One Hat Cyber Team

Edit File: 0704.1549

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%%Title: arXiv:0704.1549v1  [math.OA]  12 Apr 2007
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%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
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%%CreationDate: 1997 Jul 20 21:19:17
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
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/FullName (XYATIP10) readonly def
/FamilyName (XYATIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
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%%EndFont 
%%BeginFont: XYDASH10
%!PS-AdobeFont-1.1: XYDASH10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Aug 28 05:34:12
%%RevisionDate: 1997 Sep 18 10:23:31
%
% XYDASH10: line segments for Xy-pic at 10 point
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def
/FullName (XYDASH10) readonly def
/FamilyName (XYDASH) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -300 def
/UnderlineThickness 150 def
end readonly def
/FontName /XYDASH10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 95 /d95 put
readonly def
/FontBBox{-40 -520 503 520}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
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SDict begin H.S end
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y
SDict begin H.R end
 2251 1592 a 2251 1646 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.CK) cvn H.B /ANN pdfmark
end
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f Fz(C)3082 1615 y Fy(\003)3121 1646 y Fr(-algebras)456
1745 y Fz(C)521 1715 y Fy(\003)559 1745 y Fr($)p Fz(E)5
b Fr($)27 b(asso)r(ciated)d(with)j(directed)e(graphs)g
Fz(E)31 b Fr(has)25 b(b)r(een)h(studied)h(in)f(v)-5 b(arious)25
b(directions)g(b)n(y)456 1845 y(considerably)30 b(man)n(y)i(authors)e
($for)i(example)g(see)f(the)i(bibliograph)n(y)d(in)i(the)h(b)r(o)r(ok)
f([)3219 1845 y
SDict begin H.S end
 3219 1845 a Fr(15)3302 1791 y
SDict begin H.R end
 3302 1791
a 3302 1845 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Ra) cvn H.B /ANN pdfmark
end
 3302 1845 a Fr(])g(b)n(y)456 1944 y(Raeburn$.)i(In)21
b([)976 1944 y
SDict begin H.S end
 976 1944 a Fr(9)1017 1891 y
SDict begin H.R end
 1017 1891
a 1017 1944 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.KP) cvn H.B /ANN pdfmark
end
 1017 1944 a Fr(],)i(Kumjian)d(and)h(P)n(ask)e(sho)n(w)g
(among)h(others)g(that)h(if)g Fz(\015)k Fr(is)c(the)g(gauge)e(action)
456 2044 y(on)30 b Fz(C)639 2014 y Fy(\003)678 2044 y
Fr($)p Fz(E)5 b Fr($,)33 b(then)e Fz(C)1121 2014 y
Fy(\003)1160 2044 y Fr($)p Fz(E)5 b Fr($)1290 2014
y Fp(\015)1364 2044 y Fr(is)31 b(stably)g(isomorphic)f(to)h(the)g
(crossed)f(pro)r(duct)h Fz(C)3048 2014 y Fy(\003)3087
2044 y Fr($)p Fz(E)5 b Fr($)21 b FA(\002)3303 2056
y Fp(\015)3366 2044 y Fn(T)p Fr(,)456 2144 y(whic)n(h)27
b(w)n(as)f(done)h(b)n(y)g(hiring)g(the)g(notions)g(of)g(sk)n(ew)g(pro)r
(duct)g(of)g(graphs)f(and)h(group)r(oid)f Fz(C)3378 2114
y Fy(\003)3417 2144 y Fr(-)456 2243 y(algebras.)43 b(In)30
b(Theorem)1280 2243 y
SDict begin H.S end
 1280 2243 a Fr(6.3)1386 2190 y
SDict begin H.R end

1386 2190 a 1386 2243 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (thm.6.3) cvn H.B /ANN pdfmark
end
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(sho)n(w)g(that)g(the)h(gauge)e(action)h(is)g(actually)456
2343 y(saturated)c($this)i(implies)g(that)g Fz(C)1546
2313 y Fy(\003)1585 2343 y Fr(\()p Fz(E)5 b Fr($)1715
2313 y Fp(\015)1786 2343 y Fr(and)27 b Fz(C)2012 2313
y Fy(\003)2050 2343 y Fr($)p Fz(E)5 b Fr($)19 b FA(\002)2264
2355 y Fp(\015)2325 2343 y Fn(T)27 b Fr(are)g(stably)g(isomorphic\).)
456 2466 y
SDict begin H.S end
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SDict begin 12 H.A end
 456 2466 a 456 2466
a
SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn /DEST pdfmark
end
 456 2466 a 1610 2542 a Fr(2.)41 b Fq(Preliminaries)456
2742 y FB(W)-8 b(atatani's)43 b(index)f(theory)i(for)f
Fz(C)1755 2711 y Fy(\003)1793 2742 y FB(-algebras)p Fr(.)66
b(In)37 b([)2391 2742 y
SDict begin H.S end
 2391 2742 a Fr(19)2474 2688 y
SDict begin H.R end

2474 2688 a 2474 2742 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Wa) cvn H.B /ANN pdfmark
end
 2474 2742 a Fr(],)j(W)-7 b(atatani)37
b(dev)n(elop)r(ed)g(the)456 2841 y(index)h(theory)f(for)h
Fz(C)1164 2811 y Fy(\003)1203 2841 y Fr(-algebras,)g(and)g(here)g(w)n
(e)g(brie\015y)g(review)f(the)i(basic)e(construction)456
2941 y Fz(C)521 2911 y Fy(\003)559 2941 y Fr($)p Fz(B)t(;)14
b(e)734 2953 y Fp(A)788 2941 y Fr($.)40 b(Let)29 b Fz(B)j
Fr(b)r(e)d(a)f Fz(C)1377 2911 y Fy(\003)1416 2941 y Fr(-algebra)e(and)j
Fz(A)f Fr(its)h Fz(C)2170 2911 y Fy(\003)2209 2941 y
Fr(-subalgebra)d(con)n(taining)h(the)i(unit)h(of)456
3040 y Fz(B)t Fr(.)60 b(Let)35 b Fz(E)41 b Fr(:)36 b
Fz(B)k FA(!)c Fz(A)g Fr(b)r(e)f(a)g(faithful)i(conditional)d(exp)r
(ectation.)60 b(If)36 b(there)f(exist)g(\014nitely)456
3140 y(man)n(y)27 b(elemen)n(ts)g FA(f)p Fz(v)1103 3152
y Fp(i)1130 3140 y FA(g)1172 3110 y Fp(n)1172 3162 y(i)p
Fo(=1)1311 3140 y Fr(in)h Fz(B)k Fr(satisfying)27 b(the)h(follo)n(wing)
1039 3312 y Fz(b)23 b Fr(=)1186 3233 y Fm(X)1234 3410
y Fp(i)1319 3312 y Fz(E)5 b Fr($)p Fz(bv)1493 3324 y
Fp(i)1521 3312 y Fr($)p Fz(v)1596 3278 y Fy(\003)1593
3333 y Fp(i)1658 3312 y Fr(=)1746 3233 y Fm(X)1794 3410
y Fp(i)1879 3312 y Fz(v)1919 3324 y Fp(i)1947 3312 y
Fz(E)g Fr($)p Fz(v)2088 3278 y Fy(\003)2085 3333 y Fp(i)2127
3312 y Fz(b)p Fr($)p Fz(;)69 b Fr(for)27 b(ev)n(ery)f
Fz(b)d FA(2)h Fz(B)t(;)456 3544 y(E)36 b Fr(is)c(said)f(to)h(b)r(e)g
(of)f Fl(index-\014nite)i(typ)l(e)39 b Fr(and)31 b FA(f)p
Fr($)p Fz(v)2038 3556 y Fp(i)2066 3544 y Fz(;)14 b(v)2146
3514 y Fy(\003)2143 3566 y Fp(i)2184 3544 y Fr($)p FA(g)2258
3514 y Fp(n)2258 3566 y(i)p Fo(=1)2401 3544 y Fr(is)32
b(called)f(a)h Fl(quasi-b)l(asis)39 b Fr(for)31 b Fz(E)5
b Fr(.)456 3644 y(The)26 b(p)r(ositiv)n(e)g(elemen)n(t)1236
3582 y Fm(P)1324 3669 y Fp(i)1365 3644 y Fz(v)1405 3656
y Fp(i)1433 3644 y Fz(v)1476 3614 y Fy(\003)1473 3666
y Fp(i)1541 3644 y Fr(is)g(then)g(the)h Fl(index)37 b
Fr(of)26 b Fz(E)5 b Fr(,)26 b Fz(I)7 b(ndex)p Fr($)p
Fz(E)e Fr($,)28 b(whic)n(h)e(is)g(kno)n(wn)g(to)456
3744 y(b)r(e)h(an)f(elemen)n(t)h(in)h(the)f(cen)n(ter)f(of)h
Fz(B)k Fr(and)c(do)r(es)f(not)h(dep)r(end)h(on)e(the)i(c)n(hoice)e(of)g
(quasi-bases)456 3843 y(for)h Fz(E)32 b Fr($[)731 3843
y
SDict begin H.S end
 731 3843 a Fr(19)815 3790 y
SDict begin H.R end
 815 3790 a 815 3843 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Wa) cvn H.B /ANN pdfmark
end
 815
3843 a Fr(,)c(Prop)r(osition)d(1.2.8]$.)36 b(Let)28
b FA(B)i Fr(b)r(e)e(the)g(completion)f(of)h(the)g(pre-Hilb)r(ert)f(mo)r
(dule)456 3943 y FA(B)511 3955 y Fo(0)570 3943 y Fr(=)c
FA(f)p Fz(\021)s Fr($)p Fz(b)p Fr($)g FA(j)g Fz(b)g
FA(2)g Fz(B)t FA(g)28 b Fr(o)n(v)n(er)d Fz(A)j Fr(with)g(an)g
Fz(A)p Fr(-v)-5 b(alued)27 b(inner)h(pro)r(duct)1246
4098 y FA(h)p Fz(\021)s Fr($)p Fz(x)p Fr($)p Fz(;)14
b(\021)s Fr($)p Fz(y)s Fr($)p FA(i)26 b Fr(=)c Fz(E)5
b Fr($)p Fz(x)1912 4063 y Fy(\003)1951 4098 y Fz(y)s
Fr($)p Fz(;)42 b(\021)s Fr($)p Fz(x)p Fr($)p Fz(;)14
b(\021)s Fr($)p Fz(y)s Fr($)24 b FA(2)g(B)2594 4110
y Fo(0)2631 4098 y Fz(:)456 4253 y Fr(Let)33 b FA(L)667
4265 y Fp(A)721 4253 y Fr($)p FA(B)s Fr($)f(b)r(e)i(the)f
Fz(C)1207 4222 y Fy(\003)1245 4253 y Fr(-algebra)e(of)i(all)f($righ)n
(t$)h Fz(A)p Fr(-mo)r(dule)g(homomorphisms)e(on)i FA(B)i
Fr(with)456 4352 y(adjoin)n(ts.)g(F)-7 b(or)23 b Fz(T)34
b FA(2)23 b(L)1164 4364 y Fp(A)1219 4352 y Fr($)p FA(B)s
Fr($,)h(the)g(norm)f FA(k)p Fz(T)12 b FA(k)22 b Fr(=)g(sup)p
FA(fk)p Fz(T)12 b(x)p FA(k)21 b Fr(:)i FA(k)p Fz(x)p
FA(k)g Fr(=)g(1)p FA(g)f Fr(is)i(alw)n(a)n(ys)e(b)r(ounded.)456
4452 y(Eac)n(h)30 b Fz(b)h FA(2)g Fz(B)36 b Fr(is)c(regarded)e(as)h(an)
h(op)r(erator)f Fz(L)1975 4464 y Fp(b)2040 4452 y Fr(in)h
FA(L)2198 4464 y Fp(A)2252 4452 y Fr($)p FA(B)s Fr($)g(de\014ned)g(b)
n(y)g Fz(L)2873 4464 y Fp(b)2906 4452 y Fr($)p Fz(\021)s
Fr(\()p Fz(x)p Fr($\))g(=)f Fz(\021)s Fr($)p Fz(bx)p
Fr($)456 4551 y(for)38 b Fz(\021)s Fr($)p Fz(x)p Fr($)44
b FA(2)e(B)945 4563 y Fo(0)982 4551 y Fr(.)72 b(By)38
b Fz(e)1257 4563 y Fp(A)1353 4551 y Fr(:)43 b FA(B)h(!)e(B)g
Fr(w)n(e)c(denote)h(the)h(pro)5 b(jection)38 b(in)h FA(L)2878
4563 y Fp(A)2933 4551 y Fr($)p FA(B)s Fr($)f(suc)n(h)h(that)456
4651 y Fz(e)495 4663 y Fp(A)548 4651 y Fr($)p Fz(\021)s
Fr(\()p Fz(x)p Fr($\))30 b(=)e Fz(\021)s Fr($)p Fz(E)5
b Fr(\()p Fz(x)p Fr($\),)33 b Fz(\021)s Fr($)p Fz(x)p
Fr($)d FA(2)e(B)1554 4663 y Fo(0)1591 4651 y Fr(.)46
b(Then)31 b(the)g Fz(C)2091 4621 y Fy(\003)2129 4651
y Fl(-b)l(asic)i(c)l(onstruction)k Fz(C)2912 4621 y Fy(\003)2951
4651 y Fr($)p Fz(B)t(;)14 b(e)3126 4663 y Fp(A)3180
4651 y Fr($)31 b(is)f(the)456 4751 y Fz(C)521 4721 y
Fy(\003)559 4751 y Fr(-subalgebra)24 b(of)i FA(L)1151
4763 y Fp(A)1205 4751 y Fr($)p FA(B)s Fr($)g(in)g(whic)n(h)g(the)g
(linear)f(span)h(of)f(elemen)n(ts)h Fz(L)2734 4763 y
Fp(b)2767 4751 y Fz(e)2806 4763 y Fp(A)2860 4751 y Fz(L)2917
4763 y Fp(b)2946 4746 y Fk(0)2998 4751 y Fr($)p Fz(b;)14
b(b)3139 4721 y Fy(0)3185 4751 y FA(2)23 b Fz(B)t Fr($)k(is)456
4850 y(dense.)456 5050 y FB(Finite)36 b(dimensional)d(Hopf)k
FA(\003)p FB(-algebras.)49 b Fr(As)32 b(in)g([)2254 5050
y
SDict begin H.S end
 2254 5050 a Fr(18)2337 4996 y
SDict begin H.R end
 2337 4996 a 2337 5050
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.SP) cvn H.B /ANN pdfmark
end
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FA(\003)p Fr(-)456 5149 y(algebra)d(is)i(a)g(\014nite)g(matrix)g
(pseudogroup)f(of)h([)2009 5149 y
SDict begin H.S end
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5096 y
SDict begin H.R end
 2092 5096 a 2092 5149 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Wo) cvn H.B /ANN pdfmark
end
 2092 5149 a Fr(].)47 b(W)-7
b(e)32 b(review)e(from)h([)2819 5149 y
SDict begin H.S end
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5096 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.SP) cvn H.B /ANN pdfmark
end
 2902 5149 a Fr(])h(the)f(de\014nition)
456 5249 y(and)c(some)g(basic)h(prop)r(erties)e(of)i(a)f(\014nite)i
(dimensional)e(Hopf)h FA(\003)p Fr(-algebra)d(whic)n(h)j(w)n(e)f(need)h
(in)456 5348 y(the)g(follo)n(wing)e(section.)456 5365
y
SDict begin H.S end
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SDict begin 12 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (thm.2.1) cvn /DEST pdfmark end
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SDict begin H.R end
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w Fo(SA)-5 b(TURA)g(TED)29 b(A)n(CTIONS)h(BY)f(FINITE)g(DIMENSIONAL)f
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SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.SP) cvn H.B /ANN pdfmark
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(A)g(\014nite)h(dimensional)f Fz(C)2731 420 y Fy(\003)2769
450 y Fr(-algebra)f(is)h(called)g(a)456 550 y Fl(\014nite)j
(dimensional)i(Hopf)g FA(\003)p Fl(-algebr)l(a)k Fr(if)28
b(there)f(exist)h(three)f(linear)g(maps,)1133 738 y(\001)c(:)g
Fz(A)g FA(!)h Fz(A)18 b FA(\012)g Fz(A;)70 b(\017)23
b Fr(:)g Fz(A)g FA(!)g Fn(C)15 b Fz(;)75 b Fr(and)28
b Fz(S)f Fr(:)d Fz(A)f FA(!)g Fz(A)456 927 y Fr(whic)n(h)k(satisfy)g
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e FA(\003)p Fr(-homomorphisms,)g(and)h Fz(S)5 b Fr($an)n(tip)r(o)r
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(ultiplicativ)n(e)i(in)n(v)n(olution,)592 1270 y($ii$)42
b(\001$1$)24 b(=)e(1)c FA(\012)g Fr(1,)28 b Fz(\017)p
Fr($1$)22 b(=)h(1,)k Fz(S)5 b Fr($1$)23 b(=)g(1,)569
1370 y($iii$)42 b($\001)19 b FA(\012)f Fz(id)p Fr($\001)24
b(=)e($)p Fz(id)d FA(\012)f Fr(\001$\001,)571 1470
y($iv$)42 b($)p Fz(\017)19 b FA(\012)f Fz(id)p Fr($\001)23
b(=)g(\001$)p Fz(\017)18 b FA(\012)g Fz(id)p Fr($,)594
1569 y($v$)42 b Fz(m)p Fr($)p Fz(S)6 b FA(\012)q Fz(id)p
Fr($$\001\()p Fz(a)p Fr($\))25 b(=)e Fz(\017)p Fr($)p
Fz(a)p Fr($1)f(=)h Fz(m)p Fr($)p Fz(id)q FA(\012)q
Fz(S)5 b Fr($$\001\()p Fz(a)p Fr($\))20 b(for)f Fz(a)k
FA(2)g Fz(A)p Fr(,)f(where)c Fz(m)23 b Fr(:)g Fz(A)q
FA(\012)q Fz(A)h FA(!)f Fz(A)744 1669 y Fr(is)28 b(the)g(m)n
(ultiplication.)456 1865 y
SDict begin H.S end
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SDict begin 12 H.A end
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1865 a 456 1865 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.2.2) cvn /DEST pdfmark end
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b Fr($[)1219 1949 y
SDict begin H.S end
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1895 a 1303 1949 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
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y
SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Wo) cvn H.B /ANN pdfmark
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 1484 1949 a Fr(]$)j Fl(L)l(et)e Fz(A)i Fl(b)l(e)f(a)g(\014nite)g
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2049 y(the)30 b(fol)t(lowing)i(pr)l(op)l(erties)f(hold.)615
2193 y Fr($i$)42 b Fl(F)-6 b(or)30 b Fz(a)23 b FA(2)h
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2193 y Fz(a)2273 2163 y Fp(L)2273 2214 y(i)2341 2193
y FA(\012)18 b Fz(a)2468 2163 y Fp(R)2468 2214 y(i)2522
2193 y Fz(;)30 b Fl(we)g(have)1305 2323 y Fm(X)1353 2500
y Fp(i)1439 2402 y Fz(\017)p Fr($)p Fz(a)1549 2368 y
Fp(L)1549 2423 y(i)1598 2402 y Fr($)p Fz(a)1674 2368
y Fp(R)1674 2423 y(i)1752 2402 y Fr(=)23 b Fz(a)g Fr(=)1994
2323 y Fm(X)2043 2500 y Fp(i)2128 2402 y Fz(\017)p Fr($)p
Fz(a)2238 2368 y Fp(R)2238 2423 y(i)2292 2402 y Fr($)p
Fz(a)2368 2368 y Fp(L)2368 2423 y(i)2418 2402 y Fz(;)1283
2550 y Fm(X)1331 2727 y Fp(i)1417 2629 y Fz(a)1461 2594
y Fp(L)1461 2649 y(i)1510 2629 y Fz(S)5 b Fr($)p Fz(a)1642
2594 y Fp(R)1642 2649 y(i)1697 2629 y Fr($)23 b(=)37
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2594 y Fp(L)2414 2649 y(i)2463 2629 y Fr($)p Fz(a)2539
2594 y Fp(R)2539 2649 y(i)2594 2629 y Fz(;)1283 2777
y Fm(X)1331 2953 y Fp(i)1417 2855 y Fz(a)1461 2821 y
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2821 y Fp(L)1647 2876 y(i)1697 2855 y Fr($)23 b(=)37
b Fz(\017)p Fr($)p Fz(a)p Fr($1)22 b(=)2148 2777 y
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Fj(A)1901 825 y Fz(L)1958 837 y Fp(y)1992 845 y Fj(i)2022
825 y Fr($)c(=)2165 746 y Fm(X)2213 923 y Fp(i)2298
825 y Fz(x)2345 837 y Fp(i)2373 825 y Fz(ey)2453 837
y Fp(i)2480 825 y Fz(:)456 1061 y Fr(T)-7 b(o)27 b(see)g(that)h
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Fm(P)1742 1086 y Fp(i)1784 1061 y Fz(L)1841 1073 y Fp(x)1879
1081 y Fj(i)1908 1061 y Fz(e)1947 1077 y Fp(M)2016 1060
y Fj(A)2068 1061 y Fz(L)2125 1073 y Fp(y)2159 1081 y
Fj(i)2212 1061 y Fr(=)22 b(0.)37 b(Then)27 b(for)h(eac)n(h)e
Fz(z)h FA(2)c Fz(M)9 b Fr(,)735 1240 y($)767 1161 y
Fm(X)815 1338 y Fp(i)901 1240 y Fz(L)958 1252 y Fp(x)996
1260 y Fj(i)1026 1240 y Fz(e)1065 1257 y Fp(M)1134 1240
y Fj(A)f Fz(L)1242 1252 y Fp(y)1276 1260 y Fj(i)1306
1240 y Fr($$)p Fz(\021)s Fr(\()p Fz(z)t Fr($\))24
b(=)1664 1161 y Fm(X)1712 1338 y Fp(i)1798 1240 y Fz(\021)s
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1240 y Fz(x)2591 1252 y Fp(i)2619 1240 y Fr(\()p Fz(e)c
FA(\001)f Fr(\()p Fz(y)2823 1252 y Fp(i)2851 1240 y Fz(z)t
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1476 a Fr(19)1597 1423 y
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1751 y Fr($)p Fz(z)t(e)f Fr(=)h(0)456 1983 y(in)k Fz(M)h
Fn(o)18 b Fz(A)27 b Fr(for)h(ev)n(ery)e Fz(z)g FA(2)e
Fz(M)9 b Fr(,)27 b(whic)n(h)g(then)i(implies)e(that)1369
2153 y($)1401 2074 y Fm(X)1449 2251 y Fp(i)1535 2153
y Fz(x)1582 2165 y Fp(i)1610 2153 y Fz(ey)1690 2165 y
Fp(i)1717 2153 y Fr($$)p Fz(z)t(ez)1906 2119 y Fy(0)1928
2153 y Fr($)c(=)g(0)p Fz(;)41 b(z)t(;)14 b(z)2300 2119
y Fy(0)2345 2153 y FA(2)23 b Fz(M)t(:)456 2389 y Fr(P)n(articulary)-7
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y Fz(x)1118 2401 y Fp(i)1146 2389 y Fz(ey)1226 2401 y
Fp(i)1253 2389 y Fr($$)1317 2327 y Fm(P)1405 2414 y
Fp(i)1446 2389 y Fz(x)1493 2401 y Fp(i)1521 2389 y Fz(ey)1601
2401 y Fp(i)1628 2389 y Fr($)1660 2359 y Fy(\003)1725
2389 y Fr(=)e(0,)30 b(so)f(that)2195 2327 y Fm(P)2283
2414 y Fp(i)2324 2389 y Fz(x)2371 2401 y Fp(i)2399 2389
y Fz(ey)2479 2401 y Fp(i)2532 2389 y Fr(=)d(0)j($in)h
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2489 y Fr(is)k(w)n(ell)g(de\014ned.)55 b(It)34 b(is)f(tedious)g(to)g
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Fz(C)831 2558 y Fy(\003)870 2588 y Fr(\()p Fz(M)t(;)14
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b(=)g Fz(M)9 b(eM)34 b Fr(is)26 b(an)f(ideal)h(of)g Fz(M)d
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Fr(and)g Fz(z)g FA(2)d Fz(M)35 b Fr(and)26 b Fz(a)d FA(2)g
Fz(A)p Fr(,)456 2688 y(then)1351 2812 y($)p Fz(z)t(a)p
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2923 y Fp(n)2341 2975 y(i)p Fo(=1)2478 2953 y Fr(is)25
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3051 y Fm(X)967 3228 y Fp(i)1053 3130 y Fz(L)1110 3142
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3122 y Fk(\003)1434 3161 y Fj(i)1473 3130 y Fr($)p Fz(\021)s
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3390 y Fp(i)1229 3366 y Fz(L)1286 3378 y Fp(u)1325 3386
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3366 a Fr(2.2)3319 3312 y
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3366 a Fr($iii$)456 3467 y(and)27 b(Prop)r(osition)1064
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SDict begin H.S end
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3713 y Fp(i)2921 3693 y Fr(=)23 b(1)p Fz(:)456 3929 y
Fr(Since)e Fz(')g Fr(is)f(a)h FA(\003)p Fr(-homomorphism,)f
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y(0.)36 b(But)28 b Fz(F)40 b Fr(is)27 b(faithful,)i(and)e
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4233 a 1370 4312 a Fr(4.)41 b Fq(A)n(ctions)31 b(by)g(finite)h(gr)n
(oups)555 4462 y Fr(Throughout)22 b(this)h(section)f
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4531 y Fy(\003)559 4561 y Fr(-algebra)29 b Fz(C)946 4531
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4820 y(\001$)p Fz(\025)980 4832 y Fp(g)1019 4820 y Fr($)e(=)g
Fz(\025)1210 4832 y Fp(g)1268 4820 y FA(\012)18 b Fz(\025)1399
4832 y Fp(g)1438 4820 y Fz(;)69 b(\017)p Fr($)p Fz(\025)1644
4832 y Fp(g)1683 4820 y Fr($)23 b(=)g(1)p Fz(;)69 b(S)5
b Fr($)p Fz(\025)2096 4832 y Fp(g)2135 4820 y Fr($)23
b(=)g Fz(\025)2326 4835 y Fp(g)2360 4819 y Fk(\000)p
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4820 y FA(2)23 b Fz(C)2878 4785 y Fy(\003)2917 4820 y
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Fz(C)1939 5160 y Fy(\003)1978 5190 y Fr(-algebra)d Fz(M)9
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2407 y Fm(P)2815 2494 y Fp(j)2863 2469 y Fz(b)2899 2439
y Fp(j)2899 2489 y(g)2938 2469 y Fr($)p Fz(b)3006 2439
y Fp(j)3006 2489 y(g)3044 2469 y Fr($)3076 2439 y Fy(\003)3115
2469 y FA(k)f Fz(<)h Fr(1)5 b(+)g Fz(")p Fr(.)456 2571
y(Then)769 2709 y FA(k)825 2630 y Fm(X)840 2809 y Fp(h;j)945
2709 y Fr($)977 2630 y Fm(X)1020 2805 y Fp(g)1111 2709
y Fz(b)1147 2669 y Fp(j)1147 2734 y(h)1189 2709 y Fz(\013)1242
2721 y Fp(g)1281 2709 y Fr(\(\()p Fz(b)1381 2669 y Fp(j)1381
2734 y(h)1424 2709 y Fr($)1456 2675 y Fy(\003)1495 2709
y Fr(\))p Fz(\025)1575 2721 y Fp(g)1614 2709 y Fr(\))19
b FA(\000)f(j)p Fz(G)p FA(j)c(k)677 2949 y Fr(=)27 b
FA(k)825 2870 y Fm(X)867 3045 y Fp(g)945 2949 y Fr($)977
2870 y Fm(X)992 3049 y Fp(h;j)1111 2949 y Fz(b)1147 2909
y Fp(j)1147 2974 y(h)1189 2949 y Fz(\013)1242 2961 y
Fp(g)1281 2949 y Fr(\(\()p Fz(b)1381 2909 y Fp(j)1381
2974 y(h)1424 2949 y Fr($)1456 2915 y Fy(\003)1495 2949
y Fr(\))p Fz(\025)1575 2961 y Fp(g)1632 2949 y FA(\000)18
b(j)p Fz(G)p FA(j)c(k)677 3189 y(\024)27 b(k)825 3110
y Fm(X)840 3289 y Fp(h;j)958 3189 y Fz(b)994 3149 y Fp(j)994
3214 y(h)1037 3189 y Fr($)p Fz(b)1105 3149 y Fp(j)1105
3214 y(h)1148 3189 y Fr($)1180 3155 y Fy(\003)1237 3189
y FA(\000)18 b(j)p Fz(G)p FA(j)c(k)k Fr(+)g FA(k)1644
3110 y Fm(X)1648 3289 y Fp(g)r Fy(6)p Fo(=)p Fp(\023)1763
3189 y Fr($)1795 3110 y Fm(X)1811 3289 y Fp(h;j)1929
3189 y Fz(b)1965 3149 y Fp(j)1965 3214 y(h)2008 3189
y Fz(\013)2061 3201 y Fp(g)2100 3189 y Fr(\(\()p Fz(b)2200
3149 y Fp(j)2200 3214 y(h)2243 3189 y Fr($)2275 3155
y Fy(\003)2314 3189 y Fr(\))p Fz(\025)2394 3201 y Fp(g)2433
3189 y Fr(\))p FA(k)677 3429 y(\024)783 3350 y Fm(X)824
3529 y Fp(h)917 3429 y FA(k)973 3350 y Fm(X)1017 3527
y Fp(j)1106 3429 y Fz(b)1142 3389 y Fp(j)1142 3454 y(h)1185
3429 y Fr($)p Fz(b)1253 3389 y Fp(j)1253 3454 y(h)1296
3429 y Fr($)1328 3395 y Fy(\003)1385 3429 y FA(\000)g
Fr(1)c FA(k)j Fr(+)1666 3350 y Fm(X)1671 3529 y Fp(g)r
Fy(6)p Fo(=)p Fp(\023)1800 3350 y Fm(X)1840 3529 y Fp(h)1933
3429 y FA(k)1989 3350 y Fm(X)2033 3527 y Fp(j)2122 3429
y Fz(b)2158 3389 y Fp(j)2158 3454 y(h)2201 3429 y Fz(\013)2254
3441 y Fp(g)2293 3429 y Fr($\()p Fz(b)2393 3389 y Fp(j)2393
3454 y(h)2436 3429 y Fr($)2468 3395 y Fy(\003)2507 3429
y Fr(\))p FA(k)677 3669 y Fz(<)27 b(")p FA(j)p Fz(G)p
FA(j)19 b Fr(+)1021 3590 y Fm(X)1026 3769 y Fp(g)r Fy(6)p
Fo(=)p Fp(\023)1155 3590 y Fm(X)1195 3769 y Fp(h)1288
3669 y FA(k)1344 3590 y Fm(X)1388 3767 y Fp(j)1478 3669
y Fz(b)1514 3629 y Fp(j)1514 3694 y(h)1556 3601 y Fm(\000)1594
3669 y Fz(\013)1647 3681 y Fp(g)1686 3669 y Fr($\()p
Fz(b)1786 3629 y Fp(j)1786 3694 y(h)1829 3669 y Fr($)1861
3634 y Fy(\003)1900 3669 y Fr(\))f FA(\000)g Fr($)p
Fz(b)2101 3629 y Fp(j)2101 3694 y(g)r(h)2179 3669 y Fr($)2211
3634 y Fy(\003)2249 3601 y Fm(\001)2287 3669 y FA(k)g
Fr(+)2430 3590 y Fm(X)2435 3769 y Fp(g)r Fy(6)p Fo(=)p
Fp(\023)2564 3590 y Fm(X)2605 3769 y Fp(h)2698 3669 y
FA(k)2754 3590 y Fm(X)2798 3767 y Fp(j)2887 3669 y Fz(b)2923
3629 y Fp(j)2923 3694 y(h)2966 3669 y Fr($)p Fz(b)3034
3629 y Fp(j)3034 3694 y(g)r(h)3111 3669 y Fr($)3143
3634 y Fy(\003)3181 3669 y FA(k)677 3917 y Fz(<)27 b(")808
3825 y Fm(\020)858 3917 y FA(j)p Fz(G)p FA(j)18 b Fr(+)g
FA(j)p Fz(G)p FA(j)1181 3883 y Fo(2)1233 3917 y Fr(max)1268
3969 y Fp(g)r(;j)1401 3917 y FA(k)p Fz(b)1479 3883 y
Fp(j)1479 3937 y(g)1517 3917 y FA(k)g Fr(+)g FA(j)p Fz(G)p
FA(j)1771 3883 y Fo(2)1809 3825 y Fm(\021)677 4117 y
Fz(<)27 b(")808 4025 y Fm(\020)858 4117 y FA(j)p Fz(G)p
FA(j)18 b Fr(+)g FA(j)p Fz(G)p FA(j)1181 4083 y Fo(2)1219
4117 y Fr($1)g(+)g Fz(")p Fr($)h(+)f FA(j)p Fz(G)p
FA(j)1678 4083 y Fo(2)1716 4025 y Fm(\021)1765 4117 y
Fz(:)456 4297 y Fr(Since)677 4235 y Fm(P)765 4322 y Fp(h;j)858
4297 y Fr($)890 4235 y Fm(P)978 4322 y Fp(g)1031 4297
y Fz(b)1067 4257 y Fp(j)1067 4322 y(h)1109 4297 y Fz(\013)1162
4309 y Fp(g)1201 4297 y Fr(\(\()p Fz(b)1301 4257 y Fp(j)1301
4322 y(h)1344 4297 y Fr($)1376 4267 y Fy(\003)1415 4297
y Fr(\))p Fz(\025)1495 4309 y Fp(g)1534 4297 y Fr(\))32
b FA(2)f(J)1740 4309 y Fp(\013)1820 4297 y Fr(and)h Fz(")h
Fr(can)f(b)r(e)h(c)n(hosen)e(to)h(b)r(e)h(arbitrarily)e(small,)456
4399 y(w)n(e)c(conclude)g(that)h FA(J)1155 4411 y Fp(\013)1226
4399 y Fr(=)22 b Fz(M)27 b FA(\002)1486 4411 y Fp(\013)1552
4399 y Fz(G)p Fr(.)1740 b Fi(\003)456 4566 y
SDict begin H.S end
 456 4566
a 456 4566 a
SDict begin 12 H.A end
 456 4566 a 456 4566 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.4.2) cvn /DEST pdfmark end
 456 4566 a 129 x FB(Example)41
b(4.2.)47 b Fr(Let)37 b Fz(w)43 b Fr(=)1422 4578 y Fm(\022)1525
4644 y Fz(z)1564 4656 y Fo(1)1701 4644 y Fr(0)1542 4744
y(0)100 b Fz(z)1723 4756 y Fo(2)1801 4578 y Fm(\023)1900
4695 y Fr(b)r(e)38 b(a)f(unitary)h(with)g Fz(w)2666 4665
y Fp(n)2752 4695 y Fr(=)h(1)e(and)h(de\014ne)g(an)456
4844 y(automorphism)27 b Fz(\013)j Fr(on)e Fz(M)1286
4856 y Fo(2)1323 4844 y Fr($)p Fn(C)15 b Fr($)q(\))35
b(b)n(y)29 b Fz(\013)p Fr($)p Fz(a)p Fr($)d(=)e Fz(w)r(aw)2068
4814 y Fy(\003)2108 4844 y Fr(,)29 b Fz(a)c FA(2)h Fz(M)2391
4856 y Fo(2)2427 4844 y Fr($)p Fn(C)16 b Fr($.)47 b(W)-7
b(e)29 b(will)g(sho)n(w)f(that)h Fz(\013)h Fr(is)456
4944 y(saturated)h(if)h(and)g(only)f(if)i Fz(z)1382 4956
y Fo(2)1449 4944 y Fr(=)c FA(\000)p Fz(z)1647 4956 y
Fo(1)1684 4944 y Fr(.)50 b(F)-7 b(or)31 b(this,)i(recall)e(from)h($)
2559 4944 y
SDict begin H.S end
 2559 4944 a Fr(3)2600 4891 y
SDict begin H.R end
 2600 4891 a
2600 4944 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.3) cvn H.B /ANN pdfmark
end
 2600 4944 a Fr($)h(that)f Fz(\013)g Fr(is)g(saturated)f
(if)456 5044 y(and)c(only)g(if)h(there)g(exist)f Fz(x)1331
5056 y Fp(j)1390 5044 y FA(2)c Fz(M)1549 5056 y Fo(2)1586
5044 y Fr($)p Fn(C)15 b Fr($,)34 b(1)23 b FA(\024)f
Fz(i)h FA(\024)g Fz(m)p Fr(,)k(satisfying)1331 5264 y
SDict begin H.S end

1331 5264 a 1331 5264 a
SDict begin 12 H.A end
 1331 5264 a 1331 5264 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.4.8) cvn /DEST pdfmark
end
 1331
5264 a 1345 5160 a Fp(n)p Fy(\000)p Fo(1)1348 5185 y
Fm(X)1347 5364 y Fp(k)q Fo(=0)1515 5160 y Fp(m)1484 5185
y Fm(X)1487 5362 y Fp(j)s Fo(=1)1618 5264 y Fz(x)1665
5276 y Fp(j)1701 5264 y Fz(\013)1754 5230 y Fp(k)1795
5264 y Fr($)p Fz(x)1874 5230 y Fy(\003)1874 5285 y Fp(j)1913
5264 y Fr($)p Fz(\025)1993 5276 y Fp(k)2057 5264 y Fr(=)c(1)2187
5279 y Fp(M)2250 5287 y Fh(2)2282 5279 y Fo($)p Fg(C)t
Fo($)2380 5264 y Fz(:)935 b Fr($8$)p eop end end
%%Page: 8 8
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 0 0 a 456 251
a
SDict begin H.S end
 456 251 a 456 251 a
SDict begin H.R end
 456 251 a 456 251 a
SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn /DEST pdfmark end
 456 251 a Fo(8)910
b(JA)22 b(A)g(JEONG)h(AND)f(GI)h(HYUN)f(P)-5 b(ARK)456
450 y Fr(Hence,)27 b(particularly)g(for)g Fz(k)f Fr(=)c(0)p
Fz(;)14 b Fr(1,)27 b(w)n(e)g(ha)n(v)n(e)666 554 y Fm(X)710
730 y Fp(j)799 633 y Fz(x)846 645 y Fp(j)882 633 y Fz(x)929
598 y Fy(\003)929 653 y Fp(j)991 633 y Fr(=)1078 515
y Fm(\022)1181 582 y Fr(1)82 b(0)1181 681 y(0)g(1)1388
515 y Fm(\023)1519 633 y Fr(and)1722 554 y Fm(X)1766
730 y Fp(j)1855 633 y Fz(x)1902 645 y Fp(j)1938 633 y
Fz(\013)p Fr($)p Fz(x)2070 598 y Fy(\003)2070 653 y
Fp(j)2109 633 y Fr($)23 b(=)2252 554 y Fm(X)2297 730
y Fp(j)2386 633 y Fz(x)2433 645 y Fp(j)2468 633 y Fz(w)r(x)2576
598 y Fy(\003)2576 653 y Fp(j)2616 633 y Fz(w)2677 598
y Fy(\003)2738 633 y Fr(=)2826 515 y Fm(\022)2929 582
y Fr(0)82 b(0)2929 681 y(0)g(0)3136 515 y Fm(\023)3211
633 y Fz(:)456 916 y Fr(With)28 b Fz(x)717 928 y Fp(j)776
916 y Fr(=)863 799 y Fm(\022)966 866 y Fz(a)1010 878
y Fp(j)1132 866 y Fz(b)1168 878 y Fp(j)970 965 y Fz(c)1006
977 y Fp(j)1128 965 y Fz(d)1171 977 y Fp(j)1247 799 y
Fm(\023)1336 916 y Fr(and)g Fz(z)1537 928 y Fp(i)1587
916 y Fr(=)23 b Fz(e)1714 886 y Fp(i\022)1769 894 y Fj(i)1799
916 y Fr(,)28 b Fz(i)22 b Fr(=)h(1)p Fz(;)14 b Fr(2)p
Fz(;)26 b Fr(this)i(means)1062 1083 y Fm(X)1107 1259
y Fp(j)1196 1044 y Fm(\022)1298 1111 y FA(j)p Fz(a)1365
1123 y Fp(j)1400 1111 y FA(j)1423 1081 y Fo(2)1479 1111
y Fr(+)18 b FA(j)p Fz(b)1621 1123 y Fp(j)1656 1111 y
FA(j)1679 1081 y Fo(2)1808 1111 y Fz(a)1852 1123 y Fp(j)1901
1111 y Fr(\026)-56 b Fz(c)1923 1123 y Fp(j)1976 1111
y Fr(+)18 b Fz(b)2095 1123 y Fp(j)2148 1089 y Fr(\026)2130
1111 y Fz(d)2173 1123 y Fp(j)1307 1210 y Fz(c)1343 1222
y Fp(j)1397 1210 y Fr(\026)-61 b Fz(a)1422 1222 y Fp(j)1475
1210 y Fr(+)18 b Fz(d)1601 1222 y Fp(j)1651 1189 y Fr(\026)1637
1210 y Fz(b)1673 1222 y Fp(j)1799 1210 y FA(j)p Fz(c)1858
1222 y Fp(j)1893 1210 y FA(j)1916 1180 y Fo(2)1972 1210
y Fr(+)g FA(j)p Fz(d)2121 1222 y Fp(j)2156 1210 y FA(j)2179
1180 y Fo(2)2258 1044 y Fm(\023)2342 1161 y Fr(=)2430
1044 y Fm(\022)2532 1111 y Fr(1)83 b(0)2532 1210 y(0)g(1)2740
1044 y Fm(\023)2815 1161 y Fz(;)676 1487 y
SDict begin H.S end
 676 1487 a
676 1487 a
SDict begin 12 H.A end
 676 1487 a 676 1487 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.4.9) cvn /DEST pdfmark
end
 676 1487 a 690 1408
a Fm(X)735 1585 y Fp(j)824 1370 y Fm(\022)926 1436 y
FA(j)p Fz(a)993 1448 y Fp(j)1028 1436 y FA(j)1051 1406
y Fo(2)1107 1436 y Fr(+)18 b Fz(e)1229 1406 y Fp(i)p
Fo($)p Fp(\022)1310 1414 y Fh(2)1342 1406 y Fy(\000)p
Fp(\022)1426 1414 y Fh(1)1458 1406 y Fo($)1488 1436
y FA(j)p Fz(b)1547 1448 y Fp(j)1582 1436 y FA(j)1605
1406 y Fo(2)1732 1436 y Fz(e)1771 1406 y Fp(i)p Fo($)p
Fp(\022)1852 1414 y Fh(1)1884 1406 y Fy(\000)p Fp(\022)1968
1414 y Fh(2)2000 1406 y Fo($)2030 1436 y Fz(a)2074 1448
y Fp(j)2123 1436 y Fr(\026)-56 b Fz(c)2145 1448 y Fp(j)2198
1436 y Fr(+)18 b Fz(b)2317 1448 y Fp(j)2370 1414 y Fr(\026)2352
1436 y Fz(d)2395 1448 y Fp(j)929 1540 y FA(j)p Fz(a)996
1552 y Fp(j)1031 1540 y FA(j)1054 1510 y Fo(2)1110 1540
y Fr(+)g Fz(e)1232 1510 y Fp(i)p Fo($)p Fp(\022)1313
1518 y Fh(2)1345 1510 y Fy(\000)p Fp(\022)1429 1518 y
Fh(1)1461 1510 y Fo($)1491 1540 y Fz(d)1534 1552 y Fp(j)1584
1518 y Fr(\026)1569 1540 y Fz(b)1605 1552 y Fp(j)1725
1540 y Fz(e)1764 1510 y Fp(i)p Fo($)p Fp(\022)1845 1518
y Fh(1)1877 1510 y Fy(\000)p Fp(\022)1961 1518 y Fh(2)1993
1510 y Fo($)2023 1540 y Fz(a)2067 1552 y Fp(j)2117 1540
y Fr(\026)-57 b Fz(c)2138 1552 y Fp(j)2192 1540 y Fr(+)18
b FA(j)p Fz(d)2341 1552 y Fp(j)2376 1540 y FA(j)2399
1510 y Fo(2)2478 1370 y Fm(\023)2562 1487 y Fr(=)2650
1370 y Fm(\022)2752 1436 y Fr(0)83 b(0)2752 1536 y(0)g(0)2960
1370 y Fm(\023)3035 1487 y Fz(:)280 b Fr($9$)456 1712
y(Therefore,)27 b(b)n(y)h(comparing)f($1,1$)h(en)n(tries)f(of)i(eac)n
(h)e(matrices,)h(it)g(follo)n(ws)g(that)g(if)h Fz(\013)g
Fr(is)f(satu-)456 1812 y(rated,)e(then)h(there)f(exist)g(p)r(ositiv)n
(e)g(real)g(n)n(um)n(b)r(ers)f Fz(a)14 b Fr($=)2267
1750 y Fm(P)2355 1837 y Fp(j)2404 1812 y FA(j)p Fz(a)2471
1824 y Fp(j)2506 1812 y FA(j)2529 1782 y Fo(2)2566 1812
y Fr($)23 b Fz(>)g Fr(0,)j Fz(b)14 b Fr($=)2969 1750
y Fm(P)3057 1837 y Fp(j)3106 1812 y FA(j)p Fz(b)3165
1824 y Fp(j)3200 1812 y FA(j)3223 1782 y Fo(2)3260 1812
y Fr($)23 b Fz(>)g Fr(0)456 1914 y(suc)n(h)k(that)1265
2051 y
SDict begin H.S end
 1265 2051 a 1265 2051 a
SDict begin 12 H.A end
 1265 2051 a 1265 2051
a
SDict begin [ /View [/XYZ H.V] /Dest (equation.4.10) cvn /DEST pdfmark
end
 1265 2051 a Fz(a)18 b Fr(+)g Fz(b)23 b Fr(=)f(1)55
b(and)28 b Fz(a)18 b Fr(+)g Fz(e)1999 2017 y Fp(i)p Fo($)p
Fp(\022)2080 2025 y Fh(2)2112 2017 y Fy(\000)p Fp(\022)2196
2025 y Fh(1)2228 2017 y Fo($)2258 2051 y Fz(b)23 b Fr(=)g(0)p
Fz(:)827 b Fr($10$)456 2192 y(Note)31 b(that)i Fz(b)c
FA(6)p Fr(=)h(0)h(since)h Fz(b)e Fr(=)f(0)j(implies)g
Fz(a)e Fr(=)f(0)j(from)2248 2130 y Fm(P)2336 2217 y Fp(j)2385
2125 y Fm(\000)2423 2192 y FA(j)p Fz(a)2490 2204 y Fp(j)2525
2192 y FA(j)2548 2162 y Fo(2)2606 2192 y Fr(+)21 b Fz(e)2731
2162 y Fp(i)p Fo($)p Fp(\022)2812 2170 y Fh(2)2844 2162
y Fy(\000)p Fp(\022)2928 2170 y Fh(1)2960 2162 y Fo($)2990
2192 y Fz(d)3033 2204 y Fp(j)3083 2170 y Fr(\026)3068
2192 y Fz(b)3104 2204 y Fp(j)3139 2125 y Fm(\001)3207
2192 y Fr(=)30 b(0)h(in)456 2299 y($)488 2299 y
SDict begin H.S end
 488
2299 a Fr(9)529 2245 y
SDict begin H.R end
 529 2245 a 529 2299 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.9) cvn H.B /ANN pdfmark
end
 529 2299
a Fr($.)38 b(There)27 b(are)f(three)i(p)r(ossible)f(cases)f(for)i
Fz(\022)1900 2311 y Fo(1)1937 2299 y Fr(,)f Fz(\022)2026
2311 y Fo(2)2091 2299 y Fr(as)g(follo)n(ws.)615 2417
y($i$)42 b(If)28 b Fz(\022)866 2429 y Fo(2)922 2417
y FA(\000)18 b Fz(\022)1044 2429 y Fo(1)1104 2417 y FA(\021)23
b Fr(0$mo)r(d)28 b(2)p Fz(\031)s Fr($,)f(that)h(is,)g
Fz(\013)g Fr(is)f(trivial,)g(then)i($)2570 2417 y
SDict begin H.S end
 2570
2417 a Fr(10)2653 2364 y
SDict begin H.R end
 2653 2364 a 2653 2417 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.10) cvn H.B /ANN
pdfmark end
 2653
2417 a Fr($)f(is)f(not)h(p)r(ossible.)592 2517 y($ii$)42
b(If)28 b Fz(\022)866 2529 y Fo(2)922 2517 y FA(\000)18
b Fz(\022)1044 2529 y Fo(1)1104 2517 y FA(\021)23 b Fz(\031)s
Fr($mo)r(d)28 b(2)p Fz(\031)s Fr($,)g(then)1110 2704
y Fz(x)1157 2716 y Fo(1)1217 2704 y Fr(=)1350 2647 y(1)p
1315 2685 111 4 v 1315 2701 a FA(p)p 1384 2701 42 4 v
69 x Fr(2)1450 2587 y Fm(\022)1552 2653 y Fr(1)83 b(1)1552
2753 y(0)g(0)1760 2587 y Fm(\023)1890 2704 y Fr(and)55
b Fz(x)2126 2716 y Fo(2)2187 2704 y Fr(=)2319 2647 y(1)p
2284 2685 111 4 v 2284 2701 a FA(p)p 2354 2701 42 4 v
2354 2770 a Fr(2)2419 2587 y Fm(\022)2522 2653 y Fr(0)82
b(0)2522 2753 y(1)g(1)2729 2587 y Fm(\023)744 2891 y
Fr(satisfy)28 b($)1036 2891 y
SDict begin H.S end
 1036 2891 a Fr(8)1077
2837 y
SDict begin H.R end
 1077 2837 a 1077 2891 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.8) cvn H.B /ANN pdfmark
end
 1077 2891 a Fr($)g(with)g
Fz(m)23 b Fr(=)g(2.)36 b(Th)n(us)28 b Fz(\013)g Fr(is)f(saturated.)569
2990 y($iii$)42 b(If)26 b Fz(\022)864 3002 y Fo(2)914
2990 y FA(\000)13 b Fz(\022)1031 3002 y Fo(1)1091 2990
y FA(6)p Fr(=)23 b(0)p Fz(;)k(\031)s Fr($mo)r(d)h(2)p
Fz(\031)s Fr($,)e(then)f($)1931 2990 y
SDict begin H.S end
 1931 2990 a
Fr(10)2015 2937 y
SDict begin H.R end
 2015 2937 a 2015 2990 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (equation.4.10) cvn H.B /ANN
pdfmark end
 2015 2990 a
Fr($)g(is)g(not)g(p)r(ossible)f(for)h(an)n(y)f Fz(a)p
Fr(,)h Fz(b)e(>)g Fr(0.)35 b(Hence)744 3090 y Fz(\013)28
b Fr(is)g(not)g(saturated.)456 3241 y
SDict begin H.S end
 456 3241 a 456
3241 a
SDict begin 12 H.A end
 456 3241 a 456 3241 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.4.3) cvn /DEST pdfmark end
 456 3241 a 100 x Fl(R)l(emark)48
b Fr(4.3)p Fl(.)g Fr(Let)38 b Fz(\013)p Fr(,)k Fz(\014)j
FA(2)d Fz(Aut)p Fr($)p Fz(M)9 b Fr($)38 b(satisfy)g
Fz(\013)2083 3310 y Fp(n)2170 3341 y Fr(=)i Fz(\014)2326
3310 y Fp(n)2413 3341 y Fr(=)g Fz(id)2590 3353 y Fp(M)2702
3341 y Fr(for)e(some)g Fz(n)j FA(\025)f Fr(1.)69 b(If)456
3440 y(there)37 b(is)h(a)f(unitary)h Fz(u)h FA(2)i Fz(M)46
b Fr(suc)n(h)38 b(that)g Fz(\014)44 b Fr(=)c Fz(Ad)p
Fr($)p Fz(u)p Fr($)26 b FA(\016)38 b Fz(\013)p Fr(,)j(then)d
Fz(\013)h Fr(and)e Fz(\014)42 b Fr(are)37 b(said)h(to)456
3540 y(b)r(e)30 b Fl(exterior)i(e)l(quivalent)p Fr(,)f(and)e(if)h(this)
g(is)g(the)g(case)f(the)h(crossed)e(pro)r(ducts)i(are)f(isomorphic,)456
3639 y Fz(M)11 b FA(\002)613 3651 y Fp(\013)661 3639
y Fz(G)750 3617 y FA(\030)750 3644 y Fr(=)837 3639 y
Fz(M)g FA(\002)994 3651 y Fp(\014)1040 3639 y Fz(G)p
Fr(,)22 b([)1173 3639 y
SDict begin H.S end
 1173 3639 a Fr(14)1256 3586 y
SDict begin H.R end

1256 3586 a 1256 3639 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Ph1) cvn H.B /ANN pdfmark
end
 1256 3639 a Fr(,)f(p.45].)34
b(Example)1869 3639 y
SDict begin H.S end
 1869 3639 a Fr(4.2)1975 3586 y
SDict begin H.R end

1975 3586 a 1975 3639 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (thm.4.2) cvn H.B /ANN pdfmark
end
 1975 3639 a 20 w Fr(sa)n(ys)18
b(that)h(the)h(prop)r(ert)n(y)f(of)g(b)r(eing)g(saturated)456
3739 y(ma)n(y)k(not)g(b)r(e)i(preserv)n(ed)d(under)h(exterior)g(equiv)
-5 b(alence.)35 b(Also)23 b(the)h(case)f($iii$)i(of)e(Example)3338
3739 y
SDict begin H.S end
 3338 3739 a Fr(4.2)3444 3686 y
SDict begin H.R end
 3444 3686 a 3444
3739 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (thm.4.2) cvn H.B /ANN pdfmark
end
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Fz(w)i Fr(=)d Fz(diag)s Fr($)p Fz(\025;)1325 3825 y
Fr(\026)1322 3847 y Fz(\025)p Fr($,)j Fz(\025)d Fr(=)g
Fz(e)1659 3795 y Fh(2)p Fj(\031)p 1658 3804 65 3 v 1676
3837 a Fh(3)1761 3847 y Fr($hence)i Fz(\022)2060 3859
y Fo(1)2109 3847 y FA(\000)12 b Fz(\022)2225 3859 y Fo(2)2285
3847 y Fr(=)2383 3815 y Fo(2)p Fp(\031)p 2383 3829 75
4 v 2403 3876 a Fo(3)2478 3847 y FA(\000)2565 3815 y
Fo(4)p Fp(\031)p 2565 3829 V 2585 3876 a Fo(3)2672 3847
y FA(6)p Fr(=)23 b Fz(\031)s Fr(\(mo)r(d)28 b(2)p Fz(\031)s
Fr($\),)e(sho)n(ws)456 3947 y(that)35 b Fz(I)7 b(ndex)p
Fr($)p Fz(E)e Fr($)36 b Fz(<)f FA(j)p Fz(G)p FA(j)h
Fr(is)f(p)r(ossible)g(ev)n(en)f(when)h Fz(E)41 b Fr(is)35
b(of)g(index-\014nite)g(t)n(yp)r(e.)60 b(In)35 b(fact,)i(if)456
4096 y Fz(u)504 4108 y Fo(1)564 4096 y Fr(=)653 3979
y Fm(\022)755 4046 y Fr(1)83 b(0)755 4145 y(0)g(1)963
3979 y Fm(\023)1052 4096 y Fr(and)28 b Fz(u)1262 4108
y Fo(2)1323 4096 y Fr(=)1412 3979 y Fm(\022)1514 4046
y Fr(0)83 b(1)1514 4145 y(1)g(0)1722 3979 y Fm(\023)1783
4096 y Fr(,)28 b(then)h FA(f)p Fr($)p Fz(u)2146 4108
y Fp(i)2173 4096 y Fz(;)14 b(u)2258 4066 y Fy(\003)2258
4118 y Fp(i)2295 4096 y Fr($)p FA(g)2369 4066 y Fo(2)2369
4118 y Fp(i)p Fo(=1)2509 4096 y Fr(forms)28 b(a)f(quasi-basis)g(for)g
Fz(E)5 b Fr(,)456 4246 y(but)28 b Fz(I)7 b(ndex)p Fr($)p
Fz(E)e Fr($)24 b(=)e(2)h Fz(<)g FA(j)p Fn(Z)1308 4258
y Fo(3)1339 4246 y FA(j)p Fr(.)456 4418 y
SDict begin H.S end
 456 4418 a
456 4418 a
SDict begin 12 H.A end
 456 4418 a 456 4418 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.4.4) cvn /DEST pdfmark end
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b Fr(4.4)p Fl(.)g Fr(Recall)35 b(that)h(the)h(Rokhlin)e(prop)r(ert)n(y)
g(and)h(the)g(tracial)f(Rokhlin)h(prop)r(ert)n(y)456
4596 y($w)n(eak)n(er)17 b(than)j(the)g(Rokhlin)f(prop)r(ert)n(y$)g
(are)f(de\014ned)i(as)f(follo)n(ws)f(and)i(considered)e(in)n(tensiv)n
(ely)456 4696 y(in)27 b([)575 4696 y
SDict begin H.S end
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4642 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Iz) cvn H.B /ANN pdfmark
end
 617 4696 a Fr(])h(and)f([)852
4696 y
SDict begin H.S end
 852 4696 a Fr(12)935 4642 y
SDict begin H.R end
 935 4642 a 935 4696
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.OP) cvn H.B /ANN pdfmark
end
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b($[)799 4814 y
SDict begin H.S end
 799 4814 a Fr(7)841 4761 y
SDict begin H.R end
 841 4761
a 841 4814 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.Iz) cvn H.B /ANN pdfmark
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(v)n(e)f(the)i Fl(R)l(okhlin)h(pr)l(op)l(erty)40 b Fr(if)32
b(for)f(ev)n(ery)f(\014nite)i(set)f Fz(F)41 b FA(\032)29
b Fz(M)9 b Fr(,)744 4914 y(ev)n(ery)29 b Fz(")d(>)h Fr(0,)i(there)h
(are)f(m)n(utually)h(orthogonal)d(pro)5 b(jections)29
b FA(f)p Fz(e)2854 4926 y Fp(g)2918 4914 y FA(j)e Fz(g)i
FA(2)e Fz(G)p FA(g)j Fr(in)g Fz(M)744 5013 y Fr(suc)n(h)e(that)790
5113 y($i$)43 b FA(k)p Fz(\013)1015 5125 y Fp(g)1053
5113 y Fr($)p Fz(e)1124 5125 y Fp(h)1167 5113 y Fr($)19
b FA(\000)f Fz(e)1340 5125 y Fp(g)r(h)1417 5113 y FA(k)k
Fz(<)h(")k Fr(for)g Fz(g)s(;)14 b(h)23 b FA(2)g Fz(G:)767
5213 y Fr($ii$)43 b FA(k)p Fz(e)1001 5225 y Fp(g)1038
5213 y Fz(x)19 b FA(\000)f Fz(xe)1273 5225 y Fp(g)1312
5213 y FA(k)23 b Fz(<)f(")28 b Fr(for)f Fz(g)e FA(2)f
Fz(G)k Fr(and)f(all)g Fz(x)d FA(2)f Fz(F)12 b Fr(.)744
5312 y($iii$)920 5250 y Fm(P)1007 5337 y Fp(g)r Fy(2)p
Fp(G)1156 5312 y Fz(e)1195 5324 y Fp(g)1256 5312 y Fr(=)23
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SDict begin H.R end
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/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.OP) cvn H.B /ANN pdfmark
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 883 450 a Fr(]$)34 b Fz(\013)f Fr(is)h(said)f(to)g(ha)n(v)n(e)f
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y($i$)43 b FA(k)p Fz(\013)1015 861 y Fp(g)1053 849
y Fr($)p Fz(e)1124 861 y Fp(h)1167 849 y Fr($)19 b
FA(\000)f Fz(e)1340 861 y Fp(g)r(h)1417 849 y FA(k)k
Fz(<)h(")k Fr(for)g Fz(g)s(;)14 b(h)23 b FA(2)g Fz(G)p
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1048 y($iii$)43 b(With)f Fz(e)j Fr(=)1343 986 y Fm(P)1430
1073 y Fp(g)r Fy(2)p Fp(G)1579 1048 y Fz(e)1618 1060
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y(The)j(follo)n(wing)g(prop)r(osition)f(is)i(actually)f(observ)n(ed)f
(in)i([)2296 1500 y
SDict begin H.S end
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SDict begin H.R end
 2379
1446 a 2379 1500 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
/BS <</S/S/W 1>> /Subtype /Link /Dest (cite.OP) cvn H.B /ANN pdfmark
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 2379 1500 a Fr(,)g(Lemma)g(1.13],)f(and)g(w)n(e)h
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1715 y
SDict begin H.S end
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SDict begin 12 H.A end
 456 1715 a 456 1715 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.4.5) cvn /DEST pdfmark end
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1715 a 83 x FB(Prop)s(osition)32 b(4.5.)42 b Fl(L)l(et)31
b Fz(M)41 b Fl(b)l(e)32 b(a)g(unital)g Fz(C)1916 1768
y Fy(\003)1954 1798 y Fl(-algebr)l(a)h(and)g Fz(\013)f
Fl(b)l(e)g(an)g(action)h(of)f(a)h(discr)l(ete)456 1898
y(gr)l(oup)c Fz(G)h Fl(on)g Fz(M)9 b Fl(.)38 b(Supp)l(ose)30
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2010 y Fp(g)r Fy(2)p Fp(G)1815 1998 y Fl(such)f(that)456
2040 y
SDict begin H.S end
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SDict begin 12 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (Item.5) cvn /DEST pdfmark end
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2040 a 597 2116 a Fr($1$)41 b FA(k)p Fz(\013)839 2128
y Fp(g)878 2116 y Fr($)p Fz(e)949 2128 y Fp(h)992 2116
y Fr($)18 b FA(\000)g Fz(e)1164 2128 y Fp(g)r(h)1241
2116 y FA(k)23 b Fz(<)g(":)456 2139 y
SDict begin H.S end
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2139 a
SDict begin 12 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (Item.6) cvn /DEST pdfmark end
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2483 y(Pr)l(o)l(of.)43 b Fr(Supp)r(ose)28 b(there)f(is)h(a)f(unitary)g
Fz(u)c FA(2)h Fz(M)36 b Fr(suc)n(h)28 b(that)g Fz(\013)2388
2495 y Fp(g)2427 2483 y Fr($)p Fz(x)p Fr($)c(=)f Fz(uxu)2793
2453 y Fy(\003)2859 2483 y Fr(for)k(ev)n(ery)f Fz(x)e
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Fz(u)p FA(g)g Fr(and)h(0)g Fz(<)f(")h(<)g Fr(1)p Fz(=)p
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2683 y(pro)5 b(jections)26 b FA(f)p Fz(e)964 2695 y Fp(g)1002
2683 y FA(g)1044 2695 y Fp(g)r Fy(2)p Fp(G)1207 2683
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2683 y Fr($)p Fz(e)1779 2695 y Fp(h)1822 2683 y Fr($)18
b FA(\000)h Fz(e)1995 2695 y Fp(g)r(h)2072 2683 y FA(k)k
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FA(k)p Fz(e)2852 2695 y Fp(g)2890 2683 y Fz(u)18 b FA(\000)h
Fz(ue)3127 2695 y Fp(g)3164 2683 y FA(k)k Fz(<)h(")f(<)456
2782 y Fr(1)p Fz(=)p Fr(4.)35 b(Then)28 b FA(k)p Fz(\013)952
2794 y Fp(g)990 2782 y Fr($)p Fz(e)1061 2794 y Fp(\023)1090
2782 y Fr($)19 b FA(\000)f Fz(ue)1311 2794 y Fp(\023)1338
2782 y Fz(u)1386 2752 y Fy(\003)1424 2782 y FA(k)23 b
Fr(=)f(0.)37 b(But)834 2918 y FA(k)p Fz(\013)929 2930
y Fp(g)968 2918 y Fr($)p Fz(e)1039 2930 y Fp(\023)1067
2918 y Fr($)19 b FA(\000)f Fz(ue)1288 2930 y Fp(\023)1316
2918 y Fz(u)1364 2884 y Fy(\003)1402 2918 y FA(k)k Fr(=)h
FA(k)p Fz(\013)1649 2930 y Fp(g)1687 2918 y Fr($)p Fz(e)1758
2930 y Fp(\023)1787 2918 y Fr($)c FA(\000)f Fz(e)1960
2930 y Fp(g)2016 2918 y Fr(+)g Fz(e)2138 2930 y Fp(g)2195
2918 y FA(\000)g Fz(e)2317 2930 y Fp(\023)2364 2918 y
Fr(+)g Fz(e)2486 2930 y Fp(\023)2533 2918 y FA(\000)g
Fz(ue)2703 2930 y Fp(\023)2731 2918 y Fz(u)2779 2884
y Fy(\003)2816 2918 y FA(k)1466 3048 y(\025)23 b(k)p
Fz(e)1635 3060 y Fp(g)1691 3048 y FA(\000)18 b Fz(e)1813
3060 y Fp(\023)1842 3048 y FA(k)g(\000)g(k)p Fz(\013)2080
3060 y Fp(g)2118 3048 y Fr($)p Fz(e)2189 3060 y Fp(\023)2218
3048 y Fr($)g FA(\000)g Fz(e)2390 3060 y Fp(g)2429 3048
y FA(k)g(\000)g(k)p Fz(e)2653 3060 y Fp(\023)2699 3048
y FA(\000)g Fz(ue)2869 3060 y Fp(\023)2897 3048 y Fr(1)p
Fz(u)2987 3013 y Fy(\003)3024 3048 y FA(k)1466 3225 y(\025)23
b Fr(1)18 b FA(\000)1707 3168 y Fr(1)p 1707 3206 42 4
v 1707 3282 a(4)1777 3225 y FA(\000)1870 3168 y Fr(1)p
1870 3206 V 1870 3282 a(4)1944 3225 y(=)2042 3168 y(1)p
2042 3206 V 2042 3282 a(2)2093 3225 y Fz(;)456 3384 y
Fr(whic)n(h)27 b(is)h(a)f(con)n(tradiction.)2032 b Fi(\003)456
3635 y
SDict begin H.S end
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SDict begin 12 H.A end
 456 3635 a 456 3635 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.4.6) cvn /DEST pdfmark end
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3635 a 100 x Fl(R)l(emark)35 b Fr(4.6)p Fl(.)j Fr(If)24
b Fz(M)c FA(\002)1184 3747 y Fp(\013)1241 3735 y Fz(G)25
b Fr(is)e(simple,)i Fz(\013)g Fr(is)f(ob)n(viously)e(saturated,)i(and)f
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FA(g)j Fr(for)g(all)h Fz(g)f FA(6)p Fr(=)d Fz(\023)k
Fr($[)2813 3836 y
SDict begin H.S end
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SDict begin H.R end
 2855
3783 a 2855 3836 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 1]BorderArrayPatch
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3936 y(In)f(particular,)g Fz(\013)h Fr(is)f(saturated)g(if)h
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Fz(M)34 b Fr(and)25 b Fz(u)g Fr(is)g(a)g(unitary)g(in)h
Fz(M)34 b Fr(with)25 b Fz(u)1799 4105 y Fp(n)1867 4135
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4235 a Fr(4.2)1513 4181 y
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b Fz(M)37 b Fr(is)29 b(outer)g(but)456 4334 y(not)e(saturated.)456
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y Fr(-algebra,)e(w)n(e)i(write)456 4883 y Fz(p)e FA(?)h
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b Fr($)33 b(for)f(some)g Fz(n)p Fr(,)i(with)f Fz(p)e
FA(?)g Fz(r)r Fr(,)j Fz(q)h FA(?)30 b Fz(r)r Fr(,)35
b(and)e($)p Fz(p)21 b Fr(+)h Fz(r)r Fr($)32 b FA(\030)f
Fr($)p Fz(q)25 b Fr(+)c Fz(r)r Fr($,)456 5282 y(then)28
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SDict begin H.S end
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a 456 5448 a
SDict begin 12 H.A end
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251 a
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SDict begin H.R end
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b(or)34 b(eac)n(h)456 1976 y Fz(g)25 b FA(2)f Fz(G)p
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1976 y Fr(suc)n(h)h(that)1507 2141 y FA(k)p Fz(\013)1602
2153 y Fp(k)1642 2141 y Fr($)p Fz(e)1713 2101 y Fp(g)1713
2166 y(h)1756 2141 y Fr($)19 b FA(\000)f Fz(e)1929 2101
y Fp(g)1929 2166 y(k)q(h)2008 2141 y FA(k)23 b Fz(<)2246
2084 y(")p 2170 2121 191 4 v 2170 2198 a Fr(2)p FA(j)p
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2271 y Fm(P)1084 2358 y Fp(h)p Fy(2)p Fp(G)1237 2333
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Fz(g)s Fr(,)h(then)g Fz(b)2434 2303 y Fo(1)2434 2357
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y Fp(h)p Fy(2)p Fp(G)2896 2533 y Fr(in)i Fz(M)35 b Fr(suc)n(h)25
b(that)1514 2697 y FA(k)p Fz(\013)1609 2709 y Fp(k)1649
2697 y Fr($)p Fz(f)1731 2657 y Fp(g)1722 2722 y(h)1770
2697 y Fr($)19 b FA(\000)f Fz(f)1954 2657 y Fp(g)1945
2722 y(k)q(h)2024 2697 y FA(k)k Fz(<)2262 2641 y(")p
2186 2678 191 4 v 2186 2754 a Fr(2)p FA(j)p Fz(G)p FA(j)2339
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b FA(\030)e Fr($)p Fz(e)2122 2950 y Fp(g)2161 2984 y
Fr($)2193 2950 y Fy(0)2239 2984 y Fz(<)h(e)2366 2950
y Fp(g)456 3192 y Fr(for)f(a)h(subpro)5 b(jection)23
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3295 y(follo)n(ws)e(that)i Fz(f)948 3265 y Fp(g)1009
3295 y FA(\030)1097 3228 y Fm(\000)1135 3295 y Fr(1)9
b FA(\000)g Fr($)p Fz(e)1331 3265 y Fp(g)1368 3295 y
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3700 y Fp(k)2391 3755 y(h)2440 3734 y Fz(\013)2493 3750
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3642 y Fm(\021)2837 3734 y Fz(;)42 b(g)26 b FA(2)d Fz(G:)456
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SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn /DEST pdfmark end
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a 342 w Fo(SA)-5 b(TURA)g(TED)29 b(A)n(CTIONS)h(BY)f(FINITE)g
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644 y Fp(k)2351 632 y Fz(f)2401 598 y Fp(k)2441 632 y
Fz(v)2484 598 y Fy(\003)2481 652 y Fp(k)2523 632 y Fr(\))1297
883 y(=)1429 827 y(1)p 1394 864 V 1394 940 a FA(j)p Fz(G)p
FA(j)1516 816 y Fm(\000)1568 804 y(X)1609 983 y Fp(k)1701
883 y Fz(e)1740 849 y Fp(k)1799 883 y Fr(+)1882 804 y
Fm(X)1924 983 y Fp(k)2002 883 y Fr($1)h FA(\000)f Fz(e)2217
849 y Fp(k)2257 883 y Fr($$1)h FA(\000)f Fr(\()p Fz(e)2536
849 y Fp(k)2576 883 y Fr($)2608 849 y Fy(0)2632 883
y Fr(\))2664 816 y Fm(\001)1297 1081 y Fr(=)k(1)p Fz(:)3380
1245 y Fi(\003)555 1535 y Fr(F)-7 b(or)30 b(the)g(rest)g(of)g(this)h
(section)f(w)n(e)g(consider)f(a)h(\014nite)h(group)e(action)g(on)h(a)g
(comm)n(utativ)n(e)456 1635 y Fz(C)521 1605 y Fy(\003)559
1635 y Fr(-algebra)23 b Fz(C)6 b Fr($)p Fz(X)h Fr($.)37
b(If)26 b Fz(G)g Fr(acts)f(on)g(a)g(compact)g(Hausdor\013)g(space)g
Fz(X)7 b Fr(,)25 b(it)h(induces)g(an)f(action,)456 1734
y(sa)n(y)h Fz(\013)p Fr(,)i(on)g Fz(C)6 b Fr($)p Fz(X)h
Fr($)27 b(b)n(y)1367 1864 y Fz(\013)1420 1876 y Fp(g)1459
1864 y Fr($)p Fz(f)9 b Fr($$)p Fz(x)p Fr($)24 b(=)f
Fz(f)9 b Fr($)p Fz(g)1921 1830 y Fy(\000)p Fo(1)2009
1864 y Fz(x)p Fr($)p Fz(;)43 b(f)31 b FA(2)24 b Fz(C)6
b Fr($)p Fz(X)h Fr($)p Fz(:)456 2010 y Fr(F)-7 b(or)30
b(eac)n(h)g Fz(x)e FA(2)h Fz(X)7 b Fr(,)31 b(let)g Fz(G)1275
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(group)g(of)i Fz(x)g Fr(and)g(for)f(a)456 2110 y(subgroup)c
Fz(H)34 b Fr(of)28 b Fz(G)g Fr($)p Fz(H)i(<)23 b(G)p
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2273 y Fr(=)23 b FA(f)p Fz(x)g FA(2)g Fz(X)30 b Fr(:)23
b Fz(G)2125 2285 y Fp(x)2190 2273 y Fr(=)f Fz(H)7 b FA(g)p
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y Fr(,)27 b(and)f Fz(X)33 b Fr(is)26 b(partitioned)g(as)1691
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y
SDict begin H.S end
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SDict begin 12 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (thm.4.10) cvn /DEST pdfmark end
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a 66 x FB(Theorem)h(4.10.)41 b Fl(L)l(et)30 b Fz(X)37
b Fl(b)l(e)31 b(a)g(c)l(omp)l(act)g(Hausdor\013)h(sp)l(ac)l(e)f(and)g
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Fz(X)7 b Fl(.)38 b(If)31 b Fz(\013)f Fl(is)g(the)g(induc)l(e)l(d)g
(action)g(of)h Fz(G)f Fl(on)g Fz(C)6 b Fr($)p Fz(X)h
Fr($)p Fl(,)30 b(the)g(fol)t(lowing)i(ar)l(e)e(e)l(quivalent:)615
3184 y Fr($i$)42 b Fz(E)28 b Fr(:)c Fz(C)6 b Fr($)p
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3154 y Fp(\013)1496 3184 y Fl(is)30 b(of)h(index)f(\014nite)f(typ)l(e.)
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y Fl(is)30 b(close)l(d)h(for)f(e)l(ach)h Fz(H)f(<)23
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a(j)p Fp(H)t Fy(j)2832 3428 y Fz(\037)2884 3440 y Fp(X)2938
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Fz(f)k Fr($)23 b(=)g Fz(f)t(;)456 4162 y Fr(that)k(is,)1136
4325 y
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y Fy(\003)1844 4345 y Fp(i)1882 4325 y Fr($)p Fz(g)1957
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b Fr($)p Fz(g)2250 4291 y Fy(\000)p Fo(1)2339 4325 y
Fz(x)p Fr($)2418 4233 y Fm(\021)2492 4325 y Fr(=)22
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4560 y(for)23 b Fz(f)31 b FA(2)24 b Fz(C)6 b Fr($)p
Fz(X)h Fr($)23 b(and)h Fz(x)f FA(2)h Fz(X)7 b Fr(.)35
b(F)-7 b(or)23 b(eac)n(h)f Fz(x)i FA(2)f Fz(X)7 b Fr(,)24
b(c)n(ho)r(ose)e(a)i(con)n(tin)n(uous)e(function)i Fz(f)3096
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4660 y FA(j)931 4675 y Fp(Gx)p Fy(nf)p Fp(x)p Fy(g)1187
4660 y FA(\021)c Fr(0)k(and)h Fz(f)1547 4672 y Fp(x)1588
4660 y Fr($)p Fz(x)p Fr($)c(=)f(1.)36 b(Then)28 b($)2161
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5296 y Fp(i)2103 5275 y Fr($)p Fz(x)p Fr($)d(=)f(1)p
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SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn /DEST pdfmark end
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Fz(s)p Fr($)p Fz(ds)g Fr(for)f Fz(t)e FA(2)g Fz(G)p
Fr(.)37 b(With)26 b(notations)f Fz(x)1786 2230 y Fy(0)1832
2260 y Fr(=)1920 2193 y Fm(R)1989 2260 y Fz(x)p Fr($)p
Fz(s)p Fr($)p Fz(ds)i Fr(and)e Fz(x)2454 2230 y Fy(00)2520
2260 y Fr(:=)2630 2193 y Fm(R)2700 2260 y Fz(\013)2753
2272 y Fp(s)2788 2260 y Fr($)p Fz(x)p Fr(\()p Fz(s)2938
2230 y Fy(\000)p Fo(1)3028 2260 y Fr($\))p Fz(ds)14
b Fr($)p FA(2)24 b Fz(M)9 b Fr($,)456 2360 y(w)n(e)27
b(see)g(that)1213 2543 y($)p Fz(x)19 b FA(\003)f Fz(e)g
FA(\003)g Fz(y)s Fr($$)p Fz(t)p Fr($)24 b(=)1770 2430
y Fm(Z)1853 2543 y Fr($)p Fz(x)19 b FA(\003)f Fz(e)p
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2543 y Fr($)p Fz(y)s Fr(\()p Fz(s)2421 2509 y Fy(\000)p
Fo(1)2510 2543 y Fz(t)p Fr($\))p Fz(ds)1682 2761 y Fr(=)23
b Fz(x)1817 2726 y Fy(0)1854 2648 y Fm(Z)1951 2761 y
Fz(\013)2004 2773 y Fp(s)2040 2761 y Fr($)p Fz(y)s Fr(\()p
Fz(s)2187 2726 y Fy(\000)p Fo(1)2276 2761 y Fz(t)p Fr($\))p
Fz(ds)1682 2978 y Fr(=)g Fz(x)1817 2944 y Fy(0)1840 2978
y Fz(\013)1893 2990 y Fp(t)1923 2978 y Fr($)1955 2865
y Fm(Z)2052 2978 y Fz(\013)2105 2990 y Fp(s)2140 2978
y Fr(\()p Fz(y)s Fr(\()p Fz(s)2287 2944 y Fy(\000)p Fo(1)2377
2978 y Fr($\))p Fz(ds)p Fr(\))1682 3150 y(=)g Fz(x)1817
3115 y Fy(0)1840 3150 y Fz(\013)1893 3162 y Fp(t)1923
3150 y Fr($)p Fz(y)1999 3115 y Fy(00)2041 3150 y Fr($)1682
3279 y(=)g Fz(f)1811 3291 y Fp(x)1849 3275 y Fk(0)1870
3291 y Fp(;y)1926 3275 y Fk(00)1971 3279 y Fr($)p Fz(t)p
Fr($)456 3445 y(b)r(elongs)k(to)g FA(J)911 3457 y Fp(\013)986
3445 y Fr(for)g Fz(x;)14 b(y)26 b FA(2)e Fz(C)6 b Fr($)p
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r(olarization)g(iden)n(tit)n(y)i(pro)n(v)n(es)e(the)i(last)f
(assertion.)1217 3781 y Fz(a\013)1314 3793 y Fp(t)1343
3781 y Fr($)p Fz(b)p Fr($)c(=)1564 3725 y(1)p 1564
3762 42 4 v 1564 3838 a(4)1673 3677 y Fo(3)1630 3702
y Fm(X)1629 3881 y Fp(k)q Fo(=0)1764 3781 y Fz(i)1793
3746 y Fp(k)1833 3781 y Fr($)p Fz(b)c Fr(+)f Fz(i)2032
3746 y Fp(k)2072 3781 y Fz(a)2116 3746 y Fy(\003)2154
3781 y Fr($)2186 3746 y Fy(\003)2225 3781 y Fz(\013)2278
3793 y Fp(t)2307 3781 y Fr($)p Fz(b)g Fr(+)g Fz(i)2505
3746 y Fp(k)2546 3781 y Fz(a)2590 3746 y Fy(\003)2628
3781 y Fr($)p Fz(:)3380 4015 y Fi(\003)456 4203 y
SDict begin H.S end
 456
4203 a 456 4203 a
SDict begin 12 H.A end
 456 4203 a 456 4203 a
SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn /DEST pdfmark
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 456 4203 a 976
4302 a Fr(6.)41 b Fq(The)32 b(ga)n(uge)g(a)n(ction)g
Fz(\015)k Fq(on)31 b(a)g(graph)h Fz(C)2512 4272 y Fy(\003)2550
4302 y Fq(-algebra)555 4452 y Fr(By)g(a)g($directed$)g(graph)f
Fz(E)37 b Fr(w)n(e)32 b(mean)g(a)f(quadruple)h Fz(E)j
Fr(=)30 b($)p Fz(E)2599 4422 y Fo(0)2637 4452 y Fz(;)14
b(E)2740 4422 y Fo(1)2777 4452 y Fz(;)g(r)n(;)g(s)p Fr($)32
b(consisting)g(of)456 4551 y(the)d(v)n(ertex)f(set)i
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Fz(E)1673 4521 y Fo(1)1711 4551 y Fr(,)g(and)h(the)f(range,)f(source)h
(maps)f Fz(r)n(;)14 b(s)26 b Fr(:)g Fz(E)3049 4521 y
Fo(1)3112 4551 y FA(!)g Fz(E)3287 4521 y Fo(0)3324 4551
y Fr(.)42 b(If)456 4651 y(eac)n(h)31 b(v)n(ertex)h(of)h
Fz(E)k Fr(emits)c(only)f(\014nitely)h(man)n(y)g(edges)e
Fz(E)38 b Fr(is)33 b(called)f Fl(r)l(ow)j(\014nite)j
Fr(and)33 b(a)f(ro)n(w)456 4751 y(\014nite)e(graph)e
Fz(E)35 b Fr(is)29 b Fl(lo)l(c)l(al)t(ly)k(\014nite)i
Fr(if)30 b(eac)n(h)f(v)n(ertex)f(receiv)n(es)g(only)h(\014nitely)h(man)
n(y)f(edges.)42 b(By)456 4850 y Fz(E)522 4820 y Fp(n)590
4850 y Fr(w)n(e)23 b(denote)h(the)g(set)f(of)h(all)f(\014nite)h(paths)f
Fz(\013)h Fr(=)e Fz(e)2071 4862 y Fo(1)2122 4850 y FA(\001)14
b(\001)g(\001)g Fz(e)2272 4862 y Fp(n)2340 4850 y Fr($)p
Fz(r)r Fr(\()p Fz(e)2482 4862 y Fp(i)2511 4850 y Fr($)23
b(=)g Fz(s)p Fr($)p Fz(e)2764 4862 y Fp(i)p Fo(+1)2875
4850 y Fr($)p Fz(;)14 b Fr(1)23 b FA(\024)g Fz(i)f FA(\024)h
Fz(n)10 b FA(\000)g Fr(1\))456 4950 y(of)31 b Fl(length)38
b Fz(n)31 b Fr($)p FA(j)p Fz(\013)p FA(j)g Fr(=)e Fz(n)p
Fr($.)48 b(Eac)n(h)30 b(v)n(ertex)h(is)g(regarded)e(as)i(a)g(\014nite)
h(path)g(of)f(length)g(0.)48 b(Then)456 5050 y Fz(E)522
5019 y Fy(\003)585 5050 y Fr(=)26 b FA([)731 5062 y Fp(n)p
Fy(\025)p Fo(0)861 5050 y Fz(E)927 5019 y Fp(n)1002 5050
y Fr(is)j(the)g(set)g(of)g(all)g(\014nite)h(paths)f(and)g(the)g(maps)g
Fz(r)j Fr(and)d Fz(s)g Fr(naturally)f(extend)456 5149
y(to)f Fz(E)623 5119 y Fy(\003)661 5149 y Fr(.)37 b(A)28
b(v)n(ertex)f Fz(v)k Fr(is)c(called)g(a)h Fl(sink)g Fr(if)g
Fz(s)1809 5119 y Fy(\000)p Fo(1)1898 5149 y Fr($)p Fz(v)s
Fr($)c(=)e FA(;)27 b Fr(and)h(a)f Fl(sour)l(c)l(e)34
b Fr(if)28 b Fz(r)2789 5119 y Fy(\000)p Fo(1)2879 5149
y Fr($)p Fz(v)s Fr($)c(=)e FA(;)p Fr(.)555 5249 y(If)h
Fz(E)28 b Fr(is)22 b(a)g(ro)n(w)f(\014nite)i(graph,)f(w)n(e)g(call)g(a)
g(family)h FA(f)p Fz(s)2137 5261 y Fp(e)2172 5249 y Fz(;)14
b(p)2251 5261 y Fp(v)2313 5249 y FA(j)23 b Fz(e)g FA(2)g
Fz(E)2565 5219 y Fo(1)2602 5249 y Fz(;)14 b(v)27 b FA(2)c
Fz(E)2850 5219 y Fo(0)2887 5249 y FA(g)f Fr(of)h(op)r(erators)d(a)456
5348 y Fl(Cuntz-Krie)l(ger$CK$)26 b Fz(E)5 b Fl(-family)25
b Fr(if)g FA(f)p Fz(s)1681 5360 y Fp(e)1716 5348 y FA(g)1758
5360 y Fp(e)1817 5348 y Fr(are)e(partial)g(isometries)g(and)h
FA(f)p Fz(p)2843 5360 y Fp(v)2882 5348 y FA(g)2924 5360
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573 y Fy(\003)2254 628 y Fp(e)2348 607 y Fr(if)28 b Fz(s)2463
573 y Fy(\000)p Fo(1)2552 607 y Fr($)p Fz(v)s Fr($)c
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y Fy(\003)1994 849 y Fr(-algebra)e Fz(C)2374 819 y Fy(\003)2412
849 y Fr($)p Fz(E)5 b Fr($)25 b(generated)e(b)n(y)g(a)h
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961 y Fp(v)1249 949 y FA(j)40 b Fz(e)f FA(2)h Fz(E)1551
919 y Fo(1)1589 949 y Fz(;)14 b(v)42 b FA(2)e Fz(E)1869
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1049 y Fr($)p Fz(s)630 1061 y Fp(e)666 1049 y Fz(;)14
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b(the)h(de\014nition)g(and)f(basic)h(prop)r(erties)e(of)i(graph)f
Fz(C)2760 1018 y Fy(\003)2798 1049 y Fr(-algebras,)g(see,)i(for)456
1148 y(example,)30 b([)830 1148 y
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1148 a Fr(,)1021 1148 y
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1241 1148 a Fr(,)1294 1148 y
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1271 y Fk(j)p Fj(\013)p Fk(j)2620 1248 y Fr($)p Fz(s)2691
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1557 y Fr(-algebras.)456 1673 y
SDict begin H.S end
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SDict begin 12 H.A end

456 1673 a 456 1673 a
SDict begin [ /View [/XYZ H.V] /Dest (thm.6.1) cvn /DEST pdfmark end
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Fy(\003)1807 2367 y Fp(\014)1874 2343 y FA(j)g Fz(\013;)14
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2699 y Fz(s)2338 2665 y Fy(\003)2338 2720 y Fp(\027)2379
2699 y Fz(:)744 2844 y Fr(Th)n(us)k(for)e(eac)n(h)h Fz(n)d
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1141 2944 V 1141 2990 a Fz(span)o FA(f)p Fz(s)1396 3002
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%%Trailer

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%%EOF