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L2ch1.ps
%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: book.dvi %%Pages: 12 0 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D300 -oL2book.ps book %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 2002.11.18:1952 %%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] 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Ft(:)967 2683 y Fq(1)p eop %%Page: 2 2 2 9 bop 228 113 a Fq(2)103 b(1.)14 b(ALGEBRAS)g(OF)h(OPERA)m(TORS)i(IN) d(A)g(FINITE)g(DIMENSIONAL)g(HILBER)m(T)g(SP)m(A)o(CE)228 213 y FD(Explicitly)m(,)e(this)i(isomorphism)c(is)k(giv)o(en)f(b)o(y)h (the)h(form)o(ulas)366 344 y Ft(V)411 333 y Fu(\030)411 346 y FD(=)454 344 y Fn(C)481 327 y Fr(n)507 344 y Ft(;)7 b FD(where)28 b Ft(V)21 b Fu(3)11 b Ft(x)h Fu($)832 248 y Fo(0)832 321 y(B)832 348 y(@)870 280 y Ft(x)894 286 y Fq(1)886 324 y FD(.)886 341 y(.)886 357 y(.)868 407 y Ft(x)892 413 y Fr(n)914 248 y Fo(1)914 321 y(C)914 348 y(A)958 344 y Ft(;)20 b(x)1014 350 y Fr(j)1043 344 y Fu(2)11 b Fn(C)6 b Ft(;)h FD(if)15 b Ft(x)d FD(=)f Ft(x)1271 350 y Fq(1)1290 344 y Ft(e)1309 350 y Fq(1)1337 344 y FD(+)f Fu(\001)d(\001)g(\001)g FD(+)j Ft(x)1502 350 y Fr(n)1524 344 y Ft(e)1543 350 y Fr(n)1566 344 y Ft(;)490 538 y FD(If)k Ft(Ae)582 544 y Fr(i)608 538 y FD(=)671 486 y Fr(n)652 498 y Fo(X)651 587 y Fr(k)q Fq(=1)719 538 y Ft(a)741 544 y Fr(k)q(i)773 538 y Ft(e)792 544 y Fr(k)812 538 y Ft(;)90 b FD(then)15 b Ft(A)c Fu($)1104 442 y Fo(0)1104 515 y(B)1104 541 y(@)1143 473 y Ft(a)1165 479 y Fq(11)1243 473 y Ft(:)c(:)g(:)49 b(a)1364 479 y Fq(1)p Fr(n)1165 518 y FD(.)1165 534 y(.)1165 551 y(.)1367 518 y(.)1367 534 y(.)1367 551 y(.)1141 601 y Ft(a)1163 607 y Fr(n)p Fq(1)1243 601 y Ft(:)7 b(:)g(:)47 b(a)1362 607 y Fr(nn)1405 442 y Fo(1)1405 515 y(C)1405 541 y(A)1448 538 y Ft(:)228 671 y FD(Comp)q(osition)11 b(of)j(op)q(erators)g(in)g Fu(L)p FD(\()p Ft(V)9 b FD(\))14 b(corresp)q(onds)i(to)e(matrix)e(m)o (ultipli)o(cation)f(in)i Ft(M)1627 677 y Fr(n)1650 671 y FD(\()p Fn(C)6 b FD(\))274 774 y Ft(A)12 b FD(=)g Ft(A)392 780 y Fq(1)410 774 y Ft(A)441 780 y Fq(2)472 774 y Fu(\))f Ft(a)547 780 y Fr(ij)588 774 y FD(=)651 722 y Fr(n)632 735 y Fo(X)631 824 y Fr(k)q Fq(=1)699 774 y Ft(a)721 753 y Fq(\(1\))721 787 y Fr(ik)765 774 y Ft(a)787 753 y Fq(\(2\))787 787 y Fr(k)q(j)845 774 y FD(if)j Ft(A)915 780 y Fq(1)945 774 y Fu($)998 728 y Fo(\020)1023 774 y Ft(a)1045 753 y Fq(\(1\))1045 786 y Fr(ij)1089 728 y Fo(\021)1121 774 y Ft(;)20 b(A)1184 780 y Fq(2)1214 774 y Fu($)1267 728 y Fo(\020)1292 774 y Ft(a)1314 753 y Fq(\(2\))1314 786 y Fr(ij)1359 728 y Fo(\021)1404 774 y FD(and)14 b Ft(A)e Fu($)f FD(\()p Ft(a)1619 780 y Fr(ij)1648 774 y FD(\))p Ft(:)303 890 y FB(Definition.)21 b FD(The)14 b FA(tr)n(ac)n(e)f FD(T)m(r)f(:)f Fu(L)p FD(\()p Ft(V)e FD(\))j Fu(!)f Fn(C)23 b FD(is)13 b(giv)o(en)h(b)o(y)f(the)i(form)o (ula)844 993 y(T)m(r)7 b Ft(A)k FD(=)1000 941 y Fr(n)980 954 y Fo(X)983 1042 y Fr(i)p Fq(=1)1047 993 y Ft(a)1069 999 y Fr(ii)1095 993 y Ft(:)303 1108 y FB(Exer)o(cise)k FD(1)p FB(.)75 b FD(\(a\))20 b(T)m(r\()p Ft(A)773 1114 y Fq(1)792 1108 y Ft(A)823 1114 y Fq(2)842 1108 y FD(\))12 b(=)f(T)m(r\()p Ft(A)1003 1114 y Fq(2)1022 1108 y Ft(A)1053 1114 y Fq(1)1072 1108 y FD(\))p Ft(;)20 b(A)1151 1114 y Fq(1)1170 1108 y Ft(;)7 b(A)1220 1114 y Fq(2)1250 1108 y Fu(2)k(L)p FD(\()p Ft(V)e FD(\).)279 1158 y(\(b\))22 b(T)m(r)13 b(do)q(es)i(not)f(dep)q(end)h(on)e(the)i(c)o(hoice)f(of)g (the)g(basis)g(in)f Ft(V)d FD(.)284 1208 y(\(c\))22 b(Let)14 b Ft(f)i FD(:)11 b Fu(L)p FD(\()p Ft(V)f FD(\))h Fu(!)g Fn(C)23 b FD(b)q(e)15 b(a)e(linear)h(functional)e(whic)o(h)i(has)g(the) h(prop)q(ert)o(y)623 1278 y Ft(f)t FD(\()p Ft(A)694 1284 y Fq(1)714 1278 y Ft(A)745 1284 y Fq(2)764 1278 y FD(\))d(=)f Ft(f)t FD(\()p Ft(A)906 1284 y Fq(2)926 1278 y Ft(A)957 1284 y Fq(1)976 1278 y FD(\))p Ft(;)48 b(A)1083 1284 y Fq(1)1102 1278 y Ft(;)7 b(A)1152 1284 y Fq(2)1181 1278 y Fu(2)12 b(L)p FD(\()p Ft(V)d FD(\))p Ft(:)356 1348 y FD(Then)14 b Ft(f)i FD(=)c Ft(c)7 b FD(T)m(r)14 b(with)f(a)h(constan) o(t)g Ft(c)e Fu(2)f Fn(C)6 b FD(.)303 1424 y FA(Hint)13 b(to)h(\(c\).)k FD(Consider)c([)p Fu(L)p FD(\()p Ft(V)8 b FD(\))p Ft(;)f Fu(L)p FD(\()p Ft(V)i FD(\)])i(=)h Fu(f)p FD(linear)h(com)o(binations)f(of)h([)p Ft(A;)7 b(B)r FD(])13 b(with)h Ft(A;)7 b(B)228 1474 y Fu(2)17 b(L)p FD(\()p Ft(V)9 b FD(\))p Ft(;)e FD([)p Ft(A;)g(B)r FD(])17 b(=)g Ft(AB)e Fu(\000)d Ft(B)r(A)p Fu(g)18 b(\032)g(L)p FD(\()p Ft(V)9 b FD(\).)29 b(Chec)o(k)18 b(that)g(dim)1259 1480 y Fm(C)1289 1474 y Fu(L)p FD(\()p Ft(V)9 b FD(\))p Ft(=)p FD([)p Fu(L)p FD(\()p Ft(V)g FD(\))p Ft(;)e Fu(L)p FD(\()p Ft(V)h FD(\)])18 b(=)f(1)228 1524 y(\(or)i(co)q(dim)409 1530 y Fm(C)432 1524 y FD([)p Fu(L)p FD(\()p Ft(V)9 b FD(\))p Ft(;)e Fu(L)p FD(\()p Ft(V)i FD(\)])19 b(=)i(1\))d(b)o(y)h (considering)g Ft(E)1106 1530 y Fr(ij)1155 1524 y Fu(2)g(L)p FD(\()p Ft(V)9 b FD(\))19 b(suc)o(h)h(that)f Ft(E)1540 1530 y Fr(ij)1569 1524 y Ft(e)1588 1530 y Fr(j)1625 1524 y FD(=)i Ft(e)1697 1530 y Fr(i)1711 1524 y FD(,)228 1574 y Ft(E)259 1580 y Fr(ij)288 1574 y Ft(e)307 1580 y Fr(k)339 1574 y FD(=)12 b(0)h(if)g Ft(k)g Fu(6)p FD(=)f Ft(j)k FD(for)e(a)f(giv)o(en)h(basis)f Fu(f)p Ft(e)915 1580 y Fq(1)934 1574 y Ft(;)7 b(:)g(:)g(:)e(;)i(e)1046 1580 y Fr(n)1068 1574 y Fu(g)p FD(.)303 1644 y(Hence)15 b(the)f(linear)g (functional)f(in)g(\(c\))i(is)e(unique)h(up)g(to)g(a)g(constan)o(t)g (factor.)303 1720 y FB(Exer)o(cise)h FD(2)p FB(.)21 b FD(Let)13 b(GL\()p Ft(V)c FD(\))k(denote)h(the)g(group)f(of)f(all)g(in) o(v)o(ertible)h(linear)f(generators)j(in)228 1770 y Ft(V)9 b FD(.)18 b(Pro)o(v)o(e)c(that)g(if)f(a)h(linear)f(functional)g Ft(f)k FD(:)11 b Fu(L)p FD(\()p Ft(V)e FD(\))j Fu(!)f Fn(C)22 b FD(satis\014es)578 1840 y Ft(f)t FD(\()p Ft(C)651 1823 y Fs(\000)p Fq(1)696 1840 y Ft(AC)s FD(\))11 b(=)h Ft(f)t FD(\()p Ft(A)p FD(\))p Ft(;)22 b(A)11 b Fu(2)g(L)p FD(\()p Ft(V)f FD(\))p Ft(;)d(C)13 b Fu(2)f FD(GL)o(\()p Ft(V)e FD(\))p Ft(;)228 1910 y FD(then)k Ft(f)j FD(=)11 b Ft(c)c FD(T)m(r.)303 1987 y FA(Hint.)25 b FD(Chec)o(k)17 b(that)f(the)h(condition)f(in)f(Exercise)j(1)e(\(c\))h(is)g (satis\014ed)g(for)e(all)h(in)o(v)o(ertible)228 2037 y Ft(A)259 2043 y Fq(1)278 2037 y FD(,)d(hence)i(for)f(all)e Ft(A)570 2043 y Fq(1)589 2037 y FD(.)807 2129 y Fp(1.2.)24 b(Pro)s(jections)303 2204 y FB(Definition)15 b FD(1)p FB(.)21 b Ft(P)c Fu(2)11 b(L)p FD(\()p Ft(V)f FD(\))k(is)f(called)h(a)g FA(pr)n(oje)n(ction)f FD(if)g Ft(P)1240 2189 y Fq(2)1270 2204 y FD(=)f Ft(P)6 b FD(.)303 2280 y FB(Definition)15 b FD(2)p FB(.)21 b FD(F)m(or)14 b Ft(A)d Fu(2)h(L)p FD(\()p Ft(V)d FD(\))14 b(denote)533 2350 y(Ker)8 b Ft(A)j FD(=)h Fu(f)p Ft(x)p Fu(j)p Ft(x)e Fu(2)i Ft(V)r(;)7 b(Ax)k FD(=)h(0)p Fu(g)41 b FD(\(k)o(ernel)14 b(of)g Ft(A)p FD(\))550 2414 y(Im)6 b Ft(A)11 b FD(=)h Fu(f)p Ft(Ax)p Fu(j)p Ft(x)f Fu(2)g Ft(V)e Fu(g)153 b FD(\(range)14 b(or)g(image)e(of)h Ft(A)p FD(\))494 2479 y(Cok)o(er)7 b Ft(A)k FD(=)h Ft(V)5 b(=)i FD(Im)e Ft(A)225 b FD(\(cok)o(ernel)15 b(of)e(A\))303 2558 y FB(Exer)o(cise.)74 b FD(\(a\))21 b Ft(P)e Fu(2)12 b(L)p FD(\()p Ft(V)e FD(\))15 b(is)f(a)h(pro)r (jection)g(i\013)g Ft(I)e Fu(\000)d Ft(P)21 b FD(is)14 b(a)h(pro)r(jection.)21 b(\(Here)c Ft(I)356 2608 y FD(is)c(the)i(iden)o (tit)o(y)e(op)q(erator)i(in)e Ft(V)c FD(\).)p eop %%Page: 3 3 3 10 bop 802 113 a Fq(1.3.)15 b(INNER)f(PR)o(ODUCT)547 b(3)279 213 y FD(\(b\))22 b(If)e Ft(P)28 b Fu(2)22 b(L)p FD(\()p Ft(V)10 b FD(\))20 b(is)h(a)f(pro)r(jection)h(then)g Ft(V)30 b FD(is)21 b(decomp)q(osed)g(in)o(to)e(a)i(direct)g(set)h(of) 356 262 y(subspaces)g Ft(V)31 b FD(=)22 b(Im)6 b Ft(P)761 256 y FD(_)751 262 y(+)g(Ker)i Ft(P)e FD(.)37 b(\(Here)21 b Ft(V)32 b FD(=)22 b Ft(V)1199 268 y Fq(1)1228 256 y FD(_)1218 262 y(+)p Ft(V)1274 268 y Fq(2)1313 262 y FD(means)d(that)h Ft(V)1566 268 y Fq(1)1585 262 y Ft(;)7 b(V)1628 268 y Fq(2)1667 262 y FD(are)356 312 y(subspaces)19 b(in)d Ft(V)27 b FD(suc)o(h)17 b(that)g Ft(V)864 318 y Fq(1)895 312 y Fu(\\)11 b Ft(V)958 318 y Fq(2)993 312 y FD(=)17 b Fu(f)p FD(0)p Fu(g)p Ft(;)23 b(V)1164 318 y Fq(1)1194 312 y FD(+)12 b Ft(V)1262 318 y Fq(2)1297 312 y FD(=)18 b Ft(V)9 b FD(,)17 b(where)h Ft(V)1556 318 y Fq(1)1587 312 y FD(+)11 b Ft(V)1654 318 y Fq(2)1690 312 y FD(=)356 362 y Fu(f)p Ft(x)401 368 y Fq(1)422 362 y FD(+)t Ft(x)482 368 y Fq(2)500 362 y Fu(j)p Ft(x)536 368 y Fq(1)566 362 y Fu(2)g Ft(V)629 368 y Fq(1)648 362 y Ft(;)c(x)691 368 y Fq(2)720 362 y Fu(2)k Ft(V)783 368 y Fq(2)802 362 y Fu(g)p FD(.\))17 b(Moreo)o(v)o(er,)12 b Ft(P)17 b Fu(j)11 b FD(Im)5 b Ft(P)17 b FD(=)12 b Ft(I)j Fu(j)c FD(Im)6 b Ft(P)16 b FD(and)11 b Ft(P)17 b Fu(j)11 b FD(Ker)d Ft(P)17 b FD(=)356 412 y(0)11 b Fu(j)g FD(Ker)d Ft(P)e FD(.)17 b(Besides)f(Im)n(\()p Ft(I)d Fu(\000)d Ft(P)c FD(\))11 b(=)h(Ker)7 b Ft(P)f FD(,)13 b(Ker)q(\()p Ft(I)g Fu(\000)d Ft(P)c FD(\))11 b(=)h(Im)5 b Ft(P)h FD(.)430 462 y(Vice)21 b(v)o(ersa)g(if)f Ft(V)31 b FD(=)23 b Ft(V)822 468 y Fq(1)851 456 y FD(_)841 462 y(+)p Ft(V)897 468 y Fq(2)936 462 y FD(and)d(a)g(linear)g(op)q(erator)h Ft(P)28 b Fu(2)22 b(L)p FD(\()p Ft(V)9 b FD(\))20 b(is)h(giv)o(en)356 511 y(b)o(y)15 b Ft(P)6 b FD(\()p Ft(x)488 517 y Fq(1)517 511 y FD(+)11 b Ft(x)584 517 y Fq(2)602 511 y FD(\))k(=)g Ft(x)704 517 y Fq(1)739 511 y FD(for)g(an)o(y)h Ft(x)909 517 y Fr(i)937 511 y Fu(2)f Ft(V)1004 517 y Fr(i)1018 511 y Ft(;)22 b(i)16 b FD(=)f(1)p Ft(;)7 b FD(2,)15 b(then)h Ft(P)22 b FD(is)15 b(a)h(pro)r(jection)g(with)356 561 y Ft(V)380 567 y Fq(1)410 561 y FD(=)c(Im)5 b Ft(P)17 b FD(=)12 b(Ker)q(\()p Ft(I)5 b Fu(\000)r Ft(P)h FD(\))p Ft(;)17 b(V)840 567 y Fq(2)871 561 y FD(=)12 b(Ker)c Ft(P)16 b FD(=)c(Im)o(\()p Ft(I)5 b Fu(\000)r Ft(P)h FD(\),)11 b(and)f(\()p Ft(I)5 b Fu(\000)r Ft(P)h FD(\)\()p Ft(x)1510 567 y Fq(1)1532 561 y FD(+)r Ft(x)1590 567 y Fq(2)1608 561 y FD(\))12 b(=)g Ft(x)1704 567 y Fq(2)356 611 y FD(for)h(an)o(y)h Ft(x)522 617 y Fr(i)547 611 y Fu(2)d Ft(V)610 617 y Fr(i)624 611 y Ft(;)20 b(i)12 b FD(=)g(1)p Ft(;)7 b FD(2.)284 661 y(\(c\))22 b(If)13 b Ft(P)19 b FD(is)14 b(a)g(pro)r(jection)g(then)820 736 y(T)m(r)7 b Ft(P)17 b FD(=)12 b(dim)5 b(Im)g Ft(P)q(:)780 843 y Fp(1.3.)24 b(Inner)14 b(pro)q(duct)303 918 y FB(Definition)h FD(1)p FB(.)21 b FD(An)d FA(inner)h(pr)n(o)n(duct)f FD(\()g(or)f FA(sc)n(alar)h(pr)n(o)n(duct)p FD(\))g(in)g(a)f(\014nite)h(dimensional) 228 968 y(v)o(ector)c(space)h Ft(V)22 b FD(is)14 b(a)f(map)f(\()p Fu(\001)p Ft(;)7 b Fu(\001)p FD(\))j(:)h Ft(V)18 b Fu(\002)9 b Ft(V)21 b Fu(!)11 b Fn(C)22 b FD(whic)o(h)14 b(satis\014es)g(the)g (follo)o(wing)d(conditions:)291 1030 y(\(i\))21 b FA(It)14 b(is)h(line)n(ar)f(with)g(r)n(esp)n(e)n(ct)g(to)h(the)g(1st)g (variable:)383 1105 y FD(\()p Ft(\025)423 1111 y Fq(1)442 1105 y Ft(x)466 1111 y Fq(1)494 1105 y FD(+)10 b Ft(\025)560 1111 y Fq(2)578 1105 y Ft(x)602 1111 y Fq(2)621 1105 y Ft(;)d(y)q FD(\))12 b(=)f Ft(\025)756 1111 y Fq(1)775 1105 y FD(\()p Ft(x)815 1111 y Fq(1)834 1105 y Ft(;)c(y)q FD(\))i(+)h Ft(\025)965 1111 y Fq(2)984 1105 y FD(\()p Ft(x)1024 1111 y Fq(2)1042 1105 y Ft(;)d(y)q FD(\))p Ft(;)21 b(x)1155 1111 y Fq(1)1173 1105 y Ft(;)7 b(x)1216 1111 y Fq(2)1234 1105 y Ft(;)g(y)13 b Fu(2)e Ft(V)r(;)c(\025)1394 1111 y Fq(1)1413 1105 y Ft(;)g(\025)1456 1111 y Fq(2)1486 1105 y Fu(2)k Fn(C)6 b FD(;)279 1184 y(\(ii\))21 b FA(It)14 b(is)h(hermitian:)j FD(\()p Ft(x;)7 b(y)q FD(\))k(=)p 803 1148 97 2 v 12 w(\()p Ft(y)q(;)c(x)p FD(\))14 b(\()p Ft(x;)7 b(y)13 b Fu(2)e Ft(V)f FD(\))356 1234 y(\(Hence,)15 b(it)e(is)h(an)o(tilinear)f(with)g(resp)q(ect)j(to)e(the)h(2nd)e(v)n (ariable:)356 1284 y(\()p Ft(x;)7 b(\025)439 1290 y Fq(1)457 1284 y Ft(y)477 1290 y Fq(1)505 1284 y FD(+)i Ft(\025)570 1290 y Fq(2)589 1284 y Ft(y)609 1290 y Fq(2)628 1284 y FD(\))j(=)701 1273 y(\026)700 1284 y Ft(\025)724 1290 y Fq(1)743 1284 y FD(\()p Ft(x;)7 b(y)822 1290 y Fq(1)840 1284 y FD(\))i(+)908 1273 y(\026)907 1284 y Ft(\025)931 1290 y Fq(2)950 1284 y FD(\()p Ft(x;)e(y)1029 1290 y Fq(2)1047 1284 y FD(\))p Ft(;)20 b(x;)7 b(y)1158 1290 y Fq(1)1176 1284 y Ft(;)g(y)1215 1290 y Fq(2)1245 1284 y Fu(2)12 b Ft(V)r(;)20 b(\025)1367 1290 y Fq(1)1386 1284 y Ft(;)7 b(\025)1429 1290 y Fq(2)1459 1284 y Fu(2)k Fn(C)6 b FD(\);)268 1334 y(\(iii\))20 b FA(It)14 b(is)h(p)n(ositive)f (de\014nite:)19 b FD(\()p Ft(x;)7 b(x)p FD(\))k Fu(\025)h FD(0)p Ft(;)20 b(x)11 b Fu(2)g Ft(V)f FD(;)j(\()p Ft(x;)7 b(x)p FD(\))k(=)g(0)j(if)f(and)h(only)f(if)g Ft(x)e FD(=)h(0.)228 1396 y(The)i(pair)g(\()p Ft(V)r(;)7 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FD(giv)o(en)h(b)o(y)879 2591 y(Im)5 b Ft(P)17 b Fu($)11 b Ft(P)q(:)p eop %%Page: 4 4 4 11 bop 228 113 a Fq(4)103 b(1.)14 b(ALGEBRAS)g(OF)h(OPERA)m(TORS)i (IN)d(A)g(FINITE)g(DIMENSIONAL)g(HILBER)m(T)g(SP)m(A)o(CE)479 213 y Fp(1.4.)24 b(Sp)q(ectral)15 b(theorem)f(for)h(self-adjoin)n(t)d (op)q(erators)303 287 y FD(Assume)e(that)h(w)o(e)h(are)f(giv)o(en)g(an) f(inner)i(pro)q(duct)g(on)e Ft(V)g FD(.)17 b(Let)11 b Ft(A)h Fu(2)f(L)p FD(\()p Ft(V)e FD(\))i(b)q(e)h(self-adjoin)o(t,)228 337 y(that)17 b(is,)g Ft(A)409 322 y Fs(\003)445 337 y FD(=)g Ft(A)p FD(.)28 b(Then)17 b(there)h(exists)g(an)f(orthonormal)e (basis)i Fu(f)p Ft(e)1350 343 y Fq(1)1369 337 y Ft(;)7 b(:)g(:)g(:)t(;)g(e)1480 343 y Fr(n)1503 337 y Fu(g)16 b FD(in)h Ft(V)26 b FD(suc)o(h)228 387 y(that)16 b Ft(Ae)370 393 y Fr(j)403 387 y FD(=)f Ft(\025)474 393 y Fr(j)492 387 y Ft(e)511 393 y Fr(j)528 387 y Ft(;)23 b(\025)587 393 y Fr(j)619 387 y Fu(2)15 b Fn(R)m FD(,)e(i.e.)24 b Ft(e)817 393 y Fr(j)850 387 y FD(are)17 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y Fo(\023)685 2422 y Ft(V)709 2428 y Fq(2)685 2471 y Ft(V)709 2477 y Fq(1)728 2447 y Ft(;)7 b(A)778 2453 y Fq(1)808 2447 y Fu(2)k(L)p FD(\()p Ft(V)916 2453 y Fq(1)934 2447 y FD(\))950 2388 y Fo(\033)993 2447 y FD(=)h Fu(f)p Ft(A)f Fu(2)h(L)p FD(\()p Ft(V)d FD(\))p Fu(j)p Ft(AV)1301 2453 y Fq(1)1331 2447 y Fu(\032)j Ft(V)1399 2453 y Fq(1)1418 2447 y Ft(;)7 b(AV)1492 2453 y Fq(2)1522 2447 y FD(=)k Fu(f)p FD(0)p Fu(gg)6 b Ft(:)-1439 b FD(\(2\))537 2532 y Ft(V)561 2538 y Fq(1)621 2532 y Ft(V)645 2538 y Fq(2)228 2608 y FD(Clearly)m(,)12 b Fu(M)430 2614 y Fq(2)460 2608 y Fu(63)f Ft(I)s(:)p eop %%Page: 6 6 6 13 bop 228 113 a Fq(6)103 b(1.)14 b(ALGEBRAS)g(OF)h(OPERA)m(TORS)i (IN)d(A)g(FINITE)g(DIMENSIONAL)g(HILBER)m(T)g(SP)m(A)o(CE)303 231 y FB(Exer)o(cise)h FD(3)p FB(.)21 b FD(Chec)o(k)14 b(that)g Fu(M)820 216 y Fs(0)820 242 y Fq(2)850 231 y FD(=)894 173 y Fo(\032\022)956 206 y Ft(\025I)998 212 y Fq(1)1073 206 y FD(0)976 256 y(0)61 b Ft(A)1089 262 y Fq(2)1108 173 y Fo(\023)1146 196 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Fs(0)1131 1180 y Fq(4)1161 1170 y FD(=)e Fu(f)p Ft(\025)d Fu(\012)h Ft(I)s Fu(j)p Ft(\025)h Fu(2)h Ft(M)1449 1176 y Fq(2)1468 1170 y FD(\()p Fn(C)6 b FD(\))p Fu(g)228 1265 y FD(in)13 b(the)i(represen)o(tation)g Ft(V)21 b FD(=)12 b Fn(C)735 1250 y Fq(2)766 1265 y Fu(\012)d Ft(V)831 1271 y Fq(1)850 1265 y FD(.)18 b(Hence)d Ft(Z)s FD(\()p Fu(M)1100 1271 y Fq(4)1119 1265 y FD(\))d(=)g Fu(f)p Ft(\025I)s Fu(j)p Ft(\025)g Fu(2)f Fn(C)6 b Fu(g)p FD(.)726 1350 y Fp(1.7.)24 b(Double)14 b(comm)o(utan)o(t)303 1424 y FD(Note)d(that)f Fu(M)536 1409 y Fs(0)559 1424 y FD(is)g(alw)o(a)o(ys)g(a)g(subalgebra)h (with)g Ft(I)k Fu(2)c(M)p FD(.)17 b(De\014ne)11 b Fu(M)1381 1409 y Fs(0)o(0)1414 1424 y FD(=)g(\()p Fu(M)1523 1409 y Fs(0)1535 1424 y FD(\))1551 1409 y Fs(0)1563 1424 y FD(.)17 b(Clearly)228 1474 y Fu(M)11 b(\032)h(M)383 1459 y Fs(0)o(0)404 1474 y FD(.)18 b(What)c(are)g(conditions)f(of)h (coincidence)h(of)e Fu(M)h FD(and)f Fu(M)1326 1459 y Fs(00)1347 1474 y FD(?)303 1546 y FB(Theorem)j 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b(is)e(su\016cien)o(t)i(to)f(pro)o(v)o(e)g(that)g Fu(M)1025 1970 y Fs(00)o(0)1067 1985 y FD(=)f Fu(M)1161 1970 y Fs(0)1173 1985 y FD(;)h(then)g(w)o(e)h(can)f(apply)f(this)h(result)228 2035 y(to)e Fu(M)326 2020 y Fs(0)337 2035 y Ft(;)c Fu(M)406 2020 y Fs(0)o(0)427 2035 y Ft(;)g(:)g(:)g(:)16 b FD(to)11 b(get)h(all)e(other)i(relations.)17 b(Ob)o(viously)10 b Fu(M)1214 2020 y Fs(0)1237 2035 y Fu(\032)i(M)1331 2020 y Fs(00)o(0)1362 2035 y FD(.)17 b(On)11 b(the)h(other)g(hand)228 2085 y Fu(M)f(\032)h(M)383 2070 y Fs(0)o(0)418 2085 y FD(implies)g Fu(M)609 2070 y Fs(0)632 2085 y Fu(\033)g(M)726 2070 y Fs(0)o(00)756 2085 y FD(,)i(hence)h Fu(M)947 2070 y Fs(0)970 2085 y FD(=)d Fu(M)1064 2070 y Fs(0)o(00)1094 2085 y FD(.)p 1692 2085 2 29 v 1694 2058 25 2 v 1694 2085 V 1719 2085 2 29 v 303 2159 a FB(Pr)o(oof)k(of)g(Theorem.)22 b FD(1\))17 b(Cho)q(ose)h Ft(x)f Fu(2)f Ft(V)27 b FD(and)17 b(consider)h Fu(M)p Ft(x)f FD(=)g Fu(f)p Ft(Ax)p Fu(j)p Ft(A)f Fu(2)h(Mg)p FD(,)228 2209 y(whic)o(h)12 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FD(-in)o(v)n(arian)o(t.)28 b(This)18 b(implies)e(that)228 2458 y Fu(M)278 2443 y Fs(0)o(0)299 2458 y Ft(x)11 b Fu(\032)h(M)p Ft(x)p FD(,)h(hence)i Fu(M)642 2443 y Fs(00)663 2458 y Ft(x)d FD(=)f Fu(M)p Ft(x)p FD(.)303 2508 y(2\))j(W)m(e)h(ha)o(v)o(e)g(pro)o(v)o(ed)g(that)g Fu(8)p Ft(x)e Fu(2)g Ft(V)r(;)22 b Fu(8)p Ft(B)16 b Fu(2)d(M)1073 2493 y Fs(0)o(0)1094 2508 y Ft(;)21 b Fu(9)p Ft(A)14 b Fu(2)f(M)i FD(suc)o(h)h(that)f Ft(B)r(x)e FD(=)h Ft(Ax)p FD(.)21 b(It)228 2558 y(w)o(ould)13 b(b)q(e)i(go)q(o)q(d)e(to)h(do)g (it)g(not)g(only)f(for)h(a)f(single)h(v)o(ector)h Ft(x)f FD(but)g(for)g(a)f Ft(N)5 b FD(-tuple)14 b Ft(x)1556 2564 y Fq(1)1575 2558 y Ft(;)7 b(:)g(:)g(:)e(;)i(x)1692 2564 y Fr(N)228 2608 y FD(\(e.g.)13 b(for)h(a)f(basis\).)18 b(This)c(is)g(done)g(b)o(y)g(the)g(follo)o(wing)e(tric)o(k.)p eop %%Page: 7 7 7 14 bop 533 113 a Fq(1.9.)15 b(GENERA)m(TION)f(OF)h Fs(\003)p Fq(-ALGEBRAS)f(BY)g(PR)o(OJECTIONS)279 b(7)303 213 y FD(Consider)14 b(the)g(algebra)g Fu(M)9 b(\012)g Ft(I)18 b FD(in)13 b Ft(V)19 b Fu(\012)9 b Fn(C)988 198 y Fr(N)1023 213 y FD(,)k(so)592 369 y Fu(M)c(\012)h Ft(I)15 b FD(=)770 247 y Fo(8)770 284 y(>)770 297 y(>)770 309 y(>)770 322 y(<)770 396 y(>)770 409 y(>)770 421 y(>)770 434 y(:)806 249 y(0)806 322 y(B)806 347 y(B)806 372 y(B)806 398 y(@)843 281 y Ft(A)209 b FD(0)915 330 y Ft(A)990 379 y FD(.)1007 391 y(.)1023 404 y(.)848 458 y(0)g Ft(A)1109 249 y Fo(1)1109 322 y(C)1109 347 y(C)1109 372 y(C)1109 398 y(A)1152 369 y Ft(;)7 b(A)k Fu(2)h(M)1303 247 y Fo(9)1303 284 y(>)1303 297 y(>)1303 309 y(>)1303 322 y(=)1303 396 y(>)1303 409 y(>)1303 421 y(>)1303 434 y(;)1346 369 y Ft(:)228 527 y FD(This)i(is)f(again)g(a)h Fu(\003)p FD(-algebra)f(with) g Ft(I)s FD(.)19 b(No)o(w)13 b(tak)o(e)478 659 y Ft(\030)h FD(=)554 563 y Fo(0)554 636 y(B)554 662 y(@)597 595 y Ft(x)621 601 y Fq(1)612 639 y FD(.)612 655 y(.)612 672 y(.)590 722 y Ft(x)614 728 y Fr(N)645 563 y Fo(1)645 636 y(C)645 662 y(A)693 659 y FD(=)e Ft(x)761 665 y Fq(1)789 659 y Fu(\010)d(\001)e(\001)g(\001)h(\010)h Ft(x)953 665 y Fr(N)996 659 y Fu(2)i Ft(V)19 b Fu(\012)9 b Fn(C)1146 641 y Fr(N)1192 659 y FD(=)j Ft(V)19 b Fu(\010)9 b(\001)e(\001)g(\001)h (\010)i Ft(V)1236 679 y Fo(|)p 1255 679 72 5 v 72 w({z)p 1365 679 V 72 w(})1291 719 y Fr(N)s Fq(times)1460 659 y Ft(:)303 799 y FB(Exer)o(cise)15 b FD(1)p FB(.)21 b FD(Chec)o(k)e(that)g(\()p Fu(M)13 b(\012)g Ft(I)s FD(\))941 784 y Fs(0)973 799 y FD(=)20 b Fu(M)1075 784 y Fs(0)1099 799 y Fu(\012)13 b Ft(M)1184 805 y Fr(n)1207 799 y FD(\()p Fn(C)6 b FD(\))p Ft(;)28 b FD(\()p Fu(M)13 b(\012)g Ft(I)s FD(\))1467 784 y Fs(00)1508 799 y FD(=)21 b Fu(M)1611 784 y Fs(0)o(0)1644 799 y Fu(\012)13 b Ft(I)s FD(.)228 849 y(\(Compare)g(Exercise)i(5)f(in)f(1.6\))303 925 y(Applying)g(the)i (argumen)o(t)e(in)h(the)i(\014rst)f(part)f(of)g(the)h(pro)q(of,)f(w)o (e)h(obtain)f(that)g(for)g(ev)o(ery)228 975 y Ft(B)g Fu(2)d(M)362 960 y Fs(00)397 975 y FD(there)k(exist)f Ft(A)e Fu(2)f(M)j FD(suc)o(h)h(that)f(\()p Ft(B)e Fu(\012)d Ft(I)s FD(\))p Ft(\030)14 b FD(=)e(\()p Ft(A)e Fu(\012)f Ft(I)s FD(\))p Ft(\030)r FD(,)14 b(i.e.)580 1046 y Ft(B)r(x)637 1052 y Fq(1)668 1046 y FD(=)d Ft(Ax)766 1052 y Fq(1)785 1046 y Ft(;)20 b(B)r(x)874 1052 y Fq(2)905 1046 y FD(=)11 b Ft(Ax)1003 1052 y Fq(2)1022 1046 y Ft(;)c(:)g(:)g(:)e(;)20 b(B)r(x)1185 1052 y Fr(N)1228 1046 y FD(=)12 b Ft(Ax)1327 1052 y Fr(N)1358 1046 y Ft(:)228 1116 y FD(If)h Ft(x)293 1122 y Fq(1)312 1116 y Ft(;)7 b(:)g(:)g(:)t(;)g(x)428 1122 y Fr(N)473 1116 y FD(span)14 b Ft(V)9 b FD(,)14 b(then)g(w)o(e)h(obtain)e Ft(B)h FD(=)e Ft(A)p FD(.)18 b(Hence)d Fu(M)1237 1101 y Fs(0)o(0)1270 1116 y FD(=)c Fu(M)p FD(.)p 1692 1116 2 29 v 1694 1090 25 2 v 1694 1116 V 1719 1116 2 29 v 303 1196 a FB(Remark.)22 b FD(Example)11 b Fu(M)722 1202 y Fq(3)754 1196 y FD(in)i(1.6)f(sho)o(ws)h(that)h(the)f (symmetry)f(condition)g(\()p Ft(A)g Fu(2)f(M)h(\))228 1246 y Ft(A)259 1231 y Fs(\003)290 1246 y Fu(2)f(M)p FD(\))j(is)f(essen)o(tial)i(\()p Fu(M)682 1252 y Fq(3)714 1246 y FD(is)f(an)g(algebra,)e Fu(M)1020 1252 y Fq(3)1050 1246 y Fu(3)g Ft(I)17 b FD(but)d Fu(M)1251 1231 y Fs(00)1251 1256 y Fq(3)1284 1246 y Fu(6)p FD(=)e Fu(M)1378 1252 y Fq(3)1396 1246 y FD(\).)303 1322 y FB(Exer)o(cise)j FD(2)p FB(.)21 b FD(Find)14 b(an)h(example)e(of)h(an)g(algebra)h Fu(M)d(\032)h(L)p FD(\()p Ft(V)d FD(\),)k(suc)o(h)i(that)e Fu(M)1593 1307 y Fs(00)1627 1322 y FD(=)f Fu(M)228 1372 y FD(but)h Fu(M)g FD(is)f(not)h(a)g Fu(\003)p FD(-algebra.)228 1449 y FA(Hint.)i FD(The)10 b(condition)g Fu(M)648 1434 y Fs(0)o(0)680 1449 y FD(=)i Fu(M)d FD(is)h(purely)g(algebraic)f(and)h (do)q(es)g(not)g(dep)q(end)h(on)e(the)i(c)o(hoice)228 1499 y(of)i(the)h(inner)g(pro)q(duct)h(in)e Ft(V)c FD(,)k(but)h(the)h (symmetry)c(condition)i(\(of)h(b)q(eing)f(a)g Fu(\003)p FD(-algebra\))g(do)q(es)228 1549 y(dep)q(end)i(on)f(the)g(c)o(hoice)g (of)g(an)f(inner)i(pro)q(duct.)850 1643 y Fp(1.8.)24 b(F)l(actors)303 1718 y FB(Definition.)d FD(A)11 b FA(factor)g FD(is)g(a)g Fu(\003)p FD(-algebra)g Fu(M)g(\032)h(L)p FD(\()p Ft(V)d FD(\))j(suc)o(h)g(that)f Fu(M)h(3)f Ft(I)k FD(and)c Ft(Z)s FD(\()p Fu(M)p FD(\))h(=)228 1767 y Fu(f)p Ft(\025I)j FD(:)c Ft(\025)h Fu(2)f Fn(C)6 b Fu(g)p FD(.)303 1844 y(In)18 b(other)g(w)o(ords)g(a)g(factor)g(is)g(a)g(subalgebra)g (in)f Fu(L)p FD(\()p Ft(V)9 b FD(\))18 b(whic)o(h)g(con)o(tains)g(the)h (iden)o(tit)o(y)228 1894 y(op)q(erator)14 b(and)g(has)g(a)g(trivial)e (cen)o(ter.)303 1971 y FB(Example.)22 b Fu(L)p FD(\()p Ft(V)9 b FD(\))17 b(and)g Fu(M)762 1977 y Fq(4)797 1971 y FD(are)g(factors;)i Fu(M)1071 1977 y Fq(1)1089 1971 y Ft(;)7 b Fu(M)1158 1977 y Fq(2)1176 1971 y Ft(;)g Fu(M)1245 1977 y Fq(3)1280 1971 y FD(are)17 b(not.)27 b Fu(M)1501 1977 y Fq(1)1536 1971 y FD(is)17 b(a)f(direct)228 2020 y(sum)d(of)g(2)h(factors.)303 2097 y FB(Exer)o(cise.)20 b FD(If)14 b Fu(M)f FD(is)h(a)g(factor,)f(then)i Fu(M)976 2082 y Fs(0)1001 2097 y FD(is)f(a)f(factor)h(to)q(o.)514 2191 y Fp(1.9.)24 b(Generation)13 b(of)i Fu(\003)p Fp(-algebras)f(b)o (y)i(pro)s(jection)o(s)303 2266 y FB(Definition.)21 b FD(Denote)645 2337 y(Pro)r(j\()p Fu(M)p FD(\))11 b(=)h Fu(f)p Ft(P)17 b Fu(2)11 b(Mj)p Ft(P)1061 2320 y Fq(2)1090 2337 y FD(=)h Ft(P)17 b FD(=)12 b Ft(P)1255 2320 y Fs(\003)1273 2337 y Fu(g)p Ft(:)303 2413 y FD(This)h(means)g(that)h(Pro)r(j)q(\()p Fu(M)p FD(\))g(is)f(the)i(set)g(of)e(all)g(orthogonal)f(pro)r(jections) j(in)e Fu(M)p FD(.)303 2490 y FB(Pr)o(oposition.)21 b FA(L)n(et)f Fu(M)g FA(b)n(e)h(a)g Fu(\003)p FA(-algebr)n(a)e(in)i Ft(V)30 b FA(\(p)n(ossibly)20 b Ft(I)25 b Fu(62)d(M)p FA(\).)35 b(Then)21 b Fu(M)f FA(is)228 2540 y(sp)n(anne)n(d)k(by)f FD(Pro)r(j\()p Fu(M)p FD(\))p FA(,)i(that)e(is,)h(every)f(element)f Ft(A)27 b Fu(2)e(M)e FA(c)n(an)g(b)n(e)g(written)f(as)h Ft(A)j FD(=)228 2564 y Fo(P)272 2574 y Fr(N)272 2608 y(i)p Fq(=1)334 2595 y Ft(c)352 2601 y Fr(i)366 2595 y Ft(P)393 2601 y Fr(i)407 2595 y FA(,)14 b(wher)n(e)g Ft(c)569 2601 y Fr(i)595 2595 y Fu(2)d Fn(C)6 b Ft(;)24 b(P)724 2601 y Fr(i)749 2595 y Fu(2)11 b FD(Pro)r(j)q(\()p Fu(M)p FD(\))p FA(.)p eop %%Page: 8 8 8 15 bop 228 113 a Fq(8)103 b(1.)14 b(ALGEBRAS)g(OF)h(OPERA)m(TORS)i (IN)d(A)g(FINITE)g(DIMENSIONAL)g(HILBER)m(T)g(SP)m(A)o(CE)303 213 y FB(Pr)o(oof.)21 b FD(W)m(e)e(can)h(write)g Ft(A)h FD(=)g Ft(A)873 219 y Fq(1)905 213 y FD(+)13 b Ft(iA)995 219 y Fq(2)1014 213 y FD(,)20 b(where)h Ft(A)1203 198 y Fs(\003)1203 224 y Fr(i)1243 213 y FD(=)g Ft(A)1327 219 y Fr(i)1341 213 y FD(,)g Ft(A)1405 219 y Fq(1)1444 213 y FD(=)g(\()p Ft(A)14 b FD(+)f Ft(A)1634 198 y Fs(\003)1653 213 y FD(\))p Ft(=)p FD(2,)228 262 y Ft(A)259 268 y Fq(2)289 262 y FD(=)f(\(2)p Ft(i)p FD(\))400 247 y Fs(\000)p Fq(1)445 262 y FD(\()p Ft(A)q Fu(\000)q Ft(A)557 247 y Fs(\003)576 262 y FD(\).)17 b(Hence)11 b(it)e(is)h(su\016cien)o(t)g(to)g(consider)g (a)g(self-adjoin)o(t)e Ft(A)k FD(=)f Ft(A)1546 247 y Fs(\003)1577 262 y Fu(2)g(L)p FD(\()p Ft(V)f FD(\).)228 312 y(Then)k(w)o(e)g(can)g(use)h(its)f(sp)q(ectral)h(decomp)q(osition) 807 393 y Ft(A)d FD(=)893 353 y Fo(X)917 442 y Fr(i)960 393 y Ft(\025)984 399 y Fr(i)998 393 y Ft(E)r FD(\()p Fu(f)p Ft(\025)1092 399 y Fr(i)1106 393 y Fu(g)p FD(\))228 506 y(\(sum)f(o)o(v)o(er)h(all)e(eigen)o(v)n(alues)i(of)f Ft(A)p FD(\).)18 b(It)12 b(is)f(su\016cien)o(t)i(to)e(pro)o(v)o(e)h (that)g Ft(E)r FD(\()p Fu(f)p Ft(\025)1402 512 y Fr(i)1416 506 y Fu(g)p FD(\))f Fu(2)h(M)f FD(pro)o(vided)228 556 y Ft(\025)252 562 y Fr(i)277 556 y Fu(6)p FD(=)h(0.)18 b(W)m(e)13 b(can)h(\014nd)f(a)h(p)q(olynomial)c Ft(f)870 562 y Fr(i)895 556 y FD(=)i Ft(f)959 562 y Fr(i)973 556 y FD(\()p Ft(x)p FD(\))g Fu(2)f Fn(R)m FD([)p Ft(x)o FD(])f(suc)o(h)15 b(that)e Ft(f)1373 562 y Fr(i)1387 556 y FD(\()p Ft(\025)1427 562 y Fr(i)1442 556 y FD(\))e(=)h(1,)h Ft(f)1579 562 y Fr(i)1593 556 y FD(\(0\))f(=)f(0)228 606 y(and)j Ft(f)329 612 y Fr(i)343 606 y FD(\()p Ft(\025)383 612 y Fr(j)401 606 y FD(\))d(=)h(0)i(if)f Ft(j)h Fu(6)p FD(=)d Ft(i)p FD(.)19 b(Then)14 b(ob)o(viously)f Ft(E)r FD(\()p Fu(f)p Ft(\025)1050 612 y Fr(i)1064 606 y Fu(g)p FD(\))e(=)h Ft(f)1176 612 y Fr(i)1190 606 y FD(\()p Ft(A)p FD(\))g Fu(2)f(M)p FD(.)p 1692 606 2 29 v 1694 579 25 2 v 1694 606 V 1719 606 2 29 v 685 711 a Fp(1.10.)24 b(Cancellation)14 b(prop)q(ert)o(y)303 786 y FB(Lemma)i FD(\(Cancellation)d(prop)q(ert)o(y\))p FB(.)22 b FA(L)n(et)16 b Fu(M)g FA(b)n(e)g(a)h(factor)e(in)i Ft(V)9 b FA(,)16 b Ft(A)e Fu(2)g(M)p FA(,)i Ft(A)1581 771 y Fs(0)1606 786 y Fu(2)e(M)1698 771 y Fs(0)1709 786 y FA(,)228 836 y Ft(AA)290 821 y Fs(0)313 836 y FD(=)e(0)p FA(.)19 b(Then)c(either)f Ft(A)e FD(=)g(0)i FA(or)h Ft(A)843 821 y Fs(0)866 836 y FD(=)d(0)p FA(.)303 915 y FB(Pr)o(oof.)21 b FD(Consider)c(Ker)8 b Ft(A)p FD(.)27 b(It)17 b(is)f(su\016cien)o(t)h(to)g(pro)o(v)o(e)f (that)h(either)h(Ker)8 b Ft(A)16 b FD(=)g Fu(f)p FD(0)p Fu(g)g FD(or)228 965 y(Ker)8 b Ft(A)15 b FD(=)g Ft(V)10 b FD(.)24 b(\(If)16 b(Ker)8 b Ft(A)15 b FD(=)h Fu(f)p FD(0)p Fu(g)f FD(then)i Ft(A)f FD(is)g(in)o(v)o(ertible)f(and)h Ft(AA)1289 949 y Fs(0)1316 965 y FD(=)g(0)f(implies)f Ft(A)1574 949 y Fs(0)1601 965 y FD(=)i(0;)g(if)228 1014 y(Ker)8 b Ft(A)14 b FD(=)h Ft(V)25 b FD(then)16 b Ft(A)f FD(=)f(0.\))23 b(Ob)o(viously)15 b(Ker)8 b Ft(A)15 b FD(is)h Fu(M)1114 999 y Fs(0)1125 1014 y FD(-in)o(v)n(arian)o(t.)22 b(\()p Ft(Ax)14 b FD(=)h(0,)g Ft(B)i Fu(2)d(M)1655 999 y Fs(0)1681 1014 y Fu(\))228 1064 y Ft(A)p FD(\()p Ft(B)r(x)p FD(\))e(=)g Ft(B)r(Ax)g FD(=)g(0.\))17 b(Hence)e Ft(P)764 1070 y Fq(Ker)5 b Fr(A)861 1064 y Fu(2)11 b(M)p FD(,)i(where)h Ft(P)1121 1070 y Fq(Ker)6 b Fr(A)1220 1064 y FD(is)13 b(the)h(orthogonal)e(pro)r(jection)228 1114 y(on)j(Ker)8 b Ft(A)p FD(.)22 b(\(Another)16 b(pro)q(of:)k(Ker)8 b Ft(A)14 b FD(=)g(Ker)8 b Ft(A)1003 1099 y Fs(\003)1022 1114 y Ft(A)16 b FD(and)f Ft(P)1178 1120 y Fq(Ker)5 b Fr(A)1277 1114 y FD(=)14 b Ft(E)1354 1120 y Fq(0)1372 1114 y FD(\()p Ft(A)1419 1099 y Fs(\003)1439 1114 y Ft(A)p FD(\))h(is)g(a)g(sp)q(ectral)228 1164 y(pro)r(jection)f(for)g Ft(A)520 1149 y Fs(\003)539 1164 y Ft(A)p FD(.\))303 1214 y(Consider)g(a)f(maximal)d 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FD(.)31 b(If)228 1512 y Ft(M)16 b FD(=)c Ft(V)d FD(,)14 b(then)g(Ker)8 b Ft(A)k FD(=)g Ft(V)23 b FD(and)14 b(Lemma)d(is)j(pro)o(v)o(ed.)303 1562 y(No)o(w)d(let)g(us)h(assume)f(that)g Ft(M)17 b FD(=)11 b Fu(f)p FD(0)p Fu(g)p FD(,)g(and)g(use)h(the)g(fact)g(that)f Ft(AA)1362 1547 y Fs(0)1386 1562 y FD(=)h(0.)k(Cho)q(osing)11 b Ft(f)17 b Fu(2)228 1612 y Ft(V)23 b FD(let)15 b(us)g(pro)o(v)o(e)f (that)g Ft(A)623 1597 y Fs(0)635 1612 y Ft(f)j Fu(2)12 b Ft(M)19 b FD(\(and)14 b(hence)i Ft(A)1015 1597 y Fs(0)1027 1612 y Ft(f)h FD(=)12 b(0\).)19 b(Indeed,)c Ft(AB)r(A)1418 1597 y Fs(0)1431 1612 y Ft(f)i FD(=)12 b Ft(AA)1574 1597 y Fs(0)1586 1612 y Ft(B)r(f)18 b FD(=)12 b(0)228 1662 y(for)h(ev)o(ery)i Ft(B)f Fu(2)d(M)p FD(.)18 b(Therefore)d Ft(A)784 1647 y Fs(0)808 1662 y FD(=)c(0.)p 1692 1662 V 1694 1635 25 2 v 1694 1662 V 1719 1662 2 29 v 303 1747 a FB(Remark.)22 b FD(The)10 b(Cancellation)f(prop)q(ert)o(y)i(in)e(the) i(Lemma)c(is)j(c)o(haracteristic)i(for)d(factors.)228 1797 y(Namely)m(,)k(if)i Fu(M)h FD(is)f(a)h Fu(\003)p FD(-algebra)e(with)i Ft(I)i Fu(2)c(M)h FD(and)h Fu(M)f FD(is)h(not)f(a)h(factor,)f(then)i(there)g(exist)228 1846 y Ft(A)k Fu(2)f(M)p FD(,)g Ft(A)441 1831 y Fs(0)474 1846 y Fu(2)g(M)572 1831 y Fs(0)603 1846 y FD(suc)o(h)g(that)g Ft(A)g Fu(6)p FD(=)h(0,)f Ft(A)986 1831 y Fs(0)1019 1846 y Fu(6)p FD(=)h(0)e(but)h Ft(AA)1256 1831 y Fs(0)1288 1846 y FD(=)h(0.)35 b(Indeed,)21 b(let)f Ft(C)i FD(b)q(e)228 1896 y(a)17 b(non-scalar)g(self-adjoin)o(t)f(op)q(erator)i(in)e Ft(Z)s FD(\()p Fu(M)p FD(\))i(=)f Fu(M)12 b(\\)f(M)1238 1881 y Fs(0)1249 1896 y FD(.)29 b(Let)17 b Ft(\025)h Fu(2)e Fn(R)c FD(lies)17 b(b)q(et)o(w)o(een)228 1946 y(minim)o(al)d(and)j(maxima)o(l)d(eigen)o(v)n(alues)k(of)f Ft(C)j FD(\(whic)o(h)d(do)g(not)h(coincide)g(b)q(ecause)h Ft(C)h FD(is)d(not)228 1996 y(scalar\).)g(Denote)12 b Ft(A)g FD(=)g Ft(E)635 2002 y Fr(\025)656 1996 y FD(,)g(where)h Ft(E)829 2002 y Fr(\025)862 1996 y FD(is)e(the)i(sp)q(ectral)f(pro)r (jection)g(of)f Ft(C)s FD(,)h(corresp)q(onding)g(to)228 2046 y(the)i(c)o(hosen)h Ft(\025)p FD(.)k(Then)14 b Ft(E)626 2052 y Fr(\025)659 2046 y Fu(2)d Ft(Z)s FD(\()p Fu(M)p FD(\).)19 b(No)o(w)14 b(tak)o(e)g Ft(A)1058 2031 y Fs(0)1081 2046 y FD(=)e Ft(I)h Fu(\000)c Ft(E)1228 2052 y Fr(\025)1261 2046 y Fu(2)j Ft(Z)s FD(\()p Fu(M)p FD(\))i(and)g(then)g Ft(A)e Fu(6)p FD(=)g(0,)228 2095 y Ft(A)259 2080 y Fs(0)282 2095 y Fu(6)p FD(=)g(0)i(but)g Ft(AA)499 2080 y Fs(0)522 2095 y FD(=)e(0.)703 2201 y Fp(1.11.)24 b(Structure)13 b(of)i(factors.)303 2276 y FB(Lemma)h FD(1)g(\(on)d(minima)o(l)e (subspaces\))p FB(.)23 b FA(L)n(et)c Fu(M)f FA(b)n(e)h(a)g(factor)f(in) h Ft(V)9 b FA(,)20 b Ft(f)j Fu(2)c Ft(V)9 b FA(,)19 b Ft(f)24 b Fu(6)p FD(=)19 b(0)p FA(.)228 2326 y(The)c(subsp)n(ac)n(e)g Fu(M)p Ft(f)20 b FA(is)14 b(a)h(minimal)g Fu(M)p FA(-invariant)f(subsp) n(ac)n(e)i(if)e(and)i(only)f(if)679 2400 y Fu(M)p Ft(f)f Fu(\\)9 b(M)850 2383 y Fs(0)862 2400 y Ft(f)16 b FD(=)c Fn(C)6 b Ft(f)19 b FD(=)12 b Fu(f)p Ft(\025f)t Fu(j)p Ft(\025)g Fu(2)f Fn(C)6 b Fu(g)p Ft(:)303 2479 y FB(Cor)o(ollar)m(y.)23 b Fu(M)p Ft(f)c FA(and)d Fu(M)782 2464 y Fs(0)793 2479 y Ft(f)k FA(ar)n(e)15 b(minimal)f(simultane)n(ously.)303 2558 y FB(Pr)o(oof)i(of)g(Lemma.)22 b FD(1.)17 b(Let)11 b Ft(E)r(;)c(E)891 2543 y Fs(0)913 2558 y FD(b)q(e)k(the)g(orthogonal)e (pro)r(jections)j(on)o(to)e Fu(M)1589 2543 y Fs(0)1600 2558 y Ft(f)t FD(,)i Fu(M)p Ft(f)228 2608 y FD(resp)q(ectiv)o(ely)m(,)f (so)f Ft(E)j Fu(2)e(M)f FD(and)g Ft(E)759 2593 y Fs(0)782 2608 y Fu(2)h(M)871 2593 y Fs(0)882 2608 y FD(.)17 b(Hence)11 b Ft(E)r(E)1096 2593 y Fs(0)1120 2608 y FD(=)g Ft(E)1196 2593 y Fs(0)1208 2608 y Ft(E)r FD(.)17 b(Moreo)o(v)o(er)10 b Ft(E)r(f)16 b FD(=)c Ft(E)1594 2593 y Fs(0)1606 2608 y Ft(f)k FD(=)c Ft(f)t FD(.)p eop %%Page: 9 9 9 16 bop 714 113 a Fq(1.11.)14 b(STR)o(UCTURE)g(OF)h(F)l(A)o(CTORS.)460 b(9)303 213 y FD(It)13 b(follo)o(ws)g(that)687 282 y Fu(M)p Ft(f)h Fu(\\)9 b(M)858 265 y Fs(0)870 282 y Ft(f)16 b FD(=)c Ft(E)r Fu(M)p Ft(f)k FD(=)c Ft(E)r Fu(M)p Ft(E)r(f)r(:)228 351 y FD(Indeed,)j(since)g Ft(E)r Fu(M)p Ft(f)i Fu(\032)c(M)p Ft(f)19 b FD(and,)13 b(on)i(the)g(other)g(hand,)e Ft(E)r Fu(M)p Ft(f)k Fu(\032)c Ft(E)r(V)22 b FD(=)12 b Fu(M)1516 336 y Fs(0)1528 351 y Ft(f)t FD(,)j(w)o(e)f(ha)o(v)o(e)228 401 y Ft(E)r Fu(M)p Ft(f)19 b Fu(\032)14 b(M)p Ft(f)h Fu(\\)10 b(M)569 386 y Fs(0)581 401 y Ft(f)t FD(;)16 b(vice)g(v)o(ersa)g Fu(M)p Ft(f)f Fu(\\)10 b(M)1000 386 y Fs(0)1012 401 y Ft(f)19 b FD(=)14 b Fu(M)p Ft(f)h Fu(\\)10 b Ft(E)r Fu(M)1303 386 y Fs(0)1315 401 y Ft(f)19 b Fu(\032)14 b Ft(E)r Fu(M)p Ft(f)h Fu(\\)10 b Ft(E)r Fu(M)1639 386 y Fs(0)1651 401 y Ft(f)19 b Fu(\032)228 451 y Ft(E)r Fu(M)p Ft(f)t FD(.)303 500 y(Supp)q(ose)g(that)g Fu(M)615 485 y Fs(0)627 500 y Ft(f)k FD(is)c(a)f(minim)o(al)d Fu(M)972 485 y Fs(0)984 500 y FD(-in)o(v)n(arian)o(t)i(subspace)j(in)e Ft(V)10 b FD(.)32 b(Consider)20 b(the)228 550 y(algebra)c(of)f(op)q (erators)i Ft(E)r Fu(M)p Ft(E)h FD(=)e Fu(f)p Ft(E)r(AE)r Fu(j)p Ft(A)f Fu(2)g(Mg)p FD(.)24 b(This)17 b(is)f(a)g Fu(\003)p FD(-algebra)f(and)h(in)g Ft(E)r(V)26 b FD(it)228 600 y(has)15 b(the)h(iden)o(tit)o(y)f(op)q(erator)h Ft(E)r Fu(j)745 606 y Fr(E)r(V)814 600 y FD(=)e Ft(I)s Fu(j)893 606 y Fr(E)r(V)948 600 y FD(.)23 b(No)o(w)15 b(tak)o(e)g(a)h(pro)r (jection)f Ft(P)20 b Fu(2)14 b FD(Pro)r(j\()p Ft(E)r Fu(M)p Ft(E)r FD(\))228 650 y(\(extended)k(b)o(y)e(0)g(to)g(\()p Ft(E)r(V)10 b FD(\))672 635 y Fs(?)700 650 y FD(\).)26 b(Ob)o(viously)m(,)15 b(if)h Ft(P)21 b Fu(6)p FD(=)16 b(0)g(and)g Ft(P)22 b Fu(6)p FD(=)16 b Ft(E)i FD(then)f(Im)6 b Ft(P)21 b FD(is)c(a)f Fu(M)1697 635 y Fs(0)1708 650 y FD(-)228 700 y(in)o(v)n(arian)o(t)e(subspace)j(b)q(ecause)h Ft(P)i Fu(2)14 b(M)p FD(.)24 b(Besides)17 b(Im)6 b Ft(P)g Fu(\032)p Ft(E)r(V)24 b FD(=)15 b Fu(M)1359 685 y Fs(0)1370 700 y Ft(f)21 b FD(and)16 b(Im)5 b Ft(P)20 b Fu(6)p FD(=)15 b Ft(E)r(V)10 b FD(,)228 749 y(whic)o(h)15 b(con)o(tradicts)h(minim)o (ali)o(t)o(y)c(of)j Fu(M)873 734 y Fs(0)885 749 y Ft(f)t FD(.)23 b(Hence)16 b(the)g(algebra)f Ft(E)r Fu(M)p Ft(E)i FD(do)q(es)f(not)g(con)o(tain)228 799 y(an)o(y)h(non-trivial)f(pro)r (jection.)30 b(Therefore)19 b(it)e(consists)i(only)e(of)g(scalar)g(op)q (erators)i(in)e Ft(E)r(V)10 b FD(.)228 849 y(Hence)637 918 y Fu(M)p Ft(f)j Fu(\\)c(M)807 901 y Fs(0)819 918 y Ft(f)16 b FD(=)c Ft(E)r Fu(M)p Ft(E)r(f)k FD(=)c Fu(f)p Ft(\025f)t Fu(j)p Ft(\025)g Fu(2)f Fn(C)6 b Fu(g)p Ft(:)228 987 y FD(Replacing)13 b Fu(M)470 972 y Fs(0)496 987 y FD(b)o(y)h Fu(M)p FD(,)g(w)o(e)g(come)f(to)i(the)f(same)g(conclusion,)g (pro)o(vided)g Fu(M)p Ft(f)19 b FD(is)14 b(a)g(minim)o(al)228 1037 y Fu(M)p FD(-in)o(v)n(arian)o(t)e(subspace)j(in)f Ft(V)9 b FD(.)303 1087 y(2.)16 b(Con)o(v)o(ersely)11 b(supp)q(ose)h(that)f Fu(M)p Ft(f)d Fu(\\)s(M)958 1072 y Fs(0)969 1087 y Ft(f)16 b FD(=)c Fn(C)6 b Ft(f)19 b FD(and)10 b(the)i(space)f Fu(M)p Ft(f)16 b FD(is)10 b(not)h(minim)o (al.)228 1137 y(Then)k(there)i(exists)e Fu(M)p FD(-in)o(v)n(arian)o(t)e (subspace)k Ft(L)d Fu(\032)f(M)p Ft(f)t FD(,)j Ft(L)d Fu(6)p FD(=)h Fu(M)p Ft(f)20 b FD(and)14 b Ft(L)g Fu(6)p FD(=)g Fu(f)p FD(0)p Fu(g)p FD(.)20 b(If)15 b Ft(P)20 b FD(is)228 1187 y(the)15 b(orthogonal)f(pro)r(jection)i(on)o(to)e Ft(L)p FD(,)i(then)f Ft(P)k Fu(2)14 b(M)1091 1172 y Fs(0)1102 1187 y FD(.)22 b(Let)15 b Ft(L)1239 1193 y Fq(1)1272 1187 y FD(=)e Fu(M)p Ft(f)i Fu(\011)10 b Ft(L)k FD(=)g Fu(M)p Ft(f)h Fu(\\)9 b Ft(L)1682 1172 y Fs(?)1711 1187 y FD(.)228 1236 y(Again)k(if)g Ft(P)414 1242 y Fq(1)446 1236 y FD(is)h(the)g(orthogonal)f(pro)r(jection)h(on)g Ft(L)1050 1242 y Fq(1)1069 1236 y FD(,)f(then)i Ft(P)1216 1242 y Fq(1)1245 1236 y Fu(2)d(M)1335 1221 y Fs(0)1346 1236 y FD(,)i Ft(P)1399 1242 y Fq(1)1428 1236 y Fu(6)p FD(=)e(0.)303 1286 y(No)o(w)j Ft(P)6 b(E)17 b Fu(6)p FD(=)f(0,)g Ft(P)604 1292 y Fq(1)622 1286 y Ft(E)h Fu(6)p FD(=)f(0)g(b)q(ecause)i(of)d(the)i(Cancellation)e(Lemma.)22 b(Since)17 b Ft(P)k FD(and)16 b Ft(E)228 1336 y FD(comm)o(ute,)11 b(this)j(means)f(that)h Ft(L)9 b Fu(\\)g(M)840 1321 y Fs(0)852 1336 y Ft(f)16 b Fu(6)p FD(=)c Fu(f)p FD(0)p Fu(g)h FD(and)h Ft(L)1117 1342 y Fq(1)1145 1336 y Fu(\\)9 b(M)1232 1321 y Fs(0)1243 1336 y Ft(f)17 b Fu(6)p FD(=)11 b Fu(f)p FD(0)p Fu(g)p FD(.)303 1386 y(On)j(the)g(other)h(hand)353 1455 y Ft(L)9 b Fu(\\)g(M)477 1438 y Fs(0)489 1455 y Ft(f)16 b Fu(\032)c(M)p Ft(f)i Fu(\\)9 b(M)740 1438 y Fs(0)751 1455 y Ft(f)17 b FD(=)11 b Fn(C)6 b Ft(f)50 b FD(and)41 b Ft(L)1064 1461 y Fq(1)1092 1455 y Fu(\\)9 b(M)1179 1438 y Fs(0)1191 1455 y Ft(f)16 b Fu(\032)c(M)p Ft(f)i Fu(\\)8 b(M)1441 1438 y Fs(0)1453 1455 y Ft(f)16 b FD(=)c Fn(C)6 b Ft(f)s(;)228 1524 y FD(so)15 b Ft(f)k Fu(2)14 b Ft(L)c Fu(\\)g(M)487 1509 y Fs(0)498 1524 y Ft(f)20 b FD(and)c Ft(f)i Fu(2)c Ft(L)729 1530 y Fq(1)758 1524 y Fu(\\)c(M)846 1509 y Fs(0)857 1524 y Ft(f)20 b FD(whic)o(h)c(con)o(tradicts)g(the)g(orthogonalit)o(y)d(of)i Ft(L)g FD(and)228 1574 y Ft(L)256 1580 y Fq(1)275 1574 y FD(.)p 1692 1574 2 29 v 1694 1548 25 2 v 1694 1574 V 1719 1574 2 29 v 303 1651 a FB(Example)i FD(1)p FB(.)j FD(Let)e Ft(V)27 b FD(=)18 b Ft(V)756 1657 y Fq(0)787 1651 y Fu(\012)12 b Ft(V)855 1657 y Fq(1)874 1651 y FD(,)18 b Fu(M)f FD(=)h Fu(L)p FD(\()p Ft(V)1090 1657 y Fq(0)1109 1651 y FD(\))12 b Fu(\012)g Ft(I)1199 1657 y Fr(V)1219 1661 y Fj(1)1237 1651 y FD(;)19 b(then)f Fu(M)1416 1636 y Fs(0)1445 1651 y FD(=)g Ft(I)1513 1657 y Fr(V)1533 1661 y Fj(0)1563 1651 y Fu(\012)12 b(L)p FD(\()p Ft(V)1676 1657 y Fq(1)1695 1651 y FD(\).)228 1701 y(\(Here)k Ft(I)361 1707 y Fr(V)381 1711 y Fj(0)399 1701 y FD(,)f Ft(I)444 1707 y Fr(V)464 1711 y Fj(1)497 1701 y FD(are)h(the)g(iden)o(tit)o(y)e (op)q(erators)i(in)f Ft(V)1054 1707 y Fq(0)1072 1701 y FD(,)g Ft(V)1123 1707 y Fq(1)1157 1701 y FD(resp)q(ectiv)o(ely)m(.\)) 23 b(Cho)q(osing)14 b(a)h(basis)228 1751 y Fu(f)p Ft(e)268 1757 y Fq(1)287 1751 y Ft(;)7 b(:)g(:)g(:)k(;)c(e)405 1757 y Fr(N)436 1751 y Fu(g)17 b FD(in)g Ft(V)550 1757 y Fq(1)585 1751 y FD(w)o(e)g(can)g(presen)o(t)i Ft(V)26 b FD(as)17 b(the)h(direct)f(sum)f Ft(V)26 b FD(=)17 b Ft(V)1389 1757 y Fq(0)1419 1751 y Fu(\010)12 b(\001)7 b(\001)g(\001)i(\010)j Ft(V)1590 1757 y Fq(0)1625 1751 y FD(\(with)228 1801 y Ft(N)19 b FD(copies)d(of)e Ft(V)476 1807 y Fq(0)494 1801 y FD(\).)21 b(This)15 b(is)f(an)h(orthogonal)e (decomp)q(osition)g(if)h(the)h(inner)g(pro)q(duct)h(in)e Ft(V)24 b FD(is)228 1851 y(c)o(hosen)16 b(naturally)e(compatible)f (with)i(the)h(tensor)g(pro)q(duct)g(structure,)h(whic)o(h)e(means)f (that)228 1900 y(it)f(is)h(induced)h(b)o(y)e(the)i(inner)f(pro)q(ducts) h(\()p Fu(\001)p Ft(;)7 b Fu(\001)p FD(\))948 1906 y Fq(0)965 1900 y FD(,)14 b(\()p Fu(\001)p Ft(;)7 b Fu(\001)p FD(\))1066 1906 y Fq(1)1097 1900 y FD(in)14 b Ft(V)1170 1906 y Fq(0)1188 1900 y FD(,)g Ft(V)1238 1906 y Fq(1)1270 1900 y FD(so)g(that)401 1970 y(\()p Ft(x)441 1976 y Fq(0)469 1970 y Fu(\012)9 b Ft(x)534 1976 y Fq(1)552 1970 y Ft(;)e(y)591 1976 y Fq(0)619 1970 y Fu(\012)j Ft(y)681 1976 y Fq(1)700 1970 y FD(\))h(=)h(\()p Ft(x)811 1976 y Fq(0)830 1970 y Ft(;)7 b(y)869 1976 y Fq(0)887 1970 y FD(\))903 1976 y Fq(0)922 1970 y FD(\()p Ft(x)962 1976 y Fq(1)980 1970 y Ft(;)g(y)1019 1976 y Fq(1)1038 1970 y FD(\))1054 1976 y Fq(1)1072 1970 y Ft(;)48 b(x)1156 1976 y Fq(0)1175 1970 y Ft(;)7 b(y)1214 1976 y Fq(0)1244 1970 y Fu(2)k Ft(V)1307 1976 y Fq(0)1326 1970 y Ft(;)c(x)1369 1976 y Fq(1)1387 1970 y Ft(;)g(y)1426 1976 y Fq(1)1456 1970 y Fu(2)k Ft(V)1519 1976 y Fq(1)1538 1970 y Ft(:)228 2039 y FD(No)o(w)k Fu(M)f FD(in)h(this)g(direct)h(decomp)q(osition)e (consists)i(of)e(all)g(op)q(erators)i(whic)o(h)f(ha)o(v)o(e)g(a)g(blo)q (c)o(k-)228 2089 y(matrix)d(form)593 2236 y Ft(A)g FD(=)679 2115 y Fo(0)679 2188 y(B)679 2213 y(B)679 2238 y(B)679 2264 y(@)716 2147 y Ft(A)747 2153 y Fq(0)1003 2147 y FD(0)807 2197 y Ft(A)838 2203 y Fq(0)901 2245 y FD(.)917 2258 y(.)933 2270 y(.)730 2324 y(0)237 b Ft(A)1019 2330 y Fq(0)1038 2115 y Fo(1)1038 2188 y(C)1038 2213 y(C)1038 2238 y(C)1038 2264 y(A)1081 2236 y Ft(;)48 b(A)1172 2242 y Fq(0)1203 2236 y Fu(2)11 b(L)p FD(\()p Ft(V)1311 2242 y Fq(0)1329 2236 y FD(\))p Ft(:)228 2394 y FD(In)j(the)g(same)f(decomp) q(osition)g Fu(M)779 2379 y Fs(0)804 2394 y FD(consists)i(of)e(the)i (op)q(erators)g(of)e(the)h(form)523 2535 y Ft(A)554 2518 y Fs(0)578 2535 y FD(=)621 2427 y Fo(0)621 2500 y(B)621 2525 y(B)621 2551 y(@)664 2460 y Ft(\025)688 2466 y Fq(11)724 2460 y Ft(I)742 2466 y Fq(0)815 2460 y Ft(\025)839 2466 y Fq(12)874 2460 y Ft(I)892 2466 y Fq(0)959 2460 y Ft(:)7 b(:)g(:)53 b(\025)1086 2466 y Fq(1)p Fr(N)1135 2460 y Ft(I)1153 2466 y Fq(0)664 2510 y Ft(\025)688 2516 y Fq(21)724 2510 y Ft(I)742 2516 y Fq(0)815 2510 y Ft(\025)839 2516 y Fq(22)874 2510 y Ft(I)892 2516 y Fq(0)959 2510 y Ft(:)7 b(:)g(:)53 b(\025)1086 2516 y Fq(2)p Fr(N)1135 2510 y Ft(I)1153 2516 y Fq(0)685 2560 y Ft(:)7 b(:)g(:)358 b(:)7 b(:)g(:)658 2609 y(\025)682 2615 y Fr(N)s Fq(1)730 2609 y Ft(I)748 2615 y Fq(0)808 2609 y Ft(\025)832 2615 y Fr(N)s Fq(2)881 2609 y Ft(I)899 2615 y Fq(0)959 2609 y Ft(:)g(:)g(:)47 b(\025)1080 2615 y Fr(N)s(N)1141 2609 y Ft(I)1159 2615 y Fq(0)1178 2427 y Fo(1)1178 2500 y(C)1178 2525 y(C)1178 2551 y(A)1221 2535 y Ft(;)h(\025)1305 2541 y Fr(ij)1346 2535 y Fu(2)11 b Fn(C)6 b Ft(;)p eop %%Page: 10 10 10 17 bop 228 113 a Fq(10)86 b(1.)14 b(ALGEBRAS)g(OF)h(OPERA)m(TORS)i (IN)d(A)g(FINITE)g(DIMENSIONAL)g(HILBER)m(T)g(SP)m(A)o(CE)228 213 y FD(where)c Ft(I)361 219 y Fq(0)392 213 y FD(=)i Ft(I)454 219 y Fr(V)474 223 y Fj(0)492 213 y FD(.)k(T)m(aking)8 b Ft(f)17 b FD(=)11 b Ft(f)754 219 y Fq(0)774 213 y Fu(\012)p Ft(e)825 219 y Fq(1)844 213 y FD(,)f(where)h Ft(f)1002 219 y Fq(0)1032 213 y Fu(2)h Ft(V)1096 219 y Fq(0)1115 213 y FD(,)d Ft(f)1156 219 y Fq(0)1187 213 y Fu(6)p FD(=)j(0,)d(w)o(e)h (see)h(that)e Fu(M)p Ft(f)16 b FD(=)c Ft(V)1632 219 y Fq(0)1652 213 y Fu(\012)p Ft(e)1703 219 y Fq(1)228 262 y FD(is)i(a)f(minim)o(al)e Fu(M)p FD(-in)o(v)n(arian)o(t)g(subspace.)20 b(Ob)o(viously)13 b(here)i Fu(M)p Ft(f)f Fu(\\)9 b(M)1349 247 y Fs(0)1360 262 y Ft(f)17 b FD(=)12 b Fn(C)6 b Ft(f)t FD(.)303 334 y FB(Lemma)16 b FD(2)g(\(On)e(irreducible)g(algebras\))p FB(.)22 b FA(L)n(et)17 b Fu(M)g FA(b)n(e)g(a)h Fu(\003)p FA(-algebr)n(a)e(with)h Ft(I)j Fu(2)15 b(M)i FA(in)h Ft(V)228 384 y FA(and)12 b Fu(M)e FA(do)n(es)i(not)f(have)h(any)g (non-trivial)e Fu(M)p FA(-invariant)h(subsp)n(ac)n(e)g(in)h Ft(V)d FA(.)18 b(Then)11 b Fu(M)g FD(=)h Fu(L)p FD(\()p Ft(V)d FD(\))p FA(.)303 455 y FB(Pr)o(oof.)21 b FD(By)g(the)f(Double)g (Comm)o(utan)n(t)e(Theorem,)i(it)g(is)g(su\016cien)o(t)g(to)g(pro)o(v)o (e)g(that)228 505 y Fu(M)278 490 y Fs(0)301 505 y FD(=)12 b Fu(f)p Ft(\025I)s Fu(j)p Ft(\025)f Fu(2)h Fn(C)6 b Fu(g)p FD(.)20 b(But)13 b(if)f(this)g(is)g(not)h(true)g(then)g Fu(M)1117 490 y Fs(0)1141 505 y FD(con)o(tains)g(a)f(non-trivial)f(pro) r(jection)228 555 y(whose)16 b(image)e(will)g(b)q(e)j(a)e(non-trivial)g Fu(M)p FD(-in)o(v)n(arian)o(t)f(subspace)j(whic)o(h)f(is)f(a)h(con)o (tradiction.)p 1692 604 2 29 v 1694 578 25 2 v 1694 604 V 1719 604 2 29 v 303 679 a FB(Definition.)21 b FD(Let)10 b Fu(M)676 685 y Fq(1)694 679 y Ft(;)d Fu(M)763 685 y Fq(2)790 679 y FD(b)q(e)j Fu(\003)p FD(-algebras.)16 b(An)9 b(algebra)g(homomorphism)c Ft(f)16 b FD(:)11 b Fu(M)1655 685 y Fq(1)1685 679 y Fu(!)228 729 y(M)278 735 y Fq(2)310 729 y FD(is)j(called)f(a)h Fu(\003)p FD(-)p FA(homomorphism)g FD(if)f Ft(f)t FD(\()p Ft(A)938 714 y Fs(\003)958 729 y FD(\))f(=)g Ft(f)t FD(\()p Ft(A)p FD(\))1117 714 y Fs(\003)1151 729 y FD(for)h(ev)o(ery)i Ft(A)d Fu(2)f(M)p FD(.)303 800 y FB(Example)17 b FD(2)p FB(.)j FD(If)13 b Ft(U)18 b FD(is)13 b(a)g(unitary)g(op)q(erator)h(in)f Ft(V)22 b FD(then)14 b(it)f(de\014nes)i(a)e Fu(\003)p FD(-automorphism)228 850 y(of)g Fu(L)p FD(\()p Ft(V)c FD(\))14 b(b)o(y)g(the)g(form)o(ula)e Ft(f)t FD(\()p Ft(A)p FD(\))g(=)g Ft(U)5 b(AU)905 835 y Fs(\000)p Fq(1)961 850 y FD(=)12 b Ft(U)5 b(AU)1102 835 y Fs(\003)1121 850 y FD(.)303 922 y FB(Lemma)16 b FD(3)g(\(On)e(automorphisms)d(of)i Fu(L)p FD(\()p Ft(V)d FD(\)\))p FB(.)21 b FA(Every)12 b Fu(\003)p FA(-automorphism)g(of)g Fu(L)p FD(\()p Ft(V)d FD(\))k FA(is)f(an)228 971 y(inner)k(unitary)h(e)n(quivalenc)n(e,)g (i.e.)24 b(if)16 b Ft(f)j FD(:)c Fu(L)p FD(\()p Ft(V)9 b FD(\))15 b Fu(!)f(L)p FD(\()p Ft(V)9 b FD(\))17 b FA(is)f(a)h Fu(\003)p FA(-automorphism)f(of)h Fu(L)p FD(\()p Ft(V)9 b FD(\))228 1021 y FA(then)15 b(ther)n(e)g(exists)g(a)h(unitary)f(op)n (er)n(ator)g Ft(U)i Fu(2)12 b(L)p FD(\()p Ft(V)d FD(\))15 b FA(such)h(that)f Ft(f)t FD(\()p Ft(A)p FD(\))e(=)g Ft(U)5 b(AU)1501 1006 y Fs(\000)p Fq(1)1561 1021 y FA(for)14 b(every)228 1071 y Ft(A)d Fu(2)h(L)p FD(\()p Ft(V)d FD(\))p FA(.)303 1143 y FB(Pr)o(oof.)21 b FD(Let)14 b Fu(f)p Ft(e)580 1149 y Fq(1)599 1143 y Ft(;)7 b(:)g(:)g(:)12 b(;)7 b(e)718 1149 y Fr(n)740 1143 y Fu(g)13 b FD(b)q(e)h(an)f (orthonormal)e(basis)j(in)f Ft(V)23 b FD(and)13 b Fu(f)p Ft(E)1453 1149 y Fr(ij)1481 1143 y Fu(j)p FD(1)e Fu(\024)h Ft(i;)7 b(j)14 b Fu(\024)e Ft(n)p Fu(g)228 1192 y FD(is)f(the)g (standard)h(basis)f(in)g Fu(L)p FD(\()p Ft(V)e FD(\),)i(where)h Ft(E)916 1198 y Fr(ij)957 1192 y Fu(2)f(L)p FD(\()p Ft(V)e FD(\))i(is)g(de\014ned)i(as)e(follo)o(ws:)k Ft(E)1511 1198 y Fr(ij)1540 1192 y Ft(e)1559 1198 y Fr(k)1591 1192 y FD(=)d Ft(\016)1653 1198 y Fr(k)q(j)1689 1192 y Ft(e)1708 1198 y Fr(i)228 1242 y FD(\(so)f(the)h(corresp)q(onding)h(matrix)c(has) i(1)g(in)g(the)h Ft(i)p FD(th)g(ro)o(w)f(and)g(the)h Ft(j)r FD(th)g(column)e(and)h(all)f(others)228 1292 y(elemen)o(ts)k (are)g(0\).)k(Ob)o(viously)688 1359 y Ft(E)719 1365 y Fr(ik)751 1359 y Ft(E)782 1365 y Fr(lj)821 1359 y FD(=)12 b Ft(\016)883 1365 y Fr(k)q(l)914 1359 y Ft(E)945 1365 y Fr(ij)974 1359 y Ft(;)90 b(E)1109 1342 y Fs(\003)1107 1369 y Fr(ij)1147 1359 y FD(=)12 b Ft(E)1222 1365 y Fr(j)r(i)1251 1359 y Ft(:)228 1431 y FD(In)20 b(particular)f Ft(P)510 1437 y Fr(i)545 1431 y FD(=)j Ft(E)630 1437 y Fr(ii)675 1431 y FD(are)e(comm)o(uting)c(orthogonal)j(pro)r(jections,)j Ft(P)1451 1415 y Fq(2)1445 1441 y Fr(i)1490 1431 y FD(=)g Ft(P)1571 1437 y Fr(i)1605 1431 y FD(=)g Ft(P)1692 1415 y Fs(\003)1686 1441 y Fr(i)1711 1431 y FD(,)228 1480 y Ft(P)255 1486 y Fr(i)268 1480 y Ft(P)295 1486 y Fr(j)324 1480 y FD(=)12 b(0)h(if)g Ft(i)f Fu(6)p FD(=)g Ft(j)r FD(,)i Ft(P)582 1486 y Fq(1)609 1480 y FD(+)c Ft(P)678 1486 y Fq(2)705 1480 y FD(+)g Fu(\001)d(\001)g(\001)h FD(+)h Ft(P)873 1486 y Fr(n)907 1480 y FD(=)j Ft(I)s FD(.)303 1530 y(No)o(w)19 b(de\014ne)j Ft(F)557 1536 y Fr(ij)608 1530 y FD(=)g Ft(f)t FD(\()p Ft(E)733 1536 y Fr(ij)763 1530 y FD(\).)37 b(Then)21 b Ft(F)970 1536 y Fr(ij)1019 1530 y FD(satisfy)f(the)h(same)e(relations)h(as)g Ft(E)1606 1536 y Fr(ij)1635 1530 y FD(.)37 b(In)228 1580 y(particular)18 b Ft(Q)457 1586 y Fr(i)490 1580 y FD(=)i Ft(F)569 1586 y Fr(ii)614 1580 y FD(=)f Ft(f)t FD(\()p Ft(P)732 1586 y Fr(i)747 1580 y FD(\),)g Ft(i)h FD(=)g(1)p Ft(;)7 b(:)g(:)g(:)k(;)c(n)p FD(,)19 b(are)g(m)o(utually)d(orthogonal)h (pro)r(jections)228 1630 y(suc)o(h)d(that)g Ft(Q)444 1636 y Fq(1)472 1630 y FD(+)c Ft(Q)547 1636 y Fq(2)574 1630 y FD(+)g Fu(\001)d(\001)g(\001)g FD(+)j Ft(Q)748 1636 y Fr(n)782 1630 y FD(=)i Ft(I)s FD(.)18 b(It)c(follo)o(ws)e(that)i (dim)1218 1636 y Fm(C)1248 1630 y FD(Im)5 b Ft(Q)1337 1636 y Fr(i)1362 1630 y FD(=)12 b(1,)h Ft(i)f FD(=)g(1)p Ft(;)7 b(:)g(:)g(:)k(;)c(n)p FD(.)303 1680 y(Cho)q(ose)18 b(a)f(v)o(ector)h Ft(h)641 1686 y Fq(1)678 1680 y Fu(2)f FD(Im)6 b Ft(Q)813 1686 y Fq(1)831 1680 y FD(,)18 b(\()p Ft(h)901 1686 y Fq(1)920 1680 y Ft(;)7 b(h)963 1686 y Fq(1)981 1680 y FD(\))18 b(=)g(1,)g(and)f(de\014ne)i(then)f Ft(h)1446 1686 y Fr(i)1478 1680 y FD(=)g Ft(F)1555 1686 y Fr(i)p Fq(1)1585 1680 y Ft(h)1609 1686 y Fq(1)1628 1680 y FD(,)g Ft(i)g FD(=)228 1729 y(2)p Ft(;)7 b(:)g(:)g(:)k(;)c(n)p FD(.)18 b(Let)c(us)g(c)o(hec)o(k)h(that)f Fu(f)p Ft(h)777 1735 y Fq(1)795 1729 y Ft(;)7 b(h)838 1735 y Fq(2)856 1729 y Ft(;)g(:)g(:)g(:)12 b(;)7 b(h)980 1735 y Fr(n)1002 1729 y Fu(g)14 b FD(is)f(an)h(orthonormal)e(basis)i(in)f Ft(V)d FD(.)18 b(Indeed)245 1797 y(\()p Ft(h)285 1803 y Fr(i)299 1797 y Ft(;)7 b(h)342 1803 y Fr(j)358 1797 y FD(\))12 b(=)g(\()p Ft(F)473 1803 y Fr(i)p Fq(1)503 1797 y Ft(h)527 1803 y Fq(1)546 1797 y Ft(;)7 b(F)592 1803 y Fr(j)r Fq(1)625 1797 y Ft(h)649 1803 y Fq(1)667 1797 y FD(\))12 b(=)g(\()p Ft(F)788 1780 y Fs(\003)782 1807 y Fr(j)r Fq(1)816 1797 y Ft(F)843 1803 y Fr(i)p Fq(1)873 1797 y Ft(h)897 1803 y Fq(1)915 1797 y Ft(;)7 b(h)958 1803 y Fq(1)976 1797 y FD(\))12 b(=)g(\()p Ft(F)1091 1803 y Fq(1)p Fr(j)1125 1797 y Ft(F)1152 1803 y Fr(i)p Fq(1)1182 1797 y Ft(h)1206 1803 y Fq(1)1224 1797 y Ft(;)7 b(h)1267 1803 y Fq(1)1285 1797 y FD(\))12 b(=)g Ft(\016)1375 1803 y Fr(ij)1405 1797 y FD(\()p Ft(F)1448 1803 y Fq(11)1483 1797 y Ft(h)1507 1803 y Fq(1)1525 1797 y Ft(;)7 b(h)1568 1803 y Fq(1)1586 1797 y FD(\))12 b(=)g Ft(\016)1676 1803 y Fr(ij)228 1864 y FD(Moreo)o(v)o(er)20 b Ft(F)443 1870 y Fr(ij)472 1864 y Ft(h)496 1870 y Fr(k)538 1864 y FD(=)h Ft(F)618 1870 y Fr(ij)647 1864 y Ft(F)674 1870 y Fr(k)q Fq(1)711 1864 y Ft(h)735 1870 y Fq(1)774 1864 y FD(=)h Ft(\016)846 1870 y Fr(j)r(k)882 1864 y Ft(F)909 1870 y Fr(i)p Fq(1)939 1864 y Ft(h)963 1870 y Fq(1)1003 1864 y FD(=)g Ft(\016)1075 1870 y Fr(j)r(k)1111 1864 y Ft(h)1135 1870 y Fr(i)1149 1864 y FD(,)e(that)g(is,)h(w)o(e)f(get)g(the)g(same)f (de-)228 1914 y(scription)c(of)e(the)j(action)e(of)g Ft(F)720 1920 y Fr(ij)763 1914 y FD(as)h(for)f Ft(E)910 1920 y Fr(ij)954 1914 y FD(but)g(with)h(the)g(basis)g Fu(f)p Ft(h)1346 1920 y Fq(1)1364 1914 y Ft(;)7 b(:)g(:)g(:)12 b(;)7 b(h)1488 1920 y Fr(n)1510 1914 y Fu(g)14 b FD(instead)h(of)228 1964 y Fu(f)p Ft(e)268 1970 y Fq(1)287 1964 y Ft(;)7 b(:)g(:)g(:)k(;)c(e)405 1970 y Fr(n)428 1964 y Fu(g)p FD(.)303 2013 y(No)o(w)14 b(de\014ne)i(a)f(unitary)g(op)q(erator)g Ft(U)20 b FD(in)15 b Ft(V)24 b FD(b)o(y)15 b(the)h(form)o(ula)c Ft(U)5 b(e)1354 2019 y Fr(i)1382 2013 y FD(=)14 b Ft(h)1452 2019 y Fr(i)1465 2013 y FD(,)h Ft(i)f FD(=)g(1)p Ft(;)7 b(:)g(:)g(:)k(;)c(n)p FD(.)228 2063 y(Then)13 b(ob)o(viously)e Ft(f)t FD(\()p Ft(E)588 2069 y Fr(ij)618 2063 y FD(\))g(=)h Ft(F)716 2069 y Fr(ij)757 2063 y FD(=)f Ft(U)5 b(E)864 2069 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