k)910 1079 y FN(T)34 b FO(=)23 b FN(T)12 b(A)1204 1091 y FL(k)q FM(\000)p FK(1)1329 1079 y FN(B)1392 1091 y FK(1)1453 1079 y FO(=)22 b FN(U)9 b(C)d(A)1733 1091 y FL(k)q FM(\000)p FK(1)1859 1079 y FN(B)1922 1091 y FK(1)518 1204 y FO(=)23 b FN(U)9 b(A)734 1216 y FL(k)q FM(\000)p FK(1)860 1204 y FN(B)923 1216 y FK(1)960 1204 y FN(C)q(:)-118 1386 y FO(Denote)28 b(b)n(y)g FN(P)40 b FO(the)28 b(pro)5 b(jection)26 b(on)i(\(k)n(er)13 b FN(C)6 b FO(\))1284 1356 y FM(?)1341 1386 y FO(.)38 b(Then,)28 b(since)f(k)n(er)13 b FN(C)30 b FO(=)23 b(k)n(er)13 b FN(U)-118 1486 y FO(and)30 b FN(U)9 b(P)39 b FO(=)26 b FN(U)9 b FO(,)31 b(w)n(e)f(ha)n(v)n(e)f FN(A)796 1498 y FL(k)837 1486 y FN(U)36 b FO(=)26 b FN(U)9 b(A)1149 1498 y FL(k)q FM(\000)p FK(1)1275 1486 y FN(B)1338 1498 y FK(1)1376 1486 y FO(.)44 b(Similarly)25 b(one)k(can)h(obtain)-118 1586 y(the)e(equalities)c FN(A)454 1598 y FK(1)492 1586 y FN(U)31 b FO(=)23 b FN(U)9 b(B)797 1598 y FK(1)862 1586 y FO(and)27 b FN(B)1086 1598 y FL(k)1127 1586 y FN(A)1189 1598 y FK(1)1226 1586 y FN(U)32 b FO(=)23 b FN(U)9 b(B)1532 1598 y FL(k)q FK(+1)1656 1586 y FO(.)6 1685 y(Applying)26 b(similar)e(argumen)n(ts)h(as)i(in)g(Section)g (2.2.2)g(to)h FN(A)1897 1697 y FL(k)1938 1685 y FO(,)g FN(A)2051 1697 y FL(k)q FM(\000)p FK(1)2205 1685 y FO(and)-118 1785 y FN(B)-55 1797 y FK(1)-18 1785 y FO(,)21 b(one)d(can)h(sho)n(w)f (that)h(for)g(an)n(y)f(Borel)f(set)h(\001)24 b FP(\032)e FI(R)1514 1755 y Fu(N)1585 1785 y FO(the)e(follo)n(wing)15 b(relation)-118 1884 y(holds:)342 2058 y FN(E)403 2070 y FL(B)453 2078 y Fx(1)486 2070 y FL(;)p Fm(A)566 2058 y FO(\()p FI(R)25 b FP(\002)18 b FO(\001\))p FN(U)32 b FO(=)23 b FN(U)9 b(E)1165 2070 y FL(B)1215 2078 y Fx(1)1247 2070 y FL(;)p Fm(A)1328 2058 y FO(\()p FN(F)1425 2022 y FM(\000)p FK(1)1413 2080 y(1)1514 2058 y FO(\()p FI(R)25 b FP(\002)18 b FO(\001\)\))p FN(;)262 b FO(\(2.78\))-118 2232 y(where)243 2405 y FN(F)308 2370 y FM(\000)p FK(1)296 2427 y(1)397 2405 y FO(\()p FN(y)470 2417 y FK(1)507 2405 y FN(;)14 b(x)591 2417 y FK(1)629 2405 y FN(;)g(x)713 2417 y FK(2)750 2405 y FN(;)g(x)834 2417 y FK(3)872 2405 y FN(;)g(:)g(:)g(:)f FO(\))24 b(=)e(\()p FN(x)1241 2417 y FK(1)1279 2405 y FN(;)14 b(x)1363 2417 y FK(2)1401 2405 y FN(=x)1490 2417 y FK(1)1527 2405 y FN(;)g(x)1611 2417 y FK(3)1649 2405 y FN(=x)1738 2417 y FK(1)1775 2405 y FN(;)g(:)g(:)g(:)f FO(\))p FN(:)-118 2579 y FO(Note)18 b(that)h(the)g(righ)n(t-hand)d(side)i(of)24 b(\(2.78\))18 b(can)g(b)r(e)h(de\014ned)g(ev)n(en)f(for)g FN(x)2126 2591 y FK(1)2186 2579 y FO(=)23 b(0.)-118 2678 y(Indeed,)29 b(since)e FN(E)441 2690 y FL(B)491 2698 y Fx(1)524 2690 y FL(;)p Fm(A)604 2678 y FO(\()p FP(f)p FO(0)p FP(g)18 b(\002)g FI(R)25 b FP(\002)18 b FI(R)25 b FP(\002)19 b(\001)14 b(\001)g(\001)g FO(\))28 b(is)f(a)h(pro)5 b(jection)27 b(on)h(k)n(er)13 b FN(B)2213 2690 y FK(1)2274 2678 y FO(=)-118 2778 y(k)n(er)g FN(U)c FO(,)33 b(the)g(whole)e(expression)f (is)h(zero.)50 b(Th)n(us,)34 b(for)e(con)n(v)n(enience,)f(w)n(e)h(can) -118 2878 y(set)455 2977 y FN(F)520 2942 y FM(\000)p FK(1)508 2999 y(1)609 2977 y FO(\()p FN(y)682 2989 y FK(1)720 2977 y FN(;)14 b FO(0)p FN(;)g(x)883 2989 y FK(2)920 2977 y FN(;)g(x)1004 2989 y FK(3)1041 2977 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g(\(0)p FN(;)14 b FO(0)p FN(;)g FO(0)p FN(;)g(:)g(:)g(:)e FO(\))p FN(:)6 3121 y FO(In)41 b(a)g(similar)36 b(manner,)42 b(one)f(can)f(easily)e(deriv)n (e)h(from)g(the)i(equation)-118 3221 y FN(B)-55 3233 y FL(k)-14 3221 y FN(A)48 3233 y FK(1)85 3221 y FN(U)32 b FO(=)23 b FN(U)9 b(B)391 3233 y FL(k)q FK(+1)543 3221 y FO(the)28 b(equalit)n(y)326 3394 y FN(E)387 3406 y FL(A)437 3414 y Fx(1)469 3406 y FL(;)p Fm(B)546 3394 y FO(\()p FI(R)d FP(\002)18 b FO(\001\))p FN(U)907 3360 y FM(\003)969 3394 y FO(=)k FN(U)1122 3360 y FM(\003)1160 3394 y FN(E)1221 3406 y FL(A)1271 3414 y Fx(1)1304 3406 y FL(;)p Fm(B)1381 3394 y FO(\()p FN(F)1478 3359 y FM(\000)p FK(1)1466 3416 y(2)1567 3394 y FO(\()p FI(R)j FP(\002)18 b FO(\001\)\))232 b(\(2.79\))-118 3568 y(with)266 3667 y FN(F)331 3632 y FM(\000)p FK(1)319 3690 y(2)420 3667 y FO(\()p FN(x)499 3679 y FK(1)537 3667 y FN(;)14 b(y)615 3679 y FK(1)652 3667 y FN(;)g(y)730 3679 y FK(2)767 3667 y FN(;)g(y)845 3679 y FK(3)882 3667 y FN(;)g(:)g(:)g(:)f FO(\))24 b(=)e(\()p FN(y)1245 3679 y FK(1)1283 3667 y FN(;)14 b(y)1361 3679 y FK(2)1397 3667 y FN(=y)1480 3679 y FK(1)1517 3667 y FN(;)g(y)1595 3679 y FK(3)1632 3667 y FN(=y)1715 3679 y FK(1)1751 3667 y FN(;)g(:)g(:)g(:)g FO(\))p FN(:)-118 3811 y FO(P)n(assing)37 b(to)i(the)i(adjoin)n(t)d(op) r(erators)g(in)h(\(2.79\))g(and)h(com)n(bining)c(it)j(with)-118 3911 y(\(2.78\))27 b(w)n(e)g(get)g(relations)e(\(2.77\))o(.)p eop %%Page: 187 191 187 190 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(187)6 96 y(The)28 b(fact)g(that)g FN(U)36 b FO(is)27 b(cen)n(tered)g(follo)n(ws) d(directly)i(from)g(\(2.77\))o(.)6 198 y(The)k(pro)r(of)e(that)i(the)g (collection)25 b(of)30 b(non-negativ)n(e)c(self-adjoin)n(t)h(op)r(era-) -118 297 y(tors)22 b(\()p FN(A)138 309 y FL(k)180 297 y FN(;)14 b(B)280 309 y FL(k)321 297 y FO(\))23 b(and)h(the)f(cen)n (tered)g FN(U)32 b FO(\(whic)n(h)23 b(is)f(a)h(partial)e(isometry\))g (whic)n(h)-118 397 y(satisfy)26 b(relations)f(\(2.77\))i(generates)g(a) h(cen)n(tered)f(op)r(erator)f(is)h(a)h(direct)e(cal-)-118 497 y(culation.)p 2278 497 4 57 v 2282 444 50 4 v 2282 497 V 2331 497 4 57 v 6 676 a(It)f(is)f(a)g(standard)g(argumen)n(t)e (similar)e(to)k(the)h(one)g(in)e(Section)h(2.2.2)f(that)-118 776 y(allo)n(ws)16 b(one)j(to)g(pro)n(v)n(e)f(the)i(follo)n(wing)15 b(decomp)r(osition)h(of)k(cen)n(tered)f(op)r(erators)-118 876 y(similar)k(to)k(the)h(W)-7 b(old)27 b(decomp)r(osition)d(of)k (isometries.)-118 1047 y FQ(Prop)s(osition)i(62.)41 b FB(L)l(et)e FN(T)50 b FB(b)l(e)40 b(a)g(c)l(enter)l(e)l(d)f(op)l(er)l (ator)i(in)e(a)h(Hilb)l(ert)g(sp)l(ac)l(e)-118 1146 y FN(H)7 b FB(.)49 b(The)35 b(sp)l(ac)l(e)f FN(H)40 b FB(c)l(an)33 b(b)l(e)h(de)l(c)l(omp)l(ose)l(d)h(into)e(the)h(dir)l(e)l(ct)g(sum)f (of)h(the)f(two)-118 1246 y(invariant)h(with)g(r)l(esp)l(e)l(ct)f(to)g FN(T)12 b FB(,)34 b FN(T)987 1216 y FM(\003)1057 1246 y FB(subsp)l(ac)l(es,)h FN(H)i FO(=)29 b FN(H)1723 1258 y FK(0)1781 1246 y FP(\010)21 b FN(H)1936 1258 y FK(1)2006 1246 y FB(such)34 b(that)-118 1346 y FO(k)n(er)13 b FN(T)29 b FP([)19 b FO(k)n(er)13 b FN(T)345 1316 y FM(\003)405 1346 y FO(=)23 b FP(f)p FO(0)p FP(g)28 b FB(in)i FN(H)818 1358 y FK(1)855 1346 y FB(,)g(and)g FN(H)1140 1358 y FK(0)1207 1346 y FB(is)g(gener)l(ate)l(d)h(by)f FO(k)n(er)13 b FN(T)29 b FP([)19 b FO(k)n(er)13 b FN(T)2236 1316 y FM(\003)2273 1346 y FB(.)6 1517 y FO(The)19 b(degenerate)e(represen)n (tations)f(can)i(b)r(e)h(completely)c(describ)r(ed)i(up)i(to)-118 1617 y(a)f(unitary)f(equiv)-5 b(alence)16 b(\(see)i(b)r(elo)n(w\).)33 b(The)19 b(structure)f(of)g(represen)n(tations)d(in)-118 1716 y FN(H)-49 1728 y FK(1)8 1716 y FO(is)k(more)e(complicated;)i(ho)n (w)n(ev)n(er,)h(realization)15 b(of)20 b(cen)n(tered)f(op)r(erators)e (as)-118 1816 y(\\op)r(erator-v)-5 b(alued)17 b(w)n(eigh)n(ted)i (shifts")g(follo)n(ws)e(from)i(Theorem)g(45)h(if)g(applied)-118 1915 y(to)27 b(relations)e(\(2.77\))o(.)-118 2087 y FQ(Theorem)30 b(46.)41 b FB(L)l(et)f FN(T)52 b FB(b)l(e)41 b(a)g(c)l(enter)l(e)l(d)f (op)l(er)l(ator)i(with)f(zer)l(o)g(kernel)g(and)-118 2186 y(dense)22 b(image.)37 b(Then)22 b(it)f(c)l(an)h(b)l(e)f(r)l(e)l (alize)l(d)i(in)e(the)h(sp)l(ac)l(e)g FN(L)1670 2198 y FK(2)1707 2186 y FO(\()p FI(R)1793 2156 y FM(1)1793 2207 y FK(+)1870 2186 y FP(\002)p FI(R)1989 2156 y FM(1)1989 2207 y FK(+)2065 2186 y FN(;)14 b FA(H)q FN(;)g(d\026)p FO(\))-118 2286 y FB(of)33 b(ve)l(ctor-value)l(d)f(squar)l(e-inte)l(gr) l(able)g(functions)g(having)h(their)g(values)f(in)g(a)-118 2385 y(c)l(ertain)e(Hilb)l(ert)g(sp)l(ac)l(e)g FA(H)f FB(by)i(the)e(formula)4 2583 y FO(\()p FN(T)12 b(f)d FO(\)\()p FQ(x)p FN(;)14 b FQ(y)q FO(\))24 b(=)f FN(x)540 2540 y FK(1)p FL(=)p FK(2)540 2605 y(1)644 2583 y FN(u)p FO(\()p FQ(x)p FN(;)14 b FQ(y)q FO(\)\()p FN(d\026)p FO(\()p FN(F)e FO(\()p FQ(x)p FN(;)i FQ(y)q FO(\)\))p FN(=d\026)p FO(\()p FQ(x)p FN(;)g FQ(y)q FO(\)\))1719 2549 y FK(1)p FL(=)p FK(2)1829 2583 y FN(f)9 b FO(\))p FN(F)j FO(\()p FQ(x)p FN(;)i FQ(y)q FO(\))p FN(;)2126 2683 y FO(\(2.80\))-118 2870 y FB(wher)l(e)39 b FN(F)12 b FO(\()p FP(\001)p FN(;)i FP(\001)p FO(\))38 b FB(is)h(intr)l(o)l(duc) l(e)l(d)f(ab)l(ove,)k FN(\026)p FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))39 b FB(is)g(a)f FN(F)12 b FO(\()p FP(\001)p FN(;)i FP(\001)p FO(\))p FB(-quasi-invariant)-118 2969 y(pr)l(ob)l(ability)24 b(Bor)l(el)e(me)l(asur)l(e)f(and)h FN(u)p FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))22 b FB(is)g(a)f(unitary)h(me)l(asur)l(able)g(op)l(er)l(ator-)-118 3069 y(value)l(d)30 b(function.)6 3170 y(Conversely,)45 b(any)40 b(c)l(ol)t(le)l(ction)h FA(H)q FB(,)h FN(\026)p FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))p FB(,)43 b(and)d FN(u)p FO(\()p FP(\001)p FN(;)14 b FP(\001)p FO(\))40 b FB(which)i(has)e(the)-118 3270 y(mentione)l(d)e(pr)l(op)l (erties)h(gener)l(ates)f(a)g(c)l(enter)l(e)l(d)g(op)l(er)l(ator)g(by)h (the)f(formula)-118 3369 y(ab)l(ove.)6 3541 y FO(Represen)n(tations)c (with)h(non-zero)f(k)n(ernel)g(can)h(b)r(e)h(completely)d(classi-)-118 3640 y(\014ed.)-118 3811 y FQ(Theorem)d(47.)41 b FB(A)n(l)t(l)i(irr)l (e)l(ducible)i(r)l(epr)l(esentations)e(of)62 b FO(\(2.76\))42 b FB(for)i(which)-118 3911 y FO(k)n(er)13 b FN(T)29 b FP([)19 b FO(k)n(er)13 b FN(T)345 3881 y FM(\003)405 3911 y FP(6)p FO(=)23 b FP(f)p FO(0)p FP(g)28 b FB(fal)t(l)j(into)f (the)g(fol)t(lowing)i(classes)7 b FO(:)p eop %%Page: 188 192 188 191 bop -118 -137 a FO(188)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-45 96 y FO(\()p FN(i)p FO(\))41 b FB (\014nite-dimensional)31 b(r)l(epr)l(esentations)f(in)g FI(C)1497 66 y FL(n)419 279 y FN(T)12 b(e)519 291 y FL(j)576 279 y FO(=)23 b FN(\025)712 291 y FL(j)747 279 y FN(e)786 291 y FL(j)s FK(+1)905 279 y FN(;)183 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)k FP(\000)g FO(1)p FN(;)43 b(\025)1794 291 y FL(j)1853 279 y FN(>)23 b FO(0)p FN(;)409 403 y(T)12 b(e)509 415 y FL(n)576 403 y FO(=)23 b(0;)-74 619 y(\()p FN(ii)p FO(\))41 b FB(in\014nite-dimensional)31 b(in)f FN(l)965 631 y FK(2)497 801 y FN(T)12 b(e)597 813 y FL(j)653 801 y FO(=)23 b FN(\025)789 813 y FL(j)825 801 y FN(e)864 813 y FL(j)s FK(+1)982 801 y FN(;)184 b(j)28 b FO(=)22 b(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)f(;)44 b(\025)1721 813 y FL(j)1779 801 y FN(>)23 b FO(0;)-102 1016 y(\()p FN(iii)p FO(\))40 b FB(in\014nite-dimensional)31 b(in)f FN(l)965 1028 y FK(2)299 1198 y FN(T)12 b(e)399 1210 y FK(1)458 1198 y FO(=)23 b(0)p FN(;)98 b(T)12 b(e)809 1210 y FL(j)866 1198 y FO(=)23 b FN(\025)1002 1210 y FL(j)1037 1198 y FN(e)1076 1210 y FL(j)s FM(\000)p FK(1)1196 1198 y FN(;)183 b(j)28 b FO(=)23 b(2)p FN(;)14 b FO(3)p FN(;)g(:)g(:)g(:)e(;)28 b(\025)1918 1210 y FL(j)1977 1198 y FN(>)22 b FO(0)p FN(:)-118 1414 y FB(Pr)l(o)l(of.)43 b FO(Indeed,)34 b(the)f(phase)e FN(U)42 b FO(of)32 b(the)h(op)r(erator) d FN(T)44 b FO(is)31 b(a)h(cen)n(tered)g(partial)-118 1513 y(isometry)e(\(non-unitary\);)35 b(in)e(the)g(same)f(w)n(a)n(y)g (as)h(in)f(Theorem)g(19)g(w)n(e)h(get)-118 1613 y(that)27 b(the)g(represen)n(tation)e(space)h(for)g(an)h(irreducible)c FN(T)38 b FO(is)26 b(the)i(same)d(as)h(for)-118 1713 y(an)i(irreducible)d FN(U)9 b FO(.)40 b(Use)29 b(Theorem)e(18)g(to)i (represen)n(t)e(the)i(op)r(erator)e FN(U)9 b FO(;)29 b(the)-118 1812 y(rest)e(of)h(the)g(pro)r(of)f(follo)n(ws)d (immediately)f(from)j(\(2.77\))o(.)p 2278 1812 4 57 v 2282 1760 50 4 v 2282 1812 V 2331 1812 4 57 v -118 1978 a FQ(3.)49 b FO(Finally)-7 b(,)30 b(w)n(e)h(giv)n(e)f(a)i(complete)e (list)g(of)i(\014nite-dimensional)26 b(irreducible)-118 2078 y(cen)n(tered)39 b(op)r(erators.)72 b(W)-7 b(e)40 b(ha)n(v)n(e)f(already)f(seen)h(that)i(there)e(is)g(a)h(family)-118 2178 y(of)c(\014nite-dimensional)c(represen)n(tations)h(\(Theorem)j (47\).)63 b(In)37 b(fact,)i(there)-118 2277 y(are)31 b(some)g(irreducible)e(\014nite-dimensional)e(represen)n(tations)j (with)i(a)g(non-)-118 2377 y(degenerate)26 b FN(T)12 b FO(.)-118 2543 y FQ(Theorem)30 b(48.)41 b FB(A)n(l)t(l)c(irr)l(e)l (ducible)h(\014nite-dimensional)g(r)l(epr)l(esentations)g(of)-118 2642 y FO(\(2.76\))31 b FB(ar)l(e)h(either)h(the)f (\014nite-dimensional)h(r)l(epr)l(esentations)g(describ)l(e)l(d)g(by) -118 2742 y(The)l(or)l(em)k FO(47)29 b FB(or)h(r)l(epr)l(esentations)g (of)g(the)g(form)6 b FO(:)79 2924 y FN(T)12 b(e)179 2936 y FL(j)235 2924 y FO(=)23 b FN(\025)371 2936 y FL(j)406 2924 y FN(e)445 2936 y FL(j)s FK(+1)564 2924 y FN(;)184 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)g(;)g(n)k FP(\000)g FO(1)p FN(;)68 3049 y(T)12 b(e)168 3061 y FL(n)235 3049 y FO(=)23 b FN(\013\025)424 3061 y FL(n)470 3049 y FN(e)509 3061 y FK(1)546 3049 y FN(;)184 b(\025)801 3061 y FL(j)859 3049 y FN(>)23 b FO(0)p FN(;)43 b(\025)1103 3061 y FL(j)1161 3049 y FP(6)p FO(=)23 b FN(\025)1297 3061 y FL(k)1398 3049 y FB(for)60 b FN(j)28 b FP(6)p FO(=)23 b FN(k)s(;)98 b FP(j)p FN(\013)p FP(j)24 b FO(=)e(1)p FN(:)-118 3231 y FB(Pr)l(o)l(of.)43 b FO(Indeed,)35 b(it)e(is)g(easy)f (to)i(see)f(that)g(represen)n(tation)e(\(2.80\))i(is)f(\014nite-)-118 3330 y(dimensional)d(only)j(if)h(the)h(measure)d FN(\026)p FO(\()p FP(\001)p FO(\))k(is)d(concen)n(trated)g(on)i(a)f(p)r(erio)r (dic)-118 3430 y(orbit)24 b(of)h FN(F)12 b FO(\()p FP(\001)p FO(\).)37 b(Irreducibilit)n(y)21 b(in)j(this)h(case)f(implies)e(that)j (the)h(sp)r(ectrum)f(of)-118 3530 y(the)k(op)r(erator)d FN(B)424 3542 y FK(1)490 3530 y FO(is)h(simple.)37 b(P)n(assing)25 b(to)j(a)g(unitarily)d(equiv)-5 b(alen)n(t)26 b(realiza-)-118 3629 y(tion)h(w)n(e)g(get)g(the)h(necessary)e(form)n(ulae.)p 2278 3629 V 2282 3577 50 4 v 2282 3629 V 2331 3629 4 57 v -118 3795 a FQ(Corollary)32 b(6.)40 b FB(A)n(ny)28 b(factor)i(gener)l(ate)l(d)f(by)g(a)f(c)l(enter)l(e)l(d)h(op)l(er)l (ator)g(is)g(hyp)l(er-)-118 3895 y(\014nite.)p eop %%Page: 189 193 189 192 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(189)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Indeed,)d(\(2.77\))c(imply)f(that)i(the)h (corresp)r(onding)c(v)n(on)i(Neumann)-118 196 y(algebra)28 b(is)i(either)g(of)i(t)n(yp)r(e)f FN(I)7 b FO(,)32 b(if)f(the)g (represen)n(tation)e(is)h(degenerate,)h(or)f(a)-118 296 y(crossed)f(pro)r(duct)i(of)f(a)g(comm)n(utativ)n(e)d(algebra)h(b)n(y)i (the)h(group)f FI(Z)o FO(.)40 b(By)30 b([75])-118 395 y(w)n(e)d(get)g(the)h(assertion.)p 2278 395 4 57 v 2282 343 50 4 v 2282 395 V 2331 395 4 57 v -118 595 a FQ(Corollary)k(7.)40 b FB(Sinc)l(e)33 b(for)h(a)g(p)l(air)g(of)g(self-adjoint)h(op)l(er)l (ators)f(ther)l(e)f(exists)-118 695 y(a)g(non-hyp)l(er\014nite)h (factor)g(r)l(epr)l(esentation,)g(the)g(description)h(pr)l(oblem)f(for) -118 794 y(c)l(enter)l(e)l(d)28 b(op)l(er)l(ators)h(by)g(the)f(pr)l (evious)h(statement)e(is)i(not)f(wild)h(in)f(the)h(sense)-118 894 y(discusse)l(d)h(in)g(Se)l(ction)g FO(3)p FN(:)p FO(1)p FB(.)-118 1132 y FQ(2.5.3)94 b(Represen)m(tations)30 b(of)i(Cun)m(tz)h(algebras)-118 1293 y FO(W)-7 b(e)32 b(consider)d(represen)n(tations)f(of)j(the)h(Cun)n(tz)f(algebras)e FA(O)1817 1305 y FL(n)1862 1293 y FO(.)47 b(Recall)29 b(that)-118 1393 y(the)i(Cun)n(tz)g(algebra)d(is)i(generated)g(b)n(y)h FN(n)f FO(isometries,)e FN(S)1707 1405 y FK(1)1744 1393 y FO(,)j FN(:)14 b(:)g(:)28 b FO(,)k FN(S)2029 1405 y FL(n)2074 1393 y FO(,)g(whic)n(h)-118 1492 y(satisfy)26 b(the)i(follo)n(wing)c(relations)337 1745 y FN(S)393 1711 y FM(\003)388 1766 y FL(i)431 1745 y FN(S)482 1757 y FL(i)532 1745 y FO(=)f FN(I)7 b(;)180 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)1440 1641 y FL(n)1401 1666 y Fy(X)1407 1843 y FL(i)p FK(=1)1535 1745 y FN(S)1586 1757 y FL(i)1613 1745 y FN(S)1669 1711 y FM(\003)1664 1766 y FL(i)1730 1745 y FO(=)23 b FN(I)7 b(:)242 b FO(\(2.81\))-118 2009 y(Notice)26 b(that)i(the)g(relations)d(imply)g(that)j FN(S)1272 1979 y FM(\003)1267 2030 y FL(i)1309 2009 y FN(S)1360 2021 y FL(j)1418 2009 y FO(=)23 b(0)k(for)g FN(i)c FP(6)p FO(=)g FN(j)5 b FO(.)6 2113 y(Our)22 b(goal)d(is)i(to)h (construct)f(a)h(comm)n(utativ)n(e)c(mo)r(del)i(for)i(represen)n (tations)-118 2212 y(of)29 b(the)i(Cun)n(tz)e(algebra)e(and)j(to)f(sho) n(w)g(ho)n(w)g(it)g(can)g(b)r(e)h(used)g(to)g(study)f(rep-)-118 2312 y(resen)n(tations)g(of)i(the)h(Cun)n(tz)g(algebra.)45 b(The)32 b(main)d(statemen)n(t)i(here)g(is)f(the)-118 2412 y(follo)n(wing)23 b(theorem.)-118 2590 y FQ(Theorem)30 b(49.)41 b FB(F)-6 b(or)34 b(any)g(r)l(epr)l(esentation)g(of)h(the)f (Cuntz)f(algebr)l(a)i FA(O)2152 2602 y FL(n)2230 2590 y FB(the)-118 2690 y(fol)t(lowing)d(form)f(for)f FN(S)618 2702 y FK(1)655 2690 y FB(,)h FN(:)14 b(:)g(:)27 b FB(,)j FN(S)941 2702 y FL(n)1016 2690 y FB(holds)7 b FO(:)442 2934 y FN(H)30 b FO(=)629 2821 y Fy(Z)712 2842 y FM(\010)675 3010 y Fu(Z)720 2993 y Fw(1)720 3026 y Fv(n)792 2934 y FN(H)861 2946 y FL(x)917 2934 y FN(d\026)p FO(\()p FN(x)p FO(\))p FN(;)-93 3129 y FO(\()p FN(S)-10 3141 y FL(i)18 3129 y FN(f)9 b FO(\)\()p FN(x)179 3141 y FK(1)217 3129 y FN(;)14 b(x)301 3141 y FK(2)338 3129 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(\016)666 3141 y FL(i)693 3129 y FO(\()p FN(x)772 3141 y FK(1)810 3129 y FO(\))14 b FN(U)913 3141 y FL(i)941 3129 y FO(\()p FN(x)1020 3141 y FK(2)1058 3129 y FN(;)g(x)1142 3141 y FK(3)1179 3129 y FN(;)g(:)g(:)g(:)g FO(\))621 3336 y FP(\002)704 3219 y Fy(\022)775 3280 y FN(d)p FO(\()p FN(\016)887 3292 y FL(i)915 3280 y FO(\()p FN(x)994 3292 y FK(1)1032 3280 y FO(\))19 b FP(\012)f FN(\026)p FO(\()p FN(x)1295 3292 y FK(2)1333 3280 y FN(;)c(x)1417 3292 y FK(3)1455 3280 y FN(;)g(:)g(:)g(:)f FO(\))p 775 3317 860 4 v 949 3393 a FN(d\026)p FO(\()p FN(x)1121 3405 y FK(1)1159 3393 y FN(;)h(x)1243 3405 y FK(2)1281 3393 y FN(;)g(:)g(:)g(:)g FO(\))1645 3219 y Fy(\023)1706 3236 y FK(1)p FL(=)p FK(2)1824 3336 y FN(f)9 b FO(\()p FN(x)1953 3348 y FK(2)1991 3336 y FN(;)14 b(x)2075 3348 y FK(3)2112 3336 y FN(;)g(:)g(:)g(:)g FO(\))p FN(;)-108 3518 y FO(\()p FN(S)-20 3483 y FM(\003)-25 3538 y FL(i)18 3518 y FN(f)9 b FO(\)\()p FN(x)179 3530 y FK(1)217 3518 y FN(;)14 b(x)301 3530 y FK(2)338 3518 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(U)695 3483 y FM(\003)686 3538 y FL(i)733 3518 y FO(\()p FN(x)812 3530 y FK(1)850 3518 y FN(;)14 b(x)934 3530 y FK(2)971 3518 y FN(;)g(:)g(:)g(:)g FO(\))621 3724 y FP(\002)704 3607 y Fy(\022)775 3668 y FN(d\026)p FO(\()p FN(i;)g(x)1013 3680 y FK(1)1051 3668 y FN(;)g(x)1135 3680 y FK(2)1173 3668 y FN(;)g(:)g(:)g(:)f FO(\))p 775 3705 578 4 v 808 3781 a FN(d\026)p FO(\()p FN(x)980 3793 y FK(1)1018 3781 y FN(;)h(x)1102 3793 y FK(2)1140 3781 y FN(;)g(:)g(:)g(:)g FO(\))1363 3607 y Fy(\023)1424 3624 y FK(1)p FL(=)p FK(2)1542 3724 y FN(f)9 b FO(\()p FN(i;)14 b(x)1737 3736 y FK(1)1774 3724 y FN(;)g(x)1858 3736 y FK(2)1896 3724 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)2126 3881 y FO(\(2.82\))p eop %%Page: 190 194 190 193 bop -118 -137 a FO(190)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(Her)l(e,)k FN(\026)p FO(\()p FP(\001)p FO(\))f FB(is)g(a)h(pr)l(ob)l(ability)h(me)l(asur)l(e)d (de\014ne)l(d)h(on)g(the)g(cylinder)i FN(\033)s FB(-algebr)l(a,)-118 196 y(quasi-invariant)k(with)g(r)l(esp)l(e)l(ct)f(to)g(the)h(tr)l (ansformations)g FN(\026)p FO(\()p FN(x)1889 208 y FK(1)1927 196 y FN(;)14 b(x)2011 208 y FK(2)2048 196 y FN(;)g(:)g(:)g(:)g FO(\))28 b FP(7!)-118 296 y FN(\016)-81 308 y FL(i)-54 296 y FO(\()p FN(x)25 308 y FK(1)63 296 y FO(\))d FP(\012)e FN(\026)p FO(\()p FN(x)337 308 y FK(2)375 296 y FN(;)14 b(x)459 308 y FK(3)497 296 y FN(;)g(:)g(:)g(:)f FO(\))p FB(,)40 b FN(i)c FO(=)h(1)p FB(,)i FO(2)p FB(,)e FN(:)14 b(:)g(:)51 b FO(;)41 b FN(H)1399 308 y FL(x)1478 296 y FB(is)d(a)f(me)l(asur)l(able)h(\014eld)g(of)-118 395 y(Hilb)l(ert)32 b(sp)l(ac)l(es)h(such)g(that)f FN(d)p FO(\()p FN(x)p FO(\))d(=)f(dim)12 b FN(H)1271 407 y FL(x)1345 395 y FB(is)33 b(invariant)g FN(\026)p FB(-a.e.)48 b(with)33 b(r)l(e-)-118 495 y(sp)l(e)l(ct)e(to)g(the)h(tr)l(ansformations)g FN(d)p FO(\()p FN(x)1050 507 y FK(1)1088 495 y FN(;)14 b(x)1172 507 y FK(2)1210 495 y FN(;)g(:)g(:)g(:)f FO(\))27 b FP(7!)f FN(d)p FO(\()p FN(i;)14 b(x)1713 507 y FK(1)1750 495 y FN(;)g(x)1834 507 y FK(2)1872 495 y FN(;)g(:)g(:)g(:)g FO(\);)32 b FN(U)2164 507 y FK(1)2201 495 y FO(\()p FN(x)p FO(\))p FB(,)-118 595 y FN(:)14 b(:)g(:)27 b FB(,)k FN(U)119 607 y FL(n)163 595 y FO(\()p FN(x)p FO(\))g FB(ar)l(e)f(me)l(asur)l (able)h(unitary)e(op)l(er)l(ator-value)l(d)i(functions.)6 694 y(The)f(er)l(go)l(dicity)h(of)f(the)f(sp)l(e)l(ctr)l(al)h(me)l (asur)l(e)e FN(\026)h FB(is)h(a)f(ne)l(c)l(essary)g(c)l(ondition)-118 794 y(of)40 b(the)g(irr)l(e)l(ducibility)i(of)f(the)e(r)l(epr)l (esentation)6 b FO(;)46 b FB(in)40 b(the)g(c)l(ase)g(of)g(a)g(simple) -118 893 y(joint)32 b(sp)l(e)l(ctrum)38 b FO(\()p FB(if)33 b FN(H)623 905 y FL(x)696 893 y FB(ar)l(e)g(one-dimensional)g FN(\026)p FB(-a.e.)p FO(\))p FB(,)i(the)d(er)l(go)l(dicity)h(is)-118 993 y(also)e(su\016cient)e(for)i(the)f(irr)l(e)l(ducibility.)6 1093 y(Two)j(r)l(epr)l(esentations)f(of)g(the)g(form)39 b FO(\(2.82\))30 b FB(ar)l(e)i(unitary)g(e)l(quivalent)g(if)-118 1192 y(and)e(only)g(if)19 b FO(:)6 1292 y FB(i.)40 b(the)29 b(sp)l(e)l(ctr)l(al)h(me)l(asur)l(es)g FN(\026)g FB(and)36 b FO(~)-48 b FN(\026)29 b FB(ar)l(e)i(e)l(quivalent)8 b FO(;)6 1392 y FB(ii.)47 b(the)32 b(multiplicity)h(functions)f FN(d)p FO(\()p FN(x)p FO(\))c(=)e(dim)13 b FN(H)1565 1404 y FL(x)1638 1392 y FB(and)1816 1370 y FO(~)1802 1392 y FN(d)p FO(\()p FN(x)p FO(\))28 b(=)f(dim)2250 1371 y(~)2228 1392 y FN(H)2297 1404 y FL(x)-118 1491 y FB(c)l(oincide)k FN(\026)p FB(-a.e.)p FO(;)6 1591 y FB(iii.)87 b(the)45 b(c)l(ol)t(le)l(ctions)i(of)f(unitary)f(op)l(er)l (ator)h(functions)f FO(\()p FN(U)1990 1603 y FL(i)2018 1591 y FO(\()p FN(x)p FO(\)\))h FB(and)-118 1690 y FO(\()-71 1669 y(~)-86 1690 y FN(U)-29 1702 y FL(i)-1 1690 y FO(\()p FN(x)p FO(\)\))40 b FB(ar)l(e)g(e)l(quivalent)g(in)g(the)g(fol)t (lowing)i(sense)6 b FO(:)58 b FB(ther)l(e)39 b(exists)h(a)g(me)l(a-) -118 1790 y(sur)l(able)30 b(unitary)g(op)l(er)l(ator-value)l(d)h (function)f FN(V)18 b FO(\()p FN(x)p FO(\))31 b FB(such)f(that)-35 1995 y FN(V)32 1961 y FM(\003)70 1995 y FO(\()p FN(x)149 2007 y FK(1)187 1995 y FN(;)14 b(x)271 2007 y FK(2)309 1995 y FN(;)g(:)g(:)g(:)f FO(\))h FN(U)568 1961 y FM(\003)559 2016 y FL(i)606 1995 y FO(\()p FN(x)685 2007 y FK(1)723 1995 y FN(;)g(x)807 2007 y FK(2)845 1995 y FN(;)g(:)g(:)g(:)f FO(\))h FN(V)19 b FO(\(1)p FN(;)14 b(x)1263 2007 y FK(1)1301 1995 y FN(;)g(x)1385 2007 y FK(2)1422 1995 y FN(;)g(:)g(:)g(:)g FO(\))1001 2135 y(=)1103 2114 y(~)1088 2135 y FN(U)1154 2101 y FM(\003)1145 2155 y FL(i)1192 2135 y FO(\()p FN(x)1271 2147 y FK(1)1309 2135 y FN(;)g(x)1393 2147 y FK(2)1431 2135 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)184 b(i)23 b FO(=)g(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n:)6 2315 y FB(Conversely,)27 b(a)c(family)i(c)l(onsisting)e(of)h(a)f(quasi-invariant)h(me)l(asur)l (e)e FN(\026)p FO(\()p FP(\001)p FO(\))p FB(,)-118 2415 y(a)g(dimension)i(function)e FN(d)p FO(\()p FP(\001)p FO(\))p FB(,)j(and)e(a)f(c)l(ol)t(le)l(ction)i(of)f(unitary)f(op)l(er)l (ator-value)l(d)-118 2514 y(functions)31 b FO(\()p FN(U)333 2526 y FL(i)361 2514 y FO(\()p FP(\001)p FO(\)\))p FB(,)i(p)l (ossessing)f(the)g(liste)l(d)g(pr)l(op)l(erties,)h(determines)f(a)f(r)l (ep-)-118 2614 y(r)l(esentation)f(of)g(the)g(Cuntz)f(algebr)l(a)i FA(O)1129 2626 y FL(n)1174 2614 y FB(.)-118 2778 y(Pr)l(o)l(of.)43 b FO(First)27 b(w)n(e)h(in)n(tro)r(duce)f(some)g(notations.)37 b(F)-7 b(or)27 b(an)n(y)h(m)n(ulti-index)c FN(\013)h FO(=)-118 2878 y(\()p FN(\013)-33 2890 y FK(1)5 2878 y FN(;)14 b(:)g(:)g(:)f(;)h(\013)242 2890 y FL(s)278 2878 y FO(\),)35 b FN(\013)421 2890 y FL(j)488 2878 y FP(2)e FI(Z)637 2890 y FL(n)676 2878 y FO(,)i FN(s)d FP(\025)g FO(0,)i(write)e FN(S)1270 2890 y FL(\013)1349 2878 y FO(=)g FN(S)1497 2890 y FL(\013)1540 2898 y Fx(1)1591 2878 y FN(:)14 b(:)g(:)f(S)1752 2890 y FL(\013)1795 2898 y Fv(s)1832 2878 y FO(,)34 b FN(P)1942 2890 y FL(\013)2022 2878 y FO(=)e FN(S)2170 2890 y FL(\013)2217 2878 y FN(S)2273 2848 y FM(\003)2268 2898 y FL(\013)2316 2878 y FO(.)-118 2978 y(The)g(set)g(of)f(all)f(\014nite)h(m)n(ulti-indices)c FN(\013)32 b FO(will)d(b)r(e)j(denoted)g(b)n(y)g(\000.)49 b(The)32 b(fol-)-118 3077 y(lo)n(wing)24 b(statemen)n(t)j(is)g (obtained)f(b)n(y)h(an)g(easy)g(direct)g(calculation.)-118 3227 y FQ(Prop)s(osition)j(63.)41 b FB(The)28 b(op)l(er)l(ators)g FN(P)1140 3239 y FL(\013)1188 3227 y FB(,)g(wher)l(e)g FN(\013)g FB(r)l(anges)g(over)g FO(\000)p FB(,)g(form)g(a)-118 3326 y(c)l(ommuting)h(family)j(of)e(pr)l(oje)l(ctions.)40 b(A)n(lso,)131 3507 y FN(S)182 3519 y FL(i)210 3507 y FN(P)263 3519 y FL(\013)306 3527 y Fx(1)339 3519 y FL(:::)o(\013)441 3527 y Fv(s)500 3507 y FO(=)23 b FN(P)641 3519 y FL(i\013)707 3527 y Fx(1)740 3519 y FL(:::)o(\013)842 3527 y Fv(s)879 3507 y FN(S)930 3519 y FL(i)957 3507 y FN(;)99 b(S)1135 3472 y FM(\003)1130 3527 y FL(i)1173 3507 y FN(P)1226 3519 y FL(\013)1269 3527 y Fx(1)1302 3519 y FL(:::)o(\013)1404 3527 y Fv(s)1463 3507 y FO(=)23 b FN(\016)1588 3519 y FL(i\013)1654 3527 y Fx(1)1691 3507 y FN(P)1744 3519 y FL(\013)1787 3527 y Fx(2)1820 3519 y FL(:::)o(\013)1922 3527 y Fv(s)1958 3507 y FN(S)2014 3472 y FM(\003)2009 3527 y FL(i)2052 3507 y FN(;)372 3631 y(i;)14 b(\013)491 3643 y FK(1)528 3631 y FN(;)g(:)g(:)g(:)g(;)g(\013)766 3643 y FL(s)824 3631 y FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)183 b(s)23 b FO(=)g(1)p FN(;)14 b FO(2)p FN(;)g(:)g(:)g(:)26 b(:)278 b FO(\(2.83\))6 3811 y(According)25 b(to)h(Theorem)f(45,)h (relations)d(\(2.83\))j(imply)d(that)k(the)g(op)r(era-)-118 3911 y(tors)i FN(S)102 3923 y FK(1)139 3911 y FO(,)h FN(:)14 b(:)g(:)28 b FO(,)i FN(S)421 3923 y FL(n)496 3911 y FO(act)g(as)f(w)n(eigh)n(ted)g(op)r(erator-v)-5 b(alued)26 b(shifts)k(on)f(the)i(join)n(t)p eop %%Page: 191 195 191 194 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(191)-118 96 y(sp)r(ectrum)30 b(of)g(the)h(comm)n(uting)c(family)h(\()p FN(P)1267 108 y FL(\013)1315 96 y FO(\))1347 108 y FL(\013)p FM(2)p FK(\000)1511 96 y FO(in)h(the)i(space)f(of)h(F)-7 b(ourier)-118 196 y(images)35 b(relativ)n(e)g(to)k(this)e(comm)n(uting) e(family)-7 b(.)67 b(Our)37 b(further)i(task)f(is)f(to)-118 296 y(describ)r(e)28 b(the)i(join)n(t)f(sp)r(ectrum)g(of)h(the)g(comm)n (uting)c(family)h(of)j(pro)5 b(jections)-118 395 y(men)n(tioned)23 b(ab)r(o)n(v)n(e,)h(and)h(to)g(study)g(the)g(corresp)r(onding)e (dynamical)e(system)-118 495 y(on)27 b(it.)6 595 y(According)19 b(to)i(the)g(sp)r(ectral)e(theorem)h(for)g(an)h(in\014nite)e(comm)n (uting)f(fam-)-118 694 y(ily)32 b(of)i(b)r(ounded)h(self-adjoin)n(t)d (op)r(erators)g(\(see,)k(for)e(example,)f([228)o(])h(etc.\),)-118 794 y(for)27 b(an)n(y)g FN(\013)c FP(2)h FO(\000)j(w)n(e)g(ha)n(v)n(e) 620 1007 y FN(P)673 1019 y FL(\013)743 1007 y FO(=)831 894 y Fy(Z)877 1083 y FM(f)p FK(0)p FL(;)p FK(1)p FM(g)1031 1066 y Fx(\000)1088 1007 y FN(\025)p FO(\()p FN(\013)p FO(\))14 b FN(dE)5 b FO(\()p FN(\025)p FO(\()p FP(\001)p FO(\)\))p FN(;)-118 1252 y FO(where)35 b FP(f)p FO(0)p FN(;)14 b FO(1)p FP(g)335 1222 y FK(\000)415 1252 y FP(3)38 b FN(\025)p FO(\()p FP(\001)p FO(\))f(is)e(a)g(set)h(of)g(measurable)c (functions)k(on)f(\000)h(taking)-118 1352 y(v)-5 b(alues)27 b(0)h(or)g(1,)g(and)h FN(E)5 b FO(\()p FP(\001)p FO(\))29 b(is)e(the)i(join)n(t)f(resolution)d(of)k(the)g(iden)n(tit)n(y)d(for)i (the)-118 1451 y(comm)n(uting)i(family)h(of)i(pro)5 b(jections)32 b(de\014ned)i(on)f(the)h(cylinder)d FN(\033)s FO(-algebra)-118 1551 y(in)c FP(f)p FO(0)p FN(;)14 b FO(1)p FP(g)184 1521 y FK(\000)227 1551 y FO(.)6 1650 y(F)-7 b(or)27 b(an)n(y)g(\()p FN(i)373 1662 y FK(1)410 1650 y FN(;)14 b(:)g(:)g(:)g(;)g(i)624 1662 y FL(k)664 1650 y FO(\))24 b FP(2)f FO(\000,)28 b(w)n(e)f(ha)n(v)n(e)75 1789 y FL(n)36 1814 y Fy(X)24 1990 y FL(m)p FK(=1)181 1893 y FN(P)234 1905 y FL(i)257 1913 y Fx(1)290 1905 y FL(:::)o(i)372 1914 y Fv(k)409 1905 y FL(m)495 1893 y FO(=)634 1789 y FL(n)595 1814 y Fy(X)583 1990 y FL(m)p FK(=1)740 1893 y FN(S)791 1905 y FL(i)814 1913 y Fx(1)865 1893 y FN(:)14 b(:)g(:)f(S)1026 1905 y FL(i)1049 1914 y Fv(k)1090 1893 y FN(S)1141 1905 y FL(m)1204 1893 y FN(S)1260 1859 y FM(\003)1255 1913 y FL(m)1318 1893 y FN(S)1374 1859 y FM(\003)1369 1913 y FL(i)1392 1922 y Fv(k)1447 1893 y FN(:)h(:)g(:)f(S)1613 1859 y FM(\003)1608 1913 y FL(i)1631 1921 y Fx(1)495 2168 y FO(=)23 b FN(S)634 2180 y FL(i)657 2188 y Fx(1)708 2168 y FN(:)14 b(:)g(:)g(S)870 2180 y FL(i)893 2189 y Fv(k)933 2051 y Fy(\022)1045 2065 y FL(n)1006 2090 y Fy(X)994 2265 y FL(m)p FK(=1)1151 2168 y FN(S)1202 2180 y FL(m)1265 2168 y FN(S)1321 2134 y FM(\003)1316 2189 y FL(m)1379 2051 y Fy(\023)1440 2168 y FN(S)1496 2134 y FM(\003)1491 2189 y FL(i)1514 2198 y Fv(k)1569 2168 y FN(:)g(:)g(:)g(S)1736 2134 y FM(\003)1731 2189 y FL(i)1754 2197 y Fx(1)1814 2168 y FO(=)22 b FN(P)1954 2180 y FL(i)1977 2188 y Fx(1)2011 2180 y FL(;:::)n(;i)2132 2189 y Fv(k)2173 2168 y FN(:)-118 2420 y FO(Then,)i(according)19 b(to)k([25)o(],)g(the)g (sp)r(ectral)e(measure)f(of)j(the)g(comm)n(uting)c(fam-)-118 2520 y(ily)26 b(is)g(concen)n(trated)g(on)i(functions)f FN(\025)p FO(\()p FP(\001)p FO(\))d FP(2)f(f)p FO(0)p FN(;)14 b FO(1)p FP(g)1490 2490 y FK(\000)1561 2520 y FO(suc)n(h)27 b(that)528 2659 y FL(n)489 2684 y Fy(X)477 2859 y FL(m)p FK(=1)634 2762 y FN(\025)p FO(\()p FN(i)743 2774 y FK(1)781 2762 y FN(;)14 b(:)g(:)g(:)f(;)h(i)994 2774 y FL(k)1035 2762 y FN(;)g(m)p FO(\))23 b(=)f FN(\025)p FO(\()p FN(i)1396 2774 y FK(1)1434 2762 y FN(;)14 b(:)g(:)g(:)g(;)g(i) 1648 2774 y FL(k)1688 2762 y FO(\))p FN(:)6 3014 y FO(Let)41 b FN(\025)p FO(\()p FP(\001)p FO(\))h(b)r(e)f(suc)n(h)g(a)f(function.) 76 b(Then)41 b FN(\025)p FO(\()p FI(?)p FO(\))h(is)d(either)h(0)g(or)g (1.)76 b(If)-118 3114 y FN(\025)p FO(\()p FI(?)p FO(\))32 b(=)f(0,)i(then)g FN(\025)p FO(\(1\))23 b(+)e FP(\001)14 b(\001)g(\001)22 b FO(+)f FN(\025)p FO(\()p FN(n)p FO(\))32 b(=)f(0)h(and)h FN(\025)p FO(\(1\))f(=)f FP(\001)14 b(\001)g(\001)31 b FO(=)g FN(\025)p FO(\()p FN(n)p FO(\))h(=)f(0,)-118 3214 y(etc.,)k(and)e(therefore,)h FN(\025)p FO(\()p FP(\001)p FO(\))f FP(\021)f FO(0.)54 b(This)32 b(implies)e(that)j FN(E)5 b FO(\()p FN(\025)p FO(\()p FP(\001)p FO(\)\))34 b(=)f(0,)h(since)-118 3313 y(the)e(latter)f(pro)5 b(jection)30 b(is)g(in)i(the)g(k)n(ernels)e(of)h(all)f FN(P)1548 3325 y FL(i)1576 3313 y FO(,)j(whic)n(h)e(is)g(imp)r(ossible)-118 3413 y(b)r(ecause)c FN(P)242 3425 y FK(1)298 3413 y FO(+)18 b FP(\001)c(\001)g(\001)k FO(+)g FN(P)632 3425 y FL(m)719 3413 y FO(=)k FN(I)7 b FO(.)6 3513 y(T)-7 b(o)24 b(an)n(y)g(function)g FN(\025)p FO(\()p FP(\001)p FO(\))i(with)e FN(\025)p FO(\()p FI(?)p FO(\))f(=)g(1)h(there)g(uniquely)f(corresp)r(onds)f(a) -118 3612 y(sequence)29 b(\()p FN(x)307 3624 y FK(1)345 3612 y FN(;)14 b(x)429 3624 y FK(2)466 3612 y FN(;)g(:)g(:)g(:)g FO(\))26 b FP(2)g(f)p FO(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)p FP(g)1113 3582 y FM(1)1208 3612 y FO(=)26 b FI(Z)1360 3582 y FM(1)1360 3633 y FL(n)1454 3612 y FO(suc)n(h)j(that)g(for)g(an)n (y)g FN(k)g FO(=)c(1,)-118 3712 y(2,)32 b FN(:)14 b(:)g(:)46 b FO(w)n(e)32 b(ha)n(v)n(e)f FN(\025)p FO(\()p FN(x)572 3724 y FK(1)611 3712 y FN(;)14 b(:)g(:)g(:)f(;)h(x)842 3724 y FL(k)883 3712 y FO(\))32 b(=)e(1)i(and)g(vice)f(v)n(ersa.)50 b(T)-7 b(o)32 b(a)g(cylinder)e(set)-118 3811 y(in)f FI(Z)42 3781 y FM(1)42 3832 y FL(n)136 3811 y FO(there)g(corresp)r(onds)e(a)i (cylinder)e(set)j(in)e FP(f)p FO(0)p FN(;)14 b FO(1)p FP(g)1631 3781 y FK(\000)1675 3811 y FO(,)30 b(whic)n(h)e(enables)g(us) -118 3911 y(to)e(de\014ne)g(the)g(image)e(of)i(the)g(pro)5 b(jection-v)-5 b(alued)23 b(measure)h FN(E)5 b FO(\()p FP(\001)p FO(\))26 b(under)g(the)p eop %%Page: 192 196 192 195 bop -118 -137 a FO(192)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(mapping)30 b FN(\036)9 b FO(:)30 b FI(Z)400 66 y FM(1)400 117 y FL(n)496 96 y FP(\000)-49 b(!)31 b(f)p FO(0)p FN(;)14 b FO(1)p FP(g)831 66 y FK(\000)907 96 y FO(whic)n(h)31 b(w)n(as)g(just)i(de\014ned;)i(this)d(image)d(will) -118 196 y(also)20 b(b)r(e)i(denoted)g(b)n(y)g FN(E)5 b FO(\()p FP(\001)p FO(\))22 b(as)g(long)e(as)h(this)g(will)f(not)h (lead)g(to)h(an)n(y)f(confusion.)6 300 y(Therefore,)33 b(the)g(sp)r(ectral)e(decomp)r(osition)e(of)j(the)h(comm)n(uting)d (family)-118 399 y(of)d(pro)5 b(jections)26 b FN(P)457 411 y FL(\013)504 399 y FO(,)i FN(\013)c FP(2)f FO(\000,)28 b(tak)n(es)e(the)i(form)194 628 y FN(P)247 640 y FL(\013)317 628 y FO(=)405 515 y Fy(Z)451 704 y Fu(Z)496 687 y Fw(1)496 720 y Fv(n)569 628 y FN(\037)621 640 y FL(\013)p FM(\002)p Fu(Z)760 624 y Fw(1)760 657 y Fv(n)819 628 y FO(\()p FN(x)p FO(\))14 b FN(dE)5 b FO(\()p FN(x)p FO(\))25 b(=)e FN(E)5 b FO(\()p FN(\013)1428 640 y FK(1)1466 628 y FN(;)14 b(:)g(:)g(:)f(;)h(\013)1703 640 y FL(k)1763 628 y FP(\002)k FI(Z)1907 594 y FM(1)1907 649 y FL(n)1971 628 y FO(\))p FN(:)6 882 y FO(No)n(w,)28 b(Theorem)d(45)i(implies)d(the)k(statemen)n (t.)p 2278 882 4 57 v 2282 829 50 4 v 2282 882 V 2331 882 4 57 v 6 1081 a(The)35 b(established)c(theorem)i(pro)n(vides)f(a)i (w)n(a)n(y)f(to)h(construct)f(represen-)-118 1180 y(tations)c(of)h(the) g(Cun)n(tz)h(algebra.)41 b(Indeed,)31 b(one)f(should)f(tak)n(e)h(an)f (appropri-)-118 1280 y(ate)e(quasi-in)n(v)-5 b(arian)n(t)21 b(measure)k(on)i FI(Z)1081 1250 y FM(1)1081 1301 y FL(n)1146 1280 y FO(,)g(c)n(ho)r(ose)f(an)g(in)n(v)-5 b(arian)n(t)24 b(m)n(ultiplicit)n(y)-118 1380 y(function,)j(and)h(a)f(unitary)f(op)r (erator-v)-5 b(alued)24 b(function.)-118 1521 y FB(R)l(emark)30 b(48.)42 b FO(The)22 b(constructed)g(comm)n(utativ)n(e)c(mo)r(del)h (can)j(also)d(b)r(e)k(rewrit-)-118 1620 y(ten)29 b(in)f(the)h(space)f (of)h(\(v)n(ector-v)-5 b(alued,)27 b(in)h(general\))e(functions)i(on)h (the)g(unit)-118 1720 y(in)n(terv)-5 b(al)25 b(as)i(follo)n(ws.)6 1824 y(T)-7 b(o)27 b(eac)n(h)g(p)r(oin)n(t)f FN(x)i FO(of)f(the)h(unit) f(in)n(terv)-5 b(al)24 b([0)p FN(;)14 b FO(1\))27 b(w)n(e)g(put)h(in)n (to)e(corresp)r(on-)-118 1923 y(dence)k(its)f FN(n)p FO(-ary)f(represen)n(tation)f FN(x)h FO(=)e(0)p FN(:x)1281 1935 y FK(1)1319 1923 y FN(x)1366 1935 y FK(2)1417 1923 y FN(:)14 b(:)g(:)g FO(.)44 b(This)28 b(corresp)r(ondence)-118 2023 y(is)g(one-to-one)f(except)i(for)f(the)i(p)r(oin)n(ts)d(with)i (tails)e(consisting)f(of)j FN(n)p FO('s)f(\(they)-118 2123 y(are)k(iden)n(ti\014ed)f(with)h(p)r(oin)n(ts)g(ha)n(ving)f (\014nite)h(decomp)r(osition\).)49 b(With)33 b(suc)n(h)-118 2222 y(a)25 b(corresp)r(ondence,)f(cylinder)f(sets)j(in)f FI(Z)1181 2192 y FM(1)1181 2243 y FL(n)1271 2222 y FO(corresp)r(ond)f (to)h(Borel)f(sets)h(from)-118 2322 y([0)p FN(;)14 b FO(1\),)27 b(and)g(the)h(represen)n(tation)d(\(2.82\))i(tak)n(es)g(the) h(form)269 2555 y FN(H)i FO(=)456 2442 y Fy(Z)502 2631 y FK([0)p FL(;)p FK(1\))650 2555 y FN(H)719 2567 y FL(\025)777 2555 y FN(dE)5 b FO(\()p FN(\025)p FO(\))p FN(;)39 2748 y FO(\()p FN(S)122 2760 y FL(i)150 2748 y FN(f)k FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(\037)508 2782 y FO([)541 2750 y Fv(i)p Fw(\000)p Fx(1)p 541 2763 96 3 v 570 2796 a Fv(n)646 2776 y FL(;)683 2754 y Fv(i)p 676 2763 37 3 v 676 2796 a(n)722 2782 y FO(\))759 2748 y(\()p FN(\025)p FO(\))14 b FN(U)942 2760 y FL(i)970 2748 y FO(\()p FN(n\025)19 b FP(\000)f FO(\()p FN(i)g FP(\000)g FO(1\)\))446 2979 y FP(\002)529 2862 y Fy(\022)601 2923 y FN(d\026)p FO(\()p FN(n\025)h FP(\000)f FO(\()p FN(i)g FP(\000)g FO(1\)\))p 601 2960 594 4 v 794 3036 a FN(d\026)p FO(\()p FN(\025)p FO(\))1204 2862 y Fy(\023)1265 2880 y FK(1)p FL(=)p FK(2)1370 2979 y FN(f)9 b FO(\()p FN(n\025)18 b FP(\000)g FO(\()p FN(i)h FP(\000)f FO(1\)\))p FN(;)24 3235 y FO(\()p FN(S)112 3201 y FM(\003)107 3255 y FL(i)150 3235 y FN(f)9 b FO(\)\()p FN(\025)p FO(\))24 b(=)f FN(U)522 3201 y FM(\003)513 3255 y FL(i)560 3235 y FO(\()p FN(\025)p FO(\))686 3118 y Fy(\022)757 3179 y FN(d\026)p FO(\(\()p FN(\025)d FO(+)e FN(i)g FP(\000)g FO(1\))p FN(=n)p FO(\))p 757 3216 636 4 v 972 3292 a FN(d\026)p FO(\()p FN(\025)p FO(\))1403 3118 y Fy(\023)1464 3135 y FK(1)p FL(=)p FK(2)1582 3235 y FN(f)9 b FO(\(\()p FN(\025)19 b FO(+)f FN(i)g FP(\000)g FO(1\))p FN(=n)p FO(\))p FN(;)-118 3471 y FO(whic)n(h)27 b(holds)g(for)g(all)f(represen)n(tations)f(that)j(do)g(not)g(con)n (tain)e(a)i(single)d(rep-)-118 3571 y(resen)n(tation)34 b(related)h(to)h(the)h(orbit)e(of)h(the)h(p)r(oin)n(t)e(\()p FN(n;)14 b(n;)g(:)g(:)g(:)g FO(\))38 b FP(2)g FI(Z)2099 3541 y FM(1)2099 3591 y FL(n)2200 3571 y FO(\(see)-118 3670 y(example)25 b(b)r(elo)n(w\).)-118 3811 y FB(Example)31 b(18.)42 b FO(\(Represen)n(tations)33 b(related)g(to)h(a)g(single)e (orbit\).)56 b(The)35 b(sim-)-118 3911 y(plest)27 b(class)g(of)h (irreducible)c(represen)n(tations)i(can)h(b)r(e)i(obtained)e(as)h (follo)n(ws.)p eop %%Page: 193 197 193 196 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(193)-118 96 y(T)-7 b(ak)n(e)26 b(an)h(arbitrary)d(p)r(oin)n(t)i FN(x)e FO(=)f(\()p FN(x)1004 108 y FK(1)1042 96 y FN(;)14 b(x)1126 108 y FK(2)1163 96 y FN(;)g(:)g(:)g(:)g FO(\))23 b FP(2)h FI(Z)1506 66 y FM(1)1506 117 y FL(n)1570 96 y FO(.)37 b(The)27 b(simplest)e(quasi-)-118 196 y(in)n(v)-5 b(arian)n(t)29 b(ergo)r(dic)i(measure)f(is)i(an)g(atomic)e(measure)h FN(\026)i FO(concen)n(trated)e(on)-118 296 y(the)39 b(orbit)d(of)j(a)f (p)r(oin)n(t)f FN(x)i FO(with)f(resp)r(ect)g(to)g(the)h(action)d(giv)n (en)h(in)g(\(2.82\).)-118 395 y(The)32 b(orbit)f(consists)f(of)i(all)e (p)r(oin)n(ts)h(ha)n(ving)f(similar)e(tails,)j(i.e.,)h(the)h(p)r(oin)n (ts)-118 495 y FN(y)50 b FO(=)d(\()p FN(y)158 507 y FK(1)195 495 y FN(;)14 b(y)273 507 y FK(2)310 495 y FN(;)g(:)g(:)g(:)g FO(\))42 b(suc)n(h)g(that)h(there)f(exist)f FN(l)49 b FP(\025)e FO(1,)f FN(m)h FP(2)h FI(Z)35 b FO(suc)n(h)42 b(that)-118 595 y FN(x)-71 607 y FL(k)-3 595 y FO(=)26 b FN(y)129 607 y FL(k)q FK(+)p FL(m)309 595 y FO(for)j(all)e FN(k)i(>)d(l)r FO(.)43 b(Denote)30 b(suc)n(h)f(an)g(orbit)f(b)n(y)i FN(O)1790 607 y FL(x)1832 595 y FO(.)43 b(In)30 b(this)e(case,)-118 694 y(an)n(y)e(op)r(erator-v)-5 b(alued)24 b(function)j FN(U)9 b FO(\()p FN(x)p FO(\))27 b(is)f(equiv)-5 b(alen)n(t)25 b(to)i(the)g(iden)n(tit)n(y)-7 b(,)26 b(and)-118 794 y(the)i(irreducibilit)n(y)22 b(implies)i(that)k FN(d)p FO(\()p FN(x)p FO(\))c(=)e(1)28 b FN(\026)p FO(-a.e.)6 893 y(No)n(w,)d FN(\016)s FO(-functions)f(concen)n(trated)f(at)h(p)r (oin)n(ts)g(of)g(the)h(orbit)e(form)g(a)h(basis)-118 993 y(of)j FN(H)7 b FO(,)28 b(and)f(the)h(represen)n(tation)d(acts)i (as)g(follo)n(ws:)163 1144 y FN(S)214 1156 y FL(i)255 1144 y FN(\016)292 1159 y FK(\()p FL(x)356 1167 y Fx(1)388 1159 y FL(;x)446 1167 y Fx(2)478 1159 y FL(;:::)10 b FK(\))621 1144 y FO(=)23 b FN(\016)746 1159 y FK(\()p FL(i;x)853 1167 y Fx(1)885 1159 y FL(;x)943 1167 y Fx(2)975 1159 y FL(;:::)10 b FK(\))1095 1144 y FN(;)147 1269 y(S)203 1235 y FM(\003)198 1290 y FL(i)255 1269 y FN(\016)292 1284 y FK(\()p FL(x)356 1292 y Fx(1)388 1284 y FL(;x)446 1292 y Fx(2)478 1284 y FL(;:::)g FK(\))621 1269 y FO(=)23 b FN(\016)746 1281 y FL(i;x)827 1289 y Fx(1)877 1269 y FN(\016)914 1284 y FK(\()p FL(x)978 1292 y Fx(2)1010 1284 y FL(;x)1068 1292 y Fx(3)1100 1284 y FL(;:::)10 b FK(\))1220 1269 y FN(;)180 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n:)265 b FO(\(2.84\))-118 1420 y(Tw)n(o)25 b(suc)n(h)g(represen)n(tations)e(are)h(unitarily)e(equiv)-5 b(alen)n(t)24 b(if)h(and)g(only)f(if)h(they)-118 1520 y(corresp)r(ond)h(to)h(the)h(same)e(orbit.)6 1620 y(As)k(one)f(can)g (easily)e(see,)j(all)d(suc)n(h)i(represen)n(tations)e(fall)g(in)n(to)i (the)h(class)-118 1719 y(of)d(p)r(erm)n(utativ)n(e)f(ones)g([46].)6 1819 y(Dep)r(ending)20 b(on)g(the)g(orbit)f(tak)n(en,)i(the)f(follo)n (wing)c(situations)i(ma)n(y)g(o)r(ccur.)6 1919 y(i.)42 b(The)30 b(p)r(oin)n(t)e FN(x)f FO(=)f(\()p FN(x)730 1931 y FK(1)768 1919 y FN(;)14 b(x)852 1931 y FK(2)889 1919 y FN(;)g(:)g(:)g(:)g FO(\))30 b(tak)n(en)f(to)g(construct)g(the)h (orbit)e(has)h(a)-118 2018 y(constan)n(t)k(tail)f(of)i(the)g(form)e(\() p FN(:)14 b(:)g(:)g(;)g(k)s(;)g(k)s(;)g(k)s(;)g(:)g(:)g(:)f FO(\).)56 b(Then)34 b FN(O)1765 2030 y FL(x)1841 2018 y FO(is)f(exactly)f(the)-118 2118 y(orbit)c(of)i(the)g(p)r(oin)n(t)f (\()p FN(k)s(;)14 b(k)s(;)g(k)s(;)g(:)g(:)g(:)f FO(\).)44 b(The)30 b(op)r(erator)e FN(S)1599 2130 y FL(k)1669 2118 y FO(has)i(a)f(unitary)f(part)-118 2217 y(formed)23 b(b)n(y)i(its)f (restriction)e(to)i(the)h(subspace)g(spanned)f(b)n(y)h(the)g(eigen)n(v) n(ector)-118 2317 y FN(\016)-81 2332 y FK(\()p FL(k)q(;k)q(;:::)11 b FK(\))158 2317 y FO(,)28 b(while)e(the)i(others)e(are)h(m)n(ultiples) d(of)j(the)h(unilateral)c(shift.)6 2417 y(F)-7 b(or)28 b(suc)n(h)g(a)f(represen)n(tation,)f(its)h(restriction)e(to)j(UHF)1774 2429 y FL(n)1848 2417 y FO(is)f(irreducible,)-118 2516 y(since)20 b(the)i(action)e(of)h(elemen)n(ts)f(of)h(the)h(form)e FN(S)1347 2528 y FL(i)1370 2536 y Fx(1)1421 2516 y FN(:)14 b(:)g(:)g(S)1583 2528 y FL(i)1606 2537 y Fv(k)1646 2516 y FN(S)1702 2486 y FM(\003)1697 2538 y FL(j)1724 2546 y Fx(1)1775 2516 y FN(:)g(:)g(:)g(S)1942 2486 y FM(\003)1937 2538 y FL(j)1964 2547 y Fv(k)2026 2516 y FO(act)21 b(tran-)-118 2616 y(sitiv)n(ely)j(on)j(the)h(orbit.)6 2716 y(ii.)87 b(Orbits)43 b(of)i(p)r(erio)r(dic)d(p)r(oin)n(ts)i(\()p FN(x)1233 2728 y FK(1)1271 2716 y FN(;)14 b(x)1355 2728 y FK(2)1393 2716 y FN(;)g(:)g(:)g(:)f(;)h(x)1624 2728 y FL(m)1688 2716 y FN(;)g(x)1772 2728 y FK(1)1809 2716 y FN(;)g(:)g(:)g(:)g FO(\))45 b(generate)-118 2815 y(another)38 b(class)g(of)i(irreducible)35 b(represen)n(tations)i(of)i FA(O)1687 2827 y FL(n)1732 2815 y FO(.)73 b(The)40 b(action)e(of)-118 2915 y(UHF)60 2927 y FL(n)140 2915 y FO(splits)33 b(the)i(orbit)e(in)n (to)g FN(m)h FO(di\013eren)n(t)g(sub-orbits)e(\()p FN(m)j FO(is)e(a)h(minimal)-118 3014 y(p)r(erio)r(d\))g(corresp)r(onding)d(to) k(\()p FN(x)908 3026 y FK(1)946 3014 y FN(;)14 b(x)1030 3026 y FK(2)1068 3014 y FN(;)g(:)g(:)g(:)f(;)h(x)1299 3026 y FL(m)1363 3014 y FN(;)g(:)g(:)g(:)f FO(\),)37 b(\()p FN(x)1681 3026 y FK(2)1719 3014 y FN(;)14 b(x)1803 3026 y FK(3)1841 3014 y FN(;)g(:)g(:)g(:)f(;)h(x)2072 3026 y FL(m)2136 3014 y FN(;)g(:)g(:)g(:)f FO(\),)-118 3114 y FN(:)h(:)g(:)27 b FO(,)34 b(\()p FN(x)142 3126 y FL(m)206 3114 y FN(;)14 b(x)290 3126 y FK(1)327 3114 y FN(;)g(:)g(:)g(:)g(;)g(x)559 3126 y FL(m)p FM(\000)p FK(1)707 3114 y FN(;)g(:)g(:)g(:)g FO(\);)35 b(therefore,)d(suc)n(h)g (a)g(represen)n(tation)d(of)k FA(O)2294 3126 y FL(n)-118 3214 y FO(splits)c(in)n(to)h(a)h(direct)g(sum)f(of)h FN(m)h FO(inequiv)-5 b(alen)n(t)28 b(represen)n(tations)g(of)k(UHF)2293 3226 y FL(n)-118 3313 y FO(corresp)r(onding)25 b(to)i(these)h (sub-orbits.)6 3413 y(iii.)70 b(The)40 b(non-p)r(erio)r(dic)d (\(irrational\))e(p)r(oin)n(ts)k FN(x)k FO(=)g(\()p FN(x)1826 3425 y FK(1)1864 3413 y FN(;)14 b(x)1948 3425 y FK(2)1985 3413 y FN(;)g(:)g(:)g(:)g FO(\))40 b(and)-118 3513 y FN(x)-71 3482 y FM(0)-13 3513 y FO(=)33 b(\()p FN(x)164 3482 y FM(0)164 3533 y FK(1)202 3513 y FN(;)14 b(x)286 3482 y FM(0)286 3533 y FK(2)324 3513 y FN(;)g(:)g(:)g(:)f FO(\))35 b(b)r(elong)e(to)h(the)g(same)f(orbit)g(of)h(UHF)1771 3525 y FL(n)1851 3513 y FO(if)f(and)h(only)f(if)-118 3612 y FN(x)-71 3624 y FL(i)-20 3612 y FP(6)p FO(=)23 b FN(x)115 3582 y FM(0)115 3634 y FL(i)171 3612 y FO(for)k(a)g (\014nite)g(n)n(um)n(b)r(er)g(of)g FN(i)p FO('s.)37 b(This)27 b(implies)d(that)k(the)g(corresp)r(ond-)-118 3712 y(ing)k(represen)n (tation)e(of)j FA(O)733 3724 y FL(n)812 3712 y FO(decomp)r(oses)e(in)n (to)h(an)g(in\014nite)g(direct)g(sum)h(of)-118 3811 y(inequiv)-5 b(alen)n(t)33 b(irreducible)g(represen)n(tations)g(of)k(UHF)1640 3823 y FL(n)1722 3811 y FO(corresp)r(onding)d(to)-118 3911 y(the)28 b(sub-orbits.)p eop %%Page: 194 198 194 197 bop -118 -137 a FO(194)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FB(Example)31 b(19.)42 b FO(Another)34 b(class)d(of)i(irreducible)d(represen)n(tations)g(is)i(related)-118 196 y(to)38 b(pro)r(duct)g(measures.)67 b(An)n(y)38 b(pro)r(duct)h (measure)d FN(\026)i FO(on)g FI(Z)1858 166 y FM(1)1858 217 y FL(n)1960 196 y FO(is)g(ergo)r(dic,)-118 296 y(and)27 b(c)n(ho)r(osing)f FN(H)k FO(=)22 b FN(L)624 308 y FK(2)661 296 y FO(\()p FI(Z)755 266 y FM(1)755 316 y FL(n)819 296 y FN(;)14 b(d\026)p FO(\))28 b(with)f FN(U)9 b FO(\()p FN(x)p FO(\))24 b FP(\021)f FO(1,)k(w)n(e)h(get)f(an)g(irreducible)-118 395 y(represen)n(tation)h(related)i(to)h(the)h(measure)d FN(\026)p FO(.)48 b(Tw)n(o)31 b(suc)n(h)g(represen)n(tations)-118 495 y(are)21 b(unitarily)e(equiv)-5 b(alen)n(t)20 b(if)h(and)h(only)f (if)g(the)i(corresp)r(onding)c(measures)h(are)-118 595 y(equiv)-5 b(alen)n(t.)6 696 y(Since)20 b(an)n(y)f(pro)r(duct)h (measure)e(is)h(ergo)r(dic)f(with)i(resp)r(ect)g(to)g(the)g(action)f (of)-118 795 y(UHF)60 807 y FL(n)106 795 y FO(,)28 b(the)g(restriction) d(of)j(the)g(corresp)r(onding)d(represen)n(tation)g(to)j(UHF)2293 807 y FL(n)-118 895 y FO(remains)d(irreducible.)-118 1030 y FB(Example)31 b(20.)42 b FO(\(A)25 b(family)d(of)i(irreducible)d (represen)n(tations\).)33 b(F)-7 b(or)23 b(eac)n(h)h FN(\013)f FP(2)-118 1130 y FI(T)p FO(,)e(consider)d(the)i(follo)n(wing) c(represen)n(tation)h(of)j(the)h(Cun)n(tz)f(algebra)d FA(O)2115 1142 y FL(n)2160 1130 y FO(.)35 b(Let)-118 1230 y FN(H)g FO(=)28 b FN(L)136 1242 y FK(2)172 1230 y FO(\()p FI(Z)266 1200 y FM(1)266 1250 y FL(n)330 1230 y FN(;)14 b(d\026)p FO(\),)33 b(and)d FN(d\026)p FO(\()p FN(x)p FO(\))g(=)1038 1167 y Fy(N)1131 1188 y FM(1)1131 1255 y FL(k)q FK(=1)1269 1230 y FN(d\026)1362 1242 y FL(k)1403 1230 y FO(\()p FN(x)1482 1242 y FL(k)1524 1230 y FO(\),)i FN(\026)1661 1242 y FL(k)1702 1230 y FO(\()p FN(j)5 b FO(\))29 b(=)e(1)p FN(=n)p FO(,)k FN(j)i FO(=)28 b(1,)-118 1329 y FN(:)14 b(:)g(:)27 b FO(,)h FN(n)p FO(,)g(and)f(the)h (op)r(erators)e(b)r(e)-39 1526 y(\()p FN(S)49 1483 y FK(\()p FL(\013)p FK(\))44 1549 y FL(j)148 1526 y FN(f)9 b FO(\)\()p FN(x)309 1538 y FK(1)347 1526 y FN(;)14 b(x)431 1538 y FK(2)468 1526 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(\013\016)849 1538 y FL(j)884 1526 y FO(\()p FN(x)963 1538 y FK(1)1001 1526 y FO(\))p FN(n)1083 1492 y FK(1)p FL(=)p FK(2)1187 1526 y FN(f)9 b FO(\()p FN(x)1316 1538 y FK(2)1354 1526 y FN(;)14 b(x)1438 1538 y FK(3)1476 1526 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)-78 1687 y FO(\()p FN(S)10 1644 y FK(\()p FL(\013)p FK(\))5 1710 y FL(j)110 1621 y FM(\003)148 1687 y FN(f)c FO(\)\()p FN(x)309 1699 y FK(1)347 1687 y FN(;)14 b(x)431 1699 y FK(2)468 1687 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(\013)812 1653 y FM(\000)p FK(1)901 1687 y FN(n)951 1653 y FM(\000)p FK(1)p FL(=)p FK(2)1107 1687 y FN(f)9 b FO(\()p FN(j;)14 b(x)1307 1699 y FK(1)1345 1687 y FN(;)g(x)1429 1699 y FK(2)1467 1687 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)181 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n:)2126 1800 y FO(\(2.85\))-118 2017 y(A)39 b(direct)f(computation)f(sho)n(ws)h (that)h(the)g(expression)e(for)h FN(S)1927 1974 y FK(\()p FL(\013)p FK(\))1922 2040 y FL(j)2026 1951 y FM(\003)2103 2017 y FO(indeed)-118 2146 y(giv)n(es)22 b(an)i(op)r(erator)f(adjoin)n (t)g(to)h FN(S)958 2103 y FK(\()p FL(\013)p FK(\))953 2170 y FL(j)1057 2146 y FO(,)h(and)g(that)f(\(2.85\))g(is)f(a)h (represen)n(tation)-118 2246 y(of)j(the)h(Cun)n(tz)g(algebra.)6 2347 y(Notice)33 b(that)i(the)f(latter)f(represen)n(tations)e(admit)h (the)j(follo)n(wing)30 b(real-)-118 2447 y(ization)25 b(on)i FN(L)326 2459 y FK(2)363 2447 y FO(\([0)p FN(;)14 b FO(1])p FN(;)g(dt)p FO(\):)84 2648 y(\()p FN(S)172 2605 y FK(\()p FL(\013)p FK(\))167 2671 y FL(j)271 2648 y FN(f)9 b FO(\)\()p FN(t)p FO(\))24 b(=)f FN(\013\037)664 2663 y FK([\()p FL(j)s FM(\000)p FK(1\))p FL(=n;j)s(=n)p FK(])1074 2648 y FO(\()p FN(t)p FO(\))p FN(n)1218 2614 y FK(1)p FL(=)p FK(2)1323 2648 y FN(f)9 b FO(\()p FN(nt)18 b FP(\000)g FO(\()p FN(j)24 b FP(\000)18 b FO(1\)\))p FN(;)46 2809 y FO(\()p FN(S)134 2766 y FK(\()p FL(\013)p FK(\))129 2832 y FL(j)233 2743 y FM(\003)271 2809 y FN(f)9 b FO(\)\()p FN(t)p FO(\))24 b(=)f FN(\013)612 2775 y FM(\000)p FK(1)701 2809 y FN(n)751 2775 y FM(\000)p FK(1)p FL(=)p FK(2)907 2809 y FN(f)9 b FO(\(\()p FN(t)19 b FO(+)f(\()p FN(j)23 b FP(\000)18 b FO(1\))p FN(=n)p FO(\))p FN(;)180 b(j)28 b FO(=)22 b(1)p FN(;)14 b(:)g(:)g(:)g(;)g(n:)-118 3026 y FQ(Prop)s(osition)30 b(64.)41 b FB(The)35 b(r)l(epr)l (esentations)f FO(\()p FN(S)1403 2983 y FK(\()p FL(\013)p FK(\))1398 3049 y FL(j)1502 3026 y FO(\))g FB(ar)l(e)h(unitarily)f(ine) l(quiv-)-118 3125 y(alent)c(irr)l(e)l(ducible)h(r)l(epr)l(esentations)f (of)g(the)g(Cuntz)f(algebr)l(a)i FA(O)1883 3137 y FL(n)1928 3125 y FB(.)-118 3296 y(Pr)l(o)l(of.)43 b FO(The)21 b(irreducibilit)n (y)14 b(of)21 b(represen)n(tations)c(of)k(the)g(form)e(\(2.85\))h (follo)n(ws)-118 3395 y(immediately)e(from)j(the)j(ergo)r(dicit)n(y)19 b(of)k(the)g(pro)r(duct)g(measure)e FN(\026)p FO(.)35 b(W)-7 b(e)24 b(sho)n(w)-118 3495 y(that)k(the)g(represen)n(tations)c (are)j(di\013eren)n(t)g(for)g(di\013eren)n(t)f FN(\013)p FO(.)6 3596 y(Indeed,)j(tak)n(e)e FN(\013)533 3608 y FK(1)599 3596 y FO(and)h FN(\013)814 3608 y FK(2)851 3596 y FO(,)g(and)g(form)f(the)i(corresp)r(onding)c(represen)n(ta-)-118 3696 y(tions)e(of)i FA(O)233 3708 y FL(n)278 3696 y FO(.)36 b(Then)25 b(there)f(exists)g(a)g(unitary)f(op)r(erator)g FN(V)28 b FO(:)g FN(L)1859 3708 y FK(2)1896 3696 y FO(\()p FI(Z)1989 3666 y FM(1)1989 3716 y FL(n)2053 3696 y FN(;)14 b(d\026)p FO(\))24 b FP(\000)-48 b(!)-118 3811 y FN(L)-61 3823 y FK(2)-24 3811 y FO(\()p FI(Z)69 3781 y FM(1)69 3832 y FL(n)134 3811 y FN(;)14 b(d\026)p FO(\))32 b(suc)n(h)g(that)h FN(V)772 3781 y FM(\003)810 3811 y FN(S)866 3768 y FK(\()p FL(\013)935 3776 y Fx(1)967 3768 y FK(\))861 3835 y FL(j)997 3811 y FN(V)50 b FO(=)31 b FN(S)1247 3768 y FK(\()p FL(\013)1316 3776 y Fx(2)1348 3768 y FK(\))1242 3835 y FL(j)1378 3811 y FO(,)i FN(j)j FO(=)31 b(1,)h FN(:)14 b(:)g(:)27 b FO(,)34 b FN(n)p FO(.)51 b(Since)31 b(the)-118 3911 y(same)25 b(comm)n(utativ)n(e)d(family)i(of)i(pro)5 b(jections)25 b(on)h FN(L)1527 3923 y FK(2)1564 3911 y FO(\()p FI(Z)1657 3881 y FM(1)1657 3932 y FL(n)1722 3911 y FN(;)14 b(d\026)p FO(\))27 b(corresp)r(onds)p eop %%Page: 195 199 195 198 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(195)-118 96 y(to)38 b(b)r(oth)h(represen)n(tations,)g(the)g(op)r(erator)e FN(V)57 b FO(has)38 b(the)h(form)f(\()p FN(V)19 b(f)9 b FO(\)\()p FN(x)p FO(\))42 b(=)-118 196 y FN(v)s FO(\()p FN(x)p FO(\))14 b FN(f)9 b FO(\()p FN(x)p FO(\),)29 b(where)e FP(j)p FN(v)s FO(\()p FN(x)p FO(\))p FP(j)e FO(=)d(1)28 b FN(\026)p FO(-a.e.)6 318 y(Since)f FN(V)290 288 y FM(\003)328 318 y FN(S)384 275 y FK(\()p FL(\013)453 283 y Fx(1)485 275 y FK(\))379 341 y FL(j)516 252 y FM(\003)554 318 y FN(V)42 b FO(=)22 b FN(S)787 275 y FK(\()p FL(\013)856 283 y Fx(2)888 275 y FK(\))782 341 y FL(j)919 252 y FM(\003)957 318 y FO(,)27 b FN(j)h FO(=)23 b(1,)k FN(:)14 b(:)g(:)28 b FO(,)g FN(n)p FO(,)f(as)g(w)n(ell,)f(w)n(e)h(ha)n(v)n(e)-35 528 y FN(v)8 493 y FM(\000)p FK(1)97 528 y FO(\()p FN(x)176 540 y FK(1)214 528 y FN(;)14 b(x)298 540 y FK(2)336 528 y FN(;)g(:)g(:)g(:)f FO(\))p FN(\013)568 492 y FM(\000)p FK(1)568 550 y(1)658 528 y FN(n)708 493 y FM(\000)p FK(1)p FL(=)p FK(2)864 528 y FN(v)s FO(\()p FN(j;)h(x)1057 540 y FK(1)1096 528 y FN(;)g(x)1180 540 y FK(2)1217 528 y FN(;)g(:)g(:)g(:)g FO(\))p FN(f)9 b FO(\()p FN(j;)14 b(x)1597 540 y FK(1)1635 528 y FN(;)g(x)1719 540 y FK(2)1757 528 y FN(;)g(:)g(:)g(:)f FO(\))615 669 y(=)22 b FN(\013)755 633 y FM(\000)p FK(1)755 691 y(2)845 669 y FN(n)895 634 y FM(\000)p FK(1)p FL(=)p FK(2)1051 669 y FN(f)9 b FO(\()p FN(j;)14 b(x)1251 681 y FK(1)1289 669 y FN(;)g(x)1373 681 y FK(2)1410 669 y FN(;)g(:)g(:)g(:)g FO(\))p FN(;)180 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)-118 832 y FO(whic)n(h)26 b(implies)e(that)k(for)f(all)e FN(j)k FO(=)22 b(1,)27 b FN(:)14 b(:)g(:)28 b FO(,)g FN(n)175 1012 y(v)s FO(\()p FN(j;)14 b(x)368 1024 y FK(1)407 1012 y FN(;)g(x)491 1024 y FK(2)528 1012 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)829 956 y FN(\013)882 968 y FK(1)p 829 993 91 4 v 829 1069 a FN(\013)882 1081 y FK(2)943 1012 y FN(v)s FO(\()p FN(x)1065 1024 y FK(1)1103 1012 y FN(;)14 b(x)1187 1024 y FK(2)1225 1012 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)181 b(\026)p FO(-a.e.)o FN(:)293 b FO(\(2.86\))-118 1212 y(This)26 b(implies,)e(in)j(turn,)h(that)81 1375 y FN(v)s FO(\(1)p FN(;)14 b(x)282 1387 y FK(1)320 1375 y FN(;)g(x)404 1387 y FK(2)442 1375 y FN(;)g(:)g(:)g(:)f FO(\))24 b(=)e FN(v)s FO(\(2)p FN(;)14 b(x)933 1387 y FK(1)971 1375 y FN(;)g(x)1055 1387 y FK(2)1092 1375 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FP(\001)14 b(\001)g(\001)23 b FO(=)g FN(v)s FO(\()p FN(n;)14 b(x)1800 1387 y FK(1)1838 1375 y FN(;)g(x)1922 1387 y FK(2)1959 1375 y FN(;)g(:)g(:)g(:)g FO(\))-118 1539 y FN(\026)19 b FO(-a.e.,)i(i.e.,)f(the)g(function)f FN(v)s FO(\()p FP(\001)p FO(\))h(is)e(indep)r(enden)n(t)h(of)h FN(x)1571 1551 y FK(1)1608 1539 y FO(.)34 b(But)20 b(then)g(the)g(righ) n(t-)-118 1638 y(hand)25 b(side)f(of)31 b(\(2.86\))25 b(is)f(in)n(v)-5 b(arian)n(t)21 b(with)k(resp)r(ect)g(to)g FN(x)1628 1650 y FK(1)1665 1638 y FO(,)h(and)f(the)g(left-hand)-118 1738 y(side)i(is)f(indep)r(enden)n(t)i(of)f FN(x)744 1750 y FK(1)782 1738 y FO(,)h FN(x)880 1750 y FK(2)918 1738 y FO(,)g(etc.)37 b(Therefore,)26 b FN(v)s FO(\()p FN(x)p FO(\))k(is)c(indep)r(enden)n(t)i(of)-118 1838 y(an)n(y)34 b(\014nite)f(n)n(um)n(b)r(er)h(of)g(v)-5 b(ariables;)35 b(since)e(a)h(pro)r(duct)g(measure)e(is)i(ergo)r(dic) -118 1937 y(with)f(resp)r(ect)g(to)g(suc)n(h)g(transformations,)e(w)n (e)i(get)g(that)h FN(v)s FO(\()p FN(x)p FO(\))g(is)e(constan)n(t)-118 2037 y FN(\026)p FO(-a.e.,)27 b(and)g(th)n(us,)h FN(\013)555 2049 y FK(1)615 2037 y FO(=)23 b FN(\013)756 2049 y FK(2)793 2037 y FO(.)p 2278 2037 4 57 v 2282 1984 50 4 v 2282 2037 V 2331 2037 4 57 v -118 2199 a FB(Example)31 b(21.)42 b FO(No)n(w)32 b(consider)f(a)h(sligh)n(tly)d(more)i(complicated)e (example)h(of)-118 2299 y(a)d(represen)n(tation)e(of)j FA(O)648 2311 y FL(n)720 2299 y FO(in)f(the)h(space)f(of)g(v)n(ector-v) -5 b(alued)25 b(functions.)36 b(Let)-20 2524 y FN(H)30 b FO(=)167 2411 y Fy(Z)213 2600 y Fu(Z)257 2583 y Fw(1)257 2616 y Fv(n)330 2524 y FN(l)355 2536 y FK(2)392 2524 y FO(\()p FI(Z)p FO(\))14 b FN(d\026)p FO(\()p FN(x)p FO(\))k(=)871 2420 y FM(1)842 2445 y Fy(M)845 2620 y FM(\0001)981 2524 y FN(L)1038 2536 y FK(2)1075 2524 y FO(\()p FI(Z)1168 2490 y FM(1)1168 2545 y FL(n)1233 2524 y FN(;)c(d\026)p FO(\))23 b(=)g FN(l)1531 2536 y FK(2)1568 2524 y FO(\()p FI(Z)p FO(\))12 b FP(\012)18 b FN(L)1846 2536 y FK(2)1883 2524 y FO(\()p FI(Z)1977 2490 y FM(1)1977 2545 y FL(n)2041 2524 y FN(;)c(d\026)p FO(\))p FN(;)138 2805 y(d\026)p FO(\()p FN(x)p FO(\))24 b(=)483 2701 y FM(1)454 2726 y Fy(O)456 2905 y FL(k)q FK(=1)593 2805 y FN(d\026)686 2817 y FL(k)727 2805 y FO(\()p FN(x)806 2817 y FL(k)848 2805 y FO(\))p FN(;)97 b(\026)1050 2817 y FL(k)1091 2805 y FO(\()p FN(i)p FO(\))23 b(=)g(1)p FN(=n;)179 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n;)-74 3016 y(u)-26 3028 y FK(1)10 3016 y FO(\()p FN(x)p FO(\))24 b(=)f FP(\001)14 b(\001)g(\001)23 b FO(=)g FN(u)489 3028 y FL(n)533 3016 y FO(\()p FN(x)p FO(\))h(=)f FN(u)p FO(\()p FN(x)p FO(\))h(=)e FN(S)h FP(\012)18 b FO(1\()p FN(x)p FO(\))p FN(;)98 b(S)5 b(e)1552 3028 y FL(k)1615 3016 y FO(=)23 b FN(e)1742 3028 y FL(k)q FK(+1)1866 3016 y FN(;)180 b(k)26 b FP(2)e FI(Z)o FN(:)6 3180 y FO(F)-7 b(or)19 b(v)n(ector-v)-5 b(alued)17 b(functions)i FQ(f)9 b FO(\()p FN(x)p FO(\))25 b(=)d(\()p FN(:)14 b(:)g(:)g(;)g(f)1481 3192 y FM(\000)p FK(1)1570 3180 y FO(\()p FN(x)p FO(\))p FN(;)g(f)1759 3192 y FK(0)1797 3180 y FO(\()p FN(x)p FO(\))p FN(;)g(f)1986 3192 y FK(1)2024 3180 y FO(\()p FN(x)p FO(\))p FN(;)g(:)g(:)g(:)g FO(\),)-118 3280 y(the)28 b(represen)n(tation)d(acts)i(as)g(follo)n(ws:)21 3443 y(\()p FN(S)104 3455 y FL(j)139 3443 y FQ(f)9 b FO(\))209 3455 y FL(k)250 3443 y FO(\()p FN(x)329 3455 y FK(1)367 3443 y FN(;)14 b(x)451 3455 y FK(2)489 3443 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(\016)817 3455 y FL(j)851 3443 y FO(\()p FN(x)930 3455 y FK(1)968 3443 y FO(\))14 b FN(n)1064 3409 y FK(1)p FL(=)p FK(2)1183 3443 y FQ(f)9 b FO(\()p FN(x)1300 3455 y FK(2)1338 3443 y FN(;)14 b(x)1422 3455 y FK(3)1459 3443 y FN(;)g(:)g(:)g(:)g FO(\))1639 3455 y FL(k)q FM(\000)p FK(1)1765 3443 y FN(;)180 b(k)26 b FP(2)d FI(Z)p FN(;)13 3584 y FO(\()p FN(S)101 3550 y FM(\003)96 3605 y FL(j)139 3584 y FQ(f)9 b FO(\))209 3596 y FL(k)250 3584 y FO(\()p FN(x)329 3596 y FK(1)367 3584 y FN(;)14 b(x)451 3596 y FK(2)489 3584 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)g FN(n)830 3550 y FM(\000)p FK(1)p FL(=)p FK(2)999 3584 y FQ(f)9 b FO(\()p FN(j;)14 b(x)1187 3596 y FK(1)1226 3584 y FN(;)g(x)1310 3596 y FK(2)1347 3584 y FN(;)g(:)g(:)g(:)g FO(\))1527 3596 y FL(k)q FK(+1)1652 3584 y FN(;)180 b(k)26 b FP(2)e FI(Z)o FN(:)6 3748 y FO(Cho)r(ose)j(a)g(v)n(ector)g(\012)c(=)f FN(e)823 3760 y FK(0)879 3748 y FP(\012)c FO(1\()p FN(x)p FO(\).)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)363 3911 y(\()p FN(S)446 3923 y FL(j)481 3911 y FO(\012\)\()p FN(x)p FO(\))d(=)f FN(e)835 3923 y FK(1)890 3911 y FP(\012)18 b FN(\016)1010 3923 y FL(i)1038 3911 y FO(\()p FN(x)1117 3923 y FK(1)1155 3911 y FO(\))1187 3847 y FP(p)p 1256 3847 50 4 v 64 x FN(n)c FO(1\()p FN(x)1441 3923 y FK(2)1478 3911 y FN(;)g(x)1562 3923 y FK(3)1600 3911 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)p eop %%Page: 196 200 196 199 bop -118 -137 a FO(196)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)355 96 y FO(\()p FN(S)443 62 y FM(\003)438 117 y FL(j)481 96 y FO(\012\)\()p FN(x)p FO(\))g(=)f FN(e)835 108 y FM(\000)p FK(1)942 96 y FP(\012)18 b FO(\()1057 33 y FP(p)p 1126 33 50 4 v 63 x FN(n)p FO(\))1208 62 y FM(\000)p FK(1)1312 96 y FO(1\()p FN(j;)c(x)1504 108 y FK(1)1541 96 y FN(;)g(x)1625 108 y FK(2)1663 96 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)-118 260 y FO(and)28 b(writing)e FN(S)382 272 y FL(\013)454 260 y FO(=)e FN(S)594 272 y FL(\013)637 280 y Fx(1)687 260 y FN(:)14 b(:)g(:)g(S)849 272 y FL(\013)892 280 y Fv(s)956 260 y FO(for)28 b(a)g(m)n(ulti-index)d FN(\013)f FO(=)g(\()p FN(\013)1849 272 y FK(1)1887 260 y FN(;)14 b(:)g(:)g(:)g(;)g(\013)2125 272 y FL(s)2160 260 y FO(\),)29 b(w)n(e)-118 359 y(get)-103 523 y(\()p FN(S)-20 535 y FL(\013)27 523 y FN(S)83 488 y FM(\003)78 543 y FL(\014)123 523 y FO(\012\)\()p FN(x)p FO(\))24 b(=)f FN(e)477 538 y FM(j)p FL(\013)p FM(j\000j)p FL(\014)s FM(j)714 523 y FP(\012)18 b FN(\016)834 535 y FL(\013)877 543 y Fx(1)913 523 y FO(\()p FN(x)992 535 y FK(1)1030 523 y FO(\))c FN(:)g(:)g(:)g(\016)1224 535 y FL(\013)1267 543 y Fv(s)1303 523 y FO(\()p FN(x)1382 535 y FL(s)1419 523 y FO(\))g FN(n)1515 488 y FK(\()p FM(j)p FL(\013)p FM(j\000j)p FL(\014)s FM(j)p FK(\))p FL(=)p FK(2)1866 523 y FO(1\()p FN(x)1987 535 y FL(s)p FK(+1)2107 523 y FN(;)g(:)g(:)g(:)f FO(\))p FN(;)2126 632 y FO(\(2.87\))-118 797 y(where)38 b FP(j)p FN(\013)p FP(j)g FO(is)f(the)i(length)e(of)h (the)h(m)n(ulti-index.)65 b(Also,)39 b(the)g(v)n(ector)e(\012)h(is)-118 896 y(cyclic.)6 996 y(The)20 b(corresp)r(onding)d(state)j(on)f FA(O)1057 1008 y FL(n)1122 996 y FO(is)g(de\014ned)h(b)n(y)g(the)g (follo)n(wing)c(form)n(ula)720 1159 y FN(!)s FO(\()p FN(S)858 1171 y FL(\013)905 1159 y FN(S)961 1125 y FM(\003)956 1180 y FL(\014)1001 1159 y FO(\))23 b(=)g FN(\016)1181 1171 y FL(\013;\014)1289 1159 y FN(n)1339 1125 y FM(\000j)p FL(\013)p FM(j)1477 1159 y FN(;)-118 1322 y FO(where)k FN(\016)159 1334 y FL(\013;\014)294 1322 y FO(is)g(one)g(if)g FP(j)p FN(\013)p FP(j)d FO(=)e FP(j)p FN(\014)t FP(j)p FO(,)28 b(and)g(zero)e(otherwise.)6 1422 y(Indeed,)k(since)e FN(S)557 1434 y FL(\013)604 1422 y FN(S)660 1392 y FM(\003)655 1446 y FL(\014)700 1422 y FO(\012)e FP(2)g FN(H)936 1437 y FM(j)p FL(\013)p FM(j\000j)p FL(\014)s FM(j)1154 1422 y FO(,)k(w)n(e)f(ha)n(v)n(e)f(\()p FN(S)1607 1434 y FL(\013)1654 1422 y FN(S)1710 1392 y FM(\003)1705 1446 y FL(\014)1750 1422 y FO(\012)p FN(;)14 b FO(\012\))26 b(=)f(0,)k FP(j)p FN(\013)p FP(j)d(6)p FO(=)-118 1522 y FP(j)p FN(\014)t FP(j)p FO(,)i(and)g(\(2.87\))e(implies)-36 1727 y(\()p FN(S)47 1739 y FL(\013)95 1727 y FN(S)151 1693 y FM(\003)146 1748 y FL(\013)193 1727 y FO(\012)p FN(;)14 b FO(\012\))23 b(=)493 1614 y Fy(Z)539 1803 y Fu(Z)584 1786 y Fv(s)584 1820 y(n)637 1727 y FN(\016)674 1739 y FL(\013)717 1747 y Fx(1)753 1727 y FO(\()p FN(x)832 1739 y FK(1)870 1727 y FO(\))14 b FN(:)g(:)g(:)g(\016)1064 1739 y FL(\013)1107 1747 y Fv(s)1143 1727 y FO(\()p FN(x)1222 1739 y FL(s)1258 1727 y FO(\))g FN(d\026)1397 1739 y FK(1)1435 1727 y FO(\()p FN(x)1514 1739 y FK(1)1552 1727 y FO(\))g FN(:)g(:)g(:)g(d\026) 1802 1739 y FL(s)1838 1727 y FO(\()p FN(x)1917 1739 y FL(s)1953 1727 y FO(\))23 b(=)g FN(n)2146 1693 y FM(\000)p FL(s)2233 1727 y FN(;)-118 1954 y FO(where)k FN(s)c FO(=)g FP(j)p FN(\013)p FP(j)p FO(,)28 b(and)f(the)h(form)n(ula)d(follo)n(ws.) 6 2054 y(P)n(assing)17 b(to)j(a)g(unitarily)c(equiv)-5 b(alen)n(t)18 b(realization,)f(one)j(has)f(the)i(follo)n(wing)-118 2153 y(form)n(ulae)j(for)k(the)f(op)r(erators)f(in)h FN(H)j FO(=)22 b FN(L)1201 2165 y FK(2)1238 2153 y FO(\()p FI(T)c FP(\002)g FI(Z)1489 2123 y FM(1)1489 2174 y FL(n)1553 2153 y FN(;)c(dz)22 b FP(\012)c FN(d\026)p FO(\),)212 2316 y(\()p FN(S)295 2328 y FL(j)330 2316 y FN(f)9 b FO(\)\()p FN(z)t(;)14 b(x)571 2328 y FK(1)608 2316 y FN(;)g(x)692 2328 y FK(2)730 2316 y FN(;)g(:)g(:)g(:)g FO(\))23 b(=)f FN(z)t(n)1113 2282 y FK(1)p FL(=)p FK(2)1217 2316 y FN(\016)1254 2328 y FL(j)1289 2316 y FO(\()p FN(x)1368 2328 y FK(1)1406 2316 y FO(\))p FN(f)9 b FO(\()p FN(z)t(;)14 b(x)1647 2328 y FK(2)1684 2316 y FN(;)g(x)1768 2328 y FK(3)1805 2316 y FN(;)g(:)g(:)g(:)g FO(\))p FN(:)-118 2480 y FO(The)33 b(latter)e(form)h(giv)n(es)e(an)j(explicit)d(decomp)r (osition)f(of)k(the)g(constructed)-118 2579 y(represen)n(tation)28 b(in)n(to)i(a)g(direct)g(in)n(tegral)d(of)k(irreducible)d(inequiv)-5 b(alen)n(t)27 b(rep-)-118 2679 y(resen)n(tations)e(\()p FN(S)430 2649 y FL(\013)425 2701 y(j)477 2679 y FO(\))j(constructed)f (ab)r(o)n(v)n(e.)-118 2804 y FB(Example)k(22.)42 b FO(\(The)25 b(KMS)f(represen)n(tation\))e(W)-7 b(e)25 b(giv)n(e)d(an)i(explicit)e (form)n(ula)-118 2904 y(for)h(the)i(represen)n(tation)c(of)i(the)i(Cun) n(tz)f(algebra)d FA(O)1503 2916 y FL(n)1548 2904 y FO(,)k(corresp)r (onding)c(to)i(the)-118 3003 y(unique)k(KMS)g(state)h(related)e(to)h (the)h(action)e(of)i(the)g(gauge)e(group.)6 3103 y(In)n(tro)r(duce)36 b(some)f(notations.)62 b(Let)37 b FI(Z)1251 3073 y FM(1)1251 3124 y FL(n;)p FK(0)1380 3103 y FO(b)r(e)g(the)g(set)f(of)h(all)d (\014nite)i(se-)-118 3214 y(quences)c FN(\013)g FO(=)f(\()p FN(\013)460 3226 y FK(1)497 3214 y FN(;)14 b(:)g(:)g(:)g(;)g(\013)735 3226 y FL(n)780 3214 y FN(;)g FO(0)p FN(;)g FO(0)p FN(;)g(:)g(:)g(:)e FO(\);)36 b FI(Z)1236 3184 y FM(1)1236 3234 y FL(n;)p FK(0)1359 3214 y FO(=)1455 3151 y Fy(S)1524 3239 y FL(k)1579 3214 y FI(Z)1641 3184 y FL(k)1641 3234 y(n)1680 3214 y FO(,)e(with)e FI(Z)1992 3184 y FL(k)1992 3234 y(n)2062 3214 y FP(\032)f FI(Z)2220 3184 y FL(k)q FK(+1)2220 3234 y FL(n)-118 3313 y FO(giv)n(en)h(b)n(y)h(\()p FN(\013)311 3325 y FK(1)349 3313 y FN(;)14 b(:)g(:)g(:)f(;)h(\013)586 3325 y FL(n)631 3313 y FO(\))34 b FP(7!)f FO(\()p FN(\013)898 3325 y FK(1)936 3313 y FN(;)14 b(:)g(:)g(:)f(;)h(\013)1173 3325 y FL(n)1218 3313 y FN(;)g FO(0\).)55 b(Let)34 b FP(j)p FN(\013)p FP(j)g FO(b)r(e)g(the)g(n)n(um)n(b)r(er)f(of)-118 3413 y(the)27 b(tailing)d(non-zero)h FN(\013)671 3425 y FL(j)733 3413 y FO(in)i FN(\013)p FO(.)37 b(W)-7 b(rite)26 b FN(j)1207 3425 y FL(k)1271 3413 y FO(=)d(\(0)p FN(;)14 b(:)g(:)g(:)f(;)h FO(0)p FN(;)g(j;)g FO(0)p FN(;)g(:)g(:)g(:)f FO(\))27 b(with)f FN(j)32 b FO(in)-118 3513 y(the)f FN(k)s FO(-th)f(place,)f FN(j)k FP(2)27 b FI(Z)659 3525 y FL(n)698 3513 y FO(,)k FN(k)f FO(=)d(1,)k(2,)f FN(:)14 b(:)g(:)27 b FO(.)45 b(The)30 b(group)g(op)r(eration)e(in)h FI(Z)2247 3482 y FM(1)2247 3533 y FL(n;)o FK(0)-118 3612 y FO(will)c(b)r(e)j (denoted)f(b)n(y)h(the)g(addition)d(sign.)6 3712 y FI(Z)68 3682 y FM(1)68 3732 y FL(n;)p FK(0)188 3712 y FO(is)i(a)g(coun)n(table) f(set,)i(so)g(w)n(e)f(can)h(consider)d(the)k(separable)c(Hilb)r(ert) -118 3811 y(space)j FN(l)130 3823 y FK(2)167 3811 y FO(\()p FI(Z)261 3781 y FM(1)261 3832 y FL(n;)p FK(0)353 3811 y FO(\).)41 b(F)-7 b(or)29 b FN(x)d FO(=)f(\()p FN(x)842 3823 y FK(1)880 3811 y FN(;)14 b(x)964 3823 y FK(2)1001 3811 y FN(;)g(:)g(:)g(:)g FO(\))26 b FP(2)f FI(Z)1349 3781 y FM(1)1349 3832 y FL(n)1413 3811 y FO(,)30 b(put)f FN(\033)s FO(\()p FN(x)p FO(\))e(=)e(\()p FN(x)1976 3823 y FK(2)2014 3811 y FN(;)14 b(x)2098 3823 y FK(3)2136 3811 y FN(;)g(:)g(:)g(:)f FO(\),)-118 3911 y FN(\033)-71 3923 y FL(j)-36 3911 y FO(\()p FN(x)p FO(\))24 b(=)f(\()p FN(j;)14 b(x)337 3923 y FK(1)375 3911 y FN(;)g(x)459 3923 y FK(2)497 3911 y FN(;)g(:)g(:)g(:)f FO(\),)28 b FN(j)g FO(=)23 b(0,)k FN(:)14 b(:)g(:)28 b FO(,)g FN(n)18 b FP(\000)g FO(1.)p eop %%Page: 197 201 197 200 bop -118 -137 a FJ(2.5.)36 b(Represen)n(tations)25 b(of)j(some)e(n)n(uclear)f(algebras)672 b FO(197)6 96 y(The)24 b(represen)n(tation)d(space)h(in)h(our)g(example)e(will)f(ha)n (v)n(e)j(the)h(form)e FN(H)2227 108 y FK(0)2274 96 y FP(\012)-118 196 y FN(L)-61 208 y FK(2)-24 196 y FO(\()p FI(Z)69 166 y FM(1)69 217 y FL(n)134 196 y FN(;)14 b(d\026)p FO(\()p FN(x)p FO(\)\),)32 b(where)e FN(\026)g FO(is)f(an)h(in\014nite) f(pro)r(duct)h(of)g(Haar)f(measures)f(on)-118 296 y FI(Z)-57 308 y FL(n)-18 296 y FO(.)6 395 y(The)k(space)g FN(H)477 407 y FK(0)546 395 y FO(is)e(generated)h(b)n(y)h(its)f(orthogonal)e (basis)h(consisting)f(of)-118 495 y(v)n(ectors)d FN(e)p FO(\()p FN(i;)14 b(k)s(;)g(\013)p FO(\),)27 b(where)g FN(\013)c FP(2)h FO(\000)j(is)f(an)n(y)h(m)n(ulti-index,)c FN(i)g FP(2)g FI(Z)1890 507 y FL(n)1929 495 y FO(,)k(and)h FN(k)d FO(=)e(0)-118 595 y(if)28 b FN(\013)d FP(6)p FO(=)f FI(?)k FO(and)g FN(i)c FO(=)h(0,)j(and)g FN(k)g FO(=)c(0,)k(1,)g FN(:)14 b(:)g(:)28 b FO(,)g(elsewhere,)f(i.e.,)h(for)g FN(\013)d FO(=)f FI(?)p FO(,)29 b(or)-118 694 y FN(i)23 b FP(6)p FO(=)f(0.)6 794 y(The)28 b(op)r(erators)e FN(S)596 806 y FK(0)633 794 y FO(,)i FN(:)14 b(:)g(:)27 b FO(,)h FN(S)910 806 y FL(n)p FM(\000)p FK(1)1068 794 y FO(act)f(as)g(follo)n (ws)-48 1025 y FN(S)3 1037 y FL(j)38 1025 y FN(e)p FO(\()p FN(i;)14 b(k)s(;)g(\013)p FO(\))k FP(\012)g FN(f)9 b FO(\()p FN(x)p FO(\))24 b(=)f FN(n)767 990 y FK(1)p FL(=)p FK(2)885 908 y Fy(\032)988 974 y FN(e)p FO(\()p FN(j)h FP(\000)18 b FN(\013)1253 986 y FK(1)1290 974 y FN(;)c FO(0)p FN(;)g(\033)s FO(\()p FN(\013)p FO(\)\))p FN(;)84 b(i)23 b FO(=)g(0)p FN(;)14 b(\013)22 b FP(6)p FO(=)h FI(?)988 1074 y FN(e)p FO(\()p FN(i;)14 b(k)21 b FO(+)d(1)p FN(;)c(\013)p FO(\))p FN(;)253 b FO(otherwise)2200 908 y Fy(\033)790 1207 y FP(\012)19 b FN(\016)911 1219 y FL(j)945 1207 y FO(\()p FN(x)1024 1219 y FK(1)1062 1207 y FO(\))p FN(f)9 b FO(\()p FN(\033)s FO(\()p FN(x)p FO(\)\))p FN(;)-56 1390 y(S)0 1356 y FM(\003)-5 1411 y FL(j)38 1390 y FN(e)p FO(\()p FN(i;)14 b(k)s(;)g(\013)p FO(\))k FP(\012)g FN(f)9 b FO(\()p FN(x)p FO(\))24 b(=)f FN(n)767 1356 y FM(\000)p FK(1)p FL(=)p FK(2)937 1273 y Fy(\032)1040 1339 y FN(e)p FO(\(0)p FN(;)14 b FO(0)p FN(;)g(\033)1316 1351 y FL(j)s FM(\000)p FL(i)1426 1339 y FO(\()p FN(\013)p FO(\)\))p FN(;)84 b(k)26 b FO(=)d(0)1040 1439 y FN(e)p FO(\()p FN(i;)14 b(k)21 b FP(\000)d FO(1)p FN(;)c(\013)p FO(\))p FN(;)171 b(k)26 b FP(6)p FO(=)d(0)1922 1273 y Fy(\033)790 1572 y FP(\012)c FN(f)9 b FO(\()p FN(\033)1003 1584 y FL(j)1038 1572 y FO(\()p FN(x)p FO(\)\))p FN(:)6 1754 y FO(The)27 b(v)n(ector)e(\012)e(=)g FN(e)p FO(\(0)p FN(;)14 b FO(0)p FN(;)g FI(?)p FO(\))h FP(\012)h FO(1\()p FN(x)p FO(\))27 b(is)f(cyclic,)e(and)j(the)g(corresp)r(onding)-118 1854 y(state)g(is)457 2036 y FN(!)s FO(\()p FN(S)595 2048 y FL(\013)642 2036 y FN(S)698 2002 y FM(\003)693 2057 y FL(\014)738 2036 y FO(\))c(=)g(\()p FN(S)964 2048 y FL(\013)1012 2036 y FN(S)1068 2002 y FM(\003)1063 2057 y FL(\014)1107 2036 y FO(\012)p FN(;)14 b FO(\012\))23 b(=)g FN(\016)1444 2048 y FL(\013;\014)1552 2036 y FN(n)1602 2002 y FM(\000j)p FL(\013)p FM(j)1740 2036 y FN(:)6 2219 y FO(Notice)32 b(that)g(the)h(v)n(ector)e(\012)h(is)f(not)h(cyclic)e (with)i(resp)r(ect)g(to)g(the)h(UHF)-118 2318 y(subalgebra)23 b(in)j FA(O)451 2330 y FL(n)496 2318 y FO(,)g(and)g(the)h(corresp)r (onding)c(cyclic)h(subspace)h(is)g(a)h(prop)r(er)-118 2418 y(subspace)33 b(of)h FN(H)7 b FO(.)55 b(This)33 b(means,)h(in)f(particular,)f(that)i(the)g(represen)n(tation)-118 2518 y(corresp)r(onding)26 b(to)j(the)g(tracial)d(state)i(on)h(UHF)h (cannot)e(b)r(e)h(extended)g(to)g(a)-118 2617 y(represen)n(tation)c(of) i FA(O)578 2629 y FL(n)651 2617 y FO(in)g(the)h(same)e(space.)6 2717 y(Also,)35 b(c)n(ho)r(osing)c(another)i(pro)r(duct)h(measure)e(on) h FI(Z)1710 2687 y FM(1)1710 2738 y FL(n)1774 2717 y FO(,)j(corresp)r(onding)-118 2817 y(to)k(w)n(eigh)n(ts)f FN(p)347 2829 y FK(0)384 2817 y FO(,)i FN(:)14 b(:)g(:)27 b FO(,)45 b FN(p)682 2829 y FL(n)p FM(\000)p FK(1)811 2817 y FO(,)879 2754 y Fy(P)980 2817 y FN(p)1022 2829 y FL(i)1094 2817 y FO(=)g(1,)e(the)e(form)n(ula)d(similar)e(to)k(that) -118 2932 y(giv)n(en)33 b(ab)r(o)n(v)n(e,)i(but)g(with)f FN(p)770 2889 y FK(1)p FL(=)p FK(2)770 2956 y FL(j)909 2932 y FO(replacing)d FN(n)1321 2902 y FM(\000)p FK(1)p FL(=)p FK(2)1477 2932 y FO(,)37 b(giv)n(es)32 b(a)i(represen)n(tation) -118 3032 y(corresp)r(onding)18 b(to)i(states)h(constructed)f(as)g(an)g (extension)g(of)g(pro)r(duct)h(states)-118 3132 y(on)27 b(UHF)i(whic)n(h)d(ha)n(v)n(e)h(the)h(form)405 3314 y FN(!)s FO(\()p FN(S)543 3326 y FL(\013)590 3314 y FN(S)646 3280 y FM(\003)641 3335 y FL(\014)686 3314 y FO(\))23 b(=)g FN(\016)866 3326 y FL(\013;\014)973 3314 y FN(p)1015 3326 y FL(\013)1058 3334 y Fx(1)1109 3314 y FN(:)14 b(:)g(:)f(p)1261 3326 y FL(\013)1304 3334 y Fv(s)1340 3314 y FN(;)180 b(s)23 b FO(=)g FP(j)p FN(\013)p FP(j)p FN(:)-118 3497 y FB(Example)31 b(23.)42 b FO(W)-7 b(e)25 b(giv)n(e)e(another)g (example)f(of)i(a)g(represen)n(tation.)34 b(The)24 b(cor-)-118 3596 y(resp)r(onding)i(state)h(is)287 3849 y FN(!)s FO(\()p FN(S)425 3861 y FL(\013)473 3849 y FN(S)529 3814 y FM(\003)524 3869 y FL(\014)568 3849 y FO(\))d(=)711 3707 y Fy(\()778 3792 y FN(n)828 3762 y FM(\000j)p FL(\013)p FM(j)967 3792 y FN(;)83 b FP(j)p FN(\013)p FP(j)23 b FO(=)g FP(j)p FN(\014)t FP(j)p FN(;)1431 3730 y Fy(P)1533 3792 y FN(\013)1586 3804 y FL(i)1636 3792 y FO(=)1724 3730 y Fy(P)1825 3792 y FN(\014)1872 3804 y FL(i)1900 3792 y FN(;)778 3912 y FO(0)p FN(;)230 b FO(otherwise.)p eop %%Page: 198 202 198 201 bop -118 -137 a FO(198)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(The)j(represen)n(tation)e(space)i(is)f FI(Z)962 108 y FL(n)1019 96 y FP(\012)18 b FN(l)1127 108 y FK(2)1164 96 y FO(\()p FI(Z)o FO(\))13 b FP(\012)18 b FN(L)1442 108 y FK(2)1478 96 y FO(\()p FI(Z)1572 66 y FM(1)1572 117 y FL(n)1636 96 y FN(;)c(d\026)p FO(\))28 b(with)f(the)h(stan-)-118 196 y(dard)f(pro)r(duct)g(measure)f FN(\026)p FO(,)i(and)f(the)h(form)n(ulas)d(are)157 377 y FN(S)208 389 y FL(j)243 377 y FN(e)282 389 y FL(l)325 377 y FP(\012)18 b FN(e)447 389 y FL(k)506 377 y FP(\012)g FN(f)9 b FO(\()p FN(x)p FO(\))24 b(=)f FN(n)912 343 y FK(1)p FL(=)p FK(2)1016 377 y FN(e)1055 389 y FL(l)p FM(\000)p FL(j)1181 377 y FP(\012)18 b FN(e)1303 389 y FL(k)q FK(+1)1446 377 y FP(\012)g FN(\016)1566 389 y FL(j)1601 377 y FO(\()p FN(x)1680 389 y FK(1)1718 377 y FO(\))p FN(f)9 b FO(\()p FN(\033)s FO(\()p FN(x)p FO(\)\))p FN(;)166 518 y(s)205 484 y FM(\003)205 538 y FL(j)243 518 y FN(e)282 530 y FL(l)325 518 y FP(\012)18 b FN(e)447 530 y FL(k)506 518 y FP(\012)g FN(f)9 b FO(\()p FN(x)p FO(\))24 b(=)f FN(n)912 484 y FM(\000)p FK(1)p FL(=)p FK(2)1068 518 y FN(e)1107 530 y FL(x)1145 538 y Fx(1)1176 530 y FK(+)p FL(l)1271 518 y FP(\012)18 b FN(e)1393 530 y FL(k)q FM(\000)p FK(1)1537 518 y FP(\012)g FN(f)9 b FO(\()p FN(\033)1749 530 y FL(j)1784 518 y FO(\()p FN(x)p FO(\)\))p FN(:)-118 699 y FO(The)33 b(state)g(whic)n(h)f(is)g(written)g (ab)r(o)n(v)n(e)g(is)f(obtained)h(from)g(the)h(v)n(ector)f FN(e)2215 711 y FK(0)2274 699 y FP(\012)-118 798 y FN(e)-79 810 y FK(0)-24 798 y FP(\012)18 b FO(1\()p FN(x)p FO(\),)29 b(whic)n(h)d(is)h(cyclic.)-118 1039 y FG(Commen)m(ts)38 b(to)f(Chapter2)-118 1221 y FQ(Section)31 b(2.1.)6 1321 y FO(2.1.1.)36 b(The)26 b(study)h(of)g(a)f(class)f(of)i(the)g (relations)c FN(X)7 b(X)1714 1291 y FM(\003)1774 1321 y FO(=)23 b FN(F)12 b FO(\()p FN(X)2035 1291 y FM(\003)2072 1321 y FN(X)7 b FO(\))27 b(w)n(as)-118 1420 y(motiv)-5 b(ated)33 b(b)n(y)h(n)n(umerous)e(examples)g(arising)f(in)j (mathematical)c(ph)n(ysics,)-118 1520 y(including)k(the)j(Hermitian)d FN(q)s FO(-plane,)j FN(q)s FO(-CCR,)g(quan)n(tum)e(disk,)j(etc.)65 b(\(see)-118 1620 y([158)n(,)23 b(36)o(,)f(62)o(,)h(151)o(,)f(135)o(,)g (136)o(])g(etc.\).)36 b(In)22 b([63)o(],)i(suc)n(h)e(relations)d(are)i (treated)h(as)-118 1719 y(a)k(general)e(deformation)g(of)j(CCR.)f(Note) h(that)g(these)g(relations)c(are)j(\\singly)-118 1819 y(generated")g(\(connect)i(a)f(single)e(op)r(erator)h FN(X)34 b FO(and)27 b(its)g(adjoin)n(t\).)6 1919 y(In)19 b([273)o(],)h(represen)n(tations)c(of)i(suc)n(h)g(relations)d(w)n(ere)j (studied.)33 b(The)18 b(study)-118 2018 y(of)30 b(the)g(relation)d(is)h (based)h(on)h(the)g(study)g(of)g(relations)c(of)k(the)g(form)e FN(AU)36 b FO(=)-118 2118 y FN(U)9 b(F)j FO(\()p FN(A)p FO(\))32 b(with)g(a)f(self-adjoin)n(t)e(\(or)j(normal\))d(op)r(erator)h FN(A)p FO(.)50 b(Prop)r(erties)29 b(and)-118 2217 y(represen)n(tations) 35 b(of)i(suc)n(h)h(relations)c(w)n(ere)j(discussed)f(in)h([186)o(,)h (273)n(,)g(275)o(,)-118 2317 y(276)n(],)28 b(etc.)6 2417 y(F)-7 b(or)19 b(a)g(one-to-one)f(con)n(tin)n(uous)f(mapping)g FN(F)12 b FO(\()p FP(\001)p FO(\))20 b(of)f(a)g(compact)f(set)h(\001,)i (the)-118 2516 y(mapping)d FN(F)12 b FO(\()p FP(\001)p FO(\))21 b(de\014nes)f(an)h(automorphism)16 b(of)k FN(C)6 b FO(\(\001\),)23 b(and)e(represen)n(tations)-118 2616 y(of)32 b(the)g(relation)d(are)i(represen)n(tations)e(of)j(the)g (corresp)r(onding)d FN(C)2009 2586 y FM(\003)2048 2616 y FO(-algebra)-118 2716 y(whic)n(h)c(is)g(the)i(crossed)e(pro)r(duct)h FN(C)6 b FO(\(\003\))16 b FI(o)f(Z)p FO(.)30 b(In)c(this)g(case,)g (metho)r(ds)f(of)h(the)-118 2815 y(study)j(of)h(represen)n(tations)c (go)i(bac)n(k)h(to)g([159)o(],)g(see)g(also)f([77)o(,)h(263)o(])g(and)g (the)-118 2915 y(bibliograph)n(y)23 b(therein.)6 3014 y(In)36 b(our)f(exp)r(osition,)g(w)n(e)g(do)g(not)g(assume)f(that)i (the)g(mapping)d FN(F)12 b FO(\()p FP(\001)p FO(\))36 b(is)-118 3114 y(con)n(tin)n(uous)23 b(and)j(one-to-one;)e(this)h(fact) h(mak)n(es)e(the)i(problem)d(of)i(in)n(tro)r(duc-)-118 3214 y(ing)20 b(and)h(studying)f(the)i(corresp)r(onding)c FP(\003)p FO(-algebra)g(more)i(complicated)e(\(see,)-118 3313 y(e.g.,)27 b([221)o(,)g(12],)g(etc.\).)6 3413 y(W)-7 b(e)25 b(sho)n(w)f(in)g(Prop)r(osition)d(29)j(that)h(the)g(considered)d (class)h(of)i(op)r(erators)-118 3513 y(is)36 b(a)g(subset)h(of)g(the)g (class)e(of)i(cen)n(tered)f(op)r(erators)f(studied)i(in)f([168)o(].)64 b(In)-118 3612 y(Section)26 b(2.4.2)h(w)n(e)g(discuss)f(cen)n(tered)h (op)r(erators)f(again.)6 3712 y(In)j(the)f(case)f(of)h(the)h(crossed)d (pro)r(duct,)i(the)h(relationships)24 b(b)r(et)n(w)n(een)j(er-)-118 3811 y(go)r(dic)39 b(measures)g(and)i(irreducible)c(represen)n(tations) h(w)n(ere)i(discussed)f(b)n(y)-118 3911 y(man)n(y)27 b(authors)h(\(see,)h(e.g.,)h([263)n(])f(and)g(the)g(bibliograph)n(y)c (therein\).)40 b(In)29 b(the)p eop %%Page: 199 203 199 202 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(2)1452 b FO(199)-118 96 y(non-bijectiv)n(e)22 b(case,)j(the)h(dynamical)21 b(system)j(admits)f(a)h(standard)g(in\014nite-)-118 196 y(dimensional)h(one-to-one)k(realization)c(\(see,)31 b(e.g.,)g([275)n(]\);)h(di\013eren)n(t)d(condi-)-118 296 y(tions)21 b(for)h(a)f(non-bijectiv)n(e)f(dynamical)f(system)i(to)h (b)r(e)h(simple)c(are)i(discussed)-118 395 y(in)27 b([276)n(],)h([274)o (])f(and)h(the)g(bibliograph)n(y)23 b(cited)k(there.)6 525 y(2.1.2.)43 b(F)-7 b(or)29 b(the)h(relationship)c(b)r(et)n(w)n(een) k(cycles)e(of)i(the)g(dynamical)d(sys-)-118 625 y(tem)k(and)g (\014nite-dimensional)26 b(represen)n(tations)j(of)j(the)g(relation,)d (see)i([77)o(].)-118 724 y(The)h(connection)f(with)h(the)g(Shark)n(o)n (vsky)e(theorem)g(is)i(discussed)e(in)i([276)n(].)-118 824 y(Basic)h(facts)j(ab)r(out)f(cycles)f(of)h(second)g(order)f(and)h (more)e(general)g(con)n(tin-)-118 924 y(uous)k(mappings)f(of)i(an)f(in) n(terv)-5 b(al)36 b(and)i(their)f(detailed)f(in)n(v)n(estigation)d(can) -118 1023 y(b)r(e)f(found,)h(e.g.,)g(in)e([245)n(],)i([246)o(];)h (represen)n(tations)28 b(of)k(algebraic)c(relations)-118 1123 y(arising)c(from)i(con)n(tin)n(uous)g(fraction)g(mapping)f(w)n (ere)i(discussed)f(in)h([252)n(].)6 1253 y(2.1.3.)63 b(The)37 b(partition)d(of)j(all)d(irreducible)f(represen)n(tations)h (in)n(to)h(t)n(w)n(o)-118 1352 y(classes,)25 b(degenerate)i(\(with)g FN(U)37 b FO(or)27 b FN(U)1068 1322 y FM(\003)1134 1352 y FO(ha)n(ving)e(non-zero)h(k)n(ernel\))h(and)g(non-)-118 1452 y(degenerate)k(\(with)h(unitary)e FN(U)9 b FO(\),)33 b(is)e(similar)d(to)k(the)g(W)-7 b(old)31 b(decomp)r(osition)-118 1551 y(for)f(isometries.)41 b(In)31 b(fact,)g(one)f(can)g(sho)n(w)f (that)i(an)n(y)f(represen)n(tation)d(space)-118 1651 y(can)c(b)r(e)i(decomp)r(osed)d(in)n(to)h(the)h(orthogonal)d(direct)i (sum)g(of)g(t)n(w)n(o)h(subspaces,)-118 1751 y FN(H)30 b FO(=)22 b FN(H)137 1763 y FK(0)188 1751 y FP(\010)14 b FN(H)336 1763 y FK(1)373 1751 y FO(,)26 b(suc)n(h)f(that)g(the)h(op)r (erator)e FN(U)34 b FO(is)24 b(a)h(completely)d(non-unitary)-118 1850 y(cen)n(tered)h(partial)e(isometry)g(on)j FN(H)989 1862 y FK(0)1050 1850 y FO(and)g(unitary)f(on)g FN(H)1678 1862 y FK(1)1715 1850 y FO(.)36 b(The)24 b(description)-118 1950 y(of)36 b(cen)n(tered)g(partial)e(isometries)f(is)i(essen)n (tially)e(con)n(tained)i(in)g([168)o(];)41 b(the)-118 2050 y(application)22 b(of)k(this)f(description)e(to)j(the)h(study)f (of)f(op)r(erator)g(relations)d(is)j(a)-118 2149 y(mo)r(di\014ed)h(v)n (ersion)f(of)i([182)o(].)6 2251 y(The)40 b(description)c(of)k(the)f(an) n(ti-F)-7 b(o)r(c)n(k)38 b(represen)n(tations)e(for)j(a)g(second-)-118 2351 y(order)c(mapping)f(relies)g(on)i(the)h(formalism)32 b(of)37 b(sym)n(b)r(olic)c(dynamics,)j(see,)-118 2451 y(e.g.,)27 b([246)o(].)6 2553 y(The)h(exp)r(osition)d(for)i(a)g (unitary)g FN(U)36 b FO(follo)n(ws)25 b([275)n(].)-118 2710 y FQ(Section)50 b(2.2.)83 b FO(In)43 b(this)g(section)f(w)n(e)i (giv)n(e)d(a)i(n)n(um)n(b)r(er)f(of)i(examples)d(of)-118 2810 y FP(\003)p FO(-algebras)33 b(kno)n(wn)k(from)e(pap)r(ers)i(on)f (mathematical)d(ph)n(ysics)j(\(see,)j(e.g.,)-118 2910 y([293)n(])e(and)f(the)h(bibliograph)n(y)31 b(therein\).)62 b(Our)36 b(purp)r(ose)g(here)g(is)f(to)h(illus-)-118 3009 y(trate)30 b(ho)n(w)g(the)h(metho)r(ds)f(dev)n(elop)r(ed)f(in)h (Section)f(2.1)h(can)g(b)r(e)h(generalized)-118 3109 y(to)c(a)h(wider)e(class)f(of)j(relations)c(connecting)i(sev)n(eral)f (op)r(erators.)6 3211 y(2.2.1.)59 b(T)-7 b(riples)33 b(of)i(op)r(erators)f FN(A)1083 3223 y FK(1)1120 3211 y FO(,)k FN(A)1243 3223 y FK(2)1280 3211 y FO(,)g FN(A)1403 3223 y FK(3)1440 3211 y FO(,)f(satisfying)c(the)j(relations)-118 3311 y FP(f)p FN(A)-14 3323 y FK(1)23 3311 y FN(;)14 b(A)122 3323 y FK(2)159 3311 y FP(g)28 b FO(=)f FN(A)383 3323 y FK(3)420 3311 y FO(,)32 b FP(f)p FN(A)579 3323 y FK(2)616 3311 y FN(;)14 b(A)715 3323 y FK(3)752 3311 y FP(g)27 b FO(=)h FN(A)976 3323 y FK(1)1013 3311 y FO(,)k FP(f)p FN(A)1172 3323 y FK(3)1209 3311 y FN(;)14 b(A)1308 3323 y FK(1)1345 3311 y FP(g)27 b FO(=)h FN(A)1569 3323 y FK(2)1637 3311 y FO(arise)g(as)i(represen)n(ta-)-118 3410 y(tions)f(of)i(a)f(natural)f(graded)h(analogue)e(of)i(the)h(Lie)f (algebra)e FN(so)p FO(\(3\))j(\(on)f(the)-118 3510 y(de\014nition)d(of) i(graded)f(Lie)g FP(\003)p FO(-algebras,)d(see,)k(e.g.,)g([228)o(]\).) 41 b(The)29 b(irreducible)-118 3610 y(represen)n(tations)23 b(of)i(this)g(algebra)e(w)n(ere)i(studied)g(in)g([101)n(];)i(represen)n (tations)-118 3709 y(of)g(these)h(relations)c(with)k(other)f(in)n(v)n (olutions)c(w)n(ere)k(studied)g(in)g([192)n(].)6 3811 y(Algebras)40 b(generated)g(b)n(y)h(families)c(of)42 b(pro)5 b(jections)39 b(satisfying)f(linear)-118 3911 y(relations)26 b(w)n(ere)i(considered)f(in)i([32)o(].)42 b(In)29 b([88)o(],)h(follo)n(wing)25 b(the)30 b(lines)d(of)i([32)o(],)p eop %%Page: 200 204 200 203 bop -118 -137 a FO(200)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)-118 96 y FO(the)35 b(results)e(of)i([101)o(])f(are)g (applied)f(to)h(the)i(description)c(of)i(four-tuples)g(of)-118 205 y(pro)5 b(jections)38 b(suc)n(h)h(that)714 142 y Fy(P)801 163 y FK(4)801 230 y FL(i)p FK(=1)927 205 y FN(P)980 217 y FL(i)1052 205 y FO(=)k FN(\013I)48 b FO(\(see)40 b(also)e([89)o(]\).)74 b(Notice)39 b(that)-118 304 y(solutions)d(of)j (the)h(latter)e(relation)e(exist)i(only)g(for)h(a)f(coun)n(table)g(n)n (um)n(b)r(er)-118 404 y(of)e(the)h(parameter)e FN(\013)p FO(;)41 b(see)36 b(also)f([218)o(],)k(where)d(the)h(authors)e(studied)h (the)-118 504 y(sets)d FN(\013)g FP(2)g FI(R)40 b FO(for)32 b(whic)n(h)h(there)g(are)f(orthogonal)e(pro)5 b(jections)32 b FN(P)1964 516 y FK(1)2001 504 y FO(,)i FN(:)14 b(:)g(:)27 b FO(,)35 b FN(P)2293 516 y FL(n)-118 603 y FO(satisfying)252 541 y Fy(P)339 561 y FL(n)339 628 y(i)p FK(=1)465 603 y FN(P)518 615 y FL(i)569 603 y FO(=)23 b FN(\013I)7 b FO(.)6 723 y(2.2.2.)36 b(The)28 b(exp)r(osition)d(follo)n(ws)f([181)o (].)6 842 y(2.2.3.)36 b(The)26 b(class)e(of)i(represen)n(tations)d (describ)r(ed)i(in)h(Section)f(2.2.2)g(can)-118 942 y(easily)33 b(b)r(e)j(extended)h(to)e(more)f(general)g(relations)f(whic)n(h)i (arise)e(from)i(dy-)-118 1042 y(namical)25 b(systems)i(on)h(the)g (plane;)g(the)h(metho)r(ds)e(and)i(ideas)d(used)j(here)f(are)-118 1141 y(the)36 b(same)e(as)h(in)g(Section)g(2.1.)60 b(Notice)35 b(that)h(some)e(facts)i(from)e(the)i(one-)-118 1241 y(dimensional)23 b(dynamics)i(fail)h(to)i(hold)e(in)h(the)h(t)n(w)n(o-dimensional)22 b(case.)37 b(The)-118 1341 y(exp)r(osition)25 b(essen)n(tially)f(follo) n(ws)g([275)o(].)6 1460 y(2.2.4.)36 b(The)28 b(exp)r(osition)d(follo)n (ws)f([203)o(].)6 1580 y(2.2.5.)38 b(The)28 b(Skly)n(anin)e(algebra)f (w)n(as)i(in)n(tro)r(duced)g(and)h(studied)g(in)f([254)o(,)-118 1679 y(255)n(];)34 b(these)e(pap)r(ers)f(also)e(con)n(tain)h(classes)f (of)j(represen)n(tations)c(of)k(its)f(real)-118 1779 y(forms.)6 1879 y(F)-7 b(or)33 b(represen)n(tations)c(of)k(the)g(quan)n (tum)f(algebra)e FN(U)1698 1891 y FL(q)1734 1879 y FO(\()p FN(sl)r FO(\(2\)\),)k(see,)g(e.g.,)-118 1978 y([137)n(,)29 b(140)o(],)g(and)f(the)h(bibliograph)n(y)23 b(therein.)39 b(In)29 b(man)n(y)e(pap)r(ers,)h(the)g(nota-)-118 2078 y(tion)i FN(U)113 2090 y FL(q)149 2078 y FO(\()p FN(sl)r FO(\(2\)\))i(is)e(used)h(for)g(another)f(deformation)e(of)j FN(sl)r FO(\(2\);)i(the)e(algebra)-118 2177 y(considered)f(in)h(the)h (b)r(o)r(ok)g(is)e(also)g(denoted)i(b)n(y)1461 2156 y(\024)1446 2177 y FN(U)1503 2189 y FL(q)1540 2177 y FO(\()p FN(sl)1636 2189 y FK(2)1673 2177 y FO(\))g(\(see,)h(e.g.,)g([137)n(]\).)-118 2277 y(Our)27 b(exp)r(osition)e(follo)n(ws)f([272)o(].)6 2377 y(A)k(more)e(general)f(class)h(of)i(algebras)c(w)n(as)j(in)n(tro)r (duced)f(in)h([178)n(].)-118 2516 y FQ(Section)40 b(2.3.)59 b FO(The)36 b(non-standard)d FN(q)s FO(-deformed)h(algebras)f FN(U)1956 2528 y FL(q)1992 2516 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))-118 2616 y(de\014ned)34 b(in)f(terms)f(of)i (trilinear)c(relations)h(for)i(generating)e(elemen)n(ts)h(w)n(ere)-118 2716 y(in)n(tro)r(duced)e(b)n(y)i(D.)g(F)-7 b(airlie)29 b([80)o(].)50 b(An)32 b(algebra)d(whic)n(h)i(can)g(b)r(e)h(reduced)g (to)-118 2815 y FN(U)-61 2827 y FL(q)-25 2815 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))27 b(w)n(as)20 b(de\014ned)i(in)e([177)o(].)35 b(It)21 b(is)f(kno)n(wn)h(that)g(Lie)f (algebras)e(of)j(the)-118 2915 y(Lie)27 b(groups)h FN(S)5 b(L)p FO(\(2)p FN(;)14 b FI(C)g FO(\))35 b(and)28 b FN(S)5 b(O)r FO(\(3\))29 b(are)f(isomorphic,)d(but)k(the)g FN(q)s FO(-deformed)-118 3014 y(algebras)d FN(U)264 3026 y FL(q)300 3014 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))q(\))35 b(di\013er)29 b(from)f(the)h(quan)n(tum)f(algebras)e FN(U)1969 3026 y FL(q)2006 3014 y FO(\()p FN(sl)r FO(\(2)p FN(;)14 b FI(C)g FO(\)\))-118 3114 y(in)n(tro)r(duced)25 b(b)n(y)h(V.)h(Drinfeld)f(and)g(M.)h(Jim)n(b)r(o.)34 b(Moreo)n(v)n(er,)24 b(they)j(ha)n(v)n(e)e(non-)-118 3214 y(coinciding)33 b(sets)j(of)h(irreducible)c(\014nite)j(and)g (in\014nite)g(dimensional)c(repre-)-118 3313 y(sen)n(tations.)i (Finite-dimensional)18 b(irreducible)j(represen)n(tations)g(whic)n(h,)j (for)-118 3413 y FN(q)48 b FP(!)d FO(1,)f(yield)39 b(the)j(w)n(ell-kno) n(wn)37 b(\014nite-dimensional)f(irreducible)h(repre-)-118 3513 y(sen)n(tations)d(of)i(the)g(Lie)f(algebra)e FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))42 b(w)n(ere)35 b(describ)r(ed)g(in)g ([80)o(].)63 b(The)-118 3612 y(complete)30 b(classi\014cation)d(of)32 b(\014nite-dimensional)26 b(represen)n(tations)j(when)j FN(q)-118 3712 y FO(is)e(not)h(a)f(ro)r(ot)g(of)h(unit)n(y)f(ha)n(v)n (e)g(b)r(een)h(obtained)f(in)g([108)o(].)46 b(This)30 b(pap)r(er)h(also)-118 3811 y(con)n(tains)d(a)h(description)e(of)j (some)e(classes)f(of)j(irreducible)c(represen)n(tations)-118 3911 y(when)i FN(q)i FO(is)d(a)g(ro)r(ot)g(of)g(unit)n(y)g(\(see)h (also)d([107)o(]\).)p eop %%Page: 201 205 201 204 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(2)1452 b FO(201)6 96 y(Irreducible)36 b(represen)n(tations)g(of)j(the)g FP(\003)p FO(-algebras)c FN(U)1743 108 y FL(q)1779 96 y FO(\()p FN(so)p FO(\(3\)\))40 b(\()p FN(q)45 b FP(2)d FI(R)-118 196 y FO(and)d FP(j)p FN(q)s FP(j)k FO(=)f(1\))e(determined)d (b)n(y)i(the)h(in)n(v)n(olution)c FN(I)1577 166 y FM(\003)1570 217 y FK(1)1658 196 y FO(=)42 b FP(\000)p FN(I)1866 208 y FK(1)1903 196 y FO(,)h FN(I)2012 166 y FM(\003)2005 217 y FK(2)2093 196 y FO(=)f FP(\000)p FN(I)2301 208 y FK(2)-118 296 y FO(w)n(ere)24 b(describ)r(ed)g(in)g([231)o(].)36 b(It)25 b(w)n(as)g(sho)n(wn)f(there)h(that)g(all)e(suc)n(h)i(represen)n (ta-)-118 395 y(tions)j(are)f(analogues)f(of)j(irreducible)c FP(\003)p FO(-represen)n(tations)g(of)k(the)g(real)e(com-)-118 495 y(pact)32 b(form)f(of)i FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))39 b(for)32 b FN(q)i(>)d FO(0)h(and)g(its)g(graded)f (analogue)f(for)i FN(q)i(<)d FO(0.)-118 595 y(The)36 b(later)e(w)n(ere)h(classi\014ed)f(in)h([101)n(].)63 b(The)36 b(classi\014cation)31 b(of)36 b(irreducible)-118 694 y FP(\003)p FO(-represen)n(tations)18 b(of)23 b FN(U)668 706 y FL(q)704 694 y FO(\()p FN(so)p FO(\(3\)\),)i(when)d FN(q)k FO(is)21 b(a)h(ro)r(ot)f(of)i(unit)n(y)-7 b(,)23 b(w)n(as)e(also)f(ob-)-118 794 y(tained)i(indep)r(enden)n(tly)f(in)i ([17)o(].)35 b(It)23 b(w)n(as)f(sho)n(wn)g(in)h([17)o(,)g(107)n(,)g (108)o(,)g(231)n(])g(that)-118 893 y(the)34 b(algebras)d FN(U)418 905 y FL(q)455 893 y FO(\()p FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\)\))40 b(ha)n(v)n(e)33 b(irreducible)d (\014nite-dimensional)f(repre-)-118 993 y(sen)n(tations)23 b(whic)n(h)i(ha)n(v)n(e)f(no)h(analogue)e(for)i(the)g(Lie)g(algebra)d FN(so)p FO(\(3)p FN(;)14 b FI(C)h FO(\))32 b(or)24 b(its)-118 1093 y(graded)i(analogue,)f(that)j(is,)f(whic)n(h)f(do)i(not)f(admit)f (the)i(limits)c(as)j FN(q)f FP(!)d FO(1)28 b(or)-118 1192 y FN(q)e FP(!)d(\000)p FO(1.)35 b(Some)25 b(classes)e(of)i (irreducible)d(represen)n(tations)h(of)i(the)h FP(\003)p FO(-algebra)-118 1292 y FN(U)-61 1304 y FL(q)-25 1292 y FO(\()p FN(so)p FO(\(2)p FN(;)14 b FO(1\)\))32 b(corresp)r(onding)d (to)i(the)h(in)n(v)n(olution)c FN(I)1562 1262 y FM(\003)1555 1313 y FK(1)1629 1292 y FO(=)h FP(\000)p FN(I)1824 1304 y FK(1)1862 1292 y FO(,)j FN(I)1960 1262 y FM(\003)1953 1313 y FK(2)2028 1292 y FO(=)d FN(I)2158 1304 y FK(2)2228 1292 y FO(are)-118 1392 y(giv)n(en)f(in)h([90)o(].)44 b(Note)30 b(that)h(all)c(suc)n(h)j(non-trivial)c(represen)n(tations)h (are)i(un-)-118 1491 y(b)r(ounded.)35 b(Un)n(b)r(ounded)21 b(represen)n(tations)d(of)j(the)g FP(\003)p FO(-algebra)c(de\014ned)22 b(b)n(y)e(the)-118 1591 y(in)n(v)n(olution)k FN(I)315 1561 y FM(\003)308 1611 y FK(1)376 1591 y FO(=)f FP(\000)p FN(I)565 1603 y FK(1)602 1591 y FO(,)28 b FN(I)696 1561 y FM(\003)689 1611 y FK(2)757 1591 y FO(=)22 b FP(\000)p FN(I)945 1603 y FK(2)1010 1591 y FO(w)n(ere)27 b(studied)g(in)g([231)o (].)-118 1760 y FQ(Section)k(2.4)6 1866 y FO(2.4.1.)37 b(F)-7 b(ollo)n(wing)24 b([124)o(])k(w)n(e)g(consider)e(the)i(Wic)n(k)f (analogue)f(of)i(the)g(\\di-)-118 1966 y(rect)23 b(pro)r(duct")g(of)g FN(q)s FO(-CCR)g(whic)n(h)g(is)f(an)h(example)e(of)i FN(q)1627 1978 y FL(ij)1686 1966 y FO(-CCR)g(constructed)-118 2065 y(in)28 b([43)o(].)41 b(All)28 b(represen)n(tations)e(of)j(the)g (\\direct)e(pro)r(duct")i(are)f(describ)r(ed)f(in)-118 2165 y(terms)f(of)i(the)g(one-dimensional)22 b FN(q)s FO(-CCR.)6 2303 y(2.4.2.)35 b(W)-7 b(e)25 b(consider)e(Wic)n(k)g (analogues)f(of)j(the)g(t)n(wisted)f(canonical)d(com-)-118 2402 y(m)n(utation)i(relations)f(in)n(tro)r(duced)i(and)h(studied)g(b)n (y)g(Pusz)f(and)h(W)-7 b(orono)n(wicz)-118 2502 y([212)n(],)38 b(and)e(t)n(wisted)f(canonical)d(an)n(ticomm)n(utation)f(relations)i (studied)i(b)n(y)-118 2601 y(Pusz)h([211)n(].)64 b(W)-7 b(e)37 b(denote)g(these)f(algebras)e(b)n(y)i FN(\026)p FO(-CCR)g(and)h FN(\026)p FO(-CAR,)f(re-)-118 2701 y(sp)r(ectiv)n(ely) -7 b(.)44 b(The)31 b(pro)r(of)f(of)h(Prop)r(osition)c(51)j(is)g(giv)n (en)f(according)f(to)j([124)n(].)-118 2801 y(The)20 b(same)f(prop)r (osition)e(for)j FN(\026)t FP(\000)t FN(C)6 b(AR)21 b FO(\(Theorem)e(36\))g(w)n(as)h(pro)n(v)n(ed)e(in)i([208)n(].)6 2938 y(2.4.3.)36 b(The)28 b(exp)r(osition)d(follo)n(ws)f([193)o(].)6 3076 y(2.4.4.)73 b(Algebra)37 b(of)j(functions)f(on)h(the)g (non-standard)e(three-dimen-)-118 3176 y(sional)e(real)h(quan)n(tum)h (sphere)g(w)n(as)g(in)n(tro)r(duced)g(b)n(y)h(M.)g(Noumi)e(and)i(K.) -118 3275 y(Mimac)n(hi)21 b([176)o(].)36 b(The)25 b(description)d(of)j (irreducible)c(represen)n(tations)h(of)i(this)-118 3375 y(algebra)h(follo)n(ws)f([193)o(].)6 3513 y(2.4.5.)51 b(The)33 b(Heisen)n(b)r(erg)e(relations)f(for)i(the)h(quan)n(tum)f FN(E)1896 3525 y FK(2)1966 3513 y FO(group)g(w)n(as)-118 3612 y(in)n(tro)r(duced)37 b(b)n(y)h(W)-7 b(orono)n(wicz)36 b([289)o(].)69 b(The)39 b(classi\014cation)34 b(of)k(irreducible)-118 3712 y(represen)n(tations)33 b(of)j(this)g(algebra)d(is)i(due)h(to)g ([189)o(].)63 b(Notice)35 b(that)h(w)n(e)g(do)-118 3811 y(not)28 b(assume)e(the)i(natural)e(sp)r(ectral)h(condition)e(of)j ([289)o(],)g(th)n(us)g(obtaining)d(a)-118 3911 y(wider)h(class)g(of)h (represen)n(tations.)p eop %%Page: 202 206 202 205 bop -118 -137 a FO(202)443 b FJ(Chapter)27 b(2.)37 b(Represen)n(tations)25 b(of)j(dynamical)c FP(\003)p FJ(-algebras)6 96 y FO(2.4.6.)37 b(The)28 b(notions)e(of)i(a)g(Wic)n(k) e(algebra)g(and)h(Wic)n(k)g(ideal)f(w)n(ere)h(in)n(tro-)-118 196 y(duced)e(in)f([124)n(].)36 b(Prop)r(osition)21 b(55)j(w)n(as)f (pro)n(v)n(ed)g(for)h FN(n)f FO(=)g(2)h(in)g([124)n(],)i(and)e(for)-118 296 y(general)h FN(n)j FO(in)e([208)o(].)-118 445 y FQ(Section)31 b(2.5.)6 545 y FO(2.5.1.)j(The)22 b(idea)e(of)i(decomp)r(osition)c(of)j (the)h(represen)n(tation)d(space)i(with)-118 644 y(resp)r(ect)36 b(to)g(some)e(comm)n(utativ)n(e)e(family)h(w)n(as)i(used)h(b)n(y)g(G.)g (Mac)n(k)n(ey)f(in)g(a)-118 744 y(general)i(framew)n(ork)g(of)j (imprimitivit)n(y)34 b(systems)k(whic)n(h)h(resulted)f(in)i(his)-118 844 y(metho)r(d)31 b(of)h(description)d(of)j(represen)n(tations)d(of)j (semi-direct)c(pro)r(ducts)k(of)-118 943 y(lo)r(cally)25 b(compact)h(groups)h([159)o(,)h(160)o(].)39 b(The)28 b(non-group)f(examples)f(of)i(suc)n(h)-118 1043 y(decomp)r(osition)33 b(are)k(the)g(G)-10 b(\027)-52 b(arding{Wigh)n(tman)33 b(decomp)r(osition)g(of)38 b(CCR)-118 1143 y(and)33 b(CAR)g(with)g (in\014nite)f(n)n(um)n(b)r(er)f(of)i(degrees)f(of)h(freedom)f([84)o(,)h (85)o(];)j(also)-118 1242 y(w)n(e)g(men)n(tion)e([6)o(,)i(93)o(,)h(98)o (,)f(110)n(])h(on)e(comm)n(utativ)n(e)e(mo)r(dels)h(for)h(CAR)i(and) -118 1342 y(CCR,)j([258)n(])g(for)f(AF-algebras,)h([278)o(,)f(166)o(,)h (97)o(,)g(115)n(],)j(and)c(the)h(bibliog-)-118 1441 y(raph)n(y)29 b(therin,)i(for)f(curren)n(t)g(and)g(lo)r(cal)e(curren)n(t)i(algebras,) e(etc.)46 b(The)31 b(term)-118 1541 y(\\comm)n(utativ)n(e)17 b(mo)r(del")j(w)n(as)g(in)n(tro)r(duced)h(in)g([279)n(].)35 b(As)22 b(a)f(rule,)h(the)g(general)-118 1641 y(comm)n(utativ)n(e)i(mo) r(del)h(do)r(es)i(not)h(giv)n(e)d(a)i(unitary)f(description)f(of)j(all) d(repre-)-118 1740 y(sen)n(tations)19 b(of)i(the)g(corresp)r(onding)d (op)r(erator)i(relations;)g(ho)n(w)n(ev)n(er,)g(the)h(exis-)-118 1840 y(tence)k(of)f(a)g(comm)n(utativ)n(e)d(mo)r(del)i(is)g(a)i(prop)r (ert)n(y)e(of)i(the)g(relation)c(re\015ecting)-118 1940 y(the)34 b(structure)f(of)g(its)f(represen)n(tations.)52 b(General)31 b(op)r(erator)h(relations)e(for)-118 2039 y(whic)n(h)h(there)g(exists)g(a)g(comm)n(utativ)n(e)d(mo)r(del)i(in)h (terms)g(of)h(m)n(ultiplicatio)o(n)-118 2139 y(and)21 b(w)n(eigh)n(ted)f(op)r(erator-v)-5 b(alued)19 b(shift)i(w)n(ere)g (studied)g(in)g([28)o(,)h(183)o(,)f(186)o(])h(\(see)-118 2238 y(also)j([26)o(]\).)6 2338 y(2.5.2.)52 b(The)32 b(comm)n(utativ)n(e)d(mo)r(del)i(for)i(cen)n(tered)f(op)r(erators)f(w)n (as)g(con-)-118 2438 y(structed)e(in)f([168)o(].)41 b(W)-7 b(e)29 b(rewrite)f(the)h(relations)d(whic)n(h)i(de\014ne)h(the)h(class) d(of)-118 2537 y(cen)n(tered)h(op)r(erators)f(in)h(the)i(form)d(that)i (enables)f(us)g(to)h(apply)f(the)h(general)-118 2637 y(theorem)d(from)g(the)i(previous)e(section.)6 2737 y(2.5.3.)45 b(Represen)n(tations)28 b(of)j(Cun)n(tz)f(algebras)e(o)r(ccup)n(y)i(a)g (sp)r(ecial)e(place)-118 2836 y(due)c(to)f(their)g(relation)e(to)i (endomorphisms)d(of)j FN(L)p FO(\()p FN(H)7 b FO(\))24 b(preserving)d(the)j(iden-)-118 2936 y(tit)n(y)33 b([11)o(,)h(153)o(],) h(and)f(other)f(applications)d(\(see,)35 b(e.g.,)g([45)o(]\).)56 b(Their)33 b(repre-)-118 3035 y(sen)n(tations)g(w)n(ere)i(studied)g(in) g(n)n(umerous)e(pap)r(ers)i(\(see,)i(e.g.,)g([47)o(,)f(44)o(,)f(46]) -118 3135 y(etc.\).)46 b(W)-7 b(e)30 b(c)n(ho)r(ose)g(appropriate)d (elemen)n(ts)h(in)i(the)h(Cun)n(tz)f(algebra,)f(whic)n(h)-118 3235 y(enables)21 b(us)i(to)g(apply)f(the)h(results)e(of)i(Section)f (2.5.1,)h(and)f(construct)h(a)f(com-)-118 3334 y(m)n(utativ)n(e)29 b(mo)r(del)h(for)i(its)f(represen)n(tations;)g(w)n(e)g(apply)g(this)g (mo)r(del)f(to)h(the)-118 3434 y(construction)26 b(and)h(study)h(of)f (represen)n(tations)e(of)j(the)g(Cun)n(tz)f(algebras.)p eop %%Page: 203 207 203 206 bop -118 668 a FR(Chapter)45 b(3)-118 990 y(On)f(the)i (complexit)l(y)d(of)i(the)g(description)f(of)-118 1139 y(represen)l(tations)i(of)f Fl(\003)p FR(-algebras)-118 1594 y FG(3.1)112 b FF(\003)p FG(-Wild)38 b(algebras)h(and)f(relations) -118 1788 y FO(Before)31 b(passing)e(to)j(the)g(complexit)n(y)c (problem)h(of)i(unitary)g(description)e(of)-118 1888 y(represen)n(tations)23 b(for)i(concrete)f(classes)g(of)h FP(\003)p FO(-algebras,)e(w)n(e)i(giv)n(e)f(de\014nitions)-118 1987 y(and)33 b(some)f(results)g(concerning)f(the)j(ideology)29 b(and)34 b(metho)r(dology)c(of)j(ma-)-118 2087 y(jorization)f(of)i FP(\003)p FO(-algebras)d(and)k FP(\003)p FO(-wildness)d(in)i(Sections)g (3.1.1)f(and)i(3.1.2.)-118 2187 y(Then,)e(in)e(Sections)f(3.1.3{3.1.6)f (w)n(e)i(will)e(giv)n(e)g(a)j(n)n(um)n(b)r(er)e(of)i(examples)d(of)-118 2286 y FP(\003)p FO(-wild)19 b(algebras)f(generated)i(b)n(y)h(pro)5 b(jections)19 b(and)i(idemp)r(oten)n(ts,)g(generated)-118 2386 y(b)n(y)29 b(quadratic,)e(cubic)h(and)h(semilinear)24 b(relations,)i FP(\003)p FO(-wild)h(group)h(algebras,)-118 2485 y(etc.)6 2592 y(W)-7 b(e)20 b(emphasize)d(once)i(again)f(that)i (the)g(fact)f(that)h(some)e(algebra)f(is)h FP(\003)p FO(-wild)-118 2691 y(implies)h(that)k(the)g(problem)e(of)h(unitary)g (description)e(of)j FB(al)t(l)h FO(represen)n(tations)-118 2791 y(is)29 b(v)n(ery)g(complicated.)42 b(Ho)n(w)n(ev)n(er,)29 b(the)i(problem)d(of)i(construction)f(and)h(de-)-118 2891 y(scription)i(of)j(di\013eren)n(t)f(classes)f(of)i(represen)n (tations)c(b)r(ecomes)j(ev)n(en)g(more)-118 2990 y(attractiv)n(e.)6 3096 y(Belo)n(w)40 b(in)g(this)h(c)n(hapter,)j(w)n(e)d(will)d(write)j (Rep)14 b Fz(A)41 b FO(for)f(a)h FP(\003)p FO(-algebra)d Fz(A)-118 3196 y FO(instead)26 b(of)i FP(\003)p FO(-Rep)13 b Fz(A)p FO(,)27 b(to)h(simplify)c(the)k(notations.)-118 3447 y FQ(3.1.1)94 b(Ma)5 b(jorization)33 b(of)g FP(\003)p FQ(-algebras)f(with)g(resp)s(ect)h(to)f(the)h(com-)174 3546 y(plexit)m(y)f(of)f(their)h(represen)m(tations)-118 3712 y(1.)52 b FO(Let)33 b Fz(A)g FO(b)r(e)g(a)f FP(\003)p FO(-algebra.)50 b(W)-7 b(e)33 b(recall)d(\(see)j(Section)f(1.1.3\))g (that)h(a)f(pair)-118 3811 y(\()-77 3790 y(~)-86 3811 y Fz(A)p FN(;)14 b(\036)9 b FO(:)28 b Fz(A)23 b FP(\000)-49 b(!)334 3790 y FO(~)325 3811 y Fz(A)p FO(\),)27 b(where)715 3790 y(~)706 3811 y Fz(A)f FO(is)f(a)h FP(\003)p FO(-algebra)d(and)j FN(\036)h FO(is)f(a)g FP(\003)p FO(-homomorphism,)-118 3911 y(is)f(called)f(an)j(en)n(v)n(eloping)c FP(\003)p FO(-algebra)f(if)k(for)g(an)n(y)g FP(\003)p FO(-represen)n(tation)d FN(\031)12 b FO(:)28 b Fz(A)23 b FP(\000)-48 b(!)1048 4121 y FO(203)p eop %%Page: 204 208 204 207 bop -118 -137 a FO(204)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FN(L)p FO(\()p FN(H)7 b FO(\))21 b(of)f(the)h(algebra)d Fz(A)j FO(there)f(exists)g(a)g(unique)g FP(\003)p FO(-represen)n (tation)h(~)-46 b FN(\031)13 b FO(:)2165 75 y(~)2156 96 y Fz(A)23 b FP(\000)-48 b(!)-118 196 y FN(L)p FO(\()p FN(H)7 b FO(\))27 b(suc)n(h)h(that)g(the)g(diagram)874 722 y(~)865 744 y Fz(A)286 b FN(L)p FO(\()p FN(H)7 b FO(\))p 949 718 239 4 v 1104 716 a Ft(-)1047 787 y FO(~)-47 b FN(\031)1154 526 y(\031)953 442 y Ft(@)1036 525 y(@)1119 608 y(@)1169 658 y(@)-83 b(R)865 330 y Fz(A)p 893 658 4 299 v 895 658 a Ft(?)812 529 y FN(\036)-118 898 y FO(is)26 b(comm)n(utativ)n(e.)-118 1038 y FQ(2.)58 b FO(T)-7 b(o)34 b(v)n(erify)f(whether)i(or)f(not)h(a)f(pair)f(\()p Fz(A)p FN(;)14 b(\036)9 b FO(:)31 b Fz(A)k FP(\000)-49 b(!)1706 1017 y FO(~)1697 1038 y Fz(A)o FO(\))35 b(determines)e(an)-118 1138 y(en)n(v)n(eloping)21 b(algebra,)h(it)i(is)g(only)f(su\016cien)n (t)g(to)i(consider)d(unital)h(represen)n(ta-)-118 1238 y(tions)k(of)i(the)g(algebra)d Fz(A)p FO(.)39 b(Indeed,)29 b(for)f(an)n(y)g(unital)f(represen)n(tation)f FN(\031)31 b FO(of)e Fz(A)-118 1337 y FO(let)24 b(there)i(exist)e(a)h(represen)n (tation)h(~)-46 b FN(\031)29 b FO(of)1188 1316 y(~)1179 1337 y Fz(A)c FO(making)d(the)k(follo)n(wing)21 b(diagram)-118 1437 y(comm)n(utativ)n(e)874 1954 y(~)865 1976 y Fz(A)286 b FN(L)p FO(\()p FN(H)7 b FO(\))p 949 1950 239 4 v 1104 1948 a Ft(-)1047 2020 y FO(~)-47 b FN(\031)1154 1758 y(\031)953 1674 y Ft(@)1036 1757 y(@)1119 1840 y(@)1169 1890 y(@)-83 b(R)865 1562 y Fz(A)p 893 1890 4 299 v 895 1890 a Ft(?)812 1761 y FN(\036)6 2149 y FO(Supp)r(ose)26 b(that)f FN(\031)12 b FO(:)28 b Fz(A)23 b FP(\000)-48 b(!)23 b FN(L)p FO(\()p FN(H)7 b FO(\))25 b(is)f(not)h(a)g(unital)e (represen)n(tation.)33 b(Then)-118 2249 y FN(\031)s FO(\()p FN(e)p FO(\))46 b(=)f FN(E)i FO(is)40 b(a)g(pro)5 b(jection)40 b(in)g FN(L)p FO(\()p FN(H)7 b FO(\),)45 b(and)c(for)f(an)n(y)h FN(a)k FP(2)h Fz(A)41 b FO(w)n(e)g(ha)n(v)n(e)-118 2348 y FN(\031)s FO(\()p FN(a)p FO(\))26 b(=)f FN(\031)s FO(\()p FN(eae)p FO(\))h(=)f FN(E)5 b(\031)s FO(\()p FN(a)p FO(\))p FN(E)g FO(.)43 b(The)29 b(space)f FN(H)36 b FO(decomp)r(oses)27 b(in)n(to)h(the)i(direct)-118 2448 y(sum)25 b FN(H)30 b FO(=)22 b(Im)13 b FN(E)21 b FP(\010)14 b FO(k)n(er)f FN(E)5 b FO(,)27 b(and)e(the)i(subspaces)e(Im)13 b FN(E)5 b FO(,)26 b(k)n(er)13 b FN(E)31 b FO(are)25 b(in)n(v)-5 b(arian)n(t)-118 2548 y(with)30 b(resp)r(ect)g(to)g(the)h(op)r(erators) e FN(\031)s FO(\()p FN(a)p FO(\);)j(moreo)n(v)n(er)27 b(k)n(er)13 b FN(E)33 b FP(\032)27 b FO(k)n(er)13 b FN(\031)s FO(\()p FN(a)p FO(\),)32 b(and)-118 2647 y(according)25 b(to)i(this)g(inclusion,)e(w)n(e)i(ha)n(v)n(e)710 2854 y FN(\031)s FO(\()p FN(a)p FO(\))d(=)979 2737 y Fy(\022)1040 2805 y FN(\031)1090 2775 y FK(\(1\))1180 2805 y FO(\()p FN(a)p FO(\))83 b(0)1143 2905 y(0)186 b(0)1413 2737 y Fy(\023)1488 2854 y FN(;)-118 3076 y FO(where)25 b FN(\031)170 3046 y FK(\(1\))285 3076 y FO(is)g(a)g(unital)f(represen)n(tation)f(of) i Fz(A)p FO(.)36 b(Since)25 b(the)h(homomorphism)-118 3176 y FN(\036)i FO(is)f(unital,)e(w)n(e)j(ha)n(v)n(e)e FN(\031)669 3145 y FK(\(1\))758 3176 y FO(\()p FN(e)p FO(\))e(=)j(~)-47 b FN(\031)1022 3145 y FK(\(1\))1112 3176 y FO(\()p FN(\036)p FO(\()p FN(e)p FO(\)\),)29 b(and)733 3386 y(~)-47 b FN(\031)s FO(\()p FN(a)p FO(\))24 b(=)998 3269 y Fy(\022)1063 3338 y FO(~)-46 b FN(\031)1109 3308 y FK(\(1\))1198 3338 y FO(\()p FN(a)p FO(\))84 b(0)1162 3437 y(0)186 b(0)1431 3269 y Fy(\023)-118 3612 y FO(is)28 b(the)i(needed)f(homomorphism)24 b(from)k Fz(A)h FO(to)1377 3591 y(~)1368 3612 y Fz(A)p FO(.)42 b(Since)32 b(~)-46 b FN(\031)s FO(\()p FN(e)p FO(\))26 b(=)k(~)-46 b FN(\031)s FO(\()p FN(\036)p FO(\()p FN(e)p FO(\)\))27 b(=)-118 3712 y FN(E)5 b FO(,)23 b(an)n(y)e(represen)n(tation)e(of)780 3690 y(~)771 3712 y Fz(A)j FO(that)g(mak)n(es)e(the)i(previous)e (diagram)e(comm)n(u-)-118 3811 y(tativ)n(e,)j(coincides)e(with)j(the)g (one)f(presen)n(ted)g(ab)r(o)n(v)n(e.)34 b(Th)n(us)25 b(~)-46 b FN(\031)25 b FO(is)20 b(determined)-118 3911 y(uniquely)-7 b(.)p eop %%Page: 205 209 205 208 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(205)-118 96 y FQ(3.)38 b FO(If)29 b(\()120 75 y(~)111 96 y Fz(A)p FN(;)14 b(\036)p FO(\))29 b(is)e(an)h(en)n(v)n(eloping)d(algebra)g(of)k(the)f (algebra)e Fz(A)i FO(and)g(\()2040 75 y(~)2031 96 y Fz(A)2091 108 y FK(1)2128 96 y FN(;)14 b( )s FO(\))29 b(is)-118 196 y(an)c(en)n(v)n(eloping)d(algebra)g(of)790 174 y(~)781 196 y Fz(A)p FO(,)k(then)f(\()1117 174 y(~)1108 196 y Fz(A)1168 208 y FK(1)1206 196 y FN(;)14 b( )i FP(\016)e FN(\036)p FO(\))26 b(is)e(an)h(en)n(v)n(eloping)c(algebra)-118 296 y(of)27 b(the)h(algebra)d Fz(A)p FO(.)37 b(The)27 b(pro)r(of)g(is)g(eviden)n(t.)-118 448 y FQ(4.)69 b FO(No)n(w)38 b(w)n(e)g(in)n(tro)r(duce)f(the)i(relation)d(of)j(ma)5 b(jorization)33 b(for)38 b FP(\003)p FO(-algebras.)-118 548 y(Denote)31 b(b)n(y)f Fz(K)h FO(the)g(algebra)c(of)k(compact)e(op)r (erators)f(in)i(a)g(separable)e(\(p)r(os-)-118 647 y(sibly)i (\014nite-dimensional\))d(Hilb)r(ert)j(space)h FN(H)1374 659 y FK(0)1412 647 y FO(.)50 b(The)32 b(algebra)d Fz(K)22 b FP(\012)f Fz(A)31 b FO(will)-118 747 y(o)r(ccasionally)23 b(b)r(e)28 b(denoted)f(b)n(y)h FN(M)967 759 y FL(n)1011 747 y FO(\()p Fz(A)p FO(\),)g FN(n)23 b FP(2)g FI(N)29 b FP([)19 b(f1g)p FO(.)6 847 y(Let)31 b FN(\031)12 b FO(:)29 b Fz(A)d FP(\000)-48 b(!)27 b FN(L)p FO(\()p FN(H)7 b FO(\))30 b(b)r(e)g(a)f(represen)n(tation)f(of)i Fz(A)p FO(.)43 b(It)30 b(induces)f(a)h(repre-)-118 947 y(sen)n(tation,)499 1132 y(^)-47 b FN(\031)27 b FO(=)22 b(id)c FP(\012)g FN(\031)12 b FO(:)28 b Fz(K)19 b FP(\012)f Fz(A)23 b FP(\000)-49 b(!)24 b FN(L)p FO(\()p FN(H)1457 1144 y FK(0)1512 1132 y FP(\012)18 b FN(H)7 b FO(\))p FN(;)-118 1339 y FO(of)22 b(the)g(algebra)d Fz(K)7 b FP(\012)g Fz(A)p FO(.)35 b(If)22 b(\()765 1320 y FI(^)754 1339 y Fz(K)e FP(\012)e Fz(A)o FN(;)c(\036)p FO(\))23 b(is)e(an)g(en)n(v)n(eloping)e FP(\003)p FO(-algebra)f(of)k Fz(K)7 b FP(\012)g Fz(A)-118 1439 y FO(then,)22 b(b)n(y)e (de\014nition,)k(^)-46 b FN(\031)24 b FO(uniquely)18 b(determines)g(the)i(represen)n(tation)i(~)-46 b FN(\031)23 b FO(of)d(the)-118 1557 y(en)n(v)n(eloping)f(algebra)581 1538 y FI(^)570 1557 y Fz(K)g FP(\012)f Fz(A)j FO(in)h(the)g(same)f (Hilb)r(ert)f(space)i FN(L)p FO(\()p FN(H)1895 1569 y FK(0)1939 1557 y FP(\012)7 b FN(H)g FO(\).)35 b(No)n(w)-118 1680 y(let)28 b FN( )k FO(b)r(e)d(a)f(homomorphism)23 b(of)29 b(the)g(algebra)d Fz(B)j FO(in)n(to)e(the)i(algebra)2110 1661 y FI(^)2099 1680 y Fz(K)19 b FP(\012)f Fz(A)p FO(;)-118 1780 y(then)688 1964 y(~)-46 b FN(\031)22 b FP(\016)c FN( )12 b FO(:)28 b Fz(B)23 b FP(7!)g FN(L)p FO(\()p FN(H)1290 1976 y FK(0)1346 1964 y FP(\012)18 b FN(H)7 b FO(\))-118 2149 y(is)26 b(a)i(represen)n(tation)c(of)k Fz(B)p FO(.)-118 2336 y FQ(Lemma)h(13.)41 b FB(L)l(et)26 b FO(\()567 2317 y FI(^)556 2336 y Fz(K)19 b FP(\012)f Fz(A)p FN(;)c(\036)p FO(\))27 b FB(b)l(e)f(an)h(enveloping)h FP(\003)p FB(-algebr)l(a)f(of)g Fz(K)11 b FP(\012)g Fz(A)26 b FB(and)-118 2436 y FN(\031)-71 2448 y FL(j)-3 2436 y FP(2)33 b FO(Rep\()p Fz(A)p FO(\))p FB(,)k FN(j)h FO(=)32 b(1)p FB(,)37 b FO(2)p FB(.)54 b(L)l(et)39 b FO(~)-46 b FN(\031)1005 2448 y FL(j)1075 2436 y FB(denote)35 b(its)g(lifting)i (to)e(the)g(enveloping)-118 2535 y FP(\003)p FB(-algebr)l(a)h(as)g (describ)l(e)l(d)i(ab)l(ove.)58 b(If)36 b FN(V)53 b FP(2)35 b FN(L)p FO(\()p FN(H)7 b FO(\))35 b FB(intertwines)h FN(\031)1988 2547 y FK(1)2061 2535 y FB(and)h FN(\031)2276 2547 y FK(2)2313 2535 y FB(,)-118 2635 y(then)23 b FN(I)11 b FP(\012)t FN(V)42 b FB(intertwines)27 b FO(~)-46 b FN(\031)736 2647 y FK(1)797 2635 y FB(and)28 b FO(~)-46 b FN(\031)999 2647 y FK(2)1059 2635 y FO(\()p FN(I)31 b FB(is)23 b(the)h(identity)g(op)l(er)l(ator)g(in)f FN(L)p FO(\()p FN(H)7 b FO(\)\))p FB(.)-118 2804 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)31 b(will)c(pro)n(v)n(e)h(the)j(lemma)c(for)i FN(V)49 b FO(in)n(tert)n(wining)27 b(one)j(represen)n(ta-)-118 2904 y(tion)d FN(\031)s FO(,)h(since,)e(b)n(y)h(setting)824 3133 y FN(\031)f FO(=)985 3016 y Fy(\022)1046 3083 y FN(\031)1093 3095 y FK(1)1235 3083 y FO(0)1068 3182 y(0)104 b FN(\031)1261 3194 y FK(2)1298 3016 y Fy(\023)1373 3133 y FN(;)-118 3363 y FO(and)835 3495 y FN(V)42 b FO(=)1012 3378 y Fy(\022)1085 3444 y FO(0)95 b FN(U)1073 3544 y(U)104 b FO(0)1288 3378 y Fy(\023)1363 3495 y FN(;)-118 3692 y FO(w)n(e)27 b(obtain)f(the)i(result)f(for)g FN(U)36 b FO(in)n(tert)n(wining)24 b FN(\031)1370 3704 y FK(1)1436 3692 y FO(and)j FN(\031)1644 3704 y FK(2)1682 3692 y FO(.)6 3811 y(Let)20 b FA(A)f FO(denote)g(the)g FN(C)688 3781 y FM(\003)727 3811 y FO(-algebra)c(generated)j(b)n(y)23 b(~)-46 b FN(\031)s FO(\()1606 3792 y FI(^)1595 3811 y Fz(K)20 b FP(\012)e Fz(A)o FO(\))i(and)e FA(B)i FO(denote)-118 3911 y(its)k FN(C)60 3881 y FM(\003)98 3911 y FO(-subalgebra)e (generated)i(b)n(y)29 b(^)-46 b FN(\031)s FO(\()p Fz(K)13 b FP(\012)g Fz(A)p FO(\);)26 b(then)g FA(B)f FO(is)f(a)g FN(C)1884 3881 y FM(\003)1923 3911 y FO(-subalgebra)p eop %%Page: 206 210 206 209 bop -118 -137 a FO(206)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FO(in)k FA(A)p FO(.)54 b(Let)34 b(us)f(sho)n(w)f(that)i(if)f FN(\031)918 108 y FK(1)955 96 y FO(,)i FN(\031)1060 108 y FK(2)1130 96 y FP(2)e FO(Rep\()p FA(A)p FO(\))h(and)f FN(\031)1738 108 y FK(1)1808 96 y Fr(\026)f FA(B)h FO(=)f FN(\031)2115 108 y FK(2)2185 96 y Fr(\026)g FA(B)p FO(,)-118 196 y(then)41 b FN(\031)131 208 y FK(1)212 196 y FO(=)j FN(\031)368 208 y FK(2)406 196 y FO(.)75 b(Indeed,)44 b(the)c(represen)n(tations)d FN(\031)1603 208 y FK(1)1668 196 y FP(\016)31 b FO(~)-47 b FN(\031)44 b FO(and)c FN(\031)2048 208 y FK(2)2113 196 y FP(\016)30 b FO(~)-46 b FN(\031)44 b FO(of)-107 295 y FI(^)-118 314 y Fz(K)19 b FP(\012)f Fz(A)27 b FO(are)g(liftings)d(of)k(the)g(represen)n(tation)d FN(\031)1367 326 y FK(1)1423 314 y FP(\016)d FO(~)-46 b FN(\031)21 b FP(\016)d FN(\036)24 b FO(=)e FN(\031)1818 326 y FK(2)1874 314 y FP(\016)h FO(~)-47 b FN(\031)22 b FP(\016)c FN(\036)p FO(,)28 b(since)-118 414 y FA(B)34 b FO(is)f(the)i(closure)d(of)38 b(^)-46 b FN(\031)s FO(\()p Fz(K)24 b FP(\012)e Fz(A)p FO(\))34 b(=)k(~)-47 b FN(\031)t FO(\()p FN(\036)p FO(\()p Fz(K)24 b FP(\012)e Fz(A)p FO(\)\).)57 b(By)34 b(the)g(de\014nition)f(of)-118 513 y(en)n(v)n(eloping)f FP(\003)p FO(-algebra,)i(suc)n(h)h(a)g(lifting)e (is)i(unique,)h(hence)g FN(\031)1893 525 y FK(1)1967 513 y FO(=)g FN(\031)2115 525 y FK(2)2152 513 y FO(.)61 b(By)-118 613 y(Lemma)30 b(11.)48 b(in)32 b(Section)f(1.1.3,)g FA(B)g FO(=)f FA(A)p FO(.)49 b(By)32 b(the)g(assumption,)f FN(I)d FP(\012)21 b FN(V)49 b FP(2)-114 713 y FO(^)-46 b FN(\031)s FO(\()p Fz(K)26 b FP(\012)f Fz(A)p FO(\))227 683 y FM(0)250 713 y FO(,)40 b(so)e FN(I)32 b FP(\012)25 b FN(V)58 b FP(2)41 b FA(B)849 683 y FM(0)872 713 y FO(,)f(hence)e FN(I)32 b FP(\012)25 b FN(V)59 b FP(2)40 b FA(A)1600 683 y FM(0)1624 713 y FO(,)g(and)d(consequen)n(tly)-7 b(,)-118 835 y FN(I)25 b FP(\012)18 b FN(V)42 b FP(2)28 b FO(~)-46 b FN(\031)s FO(\()288 816 y FI(^)277 835 y Fz(K)19 b FP(\012)f Fz(A)p FO(\))526 805 y FM(0)549 835 y FO(.)p 2278 835 4 57 v 2282 783 50 4 v 2282 835 V 2331 835 4 57 v -118 1001 a FQ(De\014nition)31 b(13.)40 b FB(We)h(say)f(that)h(a)f FP(\003)p FB(-algebr)l(a)h Fz(B)f FB(majorizes)i(a)f FP(\003)p FB(-algebr)l(a)-118 1101 y Fz(A)g FO(\()p FB(and)i(denote)f(it)g(by)g Fz(B)j FP(\037)f Fz(A)p FO(\),)h FB(if)e(ther)l(e)e(exists)h(an)g(enveloping)h(alge-) -118 1224 y(br)l(a)d FO(\()76 1205 y FI(^)65 1224 y Fz(K)19 b FP(\012)f Fz(A)p FN(;)c(\036)p FO(\))40 b FB(of)h(the)e(algebr)l(a)j Fz(K)26 b FP(\012)f Fz(A)p FB(,)42 b(and)e(a)g(unital)g(homomorphism) -118 1347 y FN( )12 b FO(:)30 b Fz(B)i FP(\000)-46 b(!)252 1328 y FI(^)240 1347 y Fz(K)19 b FP(\012)f Fz(A)35 b FB(such)f(that)h(the)g(functor)g FN(F)21 b FO(:)43 b(Rep)14 b Fz(A)32 b FP(\000)-46 b(!)32 b FO(Rep)14 b Fz(B)35 b FB(de\014ne)l(d)-118 1446 y(by)30 b(the)g(fol)t(lowing)i(rule)6 b FO(:)-16 1629 y FN(F)12 b FO(\()p FN(\031)s FO(\))23 b(=)k(~)-46 b FN(\031)22 b FP(\016)c FN( )s(;)183 b FB(for)31 b(any)f FN(\031)d FP(2)c FO(Rep)14 b Fz(A)p FN(;)-28 1753 y(F)e FO(\()p FN(A)p FO(\))23 b(=)g FN(I)j FP(\012)18 b FN(A;)183 b FB(for)31 b(any)f(op)l(er)l(ator)h FN(A)f FB(intertwining)g FN(\031)1913 1765 y FK(1)1980 1753 y FB(and)g FN(\031)2188 1765 y FK(2)2226 1753 y FN(;)-118 1936 y FB(is)g(ful)t(l.)-118 2102 y(R)l(emark)g(49.)42 b FO(It)25 b(follo)n(ws)d(from)h(the)i(pro)r(of)f(of)h(the)g(previous)d (lemma)g(that,)j(in)-118 2201 y(order)i(to)h(v)n(erify)f(that)i FN(F)41 b FO(is)27 b(full,)h(it)g(is)f(su\016cien)n(t)h(to)g(v)n(erify) f(that,)i(for)f(ev)n(ery)-118 2301 y FN(\031)43 b FP(2)e FO(Rep\()p Fz(A)p FO(\))d(in)f FN(L)p FO(\()p FN(H)7 b FO(\))38 b(and)f(an)n(y)g(op)r(erator)f FN(A)i FO(in)f FN(L)p FO(\()p FN(H)7 b FO(\),)41 b(the)d(inclusion)-118 2400 y FN(I)25 b FP(\012)18 b FN(A)24 b FP(2)f FN(F)12 b FO(\()p FN(\031)s FO(\)\()p Fz(B)p FO(\))506 2370 y FM(0)559 2400 y FO(implies)24 b FN(I)h FP(\012)18 b FN(A)23 b FP(2)h FN(\031)s FO(\()p Fz(A)p FO(\))1323 2370 y FM(0)1347 2400 y FO(.)6 2533 y(F)-7 b(or)20 b(some)f(commen)n(ts)f(on)i (De\014nition)f(13,)i(see)f(also)e(Section)i(3.2.3)f(b)r(elo)n(w.)-118 2683 y FQ(5.)38 b FO(F)-7 b(or)28 b(a)f(class)g(of)34 b FN(C)576 2652 y FM(\003)615 2683 y FO(-algebras,)25 b(it)j(is)f(p)r(ossible)f(to)i(only)e(consider)h(the)h(ho-)-118 2782 y(momorphisms)17 b FN( )12 b FO(:)28 b Fz(B)23 b FP(\000)-48 b(!)23 b Fz(K)818 2769 y FO(^)806 2782 y FP(\012)8 b Fz(A)21 b FO(in)h(De\014nition)e(13,)i(since)f(for)h(a)f FN(C)2009 2752 y FM(\003)2048 2782 y FO(-algebra)-118 2882 y Fz(A)p FO(,)27 b Fz(K)77 2869 y FO(^)66 2882 y FP(\012)17 b Fz(A)28 b FO(is)e(also)f(a)i FN(C)619 2852 y FM(\003)658 2882 y FO(-algebra,)d(and)j(the)h(unique)f(en)n(v)n (eloping)d(algebra)h(of)-118 2981 y(a)36 b FN(C)25 2951 y FM(\003)63 2981 y FO(-algebra)e(is)h(the)i(algebra)d(itself)h(\()p Fz(K)1269 2968 y FO(^)1257 2981 y FP(\012)24 b Fz(A)36 b FO(is)g(the)h(completion)c(of)j(the)-118 3081 y(algebraic)28 b(tensor)k(pro)r(duct)g(in)g(the)g FN(C)1122 3051 y FM(\003)1161 3081 y FO(-prenorm.)48 b(It)33 b(is)e(kno)n(wn)h(to)g(b)r(e)h(in-)-118 3181 y(dep)r(enden)n(t)38 b(of)f(the)h(c)n(hoice)e(of)h(the)h (prenorm\).)64 b(Moreo)n(v)n(er,)37 b(w)n(e)g(ha)n(v)n(e)f(the)-118 3280 y(follo)n(wing)23 b(fact.)-118 3446 y FQ(Lemma)29 b(14.)41 b FB(L)l(et)h FN( )12 b FO(:)33 b Fz(B)47 b FP(\000)-46 b(!)46 b Fz(A)c FB(b)l(e)h(a)g(morphism)h(of)61 b FN(C)1825 3416 y FM(\003)1864 3446 y FB(-algebr)l(as.)78 b(If)-118 3546 y FN(\031)s FO(\()p FN( )s FO(\()p Fz(B)p FO(\)\))190 3516 y FM(0)238 3546 y FO(=)23 b FN(\031)s FO(\()p Fz(A)p FO(\))500 3516 y FM(0)553 3546 y FB(for)31 b(any)f FN(\031)c FP(2)e FO(Rep)14 b Fz(A)p FB(,)29 b(then)h FN( )j FB(is)d(a)g(surje)l(ction.)-118 3712 y(Pr)l(o)l(of.)43 b FO(Denote)i Fz(B)517 3724 y FK(1)606 3712 y FO(=)52 b FN( )s FO(\()p Fz(B)p FO(\).)89 b(Assuming)43 b(that)i Fz(B)1698 3724 y FK(1)1787 3712 y FP(6)p FO(=)52 b Fz(A)p FO(,)c(w)n(e)d(ha)n(v)n(e)-118 3811 y(that)40 b(there)g(is)f(a)h (non-zero)e(con)n(tin)n(uous)g(functional)g FN(f)53 b FP(2)44 b Fz(A)1909 3781 y FM(\003)1987 3811 y FO(suc)n(h)39 b(that)-118 3911 y FN(f)9 b FO(\()p FN(b)p FO(\))48 b(=)g(0)42 b(for)g(all)e FN(b)48 b FP(2)h Fz(B)810 3923 y FK(1)847 3911 y FO(.)82 b(Then)43 b(one)g(can)f(\014nd)h(\(see)g(the)g(pro)r(of) f(of)p eop %%Page: 207 211 207 210 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(207)-118 96 y(Lemma)33 b(3.9)i(in)h([150)n(]\))g(a)g(represen)n(tation)d FN(\031)12 b FO(:)31 b Fz(A)36 b FP(\000)-48 b(!)37 b FN(L)p FO(\()p FN(H)7 b FO(\))35 b(and)h(a)f(\014nite-)-118 196 y(dimensional)28 b(op)r(erator)i FN(\032)h FP(2)h FN(L)p FO(\()p FN(H)7 b FO(\))32 b(suc)n(h)g(that)h FN(f)9 b FO(\()p FN(a)p FO(\))31 b(=)g(T)-7 b(r)o(\()p FN(\032\031)s FO(\()p FN(a)p FO(\)\))34 b(for)e(all)-118 296 y FN(a)h FP(2)h Fz(A)p FO(.)55 b(Since)33 b FN(\031)s FO(\()p FN( )s FO(\()p Fz(B)p FO(\)\))717 266 y FM(0)775 296 y FO(=)g FN(\031)s FO(\()p Fz(A)p FO(\))1047 266 y FM(0)1071 296 y FO(,)i(w)n(e)f(ha)n(v)n(e)e FN(\031)s FO(\()p Fz(B)1610 308 y FK(1)1648 296 y FO(\))1680 266 y FM(0)q(0)1756 296 y FO(=)h FN(\031)s FO(\()p Fz(A)p FO(\))2028 266 y FM(0)q(0)2071 296 y FO(.)55 b(Then)-118 423 y FN(\031)s FO(\()p Fz(A)p FO(\))33 b FP(\022)p 185 351 226 4 v 31 w FN(\031)s FO(\()p Fz(B)340 435 y FK(1)378 423 y FO(\))411 364 y FL(W)9 b(O)r(T)620 423 y FO(b)n(y)33 b(the)g(v)n(on)f(Neumann)h (densit)n(y)f(theorem.)51 b(Hence,)-118 522 y(for)32 b(ev)n(ery)f FN(a)f FP(2)h Fz(A)h FO(there)g(is)f(a)h(generalized)d (sequence)j FN(b)1688 534 y FL(\013)1765 522 y FP(2)g Fz(B)1925 534 y FK(1)1994 522 y FO(suc)n(h)g(that)-118 622 y FN(\031)s FO(\()p FN(a)p FO(\))24 b(=)e(w-lim)11 b FN(\031)s FO(\()p FN(b)486 634 y FL(\013)534 622 y FO(\).)37 b(So)410 784 y FN(f)9 b FO(\()p FN(a)p FO(\))23 b(=)f(T)-7 b(r\()p FN(\032)14 b FO(w-lim)d FN(\031)s FO(\()p FN(b)1188 796 y FL(\013)1235 784 y FO(\)\))591 909 y(=)22 b(lim)12 b(T)-7 b(r)o(\()p FN(\032\031)s FO(\()p FN(b)1086 921 y FL(\013)1134 909 y FO(\)\))24 b(=)e(lim)11 b FN(f)e FO(\()p FN(b)1556 921 y FL(\013)1603 909 y FO(\))24 b(=)e(0)p FN(:)-118 1071 y FO(This)k(con)n(tradiction)f(pro)n(v)n(es)g (that)j Fz(B)1086 1083 y FK(1)1147 1071 y FO(=)23 b Fz(A)p FO(.)p 2278 1071 4 57 v 2282 1018 50 4 v 2282 1071 V 2331 1071 4 57 v -118 1233 a FQ(Theorem)30 b(50.)41 b FB(A)22 b FN(C)600 1203 y FM(\003)639 1233 y FB(-algebr)l(a)h Fz(B)g FB(majorizes)i(a)e FN(C)1536 1203 y FM(\003)1574 1233 y FB(-algebr)l(a)h Fz(A)e FB(if)i(and)f(only)-118 1332 y(if)48 b Fz(B)30 b FB(c)l(ontains)g(an)g(ide)l(al)40 b Fz(I)29 b FB(such)h(that)38 b Fz(B)p FN(=)p Fz(I)23 b FP(')f Fz(K)d FP(\012)f Fz(A)p FB(.)-118 1482 y(Pr)l(o)l(of.)43 b FO(Let)28 b(us)f(note)h(that)g(an)n(y)f(represen)n(tation)e FN(\031)31 b FO(of)c Fz(K)1720 1469 y FO(^)1708 1482 y FP(\012)19 b Fz(A)27 b FO(has)g(the)h(form)-118 1582 y FN(\031)e FO(=)d FN(U)9 b FO(id)k FP(\012)h FN(\031)318 1594 y FK(0)356 1582 y FN(U)422 1552 y FM(\003)485 1582 y FO(for)25 b(some)f FN(\031)863 1594 y FK(0)924 1582 y FP(2)f FO(Rep\()p Fz(A)p FO(\))j(and)g(unitary)e FN(U)9 b FO(.)36 b(Indeed,)26 b(since)-118 1681 y FN(C)-53 1651 y FM(\003)-15 1681 y FO(-algebra)g Fz(K)j FO(is)e(of)i(t)n(yp)r(e)f FN(I)36 b FO(and)28 b(an)n(y)g(irreducible)d(represen)n(tation)g(of)k Fz(K)g FO(is)-118 1781 y(unitary)e(equiv)-5 b(alen)n(t)27 b(to)i(id)o(,)h(the)f(ab)r(o)n(v)n(e)f(statemen)n(t)g(is)g(true)h(for)f (irreducible)-118 1881 y FN(\031)k FO(\(see,)e(e.g.,)f([259)n(]\).)42 b(An)30 b(arbitrary)c(represen)n(tation)g(can)j(b)r(e)g(decomp)r(osed) -118 1980 y(as)f FN(\031)f FO(=)148 1918 y Fy(L)240 2005 y FL(\013)302 1980 y FN(\031)349 1992 y FL(\013)396 1980 y FO(,)i(where)f FN(\031)736 1992 y FL(\013)812 1980 y FO(is)f(irreducible,)e(and)j(so)g FN(\031)1643 1992 y FL(\013)1715 1980 y FO(=)c FN(U)1861 1992 y FL(\013)1908 1980 y FO(id)18 b FP(\012)h FN(\031)2127 1992 y FL(\013)2174 1997 y FK(0)2212 1980 y FN(U)2278 1950 y FM(\003)2269 2001 y FL(\013)2316 1980 y FO(.)-118 2080 y(Setting)37 b FN(\031)225 2092 y FK(0)302 2080 y FO(=)406 2018 y Fy(L)498 2105 y FL(\013)559 2080 y FN(\031)606 2092 y FL(\013)654 2096 y FK(0)728 2080 y FO(and)h FN(U)48 b FO(=)1109 2018 y Fy(L)1201 2105 y FL(\013)1262 2080 y FN(U)1319 2092 y FL(\013)1404 2080 y FO(w)n(e)37 b(get)g FN(\031)43 b FO(=)c FN(U)9 b FO(id)23 b FP(\012)i FN(\031)2174 2092 y FK(0)2212 2080 y FN(U)2278 2050 y FM(\003)2316 2080 y FO(.)-118 2179 y(Let)k FN(F)85 2191 y FL( )145 2179 y FO(:)42 b(Rep)q(\()p Fz(A)p FO(\))26 b FP(\000)-49 b(!)27 b FO(Rep\()p Fz(B)p FO(\))j(b)r(e)g(full.)41 b(W)-7 b(e)30 b(will)d(sho)n(w)h(that)i(the)g(functor)-118 2279 y FN(F)-65 2291 y FL( )-5 2279 y FO(:)43 b(Rep\()p Fz(K)22 b FP(\012)e Fz(A)p FO(\))30 b FP(\000)-48 b(!)29 b FO(Rep\()p Fz(B)p FO(\))k(is)e(also)e(full.)48 b(Indeed,)33 b(tak)n(e)d(an)i (arbitrary)-118 2379 y FN(\031)44 b FP(2)d FO(Rep\()p Fz(K)26 b FP(\012)g Fz(A)p FO(\).)68 b(Then)39 b FN(\031)44 b FO(=)c FN(U)9 b FO(\(id)25 b FP(\012)g FN(\031)1354 2391 y FK(0)1391 2379 y FO(\))p FN(U)1489 2349 y FM(\003)1528 2379 y FO(.)68 b(Let)39 b FN(V)59 b FP(2)41 b FN(\031)s FO(\()p FN( )s FO(\()p Fz(B)p FO(\)\))2290 2349 y FM(0)2316 2379 y FO(,)-118 2478 y(then)f FN(U)149 2448 y FM(\003)187 2478 y FN(V)18 b(U)52 b FP(2)42 b FO(\(id)26 b FP(\012)g FN(\031)725 2490 y FK(0)762 2478 y FO(\()p FN( )s FO(\()p Fz(B)p FO(\)\)\))1052 2448 y FM(0)1077 2478 y FO(,)43 b(and)c(so)f FN(U)1495 2448 y FM(\003)1533 2478 y FN(V)19 b(U)51 b FO(=)42 b FN(I)33 b FP(\012)26 b FN(V)2023 2490 y FK(0)2061 2478 y FO(,)42 b(where)-118 2578 y FN(V)-70 2590 y FK(0)-1 2578 y FP(2)31 b FN(\031)132 2590 y FK(0)169 2578 y FO(\()p FN( )s FO(\()p Fz(B)p FO(\)\))427 2548 y FM(0)452 2578 y FO(.)51 b(Then)33 b FN(V)796 2590 y FK(0)865 2578 y FP(2)e FN(\031)998 2590 y FK(0)1035 2578 y FO(\()p Fz(A)p FO(\))1159 2548 y FM(0)1183 2578 y FO(,)j(since)d FN(F)1501 2590 y FL( )1561 2578 y FO(:)43 b(Rep\()p Fz(A)p FO(\))31 b FP(\000)-48 b(!)30 b FO(Rep)q(\()p Fz(B)p FO(\))-118 2678 y(is)g(full.)45 b(This)30 b(pro)n(v)n(es)f(that)i FN(U)858 2647 y FM(\003)896 2678 y FN(V)19 b(U)38 b FP(2)29 b FO(\(id)20 b FP(\012)g FN(\031)1396 2690 y FK(0)1434 2678 y FO(\()p Fz(A)p FO(\)\))1590 2647 y FM(0)1613 2678 y FO(,)32 b(and)f(so)f FN(V)48 b FP(2)29 b FN(\031)s FO(\()p Fz(A)p FO(\))2292 2647 y FM(0)2316 2678 y FO(.)-118 2777 y(No)n(w)e(w)n(e)g(can)g(apply)g(the)h(previous)d(lemma)g(to)i (\014nish)g(the)h(pro)r(of.)p 2278 2777 V 2282 2724 50 4 v 2282 2777 V 2331 2777 4 57 v 6 2939 a(In)33 b(what)e(follo)n(ws,)f (the)j FN(C)840 2909 y FM(\003)878 2939 y FO(-algebra)c Fz(K)22 b FP(\012)f Fz(A)31 b FO(will)e(also)h(b)r(e)j(o)r(ccasionally) -118 3039 y(denoted)28 b(b)n(y)f FN(M)392 3051 y FL(n)437 3039 y FO(\()p Fz(A)p FO(\),)g FN(n)c FP(2)h FI(N)k FP([)19 b(f1g)p FO(.)-118 3180 y FQ(6.)50 b FO(No)n(w)32 b(consider)e(the)j (general)d(case)i(of)g(arbitrary)d FP(\003)p FO(-algebras.)48 b(The)32 b(fol-)-118 3280 y(lo)n(wing)24 b(lemma)h(is)h(a)i(simple)c (mo)r(di\014cation)h(of)i(Theorem)f(6.3.5)g(in)h([169)o(].)-118 3429 y FQ(Lemma)i(15.)41 b FB(L)l(et)52 b Fz(A)45 b FB(b)l(e)g(a)h (unital)53 b FP(\003)p FB(-algebr)l(a)46 b(and)g(let)53 b FA(B)45 b FB(b)l(e)g(any)g FN(C)2270 3399 y FM(\003)2309 3429 y FB(-)-118 3529 y(algebr)l(a.)60 b(L)l(et)43 b FN(\031)c FP(2)c FO(Rep\()p FA(B)24 b FP(\012)f Fz(A)p FO(\))36 b FB(b)l(e)g(a)h(r)l(epr)l(esentation)f(in)h FN(L)p FO(\()p FN(H)7 b FO(\))p FB(.)58 b(Then)-118 3629 y(ther)l(e)38 b(ar)l(e)g(r)l(epr)l(esentations)h FN(\036)9 b FO(:)31 b Fz(A)38 b FP(\000)-47 b(!)38 b FN(L)p FO(\()p FN(H)7 b FO(\))38 b FB(and)h FN( )12 b FO(:)31 b FA(B)38 b FP(\000)-46 b(!)38 b FN(L)p FO(\()p FN(H)7 b FO(\))37 b FB(such)-118 3728 y(that)30 b(for)g(al)t(l)h FN(a)23 b FP(2)g Fz(A)30 b FB(and)g FN(b)23 b FP(2)g FA(B)p FB(,)525 3890 y FN(\031)s FO(\()p FN(b)c FP(\012)f FN(a)p FO(\))23 b(=)f FN( )s FO(\()p FN(b)p FO(\))p FN(\036)p FO(\()p FN(a)p FO(\))j(=)d FN(\036)p FO(\()p FN(a)p FO(\))p FN( )s FO(\()p FN(b)p FO(\))p FN(:)p eop %%Page: 208 212 208 211 bop -118 -137 a FO(208)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Let)26 b FN(a)333 66 y FM(0)380 96 y FP(2)d Fz(A)j FO(and)g(de\014ne)g FN( )996 108 y FL(a)1032 92 y Fw(0)1068 96 y FO(:)i FA(B)23 b FP(\000)-48 b(!)23 b FN(L)p FO(\()p FN(H)7 b FO(\),)26 b FN(b)d FP(7!)g FN(\031)s FO(\()p FN(b)16 b FP(\012)f FN(a)1997 66 y FM(0)2021 96 y FO(\).)36 b(Let)27 b(us)-118 196 y(sho)n(w)k(that)h FN( )330 208 y FL(a)366 192 y Fw(0)424 196 y FO(is)e(con)n(tin)n(uous)g (for)h(all)e FN(a)1225 166 y FM(0)1278 196 y FP(2)h Fz(A)p FO(.)49 b(Since)31 b FN( )1770 208 y FL(a)1806 192 y Fw(0)1864 196 y FO(is)g(a)g(mapping)-118 296 y(b)r(et)n(w)n(een)24 b(Banac)n(h)e(spaces,)i(w)n(e)f(can)h(apply)e(the)i(closed)e(graph)h (theorem.)34 b(W)-7 b(e)-118 395 y(need)33 b(to)h(sho)n(w)e(that)i(if)f FN(b)704 407 y FL(n)782 395 y FP(!)g FO(0)g(in)f FA(B)i FO(and)f FN( )1393 407 y FL(a)1429 391 y Fw(0)1456 395 y FO(\()p FN(b)1524 407 y FL(n)1569 395 y FO(\))g FP(!)g FN(c)h FO(in)e FN(L)p FO(\()p FN(H)7 b FO(\),)35 b(then)-118 495 y FN(c)30 b FO(=)f(0.)49 b(Replacing)29 b FN(b)580 507 y FL(n)656 495 y FO(with)i FN(b)885 465 y FM(\003)885 515 y FL(n)930 495 y FN(b)966 507 y FL(n)1011 495 y FO(,)i FN(a)1111 465 y FM(0)1166 495 y FO(with)e(\()p FN(a)1435 465 y FM(0)1458 495 y FO(\))1490 465 y FM(\003)1529 495 y FN(a)1573 465 y FM(0)1596 495 y FO(,)i(and)e FN(c)h FO(with)f FN(c)2114 465 y FM(\003)2152 495 y FN(c)p FO(,)i(w)n(e)-118 595 y(can)27 b(assume)f(that)i FN(c)23 b FP(\025)g FO(0.)37 b(Let)27 b FN(\034)38 b FO(b)r(e)28 b(an)f(arbitrary)e(p)r(ositiv)n(e)g (functional)h(on)-118 694 y FN(L)p FO(\()p FN(H)7 b FO(\).)37 b(Then)27 b(the)h(functional)526 876 y FN(\032)9 b FO(:)28 b FA(B)23 b FP(\000)-48 b(!)23 b FI(C)15 b FN(;)186 b(b)22 b FP(7!)h FN(\034)9 b FO(\()p FN(\031)s FO(\()p FN(b)20 b FP(\012)e FN(a)1607 842 y FM(0)1630 876 y FO(\)\))-118 1058 y(is)24 b(p)r(ositiv)n(e)f(and,)j(therefore,)f(con)n(tin)n(uous.) 35 b(Hence)25 b FN(\034)9 b FO(\()p FN(c)p FO(\))25 b(=)d(lim)11 b FN(\034)e FO(\()p FN(b)2012 1070 y FL(n)2073 1058 y FP(\012)k FN(a)2195 1028 y FM(0)2219 1058 y FO(\))23 b(=)-118 1157 y(lim)11 b FN(\032)p FO(\()p FN(b)122 1169 y FL(n)167 1157 y FO(\))37 b(=)f(0,)h(since)e(lim)11 b FN(b)816 1169 y FL(n)897 1157 y FO(=)36 b(0.)61 b(So)36 b FN(c)g FO(=)g(0,)i(whic)n(h)c(pro)n(v)n(es)g(that)i FN( )2276 1169 y FL(a)2312 1153 y Fw(0)-118 1257 y FO(is)c(con)n(tin)n (uous.)52 b(W)-7 b(e)34 b(can)f(assume)e(that)j FN(\031)i FO(is)d(non-degenerate.)52 b(Consider)-118 1357 y(the)38 b(linear)d(subset)j FA(L)j FO(=)f FN(\031)s FO(\()p FA(B)26 b FP(\012)f Fz(A)p FO(\))p FN(H)7 b FO(.)67 b(Ev)n(ery)36 b FN(z)44 b FP(2)c FA(L)f FO(can)e(b)r(e)h(written)-118 1456 y(as)e FN(z)41 b FO(=)176 1394 y Fy(P)264 1414 y FL(n)264 1481 y(i)p FK(=1)389 1456 y FN(\031)s FO(\()p FN(b)507 1468 y FL(i)559 1456 y FP(\012)25 b FN(a)693 1468 y FL(i)720 1456 y FO(\)\()p FN(x)831 1468 y FL(i)860 1456 y FO(\),)39 b(where)d FN(b)1239 1468 y FL(i)1305 1456 y FP(2)i FA(B)p FO(,)h FN(a)1567 1468 y FL(i)1633 1456 y FP(2)f Fz(A)p FO(,)h FN(x)1895 1468 y FL(i)1961 1456 y FP(2)g FN(H)7 b FO(.)64 b(Let)-118 1556 y FN(z)48 b FO(=)79 1494 y Fy(P)166 1514 y FL(m)166 1581 y(j)s FK(=1)299 1556 y FN(\031)s FO(\()p FN(b)417 1526 y FM(0)417 1577 y FL(i)472 1556 y FP(\012)27 b FN(a)608 1526 y FM(0)608 1577 y FL(i)636 1556 y FO(\)\()p FN(x)747 1526 y FM(0)747 1577 y FL(i)775 1556 y FO(\))41 b(b)r(e)g(another)f(presen)n(tation.)74 b(Let)41 b FN(v)2039 1568 y FL(\026)2125 1556 y FO(b)r(e)g(an)-118 1655 y(appro)n(ximate)24 b(iden)n(tit)n(y)i(for)h FA(B)g FO(and)h FN(a)23 b FP(2)g Fz(A)p FO(.)37 b(Then)9 1899 y FN(\031)s FO(\()p FN(v)131 1911 y FL(\026)195 1899 y FP(\012)18 b FN(a)p FO(\)\()p FN(z)t FO(\))23 b(=)611 1795 y FL(n)571 1820 y Fy(X)578 1997 y FL(i)p FK(=1)705 1899 y FN(\031)s FO(\()p FN(v)827 1911 y FL(\026)873 1899 y FN(b)909 1911 y FL(i)954 1899 y FP(\012)18 b FN(aa)1125 1911 y FL(i)1153 1899 y FO(\)\()p FN(x)1264 1911 y FL(i)1292 1899 y FO(\))24 b(=)1466 1795 y FL(m)1435 1820 y Fy(X)1438 1997 y FL(j)s FK(=1)1569 1899 y FN(\031)s FO(\()p FN(v)1691 1911 y FL(\026)1736 1899 y FN(b)1772 1865 y FM(0)1772 1920 y FL(i)1818 1899 y FP(\012)18 b FN(aa)1989 1865 y FM(0)1989 1920 y FL(i)2017 1899 y FO(\)\()p FN(x)2128 1865 y FM(0)2128 1920 y FL(i)2156 1899 y FO(\))p FN(:)-118 2165 y FO(Since)27 b FN( )153 2177 y FL(a)221 2165 y FO(is)f(con)n(tin)n(uous,)g(w)n(e)h(can)g(pass)g(to)g(the)h(limit)29 2408 y(lim)67 2458 y FL(\026)158 2408 y FN(\031)s FO(\()p FN(v)280 2420 y FL(\026)344 2408 y FP(\012)18 b FN(a)p FO(\)\()p FN(z)t FO(\))23 b(=)760 2305 y FL(n)721 2330 y Fy(X)727 2506 y FL(i)p FK(=1)855 2408 y FN(\031)s FO(\()p FN(b)973 2420 y FL(i)1019 2408 y FP(\012)18 b FN(aa)1190 2420 y FL(i)1217 2408 y FO(\)\()p FN(x)1328 2420 y FL(i)1357 2408 y FO(\))23 b(=)1530 2305 y FL(m)1500 2330 y Fy(X)1502 2506 y FL(j)s FK(=1)1634 2408 y FN(\031)s FO(\()p FN(b)1752 2374 y FM(0)1752 2429 y FL(i)1798 2408 y FP(\012)18 b FN(aa)1969 2374 y FM(0)1969 2429 y FL(i)1996 2408 y FO(\)\()p FN(x)2107 2374 y FM(0)2107 2429 y FL(i)2136 2408 y FO(\))p FN(:)-118 2683 y FO(So)25 b(the)h(mapping)c FN(\036)p FO(\()p FN(a)p FO(\))9 b(:)29 b FA(L)24 b FP(\000)-49 b(!)24 b FA(L)p FO(,)i FN(z)g FP(7!)1177 2621 y Fy(P)1265 2641 y FL(n)1265 2708 y(i)p FK(=1)1390 2683 y FN(\031)s FO(\()p FN(b)1508 2695 y FL(i)1550 2683 y FP(\012)14 b FN(aa)1717 2695 y FL(i)1744 2683 y FO(\)\()p FN(x)1855 2695 y FL(i)1883 2683 y FO(\))26 b(is)e(correctly)-118 2783 y(de\014ned.)55 b(Since)33 b FN(\036)p FO(\()p FN(a)p FO(\))h(=)e(lim)844 2795 y FL(\026)903 2783 y FN(\031)s FO(\()p FN(v)1025 2795 y FL(\026)1093 2783 y FP(\012)22 b FN(a)p FO(\)\()p FN(z)t FO(\),)35 b(it)e(is)f(ob)n(vious)f(that)j FN(\036)p FO(\()p FN(a)p FO(\))h(is)-118 2882 y(linear.)43 b(Since)30 b FN( )428 2894 y FL(a)499 2882 y FO(is)g(con)n(tin)n(uous,) f(there)h(is)g FN(M)37 b FO(=)28 b FN(M)9 b FO(\()p FN(a)p FO(\))28 b FP(2)g FI(R)1904 2894 y FK(+)1996 2882 y FO(suc)n(h)i(that) -118 2982 y FP(k)p FN(\031)s FO(\()p FN(b)21 b FP(\012)h FN(a)p FO(\))p FP(k)31 b(\024)f FN(M)9 b FP(k)p FN(b)p FP(k)p FO(,)32 b FN(b)f FP(2)h FA(B)p FO(.)52 b(So)32 b FN(\036)p FO(\()p FN(a)p FO(\))h(is)f(a)g(b)r(ounded)h(op)r(erator.) 49 b(Since)-118 3082 y FN(\031)31 b FO(is)c(non-degenerate,)g FA(L)h FO(is)f(dense)h(in)f FN(H)35 b FO(and)28 b FN(\036)p FO(\()p FN(a)p FO(\))h(is)e(uniquely)f(extended)-118 3181 y(to)i(a)f(b)r(ounded)i(op)r(erator)d(on)i FN(H)34 b FO(\(whic)n(h)28 b(is)f(also)e(denoted)j(b)n(y)g FN(\036)p FO(\()p FN(a)p FO(\)\).)39 b(Then)-118 3281 y FN(\036)9 b FO(:)28 b Fz(A)23 b FP(\000)-48 b(!)23 b FN(L)p FO(\()p FN(H)7 b FO(\))24 b(and)h FN( )h FO(=)c FN( )798 3293 y FK(1)861 3281 y FO(\()p FN(a)937 3251 y FM(0)983 3281 y FO(=)h(1\))h(are)g(the)h(required)e(represen)n(tations.)-118 3381 y(Belo)n(w)i(w)n(e)j(use)f(the)h(notations)e FN(\036)e FO(=)e FN(\031)1108 3393 y Fs(A)1189 3381 y FO(and)27 b FN( )f FO(=)d FN(\031)1565 3393 y Fi(B)p 2278 3381 4 57 v 2282 3328 50 4 v 2282 3381 V 2331 3381 4 57 v -118 3546 a FQ(Prop)s(osition)30 b(65.)41 b FB(L)l(et)29 b Fz(A)g FB(b)l(e)h(a)g(unital)g FP(\003)p FB(-algebr)l(a.)39 b(Then)-17 3712 y FO(1.)i FB(for)28 b(every)f FN(\031)f FP(2)e FO(Rep\()p Fz(K)11 b FP(\012)g Fz(A)p FO(\))26 b FB(in)h FN(L)p FO(\()p FN(H)7 b FO(\))26 b FB(ther)l(e)h(ar)l(e)f (unitary)h FN(U)k FP(2)24 b FN(L)p FO(\()p FN(H)7 b FO(\))89 3811 y FB(and)39 b(a)f(r)l(epr)l(esentation)g FN(\031)935 3823 y FK(0)1011 3811 y FP(2)g FO(Rep\()p Fz(A)p FO(\))g FB(in)g FN(L)p FO(\()p FN(H)1678 3823 y FK(1)1715 3811 y FO(\))g FB(such)g(that)g FN(H)45 b FO(=)89 3911 y FN(H)158 3923 y FK(0)214 3911 y FP(\012)18 b FN(H)366 3923 y FK(1)433 3911 y FB(and)30 b FN(U)660 3881 y FM(\003)698 3911 y FN(\031)s(U)i FO(=)23 b(id)18 b FP(\012)g FN(\031)1143 3923 y FK(0)1180 3911 y FB(.)p eop %%Page: 209 213 209 212 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(209)-17 96 y(2.)41 b(pro-)o FN(C)302 66 y FM(\003)341 96 y FO(\()p Fz(K)24 b FP(\012)g Fz(A)p FO(\))36 b FP(')g Fz(K)861 83 y FO(^)849 96 y FP(\012)24 b FO(pro-)o FN(C)1151 66 y FM(\003)1189 96 y FO(\()p Fz(A)p FO(\))p FB(,)40 b(wher)l(e)d Fz(K)1710 83 y FO(^)1699 96 y FP(\012)23 b FO(pro-)o FN(C)2000 66 y FM(\003)2038 96 y FO(\()p Fz(A)p FO(\))38 b FB(is)f(a)89 196 y(unique,sinc)l(e)29 b Fz(K)g FB(is)f(nucle)l(ar,)h FO(pro-)o FN(C)1259 166 y FM(\003)1298 196 y FB(-algebr)l(a)g(which)h (is)e(the)h(c)l(omple-)89 296 y(tion)h(of)h(the)f(algebr)l(aic)i (tensor)d(pr)l(o)l(duct)38 b FO(\()p FB(se)l(e)e FO([199)o(]\))p FB(.)-118 471 y(Pr)l(o)l(of.)43 b FO(1.)57 b(By)35 b(the)g(previous)d (lemma,)h FN(\031)38 b FO(=)d FN(\031)1420 483 y Fs(K)1493 471 y FP(\012)22 b FN(\031)1627 483 y Fs(A)1680 471 y FO(.)58 b(Denote)35 b(b)n(y)f FA(A)h FO(=)p -118 498 225 4 v -118 570 a FN(\031)-71 582 y Fs(A)-18 570 y FO(\()p Fz(A)p FO(\))23 b(the)h FN(C)333 540 y FM(\003)371 570 y FO(-subalgebra)d(in)h FN(L)p FO(\()p FN(H)7 b FO(\))23 b(generated)f(b)n(y)h(the)h(range)e(of)h FN(\031)2106 582 y Fs(A)2159 570 y FO(.)36 b(Let)-118 670 y FN(j)14 b FO(:)28 b FA(A)23 b FP(\000)-49 b(!)24 b FN(L)p FO(\()p FN(H)7 b FO(\))23 b(denote)h(the)g(natural)e(em)n(b)r(edding.)34 b(If)24 b FN(\031)i FP(6)p FO(=)d(0,)h(then)g FN(\031)2113 682 y Fs(K)2186 670 y FP(6)p FO(=)f(0,)-118 770 y(and)18 b(b)n(y)h(the)g(simplicit)n(y)13 b(of)19 b Fz(K)g FO(w)n(e)f(ha)n(v)n (e)g FN(\031)1147 782 y Fs(K)1197 770 y FO(\()p Fz(K)p FO(\))24 b FP(')e Fz(K)p FO(.)35 b(The)19 b(follo)n(wing)14 b(diagram)-118 869 y(is)26 b(comm)n(utativ)n(e.)586 1063 y Fz(K)19 b FP(\012)f Fz(A)610 b(K)19 b FP(\012)f FA(A)p 827 1039 562 4 v 1306 1037 a Ft(-)972 1009 y FO(id)g FP(\012)g FN(\031)1190 1021 y Fs(A)1011 1473 y FN(L)p FO(\()p FN(H)7 b FO(\))800 1263 y FN(\031)752 1179 y Ft(@)835 1262 y(@)918 1345 y(@)968 1395 y(@)-83 b(R)1368 1265 y FN(\031)1415 1277 y Fs(K)1483 1265 y FP(\012)18 b FN(j)1383 1179 y Ft(\000)1300 1262 y(\000)1217 1345 y(\000)1167 1395 y(\000)-83 b(\011)-118 1623 y FO(Since)27 b Fz(K)i FO(is)e(of)i(t)n(yp)r(e)f FN(I)7 b FO(,)29 b(the)f(represen)n (tation)e FN(\031)1379 1635 y Fs(K)1448 1623 y FP(\012)18 b FN(j)29 b FO(=)24 b FN(U)1749 1593 y FM(\003)1787 1623 y FO(id)18 b FP(\012)23 b FO(^)-46 b FN(\031)s(U)9 b FO(,)28 b(where)-114 1723 y(^)-46 b FN(\031)39 b FP(2)c FO(Rep\()p FA(A)p FO(\))h(\(see)f(the)g(pro)r(of)g(of)g(Theorem)e (50\).)58 b(So)35 b FN(\031)k FO(=)c FN(U)1947 1693 y FM(\003)1985 1723 y FO(id)22 b FP(\012)h FN(\031)2212 1735 y FK(0)2250 1723 y FN(U)9 b FO(,)-118 1822 y(where)27 b FN(\031)169 1834 y FK(0)230 1822 y FO(=)g(^)-47 b FN(\031)27 b Fr(\026)22 b Fz(A)p FO(.)6 1925 y(2.)34 b(T)-7 b(ak)n(e)20 b FN(\031)26 b FP(2)e FO(Rep\()p Fz(K)t FP(\012)t Fz(A)p FO(\).)34 b(Then)20 b FN(H)30 b FO(=)23 b FN(H)1369 1937 y FK(0)1410 1925 y FP(\012)t FN(H)1548 1937 y FK(1)1605 1925 y FO(and)d FN(\031)26 b FO(=)d FN(U)1986 1895 y FM(\003)2024 1925 y FO(id)s FP(\012)t FN(\031)2213 1937 y FK(0)2250 1925 y FN(U)9 b FO(.)-118 2025 y(Denote)37 b(the)h(canonical)33 b(homomorphism)f(b)n(y)37 b FN(\036)9 b FO(:)32 b Fz(A)38 b FP(\000)-49 b(!)39 b FO(pro-)o FN(C)1985 1995 y FM(\003)2024 2025 y FO(\()p Fz(A)p FO(\).)65 b(By)-118 2124 y(the)24 b(de\014nition)f(of)h(an)g(en)n(v)n(eloping)d (algebra,)h(the)i(represen)n(tation)e FN(\031)2032 2136 y FK(0)2094 2124 y FO(admits)-118 2224 y(a)36 b(unique)g(lifting)j(~) -47 b FN(\031)537 2236 y FK(0)613 2224 y FP(2)39 b FO(Rep\(pro-)o FN(C)1096 2194 y FM(\003)1135 2224 y FO(\()p Fz(A)p FO(\)\))e(in)f FN(L)p FO(\()p FN(H)1592 2236 y FK(1)1629 2224 y FO(\).)64 b(Then)37 b(de\014ne)g(the)-118 2324 y(represen)n(tation)k(~)-46 b FN(\031)47 b FP(2)c FO(Rep\()p Fz(K)899 2311 y FO(^)888 2324 y FP(\012)26 b FO(pro-)o FN(C)1192 2293 y FM(\003)1230 2324 y FO(\()p Fz(A)p FO(\)\))40 b(via)f(the)h(rule)j(~)-47 b FN(\031)t FO(\()p FN(k)29 b FP(\012)d FN(a)p FO(\))44 b(=)-118 2423 y FN(U)-52 2393 y FM(\003)-14 2423 y FO(id)23 b FP(\012)28 b FO(~)-46 b FN(\031)215 2435 y FK(0)253 2423 y FN(U)9 b FO(.)62 b(Since)35 b FN(\036)p FO(\()p Fz(A)p FO(\))i(is)e(quasi-dense)f(in)i(pro-)p FN(C)1701 2393 y FM(\003)1738 2423 y FO(\()p Fz(A)p FO(\),)j(the)d(algebra)-118 2523 y Fz(K)26 b FP(\012)g FN(\036)p FO(\()p Fz(A)p FO(\))40 b(is)d(quasi-dense)g(in)h Fz(K)1015 2510 y FO(^)1004 2523 y FP(\012)26 b FO(pro-)n FN(C)1307 2493 y FM(\003)1346 2523 y FO(\()p Fz(A)p FO(\).)71 b(Th)n(us)39 b(the)g(lifting)i(~)-47 b FN(\031)43 b FO(is)-118 2622 y(unique.)p 2278 2622 4 57 v 2282 2570 50 4 v 2282 2622 V 2331 2622 4 57 v 6 2813 a(No)n(w)30 b(w)n(e)f(are)g(in)f(a)i(p)r(osition)d(to)j(pro)n(v) n(e)e(the)i(main)d(theorem)i(ab)r(out)g(ma-)-118 2912 y(jorization.)-118 3087 y FQ(Theorem)h(51.)41 b FO(1)p FB(.)60 b(If)38 b FO(\()714 3066 y(~)705 3087 y Fz(A)p FN(;)14 b(\036)p FO(\))37 b FB(is)h(an)f(enveloping)i FP(\003)p FB(-algebr)l(a)f(of)75 b Fz(A)p FB(,)39 b(then)-118 3187 y FO(pro-)o FN(C)95 3157 y FM(\003)133 3187 y FO(\()p Fz(A)p FO(\))23 b FP(')g FO(pro-)o FN(C)581 3157 y FM(\003)619 3187 y FO(\()660 3165 y(~)651 3187 y Fz(A)p FO(\))p FB(.)6 3290 y FO(2)p FB(.)37 b FP(\003)p FB(-A)n(lgebr)l(a)26 b Fz(B)h FB(majorizes)h(a)e FP(\003)p FB(-algebr)l(a)h Fz(A)e FB(if)i(and)g(only)g(if)g(ther)l(e)f(exists)-118 3389 y(a)h(c)l(ontinuous)e(morphism)j FN( )12 b FO(:)28 b(pro-)o FN(C)1079 3359 y FM(\003)1117 3389 y FO(\()p Fz(B)p FO(\))c FP(\000)-46 b(!)23 b FO(pro-)o FN(C)1616 3359 y FM(\003)1655 3389 y FO(\()p Fz(K)11 b FP(\012)g Fz(A)p FO(\))26 b FB(with)h(quasi-)-118 3489 y(dense)k(image)37 b FO(\()p FB(such)31 b(that)f(for)h(any)g(r)l(epr)l(esentation)f FN(\031)d FP(2)e FO(Rep\(pro-)o FN(C)2129 3459 y FM(\003)2167 3489 y FO(\()p Fz(K)20 b FP(\012)-118 3588 y Fz(A)p FO(\)\))30 b FB(the)g(set)f FN(\031)s FO(\()p FN( )s FO(\(pro-)p FN(C)688 3558 y FM(\003)726 3588 y FO(\()p Fz(B)p FO(\)\)\))j FB(is)e(dense)g(in)g FO(Im)13 b FN(\031)s FO(\))p FB(.)6 3691 y FO(3)p FB(.)50 b(If)34 b FO(pro-)o FN(C)427 3661 y FM(\003)466 3691 y FO(\()p Fz(A)p FO(\))f FB(is)h(a)g FN(\033)s FB(-)p FN(C)937 3661 y FM(\003)976 3691 y FB(-algebr)l(a)41 b FO(\()p FB(for)35 b(example)f(if)68 b Fz(A)33 b FB(is)h(\014nitely) -118 3791 y(gener)l(ate)l(d)9 b FO(\))p FB(,)45 b(then)d Fz(B)j FP(\037)g Fz(A)c FB(if)i(and)f(only)g(if)h(ther)l(e)f(exists)f (a)h(c)l(ontinuous)-118 3890 y(morphism)31 b FN( )12 b FO(:)28 b(pro-)o FN(C)599 3860 y FM(\003)637 3890 y FO(\()p Fz(B)p FO(\))c FP(\000)-46 b(!)23 b FO(pro-)o FN(C)1136 3860 y FM(\003)1175 3890 y FO(\()p Fz(K)c FP(\012)f Fz(A)p FO(\))29 b FB(with)i(dense)f(image.)p eop %%Page: 210 214 210 213 bop -118 -137 a FO(210)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FB(Pr)l(o)l(of.)43 b FO(1.)33 b(Let)19 b FN( )j FO(denote)c(the)h(canonical)c(morphism)g(form)1779 75 y(~)1770 96 y Fz(A)j FO(to)g(pro-)p FN(C)2154 66 y FM(\003)2191 96 y FO(\()2232 75 y(~)2223 96 y Fz(A)p FO(\).)-118 196 y(F)-7 b(or)29 b(ev)n(ery)g FN(\031)i FP(2)d FO(Rep\()p Fz(A)p FO(\))i(there)g(is)f(a)h(unique)g(lifting)i(~)-47 b FN(\031)31 b FP(2)d FO(Rep\()1950 174 y(~)1941 196 y Fz(A)p FO(\))i(and,)h(b)n(y)-118 296 y(the)f(de\014nition)e(of)i(the) h(en)n(v)n(eloping)26 b(pro-)p FN(C)1265 266 y FM(\003)1302 296 y FO(-algebra,)32 b(~)-46 b FN(\031)33 b FO(is)c(uniquely)f(lifted) -118 395 y(to)j(a)f(represen)n(tation)j(^)-47 b FN(\031)32 b FP(2)d FO(Rep\(pro-)o FN(C)1157 365 y FM(\003)1196 395 y FO(\()1237 374 y(~)1228 395 y Fz(A)p FO(\)\).)47 b(Hence)31 b(\(pro-)p FN(C)1918 365 y FM(\003)1956 395 y FO(\()1997 374 y(~)1988 395 y Fz(A)p FO(\))p FN(;)14 b( )24 b FP(\016)c FN(\036)p FO(\))-118 495 y(is)k(an)i(en)n(v)n (eloping)c(pro-)p FN(C)697 465 y FM(\003)734 495 y FO(-algebra)h(of)i Fz(A)p FO(.)36 b(Then)26 b(pro-)p FN(C)1691 465 y FM(\003)1728 495 y FO(\()p Fz(A)p FO(\))e FP(')e FO(pro-)o FN(C)2176 465 y FM(\003)2214 495 y FO(\()2255 473 y(~)2246 495 y Fz(A)p FO(\))-118 595 y(b)n(y)27 b(the)h(uniqueness)f(of)g(the)h (pro-)p FN(C)1010 564 y FM(\003)1047 595 y FO(-en)n(v)n(eloping)c (algebra.)6 713 y(2.)69 b(a\))38 b(Let)g(\()454 694 y FI(^)443 713 y Fz(K)19 b FP(\012)f Fz(A)p FN(;)c(\036)p FO(\))39 b(b)r(e)f(an)g(en)n(v)n(eloping)d FP(\003)p FO(-algebra)g(of)j Fz(K)26 b FP(\012)f Fz(A)p FO(.)68 b(If)-118 835 y FN(\031)26 b FP(2)e FO(Rep\()221 816 y FI(^)210 835 y Fz(K)19 b FP(\012)f Fz(A)p FO(\),)k(then)f FN(\031)27 b FO(=)22 b FN(U)13 b FO(~)-46 b FN(\031)960 847 y FK(0)998 835 y FN(U)1064 805 y FM(\003)1122 835 y FO(for)20 b(some)g FN(\031)1491 847 y FK(0)1551 835 y FP(2)k FO(Rep\()p Fz(A)p FO(\))d(and)f(unitary)-118 935 y FN(U)50 b FP(2)42 b FN(L)p FO(\()p FN(H)7 b FO(\).)69 b(Indeed,)42 b(since)37 b(the)i(mapping)e(Rep\()p Fz(K)26 b FP(\012)g Fz(A)p FO(\))41 b FP(3)h FN(\024)f FP(7!)j FO(~)-45 b FN(\024)41 b FP(2)-118 1058 y FO(Rep\()69 1039 y FI(^)58 1058 y Fz(K)19 b FP(\012)f Fz(A)p FO(\))34 b(is)f(bijectiv)n(e,)h FN(\031)j FO(=)f(~)-45 b FN(\024)34 b FO(for)f(some)g FN(\024)g FP(2)h FO(Rep\()p Fz(K)24 b FP(\012)e Fz(A)p FO(\).)56 b(By)33 b(the)-118 1157 y(previous)39 b(prop)r(osition,)i FN(\024)k FO(=)g FN(U)9 b FO(id)26 b FP(\012)h FN(\031)1206 1169 y FK(0)1244 1157 y FN(U)1310 1127 y FM(\003)1348 1157 y FO(,)44 b(where)d(id)f(is)g (the)h(iden)n(tical)-118 1257 y(represen)n(tation)d(of)k Fz(K)f FO(in)g FN(L)p FO(\()p FN(H)912 1269 y FK(0)949 1257 y FO(\))h(and)f FN(\031)1245 1269 y FK(0)1324 1257 y FO(is)f(a)h(represen)n(tation)d(of)j Fz(A)g FO(in)-118 1357 y FN(L)p FO(\()p FN(H)40 1369 y FK(1)77 1357 y FO(\),)e FN(H)45 b FO(=)37 b FN(H)456 1369 y FK(0)518 1357 y FP(\012)24 b FN(H)676 1369 y FK(1)713 1357 y FO(.)64 b(Notice)36 b(that)g FN(U)31 b FO(~)-64 b FN(\031)1371 1369 y FK(0)1409 1357 y FN(U)1475 1327 y FM(\003)1549 1357 y FO(is)36 b(a)g(represen)n(tation)e(of)-107 1460 y FI(^)-118 1479 y Fz(K)19 b FP(\012)f Fz(A)24 b FO(whic)n(h)g(is)g(a)g(lifting)e(of)j FN(\024)p FO(,)g(and)g(b)n(y)f(the)h(uniqueness)f(of)g(suc)n(h)h(a)f (lifting,)812 1651 y FN(U)13 b FO(~)-46 b FN(\031)925 1663 y FK(0)962 1651 y FN(U)1028 1617 y FM(\003)1089 1651 y FO(=)26 b(~)-45 b FN(\024)23 b FO(=)f FN(\031)s(:)6 1823 y FO(b\).)63 b(Assume)35 b(that)h Fz(B)i FP(\037)e Fz(A)p FO(,)i(i.e.,)g(there)d(exists)g(a)h FP(\003)p FO(-homomorphism)-118 1946 y FN( )12 b FO(:)28 b Fz(B)23 b FP(\000)-48 b(!)229 1927 y FI(^)218 1946 y Fz(K)19 b FP(\012)f Fz(A)i FO(suc)n(h)h(that)g(the)g(functor)g FN(F)1280 1958 y FL( )1340 1946 y FO(:)41 b(Rep\()p Fz(A)p FO(\))24 b FP(\000)-49 b(!)23 b FO(Rep\()p Fz(B)p FO(\))f(is)e(full.) -118 2071 y(Denote)31 b FA(A)c FO(=)h(pro-)o FN(C)567 2041 y FM(\003)605 2071 y FO(\()648 2052 y FI(^)637 2071 y Fz(K)19 b FP(\012)f Fz(A)p FO(\))28 b FP(')g FO(pro-)n FN(C)1219 2041 y FM(\003)1258 2071 y FO(\()p Fz(K)21 b FP(\012)f Fz(A)p FO(\),)31 b(and)f(let)g FN(\036)9 b FO(:)2006 2052 y FI(^)1995 2071 y Fz(K)19 b FP(\012)f Fz(A)27 b FP(\000)-48 b(!)-118 2171 y FA(A)39 b FO(b)r(e)h(the)g (canonical)c(morphism.)69 b(Then)40 b(the)g(functor)f FN(F)21 b FO(:)46 b(Rep\()p FA(A)p FO(\))d FP(\000)-48 b(!)-118 2271 y FO(Rep\()p Fz(B)p FO(\),)37 b(Rep\()p FA(A)p FO(\))d FP(3)g FN(\031)j FP(7!)d FN(\031)26 b FP(\016)c FN(\036)i FP(\016)e FN( )37 b FP(2)d FO(Rep\()p Fz(B)p FO(\),)j(is)c(also)f(full.)54 b(Indeed,)-118 2393 y FN(\031)27 b FP(\016)c FN(\036)36 b FP(2)g FO(Rep\()384 2374 y FI(^)373 2393 y Fz(K)19 b FP(\012)f Fz(A)p FO(\),)37 b(so)d FN(\031)27 b FP(\016)c FN(\036)36 b FO(=)f FN(U)13 b FO(~)-46 b FN(\031)1228 2405 y FK(0)1266 2393 y FN(U)1332 2363 y FM(\003)1405 2393 y FO(for)34 b(some)g FN(\031)1802 2405 y FK(0)1875 2393 y FP(2)i FO(Rep\()p Fz(A)p FO(\))f(in)-118 2493 y FN(L)p FO(\()p FN(H)7 b FO(\).)37 b(Th)n(us,)272 2665 y FN(F)12 b FO(\()p FN(\031)s FO(\)\()p Fz(B)p FO(\))588 2630 y FM(0)636 2665 y FO(=)23 b FN(F)777 2677 y FL( )827 2665 y FO(\()p FN(\031)f FP(\016)c FN(\036)p FO(\)\()p Fz(B)p FO(\))1206 2630 y FM(0)1254 2665 y FO(=)23 b FN(F)1395 2677 y FL( )1445 2665 y FO(\()p FN(U)13 b FO(~)-46 b FN(\031)1590 2677 y FK(0)1628 2665 y FN(U)1694 2630 y FM(\003)1732 2665 y FO(\)\()p Fz(B)p FO(\))1901 2630 y FM(0)1926 2665 y FN(;)-118 2863 y(F)-65 2875 y FL( )-15 2863 y FO(\()p FN(U)13 b FO(~)-46 b FN(\031)130 2875 y FK(0)168 2863 y FN(U)234 2832 y FM(\003)272 2863 y FO(\)\()p Fz(B)p FO(\))441 2832 y FM(0)489 2863 y FO(=)22 b(\()p FN(U)13 b FO(~)-46 b FN(\031)721 2875 y FK(0)759 2863 y FO(\()p Fz(B)p FO(\))p FN(U)962 2832 y FM(\003)1001 2863 y FO(\))1033 2832 y FM(0)1079 2863 y FO(=)23 b(\()p FN(U)13 b FO(~)-46 b FN(\031)1312 2875 y FK(0)1350 2863 y FO(\()1393 2844 y FI(^)1382 2863 y Fz(K)19 b FP(\012)f Fz(A)o FO(\))p FN(U)1696 2832 y FM(\003)1735 2863 y FO(\))1767 2832 y FM(0)1815 2863 y FO(since)24 b(the)h(func-)-118 2988 y(tor)40 b FN(F)82 3000 y FL( )174 2988 y FO(is)g(full.)76 b(\()p FN(U)13 b FO(~)-46 b FN(\031)634 3000 y FK(0)672 2988 y FO(\()715 2969 y FI(^)704 2988 y Fz(K)19 b FP(\012)f Fz(A)p FO(\))p FN(U)1019 2958 y FM(\003)1057 2988 y FO(\))1089 2958 y FM(0)1158 2988 y FO(=)45 b FN(\031)s FO(\()p FN(\036)p FO(\()1442 2969 y FI(^)1431 2988 y Fz(K)21 b FP(\012)d Fz(A)o FO(\)\))1713 2958 y FM(0)1783 2988 y FO(=)45 b FN(\031)s FO(\()p FA(A)p FO(\))2071 2958 y FM(0)2095 2988 y FO(,)g(since)-118 3114 y FN(\036)p FO(\()-26 3095 y FI(^)-37 3114 y Fz(K)20 b FP(\012)e Fz(A)o FO(\))28 b(is)f(quasi-dense)e(in)i FA(A)p FO(.)6 3214 y(The)c(morphism)18 b FN(\036)7 b FP(\016)g FN( )26 b FO(can)c(b)r(e)g(extended)h(to)f(a)f (con)n(tin)n(uous)f(morphism)f FN(\034)-118 3313 y FO(of)g(the)h(pro-)p FN(C)317 3283 y FM(\003)355 3313 y FO(\()p Fz(B)p FO(\),)i(since)c(if)h FN(\036)849 3325 y FK(1)896 3313 y FO(:)28 b Fz(B)23 b FP(\000)-48 b(!)23 b FO(pro-)o FN(C)1379 3283 y FM(\003)1417 3313 y FO(\()p Fz(B)p FO(\))e(denotes)e(the)h(canonical)-118 3413 y(morphism,)d(then)i FN(\036)515 3425 y FK(1)553 3413 y FO(\()p Fz(B)p FO(\))h(is)e(quasi-dense)e(in)i(pro-)p FN(C)1519 3383 y FM(\003)1556 3413 y FO(\()p Fz(B)p FO(\),)k(and)d(the) g(top)r(ology)-118 3513 y(on)25 b FN(\036)44 3525 y FK(1)82 3513 y FO(\()p Fz(B)p FO(\))h(induced)f(from)g(pro-)p FN(C)961 3482 y FM(\003)998 3513 y FO(\()p Fz(B)p FO(\))h(\(the)h(pro)5 b(jectiv)n(e)23 b(top)r(ology)g(induced)-118 3612 y(b)n(y)f(all)e (represen)n(tations)f(of)j Fz(B)p FO(\))h(is)e(stronger)g(than)h(the)g (top)r(ology)e(induced)i(b)n(y)-118 3712 y(the)j(map)f FN(\036)13 b FP(\016)g FN( )28 b FO(\(pro)5 b(jectiv)n(e)23 b(top)r(ology)g(induced)h(b)n(y)h(those)g(represen)n(tations)-118 3811 y(of)40 b Fz(B)h FO(whic)n(h)e(come)g(through)h FA(A)p FO(\).)75 b(Let)41 b(us)f(sho)n(w)f(that)i(the)g(range)e(of)h FN(\034)-118 3911 y FO(is)33 b(dense)i(in)f(ev)n(ery)f(represen)n (tation.)56 b(T)-7 b(ak)n(e)33 b(an)i(arbitrary)c(represen)n(tation)p eop %%Page: 211 215 211 214 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(211)-118 96 y FN(\031)26 b FP(2)e FO(Rep\()p FA(A)p FO(\).)34 b(Then)20 b(the)f(natural)f(injection)f FN(j)d FO(:)p 1418 24 299 4 v 28 w FN(\031)s FO(\()p FN(\034)9 b FO(\()p Fz(B)p FO(\)\))26 b FP(\000)-49 b(!)23 b FN(\031)s FO(\()p FA(A)p FO(\))d(satis\014es)-118 196 y(the)29 b(conditions)d(of)i(the)h (previous)d(lemma.)36 b(So)29 b FN(\031)s FO(\()p FN(\034)9 b FO(\(pro-)p FN(C)1786 166 y FM(\003)1824 196 y FO(\()p Fz(B)p FO(\)\)\))30 b(is)e(dense)-118 296 y(in)f(Im)13 b FN(\031)s FO(.)37 b(The)28 b(con)n(v)n(erse)d(statemen)n(t)i(is)f(ob) n(vious.)6 395 y(3.)60 b(If)36 b(pro-)p FN(C)436 365 y FM(\003)473 395 y FO(\()p Fz(A)p FO(\))g(is)e(a)h FN(\033)s FO(-)p FN(C)944 365 y FM(\003)982 395 y FO(-algebra,)g(then)g(pro-)p FN(C)1744 365 y FM(\003)1782 395 y FO(\()p Fz(K)24 b FP(\012)f Fz(A)p FO(\))36 b(is)e(also)-118 495 y(a)42 b FN(\033)s FO(-)p FN(C)109 465 y FM(\003)148 495 y FO(-algebra.)79 b(Therefore)41 b(its)h(top)r(ology)e(can)j(b)r(e)g(determined)e(b)n(y)h (a)-118 595 y(coun)n(table)24 b(increasing)f(family)h FN(p)938 607 y FL(n)983 595 y FO(\()p FP(\001)p FO(\))j(of)f FN(C)1255 564 y FM(\003)1293 595 y FO(-seminorms.)32 b(W)-7 b(e)27 b(ha)n(v)n(e)e(pro)n(v)n(ed)-118 694 y(that)33 b FN(\034)9 b FO(\(pro-)p FN(C)358 664 y FM(\003)396 694 y FO(\()p Fz(B)p FO(\)\))34 b(is)d(dense)i(in)f(pro-)p FN(C)1235 664 y FM(\003)1272 694 y FO(\()p Fz(K)23 b FP(\012)e Fz(A)p FO(\))33 b(in)e(an)n(y)h FN(C)1921 664 y FM(\003)1960 694 y FO(-seminorm)-118 794 y(\(eac)n(h)j(seminorm)d FN(p)p FO(\()p FP(\001)p FO(\))37 b(de\014nes)e(a)h(represen)n(tation)d FN(\031)1618 806 y FL(p)1692 794 y FO(of)j(pro-)p FN(C)2009 764 y FM(\003)2047 794 y FO(\()p Fz(K)24 b FP(\012)g Fz(A)p FO(\))-118 893 y(and)j FN(\031)90 905 y FL(p)129 893 y FO(\()p FN(\034)9 b FO(\(pro-)p FN(C)452 863 y FM(\003)491 893 y FO(\()p Fz(B)p FO(\)\)\))29 b(is)d(dense)i(in)f(Im)o (\()p FN(\031)s FO(\)\).)6 993 y(Using)40 b(argumen)n(ts)e(similar)e (to)k(Can)n(tor's)g(diagonal)d(metho)r(d)j(w)n(e)g(can)-118 1093 y(pro)n(v)n(e)25 b(that)j(it)e(is)g(also)f(dense)i(in)g(the)g(top) r(ology)e(determined)g(b)n(y)i(the)h(family)-118 1192 y FN(p)-76 1204 y FL(n)-31 1192 y FO(\()p FP(\001)p FO(\).)37 b(This)27 b(completes)e(the)j(pro)r(of.)p 2278 1192 4 57 v 2282 1140 50 4 v 2282 1192 V 2331 1192 4 57 v -118 1355 a FQ(Corollary)k(8.)40 b FB(The)33 b(r)l(elation)f FP(\037)f FB(is)g(a)h(quasi-or)l(der)g(r)l(elation,)h(i.e.,)h Fz(C)26 b FP(\037)f Fz(B)-118 1454 y FB(and)39 b Fz(B)23 b FP(\037)g Fz(A)29 b FB(imply)i Fz(C)23 b FP(\037)g Fz(A)p FB(.)6 1606 y FO(Indeed,)32 b(if)e FN( )436 1618 y FK(1)483 1606 y FO(:)e(pro-)o FN(C)747 1576 y FM(\003)786 1606 y FO(\()p Fz(C)p FO(\))g FP(\000)-48 b(!)28 b FO(pro-)o FN(C)1270 1576 y FM(\003)1308 1606 y FO(\()p Fz(K)1395 1618 y FK(1)1454 1606 y FP(\012)20 b Fz(B)p FO(\))28 b(=)g Fz(K)1820 1618 y FK(1)1890 1593 y FO(^)1878 1606 y FP(\012)20 b FO(pro-)o FN(C)2176 1576 y FM(\003)2214 1606 y FO(\()p Fz(A)p FO(\))-118 1706 y(and)i FN( )92 1718 y FK(2)139 1706 y FO(:)27 b(pro-)o FN(C)402 1676 y FM(\003)441 1706 y FO(\()p Fz(B)p FO(\))d FP(\000)-49 b(!)23 b FO(pro-)o FN(C)937 1676 y FM(\003)976 1706 y FO(\()p Fz(K)1063 1718 y FK(2)1108 1706 y FP(\012)8 b Fz(A)p FO(\))23 b(=)f Fz(K)1438 1718 y FK(2)1495 1693 y FO(^)1484 1706 y FP(\012)7 b FO(pro-)o FN(C)1769 1676 y FM(\003)1807 1706 y FO(\()p Fz(A)p FO(\))23 b(ha)n(v)n(e)e(dense)-118 1805 y(ranges)26 b(in)h(ev)n(ery)f(represen)n(tation,)f(then)j(the)g (comp)r(osed)e(homomorphism)153 1970 y(\(id)254 1982 y Fs(K)299 1990 y Fx(1)355 1970 y FP(\012)18 b FN( )492 1982 y FK(2)529 1970 y FO(\))h FP(\016)f FN( )694 1982 y FK(1)740 1970 y FO(:)28 b(pro-)o FN(C)1004 1936 y FM(\003)1042 1970 y FO(\()p Fz(C)p FO(\))c FP(\000)-48 b(!)23 b Fz(K)1359 1982 y FK(1)1426 1957 y FO(^)1415 1970 y FP(\012)18 b Fz(K)1553 1982 y FK(2)1621 1957 y FO(^)1609 1970 y FP(\012)g FO(pro-)o FN(C)1905 1936 y FM(\003)1943 1970 y FO(\()p Fz(A)p FO(\))-118 2135 y(also)30 b(has)j(a)f(dense)g(range)g(in)g(ev)n (ery)f(represen)n(tation)f(and)j Fz(K)1840 2147 y FK(1)1911 2122 y FO(^)1899 2135 y FP(\012)22 b Fz(K)2041 2147 y FK(2)2111 2135 y FO(is)32 b(also)-118 2234 y(the)c(algebra)d(of)i (compact)f(op)r(erators.)-118 2360 y FB(R)l(emark)k(50.)42 b FO(If)23 b Fz(B)g FP(\037)g Fz(A)e FO(then)i(if)e FN(\031)985 2372 y FK(1)1045 2360 y FO(and)h FN(\031)1248 2372 y FK(2)1307 2360 y FO(are)f(distinct)g(represen)n(tation)e(of)-118 2460 y Fz(A)27 b FO(in)h(the)g(same)e(space)i FN(H)7 b FO(,)27 b(then)i(the)f(represen)n(tation)d FN(F)1696 2472 y FL( )1747 2460 y FO(\()p FN(\031)1826 2472 y FK(1)1864 2460 y FO(\))j(and)g FN(F)2139 2472 y FL( )2189 2460 y FO(\()p FN(\031)2268 2472 y FK(2)2306 2460 y FO(\))-118 2559 y(are)e(also)g(distinct.)6 2685 y(Indeed,)g(let)f FN( )12 b FO(:)28 b(pro-)o FN(C)745 2655 y FM(\003)783 2685 y FO(\()p Fz(B)p FO(\))c FP(\000)-48 b(!)23 b FO(pro-)o FN(C)1280 2655 y FM(\003)1318 2685 y FO(\()p Fz(K)13 b FP(\012)g Fz(A)p FO(\))26 b(b)r(e)g(a)f(morphism)c(with)-118 2785 y(quasi-dense)36 b(range,)j(and)g(let)e FN(\031)946 2797 y FK(1)984 2785 y FO(,)k FN(\031)1095 2797 y FK(2)1173 2785 y FP(2)g FO(Rep\()p Fz(A)p FO(\))e(b)r(e)f(distinct)f(represen-) -118 2884 y(tations)e(on)i(the)g(same)e(space)h FN(H)7 b FO(.)64 b(Let)41 b(^)-46 b FN(\031)1261 2896 y FL(i)1327 2884 y FO(=)38 b(id)23 b FP(\012)i FN(\031)1660 2896 y FL(i)1687 2884 y FO(,)40 b FN(i)d FO(=)h(1,)h(2)d(b)r(e)h(the)-118 2984 y(corresp)r(onding)c(represen)n(tations)g(of)i Fz(K)25 b FP(\012)e Fz(A)36 b FO(on)f FN(H)1566 2996 y FK(0)1627 2984 y FP(\012)24 b FN(H)7 b FO(.)61 b(It)36 b(is)f(ob)n(vious)-118 3083 y(that)g(^)-46 b FN(\031)112 3095 y FK(1)178 3083 y FP(6)p FO(=)33 b(^)-47 b FN(\031)318 3095 y FK(2)356 3083 y FO(.)47 b(By)30 b(the)i(de\014nition)d(of)i(an)f(en)n(v)n (eloping)e(algebra,)h(they)i(can)-118 3183 y(b)r(e)25 b(lifted)e(to)h(distinct)f(represen)n(tations)j(~)-47 b FN(\031)1217 3195 y FK(1)1255 3183 y FO(,)29 b(~)-46 b FN(\031)1350 3195 y FK(2)1412 3183 y FO(of)24 b(pro-)p FN(C)1717 3153 y FM(\003)1755 3183 y FO(\()p Fz(K)12 b FP(\012)g Fz(A)p FO(\).)35 b(Denote)-118 3283 y FA(A)23 b FO(=)g(pro-)n FN(C)269 3253 y FM(\003)308 3283 y FO(\()p Fz(K)c FP(\012)f Fz(A)p FO(\).)37 b(F)-7 b(or)27 b(ev)n(ery)f FN(x)e FP(2)f FA(A)28 b FO(de\014ne)559 3447 y FN(p)p FO(\()p FN(x)p FO(\))c(=)e(max)o(\()p FP(k)p FN(\031)1099 3459 y FK(1)1136 3447 y FO(\()p FN(x)p FO(\))p FP(k)p FN(;)14 b FP(k)p FN(\031)1415 3459 y FK(2)1453 3447 y FO(\()p FN(x)p FO(\))p FP(k)p FO(\))p FN(:)-118 3612 y FO(Then)26 b FN(p)p FO(\()p FP(\001)p FO(\))h(is)d(a)i FN(C)467 3582 y FM(\003)506 3612 y FO(-norm)e(on)h FA(A)p FO(,)i(and)f(w)n(e)f(denote)h(the)h(completion)22 b(of)k FA(A)g FO(in)-118 3712 y(this)i(norm)f(b)n(y)h FA(A)443 3724 y FL(p)481 3712 y FO(.)40 b(If)29 b FN(j)k FO(denotes)28 b(the)h(natural)e(morphism)e(from)i FA(A)i FO(to)f FA(A)2277 3724 y FL(p)2316 3712 y FO(,)-118 3811 y(then)d(there)g(are)f(represen) n(tations)e FN(\031)1034 3823 y FL(i;p)1139 3811 y FP(2)h FO(Rep\()p FA(A)1457 3823 y FL(p)1496 3811 y FO(\),)j FN(i)d FO(=)f(1,)j(2,)h(and)e(w)n(e)h(ha)n(v)n(e)-118 3911 y(the)j(comm)n(utativ)n(e)c(diagram)p eop %%Page: 212 216 212 215 bop -118 -137 a FO(212)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)786 117 y FA(A)331 b(A)1245 129 y FL(p)p 874 101 283 4 v 1074 99 a Ft(-)996 68 y FN(j)903 329 y FO(~)-46 b FN(\031)946 341 y FL(i)876 241 y Ft(@)959 324 y(@)1042 407 y(@)1092 456 y(@)-83 b(R)1030 535 y FN(L)p FO(\()p FN(H)1188 547 y FK(0)1244 535 y FP(\012)18 b FN(H)7 b FO(\))p 1231 456 4 299 v 1233 456 a Ft(?)1266 313 y FN(\031)1313 325 y FL(i;p)-118 699 y FO(Since)38 b FN( )k FO(has)c(a)h(quasi-dense)d (range)i(and)h FN(\036)p FO(\()p Fz(B)p FO(\))h(is)e(quasi-dense)f(in)h (pro-)-118 799 y FN(C)-53 769 y FM(\003)-15 799 y FO(\()p Fz(B)p FO(\))31 b(\(where)f FN(\036)9 b FO(:)29 b Fz(B)e FP(\000)-48 b(!)26 b FO(pro-)o FN(C)977 769 y FM(\003)1016 799 y FO(\(\()p FN(B)t FO(\)\)\))31 b(is)e(the)h(natural)e(morphism\),) f(w)n(e)-118 898 y(ha)n(v)n(e)h(that)i(the)h FP(\003)p FO(-subalgebra)26 b FA(B)953 910 y FL(p)1018 898 y FO(=)g FN(j)f FP(\016)19 b FN( )k FP(\016)d FN(\036)p FO(\()p Fz(B)p FO(\))31 b(is)e(quasi-dense)e(in)i FA(A)2277 910 y FL(p)2316 898 y FO(,)-118 998 y(and)20 b(since)e(the)i(latter)f(is)g (a)g FN(C)787 968 y FM(\003)825 998 y FO(-algebra,)g(the)h (quasi-densit)n(y)d(implies)f(densit)n(y)-7 b(.)-118 1098 y(If)30 b FN(F)20 1110 y FL( )71 1098 y FO(\()p FN(\031)150 1110 y FK(1)187 1098 y FO(\))d(=)f FN(F)390 1110 y FL( )441 1098 y FO(\()p FN(\031)520 1110 y FK(2)558 1098 y FO(\),)k(then)h FN(\031)882 1110 y FK(1)p FL(;p)1000 1098 y FO(=)26 b FN(\031)1138 1110 y FK(2)p FL(;p)1259 1098 y FO(on)j(the)h(dense)g FP(\003)p FO(-subalgebra)c FA(B)2300 1110 y FL(p)-118 1197 y FO(and)i(as)g(suc)n(h)f(m)n(ust)h (coincide.)36 b(Then)29 b(it)e(follo)n(ws)f(from)h(the)h(comm)n (utativit)n(y)-118 1297 y(of)f(the)h(diagram)d(that)32 b(~)-47 b FN(\031)669 1309 y FK(1)730 1297 y FO(=)27 b(~)-46 b FN(\031)865 1309 y FK(2)902 1297 y FO(,)28 b(whic)n(h)f(con)n(tradicts)e(the)j(assertion.)-118 1434 y FB(R)l(emark)i(51.)42 b FO(Belo)n(w)37 b(in)h(this)g(section,)j(w)n (e)d(giv)n(e)f(a)h(n)n(um)n(b)r(er)g(of)g(examples)-118 1533 y(of)32 b FP(\003)p FO(-algebras)c(and)j(mappings)f FN( )12 b FO(:)29 b Fz(A)h FP(\000)-49 b(!)30 b FN(M)1340 1545 y FL(n)1385 1533 y FO(\()p Fz(B)p FO(\))j(suc)n(h)e(that)i(the)f (functor)-118 1633 y FN(F)-65 1645 y FL( )-5 1633 y FO(:)45 b(Rep)14 b Fz(B)42 b FP(\000)-48 b(!)42 b FO(Rep)14 b Fz(A)38 b FO(is)g(full.)70 b(But)39 b(w)n(e)g(do)f(not)h(discuss)f (metho)r(ds)g(of)-118 1732 y(construction)29 b(of)j(suc)n(h)f(mappings) d(here.)48 b(This)30 b(is)g(a)h(separate)f(topic)g(to)h(b)r(e)-118 1832 y(discussed)26 b(elsewhere.)6 1934 y(W)-7 b(e)26 b(also)d(do)i(not)g(discuss)f(the)h(question)f(of)h(what)g(is)f(the)h (minimal)c(n)n(um-)-118 2033 y(b)r(er)k FN(n)p FO(,)g(for)f(whic)n(h)g (there)h(exists)e(a)h(homomorphism)c FN( )12 b FO(:)28 b Fz(A)23 b FP(\000)-49 b(!)23 b FN(M)1971 2045 y FL(n)2016 2033 y FO(\()p Fz(B)p FO(\))j(suc)n(h)-118 2133 y(that)36 b(the)f(corresp)r(onding)e(functor)i FN(F)1111 2145 y FL( )1197 2133 y FO(is)f(full.)59 b(W)-7 b(e)36 b(only)e(notice)g(that) h(in)-118 2233 y([179)n(],)j(it)d(w)n(as)f(sho)n(wn)h(that)h(for)f(the) h FN(C)1170 2202 y FM(\003)1208 2233 y FO(-algebra)c FA(A)k FO(with)f FN(m)g FO(self-adjoin)n(t)-118 2332 y(generators)g FN(a)338 2344 y FK(1)375 2332 y FO(,)i FN(:)14 b(:)g(:)28 b FO(,)40 b FN(a)667 2344 y FL(m)730 2332 y FO(,)g(for)c FN(n)j FP(\025)g(\000)p FO(3)24 b(+)1343 2267 y FP(p)p 1412 2267 258 4 v 65 x FO(9)18 b(+)g(8)p FN(m)o FO(,)40 b FN(M)1813 2344 y FL(n)1858 2332 y FO(\()p FA(A)p FO(\))e(is)e(gener-)-118 2432 y(ated)28 b(b)n(y)g(a)g(pair)f(of) h(self-adjoin)n(t)e(generators.)37 b(It)29 b(is)f(sho)n(wn)f(in)h(pap)r (er)g([215)n(],)-118 2531 y(that)34 b(this)f(statemen)n(t)g(holds)g (for)h FN(n)f FP(\025)1167 2466 y(p)p 1237 2466 216 4 v 1237 2531 a FN(m)18 b FP(\000)g FO(1,)35 b(and)f(that)g(this)g (estimate)-118 2631 y(is)d(exact,)i(i.e.,)g(there)f(exists)f(a)h(comm)n (utativ)n(e)c FN(C)1471 2601 y FM(\003)1510 2631 y FO(-algebra)h FN(C)6 b FO(\()p FN(K)g FO(\))33 b(with)f FN(m)-118 2731 y FO(self-adjoin)n(t)27 b(generators)g(suc)n(h)i(that)h FN(M)1167 2743 y FL(n)1212 2731 y FO(\()p FN(C)6 b FO(\()p FN(K)g FO(\)\))31 b(is)d(not)i(singly)d(generated)-118 2830 y(for)j FN(n)e(<)183 2765 y FP(p)p 252 2765 V 65 x FN(m)19 b FP(\000)f FO(1)o(,)32 b(i.e.,)f FN(M)765 2842 y FL(n)809 2830 y FO(\()p FN(C)6 b FO(\()p FN(K)g FO(\)\))32 b(is)e(not)h(generated)e(b)n(y)i(a)f(pair)f(of)i(self-)-118 2930 y(adjoin)n(t)26 b(elemen)n(ts.)-118 3156 y FQ(3.1.2)94 b FP(\003)p FQ(-Wildness)29 b(of)i FP(\003)p FQ(-algebras)-118 3313 y(1.)k FO(In)26 b(the)g(theory)f(of)h(represen)n(tations)c(of)k (algebras,)d(it)i(w)n(as)g(suggested)f([70])-118 3413 y(to)30 b(consider)e(the)j(represen)n(tation)c(problem)h(to)i(b)r(e)g (wild)f(if)g(it)h(con)n(tains)e(the)-118 3513 y(classical)18 b(unsolv)n(ed)j(problem)f(of)j(represen)n(tation)d(theory)-7 b(,)23 b(i.e.,)g(the)g(problem)-118 3612 y(to)36 b(describ)r(e,)i(up)f (to)f(similarit)n(y)-7 b(,)33 b(a)k(pair)d(of)j(matrices)d(without)i (relations.)-118 3712 y(T)-7 b(o)26 b(de\014ne)h(an)f(analogue)e(of)j (wildness)d(for)i FP(\003)p FO(-algebras)d(\()p FP(\003)p FO(-wildness\),)i(it)h(w)n(as)-118 3811 y(suggested)18 b([144)n(])i(to)f(c)n(ho)r(ose,)g(for)g(a)f(standard)h FP(\003)p FO(-wild)d(problem)h(in)h(the)h(theory)-118 3911 y(of)33 b FP(\003)p FO(-represen)n(tations,)d(the)j(problem)e(of)h (describing)e(pairs)h(of)i(self-adjoin)n(t)p eop %%Page: 213 217 213 216 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(213)-118 96 y(\(or)30 b(unitary\))g(op)r(erators)f(up)j(to)f(a)f(unitary)g(equiv)-5 b(alence)28 b(\(represen)n(tations)-118 196 y(of)k(the)g FP(\003)p FO(-algebra)d Fz(S)562 208 y FK(2)631 196 y FO(\(or)i Fz(U)823 208 y FK(2)860 196 y FO(\))i(generated)e(b)n(y)g(a)h (pair)e(of)i(free)g(self-adjoin)n(t)-118 296 y(\(or)f(unitary\))g (generators\))f(and)i(to)g(regard)e(the)i(problems)d(whic)n(h)i(con)n (tain)-118 395 y(the)d(standard)e FP(\003)p FO(-wild)g(problem)f(as)i FP(\003)p FO(-wild.)-118 535 y FQ(2.)33 b FO(One)19 b(can)g(pro)n(v)n (e)e(that)i(the)h(standard)e FP(\003)p FO(-wild)e(problem)h(con)n (tains)g(as)h(a)h(sub-)-118 635 y(problem)29 b(the)j(problem)d(of)j (describing)d FP(\003)p FO(-represen)n(tations)f(of)j(an)n(y)g (\014nitely)-118 735 y(generated)26 b FP(\003)p FO(-algebra.)34 b(The)27 b(follo)n(wing)d(theorem)i(holds.)-118 882 y FQ(Theorem)k(52.)41 b Fz(S)520 894 y FK(2)580 882 y FO(=)23 b FI(C)15 b FP(h)p FN(a;)f(b)29 b FP(j)23 b FN(a)h FO(=)e FN(a)1145 852 y FM(\003)1183 882 y FN(;)14 b(b)23 b FO(=)g FN(b)1403 852 y FM(\003)1441 882 y FP(i)h(\037)e Fz(S)1653 894 y FL(m)1740 882 y FO(=)g FI(C)15 b FP(h)q FN(a)1958 894 y FK(1)2001 882 y FN(;)f(:)g(:)g(:)f(;)h(a)2229 894 y FL(m)2316 882 y FP(j)-118 981 y FN(a)-74 993 y FL(i)-24 981 y FO(=)23 b FN(a)108 951 y FM(\003)108 1003 y FL(i)146 981 y FN(;)28 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)p FP(i)30 b FB(for)h(any)f FN(m)23 b FO(=)f(1)p FB(,)30 b FO(2)p FB(,)g FN(:)14 b(:)g(:)27 b FB(.)-118 1129 y(Pr)l(o)l(of.)43 b FO(F)-7 b(or)29 b(the)i(algebra)746 1107 y(~)732 1129 y Fz(S)801 1141 y FL(m)864 1129 y FO(,)g(tak)n(e)e(the)h(algebra)e Fz(S)1608 1141 y FL(m)1701 1129 y FO(itself,)h FN(n)e FO(=)f FN(m)20 b FO(+)g(2.)-118 1228 y(De\014ne)28 b(a)f(homomorphism)c FN( )12 b FO(:)28 b Fz(S)981 1240 y FK(2)1041 1228 y FP(\000)-49 b(!)23 b FN(M)1244 1240 y FL(m)p FK(+2)1391 1228 y FO(\()p Fz(S)1492 1240 y FL(m)1555 1228 y FO(\))28 b(as)f(follo)n(ws:)221 1720 y FN( )s FO(\()p FN(a)p FO(\))c(=)497 1329 y Fy(2)497 1475 y(6)497 1525 y(6)497 1575 y(6)497 1625 y(6)497 1675 y(6)497 1725 y(6)497 1774 y(6)497 1824 y(6)497 1874 y(6)497 1927 y(4)552 1386 y FN(e)684 1454 y FK(1)p 684 1468 34 4 v 684 1515 a(2)727 1486 y FN(e)739 b Fo(0)859 1554 y FK(1)p 859 1568 V 859 1615 a(3)902 1586 y FN(e)1028 1683 y FO(.)1060 1708 y(.)1093 1733 y(.)1226 1809 y FK(1)p 1213 1823 59 4 v 1213 1870 a FL(m)1282 1841 y FN(e)836 1941 y Fo(0)1469 1909 y FK(1)p 1414 1923 143 4 v 1414 1970 a FL(m)p FK(+1)1567 1941 y FN(e)1753 2014 y FK(1)p 1698 2028 V 1698 2076 a FL(m)p FK(+2)1851 2047 y FN(e)1890 1329 y Fy(3)1890 1475 y(7)1890 1525 y(7)1890 1575 y(7)1890 1625 y(7)1890 1675 y(7)1890 1725 y(7)1890 1774 y(7)1890 1824 y(7)1890 1874 y(7)1890 1927 y(5)1959 1720 y FN(;)229 2568 y( )s FO(\()p FN(b)p FO(\))23 b(=)497 2127 y Fy(2)497 2273 y(6)497 2323 y(6)497 2373 y(6)497 2423 y(6)497 2473 y(6)497 2523 y(6)497 2572 y(6)497 2622 y(6)497 2672 y(6)497 2722 y(6)497 2772 y(6)497 2825 y(4)572 2186 y FO(0)124 b FN(e)112 b(a)933 2198 y FK(1)574 2285 y FN(e)123 b FO(0)132 b FN(e)167 b(a)1160 2297 y FK(2)1621 2285 y Fo(0)552 2440 y FN(a)596 2452 y FK(1)738 2440 y FN(e)131 b FO(0)187 b FN(e)1345 2382 y FO(.)1377 2407 y(.)1410 2432 y(.)717 2595 y FN(a)761 2607 y FK(2)910 2595 y FN(e)1113 2537 y FO(.)1145 2562 y(.)1177 2587 y(.)1525 2595 y FN(a)1569 2607 y FL(m)p FM(\000)p FK(1)885 2692 y FO(.)918 2717 y(.)950 2742 y(.)1368 2750 y(0)192 b FN(e)159 b(a)1844 2762 y FL(m)699 2849 y Fo(0)303 b FN(a)1105 2861 y FL(m)p FM(\000)p FK(1)1370 2849 y FN(e)192 b FO(0)g FN(e)1336 2949 y(a)1380 2961 y FL(m)1602 2949 y FN(e)g FO(0)1907 2127 y Fy(3)1907 2273 y(7)1907 2323 y(7)1907 2373 y(7)1907 2423 y(7)1907 2473 y(7)1907 2523 y(7)1907 2572 y(7)1907 2622 y(7)1907 2672 y(7)1907 2722 y(7)1907 2772 y(7)1907 2825 y(5)1977 2568 y FN(:)6 3104 y FO(One)28 b(can)f(directly)e(c)n(hec)n(k)i(that)h(the)g(functor)f FN(F)1524 3116 y FL( )1603 3104 y FO(is)f(full.)p 2278 3104 4 57 v 2282 3052 50 4 v 2282 3104 V 2331 3104 4 57 v 6 3266 a(Theorem)e(52)g(p)r(ermits)g(one)h(to)g(sa)n(y)f(that)i (the)g(problem)d(of)i(unitary)f(clas-)-118 3365 y(si\014cation)33 b(of)k(pairs)d(of)i(self-adjoin)n(t)e(op)r(erators)g(con)n(tains,)i(as) g(a)g(subprob-)-118 3465 y(lem,)g(the)g(problem)e(of)i(unitary)e (classi\014cation)e(of)j(represen)n(tation)f(of)h(an)n(y)-118 3565 y FP(\003)p FO(-algebra)30 b(with)k(a)g(coun)n(table)e(n)n(um)n(b) r(er)h(of)h(generators)e(\(b)r(ecause)i(it)f(is)g(al-)-118 3664 y(w)n(a)n(ys)26 b(p)r(ossible)f(to)j(c)n(ho)r(ose)e(these)i (generators)d(to)i(b)r(e)h(self-adjoin)n(t\).)-118 3811 y FQ(Corollary)k(9.)40 b FB(F)-6 b(or)37 b(any)f FN(m)f FO(=)g(1)p FB(,)i FO(2)p FB(,)g FN(:)14 b(:)g(:)27 b FB(,)38 b Fz(S)1421 3823 y FK(2)1493 3811 y FP(\037)d Fz(U)1647 3823 y FL(m)1744 3811 y FO(=)f FI(C)15 b FP(h)q FN(u)1978 3823 y FK(1)2020 3811 y FN(;)f(:)g(:)g(:)g(;)g(u)2253 3823 y FL(m)2316 3811 y FN(;)-118 3911 y(u)-70 3881 y FM(\003)-70 3932 y FK(1)-32 3911 y FN(;)g(:)g(:)g(:)f(;)h(u)200 3881 y FM(\003)200 3932 y FL(m)286 3911 y FP(j)23 b FN(u)380 3923 y FL(i)407 3911 y FN(u)455 3881 y FM(\003)455 3933 y FL(i)516 3911 y FO(=)f FN(u)651 3881 y FM(\003)651 3933 y FL(i)689 3911 y FN(u)737 3923 y FL(i)787 3911 y FO(=)h FN(e;)14 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)p FP(i)p FB(.)p eop %%Page: 214 218 214 217 bop -118 -137 a FO(214)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FQ(Theorem)i(53.)41 b Fz(U)505 108 y FK(2)565 96 y FP(\037)22 b Fz(S)721 108 y FK(2)758 96 y FB(.)-118 266 y(Pr)l(o)l(of.)43 b FO(Cho)r(ose)29 b(the)i(en)n(v)n(eloping)c (algebra)1297 244 y(~)1283 266 y Fz(S)1352 278 y FK(2)1419 266 y FO(to)j(b)r(e)h(the)f(algebra)e(of)i(frac-)-118 365 y(tions)f(of)h(the)h(algebra)d Fz(S)694 377 y FK(2)761 365 y FO(\(see,)j(e.g.,)g([87)o(]\))g(with)f(resp)r(ect)g(to)g(the)h (set)f(\006)e(=)-118 465 y FP(f)p FN(a)10 b FP(\000)g FN(ie;)k(a)c FO(+)g FN(ie;)k(b)c FP(\000)g FN(ie;)k(b)c FO(+)g FN(ie)p FP(g)p FO(.)32 b(De\014ne)24 b(the)g(homomorphism)19 b FN( )12 b FO(:)28 b Fz(U)2050 477 y FK(2)2110 465 y FP(\000)-49 b(!)2246 443 y FO(~)2233 465 y Fz(S)2302 477 y FK(2)-118 565 y FO(as)27 b(follo)n(ws:)98 750 y FN( )s FO(\()p FN(u)235 762 y FK(1)272 750 y FO(\))c(=)g(\()p FN(a)c FP(\000)f FN(ie)p FO(\)\()p FN(a)g FO(+)g FN(ie)p FO(\))970 715 y FM(\000)p FK(1)1059 750 y FN(;)96 b( )s FO(\()p FN(u)1315 762 y FK(2)1353 750 y FO(\))23 b(=)g(\()p FN(b)18 b FP(\000)g FN(ie)p FO(\)\()p FN(b)g FO(+)g FN(ie)p FO(\))2034 715 y FM(\000)p FK(1)-118 934 y FO(\(the)28 b(Ca)n(yley)e(transformation\).)33 b(The)28 b(rest)f(is)f(ob)n(vious.)p 2278 934 4 57 v 2282 882 50 4 v 2282 934 V 2331 934 4 57 v -118 1110 a FQ(3.)36 b FO(Theorems)25 b(52,)h(53)h(allo)n(w,)d(as) j(a)f(mo)r(del)g(of)h(complexit)n(y)d(for)i(problems)f(of)-118 1209 y(unitary)j(classi\014cation)e(of)j(represen)n(tations)e(of)j FP(\003)p FO(-algebras,)c(to)k(c)n(ho)r(ose)e(the)-118 1309 y(problem)36 b(of)j(unitary)e(classi\014cation)e(of)k(represen)n (tations)c(of)k(the)g(algebra)-118 1408 y Fz(U)-64 1420 y FK(2)5 1408 y FO(or,)32 b(whic)n(h)f(is)g(the)h(same)f(thing,)h(the)g (en)n(v)n(eloping)d FN(C)1695 1378 y FM(\003)1733 1408 y FO(-algebra)g FN(C)2121 1378 y FM(\003)2160 1408 y FO(\()p FA(F)2248 1420 y FK(2)2283 1408 y FO(\),)-118 1508 y(where)e FA(F)178 1520 y FK(2)241 1508 y FO(is)f(the)i(free)g (group)e(with)h(t)n(w)n(o)g(generators,)e FN(u)p FO(,)j FN(v)s FO(.)-118 1677 y FQ(De\014nition)j(14.)40 b FB(A)30 b FP(\003)p FB(-algebr)l(a)g Fz(A)f FB(is)h(c)l(al)t(le)l(d)i FP(\003)p FB(-wild)e(if)48 b Fz(A)23 b FP(\037)f FN(C)1921 1647 y FM(\003)1960 1677 y FO(\()p FA(F)2048 1689 y FK(2)2083 1677 y FO(\))p FB(.)6 1847 y FO(F)-7 b(or)27 b(commen)n(ts)f(on)h (De\014nition)f(14,)h(also)e(see)j(Section)e(3.2.3)h(b)r(elo)n(w.)-118 2000 y FQ(4.)36 b FO(Theorem)26 b(50)h(immediately)22 b(implies)i(the)k(follo)n(wing)c(statemen)n(t.)-118 2169 y FQ(Prop)s(osition)30 b(66.)41 b FB(A)26 b FN(C)715 2139 y FM(\003)754 2169 y FB(-algebr)l(a)i Fz(A)f FB(is)g FP(\003)p FB(-wild)h(if)g(and)g(only)g(if)g(ther)l(e)f(exist)-118 2269 y FN(n)c FP(2)g FI(N)29 b FP([)18 b(f1g)29 b FB(and)h(a)h FN(C)680 2239 y FM(\003)718 2269 y FB(-ide)l(al)39 b Fz(I)24 b FP(\032)e Fz(A)29 b FB(such)h(that)38 b Fz(A)p FN(=)p Fz(I)22 b FP(')h FN(M)1903 2281 y FL(n)1947 2269 y FO(\()p FN(C)2044 2239 y FM(\003)2083 2269 y FO(\()p FA(F)2171 2281 y FK(2)2207 2269 y FO(\)\))p FB(.)-118 2438 y FQ(5.)59 b FO(Nuclear)34 b FP(\003)p FO(-algebras)d(ha)n(v)n(e)j (only)g(h)n(yp)r(er\014nite)g(factor)h(represen)n(tations)-118 2538 y(and,)k(consequen)n(tly)-7 b(,)38 b(they)f(can)g(not)g(b)r(e)h FP(\003)p FO(-wild.)63 b(There)36 b(also)f(exist)h(non-)-118 2637 y(n)n(uclear)h FN(C)247 2607 y FM(\003)285 2637 y FO(-algebras)f(whic)n(h)i(are)g(not)h(wild.)70 b(F)-7 b(or)39 b(example,)g(the)h(group)-118 2737 y FN(C)-53 2707 y FM(\003)-15 2737 y FO(-algebra)26 b FN(C)370 2707 y FM(\003)409 2737 y FO(\()p FN(B)t FO(\()p FN(m;)14 b FO(2\)\))30 b(of)f(the)g(Burnside)f(group)g(with)h(t)n(w)n(o)f (generators)-118 2837 y(and)34 b(su\016cien)n(tly)e(large)f(o)r(dd)j FN(m)g FO(is)f(not)h(n)n(uclear,)g(but)g(it)g(is)f(also)e(not)j(wild) -118 2936 y(\(see)27 b(Section)g(3.1.6\).)-118 3158 y FQ(3.1.3)94 b FP(\003)p FQ(-Wild)23 b(algebras)j(generated)f(b)m(y)i (orthogonal)e(pro)5 b(jections)174 3258 y(and)32 b(idemp)s(oten)m(ts) -118 3413 y(1.)52 b FO(F)-7 b(or)33 b(represen)n(tations)d(of)j(the)g FP(\003)p FO(-algebra)d FA(P)1414 3425 y FK(2)1483 3413 y FO(=)i FI(C)15 b FP(h)p FN(p)1708 3425 y FK(1)1751 3413 y FN(;)f(p)1830 3425 y FK(2)1899 3413 y FP(j)32 b FN(p)1996 3383 y FM(\003)1996 3434 y FK(1)2066 3413 y FO(=)g FN(p)2205 3425 y FK(1)2274 3413 y FO(=)-118 3513 y FN(p)-76 3482 y FK(2)-76 3533 y(1)-39 3513 y FN(;)c(p)54 3482 y FM(\003)54 3533 y FK(2)128 3513 y FO(=)36 b FN(p)271 3525 y FK(2)345 3513 y FO(=)g FN(p)488 3482 y FK(2)488 3533 y(2)525 3513 y FP(i)g FO(\(a)g(pair)e(of)i(orthogonal)c(pro)5 b(jections)34 b FN(P)1897 3525 y FK(1)1934 3513 y FO(,)k FN(P)2048 3525 y FK(2)2086 3513 y FO(\))e(there)-118 3612 y(is)24 b(a)i(structure)f(theorem)f(that)i(giv)n(es)e(a)h(decomp)r (osition)d(of)k(represen)n(tations)-118 3712 y(in)n(to)32 b(a)g(direct)g(sum)g(\(or)g(in)n(tegral\))e(of)j(irreducible)d (represen)n(tations)g(whic)n(h)-118 3811 y(are)35 b(either)h(one-)f(or) h(t)n(w)n(o-dimensional,)d(and)j(their)g(description,)g(up)h(to)f(a) -118 3911 y(unitary)26 b(equiv)-5 b(alence)25 b(\(see)j(Section)e (1.2.2\).)p eop %%Page: 215 219 215 218 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(215)-118 96 y FQ(2.)35 b FO(The)25 b(problem)e(to)i(describ)r(e,)f(up)h(to)g(a)f (unitary)g(equiv)-5 b(alence,)23 b(a)i(family)d(of)-118 196 y(orthogonal)k(pro)5 b(jections)26 b FN(P)780 208 y FK(1)818 196 y FO(,)j FN(P)923 208 y FK(2)961 196 y FO(,)g FN(:)14 b(:)g(:)27 b FO(,)i FN(P)1242 208 y FL(n)1317 196 y FO(for)f FN(n)e FP(\025)e FO(3)29 b(is)f FP(\003)p FO(-wild.)38 b(W)-7 b(e)30 b(giv)n(e)-118 296 y(a)d(fairly)e(simple)g (pro)r(of)i(that)h(this)f(problem)e(is)h FP(\003)p FO(-wild)f(for)i FN(n)c FO(=)g(3.)-118 445 y FQ(Theorem)30 b(54.)41 b FB(L)l(et)h FA(P)662 457 y FK(3)745 445 y FO(=)j FI(C)15 b FP(h)p FN(p)983 457 y FK(1)1026 445 y FN(;)f(p)1105 457 y FK(2)1142 445 y FN(;)g(p)1221 457 y FK(3)1272 445 y FP(j)g FN(p)1351 415 y FK(2)1351 467 y FL(i)1434 445 y FO(=)45 b FN(p)1586 415 y FM(\003)1586 467 y FL(i)1670 445 y FO(=)g FN(p)1822 457 y FL(i)1849 445 y FN(;)28 b(i)46 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g FO(3)p FP(i)p FN(:)-118 545 y FB(Then)30 b FA(P)153 557 y FK(3)213 545 y FP(\037)23 b FN(C)366 515 y FM(\003)405 545 y FO(\()p FA(F)493 557 y FK(2)528 545 y FO(\))p FB(,)30 b(i.e.,)i FA(P)840 557 y FK(3)907 545 y FB(is)e FP(\003)p FB(-wild.)-118 695 y(Pr)l(o)l(of.)43 b FO(Let)20 b(us)g(de\014ne)g(the)g(homomorphism) 15 b FN( )d FO(:)28 b FA(P)1500 707 y FK(3)1560 695 y FP(\000)-48 b(!)23 b FN(M)1764 707 y FK(4)1801 695 y FO(\()p FA(F)1889 707 y FK(2)1924 695 y FO(\))d(as)g(follo)n(ws:)242 1007 y FN( )s FO(\()p FN(p)373 1019 y FK(1)410 1007 y FO(\))k(=)553 790 y Fy(0)553 936 y(B)553 986 y(B)553 1039 y(@)628 856 y FN(e)84 b FO(0)e(0)h(0)626 956 y(0)h FN(e)g FO(0)f(0)626 1056 y(0)g(0)f(0)h(0)626 1155 y(0)g(0)f(0)h(0)1041 790 y Fy(1)1041 936 y(C)1041 986 y(C)1041 1039 y(A)1128 1007 y FN(;)242 1439 y( )s FO(\()p FN(p)373 1451 y FK(2)410 1439 y FO(\))24 b(=)553 1223 y Fy(0)553 1369 y(B)553 1418 y(B)553 1472 y(@)636 1256 y FK(1)p 636 1270 34 4 v 636 1317 a(2)679 1289 y FN(e)108 b FO(0)985 1256 y FK(1)p 985 1270 V 985 1317 a(2)1029 1289 y FN(e)f FO(0)651 1389 y(0)811 1356 y FK(1)p 811 1370 V 811 1417 a(2)854 1389 y FN(e)h FO(0)1160 1356 y FK(1)p 1160 1370 V 1160 1417 a(2)1203 1389 y FN(e)636 1456 y FK(1)p 636 1470 V 636 1517 a(2)679 1489 y FN(e)g FO(0)985 1456 y FK(1)p 985 1470 V 985 1517 a(2)1029 1489 y FN(e)f FO(0)651 1589 y(0)811 1556 y FK(1)p 811 1570 V 811 1617 a(2)854 1589 y FN(e)h FO(0)1160 1556 y FK(1)p 1160 1570 V 1160 1617 a(2)1203 1589 y FN(e)1242 1223 y Fy(1)1242 1369 y(C)1242 1418 y(C)1242 1472 y(A)1328 1439 y FN(;)242 1931 y( )s FO(\()p FN(p)373 1943 y FK(3)410 1931 y FO(\))24 b(=)553 1664 y Fy(0)553 1810 y(B)553 1860 y(B)553 1910 y(B)553 1960 y(B)553 2013 y(@)708 1706 y FK(3)p 708 1720 V 708 1767 a(8)752 1738 y FN(e)988 1657 y FM(p)p 1042 1657 34 3 v 1042 1705 a FK(3)p 988 1719 88 4 v 1015 1767 a(8)1085 1738 y FN(uv)1176 1708 y FM(\003)1340 1657 y(p)p 1394 1657 34 3 v 1394 1705 a FK(3)p 1340 1719 88 4 v 1367 1767 a(8)1437 1738 y FN(u)1682 1706 y FK(3)p 1682 1720 34 4 v 1682 1767 a(8)1725 1738 y FN(e)636 1791 y FM(p)p 691 1791 34 3 v 48 x FK(3)p 636 1853 88 4 v 663 1900 a(8)734 1872 y FN(v)s(u)825 1841 y FM(\003)1060 1839 y FK(5)p 1060 1853 34 4 v 1060 1900 a(8)1103 1872 y FN(e)1369 1839 y FK(3)p 1369 1853 V 1369 1900 a(8)1412 1872 y FN(v)116 b FP(\000)1643 1791 y FM(p)p 1697 1791 34 3 v 1697 1839 a FK(3)p 1642 1853 88 4 v 1670 1900 a(8)1740 1872 y FN(v)s(u)1831 1841 y FM(\003)658 1924 y(p)p 712 1924 34 3 v 712 1972 a FK(3)p 658 1986 88 4 v 685 2033 a(8)755 2005 y FN(u)803 1975 y FM(\003)1039 1972 y FK(3)p 1039 1986 34 4 v 1039 2033 a(8)1082 2005 y FN(v)1125 1975 y FM(\003)1371 1972 y FK(1)p 1371 1986 V 1371 2033 a(4)1414 2005 y FN(e)245 b FO(0)708 2105 y FK(3)p 708 2119 V 708 2167 a(8)752 2138 y FN(e)155 b FP(\000)1021 2057 y FM(p)p 1075 2057 34 3 v 1075 2105 a FK(3)p 1020 2119 88 4 v 1047 2167 a(8)1118 2138 y FN(uv)1209 2108 y FM(\003)1386 2138 y FO(0)1682 2105 y FK(3)p 1682 2119 34 4 v 1682 2167 a(4)1725 2138 y FN(e)1869 1664 y Fy(1)1869 1810 y(C)1869 1860 y(C)1869 1910 y(C)1869 1960 y(C)1869 2013 y(A)1955 1931 y FN(:)-118 2301 y FO(It)28 b(is)f(easy)g(to)g(see)h(that)g(the)g (corresp)r(onding)d(functor)j FN(F)1674 2313 y FL( )1734 2301 y FO(:)41 b(Rep)14 b FN(C)2021 2271 y FM(\003)2060 2301 y FO(\()p FA(F)2148 2313 y FK(2)2183 2301 y FO(\))24 b FP(\000)-48 b(!)-118 2401 y FO(Rep)14 b FA(P)95 2413 y FK(3)160 2401 y FO(is)26 b(full.)p 2278 2401 4 57 v 2282 2348 50 4 v 2282 2401 V 2331 2401 4 57 v -118 2563 a FB(R)l(emark)k(52.)42 b FO(The)28 b FP(\003)p FO(-algebra)c FA(P)927 2575 y FK(3)992 2563 y FO(coincides)i(with)h(the)h(group)f FP(\003)p FO(-algebra)d(of)-118 2705 y(the)k(Co)n(xeter)e(group)611 2687 y Fn(r)p 611 2688 167 4 v 141 w(r)694 2604 y(r)611 2687 y Ft(\000@)661 2728 y FM(1)578 2645 y(1)100 b(1)819 2705 y FO(,)27 b(see)h(Section)e(1.2.2,)h(i.e.,)65 2879 y FA(P)120 2891 y FK(3)180 2879 y FO(=)c FI(C)15 b FP(h)p FN(w)414 2891 y FL(i)470 2879 y FO(=)23 b FN(w)619 2845 y FM(\003)617 2900 y FL(i)681 2879 y FO(=)g(2)p FN(p)853 2891 y FL(i)898 2879 y FP(\000)18 b FN(e;)c(i)22 b FO(=)h(1)p FN(;)14 b FO(2)p FN(;)g FO(3)21 b FP(j)i FN(w)1524 2845 y FK(2)1522 2900 y FL(i)1585 2879 y FO(=)g FN(e;)k(i)c FO(=)g(1)p FN(;)14 b FO(2)p FN(;)g FO(3)p FP(i)p FN(:)-118 3042 y FO(Th)n(us,)40 b(the)f(group)e FN(C)593 3012 y FM(\003)631 3042 y FO(-algebra)e(of)j(this)g(group)f(is)f FP(\003)p FO(-wild)g(\(see)i(also)e(Sec-)-118 3141 y(tion)27 b(3.1.6\).)-118 3266 y FQ(3.)36 b FO(Ev)n(en)26 b(with)h(additional)c (conditions)i(imp)r(osed)g(on)i(the)g(triple)e(of)i(orthog-)-118 3366 y(onal)f(pro)5 b(jections,)25 b FN(p)554 3378 y FK(1)591 3366 y FO(,)i FN(p)683 3378 y FK(2)720 3366 y FO(,)h FN(p)813 3378 y FK(3)878 3366 y FO(\(the)g(triple)d(of)i (\015ips)g FN(w)1603 3378 y FL(i)1654 3366 y FO(=)c(2)p FN(p)1826 3378 y FL(i)1871 3366 y FP(\000)17 b FN(e)p FO(,)28 b FN(i)22 b FO(=)h(1,)k(2,)-118 3465 y(3\),)g(the)h(problem)e (of)h(their)g(description)e(ma)n(y)h(still)e(b)r(e)k FP(\003)p FO(-wild.)-118 3615 y FQ(Theorem)i(55.)41 b FB(The)31 b FP(\003)p FB(-algebr)l(a)-32 3777 y FA(P)23 3789 y FK(3)p FL(;)p FK(2an)n(ti)249 3777 y FO(=)22 b FI(C)390 3710 y Fy(\012)435 3777 y FN(w)494 3789 y FL(i)545 3777 y FO(=)h FN(w)694 3743 y FM(\003)692 3798 y FL(i)733 3777 y FN(;)14 b(i)23 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g FO(3)22 b FP(j)h FN(w)1238 3743 y FK(2)1236 3798 y FL(i)1299 3777 y FO(=)f FN(e;)14 b(i)23 b FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g FO(3;)1107 3911 y FP(f)p FN(w)1208 3923 y FK(1)1246 3911 y FN(;)g(w)1342 3923 y FK(2)1379 3911 y FP(g)23 b FO(=)f FN(w)1590 3923 y FK(1)1628 3911 y FN(w)1687 3923 y FK(2)1743 3911 y FO(+)c FN(w)1885 3923 y FK(2)1923 3911 y FN(w)1982 3923 y FK(1)2043 3911 y FO(=)k(0)p FP(g)2214 3844 y Fy(\013)p eop %%Page: 216 220 216 219 bop -118 -137 a FO(216)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FB(is)i FP(\003)p FB(-wild.)-118 236 y(Pr)l(o)l(of.)43 b FO(De\014ne)e(the)g(homomorphism)35 b FN( )43 b FO(of)d(the)h FP(\003)p FO(-algebra)c FA(P)1955 248 y FK(3)p FL(;)p FK(2an)n(ti)2198 236 y FO(in)n(to)-118 336 y FN(M)-37 348 y FK(4)0 336 y FO(\()p Fz(U)86 348 y FK(2)123 336 y FO(\),)28 b(where)g Fz(U)501 348 y FK(2)561 336 y FO(=)23 b FI(C)15 b FP(h)p FN(u;)f(u)868 306 y FM(\003)911 336 y FN(;)g(v)s(;)g(v)1071 306 y FM(\003)1133 336 y FP(j)24 b FN(uu)1276 306 y FM(\003)1337 336 y FO(=)f FN(u)1473 306 y FM(\003)1511 336 y FN(u)g FO(=)g FN(v)s(v)1756 306 y FM(\003)1818 336 y FO(=)g FN(v)1949 306 y FM(\003)1987 336 y FN(v)k FO(=)c FN(e)p FP(i)p FO(,)28 b(as)-118 436 y(follo)n(ws:)344 715 y FN( )s FO(\()p FN(w)492 727 y FK(1)530 715 y FO(\))23 b(=)673 498 y Fy(0)673 644 y(B)673 694 y(B)673 747 y(@)747 564 y FN(e)84 b FO(0)1046 622 y Fo(0)746 664 y FO(0)f FN(e)996 764 y(e)115 b FO(0)808 822 y Fo(0)995 863 y FO(0)82 b FP(\000)p FN(e)1222 498 y Fy(1)1222 644 y(C)1222 694 y(C)1222 747 y(A)1309 715 y FN(;)344 1146 y( )s FO(\()p FN(w)492 1158 y FK(2)530 1146 y FO(\))23 b(=)673 930 y Fy(0)673 1076 y(B)673 1126 y(B)673 1179 y(@)996 996 y FN(e)84 b FO(0)808 1054 y Fo(0)995 1096 y FO(0)g FN(e)747 1195 y(e)g FO(0)746 1295 y(0)f FN(e)1057 1253 y Fo(0)1161 930 y Fy(1)1161 1076 y(C)1161 1126 y(C)1161 1179 y(A)1247 1146 y FN(;)344 1578 y( )s FO(\()p FN(w)492 1590 y FK(3)530 1578 y FO(\))23 b(=)717 1522 y(1)p 683 1559 111 4 v 683 1576 a FP(p)p 752 1576 42 4 v 68 x FO(3)817 1361 y Fy(0)817 1507 y(B)817 1557 y(B)817 1610 y(@)894 1428 y FN(e)164 b FO(0)212 b FN(u)1399 1398 y FM(\003)1630 1428 y FN(v)1673 1398 y FM(\003)893 1527 y FO(0)131 b FP(\000)p FN(e)204 b(e)161 b FP(\000)p FN(uv)1730 1497 y FM(\003)890 1627 y FN(u)f(e)236 b FO(0)159 b FP(\000)p FN(uv)1730 1597 y FM(\003)892 1727 y FN(v)89 b FP(\000)p FN(v)s(u)1177 1697 y FM(\003)1297 1727 y FP(\000)p FN(v)s(u)1453 1697 y FM(\003)1650 1727 y FO(0)1767 1361 y Fy(1)1767 1507 y(C)1767 1557 y(C)1767 1610 y(A)1854 1578 y FN(:)-118 1873 y FO(It)35 b(is)f(easy)g(to)h(c)n (hec)n(k)f(that)i(the)f(corresp)r(onding)d(functor)j FN(F)1828 1885 y FL( )1888 1873 y FO(:)44 b(Rep)14 b Fz(U)2167 1885 y FK(2)2239 1873 y FP(\000)-48 b(!)-118 1973 y FO(Rep)14 b FA(P)95 1985 y FK(3)p FL(;)p FK(2an)n(ti)325 1973 y FO(is)27 b(full.)p 2278 1973 4 57 v 2282 1920 50 4 v 2282 1973 V 2331 1973 4 57 v -118 2132 a FQ(4.)66 b FO(It)39 b(is)d(a)i(more)d(complicated)g(fact)j(that)g(the)g(group)e FP(\003)p FO(-algebra)f(of)i(the)-118 2232 y(Co)n(xeter)23 b(group)462 2214 y Fn(r)p 462 2216 125 4 v 100 w(r)100 b(r)p 587 2216 V 492 2197 a FM(1)58 b(1)778 2232 y FO(\(i.e.,)25 b(the)g FP(\003)p FO(-algebra)c FA(P)1517 2244 y FK(3)p FL(;)p FK(2comm)1802 2232 y FO(=)h FI(C)15 b FP(h)q FN(p)2018 2244 y FK(1)2061 2232 y FN(;)f(p)2140 2244 y FK(2)2177 2232 y FN(;)g(p)2256 2244 y FK(3)2316 2232 y FP(j)-118 2331 y FN(p)-76 2301 y FM(\003)-76 2353 y FL(i)2 2331 y FO(=)40 b FN(p)149 2301 y FK(2)149 2353 y FL(i)227 2331 y FO(=)g FN(p)374 2343 y FL(i)401 2331 y FN(;)28 b(i)40 b FO(=)g(1)p FN(;)14 b FO(2)p FN(;)g FO(3;)26 b([)p FN(p)940 2343 y FK(1)977 2331 y FN(;)14 b(p)1056 2343 y FK(2)1093 2331 y FO(])40 b(=)g(0)p FP(i)h FO(=)f FI(C)15 b FP(h)p FN(w)1626 2343 y FL(i)1700 2331 y FO(=)40 b FN(w)1866 2301 y FM(\003)1864 2353 y FL(i)1946 2331 y FO(=)g(2)p FN(p)2135 2343 y FL(i)2187 2331 y FP(\000)25 b FN(e;)-118 2431 y(i)e FO(=)f(1)p FN(;)14 b FO(2)p FN(;)g FO(3)22 b FP(j)h FN(w)350 2401 y FK(2)348 2453 y FL(i)411 2431 y FO(=)f FN(e;)28 b(i)22 b FO(=)h(1)p FN(;)14 b FO(2)p FN(;)g FO(3;)26 b([)p FN(w)1058 2443 y FK(1)1096 2431 y FN(;)14 b(w)1192 2443 y FK(2)1229 2431 y FO(])24 b(=)e(0)p FP(i)p FO(\))28 b(is)e(also)g FP(\003)p FO(-wild.)-118 2571 y FQ(Theorem)k(56.)41 b FB(The)31 b FP(\003)p FB(-algebr)l(a)f FA(P)1028 2583 y FK(3)p FL(;)p FK(2comm)1319 2571 y FB(is)g FP(\003)p FB(-wild.)6 2711 y FO(Moreo)n(v)n(er,)39 b(ev)n(en)g(a)f (stronger)f(statemen)n(t)h(of)h(the)g(follo)n(wing)c(theorem)-118 2811 y(holds)26 b([144)o(].)-118 2951 y FQ(Theorem)k(57.)41 b FB(L)l(et)104 3101 y FA(P)159 3113 y FK(3)p FL(;)p FM(?)p FK(2)324 3101 y FO(=)22 b FI(C)15 b FP(h)p FN(p;)f(p)618 3113 y FK(1)661 3101 y FN(;)g(p)740 3113 y FK(2)800 3101 y FP(j)23 b FN(p)888 3067 y FM(\003)949 3101 y FO(=)g FN(p;)k(p)1171 3067 y FK(2)1231 3101 y FO(=)c FN(p;)862 3236 y(p)904 3202 y FM(\003)904 3256 y FL(i)965 3236 y FO(=)f FN(p)1094 3248 y FL(i)1122 3236 y FN(;)28 b(p)1215 3202 y FK(2)1215 3256 y FL(i)1275 3236 y FO(=)22 b FN(p)1404 3248 y FL(i)1432 3236 y FN(;)27 b(p)1524 3248 y FK(1)1562 3236 y FN(p)1604 3248 y FK(2)1664 3236 y FO(=)22 b FN(p)1793 3248 y FK(2)1830 3236 y FN(p)1872 3248 y FK(1)1932 3236 y FO(=)h(0)p FP(i)p FN(:)-118 3386 y FB(Then)30 b FA(P)153 3398 y FK(3)p FL(;)p FM(?)p FK(2)318 3386 y FP(\037)23 b FN(C)471 3356 y FM(\003)509 3386 y FO(\()p FA(F)597 3398 y FK(2)633 3386 y FO(\))p FB(,)30 b(i.e.)40 b FA(P)928 3398 y FK(3)p FL(;)p FM(?)p FK(2)1099 3386 y FB(is)30 b FP(\003)p FB(-wild.)-118 3526 y(Pr)l(o)l(of.)43 b FO(Let)-39 3799 y FN(E)22 3811 y FL(k)86 3799 y FO(=)174 3607 y Fy(2)174 3753 y(6)174 3806 y(4)231 3671 y FN(e)264 b FO(0)358 3768 y(.)391 3793 y(.)423 3818 y(.)229 3925 y(0)g FN(e)229 3990 y Fy(|)p 266 3990 99 10 v 99 w({z)p 439 3990 V 99 w(})295 4069 y FL(k)24 b FK(times)575 3607 y Fy(3)575 3753 y(7)575 3806 y(5)644 3799 y FN(;)180 b(e)27 b FO(is)g(the)h(iden)n(tit)n(y)e(in)h(the)h(algebra)c FN(C)2042 3765 y FM(\003)2081 3799 y FO(\()p FA(F)2169 3811 y FK(2)2204 3799 y FO(\),)p eop %%Page: 217 221 217 220 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(217)281 184 y FN(J)327 196 y FK(1)387 184 y FO(=)475 17 y Fy(2)475 166 y(4)563 83 y FN(E)624 95 y FK(4)530 183 y FO(0)572 195 y FK(3)p FM(\002)p FK(4)530 282 y FO(0)572 294 y FK(5)p FM(\002)p FK(4)694 17 y Fy(3)694 166 y(5)763 184 y FN(;)97 b(J)929 196 y FK(2)989 184 y FO(=)1077 17 y Fy(2)1077 166 y(4)1132 83 y FO(0)1174 95 y FK(4)p FM(\002)p FK(3)1165 183 y FN(E)1226 195 y FK(3)1132 282 y FO(0)1174 294 y FK(5)p FM(\002)p FK(3)1296 17 y Fy(3)1296 166 y(5)1365 184 y FN(;)g(J)1531 196 y FK(3)1591 184 y FO(=)1679 17 y Fy(2)1679 166 y(4)1734 83 y FN(A)1796 95 y FK(1)1734 183 y FN(A)1796 195 y FK(2)1734 282 y FN(A)1796 294 y FK(3)1834 17 y Fy(3)1834 166 y(5)1903 184 y FN(;)-118 470 y FO(where)-45 790 y FN(A)17 802 y FK(1)78 790 y FO(=)192 734 y(1)p 175 771 76 4 v 175 847 a FN(N)275 573 y Fy(2)275 719 y(6)275 769 y(6)275 822 y(4)332 639 y FN(e)84 b FO(0)101 b(0)122 b(0)102 b(0)330 739 y(0)84 b FN(e)103 b FO(0)122 b(0)102 b(0)330 839 y(0)83 b(0)f(2)p FN(e)102 b FO(0)g(0)330 938 y(0)83 b(0)101 b(0)h(3)p FN(e)83 b FO(0)947 573 y Fy(3)947 719 y(7)947 769 y(7)947 822 y(5)1016 790 y FN(;)97 b(A)1198 802 y FK(2)1259 790 y FO(=)1373 734 y(1)p 1356 771 V 1356 847 a FN(N)1456 623 y Fy(2)1456 772 y(4)1513 689 y FN(e)103 b FO(0)f FN(e)k(e)128 b(e)1511 789 y FO(0)83 b(2)p FN(e)f(e)h(u)1969 801 y FK(1)2110 789 y FO(0)1511 889 y(0)102 b(0)g FN(e)i FO(0)g FN(u)2136 901 y FK(2)2173 623 y Fy(3)2173 772 y(5)2242 790 y FN(;)-118 1130 y(u)-70 1142 y FK(1)-33 1130 y FO(,)27 b FN(u)65 1142 y FK(2)130 1130 y FO(are)f(the)i (generators)e(of)h(the)h(algebra)d FN(C)1407 1100 y FM(\003)1445 1130 y FO(\()p FA(F)1533 1142 y FK(2)1569 1130 y FO(\),)601 1322 y FN(A)663 1334 y FK(3)724 1322 y FO(=)812 1250 y Fy(p)p 895 1250 702 4 v 72 x FN(E)956 1334 y FK(5)1012 1322 y FP(\000)18 b FN(A)1157 1293 y FM(\003)1157 1344 y FK(1)1195 1322 y FN(A)1257 1334 y FK(1)1313 1322 y FP(\000)g FN(A)1458 1293 y FM(\003)1458 1344 y FK(2)1496 1322 y FN(A)1558 1334 y FK(2)1596 1322 y FN(;)-118 1513 y(N)48 b FO(is)38 b(c)n(hosen)g(so)h(that)g FP(k)p FN(A)778 1483 y FM(\003)778 1533 y FK(1)816 1513 y FN(A)878 1525 y FK(1)942 1513 y FO(+)26 b FN(A)1095 1483 y FM(\003)1095 1533 y FK(2)1133 1513 y FN(A)1195 1525 y FK(2)1233 1513 y FP(k)42 b FN(<)g FO(1)d(in)f FN(M)1694 1525 y FK(5)1731 1513 y FO(\()p FN(C)1828 1483 y FM(\003)1867 1513 y FO(\()p FA(F)1955 1525 y FK(2)1990 1513 y FO(\)\).)72 b(Then)-118 1612 y FN(J)-64 1582 y FM(\003)-72 1633 y FK(1)-26 1612 y FN(J)20 1624 y FK(1)80 1612 y FO(=)23 b FN(E)229 1624 y FK(4)267 1612 y FO(,)h FN(J)368 1582 y FM(\003)360 1633 y FK(2)406 1612 y FN(J)452 1624 y FK(2)512 1612 y FO(=)f FN(E)661 1624 y FK(3)698 1612 y FO(,)i(and)d FN(J)956 1582 y FM(\003)948 1633 y FK(3)995 1612 y FN(J)1041 1624 y FK(3)1101 1612 y FO(=)h FN(E)1250 1624 y FK(4)1287 1612 y FO(.)36 b(This)22 b(implies)d(that)k(\()p FN(J)2061 1624 y FL(i)2089 1612 y FN(J)2143 1582 y FM(\003)2135 1634 y FL(i)2181 1612 y FO(\))2213 1582 y FK(2)2274 1612 y FO(=)-118 1712 y FN(J)-72 1724 y FL(i)-44 1712 y FN(J)10 1682 y FM(\003)2 1734 y FL(i)48 1712 y FO(,)49 b FN(i)i FO(=)f(1,)f(2,)f(3.)88 b(Moreo)n(v)n(er,)46 b(since)d FN(J)1378 1682 y FM(\003)1370 1733 y FK(1)1416 1712 y FN(J)1462 1724 y FK(2)1551 1712 y FO(=)51 b(0)44 b(and)g FN(J)1977 1724 y FK(2)2015 1712 y FN(J)2069 1682 y FM(\003)2061 1733 y FK(1)2158 1712 y FO(=)51 b(0,)-118 1812 y(w)n(e)35 b(see)f(that)i(\()p FN(J)419 1824 y FK(1)456 1812 y FN(J)510 1782 y FM(\003)502 1832 y FK(1)549 1812 y FO(\)\()p FN(J)659 1824 y FK(2)696 1812 y FN(J)750 1782 y FM(\003)742 1832 y FK(2)789 1812 y FO(\))f(=)h(\()p FN(J)1035 1824 y FK(2)1072 1812 y FN(J)1126 1782 y FM(\003)1118 1832 y FK(2)1164 1812 y FO(\)\()p FN(J)1274 1824 y FK(1)1312 1812 y FN(J)1366 1782 y FM(\003)1358 1832 y FK(1)1404 1812 y FO(\))g(=)g(0)p FN(:)e FO(Set)i FN( )s FO(\()p FN(p)1954 1824 y FL(i)1982 1812 y FO(\))f(=)h FN(J)2196 1824 y FL(i)2223 1812 y FN(J)2277 1782 y FM(\003)2269 1833 y FL(i)2316 1812 y FO(,)-118 1911 y FN(i)28 b FO(=)f(1,)k(2,)g FN( )s FO(\()p FN(p)p FO(\))e(=)e FN(J)553 1923 y FK(3)590 1911 y FN(J)644 1881 y FM(\003)636 1932 y FK(3)683 1911 y FO(;)32 b FN( )i FO(de\014nes)c(a)g(homomorphism)c(of)k(the)h FP(\003)p FO(-algebra)-118 2011 y FA(P)-63 2023 y FK(3)p FL(;)p FM(?)p FK(2)115 2011 y FO(in)n(to)j FN(M)372 2023 y FK(12)442 2011 y FO(\()p FN(C)539 1981 y FM(\003)578 2011 y FO(\()p FA(F)666 2023 y FK(2)701 2011 y FO(\)\).)63 b(One)36 b(can)f(directly)f(c)n(hec)n(k)h(that)h(the)h(functor)-118 2111 y FN(F)-65 2123 y FL( )-5 2111 y FO(:)k(Rep\()p FA(P)290 2123 y FK(3)p FL(;)p FM(?)p FK(2)432 2111 y FO(\))24 b FP(\000)-49 b(!)23 b FO(Rep)q(\()p FN(C)852 2080 y FM(\003)890 2111 y FO(\()p FA(F)978 2123 y FK(2)1014 2111 y FO(\)\))28 b(is)f(full.)p 2278 2111 4 57 v 2282 2058 50 4 v 2282 2111 V 2331 2111 4 57 v -118 2311 a FQ(Corollary)32 b(10.)40 b FB(The)f(pr)l(oblem)g(of)g(a)g(unitary)f (classi\014c)l(ation)h(of)g(\\al)t(l)g(but)-118 2410 y(one")d(ortho)l(gonal)j(pr)l(oje)l(ctions)f FN(p)p FB(,)g FN(p)1075 2422 y FK(1)1112 2410 y FB(,)f FN(:)14 b(:)g(:)28 b FB(,)38 b FN(p)1404 2422 y FL(n)1449 2410 y FB(,)h FN(p)1555 2422 y FL(i)1582 2410 y FN(p)1624 2422 y FL(j)1695 2410 y FO(=)c(0)h FB(for)h FN(i)e FP(6)p FO(=)g FN(j)5 b FB(,)39 b(is)-118 2510 y FP(\003)p FB(-wild)30 b(if)48 b FN(n)23 b FP(\025)g FO(2:)32 2701 y FA(P)87 2713 y FL(n)p FK(+1)p FL(;)p FM(?)p FL(n)352 2701 y FO(=)g FI(C)493 2634 y Fy(\012)539 2701 y FN(p;)14 b(p)660 2713 y FK(1)696 2701 y FN(;)g(:)g(:)g(:)g(;)g(p)923 2713 y FL(n)991 2701 y FP(j)23 b FN(p;)14 b(p)1158 2713 y FL(i)1215 2701 y FB(ar)l(e)30 b(ortho)l(gonal)h(pr)l(oje)l(ctions)q FN(;)1053 2835 y(p)1095 2847 y FL(i)1122 2835 y FN(p)1164 2847 y FL(j)1222 2835 y FO(=)23 b(0)p FN(;)14 b(i)22 b FP(6)p FO(=)g FN(j)1566 2768 y Fy(\013)1628 2835 y FP(\037)h FN(C)1781 2801 y FM(\003)1820 2835 y FO(\()p FA(F)1908 2847 y FK(2)1943 2835 y FO(\))p FN(:)-118 3030 y FQ(5.)35 b FO(The)25 b(problem)e(of)i(unitary)f(classi\014cation)d(of)k (quadruples)e(of)i(orthogonal)-118 3130 y(pro)5 b(jections)25 b FN(p)351 3142 y FK(1)388 3130 y FO(,)j FN(p)481 3142 y FK(2)518 3130 y FO(,)g FN(p)611 3142 y FK(3)648 3130 y FO(,)f FN(p)740 3142 y FK(4)805 3130 y FO(suc)n(h)g(that)372 3321 y FN(\013)p FO(\()p FN(p)499 3333 y FK(1)555 3321 y FO(+)18 b FN(p)680 3333 y FK(2)736 3321 y FO(+)g FN(p)861 3333 y FK(3)916 3321 y FO(+)g FN(p)1041 3333 y FK(4)1078 3321 y FO(\))24 b(=)e FN(I)7 b(;)180 b FO(0)23 b FN(<)g(\013)g(<)g FO(1)p FN(;)-118 3513 y FO(for)30 b(a)g(\014xed)h FN(\013)d FP(6)p FO(=)472 3480 y FK(1)p 472 3494 34 4 v 472 3541 a(2)515 3513 y FO(,)k(has)e(only)f(a)h(\014nite)g(n)n(um)n(b)r(er)g(of) g(irreducible)d(represen-)-118 3612 y(tations,)37 b(the)g(dimension)d (of)j(whic)n(h)e(dep)r(ends)i(on)f(the)h(parameter)e FN(\013)i FO(\(see)-118 3712 y(Section)g(2.2.1\).)67 b(If)39 b FN(\013)i FO(=)780 3679 y FK(1)p 780 3693 V 780 3740 a(2)823 3712 y FO(,)g(then)d(there)g(is)f(an)h(uncoun)n(table) f(family)e(of)-118 3811 y(irreducible)30 b(represen)n(tations,)j(and)i (their)e(dimension)e(is)i(one)h(or)f(t)n(w)n(o)h(\(see)-118 3911 y(Section)26 b(2.2.1\).)p eop %%Page: 218 222 218 221 bop -118 -137 a FO(218)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)6 96 y FO(It)j(follo)n(ws)26 b(directly)i(from)g(Theorem)g(57)h(that)h FB(the)i(pr)l(oblem)h(of)g(unitary)-118 196 y(classi\014c)l(ation)25 b(of)f(\014ve)g(ortho)l(gonal)h(pr)l(oje)l(ctions)g FN(r)1454 208 y FK(1)1492 196 y FB(,)g FN(r)1579 208 y FK(2)1617 196 y FB(,)g FN(r)1704 208 y FK(3)1742 196 y FB(,)g FN(r)1829 208 y FK(4)1867 196 y FB(,)g FN(r)1954 208 y FK(5)2016 196 y FB(such)f(that)-118 296 y FN(r)-81 308 y FK(1)-25 296 y FO(+)18 b FN(r)95 308 y FK(2)151 296 y FO(+)g FN(r)271 308 y FK(3)327 296 y FO(+)g FN(r)447 308 y FK(4)504 296 y FO(+)g FN(r)624 308 y FK(5)684 296 y FO(=)23 b(2)p FN(e)29 b FB(is)h FP(\003)p FB(-wild)p FO(:)37 b FA(R)1306 308 y FK(5)p FL(;)p FK(2)1420 296 y FP(\037)22 b FA(P)1562 308 y FK(3)p FL(;)p FM(?)p FK(2)1704 296 y FO(,)28 b(where)30 561 y FA(R)90 573 y FK(5)p FL(;)p FK(2)203 561 y FO(=)23 b FI(C)345 469 y Fy(D)402 561 y FN(r)439 573 y FK(1)476 561 y FN(;)14 b(:)g(:)g(:)g(;)g(r)698 573 y FK(5)759 561 y FP(j)23 b FN(r)844 527 y FK(2)842 582 y FL(i)905 561 y FO(=)f FN(r)1029 573 y FL(i)1081 561 y FO(=)g FN(r)1207 527 y FM(\003)1205 582 y FL(i)1246 561 y FN(;)14 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)1770 458 y FK(5)1727 482 y Fy(X)1733 659 y FL(i)p FK(=1)1861 561 y FN(r)1898 573 y FL(i)1949 561 y FO(=)22 b(2)p FN(e)2117 469 y Fy(E)2167 561 y FN(:)6 819 y FO(Indeed,)28 b(let)f FN( )12 b FO(:)28 b FA(R)596 831 y FK(5)p FL(;)p FK(2)709 819 y FP(\000)-48 b(!)23 b FA(P)887 831 y FK(3)p FL(;)p FM(?)p FK(2)1057 819 y FO(b)r(e)28 b(de\014ned)f(b)n(y)h(the)g(form)n (ulas)613 1002 y FN( )s FO(\()p FN(r)739 1014 y FK(1)777 1002 y FO(\))23 b(=)g FN(p;)97 b( )s FO(\()p FN(r)1208 1014 y FK(2)1246 1002 y FO(\))23 b(=)g FN(e)18 b FP(\000)g FN(p;)239 1127 y( )s FO(\()p FN(r)365 1139 y FK(3)403 1127 y FO(\))24 b(=)e FN(p)588 1139 y FK(1)625 1127 y FN(;)97 b( )s FO(\()p FN(r)871 1139 y FK(4)909 1127 y FO(\))24 b(=)e FN(p)1094 1139 y FK(2)1131 1127 y FN(;)97 b( )s FO(\()p FN(r)1377 1139 y FK(5)1415 1127 y FO(\))24 b(=)e FN(e)c FP(\000)g FN(p)1740 1139 y FK(1)1796 1127 y FP(\000)g FN(p)1921 1139 y FK(2)1958 1127 y FN(:)-118 1310 y FO(One)30 b(directly)d(c)n(hec)n(ks)i(that)i(the)f(functor)g FN(F)1295 1322 y FL( )1355 1310 y FO(:)42 b(Rep\()p FA(P)1651 1322 y FK(3)p FL(;)p FM(?)p FK(2)1794 1310 y FO(\))27 b FP(\000)-48 b(!)27 b FO(Rep\()p FA(R)2216 1322 y FK(5)p FL(;)p FK(2)2306 1310 y FO(\))-118 1410 y(is)f(full.)6 1510 y(It)k(follo)n(ws)d(directly)g(from)h(Theorem)g(55)h(that)g FB(the)j(fol)t(lowing)i FP(\003)p FB(-algebr)l(a)-118 1610 y(is)c FP(\003)p FB(-wild)p FO(,)1 1868 y FA(R)61 1888 y FK(5)p FL(;)124 1866 y Fx(5)p 124 1875 29 3 v 124 1908 a(2)189 1868 y FO(=)23 b FI(C)331 1776 y Fy(D)388 1868 y FN(r)425 1880 y FK(1)462 1868 y FN(;)14 b(:)g(:)g(:)g(;)g(r)684 1880 y FK(5)745 1868 y FP(j)23 b FN(r)830 1834 y FK(2)828 1888 y FL(i)891 1868 y FO(=)g FN(r)1016 1880 y FL(i)1067 1868 y FO(=)f FN(r)1193 1834 y FM(\003)1191 1888 y FL(i)1232 1868 y FN(;)28 b(i)23 b FO(=)f(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)1770 1764 y FK(5)1727 1789 y Fy(X)1733 1966 y FL(i)p FK(=1)1860 1868 y FN(r)1897 1880 y FL(i)1949 1868 y FO(=)2046 1812 y(5)p 2046 1849 42 4 v 2046 1925 a(2)2112 1868 y FN(e)2151 1776 y Fy(E)189 2103 y FO(=)23 b FI(C)331 2011 y Fy(D)388 2103 y FN(w)447 2115 y FL(i)498 2103 y FO(=)g(2)p FN(r)665 2115 y FL(i)711 2103 y FP(\000)18 b FN(e;)c(i)22 b FO(=)g(1)p FN(;)14 b(:)g(:)g(:)g(;)g FO(5)22 b FP(j)h FN(w)1404 2115 y FL(i)1455 2103 y FO(=)g FN(w)1604 2069 y FM(\003)1602 2123 y FL(i)1643 2103 y FN(;)14 b(w)1741 2069 y FK(2)1739 2123 y FL(i)1802 2103 y FO(=)22 b FN(e;)1329 2340 y(i)g FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)1816 2236 y FK(5)1772 2261 y Fy(X)1779 2437 y FL(i)p FK(=1)1906 2340 y FN(w)1965 2352 y FL(i)2016 2340 y FO(=)23 b(0)2146 2247 y Fy(E)2196 2340 y FN(:)6 2609 y FO(Indeed,)30 b(its)e(quotien)n (t)g FP(\003)p FO(-algebra)d FA(R)1168 2630 y FK(5)p FL(;)1231 2607 y Fx(3)p 1231 2616 29 3 v 1231 2650 a(2)1269 2630 y FK(+1)1382 2609 y FO(=)g FI(C)1526 2542 y Fy(\012)1571 2609 y FN(r)1608 2621 y FK(1)1646 2609 y FN(;)14 b(:)g(:)g(:)g(;)g(r) 1868 2621 y FK(5)1931 2609 y FP(j)25 b FN(r)2018 2579 y FK(2)2016 2631 y FL(i)2081 2609 y FO(=)g FN(r)2210 2579 y FM(\003)2208 2631 y FL(i)2274 2609 y FO(=)-118 2737 y FN(r)-81 2749 y FL(i)-53 2737 y FN(;)j(i)k FO(=)g(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)461 2675 y Fy(P)548 2695 y FK(3)548 2762 y FL(i)p FK(=1)673 2737 y FN(r)710 2749 y FL(i)771 2737 y FO(=)878 2704 y FK(3)p 878 2718 34 4 v 878 2766 a(2)921 2737 y FN(e;)28 b(r)1048 2749 y FK(4)1107 2737 y FO(+)22 b FN(r)1231 2749 y FK(5)1301 2737 y FO(=)33 b FN(e)1438 2670 y Fy(\013)1509 2737 y FO(=)f FI(C)1660 2670 y Fy(\012)1705 2737 y FN(w)1764 2749 y FL(i)1825 2737 y FO(=)g(2)p FN(r)2001 2749 y FL(i)2051 2737 y FP(\000)21 b FN(e;)14 b(i)32 b FO(=)-118 2854 y(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5)22 b FP(j)i FN(w)278 2866 y FL(i)329 2854 y FO(=)e FN(w)477 2824 y FM(\003)475 2875 y FL(i)516 2854 y FN(;)14 b(w)614 2824 y FK(2)612 2875 y FL(i)675 2854 y FO(=)23 b FN(e;)14 b(i)22 b FO(=)h(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(5;)1283 2791 y Fy(P)1370 2812 y FK(3)1370 2879 y FL(i)p FK(=1)1495 2854 y FN(w)1554 2866 y FK(1)1615 2854 y FO(=)23 b(0)p FN(;)14 b(w)1841 2866 y FK(4)1888 2854 y FO(+)c FN(w)2022 2866 y FK(5)2083 2854 y FO(=)22 b(0)2212 2786 y Fy(\013)2274 2854 y FO(=)-118 2962 y FI(C)-64 2894 y Fy(\012)-19 2962 y FN(z)20 2974 y FK(1)84 2962 y FO(=)27 b FN(w)235 2974 y FK(1)293 2962 y FO(+)20 b FN(w)437 2974 y FK(2)474 2962 y FN(;)14 b(z)550 2974 y FK(2)614 2962 y FO(=)744 2929 y FK(1)p 716 2943 88 4 v 716 2951 a FM(p)p 771 2951 34 3 v 49 x FK(3)814 2962 y FO(\()p FN(w)905 2974 y FK(1)963 2962 y FP(\000)20 b FN(w)1107 2974 y FK(2)1144 2962 y FO(\))p FN(;)14 b(z)1252 2974 y FK(3)1317 2962 y FO(=)27 b FN(w)1468 2974 y FK(4)1533 2962 y FP(j)g FN(z)1626 2931 y FM(\003)1622 2985 y FL(k)1691 2962 y FO(=)g FN(z)1822 2974 y FL(k)1862 2962 y FN(;)h(z)1956 2931 y FK(2)1952 2985 y FL(k)2020 2962 y FO(=)e FN(e;)i(k)i FO(=)-118 3086 y(1)p FN(;)14 b(:)g(:)g(:)f(;)h FO(3;)g FP(f)p FN(z)268 3098 y FK(1)304 3086 y FN(;)g(z)380 3098 y FK(2)416 3086 y FP(g)23 b FO(=)g(0)611 3018 y Fy(\013)672 3086 y FO(=)g FA(P)815 3098 y FK(3)p FL(;)p FK(2an)n(ti)1045 3086 y FO(is)k FP(\003)p FO(-wild.)-118 3237 y FQ(6.)33 b FO(F)-7 b(or)18 b(a)g(single)e(idemp)r(oten)n(t,)j(the)g(situation)d (is)i(similar)c(to)k(the)h(situation)d(for)-118 3336 y(t)n(w)n(o)25 b(orthogonal)e(pro)5 b(jections.)34 b(Consider)24 b(the)j FP(\003)p FO(-algebra)22 b FA(Q)1810 3348 y FK(1)1874 3336 y FO(generated)j(b)n(y)-118 3436 y(an)20 b(idemp)r(oten)n(t)g(and) h(its)f(adjoin)n(t,)h FN(q)1022 3448 y FK(1)1059 3436 y FO(,)h FN(q)1144 3406 y FM(\003)1141 3456 y FK(1)1182 3436 y FO(.)35 b(Let)21 b FN(q)1419 3448 y FK(1)1480 3436 y FO(=)h FN(a)1611 3448 y FK(1)1653 3436 y FO(+)5 b FN(ib)1788 3448 y FK(1)1824 3436 y FO(,)23 b FN(q)1910 3406 y FM(\003)1907 3456 y FK(1)1971 3436 y FO(=)f FN(a)2102 3448 y FK(1)2145 3436 y FP(\000)5 b FN(ib)2280 3448 y FK(1)2316 3436 y FO(,)-118 3535 y(where)24 b FN(a)163 3505 y FM(\003)163 3556 y FK(1)224 3535 y FO(=)f FN(a)356 3547 y FK(1)393 3535 y FO(,)j FN(b)478 3505 y FM(\003)478 3556 y FK(1)539 3535 y FO(=)c FN(b)662 3547 y FK(1)699 3535 y FO(.)36 b(The)25 b FP(\003)p FO(-algebra)c FA(Q)1338 3547 y FK(1)1401 3535 y FO(coincides)h(with)i(the)h(algebra)82 3719 y FI(C)136 3652 y Fy(\012)181 3719 y FN(a;)14 b(b)23 b FP(j)g FN(a)g FO(=)g FN(a)566 3685 y FM(\003)604 3719 y FN(;)14 b(b)22 b FO(=)h FN(b)823 3685 y FM(\003)861 3719 y FO(;)14 b FP(f)p FN(a;)g(b)p FP(g)21 b FO(=)i FN(ab)18 b FO(+)g FN(ba)k FO(=)h(0)p FN(;)k(a)1715 3685 y FK(2)1771 3719 y FP(\000)18 b FN(b)1890 3685 y FK(2)1950 3719 y FO(=)k FN(e)2076 3652 y Fy(\013)2115 3719 y FN(;)-118 3911 y FO(where)27 b FN(a)c FO(=)g(2)319 3844 y Fy(\000)356 3911 y FN(a)400 3923 y FK(1)456 3911 y FP(\000)549 3878 y FK(1)p 549 3892 34 4 v 549 3940 a(2)592 3911 y FN(e)631 3844 y Fy(\001)668 3911 y FO(,)28 b FN(b)23 b FO(=)f(2)p FN(b)943 3923 y FK(1)980 3911 y FO(.)p eop %%Page: 219 223 219 222 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(219)6 96 y(Irreducible)26 b(represen)n(tations)g(of)i(the)h(algebra)d FA(Q)1588 108 y FK(1)1654 96 y FO(\(see)i(Section)g(1.2.2\),)-118 196 y(up)g(to)f(a)g(unitary)g(equiv)-5 b(alence,)25 b(coincide)g(with)i (one)g(of)h(the)g(follo)n(wing:)6 296 y(1\))i(t)n(w)n(o)f (one-dimensional)24 b(represen)n(tations)i(giv)n(en)i(b)n(y)h FN(\031)1840 308 y FK(0)1878 296 y FO(\()p FN(q)1947 308 y FK(1)1984 296 y FO(\))e(=)f(0)j(and)-118 395 y FN(\031)-71 407 y FK(1)-34 395 y FO(\()p FN(q)35 407 y FK(1)73 395 y FO(\))23 b(=)g(1;)6 495 y(2\))i(a)f(family)-7 b(,)22 b(dep)r(ending)i(on)h(a)f(parameter)e FN(\013)h(>)g FO(0,)i(of)f(t)n(w)n(o-dimensional)-118 595 y(represen)n(tations:)770 810 y FN(\031)817 822 y FL(\013)864 810 y FO(\()p FN(q)933 822 y FK(1)971 810 y FO(\))f(=)1114 692 y Fy(\022)1175 759 y FO(1)82 b FN(\013)1175 859 y FO(0)88 b(0)1353 692 y Fy(\023)1428 810 y FN(:)6 1025 y FO(By)39 b(decomp)r(osing)d(a)j (represen)n(tation)e(of)i(the)g(algebra)e FA(Q)1905 1037 y FK(1)1981 1025 y FO(in)n(to)h(a)h(di-)-118 1125 y(rect)26 b(sum)f(of)h(irreducible)c(represen)n(tations)h(on)j(a)f(\014nite)h (dimensional)21 b(space)-118 1224 y FN(H)7 b FO(,)34 b(w)n(e)f(obtain)f(the)h(structure)f(theorem)g(\(see)h([69)o(,)g(113)o (]\))g(for)g(the)g(unitary)-118 1324 y(description)25 b(of)i(idemp)r(oten)n(ts)f(in)h(the)h(\014nite)f(dimensional)c(case.)6 1424 y(There)43 b(is)g(a)g(structure)g(theorem)f(that)i(giv)n(es)d(a)i (description)e(of)i(an)n(y)-118 1523 y(b)r(ounded)31 b(idemp)r(oten)n(t)e(on)i(an)n(y)f(separable)e(Hilb)r(ert)h(space)h(in) g(the)h(form)f(of)-118 1623 y(an)d(in)n(tegral)e(of)i(irreducible)d (represen)n(tations.)-118 1767 y FQ(7.)55 b FO(F)-7 b(urther,)36 b(consider)c(the)i(problem)e(of)i(a)g(unitary)e(description)g(of)i (pairs)-118 1867 y(of)j(idemp)r(oten)n(ts)e FN(Q)532 1879 y FK(1)569 1867 y FO(,)k FN(Q)697 1879 y FK(2)771 1867 y FO(\()p FN(Q)869 1837 y FK(2)869 1887 y(1)945 1867 y FO(=)f FN(Q)1114 1879 y FK(1)1151 1867 y FO(,)h FN(Q)1279 1837 y FK(2)1279 1887 y(2)1355 1867 y FO(=)f FN(Q)1524 1879 y FK(2)1561 1867 y FO(\).)65 b(The)37 b(fact)g(that)g(the)-118 1966 y(problem)20 b(of)i(a)h(unitary)e (description)f(of)i(pairs)f(of)h(idemp)r(oten)n(ts)f(is)g(di\016cult)h (is)-118 2066 y(just)g(a)g(mathematical)17 b(folklore.)32 b(W)-7 b(e)22 b(will)e(pro)n(v)n(e)g(a)h(corresp)r(onding)e(theorem) -118 2166 y(and)26 b(sho)n(w)g(that,)h(ev)n(en)f(if)h(an)f(additional)d (restriction)h(of)i(self-adjoin)n(tness)e(is)-118 2265 y(imp)r(osed)34 b(on)h(one)h(of)g(the)g(idemp)r(oten)n(ts)e(\(one)i(of) g(the)g(idemp)r(oten)n(ts)e(is)h(an)-118 2365 y(orthogonal)24 b(pro)5 b(jection\),)26 b(the)i(problem)d(do)r(es)j(not)f(b)r(ecome)g (easer.)-118 2521 y FQ(Theorem)j(58.)41 b FB(L)l(et)33 b FA(Q)653 2533 y FK(2)719 2521 y FO(=)c FI(C)15 b FP(h)q FN(q)937 2533 y FK(1)980 2521 y FN(;)f(q)1054 2533 y FK(2)1091 2521 y FN(;)g(q)1168 2491 y FM(\003)1165 2541 y FK(1)1206 2521 y FN(;)g(q)1283 2491 y FM(\003)1280 2541 y FK(2)1351 2521 y FP(j)29 b FN(q)1443 2491 y FK(2)1440 2541 y(1)1510 2521 y FO(=)g FN(q)1641 2533 y FK(1)1679 2521 y FN(;)14 b(q)1756 2491 y FK(2)1753 2541 y(2)1822 2521 y FO(=)29 b FN(q)1956 2491 y FK(2)1994 2521 y FP(i)p FB(,)35 b FA(D)2155 2533 y FK(1)p FL(;)p FK(1)2274 2521 y FO(=)-118 2620 y FI(C)15 b FP(h)p FN(q)s(;)f(q)85 2590 y FM(\003)129 2620 y FN(;)g(p)27 b FP(j)g FN(q)325 2590 y FK(2)390 2620 y FO(=)f FN(q)s(;)14 b(p)600 2590 y FK(2)664 2620 y FO(=)27 b FN(p)g FO(=)f FN(p)958 2590 y FM(\003)996 2620 y FP(i)p FB(,)34 b Fz(S)1156 2632 y FK(2)1220 2620 y FO(=)26 b FI(C)15 b FP(h)q FN(a)1442 2632 y FK(1)1485 2620 y FN(;)f(a)1566 2632 y FK(2)1630 2620 y FP(j)27 b FN(a)1724 2632 y FK(1)1788 2620 y FO(=)g FN(a)1924 2590 y FM(\003)1924 2641 y FK(1)1962 2620 y FN(;)14 b(a)2043 2632 y FK(2)2107 2620 y FO(=)27 b FN(a)2243 2590 y FM(\003)2243 2641 y FK(2)2281 2620 y FP(i)p FB(.)-118 2720 y(Then)e FA(Q)148 2732 y FK(2)209 2720 y FP(\037)d FA(D)365 2732 y FK(1)p FL(;)p FK(1)478 2720 y FP(\037)h Fz(S)635 2732 y FK(2)672 2720 y FB(,)j(so)g(that)f(the)g FP(\003)p FB(-algebr)l(as)g FA(Q)1559 2732 y FK(2)1621 2720 y FB(and)h FA(D)1847 2732 y FK(1)p FL(;)p FK(1)1961 2720 y FB(ar)l(e)g FP(\003)p FB(-wild.)-118 2876 y(Pr)l(o)l(of.)43 b FO(Because)23 b FA(D)525 2888 y FK(1)p FL(;)p FK(1)640 2876 y FO(is)g(a)h(quotien)n (t)f(algebra)e(of)j(the)h(algebra)d FA(Q)1971 2888 y FK(2)2008 2876 y FO(,)j(w)n(e)f(ha)n(v)n(e)-118 2975 y(that)29 b FA(Q)118 2987 y FK(2)179 2975 y FP(\037)24 b FA(D)337 2987 y FK(1)p FL(;)p FK(1)456 2975 y FO(\(w)n(e)k(c)n(ho)r (ose)f(an)h(en)n(v)n(eloping)d(algebra)h(for)i FA(D)1889 2987 y FK(1)p FL(;)p FK(1)2007 2975 y FO(to)g(b)r(e)h(the)-118 3075 y(algebra)22 b FA(D)239 3087 y FK(1)p FL(;)p FK(1)353 3075 y FO(itself,)i FN(n)f FO(=)g(1,)h FN( )12 b FO(:)28 b FA(Q)997 3087 y FK(2)1058 3075 y FP(\000)-49 b(!)23 b FA(D)1249 3087 y FK(1)p FL(;)p FK(1)1364 3075 y FO(is)g(the)i (natural)e(epimorphism)-118 3175 y(of)k(the)h(algebra)d(on)n(to)i(the)h (quotien)n(t)e(algebra\).)6 3274 y(Let)42 b(us)f(sho)n(w)f(that)h FA(D)770 3286 y FK(1)p FL(;)p FK(1)905 3274 y FP(\037)46 b Fz(S)1085 3286 y FK(2)1122 3274 y FO(.)77 b(Construct)40 b(the)i(homomorphism)-118 3374 y FN( )12 b FO(:)28 b FA(D)68 3386 y FK(1)p FL(;)p FK(1)181 3374 y FP(\000)-48 b(!)23 b FN(M)385 3386 y FK(2)421 3374 y FO(\()p Fz(S)522 3386 y FK(2)560 3374 y FO(\):)303 3596 y FN( )s FO(\()p FN(q)s FO(\))h(=)576 3479 y Fy(\022)638 3546 y FN(e)84 b(a)805 3558 y FK(1)861 3546 y FO(+)18 b FN(ia)1017 3558 y FK(2)637 3645 y FO(0)208 b(0)1054 3479 y Fy(\023)1129 3596 y FN(;)96 b( )s FO(\()p FN(p)p FO(\))24 b(=)1532 3540 y(1)p 1532 3577 42 4 v 1532 3653 a(2)1598 3479 y Fy(\022)1659 3546 y FN(e)83 b(e)1659 3645 y(e)g(e)1819 3479 y Fy(\023)1894 3596 y FN(:)-118 3811 y FO(It)34 b(is)e(easy)h(to)h(c)n(hec)n(k)e(that)i(the)g(corresp)r(onding)d (functor)i FN(F)1815 3823 y FL( )1875 3811 y FO(:)44 b(Rep)14 b Fz(S)2169 3823 y FK(2)2239 3811 y FP(\000)-48 b(!)-118 3911 y FO(Rep)14 b FA(D)109 3923 y FK(1)p FL(;)p FK(1)227 3911 y FO(is)26 b(full.)p 2278 3911 4 57 v 2282 3858 50 4 v 2282 3911 V 2331 3911 4 57 v eop %%Page: 220 224 220 223 bop -118 -137 a FO(220)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FQ(Corollary)k(11.)40 b FB(The)c(algebr)l(a)g FA(Q)986 108 y FL(n)1031 96 y FB(,)g(for)f FN(n)d FP(\025)f FO(2)j(\()p FB(the)h(pr)l(oblem)h(of)g(unitary)-118 196 y(description)31 b(of)g FN(n)f FB(idemp)l(otents)g(if)g FN(n)23 b FP(\025)g FO(2\))p FB(,)30 b(is)g FP(\003)p FB(-wild.)-118 390 y FQ(8.)35 b FO(Finally)-7 b(,)23 b(w)n(e)i(sho)n(w)g(that)h(the)f FP(\003)p FO(-algebra)d FA(Q)1344 402 y FL(n;)p FM(?)1487 390 y FO(\(the)k(problem)d(of)i(unitary)-118 490 y(classi\014cation)20 b(of)k(a)g(family)d(of)j(pairwise)d(orthogonal)g(idemp)r(oten)n(ts)i FN(Q)2128 502 y FK(1)2165 490 y FO(,)i FN(Q)2279 502 y FK(2)2316 490 y FO(,)-118 590 y FN(:)14 b(:)g(:)27 b FO(,)h FN(Q)123 602 y FL(n)168 590 y FO(,)g FN(Q)285 602 y FL(i)312 590 y FN(Q)378 602 y FL(j)436 590 y FO(=)22 b(0)27 b(for)h FN(i)22 b FP(6)p FO(=)h FN(j)5 b FO(\))28 b(is)e FP(\003)p FO(-wild)f(for)i FN(n)c FP(\025)g FO(2.)-118 784 y FQ(Theorem)30 b(59.)41 b FB(L)l(et)57 985 y FA(Q)112 997 y FK(2)p FL(;)p FM(?)244 985 y FO(=)22 b FI(C)385 918 y Fy(\012)430 985 y FN(q)467 997 y FK(1)505 985 y FN(;)14 b(q)579 997 y FK(2)616 985 y FN(;)g(q)693 951 y FM(\003)690 1006 y FK(1)731 985 y FN(;)g(q)808 951 y FM(\003)805 1006 y FK(2)869 985 y FP(j)23 b FN(q)955 951 y FK(2)952 1006 y(1)1016 985 y FO(=)f FN(q)1140 997 y FK(1)1178 985 y FN(;)27 b(q)1268 951 y FK(2)1265 1006 y(2)1329 985 y FO(=)22 b FN(q)1453 997 y FK(2)1491 985 y FN(;)27 b(q)1578 997 y FK(1)1616 985 y FN(q)1653 997 y FK(2)1713 985 y FO(=)c FN(q)1838 997 y FK(2)1875 985 y FN(q)1912 997 y FK(1)1972 985 y FO(=)g(0)2102 918 y Fy(\013)2141 985 y FN(:)-118 1187 y FB(Then)37 b FA(Q)160 1199 y FK(2)p FL(;)p FM(?)292 1187 y FP(\037)22 b Fz(S)448 1199 y FK(2)485 1187 y FB(,)30 b(i.e.,)i FA(Q)765 1199 y FK(2)p FL(;)p FM(?)904 1187 y FB(is)e(a)g(wild)h FP(\003)p FB(-algebr)l(a.)-118 1381 y(Pr)l(o)l(of.)43 b FO(Let)36 b(us)f(de\014ne)h(a)f(homomorphism)30 b FN( )12 b FO(:)31 b FA(Q)1507 1393 y FK(2)p FL(;)p FM(?)1652 1381 y FP(\000)-49 b(!)36 b FN(M)1868 1393 y FK(3)1905 1381 y FO(\()p Fz(S)2006 1393 y FK(2)2043 1381 y FO(\))g(as)f(fol-)-118 1480 y(lo)n(ws:)130 1761 y FN( )s FO(\()p FN(q)256 1773 y FK(1)294 1761 y FO(\))23 b(=)437 1594 y Fy(2)437 1743 y(4)493 1660 y FN(e)86 b(e)e(a)785 1672 y FK(1)840 1660 y FO(+)19 b FN(ia)997 1672 y FK(2)492 1760 y FO(0)82 b(0)208 b(0)492 1860 y(0)82 b(0)208 b(0)1033 1594 y Fy(3)1033 1743 y(5)1102 1761 y FN(;)97 b( )s FO(\()p FN(q)1348 1773 y FK(2)1386 1761 y FO(\))23 b(=)1529 1594 y Fy(2)1529 1743 y(4)1584 1660 y FO(0)83 b FP(\000)p FN(e)f FP(\000)p FN(e)1584 1760 y FO(0)115 b FN(e)147 b(e)1584 1860 y FO(0)114 b(0)144 b(0)1998 1594 y Fy(3)1998 1743 y(5)2068 1761 y FN(:)-118 2067 y FO(One)20 b(can)g(directly)d(c)n(hec)n(k)j(that)g([)p FN( )s FO(\()p FN(q)1027 2079 y FL(k)1068 2067 y FO(\)])1123 2037 y FK(2)1184 2067 y FO(=)j FN( )s FO(\()p FN(q)1398 2079 y FL(k)1439 2067 y FO(\),)f FN(k)k FO(=)d(1,)e(2,)g FN( )s FO(\()p FN(q)1971 2079 y FK(1)2009 2067 y FO(\))14 b FN( )s FO(\()p FN(q)2181 2079 y FK(2)2219 2067 y FO(\))23 b(=)-118 2167 y FN( )s FO(\()p FN(q)8 2179 y FK(2)46 2167 y FO(\))14 b FN( )s FO(\()p FN(q)218 2179 y FK(1)255 2167 y FO(\))34 b(=)f(0,)i(and)f(that)g(the)g(functor)g FN(F)1370 2179 y FL( )1430 2167 y FO(:)43 b(Rep)14 b Fz(S)1723 2179 y FK(2)1794 2167 y FP(\000)-48 b(!)33 b FO(Rep)14 b FA(Q)2140 2179 y FK(2)p FL(;)p FM(?)2283 2167 y FO(is)-118 2266 y(full.)p 2278 2266 4 57 v 2282 2214 50 4 v 2282 2266 V 2331 2266 4 57 v -118 2508 a FQ(Corollary)32 b(12.)40 b FB(The)23 b(pr)l(oblem)f(of)h(unitary)e (classi\014c)l(ation)i(of)g(p)l(airs)f(of)h(c)l(om-)-118 2607 y(muting)29 b(idemp)l(otents)h(is)g FP(\003)p FB(-wild.)-118 2801 y FQ(Corollary)i(13.)40 b FB(The)h FP(\003)p FB(-algebr)l(a)g FA(Q)1068 2813 y FL(n;)p FM(?)1227 2801 y FO(=)h FI(C)15 b FP(h)q FN(q)1458 2813 y FK(1)1501 2801 y FN(;)f(:)g(:)g(:)f(;)h(q) 1722 2813 y FL(n)1810 2801 y FP(j)42 b FN(q)1915 2771 y FK(2)1912 2823 y FL(i)1995 2801 y FO(=)g FN(q)2139 2813 y FL(i)2166 2801 y FN(;)14 b(i)42 b FO(=)-118 2901 y(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)p FO(;)27 b FN(q)245 2913 y FL(i)273 2901 y FN(q)310 2913 y FL(j)370 2901 y FO(=)d(0)29 b FB(for)i FN(i)24 b FP(6)p FO(=)g FN(j)5 b FP(i)30 b FO(\()p FB(the)h(pr)l(oblem)h(of)f(unitary)g(classi\014c)l (ation)h(of)-118 3001 y FN(n)d FB(p)l(airwise)j(ortho)l(gonal)f(idemp)l (otents)7 b FO(\))31 b FB(is)f FP(\003)p FB(-wild)g(for)h FN(n)22 b FP(\025)h FO(2)p FB(.)-118 3195 y FQ(Corollary)32 b(14.)40 b FB(The)f FP(\003)p FB(-algebr)l(a)f FI(C)15 b FP(h)p FN(q)1131 3207 y FK(1)1175 3195 y FN(;)f(:)g(:)g(:)f(;)h(q) 1396 3207 y FL(n)1479 3195 y FP(j)37 b FN(q)1579 3165 y FK(2)1576 3217 y FL(i)1654 3195 y FO(=)g FN(q)1793 3207 y FL(i)1821 3195 y FN(;)28 b(i)37 b FO(=)g(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)p FO(;)-118 3295 y FN(q)-81 3307 y FK(1)-24 3295 y FO(+)20 b FP(\001)14 b(\001)g(\001)19 b FO(+)h FN(q)299 3307 y FL(n)371 3295 y FO(=)26 b FN(e)p FP(i)32 b FO(\()p FB(the)g(pr)l(oblem)h(of)g(unitary)f(classi\014c)l (ation)h(of)50 b FN(n)32 b FB(idem-)-118 3394 y(p)l(otents)38 b FN(Q)241 3406 y FK(1)277 3394 y FB(,)h FN(:)14 b(:)g(:)27 b FB(,)79 b FN(Q)635 3406 y FL(n)718 3394 y FB(such)38 b(that)g FN(Q)1159 3406 y FK(1)1220 3394 y FO(+)24 b FP(\001)14 b(\001)g(\001)25 b FO(+)f FN(Q)1586 3406 y FL(n)1669 3394 y FO(=)37 b FN(I)7 b FO(\))39 b FB(is)f FP(\003)p FB(-wild)h(for)-118 3494 y FN(n)23 b FP(\025)f FO(3)p FB(.)-118 3688 y(Pr)l(o)l(of.)43 b FO(If)37 b FN(m)h FO(=)g(3,)h(the)e(condition)d FN(q)1114 3700 y FK(1)1176 3688 y FO(+)24 b FN(q)1302 3700 y FK(2)1364 3688 y FO(+)g FN(q)1490 3700 y FK(3)1565 3688 y FO(=)38 b FN(e)e FO(implies)d(that)k(the)-118 3788 y(idemp)r(oten)n(ts)e FN(q)399 3800 y FK(1)436 3788 y FO(,)k FN(q)535 3800 y FK(2)572 3788 y FO(,)g FN(q)671 3800 y FK(3)744 3788 y FO(are)d(pairwise)d(orthogonal.)60 b(Then)37 b(the)f(algebra)-118 3887 y(under)27 b(consideration)d(coincides)h(with)j(the)f(algebra)e FA(Q)1654 3899 y FK(2)p FL(;)p FM(?)1763 3887 y FO(.)p 2278 3887 V 2282 3835 50 4 v 2282 3887 V 2331 3887 4 57 v eop %%Page: 221 225 221 224 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(221)-118 96 y FQ(3.1.4)94 b FP(\003)p FQ(-Wild)30 b(semilinear)e(relations)-118 252 y(1.)58 b FO(In)35 b(Sections)f(1.3.2-1.3.5)e(w)n(e)i(studied)h (represen)n(tations)d(of)j(semilinear)-118 351 y(relations.)j(In)29 b(particular,)d(the)k(structure)e(of)h(pairs)e(of)i(op)r(erators)e FN(A)f FO(=)f FN(A)2277 321 y FM(\003)2316 351 y FO(,)-118 451 y FN(B)i FO(=)c FN(B)127 421 y FM(\003)193 451 y FO(whic)n(h)j(satisfy)g(the)i(semilinear)23 b(relation)751 594 y FL(n)712 619 y Fy(X)712 798 y FL(k)q FK(=1)846 698 y FN(f)887 710 y FL(k)928 698 y FO(\()p FN(A)p FO(\))14 b FN(B)k(g)1189 710 y FL(k)1230 698 y FO(\()p FN(A)p FO(\))24 b(=)e(0)659 b(\(3.1\))-118 957 y(w)n(as)26 b(studied.)6 1058 y(This)h(relation)e(corresp)r(onds)g(to)j(the)g(c)n(haracteristic) 23 b(function)609 1305 y(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)990 1201 y FL(n)951 1226 y Fy(X)950 1405 y FL(k)q FK(=1)1085 1305 y FN(f)1126 1317 y FL(k)1167 1305 y FO(\()p FN(t)p FO(\))14 b FN(g)1315 1317 y FL(k)1355 1305 y FO(\()p FN(s)p FO(\))24 b(=)f(0)556 b(\(3.2\))-118 1585 y(\(w)n(e)30 b(supp)r(ose)f(that)i(\010\()p FN(t;)14 b(s)p FO(\))27 b(=)p 885 1513 231 4 v 27 w(\010\()p FN(s;)14 b(t)p FO(\),)31 b FN(t)p FO(,)f FN(s)d FP(2)h FI(R)p FO(\).)50 b(If)30 b(the)h(graph)e(\()p FI(R)2115 1555 y FK(1)2158 1585 y FN(;)14 b FO(\000)27 b(=)-118 1685 y FP(f)p FO(\()p FN(t;)14 b(s)p FO(\))34 b FP(2)h FI(R)272 1655 y FK(2)349 1685 y FO(:)g(\010\()p FN(t;)14 b(s)p FO(\))34 b(=)g(0)p FP(g)p FO(\))g(has)g(only)f(connected)h(comp)r(onen)n(ts)f(of)h(the) -118 1823 y(form)123 1805 y Fn(r)181 1823 y FO(,)278 1805 y Fn(r)p 278 1807 4 4 v 274 1802 V 269 1797 V 265 1793 V 261 1788 V 258 1784 V 255 1779 V 253 1775 V 250 1771 V 249 1767 V 247 1763 V 246 1760 V 245 1756 V 245 1753 V 245 1749 V 246 1746 V 246 1743 V 248 1740 V 249 1737 V 251 1735 V 253 1732 V 253 1732 V 256 1730 V 258 1728 V 261 1726 V 263 1724 V 266 1723 V 268 1722 V 271 1721 V 273 1720 V 276 1720 V 278 1720 V 281 1720 V 283 1720 V 286 1721 V 288 1722 V 291 1723 V 293 1724 V 296 1726 V 298 1728 V 301 1730 V 303 1732 V 278 1807 V 283 1802 V 288 1797 V 292 1793 V 295 1788 V 299 1784 V 301 1779 V 304 1775 V 306 1771 V 308 1767 V 309 1763 V 310 1760 V 311 1756 V 311 1753 V 311 1749 V 311 1746 V 310 1743 V 309 1740 V 307 1737 V 306 1735 V 303 1732 V 336 1823 a FO(,)f(or)539 1805 y Fn(r)p 539 1807 125 4 v 99 w(r)721 1823 y FO(,)g(then)e(irreducible)d(represen)n(tations)g(of)k (relation)-118 1923 y(\(3.1\))27 b(are)g(one-)g(and)g(t)n(w)n (o-dimensional)22 b(and)27 b(w)n(ere)g(describ)r(ed)f(in)h(1.3.5.)-118 2076 y FQ(2.)36 b FO(W)-7 b(e)28 b(sho)n(w)f(that)h(all)d(other)i (relations)d(are)j FP(\003)p FO(-wild.)-118 2245 y FQ(Prop)s(osition)j (67.)41 b FB(If)32 b(the)h(gr)l(aph)g(of)h(semiline)l(ar)f(r)l(elation) 39 b FO(\(3.1\))32 b FB(c)l(ontains)-118 2383 y(a)e(sub)l(gr)l(aph)334 2365 y Fn(r)p 334 2367 4 4 v 329 2362 V 325 2357 V 321 2353 V 317 2348 V 314 2344 V 311 2339 V 308 2335 V 306 2331 V 304 2327 V 303 2323 V 302 2320 V 301 2316 V 301 2313 V 301 2309 V 301 2306 V 302 2303 V 303 2300 V 305 2297 V 307 2295 V 309 2292 V 309 2292 V 312 2290 V 314 2288 V 317 2286 V 319 2284 V 321 2283 V 324 2282 V 326 2281 V 329 2280 V 331 2280 V 334 2280 V 336 2280 V 339 2280 V 341 2281 V 344 2282 V 346 2283 V 349 2284 V 351 2286 V 354 2288 V 356 2290 V 359 2292 V 334 2367 V 339 2362 V 343 2357 V 347 2353 V 351 2348 V 354 2344 V 357 2339 V 360 2335 V 362 2331 V 364 2327 V 365 2323 V 366 2320 V 367 2316 V 367 2313 V 367 2309 V 367 2306 V 366 2303 V 365 2300 V 363 2297 V 361 2295 V 359 2292 V 334 2367 125 4 v 99 w(r)517 2383 y FB(or)665 2365 y Fn(r)p 665 2367 V 100 w(r)p 790 2367 V 99 w(r)956 2383 y FB(,)g(then)g(the)f (r)l(elation)i(is)f FP(\003)p FB(-wild.)-118 2564 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)23 b(assume)f(that)h(the)h(functions)e FN(f)1271 2576 y FL(k)1312 2564 y FO(\()p FP(\001)p FO(\))h(and)g FN(g)1619 2576 y FL(k)1660 2564 y FO(\()p FP(\001)p FO(\))g(are)g(p)r (olynomials)-118 2663 y(and)k(pro)n(v)n(e)f(that)i(the)g FP(\003)p FO(-algebra)276 2910 y Fz(A)336 2922 y FK(\000)404 2910 y FO(=)22 b FI(C)545 2818 y Fy(D)602 2910 y FN(a)h FO(=)g FN(a)801 2876 y FM(\003)839 2910 y FN(;)14 b(b)22 b FO(=)h FN(b)1058 2876 y FM(\003)1119 2910 y FP(j)1205 2806 y FL(n)1165 2831 y Fy(X)1165 3010 y FL(k)q FK(=1)1299 2910 y FN(f)1340 2922 y FL(k)1381 2910 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)1593 2922 y FL(k)1633 2910 y FO(\()p FN(a)p FO(\))24 b(=)e(0)1894 2818 y Fy(E)-118 3221 y FO(is)k FP(\003)p FO(-wild)f(if)j(\000)22 b FP(\033)499 3203 y Fn(r)p 499 3205 4 4 v 494 3200 V 490 3195 V 486 3190 V 482 3186 V 479 3181 V 476 3177 V 474 3173 V 471 3169 V 470 3165 V 468 3161 V 467 3158 V 466 3154 V 466 3151 V 466 3147 V 467 3144 V 467 3141 V 469 3138 V 470 3135 V 472 3133 V 474 3130 V 474 3130 V 477 3128 V 479 3126 V 482 3124 V 484 3122 V 487 3121 V 489 3120 V 492 3119 V 494 3118 V 497 3118 V 499 3118 V 502 3118 V 504 3118 V 507 3119 V 509 3120 V 512 3121 V 514 3122 V 517 3124 V 519 3126 V 522 3128 V 524 3130 V 499 3205 V 504 3200 V 508 3195 V 513 3190 V 516 3186 V 519 3181 V 522 3177 V 525 3173 V 527 3169 V 529 3165 V 530 3161 V 531 3158 V 532 3154 V 532 3151 V 532 3147 V 532 3144 V 531 3141 V 530 3138 V 528 3135 V 526 3133 V 524 3130 V 499 3205 125 4 v 100 w(r)682 3221 y FO(\()p FN(\025)762 3233 y FK(1)800 3221 y FN(;)14 b(\025)885 3233 y FK(2)945 3221 y FP(2)24 b FI(R)p FO(,)34 b FN(\025)1183 3233 y FK(1)1243 3221 y FP(6)p FO(=)23 b FN(\025)1379 3233 y FK(2)1417 3221 y FO(\).)6 3322 y(De\014ne)29 b(a)e FP(\003)p FO(-homomorphism)21 b FN( )12 b FO(:)28 b Fz(A)1165 3334 y FK(\000)1233 3322 y FP(\000)-48 b(!)23 b FN(M)1437 3334 y FK(3)1474 3322 y FO(\()p Fz(S)1575 3334 y FK(2)1612 3322 y FO(\))28 b(as)f(follo)n(ws:)153 3606 y FN( )s FO(\()p FN(a)p FO(\))d(=)429 3439 y Fy(0)429 3589 y(@)502 3506 y FN(\025)550 3518 y FK(1)588 3506 y FN(e)124 b FO(0)165 b(0)543 3605 y(0)124 b FN(\025)757 3617 y FK(1)795 3605 y FN(e)g FO(0)543 3705 y(0)166 b(0)124 b FN(\025)965 3717 y FK(2)1002 3705 y FN(e)1041 3439 y Fy(1)1041 3589 y(A)1127 3606 y FN(;)97 b( )s FO(\()p FN(b)p FO(\))24 b(=)1515 3439 y Fy(0)1515 3589 y(@)1588 3506 y FN(a)1632 3518 y FK(1)1773 3506 y FN(e)106 b(e)1609 3605 y(e)e(a)1796 3617 y FK(2)1916 3605 y FO(0)1609 3705 y FN(e)124 b FO(0)102 b(0)1958 3439 y Fy(1)1958 3589 y(A)2044 3606 y FN(:)6 3887 y FO(It)28 b(is)f(easy)f(to)i(c)n(hec)n(k)f(that)h(the)g(functor)f FN(F)1352 3899 y FL( )1430 3887 y FO(is)g(full.)p 2278 3887 4 57 v 2282 3835 50 4 v 2282 3887 V 2331 3887 4 57 v eop %%Page: 222 226 222 225 bop -118 -137 a FO(222)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FQ(Prop)s(osition)i(68.)41 b FB(If)22 b(the)g(gr)l(aph)h FO(\000)e FB(of)i(semiline)l(ar)g(r)l(elation)29 b FO(\(3.1\))21 b FB(c)l(ontains)-118 196 y(a)31 b(sub)l(gr)l(aph)336 178 y Fn(r)p 336 180 125 4 v 99 w(r)p 460 180 V 100 w(r)300 253 y FL(\025)339 261 y Fx(1)425 253 y FL(\025)464 261 y Fx(2)549 253 y FL(\025)588 261 y Fx(3)657 196 y FB(,)h FN(\025)762 208 y FK(1)824 196 y FP(6)p FO(=)25 b FN(\025)962 208 y FK(2)1024 196 y FP(6)p FO(=)g FN(\025)1162 208 y FK(3)1224 196 y FP(6)p FO(=)g FN(\025)1362 208 y FK(1)1399 196 y FO(;)32 b FN(\025)1502 208 y FK(1)1539 196 y FB(,)g FN(\025)1644 208 y FK(2)1681 196 y FB(,)g FN(\025)1786 208 y FK(3)1848 196 y FP(2)25 b FI(R)p FB(,)38 b(then)30 b(the)-118 328 y(r)l(elation)g(is)g FP(\003)p FB(-wild.)-118 510 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)29 b(pro)n(v)n(e)e(that)i Fz(A)752 522 y FK(\000)821 510 y FO(=)24 b FI(C)964 443 y Fy(\012)1009 510 y FN(a)h FO(=)f FN(a)1211 480 y FM(\003)1249 510 y FN(;)14 b(b)24 b FO(=)g FN(b)1471 480 y FM(\003)1533 510 y FP(j)1581 448 y Fy(P)1669 469 y FL(n)1669 535 y(k)q FK(=1)1807 510 y FN(f)1848 522 y FL(k)1889 510 y FO(\()p FN(a)p FO(\))14 b FN(b)g(g)2101 522 y FL(k)2141 510 y FO(\()p FN(a)p FO(\))25 b(=)-118 618 y(0)-76 551 y Fy(\013)-10 618 y FO(is)i FP(\003)p FO(-wild)e(if)i(\000)c FP(\033)580 600 y Fn(r)p 580 602 V 99 w(r)p 704 602 V 100 w(r)545 675 y FL(\025)584 683 y Fx(1)669 675 y FL(\025)708 683 y Fx(2)793 675 y FL(\025)832 683 y Fx(3)898 618 y FO(\()p FN(\025)978 630 y FK(1)1016 618 y FN(;)14 b(\025)1101 630 y FK(2)1139 618 y FN(;)g(\025)1224 630 y FK(3)1284 618 y FP(2)24 b FI(R)p FO(,)33 b FN(\025)1521 630 y FK(1)1582 618 y FP(6)p FO(=)23 b FN(\025)1718 630 y FK(2)1779 618 y FP(6)p FO(=)f FN(\025)1914 630 y FK(3)1975 618 y FP(6)p FO(=)h FN(\025)2111 630 y FK(1)2148 618 y FO(\).)6 761 y(W)-7 b(e)24 b(construct)e(the)h FP(\003)p FO(-homomorphism)17 b FN( )12 b FO(:)28 b Fz(A)1471 773 y FK(\000)1539 761 y FP(\000)-48 b(!)23 b FN(M)1743 773 y FK(7)1780 761 y FO(\()p Fz(S)1881 773 y FK(2)1918 761 y FO(\))g(as)g(follo)n(ws:)190 1253 y FN( )s FO(\()p FN(a)p FO(\))g(=)466 887 y Fy(0)466 1033 y(B)466 1083 y(B)466 1132 y(B)466 1182 y(B)466 1232 y(B)466 1282 y(B)466 1332 y(B)466 1382 y(B)466 1435 y(@)539 953 y FN(\025)587 965 y FK(1)624 953 y FN(e)125 b FO(0)165 b(0)g(0)h(0)f(0)g(0)580 1053 y(0)124 b FN(\025)794 1065 y FK(1)832 1053 y FN(e)g FO(0)165 b(0)h(0)f(0)g(0)580 1152 y(0)h(0)123 b FN(\025)1001 1164 y FK(2)1039 1152 y FN(e)h FO(0)166 b(0)f(0)g(0)580 1252 y(0)h(0)f(0)124 b FN(\025)1209 1264 y FK(2)1246 1252 y FN(e)h FO(0)165 b(0)g(0)580 1352 y(0)h(0)f(0)g(0)124 b FN(\025)1416 1364 y FK(2)1454 1352 y FN(e)g FO(0)165 b(0)580 1451 y(0)h(0)f(0)g(0)h(0)124 b FN(\025)1624 1463 y FK(3)1661 1451 y FN(e)g FO(0)580 1551 y(0)166 b(0)f(0)g(0)h(0)f(0)124 b FN(\025)1831 1563 y FK(3)1869 1551 y FN(e)1907 887 y Fy(1)1907 1033 y(C)1907 1083 y(C)1907 1132 y(C)1907 1182 y(C)1907 1232 y(C)1907 1282 y(C)1907 1332 y(C)1907 1382 y(C)1907 1435 y(A)1994 1253 y FN(;)201 1983 y( )s FO(\()p FN(b)p FO(\))24 b(=)469 1617 y Fy(0)469 1763 y(B)469 1813 y(B)469 1863 y(B)469 1913 y(B)469 1963 y(B)469 2013 y(B)469 2062 y(B)469 2112 y(B)469 2165 y(@)667 1684 y FO(0)208 b(0)84 b FN(e)105 b(e)f(a)1374 1696 y FK(1)1429 1684 y FO(+)18 b FN(ia)1585 1696 y FK(2)1705 1684 y FO(0)102 b(0)667 1783 y(0)208 b(0)83 b(0)103 b FN(e)230 b(e)210 b FO(0)102 b(0)669 1883 y FN(e)209 b FO(0)83 b(0)102 b(0)227 b(0)209 b FN(e)104 b FO(0)669 1983 y FN(e)211 b(e)84 b FO(0)102 b(0)227 b(0)208 b(0)82 b(2)p FN(e)542 2082 y(a)586 2094 y FK(1)642 2082 y FP(\000)18 b FN(ia)798 2094 y FK(2)919 2082 y FN(e)84 b FO(0)102 b(0)227 b(0)208 b(0)102 b(0)667 2182 y(0)208 b(0)84 b FN(e)104 b FO(0)227 b(0)208 b(0)102 b(0)667 2281 y(0)208 b(0)83 b(0)f(2)p FN(e)208 b FO(0)g(0)102 b(0)1910 1617 y Fy(1)1910 1763 y(C)1910 1813 y(C)1910 1863 y(C)1910 1913 y(C)1910 1963 y(C)1910 2013 y(C)1910 2062 y(C)1910 2112 y(C)1910 2165 y(A)1996 1983 y FN(:)6 2476 y FO(It)28 b(is)f(easy)f(to)i(c)n(hec)n(k)f(that)h(the)g(functor)f FN(F)1352 2488 y FL( )1430 2476 y FO(is)g(full.)p 2278 2476 4 57 v 2282 2424 50 4 v 2282 2476 V 2331 2476 4 57 v -118 2720 a FQ(3.1.5)94 b FP(\003)p FQ(-Wild)30 b(quadratic)j(and)f(cubic)h(relations)-118 2883 y(1.)80 b FO(The)42 b(relation)d(\()p FN(I)609 2895 y FK(0)647 2883 y FO(\))k(0)j(=)h(0)42 b(de\014nes)g(the)h(standard)e(wild)f FP(\003)p FO(-algebra)-118 2983 y Fz(S)-49 2995 y FK(2)23 2983 y FO(=)34 b FI(C)15 b FP(h)q FN(a;)f(b)40 b FP(j)35 b FN(a)468 2953 y FM(\003)541 2983 y FO(=)g FN(a;)27 b(b)771 2953 y FM(\003)844 2983 y FO(=)34 b FN(b)p FP(i)p FO(.)59 b(By)34 b(Theorem)f(52,)j(the)f(theory)f(of)g(its)-118 3083 y FP(\003)p FO(-represen)n(tations)28 b(con)n(tains)i FP(\003)p FO(-represen)n(tations)f(of)j(ev)n(ery)e(\014nitely)h(gener-) -118 3182 y(ated)c FP(\003)p FO(-algebra.)-118 3348 y FQ(2.)42 b FO(The)30 b(relation)c(\()p FN(I)546 3360 y FK(1)584 3348 y FO(\))k FN(a)690 3318 y FK(2)754 3348 y FO(=)c FN(e)j FO(de\014nes)h(the)g FP(\003)p FO(-algebra)c Fz(D)g FO(=)g FI(C)15 b FP(h)p FN(a;)f(b)32 b FP(j)27 b FN(a)2210 3318 y FM(\003)2274 3348 y FO(=)-118 3447 y FN(a;)h(b)13 3417 y FM(\003)73 3447 y FO(=)23 b FN(b;)k(a)291 3417 y FK(2)352 3447 y FO(=)22 b FN(e)p FP(i)p FO(.)-118 3629 y FQ(Prop)s(osition)30 b(69.)41 b FB(The)30 b FP(\003)p FB(-algebr)l(a)h Fz(D)e FB(is)i FP(\003)p FB(-wild.)-118 3811 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)32 b(will)d(sho)n(w)h(that)i Fz(D)e FP(\037)f FA(P)1091 3823 y FK(3)p FL(;)p FM(?)p FK(2)1262 3811 y FO(=)g FI(C)15 b FP(h)q FN(p)1485 3823 y FK(1)1528 3811 y FN(;)f(p)1607 3823 y FK(2)1643 3811 y FN(;)g(p)1722 3823 y FK(3)1789 3811 y FP(j)30 b FN(p)1884 3781 y FM(\003)1884 3833 y FL(i)1951 3811 y FO(=)f FN(p)2087 3823 y FL(i)2115 3811 y FN(;)e(p)2207 3781 y FK(2)2207 3833 y FL(i)2274 3811 y FO(=)-118 3911 y FN(p)-76 3923 y FL(i)-49 3911 y FN(;)h(p)44 3923 y FK(1)81 3911 y FN(p)123 3923 y FK(2)193 3911 y FO(=)k FN(p)332 3923 y FK(2)369 3911 y FN(p)411 3923 y FK(1)481 3911 y FO(=)h(0)p FP(i)p FO(.)54 b(T)-7 b(o)33 b(sho)n(w)g(this,)h(w)n(e)f(de\014ne)h(a)f FP(\003)p FO(-homomorphism)p eop %%Page: 223 227 223 226 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(223)-118 96 y FN( )26 b FO(:)d Fz(D)g FP(\000)-48 b(!)23 b FA(P)278 108 y FK(3)p FL(;)p FM(?)p FK(2)447 96 y FO(as)k(follo)n(ws:)487 256 y FN( )s FO(\()p FN(a)p FO(\))c(=)g FN(p)18 b FP(\000)g FO(\()p FN(e)g FP(\000)g FN(p)p FO(\))24 b(=)e(2)p FN(p)c FP(\000)g FN(e;)495 429 y( )s FO(\()p FN(b)p FO(\))23 b(=)g FN(p)805 441 y FK(1)860 429 y FO(+)953 373 y(1)p 953 410 42 4 v 953 486 a(2)1005 429 y FN(p)1047 441 y FK(2)1102 429 y FO(+)1195 373 y(1)p 1195 410 V 1195 486 a(3)1247 429 y(\()p FN(e)18 b FP(\000)g FN(p)1461 441 y FK(1)1517 429 y FP(\000)g FN(p)1642 441 y FK(2)1679 429 y FO(\))p FN(:)-118 612 y FO(It)26 b(is)e(easy)g(to)i(c)n(hec)n(k)e (that)i(the)g(corresp)r(onding)c(functor)j FN(F)1741 624 y FL( )1815 612 y FO(:)e(Rep)14 b FA(P)2074 624 y FK(3)p FL(;)p FM(?)p FK(2)2239 612 y FP(\000)-48 b(!)-118 712 y FO(Rep)14 b Fz(D)28 b FO(is)e(full.)p 2278 712 4 57 v 2282 659 50 4 v 2282 712 V 2331 712 4 57 v -118 873 a FQ(3.)75 b FO(No)n(w)40 b(w)n(e)g(giv)n(e)f(a)h(criterion,)g(in)g (terms)f(of)i(the)g(co)r(e\016cien)n(ts,)h(for)e(the)-118 973 y(quadratic)32 b FP(\003)p FO(-algebra)e Fz(A)j FO(=)g FI(C)14 b FP(h)q FN(a;)g(b)39 b FP(j)33 b FN(a)g FO(=)g FN(a)1334 943 y FM(\003)1372 973 y FN(;)28 b(b)33 b FO(=)g FN(b)1626 943 y FM(\003)1663 973 y FN(;)28 b(P)1767 985 y FK(2)1805 973 y FO(\()p FN(a;)14 b(b)p FO(\))33 b(=)g FN(\013a)2214 943 y FK(2)2274 973 y FO(+)-118 1072 y FN(\014)t(b)-31 1042 y FK(2)26 1072 y FO(+)19 b FN(q)s(=i)p FO([)p FN(a;)14 b(b)p FO(])k(+)h FN(\015)5 b FP(f)p FN(a;)14 b(b)p FP(g)k FO(+)h FN(\016)s(a)h FO(+)f FN(\017b)g FO(+)g FN(\037I)33 b FO(=)25 b(0)p FP(i)p FO(,)30 b FN(\013)p FO(,)g FN(\014)t FO(,)h FN(q)s FO(,)e FN(\015)5 b FO(,)30 b FN(\016)s FO(,)g FN(\017)p FO(,)f FN(\037)d FP(2)h FI(R)-118 1172 y FO(to)g(b)r(e)h FP(\003)p FO(-wild.)-118 1320 y FQ(Theorem)i(60.)41 b FB(A)30 b FP(\003)p FB(-algebr)l(a)h Fz(A)e FB(is)i FP(\003)p FB(-wild)g(if)g(and)g(only)g(if)g(one)f(of)h (the)g(fol-)-118 1419 y(lowing)g(c)l(onditions)f(holds)7 b FO(:)-19 1567 y(1)p FB(.)41 b FN(\013)24 b FO(=)f FN(\014)k FO(=)c FN(\015)k FO(=)c FN(q)j FO(=)d FN(\016)j FO(=)c FN(\017)h FO(=)g FN(\037)g FO(=)f(0;)-19 1733 y(2)p FB(.)89 1666 y Fy(\000)127 1733 y FN(\037)d FP(\000)297 1700 y FL(\016)329 1675 y Fx(2)p 291 1714 77 4 v 291 1761 a FK(4)p FL(\013)377 1666 y Fy(\001)429 1733 y FN(\013)k(<)g FO(0)p FB(,)85 b FN(\014)27 b FO(=)c FN(\015)28 b FO(=)22 b FN(q)k FO(=)d FN(\017)g FO(=)f(0;)-19 1912 y(3)p FB(.)89 1845 y Fy(\000)127 1912 y FN(\037)d FP(\000)298 1879 y FL(\017)326 1854 y Fx(2)p 291 1893 74 4 v 291 1940 a FK(4)p FL(\014)375 1845 y Fy(\001)426 1912 y FN(\014)28 b(<)22 b FO(0)p FB(,)85 b FN(\013)24 b FO(=)e FN(\015)28 b FO(=)23 b FN(q)j FO(=)c FN(\016)27 b FO(=)22 b(0;)-19 2102 y(4)p FB(.)41 b FN(\015)137 2072 y FK(2)197 2102 y FO(=)23 b FN(\014)t(\013)h FP(6)p FO(=)e(0)p FN(;)99 b(\013)p FO(\()p FN(\037)19 b FP(\000)919 2069 y FL(\016)951 2044 y Fx(2)p 913 2083 77 4 v 913 2130 a FK(4)p FL(\013)999 2102 y FO(\))k FN(<)g FO(0)p FB(,)1304 2069 y FL(\016)1336 2044 y Fx(2)p 1304 2083 65 4 v 1315 2130 a FL(\013)1401 2102 y FO(=)1499 2069 y FL(\017)1527 2044 y Fx(2)p 1499 2083 61 4 v 1509 2130 a FL(\014)1569 2102 y FB(,)85 b FN(q)27 b FO(=)22 b(0)p FB(.)-118 2249 y(Pr)l(o)l(of.)43 b FO(The)34 b FP(\003)p FO(-algebra)c(with)j(t)n(w)n(o)g(self-adjoin)n (t)f(v)-5 b(ariables)30 b(and)k(quadratic)-118 2349 y(relations)28 b(is)j(wild)f(i\013)i(there)f(exists)g(a)g(c)n(hange)g(of)g(v)-5 b(ariables)29 b(suc)n(h)i(that)h(the)-118 2448 y(algebra)38 b(can)j(b)r(e)g(transformed)e(to)i(the)g FP(\003)p FO(-algebra)d Fz(S)1670 2460 y FK(2)1752 2448 y FO(=)45 b FI(C)15 b FP(h)p FN(a;)f(b)51 b FP(j)46 b FN(a)f FO(=)-118 2548 y FN(a)-74 2518 y FM(\003)-36 2548 y FN(;)28 b(b)k FO(=)g FN(b)216 2518 y FM(\003)254 2548 y FP(i)i FO(or)e(the)i FP(\003)p FO(-algebra)29 b Fz(D)k FO(=)f FI(C)15 b FP(h)q FN(a;)f(b)38 b FP(j)32 b FN(a)h FO(=)f FN(a)1655 2518 y FM(\003)1693 2548 y FN(;)c(b)k FO(=)g FN(b)1945 2518 y FM(\003)1983 2548 y FN(;)c(a)2078 2518 y FK(2)2147 2548 y FO(=)33 b FN(e)p FP(i)p FO(.)-118 2648 y(One)e(can)f(de\014ne)i (suc)n(h)e(a)h(quadratic)e FP(\003)p FO(-algebras)e(b)n(y)k(imp)r (osing)d(one)i(of)h(the)-118 2747 y(conditions)25 b(1{4.)p 2278 2747 4 57 v 2282 2695 50 4 v 2282 2747 V 2331 2747 4 57 v -118 2909 a FQ(4.)35 b FO(No)n(w)24 b(w)n(e)g(consider)f(a)h (pair)f(of)i(self-adjoin)n(t)d(op)r(erators)g(whic)n(h)i(satisfy)f(the) -118 3008 y(cubic)k(semilinear)22 b(relation:)-68 3168 y FN(\017)p FP(f)p FN(A)70 3134 y FK(2)107 3168 y FN(;)14 b(B)t FP(g)k FO(+)g FN(i\016)s FO([)p FN(A)508 3134 y FK(2)545 3168 y FN(;)c(B)t FO(])19 b(+)f(2)p FN(\026AB)t(A)h FO(+)f FN(i\015)5 b FO([)p FN(A;)14 b(B)t FO(])k(+)g(2)p FN(\014)t FP(f)p FN(A;)c(B)t FP(g)j FO(+)h FN(\013B)28 b FO(=)23 b(0)p FN(:)2168 3267 y FO(\(3.3\))-118 3428 y(T)-7 b(o)30 b(relation)d(\(3.3\))k(there)f(corresp)r(onds)e(the)j FP(\003)p FO(-algebra)c Fz(A)1758 3440 y FK(3)1823 3428 y FO(=)g FI(C)15 b FP(h)p FN(a;)f(b)33 b FP(j)28 b FN(a)g FO(=)-118 3527 y FN(a)-74 3497 y FM(\003)-36 3527 y FN(;)g(b)22 b FO(=)h FN(b)197 3497 y FM(\003)235 3527 y FN(;)28 b(\017)p FP(f)p FN(a)406 3497 y FK(2)442 3527 y FN(;)14 b(b)p FP(g)f FO(+)g FN(i\016)s FO([)p FN(a)784 3497 y FK(2)820 3527 y FN(;)h(b)p FO(])f(+)g(2)p FN(\026)h(aba)f FO(+)g FN(i\015)5 b FO([)p FN(a;)14 b(b)p FO(])f(+)g(2)p FN(\014)t FP(f)p FN(a;)h(b)p FP(g)f FO(+)g FN(\013b)21 b FO(=)i(0)p FP(i)p FO(.)-118 3627 y(The)28 b(corresp)r(onding)c(c)n(haracteristic)g (function)j(is)f(the)i(follo)n(wing:)347 3787 y(\010\()p FN(t;)14 b(s)p FO(\))24 b(=)e FN(\017)p FO(\()p FN(t)784 3752 y FK(2)840 3787 y FO(+)c FN(s)962 3752 y FK(2)999 3787 y FO(\))h(+)f FN(i\016)s FO(\()p FN(t)1264 3752 y FK(2)1320 3787 y FP(\000)g FN(s)1442 3752 y FK(2)1479 3787 y FO(\))596 3911 y(+)g(2)p FN(\026)c(ts)k FO(+)g FN(i\015)5 b FO(\()p FN(t)18 b FP(\000)g FN(s)p FO(\))g(+)g(2)p FN(\014)t FO(\()p FN(t)h FO(+)f FN(s)p FO(\))h(+)f FN(\013;)p eop %%Page: 224 228 224 227 bop -118 -137 a FO(224)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FN(\013)p FO(,)g FN(\014)t FO(,)g FN(\015)5 b FO(,)27 b FN(\016)g FP(2)c FI(R)p FO(.)6 196 y(It)40 b(follo)n(ws)d(from)h(the) i(general)d(theory)h(of)i(semilinear)34 b(relations)i(that)-118 296 y(the)42 b(corresp)r(onding)c FP(\003)p FO(-algebra)g(is)i FP(\003)p FO(-wild)f(if)i(and)g(only)f(if)h(the)g(equation)-118 395 y(\010\()p FN(t;)14 b(s)p FO(\))23 b(=)g(0)e(has)g(either)f(t)n(w)n (o)g(solutions)f(of)i(the)h(form)e(\()p FN(t)1625 407 y FK(1)1662 395 y FN(;)14 b(t)1729 407 y FK(1)1767 395 y FO(\),)23 b(\()p FN(t)1907 407 y FK(1)1944 395 y FN(;)14 b(t)2011 407 y FK(2)2048 395 y FO(\),)23 b(where)-118 495 y FN(t)-88 507 y FK(1)-27 495 y FP(6)p FO(=)g FN(t)91 507 y FK(2)128 495 y FO(,)28 b(or)f(t)n(w)n(o)g(solutions)e(of)j(the)h (form)d(\()p FN(t)1282 507 y FK(1)1320 495 y FN(;)14 b(t)1387 507 y FK(2)1424 495 y FO(\),)28 b(\()p FN(t)1569 507 y FK(1)1607 495 y FN(;)14 b(t)1674 507 y FK(3)1711 495 y FO(\),)28 b(where)g FN(t)2065 507 y FK(1)2102 495 y FO(,)g FN(t)2183 507 y FK(2)2220 495 y FO(,)g FN(t)2301 507 y FK(3)-118 595 y FO(are)e(distinct.)6 694 y(The)32 b(equation)f(\010\()p FN(t;)14 b(s)p FO(\))30 b(=)g(0)h(decomp)r(oses)f (in)h(a)h(natural)e(w)n(a)n(y)h(in)n(to)f(the)-118 794 y(system)c(of)i(t)n(w)n(o)f(equations:)149 876 y Fy(\032)252 942 y FO(\010)312 954 y FK(1)350 942 y FO(\()p FN(t;)14 b(s)p FO(\))23 b(=)g FN(\017)14 b(t)709 912 y FK(2)764 942 y FO(+)k(2)p FN(\026)c(ts)k FO(+)g FN(\017s)1196 912 y FK(2)1251 942 y FO(+)g(2)p FN(\014)t(t)g FO(+)g(2)p FN(\014)t(s)h FO(+)f FN(\013)23 b FO(=)g(0)p FN(;)252 1042 y FO(\010)312 1054 y FK(2)350 1042 y FO(\()p FN(t;)14 b(s)p FO(\))23 b(=)g FN(\016)s FO(\()p FN(t)733 1012 y FK(2)789 1042 y FP(\000)18 b FN(s)911 1012 y FK(2)948 1042 y FO(\))h(+)f FN(\015)5 b FO(\()p FN(t)18 b FP(\000)g FN(s)p FO(\))23 b(=)g(0)p FN(:)6 1197 y FO(First)34 b(w)n(e)g(assume)g (that)h FN(\016)j FO(=)c FN(\015)40 b FO(=)34 b(0)h(\(\010)1352 1209 y FK(2)1389 1197 y FO(\()p FN(t;)14 b(s)p FO(\))36 b FP(\021)e FO(0\).)58 b(The)35 b(equation)-118 1296 y(\010)-58 1308 y FK(1)-21 1296 y FO(\()p FN(t;)14 b(s)p FO(\))39 b(=)f(0)e(de\014nes)h(a)g(curv)n(e)f(of)g(degree)g(t)n(w)n(o.) 64 b(Suc)n(h)37 b(curv)n(es)f(ha)n(v)n(e)f(the)-118 1396 y(follo)n(wing)23 b(in)n(v)-5 b(arian)n(ts:)505 1595 y FN(I)541 1607 y FK(1)602 1595 y FO(=)23 b(2)p FN(\017;)96 b(I)921 1607 y FK(2)982 1595 y FO(=)1069 1474 y Fy(\014)1069 1524 y(\014)1069 1574 y(\014)1069 1624 y(\014)1105 1544 y FN(\017)91 b(\026)1097 1644 y(\026)g(\017)1280 1474 y Fy(\014)1280 1524 y(\014)1280 1574 y(\014)1280 1624 y(\014)1331 1595 y FO(=)23 b FN(\017)1453 1561 y FK(2)1508 1595 y FP(\000)18 b FN(\026)1641 1561 y FK(2)1678 1595 y FN(;)319 1877 y(I)355 1889 y FK(3)416 1877 y FO(=)503 1707 y Fy(\014)503 1757 y(\014)503 1807 y(\014)503 1856 y(\014)503 1906 y(\014)503 1956 y(\014)540 1777 y FN(\017)92 b(\026)85 b(\014)532 1876 y(\026)92 b(\017)h(\014)531 1976 y(\014)88 b(\014)f(\013)853 1707 y Fy(\014)853 1757 y(\014)853 1807 y(\014)853 1856 y(\014)853 1906 y(\014)853 1956 y(\014)904 1877 y FO(=)23 b(\()p FN(\017)1058 1843 y FK(2)1113 1877 y FP(\000)18 b FN(\026)1246 1843 y FK(2)1284 1877 y FO(\))c FN(\013)19 b FP(\000)f FO(2)p FN(\014)1578 1843 y FK(2)1629 1877 y FO(\()p FN(\017)g FP(\000)g FN(\026)p FO(\))p FN(:)-118 2131 y FO(If)28 b FN(I)1 2143 y FK(2)62 2131 y FP(6)p FO(=)c(0,)j(then)i(the)f(equation)e(\010)975 2143 y FK(1)1012 2131 y FO(\()p FN(t;)14 b(s)p FO(\))24 b(=)f(0)28 b(de\014nes)g(a)f(cen)n(tral)f(curv)n(e.)37 b(By)-118 2230 y(an)d(a\016ne)g(transformation)d(of)j(v)-5 b(ariables,)33 b(the)i(equation)e(can)h(b)r(e)h(reduced)-118 2330 y(in)n(to)26 b(the)i(follo)n(wing)c(form:)539 2521 y(\()p FN(\017)18 b FO(+)g FN(\026)p FO(\))804 2506 y(^)802 2521 y FN(t)832 2487 y FK(2)888 2521 y FO(+)g(\()p FN(\017)h FP(\000)f FN(\026)p FO(\))f(^)-45 b FN(s)1274 2487 y FK(2)1330 2521 y FO(+)1422 2465 y FN(I)1458 2477 y FK(3)p 1422 2502 74 4 v 1422 2578 a FN(I)1458 2590 y FK(2)1529 2521 y FO(=)23 b(0)p FN(:)486 b FO(\(3.4\))6 2716 y(a\))28 b(Let)g FN(I)293 2728 y FK(2)353 2716 y FN(>)23 b FO(0.)6 2815 y(If)28 b FN(I)125 2827 y FK(3)186 2815 y FO(=)23 b(0,)k(then)h(the)g FP(\003)p FO(-algebra)c Fz(A)1118 2827 y FK(3)1183 2815 y FO(is)i(not)i(wild.)6 2915 y(If)k FN(I)129 2927 y FK(3)197 2915 y FN(>)d FO(0,)j(then)g(the)g(set)g(of)f (solutions)e(of)i(\(3.4\))h(is)e(empt)n(y)-7 b(.)48 b(Therefore)-118 3014 y(the)28 b FP(\003)p FO(-algebra)c Fz(A)445 3026 y FK(3)509 3014 y FO(is)j(not)g(wild.)6 3114 y(If)i FN(I)126 3126 y FK(3)188 3114 y FN(<)23 b FO(0,)28 b(then)g(equation)f(\(3.4\))h (b)r(ecomes)f(an)g(equation)g(of)h(an)g(ellipse.)-118 3214 y(Therefore)h(there)i(are)f(t)n(w)n(o)g(solutions)e(\()p FN(t)1191 3226 y FK(1)1228 3214 y FN(;)14 b(t)1295 3226 y FK(2)1332 3214 y FO(\),)32 b(\()p FN(t)1481 3226 y FK(1)1519 3214 y FN(;)14 b(t)1586 3226 y FK(3)1623 3214 y FO(\))31 b(suc)n(h)g(that)g FN(t)2090 3226 y FK(2)2155 3214 y FP(6)p FO(=)d FN(t)2278 3226 y FK(3)2316 3214 y FO(.)-118 3313 y(Hence,)g(the)g FP(\003)p FO(-algebra)c Fz(A)715 3325 y FK(3)779 3313 y FO(is)j FP(\003)p FO(-wild.)6 3413 y(b\))f(Let)f FN(I)292 3425 y FK(2)353 3413 y FN(<)e FO(0)i(then)g(the)h(equation)e(\010\()p FN(t;)14 b(s)p FO(\))23 b(=)g(0)h(is)h(of)g(h)n(yp)r(erb)r(olic)d(t)n(yp)r(e.)6 3513 y(If)h FN(I)120 3525 y FK(3)180 3513 y FO(=)g(0,)g(then)f(w)n(e)f (ha)n(v)n(e)g(a)g(pair)f(of)i(in)n(tersecting)d(straigh)n(t)h(lines.)32 b(Hence)-118 3612 y(it)27 b(is)f(easy)h(to)h(see)f(that)h(the)g FP(\003)p FO(-algebra)c Fz(A)1210 3624 y FK(3)1274 3612 y FO(is)j FP(\003)p FO(-wild.)6 3712 y(If)22 b FN(I)119 3724 y FK(3)179 3712 y FP(6)p FO(=)h(0,)f(then)f(w)n(e)g(ha)n(v)n(e)e (an)i(equation)e(of)h(a)h(h)n(yp)r(erb)r(ola.)33 b(This)19 b(equation)-118 3811 y(has)27 b(no)g(more)f(than)i(one)f(solution)e (only)h(in)h(the)h(case)f(where)g FN(\017)22 b FO(=)h(0.)36 b(Hence,)-118 3911 y(if)27 b FN(\017)c FP(6)p FO(=)f(0,)28 b(then)g(the)g FP(\003)p FO(-algebra)c Fz(A)947 3923 y FK(3)1011 3911 y FO(is)j(wild.)p eop %%Page: 225 229 225 228 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(225)6 96 y(c\))26 b(Let)g FN(I)284 108 y FK(2)345 96 y FO(=)d(0.)36 b(Then)26 b(the)g(equation)e(\010)1287 108 y FK(1)1324 96 y FO(\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0)k(has)f(parab)r(olic)d(t)n(yp)r(e)-118 196 y(\(the)28 b(curv)n(e)f(is)f(not)i(cen)n(tral\).)6 296 y(Let)g FN(I)191 308 y FK(3)252 296 y FO(=)23 b(0.)36 b(Then)28 b FN(\017)23 b FO(=)f FN(\026)28 b FO(and)388 480 y(\010)448 492 y FK(1)486 480 y FO(\()p FN(t;)14 b(s)p FO(\))23 b(=)g FN(\017)p FO(\()p FN(t)18 b FO(+)g FN(s)p FO(\))1035 446 y FK(2)1091 480 y FO(+)g(2)p FN(\014)t FO(\()p FN(t)h FO(+)f FN(s)p FO(\))g(+)g FN(\013)24 b FO(=)e(0)p FN(:)6 665 y FO(If)33 b FN(\014)145 635 y FK(2)204 665 y FP(\000)21 b FN(\013\017)30 b(<)g FO(0,)j(the)f (equation)f(\010)1151 677 y FK(1)1188 665 y FO(\()p FN(t;)14 b(s)p FO(\))31 b(=)f(0)h(has)h(no)f(solutions,)g(and)-118 765 y(the)d FP(\003)p FO(-algebra)c Fz(A)445 777 y FK(3)509 765 y FO(is)j(not)g(wild.)6 865 y(If)36 b FN(\014)148 835 y FK(2)208 865 y FP(\000)23 b FN(\013\017)35 b(<)g FO(0,)h(then)f FN(t)23 b FO(+)g FN(s)35 b FO(=)f FP(\000)p FN(\014)t(=\017)p FO(,)i(and)e(it)g(is)g(ob)n(vious)e(that)j(the)-118 965 y FP(\003)p FO(-algebra)24 b Fz(A)302 977 y FK(3)366 965 y FO(is)j(not)g(wild.)6 1065 y(F)-7 b(or)41 b FN(\014)220 1035 y FK(2)285 1065 y FP(\000)27 b FN(\013\017)46 b(>)f FO(0,)g(the)c(equation)f(\010)1299 1077 y FK(1)1336 1065 y FO(\()p FN(t;)14 b(s)p FO(\))46 b(=)f(0)c(describ)r(es)f(a)g(pair) -118 1165 y(of)32 b(parallel)d(lines.)49 b(Hence)33 b(the)g(set)g(of)f (solutions)e(con)n(tains)h(t)n(w)n(o)g(solutions:)-118 1265 y(\()p FN(t)-56 1277 y FK(1)-19 1265 y FN(;)14 b(t)48 1277 y FK(2)86 1265 y FO(\),)43 b(\()p FN(t)246 1277 y FK(1)283 1265 y FN(;)14 b(t)350 1277 y FK(3)387 1265 y FO(\))40 b(suc)n(h)f(that)h FN(t)880 1277 y FK(2)960 1265 y FP(6)p FO(=)j FN(t)1098 1277 y FK(3)1135 1265 y FO(.)73 b(Therefore)38 b(the)i FP(\003)p FO(-algebra)c Fz(A)2206 1277 y FK(3)2283 1265 y FO(is)-118 1364 y FP(\003)p FO(-wild.)6 1464 y(If)43 b FN(I)140 1476 y FK(2)224 1464 y FO(=)k(0,)e FN(I)482 1476 y FK(3)566 1464 y FP(6)p FO(=)h(0,)f(then)e(the)f(equation)e(\010)1561 1476 y FK(1)1598 1464 y FO(\()p FN(t;)14 b(s)p FO(\))48 b(=)e(0)41 b(de\014nes)h(a)-118 1564 y(parab)r(ola.)67 b(In)39 b(this)f(case,)j (if)d FN(\014)930 1534 y FK(2)993 1564 y FP(\000)26 b FN(\017\013)g FP(\000)f FO(4)p FN(\014)t(t)42 b(>)f FO(0,)g(w)n(e)d(ha) n(v)n(e)g(t)n(w)n(o)f(solu-)-118 1664 y(tions)28 b(of)g(the)i(equation) d(\010)727 1676 y FK(1)764 1664 y FO(\()p FN(t;)14 b(s)p FO(\))26 b(=)e(0)29 b(for)f(one)g(v)-5 b(alue)28 b FN(t)p FO(,)h(and)g(the)g FP(\003)p FO(-algebra)-118 1763 y Fz(A)-58 1775 y FK(3)6 1763 y FO(is)e FP(\003)p FO(-wild.)6 1864 y(No)n(w)h(consider)d(the)j(case)f(\010)898 1876 y FK(1)935 1864 y FO(\()p FN(t;)14 b(s)p FO(\))23 b FP(\021)g FO(0,)k FN(\016)f FP(6)p FO(=)d(0.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)484 2048 y(\010)544 2060 y FK(2)581 2048 y FO(\()p FN(t;)14 b(s)p FO(\))24 b(=)e(\()p FN(t)d FP(\000)f FN(s)p FO(\)\()p FN(\016)s(t)h FO(+)f FN(\016)s(s)g FO(+)g FN(\015)5 b FO(\))23 b(=)g(0)p FN(:)-118 2232 y FO(In)37 b(that)g(case)g(the)g (equation)e(\010)930 2244 y FK(2)968 2232 y FO(\()p FN(t;)14 b(s)p FO(\))39 b(=)f(0)f(de\014nes)g(a)f(pair)g(of)g(in)n(tersect-)-118 2332 y(ing)27 b(and)i(non)g(coinciding)c(straigh)n(t)i(lines.)38 b(Therefore)28 b(the)h FP(\003)p FO(-algebra)c Fz(A)2217 2344 y FK(3)2283 2332 y FO(is)-118 2431 y FP(\003)p FO(-wild.)6 2531 y(Consider)i(the)i(case)f(\010)735 2543 y FK(1)772 2531 y FO(\()p FN(t;)14 b(s)p FO(\))g(\010)1016 2543 y FK(2)1054 2531 y FO(\()p FN(t;)g(s)p FO(\))25 b FP(6)p FO(=)g(0.)40 b(It)29 b(is)f(ob)n(vious)e(that)j(if)f(eac)n(h)-118 2631 y(equation)36 b(\010)291 2643 y FK(1)328 2631 y FO(\()p FN(t;)14 b(s)p FO(\))40 b(=)g(0)d(and)g(\010)953 2643 y FK(2)991 2631 y FO(\()p FN(t;)14 b(s)p FO(\))40 b(=)f(0)e(generates)g(a)g(non)g FP(\003)p FO(-wild)e FP(\003)p FO(-)-118 2731 y(algebra,)25 b(then)804 2843 y Fy(\032)908 2909 y FO(\010)968 2921 y FK(1)1005 2909 y FO(\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0)p FN(;)908 3009 y FO(\010)968 3021 y FK(2)1005 3009 y FO(\()p FN(t;)14 b(s)p FO(\))24 b(=)e(0)p FN(;)2168 2960 y FO(\(3.5\))-118 3189 y(corresp)r(onds)39 b(a)i(non)g FP(\003)p FO(-wild)e FP(\003)p FO(-algebra,)i(to)r(o.)77 b(Therefore)40 b(w)n(e)h(will)d (con-)-118 3289 y(sider)c(the)h(case)g(when)g(\010)707 3301 y FK(1)744 3289 y FO(\()p FN(t;)14 b(s)p FO(\))36 b(=)g(0)f(and)g(\010)1357 3301 y FK(2)1394 3289 y FO(\()p FN(t;)14 b(s)p FO(\))36 b(=)f(0)g(generate)f FP(\003)p FO(-wild)-118 3389 y FP(\003)p FO(-algebras.)f(Then)28 b(w)n(e)f(ha)n(v)n(e)f(that)i FN(\016)s FO(\()p FN(\017)1123 3358 y FK(2)1179 3389 y FO(+)18 b FN(\026)1312 3358 y FK(2)1349 3389 y FO(\))24 b FP(6)p FO(=)e(0.)6 3489 y(By)28 b(an)g(a\016ne)g(c)n(hange)f(of)h(v)-5 b(ariables,)26 b(system)h(\(3.5\))g(can)h(b)r(e)h(reduced)f(to)-118 3589 y(the)g(follo)n(wing)23 b(form:)254 3708 y Fy(\032)367 3754 y FO(~)358 3775 y(\010)418 3787 y FK(1)455 3775 y FO(\()488 3760 y(~)487 3775 y FN(t;)17 b FO(~)-45 b FN(s)p FO(\))23 b(=)g FN(\017)785 3760 y FO(~)784 3775 y FN(t)814 3745 y FK(2)869 3775 y FO(+)18 b(2)p FN(\026)1059 3760 y FO(~)1058 3775 y FN(t)s FO(~)-45 b FN(s)18 b FO(+)g FN(\017)s FO(~)-45 b FN(s)1301 3745 y FK(2)1356 3775 y FO(+)18 b(2)1493 3753 y(~)1481 3775 y FN(\014)7 b FO(~)-45 b FN(s)19 b FO(+)26 b(~)-50 b FN(\013)23 b FO(=)g(0)p FN(;)367 3860 y FO(~)358 3881 y(\010)418 3893 y FK(2)455 3881 y FO(\()488 3866 y(~)487 3881 y FN(t;)17 b FO(~)-45 b FN(s)p FO(\))23 b(=)737 3866 y(~)736 3881 y FN(t)766 3851 y FK(2)822 3881 y FP(\000)e FO(~)-45 b FN(s)944 3851 y FK(2)1004 3881 y FO(=)23 b(0)p FN(;)p eop %%Page: 226 230 226 229 bop -118 -137 a FO(226)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FO(where)143 75 y(~)131 96 y FN(\014)42 b FO(=)c FN(\014)29 b FP(\000)24 b FO(\()p FN(\017)h FO(+)f FN(\026)p FO(\))p FN(\015)5 b(=)p FO(\(2)p FN(\016)r FO(\),)47 b(~)-50 b FN(\013)39 b FO(=)e FN(\013)25 b FO(+)f(\()p FN(\017)h FO(+)f FN(\026)p FO(\))p FN(\015)1718 66 y FK(2)1755 96 y FN(=)p FO(\(2)p FN(\016)1911 66 y FK(2)1948 96 y FO(\))g FP(\000)h FO(2)p FN(\014)t(\015)5 b(=\016)s FO(.)-118 196 y(Solutions)22 b(of)i(this)f(system)g(are)g(solutions)f (of)i(the)g(equation)f(of)1905 175 y(~)1896 196 y(\010)1956 208 y FK(1)1993 196 y FO(\()2026 181 y(~)2025 196 y FN(t;)17 b FO(~)-45 b FN(s)p FO(\))23 b(=)g(0,)-118 296 y(where)30 b(~)-45 b FN(s)23 b FO(=)30 b(~)-49 b FN(p)27 b FO(and)k(~)-46 b FN(s)23 b FO(=)g FP(\000)718 280 y FO(~)717 296 y FN(t)p FO(.)6 395 y(Therefore,)29 b(for)g Fz(A)597 407 y FK(3)663 395 y FO(to)h(b)r(e)g FP(\003)p FO(-wild)d(it)h(is)h(necessary)f(and)h (su\016cien)n(t)f(that)-118 495 y(the)g(equation)374 474 y(~)365 495 y(\010\()458 480 y(~)457 495 y FN(t;)17 b FO(~)-45 b FN(s)p FO(\))24 b(=)f(0)28 b(ha)n(v)n(e)f(t)n(w)n(o)g (solutions)e(of)j(the)g(form:)37 b(\()1974 480 y(~)1973 495 y FN(t)2003 507 y FK(0)2040 495 y FN(;)2078 480 y FO(~)2077 495 y FN(t)2107 507 y FK(0)2144 495 y FO(\))29 b(and)-118 595 y(\()-85 579 y(~)-86 595 y FN(t)-56 607 y FK(0)-19 595 y FN(;)14 b FP(\000)84 579 y FO(~)83 595 y FN(t)113 607 y FK(0)150 595 y FO(\).)43 b(It)30 b(is)f(easy)f(to)i (sho)n(w)f(that)g(this)g(condition)f(is)g(ful\014lled)g(in)h(one)g(of) -118 694 y(the)f(follo)n(wing)23 b(cases.)2 846 y(i.)40 b FN(\017)23 b(>)g FO(0,)82 b FN(\026)24 b(>)e FO(0,)83 b(2)p FN(\016)s(\014)22 b FP(\000)c FN(\017\015)27 b FO(=)c(0,)83 b FN(\013\016)1357 816 y FK(2)1413 846 y FP(\000)18 b FN(\017\015)1578 816 y FK(2)1637 846 y FN(<)23 b FO(0;)-21 1048 y(ii.)39 b FN(\026)24 b FP(6)p FO(=)e(0,)83 b(2)p FN(\016)s(\014)22 b FP(\000)c FN(\017\015)27 b FP(6)p FO(=)c(0,)83 b FN(\013\017)1059 1014 y FK(2)1115 1048 y FP(\000)18 b FN(I)1234 1060 y FK(3)1290 1048 y FO(+)1383 992 y FN(\017\015)p 1383 1029 82 4 v 1403 1105 a(\016)1484 992 y FO(\()p FN(\017)g FO(+)g FN(\026)p FO(\))p FN(\015)24 b FP(\000)18 b FO(4)p FN(\014)t(\016)p 1484 1029 532 4 v 1709 1105 a FO(2)p FN(\016)2049 1048 y FO(=)k(0.)-118 1224 y(Th)n(us)27 b(w)n(e)g(pro)n(v)n(ed)f(the)i (follo)n(wing)c(theorem.)-118 1375 y FQ(Theorem)30 b(61.)41 b FB(The)25 b FP(\003)p FB(-algebr)l(a)f Fz(A)1021 1387 y FK(3)1081 1375 y FO(=)f FI(C)1223 1308 y Fy(\012)1268 1375 y FN(a;)14 b(b)22 b FP(j)h FN(a)g FO(=)g FN(a)1652 1345 y FM(\003)1690 1375 y FN(;)14 b(b)23 b FO(=)f FN(b)1909 1345 y FM(\003)1947 1375 y FN(;)28 b(\017)p FP(f)p FN(a)2118 1345 y FK(2)2154 1375 y FN(;)14 b(b)p FP(g)6 b FO(+)-118 1483 y FN(i\016)s FO([)p FN(a)18 1453 y FK(2)55 1483 y FN(;)14 b(b)p FO(])i(+)h(2)p FN(\026)d(aba)i FO(+)g FN(i\015)5 b FO([)p FN(a;)14 b(b)p FO(])i(+)h(2)p FN(\014)t FP(f)p FN(a;)d(b)p FP(g)h FO(+)i FN(\013b)1394 1416 y Fy(\013)1433 1483 y FB(,)30 b FN(\017)22 b FP(\025)h FO(0)p FB(,)29 b FN(\017)1762 1453 y FK(2)1816 1483 y FO(+)17 b FN(\026)1948 1453 y FK(2)2002 1483 y FO(+)g FN(\016)2124 1453 y FK(2)2184 1483 y FP(6)p FO(=)23 b(0)p FB(,)-118 1583 y(is)30 b FP(\003)p FB(-wild)g(if)h(and)f(only)g(if)h (one)f(of)h(the)f(c)l(onditions)37 b FO(1\))p FB({)p FO(3\))30 b FB(holds)7 b FO(:)-26 1735 y(1\))41 b FN(\016)27 b FO(=)22 b FN(\015)28 b FO(=)22 b(0)p FB(,)30 b(and)g(one)g(of)h(the)f (fol)t(lowing)i(c)l(onditions)f(holds)7 b FO(:)122 1894 y(\()p FN(a)p FO(\))42 b FN(I)308 1906 y FK(1)369 1894 y FN(>)23 b FO(0)p FB(,)29 b FN(I)589 1906 y FK(2)650 1894 y FN(>)23 b FO(0)p FB(,)29 b FN(I)870 1906 y FK(3)931 1894 y FN(<)23 b FO(0)p FB(,)130 2020 y FO(\()p FN(b)p FO(\))42 b FN(I)308 2032 y FK(2)369 2020 y FN(<)23 b FO(0)p FB(,)29 b FN(I)589 2032 y FK(3)650 2020 y FO(=)23 b(0)p FB(,)130 2145 y FO(\()p FN(c)p FO(\))42 b FN(I)308 2157 y FK(1)369 2145 y FN(>)23 b FO(0)p FB(,)29 b FN(I)589 2157 y FK(2)650 2145 y FN(<)23 b FO(0)p FB(,)29 b FN(I)870 2157 y FK(3)931 2145 y FP(6)p FO(=)23 b(0)p FB(,)123 2271 y FO(\()p FN(d)p FO(\))42 b FN(I)308 2283 y FK(2)369 2271 y FO(=)23 b(0)p FB(,)29 b FN(I)589 2283 y FK(3)650 2271 y FO(=)23 b(0)p FB(,)29 b FN(\014)885 2241 y FK(2)941 2271 y FP(\000)18 b FO(8)p FN(\013\017)23 b(>)g FO(0)p FB(,)127 2397 y FO(\()p FN(e)p FO(\))42 b FN(I)308 2409 y FK(2)369 2397 y FO(=)23 b(0)p FB(.)38 b FN(I)598 2409 y FK(3)658 2397 y FP(6)p FO(=)23 b(0;)-26 2556 y(2\))41 b FN(\017)23 b FO(=)g FN(\026)g FO(=)g FN(\014)k FO(=)c FN(\013)g FO(=)g(0)p FB(,)29 b FO(\()p FB(then)h FN(\016)c FP(6)p FO(=)d(0\))p FB(.)-26 2715 y FO(3\))41 b FN(\016)s FO(\()p FN(\017)195 2685 y FK(2)251 2715 y FO(+)18 b FN(\026)384 2685 y FK(2)421 2715 y FO(\))24 b FP(6)p FO(=)e(0)30 b FB(and)g(one)g(of)g(the)g(fol)t(lowing)j(c)l(onditions)d (holds)7 b FO(:)122 2874 y(\()p FN(a)p FO(\))42 b FN(\017)23 b(>)f FO(0)p FB(,)30 b FN(\026)23 b FO(=)g(0)p FB(,)30 b FO(2)p FN(\016)s(\014)22 b FP(\000)c FN(\017\015)27 b FO(=)c(0)p FB(,)30 b FN(\013\016)1387 2844 y FK(2)1443 2874 y FP(\000)18 b FN(\017\015)1608 2844 y FK(2)1667 2874 y FN(<)23 b FO(0)p FB(,)130 3048 y FO(\()p FN(b)p FO(\))42 b FN(\026)23 b FP(6)p FO(=)g(0)p FB(,)29 b FO(2)p FN(\016)s(\014)20 b FP(\000)c FN(\017\015)27 b FP(6)p FO(=)c(0)p FB(,)29 b FN(\013\017)1134 3017 y FK(2)1187 3048 y FP(\000)16 b FN(I)1304 3060 y FK(3)1358 3048 y FO(+)1449 2991 y FN(\017\015)p 1449 3029 82 4 v 1469 3105 a(\016)1550 2991 y(\017)i FO(+)g FN(\026)p FO(\))h FP(\000)f FN(\015)23 b FP(\000)18 b FO(4)p FN(\014)t(\016)p 1550 3029 601 4 v 1810 3105 a FO(2)p FN(\016)2184 3048 y FO(=)23 b(0)p FB(.)-118 3230 y FQ(5)p FO(.)59 b(F)-7 b(or)34 b(a)h(non-semilinear)30 b(cubic)k(relation,)g(w)n(e)h(giv)n(e)e (only)g(the)j(follo)n(wing)-118 3330 y(prop)r(osition.)-118 3481 y FQ(Prop)s(osition)30 b(70.)41 b FB(The)30 b FP(\003)p FB(-algebr)l(a)486 3646 y FA(B)549 3658 y FK(2)610 3646 y FO(=)22 b FI(C)15 b FP(h)q FN(a)29 b FO(=)22 b FN(a)988 3612 y FM(\003)1026 3646 y FN(;)14 b(b)23 b FO(=)g FN(b)1246 3612 y FM(\003)1306 3646 y FP(j)h FN(aba)e FO(=)h FN(bab)p FP(i)-118 3811 y FB(is)35 b FP(\003)p FB(-wild.)56 b(Mor)l(e)l(over,)38 b(its)d(quotient)g(algebr)l(a)i FI(C)15 b FP(h)p FN(a)39 b FO(=)32 b FN(a)1714 3781 y FM(\003)1752 3811 y FN(;)14 b(b)33 b FO(=)f FN(b)1991 3781 y FM(\003)2062 3811 y FP(j)h FN(aba)f FO(=)-118 3911 y FN(bab)22 b FO(=)h(0)p FP(i)29 b FB(is)h FP(\003)p FB(-wild.)p eop %%Page: 227 231 227 230 bop -118 -137 a FJ(3.1.)36 b FP(\003)p FJ(-Wild)25 b(algebras)g(and)i(relations)1094 b FO(227)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Let)20 b(a)f(homomorphism)14 b FN( )e FO(:)28 b FA(B)1102 108 y FK(2)1163 96 y FP(\000)-48 b(!)23 b FN(M)1367 108 y FK(4)1403 96 y FO(\()p Fz(S)1504 108 y FK(2)1542 96 y FO(\))d(b)r(e)g(de\014ned)g(as)f(follo)n(ws:)207 436 y FN( )s FO(\()p FN(a)p FO(\))k(=)483 269 y Fy(0)483 419 y(@)555 332 y FO(0)84 b FN(e)557 432 y(e)g FO(0)804 383 y Fo(0)609 539 y(0)136 b(0)863 269 y Fy(1)863 419 y(A)949 436 y FN(;)97 b( )s FO(\()p FN(b)p FO(\))24 b(=)1337 220 y Fy(0)1337 366 y(B)1337 415 y(B)1337 469 y(@)1464 337 y Fo(0)1700 286 y FN(e)84 b FO(0)1699 386 y(0)e(0)1411 485 y FN(e)j FO(0)1410 585 y(0)e(0)1659 485 y FN(a)1703 497 y FK(1)1844 485 y FN(e)1680 585 y(e)104 b(a)1867 597 y FK(2)1904 220 y Fy(1)1904 366 y(C)1904 415 y(C)1904 469 y(A)1991 436 y FN(:)-118 776 y FO(One)30 b(can)g(c)n(hec)n(k)f (that)i FN( )s FO(\()p FN(aba)p FO(\))c(=)g(0,)k(and)f FN( )j FO(de\014nes)e(a)f(homomorphism)25 b(of)-118 876 y(the)j(quotien)n(t)e(algebra.)34 b(The)28 b(constructed)f(functor)h FN(F)1636 888 y FL( )1714 876 y FO(is)e(full.)p 2278 876 4 57 v 2282 823 50 4 v 2282 876 V 2331 876 4 57 v -118 1073 a FB(R)l(emark)k(53.)42 b FO(The)36 b FP(\003)p FO(-wild)d FP(\003)p FO(-algebra)f FA(B)1208 1085 y FK(2)1281 1073 y FO(=)k FI(C)15 b FP(h)p FN(a)42 b FO(=)35 b FN(a)1698 1043 y FM(\003)1736 1073 y FN(;)14 b(b)36 b FO(=)g FN(b)1982 1043 y FM(\003)2055 1073 y FP(j)h FN(aba)e FO(=)-118 1172 y FN(bab)p FP(i)24 b FO(is)f(obtained)g(b)n(y)i(in)n(tro)r(ducing) c(the)k(in)n(v)n(olution)c(in)j(the)h(algebra)c FI(C)15 b FP(h)p FN(x)q(;)f(y)32 b FP(j)-118 1272 y FN(xy)s(x)24 b FO(=)e FN(y)s(xy)s FP(i)28 b FO(b)n(y)f(setting)g FN(x)c FO(=)g FN(x)919 1242 y FM(\003)958 1272 y FO(,)k FN(y)f FO(=)d FN(y)1207 1242 y FM(\003)1245 1272 y FO(.)6 1375 y(The)g FP(\003)p FO(-structure)e(on)h(the)g(algebra)e FI(C)15 b FP(h)p FN(x;)g(y)31 b FP(j)23 b FN(xy)s(x)h FO(=)f FN(y)s(xy)s FP(i)f FO(is)f(not)h(unique.)-118 1475 y(If)d(one)f(in)n(tro)r(duces)f(an)h(in)n(v)n(olution)d(in)j(this) g(algebra)d(b)n(y)k FN(x)1655 1445 y FL(?)1716 1475 y FO(=)k FN(y)s FO(,)d(the)f(obtained)-118 1575 y FP(\003)p FO(-algebra)26 b(is)i Fz(C)380 1587 y FK(2)443 1575 y FO(=)d FI(C)15 b FP(h)p FN(x)q(;)f(x)751 1545 y FL(?)821 1575 y FP(j)26 b FN(xx)964 1545 y FL(?)1003 1575 y FN(x)g FO(=)f FN(x)1213 1545 y FL(?)1252 1575 y FN(xx)1346 1545 y FL(?)1385 1575 y FP(i)p FO(.)42 b(The)30 b FP(\003)p FO(-algebra)25 b Fz(C)2067 1587 y FK(2)2134 1575 y FO(is)j(not)-118 1674 y FP(\003)p FO(-wild)d(\(see)i(Section)g(3.2.1\).)-118 1911 y FQ(3.1.6)94 b FP(\003)p FQ(-Wild)30 b(groups.)41 b(P)m(erio)s(dic)31 b(groups)h(are)g(not)g FP(\003)p FQ(-wild)-118 2071 y FO(In)23 b(this)f(section)f(w)n(e)i(study)g(the)g (complexit)n(y)c(of)k(description)d(of)j(unitary)e(rep-)-118 2171 y(resen)n(tations)28 b(\()p FP(\003)p FO(-represen)n(tations)f(of) k(group)e(algebras\))f(for)i(some)f(discrete)-118 2270 y(coun)n(table)k(groups)h FN(G)p FO(.)59 b(F)-7 b(or)34 b(groups)g FN(G)1185 2282 y FK(1)1258 2270 y FO(and)g FN(G)1491 2282 y FK(2)1529 2270 y FO(,)j(for)d(whic)n(h)g FN(C)2033 2240 y FM(\003)2072 2270 y FO(\()p FN(G)2169 2282 y FK(1)2206 2270 y FO(\))i FP(\037)-118 2370 y FN(C)-53 2340 y FM(\003)-15 2370 y FO(\()p FN(G)82 2382 y FK(2)120 2370 y FO(\),)42 b(w)n(e)c(will)e(write)h(b)r(elo)n(w)g FN(G)1052 2382 y FK(1)1131 2370 y FP(\037)j FN(G)1301 2382 y FK(2)1339 2370 y FO(.)69 b(If)39 b FN(C)1590 2340 y FM(\003)1629 2370 y FO(\()p FN(G)p FO(\))g(is)e FP(\003)p FO(-wild,)i(then)-118 2470 y(the)30 b(description)d(of)i(unitary)g (represen)n(tations)d(of)k(suc)n(h)f(a)g(group)g(con)n(tains,)-118 2569 y(as)37 b(a)h(subproblem,)g(the)h(description)c(of)j(represen)n (tations)d(of)j(an)n(y)f(coun)n(t-)-118 2669 y(able)31 b(group;)j(further)f(in)e(the)i(b)r(o)r(ok)g(w)n(e)f(will)d(call)i (them)h FP(\003)p FO(-wild)e(\(from)h(the)-118 2768 y(viewp)r(oin)n(t) 25 b(of)j(complexit)n(y)c(of)k(their)e(unitary)g(represen)n(tations\).) 6 2872 y(Belo)n(w,)f(w)n(e)g(giv)n(e)f(a)i(n)n(um)n(b)r(er)e(of)i (examples)d(of)j(b)r(oth)h FP(\003)p FO(-wild)c(groups,)i(and)-118 2972 y(groups)h(that)i(are)f(not)g FP(\003)p FO(-wild.)-118 3133 y FQ(1.)36 b FO(Let)28 b(us)f(giv)n(e)f(a)h(n)n(um)n(b)r(er)g(of)g (examples)e(of)j FP(\003)p FO(-wild)d(groups.)-118 3273 y FB(Example)31 b(24.)42 b FO(It)h(follo)n(ws)c(directly)h(from)h(the)h (list)f(of)h FP(\003)p FO(-wild)e FP(\003)p FO(-algebras)-118 3373 y(giv)n(en)24 b(ab)r(o)n(v)n(e,)h(that)h(the)g(groups)f FA(F)996 3385 y FL(n)1039 3373 y FO(,)h FN(n)d FP(\025)g FO(2,)j(and)f FI(Z)1561 3385 y FL(n)1615 3373 y FP(\003)14 b FI(Z)1733 3385 y FL(m)1790 3373 y FO(,)26 b FN(n)d FP(\025)g FO(2,)i FN(m)e FP(\025)g FO(3,)-118 3472 y(are)j FP(\003)p FO(-wild.)-118 3613 y FQ(2.)36 b FO(The)28 b(follo)n(wing)23 b(simple)i(statemen)n(t)i(holds.)-118 3791 y FQ(Prop)s(osition)j(71.)41 b FB(If)30 b(a)h(gr)l(oup)f FN(G)g FB(c)l(ontains)h(a)f(normal)h(sub)l(gr)l(oup)f FN(N)38 b FB(such)-118 3890 y(that)30 b FN(G=)-5 b(N)32 b FO(=)22 b FN(G)405 3902 y FK(1)443 3890 y FB(,)30 b(then)g FN(G)23 b FP(\037)f FN(G)923 3902 y FK(1)961 3890 y FB(.)p eop %%Page: 228 232 228 231 bop -118 -137 a FO(228)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FB(Pr)l(o)l(of.)43 b FO(Denote)29 b(b)n(y)f FN(\036)9 b FO(:)29 b FN(G)c FP(\000)-48 b(!)25 b FN(G)934 108 y FK(1)1000 96 y FO(the)k(mapping)d(whic)n(h)i(maps)f(an)i(elemen)n(t) -118 196 y FN(g)i FP(2)d FN(G)j FO(to)f(its)g(conjugacy)g(class)e FN(\036)p FO(\()p FN(g)s FO(\))h FP(2)f FN(G)1276 208 y FK(1)1342 196 y FO(=)f FN(G=)-5 b(N)9 b FO(.)46 b(Then,)32 b(in)n(tro)r(duce)d(a)-118 296 y(unitary)d FP(\003)p FO(-homomorphism)-50 472 y FN( )12 b FO(:)28 b FN(L)p FO(\()p FN(G)p FO(\))c FP(3)f FN(f)9 b FO(\()p FP(\001)p FO(\))23 b FP(7!)g FN( )s FO(\()p FN(f)9 b FO(\)\()p FN(g)864 484 y FK(1)902 472 y FO(\))23 b(=)1148 394 y Fy(X)1045 576 y FL(g)10 b FK(:)22 b FL(\036)p FK(\()p FL(g)r FK(\)=)p FL(g)1337 584 y Fx(1)1384 472 y FN(f)9 b FO(\()p FN(g)s FO(\))23 b FP(2)g FN(L)p FO(\()p FN(G)1796 484 y FK(1)1834 472 y FO(\))g FP(\032)g FN(C)2042 438 y FM(\003)2080 472 y FO(\()p FN(G)2177 484 y FK(1)2215 472 y FO(\))p FN(;)-118 729 y FO(and)37 b(use)h(the)g(same)e(notation)g FN( )41 b FO(for)c(its)g(unique)g(extension)g(to)g(a)g(unital)-118 829 y FP(\003)p FO(-homomorphism)22 b(from)27 b FN(C)800 799 y FM(\003)839 829 y FO(\()p FN(G)p FO(\))i(to)f FN(C)1164 799 y FM(\003)1203 829 y FO(\()p FN(G)1300 841 y FK(1)1337 829 y FO(\).)40 b(It)29 b(is)e(clear)f(that)j(the)g(corre-)-118 929 y(sp)r(onding)d(functor)i FN(F)577 941 y FL(\036)630 929 y FO(:)42 b(Rep)14 b FN(C)918 899 y FM(\003)956 929 y FO(\()p FN(G)1053 941 y FK(1)1091 929 y FO(\))24 b FP(\000)-49 b(!)23 b FO(Rep)14 b FN(C)1492 899 y FM(\003)1531 929 y FO(\()p FN(G)p FO(\))28 b(is)f(full.)p 2278 929 4 57 v 2282 876 50 4 v 2282 929 V 2331 929 4 57 v 6 1091 a(The)h(prop)r(osition)d(ab)r(o)n(v)n(e)h(implies)e(the)k(follo)n(wing) 23 b(statemen)n(t.)-118 1243 y FQ(Corollary)32 b(15.)40 b FB(A)26 b(gr)l(oup)g FN(G)g FB(that)g(p)l(ossesses)g(a)h(normal)f (sub)l(gr)l(oup)g FN(N)34 b FB(such)-118 1343 y(that)c FN(G=)-5 b(N)32 b FO(=)22 b FA(F)396 1355 y FK(2)461 1343 y FB(is)30 b FP(\003)p FB(-wild.)-118 1494 y(Example)h(25.)42 b FO(An)d(extension)f FN(G)h FO(of)f(an)n(y)g(group)f FN(G)1584 1506 y FK(1)1661 1494 y FO(b)n(y)h FA(F)1843 1506 y FK(2)1878 1494 y FO(,)k(is)37 b(a)h FP(\003)p FO(-wild)-118 1594 y(group.)6 1720 y(Notice)22 b(that)h(the)g(pro)r(of) f(of)g(wildness)f(of)h FI(Z)1351 1732 y FK(2)1390 1720 y FP(\003)8 b FI(Z)1502 1732 y FK(3)1556 1720 y FO(giv)n(en)20 b(in)i(Section)g(3.1.3,)-118 1819 y(is)k(not)i(reduced)f(to)h(the)g(c)n (hec)n(k)e(of)i(the)g(conditions)d(of)j(the)g(corollary)23 b(ab)r(o)n(v)n(e.)6 1919 y(Since)33 b(the)h(ma)5 b(jorization)29 b(is)j(a)h(quasi-order)d(relation)h(\(Section)i(3.1.1\),)-118 2018 y(the)28 b(follo)n(wing)23 b(corollary)g(from)k(Prop)r(osition)d (71)i(holds.)-118 2170 y FQ(Corollary)32 b(16.)40 b FB(If)e(a)f(gr)l (oup)h FN(G)f FB(p)l(ossesses)h(a)g(normal)g(sub)l(gr)l(oup)f FN(N)45 b FB(such)-118 2270 y(that)30 b FN(G=)-5 b(N)32 b FO(=)22 b FN(G)405 2282 y FK(1)472 2270 y FB(is)31 b FP(\003)p FB(-wild,)f(then)g FN(G)g FB(is)g FP(\003)p FB(-wild.)-118 2422 y(Example)h(26.)42 b FO(The)22 b(group)e FN(S)5 b(L)p FO(\(2)p FN(;)14 b FI(Z)n FO(\))i(is)k FP(\003)p FO(-wild,)g(since)g FN(S)5 b(L)p FO(\(2)p FN(;)14 b FI(Z)n FO(\))p FN(=)p FP(f)p FN(e)o(;)g FP(\000)p FN(e)o FP(g)j FO(=)-118 2521 y FN(P)12 b(S)5 b(L)p FO(\(2)p FN(;)14 b FI(Z)n FO(\))j(=)23 b FI(Z)429 2533 y FK(2)479 2521 y FP(\003)18 b FI(Z)600 2533 y FK(3)632 2521 y FO(.)-118 2647 y FB(Example)31 b(27.)42 b FO(The)24 b(braid)d(group)h FN(B)1043 2659 y FK(2)1103 2647 y FO(is)g FP(\003)p FO(-wild,)g(since)g FN(B)1713 2659 y FK(2)1750 2647 y FN(=)p FI(Z)16 b FO(=)23 b FI(Z)2019 2659 y FK(2)2060 2647 y FP(\003)9 b FI(Z)2172 2659 y FK(3)2203 2647 y FO(,)24 b(its)-118 2746 y(group)30 b FP(\003)p FO(-algebra)e(is)j FI(C)15 b FO([)p FN(B)713 2758 y FK(2)756 2746 y FO(])30 b(=)f FI(C)15 b FP(h)p FN(u;)f(v)39 b FP(j)30 b FN(u)p FO(,)d FN(v)k FO(are)26 b(unitary)o FN(;)14 b(uv)s(u)29 b FO(=)g FN(v)s(uv)s FP(i)h FO(=)-118 2846 y FI(C)15 b FP(h)p FN(w)45 b FO(=)36 b FN(uv)s(u;)14 b(z)39 b FO(=)d FN(uv)j FP(j)d FN(w)r FO(,)28 b FN(z)j FO(are)c(unitary)n FN(;)14 b(w)1398 2816 y FK(2)1472 2846 y FO(=)36 b FN(z)1616 2816 y FK(3)1653 2846 y FP(i)p FO(,)i(and)d(its)g(quotien)n(t)-118 2946 y(algebra)25 b(is)h FI(C)15 b FP(h)p FN(w)s(;)f(z)32 b FP(j)23 b FN(w)r FO(,)29 b FN(z)i FO(are)26 b(unitary)o FN(;)14 b(w)1243 2916 y FK(2)1304 2946 y FO(=)22 b FN(z)1434 2916 y FK(3)1494 2946 y FO(=)h FN(e)p FP(i)g FO(=)f FI(C)15 b FO([)p FI(Z)1902 2958 y FK(2)1957 2946 y FP(\003)j FI(Z)2079 2958 y FK(3)2110 2946 y FO(].)-118 3114 y FQ(3.)35 b FO(It)23 b(lo)r(oks)e(attractiv)n(e)g(to)i(in)n(v)n(estigate)d (whether)j(the)h(follo)n(wing)19 b(groups)j(are)-118 3214 y FP(\003)p FO(-wild)j(or)i(not:)6 3313 y(a\))20 b(Co)n(xeter)e(groups)g(whic)n(h)g(are)h(not)g(a\016ne)g(\(except)h (for)1819 3295 y Fn(r)p 1819 3297 125 4 v 99 w(r)100 b(r)p 1943 3297 V 1848 3279 a FM(1)58 b(1)2129 3313 y FO(whic)n(h)-118 3413 y(is)26 b FP(\003)p FO(-wild,)g(see)h(Section)f (3.1.3\);)6 3513 y(b\))j(non-elemen)n(tary)24 b(h)n(yp)r(erb)r(olic)h (groups;)-118 3612 y(and)i(man)n(y)f(other)h(kno)n(wn)g(groups,)g(whic) n(h)f(are)h(not)g(amenable.)6 3712 y(The)34 b(structure)g(of)g(quotien) n(t)f(groups)f(for)i(the)g(groups)f(listed)f(ab)r(o)n(v)n(e)h(is)-118 3811 y(a)28 b(sub)5 b(ject)29 b(the)h(authors)d(are)h(not)h(familiar)24 b(enough)29 b(with;)g(th)n(us)g(w)n(e)f(cannot)-118 3911 y(estimate)d(prop)r(erly)h(ho)n(w)h(complicated)d(the)k(listed)e(tasks) h(are.)p eop %%Page: 229 233 229 232 bop -118 -137 a FJ(3.2.)36 b(Classes)25 b(of)j(non-self-adjoin) n(t)c(op)r(erators)854 b FO(229)-118 96 y FQ(4.)36 b FO(Consider)26 b(some)g(examples)f(of)i(groups)f(whic)n(h)h(are)f(not)i FP(\003)p FO(-wild.)6 196 y(If)41 b FN(G)f FO(is)e(amenable,)j(then)f FN(C)976 166 y FM(\003)1014 196 y FO(\()p FN(G)p FO(\))h(is)e(n)n (uclear.)71 b(Th)n(us,)42 b(w)n(e)e(ha)n(v)n(e)e(the)-118 296 y(follo)n(wing)23 b(statemen)n(t.)-118 447 y FQ(Prop)s(osition)30 b(72.)41 b FB(If)30 b FN(G)g FB(is)g(an)g(amenable)h(gr)l(oup,)f(then)g FN(G)g FB(is)g(not)f FP(\003)p FB(-wild.)6 598 y FO(Also,)e(the)h (follo)n(wing)23 b(theorem)j(holds.)-118 749 y FQ(Theorem)k(62.)41 b FB(L)l(et)27 b FN(G)g FB(b)l(e)h(a)f(p)l(erio)l(dic)j(gr)l(oup,)e (i.e.,)i(any)d(element)h FN(g)d FP(2)f FN(G)j FB(is)-118 849 y(p)l(erio)l(dic.)41 b(Then)30 b FN(G)g FB(is)g(not)g FP(\003)p FB(-wild.)-118 1000 y(Pr)l(o)l(of.)43 b FO(Supp)r(ose)29 b(that)h FN(G)g FO(is)e FP(\003)p FO(-wild,)f(i.e.,)i(there)g(exists)f (a)h(homomorphism)-118 1100 y FN(\036)9 b FO(:)29 b FN(G)g FP(\000)-48 b(!)28 b FN(U)271 1112 y FL(n)316 1100 y FO(\()p FN(C)413 1070 y FM(\003)452 1100 y FO(\()p FA(F)540 1112 y FK(2)575 1100 y FO(\)\))k(suc)n(h)f(that)g(the)g(functor)g FN(F)21 b FO(:)43 b(Rep)14 b FA(F)1837 1112 y FK(2)1900 1100 y FP(\000)-48 b(!)29 b FO(Rep)14 b FN(G)31 b FO(is)-118 1199 y(full.)40 b(Consider)27 b(a)h(family)e(of)j(one-dimensional)24 b(represen)n(tations)i FN(h)2069 1211 y FL(t)2127 1199 y FO(of)j(the)-118 1299 y(group)i FA(F)178 1311 y FK(2)243 1299 y FO(=)e FP(h)p FN(u;)14 b(v)s FP(i)33 b FO(in)e(the)h(space)f FI(C)53 b FO(suc)n(h)31 b(that)h FN(h)1551 1311 y FL(t)1580 1299 y FO(\()p FN(u)p FO(\))e(=)g(1,)j FN(h)1963 1311 y FL(t)1992 1299 y FO(\()p FN(v)s FO(\))e(=)e FN(e)2263 1269 y FL(it)2316 1299 y FO(,)-118 1408 y FN(t)23 b FP(2)h FO(\(0)p FN(;)14 b FO(2)p FN(\031)s FO(],)27 b(and)h(denote)g(b)n(y)f FN(U)892 1420 y FL(t)921 1408 y FO(\()p FN(g)s FO(\))d(=)1140 1386 y(^)1140 1408 y FN(h)1188 1420 y FL(t)1235 1408 y FP(\016)18 b FN(\036)p FO(\()p FN(g)s FO(\),)29 b FN(g)d FP(2)e FN(G)p FO(,)k(the)g(matrix)d(with)-118 1507 y(en)n(tries)c(that) i(are)f(con)n(tin)n(uous)f(functions)h(in)h FN(t)p FO(.)35 b(Since)22 b(the)h(functor)g FN(F)35 b FO(is)22 b(full,)-118 1607 y(the)30 b(represen)n(tations)606 1585 y(^)605 1607 y FN(h)653 1619 y FL(t)678 1627 y Fx(1)735 1607 y FP(\016)19 b FN(\036)31 b FO(and)1041 1585 y(^)1040 1607 y FN(h)1088 1619 y FL(t)1113 1627 y Fx(2)1169 1607 y FP(\016)20 b FN(\036)30 b FO(of)g(the)h(group)e FN(G)h FO(are)f(unitarily)-118 1707 y(inequiv)-5 b(alen)n(t)27 b(for)i(all)f FN(t)625 1719 y FK(1)689 1707 y FP(6)p FO(=)f FN(t)811 1719 y FK(2)848 1707 y FO(.)45 b(Since)29 b(the)h(irreducible)d(represen)n (tations)g(of)-118 1806 y(the)c(group)f FN(G)h FO(in)g(a)f (\014nite-dimensional)c(space)k(are)g(uniquely)f(de\014ned,)j(up)g(to) -118 1906 y(a)30 b(unitary)g(equiv)-5 b(alence,)29 b(b)n(y)i(their)f(c) n(haracters)f(\(see,)j(for)e(example,)g([132)n(]\),)-118 2005 y(there)i(exists)g FN(g)i FP(2)e FN(G)h FO(suc)n(h)g(that)g(T)-7 b(r)13 b FN(U)1127 2017 y FL(t)1152 2025 y Fx(1)1189 2005 y FO(\()p FN(g)s FO(\))31 b FP(6)p FO(=)h(T)-7 b(r)13 b FN(U)1580 2017 y FL(t)1605 2025 y Fx(2)1641 2005 y FO(\()p FN(g)s FO(\).)53 b(Then)33 b(T)-7 b(r)14 b FN(U)2203 2017 y FL(t)2231 2005 y FO(\()p FN(g)s FO(\))-118 2105 y(is)35 b(a)h(con)n(tin)n(uous)e(function)i(in)g FN(t)p FO(,)j(whic)n(h)c(is)g(not)i(a)f(constan)n(t.)62 b(Order)35 b(the)-118 2205 y(eigen)n(v)-5 b(alues)27 b FN(k)357 2217 y FL(i)384 2205 y FO(\()p FN(t)p FO(\),)32 b FN(i)26 b FO(=)g(1,)k FN(:)14 b(:)g(:)27 b FO(,)j FN(n)p FO(,)h(of)f(the)g (matrix)d FN(U)1626 2217 y FL(t)1655 2205 y FO(\()p FN(g)s FO(\))j(b)n(y)g(the)g(v)-5 b(alue)29 b(of)-118 2304 y(the)35 b(argumen)n(t.)55 b(Then)35 b(there)f(exists)g FN(i)g FO(suc)n(h)g(that)h FN(k)1623 2316 y FL(i)1651 2304 y FO(\()p FN(t)p FO(\))g FP(6)p FO(=)f(const.)57 b(Since)-118 2404 y(for)38 b(a)g(unitary)g(matrix)e(the)j(problem)d(of)j(\014nding)f (eigen)n(v)-5 b(alues)35 b(is)j(stable,)-118 2504 y(the)26 b(eigen)n(v)-5 b(alue)22 b FN(k)460 2516 y FL(i)488 2504 y FO(\()p FN(t)p FO(\))k(is)e(a)h(non-trivial)c(con)n(tin)n(uous)i (function)i(of)g FN(t)g FO(on)g(some)-118 2603 y(in)n(terv)-5 b(al)19 b(\()p FN(t)238 2615 y FK(1)275 2603 y FN(;)14 b(t)342 2615 y FK(2)379 2603 y FO(\).)35 b(But)22 b(then)g(there)f (exists)f FN(t)1270 2615 y FK(0)1328 2603 y FO(suc)n(h)h(that)h FN(k)1726 2615 y FL(i)1754 2603 y FO(\()p FN(t)1816 2615 y FK(0)1853 2603 y FO(\))g(is)e(not)i(a)e(ro)r(ot)-118 2703 y(of)30 b(unit)n(y)-7 b(.)44 b(Since)29 b FN(G)h FO(is)f(a)h(p)r(erio)r(dic)e(group,)i(for)f(eac)n(h)h FN(g)f FP(2)f FN(G)i FO(there)g(exists)f(a)-118 2814 y(p)r(o)n(w)n(er)d FN(N)9 b FO(\()p FN(g)s FO(\))27 b(suc)n(h)g(that)g FN(g)743 2784 y FL(N)6 b FK(\()p FL(g)r FK(\))915 2814 y FO(=)23 b FN(e)p FO(;)k(but)h(on)f(the)g(other)g(hand,)g FN(U)2014 2771 y FL(N)6 b FK(\()p FL(g)r FK(\))2005 2834 y FL(t)2030 2842 y Fx(0)2186 2814 y FP(6)p FO(=)23 b(1.)-118 2914 y(The)28 b(obtained)e(con)n(tradiction)e(completes)h(the)j(pro)r (of.)p 2278 2914 4 57 v 2282 2861 50 4 v 2282 2914 V 2331 2914 4 57 v -118 3076 a FQ(Corollary)k(17.)40 b FB(Ther)l(e)23 b(ar)l(e)f(gr)l(oups)g(which)h(ar)l(e)f(not)g FP(\003)p FB(-wild)g(and)g(not)g(amen-)-118 3176 y(able.)37 b(Those)24 b(ar)l(e,)h(for)e(example,)i(Burnside)e(gr)l(oups)g FN(B)t FO(\()p FN(m;)14 b(n)p FO(\))23 b FB(which)h(ar)l(e)f(not)-118 3275 y(amenable)31 b(for)f(o)l(dd)h FN(n)23 b FP(\025)g FO(665)28 b FB(and)i FN(m)23 b FP(\025)g FO(2)29 b(\()p FB(se)l(e)36 b FO([180)o(,)28 b(2)o(]\))p FB(.)-118 3513 y FG(3.2)112 b(On)26 b(the)e(complexit)m(y)i(of)f(the)g(description)h (of)f(classes)137 3630 y(of)38 b(non-self-adjoin)m(t)i(op)s(erators) -118 3811 y FO(The)35 b(b)r(orderline)d(b)r(et)n(w)n(een)j(the)g (theory)f(of)h(op)r(erators)e(and)h(the)i(theory)e(of)-118 3911 y(op)r(erator)25 b(algebras)e(and)j(their)f(represen)n(tations)f (can)i(b)r(e)h(view)n(ed)d(as)i(a)g(riv)n(er)p eop %%Page: 230 234 230 233 bop -118 -137 a FO(230)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FO(with)d(n)n(umerous)g(t)n(w)n(o-w)n(a)n(y)e(bridges)h(joining)g (the)j(banks)e(\(see,)h(for)g(example,)-118 196 y([79)o(],)c(and)e (others\).)34 b(One)21 b(of)f(these)h(bridges)d(is)h(discussed)h(in)f (this)h(section:)32 b(w)n(e)-118 296 y(consider)21 b(an)i(application)d (of)j(the)g(theory)g(of)g(represen)n(tations)d(of)k FP(\003)p FO(-algebras)-118 395 y(to)j(a)h(study)f(of)h(classes)d(of)j(op)r (erators)d(that)j(are)f(singled)e(out)j(algebraicall)o(y)-7 b(.)6 495 y(Let)24 b FN(X)31 b FO(b)r(e)24 b(a)f(b)r(ounded)h (non-self-adjoin)n(t)d(op)r(erator)h(acting)h(in)g(a)g(Hilb)r(ert)-118 595 y(space)k FN(H)7 b FO(.)39 b(W)-7 b(e)29 b(consider)d(classes)g(of) j(op)r(erators)d(whic)n(h)h(satisfy)g(p)r(olynomial)-118 694 y(relations)d FN(P)272 706 y FL(j)308 694 y FO(\()p FN(X)r(;)14 b(X)524 664 y FM(\003)561 694 y FO(\))23 b(=)g(0,)k FN(j)h FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)27 b(;)14 b(m)p FO(,)27 b(and)h(more)d(general)h(relations.)6 794 y(F)-7 b(or)36 b(ev)n(ery)g(suc)n(h)g(class)e(of)j(relations)c (there)k(corresp)r(onds)d(a)i FP(\003)p FO(-algebra)-118 893 y Fz(A)i FO(=)g FI(C)15 b FP(h)p FN(x;)f(x)301 863 y FM(\003)383 893 y FP(j)39 b FN(P)498 905 y FL(j)533 893 y FO(\()p FN(x;)14 b(x)696 863 y FM(\003)735 893 y FO(\))39 b(=)f(0)p FN(;)27 b(j)43 b FO(=)38 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m)p FP(i)p FO(.)65 b(If)37 b(the)g(class)e(of)i(op)r (er-)-118 993 y(ators)31 b(is)h(giv)n(en)f(b)n(y)i(non-p)r(olynomial)28 b(relations,)j(then)i(the)h(corresp)r(onding)-118 1093 y FP(\003)p FO(-algebra)20 b(is)k(giv)n(en)e(in)h(a)h(more)e (complicated)f(w)n(a)n(y)-7 b(.)35 b(Eac)n(h)23 b(represen)n(tation)e FN(\031)-118 1192 y FO(of)27 b(the)g FP(\003)p FO(-algebra)c Fz(A)j FO(determines)f(the)i(b)r(ounded)g(op)r(erators)e FN(X)k FO(=)23 b FN(\031)s FO(\()p FN(x)p FO(\))28 b(and)-118 1292 y FN(X)-42 1262 y FM(\003)18 1292 y FO(=)23 b FN(\031)s FO(\()p FN(x)235 1262 y FM(\003)274 1292 y FO(\))28 b(suc)n(h)f(that) 510 1471 y FN(P)563 1483 y FL(j)598 1471 y FO(\()p FN(X)r(;)14 b(X)814 1437 y FM(\003)852 1471 y FO(\))23 b(=)g(0)p FN(;)179 b(j)28 b FO(=)23 b(1)p FN(;)14 b(:)g(:)g(:)f(;)h(m:)457 b FO(\(3.6\))6 1650 y(Con)n(v)n(ersely)-7 b(,)24 b(giv)n(en)g(op)r (erators)h FN(X)32 b FO(and)26 b FN(X)1361 1620 y FM(\003)1425 1650 y FO(suc)n(h)g(that)g FN(P)1842 1662 y FL(j)1878 1650 y FO(\()p FN(X)r(;)14 b(X)2094 1620 y FM(\003)2131 1650 y FO(\))23 b(=)g(0,)-118 1749 y FN(j)28 b FO(=)23 b(1,)i FN(:)14 b(:)g(:)27 b FO(,)f FN(m)p FO(,)g(one)f(can)g(uniquely)f (de\014ne)i(a)f(represen)n(tation)e(of)i(the)h(whole)-118 1849 y(algebra)f Fz(A)p FO(.)6 1949 y(The)30 b(problem)e(to)h(describ)r (e)g(the)h(class)e(of)i(op)r(erators)d(whic)n(h)i(satisfy)f(re-)-118 2048 y(lations)23 b(\(3.6\),)j(up)h(to)e(a)h(unitary)e(equiv)-5 b(alence,)24 b(is)h(equiv)-5 b(alen)n(t)24 b(to)i(the)g(one)g(of)-118 2148 y(describing)f(represen)n(tations)f(of)k(the)g(corresp)r(onding)c FP(\003)p FO(-algebra)g Fz(A)p FO(.)6 2248 y(F)-7 b(or)24 b(suc)n(h)g(algebras,)d(w)n(e)j(estimate)e(the)j(complexit)n(y)20 b(of)k(the)h(corresp)r(ond-)-118 2347 y(ing)31 b(problem)f(of)j(the)g FP(\003)p FO(-represen)n(tations)c(theory)-7 b(,)33 b(i.e.,)g(the)g (complexit)n(y)c(of)-118 2447 y(the)f(unitary)e(description)f(of)j(the) g(corresp)r(onding)c(class)i(of)h(op)r(erators.)6 2546 y(W)-7 b(e)26 b(consider)c(classes)h(of)i(op)r(erators)e(connected)h (with)h(quadratic,)e(semi-)-118 2646 y(linear)18 b(cubic,)j(and)f(some) g(other)g(relations)d(\(Section)j(3.2.1\).)33 b(Then)21 b(w)n(e)f(study)-118 2746 y(complexit)n(y)31 b(of)k(the)g(unitary)e (description)f(of)j(partial)c(isometries,)i(w)n(eakly)-118 2845 y(cen)n(tered)c(op)r(erators,)f(and)h(algebraic)d(op)r(erators)h (\(Section)i(3.2.2\).)41 b(In)30 b(Sec-)-118 2945 y(tion)23 b(3.2.3,)h(w)n(e)g(sp)r(eak)f(ab)r(out)i(the)f(complexit)n(y)d(of)j (description)e(of)i(classes)e(of)-118 3045 y(op)r(erators)j(whic)n(h)h (are)g(de\014ned)h(not)g(b)n(y)f(p)r(olynomial)d(equalities)g(but)28 b(rather)-118 3144 y(b)n(y)34 b(conditions)e(similar)e(to)35 b(inequalities,)d(or)i(other)g(non-algebraic)c(condi-)-118 3244 y(tions;)f(namely)-7 b(,)28 b(w)n(e)h(consider)f(h)n(yp)r(onormal) e(op)r(erators)h(and)i(pairs)f(of)h(com-)-118 3343 y(m)n(uting)d (completely)e(non-unitary)i(isometries.)-118 3559 y FQ(3.2.1)94 b(Classes)35 b(of)i(non-self-adjoin)m(t)f(op)s(erators)h(singled)f(out) h(b)m(y)174 3658 y(a)32 b(quadratic)h(or)f(a)g(cubic)h(relation)-118 3811 y FO(A)20 b(normal)d(op)r(erator)h FN(X)27 b FO(is)18 b(an)i(op)r(erator)e(suc)n(h)i(that)g FN(X)7 b(X)1675 3781 y FM(\003)1735 3811 y FO(=)22 b FN(X)1898 3781 y FM(\003)1936 3811 y FN(X)7 b FO(.)34 b(Normal)-118 3911 y(op)r(erators)e(mak)n(e)h(the)h(most)g(studied)f(region)f(in)i(the)h (terrain)d(of)i(b)r(ounded)p eop %%Page: 231 235 231 234 bop -118 -137 a FJ(3.2.)36 b(Classes)25 b(of)j(non-self-adjoin) n(t)c(op)r(erators)854 b FO(231)-118 96 y(linear)27 b(op)r(erators.)40 b(Irreducible)27 b(normal)g(op)r(erators)g(are)i(one-dimensional.)-118 196 y(The)j(sp)r(ectral)e(theorem)g(giv)n(es)f(a)j(pro)r(cedure)e(for)h (assem)n(bling)d(an)n(y)j(normal)-118 296 y(op)r(erator)26 b(from)g(irreducible)e(ones.)-118 445 y FQ(1.)33 b FO(Let)19 b(us)g(ha)n(v)n(e)e(a)h(pair)f(of)i(op)r(erators)e FN(X)25 b FO(and)18 b FN(X)1399 415 y FM(\003)1456 445 y FO(whic)n(h)f(satisfy) g(a)i(quadratic)-118 545 y(relation)25 b(of)i(the)h(form)585 728 y FN(P)638 740 y FK(2)675 728 y FO(\()p FN(X)r(;)14 b(X)891 693 y FM(\003)929 728 y FO(\))23 b(=)g FN(P)1137 693 y FM(\003)1125 748 y FK(2)1175 728 y FO(\()p FN(X)r(;)14 b(X)1391 693 y FM(\003)1428 728 y FO(\))23 b(=)g(0)p FN(:)532 b FO(\(3.7\))-118 910 y(A)28 b(common)d(form)h(of)i(suc)n(h)f (a)g(relation)e(is)h(the)i(follo)n(wing:)21 1093 y FN(P)74 1105 y FK(2)112 1093 y FO(\()p FN(X)r(;)14 b(X)328 1059 y FM(\003)365 1093 y FO(\))24 b(=)e FN(a)p FO(\()p FN(X)660 1059 y FK(2)715 1093 y FO(+)c(\()p FN(X)906 1059 y FM(\003)944 1093 y FO(\))976 1059 y FK(2)1014 1093 y FO(\))g(+)g FN(i)1176 1059 y FM(\000)p FK(1)1265 1093 y FN(b)p FO(\()p FN(X)1409 1059 y FK(2)1464 1093 y FP(\000)g FO(\()p FN(X)1655 1059 y FM(\003)1693 1093 y FO(\))1725 1059 y FK(2)1762 1093 y FO(\))h(+)f FN(c)p FO([)p FN(X)r(;)c(X)2139 1059 y FM(\003)2176 1093 y FO(])72 1228 y(+)k FN(d)p FP(f)p FN(X)r(;)c(X)424 1194 y FM(\003)460 1228 y FP(g)k FO(+)g FN(e)p FO(\()p FN(X)25 b FO(+)18 b FN(X)927 1194 y FM(\003)965 1228 y FO(\))g(+)g FN(i)1127 1194 y FM(\000)p FK(1)1216 1228 y FN(f)9 b FO(\()p FN(X)25 b FP(\000)18 b FN(X)1551 1194 y FM(\003)1588 1228 y FO(\))h(+)f FN(g)s(I)30 b FO(=)22 b(0;)185 b(\(3.8\))-118 1411 y(here)28 b FN(a)p FO(,)h FN(b)p FO(,)g FN(c)p FO(,)g FN(d)p FO(,)h FN(e)p FO(,)f FN(f)9 b FO(,)28 b FN(g)g FP(2)d FI(R)p FO(.)47 b(W)-7 b(e)29 b(no)n(w)f(giv)n(e)f(a)h(criterion)e(for)i(the)i (relation)-118 1510 y(\(that)36 b(is)f(the)h(corresp)r(onding)d FP(\003)p FO(-algebra\))g(to)i(b)r(e)i FP(\003)p FO(-wild)c(in)i(terms) f(of)i(the)-118 1610 y(co)r(e\016cien)n(ts.)-118 1776 y FQ(Theorem)30 b(63.)41 b FB(The)35 b(c)l(orr)l(esp)l(onding)g FP(\003)p FB(-algebr)l(a)g(is)f FP(\003)p FB(-wild)h(if)g(and)f(only)h (if)-118 1876 y(one)30 b(of)g(the)g(fol)t(lowing)j(c)l(onditions)d (hold)9 b FO(:)-17 2042 y(1)p FN(:)41 b(a)23 b FO(=)g FN(b)g FO(=)f FN(c)h FO(=)g FN(d)g FO(=)g FN(e)g FO(=)f FN(f)32 b FO(=)23 b FN(g)i FO(=)e(0;)-17 2257 y(2)p FN(:)89 2165 y Fy(\020)139 2257 y FN(g)e FP(\000)402 2201 y FN(e)441 2171 y FK(2)p 293 2238 295 4 v 293 2314 a FO(2\()p FN(a)d FO(+)g FN(d)p FO(\))598 2165 y Fy(\021)647 2257 y FO(\()p FN(a)h FO(+)f FN(d)p FO(\))23 b FN(<)g FO(0)p FB(,)85 b FN(d)19 b FP(\000)f FN(a)23 b FO(=)f FN(b)h FO(=)f FN(c)h FO(=)g FN(f)32 b FO(=)22 b(0;)-17 2534 y(3)p FN(:)89 2441 y Fy(\020)139 2534 y FN(g)f FP(\000)397 2477 y FN(f)447 2447 y FK(2)p 293 2514 V 293 2590 a FO(2\()p FN(d)d FP(\000)h FN(a)p FO(\))598 2441 y Fy(\021)647 2534 y FO(\()p FN(d)g FP(\000)f FN(a)p FO(\))23 b FN(<)g FO(0)p FB(,)85 b FN(a)18 b FO(+)g FN(d)24 b FO(=)e FN(b)h FO(=)f FN(c)h FO(=)g FN(e)g FO(=)f(0;)-17 2810 y(4)p FN(:)41 b(b)125 2776 y FK(2)185 2810 y FO(=)23 b(\()p FN(d)348 2776 y FK(2)404 2810 y FP(\000)18 b FN(a)531 2776 y FK(2)568 2810 y FO(\))24 b FP(6)p FO(=)e(0)p FN(;)99 b FO(\()p FN(a)18 b FO(+)g FN(d)p FO(\))1127 2718 y Fy(\020)1177 2810 y FN(g)j FP(\000)1441 2754 y FN(e)1480 2724 y FK(2)p 1331 2791 V 1331 2867 a FO(2\()p FN(a)d FO(+)h FN(d)p FO(\))1636 2718 y Fy(\021)1709 2810 y FN(<)j FO(0)p FB(,)188 2997 y FN(e)227 2967 y FK(2)p 99 3034 254 4 v 99 3110 a FO(\()p FN(a)d FO(+)f FN(d)p FO(\))386 3053 y(=)566 2997 y FN(f)616 2967 y FK(2)p 483 3034 V 483 3110 a FO(\()p FN(d)h FP(\000)f FN(a)p FO(\))746 3053 y FB(,)115 b FN(c)23 b FO(=)g(0)p FB(.)6 3263 y FO(This)c(theorem)f(follo)n(ws)f(from)h(Theorem)g(60,)i (b)n(y)g(the)g(c)n(hange)e(of)i(v)-5 b(ariables)-118 3363 y FN(X)29 b FO(=)23 b FN(A)7 b FO(+)g FN(iB)t FO(,)24 b FN(X)428 3333 y FM(\003)488 3363 y FO(=)f FN(A)7 b FP(\000)g FN(iB)t FO(.)36 b(It)22 b(is)f(easy)g(to)h(see)g(that)g(the)h (relation)c(satis\014ed)-118 3463 y(b)n(y)32 b(the)h(co)r(e\016cien)n (ts)e(is)g(the)i(follo)n(wing:)42 b FN(\013)31 b FO(=)g FN(a)22 b FO(+)f FN(d)p FO(,)34 b FN(\014)h FO(=)c FN(d)21 b FP(\000)h FN(a)p FO(,)33 b FN(\015)j FO(=)30 b(2)p FN(b)p FO(,)-118 3562 y FN(q)c FO(=)d(2)p FN(c)p FO(,)k FN(\017)c FO(=)f(2)p FN(f)9 b FO(,)27 b FN(\037)c FO(=)g FN(g)s FO(.)-118 3712 y FQ(2.)52 b FO(No)n(w)32 b(w)n(e)h(will)d (consider)h(some)g(classes)g(of)i(non-self-adjoin)n(t)d(op)r(erators) -118 3811 y(whic)n(h)d(satisfy)f(a)i(cubic)f(relation.)34 b(A)n(t)28 b(\014rst)g(w)n(e)g(will)d(pass)i(to)g(a)h(pair)e(of)i (self-)-118 3911 y(adjoin)n(t)23 b(op)r(erators,)f FN(A)p FO(,)k FN(B)t FO(,)e(b)n(y)g(the)h(c)n(hange)d(of)i(v)-5 b(ariables,)22 b FN(X)29 b FO(=)23 b FN(A)11 b FO(+)g FN(iB)28 b FO(and)p eop %%Page: 232 236 232 235 bop -118 -137 a FO(232)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FN(X)-42 66 y FM(\003)18 96 y FO(=)23 b FN(A)13 b FP(\000)g FN(iB)t FO(.)35 b(Let)25 b(the)h(self-adjoin)n(t)c(op)r (erators)h FN(A)i FO(and)g FN(B)k FO(satisfy)23 b(a)h(cubic)-118 196 y(semi-linear)e(relation)j(\(linear)g(in)i FN(B)t FO(\).)37 b(The)28 b(usual)e(form)h(of)g(suc)n(h)g(a)h(relation)-118 296 y(with)f(the)h(condition)d FN(P)631 308 y FK(3)669 296 y FO(\()p FN(A;)14 b(B)t FO(\))24 b(=)e FN(P)1075 266 y FM(\003)1063 316 y FK(3)1113 296 y FO(\()p FN(A;)14 b(B)t FO(\))29 b(is)e(the)g(follo)n(wing:)209 480 y FN(P)262 492 y FK(3)299 480 y FO(\()p FN(A;)14 b(B)t FO(\))24 b(=)f FN(\013B)g FO(+)18 b(2)p FN(\014)t FP(f)p FN(A;)c(B)t FP(g)k FO(+)g FN(\017)p FP(f)p FN(A)1445 446 y FK(2)1481 480 y FN(;)c(B)t FP(g)k FO(+)g(2)p FN(\026AB)t(A)629 615 y FO(+)g FN(i\015)5 b FO([)p FN(A;)14 b(B)t FO(])k(+)g FN(i\016)s FO([)p FN(A)1256 580 y FK(2)1294 615 y FN(;)c(B)t FO(])23 b(=)f(0)p FN(;)572 b FO(\(3.9\))-118 799 y FN(\013)p FO(,)28 b FN(\014)t FO(,)g FN(\015)5 b FO(,)27 b FN(\016)g FP(2)c FI(R)382 769 y FK(1)425 799 y FO(.)6 899 y(It)33 b(is)e(easy)h(to)g(see)g(that,)h(in)f(terms)f(of)h FN(X)39 b FO(and)32 b FN(X)1617 869 y FM(\003)1654 899 y FO(,)i(the)f(relation) c(\(3.9\))-118 999 y(tak)n(es)e(the)h(form:)17 1183 y FN(P)70 1195 y FK(3)108 1183 y FO(\()p FN(X)r(;)14 b(X)324 1149 y FM(\003)361 1183 y FO(\))23 b(=)g FN(i)533 1149 y FM(\000)p FK(1)622 1183 y FN(a)p FO(\()p FN(X)i FP(\000)18 b FN(X)951 1149 y FM(\003)988 1183 y FO(\))h(+)f FN(i)1151 1149 y FM(\000)p FK(1)1240 1183 y FN(b)p FO(\()p FN(X)1384 1149 y FK(2)1439 1183 y FP(\000)g FO(\()p FN(X)1630 1149 y FM(\003)1667 1183 y FO(\))1699 1149 y FK(2)1737 1183 y FO(\))412 1318 y(+)g FN(i)524 1283 y FM(\000)p FK(1)613 1318 y FO(\()p FN(c)g FO(+)g FN(d)p FO(\)\()p FN(X)965 1283 y FK(3)1021 1318 y FP(\000)g FO(\()p FN(X)1212 1283 y FM(\003)1250 1318 y FO(\))1282 1283 y FK(3)1319 1318 y FO(\))412 1453 y(+)g FN(i)524 1418 y FM(\000)p FK(1)613 1453 y FO(\()p FN(c)g FP(\000)g FN(d)p FO(\)\()p FN(X)7 b(X)1041 1418 y FM(\003)1079 1453 y FN(X)25 b FP(\000)18 b FN(X)1332 1418 y FM(\003)1369 1453 y FN(X)7 b(X)1521 1418 y FM(\003)1558 1453 y FO(\))412 1587 y(+)18 b FN(i)524 1553 y FM(\000)p FK(1)613 1587 y FN(d)656 1520 y Fy(\000)694 1587 y FP(f)p FN(X)812 1553 y FK(2)848 1587 y FN(;)c(X)961 1553 y FM(\003)998 1587 y FP(g)k FO(+)g FP(f)p FN(X)r(;)c FO(\()p FN(X)1399 1553 y FM(\003)1436 1587 y FO(\))1468 1553 y FK(2)1506 1587 y FP(g)1548 1520 y Fy(\001)412 1722 y FO(+)k FN(f)9 b FO([)p FN(X)r(;)14 b(X)752 1688 y FM(\003)789 1722 y FO(])k(+)g FN(g)956 1655 y Fy(\000)994 1722 y FO([)p FN(X)1093 1688 y FM(\003)1130 1722 y FN(;)c(X)1243 1688 y FK(2)1280 1722 y FO(])k(+)g([\()p FN(X)1535 1688 y FM(\003)1573 1722 y FO(\))1605 1688 y FK(2)1643 1722 y FN(;)c(X)7 b FO(])1779 1655 y Fy(\001)1839 1722 y FO(=)23 b(0)p FN(;)148 b FO(\(3.10\))-118 1906 y(where)27 b FN(a)c FO(=)g FN(\013=)p FO(2,)k FN(b)22 b FO(=)h FN(\014)t FO(,)28 b FN(c)23 b FO(=)g FN(\017=)p FO(4,)j FN(d)d FO(=)g FN(\026=)p FO(4,)k FN(f)32 b FO(=)22 b FN(\015)5 b(=)p FO(2,)27 b FN(g)e FO(=)e FN(\016)s(=)p FO(4.)6 2007 y(W)-7 b(rite)37 b FN(I)283 2019 y FK(1)360 2007 y FO(=)i(8)p FN(c)p FO(,)g FN(I)640 2019 y FK(2)717 2007 y FO(=)g FN(c)857 1977 y FK(2)919 2007 y FP(\000)25 b FO(4)p FN(d)1094 1977 y FK(2)1131 2007 y FO(,)40 b FN(I)1230 2019 y FK(3)1307 2007 y FO(=)e FN(a)p FO(\()p FN(c)1522 1977 y FK(2)1585 2007 y FP(\000)24 b FO(4)p FN(d)1759 1977 y FK(2)1796 2007 y FO(\))h FP(\000)g FN(b)1979 1977 y FK(2)2016 2007 y FO(\()p FN(c)g FP(\000)g FO(4)p FN(d)p FO(\),)-118 2106 y FN(I)-82 2118 y FK(4)-21 2106 y FO(=)d(\(2)p FN(g)s(b)5 b FP(\000)g FN(cf)k FO(\);)21 b(then)g(the)g(follo)n(wing)c (theorem)i(follo)n(ws)e(from)j(Theorem)e(61.)-118 2275 y FQ(Theorem)30 b(64.)41 b FB(R)l(elation)c FO(\(3.10\))30 b FB(with)i(the)f(c)l(onditions)h FN(c)24 b FP(\025)h FO(0)p FB(,)31 b FN(c)2034 2245 y FK(2)2091 2275 y FO(+)18 b FN(d)2217 2245 y FK(2)2274 2275 y FO(+)-118 2374 y FN(g)-75 2344 y FK(2)-15 2374 y FP(6)p FO(=)k(0)27 b FB(is)g FP(\003)p FB(-wild)g(if)h(and)g(only)f(if)h(one)f(of)h(the)f (fol)t(lowing)i(c)l(ases)34 b FO(1\){3\))26 b FB(holds)7 b FO(:)-26 2543 y(1\))41 b FN(f)32 b FO(=)23 b FN(g)i FO(=)e(0)p FB(,)30 b(and)g(one)g(of)h(the)e(fol)t(lowing)k(holds)7 b FO(:)122 2712 y(\()p FN(a)p FO(\))42 b FN(I)308 2724 y FK(1)369 2712 y FN(>)23 b FO(0)p FB(,)29 b FN(I)589 2724 y FK(2)650 2712 y FN(>)23 b FO(0)p FB(,)29 b FN(I)870 2724 y FK(3)931 2712 y FN(<)23 b FO(0)p FB(,)130 2846 y FO(\()p FN(b)p FO(\))42 b FN(I)308 2858 y FK(2)369 2846 y FN(<)23 b FO(0)p FB(,)29 b FN(I)589 2858 y FK(3)650 2846 y FO(=)23 b(0)p FB(,)130 2980 y FO(\()p FN(c)p FO(\))42 b FN(I)308 2992 y FK(1)369 2980 y FN(>)23 b FO(0)p FB(,)29 b FN(I)589 2992 y FK(2)650 2980 y FN(<)23 b FO(0)p FB(,)29 b FN(I)870 2992 y FK(3)931 2980 y FP(6)p FO(=)23 b(0)p FB(,)123 3115 y FO(\()p FN(d)p FO(\))42 b FN(I)308 3127 y FK(2)369 3115 y FO(=)23 b(0)p FB(,)29 b FN(I)589 3127 y FK(3)650 3115 y FO(=)23 b(0)p FB(,)29 b FN(b)870 3084 y FK(2)926 3115 y FP(\000)18 b FO(2)p FN(ac)k(>)h FO(0)p FB(,)127 3249 y FO(\()p FN(e)p FO(\))42 b FN(I)308 3261 y FK(2)369 3249 y FO(=)23 b(0)p FB(,)29 b FN(I)589 3261 y FK(3)650 3249 y FP(6)p FO(=)23 b(0)p FB(.)-26 3418 y FO(2\))41 b FN(c)23 b FO(=)g FN(d)g FO(=)g FN(b)g FO(=)f FN(a)h FO(=)g(0)p FB(,)30 b FO(\()p FB(then)f FN(g)d FP(6)p FO(=)d(0\))p FB(.)-26 3587 y FO(3\))41 b FN(g)s FO(\()p FN(c)200 3557 y FK(2)256 3587 y FO(+)18 b FN(d)382 3557 y FK(2)419 3587 y FO(\))24 b FP(6)p FO(=)e(0)30 b FB(and)g(one)g(of)g(the)g(fol)t(lowing)j(c)l(onditions)d(holds)7 b FO(:)122 3756 y(\()p FN(a)p FO(\))42 b FN(c)23 b(>)g FO(0)p FB(,)29 b FN(d)24 b FO(=)e(0)p FB(,)30 b FN(g)s(b)18 b FP(\000)g FO(2)p FN(cf)31 b FO(=)22 b(0)p FB(,)30 b FO(2)p FN(ag)1409 3726 y FK(2)1464 3756 y FP(\000)18 b FN(cf)1633 3726 y FK(2)1693 3756 y FN(<)k FO(0)p FB(,)130 3890 y FO(\()p FN(b)p FO(\))42 b FN(d)23 b FP(6)p FO(=)g(0)p FN(;)14 b(I)541 3902 y FK(4)601 3890 y FP(6)p FO(=)23 b(0)p FB(,)30 b FN(ac)866 3860 y FK(2)921 3890 y FP(\000)18 b FN(I)1040 3902 y FK(3)1096 3890 y FP(\000)g FO(\()p FN(d)-14 b(f)28 b FP(\000)18 b FN(g)s(b)g FP(\000)g FO(2)p FN(I)1650 3902 y FK(4)1687 3890 y FO(\)\(2)p FN(f)9 b(c)1879 3860 y FK(2)1916 3890 y FN(=g)2001 3860 y FK(2)2037 3890 y FO(\))23 b(=)g(0)p FB(.)p eop %%Page: 233 237 233 236 bop -118 -137 a FJ(3.2.)36 b(Classes)25 b(of)j(non-self-adjoin) n(t)c(op)r(erators)854 b FO(233)-118 96 y FQ(3.)56 b FO(A)35 b(kno)n(wn)e(class)f(of)j(quasi-normal)29 b(non-self-adjoin)n (t)i(op)r(erators)h(\([49)o(],)-118 196 y(see)d(also)d([102)o(]\))j(is) f(a)h(class)e(of)i(op)r(erators)e FN(X)35 b FO(whic)n(h)28 b(comm)n(ute)f(with)i FN(X)2203 166 y FM(\003)2240 196 y FN(X)7 b FO(.)-118 296 y(These)35 b(are)g(represen)n(tation)d(op)r (erators)i(of)i(the)f FP(\003)p FO(-algebra)d Fz(K)37 b FO(=)f FI(C)15 b FP(h)p FN(x)q(;)f(x)2235 266 y FM(\003)2316 296 y FP(j)-118 395 y FN(xx)-24 365 y FM(\003)15 395 y FN(x)23 b FO(=)g FN(x)220 365 y FM(\003)259 395 y FN(xx)p FP(i)p FO(.)35 b(It)21 b(follo)n(ws)c(from)i(the)h(relation)d([)p FN(x;)d(x)1566 365 y FM(\003)1605 395 y FN(x)p FO(])24 b(=)e(0)e(and)g(condition)-118 495 y FN(P)12 b FO(\()p FN(x;)i(x)110 465 y FM(\003)149 495 y FO(\))33 b(=)f FN(P)376 465 y FM(\003)414 495 y FO(\()p FN(x;)14 b(x)577 465 y FM(\003)616 495 y FO(\))33 b(that)h([)p FN(x)937 465 y FM(\003)976 495 y FN(;)14 b(x)1060 465 y FM(\003)1098 495 y FN(x)p FO(])33 b(=)f(0.)53 b(Therefore,)34 b(for)e(irreducible) -118 595 y(represen)n(tations)26 b(w)n(e)i(ha)n(v)n(e)g FN(X)851 564 y FM(\003)888 595 y FN(X)k FO(=)24 b FN(\025I)7 b FO(,)30 b FN(\025)c FP(\025)e FO(0.)41 b(Then)29 b(either)e FN(X)32 b FO(=)24 b FN(X)2211 564 y FM(\003)2274 595 y FO(=)-118 694 y(0,)33 b(or)f FN(\025)f(>)g FO(0)h(and)g FN(X)37 b FO(=)31 b FN(e)743 664 y FL(i\036)810 623 y FP(p)p 880 623 49 4 v 880 694 a FN(\025)p FO(,)j(or)d FN(X=)1202 623 y FP(p)p 1270 623 V 1270 694 a FN(\025)i FO(is)f(a)g(unilateral)c(shift.)51 b(There)-118 794 y(exists)26 b(a)h(corresp)r(onding)e(structure)i(theorem.)-118 942 y FQ(4.)43 b FO(No)n(w)30 b(w)n(e)g(consider)e(another)h(similar)c (class)j(of)i(non-self-adjoin)n(t)d(op)r(era-)-118 1041 y(tors)g FN(X)i FP(2)23 b FN(L)p FO(\()p FN(H)7 b FO(\))28 b(suc)n(h)f(that)h([)p FN(X)916 1011 y FK(2)953 1041 y FN(;)14 b(X)1066 1011 y FM(\003)1103 1041 y FO(])23 b(=)g(0,)k(i.e.,)817 1220 y FN(X)893 1185 y FK(2)930 1220 y FN(X)1006 1185 y FM(\003)1066 1220 y FO(=)c FN(X)1230 1185 y FM(\003)1268 1220 y FN(X)1344 1185 y FK(2)1380 1220 y FN(:)-118 1398 y FO(T)-7 b(aking)30 b(the)i(adjoin)n(ts,)f(w)n (e)g(get)h(\()p FN(X)1029 1368 y FM(\003)1066 1398 y FO(\))1098 1368 y FK(2)1136 1398 y FN(X)k FO(=)30 b FN(X)7 b FO(\()p FN(X)1520 1368 y FM(\003)1557 1398 y FO(\))1589 1368 y FK(2)1626 1398 y FO(.)49 b(Let)32 b FN(X)k FO(=)30 b FN(A)21 b FO(+)g FN(iB)t FO(,)-118 1498 y FN(A)30 b FO(=)f FN(A)130 1467 y FM(\003)168 1498 y FO(,)k FN(B)g FO(=)c FN(B)481 1467 y FM(\003)520 1498 y FO(.)48 b(Then)32 b(the)g(op)r(erators)d(in)i(this)g(class)e(are)i(selected)f(b)n(y)-118 1597 y(the)e(follo)n(wing)23 b(relation)718 1776 y([)p FN(A)803 1741 y FK(2)840 1776 y FN(;)14 b(B)t FO(])23 b(=)g([)p FN(B)t(;)14 b(A)1267 1741 y FK(2)1305 1776 y FO(])23 b(=)f(0)p FN(:)623 b FO(\(3.11\))-118 1954 y(Irreducible)36 b(represen)n(tations)f(of)k(a)f(pair)e FN(A)p FO(,)42 b FN(B)h FO(whic)n(h)37 b(satis\014es)g(relation)-118 2054 y(\(3.11\))o(,)32 b(are)e(one-)g(and)h(t)n(w)n(o-dimensional.)41 b(These)31 b(represen)n(tations,)e(up)i(to)-118 2153 y(a)37 b(unitary)g(equiv)-5 b(alence,)38 b(are)f(the)h(follo)n(wing:)54 b(one-dimensional)32 b FN(A)41 b FO(=)f FN(a)p FO(,)-118 2253 y FN(B)30 b FO(=)c FN(b)p FO(,)k FN(a)p FO(,)g FN(b)c FP(2)h FI(R)p FO(;)36 b(t)n(w)n(o-dimensional)24 b FN(A)j FO(=)f FN(a)1349 2186 y Fy(\000)1401 2222 y FK(1)49 b(0)1401 2272 y(0)23 b FM(\000)p FK(1)1555 2186 y Fy(\001)1594 2253 y FO(,)30 b FN(B)g FO(=)c FN(b)1867 2186 y Fy(\000)1919 2226 y FK(0)d(1)1919 2276 y(1)g(0)2022 2186 y Fy(\001)2060 2253 y FO(,)30 b FN(a)c(>)g FO(0,)-118 2352 y FN(b)d(>)f FO(0.)-118 2500 y FQ(5.)45 b FO(In)30 b(Section)g(3.1.5)f(w)n(e)h (considered)f(the)i(algebra)c Fz(B)1650 2512 y FK(2)1716 2500 y FO(=)g FI(C)15 b FP(h)q FN(x;)f(y)37 b FP(j)28 b FN(xy)s(x)g FO(=)-118 2600 y FN(y)s(xy)s FP(i)p FO(.)54 b(In)n(tro)r(duce)33 b(an)h(in)n(v)n(olution)29 b(b)n(y)k(setting)g FN(x)1472 2570 y FL(?)1543 2600 y FO(=)g FN(y)s FO(;)j(then)e(represen) n(ta-)-118 2699 y(tions)26 b(of)h(the)h(arising)c FP(\003)p FO(-algebra)f(are)j(related)g(to)h(the)h(class)d(of)i(op)r(erators)e FN(X)-118 2799 y FO(suc)n(h)i(that)760 2977 y FN(X)7 b(X)912 2943 y FM(\003)949 2977 y FN(X)29 b FO(=)23 b FN(X)1211 2943 y FM(\003)1248 2977 y FN(X)7 b(X)1400 2943 y FM(\003)1437 2977 y FN(:)-118 3156 y FO(Let)28 b FN(X)j FO(=)24 b FN(U)9 b(C)34 b FO(\()p FN(U)j FO(is)27 b(a)h(partial)e(isometry)-7 b(,)25 b FN(C)31 b FP(\025)23 b FO(0,)29 b(k)n(er)12 b FN(U)33 b FO(=)24 b(k)n(er)13 b FN(C)6 b FO(\))29 b(b)r(e)f(the)-118 3255 y(p)r(olar)e(decomp)r (osition)e(of)j(the)h(op)r(erator)e FN(X)7 b FO(.)36 b(Then)558 3434 y FN(U)9 b(C)689 3400 y FK(3)749 3434 y FO(=)23 b FN(C)902 3400 y FK(3)939 3434 y FN(U)1005 3400 y FM(\003)1043 3434 y FN(;)97 b(U)1229 3400 y FM(\003)1267 3434 y FN(C)1332 3400 y FK(3)1393 3434 y FO(=)22 b FN(C)1545 3400 y FK(3)1583 3434 y FN(U;)-118 3612 y FO(whic)n(h)h(implies)e(that) j FN(X)31 b FO(is)23 b(a)h(quasi-normal)19 b(op)r(erator,)24 b(and)g(therefore,)g(self-)-118 3712 y(adjoin)n(t,)35 b FN(X)271 3682 y FM(\003)344 3712 y FO(=)g FN(X)7 b FO(.)58 b(Irreducible)32 b FP(\003)p FO(-represen)n(tations)f(of)k(the) g(algebra)d Fz(B)2301 3724 y FK(2)-118 3811 y FO(equipp)r(ed)23 b(with)g(the)h(in)n(v)n(olution)c FN(x)990 3781 y FL(?)1051 3811 y FO(=)j FN(y)j FO(are)d(all)e(one-dimensional,)e FN(X)29 b FO(=)23 b FN(\025)p FO(,)-118 3911 y FN(\025)g FP(2)h FI(R)p FO(.)p eop %%Page: 234 238 234 237 bop -118 -137 a FO(234)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FQ(3.2.2)94 b(P)m(artial)24 b(isometries,)e(w)m(eakly)j(cen)m (tered)f(op)s(erators)g(and)g(al-)174 196 y(gebraic)32 b(op)s(erators)-118 350 y(1.)j FO(P)n(assing)21 b(to)j(relations)d(of)j (degree)f(four,)h(w)n(e)g(consider)e(only)g(the)j(follo)n(wing)-118 450 y FP(\003)p FO(-algebras)36 b(\(classes)h(of)i(op)r(erators\):)59 b Fz(W)43 b FO(=)f FI(C)15 b FP(h)q FN(x;)f(x)1612 420 y FM(\003)1699 450 y FP(j)43 b FO([)p FN(xx)1882 420 y FM(\003)1921 450 y FN(;)14 b(x)2005 420 y FM(\003)2044 450 y FN(x)p FO(])43 b(=)g(0)p FP(i)-118 549 y FO(\(w)n(eakly)31 b(cen)n(tered)h(op)r(erators\),)h Fz(P)e FO(=)h FI(C)15 b FP(h)p FN(x;)f(x)1374 519 y FM(\003)1450 549 y FP(j)32 b FO(\()p FN(x)1584 519 y FM(\003)1623 549 y FN(x)p FO(\))1702 519 y FK(2)1771 549 y FO(=)g FN(x)1915 519 y FM(\003)1954 549 y FN(x)p FP(i)h FO(\(partial)-118 649 y(isometries\),)27 b Fz(WP)h FO(=)f FI(C)15 b FP(h)q FN(x;)f(x)823 619 y FM(\003)896 649 y FP(j)28 b FO([)p FN(xx)1064 619 y FM(\003)1103 649 y FN(;)14 b(x)1187 619 y FM(\003)1225 649 y FN(x)p FO(])29 b(=)e(0)p FN(;)g FO(\()p FN(x)1587 619 y FM(\003)1626 649 y FN(x)p FO(\))1705 619 y FK(2)1771 649 y FO(=)h FN(x)1911 619 y FM(\003)1949 649 y FN(x)p FP(i)k FO(\(w)n(eakly)-118 749 y(cen)n(tered)27 b(op)r(erators)f(whic)n(h)g(are)h(partial)d (isometries\).)6 849 y(The)k(follo)n(wing)c(theorem)i(holds.)-118 1016 y FQ(Theorem)k(65.)41 b FB(The)31 b FP(\003)p FB(-algebr)l(a)f Fz(W)f FB(is)i FP(\003)p FB(-wild.)-118 1184 y(Pr)l(o)l(of.)43 b FO(De\014ne)28 b(a)f(homomorphism)c FN( )12 b FO(:)28 b Fz(W)23 b FP(\000)-49 b(!)23 b FN(M)1485 1196 y FK(3)1522 1184 y FO(\()p FA(F)1610 1196 y FK(2)1646 1184 y FO(\))28 b(b)n(y)431 1472 y FN( )s FO(\()p FN(x)p FO(\))c(=)711 1306 y Fy(0)711 1455 y(@)916 1366 y FO(0)366 b(0)234 b(2)p FN(e)822 1472 y FO(\(1)p FN(=)p FO(2\))p FN(e)143 b FO(\()1226 1403 y FP(p)p 1295 1403 42 4 v 69 x FO(3)p FN(=)p FO(2\))p FN(v)126 b FO(0)783 1577 y(\()815 1508 y FP(p)p 885 1508 V 885 1577 a FO(3)o FN(=)p FO(2\))p FN(u)82 b FP(\000)p FO(\(1)p FN(=)p FO(2\))p FN(uv)104 b FO(0)1680 1306 y Fy(1)1680 1455 y(A)1766 1472 y FN(:)6 1757 y FO(T)-7 b(o)28 b(sho)n(w)e(that)i(this)f(is)g(a)g(homomorphism,) 22 b(w)n(e)28 b(calculate)c(that)50 2036 y FN( )s FO(\()p FN(x)p FO(\))14 b FN( )s FO(\()p FN(x)368 2001 y FM(\003)407 2036 y FO(\))24 b(=)550 1869 y Fy(0)550 2018 y(@)623 1935 y FO(2)p FN(e)82 b FO(0)h(0)642 2035 y(0)104 b FN(e)84 b FO(0)642 2135 y(0)102 b(0)84 b FN(e)952 1869 y Fy(1)952 2018 y(A)1039 2036 y FN(;)97 b( )s FO(\()p FN(x)1295 2001 y FM(\003)1334 2036 y FO(\))14 b FN( )s FO(\()p FN(x)p FO(\))24 b(=)1660 1869 y Fy(0)1660 2018 y(@)1734 1935 y FN(e)84 b FO(0)102 b(0)1732 2035 y(0)84 b FN(e)104 b FO(0)1732 2135 y(0)83 b(0)f(2)p FN(e)2061 1869 y Fy(1)2061 2018 y(A)2148 2036 y FN(:)-118 2319 y FO(Therefore)26 b([)p FN( )s FO(\()p FN(x)p FO(\))14 b FN( )s FO(\()p FN(x)599 2289 y FM(\003)639 2319 y FO(\))p FN(;)g( )s FO(\()p FN(x)844 2289 y FM(\003)883 2319 y FO(\))g FN( )s FO(\()p FN(x)p FO(\)])25 b(=)d(0.)6 2419 y(The)34 b(homomorphism)28 b FN( )37 b FO(induces)32 b(the)i(functor)f FN(F)1663 2431 y FL( )1723 2419 y FO(:)43 b(Rep)14 b FN(C)2012 2389 y FM(\003)2051 2419 y FO(\()p FA(F)2139 2431 y FK(2)2174 2419 y FO(\))33 b FP(\000)-48 b(!)-118 2519 y FO(Rep)14 b Fz(W)p FO(.)37 b(W)-7 b(e)28 b(will)c(sho)n(w)j(that)h FN(F)925 2531 y FL( )1003 2519 y FO(is)f(full.)6 2630 y(It)h(follo)n(ws)d(from)h FN(C)6 b(X)706 2600 y FM(\003)744 2630 y FN(X)29 b FO(=)954 2609 y(^)930 2630 y FN(X)1006 2600 y FM(\003)1067 2609 y FO(^)1043 2630 y FN(X)7 b(C)34 b FO(that)654 2892 y FN(C)29 b FO(=)830 2725 y Fy(0)830 2875 y(@)902 2792 y FN(C)961 2804 y FK(11)1115 2792 y FN(C)1174 2804 y FK(12)1372 2792 y FO(0)902 2891 y FN(C)961 2903 y FK(21)1115 2891 y FN(C)1174 2903 y FK(22)1372 2891 y FO(0)946 2991 y(0)171 b(0)127 b FN(C)1387 3003 y FK(33)1457 2725 y Fy(1)1457 2875 y(A)1544 2892 y FN(:)-118 3192 y FO(F)-7 b(rom)31 b(the)i(relations)d FN(C)6 b(X)38 b FO(=)888 3171 y(^)864 3192 y FN(X)6 b(C)g FO(,)35 b FN(C)6 b(X)1203 3162 y FM(\003)1272 3192 y FO(=)1392 3171 y(^)1368 3192 y FN(X)1444 3162 y FM(\003)1482 3192 y FN(C)g FO(,)34 b(w)n(e)e(ha)n(v)n(e)g(that)h FN(C)2172 3204 y FK(12)2274 3192 y FO(=)-118 3292 y FN(C)-59 3304 y FK(21)36 3292 y FO(=)24 b(0,)29 b FN(C)278 3304 y FK(11)373 3292 y FO(=)24 b FN(C)521 3304 y FK(22)616 3292 y FO(=)g FN(C)764 3304 y FK(33)860 3292 y FO(=)968 3271 y(~)949 3292 y FN(C)6 b FO(,)29 b(and)1247 3271 y(~)1228 3292 y FN(C)6 b(U)34 b FO(=)1487 3271 y(^)1473 3292 y FN(U)1557 3271 y FO(~)1539 3292 y FN(C)6 b FO(,)1675 3271 y(~)1656 3292 y FN(C)g(V)43 b FO(=)1914 3271 y(^)1902 3292 y FN(V)1987 3271 y FO(~)1969 3292 y FN(C)6 b FO(.)39 b(Hence,)-118 3391 y(w)n(e)30 b(can)f(conclude)g(that)h(the)g(functor)g FN(F)1176 3403 y FL( )1257 3391 y FO(is)f(full.)43 b(Therefore,)29 b(the)h(algebra)-118 3491 y Fz(W)d FO(is)g FP(\003)p FO(-wild.)p 2278 3491 4 57 v 2282 3438 50 4 v 2282 3491 V 2331 3491 4 57 v 6 3661 a(Therefore,)e(the)h(problem)d(of)i(unitary)f (description)f(of)i(w)n(eakly)e(cen)n(tered)-118 3760 y(op)r(erators)i(is)i FP(\003)p FO(-wild.)-118 3911 y FQ(2.)36 b FO(F)-7 b(or)27 b(partial)e(isometries,)e(the)28 b(follo)n(wing)c(theorem)i(holds.)p eop %%Page: 235 239 235 238 bop -118 -137 a FJ(3.2.)36 b(Classes)25 b(of)j(non-self-adjoin) n(t)c(op)r(erators)854 b FO(235)-118 96 y FQ(Theorem)30 b(66.)41 b FB(The)31 b FP(\003)p FB(-algebr)l(a)f Fz(P)f FB(is)h FP(\003)p FB(-wild.)-118 258 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)20 b(will)c(sho)n(w)i(that)i Fz(P)j FP(\037)f FN(C)1038 228 y FM(\003)1077 258 y FO(\()p FA(F)1165 270 y FK(2)1200 258 y FO(\).)35 b(The)19 b(homomorphism)14 b FN( )e FO(:)28 b Fz(P)23 b FP(\000)-48 b(!)-118 358 y FN(M)-37 370 y FK(3)0 358 y FO(\()p FN(C)97 328 y FM(\003)135 358 y FO(\()p FA(F)223 370 y FK(2)259 358 y FO(\)\))28 b(is)f(constructed)g(b)n(y)482 638 y FN( )s FO(\()p FN(x)p FO(\))d(=)762 471 y Fy(0)762 621 y(@)834 472 y FP(p)p 904 472 42 4 v 904 541 a FO(3)o FN(=)p FO(4)14 b FN(u)1215 472 y FP(p)p 1285 472 V 1285 541 a FO(3)o FN(=)p FO(2)g FN(e)124 b FO(0)871 640 y(3)p FN(=)p FO(4)14 b FN(v)122 b FP(\000)p FO(1)p FN(=)p FO(2)14 b FN(v)s(u)1469 610 y FM(\003)1587 640 y FO(0)873 740 y(1)p FN(=)p FO(2)g FN(e)266 b FO(0)227 b(0)1629 471 y Fy(1)1629 621 y(A)1715 638 y FN(:)-118 914 y FO(It)28 b(is)e(easy)h(to)g(v)n(erify)f(that)623 1186 y FN( )s FO(\()p FN(x)759 1152 y FM(\003)798 1186 y FO(\))14 b FN( )s FO(\()p FN(x)p FO(\))25 b(=)1124 1019 y Fy(0)1124 1169 y(@)1198 1086 y FN(e)84 b FO(0)f(0)1197 1185 y(0)h FN(e)g FO(0)1197 1285 y(0)e(0)h(0)1488 1019 y Fy(1)1488 1169 y(A)1574 1186 y FN(:)-118 1469 y FO(Therefore)26 b(\()p FN( )s FO(\()p FN(x)426 1439 y FM(\003)466 1469 y FO(\))14 b FN( )s FO(\()p FN(x)p FO(\)\))712 1439 y FK(2)773 1469 y FO(=)23 b FN( )s FO(\()p FN(x)997 1439 y FM(\003)1036 1469 y FO(\))14 b FN( )s FO(\()p FN(x)p FO(\).)6 1569 y(The)28 b(induced)f(functor)h FN(F)828 1581 y FL( )887 1569 y FO(:)42 b(Rep)14 b FN(C)1175 1538 y FM(\003)1213 1569 y FO(\()p FA(F)1301 1581 y FK(2)1337 1569 y FO(\))23 b FP(\000)-48 b(!)23 b FO(Rep)14 b Fz(P)27 b FO(is)g(full.)p 2278 1569 4 57 v 2282 1516 50 4 v 2282 1569 V 2331 1569 4 57 v 6 1734 a(Therefore,)c(the)h(problem)d(of)i(the) g(description)e(of)i(partial)d(isometries)f(up)-118 1833 y(to)27 b(a)h(unitary)e(equiv)-5 b(alence)25 b(is)h FP(\003)p FO(-wild.)-118 1981 y FQ(3.)36 b FO(The)28 b(follo)n(wing)23 b(theorem)j(holds.)-118 2143 y FQ(Theorem)k(67.)41 b FB(The)31 b FP(\003)p FB(-algebr)l(a)f Fz(WP)f FB(is)h FP(\003)p FB(-wild.)-118 2305 y(Pr)l(o)l(of.)43 b FO(W)-7 b(e)23 b(will)d(again)g(sho)n(wn)i(that)h Fz(WP)f FP(\037)h FN(C)1401 2275 y FM(\003)1439 2305 y FO(\()p FA(F)1527 2317 y FK(2)1563 2305 y FO(\).)35 b(De\014ne)23 b(a)f(homomor-)-118 2404 y(phism)k FN( )12 b FO(:)28 b Fz(WP)22 b FP(\000)-48 b(!)23 b FN(M)626 2416 y FK(4)663 2404 y FO(\()p FA(F)751 2416 y FK(2)786 2404 y FO(\))28 b(as)f(follo)n(ws:)420 2734 y FN( )s FO(\()p FN(x)p FO(\))d(=)700 2518 y Fy(0)700 2664 y(B)700 2713 y(B)700 2767 y(@)772 2518 y FP(p)p 841 2518 42 4 v 69 x FO(3)p FN(=)p FO(4)14 b FN(u)1153 2518 y FP(p)p 1222 2518 V 69 x FO(3)p FN(=)p FO(2)g FN(e)124 b FO(0)83 b(0)809 2686 y(3)p FN(=)p FO(4)14 b FN(v)121 b FP(\000)p FO(1)p FN(=)p FO(2)14 b FN(v)s(u)1406 2656 y FM(\003)1525 2686 y FO(0)83 b(0)811 2786 y(1)p FN(=)p FO(2)14 b FN(e)265 b FO(0)228 b(0)83 b(0)879 2886 y(0)334 b(0)230 b FN(e)84 b FO(0)1691 2518 y Fy(1)1691 2664 y(C)1691 2713 y(C)1691 2767 y(A)1778 2734 y FN(:)-118 3064 y FO(It)20 b(is)f(easy)g(to)h(sho)n(w)f(that)h(the)h(corresp)r(onding)c(functor)i FN(F)1674 3076 y FL( )1734 3064 y FO(:)42 b(Rep)14 b FN(C)2022 3034 y FM(\003)2060 3064 y FO(\()p FA(F)2148 3076 y FK(2)2184 3064 y FO(\))23 b FP(\000)-48 b(!)-118 3164 y FO(Rep)14 b Fz(WI)27 b FO(is)g(full.)p 2278 3164 4 57 v 2282 3111 50 4 v 2282 3164 V 2331 3164 4 57 v 6 3329 a(Th)n(us,)g(the)f(problem)d(of)j(the)h(description)c(of)j (partial)d(isometries,)g(whic)n(h)-118 3428 y(are)j(w)n(eakly)g(cen)n (tered)h(op)r(erators,)f(is)g FP(\003)p FO(-wild.)-118 3559 y FB(R)l(emark)k(54.)42 b FO(F)-7 b(or)27 b FN(n)c(<)g FP(1)p FO(,)28 b(consider)d(the)j FP(\003)p FO(-algebra)118 3737 y Fz(WP)274 3757 y FL(n)342 3737 y FO(=)23 b FI(C)484 3670 y Fy(\012)529 3737 y FN(x;)14 b(x)660 3703 y FM(\003)722 3737 y FP(j)23 b FN(x)28 b FO(is)f(a)g(partial)e(isometry)-7 b(,)24 b(and)782 3881 y([)p FN(x)852 3847 y FL(j)887 3881 y FN(x)934 3847 y FM(\003)973 3840 y FL(j)1008 3881 y FN(;)14 b(x)1092 3847 y FM(\003)1130 3840 y FL(k)1171 3881 y FN(x)1218 3847 y FL(k)1260 3881 y FO(])23 b(=)f(0)p FN(;)28 b FP(8)p FN(k)s(;)14 b(j)26 b FO(=)d(1)p FN(;)14 b(:)g(:)g(:)f(;)h(n)2040 3814 y Fy(\013)2079 3881 y FN(:)p eop %%Page: 236 240 236 239 bop -118 -137 a FO(236)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)-118 96 y FP(\003)p FO(-Algebra)36 b(of)j(w)n(eakly)e(cen)n(tered)h(op)r (erators)f(is)h Fz(WP)j FO(=)h Fz(WP)1940 116 y FK(1)1977 96 y FO(.)71 b(In)39 b([33)o(],)-118 196 y(Theorem)26 b(67)h(w)n(as)g(extended)h(to)g(these)g(algebras:)35 b(it)28 b(w)n(as)f(pro)n(v)n(ed)f(that)i(the)-118 296 y FP(\003)p FO(-algebra)c Fz(WP)397 316 y FL(n)470 296 y FO(is)j FP(\003)p FO(-wild)e(for)i(an)n(y)g FN(n)c(<)f FP(1)p FO(.)6 395 y(On)i(the)g(other)g(hand,)g(the)h FP(\003)p FO(-algebra)20 b(of)k(cen)n(tered)f(op)r(erators)f(is)h(n)n (uclear)-118 495 y(\(see)f(Section)f(2.5.2\),)h(and)g(the)g FP(\003)p FO(-algebra)c(of)k(cen)n(tered)g(partial)d(isometries)f(is) -118 595 y(of)27 b(t)n(yp)r(e)h FN(I)34 b FO(and)28 b(admits)d(a)i (complete)f(description)e(of)k(its)e FP(\003)p FO(-represen)n(tations) -118 694 y(\(Section)h(2.1.3\))-118 826 y FQ(4.)35 b FO(W)-7 b(e)25 b(also)d(consider)g(the)j(complexit)n(y)c(problem)h(of)i (a)g(unitary)f(description)-118 926 y(for)k(algebraic)d(op)r(erators,)h (i.e.,)i(represen)n(tations)e(of)i(the)h FP(\003)p FO(-algebra)598 1107 y Fz(A)658 1119 y FL(R)708 1127 y Fv(n)776 1107 y FO(=)22 b FI(C)15 b FP(h)p FN(x)q(;)f(x)1081 1072 y FM(\003)1148 1107 y FP(j)24 b FN(R)1258 1119 y FL(n)1303 1107 y FO(\()p FN(x)p FO(\))g(=)f(0)p FP(i)p FN(;)-118 1287 y FO(where)28 b FN(R)186 1299 y FL(n)260 1287 y FO(is)f(a)h(p)r(olynomial)c(in)k(one)g(v)-5 b(ariable)25 b(with)j(complex)e(co)r(e\016cien)n(ts.)-118 1387 y(F)-7 b(or)29 b(brevit)n(y)-7 b(,)29 b(w)n(e)g(assume)g(that)h(the)g(p)r (olynomial)25 b(do)r(es)30 b(not)f(ha)n(v)n(e)g(m)n(ultiple)-118 1487 y(ro)r(ots.)-118 1651 y FQ(Prop)s(osition)h(73.)41 b FB(L)l(et)29 1852 y Fz(A)89 1864 y FL(R)139 1872 y Fx(3)199 1852 y FO(=)22 b FI(C)340 1785 y Fy(\012)385 1852 y FN(x;)14 b(x)516 1818 y FM(\003)578 1852 y FP(j)24 b FN(R)688 1864 y FK(3)725 1852 y FO(\()p FN(x)p FO(\))860 1805 y FK(def)873 1852 y FO(=)36 b(\()p FN(x)20 b FP(\000)e FN(\013)1209 1864 y FK(1)1246 1852 y FN(e)p FO(\)\()p FN(x)h FP(\000)f FN(\013)1551 1864 y FK(2)1588 1852 y FN(e)p FO(\)\()p FN(x)i FP(\000)e FN(\013)1894 1864 y FK(3)1931 1852 y FN(e)p FO(\))23 b(=)g(0)p FN(;)640 1986 y(\013)693 1998 y FK(1)731 1986 y FN(;)14 b(\013)821 1998 y FK(2)858 1986 y FN(;)g(\013)948 1998 y FK(3)1008 1986 y FP(2)24 b FI(C)15 b FN(;)33 b(\013)1250 1998 y FL(k)1314 1986 y FP(6)p FO(=)23 b FN(\013)1455 1998 y FL(l)1510 1986 y FB(for)31 b FN(k)26 b FP(6)p FO(=)c FN(l)1826 1919 y Fy(\013)1865 1986 y FN(:)-118 2167 y FB(Then)30 b Fz(A)158 2179 y FL(R)208 2187 y Fx(3)268 2167 y FP(\037)22 b FA(Q)410 2179 y FK(2)p FL(;)p FM(?)519 2167 y FB(,)30 b(and)g(c)l(onse)l(quently,)g(the)g FP(\003)p FB(-algebr)l(a)g Fz(A)1788 2179 y FL(R)1838 2187 y Fx(3)1904 2167 y FB(is)g FP(\003)p FB(-wild.)-118 2331 y(Pr)l(o)l(of.)43 b FO(De\014ne)30 b(a)f(homomorphism)24 b FN( )12 b FO(:)28 b Fz(A)1237 2343 y FL(R)1287 2351 y Fx(3)1349 2331 y FP(\000)-48 b(!)25 b FA(Q)1529 2343 y FK(2)p FL(;)p FM(?)1668 2331 y FO(as)j(follo)n(ws:)37 b FN( )s FO(\()p FN(x)p FO(\))27 b(=)-118 2431 y FN(\013)-65 2443 y FK(1)-28 2431 y FN(q)9 2443 y FK(1)60 2431 y FO(+)13 b FN(\013)191 2443 y FK(2)228 2431 y FN(q)265 2443 y FK(2)315 2431 y FO(+)g FN(\013)446 2443 y FK(3)483 2431 y FO(\()p FN(e)g FP(\000)g FN(q)682 2443 y FK(1)732 2431 y FP(\000)g FN(q)847 2443 y FK(2)884 2431 y FO(\))p FN(:)25 b FO(It)g(is)f(easy)g(to)h(c)n (hec)n(k)f(that)h(the)g(functor)g FN(F)2288 2443 y FL( )-118 2531 y FO(is)h(full.)p 2278 2531 4 57 v 2282 2478 50 4 v 2282 2531 V 2331 2531 4 57 v -118 2696 a FQ(Corollary)32 b(18.)40 b FB(If)35 b FN(R)625 2708 y FL(n)705 2696 y FB(is)g(a)g(p)l(olynomial)i(with)e(thr)l(e)l(e)f(and)h(mor)l(e)g (distinct)-118 2796 y(r)l(o)l(ots)30 b(then)f(the)h FP(\003)p FB(-algebr)l(a)g Fz(A)819 2808 y FL(R)869 2816 y Fv(n)943 2796 y FB(is)h FP(\003)p FB(-wild.)-118 3011 y FQ(3.2.3)94 b(Hyp)s(onormal)23 b(op)s(erators)i(and)h(pairs)f(of)g(comm)m(uting)d (com-)174 3111 y(pletely)31 b(non-unitary)h(isometries)-118 3264 y FO(No)n(w)i(w)n(e)f(will)f(consider)g(classes)g(of)i (non-self-adjoin)n(t)d(op)r(erators)h(that)j(are)-118 3364 y(giv)n(en)22 b(b)n(y)h(a)g(non-p)r(olynomial)18 b(equalit)n(y)-7 b(,)22 b(e.g.,)i(an)f(inequalit)n(y)-7 b(,)21 b(or)h(other)h(non-)-118 3463 y(algebraic)15 b(conditions.)31 b(F)-7 b(or)19 b(suc)n(h)f(op)r(erators)f(w)n(e)i(will)d(study)j(the)g (complexit)n(y)-118 3563 y(of)27 b(the)h(problem)e(to)h(describ)r(e)f (the)i(op)r(erators)e(up)i(to)f(unitary)f(equiv)-5 b(alence.)-118 3712 y FQ(1.)44 b FO(Let)30 b(Z)g(b)r(e)h(a)f(h)n(yp)r(onormal)c(op)r (erator,)j(i.e.,)i FN(Z)6 b(Z)1530 3682 y FM(\003)1587 3712 y FP(\000)20 b FN(Z)1735 3682 y FM(\003)1773 3712 y FN(Z)33 b FP(\025)27 b FO(0.)44 b(W)-7 b(e)30 b(will)-118 3811 y(sho)n(w)22 b(that)h(the)g(problem)d(to)j(describ)r(e)e(the)i (class)e(of)h(h)n(yp)r(onormal)e(op)r(erators)-118 3911 y(con)n(tains)32 b(the)h(description)e(problem)h(for)h(one)g (non-self-adjoin)n(t)d(op)r(erator,)p eop %%Page: 237 241 237 240 bop -118 -137 a FJ(3.2.)36 b(Classes)25 b(of)j(non-self-adjoin) n(t)c(op)r(erators)854 b FO(237)-118 96 y FN(X)45 b FP(2)39 b FN(L)p FO(\()p FN(H)7 b FO(\),)40 b(whic)n(h)c(do)r(es)h(not)g (satisfy)e(an)n(y)i(relations,)f(or)g(the)i(same)d(for)-118 196 y(a)g(pair)f(of)i(self-adjoin)n(t)d(op)r(erators.)60 b(T)-7 b(o)35 b(do)h(that,)i(w)n(e)e(use)f(W)-7 b(ogen's)35 b(con-)-118 296 y(struction)c(\(see)i([287)n(]\).)53 b(W)-7 b(e)33 b(consider)d(op)r(erators)h FN(Z)37 b FP(2)32 b FN(L)p FO(\()1807 233 y Fy(L)1899 254 y FM(1)1899 321 y FK(1)1983 296 y FN(H)7 b FO(\))33 b(of)f(the)-118 395 y(follo)n(wing)23 b(form:)159 815 y FN(Z)29 b FO(=)333 473 y Fy(0)333 620 y(B)333 669 y(B)333 719 y(B)333 769 y(B)333 819 y(B)333 869 y(B)333 918 y(B)333 972 y(@)422 537 y FO(0)421 637 y FN(I)128 b FO(0)471 b Fo(0)405 736 y FN(X)90 b FO(2)p FN(I)111 b FO(0)585 836 y(0)104 b(3)p FN(I)117 b FO(0)753 936 y(0)110 b(3)p FN(I)123 b FO(0)576 1049 y Fo(0)903 1032 y FO(.)935 1057 y(.)968 1083 y(.)1083 1032 y(.)1115 1057 y(.)1148 1083 y(.)1263 1032 y(.)1295 1057 y(.)1328 1083 y(.)1355 473 y Fy(1)1355 620 y(C)1355 669 y(C)1355 719 y(C)1355 769 y(C)1355 819 y(C)1355 869 y(C)1355 918 y(C)1355 972 y(A)1442 815 y FN(;)180 b FP(k)p FN(X)7 b FP(k)21 b(\024)i FO(1)p FN(=)p FO(2)p FN(:)-118 1234 y FO(The)34 b(op)r(erator)e FN(Z)39 b FO(is)32 b(a)i(h)n(yp)r (onormal)c(op)r(erator.)53 b(It)34 b(is)f(easy)g(to)g(pro)n(v)n(e)f (the)-118 1334 y(follo)n(wing)23 b(prop)r(osition.)-118 1471 y FQ(Prop)s(osition)30 b(74.)41 b FB(A)n(n)27 b(op)l(er)l(ator)i FA(Y)f FB(is)g(intertwining)h(for)f(the)g(p)l(airs)h FN(Z)6 b FB(,)29 b FN(Z)2301 1441 y FM(\003)-118 1571 y FB(and)63 1550 y FO(~)46 1571 y FN(Z)5 b FB(,)184 1550 y FO(~)167 1571 y FN(Z)230 1541 y FM(\003)300 1571 y FB(if)33 b(and)g(only)g(if)65 b FA(Y)28 b FO(=)f FN(Y)39 b FP(\012)20 b FN(I)1222 1583 y FM(1)1293 1571 y FB(,)34 b(wher)l(e)f FN(Y)50 b FB(is)33 b(an)f(intertwining)-118 1671 y(op)l(er)l(ator)25 b(for)h FN(X)7 b FB(,)25 b FN(X)533 1640 y FM(\003)595 1671 y FB(and)783 1650 y FO(~)760 1671 y FN(X)6 b FB(,)910 1650 y FO(~)886 1671 y FN(X)962 1640 y FM(\003)999 1671 y FB(,)26 b(c)l(orr)l(esp)l(ondingly)34 b FO(\()p FB(that)25 b(is)g FN(Y)18 b(X)30 b FO(=)2195 1650 y(~)2171 1671 y FN(X)6 b(Y)19 b FB(,)-118 1770 y FN(Y)g(X)25 1740 y FM(\003)85 1770 y FO(=)197 1749 y(~)173 1770 y FN(X)249 1740 y FM(\003)286 1770 y FN(Y)g FO(\))p FB(.)-118 1908 y FQ(2.)56 b FO(Let)34 b FN(S)219 1920 y FK(1)256 1908 y FO(,)j FN(S)367 1920 y FK(2)438 1908 y FO(b)r(e)e(isometries)30 b(without)k(unitary)e(parts,)j(and)g([)p FN(S)2055 1920 y FK(1)2092 1908 y FN(;)14 b(S)2180 1920 y FK(2)2217 1908 y FO(])34 b(=)-118 2008 y([)p FN(S)-39 1977 y FM(\003)-44 2028 y FK(1)-1 2008 y FN(;)14 b(S)92 1977 y FM(\003)87 2028 y FK(2)130 2008 y FO(])33 b(=)f(0.)55 b(W)-7 b(e)34 b(will)d(sho)n(w)i(that)h(this)e(description)f(problem)h (con)n(tains)-118 2107 y(the)22 b(description)c(problem)h(for)i(a)g (pair)e(of)j(unitary)e(op)r(erators)f FN(U)9 b FO(,)22 b FN(V)42 b FP(2)24 b FN(L)p FO(\()p FN(H)7 b FO(\).)-118 2207 y(De\014ne)28 b(the)g(op)r(erators)e FN(S)701 2219 y FK(1)738 2207 y FO(,)i FN(S)840 2219 y FK(2)900 2207 y FP(2)23 b FN(L)p FO(\()1067 2145 y Fy(L)1159 2165 y FM(1)1159 2232 y FK(1)1243 2207 y FN(H)7 b FO(\))28 b(in)f(the)h(follo) n(wing)23 b(w)n(a)n(y:)444 2685 y FN(S)495 2697 y FK(1)555 2685 y FO(=)643 2294 y Fy(0)643 2440 y(B)643 2490 y(B)643 2539 y(B)643 2589 y(B)643 2639 y(B)643 2689 y(B)643 2739 y(B)643 2789 y(B)643 2838 y(B)643 2891 y(@)716 2357 y FO(0)92 b(0)716 2457 y(0)g(0)1415 2416 y Fo(0)715 2557 y FN(I)99 b FO(0)92 b(0)84 b(0)716 2656 y(0)91 b FN(I)99 b FO(0)84 b(0)983 2756 y FN(I)91 b FO(0)111 b(0)118 b(0)984 2856 y(0)83 b FN(I)118 b FO(0)g(0)841 2969 y Fo(0)1240 2952 y FO(.)1272 2977 y(.)1304 3002 y(.)1561 2952 y(.)1593 2977 y(.)1626 3002 y(.)1653 2294 y Fy(1)1653 2440 y(C)1653 2490 y(C)1653 2539 y(C)1653 2589 y(C)1653 2639 y(C)1653 2689 y(C)1653 2739 y(C)1653 2789 y(C)1653 2838 y(C)1653 2891 y(A)1740 2685 y FN(;)8 3534 y(S)59 3546 y FK(2)119 3534 y FO(=)206 3093 y Fy(0)206 3239 y(B)206 3289 y(B)206 3339 y(B)206 3388 y(B)206 3438 y(B)206 3488 y(B)206 3538 y(B)206 3588 y(B)206 3638 y(B)206 3687 y(B)206 3737 y(B)206 3790 y(@)391 3147 y FO(0)290 b(2)765 3117 y FM(\000)p FK(1)p FL(=)p FK(2)921 3147 y FN(I)2026 3189 y Fo(0)391 3251 y FO(0)194 b FP(\000)p FO(2)734 3221 y FM(\000)p FK(1)p FL(=)p FK(2)889 3251 y FN(V)19 b(U)1022 3221 y FM(\003)280 3354 y FO(2)322 3324 y FM(\000)p FK(1)p FL(=)p FK(2)477 3354 y FN(U)288 b FO(0)390 b(0)291 b(2)1629 3324 y FM(\000)p FK(1)p FL(=)p FK(2)1784 3354 y FN(I)279 3458 y FO(2)321 3428 y FM(\000)p FK(1)p FL(=)p FK(2)477 3458 y FN(V)297 b FO(0)390 b(0)194 b FP(\000)p FO(2)1597 3428 y FM(\000)p FK(1)p FL(=)p FK(2)1753 3458 y FN(V)19 b(U)1886 3428 y FM(\003)1143 3613 y FO(2)1185 3583 y FM(\000)p FK(1)p FL(=)p FK(2)1341 3613 y FN(U)288 b FO(0)2011 3555 y(.)2043 3580 y(.)2076 3605 y(.)1143 3768 y(2)1185 3738 y FM(\000)p FK(1)p FL(=)p FK(2)1341 3768 y FN(V)297 b FO(0)2011 3710 y(.)2043 3735 y(.)2076 3760 y(.)814 3881 y Fo(0)1231 3865 y FO(.)1264 3889 y(.)1296 3915 y(.)1663 3865 y(.)1695 3889 y(.)1728 3915 y(.)2011 3865 y(.)2043 3889 y(.)2076 3915 y(.)2103 3093 y Fy(1)2103 3239 y(C)2103 3289 y(C)2103 3339 y(C)2103 3388 y(C)2103 3438 y(C)2103 3488 y(C)2103 3538 y(C)2103 3588 y(C)2103 3638 y(C)2103 3687 y(C)2103 3737 y(C)2103 3790 y(A)2190 3534 y FN(:)p eop %%Page: 238 242 238 241 bop -118 -137 a FO(238)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)6 96 y FO(It)g(is)f(easy)f(to)i(pro)n(v)n(e)e(the)i(follo)n(wing)23 b(prop)r(osition.)-118 273 y FQ(Prop)s(osition)30 b(75.)41 b FB(A)n(n)25 b(op)l(er)l(ator)36 b FA(Y)26 b FB(is)g(intertwining)g (for)h(the)f(p)l(airs)h FN(S)2165 285 y FL(i)2193 273 y FB(,)g FN(S)2301 243 y FM(\003)2296 295 y FL(i)-118 373 y FB(and)61 352 y FO(~)47 373 y FN(S)98 385 y FL(i)125 373 y FB(,)199 352 y FO(~)185 373 y FN(S)241 343 y FM(\003)236 395 y FL(i)279 373 y FB(,)35 b FN(i)29 b FO(=)g(1)p FB(,)34 b FO(2)p FB(,)g(c)l(orr)l(esp)l(ondingly,)j(if)d(and)g(only)g(if)68 b FA(Y)30 b FO(=)f FN(Y)40 b FP(\012)20 b FN(I)2242 385 y FM(1)2313 373 y FB(,)-118 484 y(wher)l(e)26 b FN(Y)45 b FB(is)26 b(an)g(intertwining)g(op)l(er)l(ator)h(for)f FN(U)9 b FB(,)27 b FN(V)45 b FB(and)1700 463 y Fy(e)1688 484 y FN(U)9 b FB(,)1816 463 y Fy(e)1806 484 y FN(V)19 b FB(,)27 b(c)l(orr)l(esp)l(ond-)-118 584 y(ingly)38 b FO(\()p FB(that)30 b(is)g FN(Y)19 b(U)31 b FO(=)639 563 y(~)625 584 y FN(U)9 b(Y)18 b FB(,)31 b FN(Y)18 b(U)945 553 y FM(\003)1006 584 y FO(=)1108 563 y(~)1094 584 y FN(U)1160 553 y FM(\003)1198 584 y FN(Y)g FB(,)31 b FN(Y)18 b(V)42 b FO(=)1576 563 y(~)1564 584 y FN(V)19 b(Y)f FB(,)30 b FN(Y)19 b(V)1886 553 y FM(\003)1947 584 y FO(=)2047 563 y(~)2035 584 y FN(V)2102 553 y FM(\003)2140 584 y FN(Y)g FO(\))p FB(.)6 761 y FO(In)45 b(Section)e(3.2.3)f(w)n(e)i(pro)n (v)n(ed)e(the)j(complexit)n(y)40 b(of)k(the)g(description)-118 860 y(of)f(some)e(op)r(erators)g(classes.)80 b(This)42 b(pro)r(of)h(is)e(similar)d(to)43 b(the)g(pro)r(of)g(of)-118 960 y FP(\003)p FO(-wildness)34 b(of)i(the)h FP(\003)p FO(-algebras)c(ab)r(o)n(v)n(e)j(in)g(this)g(c)n(hapter.)63 b(But)37 b(the)h(op)r(er-)-118 1060 y(ators)31 b FN(Z)38 b FO(and)32 b FN(S)407 1072 y FK(1)444 1060 y FO(,)i FN(S)552 1072 y FK(2)621 1060 y FO(do)e(not)g(b)r(elong)f(to)h Fz(K)22 b FP(\012)g FN(L)p FO(\()p FN(H)7 b FO(\).)50 b(It)33 b(seems)e(that)h(the)-118 1159 y(de\014nitions)37 b(of)i(ma)5 b(jorization)35 b(and)k FP(\003)p FO(-wildness)d(\(see)j (Sections)f(3.1.1)g(and)-118 1259 y(3.1.2\))31 b(are)h(restrictiv)n(e)d (in)i(the)i(case)e FN(n)g FO(=)f FP(1)p FO(.)52 b(F)-7 b(or)31 b(the)i(un)n(b)r(ounded)g(op)r(er-)-118 1358 y(ator)d(the)i(case)f FN(n)f FO(=)f FP(1)j FO(is)e(essen)n(tial)f (\(see,)k(e.g.,)f([230)o(]\).)49 b(Therefore,)32 b(these)-118 1458 y(de\014nitions)26 b(should)g(probably)f(b)r(e)j(extended.)-118 1719 y FG(Commen)m(ts)38 b(to)f(Chapter)h(3)-118 1908 y FQ(Section)31 b(3.1.)6 2011 y FO(3.1.1,)g(3.1.2.)47 b(In)32 b(the)f(theory)g(of)g(represen)n(tations)d(of)k(algebras,)d(it) i(w)n(as)-118 2110 y(suggested)d([70)o(])h(to)f(regard)f(a)i(represen)n (tation)d(problem)g(as)i(wild,)g(if)g(it)g(con-)-118 2210 y(tains)40 b(the)h(classical)c(unsolv)n(ed)i(problem)g(of)i (represen)n(tation)e(theory:)63 b(to)-118 2310 y(describ)r(e,)41 b(up)f(to)g(a)f(similarit)n(y)-7 b(,)37 b(a)i(pair)f(of)i(matrices)d (without)i(relations.)-118 2409 y(This)23 b(unsolv)n(ed)e(problem)h(of) h(represen)n(tation)e(theory)i(con)n(tains)f(in)h(itself)f(the)-118 2509 y(problem)30 b(to)j(describ)r(e,)g(up)h(to)f(similarit)n(y)-7 b(,)28 b FN(n)33 b FO(matrices)d(without)j(relations)-118 2609 y(for)26 b(an)n(y)f FN(n)e FP(2)h FI(N)t FO(,)32 b(and)26 b(therefore,)g(it)g(con)n(tains)e(the)j(problem)d(of)i (description,)-118 2708 y(up)g(to)g(similarit)n(y)-7 b(,)20 b(of)26 b(represen)n(tations)d(of)i(an)n(y)g(\014nitely)g (generated)f(algebra.)6 2811 y(Numerous)38 b(examples)e(of)j(wild)e (problems)f(of)i(represen)n(tation)e(theory)-118 2911 y(can)27 b(b)r(e)h(found,)g(e.g.,)f(in)g([13)o(,)h(86)o(],)g(also)d (see)j(bibliograph)n(y)22 b(therein.)6 3014 y(T)-7 b(o)30 b(de\014ne)h(an)f(analogue)e(of)i(wildness)e(for)i FP(\003)p FO(-algebras)d(\()p FP(\003)p FO(-wildness\),)h(it)-118 3114 y(w)n(as)j(suggested)h(in)g([144)n(])h(to)f(c)n(ho)r(ose,)h(for)f (a)g(standard)f FP(\003)p FO(-wild)f(problem)g(in)-118 3214 y(the)e(theory)e(of)i FP(\003)p FO(-represen)n(tations,)23 b(the)28 b(problem)d(of)i(description)e(of)i(a)g(pair)-118 3313 y(of)38 b(self-adjoin)n(t)d(\(or)i(unitary\))f(op)r(erators)g(up)i (to)g(a)f(unitary)f(equiv)-5 b(alence,)-118 3413 y(or)34 b(whic)n(h)h(is)f(the)i(same,)f(represen)n(tations)e(of)i(the)h(free)f FP(\003)p FO(-algebra)d Fz(S)2160 3425 y FK(2)2232 3413 y FO(\(or)-118 3513 y Fz(U)-64 3525 y FK(2)-27 3513 y FO(\))k(generated)e(b)n(y)h(a)g(pair)f(of)h(self-adjoin)n(t)e(\(or)i (unitary\))f(generators.)58 b(It)-118 3612 y(w)n(as)25 b(also)f(suggested)h(to)i(regard)d(as)i(wild)e(problems)g(the)i(ones)g (that)g(con)n(tain)-118 3712 y(a)31 b(standard)g FP(\003)p FO(-wild)e(problem;)i(it)g(w)n(as)g(pro)n(v)n(en)f(that)i(the)g (standard)e FP(\003)p FO(-wild)-118 3811 y(problem)i(con)n(tains,)h(as) h(a)g(sub-problem,)f(the)h(problem)e(of)i(description)e(of)-118 3911 y FP(\003)p FO(-represen)n(tations)24 b(of)j(an)n(y)g(\014nitely)f (or)h(coun)n(tably)f(generated)g FP(\003)p FO(-algebra.)p eop %%Page: 239 243 239 242 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(3)1452 b FO(239)6 96 y(A)27 b(n)n(um)n(b)r(er)f(of)g(pap)r(ers)g([142)o(,)h (202)n(,)g(143)n(])g(etc.)37 b(are)26 b(dev)n(oted)g(to)g(elab)r(orat-) -118 196 y(ing)20 b(the)j(meaning)c(of)i(the)i(statemen)n(t)d (\\description)f(of)j FP(\003)p FO(-represen)n(tations)c(of)-118 296 y(a)25 b FP(\003)p FO(-algebra)c Fz(A)k FO(con)n(tains,)f(as)h(a)f (sub-problem,)f(the)j(description)d(of)i FP(\003)p FO(-repre-)-118 395 y(sentations)f(of)h(a)g FP(\003)p FO(-algebra)c Fz(B)p FO(".)36 b(The)26 b(approac)n(h)d(to)i(the)h(estimation)c(of)j(the)-118 495 y(complexit)n(y)j(of)j FP(\003)p FO(-represen)n(tations)d(based)j (on)g(the)g(concepts)g(of)h(ma)5 b(joriza-)-118 595 y(tion)32 b(relation)d(for)k FP(\003)p FO(-algebras)c(\(De\014nition)i(13\),)j (and)e FP(\003)p FO(-wildness)e(\(De\014ni-)-118 694 y(tion)24 b(14\))f(Used)i(in)f(the)h(b)r(o)r(ok)f(is)g(due)g(to)h(S.)g (Krugly)n(ak)c(and)j(is)g(exp)r(ounded)g(in)-118 794 y([145)n(,)30 b(146)n(].)42 b(Theorem)28 b(50)g(on)h(ma)5 b(jorization)24 b(for)29 b FN(C)1558 764 y FM(\003)1596 794 y FO(-algebras)d(and)j(Corol-)-118 893 y(lary)17 b(8)h(establishing)d(that)k(the)g(ma)5 b(jorization)14 b(of)19 b FP(\003)p FO(-algebras)c(is)j(a)g(quasi-order)-118 993 y(relation)23 b(are)j(also)f(outlined)f(there.)37 b(Pro)r(ofs)25 b(giv)n(en)f(in)i(the)h(b)r(o)r(ok)f(are)g(due)h(to)-118 1093 y(S.)h(P)n(op)r(o)n(vyc)n(h.)6 1198 y(In)43 b(the)h(b)r(o)r(ok)e (w)n(e)h(do)f(not)h(discuss)f(relations)d(b)r(et)n(w)n(een)k(the)g (notions)-118 1298 y(of)33 b(ma)5 b(jorization)28 b(and)k(Morita)f (equiv)-5 b(alence)30 b(\(on)j(Morita)e(equiv)-5 b(alence)30 b(for)-118 1398 y FP(\003)p FO(-algebras,)24 b(see)j([223)o(,)g(51)o(,) h(154)o(],)f(etc.\))6 1503 y(F)-7 b(or)19 b(represen)n(tations)d(of)j (\014nite-dimensional)14 b(algebras)i(\(and)j(for)f(a)h(wider)-118 1603 y(class)32 b(of)i(matrix)d(problems)h(as)h(w)n(ell\),)h(it)f(w)n (as)g(sho)n(wn)g(in)g([74],)i(that)g(these)-118 1702 y(problems)d(can)j(b)r(e)h(sub)r(divided)d(in)n(to)h(\\tame")g(and)h (\\wild")d(\(for)j(accurate)-118 1802 y(de\014nitions,)30 b(see)g([70)o(]\).)47 b(W)-7 b(e)31 b(do)g(not)g(discuss)e(here)i(what) f(it)h(means)e(that)i(a)-118 1902 y FP(\003)p FO(-algebra)22 b(is)j(tame;)h(ho)n(w)n(ev)n(er,)e(if)i(one)f(c)n(ho)r(oses)g(t)n(yp)r (e)h(I)g FP(\003)p FO(-algebras)d(\(or)i(ev)n(en)-118 2001 y(n)n(uclear)30 b FP(\003)p FO(-algebras\))e(to)j(b)r(e)h(\\)p FP(\003)p FO(-tame",)e(then)j(there)e(exists)f(a)i(large)d(set)j(of) -118 2101 y(in)n(termediate)d FP(\003)p FO(-algebras,)h(whic)n(h)i(are) f(neither)h FP(\003)p FO(-tame,)g(nor)g FP(\003)p FO(-wild)e(\(see,) -118 2201 y(e.g.,)d(Section)g(3.1.6\).)6 2306 y(In)e(Sections)f (3.1.3{3.1.6,)e(a)j(n)n(um)n(b)r(er)e(of)i(examples)d(of)j FP(\003)p FO(-wild)d(problems)-118 2406 y(are)g(giv)n(en.)34 b(F)-7 b(or)23 b(more)e(examples)g(of)i FP(\003)p FO(-wild)e(problems,) g(see)i(also)e([243)o(],)j([16)o(],)-118 2505 y([33)o(],)k(etc.)6 2642 y(3.1.3.)35 b(The)25 b(exp)r(osition)d(of)j(topics)f(on)g FP(\003)p FO(-wildness)e(of)j FP(\003)p FO(-algebras)c(gener-)-118 2741 y(ated)33 b(b)n(y)g(orthogonal)d(pro)5 b(jections)31 b(and)i(idemp)r(oten)n(ts)e(essen)n(tially)f(follo)n(ws)-118 2841 y([144)n(],)23 b([145)o(,)e(146)n(].)35 b(The)21 b(pro)r(of)g(of)g FP(\003)p FO(-wildness)d(of)j FP(\003)p FO(-algebras)c FA(R)1883 2853 y FK(5)p FL(;)p FK(2)1973 2841 y FO(,)23 b(and)d FA(R)2233 2862 y FK(5)p FL(;)2296 2839 y Fx(5)p 2296 2848 29 3 v 2296 2882 a(2)-118 2941 y FO(in)j(Subsection)h(5)f(of)i(3.1.3)e(is)g(giv)n(en)f(b)n(y)i(S.)h (Krugly)n(ak,)c(Y)-7 b(u.)25 b(Samo)-9 b(\025)-32 b(\020lenk)n(o)19 b(and)-118 3040 y(A.)28 b(Piry)n(atinsk)-5 b(a)n(y)n(a.)6 3177 y(3.1.4.)52 b(F)-7 b(or)33 b(facts)g(on)g FP(\003)p FO(-wildness)d(of)j(semi-linear)27 b(relations)j(\(Prop)r(osi-)-118 3276 y(tions)19 b(67,)g(68\))g(see)g([35)o(,)h(233)o(].)34 b(The)20 b(pro)r(of)g(giv)n(en)e(here)h(is)g(due)h(to)g(S.)g(Krugly)n (ak.)6 3413 y(3.1.5.)35 b FP(\003)p FO(-Wildness)22 b(of)i(description) e(of)i(pairs)e(of)j(self-adjoin)n(t)d(op)r(erators,)-118 3513 y FN(A)p FO(,)39 b FN(B)t FO(,)g(suc)n(h)d(that)h FN(B)587 3482 y FK(2)662 3513 y FO(=)g FN(I)7 b FO(,)39 b(up)e(to)f(a)g(unitary)f(equiv)-5 b(alence,)36 b(follo)n(ws)e(di-)-118 3612 y(rectly)28 b(from)h([144)n(].)43 b(W)-7 b(e)31 b(giv)n(e)c(a)j(simple)c(criterion)h(of)j FP(\003)p FO(-wildness)c(for) k(pairs)-118 3712 y(of)c(self-adjoin)n(t)e(op)r(erators)h(connected)h (b)n(y)h(a)f(quadratic)e(relation)g(\(A.)j(Piry-)-118 3811 y(atinsk)-5 b(a)n(y)n(a\),)25 b(or)h(b)n(y)g(a)h(cubic)f (semi-linear)21 b(relation)j(in)j(terms)e(of)i(co)r(e\016cien)n(ts)-118 3911 y(of)g(the)h(relation.)p eop %%Page: 240 244 240 243 bop -118 -137 a FO(240)560 b FJ(Chapter)27 b(3.)37 b(On)27 b(the)h(complexit)n(y)c(of)k(represen)n(tations)6 96 y FO(3.1.6.)36 b(W)-7 b(e)27 b(giv)n(e)e(only)h(the)h(simplest)d (examples)g(of)j FP(\003)p FO(-wild)d(groups.)36 b(The)-118 196 y(pro)r(of)20 b(of)g(Theorem)e(62)h(that)i(p)r(erio)r(dic)d(groups) h(are)g(not)h FP(\003)p FO(-wild)e(can)i(b)r(e)g(found)-118 296 y(in)27 b([128)n(].)6 407 y(In)g(Section)f(3.1,)g(w)n(e)h(listed)e (a)h(n)n(um)n(b)r(er)g(of)h(examples)d(of)j FP(\003)p FO(-algebras)c(and)-118 506 y(mappings)37 b FN( )12 b FO(:)32 b Fz(A)42 b FP(\000)-48 b(!)43 b FN(M)715 518 y FL(n)760 506 y FO(\()p Fz(B)p FO(\))d(suc)n(h)f(that)h(the)g(functor) f FN(F)1836 518 y FL( )1896 506 y FO(:)46 b(Rep)14 b Fz(B)43 b FP(\000)-48 b(!)-118 606 y FO(Rep)14 b Fz(A)31 b FO(is)g(full.)48 b(Ho)n(w)n(ev)n(er,)31 b(w)n(e)h(do)f(not)h(discuss) f(metho)r(ds)g(of)g(construction)-118 705 y(of)i(suc)n(h)h(mappings.)52 b(This)32 b(is)h(a)g(separate)f(topic;)k(it)d(needs)g(the)h(adv)-5 b(anced)-118 805 y(language)34 b(of)i FP(\003)p FO(-categories)c([225)o (],)39 b(and)d FP(\003)p FO(-quiv)n(ers)d([142)o(],)39 b([240)n(],)g(etc.)63 b(W)-7 b(e)-118 905 y(also)31 b(do)j(not)f (discuss)g(the)h(question)e(of)h(what)h(is)e(the)i(minimal)29 b(n)n(um)n(b)r(er)k FN(n)-118 1004 y FO(for)e(whic)n(h)f(there)h (exists)e(a)i(homomorphism)26 b FN( )12 b FO(:)29 b Fz(A)g FP(\000)-48 b(!)29 b FN(M)1782 1016 y FL(n)1826 1004 y FO(\()p Fz(B)p FO(\))k(suc)n(h)d(that)-118 1104 y(the)c(corresp)r (onding)c(functor)k FN(F)894 1116 y FL( )970 1104 y FO(is)e(full.)35 b(W)-7 b(e)26 b(only)e(notice)g(that)i(in)f([179)n(],)h(it)-118 1204 y(w)n(as)d(sho)n(wn)h(that)g(for)g(the)h FN(C)792 1173 y FM(\003)830 1204 y FO(-algebra)d FA(A)i FO(with)g FN(m)g FO(self-adjoin)n(t)e(generators,)-118 1303 y FN(a)-74 1315 y FK(1)-37 1303 y FO(,)35 b FN(:)14 b(:)g(:)28 b FO(,)37 b FN(a)250 1315 y FL(m)313 1303 y FO(,)g(the)f(algebra)c FN(M)903 1315 y FL(n)948 1303 y FO(\()p FA(A)p FO(\))k(for)f FN(n)g FP(\025)h(\000)p FO(3)22 b(+)1650 1238 y FP(p)p 1720 1238 258 4 v 1720 1303 a FO(9)17 b(+)h(8)p FN(m)p FO(,)37 b(is)d(singly)-118 1403 y(generated,)25 b(i.e.,)h(generated)f (b)n(y)h(a)f(pair)g(of)h(self-adjoin)n(t)d(generators.)34 b(It)27 b(w)n(as)-118 1502 y(sho)n(wn)32 b(in)f([215)o(])h(that)h(the)g (estimate)d FN(n)h FP(\025)1290 1437 y(p)p 1359 1437 216 4 v 65 x FN(m)18 b FP(\000)g FO(1)32 b(holds,)g(and)h(that)f(this) -118 1602 y(estimate)26 b(is)h(exact,)h(i.e.,)g(there)g(exists)f(a)h (comm)n(utativ)n(e)d FN(C)1775 1572 y FM(\003)1813 1602 y FO(-algebra)g FN(C)6 b FO(\()p FN(K)g FO(\))-118 1702 y(with)39 b FN(m)g FO(self-adjoin)n(t)e(generators)f(suc)n(h)j(that)h FN(M)1519 1714 y FL(n)1564 1702 y FO(\()p FN(C)6 b FO(\()p FN(K)g FO(\)\))40 b(is)e(not)h(singly)-118 1801 y(generated)26 b(for)h FN(n)c(<)548 1736 y FP(p)p 617 1736 V 65 x FN(m)18 b FP(\000)g FO(1.)-118 1984 y FQ(Section)26 b(3.2.)33 b FO(In)23 b(Section)f(3.2)f(w)n(e)h(consider)f(an)h(application)d(of)k (the)f(theory)-118 2084 y(of)f(represen)n(tations)d(of)j FP(\003)p FO(-algebras)c(to)j(a)h(study)g(of)g(classes)e(of)h(op)r (erators)f(that)-118 2184 y(are)26 b(singled)g(out)h(algebraically)-7 b(.)6 2295 y(The)24 b(problem)d(to)i(describ)r(e)f(the)i(class)d(of)i (op)r(erators)f(whic)n(h)g(satisfy)g(rela-)-118 2394 y(tions)c(up)h(to)g(a)g(unitary)e(equiv)-5 b(alence)17 b(is)h(equiv)-5 b(alen)n(t)17 b(to)h(the)i(one)f(of)g(describing)-118 2494 y(represen)n(tations)25 b(of)i(the)h(corresp)r(onding)d FP(\003)p FO(-algebra)f Fz(A)p FO(.)6 2605 y(F)-7 b(or)27 b(suc)n(h)f(algebras)e(w)n(e)i(estimate)f(the)i(complexit)n(y)c(of)k (the)g(corresp)r(ond-)-118 2704 y(ing)35 b(problem)e(of)j FP(\003)p FO(-represen)n(tation)d(theory)-7 b(,)37 b(i.e.,)g(the)g (complexit)n(y)32 b(of)k(the)-118 2804 y(unitary)26 b(description)f(of) j(the)g(corresp)r(onding)c(class)i(of)h(op)r(erators.)6 2915 y(The)32 b(con)n(ten)n(t)f(of)h(Section)f(3.2)g(is)f(directly)g (related)g(to)h(the)h(pap)r(ers)f([49,)-118 3014 y(50)o(,)f(104)n(,)g (195)o(,)g(286)n(,)g(287)n(,)g(21)o(,)g(22)o(,)g(23)o(,)g(53)o(])g(and) f(the)i(bibliograph)n(y)25 b(therein.)-118 3114 y(In)k(particular,)d (it)i(w)n(as)g(pro)n(v)n(en)f(that)i(in)f(an)n(y)g(factor)g(of)h (in\014nite)f(t)n(yp)r(e)h(there)-118 3214 y(exists)f(a)i(generator,)e (whic)n(h)h(is)g(a)h(partial)d(isometry)-7 b(,)27 b(and)j(whic)n(h)f (is,)g(at)h(the)-118 3313 y(same)21 b(time,)h(a)g(w)n(eakly)e(cen)n (tered,)j(or)f(h)n(yp)r(onormal,)e(or)h(subnormal,)g(etc.)h(op-)-118 3413 y(erator;)29 b(this)h(indicates)d(that)j(the)h(description)c(of)j (suc)n(h)g(op)r(erators)e(is)g(com-)-118 3513 y(plicated.)59 b(Moreo)n(v)n(er,)35 b(the)h(constructions)d(used)j(in)f(the)h(pro)r (ofs,)g(in)f(some)-118 3612 y(\(but)d(not)g(all\))d(cases)h(can)h(b)r (e)h(applied)d(to)j(the)f(pro)r(of)g(of)h FP(\003)p FO(-wildness)c(of)j (the)-118 3712 y(corresp)r(onding)38 b(class)g(of)j(op)r(erators)e (\(see)i(Section)e(3.2.3,)k(item)c(1\).)77 b(Es-)-118 3811 y(sen)n(tially)-7 b(,)26 b(the)j FP(\003)p FO(-wildness)d(of)j (the)g(corresp)r(onding)d(class)h(of)i(op)r(erators)e(\(for)-118 3911 y(example,)f(h)n(yp)r(onormal\))f(means)i(that)h(one)g(can)g(c)n (ho)r(ose)f(suc)n(h)h(an)f(op)r(erator)p eop %%Page: 241 245 241 244 bop -118 -137 a FJ(Commen)n(ts)25 b(to)j(Chapter)f(3)1452 b FO(241)-118 96 y(\(h)n(yp)r(onormal\))25 b(to)i(b)r(e)h(a)f (generator)f(of)h(the)h FP(\003)p FO(-wild)d FN(C)1610 66 y FM(\003)1649 96 y FO(-algebra.)6 221 y(3.2.1.)37 b(Simple)26 b(criteria)e(of)k FP(\003)p FO(-wildness)d(for)j(classes)d (of)j(non)g(self-adjoin)n(t)-118 321 y(op)r(erators,)h(for)h(whic)n(h)f FN(X)37 b FO(and)30 b FN(X)992 290 y FM(\003)1060 321 y FO(are)f(related)g(b)n(y)h(a)g(quadratic)e(or)h(cubic)-118 420 y(semilinear)23 b(relation,)h(follo)n(w)h(the)j(results)e(exp)r (ounded)i(in)f(Section)f(3.1.5.)6 520 y(Quasi-normal)c(op)r(erators)j (ha)n(v)n(e)h(a)g(rather)g(simple)e(structure)j([49)o(,)g(102)n(],)-118 619 y(etc.)37 b(The)27 b(study)h(of)g(the)g(classes)d(of)i(op)r (erators)f(considered)f(in)i(items)f(4)h(and)-118 719 y(5)g(of)h(3.2.1)e(is)g(also)g(not)i(to)r(o)f(complicated.)6 844 y(3.2.2.)65 b(On)38 b(the)f(complexit)n(y)d(of)k(the)f(unitary)f (description)f(of)j(partial)-118 943 y(isometries,)33 b(see)i([104)o(,)g(102)n(].)60 b(The)36 b FP(\003)p FO(-wildness)c(of)j (the)h(classes)d(of)i(w)n(eakly)-118 1043 y(cen)n(tered)21 b(op)r(erators)f(and)i(w)n(eakly)e(cen)n(tered)i(partial)d(isometries)f (are)j(pro)n(v)n(ed)-118 1143 y(in)27 b([143)n(,)h(201)o(].)6 1242 y(On)h(algebraic)d(op)r(erators,)i(see)h([34)o(].)42 b(In)29 b([33)o(],)h(the)g(pro)r(of)e(of)i FP(\003)p FO(-wildness)-118 1342 y(of)36 b(the)g(description)e(of)i(non)g (self-adjoin)n(t)d(op)r(erators,)j FN(X)7 b FO(,)38 b FN(X)1892 1312 y FM(\003)1929 1342 y FO(,)h(suc)n(h)c(that)-118 1441 y([)p FN(X)-19 1411 y FL(j)15 1441 y FN(X)91 1411 y FM(\003)129 1405 y FL(j)164 1441 y FN(;)14 b(X)277 1411 y FM(\003)314 1405 y FL(k)355 1441 y FN(X)431 1411 y FL(k)471 1441 y FO(])32 b(=)f([)p FN(X)721 1411 y FL(j)755 1441 y FN(X)831 1411 y FM(\003)869 1405 y FL(j)904 1441 y FN(;)14 b(X)1017 1411 y FL(k)1057 1441 y FN(X)1133 1411 y FM(\003)1170 1405 y FL(k)1211 1441 y FO(])32 b(=)f([)p FN(X)1461 1411 y FM(\003)1499 1405 y FL(j)1534 1441 y FN(X)1610 1411 y FL(j)1644 1441 y FN(;)14 b(X)1757 1411 y FM(\003)1794 1405 y FL(k)1835 1441 y FN(X)1911 1411 y FL(k)1951 1441 y FO(])32 b(=)f(0,)j(1)d FP(\024)-118 1541 y FN(j;)14 b(k)29 b FP(\024)c FN(n)p FO(,)30 b(is)e(giv)n(en)f (for)i(a)g(\014xed)g FN(n)c FP(\025)h FO(1.)41 b(On)29 b(the)h(other)e(hand,)i(the)f(class)f(of)-118 1641 y(cen)n(tered)i(op)r (erators)f(\(for)h(whic)n(h)g(these)h(relations)c(hold)j(for)g(all)e FN(j)5 b FO(,)32 b FN(k)f FP(\025)d FO(1\),)-118 1740 y(is)e(not)i FP(\003)p FO(-wild)d(\(see)j(2.5.2\).)6 1865 y(3.2.3.)79 b(As)42 b(A.)h(Piry)n(atinsk)-5 b(a)n(y)n(a)37 b(noticed,)44 b(the)f(construction)d(in)h([287)o(])-118 1964 y(means)26 b(essen)n(tially)e(that)j(the)h(class)e(of)i(h)n(yp)r (onormal)c(op)r(erators)h(is)i FP(\003)p FO(-wild.)6 2064 y(The)33 b FP(\003)p FO(-wildness)d(of)j(pairs)d(of)j(comm)n (uting)c(partial)h(isometries)f(can)j(b)r(e)-118 2164 y(obtained)f(directly)f(from)h([31)o(].)50 b(The)33 b(pro)r(of)e(giv)n (en)g(in)g(the)i(b)r(o)r(ok)e(is)g(due)i(to)-118 2263 y(D.)28 b(Proskurin.)p eop %%Page: 242 246 242 245 bop -118 -137 a FO(242)p eop %%Page: 243 247 243 246 bop -118 664 a FR(Bibliograph)l(y)-48 1077 y FD([1])41 b(B.)26 b(Ab)r(desselam,)i(J.)e(Bec)n(k)n(ers,)g(A.)g(Chak)l (abarti,)i(and)f(N.)f(Deb)r(ergh,)h Fh(On)g(a)h(defor-)68 1156 y(mation)i(of)f Fp(sl)q FD(\(2\))h Fh(with)f(p)l(ar)l(agr)l (assmanian)34 b(variables)p FD(,)28 b(J.)f(Ph)n(ys.)g(A:)g(Math.)g (Gen.)68 1235 y Fg(29)c FD(\(1996\),)i(6729{6736.)-48 1319 y([2])41 b(S.)30 b(I.)h(Adian,)h Fh(The)g(Burnside)h(pr)l(oblem)g (and)g(identities)e(in)g(gr)l(oups)p FD(,)i(Ergebnisse)68 1398 y(der)24 b(Math.)g(und)g(ihrer)g(Grenzb)r(eb)r(eite,)i(v)n(ol.)e (95,)g(Springer,)g(Berlin,)g(1979,)g(T)-6 b(ransl.)68 1477 y(from)23 b(Russian)i(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)f (1975.)-48 1561 y([3])41 b(N.)22 b(I.)g(Akhiezer,)i Fh(Classic)l(al)h (moment)g(pr)l(oblem)p FD(,)f(Oliv)n(er)f(and)g(Bo)n(yd,)f(1965,)h(T)-6 b(ransl.)68 1640 y(from)23 b(Russian)i(edn:)31 b(Fizmatgiz,)26 b(Mosco)n(w,)e(1961.)-48 1724 y([4])41 b(N.)18 b(I.)f(Akhiezer)i(and)g (I.)e(M.)g(Glazman,)k Fh(The)f(the)l(ory)g(of)h(line)l(ar)g(op)l(er)l (ators)h(in)e(Hilb)l(ert)68 1803 y(sp)l(ac)l(e)p FD(,)28 b(Ungar,)e(New)g(Y)-6 b(ork,)26 b(1961,)h(T)-6 b(ransl.)26 b(from)f(Russian)h(edn.:)36 b(Gostekhizdat,)68 1882 y(Mosco)n(w,)24 b(1950.)-48 1966 y([5])41 b(S.)18 b(A.)e(Amitsur)i(and)g(J.)f (Levitski,)k Fh(Minimal)f(identities)f(for)h(algebr)l(as)p FD(,)f(Pro)r(c.)e(Amer.)68 2045 y(Math.)24 b(So)r(c.)g Fg(1)f FD(\(1950\),)i(449{463.)-48 2129 y([6])41 b(H.)19 b(Araki,)g Fh(Hamiltonian)k(formalism)f(and)g(c)l(anonic)l(al)h(r)l (elations)f(in)f(quantum)h(\014eld)68 2208 y(the)l(ory)p FD(,)i(J.)f(Math.)h(Ph)n(ys.)f Fg(1)h FD(\(1960\),)h(no.)e(4,)h (492{504.)-48 2292 y([7])41 b(R.)28 b(Arens,)g Fh(R)l(epr)l (esentations)j(of)f FC(\003)p Fh(-algebr)l(as)p FD(,)g(Duk)n(e)e(Math.) h(J.)e Fg(14)h FD(\(1947\),)i(269{)68 2371 y(282.)-48 2455 y([8])41 b(W.)33 b(B.)e(Arv)n(eson,)j Fh(Op)l(er)l(ator)h(algebr)l (as)g(and)f(invariant)g(subsp)l(ac)l(es)p FD(,)h(Ann)d(Math.)68 2534 y Fg(100)23 b FD(\(1974\),)i(433{532.)-48 2618 y([9])p 68 2618 212 4 v 265 w(,)36 b Fh(A)n(n)g(invitation)f(to)h Fp(C)960 2594 y Ff(\003)996 2618 y Fh(-algebr)l(as)p FD(,)h(Graduate)f(texts)f(in)h(mathematics,)68 2697 y(v)n(ol.)25 b(39,)e(Springer,)h(Berlin,)h(1976.)-83 2781 y([10])p 68 2781 V 265 w(,)37 b Fh(The)g(harmonic)g(analysis)g(of)g (automorphism)i(gr)l(oups)p FD(,)g(Pro)r(c.)c(Symp.)68 2860 y(Pure)24 b(Math.)g Fg(38)f FD(\(1982\),)i(199{269.)-83 2944 y([11])p 68 2944 V 265 w(,)33 b Fh(Continuous)h(analo)l(gues)h(of) f(Fo)l(ck)g(sp)l(ac)l(e)p FD(,)g(Mem.)d(Amer.)g(Math.)h(So)r(c.,)68 3022 y(v)n(ol.)25 b(80,)e(Amer.)g(Math.)h(So)r(c.,)f(Pro)n(vidence,)j (R.I.,)c(1989.)-83 3107 y([12])41 b(V.)30 b(Arzumanian)i(and)f(A.)f(V) -6 b(ershik,)33 b Fh(Star-algebr)l(as)f(asso)l(ciate)l(d)i(with)f (endomor-)68 3185 y(phisms)p FD(,)22 b(Op)r(erator)f(algebras)g(and)g (group)f(represen)n(tations,)j(Pro)r(c)d(In)n(t.)h(Conf.,)f(v)n(ol.)68 3264 y(I.)k(\(Boston\),)h(Pitman,)f(1984,)h(pp.)e(17{27.)-83 3348 y([13])41 b(M.)22 b(Auslander,)i(I.)e(Reiten,)j(and)e(S.)g(O.)f (Smalo,)h Fh(R)l(epr)l(esentation)j(the)l(ory)f(of)g(Artin)68 3427 y(algebr)l(as)p FD(,)19 b(Cam)n(bridge)e(Studies)h(in)f(Adv)l (anced)g(Mathematics,)j(v)n(ol.)d(36,)g(Cam)n(bridge)68 3506 y(Univ.)24 b(Press,)f(1995.)-83 3590 y([14])41 b(T.)27 b(Y)-6 b(a.)26 b(Azizo)n(v)j(and)e(I.)g(S.)g(Ioh)n(vido)n(v,)i Fh(Line)l(ar)g(op)l(er)l(ators)i(in)d(sp)l(ac)l(es)i(with)f(an)g(in-)68 3669 y(de\014nite)c(mertic)p FD(,)d(J.)g(Wiley)j(and)e(Sons,)g(New)g(Y) -6 b(ork,)23 b(1989,)g(T)-6 b(ransl.)23 b(from)f(Russian)68 3748 y(edn.:)32 b(Nauk)l(a,)24 b(Mosco)n(w,)f(1986.)-83 3832 y([15])41 b(O.)21 b(V.)g(Bagro,)h Fh(Pairs)i(of)g(self-adjoint)g (op)l(er)l(ators)i(c)l(onne)l(cte)l(d)e(by)f(a)h(cubic)f(r)l(elation)p FD(,)68 3911 y(Ukrain.)h(Mat.)g(Zh.)f Fg(47)g FD(\(1995\),)i(no.)f(5,)f (600{602,)i(\(Russian\).)1048 4121 y FO(243)p eop %%Page: 244 248 244 247 bop -118 -137 a FO(244)1866 b FJ(Bibliograph)n(y)-83 96 y FD([16])41 b(O.)18 b(V.)g(Bagro)h(and)g(S.)f(A.)g(Krugly)n(ak,)j Fh(R)l(epr)l(esentations)g(of)h(involutive)e(quivers)h(and)68 175 y(wild)27 b(pr)l(oblems)p FD(,)e(Preprin)n(t)f(KVIUS,)f(Kiev,)h (1995,)h(\(Russian\).)-83 265 y([17])p 68 265 212 4 v 265 w(,)34 b Fh(R)l(epr)l(esentations)g(of)g(D.)f(Fairlie)h(algebr)l (as)p FD(,)g(Preprin)n(t)f(KVIUS,)f(Kiev,)68 343 y(1996,)25 b(\(Russian\).)-83 433 y([18])41 b(B.)36 b(A.)f(Barnes,)k Fh(A)e(note)g(on)h(sep)l(ar)l(ating)g(families)g(of)f(r)l(epr)l (esentations)p FD(,)j(Pro)r(c.)68 512 y(Amer.)23 b(Math.)h(So)r(c.)g Fg(87)f FD(\(1983\),)i(95{98.)-83 601 y([19])41 b(H.)29 b(Bart,)h(T.)e(Ehrhardt,)i(and)g(B.)e(Silb)r(ermann,)k Fh(Zer)l(o)f(sums)g(of)g(idemp)l(otents)h(in)68 680 y(Banach)27 b(algebr)l(as)p FD(,)d(In)n(tegr.)h(Equat.)f(Op)r(er.)f(Theory)i Fg(19)e FD(\(1994\),)i(125{134.)-83 769 y([20])41 b(A.)22 b(O.)f(Barut)i(and)f(R.)g(Raczk)l(a,)h Fh(The)l(ory)i(of)f(gr)l(oup)i (r)l(epr)l(esentations)f(and)g(applic)l(a-)68 848 y(tions)p FD(,)f(PWN,)f(W)-6 b(arsza)n(w)n(a,)25 b(1977.)-83 937 y([21])41 b(H.)24 b(Benc)n(k)n(e,)i Fh(Gener)l(ators)h(of)f Fp(W)956 914 y Ff(\003)992 937 y Fh(-algebr)l(as)p FD(,)e(T)-6 b(ohoku)26 b(Math.)e(J.)g Fg(22)f FD(\(1970\),)j(541{)68 1016 y(546.)-83 1105 y([22])p 68 1105 V 265 w(,)f Fh(Gener)l(ators)j (of)f Fp(W)869 1082 y Ff(\003)905 1105 y Fh(-algebr)l(as)g(II)p FD(,)e(T)-6 b(ohoku)27 b(Math.)e(J.)g Fg(24)f FD(\(1972\),)j(371{)68 1184 y(381.)-83 1273 y([23])p 68 1273 V 265 w(,)22 b Fh(Gener)l(ators)j(of)g Fp(W)861 1250 y Ff(\003)896 1273 y Fh(-algebr)l(as)g(III)p FD(,)e(T)-6 b(ohoku)24 b(Math.)e(J.)h Fg(24)e FD(\(1972\),)j(383{)68 1352 y(388.)-83 1441 y([24])41 b(Y)-6 b(u.)22 b(M.)g(Berezansky)-6 b(,)23 b Fh(Exp)l(ansion)j(in)e (eigenfunctions)g(of)g(self-adjoint)h(op)l(er)l(ators)p FD(,)68 1520 y(T)-6 b(ransl.)22 b(Math.)f(Monogr.,)h(v)n(ol.)g(17,)g (Amer.)e(Math.)i(So)r(c.,)f(Pro)n(vidence,)j(R.I.,)d(1968,)68 1599 y(T)-6 b(ransl.)24 b(from)f(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a) 25 b(Dumk)l(a,)e(Kiev,)h(1965.)-83 1688 y([25])p 68 1688 V 265 w(,)32 b Fh(Self-adjoint)h(op)l(er)l(ators)i(in)d(sp)l(ac)l(es)i (of)f(functions)g(of)f(in\014nitely)g(many)68 1767 y(variables)p FD(,)20 b(T)-6 b(rans.)18 b(Math.)g(Monographs,)i(v)n(ol.)f(63,)h(AMS,) d(Pro)n(vidence,)k(R.I.,)e(1986,)68 1846 y(T)-6 b(ransl.)24 b(from)f(Russian)h(edn.:)32 b(Kiev,)24 b(Nauk)n(o)n(v)l(a)h(Dumk)l(a,)e (1978.)-83 1935 y([26])41 b(Y)-6 b(u.)30 b(M.)e(Berezansky)j(and)f(Y)-6 b(u.)29 b(G.)g(Kondrat'ev,)i Fh(Sp)l(e)l(ctr)l(al)i(metho)l(ds)f(in)f (in\014nite)68 2014 y(dimensional)e(analysis)p FD(,)d(Klu)n(w)n(er)f (Acad.)h(Publ.,)f(Dordrec)n(h)n(t,)h(1992,)g(T)-6 b(ransl.)25 b(from)68 2093 y(Russian)g(edn.:)31 b(Nauk)n(o)n(v)l(a)25 b(Dumk)l(a,)f(Kiev,)g(1988.)-83 2182 y([27])41 b(Y)-6 b(u.)32 b(M.)f(Berezansky)-6 b(,)34 b(G.)e(Lassner,)h(and)f(V.)f(S.)h (Y)-6 b(ak)n(o)n(vlev,)35 b Fh(On)e(de)l(c)l(omp)l(osition)68 2261 y(of)f(p)l(ositive)f(functionals)g(on)h(c)l(ommutative)f(nucle)l (ar)h FC(\003)p Fh(-algebr)l(as)p FD(,)f(Ukrain.)f(Mat.)68 2340 y(Zh)n(urn.)24 b Fg(39)f FD(\(1987\),)i(no.)f(5,)f(638{641,)i (\(Russian\).)-83 2429 y([28])41 b(Y)-6 b(u.)25 b(M.)f(Berezansky)-6 b(,)27 b(V.)d(L.)h(Ostro)n(vsky)-8 b(\025)-27 b(\020,)26 b(and)g(Y)-6 b(u.)25 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)28 b Fh(Eigenfunc-)68 2508 y(tion)e(exp)l(ansion)g(of)g(families)f(of)h(c) l(ommuting)g(op)l(er)l(ators)h(and)f(r)l(epr)l(esentations)g(of)68 2587 y(c)l(ommutation)31 b(r)l(elations)p FD(,)d(Ukr.)f(Math.)g(Zh.)g Fg(40)f FD(\(1988\),)j(no.)e(1,)h(106{109,)i(\(Rus-)68 2666 y(sian\).)-83 2755 y([29])41 b(Y)-6 b(u.)23 b(M.)e(Berezansky)-6 b(,)24 b(Z.)e(G.)g(Sheftel,)i(and)f(G.)g(F.)f(Us,)g Fh(F)-5 b(unctional)25 b(analysis)g(I,)g(II)p FD(,)68 2834 y(Op)r(erator)f (Theory)-6 b(,)23 b(Adv.)g(Appl.,)g(v)n(ol.)h(85,)f(86,)h(Birkh\177)-35 b(auser)23 b(V)-6 b(erlag,)24 b(Basel,)g(1996,)68 2913 y(T)-6 b(ransl.)24 b(from)f(Russian)h(edn.:)32 b(Vyshc)n(ha)24 b(Shk)n(ola,)h(Kiev,)f(1990.)-83 3002 y([30])41 b(F.)29 b(A.)f(Berezin)h(and)h(G.)e(I.)h(Kats,)g Fh(Lie)h(gr)l(oups)i(with)e(c) l(ommuting)h(and)g(antic)l(om-)68 3081 y(muting)24 b(p)l(ar)l(ameters)p FD(,)e(Matem.)f(Sb)r(ornik)i Fg(82)d FD(\(1970\),)j(no.)e(3,)g (343{359,)i(\(Russian\).)-83 3170 y([31])41 b(C.)33 b(A.)g(Berger,)h (L.)f(A.)g(Coburn,)i(and)f(A.)e(Leb)r(o)n(w,)k Fh(R)l(epr)l(esentation) f(and)h(index)68 3249 y(the)l(ory)21 b(for)g Fp(C)443 3226 y Ff(\003)479 3249 y Fh(-algebr)l(as)g(gener)l(ate)l(d)g(by)f(c)l (ommuting)h(isometries)p FD(,)e(J.)f(F)-6 b(unct.)19 b(Anal.)68 3328 y Fg(27)k FD(\(1978\),)i(51{99.)-83 3417 y([32])41 b(Y)-6 b(u.)23 b(N.)f(Bespalo)n(v,)i Fh(Col)t(le)l(ctions)h (of)g(orthopr)l(oje)l(ctions)i(satisfying)d(r)l(elations)p FD(,)f(Ukr.)68 3496 y(Mat.)h(Zh)n(urn.)f Fg(44)g FD(\(1992\),)i(no.)f (3,)f(309{317,)j(\(Russian\).)-83 3585 y([33])p 68 3585 V 265 w(,)21 b Fh(A)n(lgebr)l(aic)j(op)l(er)l(ators,)i(p)l(artial)f (isometries,)f(and)g(wild)h(pr)l(oblems)p FD(,)d(Meth-)68 3664 y(o)r(ds)i(F)-6 b(unct.)25 b(Anal.)f(T)-6 b(op)r(ol.)24 b Fg(3)g FD(\(1997\),)h(no.)e(1,)h(28{45.)-83 3753 y([34])41 b(Y)-6 b(u.)25 b(N.)f(Bespalo)n(v)j(and)e(Y)-6 b(u.)25 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)28 b Fh(A)n(lgebr)l(aic)f(op) l(er)l(ators)i(and)e(p)l(airs)h(of)68 3832 y(selfadjoint)34 b(op)l(er)l(ators)h(c)l(onne)l(cte)l(d)f(by)f(a)g(p)l(olynomial)i(r)l (elation)p FD(,)f(F)-6 b(unct.)33 b(Anal.)f(i)68 3911 y(Prilozhen.)26 b Fg(25)d FD(\(1991\),)i(no.)e(4,)h(72{74,)g (\(Russian\).)p eop %%Page: 245 249 245 248 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(245)-83 96 y FD([35])41 b(Y)-6 b(u.)34 b(N.)f(Bespalo)n(v,)k(Y)-6 b(u.)33 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)38 b(and)d(V.)d(S.)i(Sh)n(ul'man,)j Fh(On)d(families)68 175 y(of)29 b(op)l(er)l(ators)i(c)l(onne)l(cte)l(d)f(by)e(semi-line)l (ar)h(r)l(elations)p FD(,)f(Applications)j(of)c(Metho)r(ds)68 254 y(of)k(F)-6 b(unctional)35 b(Analysis)d(in)g(Mathematical)i(Ph)n (ysics,)g(Inst.)d(Math.)g(Acad.)h(Sci.)68 333 y(Ukraine,)25 b(Kiev,)f(1991,)g(\(Russian\),)h(pp.)f(28{51.)-83 413 y([36])41 b(L.)34 b(C.)e(Biedenharn,)38 b Fh(The)d(quantum)g(gr)l(oup)h Fp(su)1413 421 y Fe(q)1447 413 y FD(\(2\))g Fh(and)g(a)f Fp(q)r Fh(-analo)l(gue)h(of)f(the)68 492 y(b)l(oson)27 b(op)l(er)l(ators)p FD(,)f(J.)d(Ph)n(ys.)g(A)h Fg(22)f FD(\(1989\),)i(L873{L878.)-83 572 y([37])41 b(M.)29 b(Sh.)h(Birman)f (and)i(M.)d(A.)h(Solom)n(y)n(ak,)k Fh(The)e(sp)l(e)l(ctr)l(al)i(the)l (ory)e(of)g(self-adjoint)68 651 y(op)l(er)l(ators)d(in)e(Hilb)l(ert)f (sp)l(ac)l(e)p FD(,)f(Izdat.)h(Leningrad.)g(Univ,)f(Leningrad,)h(1980.) -83 731 y([38])41 b(B.)23 b(E.)g(Blac)n(k)l(adar,)h Fh(A)h(simple)h (unital)g(pr)l(oje)l(ctionless)g Fp(C)1597 708 y Ff(\003)1633 731 y Fh(-algebr)l(a)p FD(,)d(J.)g(Op)r(er.)g(The-)68 810 y(ory)i Fg(5)e FD(\(1981\),)i(63{71.)-83 890 y([39])41 b(V.)17 b(M.)f(Bondarenk)n(o)i(and)g(Y)-6 b(u.)17 b(A.)f(Drozd,)i Fh(R)l(epr)l(esentations)j(typ)l(e)e(of)h(\014nite)e(gr)l(oups)p FD(,)68 969 y(Zap.)24 b(Nauc)n(hn.)g(Sem.)g(LOMI)g Fg(71)e FD(\(1977\),)k(24{42,)e(\(Russian\).)-83 1049 y([40])41 b(A.)27 b(B\177)-35 b(ottc)n(her,)30 b(I.)d(Goh)n(b)r(erg,)i(Y)-6 b(u.)27 b(Karlo)n(vic)n(h,)j(N.)d(Krupnik,)h(S.)f(Ro)r(c)n(h,)i(B.)e (Silb)r(er-)68 1128 y(man,)c(and)f(I.)g(Spitk)n(o)n(vsky)-6 b(,)24 b Fh(Banach)i(algebr)l(as)f(gener)l(ate)l(d)f(by)g Fp(N)31 b Fh(idemp)l(otents)25 b(and)68 1207 y(applic)l(ations)p FD(,)h(Op)r(erator)e(Theory)g(Adv.)f(Appl.)h Fg(90)f FD(\(1996\),)i(19{54.)-83 1287 y([41])41 b(N.)23 b(Bourbaki,)i Fh(Gr)l(oup)l(es)i(et)e(algebr)l(es)h(de)g(Lie)f(IV{VI)p FD(,)f(Hermann,)f(P)n(aris,)h(1968.)-83 1367 y([42])41 b(M.)25 b(Bo)6 b(_)-25 b(zejk)n(o)26 b(and)g(R.)f(Sp)r(eic)n(her,)j Fh(A)n(n)g(example)g(of)g(a)f(gener)l(alize)l(d)i(Br)l(ownian)g(mo-)68 1446 y(tion.)p FD(,)24 b(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fg(37)g FD(\(1991\),)i(519{531.)-83 1526 y([43])p 68 1526 212 4 v 265 w(,)i Fh(Completely)i(p)l(ositive)g(maps)h(on)f (Coxeter)f(gr)l(oups,)j(deforme)l(d)f(c)l(ommu-)68 1605 y(tation)c(r)l(elations,)h(and)f(op)l(er)l(ator)i(sp)l(ac)l(es)p FD(,)c(Math.)g(Ann.)f Fg(300)g FD(\(1994\),)i(97{120.)-83 1685 y([44])41 b(O.)22 b(Bratteli)i(and)f(P)-6 b(.)22 b(E.)g(T.)g(J\034rgensen,)h Fh(Endomorphisms)28 b(of)c Fd(B)p FD(\()p Fd(H)q FD(\))p Fh(.)g(II.)h(Finitely)68 1764 y(c)l(orr)l(elate)l(d)j(states)d(on)h Fd(O)746 1772 y Fe(n)789 1764 y FD(,)d(J.)g(F)-6 b(unct.)25 b(Anal.)f Fg(145)e FD(\(1997\),)k(no.)d(2,)h(323{373.)-83 1844 y([45])p 68 1844 V 265 w(,)39 b Fh(Isometries,)i(shifts,)g(Cuntz)d (algebr)l(as)h(and)f(multir)l(esolution)h(wavelet)68 1923 y(analysis)27 b(of)f(sc)l(ale)g Fp(N)7 b FD(,)23 b(In)n(t.)h(Equat.)h(Op)r(er.)e(Theory)i Fg(28)e FD(\(1997\),)i (382{443.)-83 2003 y([46])p 68 2003 V 265 w(,)d Fh(Iter)l(ate)l(d)j (function)g(systems)f(and)h(p)l(ermutation)h(r)l(epr)l(esentations)f (of)g(the)68 2082 y(Cuntz)h(algebr)l(a)p FD(,)e(Mem.)f(Amer.)g(Math.)g (So)r(c.,)h(AMS,)f(1998.)-83 2162 y([47])41 b(O.)19 b(Bratteli,)j(P)-6 b(.)18 b(E.)h(T.)g(J\034rgensen,)h(and)g(G.)f(L.)g(Price,)h Fh(Endomorphisms)25 b(of)c Fd(B)p FD(\()p Fd(H)q FD(\),)68 2241 y(Pro)r(c.)j(Symp.)f(Pure)h(Math.)g Fg(59)e FD(\(1996\),)k (93{138.)-83 2321 y([48])41 b(O.)17 b(Bratteli)i(and)e(D.)f(W.)h (Robinson,)i Fh(Op)l(er)l(ator)i(algebr)l(as)f(and)g(quantum)g (statistic)l(al)68 2400 y(me)l(chanics)p FD(,)25 b(Bo)r(oks)f(and)g (Monographs)g(in)g(Ph)n(ysics,)h(Springer-V)-6 b(elag,)25 b(1979.)-83 2480 y([49])41 b(A.)22 b(Bro)n(wn,)f Fh(On)j(a)g(class)h (of)f(op)l(er)l(ators)p FD(,)g(Pro)r(c)e(Amer.)f(Math.)h(So)r(c.)g Fg(4)g FD(\(1953\),)h(723{)68 2559 y(728.)-83 2639 y([50])p 68 2639 V 265 w(,)30 b Fh(The)g(unitary)g(e)l(quivalenc)l(e)h(of)g (binormal)g(op)l(er)l(ators)p FD(,)h(Amer.)c(J.)g(Math.)68 2718 y Fg(76)23 b FD(\(1954\),)i(no.)f(2,)g(414{434.)-83 2798 y([51])41 b(L.)32 b(Bro)n(wn,)i(P)-6 b(.)31 b(Green,)k(and)e(M.)e (A.)g(Rie\013el,)36 b Fh(Stable)d(isomorphism)j(ans)e(str)l(ong)68 2877 y(Morita)26 b(e)l(qivalenc)l(e)g(of)g Fp(C)777 2854 y Ff(\003)813 2877 y Fh(-algebr)l(as)p FD(,)d(P)n(aci\014c)j(J.)d (Math.)h Fg(71)f FD(\(1977\),)i(349{363.)-83 2957 y([52])41 b(I.)34 b(M.)f(Burban)h(and)g(A.)f(U.)g(Klim)n(yk,)k Fh(On)d(sp)l(e)l(ctr)l(al)j(pr)l(op)l(erties)f(of)f Fp(q)r Fh(-oscil)t(lator)68 3036 y(op)l(er)l(ators)p FD(,)26 b(Lett.)e(Math.)g(Ph)n(ys.)f Fg(29)g FD(\(1993\),)i(13{18.)-83 3116 y([53])41 b(S.)25 b(L.)f(Campb)r(ell,)i Fh(Line)l(ar)h(op)l(er)l (ators)h(for)f(which)g Fp(T)1486 3093 y Ff(\003)1522 3116 y Fp(T)36 b Fh(and)28 b Fp(T)10 b(T)1841 3093 y Ff(\003)1903 3116 y Fh(c)l(ommute)27 b(\(II\))p FD(,)68 3195 y(P)n(aci\014c)f(Journ.)d(Math.)h Fg(53)f FD(\(1974\),)i(no.)f(2,) f(355{361.)-83 3275 y([54])41 b(V.)21 b(Chari)i(and)f(A.)f(N.)g (Pressly)-6 b(,)22 b Fh(A)h(guide)h(to)g(quantum)g(gr)l(oups)p FD(,)f(Cam)n(bridge)g(Univ.)68 3354 y(Press,)g(Cam)n(bridge,)i(1994.) -83 3434 y([55])41 b(M.)17 b(D.)h(Choi,)h Fh(The)i(ful)t(l)f Fp(C)761 3411 y Ff(\003)797 3434 y Fh(-algebr)l(a)g(of)h(the)f(fr)l(e)l (e)g(gr)l(oup)i(of)e(two)h(gener)l(ators)p FD(,)e(P)n(aci\014c)68 3513 y(J.)24 b(Math.)f Fg(87)g FD(\(1980\),)j(no.)d(1,)h(41{48.)-83 3593 y([56])41 b(J.)22 b(M.)f(Cohen,)h Fp(C)544 3570 y Ff(\003)580 3593 y Fh(-algebr)l(as)j(without)f(idemp)l(otents)p FD(,)f(J.)e(F)-6 b(unct.)23 b(Anal.)f Fg(33)f FD(\(1979\),)68 3672 y(211{216.)-83 3752 y([57])41 b(A.)23 b(Connes,)h Fh(Non-c)l(ommutative)j(ge)l(ometry)p FD(,)c(Acad.)g(Press,)g(New)h(Y) -6 b(ork,)23 b(1994.)-83 3832 y([58])41 b(J.)32 b(Cun)n(tz,)i Fh(Simple)f Fp(C)690 3809 y Ff(\003)726 3832 y Fh(-algebr)l(as)g(gener) l(ate)l(d)g(by)f(isometries)p FD(,)h(Comm)n(un.)e(Math.)68 3911 y(Ph)n(ys.)24 b Fg(57)f FD(\(1977\),)i(173{185.)p eop %%Page: 246 250 246 249 bop -118 -137 a FO(246)1866 b FJ(Bibliograph)n(y)-83 96 y FD([59])41 b(Ch.)28 b(W.)g(Curtis)g(and)h(I.)f(Reiner,)h Fh(R)l(epr)l(esentation)i(the)l(ory)f(of)g(\014nite)f(gr)l(oups)i(and) 68 175 y(asso)l(ciative)c(algebr)l(as)p FD(,)d(Wiley)-6 b(,)25 b(New)f(Y)-6 b(ork,)23 b(1962.)-83 263 y([60])41 b(Y)-6 b(u.)20 b(L.)f(Daletski)-8 b(\025)-27 b(\020,)23 b Fh(F)-5 b(unctional)23 b(inte)l(gr)l(als)g(r)l(elate)l(d)g(with)g(op) l(er)l(ator)h(evolution)e(e)l(qua-)68 342 y(tions)p FD(,)i(Usp)r(ekhi)h (Mat.)e(Nauk)h Fg(17)f FD(\(1962\),)i(no.)f(5,)f(3{115,)i(\(Russian\).) -83 430 y([61])41 b(A.)20 b(Y)-6 b(u.)20 b(Daletsky)h(and)g(Y)-6 b(u.)20 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)24 b Fh(A)e(nonc)l(ommutative)h(moment)g(pr)l(ob-)68 509 y(lem)p FD(,)h(F)-6 b(un)n(ts.)24 b(Anal.)g(Prilozh.)h Fg(21)e FD(\(1987\),)i(no.)e(2,)h(72{73,)g(\(Russian\).)-83 596 y([62])41 b(E.)26 b(V.)f(Damaskinsky)j(and)e(P)-6 b(.)26 b(P)-6 b(.)25 b(Kulish,)j Fh(Deforme)l(d)g(oscil)t(lators)h(and) g(their)e(ap-)68 675 y(plic)l(ations)p FD(,)e(Zap.)f(Nauc)n(hn.)g(Sem.) f(LOMI)h Fg(189)f FD(\(1991\),)i(37{74,)f(\(Russian\).)-83 763 y([63])41 b(C.)25 b(Dask)l(alo)n(y)n(anis,)k Fh(Gener)l(alize)l(d)f (deforme)l(d)h(oscil)t(lator)f(and)g(nonline)l(ar)h(algebr)l(as)p FD(,)68 842 y(J.)24 b(Ph)n(ys.)f(A)h Fg(24)f FD(\(1991\),)i(L789{L794.) -83 930 y([64])41 b(K.)26 b(R.)g(Da)n(vidson,)i Fp(C)654 906 y Ff(\003)690 930 y Fh(-algebr)l(as)g(by)g(example)p FD(,)f(Amer.)f(Math.)g(So)r(c.,)h(Pro)n(vidence,)68 1008 y(R.I.,)c(1997.)-83 1096 y([65])41 b(C.)26 b(Da)n(vis,)i Fh(Sep)l(ar)l(ation)i(of)e(two)h(line)l(ar)g(subsp)l(ac)l(es)p FD(,)f(Acta)g(Sci.)f(Math.)f(Szeged)i Fg(19)68 1175 y FD(\(1958\),)e(172{187.)-83 1263 y([66])41 b(C.)26 b(Delb)r(ecq)h(and)f (C.)f(Quesne,)i Fh(A)g(cubic)g(deformation)i(of)f Fp(su)p FD(\(2\),)e(Mo)r(dern)g(Ph)n(ys.)68 1342 y(Lett.)f(A)e Fg(8)g FD(\(1993\),)j(961{966.)-83 1429 y([67])p 68 1429 212 4 v 265 w(,)d Fh(R)l(epr)l(esentation)k(the)l(ory)f(and)h Fp(q)r Fh(-b)l(oson)f(r)l(e)l(alizations)i(of)e(Witten)-7 b('s)25 b Fp(su)p FD(\(2\))68 1508 y Fh(and)i Fp(su)p FD(\(1)p Fp(;)12 b FD(1\))26 b Fh(deformations)p FD(,)e(Ph)n(ys.)g (Lett.)g(B)g Fg(300)e FD(\(1993\),)k(227{233.)-83 1596 y([68])41 b(J.)g(Dixmier,)46 b Fh(L)l(es)c Fp(C)678 1573 y Ff(\003)714 1596 y Fh(-algebr)l(as)g(et)f(leur)h(r)l(epr)l (esentations)p FD(,)k(Gauthier-Villars,)68 1675 y(P)n(aris,)24 b(1969.)-83 1763 y([69])41 b(D.)179 1746 y(\024)175 1763 y(Z.)33 b(Dok)n(o)n(vi)n(\024)-33 b(c,)37 b Fh(Unitary)e(similarity)f (of)h(pr)l(oje)l(ctors)p FD(,)i(Aequationes)f(Math.)d Fg(42)68 1842 y FD(\(1991\),)26 b(220{224.)-83 1929 y([70])41 b(P)-6 b(.)41 b(Dono)n(v)l(an)h(and)g(M.)e(R.)g(F)-6 b(reislic)n(h,)47 b Fh(Some)42 b(evidenc)l(e)f(for)h(an)g(extension)f (of)68 2008 y(the)32 b(Br)l(auer{Thr)l(al)t(l)i(c)l(onje)l(ctur)l(e)p FD(,)d(Sonderforsc)n(h)n(ungsb)r(ereic)n(h)i(Theor.)c(Math.)h Fg(40)68 2087 y FD(\(1972\),)c(24{26.)-83 2175 y([71])41 b(R.)k(S.)f(Doran)h(and)g(V.)f(A.)g(Bel\014,)51 b Fh(Char)l (acterization)46 b(of)g Fp(C)1845 2151 y Ff(\003)1881 2175 y Fh(-algebr)l(as.)f(The)68 2254 y(Gelfand{Naimark)23 b(the)l(or)l(ems)p FD(,)d(Pure)e(and)h(applied)i(mathematics,)g(v)n (ol.)e(101,)h(Mar-)68 2333 y(cel)25 b(Dekk)n(er,)f(Inc.,)g(New)f(Y)-6 b(ork,)24 b(Basel,)g(1986.)-83 2420 y([72])41 b(R.)19 b(G.)g(Douglas,)i Fh(Banach)i(algebr)l(a)f(te)l(chniques)g(in)f(op)l (er)l(ator)j(the)l(ory)p FD(,)c(Acad.)f(Press,)68 2499 y(New)24 b(Y)-6 b(ork,)23 b(London,)i(1972.)-83 2587 y([73])41 b(V.)32 b(G.)h(Drinfeld,)i Fh(Hopf)f(algebr)l(as)h(and)g(the) e(quantum)i(Yang{Baxter)f(e)l(quation)p FD(,)68 2666 y(So)n(viet)26 b(Math.)e(Dokl.)g Fg(32)f FD(\(1985\),)i(no.)f(1,)f (254{258.)-83 2754 y([74])41 b(Y)-6 b(u.)41 b(A.)f(Drozd,)45 b Fh(T)-5 b(ame)42 b(and)g(wild)g(matrix)g(pr)l(oblems)p FD(,)k(Represen)n(tations)e(and)68 2832 y(quadratic)22 b(forms,)c(Inst.)i(Mat.)f(AN)g(UkrSSR,)g(Kiev,)h(1979,)h(pp.)e(39{74,)i (\(Russian\).)-83 2920 y([75])41 b(H.)23 b(A.)f(Dy)n(e,)h Fh(On)h(gr)l(oups)i(of)f(me)l(asur)l(e)i(pr)l(eserving)e(tr)l (ansformations.)h(I)p FD(,)d(Amer.)f(J.)68 2999 y(Math.)i Fg(81)f FD(\(1959\),)i(119{159.)-83 3087 y([76])41 b(K.)18 b(Dyk)n(ema)h(and)g(A.)e(Nica,)j Fh(On)g(the)h(Fo)l(ck)g(r)l(epr)l (esentation)g(of)g(the)g Fp(q)r Fh(-c)l(ommutation)68 3166 y(r)l(elations)p FD(,)k(J.)e(Reine)i(Angew.)f(Math.)f Fg(440)g FD(\(1993\),)i(201{212.)-83 3253 y([77])41 b(E.)20 b(G.)f(E\013ros)h(and)g(F.)g(Hahn,)g Fh(L)l(o)l(c)l(al)t(ly)j(c)l(omp)l (act)h(tr)l(ansformation)g(gr)l(oups)f(and)g Fp(C)2277 3230 y Ff(\003)2313 3253 y Fh(-)68 3332 y(algebr)l(as)p FD(,)j(Mem.)e(Amer.)f(Math.)i(So)r(c,)g(v)n(ol.)h(75,)f(Amer.)e(Math.)i (So)r(c.,)g(Pro)n(vidence,)68 3411 y(R.I.,)e(1967.)-83 3499 y([78])41 b(G.)20 b(G.)f(Emc)n(h,)h Fh(A)n(lgebr)l(aic)i(metho)l (ds)h(in)f(statistic)l(al)g(me)l(chanics)g(and)h(quantum)f(\014eld)68 3578 y(the)l(ory)p FD(,)i(Wiley{In)n(terscience,)29 b(1972.)-83 3666 y([79])41 b(J.)18 b(Ernest,)h Fh(Charting)i(the)g(op)l(er)l(ator)h (terr)l(ain)p FD(,)d(Mem.)f(Amer.)f(Math.)h(So)r(c.,)h(v)n(ol.)g(171,) 68 3744 y(Amer.)k(Math.)h(So)r(c.,)f(Pro)n(vidence,)j(R.I.,)c(1976.)-83 3832 y([80])41 b(D.)31 b(B.)g(F)-6 b(airlie,)35 b Fh(Quantum)f (deformations)g(of)f Fp(S)t(U)7 b FD(\(2\),)34 b(J.)d(Ph)n(ys.)g(A:)g (Math.)h(and)68 3911 y(Gen.)25 b Fg(23)d FD(\(1990\),)k(L183{L186.)p eop %%Page: 247 251 247 250 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(247)-83 96 y FD([81])41 b(T.)35 b(Finc)n(k,)k(S.)c(Ro)r(c)n(h,)k(and)d(B.)e (Silb)r(ermann,)40 b Fh(Two)d(pr)l(oje)l(ctions)h(the)l(or)l(ems)g(and) 68 175 y(symb)l(ol)g(c)l(alculus)f(for)g(op)l(er)l(ators)h(with)f (massive)g(lo)l(c)l(al)h(sp)l(e)l(ctr)l(a)p FD(,)h(Math.)c(Nac)n(hr.)68 254 y Fg(162)23 b FD(\(1993\),)i(167{185.)-83 334 y([82])41 b(M.)23 b(Flato,)j(J.)d(Simon,)i(H.)e(Snellman,)j(and)e(D.)f (Sternheimer,)i Fh(Simple)i(facts)f(ab)l(out)68 422 y(analytic)33 b(ve)l(ctors)f(and)h(inte)l(gr)l(ability)p FD(,)e(Ann.)g(Scien)n(t.)h (de)f(l')1715 405 y(\023)1709 422 y(Ecole)h(Norm.)e(Sup.)h Fg(5)68 501 y FD(\(1972\),)26 b(423{434.)-83 581 y([83])41 b(M.)18 b(F)-6 b(ragoulopulou,)22 b Fh(A)n(n)f(intr)l(o)l(duction)g(to) g(the)g(r)l(epr)l(esentation)g(the)l(ory)g(of)g(top)l(olo)l(g-)68 660 y(ic)l(al)j FC(\003)p Fh(-algebr)l(as)p FD(,)d(Sc)n(hriftenreihne)j (des)d(Math.)f(Inst.)h(der)g(Univ.)g(M)r(\177)-37 b(unster,)21 b(v)n(ol.)g(48,)68 738 y(Univ.)j(M)r(\177)-37 b(unster,)24 b(1988.)-83 818 y([84])41 b(L.)32 b(G)-9 b(\027)-44 b(arding)32 b(and)h(A.)e(Wigh)n(tman,)k Fh(R)l(epr)l(esentations)f(of)f(the)g (antic)l(ommutation)68 897 y(r)l(elations)p FD(,)25 b(Pro)r(c.)e(Nat.)h (Acad.)f(Sci.)i(USA)e Fg(40)g FD(\(1954\),)i(no.)f(9,)f(617{622.)-83 977 y([85])p 68 977 212 4 v 265 w(,)k Fh(R)l(epr)l(esentations)i(of)g (the)f(c)l(ommutation)i(r)l(elations)p FD(,)e(Pro)r(c.)f(Nat.)f(Acad.) 68 1056 y(Sci.)f(USA)e Fg(40)g FD(\(1954\),)i(no.)f(9,)f(623{626.)-83 1136 y([86])41 b(P)-6 b(.)28 b(Gabriel)i(and)f(A.)f(V.)f(Roiter,)j Fh(R)l(epr)l(esentations)h(of)f(\014nite-dimensional)h(alge-)68 1215 y(br)l(as)p FD(,)24 b(Springer-V)-6 b(erlag,)25 b(Berlin,)f(1997.)-83 1295 y([87])41 b(P)-6 b(.)32 b(Gabriel)i(and)e (M.)g(Zisman,)i Fh(Calculus)h(of)e(fr)l(actions)h(and)h(homotopy)g(the) l(ory)p FD(,)68 1374 y(Springer,)24 b(Berlin{Heidelb)r(erg{New)k(Y)-6 b(ork,)23 b(1967.)-83 1454 y([88])41 b(D.)18 b(V.)g(Galinsky)-6 b(,)21 b Fh(R)l(epr)l(esentations)h(of)e FC(\003)p Fh(-algebr)l(as)i (gener)l(ate)l(d)f(by)f(ortho)l(gonal)j(pr)l(o-)68 1532 y(je)l(ctions)e(satisfying)f(a)h(line)l(ar)h(r)l(elation)p FD(,)e(Metho)r(ds)f(F)-6 b(unct.)19 b(Anal.)g(T)-6 b(op)r(ol.)19 b Fg(4)f FD(\(1998\),)68 1611 y(no.)24 b(3,)f(27{32.)-83 1691 y([89])41 b(D.)21 b(V.)f(Galinsky)j(and)f(M.)e(A.)g(Murato)n(v,)h Fh(On)i(r)l(epr)l(esentations)h(of)g(algebr)l(as)g(gener-)68 1770 y(ate)l(d)j(by)e(sets)g(of)h(thr)l(e)l(e)g(and)h(four)f(orthopr)l (oje)l(ctions)p FD(,)f(Sp)r(ectral)h(and)e(ev)n(olutionary)68 1849 y(problems.)g(v)n(ol.)h(8,)e(T)-6 b(a)n(vria,)24 b(Simferop)r(ol,)h(1998,)f(pp.)g(15{22.)-83 1929 y([90])41 b(O.)26 b(M.)g(Ga)n(vrilik)i(and)f(A.)f(U.)f(Klim)n(yk,)j Fh(R)l(epr)l(esentations)h(of)g(the)f Fp(q)r Fh(-deforme)l(d)g(al-)68 2008 y(gebr)l(as)e Fp(U)332 2016 y Fe(q)366 2008 y FD(\()p Fp(so)460 2017 y Fc(2)p Fe(;)p Fc(1)545 2008 y FD(\))p Fh(,)f Fp(U)667 2016 y Fe(q)702 2008 y FD(\()p Fp(so)796 2017 y Fc(3)p Fe(;)p Fc(1)880 2008 y FD(\),)e(J.)h(Math.)f(Ph)n(ys.)h Fg(35)f FD(\(1994\),)i(no.)e(2,)h(3670{3686.)-83 2088 y([91])41 b(I.)25 b(M.)e(Gel'fand)i(and)g(V.)f(A.)g(P)n(onomarev,)g Fh(R)l(emarks)j(on)g(classi\014c)l(ation)g(of)g(a)f(p)l(air)68 2166 y(of)d(c)l(ommuting)g(line)l(ar)g(tr)l(ansformations)h(in)e(a)h (\014nite-dimensional)g(sp)l(ac)l(e)p FD(,)f(F)-6 b(unct.)68 2245 y(Anal.)25 b(i)f(Prilozh.)g Fg(3)g FD(\(1969\),)h(no.)e(4,)h (81{82,)g(\(Russian\).)-83 2325 y([92])p 68 2325 V 265 w(,)41 b Fh(Quadruples)f(of)g(subsp)l(ac)l(es)g(of)f(a)g (\014nite-dimensional)h(ve)l(ctor)f(sp)l(ac)l(e)p FD(,)68 2404 y(Dokl.)25 b(Ak)l(ad.)e(Nauk)i(SSSR)f Fg(197)e FD(\(1971\),)j(no.) f(4,)f(762{765,)i(\(Russian\).)-83 2484 y([93])41 b(I.)19 b(M.)e(Gelfand)j(and)f(N.)f(Y)-6 b(a.)18 b(Vilenkin,)j Fh(Gener)l(alize)l(d)h(functions)p FD(,)e(v)n(ol.)f(4,)g(Academic)68 2563 y(Press,)31 b(New)f(Y)-6 b(ork,)32 b(1964,)g(T)-6 b(ransl.)30 b(from)f(Russian)i(edn.:)45 b(Fizmatgiz,)34 b(Mosco)n(w,)68 2642 y(1961.)-83 2722 y([94])41 b(J.)25 b(Glimm,)g Fh(A)h(Stone{Weierstr)l(ass)h(the)l(or)l(em)h(for)e Fp(C)1521 2698 y Ff(\003)1557 2722 y Fh(-algebr)l(as)p FD(,)f(Ann.)f(Math.)g Fg(72)68 2801 y FD(\(1960\),)i(216{144.)-83 2880 y([95])41 b(I.)25 b(Goh)n(b)r(erg,)h(P)-6 b(.)25 b(Lancaster,)g(and)h(L.)e(Ro)r(dman,)i Fh(Matric)l(es)g(and)i (inde\014nite)e(sc)l(alar)68 2959 y(pr)l(o)l(ducts)p FD(,)g(Op)r(er.)d(Theory)h(Adv.)f(Appl.,)h(v)n(ol.)g(8,)f(Birkhauser)i (V)-6 b(erlag,)24 b(1983.)-83 3039 y([96])41 b(I.)24 b(Goh)n(b)r(erg)h(and)g(B.)e(Reic)n(hstein,)k Fh(On)e(classi\014c)l (ation)j(of)e(normal)h(matric)l(es)f(in)g(an)68 3118 y(inde\014nite)31 b(sc)l(alar)i(pr)l(o)l(duct)p FD(,)g(In)n(tegral)f (Equat.)e(Op)r(er.)f(Theory)j Fg(13)d FD(\(1990\),)j(365{)68 3197 y(394.)-83 3277 y([97])41 b(G.)27 b(A.)g(Goldin,)j(R.)c(Menik)n (o\013,)j(and)f(D.)e(H.)h(Sharp,)g Fh(Particle)i(statistics)f(fr)l(om)i (in-)68 3356 y(duc)l(e)l(d)e(r)l(epr)l(esentations)g(of)e(a)h(lo)l(c)l (al)h(curr)l(ent)f(gr)l(oup)p FD(,)e(J.)f(Math.)h(Ph)n(ys.)f Fg(21)g FD(\(1980\),)68 3435 y(no.)g(4,)f(650{664.)-83 3515 y([98])41 b(V.)22 b(Y)-6 b(a.)22 b(Golo)r(dets,)i Fh(Classi\014c)l(ation)i(of)e(r)l(epr)l(esentations)h(of)g(the)f(antic) l(ommutation)68 3593 y(r)l(elations)p FD(,)h(Russ.)e(Math.)g(Surv)n (eys)h Fg(24)f FD(\(1969\),)i(1{63.)-83 3673 y([99])41 b(K.)24 b(R.)f(Go)r(o)r(dearl)j(and)e(P)-6 b(.)24 b(Menal,)g Fh(F)-5 b(r)l(e)l(e)26 b(and)h(r)l(esidual)t(ly)g(\014nite-dimensional) f Fp(C)2277 3650 y Ff(\003)2313 3673 y Fh(-)68 3752 y(algebr)l(as)p FD(,)f(J.)e(F)-6 b(unct.)24 b(Anal.)h Fg(9)e FD(\(1990\),)i(no.)f(2,)f (391{410.)-118 3832 y([100])41 b(R.)29 b(W.)h(Go)r(o)r(dman,)i Fh(A)n(nalytic)f(and)h(entir)l(e)e(ve)l(ctors)h(for)h(r)l(epr)l (esentations)g(of)f(Lie)68 3911 y(gr)l(oups)p FD(,)25 b(T)-6 b(rans.)23 b(Amer.)g(Math.)g(So)r(c.)h Fg(143)f FD(\(1969\),)i(55{76.)p eop %%Page: 248 252 248 251 bop -118 -137 a FO(248)1866 b FJ(Bibliograph)n(y)-118 96 y FD([101])41 b(M.)c(F.)g(Goro)r(dni)-8 b(\025)-27 b(\020)39 b(and)e(G.)g(B.)g(P)n(o)r(dk)n(olzin,)43 b Fh(Irr)l(e)l(ducible)c(r)l(epr)l(esentations)h(of)e(a)68 175 y(gr)l(ade)l(d)25 b(Lie)e(algebr)l(a)p FD(,)f(Sp)r(ectral)h(Theory) f(of)f(Op)r(erators)h(and)g(In\014nite-dimensional)68 254 y(Analysis,)e(Inst.)e(Math.)f(Acad.)h(Sci.)g(UkrSSR,)e(Kiev,)j (1984,)h(pp.)d(66{76,)i(\(Russian\).)-118 338 y([102])41 b(P)-6 b(.)23 b(Halmos,)h Fh(A)h(Hilb)l(ert)g(sp)l(ac)l(e)h(pr)l(oblem) h(b)l(o)l(ok)p FD(,)d(V)-6 b(an)24 b(Nostrand,)f(Princeton,)i(1967.) -118 423 y([103])p 68 423 212 4 v 265 w(,)e Fh(Two)j(subsp)l(ac)l(es)p FD(,)f(T)-6 b(rans.)23 b(Amer.)g(Math.)h(So)r(c.)g Fg(144)e FD(\(1969\),)j(381{389.)-118 507 y([104])41 b(P)-6 b(.)26 b(R.)f(Halmos)i(and)g(J.)e(E.)h(McLaughlin,)i Fh(Partial)h(isometries)p FD(,)d(P)n(aci\014c)i(J.)d(Math.)68 586 y Fg(13)e FD(\(1963\),)i (585{596.)-118 670 y([105])41 b(P)-6 b(.)28 b(de)h(la)g(Harp)r(e,)g Fh(Op)l(er)l(ator)i(algebr)l(as,)h(fr)l(e)l(e)e(gr)l(oups)h(and)g (other)f(gr)l(oups)p FD(,)h(Recen)n(t)68 749 y(adv)l(ances)i(in)f(op)r (erator)f(algebras,)j(Orl)n(\023)-33 b(eans,)33 b(1992,)g(Asterisque,)h (v)n(ol.)d(232,)i(So)r(c.)68 828 y(Math.)24 b(F)-6 b(rance,)24 b(1995,)g(pp.)g(121{153.)-118 912 y([106])p 68 912 V 265 w(,)40 b Fh(T)-5 b(opics)38 b(on)h(ge)l(ometric)e(gr)l(oup)j(the)l (ory)p FD(,)g(Preliminary)f(v)n(ersion,)i(1998,)68 991 y(h)n(ttp://www.unige.c)n(h/math/bibli)q(o/p)q(reprin)n(t)q(/199)q(8/g) q(eogroup)q(/.)-118 1075 y([107])g(M.)26 b(Ha)n(vli)n(\024)-33 b(cek,)28 b(A.)e(U.)f(Klim)n(yk,)i(and)g(E.)e(P)n(elan)n(to)n(v\023)-35 b(a,)29 b Fh(Nonstandar)l(d)i Fp(U)2008 1083 y Fe(q)2042 1075 y FD(\()p Fp(so)2136 1084 y Fc(3)2170 1075 y FD(\))e Fh(and)68 1154 y Fp(U)116 1162 y Fe(q)151 1154 y FD(\()p Fp(so)245 1163 y Fc(4)279 1154 y FD(\))p Fh(:)53 b(tensor)35 b(pr)l(o)l(ducts)i(of)e(r)l(epr)l(esentations,)k(oscil)t(lator)d(r)l(e) l(alizations)h(and)68 1233 y(r)l(o)l(ots)27 b(of)f(unity)p FD(,)d(Czec)n(h)i(J.)e(Ph)n(ys.)h Fg(47)e FD(\(1997\),)k(no.)d(1,)h (13{16.)-118 1317 y([108])41 b(M.)20 b(Ha)n(vli)n(\024)-33 b(cek,)22 b(A.)e(U.)f(Klim)n(yk,)j(and)e(S.)g(P)n(o)l(\024)-32 b(sta,)22 b Fh(R)l(epr)l(esentations)h(of)g(the)f(cyclic)l(al)t(ly)68 1396 y(symmetric)33 b Fp(q)r Fh(-deforme)l(d)h(algebr)l(a)f Fp(so)1112 1404 y Fe(q)1147 1396 y FD(\(3\),)h(J.)d(Math.)h(Ph)n(ys.)f Fg(40)g FD(\(1999\),)k(no.)d(4,)68 1475 y(1365{1382.)-118 1559 y([109])41 b(A.)24 b(Heb)r(ec)n(k)n(er,)h(S.)e(Sc)n(hrec)n(k)n(en) n(b)r(erg,)j(J.)e(Sc)n(h)n(w)n(enk,)h(W.)f(W)-6 b(eic)n(h,)25 b(and)g(J.)e(W)-6 b(ess,)25 b Fh(R)l(ep-)68 1638 y(r)l(esentations)33 b(of)f(a)h Fp(q)r Fh(-deforme)l(d)g(Heisenb)l(er)l(g)f(algebr)l(a)p FD(,)h(Z.)d(Ph)n(ys.)g(C)h Fg(64)f FD(\(1994\),)68 1717 y(355{359.)-118 1801 y([110])41 b(G.)21 b(C.)g(Hegerfeldt)i(and)e(O.)g (Melsheimer,)h Fh(The)h(form)h(of)g(r)l(epr)l(esentations)g(of)g(CCR)68 1880 y(for)d(Bose)g(\014elds)g(and)h(c)l(onne)l(ction)f(with)g (\014nitely)f(many)g(de)l(gr)l(e)l(es)i(of)e(fr)l(e)l(e)l(dom)p FD(,)h(Com-)68 1959 y(m)n(un.)j(Math.)g(Ph)n(ys.)f Fg(12)g FD(\(1969\),)i(no.)f(4,)f(304{323.)-118 2043 y([111])41 b(I.)35 b(Hernstein,)k Fh(Nonc)l(ommutative)f(rings)p FD(,)f(Math.)e(Asso)r(c.)g(Amer.,)i(Wiley)-6 b(,)39 b(New)68 2122 y(Y)-6 b(ork,)24 b(1968.)-118 2206 y([112])41 b(A.)20 b(S.)g(Holev)n(o,)i Fh(Pr)l(ob)l(abilistic)i(and)f(staistic)l(al)g(asp) l(e)l(cts)h(of)e(quantum)h(the)l(ory)p FD(,)e(North)68 2285 y(Holand,)34 b(Amsterdam,)d(1982,)i(T)-6 b(ransl.)30 b(from)g(Russian)h(edn.:)45 b(Nauk)l(a,)33 b(Mosco)n(w,)68 2364 y(1980.)-118 2448 y([113])41 b(Kh.)34 b(D.)f(Ikramo)n(v,)j Fh(On)f(a)g(c)l(anonic)l(al)i(form)f(of)f(pr)l(oje)l(ctions)h(with)f(r) l(esp)l(e)l(ct)h(to)f(a)68 2527 y(unitary)25 b(similarity)p FD(,)e(Zh)n(urn.)g(Vyc)n(hisl.)h(Mat.)f(i)h(Matem.)f(Fiziki)j Fg(36)c FD(\(1980\),)j(no.)e(1,)68 2606 y(3{5,)h(\(Russian\).)-118 2690 y([114])41 b(A.)25 b(Inoue,)h Fh(L)l(o)l(c)l(al)t(ly)i Fp(C)677 2667 y Ff(\003)713 2690 y Fh(-algebr)l(as)p FD(,)d(Mem.)f(F)-6 b(acult)n(y)27 b(Sci.)f(Kyush)n(u)f(Univ.)h(\(Ser.)f (A.\))68 2769 y Fg(25)e FD(\(1971\),)i(197{235.)-118 2853 y([115])41 b(R.)31 b(S.)h(Ismagilo)n(v,)j Fh(R)l(epr)l (esentations)f(of)f(in\014nite-dimensional)h(gr)l(oups)p FD(,)g(T)-6 b(ransl.)68 2932 y(Math.)24 b(Monogr.,)f(v)n(ol.)h(152,)g (Amer.)f(Math.)h(So)r(c.,)f(Pro)n(vidence,)j(R.I.,)c(1996.)-118 3017 y([116])41 b(N.)h(Jacobson,)48 b Fh(Structur)l(e)43 b(of)g(rings)p FD(,)j(Amer.)c(Math.)g(So)r(c.)h(Coll.)g(Publ.,)k(v)n (ol.)68 3095 y(XXXVI)r(I,)24 b(Amer.)e(Math.)i(So)r(c.,)g(Pro)n (vidence,)h(R.I.,)e(1956.)-118 3180 y([117])41 b(A.)26 b(Jan)n(tzen,)i Fh(L)l(e)l(ctur)l(es)h(on)f(quantum)h(gr)l(oups)p FD(,)f(Amer.)d(Math.)h(So)r(c.,)h(Pro)n(vidence,)68 3259 y(R.I.,)c(1996.)-118 3343 y([118])41 b(M.)18 b(Jim)n(b)r(o,)i Fh(A)g Fp(q)r Fh(-di\013er)l(enc)l(e)h(analo)l(gue)h(of)f Fp(U)7 b FD(\()p Fg(g)q FD(\))21 b Fh(and)h(the)e(Yang{Baxter)h(e)l (quation)p FD(,)68 3422 y(Lett.)k(Math.)f(Ph)n(ys.)f Fg(10)g FD(\(1985\),)i(no.)f(1,)f(63{69.)-118 3506 y([119])41 b(V.)20 b(Jones)i(and)f(V.)f(S.)g(Sunder,)i Fh(Intr)l(o)l(duction)j(to) d(subfactors)p FD(,)g(London)f(Math.)g(So)r(c.)68 3585 y(Lect.)k(Note.)f(Ser.,)f(v)n(ol.)h(234,)g(Cam)n(bridge)h(Univ.)f (Press,)e(Cam)n(bridge,)j(1994.)-118 3669 y([120])41 b(C.)19 b(Jordan,)h Fh(Essai)i(sur)f(la)h(ge)l(ometrie)f(\022)-36 b(a)22 b Fp(n)e Fh(dimensions)p FD(,)h(Bull.)f(So)r(c.)f(Math.)g(F)-6 b(rance)68 3748 y Fg(3)24 b FD(\(1875\),)h(103{174.)-118 3832 y([121])41 b(P)-6 b(.)29 b(E.)f(T.)g(J\034rgensen,)j Fh(Op)l(er)l(ators)g(and)h(r)l(epr)l(esentation)f(the)l(ory)p FD(,)f(North-Holland)68 3911 y(\(Elsevier\),)c(Amsterdam,)d(1988.)p eop %%Page: 249 253 249 252 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(249)-118 96 y FD([122])41 b(P)-6 b(.)24 b(E.)f(T.)g(J\034rgensen)h(and)g(R.)f (T.)g(Mo)r(ore,)g Fh(Op)l(er)l(ator)k(c)l(ommutation)g(r)l(elations)p FD(,)d(D.)68 175 y(Reidel)i(Publ.)e(Comp.,)f(Dordrec)n(h)n(t,)h(1984.) -118 259 y([123])41 b(P)-6 b(.)27 b(E.)f(T.)h(J\034rgensen,)h(L.)f(M.)f (Sc)n(hmitt,)j(and)f(R.)e(F.)h(W)-6 b(erner,)27 b Fp(q)r Fh(-Canonic)l(al)j(c)l(om-)68 338 y(mutation)23 b(r)l(elations)h(and)f (stability)e(of)i(the)f(Cuntz)g(algebr)l(a)p FD(,)f(P)n(aci\014c)h(J.)d (Math.)i Fg(165)68 417 y FD(\(1994\),)26 b(131{151.)-118 501 y([124])p 68 501 212 4 v 265 w(,)19 b Fh(Positive)h(r)l(epr)l (esentation)h(of)g(gener)l(al)f(c)l(ommutation)i(r)l(elations)g(al)t (lowing)68 580 y(Wick)k(or)l(dering)p FD(,)e(J.)f(F)-6 b(unct.)24 b(Anal.)g Fg(134)f FD(\(1995\),)i(33{99.)-118 664 y([125])41 b(P)-6 b(.)26 b(E.)g(T.)f(J\034rgensen)j(and)e(R.)g(F.)g (W)-6 b(erner,)26 b Fh(Coher)l(ent)j(states)e(of)i(the)e Fp(q)r Fh(-c)l(anonic)l(al)68 743 y(c)l(ommutation)h(r)l(elations)p FD(,)c(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fg(164)g FD(\(1994\),)i (455{471.)-118 827 y([126])41 b(V.)35 b(Kac,)i Fh(R)l(o)l(ot)g (systems,)i(r)l(epr)l(esentations)e(of)f(gr)l(aphs)i(and)e(invariant)h (the)l(ory)p FD(,)68 906 y(Lect.)25 b(Notes)f(Math.,)f(v)n(ol.)i(996,)f (pp.)f(74{108,)i(Springer,)f(Berlin,)g(1983.)-118 990 y([127])41 b(R.)21 b(V.)f(Kadison)i(and)f(J.)g(R.)f(Ringrose,)i Fh(F)-5 b(undamentals)25 b(of)e(the)g(the)l(ory)g(of)h(op)l(er)l(ator) 68 1069 y(algebr)l(as,)j(I,)f(II)p FD(,)e(Acad.)f(Press,)g(1983,)h (1986.)-118 1153 y([128])41 b(S.)23 b(A.)f(Kalutsky)i(and)g(Y)-6 b(u.)22 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)25 b Fh(Perio)l(dic)h(gr)l(oups)g(ar)l(e)f(not)g(wild)p FD(,)e(Ukr.)68 1231 y(Mat.)h(Zh.)f Fg(49)g FD(\(1997\),)i(no.)f(5,)f (729{730,)j(\(Russian\).)-118 1315 y([129])41 b(D.)27 b(Kazhdan,)j Fh(Conne)l(ction)g(of)f(the)g(dual)i(sp)l(ac)l(e)f(of)g(a) f(gr)l(oup)i(with)e(the)g(structur)l(e)68 1394 y(of)d(its)f(close)l(d)i (sub)l(gr)l(oups)p FD(,)e(F)-6 b(unct.)25 b(Anal.)f(Appl.)g Fg(1)f FD(\(1957\),)i(63{65.)-118 1478 y([130])41 b(A.)32 b(Y)-6 b(a.)32 b(Khelemski)-8 b(\025)-27 b(\020,)37 b Fh(Banach)d(algebr)l(as)h(and)g(p)l(oly-norme)l(d)g(algebr)l(as:)50 b(gener)l(al)68 1557 y(the)l(ory,)26 b(r)l(epr)l(esentations,)h(homolo) l(gies)p FD(,)e(Nauk)l(a,)f(Mosco)n(w,)g(1989,)g(\(Russian\).)-118 1641 y([131])41 b(A.)23 b(A.)f(Kirillo)n(v,)j Fh(Dynamic)l(al)h (systems,)e(factors)h(and)h(r)l(epr)l(esentations)g(of)f(gr)l(oups)p FD(,)68 1720 y(Usp)r(ekhi)g(Mat.)f(Nauk)g Fg(22)f FD(\(1967\),)i(no.)f (5,)f(67{80,)h(\(Russian\).)-118 1804 y([132])p 68 1804 V 265 w(,)e Fh(Elements)i(of)h(the)f(the)l(ory)g(of)h(r)l(epr)l (esentations)p FD(,)e(Springer,)g(Berlin,)g(1970.)-118 1888 y([133])41 b(E.)25 b(Kissin)h(and)f(V.)g(Sh)n(ul'man,)h Fh(R)l(epr)l(esentations)i(on)f(Kr)l(ein)g(sp)l(ac)l(es)h(and)g (deriva-)68 1967 y(tions)d(of)f Fp(C)381 1943 y Ff(\003)417 1967 y Fh(-algebr)l(as)p FD(,)f(Pitman)g(Monographs)g(and)g(Surv.)f (Pure)g(Applied)i(Math.,)68 2046 y(v)n(ol.)h(89,)e(Addison)i(W)-6 b(esley)g(,)25 b(Longman,)g(1997.)-118 2130 y([134])41 b(D.)18 b(Kleinec)n(k)n(e,)23 b Fh(On)d(op)l(er)l(ator)j(c)l (ommutators)p FD(,)e(Pro)r(c.)d(Amer.)g(Math.)h(So)r(c.)g Fg(8)f FD(\(1957\),)68 2209 y(535{536.)-118 2293 y([135])41 b(S.)32 b(Klimek)g(and)g(A.)f(Lesniewski,)k Fh(Quantum)e(Riemann)g (surfac)l(es.)h(I.)f(The)g(unit)68 2371 y(disc)p FD(,)24 b(Comm)n(un.)f(Math.)h(Ph)n(ys.)f Fg(146)g FD(\(1992\),)i(103{122.)-118 2455 y([136])p 68 2455 V 265 w(,)30 b Fh(A)h(two-p)l(ar)l(ameter)i (quantum)e(deformation)h(of)g(the)e(unit)h(disc)p FD(,)g(Journ.)68 2534 y(F)-6 b(unct.)25 b(Anal.)f Fg(115)f FD(\(1993\),)i(no.)e(1,)h (1{23.)-118 2618 y([137])41 b(A.)30 b(U.)g(Klim)n(yk)i(and)f(K.)f(Sc)n (hm)r(\177)-37 b(udgen,)34 b Fh(Quantum)e(gr)l(oups)i(and)e(their)g(r)l (epr)l(esen-)68 2697 y(tations)p FD(,)f(T)-6 b(exts)29 b(and)h(Monographs)f(in)h(Ph)n(ysics,)h(Springer,)g(Berlin,)g(Heidelb)r (erg,)68 2776 y(1997.)-118 2860 y([138])41 b(H.)22 b(T.)f(Ko)r(elink,)j Fh(On)f FC(\003)p Fh(-r)l(epr)l(esentations)i(of)f(the)g(Hopf)g FC(\003)p Fh(-algebr)l(a)h(asso)l(ciate)l(d)g(with)68 2939 y(the)h(quantum)g(gr)l(oup)h Fp(U)724 2947 y Fe(q)758 2939 y FD(\()p Fp(N)7 b FD(\),)24 b(Comp)r(ositio)i(Math.)e Fg(77)f FD(\(1991\),)i(199{231.)-118 3023 y([139])41 b(T.)26 b(H.)f(Ko)r(orn)n(winder)h(and)g(R.)f(F.)h(Sw)n(arttou)n(w,)h Fh(On)g Fp(q)r Fh(-analo)l(gues)i(of)e(the)h(Fourier)68 3102 y(and)f(Hankel)f(tr)l(ansforms)p FD(,)e(T)-6 b(rans.)23 b(Amer.)g(Math.)h(So)r(c.)g Fg(333)e FD(\(1992\),)j(445{461.)-118 3186 y([140])41 b(L.)25 b(I.)g(Korogo)r(dski)h(and)g(Y.)e(S.)h(Soib)r (elman,)i Fh(A)n(lgebr)l(as)g(of)h(functions)e(on)i(quantum)68 3265 y(gr)l(oups.)f(Part)f(1)p FD(,)e(Amer.)f(Math.)g(So)r(c.,)h(Pro)n (vidence,)h(R.I.,)e(1998.)-118 3349 y([141])41 b(S.)18 b(A.)e(Krugly)n(ak,)k Fh(R)l(epr)l(esentations)g(of)g(fr)l(e)l(e)g (involutive)g(quivers)p FD(,)e(Represen)n(tations)68 3428 y(and)g(quadratic)i(forms,)d(Inst.)g(Math.)h(Acad.)f(Sci.)h(Ukr.)f (SSR,)g(Kiev,)i(1979,)g(pp.)e(149{)68 3506 y(151,)24 b(\(Russian\).)-118 3590 y([142])p 68 3590 V 265 w(,)44 b Fh(R)l(epr)l(esentations)d(of)g(involutive)g(quivers)p FD(,)i(VINITI)e(7266-84,)k(1984,)68 3669 y(\(Russian\).)-118 3753 y([143])c(S.)29 b(A.)f(Krugly)n(ak)i(and)f(A.)f(Y)-6 b(u.)29 b(Piry)n(atinsk)l(a)n(y)n(a,)j Fh(On)e(\\wild")g FC(\003)p Fh(-algebr)l(as)h(and)h(the)68 3832 y(unitary)d(classi\014c)l (ation)h(of)e(we)l(akly)h(c)l(enter)l(e)l(d)g(op)l(er)l(ators)p FD(,)g(Prepr.)d(ser.)g(of)g(Mittag-)68 3911 y(Le\017er)e(Inst.)g(no.)g (11,)f(1995/96.)p eop %%Page: 250 254 250 253 bop -118 -137 a FO(250)1866 b FJ(Bibliograph)n(y)-118 96 y FD([144])41 b(S.)26 b(A.)f(Krugly)n(ak)j(and)e(Y)-6 b(u.)26 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)29 b Fh(On)f(unitary)f(e)l(quivalenc)l(e)h(of)g(c)l(ol)t(le)l(c-)68 175 y(tions)f(of)f(self-adjoint)h(op)l(er)l(ators)p FD(,)f(F)-6 b(unct.)25 b(Anal.)g(i)g(Prilozhen.)h Fg(14)d FD(\(1980\),)j(no.)f(1,) 68 254 y(60{62,)g(\(Russian\).)-118 338 y([145])p 68 338 212 4 v 265 w(,)c Fh(Structur)l(e)j(the)l(or)l(ems)h(for)f (families)g(of)f(idemp)l(otents)p FD(,)g(Ukr.)d(Mat.)i(Zh)n(urn.)68 417 y Fg(50)h FD(\(1998\),)i(no.)f(4,)g(523{533,)h(\(Russian\).)-118 501 y([146])p 68 501 V 265 w(,)19 b Fh(On)i(c)l(omplexity)g(of)g (description)g(of)h(r)l(epr)l(esentations)g(of)f FC(\003)p Fh(-algebr)l(as)g(gen-)68 580 y(er)l(ate)l(d)27 b(by)e(idemp)l(otents)p FD(,)f(Pro)r(c.)f(Amer.)g(Math.)h(So)r(c.)f Fg(128)g FD(\(2000\),)i(no.)f(1.)-118 664 y([147])41 b(N.)22 b(Krupnik,)h Fh(Banach)j(algebr)l(as)g(with)f(symb)l(ol)g(and)g(singular)h(inte)l (gr)l(al)f(op)l(er)l(ators)p FD(,)68 743 y(Op)r(er.)f(Theory)g(Adv.)f (Appl.,)h(v)n(ol.)g(90,)g(Birkh\177)-35 b(auser)24 b(V)-6 b(erlag,)24 b(Basel,)g(1987.)-118 827 y([148])41 b(N.)31 b(Krupnik,)i(S.)d(Ro)r(c)n(h,)j(and)e(B.)g(Silb)r(ermann,)j Fh(On)e Fp(C)1610 803 y Ff(\003)1646 827 y Fh(-algebr)l(as)g(gener)l (ate)l(d)h(by)68 906 y(idemp)l(otents)p FD(,)25 b(J.)e(F)-6 b(unc.)24 b(Anal.)g Fg(137)f FD(\(1996\),)i(303{319.)-118 990 y([149])41 b(N.)22 b(Krupnik)h(and)f(E.)g(Spigel,)i Fh(Invertibility)f(symb)l(ols)i(for)f(a)h(Banach)g(algebr)l(a)g(gen-)68 1069 y(er)l(ate)l(d)d(by)d(two)i(idemp)l(otents)g(and)g(a)g(shift)p FD(,)e(In)n(t.)f(Equat.)h(Op)r(er.)e(Theory)j Fg(17)d FD(\(1993\),)68 1147 y(567{578.)-118 1231 y([150])41 b(P)-6 b(.)16 b(Kruszy)r(\023)-37 b(nski)17 b(and)f(S.)g(L.)f(W)-6 b(orono)n(wicz,)20 b Fh(A)e(nonc)l(ommutative)h(Gelfand{Naimark)68 1310 y(the)l(or)l(em)p FD(,)25 b(J.)e(Op)r(er.)g(Theory)i Fg(8)e FD(\(1982\),)j(361{389.)-118 1394 y([151])41 b(P)-6 b(.)34 b(P)-6 b(.)34 b(Kulish,)k Fh(Contr)l(action)f(of)e(quantum)i (algebr)l(as)f(and)h Fp(q)r Fh(-oscil)t(lators)p FD(,)g(T)-6 b(eor.)68 1473 y(Math.)24 b(Ph)n(ys.)g Fg(86)e FD(\(1991\),)k(108{110.) -118 1557 y([152])41 b(P)-6 b(.)21 b(P)-6 b(.)21 b(Kulish)i(and)f(N.)f (Y)-6 b(u.)21 b(Resh)n(tikhin,)j Fh(Quantum)g(line)l(ar)g(pr)l(oblem)h (for)f(the)f(sine-)68 1636 y(Gor)l(don)g(e)l(quation)f(and)f(higher)h (r)l(epr)l(esentations)p FD(,)e(Zap.)f(Nauc)n(h.)g(Sem.)f(LOMI)h Fg(101)68 1715 y FD(\(1981\),)26 b(101{110,)f(\(Russian\).)-118 1799 y([153])41 b(M.)26 b(Laca,)j Fh(Endomorphisms)j(of)c Fd(B)p FD(\()p Fd(H)q FD(\))h Fh(and)g(Cuntz)g(algebr)l(as)p FD(,)f(J.)f(Op)r(er.)f(Theory)68 1878 y Fg(30)d FD(\(1993\),)i(85{101.) -118 1962 y([154])41 b(E.)17 b(C.)g(Lance,)i Fh(Hilb)l(ert)g Fp(C)743 1938 y Ff(\003)779 1962 y Fh(-mo)l(dules:)31 b(a)20 b(to)l(olkit)f(for)h(op)l(er)l(ator)i(algebr)l(aists)p FD(,)c(London)68 2041 y(Math.)24 b(So)r(c.)g(Lect.)g(Notes)h(Ser.,)e(v) n(ol.)h(210,)g(CUP)-6 b(,)23 b(1995.)-118 2125 y([155])p 68 2125 V 265 w(,)28 b Fh(Finitely)g(pr)l(esente)l(d)i Fp(C)979 2101 y Ff(\003)1015 2125 y Fh(-algebr)l(as)p FD(,)e(Op)r(erator)g(Algebras)g(and)g(Applica-)68 2203 y(tions)f(\(A.)f(Kata)n(v)n(olos,)i(ed.\),)e(Nato)g(ASI)g(Series,)h (Ser.)e(C:)g(Math.)h(and)g(Ph)n(ys.)f(Sci.,)68 2282 y(v)n(ol.)g(495,)f (Klu)n(w)n(er)g(Acad.)g(Publ.,)f(1997,)i(pp.)e(255{266.)-118 2366 y([156])41 b(T.)30 b(A.)g(Loring,)i Fp(C)579 2343 y Ff(\003)615 2366 y Fh(-algebr)l(as)g(gener)l(ate)l(d)g(by)f(stable)h (r)l(elations)p FD(,)g(J.)e(F)-6 b(unct.)31 b(Anal.)68 2445 y Fg(112)23 b FD(\(1993\),)i(no.)f(1,)f(159{203.)-118 2529 y([157])41 b(G.)24 b(Lusztig,)h Fh(Intr)l(o)l(duction)i(to)f (quantum)g(gr)l(oups)p FD(,)f(Birkh\177)-35 b(auser,)24 b(Boston,)g(1993.)-118 2613 y([158])41 b(A.)20 b(J.)g(Macfarlane,)i Fh(On)g Fp(q)r Fh(-analo)l(gues)i(of)f(the)f(quantum)h(harmonic)h (oscil)t(lator)g(and)68 2692 y(the)i(quantum)g(gr)l(oup)h Fp(su)p FD(\(2\),)d(J.)f(Ph)n(ys.)h(A)f Fg(22)g FD(\(1989\),)i (4581{4586.)-118 2776 y([159])41 b(G.)17 b(W.)g(Mac)n(k)n(ey)-6 b(,)20 b Fh(Imprimitivity)f(for)g(r)l(epr)l(esentations)i(of)f(lo)l(c)l (al)t(ly)g(c)l(omp)l(act)h(gr)l(oups)p FD(,)68 2855 y(Pro)r(c.)j(Nat.)f (Acad.)h(Sci.)h(USA)e Fg(35)g FD(\(1949\),)i(no.)f(9,)f(537{545.)-118 2939 y([160])p 68 2939 V 265 w(,)d Fh(Induc)l(e)l(d)j(r)l(epr)l (esentations)f(of)g(lo)l(c)l(al)t(ly)h(c)l(omp)l(act)g(gr)l(oups)p FD(,)e(Ann.)e(Math.)g Fg(55)68 3018 y FD(\(1952\),)26 b(no.)d(1,)h(101{139.)-118 3102 y([161])41 b(S.)25 b(Ma)t(jid,)h Fh(F)-5 b(oundations)29 b(of)e(quantum)h(gr)l(oup)h(the)l(ory)p FD(,)c(Cam)n(bridge)i(Univ.)e(Press,)68 3181 y(Cam)n(bridge,)g(1995.) -118 3265 y([162])41 b(Y)-6 b(u.)17 b(I.)f(Manin,)i Fh(T)-5 b(opics)19 b(in)g(non-c)l(ommutative)g(ge)l(ometry)p FD(,)e(Princeton)i(Univ.)d(Press,)68 3344 y(Princeton,)26 b(N.J.,)c(1991.)-118 3428 y([163])41 b(T)18 b(Masuda,)h(K.)f(Mimac)n (hi,)i(Y.)d(Mak)l(agami,)j(M.)d(Noumi,)j(Y.)d(Saburi,)i(and)g(K.)e (Ueno,)68 3506 y Fh(Unitary)i(r)l(epr)l(esentations)h(of)f(the)f (quantum)i(gr)l(oup)g Fp(S)t(U)1573 3514 y Fe(q)1606 3506 y FD(\(1)p Fp(;)12 b FD(1\),)18 b(Lett.)f(Math.)g(Ph)n(ys.)68 3585 y Fg(19)23 b FD(\(1990\),)i(no.)f(3,)g(187{204.)-118 3669 y([164])41 b(K.)27 b(McClanahan,)i Fp(C)662 3646 y Ff(\003)698 3669 y Fh(-algebr)l(as)h(gener)l(ate)l(d)f(by)f(elements) h(of)g(a)g(unitary)g(matrix)p FD(,)68 3748 y(J.)24 b(F)-6 b(unct.)24 b(Anal.)g Fg(107)f FD(\(1992\),)i(no.)f(2,)f(439{457.)-118 3832 y([165])41 b(S.)30 b(A.)f(McCullough)j(and)f(L.)e(Ro)r(dman,)i Fh(Two)h(self-adjoint)g(op)l(er)l(ators)i(in)d(Kr)l(ein)68 3911 y(sp)l(ac)l(es)p FD(,)25 b(In)n(t.)f(Equat.)h(Op)r(er.)e(Theory)i Fg(26)e FD(\(1996\),)i(202{209.)p eop %%Page: 251 255 251 254 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(251)-118 96 y FD([166])41 b(R.)26 b(Menik)n(o\013)h(and)f(D.)f(H.)g(Sharp,)h Fh(R)l(epr)l(esentations)j(of)e(a)h(lo)l(c)l(al)h(curr)l(ent)f(algebr)l (a:)68 175 y(their)i(dynamic)l(al)i(determination)p FD(,)d(J.)g(Math.)f (Ph)n(ys.)g Fg(16)g FD(\(1975\),)j(no.)d(12,)i(2341{)68 254 y(2360.)-118 334 y([167])41 b(M.)32 b(Misiurewicz,)k Fh(A)n(bsolutely)e(c)l(oninuous)h(me)l(asur)l(es)g(for)f(c)l(ertain)f (maps)i(of)f(an)68 413 y(interval)p FD(,)24 b(Publ.)g(Math.)f(Inst.)h (Hautes)h(Etud.)f(Sci.)g Fg(53)f FD(\(1981\),)i(17{51.)-118 493 y([168])41 b(B.)34 b(Morrel)g(and)h(P)-6 b(.)33 b(Muhly)-6 b(,)37 b Fh(Center)l(e)l(d)f(op)l(er)l(ators)p FD(,)j(Studia)c(Math.)g Fg(51)e FD(\(1974\),)68 572 y(251{263.)-118 652 y([169])41 b(G.)18 b(J.)f(Murph)n(y)-6 b(,)18 b Fp(C)566 629 y Ff(\003)602 652 y Fh(-algebr)l(as)i(and)h(op)l(er)l(ator)h(the)l(ory)p FD(,)c(Acad.)g(Press,)g(Boston,)h(1990.)-118 733 y([170])41 b(F.)25 b(Murra)n(y)g(and)g(J.)g(v)n(on)h(Neumann,)g Fh(On)g(rings)h(of)g(op)l(er)l(ators.)i(IV.)p FD(,)d(Ann.)e(Math.)68 812 y Fg(44)f FD(\(1943\),)i(716{808.)-118 892 y([171])41 b(G.)35 b(Nagy)h(and)f(A.)g(Nica,)j Fh(On)e(the)g(\\quantum)g(disk")g (and)h(\\non-c)l(ommutative)68 971 y(cir)l(cle")p FD(,)27 b(Algebraic)h(metho)r(ds)g(in)f(op)r(erator)g(theory)h(\(R.)e(E.)g (Curto)h(and)g(P)-6 b(.)26 b(E.)g(T.)68 1049 y(J\034rgensen,)f(eds.\),) e(Birkh\023)-35 b(auser)25 b(V)-6 b(elag,)24 b(Boston,)g(1994,)h(pp.)e (276{290.)-118 1130 y([172])41 b(L.)19 b(A.)g(Nazaro)n(v)l(a,)i Fh(R)l(epr)l(esentations)i(of)f(a)g(quadruple)p FD(,)f(Izv.)f(AN.)f (SSSR)g Fg(31)g FD(\(1967\),)68 1208 y(no.)24 b(6,)f(1361{1377,)j (\(Russian\).)-118 1289 y([173])41 b(E.)24 b(Nelson,)g Fh(A)n(nalytic)h(ve)l(ctors)p FD(,)e(Ann.)h(of)f(Math.)h Fg(70)f FD(\(1959\),)i(no.)e(2,)h(572{615.)-118 1369 y([174])41 b(I.)24 b(Newton,)g Fh(Enumer)l(atio)j(line)l(arum)g(p)l (ortii)f(or)l(dinis)p FD(,)f(Optics)g(\(1704\),)g(138{162.)-118 1449 y([175])41 b(L.)23 b(P)-6 b(.)22 b(Nizhnik)j(and)e(L.)f(B.)h(T)-6 b(uro)n(wsk)l(a,)22 b Fh(R)l(epr)l(esentations)k(of)f(double)h(c)l (ommutator)68 1528 y(by)32 b(matrix-di\013er)l(ential)h(op)l(er)l (ators)p FD(,)i(Metho)r(ds)d(F)-6 b(unct.)32 b(Anal.)f(T)-6 b(op)r(ol.)32 b Fg(3)f FD(\(1997\),)68 1607 y(no.)24 b(3,)f(75{80.)-118 1687 y([176])41 b(M.)18 b(Noumi)h(and)g(K.)e(Mimac)n (hi,)k Fh(Big)f Fp(q)r Fh(-Jac)l(obi)h(p)l(olynomials,)j Fp(q)r Fh(-Hahn)d(p)l(olynomials)68 1766 y(and)31 b(a)f(family)g(of)g (quantum)h FD(3)p Fh(-spher)l(es)p FD(,)f(Lett.)f(Math.)f(Ph)n(ys.)g Fg(19)g FD(\(1990\),)j(no.)d(4,)68 1845 y(299{305.)-118 1925 y([177])41 b(A.)19 b(V.)f(Odesski,)j Fh(On)g(an)h(analo)l(gue)h (of)e(the)h(Sklyanin)f(algebr)l(a)p FD(,)g(F)-6 b(unct.)20 b(Anal.)f(Appl.)68 2004 y Fg(20)k FD(\(1986\),)i(152{154.)-118 2084 y([178])41 b(A.)28 b(V.)f(Odesski)i(and)g(B.)e(L.)h(F)-6 b(eigin,)30 b Fh(El)t(liptic)g(Sklyanin)g(algebr)l(as)p FD(,)f(F)-6 b(unkt.)29 b(Anal.)68 2163 y(Prilozh.)c Fg(23)e FD(\(1989\),)i(no.)f(3,)f(45{54.)-118 2243 y([179])41 b(C.)23 b(L.)f(Olsen)i(and)f(W.)g(R.)f(Zame,)h Fh(Singly)h(gener)l(ate) l(d)h Fp(C)1582 2220 y Ff(\003)1618 2243 y Fh(-algebr)l(as)p FD(,)e(T)-6 b(rans.)22 b(Amer.)68 2322 y(Math.)i(So)r(c.)g Fg(215)f FD(\(1976\),)i(205{215.)-118 2402 y([180])41 b(A.)17 b(Y)-6 b(u.)17 b(Ol'shanski)-8 b(\025)-27 b(\020,)20 b Fh(Ge)l(ometry)g(of)f(de\014ning)h(r)l(elations)h(in)e(gr)l(oups)p FD(,)h(Klu)n(w)n(er)d(Acad.)68 2481 y(Publ.,)24 b(Dordrec)n(h)n(t,)g (1991,)g(T)-6 b(ransl)24 b(from)f(Russian)i(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)f(1989.)-118 2561 y([181])41 b(V.)28 b(L.)f(Ostro)n(vsky)-8 b(\025)-27 b(\020,)30 b Fh(R)l(epr)l(esentations)g(of)g(a)g(family)f(of)h(quadr)l(atic)h (algebr)l(as)f(with)68 2640 y(thr)l(e)l(e)c(gener)l(ators)p FD(,)e(Selecta)i(Math.)e(So)n(v.)g Fg(12)f FD(\(1993\),)i(119{127.)-118 2720 y([182])p 68 2720 212 4 v 265 w(,)32 b Fh(On)f(op)l(er)l(ator)j(r) l(elations,)g(c)l(enter)l(e)l(d)f(op)l(er)l(ators,)i(and)e(nonbije)l (ctive)e(dy-)68 2799 y(namic)l(al)25 b(systems)p FD(,)c(Metho)r(ds)i(F) -6 b(unct.)22 b(Anal.)g(T)-6 b(op)r(ol.)22 b Fg(2)f FD(\(1996\),)i(no.) f(3-4,)f(114{121.)-118 2879 y([183])41 b(V.)34 b(L.)g(Ostro)n(vsky)-8 b(\025)-27 b(\020)35 b(and)g(Y)-6 b(u.)34 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)39 b Fh(Applic)l(ation)e(of)e(the)g(pr)l (oje)l(ction)68 2958 y(sp)l(e)l(ctr)l(al)c(the)l(or)l(em)f(to)f(nonc)l (ommuting)h(families)f(of)g(op)l(er)l(ators)p FD(,)h(Ukr.)c(Math.)i (Zh.)68 3037 y Fg(40)23 b FD(\(1988\),)i(no.)f(4,)g(469{481,)h (\(Russian\).)-118 3117 y([184])p 68 3117 V 265 w(,)d Fh(F)-5 b(amilies)25 b(of)f(unb)l(ounde)l(d)j(selfadjoint)e(op)l(er)l (ators,)i(which)e(ar)l(e)g(c)l(onne)l(cte)l(d)68 3196 y(with)20 b(non-Lie)f(r)l(elations)p FD(,)h(F)-6 b(unct.)18 b(Anal.)g(Prilozh.)g Fg(23)e FD(\(1989\),)21 b(no.)c(2,)h(67{68,)i (\(Rus-)68 3275 y(sian\).)-118 3355 y([185])p 68 3355 V 265 w(,)f Fh(R)l(epr)l(esentations)k(of)e FC(\003)p Fh(-algebr)l(as)i(with)e(two)h(gener)l(ators)g(and)g(p)l(olynomial)68 3434 y(r)l(elations)p FD(,)g(Zap.)e(Nauc)n(hn.)h(Semin.)g(LOMI)f Fg(172)f FD(\(1989\),)k(no.)d(121{129,)i(\(Russian\).)-118 3514 y([186])p 68 3514 V 265 w(,)30 b Fh(Unb)l(ounde)l(d)j(op)l(er)l (ators)f(satisfying)f(non-Lie)f(c)l(ommutation)i(r)l(elations)p FD(,)68 3593 y(Repts.)24 b(math.)g(ph)n(ys.)g Fg(28)f FD(\(1989\),)i(no.)f(1,)f(91{103.)-118 3673 y([187])p 68 3673 V 265 w(,)h Fh(Structur)l(e)j(the)l(or)l(ems)h(for)f(a)g(p)l (air)g(of)g(unb)l(ounde)l(d)i(selfadjoint)e(op)l(er)l(ators)68 3752 y(satisfying)e(a)h(quadr)l(atic)h(r)l(elation)p FD(,)d(Adv.)g(So)n(v.)f(Math.)h Fg(9)f FD(\(1992\),)i(131{149.)-118 3832 y([188])p 68 3832 V 265 w(,)g Fh(On)h(p)l(airs)i(of)f (self-adjoint)g(op)l(er)l(ators)p FD(,)f(Seminar)g(Soph)n(us)g(Lie)g Fg(3)e FD(\(1993\),)68 3911 y(no.)g(2,)f(185{218.)p eop %%Page: 252 256 252 255 bop -118 -137 a FO(252)1866 b FJ(Bibliograph)n(y)-118 96 y FD([189])p 68 96 212 4 v 265 w(,)29 b Fh(On)g(r)l(epr)l (esentations)i(of)g(the)e(Heisenb)l(er)l(g)h(r)l(elations)h(for)f(the)g (quantum)68 175 y Fp(E)t FD(\(2\))d Fh(gr)l(oup)p FD(,)d(Ukr.)f(Mat.)g (Zh.)h Fg(47)f FD(\(1995\),)i(no.)e(5,)h(689{692.)-118 259 y([190])p 68 259 V 265 w(,)i Fh(R)l(epr)l(esentations)j(of)f FC(\003)p Fh(-algebr)l(as)h(and)f(dynamic)l(al)i(systems)p FD(,)c(Nonlinear)68 338 y(Math.)e(Ph)n(ys.)g Fg(2)f FD(\(1995\),)i(no.) f(2,)f(133{150.)-118 422 y([191])p 68 422 V 265 w(,)48 b Fh(R)l(epr)l(esentations)d(of)g(quadr)l(atic)g FC(\003)p Fh(-algebr)l(as)g(by)f(b)l(ounde)l(d)i(and)f(un-)68 501 y(b)l(ounde)l(d)28 b(op)l(er)l(ators)p FD(,)d(Repts.)f(Math.)g(Ph)n (ys.)f Fg(35)g FD(\(1995\),)i(no.)f(2/3,)g(283{301.)-118 585 y([192])41 b(V.)20 b(L.)g(Ostro)n(vsky)-8 b(\025)-27 b(\020)21 b(and)g(S.)f(D.)g(Silv)n(estro)n(v,)j Fh(R)l(epr)l (esentations)g(of)g(the)f(r)l(e)l(al)i(forms)f(of)68 664 y(a)h(gr)l(ade)l(d)h(analo)l(gue)h(of)d(the)h(Lie)f(algebr)l(a)h Fp(sl)q FD(\(2)p Fp(;)12 b Fb(C)d FD(\),)27 b(Ukr.)21 b(Mat.)g(Zh)n(urn.)g Fg(44)g FD(\(1992\),)68 743 y(no.)j(11,)g (1518{1524,)h(\(Russian\).)-118 827 y([193])41 b(V.)22 b(L.)f(Ostro)n(vsky)-8 b(\025)-27 b(\020)23 b(and)f(L.)f(B.)g(T)-6 b(uro)n(vsk)l(a)n(y)n(a,)23 b Fh(R)l(epr)l(esentations)i(of)f FC(\003)p Fh(-algebr)l(as)h(and)68 906 y(multidimensional)30 b(dynamic)l(al)f(systems)p FD(,)c(Ukr.)g(Mat.)g(Zh)n(urn.)h Fg(47)e FD(\(1995\),)k(no.)e(4,)68 985 y(488{497.)-118 1069 y([194])41 b(K.)e(R.)f(P)n(arthasarath)n(y)-6 b(,)44 b Fh(A)n(n)c(intr)l(o)l(duction)h(to)f(quantum)h(sto)l(chastic)f(c)l (alculus)p FD(,)68 1147 y(Birkh\177)-35 b(auser{V)-6 b(erlag,)26 b(Basel,)e(1992.)-118 1231 y([195])41 b(C.)22 b(P)n(earcy)-6 b(,)23 b Fh(On)h(c)l(ertain)g(von)g(Neumann)h(algebr)l (as)h(which)f(ar)l(e)g(gener)l(ate)l(d)f(by)g(p)l(ar-)68 1310 y(tial)i(isometries)p FD(,)d(Pro)r(c.)h(Amer.)e(Math.)i(So)r(c.)g Fg(15)f FD(\(1964\),)i(393{395.)-118 1394 y([196])41 b(G.)22 b(K.)f(P)n(edersen,)i Fh(Me)l(asur)l(e)i(the)l(ory)f(for)g Fp(C)1238 1371 y Ff(\003)1274 1394 y Fh(-algebr)l(as)p FD(,)e(Math.)g(Scand.)h Fg(22)e FD(\(1968\),)68 1473 y(63{74.)-118 1557 y([197])p 68 1557 V 265 w(,)j Fp(C)391 1534 y Ff(\003)427 1557 y Fh(-algebr)l(as)i(and)h(their)f(automorphism) j(gr)l(oups)p FD(,)d(London)f(Math.)f(So)r(c.)68 1636 y(Monographs,)g(v)n(ol.)h(14,)e(Acad.)h(Press,)f(London,)h(1979.)-118 1720 y([198])41 b(S.)e(P)n(edersen,)k Fh(A)n(ntic)l(ommuting)d (selfadjoint)h(op)l(er)l(ators)p FD(,)j(J.)38 b(F)-6 b(unct.)40 b(Anal.)f Fg(89)68 1799 y FD(\(1990\),)26 b(no.)d(2,)h(428{443.)-118 1883 y([199])41 b(N.)25 b(C.)h(Phillips,)i Fh(Inverse)f(limits)h(of)f Fp(C)1134 1859 y Ff(\003)1170 1883 y Fh(-algebr)l(as)p FD(,)g(J.)e(Op)r(er.)g(Theory)i Fg(19)e FD(\(1988\),)68 1962 y(153{195.)-118 2046 y([200])41 b(R.)g(S.)f(Pierce,)46 b Fh(Asso)l(ciative)41 b(algebr)l(as)p FD(,)46 b(Graduate)c(texts)g(in)f(math.,)k(v)n(ol.)c(88,)68 2125 y(Springer-V)-6 b(erlag,)25 b(New)f(Y)-6 b(ork{Heidelb)r (erg{Berlin,)27 b(1982.)-118 2209 y([201])41 b(A.)28 b(Piry)n(atinsk)l(a)n(y)n(a,)k Fh(On)d(unitary)h(classi\014c)l(ation)g (of)g(we)l(akly)h(c)l(enter)l(e)l(d)f(op)l(er)l(ators)p FD(,)68 2287 y(V)-6 b(estnik)26 b(T)-6 b(am)n(b)r(o)n(v)24 b(Univ.)g Fg(3)f FD(\(1998\),)j(no.)d(1,)h(79{83.)-118 2371 y([202])41 b(A.)20 b(Y)-6 b(u.)19 b(Piry)n(atinsk)l(a)n(y)n(a)k (and)e(Y)-6 b(u.)19 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)23 b Fh(Wild)f(pr)l(oblems)i(in)e(r)l(epr)l(esenta-)68 2450 y(tion)27 b(the)l(ory)f(of)h FC(\003)p Fh(-algebr)l(as)g(with)f(gener)l (ators)h(and)g(r)l(elations)p FD(,)e(Ukr.)f(Mat.)g(Zh)n(urn.)68 2529 y Fg(47)f FD(\(1995\),)i(no.)f(1,)g(70{78,)g(\(Russian\).)-118 2613 y([203])41 b(S.)24 b(P)n(op)r(o)n(vyc)n(h,)h Fh(R)l(epr)l (esentations)i(of)f(r)l(e)l(al)g(forms)h(of)f(Witten)-7 b('s)24 b(\014rst)i(deformation)p FD(,)68 2692 y(Symmetry)f(Nonlin.)f (Math.)g(Ph)n(ys.)g Fg(2)f FD(\(1997\),)i(393{396.)-118 2776 y([204])p 68 2776 V 265 w(,)i Fh(Unb)l(ounde)l(d)k(idemp)l(otents) p FD(,)d(Metho)r(ds)g(F)-6 b(unct.)27 b(Anal.)h(T)-6 b(op)r(ol.)27 b Fg(5)g FD(\(1999\),)68 2855 y(no.)d(1,)f(95{103.)-118 2939 y([205])41 b(R.)30 b(T.)g(P)n(o)n(w)n(ers,)i Fh(Selfadjoint)h (algebr)l(as)g(of)f(unb)l(ounde)l(d)i(op)l(er)l(ators.)g(I)p FD(,)d(Comm)n(un.)68 3018 y(Math.)24 b(Ph)n(ys.)g Fg(21)e FD(\(1971\),)k(85{124.)-118 3102 y([206])p 68 3102 V 265 w(,)43 b Fh(Selfadjoint)e(algebr)l(as)g(of)g(unb)l(ounde)l(d)i(op)l (er)l(ators.)f(II)p FD(,)e(T)-6 b(rans)39 b(Amer.)68 3181 y(Math.)24 b(So)r(c.)g Fg(187)f FD(\(1974\),)i(261{293.)-118 3265 y([207])p 68 3265 V 265 w(,)20 b Fh(Simplicity)i(of)g(the)f Fp(C)913 3241 y Ff(\003)949 3265 y Fh(-algebr)l(a)i(asso)l(ciate)l(d)g (with)g(the)e(fr)l(e)l(e)i(gr)l(oup)g(on)f(two)68 3344 y(gener)l(ators)p FD(,)i(Duk)n(e)g(Math.)g(J.)g Fg(42)e FD(\(1975\),)k(151{156.)-118 3428 y([208])41 b(D.)28 b(Proskurin,)h Fh(Homo)l(gene)l(ous)j(ide)l(als)e(in)g(Wick)f(algebr)l (as)p FD(,)h(Pro)r(c.)d(Amer.)g(Math.)68 3506 y(So)r(c.)d Fg(126)f FD(\(1998\),)i(no.)f(11,)f(3371{3376.)-118 3590 y([209])41 b(D.)23 b(P)-6 b(.)23 b(Proskurin,)g Fh(A)n(b)l(out)i(p)l (ositivity)g(of)g(Fo)l(ck)h(inner)f(pr)l(o)l(duct)i(of)e(a)h(c)l (ertain)f(Wick)68 3669 y(algebr)l(as)p FD(,)g(Metho)r(ds)f(F)-6 b(unct.)25 b(Anal.)f(T)-6 b(op)r(ol.)24 b Fg(5)f FD(\(1999\),)j(no.)d (1,)h(88{94.)-118 3753 y([210])41 b(D.)32 b(P)-6 b(.)32 b(Proskurin)g(and)h(Y)-6 b(u.)31 b(S.)h(Samo)-8 b(\025)-27 b(\020lenk)n(o,)37 b Fh(R)l(epr)l(esentations)d(of)g(Wick)f(CCR)68 3832 y(algebr)l(a)p FD(,)c(Sp)r(ectral)g(and)f(ev)n(olutionary)j (problems,)d(v)n(ol.)g(8)g(\(Simferop)r(ol\),)i(T)-6 b(a)n(vria,)68 3911 y(1998,)25 b(pp.)e(43{45.)p eop %%Page: 253 257 253 256 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(253)-118 96 y FD([211])41 b(W.)g(Pusz,)j Fh(Twiste)l(d)e(c)l(anonic)l(al)h (antic)l(ommutation)g(r)l(elations)p FD(,)i(Repts.)40 b(Math.)68 175 y(Ph)n(ys.)24 b Fg(27)f FD(\(1989\),)i(349{360.)-118 260 y([212])41 b(W.)18 b(Pusz)g(and)g(S.)g(L.)f(W)-6 b(orono)n(wicz,)21 b Fh(Twiste)l(d)f(se)l(c)l(ond)h(quantization)p FD(,)f(Repts.)d(Math.)68 338 y(Ph)n(ys.)24 b Fg(27)f FD(\(1989\),)i(231{257.)-118 423 y([213])41 b(I.)18 b(F.)f(Putnam,)i Fp(C)555 399 y Ff(\003)591 423 y Fh(-algebr)l(as)h(arising)g(fr)l(om)h (interval)f(exchange)g(tr)l(ansformations)p FD(,)68 502 y(J.)k(Op)r(er.)f(Theory)i Fg(27)e FD(\(1992\),)i(231{250.)-118 586 y([214])41 b(V.)23 b(I.)g(Rabano)n(vic)n(h,)i Fh(Banach)h(algebr)l (as)g(gener)l(ate)l(d)f(by)g(thr)l(e)l(e)g(idemp)l(otents)p FD(,)e(Meth-)68 665 y(o)r(ds)h(F)-6 b(unct.)25 b(Anal.)f(T)-6 b(op)r(ol.)24 b Fg(4)g FD(\(1998\),)h(no.)e(1,)h(65{67.)-118 749 y([215])p 68 749 212 4 v 265 w(,)34 b Fh(Singly)g(gener)l(ate)l(d)h Fp(C)948 725 y Ff(\003)984 749 y Fh(-algebr)l(as)p FD(,)g(Ukr.)c(Mat.)i (Zh.)g Fg(51)e FD(\(1999\),)37 b(no.)c(8,)68 828 y(\(Russian\).)-118 912 y([216])41 b(V.)f(I.)h(Rabano)n(vic)n(h)i(and)e(Y)-6 b(u.)40 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)47 b Fh(On)41 b(r)l(epr)l(esentations)h(of)g Fd(F)2273 920 y Fe(n)2313 912 y Fh(-)68 991 y(algebr)l(as)27 b(and)g(invertibility)c (symb)l(ols)p FD(,)i(Metho)r(ds)f(F)-6 b(unct.)25 b(Anal.)f(T)-6 b(op)r(ol.)25 b Fg(4)e FD(\(1998\),)68 1070 y(no.)h(4,)f(86{96.)-118 1154 y([217])p 68 1154 V 265 w(,)28 b Fh(On)g(r)l(epr)l(esentations)j (of)e Fp(F)1093 1162 y Fe(n)1135 1154 y Fh(-algebr)l(as)g(and)h(their)f (applic)l(ations)p FD(,)h(Op)r(er.)68 1233 y(Theory)25 b(Adv.)e(Appl.,)h(v)n(ol.)g(94,)g(Birkh\177)-35 b(auser)24 b(V)-6 b(erlag,)24 b(Basel,)g(1999.)-118 1317 y([218])p 68 1317 V 265 w(,)c Fh(When)j(a)g(sum)g(of)g(idemp)l(otents)g(or)g (orthopr)l(oje)l(ctions)i(is)d(multiple)h(of)g(the)68 1396 y(identity)p FD(,)g(F)-6 b(unct.)24 b(Anal.)g(Prilozh.)h Fg(34)e FD(\(2000\).)-118 1480 y([219])41 b(I.)28 b(Raeburn)g(and)h(A.) e(M.)g(Sinclair,)j Fh(The)g Fp(C)1270 1457 y Ff(\003)1306 1480 y Fh(-algebr)l(a)g(gener)l(ate)l(d)f(by)g(two)h(pr)l(oje)l(c-)68 1559 y(tions)p FD(,)24 b(Math.)f(Scand.)i Fg(65)e FD(\(1989\),)i (278{290.)-118 1643 y([220])41 b(M.)24 b(Reed)h(and)g(B.)f(Simon,)i Fh(Metho)l(ds)h(of)g(mo)l(dern)h(mathematic)l(al)g(physics)p FD(,)d(v)n(ol.)g(1,)68 1722 y(Acad.)f(Press,)f(New)h(Y)-6 b(ork,)23 b(1972.)-118 1807 y([221])41 b(J.)25 b(Renault,)i Fh(A)g(gr)l(oup)l(oid)i(appr)l(o)l(ach)h(to)d Fp(C)1249 1783 y Ff(\003)1285 1807 y Fh(-algebr)l(as)p FD(,)e(Lect.)h(Notes.)f (Math.,)g(v)n(ol.)68 1885 y(793,)f(Springer{V)-6 b(erlag,)26 b(1980.)-118 1970 y([222])41 b(M.)24 b(Rie\013el,)j Fh(Quantum)g (deformations)h(for)f(actions)g(of)g Fb(R)1658 1946 y Fe(d)1694 1970 y FD(,)e(Mem.)f(Amer.)f(Math.)68 2049 y(So)r(c.,)h(v)n(ol.)g(506,)g(Amer.)f(Math.)h(So)r(c.,)f(Pro)n (vidence,)i(RI,)f(1993.)-118 2133 y([223])p 68 2133 V 265 w(,)39 b Fh(Morita)f(e)l(quivalenc)l(e)g(for)g Fp(C)1164 2109 y Ff(\003)1200 2133 y Fh(-algebr)l(as)g(and)g Fp(W)1735 2109 y Ff(\003)1770 2133 y Fh(-algebr)l(as)p FD(,)i(J.)d(Pure)68 2212 y(Appl.)24 b(Algebra)h Fg(5)e FD(\(1974\),)i(51{96.)-118 2296 y([224])41 b(S.)32 b(Ro)r(c)n(h)h(and)f(B.)g(Sib)r(ermann,)j Fh(A)n(lgebr)l(as)f(gener)l(ate)l(d)f(by)g(idemp)l(otents)h(and)h(the) 68 2375 y(symb)l(ol)29 b(c)l(alculus)g(for)g(singular)f(inte)l(gr)l(al) h(op)l(er)l(ators)p FD(,)f(In)n(t.)f(Equat.)g(Op)r(er.)e(Theory)68 2454 y Fg(11)e FD(\(1988\),)i(385{419.)-118 2538 y([225])41 b(A.)32 b(V.)g(Roiter,)k Fh(Boxes)e(with)g(an)g(involution)p FD(,)h(Represen)n(tations)g(and)e(quadratic)68 2617 y(forms,)c(Inst.)g (Math.)g(Acad.)g(Sci.)h(Ukr.)d(SSR,)i(Kiev,)i(1979,)f(pp.)f(124{126,)j (\(Rus-)68 2696 y(sian\).)-118 2780 y([226])41 b(W.)24 b(Rudin,)g Fh(F)-5 b(unctional)27 b(analysis)p FD(,)d(McGra)n(w-Hill,)g (New)g(Y)-6 b(ork,)23 b(1973.)-118 2864 y([227])41 b(S.)17 b(Sak)l(ai,)i Fh(Op)l(er)l(ator)h(algebr)l(as)g(in)f(dynamic)l(al)h (systems.)f(The)g(the)l(ory)g(of)g(unb)l(ounde)l(d)68 2943 y(derivations)27 b(in)e Fp(C)578 2920 y Ff(\003)614 2943 y Fh(-algebr)l(as)p FD(,)f(Cam)n(bridge)g(Univ.)g(Press,)f(Cam)n (brige,)h(1991.)-118 3027 y([228])41 b(Y)-6 b(u.)33 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)37 b Fh(Sp)l(e)l(ctr)l(al)e(the)l(ory)f(of)g (families)h(of)f(self-adjoint)g(op)l(er)l(ators)p FD(,)68 3106 y(Klu)n(w)n(er)e(Academic)h(Publisher,)h(1991,)g(T)-6 b(ransl.)31 b(from)f(Russian)i(edn.:)46 b(Nauk)n(o)n(v)l(a)68 3185 y(Dumk)l(a,)24 b(Kiev,)g(1984.)-118 3269 y([229])41 b(Y)-6 b(u.)23 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o)25 b(and)f(V.)e(S.)h(Sh)n(ul'man,)h Fh(On)g(r)l(epr)l(esentations)i(of)f (r)l(elations)h(of)68 3348 y(the)f(form)h Fp(i)p FD([)p Fp(A;)11 b(B)s FD(])20 b(=)f Fp(f)7 b FD(\()p Fp(A)p FD(\))16 b(+)e Fp(g)r FD(\()p Fp(B)s FD(\),)24 b(Ukr.)e(Mat.)h(Zh.)g Fg(43)f FD(\(1991\),)j(no.)e(1,)g(110{114,)68 3427 y(\(Russian\).)-118 3511 y([230])41 b(Y)-6 b(u.)27 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o)29 b(and)e(L.)g(B.)f(T)-6 b(uro)n(wsk)l(a,)27 b Fh(On)h(r)l(epr)l(esentations)h(of)g FC(\003)p Fh(-algebr)l(as)68 3590 y(by)k(unb)l(ounde)l(d)i(op)l(er)l(ators)p FD(,)g(F)-6 b(unkt.)32 b(Anal.)g(Prilozh.)h Fg(31)e FD(\(1997\),)k(no.)c(4,)j (80{83,)68 3669 y(\(Russian\).)-118 3753 y([231])p 68 3753 V 265 w(,)d Fh(R)l(epr)l(esentations)h(of)f(cubic)g(semiline)l(ar) h(r)l(elations)g(and)g(r)l(e)l(al)g(forms)g(of)68 3832 y(the)24 b(Fairlie)h(algebr)l(a)p FD(,)e(Quan)n(tum)g(groups)f(and)h (quan)n(tum)h(spaces,)e(Banac)n(h)i(Cen)n(ter)68 3911 y(Publ.,)g(v)n(ol.)g(40,)g(Inst.)g(Math.)g(P)n(olish)h(Acad.)f(Sci.,)g (W)-6 b(arsza)n(w)n(a,)24 b(1997,)g(pp.)g(21{40.)p eop %%Page: 254 258 254 257 bop -118 -137 a FO(254)1866 b FJ(Bibliograph)n(y)-118 96 y FD([232])41 b(Y)-6 b(u.)34 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)39 b(L.)33 b(B.)h(T)-6 b(uro)n(wsk)l(a,)36 b(and)e(S.)g(P)n(op)r(o)n(vyc)n(h,)j Fh(R)l(epr)l(esentations)68 175 y(of)g(a)f(cubic)g(deformation)h(of)g Fp(su)p FD(\(2\))f Fh(and)h(p)l(ar)l(asup)l(ersymmetric)i(c)l(ommutation)68 254 y(r)l(elations)p FD(,)25 b(Symmetry)f(in)g(Nonlin.)h(Math.)f(Ph)n (ys.)f Fg(2)g FD(\(1997\),)j(272{383.)-118 338 y([233])41 b(Y)-6 b(u.)25 b(S.)f(Samo)-8 b(\025)-27 b(\020lenk)n(o,)27 b(L.)d(B.)g(T)-6 b(uro)n(wsk)l(a,)25 b(and)g(V.)f(S.)g(Sh)n(ul'man,)h Fh(Semiline)l(ar)i(r)l(ela-)68 417 y(tions)g(and)h(their)f FC(\003)p Fh(-r)l(epr)l(esentations)p FD(,)f(Metho)r(ds)f(F)-6 b(unct.)26 b(Anal.)g(T)-6 b(op)r(ol.)25 b Fg(2)g FD(\(1996\),)68 496 y(no.)f(1,)f(55{111.)-118 580 y([234])41 b(K.)31 b(Sc)n(hm)r(\177)-37 b(udgen,)34 b Fh(Unb)l(ounde)l(d)g(op)l(er)l(ator) g(algebr)l(as)f(and)g(r)l(epr)l(esentation)h(the)l(ory)p FD(,)68 659 y(Birkh\177)-35 b(auser,)24 b(Basel,)g(1990.)-118 744 y([235])p 68 744 212 4 v 265 w(,)36 b Fh(Op)l(er)l(ator)h(r)l(epr)l (esentations)f(of)g Fb(R)1302 752 y Fe(q)1336 744 y FD(,)h(Publ)e(RIMS) f Fg(28)f FD(\(1992\),)39 b(no.)34 b(6,)68 822 y(1029{1061.)-118 907 y([236])p 68 907 V 265 w(,)k Fh(Inte)l(gr)l(able)f(op)l(er)l(ator)i (r)l(epr)l(esentations)f(of)f Fb(R)1626 883 y Fc(2)1626 923 y Fe(q)1660 907 y Fh(,)j Fp(X)1780 915 y Fe(q)r(;\015)1905 907 y Fh(and)e Fp(S)t(L)p FD(\(2)p Fp(;)12 b Fb(R)p FD(\),)68 986 y(Comm)n(un.)24 b(Math.)g(Ph)n(ys)f Fg(159)g FD(\(1994\),)i (217{237.)-118 1070 y([237])p 68 1070 V 265 w(,)38 b Fh(Op)l(er)l(ator)g(r)l(epr)l(esentations)f(of)g Fp(U)1307 1078 y Fe(q)1341 1070 y FD(\()p Fp(sl)1422 1079 y Fc(2)1457 1070 y FD(\()p Fb(R)p FD(\)\),)i(Lett.)d(Math.)g(Ph)n(ys.)f Fg(37)68 1149 y FD(\(1996\),)26 b(211{222.)-118 1233 y([238])41 b(J.)31 b(Sc)n(h)n(w)n(enk)h(and)g(J.)e(W)-6 b(ess,)33 b Fh(A)f Fp(q)r Fh(-deforme)l(d)h(quantum)g(me)l(chanic)l(al) h(toy)d(mo)l(del)p FD(,)68 1312 y(Ph)n(ys.)24 b(Lett.)g(B.)f Fg(291)g FD(\(1992\),)i(273{277.)-118 1396 y([239])41 b(V.)20 b(V.)f(Sergeic)n(h)n(uk,)j Fh(Classi\014c)l(ation)i(of)e(line)l (ar)h(op)l(er)l(ators)h(in)e(a)h(\014nite-dimensional)68 1475 y(unitary)j(sp)l(ac)l(e)p FD(,)e(F)-6 b(unct.)25 b(Anal.)f(Appl.)g Fg(18)f FD(\(1984\),)i(no.)f(3,)f(224{230.)-118 1559 y([240])p 68 1559 V 265 w(,)c Fh(Classi\014c)l(ation)j(pr)l (oblems)g(for)f(systems)f(of)h(forms)h(and)f(line)l(ar)g(mappings)p FD(,)68 1638 y(Math.)j(USSR)g(Izv)n(estiy)n(a)i Fg(31)d FD(\(1988\),)i(no.)f(3,)f(481{501.)-118 1722 y([241])p 68 1722 V 265 w(,)g Fh(Classi\014c)l(ation)k(of)e(p)l(airs)i(of)e (subsp)l(ac)l(es)i(in)e(sp)l(ac)l(es)i(with)e(sc)l(alar)i(pr)l(o)l (duct)p FD(,)68 1801 y(Ukrain.)d(Math.)g(J.)f Fg(42)g FD(\(1990\),)i(no.)f(4,)f(478{491.)-118 1885 y([242])p 68 1885 V 265 w(,)41 b Fh(Symmetric)e(r)l(epr)l(esentations)h(of)f (algebr)l(as)h(with)f(involution)p FD(,)j(Math.)68 1964 y(Notes)25 b Fg(50)e FD(\(1992\),)i(no.)e(3{4,)h(1058{1061.)-118 2049 y([243])p 68 2049 V 265 w(,)17 b Fh(Unitary)i(and)g(Euclide)l(an)h (r)l(epr)l(esentations)g(of)f(a)g(quiver)p FD(,)e(Linear)g(Algebra)68 2127 y(Appl.)24 b Fg(278)f FD(\(1998\),)i(37{62.)-118 2212 y([244])41 b(H.)23 b(Shapiro,)i Fh(A)g(survey)g(of)g(c)l(anonic)l (al)j(forms)e(and)g(invariants)g(for)g(unitary)f(simi-)68 2291 y(larity)p FD(,)e(Linear)i(Algebra)g(Appl.)f Fg(147)e FD(\(1991\),)j(101{167.)-118 2375 y([245])41 b(A.)24 b(N.)f(Shark)n(o)n(vski)-8 b(\025)-27 b(\020,)26 b(Y)-6 b(u.)24 b(L.)g(Maistrenk)n(o,)h(and)g(E.)e(Y)-6 b(u.)24 b(Romanenk)n(o,)h Fh(Di\013er)l(enc)l(e)68 2454 y(e)l(quations)35 b(and)f(their)f(applic)l(ations)p FD(,)j(Klu)n(w)n(er)d(Acad.)f(Publ.,) i(Dordrec)n(h)n(t,)h(1993,)68 2533 y(T)-6 b(ransl.)24 b(from)f(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a)25 b(Dumk)l(a,)e(Kiev,)h (1986.)-118 2617 y([246])41 b(A.)25 b(N.)f(Shark)n(o)n(vsky)-6 b(,)26 b(S.)f(F.)g(Koly)n(ada,)h(A.)f(G.)f(Siv)l(ak,)j(and)f(V.)e(V.)g (F)-6 b(edorenk)n(o,)27 b Fh(Dy-)68 2696 y(namics)k(of)f (one-dimensional)i(maps)p FD(,)e(Klu)n(w)n(er)f(Acad.)f(Publ.,)i (Dordrec)n(h)n(t,)g(1997,)68 2775 y(T)-6 b(ransl.)24 b(from)f(Russian)h(edn.:)32 b(Nauk)n(o)n(v)l(a)25 b(Dumk)l(a,)e(Kiev,)h (1989.)-118 2859 y([247])41 b(F.)21 b(V.)e(Shirok)n(o)n(v,)k Fh(Pr)l(o)l(of)h(of)f(the)g(Kaplansky)g(hyp)l(othesis)p FD(,)f(Usp)r(ekhi)g(Mat.)e(Nauk)h Fg(11)68 2938 y FD(\(1956\),)26 b(167{168,)f(\(Russian\).)-118 3022 y([248])41 b(V.)f(S.)g(Sh)n(ulman,) 45 b Fh(Multiplic)l(ation)d(op)l(er)l(ators)i(and)e(sp)l(e)l(ctr)l(al)g (synthesis)p FD(,)i(Dokl.)68 3101 y(Ak)l(ad.)24 b(Nauk)g(SSSR)g Fg(313)f FD(\(1990\),)i(no.)f(5,)f(1047{1051,)j(\(Russian\).)-118 3185 y([249])41 b(S.)30 b(D.)f(Silv)n(estro)n(v,)k Fh(Hilb)l(ert)e(sp)l (ac)l(e)h(r)l(epr)l(esentations)g(of)g(the)f(gr)l(ade)l(d)h(analo)l (gue)h(of)68 3264 y(the)28 b(Lie)g(algebr)l(a)g(of)h(the)f(gr)l(oup)h (of)f(plane)h(motions)p FD(,)e(Studia)h(Math.)f Fg(117)e FD(\(1996\),)68 3343 y(no.)f(2,)f(195{203.)-118 3427 y([250])p 68 3427 V 265 w(,)i Fh(R)l(epr)l(esentations)j(of)g(c)l (ommutation)g(r)l(elations.)g(A)f(dynamic)l(al)i(systems)68 3506 y(appr)l(o)l(ach)p FD(,)e(Hadronic)e(Journ.)e(Suppl.)i Fg(11)e FD(\(1996\),)i(no.)e(1,)h(1{116.)-118 3590 y([251])41 b(S.)29 b(D.)g(Silv)n(estro)n(v)j(and)e(L.)f(B.)f(T)-6 b(uro)n(wsk)l(a,)31 b Fh(R)l(epr)l(esentations)h(of)f(the)g Fp(q)r Fh(-deforme)l(d)68 3669 y(Lie)21 b(algebr)l(a)i(of)e(the)h(gr)l (oup)h(of)e(motions)i(of)f(the)f(Euclide)l(an)i(plane)p FD(,)e(J.)e(F)-6 b(unct.)20 b(Anal.)68 3748 y Fg(160)j FD(\(1998\),)i(79{114.)-118 3832 y([252])41 b(S.)20 b(D.)e(Silv)n (estro)n(v)k(and)e(H.)f(W)-6 b(allin,)23 b Fh(R)l(epr)l(esentations)f (of)g(algebr)l(as)h(asso)l(ciate)l(d)g(with)68 3911 y(M\177)-36 b(obius)23 b(tr)l(ansformation)p FD(,)e(J.)f(Nonlin.)g(Math.)g(Ph)n (ys.)f Fg(3)h FD(\(1996\),)h(no.)f(1-2,)g(202{213.)p eop %%Page: 255 259 255 258 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(255)-118 96 y FD([253])41 b(Y)-6 b(a.)34 b(G.)g(Sinai,)j Fh(Mo)l(dern)f(pr)l (oblems)h(of)e(er)l(go)l(dic)h(the)l(ory)p FD(,)g(Mo)r(dern)e(Problems) g(of)68 175 y(Mathematics,)26 b(Fizik)n(o-Matematic)n(hesk)l(a)n(y)n(a) j(Literatura,)c(Mosco)n(w,)e(1995.)-118 263 y([254])41 b(E.)17 b(K.)f(Skly)n(anin,)21 b Fh(On)e(some)h(algebr)l(aic)g (structur)l(es)f(r)l(elate)l(d)i(to)e(Yang{Baxter)h(e)l(qua-)68 342 y(tion)p FD(,)k(F)-6 b(unkt.)24 b(Anal.)g(i)g(Prilozhen.)h Fg(16)e FD(\(1982\),)j(no.)d(4,)h(27{34,)g(\(Russian\).)-118 431 y([255])p 68 431 212 4 v 265 w(,)30 b Fh(On)h(some)g(algebr)l(aic)h (structur)l(es)f(r)l(elate)l(d)h(to)f(the)g(Yang{Baxter)g(e)l(qua-)68 509 y(tion.)k(II.)h(R)l(epr)l(esentations)g(of)g(quantum)f(algebr)l(a)p FD(,)i(F)-6 b(unct.)35 b(Anal.)f(Prilozh.)h Fg(17)68 588 y FD(\(1983\),)26 b(no.)d(4,)h(34{48,)g(\(Russian\).)-118 677 y([256])41 b(A.)25 b(S.)h(Smogorzhevski)-8 b(\025)-27 b(\020)28 b(and)e(E.)g(S.)f(Stolo)n(v)l(a,)j Fh(Handb)l(o)l(ok)i(in)d (the)g(the)l(ory)h(of)g(plane)68 755 y(curves)e(of)g(the)f(thir)l(d)h (or)l(der)p FD(,)f(Fizmatgiz,)h(Mosco)n(w,)d(1961,)h(\(Russian\).)-118 844 y([257])41 b(Y)-6 b(a.)33 b(S.)f(Soib)r(elman)k(and)d(L.)f(L.)g(V) -6 b(aksman,)36 b Fh(The)e(algebr)l(a)h(of)f(functions)g(on)h(the)68 922 y(quantum)24 b(gr)l(oup)g Fp(S)t(U)7 b FD(\()p Fp(n)i FD(+)g(1\))23 b Fh(and)h(o)l(dd-dimensional)h(quantum)f(spher)l(es)p FD(,)d(Algebra)68 1001 y(i)k(Analiz)g Fg(2)e FD(\(1990\),)i(no.)f(5,)f (101{120,)j(\(Russian\).)-118 1090 y([258])41 b(S.)26 b(Stratila)i(and)f(D.)e(V)-6 b(oiculescu,)29 b Fh(R)l(epr)l (esentations)g(of)f(AF-algebr)l(as)h(and)g(of)f(the)68 1168 y(gr)l(oup)f Fp(U)7 b FD(\()p FC(1)p FD(\),)24 b(Lect.)g(Notes.)g (Math.,)f(v)n(ol.)i(486,)f(Springer,)g(1975.)-118 1257 y([259])41 b(M.)23 b(T)-6 b(ak)n(esaki,)25 b Fh(The)l(ory)i(of)e(op)l (er)l(ator)j(algebr)l(as)p FD(,)c(Springer,)g(1979.)-118 1345 y([260])41 b(P)-6 b(.)28 b(K.)f(T)-6 b(am,)28 b Fh(On)h(the)g(unitary)h(e)l(quivalenc)l(e)f(of)h(c)l(ertain)f(classes)h (of)g(non-normal)68 1424 y(op)l(er)l(ators.)e(I)p FD(,)c(Canad.)g(J.)f (Math.)h Fg(23)f FD(\(1971\),)i(no.)e(5,)h(849{856.)-118 1512 y([261])41 b(P)-6 b(.)22 b(T)-6 b(app)r(er,)23 b Fh(Emb)l(e)l(dding)j FC(\003)p Fh(-algebr)l(as)f(into)f Fp(C)1302 1488 y Ff(\003)1338 1512 y Fh(-algebr)l(as)p FD(,)f(Ph.)f(D.)g(thesis,)h(Univ.)g(of)68 1591 y(Leeds,)h(1996.)-118 1679 y([262])41 b(E.)30 b(Thoma,)448 1662 y Fh(\177)436 1679 y(Ub)l(er)h(unit\177)-36 b(ar)l(e)32 b(Darstel)t(lungen)f(abz\177) -36 b(ahlb)l(ar)l(er,)35 b(diskr)l(etten)30 b(Grupp)l(en)p FD(,)68 1758 y(Math.)24 b(Ann.)g Fg(153)e FD(\(1964\),)j(111{138.)-118 1846 y([263])41 b(J.)f(T)-6 b(omijama,)44 b Fh(Invitation)d(to)g(the)g Fp(C)1172 1823 y Ff(\003)1208 1846 y Fh(-algebtas)f(and)i(top)l(olo)l (gic)l(al)h(dynamics)p FD(,)68 1925 y(W)-6 b(orld)25 b(Sci.)f(Publ.,)g(Singap)r(ore,)h(1987.)-118 2013 y([264])41 b(D.)26 b(M.)f(T)-6 b(opping,)27 b Fh(L)l(e)l(ctur)l(es)h(on)g(von)g (Neumann)h(algebr)l(as)p FD(,)d(V)-6 b(an)27 b(Nostrand)f(Rein-)68 2092 y(hold)f(Comp.,)e(London,)i(1971.)-118 2180 y([265])41 b(L.)21 b(T)-6 b(uro)n(vsk)l(a)n(y)n(a,)22 b Fh(R)l(epr)l(esentations)i (of)f(some)h(r)l(e)l(al)g(forms)f(of)h Fp(U)1776 2188 y Fe(q)1810 2180 y FD(\()p Fp(sl)q FD(\(3\)\),)e(Algebras,)68 2259 y(Groups)i(and)h(Geometries)g Fg(12)e FD(\(1995\),)i(321{338.)-118 2347 y([266])41 b(E.)25 b(Twietmey)n(er,)h Fh(R)l(e)l(al)i(forms)g(of)f Fp(U)1065 2355 y Fe(q)1099 2347 y FD(\()p Fa(J)p FD(\),)f(Lett.)g (Math.)f(Ph)n(ys.)g Fg(24)f FD(\(1992\),)j(no.)e(1,)68 2426 y(49{59.)-118 2514 y([267])41 b(L.)20 b(L.)g(V)-6 b(aksman)20 b(and)h(L.)e(I.)h(Korogo)r(dskii,)i Fh(The)g(algebr)l(a)h (of)g(b)l(ounde)l(d)h(functions)e(on)68 2593 y(the)29 b(quantum)g(gr)l(oup)h(of)f(motions)g(of)g(the)f(plane)i(and)f Fp(q)r Fh(-analo)l(g)h(of)e(Bessel)h(func-)68 2672 y(tions)p FD(,)24 b(Dokl.)g(Ak)l(ad.)g(Nauk)g(USSR)g Fg(304)e FD(\(1989\),)j(no.) f(5,)f(1036{1040,)j(\(Russian\).)-118 2760 y([268])41 b(F.-H.)24 b(V)-6 b(asilescu,)27 b Fh(A)n(ntic)l(ommuting)h (selfadjoint)g(op)l(er)l(ators)p FD(,)f(Rev.)e(Roum.)g(Math.)68 2839 y(Pures)f(Appl.)g Fg(28)f FD(\(1983\),)i(77{91.)-118 2927 y([269])41 b(A.)30 b(N.)f(V)-6 b(asil'ev,)32 b Fh(The)l(ory)g(of)f (r)l(epr)l(esentations)i(of)e(a)h(top)l(olo)l(gic)l(al)i (\(non-Banach\))68 3006 y(involutary)26 b(algebr)l(a)p FD(,)e(T)-6 b(eor.)24 b(Math.)f(Ph)n(ys.)h Fg(2)f FD(\(1970\),)i (113{123.)-118 3094 y([270])41 b(N.)20 b(V)-6 b(asilevski,)24 b Fp(C)560 3071 y Ff(\003)596 3094 y Fh(-algebr)l(as)f(gener)l(ate)l(d) g(by)f(pr)l(oje)l(ctions)j(and)e(their)g(applic)l(ations)p FD(,)68 3173 y(In)n(tegr.)i(Equat.)f(Op)r(er.)f(Theory)i Fg(31)e FD(\(1998\),)i(113{132.)-118 3261 y([271])41 b(N.)26 b(V)-6 b(asilevski)29 b(and)e(I.)f(Spitk)n(o)n(vski,)j Fh(On)f(algebr)l(a)g(gener)l(ate)l(d)h(by)e(two)i(pr)l(oje)l(ctions)p FD(,)68 3340 y(Dokl.)c(Ak)l(ad.)e(Nauk)i(Ukr.)d(SSR,)i(Ser.)f(A)g Fg(8)g FD(\(1981\),)j(10{13,)e(\(Russian\).)-118 3428 y([272])41 b(E.)21 b(Y)-6 b(e.)22 b(V)-6 b(a)n(ysleb,)23 b Fh(In\014nite-dimensional)i FC(\003)p Fh(-r)l(epr)l(esentations)f(of) g(Sklyanin)g(algebr)l(a)68 3507 y(in)c(de)l(gener)l(ate)g(c)l(ase)g (\(the)g(quantum)h(algebr)l(a)f Fp(U)1353 3515 y Fe(q)1387 3507 y FD(\()p Fp(sl)q FD(\(2\)\))p Fh(\))p FD(,)g(Metho)r(ds)e(of)f(F) -6 b(unctional)68 3586 y(Analysis)21 b(in)f(problems)g(of)f (Mathematical)j(Ph)n(ysics,)f(Inst.)e(Math.)g(Acad.)h(Sci.)f(Ukr.)68 3665 y(SSR,)24 b(Kiev,)g(1990,)g(pp.)g(50{62,)g(\(Russian\).)-118 3753 y([273])p 68 3753 V 265 w(,)19 b Fh(R)l(epr)l(esentations)i(of)g (r)l(elations)g(which)h(c)l(onne)l(ct)e(a)h(family)g(of)g(c)l(ommuting) 68 3832 y(op)l(er)l(ators)36 b(with)d(non-sefadjoint)h(one)p FD(,)f(Ukrain.)f(Math.)g(Zh.)f Fg(42)g FD(\(1990\),)k(1258{)68 3911 y(1262,)25 b(\(Russian\).)p eop %%Page: 256 260 256 259 bop -118 -137 a FO(256)1866 b FJ(Bibliograph)n(y)-118 96 y FD([274])41 b(E.)20 b(Y)-6 b(e.)20 b(V)-6 b(a)n(ysleb)22 b(and)e(V.)g(V.)f(F)-6 b(edorenk)n(o,)22 b Fh(R)l(epr)l(esentations)i (of)e(op)l(er)l(ator)i(r)l(elations)68 175 y(and)g(one-dimensional)g (dynamic)l(al)g(systems)p FD(,)d(Application)j(of)c(Metho)r(ds)h(of)f (F)-6 b(unc-)68 254 y(tional)21 b(Analysis)e(in)g(Mathematical)j(Ph)n (ysics,)d(Inst.)g(Math.)f(Ak)l(ad.)g(Nauk)h(UkrSSR,)68 333 y(Kiev,)25 b(1989,)f(\(Russian\),)h(pp.)e(12{20.)-118 418 y([275])41 b(E.)27 b(Y)-6 b(e.)28 b(V)-6 b(a)n(ysleb)29 b(and)f(Y)-6 b(u.)27 b(S.)g(Samo)-8 b(\025)-27 b(\020lenk)n(o,)31 b Fh(R)l(epr)l(esentations)f(of)f(op)l(er)l(ator)i(r)l(ela-)68 496 y(tions)f(by)f(unb)l(ounde)l(d)i(op)l(er)l(ators)g(and)g (multi-dimensional)f(dynamic)l(al)h(systems)p FD(,)68 575 y(Ukrain.)24 b(Math.)g(Zh.)f Fg(42)g FD(\(1990\),)i(no.)f(9,)f (1011{1019,)j(\(Russian\).)-118 660 y([276])p 68 660 212 4 v 265 w(,)21 b Fh(R)l(epr)l(esentations)j(of)f(the)g(r)l (elations)h Fp(AU)j FD(=)19 b Fp(U)7 b(F)j FD(\()p Fp(A)p FD(\))23 b Fh(by)g(unb)l(ounde)l(d)i(self-)68 739 y(adjoint)f(and)g (unitary)g(op)l(er)l(ators)p FD(,)f(Selecta)g(Math.)e(So)n(v.)h Fg(13)e FD(\(1994\),)j(no.)e(1,)h(35{54.)-118 823 y([277])41 b(A.)29 b(M.)g(V)-6 b(ershik,)30 b Fh(A)n(lgebr)l(as)i(with)f(quadr)l (atic)h(r)l(elations)p FD(,)f(Sp)r(ectral)g(theory)g(of)e(op-)68 902 y(erators)f(and)h(in\014nite-dimensional)j(analysis,)e(Inst.)f (Math)f(Acad.)g(Sci.)g(Ukraine,)68 981 y(Kiev,)d(1984,)f(\(Russian\),)h (pp.)e(32)h({)g(56.)-118 1066 y([278])41 b(A.)31 b(M.)g(V)-6 b(ershik,)33 b(I.)e(M.)g(Gelfand,)j(and)e(M.)e(I.)h(Graev,)j Fh(R)l(epr)l(esentations)g(of)f(the)68 1144 y(gr)l(oup)25 b Fp(S)t(L)p FD(\(2)p Fp(;)12 b(R)p FD(\))p Fh(,)25 b(wher)l(e)f Fp(R)g Fh(is)f(a)h(function)g(ring)p FD(,)d(Usp)r(ehi)h(Mat.)f(Nauk)h Fg(28)f FD(\(1973\),)68 1223 y(no.)j(5,)f(83{128,)i(\(Russian\).)-118 1308 y([279])p 68 1308 V 265 w(,)33 b Fh(Commutative)h(mo)l(del)h(of)e (the)g(r)l(epr)l(esentation)h(of)g(the)f(curr)l(ent)g(gr)l(oup)68 1387 y Fp(S)t(L)p FD(\(2)p Fp(;)12 b Fb(R)p FD(\))332 1363 y Fe(X)424 1387 y Fh(r)l(elate)l(d)34 b(to)f(the)h(unip)l(otent)f (sub)l(gr)l(oup)p FD(,)j(F)-6 b(unct.)33 b(Anal.)f(Priolozh.)h Fg(17)68 1466 y FD(\(1983\),)26 b(no.)d(2,)h(70{72,)g(\(Russian\).)-118 1550 y([280])41 b(D.)17 b(V.)g(V)-6 b(oiculescu,)20 b(K.)d(J.)g(Dyk)n (ema,)i(and)e(A.)g(Nica,)i Fh(F)-5 b(r)l(e)l(e)20 b(r)l(andom)i (variables)p FD(,)c(CRM)68 1629 y(Monograph)j(Ser.,)e(Cen)n(tre)i(de)f (Rec)n(herc)n(hes)h(Math.)e(Univ.)h(Motr)n(\023)-33 b(eal,)21 b(v)n(ol.)f(1,)g(Amer.)68 1708 y(Math.)k(So)r(c.,)g(Pro)n(vidence,)h (R.)e(I.,)g(1992.)-118 1792 y([281])41 b(Y.)30 b(W)-6 b(eiss,)33 b Fh(On)e(algebr)l(as)i(gener)l(ate)l(d)f(by)f(two)h(idemp)l (otents)p FD(,)g(Seminar)g(Analysis:)68 1871 y(Op)r(erator)19 b(Equations)h(and)f(Numer.)e(Anal.)i(1987/88)h(\(Berlin\),)h(Karl-W)-6 b(eierstrass-)68 1950 y(Institut)27 b(f)r(\177)-37 b(ur)22 b(Mathematik,)k(1988,)e(pp.)g(139{145.)-118 2035 y([282])41 b(H.)22 b(W)-6 b(enzl,)23 b Fh(R)l(epr)l(esentations)i(of)f(br)l(aid)h (gr)l(oups)g(and)g(the)e(quantum)i(Yang{Baxter)68 2114 y(e)l(quation)p FD(,)f(P)n(acif.)g(J.)g(Math.)f Fg(145)g FD(\(1990\),)i(153{180.)-118 2198 y([283])41 b(J.)26 b(Wic)n(hmann,)h Fh(Hermitian)g FC(\003)p Fh(-algebr)l(as)h(which)g(ar) l(e)g(not)f(symmetric)p FD(,)e(J.)g(London)68 2277 y(Math.)f(So)r(c.)g Fg(8)f FD(\(1974\),)i(109{112.)-118 2361 y([284])41 b(H.)36 b(Wielandt,)531 2345 y Fh(\177)519 2361 y(Ub)l(er)h(die)g(Unb)l (eschr\023)-36 b(anktheit)37 b(der)g(Op)l(er)l(ator)l(en)h(des)f (Quanten-)68 2440 y(me)l(chanik)p FD(,)24 b(Math.)g(Ann.)g Fg(121)e FD(\(1949\),)j(21.)-118 2525 y([285])41 b(E.)31 b(Witten,)k Fh(Gauge)f(the)l(ories,)g(vertex)e(mo)l(dels,)k(and)e (quantum)f(gr)l(oups)p FD(,)h(Repts.)68 2604 y(Nuclear)25 b(Ph)n(ys.)f Fg(b330)e FD(\(1990\),)k(285{346.)-118 2688 y([286])41 b(W.)36 b(R.)f(W)-6 b(ogen,)39 b Fh(On)d(gener)l(ators)h (for)g(von)g(Neumann)g(algebr)l(as)p FD(,)i(Bull.)d(Amer.)68 2767 y(Math.)24 b(So)r(c.)g Fg(75)f FD(\(1969\),)i(95{99.)-118 2852 y([287])p 68 2852 V 265 w(,)19 b Fh(On)h(sp)l(e)l(cial)h(gener)l (ators)g(for)g(pr)l(op)l(erly)h(in\014nite)e(von)h(Neumann)g(algebr)l (as)p FD(,)68 2931 y(Pro)r(c.)j(Amer.)e(Math.)i(So)r(c.)g Fg(28)f FD(\(1971\),)i(no.)f(1,)f(107{113.)-118 3015 y([288])41 b(S.)25 b(L.)f(W)-6 b(orono)n(wicz,)27 b Fh(Comp)l(act)h (matrix)f(pseudo)l(gr)l(oups)p FD(,)h(Comm)n(un.)d(Math.)g(Ph)n(ys.)68 3094 y Fg(111)e FD(\(1987\),)i(613{665.)-118 3179 y([289])p 68 3179 V 265 w(,)d Fh(Quantum)j Fp(E)t FD(\(2\))g Fh(gr)l(oup)g(and)g (its)f(Pontryagin)h(dual)p FD(,)e(Lett.)g(Math.)g(Ph)n(ys.)68 3257 y Fg(23)g FD(\(1991\),)i(251{263.)-118 3342 y([290])p 68 3342 V 265 w(,)c Fh(Unb)l(ounde)l(d)k(elements)e(a\016liate)l(d)h (with)f Fp(C)1481 3319 y Ff(\003)1517 3342 y Fh(-algebr)l(as)h(and)g (non-c)l(omp)l(act)68 3421 y(quantum)j(gr)l(oups)p FD(,)d(Comm)n(un.)g (Math.)g(Ph)n(ys.)f Fg(136)g FD(\(1991\),)i(399{432.)-118 3505 y([291])p 68 3505 V 265 w(,)f Fp(C)391 3482 y Ff(\003)427 3505 y Fh(-algebr)l(as)i(gener)l(ate)l(d)g(by)g(unb)l(ounde)l(d)i (elements)p FD(,)23 b(Rev.)i(Math.)f(Ph)n(ys.)68 3584 y Fg(7)g FD(\(1995\),)h(no.)e(3,)h(481{521.)-118 3669 y([292])41 b(Sh.)31 b(Y)-6 b(amagami,)33 b Fh(On)e(unitary)h(r)l(epr)l (esentation)h(the)l(ories)f(of)g(c)l(omp)l(act)i(quantum)68 3748 y(gr)l(oups)p FD(,)25 b(Comm)n(un.)f(Math.)f(Ph)n(ys.)h Fg(167)e FD(\(1995\),)j(509{529.)-118 3832 y([293])41 b(C.)25 b(Zac)n(hos,)h Fh(Elementary)i(p)l(ar)l(adigms)h(of)f(quantum)f (algebr)l(as)p FD(,)g(Con)n(t.)e(Math.)h Fg(134)68 3911 y FD(\(1992\),)g(351{377.)p eop %%Page: 257 261 257 260 bop -118 -137 a FJ(Bibliograph)n(y)1862 b FO(257)-118 96 y FD([294])41 b(S.)25 b(Zakrzewski,)h Fh(R)l(e)l(ali\014c)l(ations)i (of)f(c)l(omplex)h(quantum)f(gr)l(oups)p FD(,)f(Groups)f(and)g(re-)68 175 y(lated)32 b(topics)g(\(R.)e(I.)g(Gielerak)i(et)f(al.,)g(ed.\),)h (Klu)n(w)n(er)e(Acad.)g(Publ.,)i(Dordrec)n(h)n(t,)68 254 y(1992,)25 b(pp.)e(83{100.)-118 333 y([295])41 b(A.)19 b(S.)h(Zhedano)n(v,)i Fp(Q)p Fh(-r)l(otations)g(and)g(other)g Fp(Q)p Fh(-tr)l(ansformations)h(as)g(unitary)e(non-)68 412 y(line)l(ar)38 b(automorphisms)h(of)d(quantum)i(algebr)l(as)p FD(,)g(J.)d(Math.)g(Ph)n(ys.)g Fg(35)g FD(\(1994\),)68 491 y(3756{3764.)-118 570 y([296])41 b(D.)20 b(P)-6 b(.)20 b(Zhelob)r(enk)n(o,)k Fh(Comp)l(act)g(Lie)e(gr)l(oups)i(and)f(their)f (r)l(epr)l(esentations)p FD(,)g(T)-6 b(ransla-)68 649 y(tions)26 b(of)e(Math.)g(Monographs.,)h(v)n(ol.)g(40,)f(Amer.)g(Math.) g(So)r(c.,)g(Pro)n(vidence,)i(R.I.,)68 727 y(1973,)f(T)-6 b(rans.)23 b(from)f(Russian)j(edn.:)31 b(Nauk)l(a,)24 b(Mosco)n(w,)g(1970.)p eop %%Page: 258 262 258 261 bop -118 -137 a FO(258)p eop %%Page: 259 263 259 262 bop -118 512 a FR(Index)-118 927 y FO(algebra)48 1027 y FN(F)101 1039 y FL(n)147 1027 y FO(-algebra,)24 b(27)48 1127 y(en)n(v)n(eloping)g(pro-)p FN(C)670 1097 y FM(\003)707 1127 y FO(,)k(20)48 1227 y(en)n(v)n(eloping)c FP(\003)p FO(-algebra,)g(15)48 1327 y(en)n(v)n(eloping)g FN(\033)s FO(-)p FN(C)599 1297 y FM(\003)638 1327 y FO(,)j(17)48 1427 y(en)n(v)n(eloping)d FN(C)521 1397 y FM(\003)560 1427 y FO(,)j(16)48 1527 y(residually)14 b(\014nite)k(dimensional,)214 1627 y(13)-118 1727 y FP(\003)p FO(-algebra)48 1827 y FP(\003)p FO(-b)r(ounded,)27 b(16)48 1927 y FP(\003)p FO(-wild,)e(214)48 2027 y FN(C)113 1997 y FM(\003)151 2027 y FO(-represen)n(tatble,)g(11)48 2128 y FN(\033)s FO(-)p FN(C)191 2097 y FM(\003)229 2128 y FO(-represen)n(tatble,)g(20) 48 2228 y(\014nitely)h(generated,)g(21)48 2328 y(generated)17 b(b)n(y)h(idemp)r(oten)n(ts,)214 2427 y(29)48 2528 y(group,)26 b(11)48 2628 y(lo)r(cally)e FN(C)376 2598 y FM(\003)414 2628 y FO(,)k(20)48 2728 y(wild,)e(212)48 2828 y(with)i(generators)e (and)i(rela-)214 2928 y(tions,)e(16)-118 3119 y(Can)n(tor)g(set,)i(106) -118 3219 y FN(\026)p FO(-CAR)g(algebra,)c(158)-118 3319 y FN(\026)p FO(-CCR)j(algebra,)e(156)-118 3419 y(comm)n(utativ)n(e)f (mo)r(del,)i(180)-118 3519 y(con)n(tin)n(ued)g(fractions,)g(84,)h(95) -118 3620 y(Cun)n(tz)h(algebra,)d(189)-118 3811 y(de\014ciency)h (indices,)g(73)-118 3911 y(dynamical)e(system,)i(89)1422 927 y(cycle,)g(89)1422 1026 y(measurable)e(section,)i(90)1422 1126 y(non-bijectiv)n(e,)f(91)1422 1226 y(orbit,)h(89)1422 1325 y(p)r(erio)r(dic)f(p)r(oin)n(t,)i(89)1422 1425 y(triangular,)d (155)1256 1603 y(F)-7 b(airlie)24 b(algebra,)g(133)1256 1782 y(graded)i FN(so)p FO(\(3\),)i(107)1256 1881 y(group)1422 1981 y FP(\003)p FO(-wild,)d(227)1422 2081 y(amenable,)g(229)1422 2180 y(Burnside,)h(229)1422 2280 y(Co)n(xeter,)g(32)1505 2379 y(not)h(a\016ne,)h(228)1422 2479 y(h)n(yp)r(erb)r(olic,)d(228)1422 2579 y(p)r(erio)r(dic,)g(229)1422 2678 y(residually)e(\014nite,)k(13) 1256 2857 y(Heisen)n(b)r(erg)e(relations,)g(169)1256 3035 y(idemp)r(oten)n(ts)1422 3135 y(comm)n(uting)f(pairs,)i(220)1422 3234 y(pairwise)e(orthogonal,)h(220)1256 3334 y(in)n(v)n(olution)1422 3434 y(completely)f(prop)r(er,)j(12)1422 3533 y(prop)r(er,)f(12)1256 3633 y(isomertries)1422 3733 y(comm)n(uting)e(pair,)i(237)1256 3911 y(Jacobi)f(matrix,)g(73)1048 4121 y(259)p eop %%Page: 260 264 260 263 bop -118 -137 a FO(260)-118 96 y(Lie)26 b(algebras)48 196 y(nonlinear)15 b(transformations,)214 296 y(26)-118 477 y(ma)5 b(jorization,)23 b(203,)j(206)-118 577 y(marginally)c(n)n (ull)k(subset,)h(61)-118 676 y(measurable)d(section,)i(65)-118 776 y(measure)48 876 y(ergo)r(dic,)f(62,)i(90)48 975 y(pro)r(duct,)h(194)48 1075 y(quasi-in)n(v)-5 b(arian)n(t,)22 b(62,)27 b(89)-118 1257 y(non-comm)n(utativ)n(e)c(curv)n(es)48 1356 y(circle,)i(39)48 1456 y(h)n(yp)r(erb)r(ola,)h(39)48 1555 y(pair)33 b(of)h(in)n(tersecting)e(lines,)214 1655 y(39)-118 1837 y(op)r(erator)48 1936 y(algebraic,)24 b(236)48 2036 y(cen)n(tered,)j(86,)g(185)131 2135 y(partial)e(isometry) -7 b(,)25 b(97)48 2235 y(h)n(yp)r(onormal,)f(236)48 2335 y(in)n(tert)n(wining,)g(8)48 2434 y(non-self-adjoin)n(t,)h(230)48 2534 y(partial)g(isometry)-7 b(,)25 b(234)48 2634 y(pseudo-in)n (tegral,)f(59)48 2733 y(quasi-normal,)f(233)48 2833 y(supp)r(orting)j (set,)i(49)48 2932 y(w)n(eakly)d(cen)n(tered,)i(234)-118 3032 y(op)r(erators)48 3132 y(b)r(ounded)20 b(self-adjoin)n(t)e(pair,) 214 3231 y(22)48 3331 y(irreducible)24 b(family)-7 b(,)25 b(21)-118 3513 y(p)r(olynomial)48 3612 y(standard,)i(27)-118 3712 y(pro)5 b(jections)48 3811 y(all)16 b(but)j(one)f(orthogonal,)f (217)48 3911 y(four-tuples,)26 b(107,)g(111,)h(217)1256 96 y(quan)n(tum)f(disk,)h(84,)f(102,)h(105)1256 197 y(quan)n(tum)f (sphere,)h(166)1256 396 y(real)e(quan)n(tum)i(h)n(yp)r(erb)r(oloid,)e (75)1256 497 y(real)g(quan)n(tum)i(plane,)f(75)1256 597 y(relations)1422 698 y FN(F)1475 710 y FK(4)1512 698 y FO(-relations,)e(26)1422 798 y FN(q)s FO(-relations,)g(26)1422 899 y(quadratic,)h(24)1505 1000 y(homogeneous,)f(24)1422 1100 y(represen)n(tation,)h(21)1422 1201 y(semilinear,)d(44)1505 1301 y(c)n(haracteristic)14 b(binary)j(re-)1588 1401 y(lation,)25 b(47)1505 1501 y(c)n(haracteristic)h(function,)1588 1601 y(47)1505 1702 y(graph,)g(47)1422 1802 y FP(\003)p FO(-wild,)f(26)1505 1903 y(cubic,)h(222)1505 2003 y(quadratic,)f(222) 1505 2104 y(semilinear,)d(221)1256 2204 y(represen)n(tation,)i(7)1422 2305 y FP(\003)p FO(-complete,)g(62)1422 2406 y(an)n(ti-F)-7 b(o)r(c)n(k,)25 b(100)1422 2506 y(F)-7 b(o)r(c)n(k,)27 b(74,)g(100)1422 2607 y(indecomp)r(osable,)c(9)1422 2707 y(irreducible,)h(9)1256 2808 y(represen)n(tations)1422 2908 y(category)h FP(\003)p FO(-Rep)13 b Fz(A)p FO(,)28 b(8)1422 3009 y(equaiv)-5 b(alen)n(t,)24 b(7)1422 3109 y(residual)g(family)-7 b(,)25 b(11)1422 3210 y(unitarily)f(equiv)-5 b(alen)n(t,)25 b(8)1256 3311 y(resolution)f(of)k(the)g(iden)n(tit)n(y) 1422 3411 y(non-orthogonal,)23 b(110)1256 3610 y(second-degree)16 b(mapping,)i(83,)h(94,)1588 3710 y(100,)26 b(105,)g(106)1256 3810 y(Skly)n(anin)f(algebra,)f(123)1256 3911 y(standard)i FP(\003)p FO(-wild)f(problem,)g(213)p eop %%Page: 261 265 261 264 bop 2214 -137 a FO(261)-118 96 y(theorem)48 196 y(Amitsur{Levitski,)23 b(27)48 296 y(F)-7 b(uglede{Putnam{Rosen)n (blum,)214 395 y(75)48 495 y(Jacobson,)26 b(44)48 595 y(Kleinec)n(k)n(e{Shirok)n(o)m(v,)c(43)48 694 y(Kleinec)n(k)n(e{Shirok) n(o)m(v)30 b(t)n(yp)r(e,)214 794 y(44,)d(48)48 893 y(Shark)n(o)n(vsky) -7 b(,)25 b(93)-118 993 y(t)n(wisted)i(CAR,)h(155)-118 1093 y(t)n(wisted)f(CCR,)g(155)-118 1275 y(Wic)n(k)f(algebra,)f(173) -118 1375 y(Wic)n(k)h(ideal,)g(174)-118 1475 y(Wic)n(k)g(ordered)h (monomials,)22 b(174)-118 1574 y FP(\003)p FO(-wildness,)j(203,)h(212) -118 1674 y(Witten's)18 b(deformations)d(of)k FN(so)p FO(\(3\),)214 1773 y(119)-118 1873 y(W)-7 b(old)27 b(decomp)r(osition,) d(75)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF