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Booss/Bleccker: Topology and Analysis Charlap: BieberbachGroups and Flat Manifolds Chern: Complex Manifolds Without PotentialTheory Chorin/Marsden: A MathematicalIntroductionto Fluid Mechanics Cohn: A ClassicalInvitation to Algebraic Numbersand Class Fields Curtis: Matrix GrouPs,2nd. ed. van Dalen: Logic and Structure Devlin: Fundamentalsof ContemporarySet Theory Edwards: A Formal Backgroundto MathematicsI a/b Edwards: A Formal Backgroundto Higher MathematicsII a/b Endler: Valuation Theory Frauenthal: MathematicalModeling in Epidemiology Gardiner: A First Course in Group Theory Godbillon: Dynamical Systemson Surfaces Greub: Multilinear Algebra Hermes: Introduction to Mathematical Logic Hurwitz/Kritikos: Lectures on Number Theory Kelly/Matthews: The Non-Euclidean,The Hyperbolic Plane Kostrikin: Introduction to Algebra LueckingiRubel: Complex Analysis: A FunctionalAnalysis Approach Lu: Singularity Theory and an Introduction to CatastropheTheory Marcus: Number Fields McCarthy: Introductionto Arithmetical Functions Meyer: EssentialMathematicsfor Applied Fields Moise: Introductory Problem Course in Analysis and Topology Oksendal: Stochastic Differential Equations Porter/Woods: Extensioni of Hausdorff Spaces Rees: Notes on Geometry Reisel: Elementary Theory of Metric Spaces Rey: Introduction to Robust and Quasi-Robust Statistical Methods Rickart: Natural Function Algebras Schreiber: Differential Forms Smoryriski: Self-Referenceand Modal Logic Stanisi6: The Mathematical Theory of Turbulence Stroock: An Introduction to the Theory of Large Deviations Sunder: An Invitation to von Neumann Algebras Tolle: Optimization Methods
V. S. Sunder Indian StatisticalInstitute New Delhi-l10016 India
AMS Classification:46-01
Library of Congress Cataloging in Publication Data S u n d e r .V . S . An invitation to von Neumann algebras. ( Universitext) Bibliography: p. lncludes index. l. von Neumann algebras. I. Title. 86-10058 5 1 2 '. 5 5 Q A - 1 2 6 . S 8 61 9 8 6 e 1987 by Springer-VerlagNew York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permissionfrom Springer-Verlag,175 Fifth Avenue, New York, New York 10010' U.S.A. The use of generaldescriptivenames,trade names,trademarks,etc. in this publication,even if the former are not esp€cially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Pnnted and bound by R.R. Donnelley and Sons, Harrisonburg,Virginia' Pnnred in the United Statesof America. v t ' 6 5 4 3 2 1 ISB\ tI-jE7-96356-l Springer-VerlagNew York Berlin Heidelberg Springer-VerlagBerlin HeidelbergNew York ISB\ -1-5-1G96356-l
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3fVJsUd
Preface
vl
(i)
(ii)
(iii)
(iv)
Some theorcms, though stated in f ull generality, are only proved under additional (sometimes very severe) simplifying assumptions -- typically, to the effect that some operator is bounded. Some other results suffer a sorrier fate -- they are not even graced with an apology for a proof. Arguments of a purely set-topological nature of ten receive s t e p - m o t h e r l y t r e a t m e n t ; w h e r e t h e a r g u m e n t i s p a i n l e s s ,i t h a s been included; where it is not, the reader is entreated to a c c e p t ,i n g o o d f a i t h , t h e v a l i d i t y o f t h e r e l e v a n t s t a t e m e n t . The exercises are an integral part of the book. Several 'lemmas" have been relegated to the exercises; any exercise, which is even slightly non-obvious, is furnished with "hints", which are often more in the nature of outlines of solutions. T h e e x e r c i s e s ,r a t h e r t h a n b e i n g c o m p i l e d a t e n d s o f s e c t i o n s , punctuate the text at junctures where they seem to fit in most naturally. Both exercises and unproved results are treated just like properly established theorems, in that they are unabashedly u s e d i n s u b s e q u e n tp o r t i o n s o f t h e t e x t .
The prospective reader: T h i s b o o k i s a i m e d a t t w o c l a s s e so f r e a d e r s : g r a d u a t e s t u d e n t s w i t h a reasonably f irm background in analysis, as well as mature mathematicians working in other areas of mathematics. As a matter of fact, this book grew out of a course of (twelve) lectures given by the author while visiting the Indian Statistical Institute at Calcutta in the summer of 1984. It was largely due to the positive response of that audience -- consisting entirely of members of the second category mentioned above that the author embarked on this venture. T h e r e a d e r i s a s s u m e dt o b e f a m i l i a r w i t h e l e m e n t a r y a s p e c t so f : (a) (b)
(c) (d)
measure theory -- monotone convergence,Fubini's Theorcm, a b s o l u t e c o n t i n u i t y , I P s p a c e sf o r p = 1 , 2 , * a n a l y t i c f u n c t i o n s o f o n e c o m p l e x v a r i a b l e - - s p a r s e n e s so f z e r o - s e t s ,c o n t o u r i n t e g r a t i o n , t h e o r e m s o f C a u c h y , M o r e r a , a n d Liouville; functional analysis -- the "three principles", weak and weak* topologies; Hilbert spaces and operators -- orthonormal basis, subspaces and projections, bounded operators, self-adjoint operators. ( T h e n e c e s s a r yb a c k g r o u n d m a t e r i a l f r o m H i l b e r t s p a c e t h e o r y is rapidly surveyed in Section0.1.)
In the latter part of the book, a nodding acquaintance with abstract harmonic analysis will be helpful, although it is not essential. For the reader who has been denied such a pleasure, a
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Contents
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Chaptcr
{ Crosscd-Products 4.1 Discrete crossed-Products 4.2 The modular operator for a discrete crossed-product 4.3 E x a m p l e s o f f a c t o r s 4.4 Continuous crossed-Productsand Takesaki's duality theorem 4.5 T h e s t r u c t u r e o f p r o p e r l y i n f i n i t e von Neumann algebras
Appcndir
Topological
GrouPs
l14 ll5 122 132 148 155 l6l
Notes
r64
Bibliography
r67
Index
169
66 7 tos L6'(z'b)n '1x)Dcts 46
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List of Symbols
xlv
M", a", 102
M @dG (for general G),
I(cr), 103
H @dK, ll7
.' 9 B, 87, lo5
f , rrs
f(M), r07
r(G), l3l
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a, 150
M @dG (for discreteG), l16
149
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NOIIf,NOOUINI 0 rerdeqJ
2
0. Introduction
physical) convention whereby inner products are linear in the first variable and conjugate - linear in the second(rather than the other way around). Consistent with our resolution to totally disregard nonseparable H i l b e r t s p a c e s ,w e s h a l l o n l y c o n s i d e r m e a s u r e s p a c e s i f t h e y a r e separable. Actually, we shall only consider measure spaces (X,f ,y) where ,r is a non-negative o-finite measure space, such that L2(X,tt) is separable. Subspacesof 13 will usually be denoted by symbols such as 11 and N . If {il"}:=l "i:-&" is a sequence of closed subspaces of lf, and if 14,,l- l"t*
f or n i shall write el=ril,, for the closure or Il=rli"; thti "direct sum" notationwill be usedonly for an orthogonaldirect sum of closedsubspaces.Of course,we shall also write ef=rf" for the "external"direct sum of Hilbert spaces,in which caseeach Xf' will be of the direct sum. If 11and N naturally identified with a subspace of 1?with N g lt we shall write It 0 N for l"tn are,closedsubspaces Nr. Vectors in lf will be denoted by [,11,6,etc., while symbolssuch as a,x,y,z,e,f ,u,w will always denote bounded operators. It will be necessary,on occasion,to considerunboundedoperators,such objects being usually denotedby A, H, K, S, F, etc. Of course,it may turn out in someinstancesthat S is actuallybounded;when that happens, relief would, it is hoped,offset the conflict with our the consequent notationalconvention. The set t(Xf)of all boundedlinear operatorson lf has the structure it is,,aBanachalgebra(with respectto the of a C*-algebra:,explicitly, = sup(llxlll: t e !f, lllll = l), pointwise vector operator norm llxll operationsand compositionproduct), equipped.with..an..involution r - x*, which satisfiesthe so-calleob*-ioentlty: llx*x ll = llr ll'. The orthogonal projection associatedwith a closedsubspaceI't will usuallybe denotedby pl"t,this is the operatorsatisfyingpn= p2n=ph and ran pn = t't (Here and in the sequel,the range of an operatorx will be denotedby ran x.) Converselyany operatorp satisfyingp = p2 = p* is the orthogonalprojectiononto ran p. Such operatorswill be simply referred to as projections. We shall never consider non-self-adjointprojections. Recall that the operator x is called self-adjoint if x = x*; more generally,for any set M g t(Xf),we shall let M* = {x*: x e luU and call M self-adjoitt if M = M*. Probably the most fundamentaltheorem in Hilbert space theory is the spectral theorem for self-adjoint operators, which may be formulated thus: let x be a self-adjointoperatorwith spectrumsp x; then there exists a mapping F ' e(F) from the class of Borel subsetvof sp x to the class of projectionsin lf satisfying: (a) e(sp x) = ! (b) if F = ui=rFn and F,, fl F- = Q for n * m, then {e(F-))I-, is a sequenceof pairwise orthoS,onalprojections and ran'L1ii';'= ef-rran e(F,); and (c) for any l,n in xf, if r1,a is the finite compld'i'measure'onsp x defined by rrt,n(r) = ie(r)l,nt, t h e n < x l , , n >= J \ d g g , n ( \ ) .
s r r o l e r e d o ( p a p u n o q u n , ( l q r s s o d )y ' s J o l B r e d op a p u n o q u n E u r u r a c u o c slceJ aruos ^\ou sn :sroleJado papunoq JoJ qJnu oS lol ll€cer '(-'gl uo uollcuny / larog i(ue ro3 (l*xl)l *n = 'r(yluraua8arotu l*r *((lxl)/n) Jo uolllsodruocap reyocleql sl l*xl*n = *r ueql 'r Jo uolllsodruocep relod aql sl lxln = x JI leql /hoqs ol pruq 'lxl ,(q polouep aq ,(11ensn lou sl lI IIr^\ pue ,a11(x*x) = 17[q ua,rrE s I { I O I C e Je r r 1 1 l S OOdq a ' x r e { - { r 3 { = n n 4 p u B 0 < r / " { r l a r u o s l IBIIr€d E sr /, :suorl!puoc Eur,no11o3eql ^q poulurelep z(lanb1un sr qcrq,$ qn = x uollrsocluroJop€ sllrup€ x roleredo frorra 1eq1 sa1e1s qclq/tr 'ruaroaql uollrsodruocep t?lod aql sl llnseJ crs€q reqlouv 'n ',tlo^rloedsoJ frleurosr lsIUBd orll Jo socedsIBUIJ pue l€lllul oql 'pa11ec er€ (n UEJ =) I uer pue (*n u€J =) a uet sec€dsqns eql pue uollceforcl € osl€ sr *ilfl - / 'os€r slql ul luorlcefo:d e sr n*n = 2 Jr. fluo puu .;r frlaruosl IBIU€(I € sr n l€rIl uir\oul-lla^\ 'nTJa\ s1 lI r I r a , r e u e q ^ \"eib;a1 l€llrBd B pellEc l l l l l = "l l t n'(h.rlaurosroc l l 3 r f r l a r u o s r ..dser) '.,(11erauot sr n roldrado ue B ,(rleuosr ''dsar) u€ pell?J s! ( = I *nn | = n*tr Eur,{3s1lesn role:odo uy ('1 fq,{ldrurs pelou sl I.\ pu€'I ,(q,{ldruls p e l o u a p s r r o l e r o d o , ( 1 1 1 u e p ra q l ' e r e q ^ \ o s 1 a p u e a r o g ) . I = * n n = ?t*fr 33r frelrun sr n rol?r3do u€ l8q1 llecau .srolurado ,(relrun '1xx = Jo sselc oql sI sJoleJado lerurou Jo ssulc luelrodurr uy x*x Jl I€rurou pallec Eulaq x roleredo ue ,sroleredo I€rurou Jo sselc ro8rul eql roJ prl€^ sl 'palels el€q a^\ sB 'uaroeql lerlcads aqa 'z/rx = /t ,(q pelouep eq sf u,rle IIr^,r,pue '0 ( t roJ zt{ = Q)I eraq^\ '(xi| = d dq uazrrEsr ,{ loor arBnbs aqt irt( = x l€t{l qcns g 4 d anbrun e slslxe eroql'0 I x JI ('< e,rl + J'(unt+ !)x>nl
0l{
=.{r'1xr7
uoql ? pue (A)f I r Jr lurll slrasse r{Jlr{irr{1r1uapr uollezlr€lod eq} pue r u'l luorooql l e r l cads eql Eulsn palord sl suorllpuoJ osaql 3o acualearnba '(-'01 oql) i r ds pue *x = x JI ',{lluale^rnbe .ro ? ul I II€ JoJ 0 < < l ' l r > J I ( 0 < x p o l o u a p ) a , r r l r s o do q o l p l e s s r x r o l € r o d o u V 'J Jo uollcunJ cllsualcEJer{c ro rol€clpur aq1 (1ooq oql lnoqEnorr{l pue oreq) salouap J1 areq,n '(x)st,.=^(g)a teqt r,E€tc?q pJnoqs lI '$)! = *1r)/ 'roqtrn3 lairoqe (c) ur sBu rt qlla'(f)" itp(f),/l = ',t ul 4'l fue roJ lBql gcns (g)g otul (rfx ds)-? ruorJ'(x)/ - / ruslqctrouosl €rqeEle__crrleuosr u€ slslxe areql u5q1 ( ;1 roJ slseq I€rurouorllro ue s1 {"1} alaq/(r
'."1'ul(c)ar,-z = (ilrt i {q uaayE sr arns€a(u qcns oug) '0 = (g)a JJI 0 = (J)rt l€rll qtns x ds uo ornseatu fue eq rt lol :enr] sl aroru .1ce3 u1 '(1)ap 1l = x l e q l r { c n s r d s u o p e u l J e p ( . ) a a r n s e a r ul e r l c o d s s s l s l x o o : a r i l . * x = , ( J I : S n q l p a s € r r l d e r e df l l e n s n 3 r € s l u o r u e l B l so a l E u l p s c a r d a q a ,{roaq1 roleJedo crseg 'I'0
0. Introduction a linear map H: D - lf where D is some linear (not necessarily closed) subspace of 4 called the domain of fl and denoted by dom //. The operator f/ is called a closed operator if it has a closed graph, i.e., if G(H) = {(1,11t): I e dom Ifl is a closed subspaceof 1l @lt. For a densely defined operator H (i.e., dom f1 - lt), the adjoint 11* i s t h e u n i q u e l y d e f i n e d l i n e a r o p e r a . t o .w r ith domain given by dom H* = {n e E 3 c > 0 3 ll < cllill Vl e dom tI) and satisfying = whenever ( e dom H and n e dom I1*. An operator Z is called an extension of an operator ^S if G(^S)g G(T), i.e, if dom S C dom Z and ?"E = ,St for I in dom S. The equation,S = I is to be interpreted as S ! I and Zc.S. An operator S is said to be closable if it satisfies either of the following equivalent conditions: (a) there exists a closed operator Z such that S g 4 (b) if I I 0, then (0,() does not belong to the closure of G(S). It is clear that a closable operator admits a smallest closed extension S which is characterized by the equation G(S) = 6T3)-. It is a standard fact that if .S is a densely defined operator, then G(S*) = {(-Sl,E): ( e dom S)r (in fi e f). A consequenceof this fact is that a densely defined operator S is closable if and only if S* is d e n s e l y d e f i n e d , i n w h i c h c a s eS = S * * . The operator Il is said to be self-adjoint if I/ is densely defined and H = I/*. The spectral theorem extends to unbounded self-adjoint operators. Recall that a scalar \ belongs to the spectrum sp 11 if and only if it is not the case that ran(I/ - I) = lf, ker(F/ - I) = {0} and (// - 1;-t is a bounded operator on lf. The formulation of the spectral theorem for unbounded H is as follows: if ff - I1*, then sp H t t R a n d t h e r e e x i s t s a s p e c t r a . lm e a s u r e e ( . ) d e f i n e d o n s p I / s u c h t h a t t , e d o m H i f a n d o n l y ^i f l l x l ' d y , r ( \ ) < - ( w i t h t l r r a S b e f o r e ) in which case = lx lur r.,(l)''for all n in lfl'' As in the bounded case, there is a "fuhctionil'calculus" (./ - /(H)) f or H. The polar decomposition theorem also extends to closed densely defined operators: every closed densely defined operator .S admits a decomposition .t = uH which is uniquely determined by the conditions: a is a partial isometry and H is a positive self-adjoint operator with domain equal to dom ,S satisfying ker ,S = ker tu = ker H. As in the bounded case, fl is the unique positive self-adjoint operator satisfying f/2 = S*S. In the second chapter, we would need to study a conjugate linear operator S (i.e. 'S(rl + n; = f.Sl + Sn) which is possibly unbounded. In this case, S* is the unique conjugate linear operator defined on dom S* = {n e lf: 3 " ) O > l.st,ntl < "lltll V( e dom ,S) and satisfying <.Sl,r,> = <S*D,l> for all I e dom .S and n e dom S*. (It should have been stated that S was densely defined, for, otherwise, the asserted uniqueness is not valid. However, here and elsewhere, we shall only consider densely defined operators.) The polar decomposition is valid in this "polar part" u will context t.oo, with the modif ication that the now be a conjugate linear partial isometry. These facts may
'*(A)) eceds l e . r l pe q l J o u o l l ? J r J r l u e p l
ue o l p e a l s e s l c J e x a E u r m o l l o . y a r l l ( 7 ' u t = | Eurreprsuoc uo paurelqo sr f lrlenbaur esJoAoJ oql ollr{rr\...zre,rnqc5-f qcne3 ruoJJ s^\orroJ = {1:ea1c.ueqr I i(lrlenboutaqr) 'llrrll
lltll
llr't"ll
'= (r;tt'lr'l fq eulJep 'll ) u'i,.(rerlrqre rog *(rf)f ?. Y.'t" '' 1 = -1-drS = lxl = ll ll ll"ll leql uollonrlsuoc ul{-", o q l u o r J r € a r os ! l I , ' u u ' 3.: x l'ql qJnsx usr roJ srs€q ue sr ("1) uaql ''gn = ul JI teqt apnlcuoc.n 3o aceds I€ruJouoqlJo I€rlrur arll sl lxl Tro{ acurs "lx[ra1 = lx I ugr roJ srseqI€rurouor{Uo 'lr'utrr"* u€ sr {"t } ererl,n 3 = lxl qderEered lssl oql ^q ,oS 'rolurado ? sI x*n = lxl uor{l ,lxln - x uolllsodruocap lceduroce,r1l1sod :e1odsuq (A) ) I r .lI (;fgrrel) .(x4hul I€epr peprs-o/(lpesolc-rurou 3 sr srol€rodo lcedruoc Jo (r{)) les oqr leql pelJlra^ flrsoa sl lI ('il Jo senlez\3o aEuereql perJ.rJods lou a^eq e/r os pu€'ellulJ sI runsoql'{usr eflulJ suq r JI) .(n)(@'0)[ uer = x u€r JoJ srs€ql€ruJouor{lroue sr {"1) pue ("\ oJaz-uoufueru {1a1rurgffi areqr JI) o r t\ arerl^\'tl'tlrt'rtta = x lsql pue.alqelunoclsoru1€ sl {0}\x ds,tPql ^\ou al?lpeturulsl lI '0 <,r,-vJBeroJ l€uorsuorurp-allulJ sr (x),- ', I u€r lcql s^tolloJ ll .(r)t€'tlI u€J ol luauelels lssl eql Eulflddv 'rolerado lcedruoc ellllsod € s! r l€ql ^\ou asoddng 'Ieuorsuaturp-ellulJ dlrressacausr ll ueql .lcedruocsl r JI pu€ .0 < t ouos puB l,f uI I II€ roJ lltll, < lllxll reqt qcnsfi Jo er€dsqnspesolc € sl l,l JI leql oas ol fse5 d! r1 'rlas"lcedruoc (-rurou) € uI peulgluoc sl (I > :lx) JJ lo€druoceq ol pl€s sr x roleredouB l€ql IIBceU lllll (*x uBr ''clsar) ,( u€r Jo slsuq l€rurouor{uoue sr (r=l{{tr) flarrrlcaclsar)-r=l{{1}pue 0 <'\
<
"'
< r\
lu'{lrl\ tit =t aJoq^\ '
se alqrssardxosr tr lu€l Jo r rol€rado fra,ra lBql ueosaq u.q?l1 .fllereueE ero41 .srolcol lrun ar€ (l puE I puE llrll = \ eroqir'u lr\ sE elqrssardxa sl euo {uBr Jo x role;ado ^rorroierit reelc sl 1r .flosreauo3 .euo >luurJo.rolutacto u e s r u ' i t u e q l ' 0 r . u ' l J I ' r e l n c r l r e du r : l l u l l lltil = llu'lrll l€rtl flllenbaul zre,nqcg-fqoneJoql Jo acuonbesudc"fs"ee" ue ir 11 ,l ul ) roJ lcu'l> = lu 11 [q paur3aproleredooql eq tt'?11el .g ur g.l rog 'Orh pnpar4 ot1l ?0 & ec?dslraqllH ele8nfuoceqt olul S'ruop LuorJ rolerado J?ourl e se fEulaaeln l(q as€cr€aurl oql uorJ pa^rrep oq '(t{h Isnperd arII 'Z'O
0. Introduction
Exercises (0.2-l). If r,r e K(lf)*, there exists a unique operator t(t^l)e E(lf) such itt"t i(r)t,n> = o(/r n) for l,n e !t. Further, ,(o) is compact and I|.r(r,r)t,,i,ti . - fot'diry orthnormal basis {[,) of lf' (Hint: Consider li" u"t'it,i.d sesquilinearform [(,n] = u{rq,n)io lay hands on t(o); for a finite orthonormalset ([,t, ...,lrr), let
r = ti =ot ,' rti ' ,l i where0, are comPlexnumbersof unit modulussuchthat = lt^t(t e,u,(tt,,tJ g.,g.)l and note that
= t^{x)< ll"ll. t l.r(r)(,,E,tl I
For compactness of t(tl), f irst reduce to the case t(tl) > 0 by considering0 e K(lf)* given by 0(x) = t(xrr), where {!r) _=.nft is the potar OecJ.position of r(r,r)and noting that t(0) = 'f;,if l(t't) ) 0' use li" "tr."Av establishedinequality tll< lloll and conclude
(O-2.2\. If ot e K(lf)*, then u admits a decomposition o
'=
J,
d,,0[,,,0,,'
w h e r ec r - ) 0 , E o . . ' a n d {(,rra } n d ( n r ) a r e , a ^ . p a ior f o r t h o n o r m a l rrqu"n.&. Conne?selyeve,fi,,'!uch sysi'em defines an element of r irl- as above. Furiher, ll,,rll= E "". (Hint: if t(tl) = I o,,tn,,,E,,it = the canonicaldecompositionof the compactoperator t(r,r)and if B (8,,) e co with Br,) 0, observethat x=EBrr/qrr,arK r t( r ) and that
= o(r) < ll"ll lltll = ll"ll lloll.o, E cr,,B,, cr,,( and appeal to the classicalresult cf = !1, to conclude that E
ll,,rll.l 0
= {r(u): u € K (r)*}. Note that r,r - t(o) is a bijection Let t(1?)* 'kt$l'"nO when t(r.)-... Hence t(lf)* is a Banach.-.space urt*..n r(tf)* K(1f)' c (0'2.1), Ex. normed tttui: llr(ur)ll,='l["11. As noted in
se d go uolllsoduroJapu oq
'u''lr'o
'O"r:;:'1}ili .'(nh ? 3 = d ta.I srsrc.rexl
ll
,t pue (A)g uo srurou arll roJ r(1uopesn aq IlI,\{ 'll pu" '(nh uo urou ertl roJ Ill .ll ar1r,r ,tonbasaqr uI IIBqsea do t1 = = <1o'd>= <1'Ddt = od tl
aqt urorJ s/holloJ lI '(nh
'(o)/ = d qll,n '(g) suoltenbo lBql ) o Jl .ue,ll .0.2.0 .xg Jo uoll€lou oql ur (d)rQ=<1'd>=dr1
3ql aulJep '*(fih
'(*)f
r d rog
fq d go ecerl ul Ieapr poprs-o^\l e sr *(g)X snqa
'r(o), = (r.o)l
'.(n)to= (n.o)t
Q)
'a^oqe s€ o', qll^\ l€ql d.;rre1
'(xaln = (x)(a.o)
(r)
l(ar)o = (r)(o.a)
tal'(n)) r x pu? '(r[)I ) o *(A)) r cq Jr.:s^\olloJse *(g)) uo ernlrnrls alnpourq-(g)g ue eurJep u€c euo '(Ah uI I€epl papls-orrl B sI (A)) aculs
*o-' ?"',':[y".],"l,l"Jt*,,f uBseurrep fftt#rT","="iltti; Eurfgslles r(*(*)f) ur iQ anbrun B slsrxe eraqt .(fih ) x JI .0.2.O) ('Q'z'\ 'xa ol luadclePue 1uu'"1";, u.,u. = tt' ll l€ql eloN :lurH) 'uo3 = -G)f"r1; requnC .a^oqe sB '(tt)X 3o d luaruela ue saurJep rualsfs qcns Xriha .dlesranuo3 .socuonbes I€rurouorluo go rred e are {"u} pu€ {"1} pu€ o t'o I.0 < .o oreg,t\
''u''lr'o t3" = o uolllsoduocap € slrrup€ '(rih r d i(uy
(g-Z-O)
sosrcJoxl 'ft)f lo I€nperd aql sB ol parroJer osl€ sl *(g)g .uoseer slrll roJ '(*h ot crqdroruosr fllecrrletuosl sI *(*(Ah) acuds I€np oqt lsgl qsrlq€lsa qcrqa sasrcrexe Jo les lxeu aql fq polecrpul^ sl *(nh 'srolerado ss€lJ-rc€rl poll€J uoJlBlou orII are '(g)g ur sroleradg *(g)g
lenperd orII
'Z'O
0. Introduction (a) The series I.r,.
1i,,4,, is convergent in ll'll'
atto consequently,
for any x in E(lf), tr px = 1P,x7 = X cr,r<x(rr,rl,r>;in particular, tr P= Ecr-<(.,'->'
and hencethat llcll, = r o,, = (u) vlrirv. lrtul ;*;''= r o|rn,,,n,, tt(P* P)rlz (c) Show that tr p = E for any orthonormal basis {5,,} of r. (Hint: Appeal to Caucfiy--and Schwarz to justify an appeal to Parseval.) O
ntrace" is vindicated by part (c) of the last The use of the word exercise: in any matrix-representation of p, the main diagonal is a summable sequence (by Ex. (0.2.1))and the sum, which does not depend on the choice of co-ordinate system, is the trace of p.
0.3. Thrcc Locally
Convcx Topologics
on q10
For several reasons, the norm topology is not a very good topology on I(lf). For example, t(lf) is nonseparable. (Reason: represent lf as 1 2 [ 0 , 1 ] ,a n d f o r 0 ( t ( l , l e t p , b e t h e p r o j e c t i o n o n t o t h e s u b s p a c e P" is a non-zero of functions supported in [0,t];-if s < l, then Pt lt, is an increasing projection and hence has norm one.) Also, if if the sequence (lvtn)is not sequence of subspaces and l'1 = U , eventually constant, it is not true that Pn' P in the norm. It turns out that, in such situations, one would d-b better to consider certain other topologies on t(lf). Let us briefly recall something about locally convex topologieson vector spaces. Recall that a seminorm on a vector space M is a mapping pi M - [0,-) such that p(\x) = lrlp(x) and p(x + v) < p(x) + p(y)whenever x,y e Mand \ e C. Supposethata family {nr:i e t\ of seminorms on M is given. The induced topology ot M is the smallest vector topology on M with respect to which each pt is continuous at the origin; in this topology, a net {x; cr e A} in M for each i e /. c o n v e r g e st o x i n M i f a n d o n l y i f F { x o - x ) ' 0 Definition 0.3.1. (a) The strong topology on t(lf) is the topology induced by the family of seminorms (pq: E e lf) defined bv pq(x) =
ll' I ll.
(b) The weak topology on t(lt) is the topology induced by the family of seminorms(Pr n:1,0 e lf) defined by pt.n(x) = l<x('4>1. (c) The o-weak topoldby on t(lf) is the topolo'gyinduced by the D defined by pp(x) = ltr xPl. famity of seminorms{pp: P e S(Xf)-} In simpler language,a net {x1} in l(tf) convergesto .r with respect to the: (a) strongtopologyiff llx,( - ttll '0 in the strong toPologYon lf ;
for all ( e 1?,i.e.,iff xrl 'xl
r " ' e r ur u 1 " " ' I l ) r ! r a a a u a q mg - (1("t = : r p u u , r { J ? aJ o J Q = r g x p u e l t t = l l r l € q l q r n s r o l u r a d o {uzr ,...J]) e l r u r J B e q r l r . I ' l u e p u e d o p u rf l r e e u r l s l ( " u , " . , r t r . ' l (lt) pue ', qcee roJ r > - ! : r ; ; ( r ) E u r f g s r l e sf i r ' { , r . . . . , r r rr t c l d ;;ltr uaqJ 'l€ruJouoqgo ua^6 'luapuddepur flreeurl aq ol pounss€ oq ',(1r1e:_ouaE 'feru s.!l orll l€ql anEry .0 < r pue Jo ssol lnogll^\ ""'tl} Irs eruosro3'{a } I > I roJ r > lllt(p -- x)ll :(r{h ,i i {"1 r x) ruro3 aql Jo sr ,tEo1odo1Euorls oql ul r go pooqrnociqtrau crseq 1ec1d,{t e l(g)g ) o t?l :lulH) .(,l)f ur osuop flEuorls sl (O = { :Qit r x) 1eq1 ^\oqs .I€uorsuourp-olrulJul sl fi uaq/t\ (e)
'(e'e'o) ('a qc€a ro; frleuosr u€ sI ,rn se .{lEuorls 0 I utt alrr{^r{lEuortrs 0 -,r*z leql f3rrarr lslseq leru.rouoglroue sr r="{u1; ereq,n ur,I+u( I=u . t t1 3 =n
"e'I irJIrIs tlrrrr,ron "
'{Eo1odo1Euorls ol eq n lel;luIH) lcadser qll^\ snonurluoc lou sI cleru gurofpe oql ^\oqs uarll 'luuorsuourp-alrurJur eq g p1 (q) lrql 'deru-31assnonurluoc € s! +r e r uolluJado fulofpe eq1 .fEo1odo1 {€er\-o eql ro {€a^r eql raqlle qll^r paddrnba sr (g)g uag1\ (e)
'(z's'0) ('-(A)f ur asuoperu .aprculoc srolu:edo pue eql (e) osn :lulH) l€rll lcBJ {uur-elrurJ ',Sol pelclrlsar uaq^\ 'sorEolodol pu€ oql /noqs leql {€e^r-o {€a,tr '(- > € sl S JI (q) {S t x:llxll)dns "e'I) (A)r ul las pepunoq-rurou
\xd, srrluopu,Jr;ffij;'l':i {:rk",ilj'fl:li;l','iJri'":h (e) (r's'o) soslcraxe 'sarEoloclol 'tuorls-o aql .f .sartolodol Jeqlo *Euorls-o pue *-8uor1s leru?u aarql Jo suolllurJep aql erB sosrJrexe aql ur popnlcur osle lsasrcroxo eql uI polsll 0JB serEolodol eseql Jo soJnlBeJ {relueruelo otuog ''(nh Jo aceds lunp eqt Euleq stl Jo anlrl^ fq '(Ah pelrror,{ur ,{q fEolodol eql uI x * Ix 33r ..{11ue1errrnba '.ro '*(&h ur d fre,ra roJ *IBa1r - !x)d rll JJI ,{Eo1octo1 0 € l(x IEe^\-o (c) : ul f . r a r e r o 3 ' g u o f E o l o d o l > l u o , t ar q l u [ l x * I n , "o'l'$ ur g'! ro3 :'rc JJI 0 * l - l JJr fEolodol {?a^\ (q)
Qth uo sar8olodoaxa^uoC f11eco1oarql .€.0
0. Introduction
l0
(b) When 1f is infinite-dimensional, show that multiplication is not jointly continuous when I(19) is equipped with any of the three l o p o l o g i e s :s t r o n g , w e a k , o - w e a k . ( H i n t : F o r t h e s t r o n g t o p o l o g y , u s e ( a ) ; f o r t h e o - w e a k a n d w e a k t o p o l o g i e s ,w i t h a a s i n E x ' ' 0 (0.3.2)(b),note that il*n ' 0 o-weakly and hence weakly, so u' o-weakly and weakly; however, u*nttn = I for all n') (c) If s is a norm-bounded subset of r(xf), then multiplication is jointly continuous, when restricted to s, with respect to the itrong topology. (The example in (b) shows that even this is false in the o-weak and weak topologies). (d) In all three topologies, show that multiplication is separately xl and ytx ' continuous; i.e., if x e E(II) and y, ' /, then x!i' yx. (0.3.4). A net {x,} in l(lf) is said to converge to x in r e s p e c tt o t h e :
(ii) o-strong* topolosyo f tllt", - t)i"ll2 * ll(",- x;*q-llz;' 6 w h e n e v er rl l i " l l ' . - ; (iii)
strong* Iinlt
topology<+ (ll(xi- x)tllz * ll(t, - x)*qll')- o for each
(a) Show that the adjoint operation is continuous in the strong* and o-strong* topologies, but not in the o-strong topology. ( b ) S h o w t h a t , w h e n r e s t r i c t e d t o n o r m - b o u n d e d s e t s ,m u l t i p l i c a t i o n i s j o i n t l y c o n t i n u o u s w i t h r e s p e c tt o e a c h o f t h e t h r e e t o p o l o g i e s defined in this Problem. (c) Let {x,) be a net in [(Xt). Prove the following:
xi-0
(stronglY)
(+
xi - 0
(o-strongly)
€ xlxt - 0
(o-weakly)
xi'0
(stronglY*)
€ "F, + x,xf - 0
(weaklY)
xi - 0
(o-strongly*)
<+ xfx, + rixl-
(o-weakly)'
xFi'o
(weakly)
0
(d) when restricted to norm-bounded sets, show that the o-strong (resp., o-strong*) topology coincides with the strong (resp'' strong*) topologY. (0.3.5). (a) If r, and 12 are topologies on a set X, show that the following conditions are equivalent: (i)
ld*: (X,rr)'
(X,rr) is continuous;
? l
! I
I
' , ( 1 " 1 S )= ( r + r r ) SP U B r ( r S ) = ' S o 1 1 r ' t ' t ( 1 1 e r a u eaE: o 1 4 1 '{S r x II€ roJ .xx = x,x:
(nh r rx) = tS
aq ol peulJep sl (fih Jo S: lesqns B Jo lu€lnruuoc aql IuerosIII
lu"lnuluoS
olqnoo oql
't'0
('erroJ E ur IIIIs sr flrlrqe:edas 3o uogldunsse aqt l4 Jo II€q llun oql u r e c u e n b o so s u o p € o s n : l u r H ) ' e l q u z u l a r u s r ( 1 e a n o 1 * E u o r l s - o ) serEoloclol xrs oql Jo qcea 'sles papunoq-rurou uo lBrll /y\or{S (p) ('s1auroprsuoc ol ,(ressaJeusr t1 lop tou 11r,nsacuanbes 'snql) 'r(Eolodo1rurou eql lclacxe (q) (S'g'O) 'xg ul pauolluoru serSolodol eql Jo due ol lcedser qll/'r -- flrlrqelunoc Jo urolx€ lsrrJ eql ,(gslles uele lou seop 'lce3 ul -- elqezrrloru lou sl (a)f (c) (loldrcurrd sseupapunoqrxroJlun :lurH) 'oJoz l(11ea,vr (-x] ol soEraluoJ Jo ecuenbasqns ou ]€r{l ^\oqS (q) ('saEraltp selJes oruour€q oql oculs '0 < r l(u€ toJ 'u ,(ueu flelrulJul roJ t t-w ) ,1.-tr'": tl i
' - > r l < * & ' ' r 'tl = u l l ' t lil JI:lulH)'r=l{*"}
o"ut
r='('l) sorJsrlBs eql ol sauoleq-0tuqr froqS (e) Jo ernsolc*Euo:1s-o 'w
q'ee ro
t''- l' 1 r,r*
'lsrrls sr suolsnlcul e^oqe eql Jo rIJBe'l€uolsuerurp-olrurJul sl ll JI leql 'salclurexe ,{q 'erro:4 (c)
; D :
1 = urc lol pue ? roJ srseq lerurouor{trouB aq t=l{*t } ia1
I
D
'{BOlr\ I
EEuorts
'(996:o)
E Euo.Its
lln l l l nl ll l n i i t = = = {?e^\-o c Euorls-oc *EuoJls-oC turoN :fu1)guo sel8olodol luaJaJJrp 0rll u3e,{l3q suollEler ,,uorsnlcur,,Eur.no11o3er{l e^oJd (q) '2r.5 r.l e11r^ ' p l o q s u o l l l p u o c e s e q l ueq71;pesolc-zt sl tes pesolc-rt f:ene (rrr) : X ' q u 1 . r p u ? Xul (!x) leu fue rog '(rr)x - tr € (zr)r - !r (g)
ursroaql lu€lnuuoc olqnoc srII 't'0
II
0. Introduction
t2 Proposition 0.4.1. Let S,T g t(tf).
( a )s c r + z ' c s , . fur n 2 r; i;i ; - "" =-sw-olo sr = 5(2n-1)
(c) S is self-adjoint ) S) is self-adjoint. (d) S' is, for any S, a weakly closed subalgebraof \0 Proof. Exercise!
and I e S).
D
Before proceeding further, it would help to set up some notation and terminology. For a subset S of 13,we shall always write I S ] for the smallest closed subspaceof lf which contains S; for S g f(Xf) and S g f, we shall simply write SS for (xl: x e S, ( e S). A set S of operators on lf is said to be non-degenerate if t,Slfl = 10. Since ranrx = ker x*, it f ollows that if S is self-adjoint, then S is n o n - d e g e n e r a t ei f a n d o n l y i f , S g= ( 0 ) i m p l i e s 6 = 0 . The stage is now set f or von Neumann's double commutant theorem,whose power will be illustrated in the rest of this section. Let M be a non-degenerate self-adjoint algebra of Theoren 0.42 operators on tl. The following conditions on M are equivalent: (i) (ii) (iii)
M = M'. M is weakly closed. M is snongly closed.
Proof. The implications (i) ) (ii) + (iii) are immediate (cf. Ex. ( 0 . 3 . 5 X a ) ,( b ) ) . T o p r o v e ( i i i ) + ( i ) , i t clearly suffices to prove the following: lf a" e M', 1L,..., Er,€ 1l and e > 0, there exists a e M such
(*)
t n a t l l ( a -" o ) ( i l l . .
for I ( i ( n.
We first verify (*) in case n = l. Let J't= fMlr) and pt = pM. It is clear that M!\ g t\ and so p'xpt = xp' for all x in M. Since M is self-adjoint, if x € M, then x* e M and so p' x* p' = x* pt . Comparison of the adjoint of this equation with the previous equation yields ptx = xpt for all x in M, whence p' e M'. So a"pl = pta", and hence a"It c )t Since M(, is dense in Jvt it suffices to prove that [,, e I't For any x in M, clearly p'xlt - "l,.r and so x(l-p')Er = ilr - xpt [r = rlr - ptx\r = 0; thus M(l - r')Et) = o. The assumed non-degeneracy (and self-adjointness) of M ensures that (l - p')[, = g, and henceEt e lt Returning to (t) for genelgl n, let 1l be the direct sum of n copies of !f. Every operator on 10 corresponds naturally to an (n x n) *-algebra a t(lf)-valued matrix, this correspondence being isomorphism. With this identification, let
'1(Urt > x :x ra{) = 'Wtl = a p-I '(A)f;o Jrnf#illt U) ltUql=Woraq,r\ .$.V-O) (alurauoSop-uou ,{llrrssecau 1ou) E aq I4[ n1 ltrrofpe-g1es seslJrrxa 'osrcraxo 8 u 1 r t o 1 1 o 3 e q l ul lno lleds 'sroleredogo erqaEle p u e l u l l u o s s o u rs l e c u a r a J J r pa q l lurofpu-g1as pesolc r(14ee,n e oq ol erqaEle uuerunaN uol € eulJep ^eql l a l e r c u a E e p - u o uo q o l s e r q e g l € u u r u n e N u o l e r r n b o r l o u o p s r o r { l n € atuos 'erqe8le uuetunrN uol € sl srolerado Jo uoJlJelloc qcns f,ue '7'y'g ruoroor{I fq 'e1rq,tr'(,|)t Jo erqaEleqns Isllun tuJofpe-31es pasolc,{11ua,e n s r e r q c E l uu u e t u n e Nu o l B . ( p ) t . 1 . 0 u o 1 1 1 s o d o r 4 , ( g ",hl = tr41serJsrlBsll JI €rqeEl€ uuBr,uneN o uol B poll€c sl (A)f jo q etqaEleqns lurofpz-Jlas V .€-t-0 uoIrIuIJeC 'uerleq€'relncrlred ur 'pue 'r u r s l e r r u o u , { 1 o3do l e s a q l J o o r n s o l c Suorgs aql sJ ,,{x) ueql '*x = Jc Jr :uotluau go ,(rJ1.ro,rsr oseJ lercads '{t} n S , { q p a l e r e u o Ee r q e E l ua q l J o a r n s o l c E u o r l s o q l s r , , S u a r l t V 'sroleJodo 'snql .y Eululeluoc erqaElr J o l o s l u r o f p e - g 1 a s, { u e s r S J I lurofpe-310s (pasolc l(11ea,n 'osle) pesolc ,(1Euor1slsollErus aql sl ,,f .oS uoql 'sro1€rado e r q e E l e e l € J e u a E e p u o u , ( u e l u r o t p e g 1 o s s l 3 o f J I 'l{ 'ul pauleluoc) s1 ueql Jo arnsolc Euorls oql ol lenbo (ocuaq puB ,,,12g 'srolerodo go erqeElu l u r o f p u - 3 1 e so l e r a u e E e p - u o uB s \ n y l e q l s e l t l s f11en1ceruoroeql eql Jo goord eql ur (*) uorlrass€ or{l l€gl {re{ueU 'rueroaql E 1 _aql Jo oouorl puB (*) Jo ' 0 < I p u e '__ o . . . o r 1 , n ^ q p a c e l d a - rr p u e ; o o - r d a q 1 a l e l d r u o co l tl ? q l l ^ ( * ) . l o I = u e s € c p e q s r l q e l s ar p e e : 1 e ; e q i o r ^ - o u l e e d d y
r , u a r o e q ll u B l n r u u o J a l q n o c e q l
€l
't'0
L4
0. Introduction
(a) (b)
x = exe for all x in M; in particular, MI4 C M; if Me = (xllt x e Ml, then Me is a non-degenerate self-adjoint subalgebra of t(It);
(c) lvlt = {x' e y
: x) e MJ, y e t(l'll)}, and
W=(x" Ollld:
x"e\,
\eC).
(Thus, a degenerate von Neumann algebra, as considered by other authors, is just a von Neumann algebra -- in our sense -- of operators on a subspace.) (0.4.5). Let (X,T,y) be a separable o-finite mea-surespace (so that r2(x,tt) is a separable Hilbert space). For 0 in L-(X,1t),let m5 denote the associated multiplication operator: (rz6t,)(s) = 0(s)l,(s), for ( in L21X,1t1= Y. (a) The map O - m6 is an isometric* - isomorphism of z-(X,p) into t(lf) (where the '*' refers to the assertion mf, = m6). (b) If M=(^60e L ' 1 X , 1 t 1 1 , t h e n M =M t ^ n d ' c o n s e [ u e n t l yM i s a n abelian voh Neumann algebra. (Hint: First, consider the case of f i n i t e y ; i f x ' e M ' , s h o w t h a t x t = r ? 1 6w h e r e Q = x t | o , l o being the constant function l; the genbral case follows by decomposing X into sets of f inite measure. Is o-f initeness necessary?) (c) The o-weak and weak topologies on M coincide; under the identification mh * 0, this topology coincides with the weak* topology inheritdd by L'(X,tt) by virtue of its being the dual s p a c eo f L r ( X , t D . (d) A general von Neumann algebra M satisfies M = M' if and only if M is a maximal abelian von Neumann algebra in l(xf). (0.4.6). If M is a von Neumann algebra of operators on lt, let M1= 1p e t(13)*: tr px = 0 Vx in M). Then Mg is a closed subspace of t(Xt)*, J M, and the induced weak* topology on M agrees with GQt)-/Ml* E the restriction to M of the o-weak topology. The last exerciseshows that every von Neumann algebra admits a predual. It can be shown that such a predual is uniquely determined u p t o i s o m e t r i c i s o m o r p h i s m ,b u t w e s h a l l n o t g o i n t o a p r o o f o f t h a t here. Consequently, we may talk of 'the' predual of M, which will usually be denoted by M*. Just as L'(X,u) is generated (as a norm-closed subspace) by indicator functions, it is true that every von Neumann algebra M is generated (as a norm-closed subspace) by the set P(lul) of its projections. To obtain this and other consequencesof the double commutant theorem, it helps to establish a useful preliminary lemma. Recall that a C*-algebra of operators on lf is a norm-closed self-adjoint subalgebra of f(8). Clearly von Neumann algebras are
sr on eruJs i1,r1uI suorlcaford Jo suorleuJqruoc rueurl Jo les aql Jo ornsolc rurou oql aq "n 1a1:uoseag) .suollcoford slr Jo los eql ,{q acedsqns ptsolc-rurou € s€ pelurouaE sr trqaElu uueruneN uo,r fue leql frelloroC e^oq€ oql ruorJ s^\olloJ lI .urqaElu aql eplslno spBOI Ja^eu €JqoElu uuerunaN uo^ B Jo sluauele ol pallddu uoJlcnJlsuoc IBJruouBc f ue lnoq? lsnl l€rll sarldrur unrloqcs eql .snql 'uoluosse srql allles E ol elras (e) ul posn auo aql ol snoEoluue d.11cexalueurnErg us pue roleraclo Ierurou ? Jo uollnlosar Ierlcads aql Jo sseuonbrun eql (q) 'goord eql seleldruoc unrloqcs aq1 ',{rerlrqr? s?Arrn esurs .lxl = r- rnlxlrn pue n = ytnnln ecurH ''-,nx,n Jo uolllsodruocap J€lod (eql acueq pu€) ? oslc s l ( r _ r n l x l r n ) ( r _ r n n r n=) vrflxtz lBrll J€olc sl lJ.pu?rl Jeqlo eql uo ilxln = x = r-,nxtn uo{l 'rl{ ur rolerodo {relrun s sl In JI (?) -JooJd 'x ds lo l lasqnsTatog tuata rcl n r (x)dt uatlt,lotutou sl x /I iW > lxl'n uaqt 'x /o uotltsoduocap nlod ary n lxln = x !1 'n ) x puo otqaSlo uuvunaN uo^ o aq n ta7 n
(q) (e)
.5-g-6 itue11oro3 ioslcroxa
'Joord
'rw u! ,n to|otadottotun ttata rct x = *,nx,n ptu s! n o7 6uo1aq o1 x .ro{ uoltlpuoo Tuatuttns puo tLrossacau y .11uo s.tolotado {o otqaSp uuounaN uo^ o n puo (U)5, , x p7 T-}-0 unlloqrs 'erqaEle uueruneN uol ? ol sEuoleq roleredo uB ueq^\ tulururralep JoJ uorJelrJc InJasn Eu1,no11og eqt sp1a1f ,(tW = t, qll^\ pcrlddu) €ruruol o^oqu eql qll^\ paldnoc uaq,r 'ruaroeql lu€lnruuoc elqnop orII 'y o1 Euolaqecuaq pu€'I pu€ r dq pelerauot erqaElu-*3 E eql '(x)*, ol Euolaq 'x Jo suollcunJ snonuyluoc Euyaq 'sroleredo eseql isrolerado frelrun o^U go oEurorruuB s€ r go uorssordxr u? sl
l Q l r Q x - I ) l - x | + { t , / r- Q , x r- ) r + x \ l I = x ueql 'I > ll xll pue v ) *x = f, JI l€r,ll ecllou 'lurofpu-Jlos ol /y\ou seJrJJns lI 'f ur srol?rado ere ty tr* eragj$ 'ux| + 'x = x uolllsoduocap u€Iseu€C agl sllurp€ y ur x ,{.uy -Joord 'V u, stolorado {..rolun tnol uotlourgtao? tpauq D so alqtssatdxa st lo .o"rqa61o-*2 v /o tuawata {tatg lolun p aq el)g3 V n7 Z-tg GuE I '€ruruol eql roJ ,t\oN ('n ul osuop i(1ea,n-o sl rlcrq^\ 7g p etqatluqns-*J redord u sI {[I.g]J t Q 9w) ps 0rll'ernsEeur anEsaqel t pu€ [I'0] = X qll,n .(S.l.O).x:I Jo uoll€lou eql uI 'aldruexa rog) 'onrl ruoplos sl rsre^uoc eql 1nq .serqeEle-*3
SI
Illerooql tu€lnruruoJ elqnoc oqJ
't'0
0. Introduction
l6
self-adjoint, it suffices to verify that if x = x* € M, then x e Ms: for this, let 0r, be a sequenceof simple functions on sp .lr such that 0,,(t)' t unifoimly on sp,x, and note that by Corollary 0.4.9(b),0"(x) = e"Mnfor eachn and lim llO"(x) 'iurtherxll 0.) properties of a von Neumann BeTore discussing some algebra, let us briefly digresswith some notational conventions. If {er:i e I} is any family of projectionsin a Hilbert space,the symbols V,rre, and A,rae,will denote, respectively,the projections onto the subsiacestui61ian e,l and q€r ran ei. For a finite collectioner,.'., en, we shall also write erV -. V e, and e, A... A err. Exercises (0.4.10). If M is a von Neumann algebra and (er) c P(tr4), then Yer, P(tu}. (Thus P(M) has the structure of a complete lattice') E Mi, the above exercise is given by the following
An extension of assertion:
Proposition 0.,Lll. Every uniformly bounded monotone (incteasing or decreasing) net of self-adioint operators on tt is weakly convergent. Proof. Suppose {x,: i e /) is a net of self-adjoint operators on Xf s a t i s f y i n g ( a ) i f i , j - e f a q C 1 .{ i , t h e n x , { x r ; a n d ( b ) t h e r e e x i s t s a constant c > 0 such that llx,ll { c for all-i in"I. For a unit vector I in lf, {<x1l,i>: i e /) is a monotone increasingnet of real numbers in [-''c,c],and consequently convergent to its supremum. It follows from t h e p o l a r i z a t i o n i d e n t i t y t h a t ( c f . E x . ( 0 . 4 . 1 2 ) )f o r a n y l , n € 1 | , t h e net {<x,l,rl>: i e I) is convergent. Denoting this limit by [l,n] it is clear that [.,.] is a bounded (by c) sesquilinear form on lf. Hence there exists x in l(lf) such that <xt,4> = [(,n] for all \,n e !1. Clearly, then, the net (xt: d e 1) converges weakly to x.
Exercises ( 0 . 4 . 1 2 ) . I f [ . , . ] i s a s e s q u i l i n e a rf o r m o n a c o m p l e x v e c t o r s p a c e % then, for any \,0 in V, 3
4[!,,n]= t i\q + ikn, q,+ ikn1. k=0
(0.4.13). Let (x,: i e I\ be a monotone increasing net of self-adjoint operatorson lf and let x = lim xt (as in Prop. 0.4'll). Then,
'2 uer = p '[44r] araqn n = (a)c uer uaql '(74/)d t a pus erqaEle uu€runeN uol € sI l{ JI ( ' p u o c e so q l _ s e l l d r y lp u e . 1 e r z r r rs1J u o r l r a s s Bl s r r J
(c)
_ aqa:1urg) ',(llu n llu) = ,(a,vu I,,g)pue f/{ u Iru =-,(,,,un t,,r) uotll t{ uo turlce surqeEleuuerunoNuo^ oq z.Mpue rN ta.I (q) 'a uet Eurureluoc ocedsqns pasolc luerrelur- | y'1 lsollurus or{l .[W,N] = auer puu// u1 uollceford€:^I U > a:(U)d ) Ilv = auaqJ, (W ) a teqt peunsse tou sl U) 'Nd = a lal pu3 I go eceisqns pasolc ,{ue eq W lel .}i uo sroleraclo go erqeEls uu€runaN uo^ E oq N lo.I (e)
( st l' o ) sesrJJoxa '(ap Jo uorldltcsap eleJcuoo eloru B ol sp€el esrcrexA 'a Eu1mo11og eqa Eurleuluop uollcaforcl Ierluec lsollurus oql sI (a), 3uotllulJap tq :(n)a , a releuoqa ((I^DD4 t (a)c .(Ot.l.O).xA ,{q 'acuoH 'erqeEle uu€runaN uurloq€ ue s1 erqe8l€ uueruneN uo^ e Jo orluec eyl 'thl U n = Q,r1)7ecurs .telncrlred ur iu:qaE1e u€runeN uo^ e ure?e sr setqoEle uueruneN uo^ Jo flrrue3 fuu go uollcosrelul aql leql uaroeql luelnrutuoc elqnop oql Jo ecuenbosuoc ,{see uB sl lI 'Q[ (n)d. t s a:(n)Zv n !)v = (a)c [,q paurJep uoJlcaford aq1 sr '(a)c [,q palouep 'a ,n ur a uorlcaforct E rog (c) Jo reloc I€rluac aql '{O I r :tll = (nl)Z Jr rolc?J € peIIBc sr Jrtl (q) 'UI)Z tq pelouop pu€ n Jo el1uec arll peller sI 0{ ul ,{ 11erog xt = rtx :7,t1t xl las eqJ (e) uo sroleredo 3o erqaEl€ uueruneN uol B eq n
rc-I
'll .1 I-1q-O uoplulJaq
'raldeqc lxeu e q l u l p a p e a u e q I I I / ' \ l s q l ( t I ' t ' 0 ' d o r 4 ) s r o l J ? J E u r u r a c u o cl s E J c l s € q € pue rolc€J 8 Jo uolllurJep oql qlJ^\ uollrss slql epnlcuoJ e^,l '!r dns = x ellJrrr n II€rIs e/r\ 'uosuar srql roJ !f I x uaql 'l IIe roJ r( > !x salJsrlss (n)f L{ ;I (p) ('(p)(l'g'O) 'xg pu? (u) esn iparro-rdoq poou ecueEro,ruoc Euorls-o ,{1uo 'tu1ofp€-Jles er€ r '!x aculg) .*[18uor1s-o x - !x (c)
('ll"t.ll ', * ll"tll'*l=" '1' l.'t ''l(lx --r)tl , 1.'u'ul(lr- ")rl i N
N pu€r ' ,ll"ull"3pu, - > zll"rll"3Jr :rurH)Trlt"r::;tlt jl; LI
rueroer{I lu?lnuuoJ
elqnoc eql
(q) 'r'0
r8
0. Introduction
Lemma 0.4.f6. Let M be a von Neumannalgebra and e,f € P (luI). The following conditionsare equivalent: (i) exf = g for all x in M. (ii) c(e) c(n = 0. Proof. (i) ) (ii). The hypothesisis that MI'l c ker e, where lv1= ran ,/. Hence,by Ex. 0.4.15(c),it follows that ran cU) e ker e, whence ec(/) - 0. This meanse ( I - c(fl, and so, by the definition of the central c o v e r , c ( e )( I - c ( n . D (ii) + (i). Reversethe stepsof the proof of (i) ) (ii). Proposition 0.4.17. If e and f are non'zero projectionsin a factor M, there existsa non'zeropartial isometryu in M such that u*u 4 e and uu* < f. Proof. The assumptionsensure that c(e) = c(f) = l' Lemma 0'4'16 then guaranteesthe existenceof an x in M such that fxe I 0. Let job. I f xe = uh be the polar decompositionof f xe. This u doesthe
' { > ra n a teqr qcns (y4)4 uy Ia s1s1xeercqt y E ! | a '.1 = *nn puB ? = n*n leql qcns Jrtlur r frlaruosl l e l l r e d B s l s r x e erel{l ossc ul I - a {ldtuts ro (14 pt) t - 2 :alrr^\ IIETISoA,\ 'U4)d
> I'a p1
'I't'I
(q) (u)
uoplulJaq
'lig ur suollcsford go aclll€l alalclruoJaql (,rtl) pue '€rqeEle d uuerun?N uol € alouep sf e,n1e ilr.a n loqufs aql 'r{lroJeJueH uoJr"IrU crLL 'I'I
(n pt)
'uustuneN uo^ puu ferrn;41 fq ,uorlcunJ uolsueruJpolrlslar, ? palleo s r p e s n 1 o o 1l e d J c u r r d o q J ' s a d f 1 o e r q l o l u l s r o l 3 e J J o u o l l € c r J l s s e l c ,(reurrd € slcaJJe 'rerp€e pessncslp uolleler Japro eql go s1s,(1eue 'eruardns alrurg Eur>1elropun a^llellluenb e t1,r 'uollcos IBUrJ oql pelresercl sl ssouallulJ feql Euraq 11nseJururu eql lsuollcotordqns redord ol luele,rrnbe lou esoql suollceforcl allulJ seurrrr€xa 'parapro ^ll€lol sr JolcEJ e ur suollcoforcl egl uorlJes lxeu eql Jo sass€lcaouel€^lnbe Jo los eql 'JapJo I€rnluu e ol lcadsar qll,r 'leql sl llnser l€rJnrc er{} eraq^\ 'l'I uollces 3o lcafqns aql sl 'W rolceJ uerr,rE€ ruoJJ oruoc ol porrnber are -- frlaruosl lerp€d ot{l s€ IIoA\ s€ suollcaford eql -- peureouoc sJoleJado oql II€ uoq/$'acuel€rrrnba sIqI ',(:1aruos1 I€lU€d e go saceds I€urJ pu€ l€Illul gql ere soEuur rraql J I l u o l € ^ r n b e E u l a q s e s u o r l c e f o r d o , r l s r a p r s u o ce u o ' [ 1 1 e r a u a E o r o r u 'gr sruectctesrpuelqord slrlJ 'z/ + r! ol lualearnbe ,(1r-relruns1 za + Ia teql enrl flu?ssecou lou sl I 'tI f tI pu" ", f ', JI pue 'Z'l = t toJ '11 o1 lualerrrnbe ,{yr-rellun sr Ia leql qcns suollcaford o:e ,! pun r! 'za 'ra J r : e s u e s E u r , n o y l o g? r l l u J a ^ l l r p p ? E u r a q l o u 3 o e E e l u e r r p u s r p el{l sBq 'l€rnl€u lsour Eureq o1rq,vl'ecualerrrnbofrelrun Jo uollou aql
rJrssvlf suorf,vj Jo Norrvf, NNVlAtnSN NOn - AVUUn1^| 3Ht I rardeq]
l. The Murray-von Neumann Classification of Factors
20
It is readily verified that - is indeed an equivalence relation on P(M) and that the validity of e ! / is unimpaired by replacing either e or f by an equivalent projection. We shall adopt the notatio\ u: e - f to mean that u, e and / belong to M and are as in (a) of the above definition. We shall find it convenient, in this chapter, at least, to work with 'PM, we may s u b s p a c e sr a t h e r t h a n p r o j e c t i o n s . V i a t h e t r a n s i t i o n 1 1 (and will) use such statements as u: It - I'11g N . Since we are only concerned with ? (luI), we should only consider subspaceswhich are the ranges of projections in M. lt wil\ be useful to consider a s\ight generalization of this notion. Definition 1.1.2. A (not necessarily closed) linear subspace D of Xt is said to be affiliated to M, denoted by D n M, if a'D g D for all ar in
Mr.
n
It follows from the double commutant theorem that if l'1is a closed subspace, then I\nM if and only if py e M. In general, there exists s e v e r a l n o n - c l o s e d s u b s p a c e sa f f i l i a t i d t o M ; i f , f o r i n s t a n c e , t h e r e e x i s t s a i n M s u c h t h a t r a n a i s n o t c l o s e d ,t h e n r a n a w o u l d b e s u c h an example. To deal with such subspaces,it becomes necessary to deal with unbounded operators. In this context, the following d e f i n i t i o n s u p p l e m e n t sD e f i n i t i o n 1 . 1 . 2 . Definition 1.1.3. A closed operator I is said to be affiliated to M, d e n o t e d A n M , i f a t A e A a ' f o r e v e r y a ) e M t ; i . e . ,i f I e d o m I - atAl' I and at e M' imply arE e dom A and Aat\ Observe that f or bounded operators, (the double commutant 'affiliated to 1}4and 'belonging to t h e o r e m e n s u r e st h a t ) t h e n o t i o n s /lf coincide. The following exercises should convince the reader that this notion is a natural one and that it is possibleto deal with this notion by consideringonly bounded operators.
Exercises (1.1.4). Let A be a closed and densely defined linear operator. The iollowing conditions are equivalent: (i) A n M; (ii) A* 4 M; (iii) if A = uH-is the polar decomposition of A, then u e M and lr(H) e M for every Borel subset F of [0,'). (1.1.5). Let (X,T,tt) be a separable o-finite measure space and M = im^:.O e L'(x,tt)\ e rQ26,uD (cf. Ex. (0.4.5)). Show that a closed d e i l s e t y d e f i n e d o p e r a t o r A o n L 2 ( X , 1 t )i s a f f i l i a t e d t o M i f a n d o n l y i f t h e i e e x i s t s a p - a . e .f i n i t e - v a l u e d m e a s u r a b l ef u n c t i o n 0 s u c h t h a t d o m . 4 = ( l e L 2 1 X , 1 r 1$:l e L I ( X , P ) ) a n d A l = { l f o r ( i n d o m l ' (1.1.6). For a closed densely defined operator A, let rp(A) (called the
,(lrrueg E sr reqr,uarulecrdi(1 asoq^r 'les oql olouep t rc1 'oraz-uou arE N Pue W qloq l8ql ',(lllereuaE 3o ssol ou qllm 'aurnssy 'Joord pasop ato
y
T T N to N T w nwla uaqt'n o1 palotlt/ln sacodsqns ptto y1 t1 'toycol o s! n asoddttg 6I'I uogFodor4
""-('Ne
zil)-
D (rN O ril) - fN O ow)l?ql e^oqe pa,rordIJBJ eqt pesn a^Br{ad\ ereq/r 'N -oN = r .,,.
,,
o=u'l
r
I=u')
o'N)e J e L(',vu h) g J de L('-'t,t ( .t*u
,, . o=u I
f .r,
,,
o=u I
0 " N )e e 8e o ' l ^ I )e L('--l^r J L(-N J=w '(,{es) l*"lttil"t:tt*"] Eulleecrdy o, pur'l3.E-tfn''l s ='N'u ='l^['u .l{v
!l leql s^\oqs uollcnrlsuoJ aqJ
u lt roJ N 0 ll| 3llr^\
e^\ araq^\
oNa o}.la)ut '("N o "l^t)fN o ow):( 'O
NTW
'S'I'f uopFodord
.u11 e-'14o D :n w:q.l qcns n {r1auos1 1e1}red e o1 ,{1Euo:ls seEJa^uoc l=j{-rr=t3} lgrll .'srsaqlocl,{q orll Jaqun 'aos 01 f see sr ll 'u N 6 .acuonbeseql '}{ t', JI 'n u"N ''14 oculs n u"N e ''il o lsql a^Jesqo lsJrc 'Joord uY1 "N 'u e uatP * la ro{ -"N e: -ry til -l^l "N '""2'l = -"w /! iasuasSutuonol atti tt ro/ Pue T Pup T ur a^tttppo (1qo7unoc st Q,r1pt) uottolat aUJ 'Z'I'I uopFodord 'W ol pep1.Jgge sacedsqnspesolc olouep s,(ear1e'pagglcads osr^\Jaqlo ssolun 'lll,r\ u pu? g '1y '14sloqur{s oql 'roldBrlc slql Jo lsor aql roJ n. . .('uolllsodruocap 'Gildt - (V)dt relod:luIH) l€rll pu€ n > GVD\-'o)1 = (V)dt Wrqt /yrotls?,{ u y Jl 'y uet oluo uollreford eql eq (V Jo uollceford aEue:
tz
(n pt)
uollulag eql
'I'I
l. The Murray-von Neumann Classif ication of Factors
{(l't, t/,): i e 1} where }t, N, I (0) for each i, \ t l'1,and N, I [/. for i c N for'all i. Propoiition"0.4.l7 * j,4'-- N,for all i, and 4 g It N,-non-void. The set F is clearly I is set the above "nsrrris thdt partially ordered by inclusion, and the union of a totally ordered collection of families of the above sort is again a family of the above sort. So, by Zorn's lemma, the set f has a maximal element, say t(!q, N,): f e 1). Sincelf is separable,the index set 1 is countable' It-follo'ws'fromProposition1.1.7that t't - N where
X=,?,4
a n d!
=,?,Nt'
The maximality of the collection {(ili, l\l,): i e /} and Proposition 0.4.17 -- applied to il e ! and N 0 U-, in case these are both E n o n z e r o - - e n s u r e t h a t M =oIr' 1N = U . : i . e . , M L N o r N t l t
12
Finitc Projcctions
For the rest of this chapter, the s y m b o l M w i l l a l w a y s d e n o t e a factor of operators on !1, the p r i m a r y r e a s o n f o r t h i s b e i n g Proposition 1.1.9. Definition 1.2-1. A projection e in M is said to be finite if eo e ?(M) and e - €o ( e imply eo= e' In the contrary case'e is said to Coirespondingly] a closed subspacel'1which is affiliated be infinite. is to be finite or infinite according as PJvlis finite or M said to O infinite. If 14 L N and N is finite, then !4 is finite; Proposition 122 particular, if any infinite lviexisls, then ll is infinite-
in
Proof. Since finiteness is preserved under equivalence (check it!), assume, with no loss of generality, that l'1g N ' If lvl - lle 4 l'1,then a n d c o n s e q u e n t l y ( 0 )N= 0 N =!te(N g M)-J'tne(N0 x)gN (14^e ( N 0 M)) = l'1 0 iln, thereby establishing the finiteness of I't In pa"rticular, if lt is finite', then every J'l is finite, thus establishing the O c o n t r a p o s i t i v eo f t h e s e c o n da s s e r t i o n . The next result is very crucial in the further development of the theory; it is a sort of Euclidean algorithm. proposition 123. Let I\ N n M and N t (0). Then there exists a pairwise orthogonal subspaces of I\ and a Il of familv { lrl,: i e 'subsiaci R'of 14such that N,, R n M,ll = 1e,rr-Nr) e R, Ni - N lor all i a n d R . 1 N ( i n t h e s e n s et h a t R ' L N b u t R * N ) . If, in one strch decontpositiott, the index set I is infinite, there exists anither decomposition in which the rentainder term R = (0)'
'.,{l1u13ur scrlslrelcBrEr{o Jo Jalrlc o q l a u J r u r o l o po l u o t l l s o d e q l u l A \ o u e r € e l r , : f r e l l o r o c E u l p e c a : d eql Jo (q) lred ece.lerd uuerunall uo^ pu€ i(errn141tlolqa r{lt^\ lueuatsls oql peer o1 Eullrelgp fltueseald ll pulJ lrlElur repear orll '(q) .lo n 'dor4) s ql llnser uralsurag-rep.grrlca J I B q p u o c a s a r l l s a r l s l l q e l s e( g ' 1 ' 1 qt1,vr:eq1eEol 'qc1q,n '(q) Jo JIsrI lsrlJ oq1 soaord srqa 'parrsap s€
'wi
ltg L
Ig l g pu€
-
eug
'g"] t[*. "g'1"]tt
6
ecueq lelqelunoc sr les eql 'elq€r-Bcles sqt ocurs 'rB 1 '6 rg pue lt!o) = ,tg r{l1an e (,!g N ol €'Z'i N taa o1
uolllsodord ,(lctcle'(q) roy l"zg t=ja = g
'(e) erord oa lnd fldurrs 'u
ll?, roJ r g - ug ara{,n '"g t=j* = * uoltlsodruocap e sllrupe W leql €'Z'I ug l=je uolllsodor4 Jo ued puoras aql urorJ s/AolloJ ll tt i teqt pue "g (o) Igl e q l ' z 1 1 er o 3 r g - ' g luql reelc sl ll'(rlt 6*il)"r = * 'tnt 'W o 'Joord i JI 11!itt esoddns os l1e1arrl sr (e) ur uoJlecrldur aug 'luaptmba ato suotlcatotd a7rut[utouy tuo 'tolntrltod u1 :tt I 1. uaqt 'at1ut/u! sl W l! puo W tr N 1^t/l (q) '(g '(g O W)e g =y1uoltlsodwozapDSllLupDW O W)- g -N4t!u 'IAttr 'VZt ^rtuz11oro3 {r [yto puo I! altut{ur st W uaqJ N t 6) ta7 @) 'a t=J. = y uotll'uN lte roJ. U:..llV pue lN *, = lN pue .-.N _' -_; O .F 6 W:tt JI ''N '-;e - y legl 8'l'l uollJsodor4ruor3 apnlJuoC
'Ni"N t6"Tw 'snqllrry E sl rg orolll N T 6 ecurs'rryJooredsqns.rorrrrri '"N '6" o ru 'E"eu-'N ti"t*=* ( ' * ' N - N - ' N o c u l s") , '("''g'z'll = l€rll osllou ueql 1 lEql etunss€ feur om ? .lo ,(tlttqeredas agl Jo Arerl uI 'uollrasse puocas arll roC 'N l U lBqr eq lsnu lI'6'1'1 uotllsodord,{q ',(lluanbosuoc '(lN It''o) I ry / u teqr sernsue{!N) .lo i(lrleurxew eql 'flrodo.rd o^oqE aql ol lcedsar qll/d lErurxeru s1 ,(1$ueg O W = d JI eql ltql ^uedord eql qll/r\ 'N .ol lualurrrnbe qJ€e .14 Jo socedsqns leuo8oqlro osr,nrlecl 3o I r | :r N) fpure; e sple1{ euural s.uroz ol l€oclde,u€ '6'I'I uolllsodord Jo Joorcl eql ul sV '0N - N leql q c n s 1 5 0 1 s l s r x a e r s q l l B q l o s 1 , rT r u l € q l 6 ' I ' I u o l l r s o d o r 4 u r o r S s^\olloJ ll 'es€c e^rleuroll€ or{l uI 1,1= d 'Q = I los 'N } W JI 'JooJd s u o l l c e f o . t go l I u I C ' Z ' l
EZ
l. The Murray-von Neumann Classification of Factors
24
The next few lemmaslead up to a proof of the main result in this section-- that a supremumof two finite projectionsis again a finite projection. Someof theseintermediateresults-- particular Lemma 1.2.5-- are interestingin their own right. Lcnna I25. Lett\ N, B n M,l4tN and B cMe N' ThenI\ N a n d B a d m i t d e c o m p o s i t i o n s l t e= 1I 1112 e ! 1, N = N r @ N , o N " a n d B = Mze N2 e Bo,with \, Ni, Bo n a and satisfying: Mz=!tnB, ils=ilnBI N z = N n 8 , N s = N nB t
xr=il o (l'12e1'L) N r = N e( N z e N r ) Bo={El+Al:ledom.,{}, where A is a closed operator alfiliated = Nl, ker ,4 = (0). Further Mr|ffi r , M 1 ,l 4 a , N ' p l y d e f i n e I 1'= P r o o f . S i m'N,1.
b e 1u"e N = [
to M such that dom A = l1y Bo - Nl. N 2' N, as above, and let 8o =
piq^ia f'= pti'. Rot. that ker(elB; = B n
ret-e
further e[ Bo is one-iir-one; and 6onsequently
) ne 6 ( e 1 10 e B ( i ! , e l ' 1 a n d O = < l , e n=>< ! , , D V €)le
l,tn 8I=il..
It follows that eB = ltr e I1r, and hence that Eo = ilr. Thus el Bo D of l'tt. onto a densesubspace maDS 8^ one-to-one An exictly similar reasoningshowsthat /l 8o maps Bo one-to-one of Nr. Define A: D'Nr by ,4I = fn where n onto a densesubspace Bo s-uchthat er\.= \. It follows at once that vector in is the unique o . = { (+ A l : l e D l ' I i s o n e - t o - o n e , dAo=mF | r , i a n A = N r a n d B -N, means precisely The fact that 8o is a cloied subspacedf t"t, e 9r
-9I(
9r(:
ueql 'slf F zru :(r) ase3 's'z'I Bturua'Iul sB aq, 'og 'tN 'h i'tt 'Joord V
uaqr.N r w t , t u . No wj s p o ' N n t r ? tf .; lTf ] ; ' - l J f # H 'xg ('(s's'z)
oraqa 'g 3o qderE aql ur esuapsr oi o, o.rrrrlser 11 '0 'uollrass€lsel < H roJ leql lcEJ eql poou plnoA\nof 3o qder8 eql aql roJ l0 = 1 ro Q = 11os€cu! 'l JoJ e,r1os' 8d alnduoc ot 2(u'u*V-) + (:p,'t) = 1u*)) lcr{l qcns *ts ruop r t. pue r urop r t anblun B slslxe pue 'V*v ruop ol ereql ? I (ff) JI :lurH) 'g ul asuap,(lluonbasuoc l'ld Bd y go qclerE uortcl.llsir eqt oql sl eEuer oql t?ql opnlcuoJ 3o 3o 'lu€rrE^c,,(eur esecoql s€' NI Jo l'11se palarctJolul eq ol sl I eJeq/
;l
N a' [ :
p u € [ ; : ] =
y(Y*Y +
L vGvv + l )*v
y(v+v + t)
,lr-Gvv+ r)*vv
= g d
l]
=*
'N lerll ^{ogs e W = ;X uorgrsodruo3ep 'N aql ol ]coclsar qlll\ J,f olul Jo +y {uop ec8dsqns osuep eql ruorJ rolarodo reaurl e se aV Eu1,no1rr'(*Z ruop t u :(u'lt*y-)) = r g uoql 'pr 3o qderE eql elouap g lo'I 'rolerodo pesolc e ae N - Ci:y 'N 'O'Z'I) lel pue W Jo oc€dsqns r€eurl esuop E oq C lol e ll = fi le'I )sIcrexl 'og - tg eaord plnon ogdy E 'l(1re1gur15'oslcJexo Eur,vrolyogegl uroJJ pecnpop Jo uoll?Jeplsuoc eq osl€ u€c ll lltllncrggrp qcnu lnoqll^r po^ord fllcarrp oq u€c , ( l r y e n b ap a l r a s s eo q l ' 0 g - T 1 4l € r l l ( q ' t ' l ) ' x X r u o r J ^ \ o I I o J p l n o ^ \ l l
.og= ,ofu u", 0
lsql rr\oqs u?c e^\ J! lril =
gd,
ue, lBrll-lYroul ,(p-ea:1ee11
'n u og 'tN 'fu ecuys 7
u V leql fgrraa ol Jell€ru eullnor B sl lI
'rolerodo pasolr E sI Z l?r{l
suollcaforo 4lIuIJ'Z'l
'L
26
l. The Murray-von Neumann Classification of Factors
B = Boel'tre N2 - Itl e t'1,e N, l x r e I l 2 o l L = 1 1. Case(ii): !1, { N2. to Nt, A as-a closeddenselydefined operatorfrom 1"1r. Regardinglet A+ denote the closeddenselydefined operator from Nt to J"lt which is the adjoint of ,1. Then ,4+, viewed as an operator in 1?,is clearly affiliated to M: further, from the general fact about the graph of the adjoint, it is clear that (J'lre /Vl) 0 B0 = {-n + A+n: n e dom l+). Arguing exactlyas in the proof of Lemma 1.2.5,it may be seenthat Nl - ((q e ^Jl) 0 B0)- Mr; hence, (J'1o N) e B = ((Jvtr e Nr) 0 Bo)oM, e N, I
Nre Nr.
N 3 =N .
The proof is complete,since,by Proposition1.1.9,one of the two casesmust arise. O Irmna 12& If !\ J'le N is finite.
N n M, X l- N, and !4 and N are both finite, then
of Proof. If M e N is infinite, then, by the 'chief characteristics infinity', there exists 8 n M such that B c t"te N and (!t e N) - B - ( ( J v t eN ) e B ) . S o ,b y L e m m a 1 . 2 . 7e, i t h e r( M e N ) I M o r ( M o N ) t i/. The assumedinfinitenessof I't o N would then contradict the finitenessof It or N (cf. Prop. 1.2.2). n Lemma 129. ryll, N n M, then(tl"t+ Nl e N) I I't Proof.
tl'l+Nl 0N =pNl([x+ N]) = t(pNI l: E e ivl)l = ran P
NIPM
- r a n P M P IN gl,t
( b YE x ' ( 1 . 1 . 6 ) )
o
Il l\ N n M, and if M and N are finite, so is Thcorcn f2f0. more generally, tlte suprenunr of finitely many + slightly N l; [Jut projections is /inite. finite
. a s r c r a x es r q l Jo Joorcl e ldurall€ ol z(e,rsnor^qoauo Jo lq8ly eq1 uI I A J o N e J € d s q n sl e u o r s u o r u r p - o u o f u e r o J ' ( N ) , c r = ? a l e q 1 Y \e c = t Q u e q l ' I ' g ' I l u e r o e q l J o ( o ) - ( e ) E u r f g s l l € s u o l l c u n J B s r [ - . 0 ] - ( ( A ) f ) a : r e J l (p) l 1 ' g ' 1r u e r o a q l Jo (c) - (€) suoJlJpuor eql salJslles W rulp = (il)O uollunba aql (c) ' o > urlp !f <+ jtt pt) atlurJ sr l^l (q) l N * l p = W u l l p ( + N - W (e)
:e^ord '(Ah = n rc'I '(Z'e't) seslrrexg 'N TOJ uollounJ uorsuarurp e g Eurllec ,(3r1snfIIr^\ pue .os€c l€rJeds eldurs e u I u e r o a r l l e q l J o f l r p r l e l a g l q s r l q e l s o I I I ^ \ o s r J r a x a B u r a t o 1 1 o go q 1
tuolstto) atrtltsodn ol dn paulwralap t(lanbun st uotlcunl ,Orrn, ?:::r'j:;: '- > (W)O attu!{ st W (c) e puoi(11)o+(w)o=(N ot,t)(+ N Tl^t (q)
:( l)o = (w)o<+ N - l^t (e)
- W)d.
1o4l qcns l-,gf :q uolnunt o stswa ararlJ ..rorco/ D aq n ta7 -11gl rueroeql
'llnser E u r n o l l o ; a q l t u l l o r d ol pelo^ep aq '(W)Oelrr^\ .1Wd;gEullrr.n IIr^\ uorlJes sIrll Jo JI?rI lsrlJ aql llBr{s e,n u€gl rerll€r 'osl€ irolceJ € ol pelsrlr3ge sacedsqns pasolc elouap s,(e,n1u IIr/r\ g pue g 'N sloqru,{s or{l .suorloes Euro8erog aql J4 ul sy 'N T W .ll fluo puu y. ( Nd)q > 0,t4O Eu1,{gslresl_,gl _ (l^t1 4 :dr uorlJunJ B ecnpur plno^r qcns .[*,0] € pasolc a oluo lesqns O Jo -/(t4l)d 3o g usrqdroruosl fu lrqrqxe .Ol)a uo IIrr'A uorlJas srql | .rapro aqt ,fq pacnpur Eureq rapro eql -- n ur suorlcaford Jo sass€lJ acualezrrnbaJo - /04t)d. los perepro 1(Ilelol aqt ,{q peplnord s! n roJ l u e r r € A u r a u o f l r u a l c ' s r o l c € J J o u o l l € c l J l s s € I ce q l r o { . , { r o E o l e ce q l '(froEaluc elerrdorclde aql ul) usrqdroruosr ;o slcefqo lcro^os arll Jo o 1 d n ' u o l l e c r J r s s e l ca q l s l f r o o q l f u e u r s r u o l q o l d c r s s q e q l J o e u g rorlJunc uorsurrurq aql
'€'I
' u o r l r e s s ep u o c o so q l s p l e l f l u a r u n E r eu o r l o n p u l , ( s e eu u ! 9 . 9 . 1 O tuorJ s^\olloJ rualoeql eql Jo uorltoss€ lsrrJ oql lsacedsqns a1ru13 y e u o E o q l r of 1 1 en l n u J o r u n s l o a r r p e s e I N + W ] g o u o r s s a r d x eu e s r 'Joord N @ (N 0 tN + wl) = [w + w] leql 6'z'I eurrue'I uorJ s^\olloJ lI
LZ
uollJunJ uorsueur(J eqf
'€'I
l. The Murray-von Neumann Classification of Factors
.28
Theorem 1.3.1would lead one to seek the abstractanalogueof a one-dimensionalsubspace.One such abstractionis afforded by the following definition. j'l I (0) and if Definition 1.3.3. An 11n M is said to be minimal if N = l t O N = ( 0 ) o r NnM, Ngtlimply It is clear that minimal projections are finite and non-zero;the trouble is that such projectionsmay not even exist in M. The next definition yields a partition of the class of factors into three subclasses,depending on the availability or otherwise of certain kinds of projectionsin M. Definition 1.3.4.. A factor M is said to be of type I, II or III accordingas it satisfiesthe correspondingcondition below: M containsa minirnal projection; M contains no minimal projection, but does contain non-zero finite projections; (III) M containsno finite non-zeroprojection'
(I) (II)
It is clear from the definition that any factor is of exactly one type. We shall prove Theorem l'3'l by treating, in order, the types Iii, I and It. Before doing that, however,it will help to examine the quantitative aspectsof the Euclidean algorithm establishedearlier (cf. Prop. 1.2.3). Proposition 1.3-5. Let I\ N n M; suppose N * (0) and,I4 is finite. Il t't = (@ierN,) e R wit& Nr - N fo.r all i e I and R { N (as in Prop' l.n), ;ir; iidex set I is finite and its cardinality is independentof the particular decompositioncltosen. proof. Suppose14= (o,.-r Nl) e Rt is another such decompositionand ' J which is suppose,if possible,t'trit'ttrere exists a map T: I that R { N ' note r(1); e injective but not surjective. Let "to "I \ ' Ro. Then, R q that Nf such Rn N JJ g. So, there exists r O f
r , r =[ e ru 1n ' ) ,e iet L
'
-l
Ni1'lJe Ro L,?'
c e NJ l; c 1 4 , I j€J
contradicting the finiteness of l'1. Both assertions follow from the non-existence of a T as above for any pair of admissible n decompositions.
iZ < [,N /N] teqt qcns rN oroz-uou3trurJ € stsrxeoraql uagt,Qzlu) ecudsqnsalrurJ orez-uou€ sr N JI leql a,,rordol secrJJnslI .Joord u 11otot olaz-rtoua1ru{ lo t]t"fVi Z < [t+"N /"Nl trrtt t1cnsQ;gu) sacodsqns acuanbaso stslxa ataqt ,ll adtQ to to1cnl o q n II 2.g.1 uunucl 'etutuale ql1,r ul8eq oA{ 'pa^lo^ur ororu 0lllll € sr as?cslqt ul uollcnrlsuor aql 1 ad,{a ' NO( w)O - g'flluanbosuoc pue J,1 erlurg {rozra ro3 ( l,/)OtN /Wl = (tt)O uettl .t.€.t rurrosrlJ Jo (c) - (e) Eurfgsrl€suorlcung{ue sl O JI (q) pue.I.€.I rueroaql Jo (r) - (e) suolllpuoc serJsll€s NO (e) 1eq1{3poa 01 .,rou ,fseo ii lI 'l pre) = (W) NO osBJraqlle u1 .acue11.Ol.Z.l rusroaql fq ,elrurg sI il uaqt 'allulJ sI 1 Jl 'pueq roqlo aql uo lelrurSur sr X ecuaq^\ 'w5"Ntru-"N t6"=N @ tu'u' '( "" 'z'I) = d.es'e11urgurst ielqeredas sr oJurs olqelunoc sr / les eql 7 ' 0 = d l u q l e p n l c u o r1. 1JI€ r u r u r u r s l Jl N a i u l s _ . 11 1 r r o 3 ! 1 -y N l U l e q l qJns W go uolllsoclruoJep s eq U e (tN trte) = ll lal,n U W fue rog 'orlutJst
W JI
olrurJursl l.f JI
'[N '
/}{] -
=,*,no ]
eurJop puc .letururur eq N lol
.1 e
'CI= t O s n q l p u € { 0 } I W J I - = ( i l ) r c r l e q l s o r n s u a( c ) 'III ad^l .(q) = sr. eculs 16 ({o}), = os fq ({o)), (oD,a n rcqr Jo cI a(, pue (c) l(q - > ({O}),4' uaqt'(c) - (e) EurfSsllss uollrun3.,(ue sr ,g 'flasre^uoC 'I'€'I ruoroaqJ Jr Jo (c) - (e) suolllpuoc aql sorJsll€s Cr l€ql r€alc sl ll 'elIuIJuI sl r'{ u W oraz-uou fre,re .srseqloclfq ,(q .acurg
' ( o ) * x . l l' - l I (o)=lrJI 'o)
= (w)o
eurJap 'eJuelsrxe roC .Ut addt 'I'g'I tuaJoaql .luaprcc€ u€ Jo goord oql ol peaoord /r\ou sn la.I lou s l ( I + u > / ) u J J l u = [ l ] ) u o r l c u n ; r a t e l u r l s a l e o r Ee q l r o J u o l l e l o u 3{l qtl,n flrrelrrurs eql os ' N rulp^r lurp p333xa lou saop qclqil\ reEalul lsalearE eql sl I w /Nl ,0Ih = 7g eldurexe eql uI lsr{l ?loN ' g ' g ' 1 ' d o r 4 u r s € ' . 1 rp r e c r e E e l u r p a u r r u r e l e p flenbrun eql elouop O 'ellulJ pu€ oroz-uou rlloq dte n tl N T JI .9.€-I uolrlulJeo t ry/wl lal
uollsunc uolsualur(eqJ '€'I
6Z
30
l. The Murray-von Neumann Classification of Factors
then the l,l's can be inductively defined. Since N is not minimal (M being of type II), there exists B nM such .that (0) * B .F N;-th9 iinit".nrtr #N tnto..s finitenessof B. If t N/B I ) 2, set N' = 8;-if . I oRwithR{ B; i r u l r f = | - - n o t e t h a t l N / B l > 0 - - t h e nN = a n ds e t N ' = R ' D i u r t h e r R l ( 0 ) s i n c e B* N ; n o t et h a tI N / R ] > 2 Definition l-3.8. A sequenceS = 1N,,)l=, as in Lemma l'3.7 will be U called a fundamentalsequencefor the type II tactor M' The following bit of notation will facilitate some of the ptools: let us agreeto write kN for any subspaceof the subsequent form N, e ... e Nn,with N i - N for all i. Thus' for example,if M and N dre finite and non-zero,then
tFlN, m,[[F]. rlru Lcmna
1.3.9. Let I\
N, 8 be linite and non-zero'
(a)tFltFl [*]. t[#]. 'l[[F]. '1, < r M e BI ' l' Ll ixl -1l * 1ft1., I F ] . t +f - nl (b) il 14I 8 , then
Proof. Note that -tM/N I is precisely the largest number of pairwise orthogonal copies of N which can be f itted into l"t The f irst Turn inequality,of both (a) and (b), is an immediateconsequence. to the second: (a) The inequality tB / Nl > (tBlltl + lXtM/ Nl. + t) would implv t h L ' e x i s t e n c e ,i n s i a e B , o f ( t B l J ' l 1 + I X I M / r y 1 + 1 ) p a i r w i s e if ([ B/yJ * l)-pairwise orthogonalcopiesof N, and consequently, (tM/NI + l)N) which is a (since !t 1,1 { of copies ortholonal contradiction. (b) By the parentheticalcommentin the proof of (a)'
, , t e B . r [ r.I' -. ][ + ] . r)l r u. ( S t r i c t l y s p e a k i n g , t h i s i s v a l i d o n l y i f t h e r ee x i s t ( t M / N l + t B / N I + i) pairwise orthogonal copies of N in lf; if that is not true, then lf must be finite, with
[F]. [F].[*].,,
in which case the desired inequality follows since [R/ N] is clearly e I from that of monotone in R.) Since finiteness is inherited by 1"1 l ' 1a n d 6 , t t r e d e s i r e d i n e q u a l i t y c a n n o t b e f a l s e . E proposition f3.fg.
Let {N r,}i=r b" a fundamental seqttencefor M and
='[+],1=t[+](') ''[-*]'[#] ":Hl='[-fi] ''[*]
=
'[* ]
€ N-W
(l)
Euratroylog aqt f;sgtzs ol uaas frrs'a sr 56; uortcu"J ".,;t:""i1;p-;;; pu€ elruu are N pue g l,t .lt '(A| Ot'e't 'dor4 ,(q paaluerenE sr ecuelsrxeesoqa llruJl erll aq ol )(g /W) ourJap 'oraz-uoupue olrurJ ere g puu x JI 'w toJ ecuenbesleluourepunJE eq t=j{" ry; = -I'g'I "uua'I Jo Joord Jo puA S pue II adf] Jo rolceJ e eq n lc'l 'O <'o ruJIl€r{l ueos n f g 'ellurJ sl pue slslxo ll,,'g pue W J.o solor oqt EulEueqcrolur ,ll -rc-t-rull ecuaH '-o Jul urll > "rc clnsur1 leql opnlcuoc'tr Eurfre,r fq I
-n
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d
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l€rll (e) lred pue flrlenbeur e^oqe oql ruorJ pue 'qEnoua oErel a JoJ o > ')c > 0 lerll (e) lrrcl IuorJ s^rolloJ tI '["N / g|/t" I /Wl = 'n Eurlrr16 'l < 4 ra8alur fue ro3
[ t * " N/ ' N ]
I ' N/ e i
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(*)
'I < ["N /g I r"ql eErelos sr u JI l€ql aas e^\ '(c) 6'€'t €turuel u1 seytllenbauleql qtoq ol Eurleaclcly(q) 'perrordsr (e) puu ,1+or,..ot , . {*ou a 'r-,rzI l-"+ll#| . l-=ll-l
-UNJLW L J L W I '7 raEalur,(ue rog 'usql "N 'ou 't 'os i o r < [ " N / p ] s p : o , nr o q l o u l : t ^ tT
'[4],.",. [+]t4l. l # l
a^Br.I3 , h ' I < u f u e r o 3 ' ( e ) o ' g ' t eruuel
'raqlunu artttsod aty{ o s! puorrrr""
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ral
uollcunc uolsuelurosrII '€'I
I€
l.
32
(iii) l'1i I t
The Murray-von Neumann Classification of Factors
rlte 8'r
rll'l
LfJr=
Lt[
.
rBt
(b)) (useLemma r.3.e Lnf ;
(t4 I r8'l +lp-l^< lvl-. L'' JS L'' JS
( i v )l ' _{18
Now fix a finite non-zero I and define
II o
,
if!1=(o)
if lt is finite and non-zero Dg (x) = | (x/ I )S , if 11is infinite' ' L It is readily verified that Dg satisfies the conditions (a) - (c) of Theorem1.3.1. Conversely,if D: P(M) '[0,'] is any function satisfying(a) - (c) of Theorem1.3.1,it is clear that for finite non-zeroM and N,
. [F1. rlotruy; [h ]r,N) ( D(]t) L''
I
for finite non-zero!t and n = 1,2,..., consequently, D(M) . il{/ N.l + t ll't/ N,l -. t B / N " l +I D ( B ) 1 B l N" l '
let n - - , recall that I B/N"] ' o and conclude that D(X) = D(8 )D3 (m). Since D(!t) = DS (lv1)= o for infinite l( conclude that
D =D(x)Ds. o
Proposition l3-f f. Let M be a factor and D; P(tnO ' [0,-] as in Theorem1.3.1.Then, (a) M I ^J e D(11)< D( N); and (b) D is countably additive - i.e., t,f (Jtl")is a sequenceof pairwise (nl4)and if 14= 6t4n,then D(M)= ID(M"). orthogonalsubspaces Proof. The conditions(a) and (b) of Theorem 1.3.1imply that l'l I N ) D(l't) < D( N); coupled with (a), this yields: I't < N + D(M) < D(N ); thus, lt I N (resp.,I't I N) implies that D(x) < D(N) (resp., D(l't) > -N and M N are mutually } { N,x D ( N ) ) . S i n c et h e p o s s i b i l i t i e s M = exclusiveand exhaustive,as are the possibilitiesD(J't)< D( N )' D(1"1) (a) in follows. D(N ) and D(X) > D(N ), the reverseimplication For finite sequences,the assertion (b) is a consequenceof the assumedfinite additivity (cf. (b) of Theorem 1.3.1)of D. Assume, then, that the sequence(11")is infinite and that Xn I (0) for all n. of D show that, for all N, Finite additivity and monodonicity N
"!,
f N I D(il")= '1"9, ,'lnj< D(x)'
consequently ID(JV1")< D(M). If possible, tet Ib(il") < D(M). Then ID(Jv1,)< a in particular' for each e > 0, there e*is'dsa finite non-zero N"n M such that D(N) < e'
'a^oqo so aq j'q {o auo {1tto puo auo s, V uaqJ
la7 I
:s1as3utuo11otaql 'rl'E'l uoglpodor4 ieslJrexl
'Joord
tD3 pup 'V "rc3 "' '4o'rp , v ) € rc > p u o i yt g - r c € D > g puv vrBto
l[='o]j v
(c) (q)
(u)
'uaqJ '(A)C = b 'e '€I-€'I "unurf tal puv a^oqv so aq V ta7 '{W u W : ( W ) O )= V l e s e q l r o ; u a d o o r e l € q l s a 1 t l 1 1 q 1 s s o d a q l E u r r a p r s u o c, { q J e q l J n J e l l l l l ? s r s { 1 e u e e q l e n u r l u o c s n l a . I (eerns€atu D r€8H o^ll€leJ € Jo pr€eq re^e s€q oq/tr :Eu1aq acuoullredrul qcns roJ uolleclJllsnf ouo ',,aArg€ler,, elrlcefpe aqt qll^r osuedslp o/r\ l,,uor1cun; uorsueurp elrlelor,, B lr II€o uu€runeN uol pue ,{errnq) 'n Jo uollJunJ uorsuaurp B pallec sr -- iruarll 3o fueu ool lou 'Z,['E'I uoTllulJeq er€ oraql -- I'g'I ruaroorl.1.ul s€ cr uollcung fuy 'Joorcl eql seltlduroc E uotlotp€rluor slqJ '(W)O > ("W)Og- (W)O = r > (w)O > (y)g acuaq ' o ^ l l l p p e i ( 1 q e l u n o cs l - e c u l S pu€'N i l14e - !f lulll epnlcuoc ' p s l J r r e ^ s l u o l l r e s s €a q l p u B
'[i*'E'] oNSr+ul.[6l+ull l€ql
qJnsJrtlu t''l u al,rslsrxoaraql os
i(r+"w)o< (fw)o"if
( , h ) o ' i r(-N ) o= [ [ ' - ' E] ur ] o ::*rt#,* i*rwT^:#.; >.f>I > r ror ,rw r ,h,"r^rrr;;3'1,:rffi'r:rl - I11 y'g
'g 5
le{l qcns
u
slsrxa oreql os pue
}
so11dtu1
N 't'r^ u€rs fy (,')o'Iw(r'u)o :*,,jr:ijrl;rjr";,1"",1t#*1i"r"",::l; leuo8oqlro asymrled Jo tlW) ecuanbes € slslxe erar{I :uollressv '( '{N+'l^l},(q {"w} Eulcelcler l' )o t fw)og leql -- 1gaErel roJ fq -- ,(lryerauaEgo ssol lnoqlJ/vrorunssei(eru en 'o5 'N qcea roJ N=u N=u') ,, -l"t,t "* ("w)c"3 lo = ("w)o3 - (t,t)( (,,
o
L
r=u
o
)
@
' a ^ I l l p p e { 1 a 1 t u t 3s l leql olou Creouls '("W)O3- (W)g > , pexrJ B roJ N uE qcns >1crd uollcun{ uolsus{ul( eqI
€€
'€'I
34
(I-) (I) (IIl) (II-) (III)
l.
The Murray-von Neumann Classification of Factors
(0, e 2€, ..., nZ), where 0 . ? < - ; (n = 1,2, .'.) ( n 7 : n = 0 , 1 , 2 , . . . , * )w, h e r e0 < ? < [0,4, where0
Proof. We consider, separately, the three cases corresponding to the possible type of M. Case (i): M is of type I. Let N n M be minimal. We have already seen (in the proof of T h e o r e m 1 . 3 . 1f o r t y p e I ) t h a t a n y 1 4n M i s o f t h e f o r m J " t= o i er N i , w i t h l c o u n t a b l e a n d N , - N f o r e a c hi , a n d s o A g { n 7 : n = 0 , 1 , 2 , ...)U {-} where ? = D( N ). It is easy to see that (i) if lf is finite and l l t / N l = n , t h e n a = t k 7 : k = 0 , 1 , . . . ,n ) a n d ( i i ) i f X t i s i n f i n i t e , t h e n A,=@7:n=0,1,...,-). Casc (ii): M is of type II. S i n c e M h a s a f u n d a m e n t a l s e q u e n c e ,i t f o l l o w s t h a t A d o e s n o t contain a smallest positive number. Let d = D(Xf). Infer from L e m m-
n
Definition f3-15. A factor M is said to be of type I,, I* IIt, II- or III according as the range of the dimension function of M satisfies t h e c o r r e s p o n d i n gc o n d i t i o n o f P r o p o s i t i o n 1 . 3 . 1 4 . F a c t o r s o f t y p e I , (n < -) and II, are called finite, and the other types are called infinite. (Thus, a factor M is finite if and only if Xf is finite (rel M).) Factors of type I or II are said to be semifinite. (Thus a factor M is semifinite if and only if it contains non-zerofinite projections.)
0
Examples will be given later in Section 4.3 to show that factors of all these types exist. No treatment of the Murray-von Neumann papers would be complete without at least a passing mention of the so-called reduction theory, whereby every von Neumann algebra is expressed as a direct integral of factors. Very briefly, one uses the abelian von Neumann algebra Z(M) to represent the underlying Hilbert s p a c e a s a d i r e c t i n t e g r a l o f H i l b e r t s p a c e so v e r a m e a s u r e s p a c e i n 'scalar decomposable' operators. such a way that Z(M) acts as ( A c t u a l l y , t h e u n d e r l y i n g B o r e l s p a c em a y b e t a k e n t o b e ( a c o m p a c t
pus ;ilXT::l u€ paur€luoo-Jles sr se {roaqr aql Jo uolrgsoctxa
sE roJ p€oqur?lunoJ al{l ol ,{llcarrp oE plnoqs ropeor pelserelur srll ',{e,r,{ue uollces eql paeu lou saop repeor palelllul eql (c) pue luorlerEalurslp Jo uorssncslp snorras s oIII Eu1q1f,ue ur olqedecsoul s! teqt ,ttlllqerns€3tu-uou 3o erlcads aql qll/$ rop€ar pol€lllulun oql lualurol ol ,(resseceu ,{11eerlou sr 1r (q) i4ooq oql Jo rapul€Iuer oql ol lueuruad ,i.11ear lou sI lerraleru aql (B) :sluno3 Eut,vrollo; eql uo ,,'o1 1ou,, :o3 paldo Joqlne oql ',(roaql slqr Jo uolllsodxo al€roqsle erou € ol Jelderlc s l q l u l u o l l o e s B a l o ^ e p o l l o u r o r o q l a q ^ r u o E u l l B J a q I I e pr e U V "e'1) q3eo 'uo os pue rII oclfl JI 3o sI (\)4 ('a'e 'ecuelsur roJ 'rII ad6,1ya1 11eclerqaEl€ uu€runeN uol lereuaE B Jo ed,{l f e u r a u o ' { r o a q l s l q t E u t s n ' - > l l ( . ) ] c l l d n s ' s s a= l l r l l eql 4 e a d s J o (" Eurf .;sylespuB asuos (4ue,tr) alerrctorctcle ^.uBul eiQern$'eauEulaq ;x dsru eql'(x)n t (1)x eroq,n'(\)7,p(\)reJ = x ulroJ oql Jo srolerado '3su3s u!€u3c e uI ,,elq€rns€alu,, JO uollJsllos aql sl w wql os Euroq (1[,g - 1]ueruu8Jsse eql'((\)fih f (r)Ztt ro1ce; e'1 qcee -ro3'sr araql'(1)rlp(r)A-l = lf JI lBql ^roqs ol uo seoE^roaql eql ( ' l e r c n 1 5 ' , ( 1 1 B leoru s I l e q l l n q - - r o l € r a d o t u l o f p e - J l a s a18u1s B ^q palerauaE sI oc€ds ueqIIH alq€reclas € uo Eullce 'eut1 erqaEle uueruneN uo^ uelloqe fra,re eculs lear aql (Jo lasqns
s€
uorlJunc uolsuelulceql
'€'I
Chapter2 TH E TOMITA-TAKESAKI THEORY
Section2.1 discussesthe following question(which, in the caseM = L'1X,7,y) with g finite, is answeredaffirmatively by the existenceof the Lebesgueintegral): if m: ?(tuO- [0,1] is countably additive in the sensethat
= -["!,""] @
t
n=1
m(e^)
for any countable collection of pairwise orthogonal projections in M, does n extend to a linear functional on M which is well-behaved under monotone convergence? Section 2.2 is devoted to the celebrated GNS construction, which, in case M = L'1X,r,p1 yields the Hilbert spaee Lz(X,r,p1 and the r e p r e s e n t a t i o no f M a s m u l t i p l i c a t i o n o p e r a t o r s . S e c t i o n 2 . 3 i s c o n c e r n e dw i t h t h e ( c o n j u g a t e l i n e a r ) o p e r a t o r o n t h e GNS space which is induced by the map x - x* of M. The climax is the Tomita-Takesaki theorem which involves a thorough analysis of the antiunitary and positive factors in the polar decomposition of the above mentioned operator, and their commutation relations with the operators in M. One crucial fact emerging from this theorem is t h e e x i s t e n c e o f a c e r t a i n o n e - p a r a m e t e rg r o u p ( c a l l e d t h e m o d u l a r group) of automorphisms of M, which is of fundamental importance when M is of type III. As the proof of the theorem is long and technical, the proof is presentedonly in the very special case when t h e a b o v e - m e n t i o n e do p e r a t o r i s b o u n d e d . A l t h o u g h t h i s c a s e n e v e r arises when M is of type III (as will be established later), this option has been taken as a compromise between no proof and complete proof, both alternatives being distasteful to the author. Section 2.4 introduces weights, which are non-commutative a n a l o g u e so f i n f i n i t e m e a s u r e s . T h e r e a r e f e w p r o o f s i n t h i s s e c t i o n . The results are stated and some tentative effects are made at convincing the reader that surely the stated results are plausible enough. The statement of the Tomita-Takesaki theorem in its full
.n ut D f IIB roJ (*rx)Q = (x*x)p JI l€rcerl (lll) :(tt'l'O 'dor4 'gc)-n u\ {tx) 1euEulseorcuJ ouolouou s Jo runuerdns aql sr x rolauaq^\ '(lx)O ldns = (r)0 .ll l€rurou (ll) i0 < (x)0 serldrur-n t x I 0 Jl InJqllBJ (l) eq ot pr€s sl /,t/uo 0 leuollcunJ r€eurl errllsod v
'Z'I'Z uolllulJeq
('acuelsulroJ '[I ,rry] u1 punoJ oq feu syqt E 'f1r1uapleqf l€ rurou sll sulell€ 3o goord e lo,rrlrsod{lluclleurolne sr qJIq^\ 'n uo uollcunJ r€eull pepunoqfu€ luql 'fyesranuoc'anJl sI lI) osn pue (q) ul I = t lnd :1urg) '(t)0 '
' - 6 3 ( ' W \ f r g 1 ' f 1 l u e l e , r r n b e ' r o ' I 4 l u r .x 0 qcns Jo uollcelloc oql lle roJ 0 ( @*xh JI e^lllsod eq ol pr€s sl .jrtluo 0 leuorlcunJ r€aull v '3q plnogs foql asrnoc Jo s€'luel€,rrnbo are g 4 x pue-y,7 t n'eJu€lsur rog'snq1+n, * - d u a q , t rf l a s l c a r dr ( > r f q e r e q m q/,{ '.;ry .rolcol Iuer er{l uo Japro u€ ecuds 3o slueuala 1u1o[pe-Jlos Jo ''n potouap oq III,n.;4r ur srolerodo saulJep -yg euoc eaglgsod aqa fq 'eceds o,rll1sod Jo uollcelloc eql UaqllH alqerudes e uo srolerodo Jo erqo8le uu€runoN uol € olouop s[e,rn1e ilr.^ n loqurfs arlJ uorlsr3clul
aapslnuuocuoN'I-Z
'lsrxa suollelcedxe l?uolllpuoc lerulou qJlqa ur suollenlls esoql serJrluepr qJIq,t\ rueroeql s.lI€se{3I pu€ suollelcoclxa I€uolllpuoc (q) pu€ 'l{€sa{€J pue uesreped Jo rueroeql ur,{po>11p-uopeg e^lt?lnuuocuou eql (e) Jo uolssncsrp € qll,r\ spua ralduqc eql 'tqE1a,yrB qll/tr pelelcosse dnorE rslnpou oql go (uo11cnrlsuoJ SNC eql ol lgadd€ lou seop 1eq1) uollusrrolJer€qc crsulrlur ue sealE qclq,n 'uollypuoc f .repunoq Sn) aql pellec 'uorrelrrJ IEcruqcel InJ3sn l(Jr^ € ol sulBUad uollcos lxeu eql 'uolloes srql ul ecuereadde uE so{€ru osle [111erouaE
uollerEalulalll€lnruuocuoN'I'Z
tc
2. The Tomita-Takesaki Theory
38
Exerciscs (2.1-3) If 0 e M|, show that the f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : (i) (ii) (iii)
f is tracial; Q(.xy)= O(yx) for all x,y in M: Q@xu*) = 0(x) for all x e M and unitary u e M.
(Hint: Use polarization for (i) + (ii) and prove (iii) + (ii) by re-stating (iii) as Q(ux) = b?u) and using the fact that the unitary operatorsin M span M as a vector space.) (2-1.4) Let M = L-(X,T,p), and let 6 e W. with m,y, in the notation of Ex. (0.4.5).)
(We have identified
0
( a ) T h e e q u a t i o n v ( E ) = 0 ( l o ) d e f i n e s a f i n i t e l y a d d i t i v e m e a s u r eo n ( X , F ) , w h i c h i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c tt o p . (b) Q is normal if and.only if v (as above) is countably a9ditive, in which case 0(,f) = f lc dpfor some non-negative g e Lr(X,v); (c) 0 is faithful if ind only if p is absolutely continuous with D respect to v. Exercise (2.1.4) (b) is a special case of a more general fact: a positive linear functional on M is normal if and only if it is o-weakly continuous. We shall omit a proof of this fact '- one may be found in [Dix], for instance -- but will freely use it in the sequel. The collection of normal positive linear functionals will be denoted by M* *. It is not too hard to establish that M* and M*.* are dual conesll i.e., if x e M, then x e M, if and only if 0(x) >'0 for all 0 in M* -, and similarly, the dual statement(with the roles of 0 and x interclianged) is also valid. (As above, we shall think of the elements of M* as linear functionals on M.) Exercises (2.1.5) Let 0 e M*.*, let x e M and let Mo = {x1", the von Neumann a l g e b r a g e n e r a t e db y x a n d l . (a) If x is normal, the equationv*(E);0(lB(x)) defines a measure on the spectrum of x such that J I dv* = (/(x)) for every bounded Borel function / defined on sp x. (b) For general x, show that if 0 is tracial, then, with the notation tr of (a), vl*l = tl**l . T h e r e a d e r w i l l n o t i c e m u c h l a t e r , i n S e c t i o n4 . 3 t h a t t h e v e r i f i c a t i o n t h a t a g i v e n f i n i t e f a c t o r i s f i n i t e w i l l i n v a r i a b l y b e a c c o m p l i s h e db y f i r s t c o n s t r u c t i n ga f a i t h f u l n o r m a l t r a c i a l p o s i t i v e l i n e a r f u n c t i o n a l r on M, and then observing that the restriction of r to P(M) yields a d i m e n s i o n f u n c t i o n f o r M w h i c h a s s i g n sa f i n i t e v a l u e t o l .
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'It'Ol - (W)d :Lu rsarns€eru,, e^lllppB f lqelunoc {11,n uuql ror{lBr 'sleuorlcung r€eull o^lllsod (lerurou) rlll^r {Jo^\ qUoJocuoq II€qs a^\ .spJo^\ rarllo ur iarnseau oqr qll^\ u?rll JorllBJ lerEelur aql qllrrr {Jo^\ o} osoogc 11uqse,n ,og 'e = ll lurp uaq^\ luesard ore uralqotd eql Jo sslJ€clJlur er{l II€ 'oroq uana -- uoseelg ol enp ueJoeql (1er,rrr1-uouflalrur3ep pue) pelerqoleJ € ^q pallles flanrleurrJJ€ sl .(nh = n dse) lercads fre,r eq1 u1 '1,yuo l€uollounJ rBaurI aaltlsod I€rurou e o1 (n serrleruosr IEIIr€cl roJ Qnn)w = (n*n)u turi(gsrles flrrussecou lou) [I.0] U4I)d :u uollcunJ 3^lllppe flqelunoc leraua8 e Eurpuelxa 3o urelqord oql (q) pue 1,,11srolerado 3o sEurg,, uI lel€l teo6, e fluo pa111espue ',,sro18redogo s8ulr uO. ul I I urelqord sB pelels sr slql lBrll lc€J aql ,(q pacuopr,re s€ 'uu?runoN uo^ uo^e paxoJ flrrerodural lseel le a^€q ol suees rolc€J olrurJrruos € roJ ualqord Eurpuodserroc eql (€) 'uelqord slql eql qsrlqelso ol acrJJns plnoqs Jo .,{.1r1erarr1-uou suoll€clpul on1 Eur,nolloJ aql 'lueredsuerl -lo elslparurur sueoru ou ,,(q sr uollnlos aql 're,realo11 'e^oq€ pelsaESnsrauu€tu eql ul peulJep oq lsnu r Eugllnsor eql pue 're/rrsue e^rlerurrJJe u€ e^€q peopul seop qcler8ered eq1 Jo lrels eql le pelsa8Ens ruelqord eq1 .n uo IsuollcunJ r?aull I€rurou o^llrsod e ol spuelxa pue ,q;4ruo leu-ollcunJ r e e u l l I B a r e l a u l J e p r l € q l q s r l q e l s oo l _ l d u a l l e p u e ( 1 ) * n p 1 = ( x ) r J '((r)st)O = (g)*n fq r ds uo *n les uo_ql ernseau eqt eur3dp pynoc ouo 'q.;tg, ,( JI leql lsaE8ns sosrcrexe Eulpocard aqa .1e!c"r1 p,re 'slsrxa 'rtclq^\ 'n uo InJr{ll€J eq f11ec1}eurolneplnoa ll JI luuorlounJ r€aull alllysod I€rurou e ol spualxe rolc€J elruU B Jo uorlcunJ uolsuerulp aql reqleq^i ol sB urelqo.rd asraauoc eql /hou raprsuoJ
uollrnrlsuoSsNc eql'z'7,
6E
40
2.
The Tomita-Takesaki Theory
(a) n6 is a t-algebra homomorphism of M into f,QtQ, tt6 being a Hilbert space; (b) oO e !t5 and fr6 = ,6(Wa6 : and (c) 0(x) = <26(x)op'o6'for all x in M. Such a triple is unique in the sense that if (lft,tt ',o') is another such triple, there exists a unique unitary operator w: tl1 - !l' such that wa6 = Qt and n'(x) = wn5(x)w* for all x in M. Further, the imaie n5(M) is a von Neumann algebra of operators on tt6i n6 is norm-preserving as well as being a o-weak homeomorphism of M onio n6(M). Proof. If x,y € M, define [x,y] = Q(y*x). The positivity and faithfulness of 0 ensure that [.,.] is a genuine (positive-definite) inner producton M. Letlt5 denote the completion of Min this inner product. Let n: M -eO denote the inclusion map, and let n(l) = oO. If x,y e M, note that'
= o(,y*r**nl y*y)= llrll' llntylll', < ll'll'o( lln(xy)ll' wherewe haveusedx*x < llrll'r, and so y*x*xy < y*(ll"ll)'r)y. So, there exists a unique bounded operator z6(x) on 1f6 such that n6k)n(il = n(xy) for all y in M. It is easily verified that z6 is a *-'algebra homomorphism of M into t(lt5). As a sample, we show that n6 preserves adjoints. For this, note that n(M) is dense in Xf6, and tliat if x e M, then for all y,z in M,
; conclude that z6(x*) = z6(x)*. The assertions(b) and (c) of the theorem follow iinmediately from the definition of o6 and the inner product in 1f6. If (lfr,nt,or) is another such triple, note that for x and y in M, < t l | ( x ) Q r , z t ( y ) O ' >= < I r r ( / * x ) O ' , O t >
= Q(y*x) = i
=.116and i1@N sincei611tr;116
=^tt', deducethe existenceof a
! 166 ,-+ lf ' such that .wnq(x)o6 = (w weerlrl - odeelTr ni nee( ld, ,) 'u unri t adrryy o ppserraat tuor r w n .: n
o tn.56(\ x. \)) = n ' ( x ) o w , s i n c et h e i w o na t w Q [nI'l(' x( x) O . r rf r '.. Ir t ri ss If a ri rrlryy sc rl egaarr t h n6(M)a , b o t h o p e r a t o r s m a p p i n g set ,ld(1vllrrd the qense dense sEr agreg on tne operators agree
= nt(l)Or = Or, and the n60)A6 to r'(xy)Or; also,,. AO= w.'n6(l)O O
se'cond-partof the theorem is proved.
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'Utt)u owo n Jo rusrqclrouoeuoq e sr pue pasolofl4earrr-osl 0{)u .crJlourosl u {€e^\-o st u ucql'(,$)t olul rusrqd.rououoq-*Ieurou aaltcafu1 n Jo u€ sr (r$h * m :u Jr.l_uqtEul,rnoqs ,{q goord aql .elcldruoJIIBqseA\ ('palressB se '(x)92t (tx19yleql epnlcuoc.9lX= @)u se t((lx)u) leu aql sr os 'pcpunoq flurro.;run sr (!r) lau erll ecursffir r( 11ero3 1ea,n'(x)u'ji";; ucql'x o1 d11ea,r,soSra,ruoc rlolrt/rr,*ttl ul leu Burseercureuolouoru e sl^{!r}JI ler{l esuasaql ut 0u,Oeu,agl .raqlrng .o^rlcaful ;euriou,sl sl vz snqr ig = r lB{l 'O = ,ll9u(x)Pu;;= (x*x)Q^uorlenba etll pue 0 Jo ssaulnJr{lreJ or{l ruorJ aphlcuocu5r{l .0 = (r)Pa .;r .,{11eurg
tv
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2. The Tomita-Takesaki Theorv
A'\
It may be relevant to point out here that, more generally than was established in the above proof, it is true that an injective *-homomorphism between two (possibly abstract) C*-algebras is isometric; it may be inferred from this -- by passing through an appropriate quotient algebra -- that a *-homomorphic image of a C * - a l g e b r a i s n o r m - c l o s e d . T h e i n t e r e s t e dr e a d e r m a y c o n s u l t [ A r v l ] for details. We shall, henceforth, feel free to talk of a *-homomorphism being n o r m a l - - m e a n i n g t h a t i t p r e s e r v e sm o n o t o n e l i m i t s - - a n d t o u s e t h e the Theorem, that a f act, emerging f rom the proof of *-homomorphism is o-weakly continuous if and only if it is normal. It is true, as in the C*-case, that the image n(I() of a von Neumann algebra under a normal *-homomorphism is o-weakly closed and hence a von Neumann algebra. The proof of this assertion is outlined in the following exercise.
Excrcises (2.2-2) Let I be a o-weakly closed two sided ideal in M. (a) Let x e M have polar decomposition x = ulxl. Show that the following conditions are equivalent: (i) x e I; (ii) lxl e 1; (iii) l1n-1(lxl) e 1. Conclude from (i) <+ (ii) that / is self-adjoint. ) nd so (iii) ) ( f l i n i : l x l = u * x a n d s o ( i ) e ( i i ) ; x = x l 1 o . - 1 ( l x la 'air'appropriate = p i c k Borel ( i i ) + ( i i i ) , (i); for rn , f , , ( l x l ) ,f o r function /,, such that lxlyn = l{r7n,-1(lxl),and note that 1 1 r 7 " , - y ( l x l ) ' 1 1 6 , - y ( l x l )o - w e a k l v ' ) (b) If x e M-, then xr/n - l1n-1(x) o-strongly. (c) lf e,f e ?QO n /, then'i'V 1 = o-stronglim(e + ./)r/" and so e V .f e I: in particular, P(lrj n 1 is an upward directed net (which is non-trivial if I t (0)). (d) If e=V{f e P(M):f eIl, thende IoZ(M). (Hint:del,in view of the secondstatementin (c) and the o-weak closure of /; to see that e is central, note that if f e P(ttO n l and a is a unitary element of M, then ufu* e P 114 n 1, and conclude that udu* = d.) (e) Show that I = M€, and that conversely, if e e P (Z(U), then Me is a o-weakly closed two-sided ideal of M. (Hint: The second assertion as well as the inclusion I 2 Me are trivial; if x e I, t h e n b y ( a ) a n d t h e d e f i n i t i o n o f d , l 1 s , * ; ( l x l )( Z a n d h e n c e x =
xa.)
(f)
If n: M - t(lt') is a normal *-homomorphism, then n(lv[) is o-weakly closed and hence a von Neumann subalgebra of t(t8r). (Hint: Apply (e) to write ker n = M€, note that M(\ - O may be Ngs\r'l, trtt"'.g srr \sr L, \\l\ \\\S\ \'rg\r g\ ts'a lsrr\!\\\\\
snql pue '@)Qu qll,r\ l{ l(gr1uap1II€qs e^\ 'n uo O luuollJunJ rs?uJl anlllsocl I€rurou InJqlrBJ u uarrgEore ?^\ ueq,n '1ou ueql uelJo oroIAI 'a1du1 S N C e q l J o u o r s r e ^ e u o .s p y a t { s l q l l E I l l a l E r p e I U I u r O s l l I ' t o x J o . 9 t t o t u o l l c r r l s e r e q l e q ( x ) O u t e t p u e 1 ;@$ 3 l ( n , x :9u(t o x))l = 9g let pue (q) ul s€ eq 0U let 'esec srql u1 'snonJEA lou sl I'Z'Z rueroeql feqt os'al€ls l€urrou InJqlr€J € sllrupe oJBCls UaqIIH alqeredos e uo Eurlce erqa8le uu?uneN uo,r {rela l?ql s^\oqs slql 'n ) x rc1 xd rr=(x)0 ret.(q)ur s€sJd pue(A)f j n il(c). "lr7ln"3 = 9u 'I @x = (x)Ps puB'l o @ t+ = 94 EulDos [q peurelqo sl arnlcrd crqdroruost uff' (i>tcaqC) 'll
'("ur)e = lurr6)(x)0u'("lzlh) ti" = tu ug qtlrn 'u II€ JoJ I = T=U .tfi -e = QU @
,{q uerrE sr eldul SNg eql Jo uolsro^ B leql .(grra,t ol preq lou sl lI t3" . un = ,"' rog'uaql'(ri
'1rr1u1 x f,ue r o J s r s € q I s r u r o u o q u o u B ( " 1 ) p u e o > to3 'g < un qllr'r) tl'tlr
| = , u ' 3 1 = 8 A ) . u ' ( t l ' a ' x ) c=7r g ( l l )
71{ aP ) p u € 1 1= 0 U ' A Y = A g l Q y ' ( a ' g ' 1 g ) r 7 = Q y g ( l )
:snql ue,ryEare seldyrl g51g Olqrssod o/AI 'II uo 0 l€uorlcunJ rBeurl aaglgsod I€rurou InJqlr€J = AW uoll€nbe egl ueql irl sB sles-llnu erues aql e saurgep np/ J qll/r\ arns8aru airurJ B aq n ta1 i(rl'|'y)-7 = hl le1 (s) '6'97 alduerg
n s,0ureqr,{rrerueprrur :L'"XI"j^ii,J#'ilJ:: aroN.'.fT";fij'i!
elctrrl g5ig arq Jo'lenbesoql ul InJqIIBJ€ qlIA pelslcosse1Q64y'Qy1 )1€l II€qs et 'I'Z'Z ruoroeql ur uollressessauanblunagl Jo ?nerl uI (.uorUass€ aql n 'a,r1lca[u1 s1 Jo esBJa,rllcaful paqsllq?tsefpeerle eql ol l€edde puu uollJnrlsuoc sNc eql
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2. The Tomita-TakesakiTheory
44
assume that lil g t(D and that 0(x) = <xq(> for a vector o in lf such that [Mo] = lf. The vector o is known to mathematical,,Fhysicists as the vacuum vector or the vacuum state (in case ll0ll = l). The faithfulness of 0 translates to this separating property of fI if x e M, then x = 0 if and only if xo = 0. Thus, in the terminology of the following definition, the vector O is cyclic and separating f or M. Definition
2.2.4. A set S c lt is said to be
(i) cyclic for M if [MS ] = 1?; (ii) separating for M if for x in M, x = 0 if and only if xS
= {0}. E
Exercises (2.2.5) Let M g r(lt) and let 0(x) = tr px, where p is a positive trace class operator, given by , P = X dr, tBrr,Er, with cr,, > 0 and t[,,,) orthonormal. Show that 0 is faithful linear functional on iit if and only if {t"} is separating for M.
as a
(2.2.6) If S g f, show that S is cyclic for M if and only if S is s e p a r a t i n gf o r M ' . ( H i n t : x t e M ' a n d x r S = { 0 } ) x t l M S I = { 0 } ; so, if S is cyclic f or M, S is separating f or M) . Conversely, if S is s e p a r a t i n gf o r M ' , n o t e t h a t p t = P l u S l , M ' a n d t h a t ( l - p ' ) S {0), whence p' = l.) = <x4o> for x in M, then 0 is tracial if and only if n for all x ir M.
ftiii'= iilsii'
Thus, given a faithful normal positive linear functional 0 on M, the GNS construction leads to a realization of M as a von Neumann algebra of operators on a Hilbert space 86, in which there is a cyclic and separating vector f or n6(M)i this vector is automatically a cyclic and separatingvector for n6(M)'. We shall conclude this section with an important class of von Neumann algebras which come equipped with a natural cyclic and the so-called Sroup-von Neumann algebras separating vector associated with countable discrete groups. Let G be a countable discrete group, whose identity element we shall denote by e (the symbols e and I having already been irreversibly i d e n t i f i e d w i t h p r o j e c t i o n sa n d t h e i d e n t i t y o p e r a t o r ) . L e t ! ' ( G ) d e n o t e the Hilbert spaceof square-summablefunctions on G:
<-]. ,r"li(r)12 There is a canonical orthonormalbasis{8,: t e G')of P(G), where ! 2 ( G=) [ r ' o - o :
pue) srolceJ elrurJur 3o fpnls orll JoJ lool InJJa^\od e septrrord qclq^\ 'ruaroaql r{€se{BJ-BlruroI palerqoleJ oql ,(q pelerlsuoluep , t l E u r c u . r n u o c .s r u o g l e E l l s a n u l u e q c n s J o f l t l t q e s l a p e e q l '9u1*x;9u * 9u1x;9a rolerado eqt (3o frleruost Jo )toBI aql) oulruexa o l a ^ l l c n r l s u r a q l q t n u 1 1 ' s r o 1 c e 3a l l u l J u r , { p n l s o l r e p r o u l ' o S ' W 'Q'Z'() u r x I I B r o 3 ; 1 9 u ( * r ) Q u l= l 1 1 9 u 1 x 1 9 u3;r; i ( 1 u op u e J I I B I r B r l s \ q 'xE o l 6urpioccy 'yy' eqe?p uubunep uo,r (ler.lolce; flrressocou B uo el€ls lururou lnJr{lleJ e sl 0 teql 'ueql 'esoclclns lou)
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rfssc{sl-Glluol
oq.;. 'eZ
('peaq /t\ou IIBIIS e^\ {JErupu?l qCIq/t\ SpIB^\ol 'ruaroaql lIBseIeI-slIIuoJ eql Jo ecuanbasuocB s€ 'uoglcas lxeu eql ur eS.rauraIII/( slql i,,(g t l :rd1 = ,ou lBql assc aql 'lce3 ur 's1 11) 'palress€ sE "hl ro3 crlc{c s1 '/,{rela rog r-1, = gld aculg ',Ni u leql realc sr 11 {g t l:rd1 3cueq pu€ "\ qc?o qlla salnuuoc ld qcea leql fgrran ol IBr^Irl sJ lI '@ _ 1 8
u r , ' s I I e r o g I - ' l = ' l l d ' , { l l u a l u a r n b a ' r o ) ( 1 s ) != ( . r x l l d ) , { q p a u r g e p 'rd 's,11 I uoll€luasardar rulnEar-1qEu eql lJnJlsuoJ oslr feu en aqt ol snoEoluuy 'lO'Z'd 'xA 'Jcl thl roJ c11c,(csl U l€rlt Eurrr,ord f,q'n ro; Euylerudoss1 g leql fglroa lleqs e,,l[ 'n roJ cr1c,{csr U l€rl} ',(lluonbesuoc'pu€'t r l o e e r o 3 11 = Ul\ lerll a^rasqopue tl = U lo'I ( ' 1 ' p u o r l c e g u r ' r o 1 € 1p e q s l l q € l s ou o l l r o s s e 1e:eueE eroru € luorJ /AolIoJ III/'\ slql l- > ,l(l)xlDtl3 selJslles n * g : x a r a q / h ' l \ ( t ) r c ' t 3 = r S O T J Ol uSo E r e l u o c ] f 1 E u o r 1 s - o € J o u n s e q l se elqrsserdxa flanbrun sl .tg Jo luetualo {ra^e leql enrl 'lJ€J ur 's1 lI) 'ow 'acuaq :(tr =)t Eululeluoc ,r-1'1 = lr Jo ernsolc Euorls orll sl lg ecurs) erqeEle e s1 s,11 or{l Jo suolluulqruoJ r€ourl ellulJ lurofpe-g1os on les eqJ '@)*ll {q pelouep eq III,I\ pue g 3o erqoEle uu€runoN Jo uo,r dnorE eql pellec sI ,,{g r , :l\} = py etqeap uuBr,unaNuo^ orII '9 l'\ Jo uoll€luasardar relnEar lJoI pallec-oj aql sr 1r l(9)r1 ul (l\'\ = ''e'I) t\ - / deur oqj 'g u! s g IIe roJ Jo uolluluase:der ,{relrun e sl "l -'1'\',(lluolerr,rnba'ro'(sr-l)l = (sXllf) snql ll ^q uoll€Isu€rl lJeI o1 EulpuodsarJoc roleJedo frdlrun eql elouep t\ lel 'g il rlo€e ro{ S I ' J I
s=rJI
.:]
= 819= (s)r1
(se1e15rog) tuaro0ql I{Es0{BI-ElltuoI
9V
oql
't'Z
2. The Tomita-Takesaki Theory
46
factors of type III, in particular). Till further notice, assume that M is a von Neumann algebra acting on lf and that O is a cyclic and separating vector f or M. Proposition L!-l- Let So and Fobe the conjugate-linear operators,with domains MQ and MtCl respectively,defined (unambiguously)by So(xo) = x*$ Fo(x'a) = at*q Then So and Fo are densely defined closable operators: their closures, denoted by S and F, respectively, satisfy S = F*=FSandF=S*=S3. Proof. Since o is cyclic and separating for M as well as for M' (cf . Ex. (2.2.6)), it is clear that both ^90 and Fo are densely and u n a m b i g u o u s l y d e f i n e d . I f x e M a n d x t e M t , o b s e r v et h a t = <xr*4xo> = <(\x txo> = <$xx tO> = <x*Gxto> = <So(xQ),x'o>' B y d e f i n i t i o n o f t h e a d j o i n t o f a c o n j u g a t e - l i n e a ro p e r a t o r , t h i s s a y s that Ma ! dom Ffi and that FSlMo = So; i.e., ^to g r.3. This implies t h a t F 3 i s d e n s e l y d e f i n e d a n d h e n c e F o i s c l o s a b l e . I f F d e n o t e st h e closure of Fs, then F* = Fl I So. So So is closable and if S denotes the closure of Se, then F* = f3"s.ippose : S. For the ,eue.ie inclusion, I e dom f'fi and Ffi[' = q#. Define operators Q6 ang O6i, bottr with domairt M'Q by Qo(xto) = Nbtice that if x',y' € Mt, then x't, and Cf,{x'o; = i'l#. < O o ( xt a ) , y ' o > = < x t 8 , . / ' o > = = =
(by definition of (#)
= <x'gOJ(y ' o)>. Hence, as before, Of, ) OI and consequentlyOn is closable;if Q denotesthe closuresi Qo,ihen Q* = Ot )Ot. It-is trivial to verify t h a t i f x t € M t , t h e n Q o * ' 2 x ' Q o , a n d h e n c e ,b y a n e a s y approximationargument,we may passto the closureand conclude that the closeddenselydefined linear operatorO is affiliated to M.
= tclfl
.Joord aql selalduoc slq-l irlr_vt = tslyIt n = g nE ot = g uofwnba oql aleEnfuoc '(q) rog sy i6:-yr 'r3p€3r
eql ol lJal ^loJes eq ^Bru pu€ arnleu ul oullnor fyarnd sl uollBJIJIro^ eql izlry ol frooql lerlcecls aql Jo uoltEcrldd€ eullnor € puB = t|1yv| ruorJ pe^Irap oq u?c (c) .lo {1rpr1e'r oqa '(c) 3o 3oo-rd zlry eql ul dels arrlslcap erll Jo s€ IIa^\ se (e) 30 300rd eql selelduoJ 'z/ty = slrtl t|1yyI Pu? *f = yf = I splarf slql "{:e11un1lue sl 1 sv '*t|1yYt = zlrv pue I = { *\t saaluerent uorllsoduocep relod aql Jo ssauinbrun eql 'rolerado tulo[pe-.;1as olltlsod elqllre^ul ue sl *12lr_V/ sJuIS '*fs,1yVf = zlrYzf ecueH '*[71yY = = = = = g ra{ = r-S S 71fl1 lEqi salldul oslBr-S 5 uoltenba eqt l(g) = s puB^olqllJo^ur z l l y r e T e c u r s e l q l l r e ^ u r s r v l E r l l s e r l d r u rs r q l : r - s sl'5: rBql s/hoqs luerunSre uoll"rulxordcle oldurrs e "Ys = us aculs '(A Utoq are saceds I€urJ pue "e'l) .{re1run11uesr 'g uer 5 0S u", = g7U I€rlrur sll ,l^ leql epnlcuoc pu€ J uer i 0g uat =l)tN acurs 'S uuJ ecsdsI€urJ pu€ JLqB.l acods Ierlrur qllm ,{rlauosr lerlred reourl olsEnluoc s sI 1 osle la,rllysod sl v os puB sJ = s*S = ,Q1{) = v aA€r{ o^\ 'uolllulJep l{g 'Joord 'Ul t / II€ roJ 1,V= I1rV1 r€lncllr€d ul l(r-v)l = f(v)lt ueql '(o'Ql uo uollrunJ lorog (penler-e1rur3 ',(11ereuet = (c) ir_V erou lnq pepunoqun dlqrssod) {ue sI ./ JI .tv1= :JS yv puo sJ = v tarytnt ig /o uoltlsodtaocap tolod atp s! alyyt = g (Q) l(t-t sl g ''a'y)'slolotado papunoqun lo asuas aW u! alqlrn^u, s! qtrt!frr tolotado Turotpo-{ps atrttsod D s! V puo tolotado,ttottuntluo Tuto{po-tps o s! t (e) ary to uotltsodruocap tolod aLfi aq zltyf
'uarlJ 'g .toyotadopasop = S ta7 ?€Z uoplsodord
',{.11ecr1uapy 'pe8ueqcrelu\ D tlal pue n Jo solor aql qll^\ perrord erE 'pa8ueqcrolur s puB I rllln 'slueruolels IEnp aql '*.{ = 'ocuoq pue 'S = r€I{l i 3-{ reqr Euyqsllqelsa s 9.{ = pu€ ^qareql '*l ruop r I luql 'S Jo uolllulJap t S S '#1'a = orll ruorg*'epnlcuoJ v*Oua = ulD = (uur)o5 ollq,r\ '1
an = plQln*nuan = TSlQlu an = glub *nu an = 7SQ*nu
'puer,I reqlo oql uO 'sl * *?'a Pue I : 2*nuanl?ql apnlcuoc \1+O = U*O = c*l puc "a '((l't'l) 'ig 'Jc) lsql eloN VO = 1 acuib l@ dt I *n"an pue *0 dt I n 'a pue n
) n'uaql
'A
,
111l)t"'olilAl)n=ua6 =lb
oslv 'u qcse roJ (lAl)lu'ol | =ua er?q^ n ? Jo uolllsodurocap reloct eql aq l)ln = O :c-l
(sel€ls rog) ruo:oaql l{€sal€I-€llruol
LV
agl
'E
Z
2. The Tomita-Takesaki Theory
48
The stage is now set for the Tomita-Takesaki theorem. The proof of this powerful and difficult theorem is quite long and elaborate. As stated in the introduction to this chapter, we shall only supply the proof in the very special case when ,S (or equivalently A) is bounded. For a reasonably short proof of the general assertion, the reader may consult [BRI]. With the notation established in Theoren L3-3propositions, the following statements are valid:
the
preceding
(a) AitMA-it = M lor all t in R; and (b) JMJ = M'. Proof. Suppose that ,S is bounded. This means that .S, A and F are everywhere defined bounded operators; it also means that A-r is an everywhere defined bounded operator, since a-l = JAJ. For x,y and z in M, note that (l)
((^tx.S)yXzo)= (Sx)(z*y*Q) = yzx*CL
Apply this with ./ = I to conclude that (Sx^lXzQ) = 2*+q combined with (l), this yields: (Sx'S)yXzo) = (y(^Sx'S))(zn). Infer from the cyclicity of O that (Sx^S)y= y(SxS). Since x and y were arbitrary, we get: SMSC Mt. An entirely analogous argument, with the roles of (M,S) and (M',F) i n t e r c h a n g e d ,y i e l d s : F M t F 9 M . A combination of the two inclusions established above results in AMA-L! M (since A = FS and A-r = ,SF);conclude that (2)
L n M A ' nt M
for
n = 1,2,...
los \ for \ e (0,-), and write Ao = For z e C, let g(I) = \z = "z 8(A)' Since A is invertible, it is clear that Au is bounded, for each z in C Now, fix x e M, xt € M), l,rt e tt and consider the (clearly entire) function defined by
l(z) = llall-'".ta"" a-o, xt)l,o>,z e (L where[a,b]denotesthe commutatordefined by la,bl = ab - ba. A yields crudeestimation
- ' "" z l l a " l l . l l a - " l l ' l l ' l l ' l l "l' l nl ' l l 'r l l t f ( z )
R"", n""ll = llall2 = 116z = lla"ll'=ll(a")*a"ll lla"ll.lla-'ll
roJ rotccl Eulleredos pue c11c,{cs sv 0g uaql '1Qu'Qy'QU1 eyd1r19519 p a l e l c o s s €q \ / ^ n u o I E u o l l c u n J r E a u r l a n l l l s o d I € u r r o u I n J q l r € J e s r 0 leqt pue erqaEle uuuuneN uo.rrleraueE e sr.n leql ^\ou osodclns '79 3o rusrqdJoruolne-*.;o dnorE relouered-euo snonurluoJ *,(18uor1s-o € seurJep *_VrrrV e x leql eslcrexe etoq€ aql uoJJ s^\olloJ ll 'ruerooql r{Bsa{BJ-Blrruol or{l Jo Eurllas aql ol Eururnlag O ( ' l u e r u n E J eg / r p r e p u e l s € seJrnbtJ slql lfllnulluoc Euorls (ocueq pue *Euorls) orrord o1 s e c l J J n sl r ((p) (t'g'O) 'xg 'Jo) os 'pepunoq-rurou sl {UJ, I :(r)rn} 'l{ r x ' n u l x q c e a r o J ( * f 1 8 u o r 1 s - o )( r ) r n les arll JI :turH) (*)uto € / - t/ leql esues eql ur snonulluoo *^lEuorls-o sl rlorrl/ir 'N n go swslqdrouolnE-* 3o dno:E relaurered-auo€ seulJep ']nxrn = (r)ln uorlenbo eql 't " ler{l ^\oqs ile roJ W = ? "' UJ > | inry'n lBql qcns ,{ uo srol€redo ,(relrun go dnor8 ralaur€rBd-euo snonurluoc ,(lEuor1s e sr Utlllnl esoddns pue ($h j 79 asoctcln5(q) (1'7'7 waneqJ Jo JI€rl puoooseql Jo Joord 'leruJou sr D '-n aql ol leeddy lBql epnlcuoJ :r_D soop sE ur ernlcnrls ropJo arll so^reserd r :1ur11; 'J1esll ol:uo n Jo ursrqdroruoauorl l€ea-o crrloruosr ue sr p ueql 'n lnv t p JI (e) '/t/
Jo srusrqdrourolnu-*3o dnorE aql olouop n $y
n1
(V'E'Z)
sesltrexx 'sasrcJaxaeJolu euos oruoJ oJOrI'oS 'r(€i[ aql Jo lno slueualels ,{.reluauro1aoruos lo8 ot dleq plno^\ ll 'ruoroor{I r{€so{€I -€lrtuol aql Jo seouanbesuocoql Jo etuos r{ll/r\ Euypaecord eJoJag E 'n3Itn.I
'(q) turqsrlqetsesnql
= f61yYthl11yYt = ttWt puB
tn3
SWS = ta1yYN71{I
= Ilttt
'ecuaq 'o z u\ II€ roJ th[ = z_vrhlzv l€rll s,nolloJ oslE lI '(B) Jo uorlualuoc aql ,{lasrcerd sr srql 't! = z JoJ b rrr. z lle toJ n = z_ywzy spler,( (z-) ot srql Eurfldde $ ur z lle roJ n = z-vryrv lertl epnlJuoC a v! z II3 roJ 0 = Q)I lBr{l -- (asrcerclf11elnrq oq ol ? > I oruos qlr^\ 'O < z ad roJ (;r;,1a)0= G)/) / go qlmorE aql uo uolllpuoo ra{Eolr e Jopun prlea si qclq^{ -- (ltlf] 'gc) ruoroaql s.uoslreJ uorJ s^\olloJ "" 'Z'I'0 - u toJ '(Z) fq 'os1e 16 z eig.auetd-Jl€q eql tI < 0 = @){
u t ( l l u l ll l l l l l l , t l l l l t l l s , { q )p e p u n o sqt . / u o t l c u n Ja r r l u oo q l ' s n q a '0
6'
2. The Tomita-Takesaki Theory
50
n5(M). Here and in the sequel,we shall denoteUy SO,F6, J6 and A tlie operatorsarising, in this case,from the consideratibns'thatle theorem. to the Tomita-Takesaki Dcfinition 2.3.5. With the above notation, the operators "16 and A6 are called, respectively, the modular conjugation and ^the modulai operator associated with the pair (M,0). Denote by {of: t € R) the o - w e a k l y c o n t i n u o u s o n e - p a r a m e t e rg r o u p o f * - a u t o m o r p h i s m s o f M defined by
z6(x)A6it). oft") = n-q,(aiot ( T h e c o n t i n u i t y a s s e r t i o n f o l l o w s f r o m E x . 2 . 3 . 4( b ) a n d t h e f a c t A t h a t z 6 i s a o - w e a k h o m e o m o r p h i s m . ) T h e o n e - p a r a m e t e rg r o u p ( o f ) i s c d t t e a t h e g r o u p o f m o d u l a r a u t o m o r p h i s m s( o r s i m p l y , t h e m o d u l a r n group) associated with the positive functional 0.
Exercises (2.3.6) Let 0 be a faithful normal positive linear functional on M, and lfg, nb AO,SOFO,"16and 46 be as above. (a) Show that o6 e dom 56 t1 dom F6 and SO0O = F6oq = 04; c o n c l u d e t h a t ' O O e d o m A g a n d A O a O= " 1 6 0 6= n g ; (b) Show that nd(oP(n))ad = aitn6(x)o6. (c) Show that tlie following conditiond are equivalent: (i) (ii)
Q is tracial; 56 is antiunitary;
( i i i ) s o= ! o = F o ; (iv) a6 - lr' ;
(") of(x)=x YxeM,teR (Hint: The implications (i) <+ (ii) <+ (iii) (9 (iv) + (v) are easy; use (b) for (v) + (iv).) n Let us now look at some examplesthat will illustrate the TomitaTakesaki theorem. Exanplc 2.3.7. (a) Let M = L-(X,T,tt) and Q(fl = It a", where v is a finite measure equivalent to & The GNS tripl,i is given by 1f5 = L 2 ( X , y ) , n ; ( f ) = m r a n d a 5 = ( d v / d 1 l r l 2 . l t i s e a s y t o s e e t h a t ^ S d= ' " / d = F6 is tlie map 7 - 7 dn tz(X,tt) and that 46 = I -- note that 0 i! tracial as M is abelian. Thus, the theorem inf
'puu (r{h o t = Of(t o (a)f)Of ocuaH 'r Jo l€ql Jo al€Enluoc-xolduroc (esr,n,{:1ua)aql sl 1"1) srseq IErurouorllro oql .ol lJodso.l qll/r\ xrrleu s s o g / hr { u o r o l B J a d ol s g l s J x e r s q ^ \ ' r 6 I = Q r ( t . o r ; 9 1 l e q t s l e a ^ o r '(reou11 91 ^\ou u o l l e l n d u r o c i t s e o u y a l u S n f u o c s 1 l e r l l I J B J eql roJ "l -l = "1;91 (*l q l n q ' d s r u d l t J o q t i ( l l e r l u a s s as r s r q t ) o 1 e q 1q c n s Jot€rodo[rclrunlluu.lnblun aqr sl 94 pue ,_d o d = 0t '": o ulrrfou3 = \rU 'l @ f = (.y)vu 'll @ ll = 9J[ .alnlcrd clqdrouosr u€' uI
x IIe roJ
= (x)jo r€.qlv roJ BlnruroJe^oqe,u, u,orY
3":"#"i ,r-d"r,d ,(lrseesr li 'ttiiuge rd; 1-'il = ru]:I lcql apnlcuol ,r11vf= g oJurg z9 (S'S'Z)'x:r uI rho"iredddfe slrelepoq1) '(tr)i?y,]"= (U)(:v) ('('S'S'z) :rolerodosrql sr 'ssaulurofp€-Jlos ,(q 'l€ql epnlcuoJpte''1dr-{n){eaQ V r Jprr
t- "ll(z)3dll i 3
: ( g : M ) r rr I
)
uo paurJap r o l e r a d o 1 u 1 o [ p e - g 1 aasa r l r s o d a r l l r o J e r o c B s 1 0 g l e q l e l o g ' 0 G x
ut ? II€ rog (1)!dr- p = (ZXIV) ltrql pue V ruop i og teql opnlouoJ uol 6
'. (r u" )( J zf r r l g J= ( -u). l r
lBrll puB J urop ur poureluoc sg {g
r u'la
:1*jt) .lo og ueclsr e e u l l e g r r e q l ( i r l o p ) f S r r a r r o l p r E q o o l l o u s l l I
'(rf r * n l =(dlos ,,,15;J
'nql Pue ^ 'rtJ r * p ' ) =ou(*1."i";eu ,r,LEJ
elrq/h'(ujl = 01r1,*jx10zlEr{l alou pu€
*:'ul,r/ilo
= t_j,
ourJc.p'M ul u'tu rof ''l{*g = (Z)r-lt eroq/h'(g a u'ar:,-\1} i(q uanrE !l 911ro; srs?q lturouor{lro 1e-rnird} 'ulrlio = (r)Qu :(a)fx = ( u ) ( l ( x ) v s ) : ( A u r t p , { r l l e u l p r e cJ o l o s e , ( q p a c e l d e ia q l s n r u M ' l 6 u ;r l1efiorsuorulpellulJul sI 4 paunsse e^€rl e,r\) ("''Z't) = M aror{^\
' {- > ouo
e l l (i l )?i tta * N 3 ) = ( s : s) r a = 0 x1
: s n q l u a r r r 8s 1 a 1 d t r 1S N O o q l J o u o l s r o ^ uo3'0 tn 'll toJ sls€q < eroqrn leurouoqlro ue sr {"11 pue o r r:,r1, ,o 3 = d .
u o l l l s o c h u o c a pI E c r u o u B cr l l r / r \ ' r o l e r a d o s s B l c - e c s J l a,rrlrsod err.rlcafurue s1 d ereq^\ 'xd tl = (x)0 pu€ (r{h = hl p1 (q)
(srluls rol) uroroeql r{sse{BJ,-ullruofeqJ. 't'z
I9
52
2. The Tomita-Takesaki Theorv
consequently, (l(13) o l)' = I @ t(lf). More generally, it may be shown (see [Tak l] for details) using the Tomita-Takesaki theorem that if, for von Neumann algebras M c l(18) and N g t( K), we define M @ N = (x @ yi x e M, y e /{)" c t(Xf o K ) then (M @ nr = M' o N'. This fact is quite non-trivial and remained an open problem for a long time before Tomita resolved it. We shall, however, not go into a proof of this here. (c) Let G be a countable group and M = W(G), the group von N e u m a n n a l g e b r a d i s c u s s e da t t h e e n d o f S e c t i o n 2 . 2 . I t w a s s h o w n there that O = le is a cyclic and separating vector for M. Since rf(e = \r-rEe = (r_r, it follows that the linear sPan Do of ((r)r." is contained in dorn S and that
(s(Xr)= i(r-r) for all t in G and E in Do -- recall that S is conjqgate linear. Since this map is norm preseruing and Do is dense in Iz(G), conclude that --:(S(Xt) = [(t-1) for all ( in I2(G), that J = F =,S and that A = I (cf. Ex. (2.3.6)(c)). It follows from the definition of ,I that J\rJ = prl hence Mt = JMJ = {pr: t e G}"; thus, the commutant of the l e f t - r e g u l a r r e p r e s e n t a t i o ni s t h e v o n N e u m a n n a l g e b r a g e n e r a t e d b y theright-regularrepresentation. O
24.
Yeights and Gcncralizcd Hilbcrt
Algcbras
A positive measure p on a Borel space (X,I) can be viewed as a (necessarily faithful and normal) positive linear functional on L'1X,T,1I1,via integration, only when it is a finite measure. If p is infinite, some bounded functions -- typically the non-zero constants -- are not p-integrable. Worse still, no reasonable sense can be made of ) | dlt if, for example, | = lE - lF, where E and F are disjoint sets of inf inite measure. With non-negative f unctions, however, and the integral can always be there is no such difficulty meaningfully defined as a "number" in [0,-]. These heuristics prompt the following definition. Definition 2.4-1. A weight on a von Neumann mapping Q: M,' [0,-] such that @(\x + y) = IO(x) x , y e M - a n d \ e [ 0 , - ) - - w i t h t h e c o n v e n t i o nt h a t I = - if \ > 0, while 0 .- = 0. The weight Q is said to (i) (ii)
M algebra is a + 0(y) whenever * o= @and \.be
faithful, if x e M-and x l0 imply 0(x) > 0; normal, if 0(x) = sup O(xi) whenever x is the (o-strong*) limit of a monotone increasing net {x1} it M*: 0 (iii) a trace, if 0(x*x) = 0(xx*) for all x in'M.
: E u 1 m o l 1 oagq l o^ord 'srolerado lplur{Js-lroqlrH palluc an (y)zg Jo sraqrueru : ( , t ) x , x ) = ( 1 1 )-z=J l e . 1 f i r o J s r s e q l e r u r o u o t l l r o eqr i(- t'll"ll " '(ah t x i(ue sl {ul) araqa,l 'rll"?x1;3 JI (q) lll"ll eulJep parlddb ,{tlyenba eql qll/t\ (1*x ol snorlard sr qteul quor :a llu t*xl=lzl 3 l " l x1 1ral 8o 1'l =' r , q ll^\sr q l, ( t d cl€ = rl<-rr''lx>lff= r;1't"11f :zllu(r*xll3
:lulH) '1enbeeru seprso^u aql puu 'reqlo aql sl os "e'l -'allulJ sr sprs auo JI ,;;"rrxll3.= zll:.rrll3 l?ql a^orcl tt "e roJ seseqleurouorllro 3o rted ar't ('u) '{'l}'pue (r{h t x JI (s)
0'v'z) saslcrexg 'sesrcrexe las Eur^\oIIoJ eql Jo u r e u o p s r s r q l i s r o l e r o c l ol p r u q c s - u a q l r H p e l l E c - o s e q l l n o q ? s l c € J clseq euos loolloc ol luapnrd oq lqttu ll '(n)X uo ac?rt IEsIuouBc 'uellaq? lou sl aql Eulssncslp ol Jolrd W eJeq^\ aldruexa u€ a a s o l p o o E e q I I I / { l l ' r e A e ^ \ o q' l 8 q l . e r o J e q ' p a q s l l q e l s ee q u o o s I I I ^ \ pu€ pll€^ sfe,nle sr i(1r1enbaslqt l+7g u 9W = 99 tu{1, 'as?c slql ul '((rt'!'X){1 'eloN '(TI'X)fl u (rt'X)-1 = 9y pue f!'X)rZ v (a'y)-7 = 9y 'ernsualu alIuIJuI t { pue 'a'B 0 < { :(rl:X)^1 ) {l = vC '3s€c s5tl uI ue s1 rl eroq^\ 'TIp II ='ArA pue (rf'4f)-'I = Iy fq uar18.s1 (at1u13 '[1rea13 lou sI qclq,$) 33Erl- IBruJOu InJIIIIEJ u 3o alduruxe euo '^Ia{II suees 'pallFuo aq III/!\ p ldlrcsqns eql n uolsnJuoo ou pu€ 'uolssncslp rapun lqElo^\ euo ,(1uo sI eJeql uel{rl6
' ["'
','' = ' '011 t tt'\x ' tnj" tit] = ory 9ru = ot^t i {- > (r*r)P :7,t1t x} =Qy t x) =Qg I {- > (x)P :+7,t1 oulJap 'n uo 0 tt{Etan e to1 'E'V'ZuoPIuIJaq
'aoloq ua,rr8ere suorlrulJapesoq^\i7g secedsqns 3o ur€lreo go slsfleue uE sr 0 lqElo,n e go [pn1s aqt ol leluauepunJ +nW = luql qcns O f7g , O slslxa aroql (rrr) i- > (t)g E (gy)ierrulSsl O (I) :sdtllypuocEuptollog aqTJo ecuolu,rlnbeeql e^ord +n ur. x IIs roJ - > (x)0 JI allulJ oq ol ples sI 0 lqEIa/Y\Y G'V'C) sssltrexx 'V'Z serqaElyuaqllH pezllerauogpue s1qE1a11
gs
2.
54
The Tomita-Takesaki Theory
ll"ll < llrll, for x e CzQt). (Hint: if I is a unit vector, consider a n o r t h o n o r m a l b a s i s (' ( - ) s u c h t h a t q ' = q ' ; (ii) ll. ll, ir a norm on C'jttl with respect to which - Cr{r) is a Hil6ert space. (Hint: use (i) to locate the limit of a Cauchy sequence in Cr(Xf);the inducing inner-product is given by <x,y> = I <xlrr,yE,r>,where (En) is any,orthono-rmal basis for 1f') -llrll, (i)
(iii) x e c-i(h1-"+'x* e iJ?i)ana
= ll"*llr. (Hint:seehint to
(a).) (iv) ' ' C 2(1f)is a two-sided ideal in t(Xf): in f.act, if # f
first inequality is easy, and, together with (iii), it implies the second.) (v) Let x ) 0. Then x e Cr(lt) € x e K(xf) and x admits a ((") is d e c o m p o s i t i o n . x= I " r t f " , t " , w h e r e c r , ,) 0 , I " 3 . ' a n d
an orthonormal sequence. (Hint: f or g exteld ,,( En) to an orthonormal basis and use that basis to compute llxllr; for ), use l l r l l r . @ t o c o n c l u d e t h a t , f o r e a c h € > 0 , 1 1 6 , - y ( x ih a s f i n i t e rank, and hence that x is compact.) (vi) If x e I(lf), show that x e C "(Xf) if and only if x*x e t(Xf)*, in w h i c h c a s e l l x l l'= l = t r x * x . ( H i n t : i f x e Cr ( x f ) h a s p o l a r alxl, use (v) to conclude lxl2 e t(-Xf),;if decomposition x lxl2=ror,/[rr,[r, , x t e n d ( t , r r }t o a n with crn > 0, E dr, (. and {lrr} orthonourmale orthonormal basis for lf and compute llx llr.) (vii) Let x e t(lf)r. The following conditions are equivalent: (cr) x e f(lf)-; (B) I ix[,,,|,r> < - for every orthonormal basis {1"}; (7) -for some orthonormal basis {1,}. (Hint: for (7) + Icx(,,(.>. 0 1cr),c'irniide, x'l'). Let M = f(xf), with 1f (separable and) infinite-dimensional. Define b y 0 ( x ) = E < x E , ' ' t r , >w h e r e { 1 " } i s a n y o r t h o n o r m a l Q: f(xf)*'[0,'] basis for (("). thus 0(x) = tr.r if x is of trace class and 0(x) = -, otherwise (bV fx. Q.4.4) (b) (vii)) -- in particular, the definition is independent of {(n}. It is clear that 0 is a faithful trace; the fact that 0 is a trace follows from Ex. (2.4.4) (b) (iii). (In fact, the reason f o r u s i n g t h e w o r d t r a c e , i n t h e s e n s eo f D e f i n i t i o t 2 . 4 . 1 ( i i i ) , s t e m s from this example.) Furthermore, 0 is normal. The verification is fairly easy; if x, ) x, the cases x e l(lf)* and x f l(Xf)'- must be treated separately. Note, finally, that for this Q, Nd = C r(lf) and l'1d = I(lf)i. Let us proceednow with the analysisof a general weight. Let Q be a weight on M, with associated spaces D, Proposition 24.5. N and 14as in Definition 2.4.3.
.14 Eurrreq ra13e '1eql ^€^\ aql fllcuxe sl stql) Jo II€ ol puc]xe ueql pu€ '-N ;o ueds Jeaull I€aJ ar{l ol uolr{suJ peulJap-lle^\ s ul puolxa o t C u o 0 J o f f r ^ l l l p p B e s n : ( p ) J o a J u e n b a s u o c€ o s l € s l o c u a l s l x S '(p) -lo ecuenbesuoJ ol€rpeutul uB sl Q u qcns .;o ssauonbrun (J) '(q) &ory acuo lB s^\olloJ slql (e) '*t[t-f l e q l o s ' + N U N ) z l t x P . u " O< x u e q l ' G I r 3 r ' , ( 1 o s r a , r u o 3 'G -l,l lerll sarlsllqtlsc slql lg ) z l€ql (€) ruorJ apnlcuoc J
I g r (t,r+ fr;*11,r * f")tit, tif = {(f,r- lx)*(l,r- tx)- (f,r + fr)*(r,(+ lr)) I=!
1rrj,r+ !4*)z .ut (*z+z)7=zp 'e3ueq pue
-€,2,(o= qceeroJ I
.g r
'*W r " JI *z = z uarll
tit lrrol + tr;*1t"nl* tn)
'(q) fq 'acu1s'(p) ul uollressu puoces aql so^orcl rIJIq^\
oin tit = ro lfxnl + l.r;*1lxnl* r,(),rl lo8 o1 ,{1r1uep1 uollezlrulod eqt fldcty ' ry r l,f'11W1^ 'ttrjxl=Il = " asodctng(p) '1eap1-tqEp '(q) ? sI uor3 ,(1rsea s^\olloJ slql (c) N oculs 'N * 5 'x*xzllrll x,(*t*x = > Nr{ l€tll roJul pue (xf)*(rQ leql alou'tr47 t t'x gr'f11eurg 'N 3 N + N" l'tqr epnlcuoc i(t(*t. + x*x)7 = (r-r)*(,t-r) + (t+x)*(t+x) I (d+r)*({+x) l€ql elou '147t t('x 'uor1ec11d1t1nru releos repun pesolJ sl N fllernrra (q) 11 'fsug (e) 'Joord '0 = *Wlg L!?txs y1 tto toauq anbmn o st araql tDllt Q ' lottotluml l^t>zt(*x+Wtt'N tz'x ig lo stuawal.atno{ +W = Io ttotlourqtuoj .toaurlo st y1lo ruawala {tara puu'*w \) W = c i(t3o1odo1tuo ur pasop t(lrtossacau,ou puD 1 Ettttrtoltroc t(lltossacau tou) n to otqaSluqns tuto[po-tps o s! W '.W Lt, l0apt-{a1 o st N ig t z + x > z'-tI ) z'C t r puo'g t d + x1 € (-'01 ) \ pttn g t t'x "a'1 iauoc adltlsod tta7tpataq ? s.rC
s9
(;) (a)
(p) (c) (q) (B)
serqo8lyuaqllH pazrlBrauagpuu stqElaTg'r'Z
2.
56
The Tomita-Takesaki Theory
defined the Lebesgue integral for non-negative functions, one extendsthe notion of the integral to complex integrable functions.)D
A weight 0 can be trivial in the following sense: if x= 0,
0(x) = {: For this 0, DO = I
.tt,
{
ifx > 0 * NO = ilO = (0) and not much more can be said.
Definition 2.4.6. A weight 0 on M is said to be semifinite if l'16is O o-weakly dense in M. Loosely speaking, semifiniteness means that there are sufficiently many elementsat whigh 0 has a finite value. Ftrr example, if M = L'1i,7,1t7 and 0(,f) = l7 dv, where v is a positive measure with the same null sets as i\ then (under the standing assumption of o-finiteness of p), semifiniteness of v is equivalent to o-finiteness of v. Observe, also, that the canonical trace on f(Xf) is semifinite, since l(lt)*, by virtue of containing all finite rank operators, is o-weakly In the f ollowing exercises, some alternative dense in f(lf). characterizations of semif initeness are given, which say that s e m i fi n i t e n e s s i s t h e s a m e a s a m p l e n e s s o f D i n o n e s e n s e o r another.
Exercises (2.4.7) ( a ) I f h , k e M - s a t i s fy h < k a n d i f / r i s i n v e r t i b l e ( i . e . , f t - l e E ( X f ) ) , then k is invertible and k-r < /t-r. (Hint: Observe that if x ) 0, then (by an easy application of the spectral theorem) x is invertible iff there exists e > 0 such that n ) e.l; this takes care of the first assertion. For the seqond,h < k + h-rlzhh-rl2 <
and so llkllzh-rlz\ll > lltll for arr t; if x e r(xf) is tx-rtzkh-rt2,
i n v e r t i b l e , w i t h p o l a r . d e c o m p o s i t i o n; ; u l x l , t h e n x * = l x l u * = u*xu+.conclude*at||n-U2t<'/,\||>l|i||rorall[,whencel< pL/2;-iprlz, and so k-L - k-rl2lk-rlz 4 1'r-t'.1 (b) lf h e M* and e > 0, define he = h(l + eh)-l. Yerify that (i) h. ) 0 and /r, € M: ( i i ) h i < h r r i f e - ) e ' a n dh e T h a s e I 0 ; fte ( k.. (Hinti Use (a) to get (l + eft)-l > (l + iiii) a'<-t't ek)-', and note that I h,=!(l-(l+eft)-r).)
i(swns altul[ {o -n ut x 11oto{ tau aqt {o tlwll aLfi so patatdtalut Sutaq wns aqt) *''n3 (r)r,frrt!3 = (x)O tollt qcns {f r,t :'4t}tlttuo{ o slslcr.aaralfi '.*n q x yo .to! (x)Q (x)trt, t *'*n D slstxa araql auolouow Sutsoatcut u! tau loql qzns 0 , I :!,1,) ilotutott st Q an suotttpuoc 7utuo11ot aW 'n
uo Q tqSnu D rol
6tZ
(ltl)
(II) (l)
:ruapunba uopFodo-r4
('[ee11]u1 ';oord leulEyo 'lcalqns slql ul ulorJ ferrrre leE eql aas ol e^llcnrlsul eq fq8lu lI f11ear J3Aau u€c auo lerll sluorunEre ,,ac€ds-to1ce,r-1ec1Eo1odol,, Jo lJos 'ernlnJ erll ul ^laerJ ll eql oas ol oIII plno/$ oqr\ reps3r oql Joc) 'dlrleutrou asn llsqs e/A lnq 'goord ou tlll^\ ^\olaq llnsoJ eql alels el[ eql seqsllq€lse rIslq/'\ dnraEuell Jo suolllulJap l?ra^es Jo ecual€^lnbo 's1qE1o,tr ol enp llnsel E sI eroql Jo f lrlerurou Eulurecuo3 n
'I/txpuetlle
qcns rorr > ;;rr;; ,lill ruur c ur{!x),i: :",:lr:&iir",,'},i,1tl agrur3Suasst p
'W
(l)
aI{I (p) e:e f lqEler'r B uo suolllpuoc Eu1,t'o11oJ :1ua1e,r1nbs Jo eJnsolc {ee/r-o aql uI se\l ata os pus esuas{€elt\-o aql ul
2t2= ,,rrtr,r*Wfi leql anEre'lau papunoq '*n ) € sr V ocurs '@) S'V'Z 'dor4 ,{.qA ) zl;trlrx € V r x ueq} r( 3r 'esrarruoJ aql roC '3m2 q paul€luoc sl l^l Jo ernsolc {ue^\-o eql l€I{}epnlcuo] 'Ima ) ros?=x > x uegl'V t x JI:luIH) 'e,roq€ s€ a qll^l (J) '3 'I ! SI W Jo ornsolJ l€e^' 1,-oeql l€r{l ,vroqs ('a = a x
= x luql opntcuoc22x= x snqr7 > t"l(:-.o)ros pu€ c t (x)(-,.:ll
'v r x gr 'i(1esrarr,uo3'(x;trrI > a os pu€ '(x1t'rt leql an8:e y a f € { l o p n l c u o c '>I x > 0 o c u l s ' r > r o c u e q l x > a \ o s p u c v r a \ u e g l ' I > \ > 0 p u €G v @ ) d ) a J l : l u l H ) ' t a u ( , w U ) a :a)1 = a uollceford eql sl ,. ]€tll /$oqs V tull = x ldl'W '(e) ,(q 'eculs (q) ul leu Eursearculouolouou € s? pe^\al^ eq ,(eurV
('(ur)(q)Q'v'z)
z q r 1 1= | F I ' r - { ! t l + t l , t ' x E e s np u e r - ( { + l ) t l = r 1 n d ' ( G -t ) ! = - f x l e q l p d e ' 0 . - r y a r u o s t o 1 - l * r y ! r 1e c u l s ' g r , - 1 l x - 1 ; ! r = lr7 teqt fgrran :lulH) 'x , tx pue x > ]x leql qcns V 7 palJerlp sI v l€ql ^\oqs (e) x slslxe eroql 'v ) tx'rx JI ''e'l isp.re,ndn '(t > llxll iC t x) = sB eq il pu€ N'g la1 pue ;4 uo gqEra,ne aq"$'ta.I (8't'z) v le1lznsn
'7'7 serqa8ly lreqllH pazll€raueg pue s1q8tor11
L9
58
2.
(iv)
A is o-weakly lower semicontint ous; i.e., il xi - x a-weakly, xi,x e l+[*, then 0(x) < lim inf Q(x,). E
The Tomita-TakesakiTheorv
With very minor modifications, the GNS construction goesthrough for weights. Proposition 2-4-lO. Let Q be a faithful, normzl, sentifinite weight on lt[. Let D6, N6 and !46 be the associated subspaces of I[, as in Definitiort 2.4.3. Let us use the same syntbol Q for the extension, as a linear fturctional, to !16, as in Prop. 2.4.5 (f). Then there exists a triple (trq,n6,nq,)tvhere' (i)
lt6 is a Hitbert space; *-algebra hontontorphisntol M into t(l|.6); (ii). nO is,a (iii) n6: NO- fO is a linear map such that
= 00*.r), n6k)n6@)= nd,?x) wheneverx,y e N6and z e M, and such that n5( N6) is dense in!t5. 'such Tlte triple is uiique in the seuse that il (W',n':nt) is another ) triple, there exists a wique witary operator u: !t.6 llt such that un6ft) = nr(x) for all x irt N6 and nt(z) = un6!)u*'for every z irt M. Furthermore, 115is isometrib and is a o-weak honteomorphism of lut Y onto n5(M). Proof. The proof is a repetition of the proof for finite weights, with only minor and obvious modifications. It will suffice to start t h e r e a d e r o f f o n t h e p r o o f b y s u g g e s t i n gt h a t 1 f 6 b e t a k e n a s t h e c o m p l et i o n o f N 6 w i t h r e s p e c tt o t h e i n n e r p r o d u c i g i v e n b y < x , y > = 6 ( 1 u * x )a, n d t h a t t h e f a c t t h a t N 6 i s a l e f t i d e a l m u s t b e p e r i o d i c a l l y r e c a l l e d . ( N o t e t h a t t h i s i s i m p l i c i t i n t h e s t a t e m e n t( i i i ) . ) n F o r t h e s a k c o f b r e v i t y a n d c o n v e n i e n c eo f e x p o s i t i o n , w e s h a l l h e n c e fo r t h w r i t e ' f n s ' f o r t h e c u m b e r s o m e e x p r es s i o n ' f a i t h f u l , normal and semifinite'. Suppose 0 is a fns weight on ld with associated spaces D, N and It and GNS triple (lt,n,n). Since z is an isomorphism, we shall identify M with n(lt[) ar'd assumc luI C t(lt), n("r)=x. LetU =n(NnN*). If (.=n(x,)e U,i= 1,2,(recalling that thc faithfulness of 0 implies the injectivity of D), write (r|z =
n(xrxr)and (f = n(xf).
Proposition L4.ll. (a) (b)
is an involutit,e, associative algebra; is eEdpped with an inner product which satisfies: ( i ) < ( 4 , [ > = < r , , l s 5 > f o r a l l \ , n , 1 i n L l; (ii) for each \ in U, the nxap n - lD is a continuortslinear operator ort U, with respect to the inner product;
( ' r p u o s p u e q f e 1o 1; Xu l ( N ) U g o , { l r s u a pa s n p u e '(t ,(q) papunoq sr (x*d)rf = ruroJ r€ourlrnbsas 3rlt [(,{)&.(x)u] leql a r o u : l u r H ) ' ( e N = ) N u I t , x g e . r o ; < ( d ) t r . ( x ) u , o >- ( x * d ) ( r p u e 1 ) r p ) 0 l€rll qcns r.rr{t ,o toleJado enbrun e slsrxa eJaql lBrll ^\or{S (e)
- .n , rf asodcrns ,'V,:r'; :,',fi'.fJ$i#'$,''rr)Ior> I :tttd;
lEr{l puE (a)r j n wqt aunss?'.w no lt1E1e,r suJ B 3q o lo.I (gt.v-z) .du.$x.4fi;elorrl
S - \ C a q l S u r u r e c u o cl u e u a l e l s s s a u e n b r u ne e l o r d p u € . e l E l n r , u + o C( p )
uo,\ E sr (4/)'? leql eloN .y ur. t,x IIB roJ QJ +,(x = (1+rt)g)Ay rcr{l qcns ('t'r{)I e ly'l :t,tt^ r.usrr{clroruouorl-* IBrurou anblun .t'I E SlSrx0 S J a q l / r \ o q s = ' l , g l E r { l + lol puB.lcnpord rouur | r\oqE eql ol Qt/n;o l c e d s o r ; + r . ^ u o r l a l c t r u o re q l a q 4 4 t e 1 ( c ) ' r ' I / n e c e d sr o l c o z r eql uo lcnpord-Jeuul uB r , . i l s n o n 8 r q u e u ns) a u l 3 o p( x * f ) f l - . Q I + , ( , $ l + r > u o r l e n b aa q f ( q ) ( v N E u t p r e E a rl u e u o l e l s S u r p u o d s e r r o ce q l Jo Joorg eql alElrrul :lurH) '/.1/ur IBapr-UaI B sl {0 = (r*x)fl :1tyt x} = 01 legt ,noqg (e) ''L,'.rt'
'+'*y1 z tlt D1 ('lnJqtleJ ,{pressecaulou ere rlclq,r\ Jo sluauela roJ uollculsuoc sNc oql sI eslcraxe slqa) (zt-v-c) sasrJJtxa
' s a s r c r o x e8 u r m o 1 l o g [ aqt ur peulllno sr puu 'parr.1o,rurleq^\aruos sl (p) Jo JooJd eqJ
relncrlredur 'o1oulsolels(c) reqmueql erorut, .$;l^":i;orrtt;l"rt? lreJ ur sJ z n leql s/y\oqs srqJ .1xr,]x)u= Ir, pue (ri.*)u = Ii qtt^ ' rurl rujt = 1!r uriy= 1 ueql'N r r qll/r\ '(x)x = '(fyEuorls-o .1ce3 Ix apnlcuoo tuqt ua,re 1 ur) fl8uo:1s JI I .dsar) .* o3urs_'W ur. Wapl.(-tqElr ..dser) -Ual B sr (* ry N ecurs N u N i N*N t xzix oste'N ? x JI l+N u'N-;' ,ix *rti'atou.r qcea 'ie 'Jc) tx rr JoJ '((p) ( 8 ' t ' Z ) p u € > q c n s (!x) r ' | I r e q r t i ' i . a t Jl lt'i"rlgl 5 p l r u ossr o o u r s l 0 u E u r s e a r c u ra u o l o u o u s s l s r x a a r a q l (c) O 'r€olc ar? (q) pue (e) suorlross€oqJ .Joord 'ol
'n {o X4uor|alduo? aW ltl g to1otado pasop o ot spuatxa ^ = luS tq pauttap ' n - n :aS totorado toattq a1o7n{tioc aql (p) i p t t t a s u a pr t ( { " ' ' Z ' r = t t ' p
''l
I=! { J =)'n (c)
t!a'!1'!U!1
.V.Z serqoElylraqllH pazrl€reuogpuu s1q8ra11
6S
2. The Tomita-Takesaki Theory
60
(b) Let (1f,r-II,r*Q,r,) be a GNS triple for 0. Show that there exists a = uniquet-ilo#etric operator u: !1,1,- lf such that z n.p(x)o4,. ( H i n t : N f ( a ) . n o t e t h a t i n w i t h a ' ' a s i n N , x a ' r l z n ( x )f o r a l l l'! so that (by semifinitenessof 0) N is o-weakly dense in M; argue that ng(N )ng must be densein 1f4;observethat for all x,Y in N.) (c) In the notation of (b), show that u n4r!) = xu for all x in M; concludethat if l0 = u0,!othen ( i ) r E , , ,= a ' r l 2 n ( x )f o r x i n N ; (ii) if {x,} is any net in N such that x, - I strongly,then (9 = (i) uniquelydetermines1.1; lim a'rl2n(x,),and consequently x in M. (Hint: verify this for x'in for all (iii) (x) = <x(rhl0> '1i; and the definition of a) , and note that N *'N , usirig both sides of the desired equation depend o-weakly continuouslyon x.) (iv) tM(gl = ilfr- 7.
(Hint: lMl,ll = t NEgl = 1a'rl2n(N )).)
(2.4.14) Let 0 be a f ns weight on M. Let {ft: i e Il be a- monotone in..ruting net in Mr * SUChthat fi(x) t QQ\-for each x in M* (cf' Prop. 2.4.9 (ii)). Foi' each i e I, let a,' .r.MI and. lr (= tP) be (a) and (c), respectively. with rf, as in Ex. (2.4.13) associated (a) {al: i e /) is a monotoneincreasingnet in YI ^:d Ii- t t. (b) Stiow that ((i: f e /) is cyclic and separatingfor M, and hence, also for 2".'1Hint: if x e M, <xli,\i> = {i(t) t Q@)'and 0 is (iv)' [Mi;] = faithful; so (ft) is separatingfor M.' ily Ex.-(2.4.13) ianQ; also af 't.l bY (a)') (c) rne sit' s = (;lll2 ajl:: i,i e I, at e Mt) is total in lf i'e" [s ] = ,t. (Hint: use (a) and (b).) (d) If .i^ it ", in Prop.2.4.ll (d), show that the set S of (c) is cont;inedin dom Sfi;so, Sfi is denselydefined and So is closable. (c) (il repeatedlyto show that, f,or any x e (Hint: use Ex. (2.4.1"3) = M) and i,j € I, <son(x), alrlza'li' N O N+, ai e aa!r/2a,*l,,n1x;r.; O pcfinition 2.4.15. An involutive algebra( [J, *), equippedwith an inner product,which satisfiesthe conditions(a) - (d) of Prop' 2'4'll is callid a generalizedHilbert algebra. It is calleda Hilbert algebra to the requirementthat ,s0 is if (d) is strensthened(considerably!) o = a t t I in U. i s o m e t r i c i . e . ,i t l l i i l l l l l i l l f o r Remark that if LI = na(N ,r'n Nh) arisesfrom a fns weight @on L[, the' U is a Hilbert itgefra if ind only if 0 is a trace. If an
ruroJ eql Jo sl n erqe8le lraqllH pezrlutaueE pe^erqce d.re,ratcqt ([z ruo3l '3c) u^roqs s€q sequroJ 'uollcorlp esranuoJ eql gI '(n)Qu nnf. sl ( n )f erqaEle uueuneN uo^ Uat aql ueql '(9 N u 9w )Or, = n JI lpalerqce sr y,g erqoEle uuerunaN uol B uo lqEla/ suJ € tuorJ Eulslre erqe8ye UeqllH pazrlureuaE eql leql ^\oqs ol pr€rl ool lou sJ lI '(n )f t(q polouep puu'n go erqe8lu uu€unoN uo^ Uel D e q l p a l l € c s l , , (n ) u e r q a S l e u u € r u n e N u o n e q l l , , p = n J r p e ^ a r q J e eq ol pr€s sr p erqoSyu lroqllH pazrlereuo8 eq1 'gI'y'C uoqrrrlJa(I
ore,,11pu€rl? qloquoql:[I'0]z? " ," orl:r?;3il'fiit: Jolesqns 'u ;1 asoddns -- lJrlls aq f eru uolsnlcur eql lnq
n j n
leql realc sl lI
'{,0 ) uA l l r , l l>, l l t ( u ) , u€| ;0 < r E : * c I t } = " r ? les aql -raprsuocpu€ JeqlrnJ dals auo eureE slql enulluoJ ',( p ) u u r e s u a p f l 8 u o r l s s l ( r I ? ) , u ' , { 1 1 e u r l. (; z t r ) , 2 ( I t r ) , u = l l t r ( 6 t r ) , u ) , u -p u u r l l z r u ( z u ) , u u a . t i ' , l j t ztt'rtt JI 'IJEJ ul -- Qth 3o erqaEleqnslurofpr-Jlos € sl (,0 ),u 2I go qderE eqr ul esuap sr r n lJ go qderE oql pu€ oA 3 ,p Euruearu .g rog eroc 3 sl ul esuap st :prlB^ osl€ eIB suollrossB lc€J uI -t , t i p f i ',p E u 1 , y r o 1 1 oogq a ' * ( u ) , u = ( a t r ) 1 2 p u e t D I ) tr JI ou ueql l e r { t s / A o q s- - l ' E Z ' d o r 4 3 o ' g o o r d o q t u 1 p a s d f l p a l e a c l e r 1 r o s a q l 'suorlecllclyllnru lqErr qtlrr Eultnrutuoc Jo -- luaunEre eldrurs y suollecllcllllnu lJol go aldrcurrct plo aql -- ,(p)u 5 (,t? ),u ter{t 'n ul po{ceqr flrs€a sl lI I II€ rol uQ)u = l(u),x l€ql qcns ,t u o ( u ) , u J o l e r o c l o p e p u n o q ? o l e s r r s e z r r E, { l r e a l J r l ? u I U q c e g
'{n ul tn l l t l l ,> l l u ( t ) u€l lo < , E i q c , u } = t t 1 :slueuola sselc Eur,nolloJ aql .Arou .teprsuoC rpepunoq-lqErr. 30 '( p )z erqa8le rolerado luro[pu-310s oql Jo (ueroaql luBlnruuoc-alqnop aql ur posn sB pro^\ eql Jo asuos oql ul) fceraueEap-uou olur selBlsuBrl ll ul zll go ,{lrsuap 0r{I '*(l)u = (11)z pue (a)u(l)u = (trl)u "e'r's€.rqaBle e^rlnlo^ur 'raqlrng 'luqt pue . p ur &.1 3o rusrqdrouroruOq? sI u deur eql IIe roJ ul = tr(l)u l€rll qcns (;x)t - n :u deru e sr orarll g uogteldruoc 'leql aes ol fsEe sI rtll,Y. pezrlerauaE urqaEle e uerrS UaqllH n U ('n uJ uollerado ,,dr€r{s,,eql qll/$ luelsrsuocsr slrll'lJ r I leql lc€J snor^qo eqt acrtop) 'oc ) J I J a A O u o q ^U \ I = puB lS = , el1r,r pue'flarrrlcailser ! Rue ilG I Q O1 -U 'J pu? sulBruop or{l roJ qA pue 1C elrr^\ IIEqs orlA S Jo 'sr,urel I e J I s n u e s e q l 3 o a E e s n " o q lJ o s s a u a l E l r d o r d d e er{l lnoq€ a8els ra1e1e le ^es ol pro^\ E o^BrI IIBqs e1yflaarlcodsar 'sro1e-redo ulBIJu pus ,,dJ€qs,, aql s€ ol petJaJal sor,ulloruos ore eseql 'flarrllcodsar 0S Jo lurofpe eql pu€ 0^S Jo arnsolJ eql rog 'f laallcodser 'J pue S elrra lleqs arr pue 'p go uollaldruoJ eql elouep ,i lal IIETIS a , n ' u a a r E s r p e r q a E l € l r e q l r H p a z r l e r o u a 8l c ? r l s q e
I9
serqe8ly lraqllH pezrlsreuag pue slqElarg 'V'Z
62
2. The Tomita-Takesaki Theory
Lt = nO(NOn Nb), where,in fact, M = t(tJ ) and 0 is defined by
= Ilkll', i f x = n ( ( * t ) , o(x) t-, if x is not of the aboveform. withE€u
" If U is not achieved, one can argue with U exactly as one l e ts tt(t,)denote the i n U " , o n e argued with Utto find that if, for I = f o r r t ' ( n ) l a ll n in U', then n ( t , ) n l f t h a t o n s u c h o p e r a t o r s unique n( g') is a self-adjoint algebra of operators on lL This, in turn, " cquips g with the structure of a generalized Hilbert algebra' which can be shown to be achieved;finally, one has the fact that rtt( Ut)' = n( U )" = rt( U ")", so that both the generalized Hilbert algebras U and U n have the same left von Neumann algebra. We are now in a position to state the Tomita-Takesaki theorem in its full generality; we shall state it as a theorem about generalized Hilbert algebras, and feel f ree to interpret it as a statement concerning the GNS representation of a von Neumann algebra that is a s s o c i a t e dw i t h a f n s w e i g h t . Thcorcn L1.17. Let M = f( U ) be the left von Neumann algebra of a generalized Hilbert algebra U . Let lt denote the completion of U . Then there exist a setf-adioint antiunitary operator J and an invertible (possibly unbounded) positive self-adioint operator A in tt such that: (a) S = JALI2and F = J6-rl2 are the polar decompositions of the sharp and flat operators; (b) ta.r : A-t, and co:nsequentlyJI@)J = 7(a-t) for any Borel function I on (0,-); and E (c) AitM A-it = M for all t e E and JMJ = M). Having balked at proving the theorem even in the case of a finite weight, we shall (naturally?) say nothing about the proof other then that the reader may find it, as also the preceding facts concerning g e n e r a l i z e dH i l b e r t a l g e b r a s i n [ T a k l ] . It is implicit in the statement of (a) that D* = dom at/2 and & = 'musical" notation dom a-1l2; we rest the case for justifying the 'sharpnand "flat". As in the case of finite weights, if 0 is a fns weight on a von an application of Neumann algebra M, with GNS triple (1t6,fl6,n6), 'wliich is the left von f l A ( M ) ' the Tomita-Takesaki Theorem to = r e s u l t s in a o-weakly N D n 6 ( N 6 o f a l g e b r a u 6 Ncumann | continuous one-paramete'r gro[.rp (or9; of"'modular" automorphisms of M, defined by
z6(x)A6it). oft"l = n-or(aiot Example 2.4.1E. Let G be a locally compact group with a fixed left Haar measure, which we shall simply denote by ds. Recall that the
'slr{Elaaa r llulJ roJ ssncsrp l s r r J I I B r l se ^ r q c r q ^ \ ' u o l l r p u o c frepunoq (lanbes aql ul 'SW)) raEurmqcg-ulpel4l-oqn;1 aql ,{q uarrrE s 1 d n o r E r e l n p o t u e q l J o u o l l e z r J e l c e r e q cE q c n s ' l l e l E u o l l c n r l s u o o 'lqE1a,v.€ qlr^\ pelelcosse dnoJE SNC eqr ol l€acld€ lou seop qclq/( relnpol'u aql Jo uolldlrcsap crsurJlur ue e^€q ol 'acrlcerd ug '1n3asn s l l l ? o 1 u l e E e { c e q u o q l p v e ( 0 ' n ) . ; o a c e d s S N C e q l o 1 E u g s s e df q peurelqo se,r dnorE slql ollq1ystuslqdroutolne Jo (U, r , :,61o) clnorE € ol aslJ se4? 1q uo O lqElo/y\ sug f:ere 'uorlJas lsEI eql u'! ueas sy uoplpuoJ
f-repunog S5DI cgl
'SZ
'srol€rodo uoll€lsu€rl pu€ (V fq) uor1ec11d1l1nur uea^Ueq SUOII€Ior uOIlSlnIuuOC oql uO SasnJoJ -- n = rr_Vy'tlrrV .,,(g > -- ruaroeql r{Ese{Bl-elruol eql t:tdi'= ,('g Jo JIuq roqlo eql 'g elerosTp Jo essr agl ut sa'oJueq ig ur. 7 qc?a roJ-3 1 :'\) ?^?q a^\ rd 'oelnturog = polJuol ,(1rsea sl l€rll esaql Eursn trxt ll lrclldxo '(uorlcungrelnpou eq1 Euglouep lqEu oql uo V arll)
(s)t(s)v= (s)Gv)po€
= (sxl.r) 1('s)l z/r_(s)v
fq peurJap ore V pu€ 1 srolerado eql ]Br{t IBe^aJ suorlelndruor aaJ v ('serlrtuopl eleurxordde Surrrlolur sluaurn8re prspuels Jo esn eql s o r r n b e r- e l l q ^ \ ' r € a 1 cs r j u o r s n l c u r e q a ) . , { g t t : l \ } = ( n h l e q l uolt?luasordar 1ur8a1u1o^oq€ eql tuorJ e_pnlJuocuec auo .((C)rZ I ? acurs) esues reuqcog eql u1 ro (sp<)'tr'l>(q)l[ = < 1 . r rQ)u>)i1:1ee,r j (t)u , palordrelur Euraq srql -- sp'r(s)l l u r E e l u r n , I JJ lsr{l 'uogllulgep aql uorJ 'ree1c sr '(g)"2 I sr uorloldruoc ogl ll puB lSnt fi
sp_(s)r, (s)l J = .rr'lt Itt)t
puc
,-(t)v = (s)*l
'tp(sr-r)u(r)l J = 1s)(u1) fq'l(lolrlcodser 'paulJap oru lcnpord-rouur pu€ dreqs 'lcnpord aql JI .erqaEle lreqllH pazrlereueEE Jo ornlcnrls eql seq (9)r3 = p- las eql
j?t:;llril,i.,lllili .pour3e, = (sXr,r) .(q.rlarrlrcedsar (,:'f3',:..-1jl"r= f:elrun snonurluoc [1Euor1s oql er€ g Jo suollelueserdal reln8ar-1q.Err pue -lJol oqJ, '(C)"^: ur I roJ sp(r-s)9(s)lJ = . tp(r_s)tJ ''a'l lg ur deu uorsralur oql Jo a^llu^rrop,rufpo4rp-u
€9
uolllpuo3 [repunog Sn) aql
'S'c
2. The Tomita-Takesaki Theory
64
Definition 2.5.1. A 0 in M* * is said to satisfy the KMS boundary condition (at inverse temperiiure B = l) with respect to a o-weakly continuous one-parameter group {crr: t e R) of automorphisms of M o-weakly as t ' 0, for each x) if, for each x and y in 1 i . e . ,c r r ( x ) ' x .ll, theie exists a bounded continuous function F: {\ e C 0 ( Im \ ( l) - G which is analytic in the interior of the strip and satisfies F(r + i) = q(xcr,(l)) and F(t) = Q(cdl)x),
for all I in R . O
"F is KMS-admissible for For brevity in exposition, let us say that .x and y", whenever F, x and y are as in Definition 2.5'1. (To be precise, 0 and (cxr) must also be incorporated into the abbreviated p h r a s e ,b u t t h e s e w i l l b e c l e a r l y d e t e r m i n e d b y t h e c o n t e x t s i n w h i c h the phrase will be used.) The following Exercises list a few simple facts from complex f unction theory, which will smooth out some of the subsequent proofs.
Excrciscs (2.5.2) Let tl < tz 1t, be real numbers. For i = l,2, let /,: (\ e G r, ( i m r < ! , -' 'r1) '1- C ' b e c - o n t i n u o u sf u n c t i o n s , a n a l y t i c i n t h t i i n t e r i o r o f the strip. f t and f , agree on the common boundary Im \ = lz, s h o w t h a t t t r e l t o b a t f u n c t i o n c o n s e q u e n t l yd e f i n e d o n { \ e G / t ( Im \ ( tr) is analytic. (Hint: use Morera's converse of Cauchy's theorem.) (2.5.3) Let f: {r e G 0 ( Im r < l} 'C the interior.
be continuous and analytic in
(a) If / is real on the real line, show that the function /, 1f . C -l ( Im \ < l) 'C defined bY
[ , r { t ) ,i f o < I m \ ( l /(\)={-l(Im\(o L/(D if is analytic in the interior of the strip. (b) If f(t) = f(t + f) for all t in lR, then / extends uniquely to an entire funtion with period i. (c) If / is constant on either bounding edge, then / is identically c o n s t a n t . ( H i n t : a s s u m et h e c o n s t a n t i s z e r o , a n d a p p l y ( a ) . ) l*.nna 2-5.4. If Q (in M* -) satisfies tfte KMS condition with respect to (the o-weakly continuous die-parameter group) (ar), then 0 o dt = Q for all t. Proof. Let F be KMS-admissible for I and y. Then F(t) = F(l + i) =
apasolc s;'g
qcea leql porunssefluo sl ll JI i(es noi( uB3 l€rli6 (c) .H roJ oroJ B sr og uaql
"l 'rA 'C ': '{u fueur [1alrur; ,(1uorog r :'l e] = 0C 0I 'tH roJ oroo E sl tC 'u qceo rog 'g1 ( q ) JI pu€
,(raao ros('H)sre= (^n)sr Yrll1# ll;it reqrpuE,ulofp"-:rrj 'Fl urop r "l e JI
(e)
'("t"H)o = ("1 e)17
(-, (zll"l"nll + ull"tll) i pue
'uA
?
urop r u: : 'l e) = rl urop
:uolldrrcsardEur,no11o3 oql fq 'n e ul p rc1endo u€ eurJap'"' (Z'l = u roJ'g u1 roleredo lurofpe-gles€ sI "tt lt $.5.2)
(g=v+v=s+
'ssaulurofpe-g1es,{q loclf S 3 V Jo ernsolc oql sl f pue posolc sr g :lulH) 'E 3 oglv lsgl qcns y rcJ oG oroc € sr er3rll 3r ,{.1uopue JI g = V fiqt ,roqs g ur srolerodo lurofpe-J10s ere g pue v JI (c) 'y cls uo uollcunJ (lqEnoua sr papunoq f11eco1)snonurluoJ fue s1 8 oreq,n '(V)B tol aroc B sr
rf;" (((u)t,,u_lr) orr; l e q l ^ \ o q s ' l u r o f p e - g 1 a ss l Z J I (q) 'lVl toJ oroJ e sl ll JI f luo puu J\ v toJ aroc B sr 0g acedsqns reaull B l€rll ,noqs 'y 3o uolllsodruocap relod aql s! lVln = V Jl (e) '(U e U ur,(Eo1odo1-rurou aql ol loodsor {11,n; n 3o qderE eqt ul asuep sr oGlZ Jo rIClerE eql 0g ecedsqns rgaull s 'c leql llecau JI ,/ JoJ eroJ E pell€o sI c Jo ureurrropqt!/'\ ,l ur :oleredo pasolo paurJap ,{losuap E eq V lo-I (S.S.Z) saslJrexx 'ureql EuluraouoJ slc€J ^seo auos raqleEol reqleE ol -- speeu Jel€l puE elBrparurur roJ rlloq -- InJasn oq plno^\ ll 'srol€redo papunoqun 'qUoJ?Jueq 'lleqs rA\ eouls {lg,n f lluanber; Ieap ol e^€q
'(,f)g= (0)g = (r)t = ((4b)0
E
', l(u€ ro3 ',{lluanbasuoJ 'luelsuoc sr J l€gl uaroaql s.alll^norl ruorJ s^\ollo3 11 '(uolllugep f q) I > \ rul I 6 drrls eql uo papunoq sr / 'rerrainoq 'aculs 'r porred Jo uollcunJ errlua u? ot spuelxe J uollcunJ arll ((qX€'S'Z) 'xg fq) 'oS ', II€ roJ ((f)to)@
s9
uolllpuoC frepunog Shl) eql
'g'Z
2.
66
The Tomita-Takesaki Theory
Definition 2.5.7. By a flow on M is meant a one-parameter group - crr(x) is o-weakly {ot)tep of automorphisms of M such that / n co'niinuous, for each x in M. We shall now lead up to the main result of this section, which statesthat if 0 is a faithful normal positive linear functional on M, t t r e n { o f ) i s t h e o n l y f l o w o n M w i t h r e s p e c tt o w h i c h 0 s a t i s f i e s t h e KMS boundary condition. S u p p o s e ,t h e n , t h a t 0 i s a f i x e d f a i t h f u l n o r m a l p o s i t i v e l i n e a r functional on M and that { where o is a vector in lt which is cyclic antl separating for M. The assumption 0 o dt = S implies, then, that there exists a unitary operator u, on ll such that urxa = crr(x)o for all x in M; it is trivially verified that {ur} is a strongly continuous one-parameter group of unitary operators on lf. Hence, by Stone's theorem, there Jxists a unique self-adjoint operator H in lt such that ut = eitlJfor all t. Let B, denote the linear subspace spanned by vectors of the with x e M and f e Cl(lR). In the special case when form l(H)ig = we have u, = ait (cf. Ex. (n.6 (b)) and so H = log A; in this cr, 09, cise, *e shall simpiy write B for B,o.o. Lcmma 25.E. (a)
B u is a core for g(H), for any continuous function g on IX
(b) B;t Mq t.l in^cise % = o9, the subspaceB is invariantunder the sharp operator S.
is an everywhere defined bounded Proof. (a) Note that g(/rlk(f| operator, for any compact s,ii f g ft hence B " c dom g(/4. In view of Ex. (2.5.5) (b), it suffices to prove the following: if | = lx(Fl)( for some compact set K g R, there exist [r, . Bn such that lr,'I a n d g ( f / ) ( , , - C U r l . T o s e e t h i s , f i r s t p i c k x , , i n - M s u c h . t h a _ tx n 0 ' (for (; next, se'iect any / € C:(D such that f(t) ="1 for all / in ( the existence of such air /, see, f or example [Yos].) If K is a compact set containing the support of I observe that l. = "f(I/)x'O e 8 s for all n, that \n - .f(m\ = q (since f(H) is bounded), and that J sUI)t since g(1/)/(1/) is bounded. (We have s(/48" - c(m^il| used the fact that
f(II)\= f(H)r1(m\= lr(rDi = E.) ( b ) L e t \ = f ( m x o € 8 s'7, w i t h / ! C : ( l R )a n d x e M . N o t i c e t h a t -' this is true of even the ll(lRi is in of Fourier transform the larger class of so-called Schwartz functions (cf. [Yos]); consequently the inversion theorem of Fourier analysis is applicable:
.. ('erroqe(q) pulJe) 8'g'Z€ruruelesn:lurH)^'Or> , > 0 roJ:Hla * ' : s l , 1 e q 1q c n sH g u r ' 1 l s r x ee r e q l ' 0 . 0 / ' r o * a r u o p r 1 ' j 1 ( c ) ('(e) ul flrlenboul oql osn:lurH) '01> I > 0 roJ 0 * u1.,2 'lror, pue 'l '0 . ol'rof ruop uaql '0 J {"1)'jr (q) 0 JI pue ('(\):rp\orzao'*'J * ([0.*))1rr >
= tr)lrrpr(rrr) (r)lrp,rrr(0"'J* (r)lnprrrr(ott'J lJ pue '<:'l(ll)slt
+ ,ll:Gr)Io''-)t11 t , lllHt ' ll zlll"o,a(H)(-'o)rll
ro;'raqlrng l0l > I > 0 roJ Hraruop ) t € 0 . 0l pue
'/ qcns ,(ue "orauoPrl
(e)
G's'z) ' u o l l € l o u e^oqE eql urBlall
saslcrexa
=
.g = a ^ B r { O S I € O r n . r _ S= S ' 0 3 u l s . 1 C eu31 . g B S E g,S oS 'lroddns lceduroc Jo uollounJ;, B osle sr (t-)! - / eruls
' 8 r u * x ( VE o F U= UxsSvt(yEoFU = rJx(sEoDl.1rv1= (ux(v Eot)/)S 'W)xpuB (Ul)jf , / JI lrC = zlrY ulop 3 g ler{l (e) uror; epnlcuoc 'snonutluoc ,(1urelrec sr. ,'jra = (l)B uollcunJ eql ecurs 'V 8oI = g uaqrq (c) 'parrsop s€ ,{ = t I os pue g(x)ln = Uru*a rcql H Jo uolllulJep eql uorJ s/AolloJ lI (Z'E uortces pesnrod E u r a e q r e U B J o o r c l s l q l l l s l ^ - e r p u € ' s n o r J e d sl o u s 1 l u o u n 8 r e e q l ' u o r s s n c s r pt u l p e c a r d e q l lBql qlrsJ uo lr e{Bl ,(eru 1e fseeun sleo3 eq 3r ':apear arll lg'g uo;tca5 ur i(lsnoroSrr paluerl eq III^\ sler8alur '41 {Beln-o q3n^s) 3o r( luauolo u8 seurJep pue f 11eam-o osues se{€ru eql 'pueq raqlo eql uO 'flEuorls t p ( x ) ' n ( t ) { 1 , t , ( u Z ) , , 1 E r E e 1 u r , , = : ocueH palarcfrolur Eur5{"'ier8a1ur a\t 'tp rJx^r,ae)I !rn_@d , . . e ? 'tPyvPQ)I = (t)/ -J ut-@d
uollrpuoJ i{rupunogsW) eql
L9
'9'z
68
2. The Tomita-Takesaki Theorv
( d ) I f I f D f , t h e r e e x i s t ( ,",( H e i nBt : s u c h t h a t A t [ n - a t l a n d A t l j - At(t for 0 ( t < l/2. use (c) wittr.iz = log a and ro = l/2, to choose !,- e E such that. At(. - Atq, for t e l0,l/21. " = Jarl2ln l Note that l: lgrtlt, - ,-l*,- ,^by the above = 6 1 1 2 1 6 1 1 2 l=, J l , ' / ( I c o n v e - r g . e n c fe' 6 r t = l / 2 ; ' i l s o A 1 / 2 q "tj " = 6Ll2gt' again appeal to (c) with in place o'i q,,.) E L c m m r Z 5 - r O .I f l e B
andneD*,then
< a n q , n * >= . n , a r - i l * r
for all z in C;
the contmon value defines an entire function of the complex variable z. Proof. Note that as /(I) = \' = "z los \ is a continuous function on (0,-), it follows from Lemma 2.5.8 (a) and (c) that both ( and (s belong to dom A' for all z in Q so that the above expressions are meaningful. Further, by Lemma 2.5.8 (b), B g MA First consider the special casewhen n is also in B. Then, [ = ;sfl and n = yQ for some x and y in M. Then, =='.:',';:::^',': (since r = r*)
n-iJxr>
=
(since Jf(A) =7@-')t) (since A-1/2,/= .S;
= 'n'Al-if *t ' s o t h a t t h e a s s e r t i o ni s v a l i d w h e n n e B . r t , e D * , u s e E x . ( 2 . 5 . 9 )( d ) t o p i c k n . e B s u c h t h a t n , r - r l a n d ,If ,?l - n', use the validity of the desired equality for each n, pass to the limit and conclude that .Aoq,n*t = .n,Al-i(*t. The verification of the analyticity of the function is routine, and may be safely left E to the reader. Thcorcn Ls-ll. Let 0 be a faithful normal positive linear functional on M. The lollowing conditions on a flow {o'r} on M are equivalent: (i) dt = of for all t; (ii) Q satislies /&e KMS condition with respect to (arl. Proof. (i) + (ii): Let x,y e M and let [ = xo and n = yfL Since q, e Ds, and.at(: u s e E x . ( 2 . 5 . 9 )( d ) t o p i c k i - i n I s u c h t h a t a t l - ' 4 : [ ao(* fo? 0 a t q , sf o r 0 < , < l / 2 . C l e a r l y , t h e n , a z i n a " l a n d a " q : ( Re z ( l/2 (since A't is unitary). It is not hard to see,using the inequality in Ex. (2.5.9) (a), that the above sequencesconverge
'[(c)(S'S'Z)'xX aas]Ea toJ oroc e sr Hg aculs'Hg luaE v lerll ^\oqs-ol soorJJnsll :Ha = V i-eq1^\oqs ol peau alA" 'g'9'g eruura.Iur sE eq H g le.I u, ul / pue n u[ x roJ ur*rr.t = g(x)'n lEql qcnsu uI H rol€r3do lurotpe-31as€ slsrxo oroqt teql g'g'Z eturual Eurpece:d uoJssncsrp eql urorJ s/AolloJlI 't'g'Z €ururo.I^q 0 - lrco O uaqt ,uoyllpuocgy,111 eql selJsllssO tlrltt^\ ol lcadsor qll^\ ^\olJ e sr {rn) JI :(l) + (II) 'd pue x roJ alqrsslupe-Sn) sl jr spro^\ roqlo ul
:((r)jox)p= = = = = (t + t)l
(r(Qfo)O=
pu€
q;tyxr,-vl(r,v> = (lA u = Ut,-v ecurs) $rt-V*{ .Urlr_V>= qy*d tJr11-v>= (tU 'I
'salJsllespu€ 'rorJoluror{l ur cllfleue sr qcrq,$ fo r I ro3 > z rul > 0 duls ogl uo uollcunJ snonurluocpepunoq€ saurJep | > z tJrl> (,/l (,/l > z rul > 0
'I
. < o t r . l " , _ v >| J
= (z)r
'(Q'S'd'xA 'Jc) ooueH .(u/)
,qloq .are
o c u o n b e sa q l J o l I r u I I o q l 'z/t = z .uIoulleqruo::i3.:irll'r,'j,,t^",:l,"tifio3,,, '(oq1 aculs
posolcoql ul snonurluor pu€ papunoq's1 .i"1q^ ,.0i",-lv,ut'= (i)zl q-oll3unJ aql ol I > z url > Z,/l drrls eq1 ur flurogrun saEraruoc ("/) lerll otou-.'eAoq? su [1lc-uxa EurnEry .z wl - I--= e.l - I)eU l€r{l ocllou : = (z)"! leql enrl sI lI '0I's'z eurirol ,{g '((E) (6'S'Z) 'xfl ul elerrrrlsaaql Eulsn ureEr) Izlt'oltt
+ rlltll)> llrull(lll,vll'trns. ) > l(z)'.rl ll*!llrt'Qlllzlrvll :duls aql ul papunoq osle sl rg uollcun3 aqa 'rorrelur aql uI c1t,{1euepue Z/l > z ull > 6 dr-r1seql ur s.nonulluoJ s.r = (z)rl uollcunJ aqt leql apnlcuoc 'alrlua sr <*u''1",-v, = (z)"! a5u1g' 'C/ t > z au > 0 duls aql ur ,,{lurro3run
uolllpuo3 frepunog Sn) oql
69
'9'Z
70
2. The Tomita-Takesaki Theorv
Let q, = vq . Bu, y e M. Note that the function given by G(z) = ."-izHrq y*o> is eri^tire(use Lemma 2.5.8 (a)) and that for t e R, C(t) - <e-itHx$ y*o> = <xg eitHy*o> = <xgcrt(/*;o = (cr,(l)x). It follows (on applying Ex. (2.5.3) (c) to F - G where F is KMS-admissible for x and y (relative to cr,)) that C(t + i) = f(xcrr(y)) for all r in R. Hence, = G(t + i) _ <e-i(t+i)Hxe y*0> = <eHrg ,itEr+96 = <eHxQ crJl*Xb . Setting n = crt(y)Q we find that = <eHl, ,son> for all n in MO This means that eHl e dom F and that F eH( = Sf. Since F = F-1, dom F = ran F and so S( e dom F and A[ = f,Jl = D e H g ;i . e . , ( e d o m A a n d A [ = e H l , a n d t h e p r o o f i s c o m p l e t e . An immediate consequenceof Theorem 2.5.11 and Lemma 2.5.4 -which can also be directly verified using the fact that Aito = o -- is stated below as a corollary for convenience of reference.
C-orollary Ls-lz- 0 o o! = 6.
O
Definition 2.5.13. The fixed point algebra of a flow cr = {crr),.p on M is the von Neumann subalgebra, denoted by W, of M givei bl' W
= {x e M : ar(x) = x
If 0 is a faithful
for all t).
normal positive linear functional, we shall write
A
ttf
f or Mov: thus,
ptQ = G e u:
o!{i
= x
for all l}.
D
The next result is a very elegant characterization of M0 which, among other things, drives home the fact that the modular group oP effectively measures the lack of traciality of 0. Corollary ?J.1L on M. Let x e M. to the lixed point particular, Z(n 2
Let 0 be a faithJul normal positive linear functional A necessayyand suf ficient condition for x to belong algebra MQ is that QQfi = q1x) lor all y in M. In M9.
Proof. If x e MQ and I e M,let
F be KMS-admissible for y and x.
leql asues eql ul c1l{1rue [1>1ea,n-osI C (B) ler{l qcns N - D il uollcunJ 8 slslxe 3rorll JI iloo roJ c1lf1eur,, n Jo x lueuola uB IIBJ :s^\olloJ s3 palculsuoc s[ -- rIBsa{?J ,(q erqaEle Bllurol aqt- pe4ec -- 0p ? qJnS A V z fre,re ro3 'rv toJ aroc B aq lsnu ('n )qu (q) pue isn o1 alqup€Ae ore sluaunEre uoJlerulxordd€ lsql os ',pgur aldrue {1tue1c1y;nsoq lsnu 11 (e) :sanlrrr o^rl e^€q lsnru 0 f? B rlcns^ 'esec elrulJ eq1 Eulnord u1 posn g los aql Jo elor eql fBId uec (op )Pu teqt qcns 7g ;o 0p ececlsqnsecru e Jo plorl qerS o1 st ruerooql a^oq€ eql 8u1,rord ur dals lsrlJ ogJ 'slqEran olrurJrrues Eururecuoc sluerual?ls Jo sJooJd reqlo pue slql olul saoE leql ldacuoc auo uocln 11o,npflgorrq ll€rls o,rr .p€olsul .[Z uroC] ul punoJ eq u?c rIJIrI,n 'usroorll slql Jo Joord B lnoqe EuJqtou fes llsqs aA\ E
:Tualounbaan) W uo (\\ uo[ o uo uo y13taa su1 p st Q II .rII-SZ trraroaql
:sauoceq ^\ou ^ do dnorE relnpotu eql Jo uorl€zrrolJ€r€qo SI/{) eql 'YW uo 0 tq8le,n aql ^q pecnpul leuollJunJ reaurl oql roJ pesn Euleq osle sl -0 Ioqurfs erues-oql leql .r(11e1uaplJul.aJIloN .ln3Eurueeu are (({)rnx)p pue (x(f)b)p suoJssarclxe. oql pue 014 t (tetix .x(rf)ln os pue try u 9N ? (rfp u.eql'?N u 0N r l('x gr .og 4y = 1014;1n flluonbasuoc pue 9ry = (9N )rD ueql ? = to o O JI lpql eloN '((/)lox)Q = (, + r)J'(r(d)rn)g = (r)./ U O ul roJ 'serJsll€s pue drrls egl Jo JorJelur eql uI cllfleue s1 , IIB qclq,n b * {t }^z ru1 0 .? ) z} ii uorlcunJ snonurluoc papunoq .l B slslxe eraqt TN u v N ur f puu x slueuele go rred frane roy (il) p u e! g u I , l l e t o J ' + n u o O = t o o O (l) Jl n uo {rn} molg B ol lcectser{11,n 1, = g tu) uolllpuoc frepunoq SW) eql fgsyles ol pl€s sr J/ uo 0 lqEle,r suJ V .tl.g.Z uogIrIJaC 's^\oIIoJ s€ popuerrrusl uorlrpuoc .slq81e,vr Sn) orII suJ Jo asec srlt ol pepualxe eq uec srsfleue EuroEerog orlf Jo IIy
or o JosseurnJrrr'J paunsss eqror ,"rR" ,"; if!?;;T)t:1l"l,i:
'n .io u \ t g € r o J o 0 = ((x- 1r)jo;r;g'snqa Q =g aculYs.((x)jor)@ .luelsuoc Scuaq = (x(,f)lfo)Q = Q-)t = (0)J'= $t(.)g ,t fue'rog .og puB papunoq sl qrlq^r uollcunJ erllue u€ ol spuslxo .{ lsql t.g.z 8urru3.I '(l + t)t = ((Qjox)Q = (r(rfo)O = (/)J lugl Jo Joord eql ul se enEry sornsua r uo slseqlod^q eqr l€r1l olou ld p'ue r -ro.;alqfisrrupe-Sn) eq '1,t1 g 1e1 pue 'd xrg ur.t(. roJ etp.. = (,tx)Q leql f lesronuoc asoctdns IIU '(xt)q = (rfx)p 'relnclU€d ur .leql os .luelsuoc sr J lEql (c) (g.S.Z)'xA ruorJ opnlcuoJ U ul , IIB roJ 'gt()q = (t + t)t pu€ (,(r)0 = (rhr uoql
uolllpuoJ ,{repunog SW) eql
IL
'S'Z
2. The Tomita-Takesaki Theory
72
rl,(f(.)) is an entire function for every $ in M*, and (b) F'(t) = o9(x) for t in lR Such a function, if it exists, is unique since an entire fqnction is determined by its values on the real line; we shall write for F(z), z e C,. o!G) " L"t t,to d'enote the set of oO-analytic elements in M. It is easily verified that Mo is a self-adjoint subalgebra of M containing I and that, for x,y e Mo and I, z e q one has:
of trr + y) = lof tr) * o!{il, o! {xil = o! {*)o!{il, and
of t"*) = qh{r)*,
lf x e M and 7 > O,the integral
Iz I,p.*o (2r-y\-t t#o! {ia, of M. (This converges strongly and o-weakly to an element x1'probabilists') It process of "Gaussian smoothing" is an otd friend of i s n o t h a r d t o e s t a b l i s ht h e f o l l o w i n g f a c t s :
(i) x, e Mo anao! G) = (2rryzy-t/z Ip ,*o
l-#)o!
{iat;
(ii) x, - x o-weaklyas 7'0; (iii) x'e M6 ) x, e 146and 0(x7) = 0(x). (Prove this first for x e D6 and eitend'by lihearity.) Consequently Mo is o-weakly dense in M and Mo.l MO is.o-weakly dense in I'16. It cah, further, be shown that both NO A ll$ and l'16 are invariaht under multiplication (from left or right) by elementS of Mo. This last statement is proved using a fact about self-adjoint operaiors, which is stated below as an exercise. This fact might also convince the reader of the plausibility of the fact that if Uo= Mor\ N n Nf, then n6( tl o) is a core for Afi, for each z in C" 6 Exercises (2.5.15) Let H be a positive invertible self-adjoint operator in Xf. Let I e lf and to r 0. The following conditions are equivalent: (i)
| e dom H'io, for 0 ( Im z 4 tsi
(ii) (iii)
teaomlrto; there exists a (norm-) bounded (strongly) continuous function F: - lf which is (strongly, or equivalently, {z e Q, 0 ( Im z 4 t) i n weakly) analytic tle interior of the strip and satisfiesF.(l) = F/-itE for t in lR .
Isrurou ,(reao lcnrlsuoc u€c ouo ,147uo (tf =) O lqEroa suJ € sBq euo eJuo lerll slrass€ ruoJoaql ru[po>1r1q-uop?UIecrsselc eql .snq1 .fl ol lual€^rnbe sr n ueq^r d.lasrce:dy'g uo lqEle^\ suJ B sr 4,1 teqt acJloN
flelnlosqe ere qclq^\ 'sernseaur olIuIJ-o ocueH ^,fr = ,ft sacrog rfi 3o ,(lrlerurou leql reolJ sr 1r l(n) (p) (g.l'Z) .xg EI^ .rJr3o ssauelrurJrues e g l J o e c u o n b a s u o cu E u r a q n J o s s a u o l l u r J - o o q l - - ( 1 f ) u o n erns€aru a l r u l J o ( a r r l r p p e d l q e l u n o c ) e s e u r J a p( s t h = ( A , ) nu o J t e n b e eyl ueql 'n uo ellulJrruos leruJou B sl 0 ;r .flasraluo3 lqEJe,r '(n = U)^rl, uorlenba oql atn yg io n,l, lqara,n ) I) ^p II elrurJrues Ierurou e sourJap .t ol lJeclsal qll/rr snonurluoc {1a1n1osqe .(d, sr rfcrqrn '(J?) uo n erns€eru (alrurg-o errrlrsod) ftolg l,X)^7 = N Fl pu€ ac8(ls orns€aur (o11urg-opuu elqerudas) e aq (rt,t,X) t:,l suoltrulccdrg
luuoqrpuoJ
puc trrrrqrql
u{po1rtr1-uopg1
eql
-92
'[fa] ut goord alelduoc eql purJ uer rapeer palserelur eqt leql pue .(e) 9'y.7 .dot4 pue (l) + (ll) uolleclldrur eql uorJ /holloJ paepur seop uaroeql aql Jo luauelzls tsBI oqt l€r{l lclecxa '3oord aql lnoq€ Eurqlou frs IIBTIs aA{
,r 4w rt t 11o to! (xt)Q= qxlq=puo&,{:ff1t,;';'g"d;X i6w t x aAJ
'n
(ll) (l)
:1ua1nt1nba ato x uo suorl!puoc8uruo11ol ) x tal puo 14 uo 7t18nd{ suJ o aq Q p'I 'fi1's7 rnrrcrqJ
'[ra] ul punoJ
eq feu py9'7 fi,tegoroJ Jo uoJsuelxe Eug,vrolyogorll pue ('cla .0 n '"n uo) leql aroJaq lerreleru oql Jo lsoru 'esroroxa Eurpecord eqJ
(z)g puuzr-l/uop , I l'rn .{11ereuaE pu' .r0," errfl(olrTf; oror,u ruop , 1 lerll epnlcuoc 'lurotpe-91o, .r org pue frurlrqr€
. .lofl.lt
sB,$ I ecurg
= <1.(o/r)g>
os ' 0 = z rul aull eql uo eerEepue 'frepunoq eql uo snonurluoc.dJr1s eql ul cllfleue ete e z pue <1,(z)f> e z suorlJunJoql leql aloN 'or;/ ruop r t lal '(l) e (lll) toJ ,.|rr_H= e)i los .(rrr) € (l) roJ :(e) (0'S'Z)'xe fq (l) + (lt) egq,n 'f1rea1c(ll) + (l) :lulH) '0/ > z ru1 y 0 roJ lzr_F/= Q)I ueql .peryslleserB suolllpuoolualE^rnbaeseqlJI ruerooql urIpo1r51-uop€U aqJ '9'Z
9t
2. The Tomita-Takesaki Theory
74
semifinite weight 0 (= 0v) using 0. In case the density dv/dtt is bounded, the theorem can be reformulated thus: if 0 < cQ for some c > 0, there exists h in M, such that 0U) = Q(hf) for all f in M*. In the case of an unbounded density, the above equality is still valid, but must be made senseof (within the language of von Neumann a l g e b r a s ) ,w h e n & i s u n b o u n d e d b u t a f f i l i a t e d w i t h M . We shall prove the non-commutative Radon-Nikodym theorem of P e d e r s e na n d T a k e s a k i o n l y i n t h e c a s e c o r r e s p o n d i n gt o f i n i t e p a n d v and bounded dv/d* The result, in its full generality, will only be stated, and the reader desirous of a proof is directed to [PT]. The first problem one encounters-- even in the finite case -- is this: if $ e M** and if h, x e M*, there is no reason why Q(&x) should.be non-nigative. This is true if 0 is tracial (since,then S(/rx) = Q(hrl2xltr/') > O) but not in general. Thus, the modular group oe n a t u r a l l y m a k e s i t s p r e s e n c ef e l t . Af ter this somewhat lengthy preamble, let us set the ( n o n - c o m m u t a t i v e ? )b a l l r o l l i n g b y g e t t i n g a f e w s i m p l e o b s e r v a t i o n s , i n t h e g u i s e o f e x e r c i s e s ,o u t o f o u r w a y .
Exercises (2-6-l) Let 0 be a faithful n o r m a l p o s i t i v e l i n e a r f u n c t i o n a l o n M . If h e M, let 0(&.) denote t h e l i n e a r f u n c t i o n a l o n M d e f i n e d b y (O(&e.)Xx)= $(hx). (a)
The map h - 0(h.) is a linear map from M into M*.
(bi rf h e u! rcr'.'Def. 2.5.13) andx e M, then9(!r) = frQllzxhrl\, particular, lr ( k M\ and h,k e and consiquently Q(h.)e M* -; in implv 0(4.) < O(ft.). (c) lf h e M!,,then 0(ft.)is faitfrful if and only if ker /r = (0). (d) lf,h e U!, ttren ilh.) o ol = q1n.1for all t in lR. (Hint: 0 o
oP= o.l.E
Proposition 262 Let 0 be a laithlut normal positivelinear functional on M. The following conditionson a { in M*,* are equivalenti (i) ,t)= OQ) for someh in U!; ^ ( i i ) 0 < c 0f o rs o m e c > 0 , a nV dooY=tltforalltinlR. Proof. (i) ) (ii): This follows -- with , = llnll -- tto- parts (b) and (d) of Ex. (2.6.1). (ii) + (i): Assume,without loss of generality that 0(x) = <x$0> for x in M, where o is a cyclic and separatingvector f or M. Notice that the sesquilinearform [xQya] = {(y*x) is well-defined on the densesubspaceMQ of 4 and is bounded,since
e'Ee
'dord '3c]
'ecueqpue [(c) '/ 1 seop se tlcee roJ rrv rIll/h selnuruoc I r7 .,{11eurg '(xtt)p = = <1;rt/lrx> = = (x\f 'n vI x ,{ue rog 'aouo11 lJtI =ur4l =Utrvt =U4
pue (uraroeql l{Esal€I-€llurol orll fq) 'n
l€r{l t f ,It = U lsrll s^rolloJlI
1JtVf = tJt%alyvf = tJr4I = tJt4 luql epnlJuoc! q1;,nsolnuuoc tQ pue'In > ,rl ecurs1,.;4r ur 13 roJ u+to = ur"J pue c rrrop3 urhl t€r{l IIE3eu (. = tf'_V
\)x,I> =
(r*(uf)riolfr = 11x;jo*,f)rlr= = 4)/t\x1rv t4> :uos?3g) 'V qll^\ oJueq pus '/ 11ero3 c t 4 ! e \ l l s E J a q l o l u l s a l e l s u B r l4 r J o O J U E T J B A U T - 6 o i l V t l l l 4 s e l n u u oqfE{9g 4y pue eqf pue rog elrl/{ f.1ctrurssn la-IY v .f.J.S' 'tw ) tq leql epnlcuos i <152\St,4x> = = (t(*(z*x)\lt = (tk*z\ft = <1Sz\Sttx ,t1> '147ut z'tt'x tue roJ osle le,rltlsod s1 rfi acurs 0 I t4 l€rll eloN -n ur. t ,x toJ @*,(),1, = q.Jd!r,tt> WrIl qcns fi ao ,I .rolerodo pepunoq € stslxa oroql oS
With no f urther apology, we state below, without proof, the Radon-Nikodym theorem of Pedersen and Takesaki in its general form. Theorem 26.3. Let 0 be a fns weight ,on M. Let rb be a normal semifinite weight on M such that 4t o of = $ for all t. Then there exists a unique ,positive sel/-adioint operator H (possibly unbounded) alfiliated to MP such that 0 = 0@): where 0(H.) is defined to be the limit (as e - 0) of the increasing net {O(He.): € > 0) (directed > [email protected] of normal semifinite weights so that €r < €z + 0(1{e r.)
on ry defined by (Q@e)@) = S@!/2xn2/\, where H, = H(l + em-L.n As might be expected,this result will suffer the samefate as we shalluseit semifiniteweights: otherunprovedresultsconcerning in the future with completeequanimity. T h e r e s t o f t h i s s e c t i o n i s a d i g r e s s i o n ,a s f a r a s t h e s u b s e q u e n t trend of this book is concerned. The reader who has had no prior theory, who might consequently not exposure to probability appreciate the rest of this section may safely proceed to the next chapter. Let Mo be a von Neumann subalgebra of M. lf M = L-(X,T,1L),it follows, lrom the fact that Mo is generated by its projections, that Mo = L-(X,Fs,p) where fo (is thi o-subalgebra of f which) consists of th-osesets in- I, multiplication by whose indicator function defines a projection in Mo. When p is a probability measure (i.e., u(X) = l) the classical conditional expectation is a linear map E: M + Mo satisfying: (i) x > 0 implies Ex ) 0; (ii) E' is a projection of norm o n e ; ( i i i ) E i s n o r m a l , i n t h a t i t r e s p e c t sm o n o t o n e l i m i t s ; a n d ( i v ) Q o E - f, where 0 is the faithful normal state defined bv (,f) = ll ay to, I in M. Notice that O is a faithful normal state on M, so ltrat 0o = AlMo is a faithful normal state on I(s: tne GNS triples for (Mo,Ooi ana (ir,O) are (Lz(x,T6,F),rr.,o) and (r2(x,7,!t),m.,Q), where z. i s i t r d r e p r e s e n t a t i o nf ' m , a n d o i s t h e c o n s t a n t f u n c t i o n l . T h e GNS space 1lo for (Mo,ilo) slts naturally as a subspace of the GNS space Xl for (fr{,Q),and-it-is well-known (and easy to derive, from the properties(i) - (iv) listed above) that
= tt6o(Ex)o plto(zo(x)o) . We shall commence the non-commutative proceedings with an old result due to Tomiyama on norm one projections' The result is valid i n t h e c o n t e x t o f C * - a l g e b r a sa n d m a y b e i n f e r r e d f r o m t h e v e r s i o n given below, via the so-calledenveloping von Neumann algebra; we shall, however, be content with the result for von Neumann algebras.
'ft + pl < ll(oar + oaoroa;aa;;1
(€)
l l o r ( o r+r o r ) o a ;<; ; ; o a r+ o x ; ; uaql '0 I 1n fe{l qcns raqrunuI€ar fuu s1\ JI pue -- (r{ + ,(ft = t eU aroq^\ -- ((0a0x0a)aa)cls r n gr 'pueq roqlo eql uO
' r N+l t r oaxlaxoall = ;;oarkl+ = + ll*(oar+ faxoa)(oa1 faroa);;
a^\ snql iZ = [ 'l = t eseceql raprsudco1 (slurofpuEurraprsuoc fq) sacrJJns11'{ * ! tof 'I = r uo{A{ (1) Eurqsrlqelse fqoraqt,IAxl)Zl = l e q l s a l l d r u sl l q l ' o n t t o c u r s y = { g > ( l x I ) Z > 0 ' ( e ) , { q U x n Z ' e c u a qi I > r ) 0 e c u r s/ > / x ! ) 0 t e q l o c l l o u ' Z ' l = ! ' t 1 = ! t o g .z
> f.t> I,!!(l!*tnzt1
=(llrtryz:uollressv
(r)
.or.I x | = lt= > a = z I p o t o a= r ! n 1 > 0 pue (on) d I 0a re,reuaq,n(r)goa 1xoa\gluqt ^ror{sol seclJJns ll '((B) o1 slueqt) slurofpe se,rrosardpue l€rurou sr g acurg (q) '((c) (f'f'Z)'xg ul {rurrrerI€rlleqluor€deql'Jc) lW, Z o 0g reqr sarldrursrql 'luluole ftltuapl eqt lE rurou slr sulell€ g o oQl€gl os
, z o lg^c *'k r 0g,tuqriroqs lsnu an 'spron reqlo ul :o < T"aFO uoql '- in t "Q pue -n r r JI lsrll o^ord ol secrJJnslI (B) 'Joord 'n '(x*x)g u! x tol > @Z)r@Z) (c) 11,o p(d) :oW sq'oo.'n t x ll'oq(xg)oo = 1oq*oop > :+'on t xZ ++n ) x''a'1 2O< A (e) :Z lI
'auo uaqt 'lotarou st tt?ttltr'r turou uolna[o.rd o st o1,t1 - 141 lo '1tg otqa?pqns uuotunaN uo^ o aq oyg ta7 ?'92 uoplsodord /o ruoroarll rufpo4lp-uop€U oql
LL
'9'Z
2. The Tomita-Takesaki Theory
78
If (2) and (3) arc to be compatiblefor all r of the samesign as cr and of arbitrarily large modulus, it must be the case that a = 0' is self-adjoint and cr was an arbitrary number in its Since Re(eoxoeo) = 0. An exactly similar reasoning spectrum,doriciudethat Re(eoxses) = 0. shows that Im(eoxoeo)= 0, whence elsxs€s iir the above reaspninp,we find efand Rev,ersi48the, roles of eo -ao.loints to conclude that efxoef = 0. tnat e{rpdxe)e* = 0; take Thd coichisi6ns of the preceding paraglaphsshow that, with respect to the decompositionlf = e# @ eo{f, the operator xo is representedby a matrix of the form
ft o o ll t t . Lb 0l To complete the proof of the assertion, we must show that b = 0' Minor computations reveal that for any scalar \,
lll:,;lll = max{l\l llall,ll'lll, where s = lerxe[ 1 e$lf): ,*!,,- t&' The validity of this inequalitv for large posiiivi \ f6rcesllDll = 0, and the assertionis proved' Conclude, finally, that
.sroleradorelnporusoql elouap(0v pue 0f '0i'0g ..ctsar) V pue.f..{.S lelf '(o['on) rog e1drr1gNc e sl (u'02'0fi)leql ,(.;r.raaol l"tnrrl rt l-I 'l'Q =.r roJ (!,lh olul oy'vgo ursrqdrououoq-*Ierurou e s1tz eraq^\ 'la g ug =^ oql 9l lcadsar {ll,n 1o^ ur 0x aruosrog) f1^uoflf socttuocap 1ox)tue (-ux)oururo^Joql go sr (07g)uul roleredo qcea 'flluanbasuoc iu$O Xl = ? sl os '(u,;,,g)u erqeEletulofpe-31as eql repun luerre,rursr 0g = 0g ret puu '(O?) rog alctlrr SNC or{l eQ (u.u?) rat e^cuts 'un uo !J%)-, elels lerurou InJglleJ € sl 0/{p = 0g leql r€olc sl lI .Joord .onlQ oQ on oQ = {q ua47 ,o awts f)urou p!t1t1o! aW ot Su\puodsat.rocon swsttyd.rotuono dnot7 rulnpotu lo lo on u! 0* aLfi s! o^o llo to! (0x)olo -e atatlu \1 ut t pup "9 = 10x;jo (gl) on - (fu$o (ll) UJur I 17oto! iow otuo n [o g uoltotcadxa1ouo1tlpuoralqltodwoc4 D stslxa anqt
(l)
aro suotrrpuocEutuollo! aqa 'n uo aprs purou tntqtt;"'J':t;u; ial puo '1rgto otqaSlzqns uuounaN uor o aq oy,1ta7 gg7 uoJrlsodord ar{r sJ lf l'qr s^\orrsllnser Eur,noIIoJer{l ruoglcnrlsqo;:tlt"''Jlllir:Hr; .suoll€lcedxo ,o dnor8 J€lnpou eql'n u€rloqc-uoulerauaEB rog ieuolllpuoc alqlleduroc-o Jo oJuelsrxo eql Jo uollsenb eql solllas uollBlcedxa oqt .(rl.4.x)-7 = u?r.ll1A I€uorllpuoc lecrssBlc 1r! ".: e^Br{o,,, :l "i';it'l,:i,T#;J?'jl r€r' uo,ou:?l :Jiji ili?rXll,
sa^e oql ur ruJel eql Jo esn oql {grlsnf 11ratl, ruorlelcedxa 1uuor1rpuoc,, -- ,'9't 'dord Jo (q) lt1re1ncr1rud -- qclqa .uollelcadxa leuolllpuoc lereueE u Jo seJlredord auos slsll llnsoJ s.eu?frruoa .ocue11 'Q = g o otuo 1,r7 n 0 JI elqrl€duroJ-ooq ol pl€s sr o1,t1 1o 3r uollulcadxelBuolllpuoxe'1A1uo ol€ls l€rurou InJr{ll€J€ sl O JI (q)
o;,g €parr€c eqrrr,,r oluo (e) ^'f""::: #,# ff?ff:ir:1J",:"JttJJ"; 'pg p
'pa^orclsr (q) pue ueroarlJ ur{po>1r51-uop€U aql'9'Z
6L
2. The Tomita-Takesaki Theory
(i) t (ii).
If x e M a'nd xo e Mo, then
= f(xfx) = f(E(xf;x)) = f(xfiE(x)) = ; since z]@-n = fo, concludethat if p = Pfo, then pn(x)o = lt(Ex)Q p(n(M)n)! dom S; also,for any x in M, Sp(z(x)o)= Consequently z((Ex)*)o = n(Ex+)o= pS(lt("r)n).Since n(M)a is a core for S (bv definition of S), this impliesthat p,Sc ,Sp. It alsofollowsfrom the
above equation that Sn = Sl(domS n lto) and that in fact S = 'S0o ^tr (for an appropriate conjugate linear closed operator ^S,in ltt) with respect to the decompositionlf = lfo o 111-' the direct sum of unbounded operatorsbeing defined in the natural way (cf. Ex. (2.5.6)). (In case the reader feels he is being hoodwinked by a case of somewhatexcessive"hand-waving",he may be pleasedto know that the gruesomedetails of the verification of the preceding statementsare spelt out in Ex. (2.6.7).) It follows easilynow that all the "modularoperators"admit direct F s=: F o o F L ,J = J o Q , I , a n d A = A o o A r . I n sumdecomposition particular, if xo e Mo, t e R, z(of (xo))o = A't n(xo)A-ito = af;no(xo)aoltn (since o e 1?6)
= n(olo(xo))o. - no{ofo{"'))n Sinceo is a separatingvector for n6(M),concludethat , t of {xo) = olo(xo) e Mo. (We have actually proved(i) ) (iii), but clearly (iii) + (ii).) the assumption (ii) + (iii) rc o!@J ! Mo, then ao g o!r{uto), and^1o_, is that ot@l =' Mo-for ail t. The equatibn dt : o!\Mo clea-rlydefines a flow
araq.rr'&v = sv p1 'z'l = ! ro1'$4 =td 1a1 'C= y luop qlr/h .sl 0G'g 'ofi - ,{ ul roleredo peurJap^lrsuep pesolc e eq V p1 e.g.Z) e sosJcrcxl
Ieuorlrpuor alqJredruoc4 e s! g lEql ^\ou i(grrer "t ;:l##t11;: uB sl lI 'uorlrass€or{l ur sB prlelar lrx pue r qll/r\ .1rlx;r9u= G)z .iq on - I4l tZ aulJop {1
:,:] l'^"*,
sr f oouls of - oa!^fpue oU -= ol+t luql rlrienbe Eulpaca-rd aql uorJ :pnl3uoJ '$ ur esuap sr ltJel/{)u.1rv u(n)ourt;v 'uf,vl)u ^=) 'os pup 'alerlr3.rursr qJ!q^\ '711vn1 aroc € q lsql eloN .(s(ox)ur7rv)01 = ufr)0uzl9v01= U1!x)0u= U(!x)u = 1rJ1ox)urhy)t (le u j urop),?= og pue dv -= secroJslql .{rer1rqr, ,r't"ltj,llt ',h - crf",?let{l pue,,vd= dr,v vd sallor.irl itqr'oit ='ven)u ecuig teqt r , q l ( g ' € ' z ). x g , ( q u l e E e ) ( ( t t t ) s l s o t { l o c l f qf q )
a c o r e f o r A . ( I n t h i s e x e r c i s e ,e i t h e r a l l t h e o p e r a t o r s a r e l i n e a r , o r all the operators,but for pt and P2, ara conjugate linear') Suppose that prAo g Aopt (a)
prAo t Aop, and in particular, piD0 c Do, i = l'2. (Hint: D0 is a l i n e a r s u b s P a c ea n d P r \ = q - P l l . )
(b)
It D9 = D0 n 1f.,then Df = p.Do-andhence^Df is dense in 1f,,for I =
n 4 g D0h rr, while Do'beinga 1,2.'(Hint:of = r,tofi I r,(D01
c o r e f o r t h e d e n s e l y d e f i n e d , 4 i s n e c e s s a r i l yd e n s e i n X f . ) ( c ) p r A e A p ; a n d p , ( D ) = D n X f i ,f o r i = l r 2 . ( H i n t : b y d e f i n i t i o n o f lo[, bb, ii (', o, tttin there exist l. in D0such that [n'(, A\.) (d) Let D, = pi(D). Since l(D,) g 1f,,define_an operato\ (t i! Xfi.with dom 2, = b,, by A,l = Al'for i in D,. Show that l, is closedand t h a t D f i s i c o r e - f o r A , f o r i = 1 , 2 . ( H i n t : I d e n t i f y l f t _ ^ w i t ha closed'subspaceof if and notice that G(.4r)= G(A) n (ft e 18s).) ( e ) S h o w t h a t A = A t @ A r i n t h e s e n s et h a t d o m 1 = , { \ r o ( 2 € J f l e D I t r : \ r e d o m / , , i - = I , i | a n d l ( ( , e ( z ) = ( , 4 1 8 1e) ( A r l 2 ) . Example 2.6.8. Let M = f(Xf) with dim tt > 2. Let e be a projection of rank one, and let Mn = te)t. Let p be a positive invertible traceclass operator with tr p = 1. Then 0(x) = tr Px defines a falthful n o r m a l s t a t e o n M . R e c a l l - - f r o m E x a m . p l e2 . . 3 . 7( b ) - - t h a . ! o f ( { ) = pitxp-it for x e M, t e R. Notice that pttMop-" = Mo € pttlt{[p-'o = Ml: however, (by the remarks following the double commutant thlorem) MI = {e}" = tle + p: \,p e C}. Notice tnat. P(Mil _= (0'e' l-e, l) and- that e is the only rank one projection in Mi. Hence, pi'ep-i' = ,. Thus, there exists a O-compatible pt Mop-t' = Mo e cond"itional e*bectution of M onto M € Pi = epit Vt in lR € pe = ep. Neumann algebras, f inite-dimensional von Thus, even f or n 0 - c o m p a t i b l e c o n d i t i o n a l e x p e c t a t i o n sn e e d n o t e x i s t . Remarks 2.6.9. (a) There is an extension of Prop' 2.6.6 to the case when Q is a fns weight. The result -- which is the theme of [Tak 2] - - i s t h a t i f 0 i s a f n s w e i g h t o n M s u c h t h a t Q l M o . +i s s t i l l a semifinite weight, then the conditions (i) - (iii) of Prop. 2.6.6 ate equivalent. (Recall the o-finiteness hypothesis in the classical n a O o n - N i k o d y m t h e o r e m , w i t h o u t w h i c h h y p o t h e s i s ,t h a t t h e o r e m i s f alse.) (b) Even when the condition (iii) of Prop. 2.6.6is not satisfied, it is possible to pursue the reasoning of the proof of (iii) ) (i) as follows: with the notation of the proposition, notice that if x e M, then "/n(x)./ e n(lt[)t c n(tr{o)t. If
lni, nlrf I, Vi, olzJ
Jn(x)J = |
'uolleEllsolur D -[3V] ttnsuoc feru rapsar eql u€ qcns slr?lap rog 'og uo sroleredo erqa8lu uu€runeN uol Jo 3o 'on € lou sl (n)A ftW as€c egl eq ua^e l(eru U Jo a3€clsqnsradorcl u s r f : l e r u o s r . 1 e r 1 r e cslr q l J o e c e d s 1 e u r 3o q l l e r l l l n o u J n l u o ^ e u E c l I 95$il9y s1 9U1r)Pa spues qcrg^\ .9g eceds SNg eql uo -- uorlcoforcl B flrr€ssecou lou -- {r1auros1 1e11red? saonpur e^oqe s€ peurJep A deru ogl 'es€c lereuaE eq1 u1 'on oluo /,{ Jo uoll?looclxe l€uorlrpuoc elqlleclruoo.f enblun o-gl sJ ll asec gclq/rr ur '1eul sI (lll) uorllpuoc aql uoqA flasrcerd -- on Jo slueurole sexlg ''a.1 -- 01,'goluo uollcaford E sl ^A slr{l rerll ^roqs ol llnclJJlp lou sl lI (',,uoll€r0do-uorsserduroc,, e pue srusrqdroruoruoq IBruJou Jo olrsocluroc e Euraq .anllJsocl dlelalduoc osl€ sr g 'esues se{su slql uoq^r ol JepseJ aqt rog) -on ol Jtll tuoJJ rolerado J€eull- en_Jlcerluoc'leurrou 'Eulurrasard-flrrrrlysod = (x)g Aq on - N :E eulJac e sI ? lsq-l _r_ee-tc sl u '(lrloofir9u '(on)ou , ofr'loot l€ql acueq pue",
g8
ruaroaqlru[po>1r51-uop€U orIJ'9'Z
Chapter3 OF T H E C O N N E SC L A S S I F I C A T I O N F A C T O R S I I I TYPE
T h e f i r s t s e c t i o n d i s c u s s e st h e e x t e n t t o w h i c h t h e m o d u l a , g . o u p o o depends upon the fns weight O. The precise description is the u n i t a r y c o c y c l e t h e o r e m o f C o n n e s ,w h i c h s a y s , l o o s e l y , t h a t m o { u l o the group of inner automorphisms of M, the modular group oY is independentof 0. Stone's theorem states that every strongly continuous unitary r e D r e s e n t a t i o nt + u - o f t h e r e a l l i n e J Ri n a H i l b e r t s p a c e l f i s g i v e n by u, = eitH fo. a uhiquely determined self-adjoint operator H in tt. "the Takiirg a cue from the physicists, one may regard sp F/ as this i m i t a t e i s t o s p e c t r u m o f t h e r e p r e s e n t a t i o n{ a , } " . A r v e s o n ' s i d e a p r o ofs t h e S i n c e a l g e b r a ' v o n N e u m a n n procedure for flows on a are no harder in the more general setting of locally compact abelian g r o u p s ( r a t h e r t h a n j u s t l R ) ,t h e g e n e r a l c a s e i s t r e a t e d i n S e c t i o n 3 . 2 ' which begins with a rapid survey of the necessaryresults from abstract harmonic analysis, and goes on to the definition and some elementary propositionsconcerning the Arveson spectrum of a group "spectrum" being a action on a von Neumann algebra, the said group. certain closed subset of the dual ,rr The Arveson spectrum of the modular group (oy) would, in general, vary with the f ns weight 0; Section 3.3 introduces the Connes spectrum of a group action, which is a refinement of the Arveson spectrum and has the following pleasing properties:(a) the Connesspectrum of an action of G on M is a closed subgroup of the dual group t; and (b) if 0 an{ 0 are any two fns weights on M, the Connesspectra of (op) and (of; cpincide. Thus one may define r(14) to be the Connesspectrum of- (ofl) where Q is any fns weight on M. S i n c e t h e c l o s e d s u b g r o u p so f l R i r e e a s i l y e n u m e r a t e d ,t h e i n v a r i a n t f (MD leads to a ref inement of the Murray-von Neumann classification. The definition of l(lrt) given in Section 3.3 is somewhat u n m a n a g e a b l e ,f o r c o m p u t a t i o n a l p u r p o s e s ;S e c t i o n 3 . 4 i s d e v o t e d t o
'l tl > e (['z!) > (I.r'Il) ropro aql ol locdser qll^\ spre^\dn pc1ccr.1p s l f x / ^ = X l c s c q l _ ' f i t , , t ' ' d s a t )I , 1 r x l € r l l V c n s( ' | g j { t t t:t,{} ''dsar) 'xg 'J3) lsrxc eraql v C . 3 s {t t t :rx) 1au euolouou e ((S'l'Z) 'allulJlruos ''dsar) 'cllulJltuos s r ( { i s r a rouraqlrnl 0 aJuls :uoseag) 'lqEre,vt ' 1 n 3 q l r e g B , o u r J a p o l u a a s , { . 1 r s r : a p f0e sr If,turou lngEurueaur sr 'nII -t X '(zzx),t, + (Irx)Q = (l)g uollrnba aql ,{11uinbosuo3 (4 -u! t: pexlJ ,(lrrerodruol puu I r' I\ ,t.rerirqre roJ ,:ar e l l ' r = I e r e r { m< l ' l x > S u r u r r u e x e^ q , s r r l l l c a r { J ) . 0 = I c 0 = r r t = IIr'0 ir uar{,v'felf pue +y1 7 czv'rrx ++n r r reqr acilou .r{ uo srolerado go erqaEle uuurunaN uol € sl r'{ leql perJuo^ flrpear 31 lI
(
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eceld
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ercqm) z x e roreseqrsr(n)2p,1 6fwr^fi1#T,lli'fifi"1;li,ffii
'Z Iernlcu € sl eraql > ['l > I roJ (A)f a lrx eraqR .((,,x)) socrrlau uo sroteredo fJrluepr IIBrls 3^\ ille u = fi lo.I .Joord zx zrtrllt fl I
'A u! t'n
' '("n)$orn "*rt1 (q) = y1 ut 1's 11uto! puo u! x 11oto! ]n(x)$o rn = (x)$o (e)
toql q?ns (II)n oru! Vl taot/ rtt * 7 dow snonurfitor t13uot7s o stsffa ataqt'yrl tto syq8tau suJ aro (tt pup Q {t -t-fe urcrocql 'uorlcos srrll Jo lrBerI eql ol pOaJord sn lel'^B^r aql Jo lno er^rJl Jo llq slql qllA (.0 -.lr,r.l!n> ed Z n - tn = - r r ) l l u o q l ' J t r I J I p u e ' , { 1 4 e a an 3r :uoseog) l(rr "tuorls ' a pzilcl tul liZo c szi lrlS olodol puc I€e^\ oql ,(n)n ol pelJlrlsar 'leql lceJ crseq s sl lI 'lr{ ul sJolurado frelrun 3o dnorE aql 'lenbes s r l l u l ' a l o u s p I I I r h ( , r f ) n 1 o q u r , ( se q l ' e r q a 8 1 e u u € r u n e N u o ^ B s r / { J I ucrocql
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c{I
'I'€
'sed,{l snorr€^ eql Jo srolJ€J go saldruexe Jo uorlcnrlsuor or{l ol palo^ep sr rIOrq^\ .€.t uollJ3S ur InJesn euocoq IIr^\ r4Jrrl^\ uolldlrcsap slql sl 1r. ,.147 uo lq8rear, s u J u e r r , r Se u o q l r ^ \ p o l € r c o s s B9 9 r o l e r a d o r E l n p o u e q l J o s t u r o l u r poqucsap oq uec 'urn1 ul 'r{clqm (/,{)S tuerre^ur raqloue Jo sruJal ur s r s u o r l d r r c s a pa s o q l J o a u o ' ( n ) l p s u o l l d r r c s o pr e q l o E u r q s r l q u l s e tueroeql a1c,(co3{re1ru61 eqa 'I'€
98
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
86
and j, (^ 7r; the net {.rc,e yr: (i,)
e K} is monotone, lies in Dg and xt
@yi ) rft.) Since
- -
(ini)i,.i=
2
,!,
rilr,i,
conclude that; e Ng (i xr,x2. e N6and xrz,x2z, N,l, Thus, if i e N g and if / is the matrix obtained by setting some of the matrix entries of i equal to zero and leaving the other entries unchanged, t h e n t e N g i i n p a r t i c u l a r , N g ( l e e r 1 )I N g , a n d s o } 1 g ( l o e r r ) c M0. Since ltg is self-adjoint, as is I 6 er' this implies that also (l @ e11)MgC Mg. Finally, some simple matrix multiplication shows that
if 7,y e Ne, e((l e err)T*V)= O(xir/rr + xlrt21) = e(t*t(l o err)). 2.5.14)that (l @err) e i10;,o, for any x in M by Theorem Conclude, and t in lR,
o,9{"t err) = o,0((lo err)(x o errXl e er1)) = (l o errXoro(x o err))(l e err); t ence ore(x e err) = crr(x) @ e' for some crr(x) e M. -A routine verificafionshowi that {crr}is a flow on M, in the senseof Def.2.5.7. Assertion: 0. satisfies the KMS condition with respect to {crr} and hencecr,= o? Vt. First,if x€ltt[,, = g(crt(x)@err) = elore(xo er1))= g(x o err) = 0(x). O(cxr(x)) N e x t , i f x , y e N O n N A , n o t i c e t h a t x e u e r .oueyt t € f . l g n N [ , and that
e((oPfuo err))(:ro er1))= f,(crlr)x) and
g((x e err)@!0 o err)) = 61xcr,(l));
thus if F is KMS-admissible for x E e' and | @ en (relative to 0), then F is KMS-admissible for x and y (relative to 0), and the a s s e r t i o ni s p r o v e d . We have shown that (l o err) e ifo anO ttut oro{t o er1) = opt") t e ' f o r x e l t t [ , , € l & o I n a n e n t i r o e l ya n a l o g o u s , , m a n n e r-,i t m a y b e s6in that 1l o err) e l}y'uand that ore(-ro er2) = "K") o errfor x e M, t e lR. conclude ttrat o,e{t o err) = (l @ err)(ore(t e Since er, = €22€21€1y
( t o ) o l l u a l e r r r n b cr e l n o '("n)lnln = 8+1n .:irrEsso3cusr qcrq.t '/r' uo .nolJ e sl {tg) uaql s r r J s r l € s( r r r ) J I l e r { r . f , o r { s: j r r 1 r ) t n r n= ( x ) l g t ? l ' ( h l ) n o l U l r u o r J i e r u s n o n u r l u o c{ l t u o r l s E s r r n - / J I p v a W u o ^ \ o l J e s r { l n } 3 y ( q ) ('snonurluoc i 1 3 u o r 1 ss l l I ' O y ) n o l p r l f , r r l s e r u o r l A \ ' a c u e q p u E s n o n u r l u o J i 1 1 e a , ns l l l ' s n o n u r t u o r i l t u o r l s l o u s r s l u l o t p e E u r > 1 eq1t n o q l l e 'pue '{'e) .sir{otJ {1n} ; {rO} :(,lrr}u:qr ; {ln) :{?n) JI :lurH) (e) _to les aql uo uorlElar ::utlenrnba u€ sr acualulrnba rolno
(e'r'e ) srsr3JSxg 'sldacuoc i.;rre1c ot dlag llr^\ sosrcJaxaaldrurs eruog asaqr 'lualelrnba relno are;g uo srqtla^\ suJ Jo rred fue ol Eurpuodso-rroc s d n o r E r e l n p o u a q l l c q l s r r r s s Br u a r o e q l e 1 c , ( c o cf r e l r u n a q l a J u e H 'II ul sJoleJado frrlrun O ;o dnorE :cleure:ed-euo = 1x1rn s n o n u r l u o J iftuorls e s; ereqr\ {14) }nxrn ',(l1ue1ezrrnba 'Jo '(U l 'rl' r xA a = (x)lr ,{q ua,rrE) ^\olJ ler^rrl _1r 3 r { l o l l u e l e r r r n b ar o l n o s t t r J r r a u u r a q o l p r e s s l { l p } , n o 1 3y ( c ) ']n(x)rnrn = (x)lg pue ur ,'s IIe roJ 'leql qcns (/.{)n ol UJruorJ E slsrxearoql JI -- d : p fq palouop
"jln 1tn;rnln =
'147ur. x puE
' r r - t d u r u s n o n u l l u odcl t uU ols
-- luale^rnbeJalnoaq ol prEs rre Ul?t{tg)pue Utlllo; s"rrro13 orna (q) 'nulx IfE .roJ .nxn = (xF leql qcns (7,9)lJ ur 11slsrxs araql JI rauur poll€Js! /f Jo D tusrqdrouolne uV G) 'Z'fg uorllulJeq O
'rueroaql aqt turqsrlqelse ,(qaraql 'rza
o ("n)$orn =
((Ira o 'r)(rzt o t))rro =
( r z ae " n 1 ] o = (Izae t)g'oo jo = rza@8+1n ' ',(1EuIl ']ng)]otn = U 2-r'sJr 1x)jo l€q"l epntcuoc or 'izla @ Ixrra e xxlza e I) = (zzao x) u6rlrnba a,ji or ^toly_ ,,o snonurluocflEuorgsu s1rn - , lurll uoll3asslrll Jo qderErrecll€llrur rql urorJ .ragur'snonulluor ({11eo,r acuoq pue) d.14eo,n-o sl (rza e I)rto e / acuJg 'W)n ?,tn ''r'l 21 = ]nrn = tnln leql pulJ puy-';o ..-l_paxrJ "a_@ pue "a O r e p u r u u r r E o q ' " u a€ I = * ( ' o t o l e q l O I I I;a e = (Iea pue IIa : @ I) o 6 1) suorlenbaar{l or .fo flooy I)*(Iea rrt oruosroJ rzaI rn = (rza @ 1'1r e l)jo lcrll ooueqpue '(rrao6rt)((Iza rueJoarlJ alcfco3 ,{rellun eq1 'I'€
-S
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
88
( 3 . 1 . 4 ) L e t 0 , { , ( r t } b e a s i n ( t h e s t a t e m e n to f ) T h e o r e m 3 . 1 . 1 . (a)
(b)
(c)
group of If 1rv1)1 Re i s a s t r o n g l y c o n t i n u o u s o n e - p a r a m e t e r = in Z(M), if vt w#v show that operators and unitary {vr) is a s t r o n g l y c o n t i n u o u p p a t h o f q n i t a r y o p . e r a t o r si n M , w h i c h a l s o satisfies vr*, = vrof(vr) and of(x) = vrof{x)uf, for x e M, s,t e R. (Hint: you will need to use Z(M) t M0.) I f , c o n v e r s e l y , t - r , , i s a s t r o n g l y q o n t i n u o u s . , m - a nf r o q R t o [ (nf which also satisfies vr+, = vrof(vr) and of(x) = vrof(x)vf, s h o w t h t t h e r e e x i s t s a s t r o n g l y c o n t i n u o u s o n e - p a r a m e t e rg r o u p (wr) in UQQ()) such that vt = w{t for all t. (Hint: put w, = ufv, and verify that {wr} is as wonderful as it is claimed to be.) If M is a factor, the unitary cocycle of Theorem 3.1.1 is a continuous uniquely determined up to scaling by o n e - p a r a m e t e rg r o u p o f c o m p l e x s c a l a r s o f u n i t m o d u l u s , i . e . , i f ( a r ) i s o n e s u c h .u n i t a r y c o c y c l e , a n y o t h e r u n i t a r y c o c y c l e i s o f tr th! form rt= eit'ut for somea in R.)
In the remainder of this section, we shall discuss two results of T a k e s a k i ' s : a p a r t o f t h e f i r s t r e s u l t i s a c o n s e q u e n c eo f t h e c o c y c l e theorem, while the second result explicitly producesa cocycle which works in some casesand also explains why the cocycle theorem is sometimes referred to as Connes' Radon-Nikodym Theorem. The proofs of both these results are somewhat technical, and we shall only present the proof under some additional hypothesis invariably that some self-adjoint operator is bounded. The theorems will be stated in their full generality, while the simplif ying assumptionwill be spelt out at an appropriate juncture in the proof. B e f o r e p r o c e e d i n g t o t h e s e r e s u l t s ,h o w e v e r , w e e x t e n d t h e n o t i o n o f s e m i fi n i t e n e s s t o a g e n e r a l , p o s s i b l y n o n - f a c t o r i a l , v o n N e u m a n n algebra. Dcfinition (a) (b)
3.1.5. A von Neumann algebra is said to be:
semifinite, if it admits a fns trace; finite, if it admits a faithful normal tracial state.
n
Thcorcm 3-l-6. The following conditions on LI are equivalent: (i) (ii)
M is sentifi4ite; the ltow 1of) is inner, lor sonte f ns weight 0 on M;
(iii)
the Ttow {o!\ is inner, for every f ns weight Q on l{.
Proof. (i) + (ii): By assumption, there exists a fns trace t on l+[; ttren (of) is the trivial flow on M and hence (trivially) inner. (ii) + (iii).
Let 0 and 0 be fns weights on M, and suppos" o0 i,
i J t z / , , . t a\ r r / , r t ,
=
= at zl,a-azI rY \Jx 71.q-a 711Y> = <(tsxs tl tv t' (tJ[z/ q-a)x1{ t> 1,a-a) iJz /,t-a*x 1J71u-a*t> = (*/k)L :atnduoc pue nt ur f'x '1erce:l s r . 1 .t e q t , { ; 1 r a l o a ' l € u o l l c u n J r c e u r l e n r l r s o d I E u J o u lord ' 5 t r Vt t l o J u r s < U e l q _ a InJglrEJ e sr :' leql (I'9'Z)'xg uorJ s^\olloJ lI = (xq-a)Q = (x)r paut.;6p o^eq o^\ ',(11eurg !s7*-ar> = (z/,1-axzl,a-a)Q ' = (x)0 l e r l l q c n s n r c J U r o l c a ^ E u l l e r c d a sp u e c r l c f c € s l s l x e a r a q r ' p 3 o a c e d sS N C e q l u r a c e y dE u 1 4 e 1s l u o r l c € e r { l l € r l l E u r u n s s u a . r ea . $ a c u r s i , T g) , 4 = r H ' 5 W ) t l = H p u e ' , W r Q e s o d d n s ' o S ' ' t q E l a n na l l u l J € s r p u e p e p u n o q a r e @ 'n r H p u e H q l o q l € q l u o r l d u r n s s €e q l r a p u n s r q l q s r l q ? l s eI I € r I s o i 6 u o a J E r l s u J € s l r s l q l l B q l p o q s r l q 8 l s eu e o q s ? q l l o c u o p e r r o r d a q '€'9'Z lu?Joegl ut se ('".-a)p = l' oulJaP f'evt em'6hl IILY\uleroeql eql o l p a t B l l r J J Er o l € r o d o l u r o f p u - g 1 a sa r r l l l s o de l q r l r e ^ u r u e s r " _ a a c u r $ ('srolerado pepunoqun qlI^\ elqelroJruoc oot lou sl oq^\ repBer eql ol sralleu asaql ,{.;rre1cfeu rueroeql slql J o p u e o q l l E s a s r c r o x oJ o e l d n o c u i s u l e u r o p I B J n l B u a q l u o p e u l J a p l c n p o r d p u e t u n s a q l J o s a r n s o l ca q l s B p e u r J e p e r e s r o l s J e d ol u r o f p u -g1asEullnuuoc o/hl Jo lcnpord put runs erll JI pllul ere suorl€nba pug e ^ o q B 2 \ t , r H p u e H p e p u n o q u n r o J i t l , H a z f t i a z / , H + 2 . / n= a 711v 7 ',g+g, = 'papunoq ete '(p, ruop) r-g U dr ruop V ueql tH puu // JI) = gV wop uo fllernleu paulJap ?urcq gy 'g pue Z pepunoqun roJ -'alnuruoc z/,^az/sa rolerodo aql Jo ernsolc aql sl z/rv lerll s/ ollo3l 11 = = , ^ r rp u g l / a c u l s P u € ' , H l ! a H r ! a t t g e c u r g - 1 6 1u l t l l E r o J , u r r a f r r p u e ='n l€ql qJns til tt"t11 pue ( l 1 1 s r o l B r e d o r u r c i l p ' e - 3 l olss l x e HTt 6W eraql '(uaroaql luelnuuoc olqndp aqt pue) rueroaql s,auols ,{g '/ IIe roJ tn ? i?t lBr4l o s ' 7 . gu r : c I I B r o J x = * l n x l n J a q l r n J l X 1u l d n o r E f r e l r u n J e l e u € J e d o u o s n o n u r l u o c , ( l 8 u o - r 1u ss r ( f r r ) s n q a u , u r t ' s r c i l n l n = ' f f n u o q l = fn 31 1eq1sr [1rn11e1nruuocslql Jo eJuanbosuocr-oqlouv 'qw 11vln "n j'{1n) spro.tr,raqlo ur ll pue s IIB roJ 1eq1 rajur 1rVt{l1rrrsalnuuoc 8x p u e n u r r g B r o J r r - V x r r v= ] n x r n l e q l q o n s ol 1x laS' U ul, 7 9 u r ( l r r ) d n o r S , { . r e l r u n r a l e r r r € r e d - o u os n o n u r l i i o c , ( 1 8 u o r 1 s? s l s r x e e r s q l l E q l s u e e u J e u u r s l / , o l e q l u o l l c l u r n s s na q l ' 9 y r o J v o l r r ^ \ 'flllerauaE 'rt u\ x roj x = (r)Qu pue Q$ = ,{1durs sn la'I leql U '(l) 'arunsse pue n uo Q tqEJen suJ B xlC Jo ssol ou qrr^r e (gt) 'louut st
'xg p u u ( t o a J u a r {p u E , l e q l epntcuoc'(e) (g't'e) rto) t't'g _ fo oo snql :reuul t l l e r o a q l . \ g ' . Y \ o l JI E r ^ I r l 3ql selouep t Sreq/\\' t ; Oo 68
luoroorlJe1c{co3,{.re1ru11 aq1 'I'€
90
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
= <eh'x4 /o> ;
(*)
SO,
t(xv)
= r((xv)l*) = <eh'xyQo>
(by (*))
= <eh'Yg x*o>
( s i n c eh ) e M t )
= r(yx)
(againby (*)),
and the proof is complete. O Excrciscs (3.1.7) Let (e")i=1 and {/")l=, en) | and fn) l. (a)
(b)
be sequencesof projections such that
If e, and ,f,, commute for all n, then en A ln ) l. (Hint: en A f n = €nfn under the hypothesis, and multiplication is jointly continuous in the strong topology,on bounded sets.) Show that en A f n may be 0 for all n, if ^the hypothesis of co-mmutativify is iiropped. (Hint: in lf = 12[0,1], let ran e,., = L21l/n, ll and ran /,, = set of polynomials of degree < n; non-zeropolynomials cannot vanish too oftcn.)
(3.1.8) Let M, I Mz { be an increasing sequence of closed s u b s p a c e sw , hose union is densein lf. If Z is a closed operator such that dom f I UJytn, fMn g }tn and 7*l'1oC l't, for all rr, show that: (a) (b)
lM, is bounded for each n. (Hint: closed graph theorem.) uM,, is a core for T. (Hint: 11 pr, = p14r,then pn ) | and pnT C Tpn)
(3.1.9) Let A and B be self-adjoint operators in Xf such that l"(l) a n d l r ( B ) c o m m u t e , f o r a l l B o r e l s u b s e t sE a n d F o f l R ( f o r e x a m p l e , A = H, B = Ht as in the proof of Theorem 3.1.6). Let en = 11_.,.1(,4), /r, = l[-n,r,](B),Pn = en A fn' (a) (b)
p^ ) l, pnA I Apn and pnB { Bpn for all n. (Hint: enA is b o u n d e d a n d c o m m u t e sw i t h / , . ) Do= , ran pn is a core f or f(A) as well as for g(B), for any two continuous functions f and g on ft further, f(A)Do u g(A)D. ! D0' ( H i n t : A p p l y E x . ( 3 . 1 . 8 )t o e a c h o f I U ) a n d g ( B ) , w i t h M , = r x 1 Pn')
0r{l l€rll oloN) 'pcpunoq ete pva stoleredo e q l ( l l ) p ue +'*^ H . y , _ H :.r4t'0 (I) lerll suolldurnssu rerjl.rnJ arll ropun srql qsrlq€lsa II€qs oi6 't !, ol l J e o s o r r l l l ^ \ u o l l l p u o c S W ) e q l s o r J s r l c sf l B r l l r l s l l Q e t s aa , t r lueutrrotuoql po^ord aq rueroorll eqj ,1,_H(r)jo1rH= (r)rn acurs l7g llll'r 'iorten'bj'eql uo ^\olJ B saurJap = (x)ln lErll opnlJuoc 1r_fl1r_VxqrVilg '!11^t trSS- 'tt ug dnorE frulrun .ralauerecl_euo snonurtuoJ uH e s1 ,(ltuanbasuoC ./ pu€ s IIB roJ elnu{uoc sr11 llE_uortt ,dl"{r,v1-,//} puu rrv lerll (rolerado lurofpe-Jlas o^lllsod alqrlro,rul u€ sr g ocu!i) ,aW toJ " = leql s^\olloJ fI orurs t\ tl rr-V rrv H f}AolloJ lI 'YliIoJ .'A ile 'tr p,uY^ts .i,,,tu\ * pus alrr,vr f V S ldurs llEqs eA,\ Ile roJ .arunssy .Joord x-lV= \x)Yu pue vn = |l lBr{l flllercuaE Jo ssol lnoqll^\
{rtstyay
arrDst H alatrtr,r,rt ?i,'yt;X3'i'ri,ti
i!:'!:, "utpo7r1,7-uop,v '1 ayH\x)$oa,H = $)do uaqJ ltv rot A = io o tlt tzltt LlrnsLU tto Tq7rau .0I-I-g ltrrrocrll suJ rai.ltouL,aq 4'ta7 .t[ tto Ttl?tau suJYo aq Q ta7 D
('(llfxo) roJ lulq ees:1urg) '@)AeY roJ aroc B sl oC (gt) lolqesolcsr (g)3(y)/ os pue'(y)I@)ie*(@)aGV) uoql '3 pue soleEnfuoc-xelduoc a q l o l o u e p p u e / f 3 o g 1 ( ll) , f :paurJap ,{lesuap sI @)8(il (t) :lBrll ^\oqs 'U, u o s u o r l J u n J ( p a n l e r r _ x c l d t u o c )n o n u r l u o c s 'CFI ruop)r_) aq 3 pue n1 ulop = (Xil / ruop qlr^{ '.{11ern1eusror'rodo papunoqirn U X o/yu Jo ){// lcnpord oql eurJaa (p)
uer - "W,{ll^.lqEll ul srolerado e q l I I e o t ( g . 1 . g.)x g - 1'"d ,{1dde:1ur11)'oroc e se 06 ser{ pue lurotpe-31cssl (gtZj (gt) ielqesolcst (A + y) os pue *(A + v) j fA +n) (g) i p e u r g a p f l o s u espr . g + V (l)
uopu u ruop= (s + z) ruopr I Jr 2a+ 1v= :(;T;i::?rrE I6
(r)
uoroer.II clJ^co3 f:u1run aqa
'r'g
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
92
( f ( crQ b o u n d e d n e s so f l l a n d 1 1 - 1a m o u n t s t o t h e i n e q u a l i t y c r O f o r s o m e c o n s t a n t sc ' , c o > 0 . ) C o n s i s t e n tw i t h o u r r i S t u t i o n " f c o n v e n t i o n , l e t u s w r i t e H = h e M Q . The assumptionthat all the action is going on in xf = lfd means that there is a cyclic and separating vector Q for M such' that 0(x) = <xQb, and consequently 0(x) : 1/r.x$6. N.gy lgt x,y e M and define F: C - C by r(z) = .7'tiz'tr6iz+tro,tLlzJh-izlQ>. Since, by h y p o t h e s i s ,t h e s p e c t r a o f f t a n d A a r e a s a f e d i s t a n c e a w a y f r o m t h e origin, it follows that F is an entire function. (Note that the 'f "cancels"the conjugate linearity of the facior in the second term i n n e r p r o d u c t i n t h e s e c o n d v a r i a b l e . ) S o m e e a s y e s t i m a t e s ,o f t h e sort uied in the proof of the Tomita-Takesaki theorem for finite weights (Th. 2.3.3),show that F is bounded in every strip of the form llm zl < 7. Finally, compute: F(r) = = <&hir+lAitrg
n-tI zJh-ir xtt>
=
=
(since a-1/2"1= s;
= O(h-irhxnito!0)) = glh.xhito!(y)/r-it)
(since |r-it , 76Q1
= rp(xcxr(r))i i n o t h e r w o r d s , F i s K M S - a d m i s s i b l e for x and y (relative to rl) and the theorem is Proved, at least in the special case we have n considered. I n v i e w o f E x . ( 3 . 1 . 4 )( c ) , i f . 0 a n d r I a r e f n s w e i g h t s o n a f a c t o r , and if 0 is invariant under o9, then any u.nitary cocycle {rrr) as in Theorem 3'l'1 must be of the form u, = srt\ 11rt'for some \ in R' w h e r e I / i s t h e " R a d o n - N i k o d y m d e r i v a i i v e o f { w i t h r e s p e c tt o 0 " i n the senseof Theorem 2.6.3. It follows that if one has managed to find one unitary cocycle [ur], then {ur) is actually a one-parameter
€rqaBIE-* B sr (/)n - / lerll s^\olloJ rl '(L)ap(L)I | = U)n aurJap em JI '(g)r7 ur / rog '(;L't>- * (L't) Iq pcsscrdxa"1' p_ur g uac/hleq , { 1 r 1 e n pa { 1 ) ( L ) a p r - . L ' r r [ = r r r l e q l q c n s ( ( A ) f ) d - J S : a a r n s t o r u 'g a c e d s lrcqllH e ul g Jo uoltelucsarder IerlJeds e slslxa areqt udql ,(relrun snonurluoc ,{lEuor1se sr lrr F , JI lErll so}BlstueJoar{ls.euolS . '(t-){ = (t)J puu sp(s-r)3(s)/l = (t)B * | suolrerado agl puc tll .-[-Ti lcadser qll^\ erqoEl"e r{o€uEge^Jl€lnuuoJ ?^rlnlo^ur u€ sr.(9)rZ 'relncrlred u1 'ornsBeru r B E H o l l c a d s o r q l l / h s n o n u y l u o c f l a l n l o s q e s a r n s u a r uJ o E u l l s l s u o c ( D W u \ I € a p r p o s o l c o q l q t r ^ \ p e r J r l u o p r e q u e c ( g ) r Z a c e O sa q a 'Q)rZ ot (g)rZ tuo.t3 rolerado ^J€lIun e sr / / ur:o3suerl IererlJueld -rerrnog aql legl uasorlJ os suollEzrleuriou aql -- Lp puu tp iq , ( l d r u r sp e l o u a p - - J p u € g u o s a r n s e e t ur e B H x l J ' l l e r o J p u € o r u o '(t)apQ V)n J = (A)(n * zl) lcnpord uoltnto^uoc pue ') =t3 tll' f t 1 . l ( ' ^ ? } ,: l ( ! a )I'7_ ons = llrtllI to uolrtllBd Ieros B tt tt uJ (f l{c LL
)
rurou uorlelJE^-lBlol ' ( g - ) t I = ( S ) * T l u o r l n l o l u r e q l q l l a p o d d y n b au o q ^ \ ' e r q o E l e q c u u e g a-,r-!Ii-lo,ruruB Jo ernlcnrls aql scrl (g)ru les eql :(C)n fC, palouap sr g uo sernseaurxeldruoc 'relnEor 'allulJ Jo aceds cr{I 'JoprsuoJ IIBqs eA\ scrnseeru IIE Jo uorlrurJ3p Jo ur0luop aql oq 11r,nrrqeSlB-o slr{l 19 ur sles lcedruoc eql ll€ 3u1u1e1uocc4 e:qe81e-o lsoll€rus orll sI sles 'sles I e r o g J o s s E l ce q l larog pu? sles srreg lnoq€ ssnJ ou eq peeu eraql loql os -- sles lceduroc Jo suorun elqelunoc ''e'l -- lceduroc-o e J u J ) u l s l a s u a d o 1 1 e l € q l s r a c u e n b a s u o cJ e q l o u e l s o c e d s l r e q l l H a l q € r e d o s g o , ( r o 3 a 1 e ce q l u l , ( e 1 s y p l s l l e q s e ^ \ ' ( g ) r l E u r r a p r s u o c u r ' l e r { l o s ' o l q e r e d o s p u s e l q € z r J l a r us r g l e r { l s r o c u e n b o s u o ca u g 'i(1r1rqe1unoc ruolxe puocos aql selJsll€s g l€r{l erunsse Ileqs oA\ Jo ( ' s u o r l o uo s a q l o l p e l o ^ a p x r p u a d d e € sr aroql ' { u r o g s u u r l r c r r n o J ' a r n s e o r ur u e 1 1 ' d n o r E r a l c e r e q c 'Jdenror q r8 lcedruoc ,{11tco1) Jo losqns dldrue-uou B rlll^\ r€rllrusJun sr orl,r re pEOr e q l J o J ) ' s r s , { 1 e u ec l u o r u r s r l l o e r l s q € o r u o s u r e a l o l p € q E u r n r q r o 3 porur€rl ,(lqerudorrr pue g lurauaE e :oJ uoaq ser{ ou u[rll luapnls t r o J r e p r e q o u a r e s J o o r d o q t ( q ) p u u i e l q e ; r s o p, ( 1 r r 1 n c r 1 r e d 1ou Suyoq J rlllly\ ,) Jo uorlBJIJIluopr luanbesuoc 0rll tJ r{ll/l\ Ieclluopr souoceq o s l p J I € n p e q l t l = g u c q m ( e ) : s u o s c o rE u r m o l l o ; c q l r o J u o l l € n l l s lsEJlsqB aql qll/r\ lsrsrad IIBTIS3^\ '1uune1orsl u, = g osEc aql fluo ' p o u r e o u o os r u o l l e c r l d d e r n o s € r e J s B ' l e q l e n r l s r '1 U ellq71\ dnorE ( r a l c e r e g c r o ) l e n p q l l i r ' d n o r E u s r l a q e l c e d u r o c K 1 1 e c o(1; ; r o p s n u g ) € alouap IIIrt\ I loqrufs eql 'lxcu eql puB uortcas slqt lnoqEnorqa uopty
ua Jo rrrulccds
uosr^rv
aql
'TE
' t u e r o e q lu r , ( p o 4 r 5 1 - u o p B€ lI I'l'€ r u o r o c r { I 8 u 1 1 1 e cr o J a s e o E u o r t s € s r c r a r { } ? c u c H ' { l l ) d n o r S a q l 'rEI€Js cnrlrsod J o r o l € r o u a E l e u r r s a l r u r J u re q l J o I e l l u a u o c l x ae q l s E e , ( q u o r 1 e c 1 1 d r 1 l n rou1 d n ' p a u r r u r e l a p s y g , , , ( 1 1 s u a pe,q, l p u e d n o r E uollcv uE Jo runrlcodguoscrry aqa 'Z't
g6
94
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
homomorphism of ll(C) into l(lf); it is not hard to see that u(f) = l 1 t ' 1 u d t , i n t h e s t r o n g s e n s eo f t h e i n t e g r a l . I t c a n , f u r t h e ^ r ' b e s e e n l t r a t r a n e ( E ) = { E e X f :u ( f ) \ = 0 w h e n e v e r f e L t ( G ) a n d f v a n i s h e s o n E ) , f o r a n y c l o s e d s e t U i n f . O u r a i m , i n t h i s s e c t i o n ,i s t o c a r r y out a similar analysis of a group action on a von Neumann algebra. Dcfinition 3.2.1. (a) An action a of G on a von Neumann algebra M is a homomorphism cr from G into the group Aut M of * - a u t o m o r p h i s m so f M , w h i c h i s p o i n t w i s e o - w e a k l y c o n t i n u o u s - - i . e . ' / ' c r . ( x ) i s o - w e a k l y c o n t i n u o u s ,f o r e a c h x i n M . (b)'A dynamical system is a triple (M,G,a) consisting of a von Neunlann algebra M, a locally compact group G, and an action cx of E GonM. Notice that we have written dt rather than for 0(x) and f or'l(t). Let q be an action ol G on M. The equation a(p) = froposition 322 -the integral being interpreted in the pointwise o-weak sense lcrduQ) !-"induces an algebra homomorphism p ' a(1t) front M(G) into the algeblp Eo(!,O,,ol o-weakly continuous linear operators on M. Further, llc(p)ll < llpll. Recall that every Proof. Fix g in M(G), x in M, 0 in M* * - a u t o m o r p h i s m o f M i s i s o m e t ; i q ,( g n d o - w e a k l y c o n t i n u o u s ) ; s o / r < c r . ( x ) , O >i s a b o u n d e d ( b y l l x l l l l 0 l l ) c o n t i n u o u s f u n c t i o n o n G ( b y thi definition of an action). It follows that the assignment . ? 0 ' J dtr(t)
definesa boundedlinear functional on M* of norm at most ll"ll llpll; since(M*)* = M, there existsa unique ele,mqntin M, which we shall denoteby cr(p)x,such that llcr(p)xlls llsll llxll and = l<ar(x),O>duU) f o r a l l Q i n M ' . , C l e a r l y , [ , h q ,a s s i g n m e n t x ' c x ( p ) xi s l i n e a r a n d s o c r ( p )e t ( n { a n d l l ( p ) l l < l l p l l . I t i s e q u a l l y c l e a r t h a t t h e m a p p ' cr(p)is linear. To prove c(v * r) = cr(v)cx(tt)proceed thus: if s e G, x e M and 0 e M*, note first that Q o a" e M* ( s i n c e c r , i s o - w e a k l y c o n t i n u o u s ) a n d o a">dy(t) that = and hence ct, o c(l) = cr(p) o definition of convolution,
* p)(r)= II I s@a{u
g(t + s)dv(s)du1)
('(e) pue uaroar{l rlcuu€g-uqeH eql ol IEedd.B,,hl 3 o',r[ 0'*7g :lurH) J I p u e l o s s l q l f q p e l e r a u a Eo c r d s q n s p a s o l J o r { l s r JJ *N J o l e s q n s l ? l o l - r u r o u u s t ( ( g ) ' Jt / " w ; Q : $ ) n o Q l
(q)
(@)-z sl (o)rz Jo lenp oql puu (c)rz ut asuapsr (9)'3
: l u I H ) ' 0 = x u a q l ' ( l J o d d n s l c o d r u o cq l J / r \ s u o r l c u n J s n o n u l l u o c go eceds orll) (O)", ul ./ IIB roJ 0 = x(!F pue n ) x JI 't[ uo D Jo uorlce uE aq D l€-I
(u)
G'Z'€)
scsrcJcxS
lp @l | = snql !(Jrt)n rog (/)o e1rr,n fldrurs IIBr4s a^\ 'tp ! = trtp puz, (C)rZ > ! ll ' a l a l d r u o cs 1 'f1leurg '((S'l'O) E 3oord eqt 'xg Eur,l,o11oJsluatutuoc eq1 '3c) snonulluoc .,(14ea,n-oflluanbasuoc 'snql pue reaurl aallrsod e sl (rr)" o l€uorlcunJ, IBruJou ' o3uaH 'g ,(q peceldar y qll/K pll€^ Q sur€ruar I luotuelels o^oqe erll leql paonpap flrsee s1 ll 'lceduoc-o sl g sV y, X, (t[tn ) t J '(t)rtp<|'e)'r >
'g j X lcudruoc roJ tsql s.trolloJ t1 i9 3o slasqns lcedruoc uo ruroJrun oq lsnu acuaEraluoc eql ' u r a J o a q ls . 1 u r q { g ' < 0 ' ( x ) ' o > u o l l c u n J snonurluoJ eqt ol osr/r\lurod s o s € o r _ J u ls u o l l c u n J s n o n u r l u o r J o { . 0 ' ( ! x ) ' o > ) l a u e q l . u a q J 'x 1\x lrql pue *n u! lau ouolouoru ? sl {!r} teqt mou esoddng 'O
< Q)rtp
'+n x u l roJ acurs'0 < (tlc o O ruql 0 < '0 'a,r1l1sodsr .ool .srql < rl uorg s/{olloJ lI 0 0 l€ql otunsse ,(eu arrr toJ :;W r (rt)p o Q sorlclurr ,W ? Q lBrll ir\or{s ol poau ol[ 7' eJnseau e ^ l l r s o d € r o J s l g l o p o l s e J r J J n s { 1 , r e a 1 cl 1 l s n o n u J l u o c , { 1 1 e e m - o sl (d)p qc€e ltql ,{,grrarr ol peeu eA\ '3ootd cql alalduoc oJ
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
(c)
Let {k,: i e I) be a bounded approximate identitv for lr(G) -i . e . ,{ k 1 } i s a n e t s u c h t h a t il.
tl
s y p l l k i l l< @
ano
tim llt,-/- /ll = o
for all f in Lr(G). Then
rim ll0o c(k,)- Oll= o i
for all Q in M*. (Hint: as sup ll
l l o o ( r , , ) - o
also .11(G)is commutative.) If (ki)i€r is a bounded approximate identity for Illc), E cr(k,)r i x o-weakly for each x in 11L
then
We shall use the symbol t to denote the set t =^{l e zrlc;: i nas c o m p a c t s u p p o r t ) . I t i s h o p e d t h a t t h e n o t a t i o n c w i l l s u g g e s tt h e phrase "compactly supported Fourier ^tra.nsform"to the reader' Since (f * d^ = i ;, ^it-is clear that a is an ideal in Zr(G)' It is well-known that C tras a rich supply of f unctions. We gather together some results (which may be found in [Rud], for instance) which we shall need. (Parts (a), (b) and (c) of the f ollowing p r o p o s i t i o n a r e , r e s p e c t i v e l y ,T h e o r e m s 2 ' 6 . 2 ,2 . 6 . 8a n d 2 . 6 . 6i n [ R u d ] . ) Proposition 324. (a) (b) (c)
If K g U C t,^with K compact and U open, there exists f in C suchthatlK(/ 0, there exists k in C such t h a tl l k l l , . l + e a n dk = l o n K 'ti'i';"tt(C),
ina e > Q, there existsk in t sttchthat llt n in Lr(c). i.l[], . e;'iri partictilar,t is norm-dense
Excrciscs
(3.2.s) Let A = {k e b: llkll, . 21. (a) If k,,k, e A, say that kl I k2 if i<, = t on the support9f [.r, bi this order relation; i.e.,il SnoJ'tfiat A is directed'upwa'rds k,k, e 4,, there exists k, e A such that ki _1 k, fot i = 1,2. (ftint: apply Prop. 3.2.4(b) wit\ e = I ^and K any compact set containingthe supportsof both kt and kr.) (b) Show that A is a bounded apiroxima"seidentity for tr(C). (Hint: since llt ll . Z for all k in A, it is enoughto prove that
n
'{s
'l IETIJAdS{ p0l€rcossB 3r{l eurJap
. { 0 = & ) l + 0 = x ( / ) D : J) L\ =
i (r)Dds :yrgt x} = (gd)n ,(q ,,ecedsqns Jo lasqns posolc € sl g JI
r{O
= x(,iF :(9)J
(r)
) l\ = 1r)Dds ]aI,W 9 X JI
.{0 =
(q)
&)l c 0 = u),r :1 t Ll =T(0 = aF :@)rz r /) = n ds f q n g o r u n r l o e d su o s o ^ J v o r l l o u l J o q
(e)
'n uo 'z'z'€ uolrlulJeq I Jo uollcs u€ 3q D l3.I '[oo1] go 3'19 n = !r se/r\ se parro:d sr (e) uorlc3s ur ruoroor{l € sl (q) lorroqe 1@)t 'I ) 'i ! uary Io pootltoqqSou D uo sallstuo^ I tutlt Llrns st (C)rZ t ! lt puo (D)rl'ut papt pasop o s! I It G) -(Z)"1 = Z puo (C)rl ut papt uo s! {Z lo pootpoqq8nu D Lto saqsyorr I i(g)J ) I\ = @)ot uary'1 ut tas pasolz o s! g lt @) 9'7:g uollpodorf, '([oof] '3c) tlnsar Eutltrollog aq1 fq JoJ palesuaduroc uauo sr srsaqluz(sIErlcads Jo >Ic€l oql 'acllcerd 'slseqluf uI s lerlcods e^rq slos uolalEurs leql sel€ls r,uoroeql 'Z rol€lrqruu€ rlll^\ utrreqn€I rouell[ pel€rqolac eql @)rl ur Ieepl pesolc ,{luo aql sl (^?)131 slseqtu,(s lerlceds e^€q ol pr€s sr J ur 'srsaq1u,{s ? los posolcy I e r l c o d s J o r u o l q o r d p s l l € c - o sa q l o l p o l s l s r sr uoueruouaqd srrll 'lcrJts oq 'rarr.emoq'uec uorsnlcur srql '(-S)f 'S s7 er5qm E u l u l e l u o c J s1 leqt pue leepr pesolc lsellerus aql sl '(j))r? j 'uorl€nlrs = l€rll reels sI ll s uar{,r Ienp aql uI
(L)I (9),7 ur s1s{xa (e) areql s^\ogs idq'g rcr$ l€rll L dcns / V'Z't / ''(rh7 ur I€epl pasot3B sr (a)t reqt puB (a)1 = Q)t leql ree15-sr 11 JI '({O)= 'f1os:aauo3 @)l :@)fl ) {} = @D l:l'J Jo ? lasq-ns,{ue rog ul les pasolc e '{s ut / acurg 'tc)rl i s tes eql sellrepun posol3 ol (rh7
.J s,(en1e sl TS les a{l '(g)r7 u1 t f,ue roJ snonulluoc sl ./ II€ roJ d = &)/ :J r f} - Ts: rol€lrqruu€ eql eulJep € ro; 'asrcard ad oa ('fEoldctol leura{-llnq polleJ-os €opl slql 'lJ€J uI) '€sre^ ecr^ puB J Jo slesqns Jo sleapl pesolc ruor3 ssed ol ,(e.n IBrnlBu e sl areqJ
('ruro3suerl rerrnoC aill Jo ,(1rrr1lcofuroql osn ol poau [ n o , { ' s l q l - r o 3l 1 l 0 Z t t q f q c n s V u l l l l u r o J ! = / * Z l u q t q c n s 0Z slsrxo ereql luql alou ') ) :s../ les asuop € roJ V ul / J I J o v trt
0 = ll/ - /'211tu'y
uollJv uB Jo unrlccd5 uosarrry oq1 'Z't
L6
9E
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
l-cmma 32-t.
Let x e M and let E be a closed set in l.
x e M(q,E) <+ (,f)x = 0 whenever f e Ll(G) ts ,suclt that I vanishes on a neighborhood of E (i.e., E c ht {f)-, where lnt denotes interior); consequently M(qE) is a o-weakly closed subspace of M', (b) spcr(x)=0(9x=0; ( c ) s p ; ( x ) = { 0 ) ( + c r f x ) = x f o r a l l t i n G ( i . e . , x e M s T , a n dx * 0 . (3)
Proof. Let I* = {f e !-'(C): cr(flx = 0). Clearly 1* is a closed ideal in tt(c) -- recal-i that .r,l(C) intrerits commutativity ?rom G. Further, by definition, spo(x) = {. ( a ) T h e i m p l i c a t i o n s ( ) ) a n d ( € ) a r e e a s y c o n s e q u e n c e so f ( b ) a n d (a) of Prop. 3.2.6applied to the closed ideal f* and the closed set t, r e s p e c t i v e l y . T h e s e c o n d a s s e r t i o nf o l l o w s f r o m t h e f i r s t i n v i e w o f the o-weak continuity of the cr(/)'s. (b) If spo(x) = o,-then by (a) (applied to E = 0), we find that c(rx = 0 f o r a l l / i n L ' ( G ) , a n d s o , b y E x . ( 3 . 2 , 3 )( a ) , x = 0 ; i t i s c l e a r , using Prop, 3.2.4, that spo(0) = 6. (c) If If (= sn..(x)) = (0), it follows from Wiener's Tauberian theorem that 1* = 1({0})= ({ , L'(G): /(0) = 0}. Let A be the bounded a p p r o x i m a t e i t i - e n t i t y f o r ^ l r ( C ) c o n s t r u c t e da s i n E x . 3 . 2 . 5 . I t i s c l e a r that the set Ao= {k € A: k(0) = l} is co-final in A -- in fact, k'k, e A, kr 1 k2 and k, e Ao imply k, e Ao. Thus Ao is also a bounded a p p r o x i m a t e i d e n t i t y f o r Z ' ( G ) a n d s o , b y E x . ( 3 . 2 . 3 )( d ) , x = o-w€Bk - lim cx(k)x. k€Ao If, now,rf e Lr1G1,note that ft -i
-. .
) IQ)
- x,Q> dt = = 0.
Since / is arbitrary, conclude -- since the dual of ZI(G) is I-(G) -that = 0 almost everywhere (with respect to Haar measure). Since the f unction I - is continuous, c o n c l u d e t h a t < < r . ( x )- x , 0 > = 0 f o r e v e r y / . S i n c e 0 i s a r b i t r a r y , conclude that x e"Mq. Also, by (b), x I 0. Conversely, if crr(x) = x for all t, it follows that . l e(f)x (= ) f(t)ar(x)dt) = /(0)x
for any f in LL(G). So, if x * O, Ix= 1((0)) and so spcx(x)= 1((0))I =
t0). n
The following proposition lists some f urther elementary f acts concerning the notions introduced in Definition 3.2.7; we use the
'A[ ur. x Al. ,(yrea13 7 5 (^x)DOs IIB roJ p ds 5 (r)pcls ecurs 1o ds r t AL >1c1diE I (^x)Dcts '(q) 8'Z'€ Brurua'1 ,(q 'uaqt '(l'n)lU s ^x * 0 ',{1esrarluo3 'L JI Jo tl pooqroqq8tou ,(raro roJ {0) * U%DU esodclns r ds / L os l0 * &)3 stlr{1( 0 = (3)n snqJ 'g = x(3)n os puE '(A'o)w I x(3)n arueq^\ '71j 661")pds leqt selldur (p) uud 'm u\ x fuu ro3 'uaql 'A 3 3 lds puu I = (1.)Btcrll q c n s J u r 3 4 c 1 d ' L J o A p o o q r o q q S r a ua u o s i o ; t O ) = C 4 ! o D t / J i ( e ) .(x(/)D)Dds / l. acueq ig * (L)3 olq,r\'0 = (x(/)n)(3)n leql os O = { * 3 leqt apntcuoJ 'dor4 ,{q) slsrxe '0 = {3 pue i = G)B leql qrns ;r ut 3 ((e) V'Z't rpueq reqro e"qr ug '(x(/)n)Dds q (x)Dds os 16 oraql't tds / L JI = x(3)n(/)n = (x)(3 * t)p = (x)(I * 3)o = (r(/)nXt)p e g = x(3)n (p) '@'o)ltt a (r)to (e) €rurue'I ruorJ s^\olloJ r x leql @'byit S'Z'g e ',{lluanbasuoJ 'Q = g = (f,)/ ter{l os '(L)[,- = lI G)rI e "1r)rn(/)n (1,)t/ oste :(t - s)! = (s)r./ eioq,v' 'x(V)" = rcqr bio51 1c) r ds / L*rlt roJul '0 * (1.)/ acurs '0 = @rc lBql ernsue mou (/)n go .,{.1rnur1uoc 'S u l {eea-o aqt priu g uo uolldunssu oql 4 tte rog g = d(/)n '-((f)rds 8'nn) leql (B) g'Z'€ eruue.I uorJ s/AolloJ lI Jo pooqroqqEgau uodo uB uo ser{slue^ / pun | = (LY leql qcns (C)rZ ut ./ >tctcl
'_[{o"or[A^1 It 3r
'I1osre,ruo3
si^1eo o, -[t,o"o, 'pasolc sr p ds aculs pue 'd loE e,tr 11ero3 (f)Dds f n ds l€rll s^\olloJ t\'.AI u\ f 11erog O = tU)n serldurr O = (/)p ,(1rea13(q)
' (L-){ = (L) {
Pu€ *x(IF = *(x(/)n)
suorlenbe polJlre^ ,(lrsee or{l ruory s^\olloJ sltIl (E) 'Joord 'g = x(rl)n uary'(x)nds {o poor1toqtlStauo uo,(1ptc1yuapt saqsluv^ tl puo (g)n t TI {I 'L {o,t pootltoqqSrauttaaa rcl {O\"* Qf4n <+ D ds , f :/ lds u 1r)Dos 3 (r(/)D)Dds
(;) (e) (p)
:D ut t ilo to! (az)w = (r'nfufiin (c) :-[{r)" o sB i^ ] =o o ,
\ l)r-{ = I $s :1 uollcunJ aql Jo lroddns aql roJ / lds uollttou uorlJv uE Jo unrlcadg uosc,rry aq1 'Z't
r00
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p c I I I F a c t o r s
So, Z fi sp cx I 0, for every open neighborhood V of 7; since sp cr is closed, conclude that 7 € sp cr. = ? i vanishes on a neighborhood of (f) lf f e LL(G), tlrrn /i spcr(x) and so, by Lemma 3.2.8 (a), o(/)o(tr)x = 0. Since / was a r b i t r a r y , c o n c l u d e f r o m E x . ( 3 . 2 . 3 )( a ) t h a t c r ( p ) x= 0 . E We conclude this section with an analogue of the statement that
spt(/ * s) g sprEp-t
s.
Proposition 32f0. Let E, and E, be closedsubsetsof T and let E = Er,T, If xre M(a"E1) iori = l-,2,andx= xrxz,thenx e M(c"E). Proof. Casc(i). spo(x,)is compact,for i = 1,2. In view of Lemma 3.2.8 (a), we need to show that cr(flx = 0 wheneverf e Lr(G) is such that j vanishesin a neighborhoodof E. Also, in the case under discussion,we may assumeE, and E, are compact (by replacing fi by spo(x')); then E = Et + E, is also compact. !-et V be a neighborhoodof 0 in f such that f vanisheson E +^V + V. Appeal to Prop. 3.2.4 {a), and pick fi in C such^that /, is identicallyequal to one on a neighborhood of E, and spt /i c E, + V, f or i = 1,2. (Locally compactHausdorff spacesare regular!) Notice that, by Prop. 3.2.9(f), "(/i)xi = xr, i = 1,2. So, for any Q in M*, =
=
dt dt Ldt2 I | ! t Ult r
= JfJftrlfrts, - s)/r(s,+ s, - s) . <(crrr(xr)Xcxr ds dsrds, r+"r(* )),Q, where we have used (i) Fubini's theoremand the o-weakcontinuity of a(g) which ensuresthat c(g) may be "taken inside" a o-weakly defined integral, in the secondequality, and (ii) Fubini's theorem and the substitutionss = l, Jl = I + tt, s2 - t2 tr, in the third equality. Hence
= JJ t{r'sr)h(s'sr)ds,ds, where &(s'sr) = <(cx.r(x1))(orr+rr(xz)),0t and
k(s'sr)= J /tr)/rtr, - s)fz!z+ s, - s)ds.
'""r:)l
I=f
t.r:' !1 ' l3u > ll':"ll 3 )l z/r( zl l f:xl l l t:l>l g l ftx;1 ueql [(*: '"' 'It)] = [(ul l€ql qons les lururouorllro us sl {"1 '"''I:} JI '(O re g
) > . ; o{ 1 r n u r l u o c , ( q € r t r> I > I r o J , > l l ' : r l l l p q l q 3 n s - :I' " ' ' Il :( x ) 0 1 '(A)ru'i x 0 < r pu€,l ul {cld lsrrg:lurg) IIB roJ ( ..,- .. I=!l
-zlr\ , - l r l l ' : *{ lJl " > l(x)01 leql qcns 0 < ) lu€lsuoc u pu€ ,l ul {"1 "" "l} tos I€rurouoquo u8 slsrxo ejsql lerll ^\oqs
(€)
'snonurluoc ,{1Euor1s sJ rlclr{/AIuuollounJ reourl e aq O - (A)f :0 la"I (tt.Z.g) srsIJJexl '@'o)n zxrx t o p n louoc'((q)l'€'0 l € r l l 'xg 'JJ) oplculoc sar8olodol 'slas papunoq lean pue {Ba^\-o eql uo 'ocurs lposolc {11eaa-o sl '(e) B'Z'€ €rurue'1 ^q 'gclq ^ (Z}o)W '(p) 'ctor4 o1 sEuolaq 1 z x 1 ! A p ; 1 I x 1 ! 1 1 ns;l c n p o r d a q l J o q c e a - 6'Z'€ pue (I) ase3 [g ',(1>1eo,n oouoq pue '[1Euor1s zxrx * 1zx1!A;n;1rr1lp; 'slas papunoq uo snonurluoc [ISuorls fllurof sl uoll€clldlllnru eJurs
rrt ' - ' re!(ll"(r:)"ll ';1Ix1!1p1;1ons Gx zx1!31n pue Ix * I"(!F pue g t I;;!a;; leql -eloN 'f18uor1s '((rrtz'E) "ll7ll 'c t t3't{ ql qrns (r t l s { : l t 1 p u e l t ! 1 ' y ;s l o u l r a t a s 'xg rgc)'oprcuroo sernsolJ{€e,lr\pue tuorls oql '(fi)f Jo slasqnsxeluoJ roJ leql anrl sr tl pue 'xeluo3 sl les slql '{Z r rll 4ll 'J ) 4 lt({o} acuaq pue) {ee^\-o aql'6l"sElolaq d leql les oql Jo ernsolc(>1ea,v' (p) (€'Z'€) 'xX pue (q) (S'Z'€) 'xA uorJ si(oltoJ 1r.'147 ur.( f ue rog 'frerlrqr? ax'Ix (1g)ese3 '(r) asec goordaql sapnlJuocslql '{rerlrqre s€,r ocurs 0 '0 = <0'xU)p> leql;o'Arou 'sernsuoruoroegl s.lurqnC Jo uolleJrldde '0 = (zs'.)ry'os pu€ = raqloue la1 tttu"nbosuoJ 0 .{'"'zI.t!).{ -) 'A + zg + Ig u1 paugeruoJsr qJIq^\ u'-'a1rds + A + g I + A + t/ to, eplsul polroddnss\ (2"'z! pue r./ Jo uollnlo^uocorll {11er1uesse sr qcrq,n) (""'z!.t./) rgq^ 'A + A + ? uo s a q s l u B ^ $ ' $ {
ocllou
L
:(r-s)S = (s)r3 puE lrnpord esr,nlurod selouep l o p o q l e r a q / $ ' ( - " - ' N I ' r { ) * / lcnporcl uollnlo^uoJ aql sl (us'.)Z leql alou 'Gs pexrJ s Joc
IOI
uollcv ue Jo unrlJaclg uoselry aq;
'Z't
3. The ConnesClassificationof Typc III Factors
t02
(b)
Note that ll"i,ll = 0 fo.r all..j t Q(x)= Q. Show (by considering - r m a x , l l x ( , lwl o r k s . ) K-lx) that K i e (well-dbfin6d) There existsa boundedlinear functional 0 on Xf, where
fr=fte...olf n copiee
such that
,[,i, "r,]=o{") (c)
for all x in t(lf). There exist vectors rl1, ..., 4r, in lf such that
<xl,,n,>; 0(x)= j.t, =1
(d)
in particular, 0 is weakly continuous. (Hint: look at (b) and appcal to Riesz.) A convex subset of I(lf) is weakly closed iff it is strongly closed. (Hint: the (locally convex version of the) Hahn-Banach theorem says that a closed convex set in a locally convex topological vector space can be separated from any point n outside it by a continuouslinear functional.)
3-3. Thc Conncs Spcctrum of an Action I f a i s a n a c t i o n o f C o n M , w e s h a l l , a s i n D e f i n i t i o n 2 . 5 . 1 3( w h c r e o n l y t h e c a s e G = l R w a s c o n s i d e r e d ) ,d e n o t e b y M o t h e f i x e d p o i n t algebra: Mq = {x e M: er(x) = x Yt e Gl. Clearly^_Ma is a von Neumann subalgebra of 1L For a projection e in W, eMe may be viewed as a von Neumann algebra M" of operators on ran e. Since e € Mq, it follows that a induces an action creof G on M" such that ai@xe) = e(crt(x))e. (The invariance crr(e)= e is needed to ensure that t h i s d e f i n i t i o n i s u n a r n b i g u o u s ,a n d t h a t a e i s a n a c t i o n . V e r i f y this!) Proposition 3.3.1. Let q be an actiott of G ott M; let e, e,ez e P(l'tq) and let E ll be closed. (a)
M.(a",E) = M(q,E) i M.i
(b)
er l er+ sp c(tl c sp o"2;
(c) (d)
) a(q(tt)x)b; i f u e M ( G ) , x e M a n d a , b e I + { d , t h e nc x ( p X a x b = if x e l+[ and iI a and b are invertible operators itt l+:[q, then spo(axb) = spa(.r).
Proof.
(a) If x e M", note that cre(flx = c(f)x for all I i\ Lr(G)
'n uo g to p ttottco tto tol
.€-€-€ uollFodord
ur eJe r{Jrq^\ suorlceford oJoz-uou ,(1uo raprsuo, o, ,rrrg;nffl Ie'luec ' ( n ) 1 E u r u r 3 e pu l l € r { l s l u o r l r e s s eE u r , n o l l o ge q l J o l u o l u o o a q l . J J o l e s q n s p e s o l c e s , { e m 1 es l ( p ) . 1l u q l u o l t l u l J a p e q l u o r J r e e l c s l l I ! '(Gn)a ) a * 0:"n dslu = (D)J :snql paurJop sl '(lr)l ,{q palouop !o 3o urnrlceds sauuoJ oql 'n uo g Jo uorlce u€ sr lc JI .2.€.€ uoIfIuIJa( E
'(gxz)pds
3 (r-q(qxo)r-u;pds = 1x;pcrs e o u r s' s / I \ o l l o Ju o r s n l c u r o s r a ^ o r e q l l(r)Dcts =
ecueq pu€ ( ^ . 1 Itp a(x)' n aQ)l '| = tp(axa)' p(t)I "ll) L
g0r
uoJlJV uB Jo runrlceds sauuoJ oqJ. .€.€
104
3.
The ConncsClassification of Type III Factors
r(cr) = fl {sp ce: 0 * e e P(z(Mq))); in particular, if Mo is a factor, then t(a) = sp d. Proof.
We shall show that if 0 * e e
P(M'\, then there exists a
non-zero d in PQ@[c\) such that Sp cre= rp o{ ln fact, define d = Y(ueu*: u e U(l/qD. On the one hand, d e Mq since it is the supremum of a family of projections in ifd; on the other hand, it is (bv clear that udu* = d for all u in Lt(L{c1 and so d e (W)' Scholium 0.4.8); thus e e P(Z(MG\). I n o r d e r t o p r o v e S p q e = s p q { i t s u f f i c e s ( b y P r o p . 3 . 2 . 9( e ) a n d Prop. 3.3.1 (a)) to show that, for any closed set E in t, M(o,E) n ItI, t {0} if and only if l+{(o"E)n M; * (0). Since i{" { M; (as e < f, the " o n l y i f " p a r t i s c l e a r ; c o n v e r s e l y s u p p o s e0 I x e M ( a , E ) t 1 M u ; s i n c e x = VxV, therc cxist u,v e U(Mo) such that Qteu*)x(vev*)* 0; then y = eLfxve 10, and it is clear that / e M"; and since e, u, v € Mq, it f o l l o w s f r o m P r o p . 3 . 2 . 1 0a n d L e m m a 3 . 2 . 8 t h a t s p c r ( y )C s p o ( x ) C E , O whence M(",8) n Me t (0), as desired. We shall head towards the main result of this section via a s e q u e n c eo f l e m m a s . Irmma 3.3.4. Let {Vr: j e^A) be an open cover of t and let x e I[. If x 10, there exists f in C such that spt / c V, /or sonte i in h and d(f)x t 0. Proof. Let Io be the set of linear combinations of elements of ll(C) whose Fourier transforms are supported inside compact subsetsof members of the cover {2,}. It is clear that /n is an ideal in Zl(G); hence,the closure r of Ii in tl(G) is a closed-idcalin LrG). If 7 e ( a ) , c h o o s e/ i n f, pick j e A such that / e Vri t!,en, using Prop. 3.2.4 = I and ^spt / g Z,; thus, for each 7 in r, there C such that f(i 0. On the other hand, it is a fgct exists an f in I such that l(7\{ (cf. [Loo], Section 37) that if I is a closed id^ealin LL(G) and lf I I LL(G), then there exists a 7 in t such that i0) = 0 for all f in I. Conclusion: 1o is dense in Zl(G). .. This conclusion, together with Ex. (3.2.3) (a) and the fact that l l . ' ( g ; ; ; < l l g l l , f o r a l l g i n r t ( G ) , c o m p l e t e st h e p r o o f o f t h e l e m m a . E Lcmma 3.3.5. Il e, and ez are non-zero projections in lrt[q,wltich are equivalentrelative to I{ (as in Def. l.l.l), then r1
rra
putt 6r (x)tn = lIIa 6r r)tl,
tvqt Llans(1'1'g uatoaql 'lc) (l)ztrt = g uo ttortcu up sts:rr.aitatg uatla @ L to lootd n hl lo 'W Lto g .d o asoddttg 9-g-€ suur-I lo sttotlcv ato g puD b atal1x ;
= (r)?dpue(',r;tnrn = "rr,1,r#;rt":'r;\f Xll;tj'J""J,rlt9t3' dBlu snonurluoc .{l8uorls € sl aJor{lJI (s1oqu{s ul .g : n) luelulrnba r a l n o o r e h l u o g J o d p u e p s u o r l c e l e q l , { r s l 1 e q se , i , - - [ f = g a s e c otll ol se^los.rnopal3rrlsar e^\ eJoq^\ -- (q) Z't'g uotlrurJoq ul sv
pue 'zt^ ) zI zz! = z $qr rzelc sr l! 'l 1Ie rog (r)tn z1 = 1")to ,ruij 'o
I *(x)alP tllx'1rrn= z
'r/ leql rlcns g ur.zt Jo acualsrxe aql oJnsue t / , c ' I = d i ( 1 r 1 e n b ec q l p u e r ; g o u o r l l u l J a p e q l -tt^ , (np)n ut u(luaruele orez-uou B slsrxe aJoql'(ra)r)1 r l, puu,( 3o pooqroqqErau € s r l ? a J u r s : t a > r I l e q l a l o u ' r a ( x ) r n z 1 = 1 r a x z 1 1 r n= ( r ) r n s e 'os1e l ( o 7 g ) 4 t r ! r e e l c s l lI '(D t t:(*x)tn dll=tI I0leql t"-l 'dor4 r(q) 3 ,lL i ({)oos - (x)pcts le ql os ((p) O'Z'e lds J (x)pds le ql = nzq aruls 'taxN! = x ((r) I'€'€ 'dbra fq) teqr atou puB 'r?.,{{.zl 'O puv 1nz/161n - rc la.I * @zl)(8)p pue /14j .s lds - 3 1ds leqr qcns '0 'zi Ia V a € esooqJ ol V't't etuue'I asn 1 nz1 ec"urg *nn puu ? 'slsaqlod,(q ,{g : n*n leql qcns ,{ ur n frloruosl leltrud u sl eroql "
'rxaN - tA n*qtqcnsr p (t) roaocuadouE osoorrc \t;"j41t:'(,
qcns ,{lo,rglcadsar(: ut) 0 pu€ l. 3o spooqroqqElau dq til pue n t:-I
'{d
s0r
*'tw u (A"b)I^t 'E
uollcv uE Jo rrrulceds seuuoJ aql
€
106
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
7r@oe2)=Br(x)6e* for all t in G and x in M. Proof. If {ar}: (crt}: {Bt},simply define
',[;;:;;;]l l;;;:,, ;1.;l''l
and verify that'l does the job.
I
Thcorcm 3-3-7. Let q be an actiort of G on M. (a) (b) (c)
I(cx)+spct=spog I(cr) is a closed subgroup of t; If B is another action of G on M, wlticlt is outer equivalent to q, then T(a) = r(B).
Proof. (a) If 0 I e e P(Mo), then e is an identity for M" and hence spo"(e) = {0} (cf. Lemma 3.2.8 (c)), so that 0 e spo"(e) G sp cru; so 0 e I(o) and consequently f(cx) + sp d I sp ct Suppose, conversely, that 7L e f(cr) and 7, € sp oq we need to verify that M(q"lO I {0} for 'yr1. Pick neighborhoods Z, of any ndighborhood Z-of (7, + 7, such c v. Since 7, e sp oc there exists a non-zero element tnat ffi x , i n A 1 a , V , 7 . P u t e = V l r p c x . ( x f ) :t e G | a n d n o t i c e ( a s i n t h e p r o o f o f L e - - a i . l . S ) t t r a t 0 * e e - P i U a y S i n c e / , e t ( c x ) ,t h e r e e x i s t s a non-zero element x, in M(a,Vr) n Mo It follows from the definition of e and the fact tliat 0 * xr= ex1, that there exists t in G such that er(xr)x, * 0; set x = ar(xr)x1and notice that x e M(q"V) (as spa(x) c (spo(crr(xr))+ spo(xr))- = (spo(xr; + spo(xr))-
C(V,+ V)- tn. ( b ) F i x a n o n - z e r o e i n P ( M ' \ a n d c o n c l u d e f r o m ( a ) t h a t t ( c r e )+ sp cre= sp de. Since clearly r(cr) ! f("") ! sp cr",infer that r(ct) + f(cr) c sp cxe;allow e to vary and conclude that r(cx)+ I(cx) c f(
'E'tS s^\oqs oIII edft Jo srol3€J Jo ocualsrxocql p a l r q rqxo eq u l 'I 'III adfl 1 1 r ms c d f , 1e s o q l I I B J o s r o t J B J g o s a l c l r u t x S > \ > 0 roJ J o a r c \ 1 1 1 e d , { 1J o s r o l J B J l c r l l ( q ) 0 ' g ' g r ( r 1 1 1 o r o 3r u o r J s ^ r o l l o J l I
D
' ( - ' o )= 0t): JI'Iut (r)
l((t'o) ut \ roJ) {z r u :,1} = (tDs JI '\III : ( r ) = O t ) _ rp u e I I I a d f l g o s t 7 g J I ' 0 I I I odfl 3o aq ol prus sr 111rolceJ V
(r) (O)
.0I.€.€ uolllulJcq
'{t) = (ro)1 = .3c) pue prt = hr os :n O At)t ((c) g'Z.e eurula-r ul r pu€ U, ul , 11f ro3 x = (x)lo ueqt 'N uo er€rl sug e-sr r JI (q) 'seuocrlc^c ar{l 3r€ (1ry clnorE crqdroruosr erll Jo ',(lluanbasuoc pue) 6 3o sdnorEqns posolc 'Joord IcrlrJl-uou fluo aql lerll loeJ aql Jo eJuenbesuoce sr srql (e) '(t) = (rV): uatp'a7rutlnuas sr trI It Q) l ( 1 ' g )t t t \ a u o s t o ! ( Z
'7,9uo lqErom su3 f ue s1 Q aroqrrr '(ao)l = U,tt)l D etqe?le uu€unaN uo,r frerlrqJe ue roJ 'g'g'g'uo1t1u13cq
'N uo JEInpou,, eql $ pue Q slqEre^\ suJ qll^\ palBrcossB,rs./AolJ er€ do pue do oraq^\'(6o)1= (Oo)r tcqt (c) f'g'S pue I'l't sueroeql r u o r J s i r , r o 1 1 ot r3' e r q o 8 1 e u u e u n e l q u o l i ( r e . r l r q r B u B s r l t l J I ' t L =
.(d)J = (n)1 ecuoq
', 'zzt@r^) (B'c'n) i(,r. = ""e6I 1 $
puB 'g 'rr"@r (r, ^) = (o'c'n) '-a@IL :suels,{sleclrutufp spueqroqlo aql uO IBrnl€uor€ oreql l€r{l rBalcsl ll Jo sruslgdrouosr = (,, (,, " " a @ I ,()J . ' a 6 I L)s l s q l s p n l J u o c' S ' E ' €
uorlov ue Jo unrlcrds souuoJer{I '€'E
l0l
3 . T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
108
that the condition l(lut = ( I ), while s e m i f i n i t e n e s s ,i s b y n o m e a n s s u f f i c i e n t .
3.4- Altcrnativc
l)cscriptions
being
necessary f or
of r(liI)
As the title of this section announces, we shall derive some equivalent dcscriptions of r(fut), which will be useful when it comes to explicit computation. Lcnna
3.4.1. Let q be an aclion of G on a lactor lt[ of type llI.
Then,
r(cr)=n{spB'oi8}. j Proof. If cr 3 B, then r(cr) = r(B) g sp B and so r(cr) c n(sp 8: cx B). The reverse inclusion will be proved once we show t-hat for each j non-zero e in P (Mc\, there is an action B such that ct B and sp a" =sPB. So, suppose 0 I e e P(Mq); since M is of type III, any two non-zero projections in M are equivalent, and in particular, there exists an isometry u in M such that u*u = | and uu* = e. Since e e = t and ctr(u)crr(u)*= e for each t in G; L[d, it follows that crr(u)*cx1(a) = u * d r ( u ) defines a strongly continuous unitary v t so the equation path in M. Notice that for any s,t in C, u* a"(u)) = u* e o.r*"(tt)= Vr+, vrcrr(ur)= u*crt(u)crt( (since u*e = u*): hence, by Ex. (3.1.3) (b), the equation Br(x) = vrcrr(x)rf defines an action B of G on M, which is outer equivalent to Observe now that B.(x) = u*a,(ux u*)u; the operator u may be regarded as a unitary bperator fiom lf to ran e, which sets up an isomorphism between the dynamical system (lr[,G,B) and (M",G,cr")-is a von Neumann nlgebra explicitly, if z(x) = uxu*, then n M'M. isomorphism such that
rsplsuoJlsnu auo '( 1I'Z'E) 'x:I ur s€ uorlBlou orll r{ll/'r;snonurluoJ ,(1>luarrr-o s1 Oh uo leuortounJr€aull snonurluoc*{1Euorls-oltue lcql '(lt.Z'g) .xg u1 pedolclep auo er{l ol JelrrursluorunEreuu Eursn'moqs lsnu euo :slrBlapoql ul IIIJ ol pall^ul sr rapearaq1) 'pasolcf14eam-osl ll JI ,(1uopue pasolc*f1Euo-r1s-o sl las xe^uoc B leql IJEJ e sl lI .G.V'Z 'dor4 'gc)JrsnonuJluocluos Je^\ol f14eam-osr 'lzurrou Euraq 'Q ocurs pus sles p0punoq uo snonulluoc,(1Euor1s ,{ttulol sr uo1lec11dr11nu 'xaluoc € sr ccurs pasolJ*,(1Euor1s-o sl X las aql 'las pepunoq-r.uJou
'll"ll> > ll(z)0a;; {ll(r)0r,11 ll'll '0N t z\ = y les eql l8ql esrlou : r = t p ( t ) ! f t n q t p u e 0 1 I W , q t ( i ( l r l e r a u a EJ o s s o l o u q l 1 , n ) c r u n s s e'e1rurgu1 si 0 JI
'0ux,4v= 0u'Qv"r0,v = Ou1r)jo OJUIS
'Ou"(r{v){= tp tr!)$o(t)l =Qu$(h'o) = (x(!)ro)Qu t = 0N
'uoql acurs) I E t ^ r J l s r u o r l r o s s €l s J r J o q l ' o l l u l J
allq,\{ 'Utl s l O e s B Ju I
'(x10u1,0-y14 =gg)ro)Qa pueOry t rU)Qo= /6Ory ) x :uoruessy ',{14eaalpo}orclrolur Euraq lerEalul eql
tp ?F(t)I
t''o'1= tgoly
IBI{I pue '0g uo rolurado papunoq peulJep . , { 1 1 n . ; S u r u e aeu s l . ( r ? v ) / ( g t e p a u r g c p l o u s r / q S n o q t ) l e r l l s i ' r o l l o J 'elqrlra^ur sr 9y octrs llp,,_t(l)"/ ll | = (r)/ l(q (-'0) uo peurJep uollounJ snonurluoc pepunoq aqt s1'/ u"qi '(U/)rz r ./ Jt 'Joord (fu ,ttt^ g to drrotS pnp aLfi Sutttvuapl a r o a t u . ' € ' V L t o l l r . a sl o p u a a t i l t D s o ' o s l o ' . Q g t o t o t a d o r u 1 o { p o - l p s art|tsod ?qt Io untlcads aqt 'astnoz lo 'sa\ouap 0y ds pqw,$ atla) '(-'0) u 9y ds = ,o ds uatp'141 tto tt13pu suJ , sl O /7 'r't€ uuue-I eql uI dn ue4e1 s1
,o
'ErurualEur,rro1lo3 cls Eururuexa Jo dals lxau lernleu grll
'Z' n't pue I't'€ s€ruruo'I J p u e I ' I ' g r u a r o o r l J g o a c u a n b e s u o JB s r u o l l r o s s p e q l ' I I I e d , { l g 9 s r 7 . 9 'n uo e}eJl s suJ sr r JI {I} = ,o ds lsrlt lJeJ snor^qo eql puB (q) JI 'allulJlrues sr 'Joord 6'9'9 f:e11oroJ ruorJ s^\olloJ uo1lr5ssuaql /.t/ JI
'{1t1tro 1t18pusuJ r, :ro ds} v = (n)l 0 (n)l p suolldycseq o^rl€urollv 't'E
601
3. The Connes Classification of Type III Factors
ll0
[ =
,r=t* nlo
fr,, with lfr, = lf for
n > 0, and
xtn= 1l
(the conjugate Hilbert space) for n < 0.) Thus K is a o-weakly closed convex set which contains of(x) for all l; since f ) 0 and ) lQ)at = l, conclude that
v=!I@o!{;oatex: observe that for arbitrary z in
NO,
= iQ*y)
= I le)."*o!G),o>dt = I /tr).4'6no(x),nok)>dt = .i1a-4f)nq(x), n6k)>, the validity of the assertion. thus establishing Hence,iffeLr(R), oQ(n =o I
oO(flx = o
vx in
^Jo
= o vx in No <+l1a-4|ln6(x)
<+i1a-l)= o €* / vanisheson (so a;rrn mp (9 f vanisheson (sp AOn lRl), following from the equation JOAQJ6= n:rt. Set the last equivalen^ce I = {f e Ir(lR): av(f) = 0} and E = SP AO n,Rl; thb,above statement t r a n s l a t e si n t o / = I ( E ) , a n d h e n c e , s p o e ' = P = I ( E ) - = E , a s d e s i r e d .
Thcoren
3-45. If M is a factor, then r(ln, = RI. n (n(tp aE: Q a fns weight on luI)).
Proof. This is an immediate consequenceof Lemmas 3.4.3 and 3.44. F o l l o w i n g C o n n e s ,l e t u s i n t r o d u c e t h e n o t a t i o n S(M) = n{sp A0 Q a fns weight on lul). Then S(I4) is clearly an isomorphism-invariant;further, if DI is a factor, it follows, from Theorems 3.3.7 and 3.4.5, that S(14 is a
) a I 0 /I
'+'"nlQ- "o ttl'eN)
'yr1 lo1co{ o uo ltl?tau suJ o aq Q ta1
2-g-g uogrl$dor4
d.
'7,1uo lq8ra/rr suJ paxrJ suo fue Jo srural ur sl ll leql (g't'€ rueroorlJ ,(q ualrE leql rano) a3uluc^pe orll seq lsql Utt)l lo uogldlrcsep reqloue rllli( uollcas slql epnlcuoc IIEqs a/,1 1 - ' 6 1r o ^ ( Z r u : , , \ ) n ( 0 ) ' ( I ' 0 ) s l O t l ) ^ Ss e E u r p r o c c u tIII ro (t > r > 0) \ttI '01f1 ad,(1lo rl lt, rolceJ e 'g'v'€, uorllsodo:d g'p'E lueroor4J Jo ecuanbasuoc B sE ,leql osle aclloN pue
,{lrrrssacau sr 0v uor{r,rojfinlo rlttt}iT';'rt"#
::';:ii'::X1: ore ,9v puu 9y eculg llnsar e^oqe aql 'luslcnrnbai(1r-rellun-lluB 'alrurJlruas q n'g'l'E rueJoeqldq'ecuaq n pue':auur sl 60 iholJoql'spto,nJeqlo.ulU ul I pue g41Qyur r IIB JoJ lErll qrns (hl)vUur.{'n} dnort frelrun raleuered-auo |nx'n = H|_axH!'Ja snonulluoc ,{ltuorts E lsrxe lsntu araql 'suolllpuoc esaql flesgcord rapun leql sel?ls-- serqe8leuuerunoNuol suorle^rJep tururacuoc Jo sIJBJeruosuo sarlar ;oord asoq,n-- ([leS] 'gc) re>1eg ol onp llnsor V uJ ut r IIe roJ @)Qu = 11t.-e(14)Qu"r,a tuql qonsg;1u^oJoleradopapunoq A ^ " * r L P X S o l : U ' o S ' ( L ' 0 ) u r r a l u o sr o 3 ( r _ r ' r )5 9 y d s l c q l a p n l o u o r 'v frYvvf = Yv aJuls v9 ds / 0 teql qcnsy'{ uo Q lqEramsuJ B slsrxe araql l€ql sl (lll) uortdunsst oql '(l{)S ;o uoltyul3ap,{g :(l) e (lg) 'IIO:(III)€(II) rfi1 ry '{t} f,v = = (ru)S cls os pue ueqt 5 'W uo ac€rl suJ € sl 1' gr 'i(1es:e,ruo3'(n)S > I l€rtl g't.€ r,ueroaql ruorJ s^\olloJ ll '(t'€'€ 'qI 'Jc) (/t/)l r I ecurg :(rr) 6 (r) .Joord 'U/'t)s
(lll)
/ o '{r) = Un)s (l)
ia1luttnaas s1y,t1 (l)
atv
W
to7ct{
D tto suo!1!puo? Euruol1ot
Jualottnba '9.V.€, uolllsodor4
aqJ
'elluJJlulas sl JrtluoqiY\,(losrcard acuereJJlp ou sr aJer{l leql sr uolllsodord Eur,nollog or{l Jo lueluoc aql '0 reqrunu aql l(q lsour l€ raJJIp snql uec (f,V)Spue (79)1 stos aq1 'rolcuJ B sl ttl Jl
.IUl
u (tt)s = Uu)J
:scruoceqg't'€ r,ueroeql '0{)S Jo sural uI 'n uo 0 t q 8 l a , v ' s u 3 , ( u e r o J C Z V ) ^ S .rual q u n u e r r l l l s o d { u e { q u o l l e c r l d r t l n r u ropun luerJelur lJel sl 9v ds leql r.Icns (-'Ol Jo losqns pasolc
(.rt)r ;o suoJldJrcsaq e^lleurotlv
III
'v'E
rt2 (a)
ibi
3.
T h e C o n n e sC l a s s i f i c a t i o n o f T y p e I I I F a c t o r s
Q" is a f ns weight on M"i
;in
= Ri n dtp Lq":6 * e e P(z@qD: in particutar,if MQ is
a factor,thenr(nfi = nl n sPAO Proof. (a) 11 is clear that 0" is a faithful and normal weight on M.. Since e e Mv, it follows from Theorem 2.5.14r that eD6e c Dpi so
D6"2 eD6e. of a monotonenet {x;} the existence The semifiniteness of 0 ensures in D6 such that xi .t' I (cf. Ex. (2.4.8));then {ex,e) is a monotone net i n D i " w h i c h c o n v e r g e sw e a k l y t o e , t h e i d e n t i t y o f M " i c o n s e q u e n t l y 0 -. i s s e m i f i n i t e . ( b ) I n v i e y o f P . r o p .3 . 3 . 3 a n d L e m m a 3 . 4 . 4 , i t w o u l d s u f f i c e t o show that 1o$" = o9" for non-zeroe in P(Z(MAD. Since tO" g tO, it is trivial to^verify that 0e satisfiesthe KMS condition with respect n t o t h e f l o w ( o P ) " ,a n d t h e c d n c l u s i o n f o l l o w s . Corollary
3-4-E. If M is a factor, then S(tt4) = n{sp aO": 0 * e e
PQ@\)),
for any fns weight 0 on M,with 0eas in Proposition 3.4.7. Proof. Case (i): M is of type III. In this case, 0 € S(M), by Prop. 3.4.6. lf 0 t e e p(z(aQ)), since M is of type III, there exists an isometry u in l+[ such that u*u = | and uu+ - e: the map x - ttxtr+ is a von Neumann algebra isomorphism of M onto M"and hence M" is also a factor of type III; s o , b y P r o p . 3 . 4 . 6a n d P r o p . 3 . 4 . 7( a ) , 0 e s p 4 6 " . Case (ii): M is semifinite. In this cage, 0 | S(luI),by Prop. 3.4.6. We must exhibit a non-zero e i n ? ( Z ( M \ ) s u c h t h a t 0 I s p A O " ,o t , e q u i v a l e n t l y , s u c h t h a t A g " i s bounded. Let r be a fns trace ot M. So, by Theorem 2.6.3, there exists an invertible positive self-adjoint operator H n M such that 0 = r(H.).Pick e > 0 such that e = l6,yr1(H) 10. We know -- by for x in M and I in ft thus x Theqrem 3.1.10 -- ttrut of{r) = 11it*11-it', e uQ if and only if x^ iommutes with lB(H) for all Borel sets E; in -'e particular, e e P(Z(Mv)). It follows from ee ( l/e ( e that
"9 "0 = zl1*;o u:frv"oril tt:fivll* N u N' x zl1co'o ' 3 C u e qp u B
'(r*xlPr-r (x*r)lr_ r
=
(*r(x)lr-r ) (*xx)g +"n
>x
(n)l p suolldlrcsoqo^ll€urallv 'r't
€II
Cha p t e r 4 CROSSED-PRODUCTS
T h e c r o s s e d - p r o d u c tc o n s t r u c t i o n w a s f i r s t e m p l o y e d b y M u r r a y a n d von Neumann to exhibit examples of factors of types I, II and III. The set-up is as follows: one starts with a dynamical system (M,G,q.) -- with G not neceSsarily abelian -- and constructs an associated von Neumgpn algebra M (usually denoted by M oo G) on a larger Hilbert space Xf. Section 4.1 discussesthis construction when G is a countable discrete group, and develops some of the features of the crol;sed p r o d u c t ; f o r i n s t a n c e , a n e c e s s a r ya n d s u f f i c i e n t c o n d i t i o n f o r M t o be a factor, is given in terms of the action a. In Section 4.2, we assume that M is sgmifinite and use a fns trace on M to construct a fns weight 0 on M, whose associated modular operator is explicitly conlputed; this description is used to compute the invariant .S(14),when M is a factor. S e c t i o n 4 . 3 i s d e v o t e d t o t h e c o n s t r u c t i o n o f e x a m p l e so f f a c t o r s o f , II* III\ (0 < \ < l)' Practically all these a l l t h e t y p e s : I , , , I * I I t^crossed-p?oduct of L-(X,T,p1 by an ergodic examplej arise"as the g r o u p o f a u t o m o r p h i s m s ;t h e c o n s t r u c t i o n o f f a c t o r s o f t y p e I I I l ' \ e ic groups of automorphisms [ 0 , 1 1 ,r e q u i r e s t h e c o n s t r u c t i o n o f e r g o"dr a tio sets" in the sense of o f a m e a s u r e s p a c e , w i t h s p e c i fi e d Krieger. S e c t i o n 4 . 4 t a k e s u p t h e c o n s t r u c t i o n o f t h e c r o s s e d - p r o d u c t ,w h e 3 G i s a g e n e r a l ( n o t n e c e s s a r i l yd i s c r e t e ) l o c a l l y c o m p a c t g r o u p . f f A = M @qG, with G locally compact and abelian, an action d of I on lf is conitructed. The main reSult of this section is Takesaki's duality theorem which states that M @d t is naturally isomorphic to l4 @ r(t2(G)). This is a genuine dualiiy theorem if it is the case that M = It is shown that such is the case for a fairly large M @ f&zGD. ( t h e so-called properly inf inite) von Neumann algebras, class of which includes all infinite factors. Section 4.5 applies the results of Seption4.4 to the casewhen M is a factor of type III, G = lR and cr = o9, where 0 is a fns weight on ly'.
('(e) asn 's1ql ro3 lecuo8reluocEuorlsazr,ord o l ' ( q ) , ( q ' s a c r 3 3 nrsl t l l 4 l l ' l l t l l ) x e u r > l l u n s ; e r l r e d ( o r r u r 3 ) ,(ue;;ccurs:lulH) '*f1Euoi1$'-o Bu'iEranuoc tqSrr aqt uo s3rrasoql ' (t'tt) {(rt,s)xct" = 1l.s;z '9 ) t's uaql'tX = pue (r{h ) A Z Z,{,X JI (c) 19 r 1's4 *(s'l)I = (/'s)*{ uSql .(Ah , { JI (q) i I ur rurou ur EurEraauoc tqEJraq1uo"i;arreseq1
Dit = '(r)l(r's)t (rXlx) 'o ut s ,o, urui
" r I pue(a)r r I .lI (e) (r'r'l) sasrJraxl
,l ur U'l IIB roJ <1e;t{'11;1I> = < & ' : ( / ' s ) { > E u r f g s l l e s r o l eredo papunoq anbrun oqt s1 (l,s)1 u o { 'g ur. puE s qcee JoJ J'areq,n -- g ,(q pexepur suunloJ pu€ s1r\or t r{ll,!r -- (((l's)E)) xrrlBu enbrun e ,{q poluese:der sr (A)f ul { ^uV t { r o J s r s € q l e - t u r o u o r l u ou e s r { 9 r y . s : 1 , ! l } . r e y n f r l r e d u l ! 9 u r I puu g.trr ! 11ero3 rl o--l -- (r)l {qaroq,n tcl)-i-o $ = u uorlcrrJlluepr 'l'"9 = .g tfl (i')f,,,1 errrm IBrnlBu B sr arer4l , pue $ ur I 11uqde,n rot '- > d?ru € sl ,t Jo luauota uV ., IIB a l l ( r ) t l l 3l e y l - q o n s i l ; ' b ; l -> roJ rf = rg'5re[lh 'r$cere = arunss€ller{s eA{ rrt put (A)f nier$ I '[?tj ul uolltluosarder .ru1-nEar (-tqElr ..ctser)-lJel eql ,rou.O 9 Jo''dsar) r\ / / pue (t"g = (s)tl [.q peulgap) (g)rt to1 srseq luturouoquo ,D I B c r u o u B co q l o l o u a p ( g t 1 : , 1 ) Jo luouelo {1r1uepr eql elouap r tel IIBqs d^ '7,'(, uollcas Jo pue oql le sv .1,t1etqeEp uu€unoN uol e uo g dnoJE (uerlaqe {lrressoceu lou) aleJcsrp olq€lunoJ E Jo uorlc? ue sr rc l€ql arunsse II€rIs e^\ .uorlJes slql lnoqEno:q1 SlJnpord-ptssorJ cltrtsrq'I't'
Jo {'g} dnorE raleuerud-euo e fq 1g erqaEle uu?uneN uo^ alrurJlruas € Jo lcnpord-pessorc eql s€ alqrssardxa ,[lanbrun i(1yu11uossa,, sI adf rolceJ ? eql uI ,,Je!loq alq€uos€ar,, t l8ql llnsar III Jo 'padoq sl sgcJg^{.uerrrEa:e sluaun8re 3 qll,r\ rep€eJ orll e^Bal 11raa, ll cllslrneq euos 'peelsul isgoord ,reJ r(raa sulBluoJ uollJes sIqI
9II
s l c n p o r d - p e s s oor lce r c s r q ' I . y
4. Crossed-Products
I 16 Definition 4.1.2. With the above notation, define
irt= tx e r(i): I(s,/) e M andl(s,t) = 1-r(i(st-1,e)) for all ,,, ,n G). of M with G (by cr)and also The set ff is called the crossed-product by M @oG. O denotedsometimes Note that if 7 e ft and s,t,u e G, then i(su,tu) = cr,r_Ji(s,t)). It is easy to see that f4 !s weakly closed (in [(il);; it is not mugh harder io verify that M is a self-adjoint subalgebra of f(Xf) contlriningl, and consequently A von Neumannalgebraof operators on ii. (For instance,if 7,,! e M and V = 77, it follows from Ex' (4.1.1)(c) and the fact that M is weakly closed, that V(s,t) e M: further, z(s,t) = I i(s,u)I(u,t) = Ii(s,u)crr-r(I(ul-1,e)) = = E r(s,v/)cr,-17(v,e)
!
ct-r(i(sr-1'u)/(v,e))
= cr-.,(Z(sl-r,e)).) t '
Define n: M - ir Uy tne prescription (n(x))(s,r) = 6rrcrr-1(x). (When there is more than one action floating around, we shall sometimes *-algebra write n- for z.) It is rJpadily verified that tr is a normal isoggorpiism of M into M. Hence n(I4) is a von IJeumann subalgebra = of M: in fact it is precisely the set of those 7 in M for which i(s,r) 0whens#t. N e g t , d e f i g e \ : G ' f ( l f ) b y l e t t i n g ( \ ( u ) ) ( s , l ) = 6 r , u , ,o r e q u i v a l e n t l y , (r(,rixr) = \1u'rt1. It is clegr that \ is a unitary iepresentation of G in'f; i" the identifigation k = tt 6 !2(G), r(u) = I @ ru' It is easilv verified that I(G) g M arLd that (4.1.1)
I(ri)n(x)\(u)* = n(cr.,(x))
for all u in G and x in M. It follows from the above discussion that (n(M) u XG))' c M; the f o l l o w i n g e x e r c i s e so u t l i n e a p r o o f o f t h e r e v e r s e i n c l u s i o n .
'H > tl'((rl)r-'1o)l = (rlXl{n) uollenba 3ql leql parJrra^ flrsee sI lI
'Qr\tt'(ur9o'9no) = (tt'tDr-(>t'o u)
(*)
lcql slee^eruollelnduoc fsee ue '.(zrtrr!'(c rl)r4or tt) = ( tt,8rl)(r t!) 4,r fq uanrEsr uo11ecr1d1t1nru-dnorE aroq,n 'y x .F/sl 1asturi(lrepun osoq^r9 dnorE oql sI 'X o@H {q olouop l1eqs o/h I{cIq/$ 'lcnpoJd lcoJrpruraseql l€ql l1ecar 'usrqdrououroq e sr G/)lnV - ) : rc pup sdnorEelercsrpalq€lunoJ ere X pue H JI (q) '@)*ll erqeEle uu8unsN uo,r dnorE oql lsnf sl "(gx = n ese)srql ur .snqa .n\ qllan palJrlueprsraE(n)1 = (t\E ,(Cf,r Vl1,ttpelJlluaprfJlurnluu ,pue II sr fi uar{I ', II€ JoJ,,p! ='o 1a1'dnorEeloJcslpalq€lunocfue rog ? = .fr snql l(A)f = n pue leuorsueurp-auooq fi lo-I (€) .y.1.9sa1
cl" .1r;111n;x;u s1oy,1 {lrualerrnba ro) sluuoEuJo H:'.tllri,XrT",*?:l'j:rt:: :H
qclf,y. n q X asogl Jo les eql sl 0// JI '[1l1cJ1clxa aroru i,,((9)1 n Q;q)u)-=,2Vr€lncllr8ct ur i(q) ur sf pelerctrolurturoq uns oql '(n)1(((r'n)gfo)u"i'
=f
ler{l ^\orls 'n ) X Jl (c) 'srunselrurJ Jo lou eql Jo l1tu11*tuorls-6 oq1se palordralurEuraqlqErr oql uo serJeseql (r[c,"3 = r l€Hl ,no{S
'nt r-rsJI n = ,_ls J[
'0 '(l's)I
I = (r'sx(n){) I J
fq (t)r ) (n)X eulJap 'D ) n pue (nh r { JI (q) ('pelou uaaq ,ipuCile seq slql ') = n ueq1\) ',;igtr x auros rog (n)1(r)u = { JI [1uo pue g1 leuoturp qln,oql uo palroddns sI I l"ql A\oqs 'n * vls ra^aueq^\6 = (t's){ JI .g ul n auos rog 'leuoEelp qfr? aql uo palroddns st I l?ql 6,es,.7t9, I lo.I (e)
(e'r'r) sssrcJcxfl slcnpord-passorc el0rcsro'I't
Ltl
4. Crossed-Products
ll8
I e f(m def ines a unitary representation/< - u, of K in l21H;;if r, denotesthe left-regular representationof 1{, it is easy to check that for k in K and h in H. It follows that there aulr(lr)af = r" (cr1(/r)) eiists an action d. of K on W(H) such that d1(x) = u*xuf f or all x in W(m and k in K. 'The W(m od K acts on the Hilbert spacetf = lz(K: crossed-product !2(m) which can be natural$ identified with n21Flx K) = t2161. Under this identification, ( z6( l"(ft o))I)(h,k) = E(crk-1(&6r)/,,k) and
= i6, *-orrr'; (r(ko);x&,k) for ho e H, ko e K and i in l2(c). lf ri denote! the (clearly unitary) operatoron !2(G) defined by
(rixa,t) = f(
= iq:o'n,4 ()r/ra( rH(ft0))w*ixlr,t) and
(h,k)= l(cr,-r(/r),tortl; 1wr1&o)w*i) ko-
in view of equation (*), this says that wz6(\H(fts))w* = rc(lto,€x) and w\(ko)w* = \o(er, ks); thus w(W(H) @a K)w* = W(H oo K).
E
Remark 4.1.5. Although the construction of the crossed-product seems to depend upon the Hilbert space lf on which M acts, it is a f act -- which we shall prove in Section 4.4, when dealing wi{r that the isomorphism class of M continuous crossed-products depends only on the isomorphism class of the dynamical system (M,G,a,);explicitly, if (Mr,G,a), i = 1,2, are dynamical systems and if fl; M,' M, is a von Neumann algebra isomorphismsuch that tr o or,t = d2,r o z for all /, then
ft tluo puo Il aart st D uottlo aLlJ 6'1'g [ru11oro3 'a^oqc (e) ur se 'aorJ sr ln ursrqdrourolnu E aql 'r I I roJ JJ aarJ oq ol plBs sr ,,ryuo I Jo )c uollJB uv (q) '0 = r salldrur n u\ K Il€ roJ x({)g = {x 'g'I't uoIfIuIJeC JI aarJ aq ol ples sl .ZVJo g tusrqdrouotneuV (e) [
'oleldruoJs1 '(sr_n= I Eulltnd uo) .yoordaql pue
I ul ,A
G
't)X = ((t 'r-nn){)t-"o (+
g ul sA (r 'sr-n)x = ((r 'r-,?s)I)r-"o<+ g ul sA (r 'sr-n)E= (/,'s)I <+ 's)(1(tz)1) = (r 's)((n)19)q9 9 ut sA (r {(n)1 = (r)1x ueql 'D ) n n'I (q) 's/r\olloJuolilOssseql pu€ 'g ul ,A 9 ul tA
(e 'l)1(d)I-rn= /q) 'DX e = (r'r)((f)ug) e X(Ou = (t)uX (>'r(XQ()u) uaql
'n ) t le'I (€) 'Joord
'g u! n't yo toS ((r'l)g)"n = ({ywn)X <+ ,(g)f r I ( t ' t ) y ( t ) f - n= , { ( r ' 1 ) I ( * r ( n f u ,
igult pupnultllotol
'n t 'l'I't x 147
X
(q) (e)
"uutI
'JolcEJB s\ n IBIII ernsuaqJlg/r.(n'?'n) uralsfs luclueudp eql uo suolllpuoc ao uo$sncslpB ol lxeu urnl eA,\ el^ ssauallulJlrues aql osn 0 D ('(p) Q'7d'xg Jo 'ssaualrurgruras 'n roJ :fulH) uo suJ B sl tqEle,n 0 l€ql ^\oqs t-ry ut I ro3 ((r'r)f)9 = (XDtuJep'n uo 1qE1am suf fue sl O JI (q) '$t1lu e4e?IuqnsuuuruneNuo^ oql oluo 'InJqlleJ B sl g teql e^ord (e) W Jo uoq)aford euo-ruroulcurrou '((r'r)g)z = XZ 8q (4)u - W :Z elurJae (S't't) srslrJexg 'g'oo'I,tt
= gtort^
s l c n p o r d - p 0 s s o JeJl a J c s r c l ' l ' ,
6II
r20
4. Crossed-Products
f,tnn4!W)=IJZ(M)). Proof. Supposethe action cr is free. Then, it follows from Lemma 4.1.7(a) that 7 e MnnJM)' )f(/,e) =0 for t * e )7 = fl(i(e,e))e n(A. Since zo is l-1, tle assumptionI e nJIr|.' forces i(e,e) to belong to z(M), and so, M n not(M)t c not(z(lut); the other inclusion is obvious. If conversely,there exists , * e such that cr, is not free, then (by definition) there exists a Uon-zerox.in M such that x/ = 0, =lF e I such that F C E, y(F) > 0 and F nTF = Q. Proof of Assertion. € : lf lg = t(6)f for all g in. M, it follows th€t if f = (r,re X: l(w) l0), then ipg = lp.(S o fl; for all g in L-(X,T,1I)-- as usual, equality meansp - a.e. equality. lf f * 0 and p(E) > 0, and if F is as in the assertion,the above equality is violatedif g = lt; this contradictionestablishes that crTis f-ree. ) : If we can find Fo in I such that Fo ! E and (Fo\?*'(Fo)) r 0, the set F = Fo\fr(Fo)-does the needfutl if no such iet Fo-exists, argue that it must be the casethat u(Q@) n t)AF) = 0 wheneverF e T, F tE and r(F) > 0; hence|rf = ls.(f o T-r) a.e.,whenever f = lr with F as above;concludethat the above equation persistsfor all f in M, therebycontradictingthe assumptionthat aT is free. O
'eroq llnseJ Eur,trollo; oql el€ls oarr'1r a1uco1 ol aceld Jalleq e Jo lue/tr JoJ pue 'aouerc3er Jo acuarueluoc JoJ
'{lelerpetuturs^\olloJ(g) pue (r) eql D Jo oJuole,rrnbo :G AtilZfu = @)Z leql s^\ollo311'ree1cs! uolsnlcul Jeqlo oql acurs
'g vr. n fue ro3 '(n)Z ut r auos .rog (x)Du = ,c lerll /r\ou aclloN l€rll .(n)Z t .JIOslI oluo 6'1'y frelloroJ ruorJ s^\olloJ lI X pl e;,9)7 dew 'JooJd illz'l.n 3o usrqdrourolne {ue aculs-reo1csJ uollJasse lsJU aql 'UvDZ zo g uo lo uoltco ctpo7ta uo sl ltoTco! o s! DD@n
(ll) (l)
:1ua1ot1nba an suoltlpuoc 3utuo11o/ atil 'uallJ 'Ott)Z uo g lo (uo11cu1sat tq) uotrco paznput aqt atouap zn q i no ro{ (lDZ = (jrDZ)rp uaqJ 'n uo 9 to uollco aa$ o sr D asoddng 'gI'I't uolllsodor4 'flrcrpoEra O Jo uollou IeclssBIJeql sl ler{l -- 0 = (AW)z ro Q = (g)d serlctrul9 ul r IIB roJ 0 = (g v (g)rl)tt'1. t g 9 clpoEro sr D uollc€ eql leql (itl op) f3rran ol pr?q 1ou 'uaq1 's1 lI ('uorqsBJ slr{l ul paul€lqo sl J / uo ,) Jo uollc€ {-r3^e uoql 'alrurJ-o pu€ alqur€clossr (t'J?) JI teqt -- rarllrnJ ensrnd lou IIr^\ e^{ qclq^\ -- lo€J E sl lI) 'r\ o / = (f)rp dq uo,rrE (Tl'1.'X)-1 = JA! Lro g Jo rc uollc€ pacnpur u€ a^sq uaql o^\ i(tl'J'X);o surslqdrouoln€ go dnorE eql olul g tuorJ usrqd.rouroruoq € sl 'J - | osoddns 'i11l't elcluura
.bn Jl clpotre 3q ol plBs sl lg uo I Jo )c uollre uV o))zt <+ (tt't'l
' { or r : I ' \ ) =
'el'fi
uoIUuIJaC
.0=({o=$J:X) n eldurexg Jo uollelou eqt qt1,r) eerJ s! Jb teqt ^\ogs -rr IIE roJ
(urq)usl = 1Io;"41 t)
zo = ro
.; r ao'ro fue rog l€gl r{cns .* ul {"^?} ecuenbes e slslxe areql JI "a'1 -- paleredes alqelunoc sr (J?) aJBClslarog erll JI Qt't'V) seslcrexg
slcnpord-passore JleJcsrc'l't
tzl
4. Crossed-Products
r22
Proposition 4.1.16. Let A be an automorphism of a factor M; the following conditions on 0 are equivalent: (i) (ii)
Q is free; 0 is outer -- i.e., there does not exist a u in uxut for all x in M.
U(M) such that Q(x) =
Proof. (i) ) (ii): If 0 is inner, i.e., if there exists a in U(M) such that 0(y) = ttytt* for all y in M, then uy = Q(y)u for all y in M, whence 0 is not free. (ii)+(i): Suppose x e M a n d x s a t i s f i e sx y = 6 ( y ) x f o r a l l y i n M ; then y*x* = x*O(y*) for all y in M; so, if v e U(lvI), note that v*x*xv = x*0(v*v)x = x*x, and 0(r)xx+O(v)* = xvv*)c* = xx*i since v e U(kO was arbitrary, conclude (via Scholium 8.4,t8)that x*x, xx* e Z(lrO. Since M is a factor, it follows that x = llxllu, for some u in U(luj; it x * 0, infer that for any y in M, uy = Q(y)u, or 0(y) = u!u*, n c o n t r a d i c t i n g t h e a s s u m p t i o nt h a t 0 i s n o t i n n e r .
42
Tbic Modular
Opcrator
for a Discrctc
Crosscd-Product
Throughgut this section, we shall let (M,G,a) be a discreg dynamical system, M denole the crossed-product M @d G, and let 0 denote the f n s w e i g h t o r M J c f . E x . ( 4 . 1 . 6 )( b ) ) i n d u c e d b y a f n s w e i g h t Q o n M b y t h e e q u a t i o n 0 ( I ) = Q ( i ( e, e) ) . I n t h i s s e c t i o n , w e s h a l l a t t e m p t t o "modular operators S, F, ,I, A" determine the GNU space and the associated with (M,0) in terms of the corresponding objects for (M,0). "standard with I n v i e w o f R e m a r k 4 . 1 . 5 ,w e s h a l l a s s u m e t h a t M i s = n 6 ( x ) x for all x jn a n d M C f ; Q q , U = U A i.e.,that respectto 0" "standard with respect to 0", M. We shall see that in this case M is b y e x p l i c i t l y c o n s t r u c t i n g n 6 i f o r n o t a t i o n a l c o n v e n i e n c e ,w e s h a l l respectively; likewise, we shall write n and tI in plage.pf ,ltO and 06, 'of MO D6, NO, MO and Dp, N write D, N, l'1 and D, N, J't instead O, respectively. W e s h a l l b e g i n t h e p r o c e e d i n g sb y a t e c h n i c a l l e m m a w h i c h s t a t e s t h a t t h e l i n e a r m a p n : N - 1 3i s c l o s e d i n a c e r t a i n s e n s e . Lcmmr 421. Il {xrl is a net in N such tftar sup,llx,ll . - , "i ' o - s t r o n g l ya* n d n ( x ) - l i n ! 1 , L h e n x e N a n d 1 = 1 7 1 r " 1 .
"
Proof. Pas.gto..a .subnet (in fact of the form (x,: t > to)) and assume tnat lln(x,)l[...lllll + t for all i. Then there exists a donstant C > 0
s ,s i n t h e s u c t r ' i t t a t ' ' l l " i, l lc a n d l l n t " , l l l( C f o r a l l i ; i t , f o l l o - w a proof of Lemma3.4.4,thatthe setK = {y. rui llyll < C, lln(y)ll< Cl x e N. is o-strongly* closedandconsequently For any y in
N*, notice that
fq paur;ap uollcatord aqt sl N4 aJoq^r .N4 { = *{ ur,ll ! ( l ; t N r ' " ' ' I r 1I ( r . s ) r = ( l . s ) N { uorlenba eqt ,{q (A)f ul Ng eur3ep .... .Z.l = N roC 'g Jo uolleror,unua ue aq {"' 'ot"t) t{l pu€ 'g ur s uB xlJ .slql op oI -([ gU = ({)AX .3oo:d aql elalclruoJ u3ql t r( pue oJ X JI l€rll ^roqslsnu am U 4,
'(x*{N = <({)u'(x)u> '[ .uorlezrrelod q ,smo11o3 u l l ( 4'r fue JoJ lur{l '(*) .g ll l({*{)0 5,req a , n u o r l e n b o p u u uorlrul3op 3 o : zll(f)A;; ^q:ruaurl AIrEelJsl qcrq^\.Jl * deru B aurJep s o op ((r.s)5)tr N : g = (sx(I)lr) uoltenba aql leql Sbrnsfo (*) uorrenb" evl r { JI (q) !
i(r,rn)(t(t)Xtn) .I, * (i t(I)Xs) (bvEx.(4.1.1) (a)).
Concludefrom Lemma 4.2.1that (; t(J)Xs) = n((* 7)(s,e)) = (t(t 7)Xs); sinces was arbitrary, the proof is complete.
O
Assumefor the rest of this section that M is semifinite and that 0 is a fns trace on M. (We have chosennot to denote the trage by r for the following reasons:(a) if ? is the induced weight on M, then 7 need not be a trace, and this could be confusing; (b) we can continue to use the notation establishedso far in this section.) Recall that n always meansn6 and ?l meansn[. Irmna 42-3. Let T be anotherfns trace on M. There existsa uniqueinvertiblepositiveself-adjointoperatorHOM suchthat t = Q(H,.),as in Theorem2.6.3; (b) H n z(IO: n( N o h rurl is a core for Htlz' i;i (d) if x,y e N6n N7, then
'papunoq sl .0C 'rN, eculs ll - ("tr)rr pue (il ("rf)u uop r I ,tNal(H)/ ao-llouu'l- ("r)u l?ql. qrns,, ul "r 4crd .s1qlrc1 ,.2111y * ;;r.1tql pue "|!Hlt { "1 teqr qcns (%)u 3 {"?i srsrxeeraql uaql .14eruosrog ! i{Na r : JI leql ^\oqs ol sacrgJns\ ,(H)I .roj eroc e st 4tat='h ocut5 (711H ruoP j (rN u il)u reqr s^\oqs(p) ro.l luaurnEreoql :aloN) r(3.iii uo / uollcunJ snonulluoc due rog '(H)I toJ aroc a sl (oC)u:(c) sorldur .1/y flreolc l€ql lueual€ls € aaord_^rou II€qs eA\ u N - 0g os -i 'ril i 0o'snqypopunoq sI uag pu;'0 t s" ua17-g'\a?11 ecurs t -
'*nulx pu€NrrJI to5$)g)qoDurll = (r)r r"Ur"'Ou'"t'O= 1 r II€ roJ (n)Z t 'H $rtl slou 'r-(lir + l)H = tH ,O < r :og .leql II?caU 'N 5 og teql realc sl ll -- n ul wepl p3prs-o^rl? sr N l € g l o s ' * N = N sa11dru1 0 Jo flrlercerl eql f11enlce - n ur I B e p r - u e l € s I N a J u r s 'N 'a ti" = oC aurJap pue'((lr[)z ry 1gyt"'i, ='a la-I (c) '(n)z u l/ ecyoq.a'l 11ero3 (n)z t 'rl l?tll opnltuor 'ocerl B sl ecurs.",_HrvH= r uaroaqa ruorJ ^\ou{ 3A\ (q) reql G)lo ol.l.g 'l Jo ssaulnJqu8J3rll go acuanbasuoc? Euroqg go [1r11qr1ra^ur eql ,6.9.guraroeql uorJ 9Zl
lcnpoJd-pessorJ eleJcsrc B roJ roluradg rslnportl aqJ
.2.?
r26
4. Crossed-Products
n(x)ll2= r(x*x) < -, supllpt/2",,
since x e ffrl
= r(x*x). it followsthat n(x) e domHLlz and tnat ll?r/2n(")ll2
D
Coming back to the dynamical system(M,G,a,),it is clear that the property of being a fns trace is inherited from 0 by each 0 o cr,; for each t in G, let H, be the positive invertible self-adjoint operator qr = Q(,Ht). Let dfiligted tg Z(n (as in Lemma 4.2.3)such that 0 o 0, N, If" lt , n be as in Proposition4.2.2. Lcmma 424. Let X e irt. Then
-' (a) ; e il ++I(r,e)e lrt vr,andL lln(r(r,e))112. (b)
; e i*
(9 i(r, e) e Npoo, V/, and
< '; €))ll2 r lln6oo,(i(t, (c) ;e F nff<+i(t,e)e lrln A/6oorV i n,G , a n d
.))ll2). - . r (lln(r(r,e))ll2+ llnlt2n{o;(t, t Proof. (a) This is just Ptop. 4.2.2 (a). ( b ) S i n c e I * ( t , e ) = f ( e , / ) * = c r r - t ( t ( r - 1 , e ) + c) ,o n c l u d e f r o m ( a ) t h a t 7 e ff <+ o ,(i(t-\,
e)*) e N
for all t
. andrlln(a,-r(r(rl,e ))*)ll2 (9 crr(i(t,e)) e N
for all t
. and E lln(cx,(t(r,e)))ll' = llnfu*)ll),which is just a (since0 a trace I l'/ = N* and lln9111 reformulationof (b). (c) This is an immediateconsequence of (a) and (b) of this lemma, and Lemma a.2.3 (c) (applied to T = Q o cr* for each , in G)'
D
Proposition 425. l{ith the notation established thus far in this sectiott, the modular operator A$ is given by A$ = orroHr; i.e.,
(- - e . -J d o m a f = t ( r n t ( r ) e d o m Yt 1 1and , rllg.irrlll' r ) = ana@$i)@ 4i(r) fori tn doma[. Proof. For each t in G, it is clear that the equation r, = Q@2r.) (Faithf ulness follows f rom the def ines a fns trace on M.
'<({)u'(x)ttH> = (tH r.rop r ((r'l)1)u ecurs) <((r 'l){)tr '((r 'lX)trrH>3 = (@) e'Z'Veunual {q) <((t ,r)[)ur1lu ,(G ,t)X)uztlH>Z= (acerl € sr O aruls) ((r 'l)1*(r 'i[)rp o 0 3 = ( * ( r ' l ) 1 i ( r ' l ) { ) r po 0 3 = ( * ( r ' r - l ) 4 i ( r ' r - l ) { ) I - t oo 3 = 0 ._r ._r crr < ( ( * ( r ' r _ l X ) 'n- ) u ' ( ( * ( r ' r _ r ) 4 ) , - n ) u > - 3 = <((r 1)*1)rr'((r 'r)*4)rr>"it =.(*t)g
'(*{)u>
:s^\olloJs€ alncluoc pue { € rlcns a{el .oS '<(0{)u'G)u = l> uar{l '*N U N r ( JI leql aaord ol 'snql 'a,reqo,n '5' rog oror e sl (*N u N)u ac6rsfiH = (*t)ug leql pue J urop r (*I)tr-teqt qsrtqelse o1"luar5rt3ns oq lil? tl :ofi8 aqtralaldufoc o1 .,Sutrp , I r { l€rtl (c) V'Z'V€ruruo.IurorJ s^rolloJt.'ory r r eculS l€tll os '*Iu I : r o o 9 ' = ( . r n ) O n) O , t ) X ,(lluanbasuoc pue,,l17 urrrop r ((r'l)g)rr os irfin urop J tg r.rop r (l)l '(c) g.3.yrruuol ,i{'.1 qrce roJ 'g u\ | {utru f lalruJJlsoru le roJ 6 * (l!)f pu€ .g ul , IIB I, roJ ail,U r (t't)X'n ) ararl^\'(I)u = I snqt log r I lal'oS X I uCFf ter{l ,v.ot ^\oqs-ol '(c) 'xt ^q 'saclygnf lt l/7 rog 3v -i S'fZ eroc e sr og le(t 'enoqe ua.,rrE0g Jo uolldlJcsap puoJas aq1 put (q) g'S'Z 'xg, '(c) €'Z'V €rutuo.I tuorJ s/rrolloJ tl ,g.g.?,.xg ul se tfl e = H JI
'{l ,{ueru l(1a1ru1g lnq IIB roJ tO= ( r) J pu e ,A (rN u N )ur ( r U. I r 1 ) = o c
'{1arrr1eura11u l(c) (g't'l) 'xA ul su 0y4rqlrr'r , '{O ut /A t'rN t Q 't)y. 'ory r :(I)1a)= oC A I aulJeq (Urt)z u la uotJ s^\olloJlser aql pue)H go flrlrqrlrarrur
LZI
lcnpord-passorC olarJsrc E roJ roluraclg relnpol^l aqJ
'Z'V
4. Crossed-Products
128 and the proof is complete.
tr
We shall now head towards a "usable"description of S([z]. The setting -- for the rest of this section -- is as above: (M,G,a) is a discretedynamical system;M is gemifinite; 0 ig a fns trace on M and 0 the induced fns weight on M, given by 0(I) = 0(I(e,e)); M is "standardrelative to Q" (ioe.,M St(lt),n = tl6, n6= idy and n (= nd): lJ (= No) - lf) so that Mjs standardrelative'to 0 (by Prop. 4.2.2) w i t h t 1 : n f ) : N ( = N D - r g i v e n b y ( t ( t ) ) ( s ) = n ( i ( s , e ) )f;o r f i n G , 0 o a, = Q(H,.), wliere //. is a (uniquely determined).positive invertiLle self-a'djointoperatoraffiliated to Z(M): finally A=AX= @ H,. Y
t€G
We shall also assume henceforth-that the action cr is free and e r g o d i c , s o t h a t - - b y P r o p . 4 . 1 . 1 5- - M i s a f a c t o r . l*tnna-
126. 6 , l t
(a)
no(M) c MY:
(b) z(frr\ g nJzQuD. Proof. If I = n(x) e n(1u0,then by Prop. 4.2.5, of{;) = e Flit crr-r(x)t/,it= z(x), since11"n Z(M) for all s, and (a) is proved. tf i e z:6I\, then by (a) above, r e M n tto/.lut';the assumption that cr is free, together with Corollary 4.1.9,completesthe proof of
(b). n
L e m m t 4 . 2 . 7L. e t 0 t e
e P Q ( 1 4 ) ) a n d d= n ( e ) ( s o t h a t 0 t d
P 64\ by Lemmaa.2.6(a)); (a)
let 7 e ft; *en 7e M; €*(s, e)=crr-r(e)ei(s,e f o) r, a l l s i n G ;
(b)
i ol(M;) = M; for all t in lR:
(c)
if
K" = ttl(il nirt)l and p"= K., then F" =
"?"
(crr-t(e)e).
e
leql so^ord 9'9'Z uollrsodor4 Surrrord ul posn ouo oql ol relrturs lueunSre uu :13g'1g; ro3 alclrrl SND B sl ; e ( -e Nlg'-wp! "> )
lI
ogy=
"4 !
t e q l r E a t cs I l I '(n)Z tr , ," H cA =y
l€rJl A\ou ,(grre,r o1 ,{sea s1 t (a)r-ln 'a 'UI'I)Z r, tg eculs pue
"3t puu (a(a;r-ln; = "d aculs 'Joord
:Qy '") rH c)ra pro ot {o uoltolusat aw Wkt pa,{uuapr aq [Du 'Sgyju"g.uatlJ 'L'Z', DauaT u! so aq'9"> 'Z'a 'gZ:?uuiuiaa ta7 'perrsep sB '"d = ( ecueq la; ';Xur 3 d uet ler{l opnlcuoc E osuap sr (0n, v ry)g oculs :(c) (S't'l) 'xg ul se sr oy,g'1€nsn se 'a.r6q,n
'") 3 (onu ?, llg 3 (oryuN)u)g lcrll ((B) Eursn ure8e) ees o1 fseo sr lg 'pueq reqlo aql uO 'l uet j 'y acuaq pue '"; Jo lasqns asuap B ul rolJa^ frela soxrg 4 wrtt s^\oqs 'e,roq€ (e) qtlrrl raqlaEol 'g Jo uorlrurJap aW :WII)Z r a acurs) uollcaford € sI d leql o^rosqo puu
( r's)((1) )a(r)tro = (r's)((5)jo) fo r ir '.r)x eours os iQtg)7r ,iuvRue'(s) ,{q ."1"yr-L7g _ _ 8 (r 's)I ],ru t1a'1rn = (r 's)I rig =
's)r = ,'?r(r ,!a (r 's)(,r-I r *I) = (, 'sxft)do) 'Urrpue3^rIJI(q) '(n)z t-a lBql ocllou !9 u1 s4 a(r 's)g(a)r-"o= 1, 's)I <+ g u l s A ( r ' s X ? X A =( r ' s ) { e 2 X e = X
(e) 'Joord
6Zr
'Z'V l c n p o J d - p o s s o J Ja l o r 3 s r c o J o J J o l e J o d o J E I n p o WO q I
130
4. Crossed-Products
a 6 _ = n K "o. Proposition 429. Let o, be a free, ergodic actiorr of a discrete grottp G on a semifinite von Neuntann algebra. The following conditions on a non-negative real number \ are equivalent: (i) (ii)
\ e S(M oo G); for every e > 0 and non-zero e in P(Z(lrO), there exist t in G and non-zero f in P(Z(lvD such that 7 V er(f) ( e, and sp(Hrllran I) g(\-e,\+e).
Proof.
Let us retain the notation established so far in this section.
Since,by Lemma a.2.6,z1fvt6'1 e vQQO) E i,4A,it follows from (b) that Corollary3.4.8and Ex. (3.4.9) s(f4 = n (spa6- : 0 t d e n(P (z(IO)1. For notational convenience, let us write
K(t,e) = ran(crr_r(e)e)and
H ( t , e ) = H r l ( K ( t , e ) r t d o m H g ) ,w h e n e v e rt e G a n d 0 l e e P(Z(W). Notice that all the operators in sight commute and hence H(t,e) is a self-adjoint operator in K(t,e). It follows from Lemma 4.2.8 that A;_ (where A = n(e), e e PQ(M)) ) may be identified with Ye arcGH(t,e), so that
r
t-
Lt€G
)
= | u sp fl(r,e) | sp A;_ -e So, if \ € [0,-), conclude that
\ € . t ( i 4 ( 9 \ e s p A 6 ; w h e n e v e r0 * d = n ( e ) , e e P ( Z ( t u O ) <+V0
t e e P(Z(M)) and
Ve > 0, 3r in G
* O. such that 1 = l1\-e ,r+ey(11(l,e))
E
Before concluding this section, let us consider a special case, and for that case, reformulate Proposition 4.2.9 in a form that will be tailor-made for use in the next section. Let M = L'(X,T,u) with (X,F,,r) a separable and o-finite measure space; let t - T, be a homomorphism from G into the group of a u t o m o r p h i s m so f ( X , T , t t )( c f . D e f i n i t i o n - 4 . 1 . 1 0 ) ,a n d l e t t - o t b e t h e induced action of G on M: er(f) = '7 o \L: v let 0 be the fns traie on M
given by O0 = tt au; if E ;'r',
Q o <xr(lB)= 0(lr,1s1)= 1t(Tr(E)), and so 0 o
( 6 ' 7 ' y ' d o t 4 a s n : 1 u r g ) ' \ 1 1 1e d f l J o s l h t
(c)
('{a I t :I'\) = ,Q,t)uv n $\l ecnpap'(u) ruorg'lce; ur :lulH) 'rolceJ ')c [,q p7 n uollse u3 euJJe( (q) 2 uo leqt ,noq$ ',rg = v. n \ 1o 6 ('r = 'reuur sl g e o J' acJoJplno^\ t go ,{lrlercerl aql JI :1urg) '(e) g'l'l uolllurJoq Jo osuesarll ur oarJ sr g leql ^\oqs (e) .(t.0) ul etuosroJ l\ = g o J. \ u€ 0q g lo'I 't aoerl su3 (peurruralap leql qcns y,ggo ursrqdJoruolnc qll^r rolJBJ alrurJrruosB aq n le-l G16V) flenbrun ,{11er1uesse) 'r
I s roJ 0 ="a gr ,{1uopuB JI tnfu
= glf le{l rtoqg (q)
'(Out sA"'n , (, 's)X:4r I) =
i^ ,*ur ^ou, (e) iTytrlr=tatar,D
ul , r{ceeroJ 'e^oq€ ss aq slsoqlodfq pue uollelou arll la.I kt'Z'V)
(6'Z'V'dota esn:1u1g) '{t} = f,tt)Sreql ,{l1carrp Eurmoqs^q ssouallurJrrues eql ocnpep'ludire,rur-9sl 0 JI (q) ry Jo '9ul ,A0=t, o 0 ''e'l :lu€lr€^ur-gsl 0 JI fluo pue JI eo€rl " sl g l€ql ecnpap "i" = (*{I)g l(*(r 's)1(r 's)I)'o o
O " ou, "i" ((, 'sX *(e 's)I)9 = (C*r)O
rBqr^\ogs,ryi " ,, 'uollJes slrlr
'rH'0'(o'g'w) J o J I B I I p u o c a sa q l u I s e e q 6 O
lef
(r)
(tt'Z'l) sasrcJaxa
'C ut o IIe roJ r > l\ - (n)@p/Jottp)l D pue^?3G)t nJ'0 < G ) a l e q l q c n s! ) I p u e e ) J l s r x eo r o r l l '0 < (A)zt 'JI fluo puu leql qcns I ) g pue 0 < r [:a,ro roJ JI (0 )r t 'O '(rl't'X) 1 uarfl < \ JI i(-'0] 3 (g)t:snql ( g)/ las oll€r orll aulJaq '}I'Z't uolllulJeq 3o surslqclrouoln€ Jo dnorE alq€lunoc B eq 0 n1 '^\oloq paurJop ([lrX] '.lc) (rl'tX) go sruslqclrouoln€ Jo g dnor8 E Jo las ollBr ar.Il Jo uollou s.raEelr; sI ( 0 )/ arar{/ '( 0 )/ I (C oo (tl't'X)-7)S ueql'(g > t:rJl = 0 JI pue '(Vt't', pu€ II'I'9 solduexg 3o esires aql ur) clpoEra pue ?arJ sl (rt'JX) uo g 1o (ra) '6'Z'? uolllsoclord ruorJ eonpop ol '^\ou 'fsBa sl uollcB aql $ql lI JI ' f lp /tJod p,{q uorlecrldJllnur
I€I
l c n p o r d - p e s s o r Je l a J c s r q B r o J J o l e r e d o J B I n p o We q l
'Z'V
r32 (d)
4. Crossed-Products
Assume the fact that every automorphism of a factor of type I is inner, and show that a factor M as above, is necessarily of type II*
(4.2.14) Let M be a semifinite factor with fns trace T. Suppose cr is an action of Q -- the additive group of rational numbers -- on M such that r o crt= erT for all t in Q. Show that M @qQ is a factor o f t y p e I I I ' . - ( H i n t : a s i n E x . ( 4 . 2 . 1 3 ) ,s h o w t h a t t h e a c t i o n c r i s f r e e , deduce tnai A is a factor, notice that H, = etl, and use Prop. 4.2.9.)
4.3. Examplcs of Factors If Xf is a separable Hilbert space of dimension n (l ( t? < -) and M = I(lf), then M is clearly a factor. The function D, defined by D(M) = dim !t is a dimension functiot for M and consequently M is of type I n . T h e p r o o f o f t h e c o n v e r s ea s s e r t i o n- - t h a t a n y f a c t o r o f t y p e I , is isomorphic to l(tf) for an n-dimensional Hilbert space l8 -- is outlined in the following exercises.
Exercises (4.3.1) Let M be a factor of type I,, I < n ( -. (a) (b)
(c)
(d)
(e)
(f)
If e is a minimal projection in M, then eMe = {\e: \ e C). (Hint: if x = x* e M, consider the spectral projections of exe.) Any two minimal projectionsare equivalent; further, if e and f a r e m i n i m a l p r o j e c t i o n s a n d i f u a n d v a r e p a r t i a l i s o m e t r i e si n M such that u*u = y*y = e and uu* = yy* = /, then there exists a complex scalar r of unit modulus such that u = \v. (Hint: consider a*y and use (a).) T h e r e e x i s t s a f a m i l y { e r :i e 1 } o f p a i r w i s e o r t h o g o n a l m i n i m a l p r o j e c t i o n s i n M s u c h t h a t I = V i e l € i ,w h e r e I = { 1 , 2 , . . . , n } o r (1,2, ...)according as n is finite or infinite. W i t h { e i } a s i n ( c ) , p i c k p a r t i a l i s o m e t r i e su , i n M s u c h t h a t u f r , = e, urul = eii for any x in M, i,i in 1, show that etxe' = \r'utaf for^ sonie \,, in C. (Hint: let eixei = ult be the pbiai decompositiori of erxe;i apply (b) dnd (a) to r and h respectively.) Let I't = ran er and let ffn be an Suppose M t f(f). n-dimensional Hilbert space with a fixed orthonormal basis {(r: i e I). Show that there exists a unique unitary operator u: lln @ 1 1- l t s u c h t h a t r ( i i 6 n ) = u i n , f o ( i e I , n e I t With r.ras in (e), show that Lt*My= {xol (Hint: use (d).)
e r(lf,,oM): x€ D
f(H")).
'r11ad't1 O Eululeruar,{1uoeq1 .u elrulJ e rog '1 Jo sl (g)*l leql sl flrJrqrssoct od{l oq (D)*ll tquanbosuocpue 'ltuorsuerurp-alrurJ louu?c Jo lou sI rlaql 1e 4oo1)luapuadopurflreeurl [1rea1csr @)*ru o! pue (1sacr.r1er,u {D a t :'\) les eql :((e) [q) alrurgursr g uaql '{r) r g Jt 'ad,{}olrurJ Jo rolc€J e s\ ry leql stt\ollo J t\ 2Wuo ec€rl n rl g flluonbesuocpu€ "l( r"-s;1; 3 =
,l(s'r)rl 3 = .l(r's)*xl 3 = (,rI{)0 elIq,rt
'rl(r 's)Il3I = (I*{)0 'n , 'Jrtluo el?ls X Jl l?turou InJrlllBJ s seurJap (r.r)I = (t)O uollenba Er{t teql s,fo11o3ll b uo 0 el?ls Isrurou InJqll€J u saurgop f = (f )0 uoJlenba or{l eJuls 'rolc€J e q W lBql /hou esoddns /,ti Jo lueruelo Isrluac relecs-u6u € sl I t€rll elou pu€ . o s r A \ J a q l o. 0
I .s)5 1 = 1r I ul n oruosJoJvntn - s Jr 'l J 1 e q 1S u r r l n b e t f t q o y g r x a u r J ? p . ) ; ; a u r o s .I(3,r)I = roJ olrurJ sl {g r s :r-s/s} JI 'puerl reqlo ef1 u6 I (9 ry s roJ g = (r's)1 e U^t)Z r { lBql s^\olloJ lJ .alluJgur sr {r} ueqt raqlo sselcfceEn[uoc freia g1 acuaq t r = rllrl {ll r l ( r . s ) t l ' 3 . o s'gl v ul /'s IIE rog (a'1)g = (r'r_srs)I e @)Z t I leqt (il tt.V erurue.1ruoJJ s ^ \ o l l o J l J ' r a q l r n g i 9 u 1 / ' s 1 1 u. r o f ( a ' r _ t s ) X= ( l . s ) 1 u a q a . ( C ) r r W ( C , r : ' l ) s r s e q l ? r u J o u o q u o l € c r u o u B co l l c a d s a r g l l ^ r x J o x r J l u r r re q l alouep (((l's)f)) lol -- O uo g Jo uoJlc€ I€r^ul aql sr D eraq^\ ,D,@ O .Joord = (C)*U leql (€) 7'1'y eldurexg uorJ ll?rar -- (C)*tn= X JI 4, '{r} = C ssalun'r11ad,{1lo to1co! o s1 (g)oy uaq| 'paltsltos st tlorttpuoJ ssolc ,tco7n[uoc arrur/u!,, atry/I (q) .aTtuttut st {g r s :r_s/s) ssolc t(colnfuo? aW 'g u! > f, t ttata rcl pqt st rolzo/ p aq ot (C)*tn rc/ uoltlpltox Tuatu/lns puo [tossacau V (e) -((c) tt.Z alduoxg'lc) W = (C)*trt otqagp uuownaN atatosrp alqptuno? p f,q g taZ 7g-g uo111sodor4
uoa.dnot8 tlt!fi{'dnod
'II aclft Jo srolc€J ol t\ou ulnl eA\ srolceJ go soldruexg '€,'V
ggl
4. Crossed-Products
t34 Exercises
(4.3.3) Verif y that the following countable groups satisfy the "infinite conjugacyclasscondition"of Prop.4.3.2(a): (a) (b)
the group of permutationsof 1,,f= {1,2,...,}which move only finitely many integers; a finitely generatedfree group on two or more generators. O
Next in line are factors of type II- We shall show that if M is a Hilbert infinite-dimensional factor of type II, and if lf is a separable space, then M O t(lf) is a factor of type II* and that conversely every factor of type II- arisesin this fashion. If lt is a separableHilbert space,let -n=; -f- -tn ' t=t
where trn = ll for all n. Then, as in our discusslon of discrete crossed-pioducts, we shall identify an operator i on lt with a matrix ((i(z,n)))l --, where i(m,n) e f(fi) for all nt and n. If M is a von Neumann"i't-giUra of pperators on lf, let M = {f e t!t): V(m,n) e IuI Ym,n); it is clear that M is a von Neumann algebra of operators on !f. Proposition (a) (b)
4-3-4. Let M c t(lt), A g f;!tl be as above.
If It{ is a lactor ol type ll, then M tt o lactor ol type lI; A If M is a factor of typC ll- operating on a Hilbert space !1, there exists a factor M of_type^ll, acting o4 a HilQert space X and a unitary operator ui t? ' lt such that uMu* = M.
Proof. (a) It is not hard to show that if ni M - N is an isomorphism of von Neumann qlgebras_M g t(lf) and N g [(K ), then the von Neumann algebras M and N are isomorphic; the details are outlined in Exercises (4.3.5) and (4.3.6). Hence, we may assume that M is standard relative to a faithful normal tracial state T on M; thus, assume that O is a unit vector in lt which is cyclic and separating for M and that the equation T(x) .= <{go> defines a faithful normal
tracialstate-onM; thus ll"oll'= llx*nllfot all x in M. Define T: M- - [0,-] by
7 ( D = E
It is clear that f is a weight on M, which is faithful and normal ( R e a s o n : i f t ) 0 and i(nSt) = 0 for all n, t h e n i = 0 , s o i i s faithful; if r , a 0,,(D=
S
krr
,
then each 0r, is a normal positive linear functional on M and {* z T,
Jo r'{ fluo aq1 'I ad^l Jo lou sl /,V os pue suorlcaford leurunu ursluoc louuEJ .;41'suorlcaford lerururur ureluoc lou seop ltl ocurs 'rolJBJ olrurJ e sl r'[ l€ql s/yrolloJ ll sollurJ sr a ecurs pud ,n Jo tuelualo '.n roJ uorlcuri3 f lrluapr aql sl uorsuourp E sI (n) d lr .a eculs leql e^rasqo'77.ito ec€rl suJ € sl + JI -w urotJ rolc€J e auriq*g6 flradord aql sl'pequr n wqt -- I*ldl ul pdlJ ,(eru repei oqr qclqrr pue 'eraq 'olul oE lou llr,tr a^\ gotq^\ -- loEJ e sl lI .g uo Eullce erqaEle uucrunaN uol € sB peit\er^ ,a1ga = n pue o uEJ = g :;'-I 'l
ur'n xrg.(or.z.r l"f""ili, i,;*i iJ"""-fi:#ii"'",'l';'::# ', olrurJur sr g = = ll qll^\ 9.7.1 uollisodor4 alou uer l€rll "r3N = ? [1
^.1.il;:rl::" ,pru. rus= (u, w)2 *:;t :,',#W ",Ii'^rrri;rrJ:il# 11l,,uo1suarulp,, Ilsrus flrrurlrqre {11,n7g u1 suollcoforforoz-uou lsrxo araql 'rII od[1 go sr Jr{ aJurs.ra,re,no11-II ro bI adrit Jo sl ll1 ecueq t=t
:
-=zllull =(r)r 3
oJurs 'alrurJ lou sr rlorq/r\ rolc€J alrurJruas e 0q ol ueas sr J\l uorssncsrp EuloEaroy er{l ruou 'w wotJ Alrl€rrolcBJ slrJarlul lg ocuorl :(ueJoaql lu€lntuuoc elqnop aql Jo Joord aql ur se) (tht ) rx oruos toJ'u pua ta ll? roJ ,xt-g = (u,tu)rx:(11)g I ,{) = ,1,t1Qolcej € lou sr 1,tgy. uarc) lBql pa{coqc fpsea-r il 'alrurJrruassr /g aouoq locerl e sr .l' flluenbesuoJ pue
(r *r)r = utu
(ecer1€ sJl. acurs) zllu(,4.aEll3 = u'ur
3 zllu*(ur'a)11; u'ur
allu(rr'nr)r,{;13
= (*s r)r
l€rll acllou 'n t x gr '{11eurg 'ollulJlluas sI L letll uaessl fl lW .lo losqnse oq o-l par3ire,r,tlrpuer sl "y'ti eculs l;,9 ur esuap qcrq^\ sl rqaEleqns e pe lEuorls-o lurof *f -;1asn sl 'ZV ttrqt aesol {_seasr y. ,((u,tu)srred fueu f1et1ur3lnq IIB roJ Q = (u'ti)x :1y t ll = ul,yJl (.1eurrous! L ,6.r.2 .ctor4fq .leql os srolceJ 3o salclruexS'8,
sgl
4. Crossed-Products
136
an easy computation reveals that if V e T4o,then
u 7 u* =
a
-'l="'
v(m'n)uf,;
since sums of this sort are easifl verified to conslitute a ^o-weakly D denseself-adjoint subalgebraof M, concludethat u M u* = M. Exercises (4.3.5) (a) If n: M - N is a *-algebra isomorphismof a von Neumann algebra M onto a von Neumann algebra N, then n is a o-weak homeomorphism.(Hint: tt(M*) tn(N*) and so n preservesorder; since the same holds for z-1, if x,y i M-, then x ( y (* z(x) < r(y); concludethat z is normal and hence o-weaklycontinuous-cf. the proof of Theorem 2.2.1; the same reasoning holds for 7r-1.). Let M (c r(fi)) and N (g t( K )) be von Neumann algebras,and normal) *-isomorphismof M supposen: M 2 { i1 a_(necessarily gnto N. _ Let lf, M, K N be constructedas in Prop. 4.3.4,and let Mn and Nn be as in tle proof of Prop.4.3.4. (b) If't = ((i\m,n))) e l(f) and k = 1,2,...,define in e t(lf) by ;n{z,z) =
ifm,n4k I i1m,n'1, [0,
(c)
otherwise.
Showthat ll;rll < ll;ll ror all k and Vt - i o-strongly:. lf i(m,n) e t1t; for m,n = 1,2,-, iiefine in e s(lf) as in (a)
above. Shog that the,, matrix ((i(rn,n))) repiesents a bounded o p e r a t o r o n X fi f f s u p n l l i u l l < - . _ ( d ) For k = 1,2, ..., let lri; : (V e M : V(mg) I 0 ) mrn < &), and Then Mk (resp. Nr) may be let Nk be similarly defined. operatois on- 6(k) regard^ed,a,s a uon {eumann algebra tf (resp., K ('(r), where lf(xl is the direct sum of ! copiqs of lf. Show that there is a unique l-isomorphism F: M. - N* such that fr(;) = ((n(i(m,n)))); further i is isometric and a o-weak homeomorphism. (Hint: use (a) and the fact that an injective *-homomorphism is isometric -- cf. second half of proof of Theorem 2.2.1.) (e) Show that there e1lsts a gnique (normal and isometric) *-isomorphism i of M onto N such that n(;) = ((n(i(z,n))). (Hint: use (d), (c) and (b) above.) E Recall that g factor ff is oe type III0, III\ (0 < \ < l) or III, according as S(M) is {0,1}, {0) u {ln: n e Z} or [0,') (cf. the d i s c u s s i o nf o l l o w i n g P r o p . 3 . 4 . 6 ) . S o , i n v i e w o f P r o p o s i t i o n 4 . 2 . 9 ,i n
r q I I u J o J m l o r ' o - ,l ' r { l ^ \ o q s ' M , : t p u '
r o'uJ )o urrr"rJj tql I ' '0,)gur / ro.; N ul ? .(r")r fcl = 1r";{'!c,
gr ,{1uopue JI 8(,)= Io lBql /roqs }
r:eruaurala aqloq:t'f3rer ''- ,z,t = o jf
r urn.Iog1 (e)
f;{"{rti"t1;"d-f|.fi sasJcrexg
'sosrcrexe Eur,no11o; aql u! p3urllno sl qcrq/$Jo euo .(rt,1.;) uo flyecrpoEropue {1aar3slcu g leql ags ol sfeao,yeronaseJ€ eraql Ns, dnorEqnsc11c[caq] sl g "r3ul {us 'e1cfc.11ng e itq^palerauaE Jo '{N ) '") t o :4og) fq paluraueE dnorE aql oq O ra1:(/i4}) 3o 4 usrqdroruolneuB sl aua qcealeql realJ sl ll 'f rlJueJoJ g l = w J l ' ( ( U ) o ) o =l 'Qu)n)| 1ra;1oToz) )t * LuJl fq pourgap eg :t'ot p u € ( N " " ' I ) J o o u o l l e l n u r e dq c e ar o g u r l o l ' M X X Z '0x 0f pue lt u! u euos roJ pr = (zp :1 r ul o) rurog arll Jo las ? ueeruIIBTIS oa .; u1 tas rapurldc freluourayoue ,{g 's1es:apur1[c fq pale.rauoE erqaEle-ooql sr tr snql ,.".,Z,l = u pu€
rxr.l.,.o"=g" 'g Jo lasqnsfue s1 ereq,n{"a I ((u)rl'"',(l)o) :; r o) ruroJ orll Jo las u ueetu IIEqsolrl .1 u1 1asrapurl{c e [g 'il II€ roJ ort = ut qllrn 'ut r6t=rt ernsserulcnpord aql pu€ 4 erqa^E1r-o lcnpord eql ql; ; drnbg .{..' 'Z'tl = .(lX .(t = Id3 pue) M .areqin NI :o) =^ItX - X :;'-t lrr > .r > I roJ 0
,:"\) n {0) .(I.0) = ( 9).r ,(q uarrE las oller serl pue 'fllecypoErepue {loarg slce qolrl/r\ '(rl'4}) oceds ornsear,uellulJ-o pu€ alq€J€dase Jo sruslrldror,uolnB Jo g dnor8 elqelunoc.. B Jo elduexo u? lcnJlsuor ol lualclJJnssl ll .IIII ro (I > \ > 0) \III'0III edd.1go Jolc€Je 3o eldruexaue 1cnr1suo3 ol repro slolc€C ;o saldurexg 'E V
LEI
4. Crossed-Products
138
(c) lf To.k is as in (b), show that Tq,k acts freely' abovea ' nd Ex. (4.1.12).)
(Hint: use (a)
(4.3.7) (The aim of this exercise is to establish that G acts ergodically on X; the reader who knows some probability theory will realize that this is a special case of Kolmogorov's zero-one law.) SupposeE e T and p(TE AE) = 0 for all T in G. (a) Let F= U,nr TE;then F eT, P(F) = p(E) and TF = F for all Z in G. (Hirit: G is countable.)- xfk+l'k+2"") *:x ( b ) I f , f o r i = 1 , 2 , . . - . - ; r ; , i - i t r ' 2 ' " ' ' k ) a n d z ,'(211(t^t),n(k(o)) = for all (so 6 that denote the natural 6i'ojectionsur in X) and if F is as in (a), show that f = nil(tr(k(F)) for all k. (Hint: the hypothesison F is that lF(u) = lp(6),'whenever 6 is o b t a i n e d b y c h a n g i n g a n y f i n i t e l y m a n y c o o r d i n a t e so f t t . ) (c) Show that p(C n F) = y(C)1r(F)for everv (finite) cylinder set C in X. (d) Show that p(C n r) = u9)p(F) for all C in F. (Hint: the collection of C for which the assertion is valid is a monotone class containing the field of cylinder sets.) (e) Conclude that p(E) = 0 or l, and hence that G acts ergodically E ot X. (Hint: Put C = F in (d) and use p(E) = ,r(F).) 43.t. With the above notation, let A(G ) denote the Lcmna multiplicative group generated by {p/pi: I < i,i ( M); then r(G ) is the closure (in [0,')) of A( G ). Proof. Fixk in N, oe c.andfor j= l'"',N'letc,.n={oe x r n e n ( C 5 , 1 ) l { =i ,s a ( m e a s u i ' a b l ep) a r t i t i o n o f x a n d c l e a r l y dtr o To, dp
o(k)=i)'
=po(i) pj
on C, u. Since every element of To,*'s''(in fact, is of the form To,xrTo,Yr'
G is a product of finitely many
' ' to-,*-
w h e r e I < k , < k , < . . . < f t - ) , i t c a n b e d e d u c e dt h a t i f T e G , t h e n t h e r e i s a p a ^ r t i t i 6 n{ D r , . . . , a j ) o f X i n t o c y l i n d e r s e t s s u c h t h a t dttoT _ ! ttto''
du
'!'
where \, e A(G ) for each i; it follows immediately t h a t A ( G) . For the reverseinclusion, it clearly suffices to show that r(G) for I < i,j < N. So fix i,i < N and supposee > 0 , E p(E) > 0. Choose6 so that
r( G) c pi/ pi e e-f hnd
(rd
zdt
( r ' r -= r) lra'n J = zd pue\+I/I = rd'(Z't| = oX 't > \ > 0 lot (r) U;l;t;L-"t/;
seqsrrq€rsa slrlr'frerlrqre(e-re,vr) rrln
ttJ:ti?t:J,?Jt:r"l{ rT;}
lr = tlp/Jodp pue 0 < (A)A 'Z 3 (4)t 'Z -= I leql alou pue (0g)r-Z u (o v oil = J les '0 < (0g u (o u oz)Drt l€tll sornsua,{1r1e nbaur eloqs oql ', 3 (O v og)1. ecrc1C 3 oZ pus J = ())J suorlelar oql ocurs '(r)rt < I ( d + t d + t a t a l s- 1 s ) r f a t a l t + d ( J ) 7=r I
al!. s - (J)rt+ [g - (g-
= (q)Tt()fl
= O)J l p _
td = JoTl p = (o v ))tl
(c) (q) (e)
:Jeelc 'O ,toJ = J alrJ/r pu€ l = (.r)o l?ql aru suorlJosseEur,nolloJ eql osNJ ro {cld'[f = (Z)o X ro) =Olel'w 9 l€rll aclloN 'g > (g v ,)t r€rll qcns '3 fes 'los repull^c (atrurg) € slsJxa eraql 'a.lnseau-lJnpord e sr t acurs l4t4*14 *t4
srolceJ Jo saldur€x1 'E V
6€r
4. Crossed-Products
140
S o , w i t h G a s i n L e m m a 4 . 3 . 8( a n d n = 2 ) , w e h a v e / ( G ) = { 0 } u { \ n : n e V,). ( b ) W i t h t h e n o t a t i o n o f L e m m a 4 . 3 . 8 ,l e t N = 3 , \ 1 , I 2 > 0 a n d \, \'| I
, , =r -- l +
= ---------, .' Ua 2 tta l------lY l+\r+\z' \r+\r l+\rf\r'PE then A(G) is the multiplicativegroup generatedby I, and \r; so, if log \, and log 12 are not rational multiples of one another -- for E initarice, \1 = €, xl, = ,'tz -- then r( G) = [0,'). D.
Examplc4.3.10.Let Xo= (-!tl and let X = Xi. We shall show that there exists a sequence{rr")i=, of probability measureon Xo, such product o-algebraT on X),-and if that if E = 6l=rtrr, (defineci'cin-ttre is the g.oup generatedby {I,,}l-r, ttris case, G is bef<j'r!^ as G 1ih where = 1*t;5"ko(k) (?""ur)(k)
)
t h e nr ( 6 ; = ( 0 , 1 ) . Let tktli=, be a sequenceof positive integers satisfying the following conditions. (i)
kj"ki*,
(ii)
k j l k j + r ( i . e . ,k , d i v i d e s k , * t )
(iii) k12k+ 1...+
(*)
for all j
kj2ki
Next, define a sequence1rr")l=1 of integers thus:
mn=ki
ir
iltrt-t
. n , .i, zut,
in other words, the first zkLmn's are all equal to k' the next 2k2 n?rr'sare all equal to k2, etc. L e t 1 r rb , e t h e p r o b a b l l i t y m e a s u r eo n ( - l , l ) d e f i n e d b y 2 n - I
s"({-l}) = ----;1,
s,,((l))-
l + 2 n
l + 2 - t '
and let g be the product measure lr = 6i=rlrr, on X. StcP t. tt({tt)) = 0 for all o in Q Proof. m t fll=r(l + 2 ")
fy t z f r t = J 3u " , { t = r l c t r u l 0 " ' u > f > I r o J g = s autr.....rt lu > f > I roJ rnzt l,;l Eurfgsrles sretalulaq la-I .€ dafs 'Joord eql seloldruoc uollclp€rtuoc slql
peurruJalap sgesrapurJ{J ele s.:p eql 's.f lcullslp JoJ l€r{l ^\ou aJIloN " I r = f -= '(/rtporu)tC ttr z
lsl't{
. l u r { lo l o up u e l v . t o l g = 1 2 > f > r.f turf3sr1esf qcue roJ 0 = (!S v iltt ro Q = (lV v ilrt il
'@)rt > ,*rr"{ lerll qcns 1{ .z[ roEalugu€ lcld uoqr il, t (tC)t uaql ,lg :o lV =tJ pue r.f < { I l e q l q c n so [ . t { r a E e l u ru B { c r c tl x a u : l > f > a/l - | (> a/t) teql qcnsl. raqunu € )tcld 'a/l - l - (!g)rt eruaq pue ,:_ I
lna(l,n-Z+t) = I
=(v)rr [rn-f rzL \Z )
l e q l o p n r c u " ": : - r ; -
z t' ecu,g'Joord
g.'jiYJf ]'[,]'li o[. I raEolul uBsrsrxa r,r$:,t'nt,'.,flI',::'i
'{1+ = ( u ) o leql qcns'J ul:!E :1 r o} = ly = tg pue '(1/ ul r/ IIB roJ I= ( u ) c :qx ? o ) = ' y ' { ' l = " u t : } d r a} =,/ lal 'v'Z'l =/ JoJ 'Zdefs 'g = ({rn})d lzrll opnlcuoc ' @ =" '
I=u + 7 u Z 3 "I- Z ' z Z +r'rl- Z .'{ r Z = -lu;
ecurs srolc€J go sayclurexX'€',
lvl
t42
4. Crossed-Products
Proof. Temporarily fix an arbitrary integer j > 0, and let E = {o € X r { n ) = I f o r I ( n ( 2 k ' + . . . * 2 * j - t ) . C l e a r l yE e T a n d t r ( E ' ) > 0 . S u p p o s et h e r e e x i s t s F i n F a n d Z i n G s u c h t h a t g ( F ) > 0 a n d F u TF C E; then it must be the case that ?" is in the group generated by {Tn: mn > kr): it follows from the assumption that kilki+l that duoT
t'
=
I
^tlo' '
tI'
where tDi)i!=, is a partition of X (into cylinder sets defined by k r . coordinates after 2-1 + . . . + 2-j-t) and each \, is in the cyclic (multiplicative) group generated by 2*j. Since j was arbitrary, conclude that r(C ) c {0,1}. To prove that 0 e r(G), it suffices to show that if E e T, p(E) > 0 and e > 0, there exist F e Fand T e G such that FvTF CE, tt(F)> 0 and (dttoT/dtt)(u) I [e,e-l] for all o in F (where we haG assumed € < l, as we clearly may). (Then, either 'l ( ( duoT I €},fl {F'n{o:--;-(tt). dtt L L )' ) or (
(
(
duoT
ll
lTlFn{o: -(o)'€ll,r' L L r a p ) )
.,'l
J
will do the job.) First fix js such that 2-kj < e for j > jo. Let jo-l k. flo= 'l'2r' l=t
since p(E) > 0, there exists of,..., "lo in {-l,l)
such that if C = (u €
f t r , { n ) = o l f o r I ( n ( r ? o } ,t h e n r ( E n C ) - > 0 . B y S t e p 2 a p p l i e d t o E n C and this jo, there exists an integer j , jo such that y(E n C i A) > 0 and p(Eh C n B;) > 0. Since G actsirgodically on X, (cf. E i . ( 4 . 3 . 1 1 ) t)h e r e e x i s t s 7 " i n G s u c h t h a t p(T(E n C n A7) n (E n C n 8i)) > 0. Let F = (E n C n A) ATL(En C n8;); then F e T,p(F) > 0 (since p(TF) > 0) and F u TF c E. It follows from the definition of C that Z must belong to the group generatedby {Tn: n > no}, and also Z I 1fro e , s i n c eA i n B * = 9 . S o , t h e r e e x i s t i n t e g e r s nftl <2 < . . such that fi0. ,i and 7"= TnrTnr...Z"o. Argue (as in the proof cif r(G ) g {0,1)) that the range of duoT/dp is contained in the cyclic l(i
group generatedby 2
'o,
which in turn, is contained in (0,e) u (l) u
('Patrasse se '0 * !;7 u ecueq pue 'Iltadord uollcesralul-elrurJ eql surl ;7 Jo slosqns lcecluoc Jo (/ r r :ty) uoJlcelloc aql leyl s/roqs ruoroaql lulod poxrJ rapn€r{cs eqf Jo uollecrldde paleodar l; qcea fq g1as1rolur padderu s! rIJlrI^r las ,{lcltuo-uou xaAuoc lceduoc B sl l) leql arnsue r u e J o e q f l u r o d p e x 1 . ; J a p n e q c s a q l - p u e s . ! - 1a q l u o s r s o q l o d ^ q e q l i(x = yta :; r x) = !) lel :uoseog) .!.1qcea ,(q paxrg sl qclq/( lulod e suleluoo ) uaql '(suogleurqulooxo^uoc slcactseJIZ l"q1 su€eru eurgge) y sderu-Jlos ourJJ€ snonurluoc Eullnuruoc esl,trrred go fllrueg {uu 3o puu e^oqe se sr sl l :L) gr :Eur,no11oJ aqr sl uollezrlelaueE {l I JI ) '1u1od pexlJ € seq aceds rolce^ {sea uV lecrEolodol xeAuoc fll€Jol e Jo X lesqns xeluoc lc?cluoc e go deru-Jles snonulluoc ,{u€ slrass€ ruaroeql lulod-poxrg repn?qcs aql lgrll IIBror :€ruruel l€r{l aql aql ul pepeau oq III,r\ suoll?^rasqo Eurmollog aq1 Jo Joord '€ruruol l€cluqcel d.teururrlerd B gll^r ssarElp sn lal .ureroorll l€tll ol tulttaE aroJeg '(1L o { = (!)rp oraq,n) g Do (tt,1.,1g)^7= n JolcEJ eql Jo adfl aqt pue ((t'J?) oc€clsernseoru € Jo srusrqdioruolnE s€ dnot8 olqelunoJ € Jo) ,J e I uoJlc€ crpotro puu eorJ B u3e/ruaq uocrxsl aleldruoJ lsoruls u? seqslJq€lseqclq^l uuBrunoN uo^ Jo llnsar lue8ela [re,r E qll^\ uollcas srql 3pnlcuoc II€rIs ail
erns€oru e^rrrsod sett zJ gtrn = r ,o*,uJ',?rt;ilf".1"j 3;3i i:r:
r { J n s u l J e s l s r x o e r a q l t e q t f r o q s . 0 < ( J ) t . 0 < ( A )pr u , el t g , g 9 'fllecrpoEre slc€ g osoddnspue ,(tl,4,y) oceds arnseaurlureuoE JI E Jo sruslqdro(uolnBgo dnorE alqElunoc € sl e asoddns (tt.g.l) sssrJraxfl 'aleldluoJ .O
sr goord oql pue 'J ul o IIe JoJ 'relncrlredur .pue ,)v u\ 0t II€ JgJ I i (o)(dp/JoTtp)wrp g clets ruorJ epntruoc rV > .f ) I roJ t,,Z > l(oo)lrl [1rea1cecurs.:Jy ul o IIB roJ t- > (ol)Jr i(v p uoJlruJJapeql fq) os pue '-!1 o1 s8uoJaq!u euo lsBallE '6 = Jg U Jf ocurs.reAe^\oH r' {.t.or , 'rt = - i r3Z = ( o # * Joflp lBql epnlcuoc
'r."(u)", =($\frflp !"r, 'u f\veroJ ocurs'(d :(lr)o)I = (o)fr aurJep W > > I > I,tZ = ""'Irt) f > t pue x ul o ro; l(Nry = {"*,'.',ruwl leql qrns N esooqJ 'J ul o II€ roJ I r ($)(flp/Jotp) lBrtl /hoqs ol sacrJJns lI 'JooJd eql alaldruoc o1 'acue11 'r r orr_a acurs ,(-.r_r)
EVI
srolosC3o salduexg '€,',
144
4. Crossed-Products
Lcrnma 4.3.12 Let M be a semifinite von Neuntann algebra with a fr^s trace T. Let A be a maximal abelian von Neumann subalgebra of M. If there exists a normal norm-one projection E of M onto A, then tlA* is semifinite and t o E = T. Proof. Fix x in M- and let C(x) be the convex hull of {uxu*: u e U (A)). Since C(x) is norm-bounded, it follows that the o-weak closure K(x) of C(x) is a o-weakly compact convex subset of M. The abelian group U(l) clearly acts (via inner conjugation: Tu(!) = ulu*, y e K(x)) as a family of pairwise commuting o-weakly continuous affine self-maps of K(x). So, by the discussion preceding the lemma, there is a point xo in K(x) such that uxou,* = xo for all u in U(A). Since .l is maximal abelian, it follows that xo e A. (Note that x e M*) K(x) EM*) xo = xfi.) In particular, K(x) n A i 0. Suppose now that F is any normal norm-one projection of M onto A. Then, by Prop. 2.6.4,notice that for any a in U(A), F(uxu*) = uF(x)u* = F(x), since u4 is abelian; thus F is constant on C(x). Since the normality of F implies that F is o-weakly continuous, conclude that F is constant on K(x). Since F fixes points in l, conclude that K(x) n I = {F(x)). Since x € M- was arbitrary, conclude that any two normal norm-one projections of M onto .,{ must be equal. S u p p o s ew e c a n s h o w t h a t r l l * i s s e m i f i n i t e . S i n c e r i s a t r a c e , i t would then follow (cf. Remark 2.6.9 (a)) that there exists a normal norm-one projection F of M otrto A such that r o F = T. The u n i q u e n e s ss t a t e m e n t o f t h e l a s t p a r a g r a p h w o u l d t h e n s h o w t h a t E = F, and so T o E = T. Thus it suffices to establish the semifiniteness of TIA+. Suppbse x e Dr -- i.e., x e M* and r(x) < -. The traciality of r implies that r(y) = ?(x) for all y in C(x). Since r is o-weakly lowersemicontinuous (cf. Prop. 2.4.9), infer that r(y) < z(x) for all y in K(x); in particular, r(Ex) < -, since Ex e K(x) (we have shown, in fact, that K(x) n 1 = {ExJ). Since r is semifinite, there is an increasingnet (x,) in Dt such that x, 2 l; since E is normal, Ex, ) l; however, by the last paragraph, E Drlo_; this shows that. rlA+ is semifinite. d*r, S u p p o s en o w t h a t t - { i s a f r e e a n d e r g o d i c a c t i o n o f a c o u n t a b l e group G as automorphisms of a separable and o-finite measur-espace (x,f ,p). Let t - cr, be thg induced action of G on M = L-(X,Y.,tt). Then, the crossed-product M = M ao G is a factor. The type of ll[ -in the Murray-von Neumann classification-- is determined as below. Thcorcm 4-3-13. (a)irt is of type lll if and only il there does not exist a o-finite positive nteasure v whiclt is equivalent to lt (in the sense of mutual absolute continuity) such that v o T, = v for all t itt G. (b) Suppose there exists a G;invariant o-finite positive nteasilre v which is equivalent to tt (so that M is semifinite, by (a)). Then,
u-
'11 ad'(1
3o € sl (4)alf Jo sl ry snqlig - (( st)u)f puery roJ uorlcunJuorsuaurp *W u\ uoql'((B) aes) pecnpur uo ecerl s u J p u € e q _ l s r 7 1 / J o o r c l f J o = gl, :r 'o j("a)n pu? u 18 roJ 0 < ("^?)n leql qcns n1 apl {, J t J -g s l e s l s 1 ) r 0e t e q l ' o g ' ( a ' l X ) s r o s , , ( 1 r a a 1' ccl u o l g - u o u s l ( r f ! ? ) J I 'a = apnlJuoc '(trtl)u uI I telq.{.t '(6'I't 'ro3 ,(q) Ieutrtrurtusrc sa pue Ort)u = trtt U t(ltt)u > / relncrlred ul 7\ = /x .(Felturs pue /1 = aly = (a1)l ='axa{ = xa{ = x{ e (n)u l x ' a c u o q i ( a s r c r a x leu r l l r o J l u l q o q l l s e e l l e . r o . ( e ) ( t . g . l ) . x g . J c ) ,(trtt)uur o ul \ owos roJ a\ = axa + (n)u r x leql s^\olloJ lr. l€rururlu sr a ecurs 'a > { * 0 pue U^l)d r / asoddns H ur uollceford IBruluru € sl a Leql ^\oqs o,y.:l I ed^l Jo sl l{ t€qt ,no11o31[,tr l l ' 0 I a a c u l s ' ( s 1 ) u = a 1 e 1 ' y ' t lu r u o r l c a f o r d l r f r r u l u r e s r s l u a q l ,{1:re13'O=(.i\a)d ro0 = (C)t raqlra,Z- I pueJ r JJo^eueq^\ pu€ 0 < (g)TI '! t Z "e'l -- E wote ue sulBluoc (d'l,X) osoddns (q) ''J o n = n l B q l a p n l o u o c' l u I z , { r e r l r q r e r o J s t = / a u i i i r t
:(t1oep!!=
^pe:ro I)! = Q1t o 11t= (((/)tn)u)j = (eoerl B sr 1 ecurs) (._(l)r(/)u(l)r)f =
((Du)t=(!)r="pII g ) | pue *w > I JI leql 'uoql 'anrosqg '+q ur.! rc j = qcns (lnJqlrBJ sr teqt r aculs) 7t ol lual€^rnbe sI r{rltl,r\ U).1 lpI 'n ) a r n s t e u o ^ r l l s o d o l r u r J o € s r e Jaql leql (9'Z uollJes Jo s{Jetuol 't Suruado or{l 'Jc) saolloJ lI II€J ol onurluoJ II€qs e^\ qc!g^\ 'W uo ecsJl suJ € ol t. raJsu€rl ol Du ruslqdrouosl ar{l osn .Alfu u o e J € r l s u J B s r ( n ) " u l t - : . l € t t t Z I ' g ' t s r u r u e . Ip u e 9 ' I ' t ' x g t u o r J s^\olloJ l-[ 'oJuoq |n Jo erqaEleqns uuerunoN uo^ uerleqe Isrulxetu e s y Q ; y ) D ul e q l s a l t s u e 6 ' f , i ( r e 1 1 o r o 3 ' e e r g s r D u o r l c e a q l ? c u l s 'W 'allulJlrues 'flasreauo3 vo acerl suJ etuos sr sr I pue lr{ asocldns ' e l r f r 3 r r u a sq W aluap'W u o a r € r t s u J B s a u r J a p( ( r ' r ) 1 ) r = ( I ) t u o l l € n b e a q l r e q f ( u ) ( t t . Z . f l 'xg pu€ (q) (g't'l) 'xg ruorJ apnlruoC '^pI | = A ) L { q u a , r r E . i 4 ru o ac€rl suJ eql eq r lo.I 'slslxe n erns€eru e {cns asodclng(e) .Joord 'arnsoaw attury o s! ^ l! {1uo puo lt ad& a1rutl {o totco! o s1 1,t1 er.) istuoto dtt sutDiuoz(d'S'X) acods atnsoau atfi /t,{1uo puo /! 11 adfi Io q n (ll) ,.swolD srttvtuo?(rl't'X) acods atnsoau aUt /r tluo pup /! 1adty lo q n (l)
srolceCao salduruxg 't'i
9Vl
t46
4. Crossed-Products
Since the possibilities"(X,T,tt)has atoms"and "$,f,9) has no atoms" a5e mutually exclusive,as are the possibilities"M is of type I" and "M is of type II", the previous two paragraphsestablishthe validity of (i) and (ii). For (iii), with 7 as above,note that 7(l) = v(X).O Example 4.3.14. l,et G be a secondcountablelocally compactgroqp, with a Jeft Haar measurer (defined on the Borel o-algebraI of G). Then (G,F,p) is segarableand o-finitg. SupposeG is a countable densesubgroupof G.-Then G acts on G as left translations:?"rF= /F for t in G and fr in G. Jhen it is easily seenthat t ' Tt is a free, automorphisms. ergodic action of G on (G,T,tt)as measure-prese11,ing * cr,is the induced action of G on M-= L'(G,T,p) it follows that t !f M = M oo G is a semifinite factor. Thgn M is of (i) type I, (ii)r lvpS IIr or (ii)- type II- if and only if (i) 6 if discretea.ndG = G, (ii)r G is not discrete, but compact, or (ii)- G $ not discrete and noncompact (since p is fjnite if and only if G is compact). Example.p are given by (i). G = G = Zn, the cyclc group of order n; (i)- G = G = Z, the infini'te cyclic grciirp;(ii),'27 C = T = (z e t: lzl = l) (under where Q/2n is irrational; and n e multiglication) and G 1ei'u: (ii)- C = lRand G = Q (the rational numbers). n Example4.3.f5. In order to use Theorem4.3.13to constructexamples of factors of type III, one must have some condition which will of an equivalentinvariant measure. One ensurethe non-existence such is given by: Asscrtion: If G is an ergodic group of automorphismsof (X,T,1t),if G o = { T e G : p o T = t r l I G , a n d i f G o a l s oa c t se r g o d i c a l loy n (XJ,t), then there exists no o-finite positive measurev which is equivalentto p and G -invariant. Proof. Suppose such a measurev exists. If g = dv/dp,T e GoandI (measurable) function on X, then is any non-negative
'u''"= n',:,'," ,ii,i:l::;;::,"'=',,,'!
since / is arbitrary and ?" is an automo.phir., conclude that g o T = C a.e.(p). Since ?" was arbitrary and Go acts ergodically on (X,I,,r) c o n c l u d e ( b y o b s e r v i n g t h a t g ' r @ ) h a s f u i l o r z e r o m e a s u r ef o r e v e r y B o r e l s e t E i n l R )t h a t g i s c o n s t a n t :I = r > 0 ( s a y ) . T h e n p = r - r v i s Go I A . This G -invariant, contradicting the assumption that c o m p l e t e st h e p r o o f o f t h e a s s e r t i o n . Let (X,f,1r) be the real line with Lebesgue measure; let G be the g r o u p o f a u t o m o r p h i s m s ( T n , o :p , q e Q , p > 0 ) w h e r e f D . o ( / ) = p t + q f o r t i n l R . C l e a r l y ? " 'n i s t f , 6 i d e n t i t y e l e m e n t f o r G a h d i t i s e a s i l y s e e n t h a t Z o , oi , f r e e i T ( p , a ) I ( 1 , 0 ) . I n t h e n o t a t i o n o f t h e a s s e r t i o n ,
'tr)=(0)r (ll)
:g tt! J llD to/ a - a o a sa{sqos puo TI o7 Tuapttttba sr qznl$ A arnsDau a1rur/-o o ststxa atatil (l)
suotltprorButuo1lo!aqa .gI.€.t .do,t4ut st)aq o ta7 ,;:;iT::ir1::: '(-'01= (g adtr lo st (c) D ).r<+ 1111 ry ( l > r > 0 ) : { 2 .) u : " \ ) n { 0 ) = ( g ) / < + \ I t t a d f il o s l r y ( q ) !{I'0) = (O )r <+ 0111 ad{1lo sry @) 'e o@(fl'r'x)-Z totcol alo aq n ralpuo
{f,'!'X-)_7 uo g {o uottco pa?npu, ary a|ou{p p ta7 .t11ocrp67ta puo Alaatt sl)D alunsso tullt i(d,t,X) acods atnsoatu a|rut{_o pun alqntodas 0 'gI-€., uollpodorg o lo stusttldrotuotnp/o dnot? alqpruno? o aq O ta7
' 0 1 ' e , ' vu o l l l u r J e c ol dn uorssncslp eql pue 6'Z'V uolllsodor4 go ecuanbasuocfsea u E s l l J e c u r s ' l n o l l e d s l o u I I r ^ \ a ^ \ J o o r d a s o q / A. l l n s e r E u r n o l l o g eqt t(q uaroor{l lur{l luo{uEna lsnur a^\ .slql op oJ .srolceJ IIJ e d l ( 1 g s u o r l e J r J r s s € l 3 q n .ss a u u o 3 t u r u r o c u o c E u r q l , { u e , ( e s 1 o u s a o p 'pouJecuoc sr srolJ?J ll Jo uoll€cJJlsselc uusurne5l uo,r-,{euntrAl?ql sB rBJ sc f:olceJsrles fleloldruoc sl gl.g.t ruoroaql olrr{^\ l?rll oclloN [
'{2. ) u :u\} n {o} 3 (g)t leql roJur
nP, '(2, ) a:u\) j unrOit J onp eSurs :(g)t 3 {Z r u :u\) l?ql easol (gl.g't) .xg ul se enEre:lurH) .\lil ad,(1 3o sl O "6l (Ui)-Z te{t ,no{S .(U)_Z uo g Jo uortc€ IErnlBu oqt oqDl3lpu€'l>\>0oraq^\
{Otb,Z
tu,o'"J)=
O le.I (ff.g.l)
( . d = t i p / a o t t p p u e : d b - ' d J = J e r a r { ^ \g j f J n J ler{l e^resqo pue Z U (b + gvd) = {,-lol :0 < (? u (b + gr_d))tt l€ql qrns 'crpo8ro sr O t b lcrd {i) t b :o'rJ} ecurs l(g t t:tr_ti1 = A,_d eraq^\ '0 < ( g y d ) t t o s t € ' 0 < i(-'0) t d.yle :og (il), t d iuql moqs v G ) ! t O J I "o (ULZJeqr ^o,{S .SI.g', o1 q8ntiua sr 1r:1ur11) 'rUI actfl Jo sr O alcfruuxg ur se (g)_7 uo tre (O < rt ,O ) b,d :'?) = O rc1 (St.g.l)
sssIJJAXfl
rJo{=U)Jbareq,vr'111 oo (Ul)-z ad,{1 3o rolcsJ B sl ,nut ,^Oo,ro, lI 'C(ll) tt'e't eldruexg '3c) g uo fllecypoEraO stce {1ieo1c r{clr{,rr.{A t b 'o'rJ]r= oe e^aq a^r srolJBCgo seldruexg '8,
LVI
4. Crossed-Products
148
Proof. This is an immediate consequenceof the equalitv S(fr) = r(G ), Theorem4.3.13(a) and Proposition3.4.6. 0 4.4. Continuous
Crosscd-Products
and Takcsaki's
Duality
Thcorcm
The symbol G will, throughout this section,denote a locally compact (Hausdorff) group which, for convenience, will be assumed to satisfy the second axiom of countability: i.e., G has a countable base of open sets. It follows f rom Urysohn's metrization theorem (second countable and regular implies metrizable) that G is metrizable, separable and o-compact. It follows that L2(G) -- with respect to a fixed left Haar measure, denoted simply by d.s -- is geparable. If Xf is any separable Hilbert space, the symbol Xt will denote the Hilbert space Z2(Gilt)-a typical member 9f,-w[i^ch is a weakly measurable function [: G - lf satisfying lll((s)ll'ds < @. (As is customary, two functions will be identifieb if ttrey agree almost everywhere.) A useful fact -- which will help overcome annoying measurability questiong -- is that the space C.(G;lt), of strongly continuou; functions l: G ' lf which-are compactly supported, is dense in X?. It is a standard fact that Xf is canonically isomorphic to It @ Lz(G), whereby I o / corresponds to the fugction s - /(s)l; when convenient, we shall employ this description of Xf. As usual, we shall let t - \. denote the left-regular representation of G in t2(c): (rrD(s) = 1r'rs); we shall let I - r(t) dengte the (clearly-strongly continuous unitary) replesentationof G in lf given b y ( r ( t ) ( X s ) = q ( r - r s )- - o r , i n t h e p i c t u r e l l = t t @L ' ( G ) , \ ( t ) = 1 s 1 r . Suppose now that (M,G,a) is a dynamical system -- in the sense of where M is a von Neumann algebra of Definition 3.2.1 (b) operators on lf. It is fairJy easily established that there is a normal *-isomorphism ttd: M - t!t) such that 6
9
(2"(x)(Xs) = cx.-r(x)l(s). (fi;;f,show that zo(x) maps C"(G;lf) into itself and that llno(x)fll < l l x l l l l f l l f o r ( i n C . ( G ; l t ) ;t h e r e s t o f t h e v e r i f i c a t i o n i s r o u t i n e . ) A s in the discrete case, when there is no possibility of confusion, we will sometimes drop the qualifying subscript cc Also, as in the discrete case, we have the following basic commutation relations, which are easily verified: (4.4.1)
if t eG
and x e M,n(ar(x))= r(r)11(x)\(r)*.
Dcfinition 4.4.1. (a) Two dynamical systems (Mi,Gd), i = 1,2, are said to be isomorphic if there exists a von Neumann algebra i s o m o r p h i s m n : M r - M , s u c h t h a t n o < x f= o ! o n f o r a l l t i n G . (b) A dynamical system (M,G,a.), where I'I g t(lt), is said to be unitarily implemented if there exists a strongly continuous unitary
D
'qo[ aql seop
I @5 = (g)U,(q paurJeprry-try:4
duruaql l€rll pue
' ) o r U= ( c ) r zo 6 t u = ' I ) I E I { I ' e s B cs l t { l u I ' r e a l c s l lI ''nur. f,roJ IOX=(x)z pue ) oIa=zgasec ut uolrlsoooroaqi 'llnsar aaord ol s._reluxJq /nar^ ur .seJrJJns\ ,(rn ul x rog (x)u Jo = *nxn lsql qcns rx-t$:n r o l e r o d o f r e l r u n € s r o J o r l l J I . . e . 1 )I B I l € d s sl U usrqdroruosl eql uaq^t anrl i(1ree1csl uollJsodord aql aculs 't^ uI x lle roJ I o (x)u = *r(t e xjn teqt q.n, y o-h , -, ) E 'll in Jolztedo frelrun € pu€ ) ecEds lroqllH € sr eJoql uor{l '(4)r ryg areq,n 'surqatle uu€runeN uo^ 5 usrqclroruosr. f,g ue st Jo * 1147iu JI leql sfes qcrq,n (txlol finsuoc ieu ripear eqj qcrq^ ;o Joord 0r{l .ro3) rerruxrq Jo rrrrroaql € uo pes€q sr JooJd oql .Joord
u1r to! (r)91= ((l)Dr)upuDn u1x to! 11x)u)tbu= 11r;IDu)!,ory Orl, cn - gloo tn : Dzo@ U wsttldtotuos!uo slslxa atatfi uatfi ,Z,l = ! ,(p,g,ln) swals{s lvcltuvttt(.plo tustqd.rotuos!_ uD -saqstlqo$att?lttfrr ,iotqa7p uuotunaNuod,to tusltldlowoszuo st G1,,g * ryy :u lJ -?.r.p uolrlsodor4 trI ul osuapf1ryean-os1oy,gacuoq pue ,n go erqaEleqnslurofpr-g1as e sf 04 lerll (I't't) uorienbo go icuanOlsuocotelpa*rur u€ sl U
O
' {"'
'r', = tt '9 e tt 'n z tx
tit}= :1tl;11tx;Du
(Ol ? ,.-1
'"((c)rn @fu)
fi uo srolBrodoJo urqaEle uuErunaNuo^ aql aq g 2 N fie paurJap -@VI = y'g let 'uclsfs lecrruuufp € sl (p.Ck) .ll (e) .€.?.t uoplulJaq -
'euo poluouoldrur flrrelrun B ol orqcuoruosr sr urelsfs O pue (p,g,n) sruals{s leclueufp ^ue snql lcrqclrouosrote (9,C,(I4l)Du) leclureu{p aqf (puu (nYu uo g Jo uorlcs ue peapur sr srql) ler{l s,(es(1'y'g)uorlBnbaorll 'g uI , pue (nfu ul x -rog*(l)r r(l)1 = (xfg .erqa8la {q peur3ep uo g uorlcu eql sl uuerunaN Jo d JI -ursrqdroruosr-* !nf, .Qth otq uol € sr (;,g)Du n jo Ierurou -luql s,r{oIIoJll e st Q)f - n lu acurs ueql 'ur5is{s lejrrueufp e- sr.(n,9,141ill (q) -'(6\l'(W)vu)pue (ro!7tl) suralsfslecrureufper{l Jo usrqdroruosr u€ seqsllq€lserr'g puu' 'palueureldrurfyrrelrun sl (dt.Of)9u) uaqt U ul , pue (n)vu ur r roJ ,,?vxrtv = (x)rg fq uear8 (7g)Quuo g go uorloe oql sl g JI pu? ,n uo tqtla,n suJ E sJ 0 "ll (e) .g'y.galduerg
A consequenceof Proposition 4.4.4 and Example 4.4.2 (b) is that while dealing with crossed-products,we may assume that the underlyingdynamicalsystemis unitarily implemented. Lemna 4.4.5. Suppose the dynamical system (M,G,e,) is unitarily implemente{ by the jlnitary representationt - u, of G ig lt, Then the equation(wtXs) = z.l(s) defines a unitary operatorw on ll, satislying wn;!x)w*=xol,xeM w \ ( s ) w *= u " 8 \ r , s e G ; in particular, id i, spatialtyisomorphicto {x @ l, tr" @ \r: x € M, s e G)". Proof. The verification is routine, and left as an exercisefor the reader. O Assume,henceforth,that G is abelian,and let f denote its dual group. If 7 e f, let v., denotethe unitary operatoron tz(G) defined by (vtlXr) = 'rt(t). It is clear that 7 ' vt is a strongly contin'uousunitary representationof r in Lz(G). (Recall that r is topologizedpreciselyso as to make the map 7 ' continuousfor each t in G.) Consequently,if we set r(7) = I o vTv.then7 ' v(7) is a strongly continuousunitary representationof t in'H. Lcmrna 1-4-6 With the Ioregoing notation, (a) v(7)\(t)v(7)*= -1\(r),if 'l e r, t e G; (b) (v(Z):7 € r) t not(tuO' . Consequentfu, Jhere is an action d, of r on inI given by ,(7) v('l)iv(1)*,7 e M.
=
Proof. (a) is a consequence of the readily verified equation r7\rv-t = (b) is a routine compgtation. The last. stlrtim6nt -l\r, while follows-from (a), (b), the definition of M and (the extensionfrom R tl to generalG of) Ex. (2.3.4)(b). I*nnz 1.4-7, If n, (Mt,G,d), i = 1,2, and 7l are as in Proposition 4.4.4, then il o 4 - 4 o'11for att 7 in t; consequently, (M, o
or
G) oar I = (Mz td
O) *A, t'
it suffices to Proof. Since fl and dL are normal *-homomorphisms, verify that n(at(D) =8' 7(n@), for 7 in n(-r(Mr)u r(c); this follows
arer{^\'(/(es)07g= *4 uer{] $d!. = q '(L)1(L)a = (/)( !c) {q peurJap(U:l)zZ = & uo rolurado {relrun eql sr g JI lpql S.t.t €ruue.I ruorg smofloJ ll '{G),r) fq petuauraldur Iprelrun sr {f,p) ecu15 .Joord '(t)t7 otuo (g)rZ utoti rolerado frelrun € sl * rurogsucrl laraqcu?14-ierrnog ./ 'uo / ,(Dr7 eql pazrleruJou os J ltrtl arnseeur rBBH eql u l I J o u o l l e l u e s e d e rr e l n t e r - l J a l e q l s a l o u o pL \ - L . e s J n o J g o e r a q , t r ',(! > L'9 t 1'n r t :f1 o/,to
I'I ol\ o
In'I
o I o r) = Iltl
aror{,tr'((irZ @(D)N1o nh = 5 t^ = ,J .I dars Qh 'sursrqd-rouosr 0lelporurelul go ocuanbes? €rA usrqdrouroslperlnbar eql qsllq€lse 'g ur.t pue ur r roJ In II€qs eA\ n ]nxrn = (r)'o reqt qcns fi u1 9 go e J uoflBluesardarfrelrun snonurluoJflEuorls € sr arerll leql olunsse feru a,n '(q) Z'V'V oldurexX pua L.r.V Bruruel Jo ^\or^ uI .JooJd ' ( ( ( c ) r z ))rt ' n t x : t @ x| tq pa7otaua8(drZ o ll uo srowradolo otqa?li uuouna1l uo^ aql st (O)s?h & n ,astno?lo ,ataqu .((C)rZ)f @ n = tg T-t-t urroaqJ
'(n'J '. c)rz
=
Q)" 2o (c) r r o u = Q)rzo A = U '(g ) r 3 t 9 e N = U :(uiC)zz = (C)rz@$ = A .(g)r5 Dbs W = U 0lrr^\ IIBqs o,r !(g)g j pg anqm'(r'g'14) ualsfs lucyureu,{p€ rlll^\ Iro^\ Ilsqs e^\ :palpueq eq ol suorssordx, er{l Jo ssouauosJsqrunc?ql ezrururu IIr^\ qJlrlr$ uoJl€lou sruos xrJ ol osned sn lol 'ror{lrnJ EurpaacoJdoroJag EI
'g'g'p 'dot4 go acuanbasuor 3lsrparurur
u? sr ?rurual aql Jo uoluasse puocas eql pue .(,p.J,gfo !r'tl) sruats,{s leclueud.p aql uee^rleq rusrqdrouosl u? saqsrlqeisa U leql sffrolloJ lI
('(((x)I"u)u)h= ($)r)toDb = = 11r;rDu)U = (((x)rnu)fu)U 11r;u)zDu ,lw , r JI 'eJuelsurroJ) '9't't Burua-Ipue VV'V'dor4 uorg ,(lolelpeulul slcnpord-passorc snonurluoJ'r'i
ISI
4. Crossed-Products
152
M o ={ 7 . l " r ( r ) , v ( 7o) \ r : , 7e i * , 7 , r } ' c r ( me z 2 ( r ) ) . Now if w is the unitary operatoron il gilren uv (wixr) = iltf(t), then, again by Lemma 4.4.5, wfuw* = {x o lrr("), ur o \,: x e M, t e G)"c r(xf@L2GD. Concludethat if o !rz,s,)* c r1il o L2GD, Mr= (w o l1z11y)Mo(, then, sincew e {v(7):7 e f} t (as is easilyverified), t ' t = P t r = { x @I @l , t r @ I r e l , I @ v r @ \ r : x e M , t e G , 7 e f ) " c r(lt I L2(G)@r2(r)). Step 2. Mr= Mz c E(lt e Lz(G)@Lz(G)),where Mr= {xo I o l, ilro \ro l, I ev7€ v-/ x e M, t e G,7 € l)". Proof. Let 7: L21r1- Lz(G) be the Fourier-Planclrereltransform (on identifying G with'ttre duai of t); thus (ftxt) = l.t,7t-1((id7 if I e r21r; n zllr;. The following facts are basic(and easily established): F Iy F* = v? ord I v, I* = Pt (= \-r) for 7 in t and t in G, where of coufse v* d'enotesthe multiblicatibn operator on 12(r): (vrt)(7) = .t,7r'Ll(i). It follows from the unitarity of F that I
r = I I I o F : l t @ L z ( G )o Z , 2 ( r ) - l t @ L 2 ( G ) @ L 2 ( G ) "basic i s u n i t a r y ; i t i s e g s i l y y N e d u c e df r o m t h e a b o v e - m e n t i o n e d facts" about F that f Mrf* = Mr. SteP 3. Mr= Ms c l(Xf o Lz(G)), where Mr= {xo l, ilro It, I @vl: x e M, t € G,7 e l}". Proof. Identify L2(G) @l2(c) wittr L21G^* G) and regard Mras a vorr N e u m a n n a l g e b r a o f o p e r a t o r s o n 1 ?@ L z ( G x G ) . I t i s c l e a r - t h a t t h e equation (aiXs,t) = l(s+t,l) defines a unitary operator on Lo(G x G). A minor computation revealsthat, with the above identification, the following equationshold: u*(vt o v7)u= ,7 @ I and a*(\, @ l)u = \, @ (Reason: 1,7el,teG.
((r7 e vr)!Xs,r)= <s+r,7>-18(s,/), ( ( \ t @l ) E X s , s=' ) l ( s - t , s ' ) . ) Conclude that
puE (surqaEleuuBrunaNuol Jo slJnpord rosual Jo suorlcasratur Eururecuoc) a^oqu IJBuaJ f reururlard rno urorJ s/r\olloJ ll i N o n S l U o I = / r { ( l { oI ) * n $'V.V) 'ocuoqpue 'J u1 l. 11erc1 (L)n = u(L)t*u l€ql perJue,rfylsea osle sI ll
:((c)sz)r @n =,fi o,n)3 (n)"u (e'v'v) l€rlt parJrre,rflrsee sl ll (rAl ur rr eruosrog (s)l ,r = (s)(!,9) urroJ ar{l Jo sl I 6 rN vl JoleJado{rala ocurs).pueq-raqto aqT uo we (*)r = r(No :03(n)Du = u(l s h}*u
(Z'v'v)
'(q) '{lluonbasuoJ .rN = .'e'r :uellaq€ 9't't €rrrula.IpuB S't't ?uue1 dq rV Ieturxerusr N l?ql srsfleue cluoruJ€r{ruorJ lc€J pr€puels E sI lI .goordeql qtl^\ u o l a E s n l e l ' g t Q r u e a . rfdq L a u a ls J r l lr a U V ( i ( C )( S I ' t ' 0 ) ' x g E u r s n ' s 1 q tf g l r a n ) ' ( z , r u ur r r g ) o( z w v r n ) = ( u ozn) u (INe llg) r;qr IuoJooqlluelnuuoc elqnop oql pug 'slcnpord Josuel Jo sluelnuuoc tururacuoc lcBJ tulpecard aql 'lc€J a^oqs eql tuorJ st\olloJ 11 lserqe8luuusun?N uo^ Jo lcnpord rosuol eql Jo uolllulJep l(q rlr) o ,(zrt{n ,,(2.r4nrn)
= ,((zN @Gn) n (rN o I/,t/))
uot{l 'Z'I = I JoJ 'serqeEleuuBlunoN uol er€ ( )f i !r,Upue (A)f 5 t1a1 y :1ce1o^oq€ oqt Jo ocuonbesuoc l(sea ue ur€lqo ISJIJ sn lal '3oorcl eql qllr$ Eulpoacord aJoJog 'ry'{ @ tN = tW s n) lstll ((q) 7'g'7 aldwexg ul peuoltuaru)lcey 3ql uo i(lr^€eq flrrug flar 11eqsr,t\ /{ 6 N = floU O n)*& d^ord oJ
ytl,".Tl: = ,({er r :rr)n 19)'spron eqrpus((e)z.os, ,ri1Ttili?,,ff
sroqtuatu"e'r) suollelsu€rlIIs pue (1gJo sJaqrueu"e.1)suoyleclldlflnur IIs rtll,rr olnruuo, sJol€JadoJ€lBcsfluo l€rll IrsJ pJ€pu€ls€ Je^e^toq -,,((g)t s1 = ucn*u,g.v., 1,,({9r :11} (lt{ tr I n o n o il) .r,{ n)) €IrIua'I Jo ael^ uI 'eAerl ueql plno^r a,yr,:l{@n = or(1lg O n)*$ Wt1l /t\oqu s e c a ^ \ e s o d d n s' , { J r l , ' n ? x : f , a O l ' l O x } = i / O n ' ( q ) 'aroq,n 'd,({g lnl n (/{ o p'g'7 aldwexg s? r :11 I 6r N))*il = atn4tl II 'K')r7)x = uEn*M ler{l ^\oqs II€r{s oA\ uaql 1,,(gt :Lrl = L le.I & i{ hl 'g't't Bruruo'Iur s€ uo Jol€JedofJelrun eql eq ar lal 'uIeEV .JooJd J[
'(lilrz)t@w=82'ldats '8n=re8n= 'g t 7 ' 1 4 72 x "(J r ,( :l Of,r6 I'I
O
1\
O
tr'I
O I O r) - (n@04n*@O t)
€9r
slcnpord-pessorJ snonulluoJ't',
154
4. Crossed-Products
e q u a t i o n s ( 4 . 4 . 2 ) ,( 4 . 4 . 3 )a n d ( 4 . 4 . 4 )t h a t w*(M @ N)w c M @N. For the reverse inclusion, begin !y olserving -- after ag easy comp"$tation -- that if x e M and \ e !t, then (w(x @ l)w*()(s) = cr,(x)t(s). If "I is the unitary operator on r21G1 given by (/tXs) = l(-r), the previous equation says that w(x o l)w* = (l e /)n(x)(l @"/) e M s f(42(O), and so, w(M o l)w* c M @ E@?GD. Also, notice -from the above formula for w(x o l)wt -- that w(M @ l)wr c (l o N)' = l(tf) o N, so that, as before, w(M o l)w* c M @ N and w(l e ly')w* = I o N C M @lf, whence w(M oN)wt c M @ N, and the proof I is finally complete.
Exercises (4-4.9) Let (M,G,a,),{ur} be as in Theorem 4.4.8. (a) Show that there is an action g of G on M @ f(42(C)) such that 0, = ad(ut e^ p) for all / -- i.e., €r(x) = (u, @ pr)x(u, @ pt)* for x iri M @ f(L'(G)) and t in G -- where, as usual, t - pt denotes the r i g h t r e g u l a r r e p r e s e n t a t i o no f G i n L 2 ( C ) . (b) Since G gray be identified with t[e dual gro3p of r, the action d of t on M induces a dual action E of G on M. as in Lemma 4.4.6. Show that the dynamical systems (frt,C,6 ^na 1A 6 r(I2(G)),G,e) are isomorphil:, where 0 is as in (a). (Hint: under the isomorphisms tt = M, = M, = M, = M @ EQz(g)) established in the proof of Theorem 4.4.8, let the action d be successively transformed into dl, d2, cr3 and q4; writing ad v for the a u t o m o r p h i s m . l r - v y r * , s h o w t h a t c r ] = a d ( l 6 l € ' v", ) , c (" 3= a d ( l O 0 I apt), cf = ad(t @pr) and a.l = ad(urop,) = 9,.) Dcfinition 4-4.10. A von Neumann algebra M is said to be properly inf inite if there is a sequence {""}i=, of pairwise orthogonal projections in M such that leo = I and en - 1 (rel luI)for all n. n Observe that (by the proof of Corollary 1.2.4(a)) every infinite f a c t o r ( i . e . ,o f t y p e I * I I - o r I I I ) i s p r o p e r l y i n f i n i t e . Lcmna 4.4.11. Il M is properly infinite, and if ll is any separable Hilbert space, them M @ t(lt) = M. P r o o f . B y h y p o t h e s i s ,t h e r e e x i s t s a s e q u e n c eo f p a i r w i s e o r t h o g o n a l projections enin M and partial isometriesvnin M such that Ie,, = l, ulr,, = e,, and v.vl = I for all n. Notice that vmvl = 6-rl for all nt and n. Let (ei.;) be matrix units in f(Xf) -- i.e., pick an orthonormal basis
ros jn(x)rnr - r,n leql ur#, n = (x)tg pue (rr)snsn
[hiro:t, ? Tjr:
tl;
- / deu snonurluocflElorls € sr eraql luel€^rnbaialno aq o1 pres JI ere II uo 9 dnorE lc?druocf11eco1 B Jo g puu D suollce leqt II€coU serqcEly uuurrrnrN uo^ rlJuIJuI [1rcdor4 Jo crnfruls
JqI
-g-,
't Po (c oo n) - 1g'atolaq t^l uaU'(g'r't owwaT ut q ht sn) = uo'-1 lo'riottristrrp'rri ltt!^'U .q otqa8p uuownaN sarcuap p uot fI a|ru{At o tto g dnotE etadotd uotlaqv tcodtuot {11oco1o lo uotlao uo aq b ta7 Zl.r., uorlrsodord Eutarrollo3aqt leE arrr .Il.t't
. r u a r o e q lf t l l e n p euruo.I pue 8't't ruaroar{J Sulurqruo3
sr 'urn1ur 'qcrqm.rla o hr oruo(a)r e nDp #rr?.it"tlff3r-XT
s?ulJep,f*n - x deur eql s^rolloJ lcrrloruosrsr *r, oS .I e I = leql 1J "a *nn_pue @ I - r*l"e.Jaq^\'(Ahe t n s ol (*{lSuorls-o.loeJ ul) n *flEuo:1sse8rarruoc {r,) teqt epnlcuoJ.(,{lEuo-rtslo 1ce3 ur) ^fSuorji !!, I=! 1*
{
'
I=r
nu" r*'a q acurs
.["roti'n] e = ([,rr I4(n -tn) ol It{,1\
'rrr. [r" {l 'n ]
'[ttr tit]
= (n - ' n ) ( ! n- ! n )
', < I J I l e q l ^ \ o q s s u o r l e l n c t r u o cI B c J l u a p I
@r = ! ! a * ,
tit=
1 ! r r r r ro; 1 j , r r , r y l {=' rl n i n qll^\
.rra,
= [ral!!]
t{'t rra e lnint{' = = '1r1, 1ttr"r, o (rnin)
'(a)r
l8r{l 3^rosqopuE tit = t, o hl > rta e to
e u r J a p' " ' ' z ' I = I r o r . l l ' ' 1 , = r t r r r l o u " , t r o J ( f r ) 9EI
serqaElyuu€tunoNuo^ alrurJulflrector4 ornl3nrlSaqI .g.V Jo
4. Crossed-Products
r56
Lcmna 4.5.t. If aand B are outerequivalentactiortsoI a locally compact groupG on a vonNeuntannalgebraM, thenM @dG= M @BG, Proof. Assume A S E\ll), so that both the relevant von Neumann algebrasact gn it = t2Q:tt). It is easily checkedthgt the equation (tixr) = rr -,i(t) defines a unitary operator fr on ll, with inverse t given by (t*?Xt) = u*-.,f(t). Let x e M, t e G; then t = u,-rcrr-r(x)(ii*l)(s) 1aro1x)il*i)(s)= ur-1(7lcr(x)il*tXs) = ur-rcx.-r(x)r*-ri{r)= B.-r(x)i{s)
fr \U)il* = n3(uf)I(t), and hence,i(M ooG)fr* c M @gG. An entirely simliar computaiionshows that fr*(M eg Gry C M @dG, whenceM @dG and M @gG ate spatiallyisomorphic.' E corollary 4.52 If 0 and 0 are fns weightson a von Neuntannalgebra M '. t h eonL I @Y^ R = o M @Y, t , R . Proof. This is an immediate consequence of the unitary D theorem and the preceding Lemma.
cocycle
proposition 4.5.3. Let 0 be a fns weight on M. Identify M with n_q(M) and assunte that M c f(fi), ,t = tr6 and n6G) = x lor all x itt -M' Let a be an action of a loially contpaci abeliah group G on M sucl that Q o dt = 0 7or ait t. Then there exists a (canonically constructed) fns
tirt;/f, .x:r Jo esuas = rp a:eq,n.((g.s.z) eql ur) rs a = H ^o ;:lH
:olerecto arrrlrsodeql roJ eroJ e osl€ sl 0g teql (CI(9i.Zl lurofpu-g1os 'xg ruoJJ s,tolloJ ll ,(zlty roJ {lluanbesuoc pue)-g roJ aJoc e
sr( n)u (uorlrur3ep ,(quitFre) pue ocurs
=
,lllzrrvli ,lll{ll :r"_ur.: uG ..S l?Ju6irrui;5p ocurs'pu€q reqlo eql uo lg rog aroc E sl oql ^g lo
't-'
[1t,llsll+ ell(,)ll;{t
pu€'O ul /A ( n)u I (t)1:U I l) = oC
,^; ;u,:^ .,i, =osrr
'(-, (zll(*(r.r)1)ull + r11((r.r)r)ull)cil
pue 'O ul ,A n r (t./)I = ) 4 Xl tJ l€ql acnpop.9ut, pu? N ur r roJ = ecurs ll(x)ull ll((r)ln)u;; - t
:(*(r'r-r){)1-o= *(/' r)x = (r.l)rg osls
ci' '- t r ll( r . r 1 1 ; t r ; ; puB g ul t (),t)X <+ N r ! teql ,A i\/ .9 ut pire ((e) Z'q'V'dold 'Jc) trexor,4 t ?rn{ Jr ry ui r r#3 fGlbll = (x)urn qJns-g ul leql uolleluasordaj'r(relrun | 5l i,, are.1l JO = 1o (N;tn lBrIl pue 5 o tuqt uoitctunssB N orll -oig s^o11og 0 0 lI .f le.,rrlcaclsar .f1a.,rr1cedsar
{
,
!
puu 1IV U N roJ I? puB n elu^\ osle sn te-I
pu , Qv4 t 4 s 4 u 4 i l ro J
.N
9v 9t' 9s 9, 9r y v. r . . s. u p u gv. 1 ',S 'u elrr^\ a^\ 'eroJaqsV .((r.r)f)9 =-(90-l(f ll€qs (f.p uorlcag ur se) paurgepn uo lr{E1a,rsuJ er{l alouep0 lal .gf, uoltcog Jo JI€rl lsrrJ aql ur poJSlunocuauoll?nlls eql ur tSuq are a^ .iseJ olorcslp eql uI '9 elarcslp roJ Joord aql luasard [1uo 11eqse^\ ,([€ {sJ] .Jc) IBcruqcelpu? pa^lo^ur alrnb sl llnsar slr{l Jo 3oord aql sy .Joord DtuuaT ttt so pau{ap sl 2 atatli .J u! L tp to! Q = Lp o Og',y;yl -uo J Io p uou?olottp aW rapun uottoa,w st Q-ttl7nu pfp aW (q) ry :rrtv uoo ut I to! (,)!rt|, = e)er1\j) puo
{- > tpcll!)lrfrvlllpuo, po'o/ ttQre ruopr (,U:l r 1} = r7{v uop (e) iprlo^ arD stuauarDts 3ut,,lo11o/atfi puo Wpl = ^ 9u rct11rtou o Lllns u, ht ,tttn petlttuap aq tDu (ll:il.7 = ll tztlt qrns n uo q ryBnu
.S., serqaElyuusunaNuo^ alrurJulAlroclor4 Jo ornlcnrlsaql
LSI
4. Crossed-Products
158 It is clear that the equation tilltrl '
= , -,Ji(rr) l-L
defines an
antiunitary operatoron il; obseruethat if I e il, th.n
= n(i( e ,t)*) = n(if(t,e)
= i(;*)(r). 6
-
-
9
-
6
-
'
ThusJ-I/ [ =S I qhenever( e Do. SinceDoisgcole-forf/as well as for S, and since J is antiunitary, deduce that^S =-J H. Jnfeg,from the uniqueness of the polar decomposition that J = J and H = A't". For (b), notice that if 7 € r, then v(7) = .@^-rlft; r€G if i e inI, un easy matrix multiplication shows that for all s,t in G, (4.(3))(s,r) = <s,7>'ri(s,r),and ig particglar, (dr,(I)Xe 'e ) = I(e ,e); 'follows (Note: The f r o m t h e d e f i n i t i o n o f 0 t h a t 0 o A '^tth' = it is Q. proof of the commutativity of G is not used anywhere in discrete case of (a).) D
Exercises (4.5.4) Let (M,G,a) and 0 be as in Theorem 4.5.3. Assume discrete. If u, is as in the proof of Theorem 4.5.3, and if -u, e Z(Lf). -(Hint: use the formula obtained for show that ur' "fact that J2 = I to conclude that JurJ = ut; ana the I theorem.) O Tomita-Takesaki the appeal to
G is 9. M, ,I (= now
Thcorcn 4.5.5. If M is properly infinite (cf- Definition 4.4.10), there exists a properly infinite but semifinite yon Neumann algebra N and an action A ofRon N such that M = Nog R Further, there exists afns trace T on N such that r o 0t = e-tt lor all t in R' Proof. Let 0 be a fns weight on M, assume M ct(lt),tt
= tt1 and n6 =
,rH X tl=
@$o
acurs ' u, ul t IIe roJ 1'll = (/x lErll qcnsN ot pol€ll! JJeH rol€redo e,rrlrsodayqifiar'uruB slsrxe eJaql .uroroeqts.dirolg,(g lurotpe-g1as
= (rX t 1?;rdr'ilirl-;: srsl:*(r)1(n)1(1)1 so soordaq1selalcturoc .(,r- s)3= (/,- s)t ,flvfv=
(')(3,$vXr)"Oo f,o= .cror4 (g.e.p rq) (rXJ = ,tv(r)u)f,v (s)Q = (s)0((r)u)do) ,$v(x)u $v) .(U)r :s^\olloJ sBalnduoc pue , n.r.s r r ,I rsqr n U n JI 6V 'flrleraueE ssol i(eu e^r.(r) t{sllqetse lnoqlr/r\'erunsse ol JeprouJ Jo
'g'I'€ uraJoaql uorJ ^loIIoJ plno^\ N JO ssaualrurJrrles eql pu€ 'Jouur sr ^\oIJ oql l€ql ,{es pyno,n slql W r (l)1 acurs Oo
l*(r)rr(r)r=(r)jo
(*)
'U uI r pu' /{ u1 5 i(ue .,o, ,"Ui^or{s IIBTI' el\ '19'g'yuolllsodord ur se) 1g uo lqtrea lenp aql alou?p Q n1 'ellurJur Alraddid osle .e1rur3ur s1 flluanbosuocpyr 'r,{ sr os {l.raclord 1rr 3 sr e3urs @)u ' 'T,y'V'Vuorllsodord/,{ Aq .uaql U 0o N = n e^Bq a^\ W uo U (qtl,n {31tuepr rirr r.lcrrl^ig 3o dnorE lenp aqt) Jo -(po) ra-I .npl u o l l o el € n p e q l e l o u a po l a l p u e . U O o e N = h l = N
69r
s e r q a E l y u u e u n a N u o l a l r u r J u y{ 1 r a d o r 4 J o o r n l o n r l s e q l . g . t
4. Crossed-Products
160
for i in N an{.4 fn B it follows at once ttrat F n lfO. Hence, the equation t = Q(H-L.)defines a fns weight r on N. According to Theorem3.1.10,we have
oJ(;)= h-no?G)it"= x -
-
r
-
for all I in N, and consequently r is a trace. Observe next that if s,l e R, then S.(\(r)) = v(s)\(t)v(-s) = e-i'tr(t) [cf. 'passing to the infinitesimal genegator" Lemma 4.4.j (a)l; it fgllows -- by -- that s"(i-l) = e"H-r. (The last equation is me3ningful if*H-r is b o u n d e d ; b t h e r w i s ei t m u s t b e i n t e r p r e t e d a s 0 . ( 1 " ( H ' ' ) ) = l " ( e ' f l - ' ) f o r e v e r y B o r e l s e t E i n R . ) H e n c e ,i f i e N * ,
r og,(i):;.:;::,1_,r, t'" fui1-rx1
(by Prop.4.5.3(b))
= e-" r(7), Wbele, for convenience, we have written expressionssuch as o(a-le"(r)) in place of the more accurate(but more cumbersome)
rip &tF-f. e"(i)); €lo the proof is complete.
E
It is a fact that if cr is an action of a locally compact1p$ian group G on M, then fld(14)is preciselythe fixed point algebraMs. It can be deducedfrom- this that the dynamical system(N'.R, e) of Theorem 4.5.5 is unique in the following sense:if (Np B €') i = 1,2 are dynamicalsystems,if T, is a fns trace on N, such that Ti o 0l : N, @grlR e-tT,f or all t in R and i = \,2, and if the crossCd-products and'N, €gz R are isomorphicvon Neumannalgebras,then there exists a von \e-umann algebra isomorphismI N, - N, such that if 0r(x) = n-r(e3(z(x))for x in N, and t in R, then th-eactibns 0 and 0' of G on N, aie outer equivalerit.The detailsof this argument,and, in fact, th-eentire discussionof Sections4.4 and 4.5 may be found in [Tak 3]' Since a major portion of this book has been devoted to factors, it is only fitting that we concludewith the statementof two very beautiful and powerful structuretheorems: (l) Let M, N, g and r be as in Theorem4.5.5. Then M is a factor of type III' if and only if N is a factor of type II-; ( 2 ) l f M i s a f a c t o r o f t y p e I I I \ , 0 < \ < l , t h e r ee x i s t sa f a c t o r N of type II- and an automorphismcr, of N such that T o
aJnseerureeg lqElr e slrrupe g ler{l s^\olloJ l1 19 uo'rt arnsuau rue11 lqErr u seulJep ({g ) t :r-l} = ,-A, eragr,r) (r_Z)ttt = (Z)"rt uo_ll?nbo eql leql poIJIra^ fllsea sl lI 'g roJ erns€er,urB€H lJal e sl dzt JI 'suollBIsuBrl lJ0l Japun ecuerJelur palrnbor fluo arreq o^\ lerll lc€J eql urotg Eururruels 'g uo arnseou rB€H "Ualu rar3rlenb aql UOI e pclluc s! a^oq€ sB erns€otu v 'Tl? = A lsrll r{cns , Jaqrunu IeeJ e^rllsocl e sr a-roq1 u a g l ' o ^ o q e s u o r l r p u o J e e t q l o q l S u r f ; s r 1 r s " J u o o r n s e e t uJ o r l l o u e s r ^ JI '"I ul g fttata roJ (({3r r / :/s})/ =) (gs)d = (ilTt (11) pue l17 1as uado ,{ldura-uou f.rarra roJ 0 < (2)rl (lt) ig Jo X lasqns lcedruoc ,{raure roJ o > (X)Z (t) l€rll qcns "J uo peurJop rt erns€our a^ltlsod B slsrxa aJeql :slql sr sclnor8 lcecltuoc [11ecoy Eururocuoc lcEJ crsBq or{I ('[oof] ro [1 1eg] ul luoruleerl € rlonspurg {eu rop€er palseralur oql lsdnorE lecrEoloqlecl qcns orouEr lleqs o/r\ islos larog pue slas orleg uaatqoq qsrnEurlslp lsnu auo .,,eErey,, ool or€ l€r{l sdnorE JoC) '"J Jo lueurelo u€ ocuaq pu€ 'sles lcedruoJ Jo uolun alqelunoo B sI las uado fralo l€ql sernsua f,tluqetunoc puoJas pounsse eql 'g Jo slasqns lcedruoJ d.q palerauaE erqaSlu-o oql alouep c4 laT 'slos .ocuerualuoc uodo Jo osgq alqelunoc € o^Brl ol peunsse eq III/I{ rog 'qc1qm 'dnor8 ,{1yeco1 B sl g leql qlroJacuaq eunssv lcedruoc 'lueruale f1r1uap1eql Jo pooqroqqEreu lc€clruoce slsrxo a r e q l J I [ 1 u o p u e J I l c e d u o o l t 1 y e c o s1r d n o r E l e c r E o l o d o l e , , { 1 r e 1 r u r s lluaruala [1r1uap1ortl l_€sno_nulluoosl lI JI ,{1uo pue Jr snonulluoc sI sdno:8 lecrEolodol Jo 6g - 19 :/ ursrqdrououoq s l€r{l sr uorl€^rosqo slql 3o acuonbesuoc,(sea uV ',{yaarlrsuurlslcB surslqdrouoeuoq Jo dnorE slr l€r{l osues oql ur snooue8ouoq sr dnorE lecr8olodol e 'g1os1roluo g Jo srusrqdJouoeuoq ore -- g ul s poxrJ E roJ (s, * / 'dsar) ,s - I Sduureql 'e'l -- , Jo suolt?lsu€rl (-tqslr .dsor) -1ga1 oculs 'clnorE lecrEolodol E g llq.c {ldrurs pue .J. o1 roge-ri(11rcrldxe lou llegs a,n 'freurolsnc sl sV .snonurluoc era yt e / pue 7s - (/,s) { q p a u r g a pg - g p u € , - g x 3 r s d e r r a q l l s q i g J n s g d n o r E e u o [Eo1odo1JJropsnBH B sr L oror{^\ (r'9) rled e sr clnor8 1ec1Eo1odo1 y
s d n o u 9] v f t 9 0 1 0 d o r xtoNSddv
t62
Appendix
which is unique up to constant multiples. It may, however, be the case that the left Haar measure is not a r i g h t H a a r m e a s u r e . I n s u c h a c a s e ,s o m e t h i n g c a n s t i l l b e s a l v a g e d . Let r be a left Haar measure. If s e G, define a measure l, by p.(E) = p(Es); it is immediate that f" is also a left Haar measure and consequently there is a constant A(s) > 0 such that g(Es) = A(s)p(E) for all E in I". The definition of A implies that A is a continuous homomorphism from G into the multiplicative grup ry of positive real numbers; the function A is called the modular function of G; it is characterized, in the language of integrals, by the requirement that
I f(soas= a(/-1) J ^')a' for every I in C.(G). There are two special cases when A is trivial: (i) G is compact (since the only compact subgroup of R! is {l}); and (ii) G is abelian. A group is called unimodular if A(t) = I for all t. Another example of a unimodular group is the group GL(n,C). A s s u m e ,h e n c e f o r t h , t h a t G i s a l o c a l l y c o m p a c t a b e l i a n g r o u p ; a s i s customary, we shall employ the additive notation in G. A character of G is a continuous homomorphism from G into the compact multiplicative group I of complex numbers of unit modulus. It is clear that the set of characters of G is an abelian group f with respect to pointwise operations: if 'ly7z e t and t e G, then = ,where we write instead of 7(t). T h e g r o u p t , b e i n g a f u n c t i o n s p a c e ,m a y b e e q u i p p e d w i t h t h e s o "compact-open" topology: a typical sub-basic open called neighborhood of a point 7o in t is given by W(K,U) = {7 e t: 7$) C (f, where K is a compact set in G and U is an open subset of ?" such that 'lo(K) c U. It may then be shown that this topology equips I with the structure of a locally compact abelian group. Each I in G defines a character f, of I by the equation 0r(7) = . Since I is a locally compact abelian group, one can construct the character group G of t, which is again a locally compact abelian group. The celebrated Pontrjagin duality theorem asserts that the map I - 0, defines a homeomorphic isomorphism of G onto G . F o r t h i s r e a s o n , t i s s o m e t i m e sa l s o c a l l e d t h e d u a l o f t h e g r o u p G. ^If f e LL(G),define the Fourier-transform of f to be the function f:f-Cdefinedby
ittl = I .t,7,-r7g1at. I ^ tm a y b e s h o w n t h a t , w i t h r e s p e c t t o t h e t o p o l o g y o n I , t h e f u n c t i o n ,/ is continuous. (The^ Riemann-Lebesgue lemma goes one step furthcr and states that f is a continuous function which vanishesat infinity -- i.e.,is uniformly approximable by continuous functions of compact support.)
'Iueroaql uorsrs^uI pazrlErurou sr J uo aql ul s3 o r n s e a t ur € B H a q t 1 u q 1E u r u n s s e - - ( g ) r 7 U ( i l 2 7 u r / r o g = l L W , q t ! o s ( J ) e Z * ( g ) 2 7 : 4 . r o l u r e d o, { r e l r u n a r i b r u n e l l s r x a a r a q 1 " 1 e q 1s a 1 e 1 s qclqrl\ ruaroaql s.lereqcu€ld SI luaroaql uolsra^ul eql uo sorlar qrrq/r lrBJ ror{louv '(-r)0, or (O)r? ruor; { - / d€r,u orp -lo iilnliciful e q l s l r u e r o a q l u o J s l e ^ u l r q l . ; o - a c u a n b a s u o ca l € r p a t u u r u V 'elnseeru s r q l s e u J l " ( u Z ) s e u o s o q oe q = u o s r n s B e l ur 8 3 H a q l . o J n s e a u lsnru s e ^ l a c 3 : , r [ I . 0 ] J o s l l u n u u l l € r { l pazrlErurou sr 'arns€eur onEsaqal sJ qcrq^\ .,rUuo arnsBarurB€H orll JJ '
(r r
l=!
|.'o', {
'l
tJoxa =
,,(quaarE Euraq ,{1r1enp er{l 'uUJr{lr.r perJrluopl aq ,(uur 1 lenp aql .uU = g Jr eJu€lsur roJ 'LP(LY .L't> Jl = Q)l u o r t l ' ( 9 ) r Z t / p u e( C ) r z t ) , , , r q r ' o , J u o o r n s e a t u r€EH eql ozrl?rurou ol elqlssod s! l! lBr{l salBls rlcrq^\ luaroaql uorsra^uI aql sI sruroJsu€rl rarlnol tururacuoc lc€J clseq V ' r u r o J s u e r lp u B J l e C a q t s B p a ^ \ a r ^ eq osle feu urro3suerl rerrno{ eql 'snq1 '(C),2 ul I qcea toJ &)/ - (tL)I .;r fluo pue l q l u r L - \ L : o u o ; e o r t o l o d o l" e o s l e s r ' u o r l e c r J r l u e p l J I J J o , ( S o l o c l oe slql :(g)rZ erqaEle qcBuEg eArlBlnuuoJ ar{l Jo aceds leepr I€rurxBru oql qll/n parJrluopt aq ,{.eu J snr{l lJ ur l. sruos JoJ ()I * / r.uroJaql d-raz-uou fralo Jo sl O ol (C)J uro-r3 rusrr{drourouroqanr1ecr1d111nru lBr{l u/r\oqs oq u€c lI '((r)01 - (c)rz) serqoSle qc€u€g 3^llelnl'ullloc go rusrqdrououoq Eursearcap-urou E .snqt .sl ./ * / deu aqa
€9r
xrpuoddy
NOTES
(An attempt is made, in this section, to attribute credit (of a bibliographic nature) where due. If there are shortcomings, the a u t h o r a p o l o g i z e s ,i n a d v a n c e , f o r s u c h i n a d v e r t e n t e r r o r s . )
Chaptcr O The material in Section 0.1 may be found in any standard text on H i l b e r t s p a c et h e o r y , s u c h a s [ H a l 2 ] o r [ R S ] . The treatment in Sections0.2 and 0.3 is essentially as in [Tak 4] (Sections I and 2 of Chapter II, there); the strong, weak and o-strong t o p o l o g i e sw e r e a l r e a d y i n t r o d u c e d i n [ v N 2 ] , a l t h o u g h v o n N e u m a n n "strongest topology". referred to the o-strong topology as the The double commutant theorem was proved in [vN l], although the p r o o f p r e s e n t e dh e r e i s a s i n [ A r v l ] .
Chaptcr I T h e d i s c u s s i o nh e r e i s e s s e n t i a l l ya r e p r o d u c t i o n ( a l t h o u g h , p r o b a b l y , somewhatmore cryptic) of Part II of [MvN l]. The reduction theory that was brief ly discussed at the end of Section 1.3 has its "fountainhead" in [vN a]. Chaptcr 2 The positivity of a linear functional on a C*-algebra,which attains its norm at the identity element [cf. the parenthetical remark in Ex. 2 . 1 . 1( c ) l i s p r o v e d i n t h e C o r o l l a r y t o T h e o r e m 1 . 7 . 1i n [ A r v l ] . T h e e q u i v a l e n c e, f o r p o s i t i v e l i n e a r f u n c t i o n a l s o n a v o n N e u m a n n algebra, of normality and o-weak continuity (cf. the discussion
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Notes
Theorem3.1.10of this book is Theorem4.6 of [PT]. The materialon locally compactabeliangroups(found here in the closedinterval [Ex. (3.2.3),Prop. 3.2.6],as well as in the Appendix) may be found in any standardtext on abstractharmonic analysis, such as [Rud] or [Loo] for instance. Barring that segment,all the materialin Section3.2 up to and includingLemma3.2.8is from [Arv 21. The rest of Section 3.2 as well as all of Section 3.3 is from Sections2.1 - 2.3 of [Con]. Almost all of Section3.4 is from lConl (Sections3.1 and 3.2); the exceptionsare: (i) I-emma 3.4 here is Prop. 2.3.17in [Con]; (ii) the converse(Lemma 3.4.2)to the unitary cocycletheorem is from [Kal 2l; (iii) Sakai's result, used in the proof of Prop. 3.4.6"is Theorem 4.1.5 of [Sak]; and (iv) although [Con] contains Prop. 3.4.7,the explicit statementof Corollary 3.4.8is not to be found there. Chaptcr 4 The crossed product of M * L-(X,T,!I) by a countable group of automorphismsof (X,7,tt)was first consideredin [MvN l] and [vN 3], where versionsof Corollary4.1.9and Prop.4.l.l5 can be found. The notion of an automorphismor an action being free (cf. Definition 4.1.8)as well as Prop.4.1.1Ioriginatefrom [Kal 2]. Krieger introduced the concept of a ratio set in [Kri], where he attributes the inspiration to Araki and Woods' notion of an asymptotic ratio set associatedwith what they term an ITPFI (cf. lAWl). The equality ^S(t'(X,f,r) gcr G ) = r( G), as well as the conclusion of Ex. (4.2.13)were establishedin [Con], although the proof presentedthere is quite different from the one here. The description (in Section 4.3) of factors of type Ir, (l < r < -) and II. (in terms of II, and I-) may be found in [MvN l] and [MvN 31, respectively;further, Theorem 4.3.13and Examples4.3.14and 4.3.15had their origins in [vN 3]. However,the presentationof the segment[Lemma 4.3.12,Example4.3.15]is essentiallyas in [Tak 4]. The fact about M. being a factor when M is, (used in the proof of Prop.4.3.4(b)), is the Corollary following Prop.2 in SectionI.2.1of [Dix]. Corollary 4.3.19is explicitly stated,at leastwhen G is cyclic, in [Arn]. All the material in Sections4.4 and 4.5 first appearedin [Tak 3], exceptfor the structuretheoremfor factors of type III\, 0 < \ < I (statedas (2) at the end of Section4.5),which came in [Con1. The treatment in Section 4.4 is, however,influenced by [vD]. Dixmier's result on the structure of isomorphismsof von Neumann algebras-which was neededin the proof of Lernma 4.4.7-- is in Section1.4.4 of [Dix] (particularly,seethe Corollaryand the ConcludingRemark).
of free action,Duke Math. R. R. Kallman, A generalisation J., 36 (1969),781-789. W. Krieger, On the Araki-Woodsasymptotic ratio set and non-singulartransformationsof a measure space, Lect. Notesin Math. 160,Springer,New York, 1970,pp. 158'177. L. H. Loomis, An introduction to abstract harmonic analysis,Yan Nostrand,Princeton,1953. F. J. Murray and J. von Neumann,On rings of operators, Ann. Math., 37 (1936),l16-229. F. J. Murray and J. von Neumann, Rings of operatorsII, Trans. Amer. Math. Soc.,4l (1937),208-248. F. J. Murray and J. von Neumann,Rings of operatorsIV, Ann. Math.,44 (1943),716-808. J. von Neumann,Zur Algebra der Funktional operationen und theorie der normalen Operatoren, Math. Ann., 102 (1929),370-427. J. von Neumann, On a certain topology for rings of o p e r a t o r sA, n n . M a t h . ,3 7 ( 1 9 3 6 ) , l l l - 1 1 5 . J. von Neumann,On rings of operatorsIII, Ann. Math., 4l (1940),94-161. J. von Neumann,On rings of operators.Reductiontheory, Ann. Math.,50 (1949),401-485. G. K. Pedersenand N{. Takesaki, The Radon-Nikodym theoremfor von Neumannalgebras,Acta Math., f30 (1973), 53'88. M. Reed and B. Simon, Functional Analysis, Academic Press,New York, 1972. W. Rudin, Fourier Analysis on groups, IntersciencePubl., New York, 1962. S. Sakai, C*-algebrasand l/*-algebras,Springer,New York, 197r. M. Takesaki,Tomita's theory of modular Hilbert algebras Springer,Berlin, 1970' and its applications, M. Takesaki, Conditional expectationsin von Neumann algebras,J. of FunctionalAnalysis,9 (1972),306-321. M. Takesaki, Duality in crossedproducts and the structure of von Neumann algebras of type III, Acta Math', l3l (1973),249-310. M. Takesaki,Theory of operator algebrasI, Springer,New York, 1979. E. C. Titchmarsh, The theory of functions, Oxford University Press,Oxford, 1939. K. Yosida,Functionalanalysis,Springer,Berlin-NewYork, 1968.
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r70 GNS triple, 40 g e n e r a l i z e dH i l b e r t a l g e b r a , 6 0 G l e a s o n ' st h e o r e m , 3 9 g r o u p - v o n N e u m a n n a l g e b r a ,4 5
Hilbert algebra,60 Hilbert-Schmidt operator, 53
infinite factor, 34 projection, 22 i n n e r a u t o m o r p h i s m ,8 7 flow, 87 isomorphic dynamical systems, 148
KMS condition, for states, 64 for weights, Tl
left-regular representation, for discrete G, 45 for general G, 63 left von Neumann algebra, 6l
minimal projection, 28 modular conjugation, 50 group, for a state, 50 for a weight, 62 operator, 50
non-degenerate set of operators,12 normal state, 37 *-homomorphism, 42
Index
polar decomposition f o r b o u n d e d o p e r a t o r s ,3 f o r u n b o u n d e do p e r a t o r s ,4 polarization identity, 3 positive linear functional, 37 Predual t(f)*, 6 projection, 2 properly infinite von Neumann algebra, 154
Radon-Nikodym theorem, 74 range projection,2l ratio set, l3l right-regular representation: for discrete G, 45 for general G, 63
s e m i - d i r e c tp r o d u c t o f g r o u p s , l l 7 semifinite factor. 34 von Neumann algebra, 88 weight, 52 s e P a r a t i n gs e t , 4 4 s h a r P o P e r a t o r ,6 l o'strong topology, l0 o-strong* toPologY, l0 o-weak topology, 8 spectral synthesis, 97 spectral theorem, for normal operators, 3 f o r s e l f ' a d j o i n t o p e r a t o r s ,2 , 4 standard von Neumann algebra (w.r.t. Q), 122 state,37 strong topology, 8 s t r o n g * t o P o l o g Y ,l 0 s u b s p a c en M , 2 0
trace, 52 trace of an oPerator, 8 t r a c e - c l a s so p e r a t o r , 6 tracial state, 37 type of a von Neumann algebra, 34
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