An Invitation to von Neumann Algebras (Universitext)

souleH'H'd Ouuqeg'6'3 srol!p3

lxolrsJoA!un

Universitext Editon. F.W. Gehring.P.R. Halmos

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

ofiot sued uopuol Oreqlep;eg urpogIJoA/v\oN Oepel-re6updg

seJqeov uueunoN uo^ ol uoltPl!^ul uv ropuns's

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

:slnclrorls euos e{€l o1 f.russecause^\ lr .ezrs slql erunlol 3 ur IerralBlu psuoJlueu-O^oq8 ar{l IID alBporuuoccE o} repto uI :uo{€l slnJlJoqs '^\eu aq ,{uru (g'y I't suolloog ur .,(1re1nclu€d)luatuleorl eql Jo oluos 'pJ€puels sl l€lrelgtu ar{l qEnoqllv 'uaroaql ,{tllenp s.l>l€se{sl pue 'srolce3 go seldur€xa 'les ollsr s.JaEarr;E 'slcnpord-pessorc Jo uorssncsrp € r{ll/d sepnlcuos pu€ (seuuoc Jo uorlscrJrssBlc \III eql ol ^rooql relnporu crseq ar{l qEnorql spaacord 'srolc€J Jo uolltclJlsselc uueunaN uorr - de-rrnn eql qll^\ secuauruos {ooq 3ql 'salr€ulrul1e:cl auos relJe leql oraq uolluetu ol sacrJJns ll 'lradxa aql ' A- rJooJe q l eqt Jo sqldop eqt olul rerllrnJ e^lop ol pue 'uolllsodxo slql ul sd€E (snorc€d?J '^llecruqcal puu) snororunu eqt ul IIIJ ol peldrual aq '{ooq slql Eulsnrad ralge 'se:qa31e l{Erru rep€er aql lsrll paclor{sl lI uutruneN uol Jo plro.r Surlrcxe aql ol uollell^ul u€ oq ol ll srul€lcoJd ollrl sll leqan lsnf sl ll 'es€c ,tue uy 'sluerualrnbor eseql laalu ol ssrJl {ooq srql 'Jop€er eql Jo scuolled pu€ €urru€ls aql lsel lou ol ezrs uI qEnoua Ilsus sr (c) pue lpunort4ceq s.repeer eql uo suolldunss€ l€rururtu so{stu (q) ilelop l?cluqcel qJnru ool fq uanop petEoq EulltaE lnoqlla 'seJu€ApB lue3er eql elercaldde ol pepaau ^roeql cls€q aql Jo eruos ol rap€ar € seonporlur (e) qcrqm looq € JaqloEol Eulllnd JoJ asec Euorls € aq ol sruaas areql '^JoaqJ-)/ puB ^rleuoeE l€lluaraJJlp s€ qJns searB reqlo qlr^\ lc€luoc o{Bru ol SuruurEaq aou sI lI 'scrsfqd IBcll€ruerllBru pu€ ^roor{l olpoEre qll^\ suollcouuoc Euorls p€q slte/(le seq lI 'srea,{.gg ls€l oql ur spunoq puu scl€el ur turmorE uaaq s€q sBrqeEI€uuauneN uo^ Jo .,(rooq1eq1 :>1oog srqa l(qr1

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

'ro3 padoq e^€q plnoc roqlne eql terll 11e paqslldutoJc€ 0^8q plno^\ lJ 'arunlo^ t€rll ul pefeldsrp rur€qJ eql Jo uollcuJJ elncsrurru B uaAe ornldsc ol soEuu€ruerunlo^ slr{l JI 'serqaEle-*3 uo {ooq alllJI InJlqErlep srq Jo luecsrurrueJ ,{tqElq sr leql ellll € Jo osn aql Eullllrured ,{ypur1 JoJ uose^rv rosseJord{ueqt ol flrunlroddo slql o{sl ol a{ll plno^\ roqln€ er{I :0llll eqI 'rorJo ue r{Jns Jo peruroJur Eulaq ol€lJerdclu plno/r\ Joqln€ oqt 'Jallsur lsrll JoJ lxal aql ul asle ereq,nf ur ro ssacualeJoJ orll Jo salou oql ul fcerncceur auos slods JepBeJ eql JI '[xrq] llnsuoc lq8rru ropeer aql '^r{clerEollqlq alrsuolxo u? roJ leraq luorul€all oql uo ^llcaJrp J€oq lerll sreded pue s{ooq osor{l fluo slsrl qcrq^r 'fqclerEollqlq eql ol spuelxe osl€ ssauesral aql '[l lef] 'sll8l3p roJ aroru iesral llnsuos fuu repeer aql luqm3ruos 3re sslou ',{qderEorlqrq B pu€ 'ernleu 1 e c 1 q d e r 8 o 1 1 q reqq I lecrqde:Eorlqrq € Jo solou el,uos 'pasn srurel Jo xepul u€ 'sloqurf s Jo f sll e s€ qJns 'sEullllJ pJEpu€ls aql Jo etuos rllllh paddlnba sl arunlo^ slql :sEulddBrJ ' u o q l a s n o 1 E u l p o o c o r da r o g e q f r o e q l e q l u o r J s l l n s e r f . r u s s e c a ue q l e l l d u r o o o l ' Z ' t u o r l c e s u r ' e p 8 r u '€'g pu€ uesq osl€ seq ldurelle uv z'g suorlsss u1 ecurlulenbce lsrll Jo ocu€reqlrnJ ar{l apecordplnoqs qclqrr\'uollcnporlur fressacou eqt luroJrecl ol e^res plnoqs (sdnorE 1ec16o1odo1 uo) xrpuaclde Jolrq

II^

scBJ0rd

'palscrpep ,t1puo; sJ {ooq srql uoq^\ ol pu€ 'ol€rarunuaol o l q B l u n o c u n o o l s u o s € a l r o g . r q 1 u e , ( e f , ( 1. d 1 1 e u l J p u B lorull go porrad lroqs flEurzsrue uu ul pauroJrod flln3roeqc ;gurOfl J o q o f l u a r o r J J o . { 1 e u a r 1 x au € J o J . e u : e q g ' d . A I r q S i e E e l s l e r c n r c € l€ ?l?roru EurEEes ,(ur palsooq qclq^\ ro11al EurEeJnooue f:or e roJ uose^rV 'A,\rossoJord .'srorra IBra^os 1no Burlcrd pu? lctuJsnuEru eql Jo suollrod e8.te1 Surpeer fl8urlelsured ro; .uepe141€rlqoqS lserqeEle uuerunoN uo^ Jo ^roaql aql luJ€el 1 dqaroq,r .pezruc8ro J s r e u r r u o so s o q l I I € p a p u o l t s f l l e c r l s e r s n q l u e p u e { l l n J q t l e J E u u e q roJ 'euqsu;tr :les orler s,raEarr;tr EurutaJuoc suorssncsrp aruos roJ 'ruJ€>lp€N 'g 'hl rossaJord lesuodsar enrgrsod pue lusersnqlua rrer{l r o g ' u 1 1 n c 1 e Cl e e ^ E E I s a r n l c r l J o e s r n o c e q l r o J o o u o r p n s e r l l u a eql :€llnJIsJ lB s{oe^\ InJrapuo^\ xrs pueds ol atu pell^ur Eur.lrq rog 'fog ') '17 rossaJord :{ooq srql Jo uollcnpord aql ur pa,{.e1d aireq ,{eq1 solor aql ro3 eldoad 8ur,no11ogeql lusql ot oIII plno^\ I

stN3l^t903lMONX)V

80r e0r E6 98 t8 EL E9

zs 9V 6E tc

9e 17, 77

6I 6I II 8 I I I

( n ) l p s u o r l d l r c s e pe ^ r l B u r o l l y uorlc€ uB Jo r,unrlJadssouuo3 eq1 uollJ€ ue Jo unrlcads uosaary eqa uaroaql alcfcoc i(relrun eqa Jo uollE IJIsssIJ sJuuoC eql

srol3cJ III cd,{I

v'E t'€,

z'€ I'€ t rcldsqJ

s u o J l B l c a d x eI B u o l l r p u o o pue r.traroeqlru,(po111r1-uopeg aql 9'7 uortlpuoc frepunoq Shl) eqJ g'2 s e r q o E y el r a q l l H p e z r l e r o u a Ep u e s l q E r e / A v'z (sel€ls ro3) ruaroeql I{Bso>JEI-€lrruoJorII t'7, uollcnrlsuor sNc er{I z'z u o 1 l e t E a 1 ue1^ l l e l n t u u o c u o N t ' 7, - zfIUoI cqJ l(rorql rfcsrtEJ Z rcldsqJ uollcunJ uorsuaurp erIJ € ' I s u o l l c o f o r do l l u l J z ' r (n pt) uoJlelar eqJ I ' I sro|Jq Jo uoJl?:rIJtIssEIJ urr?runeN uo,r - .(errnlry eq; I rcldsqJ ruerooql luBlnruuoc alqnop eql (Ah uo serSoloclolxaluoc d11eco1 aarql r(fih l€npard aql frooq1 rolereclo clseg uogcnporlul

v'0 €'0

z'0 r'0 0 reldBqJ

qoqu,(s Jo lsI-I slucutpcl,r,ou{cy ccBJcrd

IIIX XI

srN3tNof

Contents

xtl

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

16 to ds 9 6 'Q E6'(9)n

(,9'og'd" 9V'(9)*11 g€'+'*n LE

-t ''*IAt

Lt,4n *n 0z'(v)dr

Iz'hl u 4 , T TA

FL,(TDQ

07,'n u v

9 g '(n )n 91,( H)Q

0L'qN 19'qA

'pl,tl '*A

gg 'suJ

6I

6r'(npt)t-a LI

,U'I)Z

9l'lav'la|

vr'*n

'Q4 €s 4y1'Q11

vt'O[)d

gg')q

'lxal aql ur acuereadcluJleql Jo repro arll u1 paEuerre eru sloqrufs erIJ 'paul?Idxa sr ro 'slnJco lsrlJ loqrufs eql qclqr$ uo eted eql ol sreJer [11ensn Jeqr,unuEurfuedurocJe eql

s]osl^r^s Jo rstl

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

.s(t[), I l0

a, 150

M @dG (for discreteG), l16

149

3ql uutll roql€r) IEJII€uraqletueql ol araqp€ IIEqs e^{ 'roqlrng 'pafo1dua osl€ erB 'uollereplsuoo repun sI W pue y sloqrufs aql e J ? d sl r a q l J H a u o u e q l e r o u e r e q a ' s u o r s € c J o ^ \ e J e u o l g , { q p a l o u a p , { 1 1 e n s na q I I I ^ \ s e c e d s l r e q l l H ' f l d d e f e q l l e q l p e u n s s e f 1 1 1 c e 1 eq III^\ ll lnq 'pal€ls aq Ja^ou lsorulg IIr^\ ,,alqgr€das,,pue ,,xa1druoc,, sarllcafpe eqt :paraplsuoJ ore soJ€clslreqllH oyqeredasxoldruoc [1uo '{ooq aql 'aceds lnoqEnorql U a q l l H € u o s r o l € J a d o3 o l e s € J o p u r { ur€lrec e s1 '4ooq slql Jo c1dol yerluac aql 'erqaEl€ uueuneN uo^ v Arocq;

rolercdo

Jlssg

'I

O

'srrqaEle uuurunaN uo^ Jo sellracloJd 'uu€runelq ,(relueuala puu 'uollruIJep ouos eql uol rueroaql Jo erll sulBluos uollcos lu€lnuruoJ-olqnop eql IsluaruEpunJ IsulJ .sesrcrsxoaql ol pslBEalor are __ serEolodol *Euorls-o pue *8uo:1s aql s€ qcns -- ser8olodol roqlo otuos 'sarEolodol 1ea,n pue Euorls peurJop i(1rseaaJou oql Jo se IIa^r. se 'fEolodol ({Ee/r\-o) slql Jo uoll€ulruexo Jerrq € sr uorlcas pJIrIl or{I 'serqo8yuuueuneN uo^ Jo .(roeql oql Jo ssauqcrr eql JoJ suos€eJ frerur:d erll Jo auo sl slqt l€ql lse8Ens ol ooEICIJo lno eq lou plno^\ lI 'lceduroc sl (;1)f ',{laur€u Jo II€q llun eql qclq^\ ol lcodsar qlla -- ,(Eo1oclo1 *I€a,$ ar{l -- fEolodol xaluoJ [11eco1 e sp1e1{ I?npercl s Jo eJuolsrxe eql 'flerrllcadsar srol€redo 'lc€druoc pepunoq pu€ ss€Ic-ocurl peo€Ider t(q Eulaq secuanbes papunoq pue'alqeuruns'lueEte^uoc-Ilnu -= *(1f) pue @tr rd !c sllnsar IEcrssEIcaql Jo ,,enEo1eu€o^llelnruuoc-uour er{l soqsllqelse uollcos lxeu eqJ 'eceds ueqllH uo srol€redo Eulurecuoc slceJ crseq aql '3oord lnoqll/tr 's1sy1pue looq gql lnoqEnorql pefoldua uorlelou eql Jo aruos seqsllqelso uollJes lsrlJ ar{I 'lredxa aqt ,(q pellruo ,{1e3eseq ,{eu pue 'froeq} aql ul pepeau stlnser l€cruqcel Jlssq aql 'slseEEnsellll aq} sV Jo auos Euldola,rep ol polo^ap sr Jelder{J srql

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 = :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 = 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

*','31"'li','.:, #),J'.:'#'"";l't'"H';T''T;ff ;l*;",t;i,!ffi operators.)

(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 = = ;in particular, tr P= Ecr- 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 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

'1

'JAl ) uo

,f":i : l:

n 0 0

= "A[

(c)

0 " D

Pue i{/'r4 !]u:((f,rp))) = ,n > ,w (q) tg uo sroleractoJo erqe8leruioJpe-31es alureuaEap-uouB sl ,t|^ (e) fnlf d.Jrre,rol ssalur€d^la^rleler sl lI tt

[^

0 0

' 0 0 " o o

= W

' 0 t )

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,

''*f ill',;1;'.',[ (a) i' ,kill,,irri-ll1,ei'" fii"-{,,)liill',1';,,qYilf applied to {x, - x}.)

'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 -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

= 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.

'alcldruoc (;,i11eur.y) sl Joord eql pu€ ,snonurluoc O i(14ee,n-o oslB sl uorssnJsrp a^oqB l€r{t oql uoJJ s/AolloJ 'urslqdrouowoq-* *I! ly o,rrlcefur uB sI Ierurou ltl - UtDu :r_u oculs ('ararl lcrll olur oE tou op a^\ puB 'tuaJoeql flrsuep s,[>1suelde; sosn slql Jo Joord auo ipasolc {y1ea,n-o sl ll JI ,{1uo pue Jr posolc fl4eam sr urqeEle lurofpe-31as € l e q l I J B J € s r l I ) ' e r q a E J eu u s u n a N u o r r e f l l u o n b e s u o c p u E p e s o l J f11eom-o sl (rI)u leql opnlcuoJ 'posolc-*{€e.n sr II€q lJun slr gr fluo pue JI pasolc-*{Be^\ sr ec?ds lunp (-qcuueg) B go ocedsqns rBouII E lBrll salsls ([so1] 'ocuelsur rog ..gc) ruaroaql ueIlnruqcg-ulalrcqg 'lJedruor f p1ec,n-o aql sr (n Ileq)u = (lU)u llsq laqt raJul 'crrloruosr sr U acurs pus lc€duroc i(14ea,n-osl (I > llxll:n t x) = 1t1 'snonulluoc f .spro_^\roqto tiJ 'l{14ee,n-o (r)z IIBq ecurs 14ca,n-osr u - ('n)u 'Arerlrqre s.e.trO acurs .((r)u)d - ((lx)z)rfi ucql .r o1 f1>1ea,n-o .oS .($-t:d .xg EurmolloJ saEraruoc 7y ur. (tx) lau € s{reruar JI '3c) lualenrnba eru f lrnulluoc pue ,(1r1uurou lEqt IIBcar ' s l e u o r l c u n J J s e u l l r o ; ' r o n e m o 1{1B' e( l^e\u- or o u arc u pu€ 0 Vtoq acurs) I E u o r l c u n J r e e u r l I E r u J o u3 s l u o 0 , u a q l . ( ( , f i ) a u o l B u o r l c u n J J s a u r l snonurluoc,(11eam-o € s E p o ^ i c r a )t . ' ( , $ h r 0 J I l e r l l e l o u . l x a N ('{(x)u ds r 1 :l(1)/l)dns = ll((x)x)/ll t snonurluoc rog l E q l l o B J a q l p u e u o l l € r u r x o r c l d ul e r u r o u f 1 b d f l u o " s o r r n b e r l c B J l s € I slql eql) '((x)u)I = (@)hu oruls ,(lrrrrlceful slclpBrluoc Jo Joorcl u 3o '0 '0 = ((r)u)/ uaql .x ds u1 oreq,nfrerre slql * G)I ellq,r\ 1ou lnq (x)u ds uo saqsrue^ qcrtl,,A.r ds uo / uoytcung IBeJ snonurluoc € slsrxe aJeql ueql llcrrls sr uorsnlcur slql osoddng .x ds j (x)u cls 'snql '(r_(f - x))u osrarul {11,r .alqrlraaur sl \ _ (r)u acuaq lruerooql lu€lnuuoc alqnop eql fq ,y t ,_(\ _ r) leql clou .x ds / f JI .slgl ro{ 'r ds = (x)u ds lerll i$or{s ol luercrJJns uer{l eroru sl 11 .snlper l e t l c a d s s l l o l l e n b e s r r o l e r a d o l u r o t p e - 3 1 a su J o r u r o u e q l o o u r s : x ue qrns xrj 'w ) *x = r ue{^r llxll = ll(r)ull reqr fJ1raa ol (llx*xll = 'becrg3rts d5rrlauosl sJu andro oi ll ellrll) f lltuapl-*C or{l ol slueql

'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 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:

= <xr*4xo> = = = <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 ) | 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 (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

and y e .,Cr(Xf)

< llyllll"llr."lHint:tt'e ri#j,in.n lly'll, < llyllll"ll,un,i'llxyll,

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
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 + iiii) a' 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 )

tltr;:*,;-""""r1i'1fr = (x)4rr'u pu' (t,truo ,ro,o.,r'#',,;

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 '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(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 \ 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 '= . 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, z rul > 0 duls ogl uo uollcunJ snonurluocpepunoq€ saurJep | > z tJrl> (,/l (,/l > z rul > 0

'| 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 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

'fn) ol tcadsat qtltt4uoltlpuor SW) atg sa{sgos 0 (lt) idrttlloto/jo=tn (l)

suottlpuol Eutuol1o/ aqt'n

: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

elrurJlruos r{lr^{uorlrollqul are1! oncodsar r€rurou trtfl Lirlt;{Lt#

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

=

os ' 0 = z rul aull eql uo eerEepue 'frepunoq eql uo snonurluoc.dJr1s eql ul cllfleue ete 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 = = = = (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 (. =

(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 = = (t(*(z*x)\lt = (tk*z\ft = '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

'llurll = lluxll2 71{t*th s11(x*xpc} ((q) (r'r'Z) 'xa ,{q)

) l(x*,{hl 711({*t)$76(x*x}tr ruarooqlu,{po>1rp-uopBU ar{I'9'Z

9L

2. The T o m i t a - T a k e s a k i T h e o r y

76 ft = 6it1r6-it= o!{D,

i.e., h e Mt

o

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 ; 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

'l(ra)oot ( lla o ooll. llooll= (r)00= (ra)00 's1ql rog l€gl elou

'in

, 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 \,

;] [,i,,,

= ro * rejxoeo= E(eoxef, + rejxoeo),

and hence,

ll max{lr+ll llall,llolll= llrt"orrf + relxoeo) ll"o"r$+ r"fxoeoll

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

eoE(x)

=r,"[,,L, rfi,) =

f ,L, 'E(f"fi)fi

= j =Ll ru,*f,) = E(f:x) = E(eox),

-BrIIuor aqlBr^((ol'on)''dse:) (0?) qrl,v, rrt;ir"rtl:Jilil?J;51t:*

.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

etqa3leqns uueruneN uol e eq oyg p1

o

'(q) rq

.S.9.2uopIuIJeC

'@s)*@g)- (x*x)g =

((xZ)*GS)+ x7+x- x*(xg) - x*x)Z = ((xz - x)*(xz - x))A > 0

(c)

'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 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

(

zz*

rzxl ')

' l n r r t * : ( {e)|r =(v)zn@ = rny 1 , . _ t,t,"IJI L L"* J 0 pu€

eceld

(r'l)

aqt

ur

I

qllrr\

xrJlt{u

Z x

Z oql

l0'I 'eJeq/\{ASIo st fla eraq^

. :rr: A €r,' : r=t'l :]f _-I T

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

c1c,(ro3.(rclrun

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> = = (*/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

(l,nntygtt)

= (,n*")r/ = ra zlrv puE (,H+H)a= v leql os

' (,u+n){a = ,H42HqZ = rlV leql sfes slql '9'I'€ rroroaqJ Jo txeluoc aql uI) '(s.+v)ra=( srayra)

leql ^\oqs'D u! z ,{ue rog

(e)

('(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) = =

= 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 = 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

'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

' = J (s)np l! 'acuoq19 uo E uoll3unJalqurnseeru pepunoqi(ue rog uollcv ue Jo r,untlccdg uoserrryoqa 'Z'e

96

96

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 = 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 1oau-o [uo s! I {I i(r)Dos-=1*x)Pds (r) '1{o '(tO)

66

lasqns pasop D aq Z puo'1,r7t x'(9){I

t t ta7 6'f€

uolrFodor4

\ 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*, = ; 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 =

-({O} + (x)Pcts+ t0}) j -((q)"cls + (x)Dds + (u)Dos)j (gxa)Dds leql (c)8'Z'€ €ruua'I pu€ OI'Z'€ 'dor4 ruorg apnlcuo3 .(p)

'((p)(e'e'o)'xs 'Jc)

,{8o1odo1{Belr\-o eql ul snonurluoc flaleredas sr uoglecrldlllnru aJurs ' ',n ul0 ^u€ JoC (c)

.

?"lrte*l

(

.2"o. ds , n ds = l(t oa _l..c)""

(( e) .;c)

'xl

_[(" rd r'" ^ ;-]

=

z"

=

^i.] r. = _[tr)"0,

=, "o o , _ [t" l '" "0 r ' "T*]

'.{1a1erperuw,r^",,;:o)'f ;i;:r11";:,{'1J%. '1x;"Pds=

1x;Dds

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 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(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 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

! e Me,+) er(y) < 0(y) < e-rr(y); SO'

('g'p'g [ru11oroJ pu€ (q) f'f'g 'dor4 3o s3oord aql al€llrul :tulH)

'tQfud ) a * o'"Oyorp = Ov)s (q) a,

:(Qw)d ) a* 0 :0v ds)uulUl= Czvl: (e) 'hl totceJ fue rog lBlll /t\oqs

(A'V'e)

sesrJJexfl

E

'perrsap s€ t > r-r t il#ott l€rll os

'zll(r)"ou;1r-r = (x*x)Qz-, (*rxP =

= all(*')"0u;;

"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 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

,L ", n4 x = (r)no '(n)Z zx :(r)Pu)= (n)Z 'snql lsql u^\oqs e^€q elr\

'g ut n lle rog (r)nr = x acueq pu€

((t'r'l) 'uba'gc)

((r)nr)Pu = (*n)1(x)Pu(n)1 = *(rr)f g(n)f = 1c= (r)P11

'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 =

= (tH r.rop r ((r'l)1)u ecurs)

:s^\olloJs€ alncluoc pue { € rlcns a{el .oS ' 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

"3" '(a1a;r-8o' =g aur;aq (c) 'f1r1enbei(q paceldareq {eru uolsnlcul o^oq€ oql ltql apnlcuocpue '/ ueql reql€r (l) roprsuoci2n 3 {:tp/'o '(e) f,q 'os pue

( 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 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) .= defines a faithful normal

tracialstate-onM; thus ll"oll'= llx*nllfot all x in M. Define T: M- - [0,-] by

7 ( D = E .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- I ' z ' r i ' l = l u ' l u u>l l,

lor{l flrlerauaEJo ssorlnoqlr^\ eunsse'ro,1"ll

t$;:;

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

ttfit > l(.?)7r (r.r)rr = ,*rr_r,!' [rr"t']n

> (.?)7t

aI€{rt > .r > ,r :rc}..o.rn #::[:ffi-'jft",1:t;:ffi|;t:td

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 .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 r > 0 ) : { 2 .) u : " \ ) n { 0 ) = ( g ) / < + \ I t t a d f il o s l r y ( q ) !{I'0) = (O )r '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 ' '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=

(n- sxj

= ,$vtfv

('xJ,8v(n)r)flv = (s)Q = (sXl(trlrl do) ,*v('r)r$v) :((t'l'l) uorrenbe,(q)

1(r)1(x)u(r)1=

((r)/o)u= ((r)u)do l€gl os

'(sX3((x) = dopl (s)f (x)8/o =

(g'g'7'ctor4{q 'ureEe) (s)t rf,v(r)"Oo = f,,v

(')(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 (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 = 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

'Joord slql Jo sasodrnd eql roJ opcru uollctrunss€ssaupapunoq turfgrldurs aql uoJl€reprsuoo olul Eull€l ,goordleql Jo uolt€ld€pe uB sr araq peluaserdgoord aql :[l Aef] Jo tI uorlcasuorJ sr {ooq slqr Jo 9'I'€ tuarooql .(sec€ldu1 c11di(rcssel .[lquqord .qEnoq11e) arar{l punoJ ouo oql Jo uollcnpotclaJ€ sl oJeq uo,rr83oord eql :[uoC] ul I'Z'l ruorooql sl (araq I.I.€ ruaJoagl)ruoroeqlelcfcocl(relruneq1 g nldBqJ '[Z lef] Jo rrreJoaqluleru 'Jc) .dord aql (@) sl stqEla,r olrurJrues roJ e'S'Z {r€tuag Jo .[1al ul punoJ oqV.g.Z uorsJe^0gI 'poreodcle lsrlJ ll oJorl/yr. f.eur (s1qEro,n olrurJrruasroJ ruaroar,{l ru^polrN-uop€U eqt) t.g.e,tuaroaql go Soord elalclurocaqI '(l'g'ltI uaroaqa) [l let] ur se dllcuxo ,oEenEuel '\ (V'9'(, 'dor4) JoJ lnq ruaroarlt s,?urefrtuo1 Jo luaruleerl aqJ '[fa] ul surErrosll s?q tII'g'Z rueroaql l;.tJBg.Z uorlJas ur I€rreleu aql IIV .Joorde rc1 17 uro3l aasisoquo3 ol onp sl (r II'g'Z ruoroar{J'gc) tqEra,nolrurJrruose go dnorE relnpotu oql uorl€zrralr€rer{c'S'n') aql .poJaptsuocere s1qE1a,n elrurg [1uo 'Jo rezro,roq'araqrrn'[t:{eI] J o € I u o l l c e sJ o u o l l € J r J r p o rBu s l . Z I . g . Z {re11oro3(3o aalsnlcul pu€) ol dn uorlcas ur uorssncslpegJ S.Z '(luaura^oJdrur-Jlas se) lerauaEJoJ sB 110,n sJoordelelduoc roJ llnsuooplnoqsropearoqt qcrq^\.[t let] Jo 0I-I suollcasJo lcErlsqBrood e lnq sl (7.7 uorlca5u1) serqeEleueqllH pazrlerauaEJo uorssnJsrp arllua aql .[Z ruo3] ur punoJ eq ol euo eql f1y^erlu:ssa'erqeEle sr pozrlerauaE B lreqllH arnlcnrls oql seq (9 N Jo u Yil )Yu l€tll qsllqBlsaol (tI't'Z - e,fVZ sasrcroxX ur) oraq pa[olctrua poqlau eql '[t uro3] ur saquoC dq pacnporlul arem s1qEra11 '[4eg] llnsuoc ,{eruJap€aroql .surqa8l€lraqlrH pazrleraua8ol IeecldBlou soop lBr{l Joord lcarrp e -ro3 l[1 {€I] ul €'ZI ruaroeql s€ punoJ oq feu ll .osle14961..nrun nqsn,(y.se1o51 'serqaEleuuerunaNuo^ prepu€1s-1sen| paqde:Eoaurq41 :ur e1ruo1 ,{q ',,s1cnpord-rosual paqsrlqelsa sE^\ roJ lsJrJ ruaroaql lI uoll€lnuuoc,, eql sB ol psrreJar ,{11erauaE ((q) sl alctruexx ur s{rBruer L.e.z eql'Jc) (rlt{ o tht) = ,(p s n) l€ql lc?J aqrh ual Jo z'S'zuollces Jo l3rll ralJB pouorrls€Jsl €'z uorlsas Jo luauloarl arllur er{I lxlol .lo pallElap t'I'I uollcosoas'slosturle:r:rtrspue sloscllcfc Jo uorssncsrp € roJ '(tI ol cln turpeal uorssnosrp aql pu€ Z.g.I ueroaql) [t nrV] ur punoJ aq feur (J'g'g tuoroaql EurrlrolloJuorssnJsrpeql .Jc) pasolJ -urrou sr erqeSle-*, e Jo aEeurrclqdrouroruoq-*(alrlcofur ,{.lrressacau lou) E l€ql lcrJ luanbasuoceql s€ lla^\ se .crrlatuosl ,{lrressecou sr serqaEle-*3 Jo usrqdrouoruoq-* aaJtcaful ue luqt IJBJ eqI 't68-S88'(tS6l) g "qcer,I 'qlehi 'f 'ecBcls pasolJ uo serns€ol^I.uos8olg fraqllH € Jo sececlsqns '14l 'V perueddu :ul (s1euo1t.unJ reourl errllgsod I€rurouof ((A)f)a uo turpuelxa {tlnqlssod aql uo) ruaroaql s.uosealg {*tql rsorns?arur Jo sl (t'l.Z .x3 Eur,rollog Jo Z't'I uorlcas ur J rueroeql ul paqsrlqelsa

s9r

saloN

166

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).

'09-eV'(tt0t) 'srsf1uuy .1 .serqaEle uuuunaN 1 leuorlcunJ uo^ Jo surslqdtoruolnurauur go sdnorg .urru11ey .-l{ 'U [t teX] 't96I '{ro^ map 'ro8urrds '{ooq ualqord oceds y .soruleH.U .d [Z teg] 1raq1r11 '?t6I '{JoA ,re51'raEurrdg.froaqa .sour1u11 'g .4 eJnsBoIAI [t tuH] '8I€-20€'(Sf6t) 'srsl(1euy 61 leuollrunJ 'I 'serqaElu-*4.1 .dnraEeeH .n uo slq8ra,r Jo leuroN [eeU] 'Ig6t'{ro^ aaN 'puBIIoH-qlroN'serqaEleuueunaN uoa .rartuxrq .1 [xlC] '816I 'aEplrqure3'ssor4 flrsrearun aEprrqrue3.surqaEle uuErunaN uo^ III ed.(l pue slJnpoJCI passorcsnonulluoJ .o1oequB^ .V [Crr] 'Z9T,-EEI 'Gt1t) ..dns .ruroN .luarrs .uuV elocA 9 'III ad,{l op srnolceJ sep uorlBclJIssBIJ aun .sauuo3 .y 'LL-6, '(tt6t) "rltBIAI'duroJ 'aqcneE [uoC] €Z e auuerlJoqlrq erqaEle eun u sarJossesprod .soquoJ .C [Z uroC]

ro selnd'{rery', 'erqaEle-*, ,""'oro.t;':o,ft:'lLtl;5tI

uroo]

[r '6t6I 'IJoI ,r,o51'raEurrdg.1 scrueqJetu unlupnb leJJlsllels pue serqaEl€ rol€rado 'uosurqoS .A\ .61 pue rla11erg .O IIUAI '0gI-I9 '(996t) 'o1o[; 'lrs 'ql€IAI'lsul 'seu y 'Iqnd 'srolc€J .spoo11 .f .g ,H puu Jo uorleclJrss€lcy ImVl 'ErT,-Ue '$15t) 'srs{1euy .1 I{ErV .serqa3le 91 IsuorlcunJgo rolerado ;o srusrqdroruoln€go sclnorEug ,uoso,rry .S .A\ [Z ,rrV] '916I '{ro^ ,ne1q'ra3urrdg'serqa8le-*J ol uollell^ul uy .uose,rry.g .r11 [t arv] '16-98'(9961) "qeg ',rre,r 6 'rlosrr{e1{'Z 'seJnseeur luerrelur elrurJ-o uO .plourv .) .l [urV] 't .(ZgOt) .srsf 97 leuy leuollcun{ 'f '11esa>1e1 LZ-SVZ Jo Jo ruaroeql e pu? serqeEle uu€tunoN uol ur suollelcadxeleuoJllpuoJ.rurqccoo.J pu€ rprrJcy ..I ICVI

AHdVU90tlgtg

168 [Kal 2] tKril [Loo] tMvN ll [MvN 2] tMvN 3l [vN l] [vN 2] tvN 3l [vN 4] tPTl lRSl [Rud] tSakl [Tak l] [Tak 2] tTak 3l [Tak 4] tTitl [Yos]

Bibliography

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.

.rol3EJ 0€ II odfl e ro3 ecuonbas l8luauepunJ 'usrr{dJorrroln€ aaJJ 6I I .uorlcEoarJ 6I I 'rt\OIJ 99 .tolerodo 1o1; 99 94'urqo8lelulod-paxrJ 'lqElam €9 gg 'erqotle uu€runaNuo^ .uorlcaford 7g .rolcsJsllulJ tg 'elqllEdruoc-p 6l 'tqEla,n ZS .el€ls l€ InJqllBJ 't > \ ' 0 ' \ 1 1 1s a d { 1. 1 o tgl 'III -I 'uI sadft t€ -II Jo 'III '11 '1 sad{1 go 8Z g1'rolceg 'uorlo?crpoEra IZ[ 6l'W 1ar)luale,rrnba 76'ruolsfslecrureufp 'ZZl'lqE1o,t LgI 'uoJlc€ 09I IEnp gg'uollcunguorsuarrrJp

tt

'las crlcfc

6rt'9

lerauaErog lcnpord-passoJJ '' 9II aloJcsrpJoJ lcnpord-passoJc gg'oroc g0I luulcods sauuo3 6t'uoll€lcedxa Iouorllpuoc g .rolerodolcgdruoc .lu8lnuuoc II p'rolerodo posolc t'JolBJedoolq€solc L I ' ( n g o )a r l u e c .Jo^oc lI l€Jluac .erqa8le-*3t 71 .f1r1uapr 96 el€urlxorcld€papunoq gg1 'eceds oJnseeru € Jo ruslqdJouroln€ 'runrlcadsuosa^rV 46 'rolcancglf1euu 14 16'(n uo g Jo) uollc€ .erqoEle 19 po^alr{ce lroqllH pazrlerauoE

'ruJrl eql Jo acuarJncco lsrrJ aql Eu'rureluoc oEud oql ol JoJOJdllerauaE sraqrunu Eu1{uedurocc, oqr

XSONI

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

weight, 52

operator nM,20 outer automorphism, 122 outer equivalent actions, 105 flows, 87

partial isometry, 3

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

g 1 ' e r q a E l eu u € r u n a N u o l 'ruarosrll ' u e r o a r { l uBrraqnEl r3ual^t alcfcoc Arelrun L6 98 'l{tla,tt gp1 'uralsfs Z9 g 'fEolodol >1ea,r l e c t u e u f p p e l u o r u o l d r u rl ( 1 r r u 1 r u n

tLl

xepul