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Booss/Bleecker: Topology Topology and and Analysis Analysis Booss/Bleccker: Charlap: Bieberbach Bieberbach Groups Groups and and Flat Flat Manifolds Manifolds Charlap: Chern: Complex Manifolds Without Potential Theory Potential Theory Without Chern: Complex Manifolds Chorin/Marsden: AA Mathematical Mathematical Introduction Introduction to to Fluid Fluid Mechanics Mechanics Chorin/Marsden: Cohn: AA Classical Classical Invitation Invitation to to Algebraic Algebraic Numbers Numbers and and Class Class Fields Fields Cohn: Curtis: Matrix Matrix GrouPs, Groups, 2nd. 2nd. ed. ed. Curtis: van Dalen: Dalen: Logic Logic and and Structure Structure van Devlin: Fundamentals Fundamentals of of Contemporary Contemporary Set Set Theory Theory Devlin: Edwards: A A Formal Formal Background Background to Mathematics Mathematics II a/b alb Edwards: Edwards: A A Formal Formal Background Background to Higher Mathematics Mathematics IIII a/b alb Edwards: Endler: Valuation Valuation Theory Theory Endler: Frauenthal: Mathematical Modeling in Epidemiology Epidemiology Modeling Mathematical Frauenthal: Gardiner: A A First First Course Course in Group Group Theory Theory Gardiner: Godbillon: Dynamical Dynamical Systems Systems on Surfaces Surfaces Godbillon: Greub: Multilinear Multilinear Algebra Algebra Greub: Hermes: Introduction to Mathematical Mathematical Logic Hermes: Hurwitz/Kritikos: Lectures Lectures on Number Theory Hurwitz/Kritikos: Kelly/Matthews: The Non-Euclidean, The Hyperbolic Plane Plane The Hyperbolic The Non-Euclidean, Kelly/Matthews: Kostrikin: Introduction to Algebra Kostrikin: Introduction Approach Analysis Approach Luecking/Rubel: FunctionalAnalysis Analysis: A Functional Complex Analysis: LueckingiRubel: Complex Theory Lu: CatastropheTheory Introduction to Catastrophe and an an Introduction Lu: Singularity Singularity Theory and Marcus: Number Fields Fields Marcus: Number McCarthy: Functions Arithmetical Functions Introductionto Arithmetical McCarthy: Introduction Meyer: Fields for Applied Applied Fields Mathematicsfor EssentialMathematics Meyer: Essential Moise: Introductory Problem Course in Analysis and and Topology in Analysis Problem Course Introductory Moise: 0ksendal: Equations StochasticDifferential Equations Oksendal: Stochastic Porter/WoodS: Spaces Hausdorff Spaces Extensioni of Hausdorff Porter/Woods: Extensions Rees: on Geometry Geometry Notes on Rees: Notes Reisel: Spaces of Metric Metric Spaces Theory of ElementaryTheory Reisel: Elementary Methods Rey: Statistical Methods and Quasi-Robust to Robust Robust and Introduction to Rey: Introduction Quasi-RobustStatistical Rickart: Algebras Function Algebras Natural Function Rickart: Natural Schreiber: Forms Differential Forms Schreiber: Differential Smorynski: Logic Modal Logic and Modal Self-Referenceand Smoryriski: Self-Reference Stanish:: Turbulence of Turbulence Theory of The Mathematical MathematicalTheory Stanisi6: The Stroock: Large Deviations Deviations of Large to the the Theory Theory of An Introduction Introduction to Stroock: An Sunder: Algebras von Neumann NeumannAlgebras to von An Invitation Invitation to Sunder: An Tolle: Methods Tolle: Optimization Optimization Methods
Springer-Verlag New York Berlin Heidelberg London Paris Tokyo
V. S. S. Sunder Sunder V. Indian Statistical Statistical Institute Institute Indian New Delhi-IWO e l h i - l 1 0 016 16 N ew D India
AMS Classification: I 46-01 Classification:46-0
Library of Congress Data Publication Data Cataloging in Publication Congress Cataloging Sunder, V . S. S u n d e r .V. S. An invitation von Neumann algebras. to von Neumannalgebras. invitation to (Universitext) ( Universitext) Bibliography: p. Bibliography:p. Includes index. lncludes index. 1. Title. I. Title. von Neumann Neumannalgebras. algebras.I. l. von 512'.55 86-10058 QA326.S86 86-10058 5 1 2 '. 5 5 1986 Q A - 1 2 6 . S 8 61986 !:e 19~7 New York Inc. Inc. 1987by by Springer-Verlag Springer-VerlagNew in any any form form without written All rights reproducedin be translated translatedor reproduced part of this this book book may may be rights reserved. reserved.No part 10010' U.S.A. permission New York 10010, New York, New 175 Fifth Fifth Avenue, Avenue, New permissionfrom from Springer-Verlag, Springer-Verlag,175 publication,even if the the in this evenif The etc. in this publication, names,trademarks, trademarks,etc. generaldescriptive names,trade tradenames, descriptivenames, The use useof of general understoodby by such names, names, as as understood former as aa sign sign that that such not to to be be taken taken as identified, is is not not especially esp€cially identified, former are are not anyone used freely freely by by anyone. the may accordingly accordingly be be used Marks Act, Act, may Merchandise Marks Marks and and Merchandise the Trade Trade Marks
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The theory of von Neumann algebras has been growing in leaps and bounds in the last 20 years. It has always had strong connections with ergodic theory and mathematical physics. It is now beginning to make contact with other areas such as differential geometry and K - Theory. There seems to be a strong case for putting together a book which (a) introduces a reader to some of the basic theory needed to appreciate the recent advances, without getting bogged down by too much technical detail; (b) makes minimal assumptions on the reader's background; and (c) is small enough in size to not test the stamina and patience of the reader. This book tries to meet these requirements. In any case, it is just what its title proclaims it to be -- an invitation to the exciting world of von Neumann algebras. It is hoped that after perusing this book, the reader might be tempted to fill in the numerous (and technically, capacious) gaps in this exposition, and to delve further into the depths of the theory. For the expert, it suffices to mention here that after some preliminaries, the book commences with the Murray - von Neumann classification of factors, proceeds through the basic modular theory to the III). classification of Connes, and concludes with a discussion of crossed-products, Krieger's ratio set, examples of factors, and Takesaki's duality theorem. Although the material is standard, some of the treatment (particularly in Sections 4.1 - 4.3) may be new.
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PREFACE
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Preface Preface
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g e n e r a l i t y , are Some S o m e theorems, t h e o r c m s , though t h o u g h stated i n full f u l l generality, a r e only only s t a t e d in p r o v e d under ( s o m e t i m e s very proved v e r y severe) u n d e r additional a d d i t i o n a l (sometimes s e v e r e ) simplifying simplifying - - typically, assumptions is a s s u m p t i o n s -t y p i c a l l y , to t o the t h e effect e f f e c t that t h a t some s o m e operator o p e r a t o r is - - they bounded. Some r e s u l t s suffer f a t e -b o u n d e d . S o m e other o t h e r results s u f f e r a sorrier s o r r i e r fate t h e y are are proof. g r a c e d with not n o t even w i t h an f o r a proof. e v e n graced a n apology a p o l o g y for ( i i ) Arguments p u r e l y set-topological (ii) A r g u m e n t s of n a t u r e often receive o f a purely s e t - t o p o l o g i c a l nature o f t e n receive p a i n l e s s ,it step-motherly w h e r e the i s painless, i t has has s t e p - m o t h e r l y treatment; t r e a t m e n t ; where t h e argument a r g u m e n t is been w h e r e it b e e n included; i n c l u d e d ; where i t is i s not, n o t , the r e a d e r is i s entreated e n t r e a t e d to to t h e reader g o o d faith, accept, i n good f a i t h , the v a l i d i t y of r e l e v a n t statement. a c c e p t ,in t h e validity o f the t h e relevant statement. ( i i i ) The p a r t of (iii) T h e exercises i n t e g r a l part o f the t h e book. b o o k . Several Several e x e r c i s e s are a r e an a n integral 'lemmas" have "lemmas" have been been relegated relegated to the exercises; exercises; any exercise, exercise, "hints", which w h i c h is i s even n o n - o b v i o u s , is f u r n i s h e d with w i t h "hints", e v e n slightly s l i g h t l y non-obvious, i s furnished which w h i c h are m o r e in n a t u r e of a r e often o f t e n more i n the t h e nature o f outlines o u t l i n e s of o f solutions. solutions. The T h e exercises, e x e r c i s e s ,rather r a t h e r than a t ends e n d s of o f sections, t h a n being b e i n g compiled c o m p i l e d at sections, u n c t u r e s where p u n c t u a t e the punctuate w h e r e they f i t in i n most most t h e text t e x t at a t jjunctures t h e y seem s e e m to t o fit naturally. naturally. u s t like ( i v ) Both (iv) B o t h exercises r e s u l t s are like e x e r c i s e s and a n d unproved u n p r o v e d results a r e treated t r e a t e d jjust p r o p e r l y established properly i n that unabashedly e s t a b l i s h e d theorems, t h e o r e m s , in t h a t they t h e y are a r e unabashedly p o r t i o n s of used i n subsequent u s e d in s u b s e q u e n tportions o f the t h e text. text. prospective reader: The prospective reader: g r a d u a t e students with This r e a d e r s : graduate s t u d e n t s with T h i s book b o o k is i s aimed a i m e d at a t two t w o classes c l a s s e sof o f readers: w e l l as a reasonably r e a s o n a b l y firm f i r m background i n analysis, a s mature mature b a c k g r o u n d in a n a l y s i s , as a s well mathematicians mathematicians working working in other areas mathematics. As a matter areas of mathematics. g r e w out ( t w e l v e ) lectures g i v e n by of f a c t , this l e c t u r e s given o f fact, t h i s book b o o k grew o u t of o f a course c o u r s e of o f (twelve) by w h i l e visiting v i s i t i n g the the t h e author a u t h o r while I n d i a n Statistical I n s t i t u t e at t h e Indian S t a t i s t i c a l Institute a t Calcutta Calcutta p o s i t i v e response in i n the 1 9 8 4 . It w a s largely response t h e summer s u m m e r of o f 1984. I t was l a r g e l y due d u e to t h e positive t o the - - consisting of m e m b e r s of o f that t h a t audience a u d i e n c e -c o n s i s t i n g entirely e n t i r e l y of o f members o f the t h e second second category c a t e g o r y mentioned m e n t i o n e d above a b o v e -- that t h a t the t h e author e m b a r k e d on o n this this a u t h o r embarked venture. venture. The T h e reader r e a d e r is i s assumed f a m i l i a r with w i t h elementary a s s u m e dto t o be b e familiar e l e m e n t a r y aspects a s p e c t sof: of: (a) (a) (b) (b)
(c) (c) (d) (d)
- - monotone measure m e a s u r e theory F u b i n i ' s Theorem, Theorcm, t h e o r y -m o n o t o n e convergence, c o n v e r g e n c e , Fubini's absolute I P spaces f o r p = 1,2,,,,,; 1,2,* a b s o l u t e continuity, c o n t i n u i t y , LP s p a c e sfor - - sparseness analytic v a r i a b l e -a n a l y t i c functions f u n c t i o n s of o n e complex c o m p l e x variable s p a r s e n e s sof of o f one zero-sets, M o r e r a , and z e r o - s e t s contour ,c o n t o u r integration, i n t e g r a t i o n , theorems t h e o r e m s of o f Cauchy, C a u c h y , Morera, and Liouville; Liouville; " t h r e e principles", - - the p r i n c i p l e s " , weak w e a k and weak* functional f u n c t i o n a l analysis a n d weak* a n a l y s i s -t h e "three topologies; topologies; - - orthonormal Hilbert H i l b e r t spaces b a s i s , subs s u b spaces paces s p a c e s and a n d operators o p e r a t o r s •• o r t h o n o r m a l basis, projections, bounded and projections, tors, self-adjoin tors. operators. bounded opera operators, self-adjointt opera ( T h e necessary (The n e c e s s a r ybackground m a t e r i a l from f r o m Hilbert H i l b e r t space s p a c e theory theory b a c k g r o u n d material is i s rapidly r a p i d l y surveyed i n Section s u r v e y e d in S e c t i o n 0.1.) 0.1.)
p a r t of with In I n the l a t t e r part o f the t h e book, b o o k , a nodding n o d d i n g acquaintance a c q u a i n t a n c e with t h e latter not abstract h a r m o n i c analysis w i l l be h e l p f u l , although i t is i s not a b s t r a c t harmonic a n a l y s i s will b e helpful, a l t h o u g h it pleasure,a essential. F o r the r e a d e r who w h o has h a s been e s s e n t i a l . For t h e reader b e e n denied d e n i e d such s u c h a pleasure,
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The author would like to take this opportunity to thank Professor Arveson for kindly permitting the use of a title that is highly reminiscent of his delightful little book on C*·algebras. If this volume manages to capture even a miniscule fraction of the charm displayed in that volume, it would have accomplished all that the author could have hoped for. The title: :0llll eqI
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This volume is equipped with some of the standard fittings, such as a list of symbols, an index of terms used, some notes of a bibliographical nature, and a bibliography. The bibliographical notes are somewhat terse; for more details, the reader may consult [Tak 4]. The terseness also extends to the bibliography, which lists only those books and papers that bear directly on the treatment here; for an extensive bibliography, the reader might consult [Dix]. If the reader spots some inaccuracy in the notes or the references, or anywhere else in the text for that matter, the author would appreciate being informed of such an error. Trappings:
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brief appendix (on topological groups) should serve to perform the necessary introduction, which should precede the furtherance of that acquaintance in Sections 3.2 and 3.3. An attempt has also been made, in Section 3.2, to compile the necessary results from the theory before proceeding to use them. Vll
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I would like to thank the following people for the roles they have played in the production of this book: Professor A. K. Roy, for having invited me to spend six wonderful weeks at Calcutta; the en tire audience for the course of lectures I gave at Calcutta, for their enthusiasm and positive response; Professor M. G. Nadkarni, for some discussions concerning Krieger's ratio set; Krishna, for having faithfully and enthusiastically attended all those seminars I organized, whereby I learnt the theory of von Neumann algebras; Shobha Madan, for painstakingly reading large portions of the manuscript and picking out several errors; Professor W. Arveson for a very encouraging letter which boosted my sagging morale at a crucial stage; Shri V. P. Sharma, for an extremely efficient job of typing, cheerfully performed in an amazingly short period of time; and finally, Vyjayanthi, for reasons too uncountable to enumerate, and to whom this book is fondly dedicated.
ACKNOWLEDGMENTS
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84 85 93 102 108
The Tomita - Takcsaki Theory Noncommutative integration The GNS construction The Tomita-Takesaki theorem (for states) Weights and generalized Hilbert algebras The KMS boundary condition The Radon-Nikodym theorem and condi tiona I expectations
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The Murray - von Neumann Classification oC Factors The relation ... - ... (reI M) Introduction Basic operator theory The predual :f(X). Three locally convex topologies on :f(X) The double commutant theorem
Chaptcr {4 Crosscd-Products Crossed-Products Chaptcr 4.1 Discrete Discrete crossed-Products crossed-products 4.1 4.2 The The modular modular operator operator for for aa discrete discrete 4.2 crossed-product crossed-product 4.3 EExamples actors f ffactors 4.3 x a m p l e s oof 4.4 Continuous Continuous crossed-Products crossed-products and and 4.4 Takesaki's duality duality theorem theorem Takesaki's r o p e r l y iinfinite 4.5 TThe nfinite f pproperly 4.5 h e sstructure t r u c t u r e oof Neumann e u m a n n aalgebras lgebras vvon on N
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Notcs Notes
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Bibliography Bibliography
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Indcx Index
1 69 169
122 122 132 132 1148 48 1155 55
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L6'(z'b)n '1x)Dcts
W*(G),45
46
-t ''*IAt
M*.+,38
sp ex, 97
spt f, 99
xxiv lv
Symbols LList i s t ooff S ymbols
M", a", 102
eneral G f o r ggeneral MOo: G), @ dG ((for ), M
r( 0:), 103 103 I(cr),
H @d 00: K, ll7 117 H .....
13,, 887, .0:' 9Q B o5 7 , l105
f4',119 , rrs
r(M), 107 f(M), r07
3l rr(G), ( G ) , l131
S(M), I110 .s(t[), l0
a,, 1150 a 50
MOo: discrete G),) , l116 for d 16 @ d G ((for iscreteG M
1149 49
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
A von Neumann algebra, the central topic of this book, is a certain kind of a set of operators on a Hilbert space. Throughout the book, only complex separable Hilbert spaces are considered; the adjectives "complex" and "separable" will almost never be stated, but it will be tacitly assumed that they apply. Hilbert spaces will be usually denoted by Jf; on a few occasions, where more than one Hilbert space is under consideration, the symbols K and M are also employed. Further, we shall adhere to the mathematical (rather than the 'I
O_L Basic Operator Theory Arocq;
rolercdo
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As the title suggests, this chapter is devoted to developing some of the basic technical results needed in the theory, and may be safely omitted by the expert. The first section establishes some of the notation employed throughout the book and lists, without proof, the basic facts concerning operators on Hilbert space. The next section establishes the "non-commutative analogue" of the classical results c~ = R1 and (R 1)* = Roo -- null-convergent, summable, and bounded sequences being replaced by compact, trace-class and bounded operators respectively. The existence of a predual yields a locally convex topology -namely, the weak* topology -- with respect to which the unit ball of :e(Jf) is compact. It would not be out of place to suggest that this is one of the primary reasons for the richness of the theory of von Neumann algebras. The third section is a brief examination of this (a-weak) topology, as well as of the more easily defined strong and weak topologies. Some other topologies -- such as the strong* and a-strong* topologies -- are relegated to the exercises. The final section contains the fundamental double-commutant theorem of von Neumann, the definition, and some elementary properties of von Neumann algebras.
Chapter 0 INTRODUCTION
NOIIf,NOOUINI 0 rerdeqJ
2
O.. IIn n ttrod r o d uuction ction 0
roducts a physical) whereby products are p h y s i c a l ) cconvention h e r e b y iinner i n e a r iin n tthe irst nner p r e llinear h e ffirst onvention w r a t h e r tthan and other i n e a r iin n tthe h e ssecond h a n tthe he o ther vvariable ariable a e c o n d ((rather n d cconjugate o n j u g a t e - llinear way around). w ay a round). Consistent with our disregard nonseparable onseparable ith o e s o l u t i o n tto isregard n o ttotally otally d C onsistent w u r rresolution Hilbert wee sshall only measure are f tthey hey a e a s u r e sspaces p a c e s iif re hall o n l y cconsider onsider m H i l b e r t sspaces, p a c e s ,w separable. Actually, Actually, we shall only only consider consider measure measure spaces spaces (X,f (X,r,JL) separable. ,y) 2(X,JL) is JL is a non-negative non-negative a-finite measure space, L2(X,tt) where ,r o-finite measure space, such such that L separable. separable. Subspaces will usually be be denoted M and 13 will as 11 denoted by symbols symbols such such as Subspacesof of le lf, 14,, l- l"t* of N. of...closed subspaces of le, and M Mm }:=1 is a sequence If {M sequence of closed subspaces N . If {il"}:=l n if n 1 "i:-&"
if
el=ril,, for the or Il=rli"; closure of for the closure Ln=1Mn; thti the f or n i~ m, we shall shall write $n=1Mn "direct sum" "direct notation will be be used used only orthogonaldirect direct sum sum only for an an orthogonal sum" notation ef=rf" for the of closed we shall write $:=1len the also write course,we shall also closedsubspaces. subspaces.Of course, "external"direct Xf' will be "external" which case be caseeach each len spaces,in which direct sum sum of Hilbert spaces, naturally identified of the the direct M and If 11 and N identified with aa subspace direct sum. sum. If naturally subspace g 11, n are of le we shall M 90 N for l"t M Ii 1?with N C lt we shall write It are,closed closedsubspaces subspaces N1 r. Vectors while symbols lf will be such as as Vectors in le etc., while symbolssuch be denoted denoted by by ~,n,t, [,11,6,etc., a,x,Y,z,e,f,u,w be bounded operators. operators. It will be always denote denote bounded a,x,y,z,e,f ,u,w will always necessary, on such objects objects necessary, unboundedoperators, operators,such on occasion, occasion,to to consider considerunbounded being F, etc. course,it may may turn A, H, H, K, K, S, S, F, etc. Of course, being usually usually denoted denotedby by A, when that happens, out that happens, instancesthat is actually bounded;when out in some someinstances that S S is actuallybounded; the our relief would, it is hoped,offset the conflict conflict with our relief would, is hoped, offset the the consequent consequent notational notationalconvention. convention. lf has t(Xf)of all bounded has the The the structure structure The set linear operators operatorson on le set :f(le) boundedlinear (with respect respectto to the the Banachal&ebra of aa C*-algebra: exrliCitly, it is a Banach is,,a algebra(with C*-algebra:,explicitly, pointwise vector vector !f, lid l), pointwise operator norm IIxl sup(llxlll: t Ee le, operator norm lllll == I}, llxll == sup{ilxtll: product),equipped operations with an involution equipped.with..an..involution operationsand and composition compositionproduct), 2 - x*, • xr .... which satisfies the so-called so-calleoC*-identity: b*-ioentlty: IIx*x x*, which satisfiesthe llx*x IIll = IIx llr 11ll'. projection I't will with The aa closed subspaceM The orthogonal associated closedsubspace orthogonalprojection associated p2n= ph pn== Pk pl"t,this usually is the the operator operatorsatisfying satisfyingPM = PM usuallybe be denoted denotedby by PM; this is (Hereand pn == 11. t't (Here rangeof an and the range an operator operatorx ran PM and in the the sequel, sequel,the and ran p satisfying p == will be satisfyingP ran x.) any operator operatorP be denoted by ran x.) Conversely Converselyany denotedby projection p. p2 the orthogonal projection onto ran p. Such operators will p2 == p* p* is ran onto Such operators is the orthogonal projections. We never consider be We shall shall never consider referred to to as as projections. simply referred be simply projections. Recall is called non-self-adjoint called Recall that the operator operator x is that the non-self-adjointprojections. t(Xf),we generally,for any we shall self-adjoint M fg :f(le), shall more generally, any set setM if xx == x*; x*; more self-adjoint if M*. let if M = M*. luUand self-adjoitt if let M* = {x*: and call call M self-adjoint {x*: xx Ee M} theory Probably Hilbert space spacetheory fundamentaltheorem theoremin Hilbert the most most fundamental Probably the which may may be be is operators,which self-adjoint operators, is the the spectral spectral theorem theorem for self-adjoint with spectrum spectrumsp sp formulated operatorwith let x be be aa self-adjoint self-adjointoperator formulated thus: thus: let Borel x; from the the class classof Borel mappingFF ....' e(F) e(F) from there exists exists aa mapping x; then then there (a) e(sp projectionsin le lf satisfying: subsetS' e(sp satisfying: (a) sp xx to to the the class classof projections subsetvof sp = = * fl for n m, then (b) ui=rFn F,, Fand F x) == 1; (b) if F = U:=1Fn and F Ii F = 4> for n ~ m, then Q m n ! projectionsand pairwise orthogonal {e(F )} ...-1 is and orthoS,onalprojections is co aa sequence sequenceof pairwise {e(F-))I-, n n(c) xf, is in the for ); ef-rran e(F and any ran e(F) = $n=1ran and (c) for any t,n in Jf, if JLt,n is the e(F,); ran'L1ii';'= r1,a l,n n ie(r)l,nt, defined by by JLt finite measure on sp n(F) == <e(FH,n>, sp xx defined finite complex compld'i'measure'on rrt,n(r) < x l , , n >== f>'dJLt,n(>'). t h e n <xt,n> then ' J\dgg,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
If x ~ 0, there exists a unique y ~ 0 such that x = y2; the square root y is given by y = f(x), where f(t) = t 1/ 2 for t ~ 0, and will always be denoted by y = x 172 . The spectral theorem, as we have stated, is valid for the larger class of normal operators, an operator x being called normal if x*x = xx*. An important class of normal operators is the class of unitary operators. Recall that an operator u is unitary iff u*u = uu* = 1. (Here and elsewhere, the identity operator is denoted simply by 1, and >'·1 is noted simply by>..) An operator u satisfying u*u = I (resp., uu* = 1) is called an isometry (resp., a coisometr:y,). More generally, an ope1f:tor u is called a partial isometry if Ilu ~ II = II ~ II whenever ~ E ker u. It is well-known that u is a partial isometry if and only if e = u*u is a projection; in this case, f = uu* is also a projection and the subspaces ran e (= ran u*) and ran f (= ran u) are called, respectively, the initial and final spaces of the partial isometry u. Another basic result is the polar decomposition theorem, which states that every operator x admits a decomposition x = uh which is uniquely determined by the following conditions: u is a partial isometry, h ~ 0 and ker u = ker h = ker x. The positive factor h is given by h = (x* X)I/2 and will usually be denoted by Ixi. It is not hard to show that if x = ulxl is the polar decomposition of x, then x* = u*lx*1 is the polar decomposition of x*; more generally, (uf(lxl»* = u* l(lx*1) for any Borel function f on [0,"'). So much for bounded operators; let us now recall some facts concerning unbounded operators. A (possibly unbounded) operator is k=O
0l{
4<x~,n> = I: ik<x(~ + ikn),~ + ikn >.)
('< e,rl + J'(unt+ !)x>nl
8
=.{r'1xr7
where gn} is an orthonormal basis for R) Then there exists an isometric algebra isomorphism f ... f(x) from L G>(sp x,j/.) into :f(Jf) such that for any t,7'1 in Jf , = ·rf(>')dj/.~ n(>'), with j/.~ n as in (c) above; further, f(x)* = l(x). It shou(d be cle'ar that e(F) .:, IF(x), where IF denotes (here and throughout the book) the indicator or characteristic function of F. An operator x is said to be positive (denoted x ~ 0) if <x~,~> ~ 0 for all ~ in Jf, or, equivalently, if x = x* and sp x f [0,"'). (The equivalence of these conditions is proved using the spectral theorem and the polarization identity which asserts that if x E :f(Jf) and E 1£, then
uoql ? pue (A)f I r Jr lurll slrasse r{Jlr{irr{1r1uapr uollezlr€lod eq} pue r u'l luorooql l e r l cads eql Eulsn palord sl suorllpuoJ osaql 3o acualearnba '(-'01 oql) i r ds pue *x = x JI ',{lluale^rnbe .ro ? ul I II€ JoJ 0 < < l ' l r > J I ( 0 < x p o l o u a p ) a , r r l r s o do q o l p l e s s r x r o l € r o d o u V 'J Jo uollcunJ cllsualcEJer{c ro rol€clpur aq1 (1ooq oql lnoqEnorr{l pue oreq) salouap J1 areq,n '(x)st,.=^(g)a teqt r,E€tc?q pJnoqs lI '$)! = *1r)/ 'roqtrn3 lairoqe (c) ur sBu rt qlla'(f)" itp(f),/l = ',t ul 4'l fue roJ lBql gcns (g)g otul (rfx ds)-? ruorJ'(x)/ - / ruslqctrouosl €rqeEle__crrleuosr u€ slslxe areql u5q1 ( ;1 roJ slseq I€rurouorllro ue s1 {"1} alaq/(r
en
n
'."1'ul(c)ar,-z = (ilrt i j/.(F) = I: 2-n<e(FHn'~n>'
{q uaayE sr arns€a(u qcns oug) '0 = (g)a JJI 0 = (J)rt l€rll qtns x ds uo ornseatu fue eq rt lol :enr] sl aroru .1ce3 u1 '(1)ap 1l = x l e q l r { c n s r d s u o p e u l J e p ( . ) a a r n s e a r ul e r l c o d s s s l s l x o o : a r i l . * x = , ( J I : S n q l p a s € r r l d e r e df l l e n s n 3 r € s l u o r u e l B l so a l E u l p s c a r d a q a
In fact, more is true: let j/. be any measure on sp x such that 0 iff e(F) = O. (One such measure is given by
j/.(F)
x*, there exists a spectral measure e(·) defined on sp x such that x = J>. de(>.).
The preceding two statements are usually paraphrased thus: if x =
,{roaq1 roleJedo crseg 'I'0
3
0.1. Basic Operator Theory
O.. IIntroduction ntroduction 0
4
D -... lf:If where D D is some some linear (not necessarily necessarily closed) closed) a linear map H: D subspace of of 4:If, called the domain of of flH and denoted denoted by dom //. H. The subspace H is called a closed closed operator ifif itit has has a closed closed graph, i.e., i.e., ifif operator f/ = {(1,11t): {(t,Ht): It e E dom Ifl H} is a closed closed subspace subspace of of 1l :If @ ~ lt. :If. G(H) = For a densely densely defined defined operator H (i.e., (i.e., dom f1 H -= lt), :If), the adjoint adjoint 11* H* For iven b is the uniquely defined linear operator with domain given byy ddom w i t h domain g om is the uniquely defined linear opera.to.r H** = = {11 E :If: , celll l itil Vtl eE ddom and I) a H ll V o m tH} n d ssatisfying atisfying E 3 c > 0 3 I1 ll < {n e E dom H Hand 11 eE dom I1*. H*. and n = whenever (t e An operator T extension of of an operator S if G(S) £; ^S if G(^S)g An Z is called an extension = ,St in The G(n, i.e, if dom S £; T n = St for t S. C Z ?"E for dom S. i.e, if dom and G(T), I equation bee iinterpreted ass S !£; T and Z c£;. SS. . I a nd T I iiss tto nterpreted a e q u a t i o n ,SS = T o b An operator S is said to be be closable closable if if it it satisfies satisfies either of of the An a ) tthere equivalent exists operator Z losed o perator T ffollowing ollowing e o n d i t i o n s : ((a) here e x i s t s a cclosed q u i v a l e n t cconditions: b ) iif 0,() d £; T; 0,, tthen does not belong ot b h e cclosure losure f It I'I- 0 oes n e l o n g tto o tthe h e n ((O,t) ssuch u c h tthat hat S g 4 ((b) of G(S). closable operator admits a smallest smallest closed closed It is clear that a closable of G(S). It extension S "S which which is characterized characterized by the equation G("S) = V[3'). G(S) = 6T3)-. extension then It f a c t that i s a densely d e f i n e d operator, o p e r a t o r , then I t is i s a standard that if S . S is d e n s e l y defined s t a n d a r d fact ( (in G(S*) t E dom S} (in:lf ~ :If). A consequence of this fact fi e f). A of e S)r consequence G(S*) = {(-Set): {(-Sl,E): is that a densely if S* is if and only if closable if densely defined operator S is closable = S**. densely w h i c h case i n which c a s e "S S = S**. d e n s e l y defined, d e f i n e d , in The operator H is said to be self-adjoint if if H I/ is densely densely defined Il be said and I/*. The spectral spectral theorem theorem extends extends to unbounded self-adjoint and H == H*. 11 if if and operators. sp H belongs to the spectrum spectrum sp scalar \ belongs operators. Recall that a scalar>. = :If, = {O} I) = I) = lf, ker(H only if ran(I/ -- >') ker(F/ -- >') if it it is not the case case that ran(H {0} and (// -- >.r 1;-t1 is a bounded formulation of the (H bounded operator operator on:lf. on lf. The formulation - H*, if H ff = I1*, then sp spectral for unbounded sp unbounded H is as as follows: if spectral theorem for I / H £; IR and there exists a spectral measure e(·) defined on sp H such m e a s u r e d e f i n e d o n s p s u ch t h e r e e x i s t s s p e c t r a . l e ( . ) a n d ttR - (with ( w i t h Ilt a S before) i f and i f rl>'12dll~ before) that d o m H if a n d only o n l y ^if t h a t t , Ee dom t l r tr as l l x l ' d y , t(>') r ( \ ) < co l f l ' ' As A s in i n the < l / ( , n > = J>. i n :If.' the w h i c h case a l l 11n in in i n which c a s e l u r 11(>'J' l x Jilt r . , ( l ) 'for ' f o r all "fuhctionil'calculus" - f(H» (./ ... f H. bounded calculus" (f for H. or is a "functiona\ bounded case, case, there is /(H)) p o l a r decomposition The e x t e n d s to t o closed c l o s e d densely densely t h e o r e m also a l s o extends T h e polar d e c o m p o s i t i o n theorem .S admits a defined operators: operator S closed densely densely defined operator operators: every closed decomposition b y the the w h i c h is i s uniquely d e t e r m i n e d by . t = uH u H which u n i q u e l y determined decomposition S conditions: and H is a positive self-adjoint conditions: ua is a partial isometry and ,S = ker u operator ,S satisfying ker S tu = with domain equal equal to dom S operator with positive ker H. self-adjoint fl is the unique H. As in the bounded bounded case, case, H operator f/22 == S*S. S*S. operator satisfying H conjugate In w e would w o u l d need n e e d to t o study s t u d y a conjugate I n the s e c o n d chapter, c h a p t e r , we t h e second possibly (i.e. S(>.t f.Sl + S11) linear operator 'S(rl + 11) n; = >:St Sn) which is possibly operator S (i.e. conjugate unbounded. In this case, S* is the unique conjugate linear is case, the unbounded. = {11 < l f : ) > ll e " O operator E :If: 3 c > 0 3 I<St,11>1 , c" lIIl ttil d e f i n e d on o n dom d o m S* S* = o p e r a t o r defined l . s t , n t l {n .S <.Sl,r,> == <S*11,t> <S*D,l> for all tI Ee dom S Vt ,S) and satisfying <St,11> V( Ee dom S} ( I t should w a s densely densely and that S S was h a v e been b e e n stated s t a t e d that s h o u l d have d o m S*. S * . (It a n d 11n Ee dom valid. i s not n o t valid. defined, u n i q u e n e s s is f o r , otherwise, t h e asserted a s s e r t e d uniqueness o t h e r w i s e , the d e f i n e d , for, densely However, we shall consider densely shall only consider here and elsewhere, elsewhere, we However, here is valid in this defined operators.) decomposition is operators.) The polar decomposition "polar part" u will will context too, the "polar with the modif ication that the the modification t.oo, with facts may now be These facts isometry. These be aa conjugate conjugate linear partial isometry.
1
'*(A)) eceds l e . r l pe q l J o u o l l ? J r J r l u e p l
Then, clearly 1IWt 711/ = lit 1/ I/nll. (The inequality ill follows from Cauchy-Schwarz, 'while the reverse inequality is obtained on considering x = t 71 t.) The following exercises lead to an identification of the dual space K(Jr)*.
ue o l p e a l s e s l c J e x a E u r m o l l o . y a r l l ( 7 ' u t = | Eurreprsuoc uo paurelqo sr f lrlenbaur esJoAoJ oql ollr{rr\...zre,rnqc5-f qcne3 ruoJJ s^\orroJ = {1:ea1c.ueqr I i(lrlenboutaqr) 'llrrll
lltll
llr't"ll
'= (r;tt'lr'l fq eulJep 'll ) u'i,.(rerlrqre rog *(rf)f ?. Y.'t" '' 1 = -1-drS = lxl = ll ll ll"ll leql uollonrlsuoc ul{-", o q l u o r J r € a r os ! l I , ' u u ' 3.: x l'ql qJnsx usr roJ srs€q ue sr ("1) uaql ''gn = ul JI teqt apnlcuoc.n 3o aceds I€ruJouoqlJo I€rlrur arll sl lxl Tro{ acurs "lx[ra1 = lx I ugr roJ srseqI€rurouor{Uo 'lr'utrr"* u€ sr {"t } ererl,n 3 = lxl qderEered lssl oql ^q ,oS 'rolurado ? sI x*n = lxl uor{l ,lxln - x uolllsodruocap lceduroce,r1l1sod :e1odsuq (A) ) I r .lI (;fgrrel) .(x4hul I€epr peprs-o/(lpesolc-rurou 3 sr srol€rodo lcedruoc Jo (r{)) les oqr leql pelJlra^ flrsoa sl lI ('il Jo senlez\3o aEuereql perJ.rJods lou a^eq e/r os pu€'ellulJ sI runsoql'{usr eflulJ suq r JI) .(n)(@'0)[ uer = x u€r JoJ srs€ql€ruJouor{lroue sr {"1) pue ("\ oJaz-uoufueru {1a1rurgffi
construction that I/x 1/ = II/xl II = sUPn >'n = >'1' For arbitrary t,71 E Jr, define Wt.n E K (Jr)* by Wt,71(x) = <xt,71>. n' n
orthonormal basis for ran I xl = ker-1xl.. Since ker1lxl is the initial space of u, conclude that if t n = un n , then {tn} is an orthonormal basis for ranx such that x = L >'n t t n ' It is clear from the n' n
are infinitely many non-zero >'n) and {tn} is an orthonormal basis for r an x = ran 1(0 OO)(x). (If x has finite rank, the sum is finite, and so we have not specified the range of values of n.) It is easily verified that the set K (Jr) of compact operators is a norm-closed two-sided ideal in l(Jr). (Verify!) If x E K (Jr) has polar decomposition x = ulxl, then Ixl = u*x is a positive compact operator. So, by the last paragraph Ixl = L >'nt n 71 where {71 n} is an "n''tn
n
areqr JI) o r t\ arerl^\'tl'tlrt'rtta = x lsql pue.alqelunoclsoru1€ sl {0}\x ds,tPql ^\ou al?lpeturulsl lI '0 <,r,-vJBeroJ l€uorsuorurp-allulJ sr (x),- ', I u€r lcql s^tolloJ ll .(r)t€'tlI u€J ol luauelels lssl eql Eulflddv 'rolerado lcedruoc ellllsod € s! r l€ql ^\ou asoddng 'Ieuorsuaturp-ellulJ dlrressacausr ll ueql .lcedruocsl r JI pu€ .0 < t ouos puB l,f uI I II€ roJ lltll, < lllxll reqt qcnsfi Jo er€dsqnspesolc € sl l,l JI leql oas ol fse5 d! r1 'rlas"lcedruoc (-rurou) € uI peulgluoc sl (I > :lx) JJ lo€druoceq ol pl€s sr x roleredouB l€ql IIBceU lllll (*x uBr ''clsar) ,( u€r Jo slsuq l€rurouor{uoue sr (r=l{{tr) flarrrlcaclsar)-r=l{{1}pue
°
and {tk}~=1 (respectively (71k}~=1) is an orthonormal basis of ran x (resp., ran x*). Recall that an operator x is said to be compact if {x t: 1/ t 1/ ill I} is contained in a (norm-) compact set. It is easy to see that if lot is a closed subspace of Jr such that I/xt II ~ E 1/ t 1/ for all t in lot and some E > 0, and if x is compact, then lot is necessarily finite-dimensional. Suppose now that x is a positive compact operator. Applying the last statement to ran IrE .4x), it follows that ran I[E OO)(x) is finite-dimensional for each 'E > O. It is immediate now that'sp x\{o} is at most countable, and that x = Ln>'ntr r , where >. ~ (if there
lu'{lrl\ tit =t aJoq^\ '
nk '
< r\
~k'
>'k t r
"'
n
where >'1 ~ ... ~ >'n >
<
k=1
= L
°
0 <'\
X
For t,71 in Jr, let tt n be the operator defined by tF. 71 t = t for t in Jr. It is an easy consequence of the Cauchy-Schwarz inequality that I/t t 7111 = lit II Ihll; in particular, if t,71 ¢. 0, then tt 71 is an operator' of rank one. Conversely, it is clear that every operator x of rank one is expressible as >'t t n, where >. = Ilx 1/ and t and n are unit vectors. More generally, it din be seen that every operator x of rank n is expressible as
se alqrssardxosr tr lu€l Jo r rol€rado fra,ra lBql ueosaq u.q?l1 .fllereueE ero41 .srolcol lrun ar€ (l puE I puE llrll = \ eroqir'u lr\ sE elqrssardxa sl euo {uBr Jo x role;ado ^rorroierit reelc sl 1r .flosreauo3 .euo >luurJo.rolutacto u e s r u ' i t u e q l ' 0 r . u ' l J I ' r e l n c r l r e du r : l l u l l lltil = llu'lrll l€rtl flllenbaul zre,nqcg-fqoneJoql Jo acuonbesudc"fs"ee" ue ir 11 ,l ul ) roJ lcu'l> = lu 11 [q paur3aproleredooql eq tt'?11el .g ur g.l rog 'Orh pnpar4 ot1l ?0
0.2 The Predual r(Jr).
& ec?dslraqllH ele8nfuoceqt olul S'ruop LuorJ rolerado J?ourl e se fEulaaeln l(q as€cr€aurl oql uorJ pa^rrep oq be derived from the linear case by viewing_S as a linear operator from dom S into the conjugate Hilbert space Jr.
'(t{h Isnperd arII 'Z'O
5
0.2. The Predual l(Jr).
0. O. Introduction Introduction
6 Exercises Exercises
(0.2.1). IfIf r,r weE K(lf)*, K(lf)*, there there exists exists aa unique unique operator operator t(t^l) t(w) eE E(lf) l(lf) such such (0.2-l). and is compact ,(o) !t. Further, that n) (Hint: Consider lf' of basis LI
I:o e.t, rx ==tij=1 , t ' ,l i, = t ' r i"j'''j 1
where 0, ej are are comPlex complex numbers numbers of of unit unit modulus modulus such such that that where
= lt^t(t e,u,(tt,,tJ g.,g.)l and note note that that and
For compactness compactness of of t(tl), t(w), ffirst reduce to the case case t(tl) t(w) > ~ 0 0 by by irst reduce For is nft = where considering 0 ~ e E K(lf)* K(lf)* given given by 0(x) ~x) = t(xrr), w(xu*), where {!r) t(w) _=. = uh is the the considering polar OecJ.position decomposition of r(r,r) t(w) and and noting noting that t(0) t(~) = = 'f;,if h; if l(t't) t(w) )~ 0' 0, use use potar conclude < and lloll tll the already established inequality LI1 , IIwll and conclude inequality li" "tr."Av established that ran[E CD)(t(W)I/2) is finite-dimensional for each positive E, and hence tliat t(W)I/2 is compact, being a norm-limit of finite-rank operators.) operators.)
(0.2.2). decomposition u admits aa decomposition (O-2.2\. If e K(lf)*, ot E K(lf)*, then w If w
... o
w 7) , d,,0[,,,0,,' '== L ocnw, n=1 "n' n J,
orthonormal ( n r ) are a r e a, a pair ^ . p a of ior f orthonormal where a o . .<' a...n dand {~n} c r -~) 0, 0 , LE ocn w h e r eocn { ( , r rand } n d {7)n} of element of an ines def sequences. Conversely every such system defines an element sysi'em rrqu"n.&. Conne?selyeve,fi,,'!uch = it o,,tn,,,E,, I = (Hint: if t(tl) "". K (If)* as above. Further, Ilwll = L ocn' (Hint: if t(w) = L oc t7) , is E Furiher, ll,,rll r irl- as above. n n'''n
if 13B == and if t(r,r)and operatort(w) the compactoperator the compact of the decompositionof canonicaldecomposition the canonical (I3 E Co with I3 ~ 0, observe that ) (8,,) e co with Br, n ) 0, observethat n xx = = EL BI3rnrt,/ q 71r r , Ea rK(lf) K r t( r ) 'tn' n
and andthat that
lloll.o, LEoccr,,B,, ' ll"ll111311<: o(r),
1, to cr,,' ( that LE ocn concludethat to conclude cf == 1!1, and resultc~ classicalresult to the the classical appealto and appeal
IIwll.) ll,,rll.l00
bijection t(o) isis aa bijection that wr,r...- t(w) (r)*}. Note Note that Let == (t(w): t(1?)* Let l(lf). 'kt$l'"nO {r(u):wu E€ KK(If)*}. when Banach.-.space a t(lf)* is between K (1f)* and l(lf).. Hence l(lf). is a Banach space when Hence t(r.)-... urt*..n r(tf)* c (0'2.1), Ex. normed 111 =='l["11. IIwll. As (If). notedinin Ex. (0.2.1), l(lf). f KK(1f)' As noted tttui: Ilt(w) llr(ur)ll, normedthus:
lenperd orII
0.2. The Predual l(Jof).
7
*(g)g
'Z'O
'ft)f lo I€nperd aql sB ol parroJer osl€ sl *(g)g .uoseer slrll roJ '(*h ot crqdroruosr fllecrrletuosl sI *(*(Ah) acuds I€np oqt lsgl qsrlq€lsa qcrqa sasrcrexe Jo les lxeu aql fq polecrpul^ sl *(nh 'srolerado ss€lJ-rc€rl poll€J uoJlBlou orII are '(g)g ur sroleradg
Opera tors in l(X). are called trace-class opera tors. The notation l(X). is vindicated by the next set of exercises which establish that the dual space (l(X).)* is isometrically isomorphic to l(X). For this reason, l(X). is also referred to as the predual of l(X). sosrcJoxl
Exercises (g-Z-O)
uolllsoduocap € slrrup€ '(rih r d i(uy
(0.2.3). Any p
l(X). admits a decomposition
E
co
''u''lr'o t3" = o p =
rant, TI ' n=l "n' n
u.,u. = tt' ll l€ql eloN :lurH) 'uo3 = -G)f"r1; requnC .a^oqe sB '(tt)X 3o d luaruela ue saurJep rualsfs qcns Xriha .dlesranuo3 .socuonbes I€rurouorluo go rred e are {"u} pu€ {"1} pu€ o t'o I.0 < .o oreg,t\
where an ~ 0, r an < co and {tn} and {TIn} are a pair of orthonormal sequences. Conversely, everx such system defines an element p of l(X). as above. Further IlplIl(X) = ran' (Hint: Note that t, TI = ~n' n
•
('Q'z'\ 'xa ol luadclePue 1uu'"1";, t(w~
TI)
n' n
and appeal to Ex. (0.2.2).)
Eurfgslles r(*(*)f) ur iQ anbrun B slsrxe eraqt .(fih ) x JI .0.2.O)
4>x in (l(X).)* satisfying for all t,TI in X. The map x ... 4>x defines an isometric isomorphism of l(X) onto (l(X).)*. 0
tal'(n)) r x pu? '(r[)I ) o *(A)) r cq Jr.:s^\olloJse *(g)) uo ernlrnrls alnpourq-(g)g ue eurJep u€c euo '(Ah uI I€epl papls-orrl B sI (A)) aculs
Since K (X) is a two-sided ideal in l(X), one can define an l(X)-bimodule structure on K(X)* as follows: if a E l(X), w E K(X)* and x E K (X), let
= w(xa);
(r)
l(ar)o = (r)(o.a) (a ·w)(x)
= w(ax).
'(xaln = (x)(a.o)
(I)
(w·a)(x)
'a^oqe s€ o', qll^\ l€ql d.;rre1
Verify that with a,w as above, t(w·a)
= t(w)a.
Q)
'.(n)to= (n.o)t
fq d go ecerl ul Ieapr poprs-o^\l e sr *(g)X snqa
Thus l(X). is a two-sided ideal in l(X). trace of p by
For p
r d rog
E
l(X)., define the 3ql aulJep '*(fih
= at(w);
'r(o), = (r.o)l
tea ·w)
(2)
'(*)f
(d)rQ=<1'd>=dr1
tr p = = 4>l(P),
'(o)/ = d qll,n '(g) suoltenbo lBql ) o Jl .ue,ll .0.2.0 .xg Jo uoll€lou oql ur
E
aqt urorJ s/holloJ lI '(nh
in the notation of Ex. (0.2.4). Then, if a equations (2), with p = t(w), that
l(X), it follows from the
do t1 = = <1o'd>= <1'Ddt = od tl = <pa,l> = = =
tr ap. ll
tr pa
,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
In the sequel, we shall write 11.11 1 for the norm on l(X). and 11·11 will be used only for the norms on l(X) and R
srsrc.rexl
Exercises .'(nh ? 3 = d ta.I
(0.2.5). Let p E l(X).. Let p = r ant ~ TI be a decomposition of p as in Ex. (0.2.3). n' n
se d go uolllsoduroJapu oq
'u''lr'o
'O"r:;:'1}ili
0. O. Introduction Introduction
8
t t o cconsequently, onsequently, n l11.11 ((a) a ) TThe s cconvergent o n v e r g e n t iin l ' l l ' 1, aand . 1I,i , , 4TI, , iis h e sseries e r i e s IL. r ,cxn 'In' n
for any any xx in in E(lf), I(X), trtr px px == 1P,x7 == XL cr,r<x(rr,rl,r>; cxn<x~n,Tln>; in in particular, particular, for L ccxn<~ r - < ( .,TIn>' ,'->' ttr r Pp == E
(b) vlrirv. Verify lrtul that ;*;''= p~p = rL o|rn,,,n,, CX~/Tln TIn and and hence hence that that llcll, IIplll == rL o,, cxn = = (u)
tr(p*p)1/2. ' tt(P* P)rlz (c) Show Show that that tr tr pp == EL for for any any orthonormal orthonormal basis basis {5,,} gn} of of r.X. (c) u s t i f y aan o Appeal Cauchy Schwarz n aappeal p p e a l tto o jjustify ((Hint: Hint: A c h w a r z tto p p e a l tto o C a u c f i y - -and and S Parseval.) 0 O Parseval.) The use use of of the the word word ntrace" "trace" is is vindicated vindicated by by part part (c) (c) of of the the last last The p, main exercise: in any matrix-representation of p, the main diagonal diagonal is a of exercise: in ot h i c h ddoes 0 . 2 . 1 ) ) aand o e s nnot Ex. which by E um, w h e ssum, n d tthe x . ((0.2.1)) e q u e n c e ((by ssummable u m m a b l e ssequence of co-ordinate system, system, is the trace of of p. p. depend on the choice of 0.3. Thrcc Three Locally Locally Convcx Convex Topologics Topologies on on q10 l(1t)
several reasons, reasons, the norm topology is not a very very good topology For several (Reason: represent on I(X). For example, I(X) is nonseparable. (Reason: represent lfX as as nonseparable. example, t(lf) on I(lf). 2[0,1], and for ( ( p r o j e c t i o n p , t h e s u b s p a ce o n t o t h e L " I " 1, let P be the projection onto the subspace l , l e t b e t f o r 0 1 2 [ 0 , 1 ] ,a n d t of functions functions supported supported in in [0,t];-if [0,/]; if s < l, I, then Pt Pt -- P" p. is a non-zero non-zero of projection and hence hence has has norm one.) one.) Also, Also, if if lt, Mn is an increasing increasing (lvtn) sequence sequence of subspaces and M = ~, if the sequence {M } is not l'1 U , if the sequence of subspaces n I t turns n o r m . It turns i n the t h e norm. eventually n o t true t r u e that t h a t PPnn ... i t is i s not c o n s t a n t , it e v e n t u a l l y constant, ' P in consider certain out that, in such d-b better to consider one would do such situations, situations, one other topologies t(lf). topologies on I(X). on t o p o l o g i e s on c o n v e x topologies Let l o c a l l y convex a b o u t locally r e c a l l something s o m e t h i n g about b r i e f l y recall L e t us u s briefly is aa vector space space M is vector seminorm on aa vector vector spaces. spaces. Recall that aa seminorm - [0,00) = 1>.lp(x) p(\x) = mapping p: pi M ... such that p(>.x) v) "< p(x) + lrlp(x) and p(x + y) [0,-) such t\ f a m i l y p(y) whenever E M and >. E ([. Suppose that a family {Pi: \ e C . S u p p o s e t h a t a M a n d p(y)w h e n e v e r x,y e x,y { n r :ii Ee I} is the the ot M is given. The induced topology on of seminorms induced topology is given. on M is seminorms on pt is is each Pi to which each smallest respect to on M with respect vector topology topology on smallest vector A} M e in cr net a continuous topology, a net {xcx= at the the origin; in this topology, continuous at {x; cx E A} f o r each e a c h ii Ee I./ . converges i f Pi(x x ) ' ... 0 0 for M if i f and a n d only o n l y if in M t o xx in c o n v e r g e sto F { x CX o -- x) t(lf) is is the the topology topology (a) The Definition on I(X) topology on The strong strong topology 0.3.1. (a) Definition 0.3.1. = pq(x) = lf) defined bv p~(x) (pq: ~E Ee X} defined by induced seminorms {p~: family of of seminorms by the the family induced by
Ilx~11. ll' I ll.
the induced by by the t(lt) is (b) is the topology induced the topology (b) The on I(X) weak topology topology on The weak pt.n(x) lf) defined by P~ (Pr TI:n:1,0 definedby family {p~ ~,TI Ee X} TI(x) == l<x~,TI>I. l<x('4>1. family of of seminorms seminorms by the the induced by (c) t(lf) is is the the topolo'gy topolo'gyinduced on I(X) (c) The o-weaktopology topoldby on The a-weak = pp(x) D xPl. S(Xf)-} by family E I(X).} defined by pp(x) = Itr xpl. 0 : p e defined ltr seminorms(p famity of of seminorms {pp: p P with respect respect to x.r with l(tf) converges In in I(X) convergesto net {xi} language,aa net In simpler simplerlanguage, {x1}in to the: to the: - ~ II ...'00 for iff xi 1?, i.e.,iff xrl~ ...'xlX ~ for all all ~( Ee X, i.e., (a) iff Ilx (a) strong topologyiff strongtopology llx,( i ~ - Xttll in the strong topology on X ; lf on in the strong toPologY ;
(a) When X is infinite-dimensional, show that (x e :e(X): x 2 = O} is strongly dense in :e(X). (Hint: let a e :e(H); a typical basic neighbourhood of a in the strong topology is of the form (x e :e(X): II(x - aHjll < e for I ~ i ~ n}, for some set {t1' ..., t n} f X and e > O. Argue that the tj's may, without loss of generality, be assumed to be linearly independent, even orthonormal. Then pick 7}1' ..., 7}n e X satisfying (i) Ilatj - 7} j ll < e for each i, and (ii) {tl' ..., tn' 7}1' ..., 7}n} is linearly independent. Let x be a finite rank operator such that xt j = 7}j and X7}j = 0 for each i and xt = o whenever t e (t 1, ..., tn' 7}1' ..., 7}n}.L.)
r " ' e r ur u 1 " " ' I l ) r ! r a a a u a q mg - (1("t = : r p u u , r { J ? aJ o J Q = r g x p u e l t t = l l r l € q l q r n s r o l u r a d o {uzr ,...J]) e l r u r J B e q r l r . I ' l u e p u e d o p u rf l r e e u r l s l ( " u , " . , r t r . ' l (lt) pue ', qcee roJ r > - ! : r ; ; ( r ) E u r f g s r l e sf i r ' { , r . . . . , r r rr t c l d ;;ltr uaqJ 'l€ruJouoqgo ua^6 'luapuddepur flreeurl aq ol pounss€ oq ',(1r1e:_ouaE 'feru s.!l orll l€ql anEry .0 < r pue Jo ssol lnogll^\ ""'tl} Irs eruosro3'{a } I > I roJ r > lllt(p -- x)ll :(r{h ,i i {"1 r x) ruro3 aql Jo sr ,tEo1odo1Euorls oql ul r go pooqrnociqtrau crseq 1ec1d,{t e l(g)g ) o t?l :lulH) .(,l)f ur osuop flEuorls sl (O = { :Qit r x) 1eq1 ^\oqs .I€uorsuourp-olrulJul sl fi uaq/t\ (e)
'(e'e'o)
(0.3.3).
('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
r
where (tn}:=1 is an orthonormal basis; verify that u*n ... 0 strongly while un 0 strongly, as un is an isometry for each n.) ur,I+u( I=u . t t1 3 =n
u=r.t , n=1 t n+1,t n 00
"e'I irJIrIs tlrrrr,ron "
(a) When :e(X) is equipped with either the weak or the a-weak topology, the adjoint operation x ... x· is a continuous self-map. (b) Let X be infinite-dimensional. Then show that the adjoint map is not continuous with respect to strong topology. (Hint: let u be a unilateral shift; i.e.,
'{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)
(0.3.2). (a) Show that a net (x) converges to x weakly if and only if tr pX j ... tr px for every operator p of finite rank. (b) If S is a norm-bounded set in :e(X) (i.e., sup{ IIx II: xeS) < (0), show that the weak and a-weak topologies, when restricted to S, coincide. (Hint: Use (a) and the fact that finite-rank operators are dense in :e(X)•.)
('-(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
'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
Some elementary features of these topologies are listed in the exercises; also included in the exercises are the definitions of three other topologies, namely, the a-strong, strong-* and a-strong* topologies.
''(nh Jo aceds lunp eqt Euleq stl Jo anlrl^ fq '(Ah pelrror,{ur ,{q fEolodol eql uI x * Ix 33r ..{11ue1errrnba '.ro '*(&h ur d fre,ra roJ *IBa1r - !x)d rll JJI ,{Eo1octo1 0 € l(x IEe^\-o (c) : ul f . r a r e r o 3 ' g u o f E o l o d o l > l u o , t ar q l u [ l x * I n , "o'l'$ ur g'! ro3 :'rc JJI 0 * l - l JJr fEolodol {?a^\ (q)
(b) weak topology iff I<Xjt,71> - <xt,7}>1 ... 0 for t,7} in X, i.e., iff Xjt ... xt in the weak topology on X, for every t in X; (c) a-weak topology iff Itr p(x j - x)1 ... 0 for every p in :e(X)., or, equivalently, iff x j ... x in the weak* topology inherited by :e(X), by virtue of its being the dual space of :e(X) •.
Qth uo sar8olodoaxa^uoC f11eco1oarql .€.0 0.3. Three Locally Convex Topologies on :e(X)
9
n t rtrod o d uuction ction 0O.. I In
l 10 0
s nnot ot u l t i p l i c a t i o n i is When multiplication hat m h o w tthat n f i n i t e - d i m e n s i o n a l , sshow ((b) b) W s i infinite-dimensional, h e n 1Iff i is jointly continuous continuous when when I(19) :e(lf) isis equipped equipped with with any any of of the the three three jointly opology, H i n t : FFor o r tthe h e sstrong t r o n g ttopology, e a k , oa-weak. - w e a k . ((Hint: t r o n g , wweak, l topologies: o p o l o g i e s :sstrong, n EEx. x' i t h au aas s i in e a k ttopologies, o p o l o g i e s , wwith n d wweak a ) ; ffor - w e a k aand o r tthe h e oa-weak uuse s e ((a); ' ' u' 00 so weakly, (0.3.2)(b), note that u*n ... 0 a-weakly and hence weakly, so un ... hence and o-weakly (0.3.2)(b),note that il*n 0 = n') a-weakly and weakly; however, u*nun = 1 for all n.) I for all u*nttn however, weakly; and o-weakly (c) IfIf sS is is aa norm-bounded norm-bounded subset subset of of r(xf), :e(lf), then then multiplication multiplication is is (c) with respect s, jointly continuous, when restricted to S, with respect to to the the to jointly continuous, when restricted strong topology. topology. (The (The example example in in (b) (b) shows shows that that even even this this is is itrong false in in the the o-weak a-weak and and weak weak topologies). topologies). false (d) In In all all three three topologies, topologies, show show that that multiplication multiplication is is separately separately (d) ' t x '... n d yYix x l aand y , x ! i ' t h e n continuous; i.e., if x € :e(H) and Yi ... Y, then xYi ... xy E ( I I ) a n d /, continuous; i.e., if x e yx. yx.
(0.3.4). A A net net {x,} {xi} in in l(lf) :e(lf) is is said to to converge converge to to xx in in t(xt) :e(:I£) with with (0.3.4). respect to the: r e s p e c tt o t h e : (i) (i)
(ii) o-strong* a-strong* topolosy topology o # f~ tllt",
00 ;
- x)tllz for each each x)*qll') (ll(xi (iii) 11 2 +* Il<x d1 2) ...- 0o for ll(t,i -- x)* ( i i i ) strong* topology#<+
Xi'" x i - 00
(strongly) (stronglY)
x F i ' 0o #( + xtxi'"
(weakly) (weakly)
xi'" xi - 00
(a-strongly) (o-strongly)
xlxt ...- 00 #€ xtxi
o-weakly) ((a-weakly)
xi'" xi'00
(strongly*) (stronglY*)
x,xf - 00 "F, ++ xixi'" #€ xtxi
(weaklY) (weakly)
xi'" xi - 00
(a-strongly*) (o-strongly*)
rixl- ... 00 xfx, #<+ xi Xi ++ xixt
((o-weakly)' a-weakly).
o-strong the a-strong that the show that sets, show (d) norm-bounded sets, to norm-bounded (d) When restricted to when restricted (resp'' strong (resp., the strong with the (resp., coincides with topology coincides (resp., a-strong*) o-strong*) topology strong*) topologY. strong*) topology. the that the X, show show that set X, on aa set (0.3.5). topologies on (a) If 12 are are topologies (0.3.5). (a) If Tr,1 and and T2 following equivalent: a r eequivalent: c o n d i t i o n sare f o l l o w i n g conditions
(i) ( X , r1r) ) isi scontinuous; ( X , r2r) ) '... (X,T continuous; ( i ) idl dx*::(X,T
' , ( 1 " 1 S )= ( r + r r ) SP U B r ( r S ) = ' S o 1 1 r ' t ' t ( 1 1 e r a u eaE: o 1 4 1 = (S')'
and S(n+l)
=
(S(n),.
'{S r x II€ roJ .xx = x,x:
for all xeS}.
(nh r rx) = tS
aq ol peulJep sl (fih Jo S: lesqns B Jo lu€lnruuoc aql IuerosIII
lu"lnuluoS
olqnoo oql
't'0
('erroJ E ur IIIIs sr flrlrqe:edas 3o uogldunsse aqt l4 Jo II€q llun oql u r e c u e n b o so s u o p € o s n : l u r H ) ' e l q u z u l a r u s r ( 1 e a n o 1 * E u o r l s - o ) serEoloclol xrs oql Jo qcea 'sles papunoq-rurou uo lBrll /y\or{S (p) ('s1auroprsuoc ol ,(ressaJeusr t1 lop tou 11r,nsacuanbes 'snql) 'r(Eolodo1rurou eql lclacxe (q) (S'g'O) 'xg ul pauolluoru serSolodol eql Jo due ol lcedser qll/'r -- flrlrqelunoc Jo urolx€ lsrrJ eql ,(gslles uele lou seop 'lce3 ul -- elqezrrloru lou sl (a)f (c) (loldrcurrd sseupapunoqrxroJlun :lurH) 'oJoz l(11ea,vr (-x] ol soEraluoJ Jo ecuenbasqns ou ]€r{l ^\oqS (q) ('saEraltp selJes oruour€q oql oculs '0 < r l(u€ toJ 'u ,(ueu flelrulJul roJ t t-w ) ,1.-tr'": tl i
?
nJrn
' - > r l < * & ' ' r 'tl = u l l ' t lil L 1< ~n,17m>12 <
co ,
l
=
!
II ~J2
I
o"ut
L
JI:lulH)'r=l{*"}
r='('l) sorJsrlBs eql ol sauoleq-0tuqr froqS (e) Jo ernsolc*Euo:1s-o
°
I
More generally, write SIt
S' = {x' e :f(Jf): x'x = xx'
The commutant of a subset S of :f(Jf) is defined to be 0.4. The Double Commutant Theorem for infinitely many m, for any e > 0, since the harmonic series diverges.) (b) Show that no subsequence of {x m } converges weakly to zero. (Hint: Uniform boundedness principle!) (c) :f(Jf) is not metrizable -- in fact, does not even satisfy the first axiom of countability -- with respect to any of the topologies mentioned in Ex. (0.3.5) (b) except the norm topology. (Thus, sequences will not do; it is necessary to consider nets.) (d) Show that on norm-bounded sets, each of the six topologies (a-strong* to weak) is metrizable. (Hint: use a dense sequence in the unit ball of Jf; the assumption of separability is still in force.) 0 ; 1<~n,17m>12 :Ii mole then
n
(a) Show that belongs to the a-strong* closure of {x m }:=l' (Hint: if {~n}:=l satisfies m' m -
(0.3.6). Let {17 m}:=l be an orthonormal basis for Jf, and let x m m l / 2t 17 17 for each m. 'w
q'ee ro
t''- l' 1 r,r*
(c) Prove, by examples, that if Jf is infinite-dimensional, each of the above inclusions is strict.
'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
strong;
'{BOlr\
EEuorts
=
'(996:o)
E Euo.Its :J
weak.
I
strong
ull
ull
ull
lln l l l nl ll l n i i t = = = {?e^\-o c Euorls-oc *EuoJls-oC turoN Norm; a-strong* ; a-strong; a-weak
(b) Prove the following "inclusion" relations between the different topologies on :fPf):
'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)
(ii) xi .... X(T 2) Xi .... x(T 1)' for any net {xi} in X and x in X; (iii) every TI-closed set is T2-closed. When these conditions hold, write TlfT2.
*
ursroaql lu€lnuuoc olqnoc srII 't'0
0.4. The Double Commutant Theorem
II
II
t12 2
ntroduction 0O.. IIntroduction
Proposition 0.4.1. 0.4.1. Let S,T g £; t(tf). :f(Je). Proposition
((a)a ) sSC.T"*T'C.S'· cr+z' cs,. (b) ;S -f "" S" =-sw-olo = S(2ii) a;d sr S I = 5(2n-1) s(2n-l) fur for n 2 ~ r; 1; i;i
"*
self-adjoint ) S) S' is self-adjoint. self-adjoint. (c) S is self-adjoint S' is, for any S, a weakly weakly closed closed subalgebra subalgebra of of \0 :f(Je) and I1 eE S). S'. (d) S' is, for Exercise! Proof. Exercise!
0 D
Before proceeding proceeding further, further, itit would would help to set set up some some notation and terminology. For a subset subset S S of of 13, Je, we shall always write write I[S S ]l for the smallest smallest closed closed subspace subspace of of lf Je which which contains contains S; for for S gC. f(Xf) :f(Je) for and £; fJe,, w wee sshall write sS E S S}. x l : xXES, ) . -A e S , (t e A sset et rite S o r ({xt: S ffor imply w h a l l ssimply a nd S g 10. Since S of operators operators on lf Je is said to be non-degenerate non-degenerate if if t,Slfl [SJf] = =:If. Since ran x = ker x*, x*, it it ffollows if S is self-adjoint, then S is ranrx ollows that if 0 ) iimplies non-degenerate and only = ({O} t = = 0O.. f ,St S g= mplies 6 f a n l y iif n o n - d e g e n e r a t eiif nd o stage is now set set ffor Neumann's double commutant or von Neumann's The stage whose power will bee illustrated ower w i n tthe e s t of h i s section. section. ill b i l l u s t r a t e d in h e rrest o f tthis hose p ttheorem, heorem,w
01
Theorem Theoren 0.4.2 0.42
of non-degenerate self-adjoint self-adjoint algebra of Let M be be a non-degenerate operators following conditions tl. The equivalent: The following conditions on M are equivalent: operators on Je.
(i) M = M'. (ii) M is weakly weakly closed. closed. (iii) (iii) M is strongly snongly closed. closed.
"* "*
"*
(iii) are Proof. The implications (i) ) (ii) + (iii) are immediate (cf. Ex. ( i ) , it p r o v e (iii) ( i i i ) + (i), ( 0 . 3 . 5 X a ) ,(b». ( b ) ) . To (0.3.5)(a), i t clearly suffices to prove the T o prove following: following:
(*) (*)
1l and exists a Ee M such such , .... tEr, If E Je and Ee > 0, 0, there there exists lf a" e M', t1L,..., a" EM', n € 1
) ( i l l <. . E that t n a tIl
for f o r 1I ,( ii ,( n. n.
= PM' pt = pM. It It is is J't= [Mt1l l. Let M and p' We verify (*) in case case nn = 1. We first first verify fMlr) and is g .M, p ' xxp' p t == xp' i n M. M . Since S i n c e M is clear f o r all a l l x in t \ and x p ' for M ! \ £; a n d so s o p' c l e a r that t h a t MM p ' x * p ' == x*p'. x* pt . self-adjoint, s o p'X*p' M , then x * Ee M and a n d so i f x E€ M, t h e n x* s e l f - a d j o i n t , if previous w i t h the Comparison t h e previous t h i s equation e q u a t i o n with t h e adjoint a d j o i n t of o f this o f the C o m p a r i s o n of p ' Ee M'. w h e n c e p' M ' . So S o a"p' a"pl equation y i e l d s p' p t xx == xp' i n M, M , whence f o r all a l l x in x p t for e q u a t i o n yields Jvtit to it suffices == ppta", 'a", and is dense in .M, suffices to M(,1 is dense in c )t Since Since Mt hence a"M a"It £;.M. and hence p ' xt x l t 1 =- xt a n d so so " l , .1r and prove p r o v e that I ' t For i n M, M , clearly F o r any c l e a r l y p' e .M. a n y x in t h a t t[ ,1, E = O. o. p t x \ 1r = 0; M ( l -- P')tl) 0 ; thus t h u s M(l x(l-p'H r l r1 -- p'xt i l r 1 -- xp' x p t t[ 1 x ( l - p ' ) E 1r = xt r')Et) = r = xt (and The M ensures ensures that of M self-adjointness) of The assumed non-degeneracy (and self-adjointness) assumednon-degeneracy g , and p ' ) [ ,1 == 0, lt (1 ""( l -- plH h e n c e tE1t Ee .M. a n d hence genelgl n, (t) for 1l be of nn copies copies Returning let Je direct sum sum of for gener3,1 n, let be the the direct Returning to to (*) (n xx n) n) of an (n 10 corresponds naturally to to an corresponds naturally !f. Every Every operator operator on on Je of Je. *-algebra :f(Je)-va1ued this t(lf)-valued matrix, being aa *-algebra matrix, this correspondence correspondence being isomorphism. let i d e n t i f i c a t i o n , let W i t h this t h i s identification, i s o m o r p h i s m . With
(0.4.4). Let M be a (not necessarily non-degenerate) self -adjoi~ subalgebra of :f(lt). Let e = PM, where M = [Mlf] = (n (ker x: x EM}) . Prove that:
'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 Exercises 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
By Proposition O.4.l(d), a von Neumann algebra is a weakly closed self-adjoint unital subalgebra of :f(lt), while, by Theorem 0.4.2, any such collection of operators is a von Neumann algebra. Some authors do not require von Neumann algebras to be non-degenerate; they define a von Neumann algebra to be a weakly closed self-adjoint algebra of operators. The difference is inessential and spelt out in the following exercise. Definition 0.4.3. A self-adjoint subalgebra M of :f(lt) is called a von Neumann algebra if it satisfies M = M'. 0
",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
Remark that the assertion (*) in the proof of the theorem actually states that if M is a non-degenerate self-adjoint algebra of operators, then M" is (contained in, and hence) equal to the strong closure of M. So, if A is any non-degenerate self-adjoint algebra of operators, then A" is the smallest strongly closed (also, weakly closed) self-adjoint algebra containing A. Thus, if S is any self-adjoint set of operators, then S" is the strong closure of the algebra generated by S U {l}. A special case is worthy of mention: if x = x*, then {x}" is the strong closure of the set of polynomials in x, and, in particular, abelian.
'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
Appeal now to the",already established case n = I of (*) with M, ~1 and E replaced by M'~l $ .. , $ ~n and E > 0, to complete the proof of (*) and hence of the theorem. 0
,f":i : 0
.
a" 0
: a"
M'
'JAl ) uo
o .
...
E
'1
n 0 0
= "A[
'" = M"
a
l:
[ [ (c)
o
(c)
a" 0" . . . 0
.
)
0 " D
'" is a non-degenerate It is relatively painless to verify thJlt (a) jvf self-adjoint algebra of operators on It; (b) M I {«a/ j)): ail j E M' Vi,j}; and
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
14 L4
o.. IIntroduction ntroduction 0
(a) (b)
exe for for all x in in M; in in particular, particular, MI4 MM C f M; Itt; xx = exe if Me = (xllt {xlM: x e E Ml, M}, then Me Me is a non-degenerate non-degenerate self-adjoint self-adjoint if subalgebra of of t(It); l(M); subalgebra
(c) (c) lvlt M' = = {x' {x' e $ y Y
x' eE MJ, M~, y eE t(l'll)}, l(M1 )}, and and : x)
M"=--( x{x" $}.1 l l lIttd1·:. xx"" eE \M~ ). W " O , , \>. eECCl:}.
(Thus, a degenerate degenerate von Neumann algebra, algebra, as as considered considered by other (Thus, authors, is just a von Neumann algebra algebra -- in in our sense sense --- of of operators operators authors, subspace.) on a subspace.) (0.4.5). Let (X,f,lt) (X,T,y) be a separable separable o-finite a-finite mea-sure measure space space (so (so that (0.4.5). L 2(X,It) is a separable separable Hilbert Hilbert space). space). For 04> in in L-(X,1t),let L ""(X, It), let m5 m~ denote denote r2(x,tt) (rz6t,)(s) = 0(s)l,(s), associated multiplication multiplication operator: operator: (m4>0(s) 4>(sH(s), for for (~ in the associated L 2(X, It) = = :If. Y. L21X,1t1 - m~ (a) z-(X,p) (a) The map 4> ""(X,It) into isomorphism of of L m6 is an isometric* - isomorphism O .. = m4». t(lf) (where the '*' l(:If) *' refers to the assertion mf, = m6). assertion m~ ( b ) If M iis s aan n (b) then ==M' M t and ^ n d ' consequently c o n s e [ u e n t l yM I f M ==( ^(m~ 6 0 e 4> E L""(X,It)}, L'1X,1t11 , t h eMn M case of abelian von consider the case voh Neumann algebra. algebra. (Hint: First, consider o , ~o = xx't |~o' = m4> y ; if r ? 1 6where w h e r e 4> finite i f x ' I Ee M', M ' , show s h o w that t h a t xx't = f i n i t e p.; lo Q= g e n b r a l f o l l o w s by c a s e follows by being f u n c t i o n 1; l ; the t h e general case c o n s t a n t function b e i n g the t h e constant o-f initeness decomposing measure. Is a-finiteness f inite measure. sets of of finite decomposing X into sets necessary?) necessary?) (c) The a-weak M coincide; under the o-weak and weak topologies topologies on M with the weak* identification coincides with identification m4> mh -* 4>, topology coincides 0, this topology ""(X,It) by virtue topology inherited virtue of its being the dual inheritdd by L L'(X,tt) space o f L rl(X,It). (X,tD. s p a c eof general von Neumann algebra M' if if and only (d) A general algebra M satisfies satisfies M = M' l(xf). if M is a maximal abelian von Neumann algebra algebra in l(:If). if a lt, let M (0.4.6).If M1= (0.4.6). 1 = (p operators on :If, If M is is aa von Neumann algebra algebra of operators 1p = 0 "Ix t(Xt)*, t(13)*:tr px = Mg E 1 is aa closed Vx in M}. M). Then M closed subspace subspace of l(:If)., e l(:If).: J M, and with (l(:If)./M1)* ~ agrees with and the induced weak* topology on M agrees GQt)-/Ml* the 0E o-weak topology. topology. the a-weak the restriction to M of the v o n Neumann a d m i t s aa The N e u m a n n algebra a l g e b r a admits l a s t exercise s h o w s that t h a t every e v e r y von T h e last e x e r c i s e shows p r e d u a l is p r e d u a l . It determined predual. i s uniquely u n i q u e l y determined t h a t such s u c h aa predual I t can c a n be b e shown s h o w n that g o into p r o o f of w e shall n o t go i n t o aa proof o f that that up i s o m e t r i c isomorphism, i s o m o r p h i s m ,but b u t we s h a l l not u p to t o isometric 'the' predual predual of M, which will will here. we may talk of 'the' here. Consequently, Consequently, we usually be denoted by M •. M*. be denoted (as aa norm-closed generated (as norm-closed subspace) subspace) by Just ""(X,It) is generated as LL'(X,u) Just as is indicator v o n Neumann N e u m a n n algebra a l g e b r a M is i t is i s true e v e r y von f u n c t i o n s , it t r u e that t h a t every i n d i c a t o r functions, (as aa norm-closed generated generated (as the set set P(M) P(lul) of its norm-closed subspace) subspace) by the double projections. To obtain this and other consequences the double projections. consequencesof the preliminary lemma. lemma. commutant theorem, helps to establish establish aa useful preliminary theorem, it helps lf is aa norm-closed norm-closed Recall that aa C*-algebra operators on :If C*-algebra of operators f(8). Clearly von Neumann algebras are self-adjoint subalgebra algebras are subalgebra of l(:If).
Thus, the scholium implies that just about any canonical construction applied to elements of a von Neumann algebra never leads outside the algebra. It follows from the above Corollary that any von Neumann algebra is generated as a norm-closed subspace by the set of its projections. (Reason: let M o be the norm closure of the set of linear combinations of projections in M; since M o is
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
Proof. (a) If u I is a unitary operator in M', then u' xu 1-1 = X = ulxl; on the other hand, it is clear that u I xu' -1 = (u' uu' -l)(u 'Ixlu 1-1) is also a (and hence the) polar decomposition of u I xu 1-1. Hence U uu,-l = u and u'lxlu,-l = Ixi. Since u' was arbitrary, the scholium ' completes the proof. (b) The uniqueness of the spectral resolution of a normal operator and an argument exactly analogous to the one used in (a) serve to 0 settle this assertion. '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
If x = ulxl is the polar decomposition of x, then u, Ixl E M; If x is normal, then IF(x) E M for every Borel subset F of sp x. 'n ) x puo otqaSlo uuvunaN uo^ o aq n ta7
(a) (b)
(q) (e)
.5-g-6 itue11oro3
Corollary 0.4.9. Let M be a von Neumann algebra and x
E
M.
0
n
'Joord
Proof. Exercise! ioslcroxa
'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
Scholium 0....8. Let x E :t:(Je) and M a von Neumann algebra of operators on Je. A necessary and sufficient condition for x to belong to M is that u I xu'· = x for every unitary operator u I in M'.
The double commutant theorem, when coupled with the above lemma (applied with A = M'), yields the following useful criterion for determining when an operator belongs to a von Neumann algebra.
'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
x = -[{x + i(l _X 2)1/2} + {x -i(I _x 2)1/2}] 2 is an expression of x as an average of two unitary operators; these operators, being continuous functions of x, belong to C·(x), the C·-algebra generated by x and I, and hence belong to A. 0
l Q l r Q x - I ) l - x | + { t , / r- Q , x r- ) r + x \ l I = x I
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
Proof. Any x in A admits the Cartesian decomposition x = xl + ix 2, where xl' are self-adjoint o,perators in A. It suffices now to notice that if x = x· E A and Ilx II " I, then
x,
'V u, stolorado {..rolun tnol lo uotlourgtao? tpauq D so alqtssatdxa st .o"rqa61o-*2 v /o tuawata {tatg lolun p aq el)g3 V n7 Z-tg GuE I
Lemma 0....7. Let A f :t:(Je) be a unital C··algebra. Every element of A is expressible as a linear combination of four unitary operators in A.
'€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
C·-algebras, notation of set {m4Y 4> E dense In M.)
but the converse is seldom true. (For example, in the Ex. (0.4.5), with X = [0,1] and If. Lebesgue measure, the qO,I]} is a proper C·-subalgebra of M which is o-weaky Now for the lemma.
't'0
0.4. The Double Commutant Theorem
15
Illerooql tu€lnruruoJ elqnoc oqJ
SI
ntroduction 0O.. IIntroduction
l16 6
self-adjoint, itit suffices suffices to to verify verify that that ifif xx == x* x* €E M, M, then then xx eE Ms: M o; self-adjoint, for this, this, let let 0r, 4>n be be aa sequence sequence of of simple simple functions functions on on sp sp .lr x such such that that for 4>n(t) '.. tt unifoimly uniformly on on sp,x, sp x, and and note note that that by by Corollary Corollary 0.4.9(b), 0.4.9(b), 0"(x) 4>n(x) 0,,(t) -- xll E Mo for each each n and and lim lim llO"(x) l14>n(x) xII == 0.) 0.) e"Mnfor 'iurther Before discussing discussing some some further properties properties of of aa von von Neumann Neumann BeTore algebra, let let us us briefly briefly digress digress with with some some notational notational conventions. conventions. If If algebra, {e i: i e E I} l} is any family of of projections projections in a Hilbert space, space, the the symbols symbols {er: ViEf'i and and A,rae, AiEf'i will will denote, denote, respectively, respectively, the projections projections onto onto the V,rre, subspaces tui61 [UiE1 ian ran e,l e i] and and q€r f'\EI ran ei. e i• For a finite collection collection er,.'., el' ..., en, en' subsiaces V A... A we shall shall also also write eerV V ... Ve and e A ... A en' e, err. -. e, and we 1 1 n Exercises Exercises (0.4.10). If If M is a von Neumann algebra algebra and (er) {e) cf P(tr4), P(M), then Yer, Ve i, (0.4.10). lattice') E Ae P(M). has of complete lattice.) E 0 (Thus a complete of P(tu}. has the structure P(M) Mi,i An extension extension of of the above above exercise exercise is given by the following following An assertion: assertion: (incteasing or Proposition 0.,Lll. 0.4.11. Every uniformly uniformly bounded bounded monotone monotone (increasing Proposition tt is weakly convergent. convergent. decreasing) on If operators on self-adioint operators net of of self-adjoint decreasing) net Xf operators on If Proof. Suppose I} is a net of self-adjoint operators e /) Suppose {Xi: {x,: i E ( b ) there exists a ( a ) if a n d (b) t h e r e exists r ; and satisfying t h e n xi x , '{ xx.; i f i,j i , j - eE If and a q C i1 .,{ ij,, then s a t i s f y i n g (a) v e c t o r ~I F o r a unit u n i t vector constant f o r all i n "J II.. For a l l - i in t h a t IIx s u c h that c o n s t a n t c > 0 such l l xi,IIl l ,{ c for in r e a l n u m b e r s in n e t o f i n c r e a s i n g in : i E I} is a monotone increasing net of real numbers m o n o t o n e l f , {<xi~'~> e / ) i s i n If, {<x1l,i>: from It [-c,c], and consequently convergent to its supremum. It follows from supremum. convergent consequently [-''c,c], 1 | , the ( 0 . 4 . 1 2 ) )for € If, the ( c f . Ex. f o r any a n y ~,77 the p o l a r i z a t i o n identity E x . (0.4.12)) i d e n t i t y that t h a t (cf. l,n E t h e polarization is limit net {<xi~,77>: i E I} is convergent. Denoting this limit by [~,77] net {<x,l,rl>: e I) is convergent. [l,n] it is l f . H e n c e ( b y f o r m o n clear is a bounded (by c) sesquilinear form on If. Hence s e s q u i l i n e a r c ) i s a b o u n d e d c l e a r that t h a t [.,.] [.,.] !1. e 1f. f o r all a l l ~,77 < x t , 4 > == [~,77] l ( l f ) such there t h a t <X~,77> i n :e(lf) s u c h that \,n E t h e r e exists e x i s t s x in [ ( , n ] for (xt: id E Clearly, then, e I} 1) converges net {Xi: converges weakly to x. then, the the net
Exercises Exercises v e c t o r space s p a c eV, (0.4.12). f o r m on o n aa complex c o m p l e x vector ( 0 . 4 . 1 2 ) . If i s aa sesquilinear s e s q u i l i n e a rform % I f [.,.] [ . , . ] is then, in V, V, for any any ~,T) then, for \,0 in 33
net of of self-adjoint self-adjoint (0.4.13). increasing net (0.4.13). Let (x,: ii E monotone increasing e I} I\ be be aa monotone Let {Xi: = lim ( a s in Then, i n Prop. P r o p . 0.4.11). 0 . 4 ' l l ) . Then, operators l f and l e t xx = l i m xi x t (as o n If a n d let o p e r a t o r son
x strongly. (Hint: if 1\x.11 , c for all i,''then Ikx-x)~ II , (a)x." 211·lkx-x)1/2 d ,i'(2c)1/2Ikx-xl/2 ~ II; use II(x-xl/ Ex. (0.3.4)(c), *f ill',;1;'.',[ ,kill,,irri-ll1,ei'" fii"-{,,)liill',1';,,qYilf applied to {Xi -
(a)
x}.) 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
Let N be a von Neumann algebra of operators on Jf; let M be any closed subspace of Jf and let e = PM' (It is not assumed that e E N.) Then = AU E P(N): e II f} is a projection in N and ran = [N' M], the smallest N '-invariant closed subspace containing ran e. (b) Let N 1 and N 2 be von Neumann algebras acting on Jf. then (N 1 U N 2 )' = N: () N 2 and (N 1 () N 2 )' = (N: U N 2)". (Hint: The first assertion is trivial, and implies the second.) (c) If M is a von Neumann algebra and e E P(M), then ran c(e) = [MM], where M = ran e. 0
(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)
e
e
(a)
( st l' o )
(0.4.15).
Exercises
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
It is an easy consequence of the double commutant theorem that the intersection of any family of von Neumann algebras is again a von Neuman algebra; in particular, since Z(M) = M () M', the centre of a von Neumann algebra is an abelian Neumann algebra. Hence, by Ex. (0.4.10), c(e) E P(Z(M» whenever e E P(M); by definition, c(e) is the smallest central projection dominating e. The following Exercise leads to a more concrete description of c(e).
'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)
The set {x E M: xy = yx for all y in M} is called the centre of M and denoted by Z(M). (b) M is called a factor if Z(M) = (>.l: ). E ct}. (c) For a projection e in M, the central cover of e, denoted by c(e), is the projection defined by c(e) = AU E P(M) () Z(M): e II f}. 0 (a)
'll .1 I-1q-O uoplulJaq
Definition 0.4.14. Let M be a von Neumann algebra of operators on K
uo sroleredo 3o erqaEl€ uueruneN uol B eq n
rc-I
'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
We conclude this section with the definition of a factor and a basic fact concerning factors (Prop. 0.4.17) that will be needed in the next chapter.
'!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) X a-strongly*. (Since xi' X are self-adjoint, only a-strong convergence need be proved; use (a) and Ex. (0.3.4)(d).) (d) If Y E :f(Jf) satisfies xi II y for all i, then X II y; for this reason, we shall write X = sup xi' 0
(c)
xi'"
n=N+l
('ll"t.ll ', * ll"tll'*l="
+ 2c
L
.
II~nll
linn II.)
'1' l.'t ''l(lx --r)tl , 1.'u'ul(lr- ")rl i N Xi'" X a-weakly. (Hint: if I)I~J2 < .. and Ln lhJ2 < .., and N any integer, then
N pu€r ' ,ll"ull"3pu, - > zll"rll"3Jr :rurH)Trlt"r::;tlt jl;
(q) (b)
'r'0
0.4. The Double Commutant Theorem
17
rueroer{I lu?lnuuoJ
LI
elqnoc eql
r188
ntroduction 0O.. IIntroduction
Lemma 0.4.f6. 0.4.16. Let Let M M be be aa von von Neumann Neumann algebra algebra and and e,f e,f €E P P(luI). (M). The The Lemma following conditions are equivalent: equivalent: are conditions following (i) (i) (ii) (ii)
exf == g0 for for all all xx in in M. M; exf = c(e) c(f) = O. 0. c(e) c(n
Proof. (i) (i):} (ii). The The hypothesis hypothesis is is that that MI'l MM cf ker ker e, e, where where lv1 M== ran ran ,/. f. ) (ii). Proof. whence ec(/) ker Hence, by Ex. OA.15(c), it follows that ran c(f) f ker e, whence ec(f) e, ran cU) Hence,by Ex. 0.4.15(c),it follows that e -= 0. O. This This means means ee (~ I1 -- c(fl, c(f), and and so, so, by by the the definition definition of of the the central central (n. ccover, o v e r ,c(e) c ( e )(~ I1 -- c c(f). D (ii) + :} (i). (i). Reverse Reverse the the steps steps of of the the proof proof of of (i) (i) ):} (ii). (ii). 0 (ii) Proposition 0.4.17. 0.4.17. If If e and and ff are are non'zero non-zero projections projections in a factor factor M, Proposition there exists non-zero partial partial isometry isometry u in M such such that that u*u u*u 4~ e and and existsa non'zero there uu* < , f. f. uu* Proof. The The assumptions assumptions ensure ensure that c(e) c(e) = = c(f) == l'1. Lemma Lemma 0'4'16 0.4.16 Proof. that fxe I'I- 0. then guarantees guarantees the the existence existence of an an x in M such such that O. Let then uh be be the the polar polar decomposition decomposition of ffxe. This u does does the the job. job. I0 xe. This xe = uh ffxe
' { > 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
e '" I (reI M) or simply e '" I in case there exists a partial isometry u in M such that u*u = e and uu* = I; e { I if there exists e 1 in P(M) such that e '" e 1 ' I. 0 > I'a p1
E
:alrr^\ IIETISoA,\ 'U4)d
'I't'I
(b)
(q)
(a)
(u)
Definition 1.1.1. Let e,f
P(M). We shall write:
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
Henceforth, the symbol M will always denote a von Neumann algebra, and P (M) the complete lattice of projections in M. uoJr"IrU crLL 'I'I
1.1. The Relation ... - _ (rei M) (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
The notion of unitary equivalence, while being most natural, has the disadvantage of not being additive in the following sense: if e 1, e 2, 11 and 12 are projections such that e i is unitarily equivalent to Ii' for i = 1,2, and if e1 1 e2 and 11 1 12, it is not necessarily true that e1 + e2 is unitarily equivalent to 11 + 12, This problem disappears if, more generally, one considers two projections as being equivalent if their ranges are the initial and final spaces of a partial isometry. This equivalence, when all the operators concerned -- the projections as well as the partial isometry _. are required to come from a given factor M, is the subject of Section l.l, where the crucial result is that, with respect to a natural order, the set of equivalence classes of the projections in a factor is totally ordered. The next section examines finite projections -- those not equivalent to proper subprojections; the main result being that finiteness is preserved under taking finite suprema. The final section, via a quantitative analysis of the order relation discussed earlier, effects a primary classification of factors into three types. The principal tool used is called a 'relative dimension function' by Murray and von Neumann.
Chapter 1 THE MURRAY - VON NEUMANN CLASSIFICATION OF FACTORS
rJrssvlf suorf,vj Jo Norrvf, NNVlAtnSN NOn - AVUUn1^| 3Ht I rardeq]
20 20
actors f FFactors Murray-von l a s s i f i c a t i o n oof e u m a n n CClassification u r r a y - v o n NNeumann he M l1. . TThe
e l a t i o n oon n n eequivalence q u i v a l e n c e rrelation s i indeed n d e e d aan e r i f i e d tthat h a t -..,. i is e a d i l y vverified IIt t i is s rreadily ither e p l a c i n g eeither y rreplacing s uunimpaired n i m p a i r e d bby a l i d i t y oof f ee !i /f iis h e vvalidity h a t tthe n d tthat PP(M) ( M ) aand or ff by by an an equivalent equivalent projection. projection. We We shall shall adopt adopt the the notatio\ notation u: u: ee ee or to mean mean that that u, u, ee and and /f belong belong to to M M and and are are as as in in (a) (a) of of the the -..,. ff to efinition. aabove b o v e ddefinition. ith ork w o w e a s t , tto Wee sshall work with h a p t e r , aatt lleast, n tthis h i s cchapter, i n d iit t cconvenient, o n v e n i e n t , iin h a l l ffind W '-+P M , w e m ay 1 1 subspaces rather than projections. Via the transition M PM, we may p r o j e c t i o n s . t r a n s i t i o n V i a t h e r a t h e r t h a n subspaces (and will) will) use use such such statements statements as as u: u: ItM -..,. I'11 M1 gf N N.. Since Since we we are are only only (and concerned with with ?P(luI), (M), we we should should only only consider consider subspaces subspaces which which are are concerned aa s\ight to consider the ranges of projections in M. It will be useful to consider slight projections wil\ be useful M. lt in the ranges of generalization of of this this notion. notion. generalization
Definition 1.1.2. 1.1.2. A A (not necessarily necessarily closed) closed) linear linear subspace subspace D J) of of Xt Je is is Definition g iin n D f o r a l l a r D i f a ' D said to be affiliated to M, denoted by J) T/ M, if a IJ) f J) for all a' n M , said to be affiliated to M, denoted by M'. M r.
n0
It follows follows from from the double commutant theorem that that ifif l'1 M is a closed closed It general, there exists py In then MT/M if and only if PM E In general, exists subspace, e M. if I\nM if only subspace, tthere here i n s t a n c e , i f , f o r M ; several non-closed subspaces affiliated to M; if, for instance, t o a f f i l i a t i d s u b s p a c e s n o n c l o s e d several ould b uch an a w h e n rran exists a iin M ssuch not would bee ssuch o t cclosed, l o s e d , tthen a n a iiss n h a t rran n M u c h tthat e xists a necessary it becomes an example. example. To deal with with such such subspaces, subspaces, it becomes necessary to an deal with unbounded operators. In this context, the following following operators. deal with definition Definition 1.1.2. e f i n i t i o n 1.1.2. s u p p l e m e n t sD d e f i n i t i o n supplements affiliated to M, be affiliated I is said said to be Definition operator A closed operator 1.1.3. A closed Definition 1.1.3. I dom A i f ~I €e dom i . e . ,if e M t ; a ) denoted A T/ M, if a' f Aa' for every a' E M'; i.e., f o r e v e r y A a ' n M , i f a t A denoted e a t A~. Al' A a t \~ =- a' and ~ €e dom 0I a n d Aa' d o m A and i m p l y a' arE M ' imply a n d a tI €e M' (the double double commutant Observe operators, (the f or bounded bounded operators, Observe that for 'belonging to ' a f f i l i a t e d to 1 } 4and a n d 'belonging t o theorem ensures that) the notions 'affiliated M' to n o t i o n s t h e t h a t ) e n s u r e s theorem reader that the reader convince the M' should convince exercisesshould following exercises The following /lf coincide. coincide. The w i t h this this p o s s i b l eto d e a l with t o deal i s possible i t is this a n d that t h a t it o n e and n a t u r a l one i s aa natural n o t i o n is t h i s notion notion by considering only bounded operators. o p e r a t o r s . b o u n d e d o n l y notion by considering
Exercises Exercises The linear operator. operator. The (1.1.4). defined linear densely defined and densely closed and (1.1.4). Let be aa closed Let AA be (iii) if (ii) 4 M; A* (i) n M; A following conditions are equivalent: (i) A T/ M; (ii) A* T/ M; (iii) if iollowing conditions are equivalent: e M lr(H) M and e u A, then AA == uH is the polar decomposition of A, then u € M and IF(H) € M' of polar decomposition uH-is the for of [0,00). Borel subset subsetFF of for every every Borel [0,'). M == and M s p a c eand m e a s u r espace o - f i n i t e measure (1.1.5). ( 1 . 1 . 5 ) . Let ( X , T , t t )be s e p a r a b l ea-finite b e aa separable L e t (X,f,/l) 2(X,/l» (cf. a c l osed ( 0 . 4 . 5 ) ) . t h a t ( c f . S h o w E x . (m~: ~ € Loo(X,/l)} f :e(L Ex. (0.4.5». Show that a closed r Q 2 6 , u D e L ' ( x , t t ) \ im^:.O e only i f and a n d only M if to M densely a f f i l i a t e d to 2 ( X , 1 tis )i s affiliated o n LL2(X,IJ,) o p e r a t o rAA on d e f i n e d operator d e i l s e t ydefined that s u c h that f u n c t i o n t/J0 such m e a s u r a b l efunction if p - a . e finite-valued .f i n i t e - v a l u e d measurable e x i s t saa IJ,-a.e. i f there t h e i e exists 2(X,/l): t/J~ € L 2(X,/l)} and = ( l ' i n d o m f o r A l = a n d dom A = (~ E L A~ = t/J~ for ~ in dom A. L I ( X , P ) ) e { l d o m . 4 ( l e L 2 1 X , 1 r 1$:l ( c a l l e dthe the r p ( A ) (called l e t rp(A) A , let (1.1.6). o p e r a t o rA, d e f i n e d operator d e n s e l ydefined ( 1 . 1 . 6 ) .For F o r aa closed c l o s e ddensely
tz
uollulag eql
(n pt)
1.1. The Relation ... ". ... (reI M)
21
'I'I
n. . .('uolllsodruocap 'Gildt - (V)dt relod:luIH) l€rll pu€ n > GVD\-'o)1 = (V)dt Wrqt /yrotls?,{ u y Jl 'y uet oluo uollreford eql eq (V Jo uollceford aEue:
range projection of A) be the projection onto ran A. If A T/ M, show that rp(A) = 1(0 co)(IA*D E M and that rp(A) ". rp(A*). (Hint: polar decomposition.)' 0
'W ol pep1.Jgge sacedsqnspesolc olouep s,(ear1e'pagglcads osr^\Jaqlo ssolun 'lll,r\ u pu? g '1y '14sloqur{s oql 'roldBrlc slql Jo lsor aql roJ
For the rest of this chapter, the symbols M, N, Band :R will, unless otherwise specified, always denote closed subspaces affiliated to M.
uY1 "N 'u e uatP * la ro{ -"N e: -ry til -l^l "N '""2'l = -"w /! iasuasSutuonol atti tt ro/ Pue Pup T T ur a^tttppo (1qo7unoc st Q,r1pt) uottolat aUJ 'Z'I'I uopFodord
Proposition 1.1.7. The relation ... ". ... (reI M) is countably additive in the following sense: if Mn ". Nn for n = 1,2, ..., and Mm .L Mn and N m .L Nn for m ~ n, then $ Mn ". $ Nn'
6 eql lgrll .'srsaqlocl,{q orll Jaqun 'aos 01 f see sr ll 'u .acuonbes N '}{ t', JI 'n u"N ''14 oculs n u"N e ''il o lsql a^Jesqo lsJrc 'Joord
Proof. First observe that
Mn, $ Nn T/ M since Mn, N n T/ M.
$
If un: Mn
". Nn' it is easy to see, under the hypothesis, that the sequence O:~=lum}:=l converges strongly to a partial isometry u such that u:
.u11 e-'14o D :n w:q.l qcns n {r1auos1 1e1}red e o1 ,{1Euo:ls seEJa^uoc l=j{-rr=t3} ". $
Nn .
0
NTW
Mn
N - W t(1dut y1l 11 puo
$
'S'I'f uopFodord
Proposition U.S. M,{ Nand N,{ M imply M". N. ull, !0N .r,n = oreqrn '-O < u , \cea JoJ tl11 6 oN :ol'ld,r,tre^BrI 'olrf,r* = " l , l p u " W = h o J aq^\'O
Proof. Let u: M ". M' f N and v: N ". N f M. Set M1 = wM, where w = vu. It is clear that w E M and that wlh is a partial isometry for each n (since w maps M isometrically into itself) such that w n: Mo ". Mn for each n ~ 0, where Mo = M and Mn = wnMo' Since No f M, we also have wnPNo: No'" Nn for each n ~ 0, where N n = w n No;
consequently, for n
~
0,
'O
oNa o}.la)ut '("N o "l^t)fN o ow):( wn(PM
o
- PN
0
): (Mo g No)'" (Mn g Nn),
where we write M g N for M ("I ~. The construction shows that M No ~ M1 ~ N1 ~ ..., and so ("InMn = ("In N n = :R (say). Appealing to Proposition 1.1.7, we get: u lt roJ N 0 ll| 3llr^\
!l leql s^\oqs uollcnrlsuoJ aqJ
e^\ araq^\
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
= Mo ~
g Nn)]
$
[n~o (N n g ,, . o=u I
f .r,
[m~o (Mn
Mn+l)] $:R ( .t*u
,,
o=u I
M=
o'N)e J e L(',vu h) g J de L('-'t,t $
[n~o (Nng
o=u'l
N n)]
,,
r
[n~l (Mn g
Mn+l)] $:R
r .,,.
I=u')
".
'N -oN =
No'" N,
D (rN O ril) - fN O ow)l?ql e^oqe pa,rordIJBJ eqt pesn a^Br{ad\ ereq/r
where we have used the fact proved above that (Mo g No) ". (M l g Nl ) 0
""-('Ne
zil)-
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
Proposition 1.1.9. Suppose M is a factor. If M and subspaces affiliated to M, then either M,{ N or N,{ M.
y
". ... •
pasop ato
". (M2 El N2)
N are closed
,(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
Proof. Assume, with no loss of generality, that both M and N are non-zero. Let f denote the set, whose typical member is a family
22
ication of The Murray-von Murray-von Neumann Neumann Classif Classification of Factors Factors l.1. The
I [/. t l'1, {(1\, t/,): Ni): ii eE 1} I} where where }t, 1\, N, Ni I'# (0) (0) for for each each i,i, \t-\.l Mj and and N, Ni.l Nj for for ii {(l't, j,4'-N N, Propoiition"0.4.l7 g i. • '# j, M. '" N for all i, and M. eM, N. c N for'all for all I. Proposition 004.17 It c N,1 for all i, and 4 1 * 1 1-non-void. ensures thdt that the the above above set set If is is non-void. The The set set Ff is is clearly clearly "nsrrris partially ordered ordered by by inclusion, inclusion, and and the the union union of of aa totally totally ordered ordered partially collection of of families families of of the the above above sort sort is is again again aa family family of of the the collection element, f maximal has a above sort. So, by Zorn's lemma, the set f has a maximal element, set lemma, the above sort. So, by Zorn's say t(!q, {(1\, N,): N): fi eE 1). I}. Since Since:lf is separable, separable, the the index index set set 1I is is countable' countable. lf is say - N It follows from Proposition Proposition 1.1.7 1.1.7 that that t't M'" ti. where where It-follo'ws'from M= = ,ED? ,M.4 X iEI 1
? ,N.. Nt' ED aand n d!ti. == ,iEI 1
The maximality maximality of of the the collection collection {(ili, {(Mi , l\l,): Ni ): ii eE /}I} and and Proposition Proposition The r e bboth oth tthese h e s e aare N i n c a s e 004.17 -applied to M g M and N g Ii , in case 0 U-, applied to il e ! and 0.4.17 E N t l t o r nonzero -ensure that M = M or N = ti ; i.e., MiN or N i M. 0 N = U . : i . e . , M L N o r nonzero--ensurethatM=I'1 1.2 Finitc Finite Projcctions Projections 12 enote a ill a lways d For the the rest rest of of this this chapter, the ssymbol M w will always denote ymbol M For eing r i m a r y rreason h i s bbeing o r tthis e a s o n ffor factor of of operators operators on !1, :If, the pprimary factor Proposition 1.1.9. Proposition 1.1.9. if eo eo Ee finite if Definition 1.2.1. A projection e in M is said to be be finite 1.2-1. A Definition s a i d tto o i s said c a s e ' e is c o n t r a r y case, I n the t h e contrary P((M) e ' In i m p l y eo e o == e. € o (~ e imply a n d e - eo ? M ) and l'1 affiliated be infinite. Correspondingly, a closed subspace M which is affiliated subspace closed Coirespondingly] be infinite. finite or as PM to infinite according as finite or infinite PJvlis finite be finite to M is said to be infinite. 0O infinite. !4 is finite; then M Proposition If finite, then finite; in 14 L N and N is finite, If MiN 122 Proposition 1.2.2 ll is is infinite. infiniteparticular, if lviexists, then :If exisls, then infinite M if any infinite (check it!), it!), preserved under equivalence equivalence (check Proof. is preserved finiteness is Proof. Since Since finiteness lle l'1, g lvl then If l'1 ' generality, N assume, with no loss of generality, that M f N. If M '" M f M, then loss no 4 assume, o g M) N= Q 0 x)g a n d consequently c o n s e q u e n (0) t l y (=0 )N N (N Q f NN and = !Mt eED( N M )'"- JM'ot nEDe(N ( NQ 0 M) N = I't In In of M. finiteness of the finiteness (1\ establishing the (14^ED (N Q iln, l'1 Q 0 M thereby establishing M)) == M e (N 0 M)) o' thereby the establishing the thus establishing J'lis is finite, finite, thus particular, every M then every pa"rticular, if is finite, finite', then if :Iflt is O contra positive of the second assertion. 0 a s s e r t i o n . s e c o n d o f t h e contrapositive the o f the d e v e l o p m e n tof f u r t h e r development The i n the t h e further v e r y crucial c r u c i a l in i s very r e s u l t is n e x t result T h e next theory; it is a sort of Euclidean algorithm. a l g o r i t h m . E u c l i d e a n o f theory; it is a sort (0). Then exists aa Then there there exists proposition 1.23. N '#t (0). Proposition M and and N I\ N N T)n M Let M, 123. Let I\ and aa of subspaces pairwise family {N i: i E I} of pairwise orthogonal subs paces of M, and orthogonal Il of familv { lrl,: i e 'subsiaci - N all R, N e Ni N Nr) M,llM == (ED lor all subspace :RR'of of M i' :RR T)n M, N ) ED:R, i '" for N,, 14such that N suchthat 1e,rriEI i N ).) . b u t R f* N i i and N but:R ( i n the t h a t:RR i' L N N (in s e n s ethat t h esense a n d :RR .~ 1 N exists there exists infinite, there set II isis infinite, If, index set the index decontpositiott,the strchdecomposition, one such If, in in one = (0)' R term rentainder another decomposition in which the remainder term :R = (0). the which in anither decomposition
'.,{l1u13ur scrlslrelcBrEr{o Jo Jalrlc o q l a u J r u r o l o po l u o t l l s o d e q l u l A \ o u e r € e l r , : f r e l l o r o c E u l p e c a : d eql Jo (q) lred ece.lerd uuerunall uo^ pu€ i(errn141tlolqa r{lt^\ lueuatsls oql peer o1 Eullrelgp fltueseald ll pulJ lrlElur repear orll
The reader might find it pleasantly diverting to read the statement with which Murray and von Neumann preface part (b) of the preceding corollary: 'we are now in the position to determine the chief characteristics of infinity'.
'(q) .lo n 'dor4) s ql llnser uralsurag-rep.grrlca J I B q p u o c a s a r l l s a r l s l l q e l s e( g ' 1 ' 1 qt1,vr:eq1eEol 'qc1q,n '(q) Jo JIsrI lsrlJ oq1 soaord srqa 'parrsap s€
as desired. This proves the first half of (b), which, together with the Schroder-Bernstein result (Prop. 1.1.8) establishes the second half of (b). 0 n
ED
B1 )
ltg
eug
6
'g"] t[*. "g'1"]tt
L;2 B
f M,
L
ED:R) 1
'wi
n
-
L;2 B
ecueq lelqelunoc sr les eql 'elq€r-Bcles sqt ocurs 'rB 1 '6 rg pue lt!o) = ,tg r{l1an e (,!g N ol €'Z'i N taa o1
n
B
Ig l g pu€
N1
ED :R, with Bi' 1 to get N = (EDiEI B B1' Since Jt is separable, the set I is counta ble; hence
uolllsodord ,(lctcle'(q) roy l"zg t=ja = g
!,.2.3 to Nand
'(e) erord oa lnd fldurrs 'u
1.2.3 that M admits a decomposition M = $:=1 Bn , where Bn - B1 for all n. _ To prove (a), simply put B = $:=1 B2n ; for (b), apply Proposition
ll?, roJ r g - ug ara{,n '"g t=j* = * uoltlsodruocap e sllrupe W leql €'Z'I
ug l=je uolllsodor4 Jo ued puoras aql urorJ s/AolloJ ll tt i teqt pue "g (o) Igl e q l ' z 1 1 er o 3 r g - ' g luql reelc sl ll'(rlt 6*il)"r = * 'tnt 'W o 'Joord i JI 11!itt esoddns os l1e1arrl sr (e) ur uoJlecrldur aug
Proof. One implication in (a) is trivial; so suppose u: M - M1 f M. If B n = un(MClO:1 g M ), it is clear that B n '" B1 for all II, that B 1 '# (0) and that $n=1 ts n f M. It follows from the second part of Proposition
'luaptmba ato suotlcatotd a7rut[utouy tuo 'tolntrltod u1 :tt I 1. uaqt 'at1ut/u! sl W l! puo W tr N 1^t/l (q) '(g '(g O W)e g =y1uoltlsodwozapDSllLupDW O W)- g -N4t!u 'IAttr 'VZt ^rtuz11oro3 {r [yto puo I! altut{ur st W uaqJ N t 6) ta7 @)
two infinite projections are equivalent.
Corollary 1.24. (a) Let (0) '# M Tl M. Then M is infinite if and ollly if M admits a decomposition M = B $ (M g B), with M '" B - (M g B). (b) If M, N Tl M and if M is illfinite, then N 1 M; ill particular, any
'a t=J. = y uotll'uN lte roJ. U:..llV pue lN *, = lN pue .-.N _' -_; O .F 6 W:tt JI ''N '-;e - y legl 8'l'l uollJsodor4ruor3 apnlJuoC
Conclude from Proposition L.1.8 that M - $:=1 Nn. If u: M and N'n = u* Nn , then M = $ n= 1 N'n and N'n - N for all n.
rj
$""=1
Nn
'Ni"N t6"Tw M -1
n=1 $
""
Nn f
M.
'snqllrry E sl rg orolll N T 6 ecurs'rryJooredsqns.rorrrrri '"N '6" o ru 'E"eu-'N ti"t*=* ( ' * ' N - N - ' N o c u l s") ,
where :R' is a suitable subspace of N1, since :R 1 N '" N1; thus, - :R'
Nn -:R
$
$
n~2 Nn'
"" N (since N n n ""
$ n=2
-
N '" N n+l)
'("''g'z'll = l€rll osllou ueql 1 lEql etunss€ feur om ? .lo ,(tlttqeredas agl Jo Arerl uI 'uollrasse puocas arll roC 'N l U lBqr eq lsnu lI'6'1'1 uotllsodord,{q ',(lluanbosuoc '(lN It''o) I ry / u teqr sernsue{!N) .lo i(lrleurxew eql 'flrodo.rd o^oqE aql ol lcedsar qll/d lErurxeru s1 ,(1$ueg O W = d JI eql ltql ^uedord eql qll/r\ 'N .ol lualurrrnbe qJ€e .14 Jo socedsqns leuo8oqlro osr,nrlecl 3o I r | :r N) fpure; e sple1{ euural s.uroz ol l€oclde,u€ '6'I'I uolllsodord Jo Joorcl eql ul sV '0N - N leql q c n s 1 5 0 1 s l s r x a e r s q l l B q l o s 1 , rT r u l € q l 6 ' I ' I u o l l r s o d o r 4 u r o r S s^\olloJ ll 'es€c e^rleuroll€ or{l uI 1,1= d 'Q = I los 'N } W JI 'JooJd
Proof. If M ~ N, set I = 4>, :R = M. In the alternative case, it follows from Proposition 1.1.9 that N 1 M, so that there exists No f M such that N - No' As in the proof of Proposition 1.1.9, an appeal to Zorn's lemma yields a family {N i: i E I} of pairwise orthogonal subspaces of M, each equivalent to N, with the property that the family is maximal with respect to the above property. If:R = M g ($iEI N), the maximality of {N i} ensures that :R J. N ; consequently, by Proposition 1.1.9, it must be that :R ~ N. For the second assertion, in view of the separability of Jt, we may assume that I = {l,2,3,...}. Then notice that 1.2. Finite Projections
23
s u o l l c e f o . t go l I u I C ' Z ' l
EZ
24 24
The Murray-von Murray-von Neumann Neumann Classification Classification of of Factors Factors l.1. The
The next next few few lemmas lemmas lead lead up up to to aa proof proof of of the the main main result result in in this this The finite projections a is again -section -that a supremum of two finite projections is again a finite finite of two supremum that a section projection. Some Some of of these these intermediate intermediate results results --- particular particular Lemma Lemma projection. -1.2.5 -- are are interesting interesting in in their their own own right. right. 1.2.5 h e nM., I\ N M e$ N Let N,, BB nn M M,, l 4 Mt1N Nand N.' TThen N a n d BB ceM e t tM.,\ N 25. L LLemma c n n a I1.2-5. nd ! g1,, N r @$ N B aadmit = sM.1 $1 N=~=N~l N2, o$ N Ng" aand d m i t ddecompo~itions e c o m p o s i t iM on l t e= IM 11212$e M aand nd B B == Mz M2 e$ N2 N2 e$ Bo, Bo' with With \,Mi , Ni, Ni' Bo /:So nn aM and and satisfying: satlsfymg: B M2z==M B,, iMlgs== M B I1 i l nn B M ! tnn B
N2z==NN()n B, Ngs==NNn n B B1t N 8, N
eg1) ' L ) xMr1 == i M l oQ ((Ml '21$2 M N 2z$e Ng)r ) N1 r==NNQe ((N Boo=={t{ E+ l At: + A tl :El edom d o A}, m.,{}, where A is is aa closed closed operator alfiliated affiliated to M such such that dom A == l1y M1, where -"" Nl. Bo Mrran A == Nl, N1, ker ,4 A = = (0). (0). Further Further M "" B NI' |ffi o 1 o == let 8 a n d let a b o v e , and a s above, , , N ', N l 4ga Proof. 2' ' N N,g as BOo I 11'= r, , M21, ,M define M S i m'N,1. p l y define P r o o f . Simply 1 N2 = B B; pti'. ker(el Rot. that piq ^ia B Q (l{, $ N ). Let e = PM and f = PN' Note that ker(el B) = /:S n n e rete b e 1u" f'= 2 further is N and consequently el B is one-to-one; further one-iir-one; e[ Bo and 6onsequently N == N; o [
( i ! t, eE l M ' 1 and ! , , D Vn ) n Ee B 6 V t( eE 1M1Q a n d0O== < l , e n==>< 0 eeBB {:9 {:9 € ) lte E
M l,n t n B1 8 I == i Mg. l..
Bo ilr. Thus Thus el el B Eoo == MI' hence that that eB I1r,and It ltr1 $e ~, and hence eB == M that eB It follows follows that o
l'tt. D of MI' maps onto D subspace densesubspace onto aa dense one-to-one maDS B 8^ o one-to-one one-to-one maps B Bo that 11 8o An 0 one-to-one showsthat reasoningshows similar reasoning /l B An exactly exictly similar o maps = where T)n ,4 D'Nr by onto a dense subspace A: of N l' Define A: D .. N by At = fn Define I fn where of Nr. onto a densesubspace 1 that at once follows It er\.= is the unique vector in B such that en = t. It follows at once that Bo that s-uch vector in \. is the unique o A l : ltED}. eDl' Bo-N, o=. =g{ (++ At: N1rand andB A i a nAA == N I is i s one-to-one, o n e - t o - o dom n e , dAAo=mFM|1,r ,ran precisely e N1 means meansprecisely t"t, The df M 8o subspaceof is aa closed cloied subspace fact that that B The fact 1 $ o is 9r
-9I(
9r(:
ueql 'slf F zru :(r) ase3 's'z'I Bturua'Iul sB aq, 'og 'tN 'h i'tt 'Joord V
Proof. Let~, Ni' 80 , A be as in Lemma 1.2.5. Case (i): N2 i M3• Then
either
8 i M or (M
$
N) g 8
i
N.
uaqr.N r w t , t u . No wj s p o ' N n t r ? tf .; lTf ] ; ' - l J f # H
Lemma 1.2.7. Let Jot, N,
8
n M and 8 f M $
N, with M1
N.
Then
'xg ('(s's'z)
this last assertion follows readily from the spectral theorem -- cf. Ex. (2.5.5).)
oraqa 'g 3o qderE aql ur esuapsr oi o, o.rrrrlser 11 'uollrass€lsel '0 < H roJ leql lcEJ eql poou plnoA\nof 3o qder8 eql aql roJ l0 = 1 ro Q = 11os€cu! 'l JoJ e,r1os' 8d alnduoc ot 2(u'u*V-) + (:p,'t) = 1u*)) lcr{l qcns *ts ruop r t. pue r urop r t anblun B slslxe pue 'V*v ruop ol ereql ? I (ff) JI :lurH) 'g ul asuap,(lluonbasuoc l'ld Bd y go qclerE uortcl.llsir eqt oql sl eEuer oql t?ql opnlcuoJ 3o 3o 'lu€rrE^c,,(eur esecoql s€' NI Jo l'11se palarctJolul eq ol sl I eJeq/
where 1 is to be interpreted as 1M of IN ' as the case may warrant. Conclude that the range of P8 PM is the graph of the r~!Iiction of A to dom A*A, and consequently dense in 8. (Hint: if U..!2) E le, there exists a unique ~ E dom A and n E dom A* such that (fv,n) = (~,A~) + (-A*n,n); to compute PB ' solve for ~, in case 11 = 0 or ~ = 0; for the last assertion, you would need the fact that for H ~ 0, the graph of H restricted to Do is dense in the graph of H, where
;l
N a' [ :
y(Y*Y +
L vGvv + l )*v
y(v+v + t)
,lr-Gvv+ r)*vv
=
p u € [ ; : ] =
PH
[ A(1 + A*Ar l
AA*(1 + AA*r l A*(1 + AA*r
= g d
(1 + A*At l
l
],
l]
=*
'N lerll ^{ogs e W = ;X uorgrsodruo3ep 'N aql ol ]coclsar qlll\ Jo +y {uop ec8dsqns osuep eql J,f olul ruorJ rolarodo reaurl e se aV Eu1,no1rr'(*Z ruop t u :(u'lt*y-)) = r g uoql 'pr 3o qderE eql elouap g lo'I 'rolerodo pesolc e ae N - Ci:y 'N 'O'Z'I) lel pue W Jo oc€dsqns r€eurl esuop E oq C lol e ll = fi le'I
(1.2.6). Let le = M $ N, let D be a dense linear subspace of M and let A: D'" N be a closed operator. Let 8 denote the graph of A. Then 8 1 = {(-A*n,n): n E dom A*}, viewing A* as a linear operator from the dense subspace dom A* of N, into M. With respect to the decomposition le = M $ N, show that Exercise )sIcrexl
it would follow from Ex. (1.1.6) that Ml "" 8 0, The asserted equality can be directly proved without much difficulty; it can also be deduced from the following exercise. Similarly, consideration of IP8 would prove 8 1 "" 8 0, 0 o
'og - tg eaord plnon ogdy E 'l(1re1gur15'oslcJexo Eur,vrolyogegl uroJJ pecnpop Jo uoll?Jeplsuoc eq osl€ u€c ll lltllncrggrp qcnu lnoqll^r po^ord fllcarrp oq u€c , ( l r y e n b ap a l r a s s eo q l ' 0 g - T 1 4l € r l l ( q ' t ' l ) ' x X r u o r J ^ \ o I I o J p l n o ^ \ l l
.og= ,ofu u", 8 0,
0
gd,
=
= Ml ;
lsql rr\oqs u?c e^\ J! lril =
J.B 0 e
ran
o
We already know that ran eP8
If we can show that
ue, lBrll-lYroul ,(p-ea:1ee11
'n u og 'tN 'fu ecuys 7
that A is a closed operator. It is a routine matter to verify that A M, since ft\, Ni' 80 n M. __ .
n
u V leql fgrraa ol Jell€ru eullnor B sl lI
'rolerodo pasolr E sI Z l?r{l
1.2. Finite Projections suollcaforo 4lIuIJ'Z'l
25 'L
26 26
The Murray-von Neumann Classification Factors actors eumann C he M urray-von N l a s s i f i c a t i o n ooff F l1.. T
B = Boo $e lM' 2t r$e N22 $ N, N2 -..,.M t'1, Itl1 e$M 2 e
lI x M r 1e $M I l 22 o$M l L3==M, 1 1. Case (ii): !1, M3 {1 N2. N2' Case Regarding A as a closed closed densely densely defined defined operator operator from 1"1r. M1. to Nt, Nl' as-a Regardinglet A+ denote denote the the closed closed densely densely defined defined operator operator from Nt N to J"lt M1 let 1 the adjoint adjoint of ,1. A. Then Then ,4+, A+, viewed viewed as as an an operator operator in 1?, 1f, is is which is the clearly affiliated affiliated to M: M; further, further, from from the the general general fact fact about about the the clearly graph of the the adjoint, adjoint, it is is clear clear that (J'lr (M1 e $ /Vl) N1) 09 B0 80 = {-n {-T) + A+n: A+T): n T) e € graph dom l+). A+}. Arguing Arguing exactly exactly as as in the the proof proof of Lemma Lemma 1.2.5, may be be 1.2.5,it may dom seen that Nl N1 -..,. ((q «M1 e $ ^Jl) N1) 0 9 B0) Bo) -..,. Mr; M1; hence, hence, seen (M $ N) N) e 9 B = ((Jvtr «M1 e $ Nr) N1) 0 9 Bo) Bo) oM, $ M3 e $ N, N3 (J'1o
1 N. . I N1r $e N2r$. N3 3= = N 1.1.9,one one of the The proof is by Proposition Proposition 1.1.9, the two two is complete, complete,since, since,by The proof cases must arise. O arise. 0 casesmust then Lemma M.l N, and M and finite, then are both both finite, !\ N T)n M, M, X l- N, and !4 and N are 12& If If 1\ Irmna 1.2.1. M $ N is finite. J'le is finite.
'chief characteristics Proof. of characteristics is infinite, infinite, then, then, by by the the 'chief Proof. If If M $e N is (!t $e N)..,. infinity', t"te Nand N) - B N and (M 8 T)n M such such that that B fc M$ infinity', there there exists exists B ( M$o N) N) ( M$e N) - «M N) 1 ..,. o r (M ( ( J v$t eN) 1 . 2 . 7either N ) 9e B). B ) . So, L e m m a1.2.7, e, i t h e r(M S o ,by b y Lemma I M or would then the 1t N. I't o N would then contradict contradict the infinitenessof M$ The assumed assumedinfiniteness i/. The (cf. Prop, finiteness It or N (cf. Prop.1.2.2). 1.2.2). 0n finitenessof Mor I't N) 1 Lemma 1\ N ([M + N] Nl Q M, then Lemma 1.2.9. 129. If N T)n M, then(tl"t+ e N) I .M. ryll, Proof. Proof.
=p l ( [ x++ N]) N]) N l 90 N [M N = PN N.l([M t l '+l + N] ivl)l == [{p N.l ~: l: ~E €e M}] t(pNI ran P == ran P N.lPM NIPM
- r a n P M P IN gl,t
(1.1.6)) ( b YEx. E x '(1.1.6» (by
o
N are so is is M and are finite, so Theorem If l\ N and N N T)n M, M, and and if if M f2f0. Il 1\ Thcorcn 1.2.10. many generally, N]; slightly more generally, the supremum of finitely many of tlte suprenunr more slightly N l; finitely finite projections is is finite. /inite. finite projections
[M [Jut++
. a s r c r a x es r q l Jo Joorcl e ldurall€ ol z(e,rsnor^qoauo Jo lq8ly eq1 uI
In the light of this exercise; one obvious way to attempt a proof of M.,.. N dim M = dim N ; M is finite (reI M) dim M < co . the equation D(M) = dim M satisfies the conditions (a) - (c) of Theorem 1.3.1; if D ': P(:e(:If)) ~ [O,co] is a function satisfying (a) - (c) of Theorem 1.3.1, then D' = cD where c = D' ( N), for any one-dimensional subspace N of :If. 0
I A J o N e J € d s q n sl e u o r s u o r u r p - o u o f u e r o J ' ( N ) , c r = ? a l e q 1 Y \e c = t Q u e q l ' I ' g ' I l u e r o e q l J o ( o ) - ( e ) E u r f g s l l € s u o l l c u n J B s r [ - . 0 ] - ( ( A ) f ) a : r e J l (p) l 1 ' g ' 1r u e r o a q l Jo (c) - (€) suoJlJpuor eql salJslles W rulp = (il)O uollunba aql (c) ' o > urlp !f <+ jtt pt) atlurJ sr l^l (q) l N * l p = W u l l p ( + N - W (e)
(d)
(a) (b) (c)
**
**
:e^ord '(Ah = n rc'I '(Z'e't)
(1.3.2). Let M = :e(:If). Prove: Exercises
seslrrexg
'N TOJ uollounJ uorsuarurp e g Eurllec ,(3r1snfIIr^\ pue .os€c l€rJeds eldurs e u I u e r o a r l l e q l J o f l r p r l e l a g l q s r l q e l s o I I I ^ \ o s r J r a x a B u r a t o 1 1 o go q 1
The following exercise will establish the validity of the theorem in a simple special case, and will justify calling D a dimension function for M.
Further, such a function is uniquely determined up to a positive constant multiple. 0
tuolstto) atrtltsodn ol dn paulwralap t(lanbun st uotlcunl ,Orrn, ?:::r'j:;:
'- > (W)O attu!{ st W (c) e puoi(11)o+(w)o=(N ot,t)(+ N Tl^t (q) (a) (b) (c)
**
M", N D(M) = D(N ); M1 N D(M $ N) = D(M) + D( N ); and M is finite D(M) < co .
** *
:( l)o = (w)o<+ N - l^t (e)
p(M) ~ - W)d.
Theorem 1.3.1. Let M be a factor. There exists a function D: [0, CO] such that 1o4l qcns l-,gf :q uolnunt o stswa ararlJ ..rorco/ D aq n ta7 -11gl rueroeql
'llnser E u r n o l l o ; a q l t u l l o r d ol pelo^ep aq '(W)Oelrr^\ .1Wd;gEullrr.n IIr^\ uorlJes sIrll Jo JI?rI lsrlJ aql llBr{s e,n u€gl rerll€r 'osl€ irolceJ € ol pelsrlr3ge sacedsqns pasolc elouap s,(e,n1u IIr/r\ g pue g 'N sloqru,{s or{l .suorloes Euro8erog aql J4 ul sy 'N T W .ll fluo puu y. ( Nd)q > 0,t4O Eu1,{gslresl_,gl _ (l^t1 4 :dr uorlJunJ B ecnpur plno^r qcns .[*,0] € pasolc a oluo lesqns O Jo -/(t4l)d 3o g usrqdroruosl fu lrqrqxe .Ol)a uo IIrr'A uorlJas srql | .rapro aqt ,fq pacnpur Eureq rapro eql -- n ur suorlcaford Jo sass€lJ acualezrrnbaJo - /04t)d. los perepro 1(Ilelol aqt ,{q peplnord s! n roJ l u e r r € A u r a u o f l r u a l c ' s r o l c € J J o u o l l € c l J l s s € I ce q l r o { . , { r o E o l e ce q l '(froEaluc elerrdorclde aql ul) usrqdroruosr ;o slcefqo lcro^os arll Jo o 1 d n ' u o l l e c r J r s s e l ca q l s l f r o o q l f u e u r s r u o l q o l d c r s s q e q l J o e u g
One of the basic problems in any theory is the classification, up to isomorphism (in the appropriate category), of the several objects of the category. For the classification of factors, clearly one invariant for M is provided by the totally ordered set P(M)/ '" of equivalence classes of projections in M -- the order being induced W the order i on p(M). This section will exhibit aJl isomorphism D of P(M)/onto a closed subset of [O,co]. Such a D would induce a function D: P (M) ~ [O,CO] satisfying D(PM) 1IO D(PN ) if and only if MiN. As in the foregoing sections, the symbols M, N, Band :R will always denote closed subspaces affiliated to a factor; also, rather than writing D(PM)' we shall write D(M). The first half of this section will be devoted to proving the following result. '€'I
1.3. The Dimension Function rorlJunc uorsurrurq aql
' u o r l r e s s ep u o c o so q l s p l e l f l u a r u n E r eu o r l o n p u l , ( s e eu u ! 9 . 9 . 1 O tuorJ s^\olloJ rualoeql eql Jo uorltoss€ lsrrJ oql lsacedsqns a1ru13 y e u o E o q l r of 1 1 en l n u J o r u n s l o a r r p e s e I N + W ] g o u o r s s a r d x eu e s r 'Joord N @ (N 0 tN + wl) = [w + w] leql 6'z'I eurrue'I uorJ s^\olloJ lI
Proof. It follows from Lemma 1.2.9 that [M + N] = ([M + N] g N) $ N is an expression of [M + N] as a direct sum of mutually orthogonal finite subspaces; the first assertion of the theorem follows from 1.2.8; an easy induction argument yields the second assertion. 0 The Dimension Function
27
LZ
uollJunJ uorsueur(J eqf
'€'I
1.3.
actors f FFactors Murray-von l a s s i f i c a t i o n oof e u m a n n CClassification u r r a y - v o n NNeumann he M l 1. . TThe
..28 28
Theorem 1.3.1 1.3.1 would would lead lead one one to to seek seek the the abstract abstract analogue analogue of of aa Theorem the by is afforded one-dimensional subspace. One such abstraction is afforded by the abstraction One such one-dimensionalsubspace. following definition. definition. following Definition 1.3.3. 1.3.3. An An 11 M nT/ M M is is said said to to be be minimal minimal ifif j'l M I~ (0) (0) and and ifif Definition N = l t N T/ M, N f M imply N = (0) or N = M. 0 O N = ( 0 ) o r NnM, Ngtlimply is clear clear that that minimal minimal projections projections are are finite finite and and non-zero; non-zero; the the ItIt is The M. in exist trouble is that such projections may not even exist in M. The next next projections not even may trouble is that such definition yields yields aa partition partition of of the the class class of of factors factors into into three three definition subclasses, depending depending on on the the availability availability or or otherwise otherwise of of certain certain subclasses, kinds of of projections projections in in M. M. kinds Definition 1.3.4.. 1.3.4.. A A factor M M is said said to be be of of type type I, II II or III III Definition according as as itit satisfies satisfies the corresponding corresponding condition condition below: below: according M contains contains a minirnal minimal projection; projection; M M contains contains no no minimal minimal projection, projection, but does does contain contain non-zero non-zero M projections; finite projections; contains no no finite non-zero non-zero projection' projection. (III) M contains
(I) (II)
is of exactly It the definition that any factor is exactly one one any factor definition that is clear clear from the It is order, the type. We We shall prove Theorem treating, in order, the types types l'3'l by treating, Theorem 1.3.1 shall prove type. examine the III, Before doing that, however, however, it will help help to examine the doing that, It. Before Iii, I and and II. earlier established quantitative of the Euclidean algorithm established earlier algorithm Euclidean quantitative aspects the aspects (cf. (cf. Prop. 1.2.3). Prop. 1.2.3). Il (0) and is finite. and,I4 Proposition M is I\ N T/n M; M; suppose suppose N ~* (0) finite. If Let M, 1.3-5. Let Proposition 1.3.5. (as Prop' R in e and M = ($j€I N :R with N j ..... N for all i € I and :R ~ N (as in Prop. ) $ { i = Nr R (@ier e wit& t't N,) fo.r j
the is independent independentof the 1.2.3), cardinality is its cardinality and its is finite set I is ;ir; index iidex set l.n), the finite and particular chosen. cltosen. particular decomposition decomposition
Nn
and proof. Suppose decompositionand suchdecomposition (o,.-r Nl) $e :RRtI is is another Proof. anothersuch 14== ($j€J SupposeM which is is T: II -+' JJ which map T: suppose, that there exists possible,t'trit'ttrere exists aa map if possible, suppose,if ' N R that note r(1); { e injective but not surjective. Let jo € J \ T(I); note that :R ~ N injective but not surjective. Let "to "I \ ' Ro. Then, R N q .1 • So, there exists :R C N .' such that :R ..... :R Then, ' that Nf such Rn N Jo . So, there exists o ~ o Jr 0O JJ g f
r , r [$ =[ j€I ,e1n e Nr1'u ' ]) $:R L iet
M=
Ni1'lJe Ro .'. L~IL,?'Nh>] -l
$:Ro
C$ Nlc 14, c e N.'cM, ~I j€J j€J
=
JJ ;
from the the follow from l'1. Both assertionsfollow contradicting Both assertions of M. finiteness of the finiteness contradicting the p a i r of admissible o f admissible a n y pair non-existence f o r any a b o v e for a s above o f aa TT as n o n - e x i s t e n c e of decompositions. decompositions. 0n
iZ < [,N /N] teqt qcns rN oroz-uou3trurJ € stsrxeoraql uagt,Qzlu) ecudsqnsalrurJ orez-uou€ sr N JI leql a,,rordol secrJJnslI .Joord
Proof. It suffices to prove that if N is a non-zero finite subspace (nM), then there exists a finite non-zero N' such that [N I NI] ~ 2; for all n.
u 11otot olaz-rtoua1ru{ lo t]t"fVi Z < [t+"N /"Nl trrtt t1cnsQ;gu) sacodsqns acuanbaso stslxa ataqt ,ll adtQ to to1cnl o q n II 2.g.1 uunucl
{N n}:=l of finite non-zero subspaces (nM) such that [N nl NnH ] ~ 2
Lemma 1.3.7.
If M is a factor of type II, there exists a sequence
'etutuale ql1,r ul8eq oA{ 'pa^lo^ur ororu 0lllll € sr as?cslqt ul uollcnrlsuor aql 1 ad,{a
Type II. the construction in this case is a little more involved. We begin with a lemma.
' NO( w)O - g'flluanbosuoc pue J,1 erlurg {rozra ro3 ( l,/)OtN /Wl = (tt)O uettl .t.€.t rurrosrlJ Jo (c) - (e) Eurfgsrl€suorlcung{ue sl O JI (q) pue.I.€.I rueroaql Jo (r) - (e) suolllpuoc serJsll€s NO (e) 1eq1{3poa 01 .,rou ,fseo ii lI 'l pre) = (W) NO osBJraqlle u1 .acue11.Ol.Z.l rusroaql fq ,elrurg sI il uaqt 'allulJ sI 1 Jl 'pueq roqlo aql uo lelrurSur sr X ecuaq^\ 'w5"Ntru-"N t6"=N @ tu'u' '( "" 'z'I) = d.es'e11urgurst ielqeredas sr oJurs olqelunoc sr / les eql 7 ' 0 = d l u q l e p n l c u o r1. 1JI€ r u r u r u r s l Jl N a i u l s _ . 11 1 r r o 3 ! 1 -y N l U l e q l qJns W go uolllsoclruoJep s eq U e (tN trte) = ll lal,n U W fue rog
whence M is infinite; on the other hand, if I is finite, then M is finite, by Theorem 1.2.10. Hence, in either case DN (M) = card I. It is easy, now, to verify that (a) DN satisfies conditions (a) - (c) of Theorem 1.3.1, and (b) if D is any function satisfying (a) - (c) of Theorem 1.3.1, then D(M) = [MI N ]D(N) for every finite M, and consequently, D = D( N )D N . n=l
n
n=2
M= eN". eN co
n
eM, =
co
For any M n M, let M = (eiEI Ni ) e :R be a decomposition of M such that :R ~ N "" Ni for all i. Since N is minimal, conclude that :R = O. The set I is countable since Jf is separable; if I is infinite, say I = {1,2, ..., }, then
'orlutJst
if Mis finite.
W JI
'[N
[MI N] ,
/}{]
co,
'
-
N
=,*,no
(M) = {
olrurJursl l.f JI
D
if Mis infinite
]
eurJop puc .letururur eq N lol
.1 e
Type I. Let N be minimal, and define
'CI= t O s n q l p u € { 0 } I W J I - = ( i l ) r c r l e q l s o r n s u a( c ) 'III ad^l .(q) = sr. ({o}), eculs 16 = fq ({o)), os (oD,a rcqr n Jo cI a(, pue (c) l(q - > ({O}),4' uaqt'(c) - (e) EurfSsllss uollrun3.,(ue sr ,g 'flasre^uoC 'I'€'I ruoroaqJ Jr Jo (c) - (e) suolllpuoc aql sorJsll€s Cr l€ql r€alc sl ll 'elIuIJuI sl r'{ u W oraz-uou fre,re .srseqloclfq ,(q .acurg
Since, by hypothesis, every non-zero M n M is infinite, it is clear that D satisfies the conditions (a) - (c) of Theorem 1.3.1. Conversely, if D I is any function satisfying (a) - (c), then D' ({O}) < co by (c) and D'({O}) = 2D'({0}) by (b), so that D'({O}) = 0; since M is of type III, (c) ensures that D' (M) = co if M ~ {OJ and thus D' = D.
' ( o ) * x . l l' - l
D(M) = {
if M ~ (0).
co ,
I (o)=lrJI 'o)
= (w)o
0 ,
if M = (0)
eurJap 'eJuelsrxe roC .Ut addt
Type III. For existence, define
'I'g'I tuaJoaql .luaprcc€ u€ Jo goord oql ol peaoord /r\ou sn la.I lou s l ( I + u > / ) u J J l u = [ l ] ) u o r l c u n ; r a t e l u r l s a l e o r Ee q l r o J u o l l e l o u 3{l qtl,n flrrelrrurs eql os ' N rulp^r lurp p333xa lou saop qclqil\ reEalul lsalearE eql sl I w /Nl ,0Ih = 7g eldurexe eql uI lsr{l ?loN
Note that in the example M = l(Jf), [MI N] is the greatest integer which does not exceed dim M/dim N, so the similarity with the notation for the greatest integer function ({t] = n iff n ~ t < n + I) is not an accident. Let us now proceed to the proof of Theorem 1.3.1.
' g ' g ' 1 ' d o r 4 u r s € ' . 1 rp r e c r e E e l u r p a u r r u r e l e p flenbrun eql elouop O 'ellulJ pu€ oroz-uou rlloq dte n tl N T JI .9.€-I uolrlulJeo t ry/wl lal
Definition 1.3.6. If M, N n M are both non-zero and finite, let [MI N ] denote the uniquely determined integer card I, as in Prop. 1.3.5. 0
uollsunc uolsualur(eqJ '€'I
1.3. The Dimension Function
29
6Z
actors f FFactors l a s s i f i c a t i o n oof e u m a n n CClassification h e MMurray-von u r r a y - v o n NNeumann l .1. TThe
330 0
then the the l,l's ~'s can can be be inductively inductively defined. defined. Since Since NN is is not not minimal minimal (M (M then B (0) * B being of type II), there exists B nM such that (0) f. B .F f N;-th9 N; the nM such exists there being of type II), .that finiteness #N of N tnto..s ensures finiteness finiteness of of B. B. IfIf t[ N/B NIB I] )~ 2, 2, set set N' NI == 8;-if B; if .. iinit".nrtr BB;; I o R w i t h R { N = [N I B ] = 1 -note that[ NIB] > 0 -then N = B $:R with :R { = | n o t e t h a t l N / B l > 0 t h e n irulrf N ' = R ' D s e t a n d further :R f. (0) since B f. N; note that [ NI:R] ~ 2 and set N I =:R. 0 i u r t h e r R l ( 0 ) s i n c e B* N ; n o t et h a tI N / R ] > 2 Definition l-3.8. 1.3.8. AA sequence sequence S S == 1N,,)l=, {N n}:=l as as in in Lemma Lemma l'3.7 1.3.7 will will be be Definition U called aa fundamental fundamental sequence sequence for for the the type type IIII tactor factor M' M. 0 called The following following bit bit of of notation notation will will facilitate facilitate some some of of the the The subsequent ptools: proofs: let let us us agree agree to to write write kN kN for for any any subspace subspace of of the the subsequent form N, Nj e$ ... ... e $ Nk , with with N Nij -.... N N for for all all i.i. Thus' Thus, for for example, example, ifif M M Nn, form and N are finite and non-zero, then then non-zero, and N dre finite and
tFlN, m,[[F]. rlru
Lemma 1.3.9. 1.3.9. Let Let I\M, N, N, 8B be linite finite and non-zero' non-zero. Lcmna
(a)tFltFl [*]. t[#]. 'l[[F]. '1, r M e B ' l' Ll ixl -1l * < [M: [~ [~ ] 2. 1ft1., I F ] . t +f - nlB]I [~ [f (b) il if M 1 B then (b) 14I 8 ,, then
] +
],
<
] +
+
pairwise number of pairwise preciselythe largest number Proof. the largest is precisely Note that that [MIN] Proof. Note -tM/N I is f irst l"t The into f itted into M. The first orthogonal be fitted which can can be copies of N which orthogonal copies Turn consequence. (b), inequality, of both (a) and (b), is an immediate consequence. Turn (a) immediate is an and inequality, both to second: to the the second: implv would imply t) would Nl. ++ I) (tBlltl lXtM/N] (a) B 1M] ++ 1)([MI (a) The The inequality inequality [B / Nl ~> ([ tB IN] pairwise 1 ) pairwise l J ' l 1++ 1)([MI I X I M /N] r y 1++ 1) the B1M] B , of o f ([( t B i n s i a e B, t h L ' eexistence, x i s t e n c e ,inside l)-pairwise B/yJ if ([([ B orthogonal if 1M] +* 1) pairwise consequently, and consequently, N, and of N, copiesof orthogonalcopies is aa which (tM/NI l)N) + (since !t orthogonal copies of M (since M { ([MI N] + 1) N) which is 1,1 { of copies ortholonal con tradiction. contradiction. (a)' proof of of (a), (b) comment in the the proof parenthetical commentin (b) By By the the parenthetical
, , t e B . r [ r.I' -. ][ + ] . r)l r u. ( t MN] / N l++ [t BIN] B / N I ++ e x i s t([MI i f there t h e r eexist (Strictly v a l i d only o n l y if i s valid t h i s is ( S t r i c t l y speaking, s p e a k i n g ,this then ~lf not true, is lf; if that 2) pairwise orthogonal copies of N in~; if that is not true, then in N of copies pairwise orthogonal i) must with f i n i t e , with m u s t be b e finite,
[F]. [F].[*].,,
N] isis clearly clearly since [:RI follows since in inequality follows [R/ N] the desired desired inequality which case case the in which that of I e from 1"1 by monotone in:R.) Since finiteness is inherited by M $ B from that of is inherited finiteness monotone in R.) Since E f a l s e . b e Ml ' 1and B, the desired inequality cannot be false. 0 c a n n o t i n e q u a l i t y d e s i r e d t t r e 6 , and M and and seqttencefor proposition 1.3.10. for M Proposition b" aa fundamental Let {N f3.fg. Let fundamental sequence {N n}:=l r,}i=r be
I€
uollcunc uolsuelurosrII '€'I 1.3. The Dimension Function
31
'uaqJ 'otaz-uou puo artutl aq n u g \
ral
let M, B n M be finite and non-zero. Then,
~
0 eventually; in fact
[+]
[;. ]
[~ ] /' + .. ; and
,",
n
n
ffi
ti i (q)
lim [MI Nn ] exists and is a finite positive number. n-+" [B INn]
'raqlunu artttsod aty{ o s! puorrrr""
a^Br.I3 , h ' I < u f u e r o 3 ' ( e ) o ' g ' t eruuel
[g
(b)
p u 2o- . .
Tcou l 1 i t l p n l u a n aro [+]
(a)
Proof. By Lemma 1.3.9 (a), for any n ;, I, we have
'[4],.",. [+]t4l. l # l [t ];, [}][: ];,
2
n l - [:
'Joord
n
J;
-UNJLW L J L W I '7 raEalur,(ue rog 'usql "N 'ou 't 'os i o r < [ " N / p ] s p : o , nr o q l o u l : t ^ tT
since { N11M] is a fixed finite integer it follows that [N n/M] = 0 for all sufficiently large n; thus, there exists an integer no such that M l N n for all n ~ no; so, if n ~ no' Nn i 11; in other words [MI Nn ] ;, 1. Then, for any integer k,
N
[N:o+k] ~ [N~ ][~] ~
,1+or,..ot , . {*ou a 'r-,rzI l-"+ll#| . l-=ll-l 0
2
k
-1,
0
'I < ["N /g I r"ql eErelos sr u JI l€ql aas e^\ '(c) 6'€'t €turuel u1 seytllenbauleql qtoq ol Eurleaclcly(q) 'perrordsr (e) puu
and (a) is proved. (b) Appealing to both the inequalities in Lemma 1.3.9 (a), we see that if n is so large that [ B I Nn ] ~ I,
I I . f 1_ t*rWinJ tffiJ'-T*ffi
[MI Nn±k] ,{[MIND] + I} . {[ NDI ND±k] + I} [ BIN n+k] [ BIN n] [ N/ Nn+k]
[ t * " N/ ' N ]
I ' N/ e i
(*)
(*)
[ { + " 1/ g yJ
l€rll (e) lred pue flrlenbeur e^oqe oql ruorJ pue 'qEnoua oErel a JoJ o > ')c > 0 lerll (e) lrrcl IuorJ s^rolloJ tI '["N / g|/t" I /Wl = 'n Eurlrr16 'l < 4 ra8alur fue ro3
for any integer k ~ 1. Writing an = [MI Nn]/[B INn]' it follows from part (a) that 0 < an < .. for n large enough, and from the above inequality and part (a) that an ;
-n
w
I
,
I
dns u11
lim sup a p p -+ ..
) -n
-
-
d
'O <'o ruJIl€r{l ueos n f g 'ellurJ sl pue slslxo ll,,'g pue W J.o solor oqt EulEueqcrolur ,ll -rc-t-rull ecuaH '-o Jul urll > "rc clnsur1 leql opnlcuoc'tr Eurfre,r fq
by varying n, conclude that lim sup a p , lim inf an' Hence limn-+..a n exists and is finite. By interchanging the roles of M and B, it is seen that lim an > O. 0
Euratroylog aqt f;sgtzs ol uaas frrs'a sr 56; uortcu"J ".,;t:""i1;p-;;; pu€ elruu are N pue g l,t .lt '(A| Ot'e't 'dor4 ,(q paaluerenE sr ecuelsrxeesoqa llruJl erll aq ol )(g /W) ourJap 'oraz-uoupue olrurJ ere g puu x JI 'w toJ ecuenbesleluourepunJE eq t=j{" ry; = -I'g'I "uua'I Jo Joord Jo puA S pue II adf] Jo rolceJ e eq n lc'l
End of Proof of Lemma 1.3.1.. Let M be a factor of type II and S N n}:=l be a fundamental sequence for M. If M and B are finite and non-zero, define (MI B)S to be the limit whose existence is guaranteed by Prop. 1.3.10 (b). If M, Band N are finite and non-zero, the function (+) S is easily seen to satisfy the following conditions: = {
]
[~]S '[*
N~
=
€ N-W M".
=
[}]S; ''[*]
(l)
(i)
='[+],1=t[+](') ''[-*]'[#] ":Hl='[-fi] (ii)
[:
Js = 1; [~]s
=
[MB Js [~ Js ; [~]s = [~ ]~l ;
l1..
332 2
actors f FFactors Murray-von Neumann Classification eumann C l a s s i f i c a t i o n oof urray-von N TThe he M
~ B]S= Lt[ [~]!S LfJr= rlte 8'r
(iii) l'1 M i1 IB t::} (iii)
[ M
rll'l N
.+
rBt
[-}]S;
(use (use Lemma Lemma r.3.e 1.3.9 (b)) (b» Lnf ;
(t4 I r8'l ^ <' l v l -]S' ((iv) i v )lM.{ ' _{18B +::} l p - l ]S . L'' JS L'' JS
[~
[~
Now fix fix a finite finite non-zero non-zero IB and define define Now
i f !M1==(0) (o) II o0 ,, i f = (x/ I ifif lt Dg D S (x) (M) = M is finite finite and and non-zero non-zero )S , |{ (. .M/ B)S', if M infinite. 11 infinite' if is ' L It is readily readily verified that Dg DS satisfies satisfies the the conditions conditions (a) (a) -- (c) (c) of It Theorem 1.3.1. Theorem1.3.1. '[0,'] Conversely, if if D: D: P(M) P(M)'" [0,"] is is any any function function satisfying satisfying (a) (a) -- (c) (c) Conversely, Theorem 1.3.1, 1.3.1, it is clear clear that that for finite non-zero non-zero M and and N, of Theorem
consequently, for finite finite non-zero non-zero !t M, and and n n = = 1,2, 1,2,..., ..., consequently, N.l N,l t [M/ N ] D(M) [M/ N ] + 1 il{/ n ll't/ .....::...-,--_nu.:...._ li. D(M) • - _ _ ,.[t Bf N n] + 1 D( B ) [ B / N n] , D ( B ) I 1 B l N" l ' B/N"l+ ' o and - - ,, recall D ( X ) == t h a t D(M) let c o n c l u d e that a n d conclude r e c a l l that t h a t [B/ ] ..... l e t n ..... I B / NN" n] o for l( conclude (lv1)= .. conclude that for infinite infinite M, D(B )DS (M). S (M) (m). Since D(!t) = D DS Since D(M) D(8 )D3 D (B)DS' x ) D s . 0o D D == D(
' [0,"] as in in P(tnO D; P Proposition factor and (M) ... and D: Let M be be aa factor l3-f f. Let Proposition 1.3.11. [0,-] as 1.3.1. Then, Theorem Theorem1.3.1.Then,
**
(a) N); and (a) M.{ D( N); and MI N D(11),< D( ^J e D(M) pairwise (Jtl") is aa sequence sequenceof pairwise (b) } is t,f {M (b) D is i.e., if additive -- i.e., countablyadditive is countably n = ID(M"). (nl4) 14 6t4n, D(M) then (7'/M) and if M = $M then D(M) = ED(M , orthogonal subspaces and if orthogonalsubspaces n ). n
l'l .{I N that M (a) and (b) of Theorem 1.3.1imply imply that Proof. Theorem1.3.1 and (b) The conditions conditions(a) Proof. The
+ D(M) yields: M D(N (a), this I't << N N ::} D(M)<< D( ::} N ); this yields: ) D(M) coupled with (a), ); D( N); N); coupled D(l't) ,< D( (resp., D(l't) < D(N) thus, M .{ N (resp., M ~ N) implies that D(M) , D( N) (resp., D(M) >> (resp., D(x) I't that implies lt N) thus, I N I - N a n dM N are mutually M ~} N a r e mutually p o s s i b i l i t i eMs M D( ~{ N, N ,M x Nand t h e possibilities D (N». N ) ) . Since S i n c ethe possibilitiesD(M) D(1"1) D( N D(J't)<< D( N ),)' D(M) exclusive == the possibilities as are are the and exhaustive, exhaustive,as exclusiveand (a) in follows. D(N) and D(M) > D(N ), the reverse implication in (a) follows. D(N ) and D(X) > D(N ), the reverseimplication N
the (b) is of the is aa consequence consequenceof For the assertion(b) the assertion For finite finite sequences, sequences, D. Assume, Assume, (cf. (b) (b) of 1.3.1)of of D. Theorem1.3.1) assumed of Theorem finite additivity additivity (cf. assumedfinite I (0) for all n. Xn (11") that M then, } is is infinite infinite and and that the sequence sequence{M that the then, that n '# (0) for all n. n N, for all N, that, D show Finite additivity and monotonicity of D show that, for all of Finite additivity and monodonicity
nt"!, D(il")= '1"9, L~l ,'lnj n]
N
D(Mn ) = D
I
M
'D(M);
consequently ) ,< D(M). D(M). ID(JV1") consequently W(M n particular' for for in particular, If Then ED(M ID(Jv1,) possible, let D(M). Then tet ED(M Ib(il") a in If possible, n ) << 00; n ) << D(M). D(N) << E.e' that D(N) M such such that each 7'/ M non-zero N N"n finite non-zero e*is'dsaa finite there exists 0, there each Ee >> 0,
Proposition 1.3.14. Let D,6. be as above. Then 6. is one and only one of the following sets: 'a^oqo so aq j'q {o auo {1tto puo auo s, V uaqJ
la7
0
I
:s1as3utuo11otaql 'rl'E'l uoglpodor4 'Joord
Proof. Exercise! ieslJrexl
tD3 pup 'V "rc3 "' '4o'rp , v ) € rc > p u o i yt g - r c € D > g puv vrBto
6. £; [D,a]; a,13 E 6. and 13 < a:} a - 13 E 6.; and al'~'''' E 6. and Lan' Lan E 6..
a:}
l[='o]j v
(a) (b) (c)
(c) (q)
(u)
'uaqJ '(A)C = b 'e '€I-€'I "unurf tal puv a^oqv so aq V ta7
Lemma 1.3.13. Let D, 6. be as above and let
a = D(:I£).
Then,
'{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
Let us continue the analysis a little further by considering the possibilities that are open for the set 6. = {D(M): M T} M}.
Definition 1.3.12. Any function D as in Theorem 1.3.1 -- there are not too many of them! -- is called a dimension function of M. (Murray and von Neumann call it a "relative dimension function"; we dispense with the adjective "relative", one justification for such impertinence being: who has ever heard of a relative Haar measure?) 0
(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
and the assertion is verified. Since .... is countably additive, conclude that M .... $ M~ £; N, and hence D(M) , D( N) < E = D(M) - LD(Mn) 'D(M). This contradiction completes the proof. 0
'[i*'E'] oNSr+ul.[6l+ull ....
M~+1 ;
Ng
qJnsJrtlu t''l u al,rslsrxoaraql os M~+1 T}
L~l Ml J.
M such that l€ql
Mn +1
so there exists
J>n
> .L D(M) ~ D(Mn+1);
i(r+"w)o< (fw)o"if j=l
J
( , 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 = D(
N ) - 1:: D(W)
Assertion: There exists a sequence {M~} of pairwise orthogonal subspaces of N such that Mn .... M~ for all n. We shall construct the M~ inductively. To start with, D(M1) < D( N) implies M1 {N and so there exists M~ T} M such that M1 .... M~ £; N. If, now, M~, ..., M~ have been chosen satisfying Mi ' 1 Ml for 1 , i < j , n, and Mi .... 1'\' £; N for 1 ' i , n, then,
'g 5
le{l qcns
slsrxa oreql os pue
u
}
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
for each N. So, we may assume without loss of generality -- by replacing {Mn } by {Mn +N }, for large N -- that LD(Mn) < D( N). N=u N=u') ,, -l"t,t "* ("w)c"3 lo = ("w)o3 - (t,t)( (,,
n]- n~ND(Mn)
M
L
r=u
n~l D(Mn) = D [n~N
o
o
)
D(M) -
@
' 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
Pick such an N for a fixed E < D(M) - LD(Mn). Since D is finitely additive, note that 33
'€'I
1.3. The Dimension Function uollcun{ uolsus{ul( eqI
actors Murray-von Neumann Classification f FFactors eumann C l a s s i f i c a t i o n oof TThe he M urray-von N
{O. eE. 2€, 2E•...• nE}. where where 00 .< ?E << -0>;; (n (n == 1,2, 1.2....) .'.) (0, ..., nZ), , 1 , 2 , . . o>}. . , * )w , h e r e00 < where ({nE: n 7 : nn == 00.1.2•...• where h e r e 00 < [[O.a]. 0,4, w -] [0. 0>] [0, {O. -). o>}. (0,
consider. separately, separately. the three cases cases corresponding to the Proof. We consider, possible type type of of M. possible Case (i): M is of of type I. Case r o o f ooff i n tthe h e pproof N nn M bbee m minimal. Wee hhave e e n ((in a v e aalready l r e a d y sseen inimal. W LLet et N i er NNi ,j' o r m JM" t= Theorem 1.3.1 = o $jEI h e fform_ n y 1M4nn M iiss ooff tthe o r ttype y p e II)) tthat h a t aany T heorem 1 . 3 . 1ffor A g N with I countable and N ~ N for each i. and so 6 f {n E : n = N , f o r i , a n d s o each w i t h l c o u n t a b l ea n d ,2, { n 7 : n = 0 , 10.1.2• j ( i ) = lX f iiss ffinite inite a i f nd ...} u {o>} where E D( N). It is easy to see that (i) if and s e e t h a t I t i s e a s y t o w h e r e ? D ( N U ...) {-} ). i i ) iif n f i n i t e , tthen hen f X [XI N l]== nn., tthen 6 == t{kk 7E:: k == 00.1 n}) a and Xt iiss iinfinite. n d ((ii) , 1•...• , . . . ,n hen a llt/N 6 ,=={n A @ E: 7 : nn ==00.1 , 1•...• , . . .o>}. ,-). Case (ii): M M is of of type II. Casc ot oes n Since has 6 d does not o l l o w s tthat hat A e q u e n c e ,iitt ffollows a s a ffundamental u n d a m e n t a l ssequence. Since M h smallest positive number. number. Let d = D(Xf). D(X). Infer Infer from from contain a smallest hat n t e g e r k ssuch u c h tthat or a n y iinteger Lemma E A A, tthen E A 6 ffor any f a e h e n kka o. e 1 . 3 . 1 3tthat h a t iif L e m m-
a
a.
Case Case (iii): M is of type III. Clearly. case 6A = {O.o>}. Clearly, in this case {0,-}.
0n
II- or IIt, 110> I,, II*coo Ill' be of type type In' Definition is said f3-15. A factor M is said to be Definition 1.3.15. f u n c t i o n of o f M satisfies satisfies III d i m e n s i o n function r a n g e of o f the t h e dimension a s the t h e range I I I according a c c o r d i n g as I, F a c t o r s of o f type t y p e In 1 . 3 . 1 4 . Factors the P r o p o s i t i o n 1.3.14. o f Proposition c o n d i t i o n of t h e corresponding c o r r e s p o n d i n gcondition -) and called are called types are the other other types (n (n << 0» finite, and and the called finite. II, are are called and III (rel f i n i t e (reI i f X f is i s finite infinite. ( T h u s , aa factor i f and a n d only o n l y if i s finite f i n i t e if f a c t o r M is i n f i n i t e . (Thus. (Thus a M).) Factors of type I or II are said to be semifinite. (Thus a factor to be are said M).) Factors projections.) f i n i t e projections.) n o n - z e r o finite M is i f it i t contains c o n t a i n s non-zero i f and o n l y if a n d only i s semifinite s e m i f i n i t e if
o0
f a c t o r s of of t h a t factors g i v e n later 4 . 3 to t o show s h o w that Examples l a t e r in i n Section S e c t i o n 4.3 w i l l be b e given E x a m p l e s will all exist. types exist. all these these types papers would would be be No Neumann papers Murray-von Neumann of the the Murray-von No treatment treatment of p a s s i n g s o c a l led t h e m e n t i o n o f complete without at least a passing mention of the so-called l e a s t a complete without at i s expressed expressed a l g e b r a is v o n Neumann N e u m a n n algebra reduction w h e r e b y every e v e r y von r e d u c t i o n theory. t h e o r y , whereby abelian t h e abelian o n e uses u s e s the as V e r y briefly. b r i e f l y , one f a c t o r s . Very i n t e g r a l of o f factors. a s aa direct d i r e c t integral Hilbert u n d e r l y i n g Hilbert von r e p r e s e n t the t h e underlying Z ( M ) to t o represent a l g e b r a Z(M) v o n Neumann N e u m a n n algebra in m e a s u r e s p a c ein a space as a direct integral of Hilbert spaces over a measure space o v e r H i l b e r t s p a c e s o f i n t e g r a l spaceas a direct ' s c a l a r decomposable' operators. d e c o m p o s a b l e 'operators. such a s 'scalar Z ( M ) acts a c t s as w a y that t h a t Z(M) s u c h aa way ( a compact compact t o be b e (a b e taken t a k e n to (Actually. m a y be B o r e l space s p a c emay ( A c t u a l l y , the u n d e r l y i n g Borel t h e underlying
subset of) the real line, since every abelian von Neumann algebra acting on a separable Hilbert space is generated by a single self-adjoint operator -- but that is not really crucial.) The theory goes on to show that if 3t = J~(>,)d J,L(>'), there is, for each >., a factor M(>.) f :f(3t(>.», the assignment >. -+ M(>') being "measurable" in a certain sense, so that M is the collection of operator~ of the form x = f$x(>')dJ,L(>.), where x(>') E M(>.), the map x( .) being measurable in an appropriate (weak) sense and satisfying IIx II = ess. sup IIx( .) II < 0>. Using this theory, one may speak of the type of a general von Neumann algebra; call M type Ill' for instance, if each (i.e., a.e.) M( >.) is of type III and so on. After deliberating on whether or not to devote a section in this chapter to a more elaborate exposition of this theory, the author opted for "not to," on the following counts: (a) the material is not really pertinent to the remainder of the book; (b) it is not really necessary to torment the uninitiated reader with the spectre of non-measurability that is inescapable in anything like a serious discussion of disintegration; and (c) the initiated reader does not need the section anyway. The interested reader should go directly to the fountainhead for as readable and self-contained an exposition of the theory as is possible.
pus ;ilXT::l u€ paur€luoo-Jles sr se {roaqr aql Jo uolrgsoctxa
sE roJ p€oqur?lunoJ al{l ol ,{llcarrp oE plnoqs ropeor pelserelur srll ',{e,r,{ue uollces eql paeu lou saop repeor palelllul eql (c) pue luorlerEalurslp Jo uorssncslp snorras s oIII Eu1q1f,ue ur olqedecsoul s! teqt ,ttlllqerns€3tu-uou 3o erlcads aql qll/$ rop€ar pol€lllulun oql lualurol ol ,(resseceu ,{11eerlou sr 1r (q) i4ooq oql Jo rapul€Iuer oql ol lueuruad ,i.11ear lou sI lerraleru aql (B) :sluno3 Eut,vrollo; eql uo ,,'o1 1ou,, :o3 paldo Joqlne oql ',(roaql slqr Jo uolllsodxo al€roqsle erou € ol Jelderlc s l q l u l u o l l o e s B a l o ^ e p o l l o u r o r o q l a q ^ r u o E u l l B J a q I I e pr e U V "e'1) q3eo 'uo os pue rII oclfl JI 3o sI (\)4 ('a'e 'ecuelsur roJ 'rII ad6,1ya1 11eclerqaEl€ uu€runeN uol lereuaE B Jo ed,{l f e u r a u o ' { r o a q l s l q t E u t s n ' - > l l ( . ) ] c l l d n s ' s s a= l l r l l eql 4 e a d s J o (" Eurf .;sylespuB asuos (4ue,tr) alerrctorctcle ^.uBul eiQern$'eauEulaq ;x dsru eql'(x)n t (1)x eroq,n'(\)7,p(\)reJ = x ulroJ oql Jo srolerado '3su3s u!€u3c e uI ,,elq€rns€alu,, JO uollJsllos aql sl w wql os Euroq (1[,g - 1]ueruu8Jsse eql'((\)fih f (r)Ztt ro1ce; e'1 qcee -ro3'sr araql'(1)rlp(r)A-l = lf JI lBql ^roqs ol uo seoE^roaql eql ( ' l e r c n 1 5 ' , ( 1 1 B leoru s I l e q l l n q - - r o l € r a d o t u l o f p e - J l a s a18u1s B ^q palerauaE sI oc€ds ueqIIH alq€reclas € uo Eullce 'eut1 erqaEle uueruneN uo^ uelloqe fra,re eculs lear aql (Jo lasqns
s€
uorlJunc uolsuelulceql
'€'I
1.3. The Dimension Function
35
Chapter Chapter2 THE TH E TOMITA-TAKESAKI TOMITA-TAKESAKI THEORY THEORY
question(which, (which, in the Section 2.1 discusses Section2.1 discussesthe following Question the following the case caseM == g finite, is LL'1X,7,y) co(X,f,IL) with IL is answered answeredaffirmatively by the the existence existenceof - [0,1] the the Lebesgue Lebesgueintegral): integral): if if m: m: P(M) is countably countably additive additive in ?(tuO.... [0,1] is the the sense sensetha thatt
-["!,""] V en] =r m(e n) @
m[
n=l
=
t
n=1 n=l
m(e^)
projections in M, for any countable countable collection of of pairwise orthogonal orthogonal projections does m n extend to a linear functional does functional on M which is well-behaved well-behaved under monotone monotone convergence? convergence? Section 2.2 2.2 is devoted Section devoted to the celebrated celebrated GNS construction, construction, which, in = L case co(X,f,IL) yields the Hilbert 2(X,f,IL) and the case M = L'1X,r,p1 Hilbert space spaee L Lz(X,r,p1 r e p r e s e n t a t i o nof representation m u l t i p l i c a t i o n operators. o f M as a s multiplication operators. ( c o n j u g a t e linear) S e c t i o n 2.3 2 . 3 is i s concerned w i t h the Section c o n c e r n e dwith t h e (conjugate l i n e a r ) operator o p e r a t o r on the o n the - x* of M. The climax is GNS space space which is induced by the map x .... t h e Tomita-Takesaki T o m i t a - T a k e s a k i theorem w h i c h involves the t h e o r e m which i n v o l v e s a thorough t h o r o u g h analysis a n a l y s i s of of p o s i t i v e factors p o l a r decomposition the t h e anti a n t i unitary u n i t a r y and a n d positive f a c t o r s in i n the t h e polar d e c o m p o s i t i o n of of the above above mentioned mentioned operator, operator, and their commutation relations relations with with the operators operators in M. One One crucial fact emerging emerging from from this theorem theorem is g r o u p (called ( c a l l e d the the t h e existence e x i s t e n c e of o f a certain c e r t a i n one-parameter o n e - p a r a m e t e rgroup t h e modular modular group) of of automorphisms automorphisms of M, which is of of fundamental importance when M is of of type III. As the proof of the theorem theorem is long and p r o o f is p r e s e n t e d only technical, t e c h n i c a l , the t h e proof i s presented i n the v e r y special when o n l y in t h e very s p e c i a l case c a s e when the t h e above-mentioned a b o v e - m e n t i o n e doperator i s bounded. A l t h o u g h this never o p e r a t o r is b o u n d e d . Although t h i s case c a s e never (as will arises III (as will be arises when M is of type III be established established later), this option p r o o f and h a s been has b e e n taken t a k e n as a s a compromise n o proof c o m p r o m i s e between b e t w e e n no a n d complete complete proof, both alternatives being distasteful to the author. weights, which are Section Section 2.4 2.4 introduces introduces weights, are non-commutative p r o o f s in analogues i n f i n i t e measures. a n a l o g u e sof o f infinite m e a s u r e s . There T h e r e are a r e few f e w proofs i n this t h i s section. section. The T h e results r e s u l t s are m a d e at a r e stated s t a t e d and a n d some s o m e tentative t e n t a t i v e effects e f f e c t s are a r e made at plausible convincing c o n v i n c i n g the t h e reader r e a d e r that r e s u l t s are t h a t surely s u r e l y the t h e stated s t a t e d results a r e plausible enough. e n o u g h . The T h e statement T o m i t a - T a k e s a k i theorem i n its i t s full full s t a t e m e n t of o f the t h e Tomita-Takesaki t h e o r e m in
'lsrxa suollelcedxe l?uolllpuoc lerulou qJlqa ur suollenlls esoql serJrluepr qJIq,t\ rueroeql s.lI€se{3I pu€ suollelcoclxa I€uolllpuoc (q) pu€ 'l{€sa{€J pue uesreped Jo rueroeql ur,{po>11p-uopeg e^lt?lnuuocuou eql (e) Jo uolssncsrp € qll,r\ spua ralduqc eql 'tqE1a,yrB qll/tr pelelcosse dnorE rslnpou oql go (uo11cnrlsuoJ SNC eql ol lgadd€ lou seop 1eq1) uollusrrolJer€qc crsulrlur ue sealE qclq,n 'uollypuoc f .repunoq Sn) aql pellec 'uorrelrrJ IEcruqcel InJ3sn l(Jr^ € ol sulBUad uollcos lxeu eql 'uolloes srql ul ecuereadde uE so{€ru osle [111erouaE
generality also makes an appearance in this section. The next section pertains to a very useful technical criterion, called the KMS boundary condition, which gives an intrinsic characterisation (that does not appeal to the GNS construction) of the modular group associated with a weight. The chapter ends with a discussion of (a) the noncommutative Radon-Nikodym theorem of Pedersen and Takesaki, and (b) conditional expectations and Takesaki's theorem which identifies those situations in which normal conditional expectations exist. uorlsr3clul
aapslnuuocuoN'I-Z
2.1. Noncommutative Integration
' - 6 3 ( ' W \ f r g 1 ' f 1 l u e l e , r r n b e ' r o ' I 4 l u r .x 0 qcns Jo uollcelloc oql lle roJ 0 ( @*xh JI e^lllsod eq ol pr€s sl .jrtluo 0 leuorlcunJ r€aull v '3q plnogs foql asrnoc Jo s€'luel€,rrnbo are g 4 x pue-y,7 t n'eJu€lsur rog'snq1+n, * - d u a q , t rf l a s l c a r dr ( > r f q e r e q m q/,{ '.;ry .rolcol Iuer er{l uo Japro u€ ecuds 3o slueuala 1u1o[pe-Jlos Jo ''n potouap oq III,n.;4r ur srolerodo saulJep -yg euoc eaglgsod aqa fq 'eceds o,rll1sod Jo uollcelloc eql UaqllH alqerudes e uo srolerodo Jo erqo8le uu€runoN uol € olouop s[e,rn1e ilr.^ n loqurfs arlJ
The symbol M will always denote a von Neumann algebra of operators on a separable Hilbert space. The collection of positive operators in M will be denoted by M+. The positive cone M+ defines an order on the real vector space M h of self-adjoint elements of M, whereby x , y precisely when y - x E M+. Thus, for instance, x E M and x ~ 0 are equivalent, as of course they should be. linear functional 4> on M is said to be positive if 4>(x*x) ~ 0 for all x in M, or, equivalently, if 4>(M+) f IR+. The collection of such 4> will be denoted by Mt. An element 4> of Mt is called a state if it is normalized so that 4>( l) = 1.
(a) The equation [x,y] = 4>(y*x) defines a sesquilinear positive semi-definite form on M. (b) 4> satisfies the Cauchy-Schwarz inequality: 76(t*ths11(r*x)0
> l(x*,q01
l4>(y*x)1 , 4>(x*x)1/24>(y*y)1/2. '
= ;1r;1 s! 0 (c) ourt;'dHlL; x*x , IIx 11 2 1.)
=
osn pue (q) ul I = t lnd :1urg) '(t)0
(c) 4> is bounded and 114>11 = 4>(1). (Hint: put y
in (b) and use
x*x
('acuelsulroJ '[I ,rry] u1 punoJ oq feu syqt E 'f1r1uapleqf l€ rurou sll sulell€ 3o goord e lo,rrlrsod{lluclleurolne sr qJIq^\ 'n uo uollcunJ r€eull pepunoqfu€ luql 'fyesranuoc'anJl sI lI)
(It is true, conversely, that any bounded linear function on M, which
attains its norm at the identity, is automatically positive; a proof of this may be found in [Arv 1], for instance.) 0 eq ot pr€s sl /,t/uo 0 leuollcunJ r€eurl errllsod v
'Z'I'Z uolllulJeq
Definition 2.1.2. A positive linear functional 4> on M is said to be
.n ut D f IIB roJ (*rx)Q = (x*x)p JI l€rcerl (lll) :(tt'l'O 'dor4 'gc)-n u\ {tx) 1euEulseorcuJ ouolouou s Jo runuerdns aql sr x rolauaq^\ '(lx)O ldns = (r)0 .ll l€rurou (ll) i0 < (x)0 serldrur-n t x I 0 Jl InJqllBJ (l) (i)
faithful if 0 'I- x E M+ implies 4>(x) > 0; normal if 4>(x) = sUPi 4>(x), whenever x is the supremum of a monotone increasing net {Xi} in M+ (cL Prop. 0.4.11); (iii) tracial if 4>(x*x) = 4>(xx*) for all x in M. 0 (ii)
heory The Tomita-Takesaki Theory he T omita-Takesaki T 22.. T
3388
Exercises Exerciscs (2.1.3) If If 04J eE M|, ~, show that that the ffollowing quivalent: o l l o w i n g cconditions r e eequivalent: (2.1-3) o n d i t i o n s aare ((i) i) (ii) (ii) (iii) (iii)
racial; f4J iiss ttracial; 4J(xy) = O(yx) 4J(yx) for for all x,y x,y in in M: M; Q(.xy) 4J(uxu*) = 0(x) 4J(x) for for all all x eE M and unitary unitary uU eE M. Q@xu*)
r o v e ((iii) iii) + i i ) bby ii) a Use 9 ((ii) and 9 ((ii) o l a r i z a t i o n ffor i) + y ((Hint: Hint: U n d pprove o r ((i) s e ppolarization (iii) as as Q(ux) 4J(ux) = b?u) 4J(xu) and using the fact that that the unitary unitary re-stating (iii) operators ass a vvector e c t o r sspace.) n M sspan pace.) pan M a o p e r a t o r s iin
(2.1.4) Let M = L-(X,T,p), L ClO(X,f,jL), and let 6 4J eE W. ~. (2-1.4) with m.p. in the notation of of Ex. (0.4.5).) (0.4.5).) with m,y,
(We (We have identified identified 0c/J
e a s u r eo The equation 4J(IE) defines additive measure onn ((a) a) T initely a dditive m (E) = 0 e f i n e s a ffinitely he e q u a t i o n vv(E) (lo) d which absolutely with i t h rrespect ((X,f), X,F), w e s p e c t tto o pjL.. h i c h iiss a b s o l u t e l y ccontinuous ontinuous w ( a s above) in (b) 4J is normal a 9 d i t i v e , in (b) Q i s countably c o u n t a b l y additive, i f vv (as a b o v e ) is is n o r m a l if i f and a n d . only o n l y if which case 4J(f) = f lc fg dpfor d jL for some non-negative g Ee Lr(X,v); L 1(X,jL); some non-negative which case 0(,f) ith continuous w (c) 4J is jL is with (c) 0 if p i s absolutely i f and a b s o l u t e l y continuous f a i t h f u l if i n d only o n l y if i s faithful respect to v. v. 0D respect
J
g e n e r a l fact: fact: a ( b ) is m o r e general Exercise ( 2 . 1 . 4 ) (b) i s a special c a s e of o f a more s p e c i a l case E x e r c i s e (2.1.4) i f it i t is is positive p o s i t i v e linear n o r m a l if i f and a n d only o n l y if i s normal f u n c t i o n a l on o n M is l i n e a r functional '- one one may a-weakly We shall omit a proof of this fact _. o-weakly continuous. continuous. We - - but i n the w i l l freely i t in t h e sequel. sequel. be f r e e l y use u s e it f o r instance i n s t a n c e -b u t will i n [Dix], f o u n d in b e found [ D i x ] , for p o s i t i v e linear w i l l be b e denoted denoted The l i n e a r functionals f u n c t i o n a l s will n o r m a l positive T h e collection c o l l e c t i o n of o f normal M*.* M* and are dual and M* by M* +. 1" are establish that M It is not too hard to establish *. It t l l i.e., f o r all a l l 4J0 i f 4J(x) cones M , if i f and a n d only o n l y If M , then i . e . , if i f x Ee M, t h e n x Ee M+ c o n e s :. 0 ( x ) ~> ''u0 for ( w i t h the r o l e s of and x t h e roles o f 4J in t h e dual d u a l statement s t a t e m e n t(with i n M* M * +, a n d similarly, s i m i l a r l y , the 0 and - , and ( A s above, w e shall t h i n k of o f the the interchanged) v a l i d . (As a b o v e , we s h a l l think i s also a l s o valid. i n t e r c l i a n g e d ) is elements of M* as linear functionals on M.) M.) as elements
Exercises Exercises = {x1", the the von Neumann (2.1.5) Let 4J0 Ee M*,+' Mo (2.1.5) and let M M*.*, let xx Ee M and o = {x}", algebra g e n e r a t e dby l. a n d 1. b y x and a l g e b r a generated measure (a) ( E ) ;=0 (4J(IE(x» l B ( x ) ) defines d e f i n e s aa measure ( a ) If t h e equation e q u a t i o n v *(E) I f x is i s normal, n o r m a l , the ( / ( x ) ) for f o r every every d vx* == 4J(f(x» on t h a t J fI dv o f x such s u c h that t h e spectrum s p e c t r u m of o n the bounded s p x. x. f u n c t i o n f/ defined d e f i n e d on o n sp B o r e l function b o u n d e d Borel w i t h the t h e notation notation (b) ( b ) For g e n e r a l x, i f 4J0 is i s tracial, t h e n , with t r a c i a l , then, s h o w that t h a t if F o r general x , show (a), vvl*l == vlx*1 tl**l . of (a), of . 0tr 1xl
J
verification 4 . 3that t h e verification The t h a t the l a t e r , in i n Section S e c t i o n4.3 w i l l notice n o t i c e much m u c h later, r e a d e r will T h e reader by w i l l invariably a c c o m p l i s h e dby that g i v e n finite i n v a r i a b l y be b e accomplished i s finite f i n i t e will f i n i t e factor f a c t o r is t h a t aa given p o s i t i v e linear f u n c t i o n a l Tr l i n e a r functional first n o r m a l tracial t r a c i a l positive f a i t h f u l normal f i r s t constructing c o n s t r u c t i n gaa faithful y i e l d s aa t o P(M) on r e s t r i c t i o n of o f Tr to P ( M ) yields t h e restriction t h a t the o b s e r v i n g that M , and a n d then t h e n observing o n M, v a l u e l . f i n i t e t o dimension function for M which assigns a finite value to 1. w h i c h M a s s i g n s a f o r f u n c t i o n dimension
'plqtlo/
, aq - ,n
E
M.,+ be faithful.
Then there exists a triple
a1du1 o stslxa apql uaqJ
Theorem 2.21. Let 4> (lf4>,n4>'04» where
) Q ta7
atatlu 1QV'QY'Q$1 -IZZ uerorql
'erqaEle uu€runoN uol 3 uo leuorlcunJ r€eull errllysod Ieurou InJqlleJ € Jo esec eq1 ,{1uo JeprsuoJ lleqs e^\ pu€ speau rno ol luelelerrr sr flrlerauaE qcng 'qEnorql oE ol uollcnJlsuoc erll roJ elenbepe Euraq 'ssay uala ro erqetle-*3 u lsnf .erqe8le uueuneN uo^ B paeu ue^a lou saop auo 'lcBJ ur lleurrou eq ua^a lou paau tl ilnJqtleJ eq lou paeu l€uollcunJ e^lllsod oql -- f111e-rauaE .uoJlcnJlsuoc (SNC relearE qcnur uI plle^ sr uollculsuoc slttl ol peler^arqq€ qpoJecuaq) leEag-{rsurr€N-pueJleg pa}€rqaloc aq} Jo lueluoc aql sI '1,y(uer.1eqe.{lrressocau1ou) lereuaE B roJ lno perrrec .sroluredo uollecrldrllnu aq f eu srsf1eue rBlrrurs frea B lerll s€ n Jo uollulu3seJdeJ pelurcosse eql pue (a,!,X)27 eceds lraqllH eql lcnrlsuoc flalerparurur u€c auo 'n B r,lJns uo^rg .IG-y.Z) 'xg 'JJl t ol '^lrnulluoc alnlosqB .luelB^Inbe lBnlnu Jo asues aql ul sl qJlqa n ornscarrr elrurJ c EuJsooqc ol slunorus n ao l€uollounJ r€aull enlllsod IBrurou InJrlll€J e Eursooqc '(tt'!'X)-.1 = l{ uerIA\
When M = L ""(X,r,p.), choosing a faithful normal positive linear functional on M amounts to choosing a finite measure v which is equivalent, in the sense of mutual absolute continuity, to p. [cf. Ex. (2.1.4)]. Given such a v, one can immediately construct the Hilbert space L 2(X,r,v) and the associated representation of M as multiplication operators. That a very similar analysis may be carried out for a general (not necessarily abelian) M, is the content of the celebrated Gelfand-Naimark-Segal (henceforth abbreviated to GNS) construction. This construction is valid in much greater generality -- the positive functional need not be faithful; it need not even be normal; in fact, one does not even need a von Neumann algebra, just a C*-algebra or even less, being adequate for the construction to go through. Such generality is irrelevant to our needs and we shall consider only the case of a faithful normal positive linear functional on a von Neumann algebra. 2.2 The GNS Construction uoprnrfsuoJ
sNc
crIJ
lrz
'It'Ol - (W)d :Lu rsarns€eru,, e^lllppB f lqelunoc {11,n uuql ror{lBr 'sleuorlcung r€eull o^lllsod (lerurou) rlll^r {Jo^\ qUoJocuoq II€qs a^\ .spJo^\ rarllo ur iarnseau oqr qll^\ u?rll JorllBJ lerEelur aql qllrrr {Jo^\ o} osoogc 11uqse,n ,og 'e = ll lurp uaq^\ luesard ore uralqotd eql Jo sslJ€clJlur er{l II€ 'oroq uana -- uoseelg ol enp ueJoeql (1er,rrr1-uouflalrur3ep pue) pelerqoleJ € ^q pallles flanrleurrJJ€ sl .(nh = n dse) lercads fre,r eq1 u1 '1,yuo l€uollounJ rBaurI aaltlsod I€rurou e o1 (n serrleruosr IEIIr€cl roJ Qnn)w = (n*n)u turi(gsrles flrrussecou lou) [I.0] U4I)d :u uollcunJ 3^lllppe flqelunoc leraua8 e Eurpuelxa 3o urelqord oql (q) pue 1,,11srolerado 3o sEurg,, uI lel€l teo6, e fluo pa111espue ',,sro18redogo s8ulr uO. ul I I urelqord sB pelels sr slql lBrll lc€J aql ,(q pacuopr,re s€ 'uu?runoN uo^ uo^e paxoJ flrrerodural lseel le a^€q ol suees rolc€J olrurJrruos € roJ ualqord Eurpuodserroc eql (€) 'uelqord slql eql qsrlqelso ol acrJJns plnoqs Jo .,{.1r1erarr1-uou suoll€clpul on1 Eur,nolloJ aql 'lueredsuerl -lo elslparurur sueoru ou ,,(q sr uollnlos aql 're,realo11 'e^oq€ pelsaESnsrauu€tu eql ul peulJep oq lsnu r Eugllnsor eql pue 're/rrsue e^rlerurrJJe u€ e^€q peopul seop qcler8ered eq1 Jo lrels eql le pelsa8Ens ruelqord eq1 .n uo IsuollcunJ r?aull I€rurou o^llrsod e ol spuelxa pue ,q;4ruo leu-ollcunJ r e e u l l I B a r e l a u l J e p r l € q l q s r l q e l s oo l _ l d u a l l e p u e ( 1 ) * n p 1 = ( x ) r J '((r)st)O = (g)*n fq r ds uo *n ernseau eqt eur3dp pynoc les uo_ql ouo 'q.;tg, ,( JI leql lsaE8ns sosrcrexe Eulpocard aqa .1e!c"r1 p,re 'slsrxa 'rtclq^\ 'n uo ll JI InJr{ll€J eq f11ec1}eurolneplnoa luuorlounJ r€aull alllysod I€rurou e ol spualxe rolc€J elruU B Jo uorlcunJ uolsuerulp aql reqleq^i ol sB urelqo.rd asraauoc eql /hou raprsuoJ
Consider now the converse problem as to whether the dimension function of a finite factor extends to a normal positive linear functional on M, which, if it exists, would automatically be faithful and tracial. The preceding exercises suggest that if x E M h , one could define the measure V x on sp x by v/E) = D(I E(x», then set T(X) = ).dvx ().) and attempt to establish that T defines a real linear functional on M h , and extends to a positive normal linear functional on M. The problem suggested at the start of the paragraph does indeed have an affirmative answer, and the resulting T must be defined in the manner suggested above. However, the solution is by no means immediate or transparent. The following two indications should suffice to establish the non-triviality of this problem. (a) The corresponding problem for a semifinite factor seems to have at least temporarily foxed even von Neumann, as evidenced by the fact that this is stated as Problem 11 in "On rings of operators", and settled only a year later in "Rings of operators II"; and (b) the problem of extending a general countably additive function m: P(M) ... [0,1] (not necessarily satisfying m(u*u) = m(uu*) for partial isometries u) to a normal positive linear functional on M, in the very special case M = I(lf), is affirmatively settled by a celebrated (and definitely non-trivial) theorem due to Gleason -- even here, all the intricacies of the problem are present when dim If = 3. So, we shall choose to work with the integral rather than with the measure; in other words, we shall henceforth work with (normal) positive linear functionals, rather than with countably additive "measures" m: P(M)'" [0,1].
J
39
2.2. The GNS Construction
uollrnrlsuoSsNc eql'z'7,
6E
40 40
2.. 2
The Tomita-Takesaki Theory heory T he T omita-Takesaki T
11 0 is a t-algebra *-algebra homomorphism homomorphism of of M (a) n6 into f,QtQ, Minto :f(lf4»' tt6 lf4> being a Hilbert space; space; Hilbert (b) oO 04> eE !t5 lf4> and fr6 lf4> = = ,6(Wa6 114>(M)04>. :; and (c) 0(x) 4>(x) = = <26(x)op'o6'for (x)04>,04» for all all x in M.
',o') is another Such a triple is unique unique in the the sense sense that that if if (lft,tt (If',l1',O') another such such Such !l' triple, there exists a unique unitary operator w: tl1 lf4> ... If' such such that that w04> wa6 = = operator w: triple, there exists unique 0' Qt and 11' (x) = = Wl14>(x)w* for all x in M. wn5(x)w* for n'(x) Further, the the imaie image 114>(M) Neumann algebra of of operators operators on on n5(M) is a von von Neumann lf norm-preserving as as well as as being a o-weak a-weak homeomorphism homeomorphism of of tt6i n6 is norm-preserving 0; 114> onto 114>(M). M onio n6(M).
If x,y € E M, define [x,y] [x,y] = Q(y*x). 4>(y*x). The positivity positivity and Proof. If positive-definite) e n u i n e ((positive-definite) faithfulness of 4> ensure that genuine f a i t h f u l n e s s o f 0 e n s u r e t h a t [.,.] [ . , . ] iiss a g r o d u c t oon n M nner product M.. Let denote off M i nin tthis h i s iinner ompletion o L e t l tlf4> 5 d e n o t e tthe h e ccompletion iinner nner p O d O. (l) = product. Let n:: M ...- elf4> denote map, and n(l) = o 04>' p roduct. L ap, a et n nclusion m n d llet et n e n o t e tthe h e iinclusion If x,y e E M, note note that' that If
= o(,y*r**nl y*y)= llrll' llntylll', < ll'll'o( lln(xy)ll' 2 2 y*x*xy ,< y*(llxl1) y*(ll"ll)'r)y. So, where l, and l)y. So, wherewe we have haveused and so so y*x*xy usedx*x x*x ,< Ilxl1 llrll'r, 1f6 z6(x) such there exists a unique bounded operator l14>(x) on lf4> such that bounded operator exists v e r i f i e d that z 6 is is a t h a t 114> l14>(x)n(y) n ( x y ) for f o r all i n M. M . It I t is i s easily e a s i l y verified n 6 k ) n ( i l = n(xy) a l l y in *-'algebra homomorphism t(lt5). As a sample, show that *-algebra M into :f(lf4>.)' sample, we show homomorphism of Minto Xf6, and preserves adjoints. n(M) is dense dense in lf4>' 114> note that n(M) n6 preserves adjoints. For this, note that for all y,z in M, if x Ee M, then for tliat if
(x)n(y),n(z»
( b ) and ( c ) of o f the the T h e assertions a n d (c) conclude z 6 ( x * ) = l14>(x)*. z 6 ( x ) * . The a s s e r t i o n s (b) t h a t l14>(x*) c o n c l u d e that o 6 and inner t h e inner theorem o f 04> a n d the f o l l o w immediately i i n m e d i a t e l y from f r o m the t h e definition d e f i n i t i o n of t h e o r e m follow 1f6. product in lf~. i n M, M, ( l f r , n t , o r ) is f o r x and a n d y in If n o t e that t h a t for i s another s u c h triple, t r i p l e , note I f (If',l1',O) a n o t h e r such < t l | ( x ) Q r , z t ( y ) O ' >=
= 4>(y* = x) Q(y*x) ;i == (x)04>,l14>(y)04»
=^tt', =.116 the existence existenceof aa i1@N deducethe since M) 04> = lf4> and (M) 0' = If', deduce i611tr;116 and 11' since114>.( ' + lf n ! 1 6 6 , that (w weerlrl - odeelTr ni nee( ld, ,) 'unitary w n . : such (well-defined) operator w: lf4> ... If' such that Wl14>(x)04> == o p p s e r r a a t t u o r r u unri t adrryy .wnq(x)o6 = l1'(x) w , since t h e two iwo n ' ( x ) 0o w, s i n c ethe l1'(x)O'. Q l14>(x) n t .56(\ x. \)) = th n a t w 0o rI t is ri ss fairly If a ri rrlryy clear sc rl egaarr that [nI'l(' x( x) O . r rf r '.. It qense mapping o p e r a t o r s mapping ,ld(1vllrrd n6(M)a , both b o t h operators operators sEr set 114>(M)00' tne dense dense set agreg on the operators agree = or, = 11'(1)0' O r , and the a n d the AO= w nt(l)Or = w.'n6(l)O l14>(y)04> 114>(1)0.4>O = also,w ,. 04> n60)A6to r'(xy)Or; also, to l1'(xy)O'; proved. second part of the is proved. theorem is the theorem se'cond-part
'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) pue ,(1r1uurou lEqt IIBcar lualenrnba eru f lrnulluoc ' 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 u Jo Joorcl 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
We shall complete the proof by showing that if n: M .... :e(JfI) is an injective normal *-homomorphism of Minto :e(Jf'), then n is isometric, n(M) is a-weakly closed and n is a a-weak homeomorphism of M onto n(M). To prove n is isometric it suffices thanks to the C*-identity to verify that Irn(x) II = IIx Ir when x = x* E M. Fix such an x; since the norm of a self-adjoint operator is equal to its spectral radius, it is more than sufficient to show that sp n(x) = sp x. For this, if >. 1 sp x, note that (x - >.r l E M, by the double commutant theorem; hence n(x) - >. is invertible, with inverse n«x - >.r l ). Thus, sp n(x) f sp x. Suppose this inclusion is strict; then there exists a continuous real function f on sp x, which vanishes on sp n(x) but not everywhere in sp x. Then f(n(x» = 0, while f(x) ~ O. This contradicts injectivity of n since n(f(x» = j(n(x». (The proof of this last fact requires only polynomial approximation and the fact that for continuous j, IIj(n(x»11 = sup{lf(>.)I: >. E sp n(x)}.) Next, note that if !/J E :e(:K").,+ (viewed as a a-weakly continuous linear functional on :e(:K"», then !/J 0 n is a normal linear functional (since both !/J and n are normal). However, for linear functionals, recall that normality and a-weak continuity are equivalent (cf. remarks following Ex. (2.1.4». So, if a net (x) in M converges a-weakly to x, then !/J(n(x) .... !/J(n(x». Since!/J was arbitrary, n(x) ... n(x) a-weaklY', in other words, n is a-weakly continuous. Since ball M = {x E M: Ix II ~ I} is a-weakly compact and since n is isometric, infer that ball n(M) = n(ball M) is a-weakly compact. The Eberlein-Schmulyan theorem (cr., for instance, [Yos)) states that a linear subspace of a (Banach-) dual space is weak*-closed if and only if its unit ball is weak*-closed. Conclude that n(M) is a-weakly closed and consequently a von Neumann algebra. (It is a fact that a self-adjoint algebra is weakly closed if and only if it is a-weakly closed; one proof of this uses Kaplansky's density theorem, and we do not go into that here.) Since n-l: n(M) ... M is an injective normal *-homomorphism, it follows from the above discussion that n- I is also a-weakly continuous, and the proof is (finally!) complete. 0
'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
nrm-
for all y in M; since the net {Xj} is uniformly bounded, so is the net (n(x)}; as = Jf¢, conclude that n¢(x j) ? n¢(x), as asserted.)
1ea,n'(x)u'ji";; ucql'x o1 d11ea,r,soSra,ruoc rlolrt/rr,*ttl ul leu Burseercureuolouoru e sl^{!r}JI ler{l esuasaql ut 0u,Oeu,agl .raqlrng .o^rlcaful ;euriou,sl sl vz snqr ig = r lB{l 'O = ,ll9u(x)Pu;;= (x*x)Q^uorlenba etll pue 0 Jo ssaulnJr{lreJ or{l ruorJ aphlcuocu5r{l .0 = (r)Pa .;r .,{11eurg Finally, if n¢(x) = 0, then conclude from the faithfulness of ¢ and the equation ¢(x*x) = Iln¢(x)n¢112 = 0, that x = 0; thus n¢. is injective. Further, the map n42 is normal in the sense that if {xi} is a monotone increasing net in M , which converges weakly to x, then T n(x j) ... n(x} weakly. (Reason: If x j ? x, then y*xjY ? y*xy for all Y in M; since ¢ is normal, this means that
tv
'z'z
2.2. The GNS Construction
41
uollsnrlsuoJsN9 eql
42
The Tomita-Takesaki Theory heorv he T omita-Takesaki T 22.. T
A'\
e n e r a l l y tthan o i n t oout as may more was han w e r e tthat, o r e ggenerally u t hhere hat, m IItt m a y bbee rrelevant e l e v a n t tto o ppoint r o o f , iitt iiss ttrue njective h a t aan n iinjective r u e tthat n tthe h e aabove b o v e pproof, eestablished s t a b l i s h e d iin **-homomorphism - h o m o m o r p h i s m bbetween p o s s i b l y aabstract) C*-algebras bstract) C * - a l g e b r a s iiss w o ((possibly e t w e e n ttwo a s s i n g tthrough may h r o u g h aan n r o m tthis h i s ---- bby y ppassing a y bbee iinferred n f e r r e d ffrom iisometric; s o m e t r i c ; iitt m - h o m o m o r p h i c iimage u o t i e n t aalgebra m a g e ooff a l g e b r a ---- tthat h a t a **-homomorphic aappropriate p p r o p r i a t e qquotient C*-algebra The may eader m a y cconsult o n s u l t [[Arv h e iinterested n t e r e s t e d rreader orm-closed. T C * - a l g e b r a iiss nnorm-closed. A r v l1]] details. ffor or d etails. - h o m o m o r p h i s m bbeing Wee sshall, off a **-homomorphism eing r e e tto alk o e n c e f o r t h , ffeel e e l ffree o ttalk W h a l l , hhenceforth, r e s e r v e sm meaning monotone and o uuse s e tthe he i m i t s ---- a n d tto o n o t o n e llimits e a n i n g tthat h a t iitt ppreserves nnormal o r m a l ---- m roof o emerging off tthe Theorem, hat a h e o r e m , tthat he T r o m tthe h e pproof m e r g i n g ffrom ffact, act, e **-homomorphism - h o m o m o r p h i s m iiss o ormal. a-weakly and only f iitt iiss nnormal. f a n l y iif nd o - w e a k l y ccontinuous o n t i n u o u s iif It as in the C*-case, C*-case, that the image n(I() n(M) of of a von Neumann It is true, as algebra under a normal *-homomorphism *-homomorphism is o-weakly a-weakly closed closed and algebra roof o hence Neumann algebra. The proof off tthis assertion his a s s e r t i o n iiss he p on N eumann a lgebra. T h e n c e a vvon outlined exercise. n tthe ollowing e xercise. h e ffollowing o u t l i n e d iin
Exercises Excrcises (2.2.2) be a a-weakly (2.2-2) Let fI be o-weakly closed closed two sided sided ideal in M. Show that the (a) (a) Let x Ee M have ulxl. Show have polar decomposition decomposition x = ulxl. (iii) ( i i ) Ixl ( i ) x Ee f; 1 ; (iii) I ; (ii) following conditions are e q u i v a l e n t : (i) f o l l o w i n g c o n d i t i o n s a r e equivalent: l x l Ee f; <+ / 1l1n (0 ClO)(lxD E f. Conclude from (i) # (ii) that f is self-adjoint. e 1. Conclude -1(lxl) = x 1l 1(0oClO)(IxD ( i ) #e (ii); (ii); x = . - 1 ( l x la ) n d so so (iii) ) ( f l i n i : Ixl (Hint: u * x and a n d so s o (i) l x l = u* ' a i r ' a p p rand o p r i a t e(iii) = Borel p i c k ( i i ) + ( i i i ) , f o r an appropriate Borel ( i ) ; for (i); (iii), pick Yr nn = In(\xD, f o r (ii) , f , , ( l x l ) ,for = note l{r7n,-1(lxl), function In' such function such that IxlYn = l(l/n,ClO)(lxD, and note that lxlyn /,,
*
*
o-weaklv') 1 1 6 , - y ( l x l )a-weakly.) 1 1 r 7 " , - y ( l x..l ) ' 1(o,ClO)(IxD 1(l/n,ClO)(lxD 1 / n .. - 1l1n-1(x) (b) (0 ClOl(x) a-strongly. o-strongly. (b) If e M-, then then xxr/n If x EM, (c) a n d so so e V ( c ) If ? Q O () l i m ( e + l)l/n n f, o - s t r o n g lim(e / , then t h e n ' e' i ' VV I1 == a-strong l f e,j e , f Ee P(M) . / ) r / " and f.f Ee f;I: in particular, P(M) P(lrj () n f1 is is an an upward directed net (which is (0)). if fI ~t (O}). non-trivial if is non-trivial (Hint:e (d) d eEl ,f,i nin ( d ) If P ( M ) I: f Ee If}, ( M ) . (Hint: l , then t h e ned eE fI o()Z Z(M). If e e ==VV{f { f Ee P(M): ( c ) and /; o - w e a k closure c l o s u r e of o f f; view t h e a-weak i n (c) a n d the v i e w of s t a t e m e n tin o f the t h e second s e c o n dstatement P(ttO n fl and is aa to (M) () and ua is if If Ee P is central, note that if central, note to see see that that e is P(M) n f, 1, and conclude that uni tary element and conclude M, then ufu* Ee P then ulu* element of M, unitary 114 ()
e
ueu* udu*
e.)
== d.)
(Z(U), then Me P (Z(M», then Me if e Ee P (e) (e) Show conversely, if M€, and and that conversely, Show that fI == Me, ( H i n t : T h e s e c o nd is a a-weakly closed two-sided ideal of M. (Hint: The second M . i d e a l o f o w e a k l y c l o s e d t w o s i d e d is a e I, i f t r i v i a l ; assertion Me are trivial; if x E f, M e a r e i n c l u s i o n fI :) w e l l as t h e inclusion a s well a s the a s s e r t i o n as 2 h e n c e x == a n d hence ( a ) and l 1 s , * ; ( l x l )II( Z and t~n o f d , ~0,4IXI) t h e definition d e f i n i t i o n of a n d the t h e n by b y (a)
e,
e
xe.) xa.) *-homomorphism, is n(lv[) is (f) homomorphism, then then n(M) t(lt')I) is (f) If is aa normal normal *n: M M ..- :e(:If If n: t(t8r). of :e(:If o-weakly '). von Neumann subalgebra of Neumann subalgebra hence aa von and hence o-weakly closed closed and may be be (Hint: M(\ -- e) (e) to note that that M(l (Hint: Apply write ker ker n n == Me, M€, note Apply (e) to write O may \sr ~, L, ~~\. \\l\ ~"I\.\\.-~ srr "'-1:.'"l\.I:.~I:.~ ~\.'61:.~'-~ lsrr\!\\\\\ Ngs\r'l, ~o;:.:\.\.~'6 trtt"'.g '\)~ \'rg\r g\ ~'" ts'a~ "l,\)~""",,1:.'\).Th.~~~ \\\S\
snql pue '@)Qu qll,r\ l{ l(gr1uap1II€qs e^\ 'n uo O luuollJunJ rs?uJl anlllsocl I€rurou InJqlrBJ u uarrgEore ?^\ ueq,n '1ou ueql uelJo oroIAI
More often than not, when we are given a faithful normal positive linear functional ~ on M, we shall identify M with Tl~(M), and thus
'a1du1 S N C e q l J o u o r s r e ^ e u o .s p y a t { s l q l l E I l l a l E r p e I U I u r O s l l I ' t o x J o . 9 t t o t u o l l c r r l s e r e q l e q ( x ) O u t e t p u e 1 ;@$ 3 l ( n , x :9u(t o x))l = 9g let pue (q) ul s€ eq 0U let 'esec srql u1 'snonJEA lou sl I'Z'Z rueroeql feqt os'al€ls l€urrou InJqlr€J € sllrupe oJBCls UaqIIH alqeredos e uo Eurlce erqa8le uu?uneN uo,r {rela l?ql s^\oqs slql 'n ) x rc1 xd rr=(x)0 ret.(q)ur s€sJd pue(A)f j n il(c). "lr7ln"3 = 9u 'I @x = (x)Ps puB'l o @ t+ = 94 EulDos [q peurelqo sl arnlcrd crqdroruost uff' (i>tcaqC)
An isomorphic picture is obtained by setting X~ = X @ X, n~ = Lna~/2~n @ ~n and Tl~(x) = x @ 1. (c) If M f I(X) and p is as in (b), let ~x) = tr px for x E M. This shows that every von Neumann algebra acting on a separable Hilbert space admits a faithful normal state, so that Theorem 2.2.1 is not vacuous. In this case, let nIP be as in (b) and let XIP = [{(x @ l)n!p: x E M)] f X @ X and let Tl~(x} be the restriction to X!p of x @ 1. It 1S immediate that this yields one version of the GNS tnple. 0 'll
(Check!)
'("ur)e = lurr6)(x)0u'("lzlh) ti" = tu n~
=
n~1 (~/2~n)' Tl~(x)($
I1 n) =
$
(Xl1 n )·
co
ug qtlrn 'u II€ JoJ I = T=U .tfi -e = QU
with Xn = X for all n, X~ =
n=1 $
Xn '
co @
,{q uerrE sr eldul SNg eql Jo uolsro^ B leql .(grra,t ol preq lou sl lI
It is not hard to verify that a version of the GNS triple is given by n=1
t3" . un = ,"'
~x) =
L an<x~n'~n>' co
'1rr1u1 x f,ue r o J s r s € q I s r u r o u o q u o u B ( " 1 ) p u e o > to3 'g < un qllr'r) n
The unitary operator, whose existence is guaranteed by Theorem 2.2.1, is defined by wi = l(dv/dJL)1/2 for I in L 2(X,f,v). (b) Let M = I(X) and let p be an injective positive trace-class operator on 'Jf. Then ~(x) = tr px defines a faithful normal positive linear functional on M. Note that if
[~~
| = , u ' 3 1 = 8 A ) . u ' ( t l ' a ' x ) c=7r g ( l l ) =
r/
|*
n'
'
(ii) X' = L 2(X,f,JL), Tl'(f)g = Ig,
71{ aP ) p u € 1 1= 0 U ' A Y = A g l Q y ' ( a ' g ' 1 g ) r 7 = Q y g ( l )
(i)
X~
Tl~(f)g
= L 2(X,f,v),
2
n~ == 1; and
= Ig,
Example 2.2.3. (a) Let M = L""(X,f,JL); let v be a finite measure with the same null-sets as JJ.; then the equation ~f) Idv defines a faithful normal positive linear functional ~ on . Two possible GNS triples are given thus:
:snql ue,ryEare seldyrl g51g Olqrssod o/AI 'II uo 0 l€uorlcunJ rBeurl aaglgsod I€rurou InJqlr€J = AW uoll€nbe egl ueql irl sB sles-llnu erues aql e saurgep np/ J qll/r\ arns8aru airurJ B aq n ta1 i(rl'|'y)-7 = hl le1 (s) '6'97 alduerg
=J
In view of the uniqueness assertion in Theorem 2.2.1, we shall talk in the sequel, of the GNS triple (X~,Tl~n~) associated with a faithful normal positive linear functional~. Note incidentally that n~ is a unit vector if and only if ~ is a state.
n s,0ureqr,{rrerueprrur :L'"XI"j^ii,J#'ilJ:: aroN.'.fT";fij'i!
elctrrl g5ig arq Jo'lenbesoql ul InJqIIBJ€ qlIA pelslcosse1Q64y'Qy1 )1€l II€qs et 'I'Z'Z ruoroeql ur uollressessauanblunagl Jo ?nerl uI (.uorUass€ aql n 'a,r1lca[u1 s1 Jo esBJa,rllcaful paqsllq?tsefpeerle eql ol l€edde puu is injective, and appeal to the already established injective case of the assertion.) 0
'z'z
2.2. The GNS Construction
uollJnrlsuoc sNc eql
43
E'
44 44
2. The Tomita-Takesaki Tomita-Takesaki Theory Theory 2.
assume that that lil Mg&; t(D l(lf) and that that 0(x) 4>(x) = <xq(> <xn,!l> for for a vector o0 in in lfIf such assume that [Mo] [MO] = = lf. If. The vector o0 is known known to mathematical,,Fhysicists mathematical physicists as as that i n ccase The ). T he a c u u m sstate e c t o r oorr tthe t a t e ((in a s e l11011 a c u u m vvector h e vvacuum tthe h e vvacuum l 0 l l = l1). faithfulness of of 04l translates translates to this separating separating property property of of fI n: ifif xx eE faithfulness = 0 ifif and only only ifif xo xO == 0. O. Thus, in in the terminology terminology of of the M, then x = following definition, definition, the vector O 0 is cyclic cyclic and separating separating ffor or M. following Definition 2.2.4. 2.2.4. A A set S c&; ltIf is said to be Definition (i) (ii) (ii)
cyclic for M ifif [MS [M S ]] = 1? If ;; for M cyclic separating for if for in M, x == 0 ifif and only only ifif xS in for M if for x separating
= = {0}. {OJ. E 0
Exercises
(2.2.5) Let M g&; r(lt) l(lf) and let 0(x) 4>(x) = tr tr px, px, where p is a positive trace (2.2.5) class operator, operator, given by class dr, tBrr,Er, X cxn p == I: t~ ~ , P n' n
faithful as as a with cxn 4l is faithful with cr,, > 0 and gn} t[,,,) orthonormal. Show that 0 f o r M. M. linear i s separating i i t if i f and i f gn} s e p a r a t i n g for f u n c t i o n a l on a n d only o n l y if l i n e a r functional on M { t " } is g If, if S is (2.2.6) S &; (2.2.6) If for M if if and only if f, show is cyclic for If show that S is = {OJ; ) x'[MS] ( H i n t : x' separating M ' a and n d x'S xtlMS I = x r S = {OJ f o r M'. M ' . (Hint: x t Ee M' s e p a r a t i n gfor {0}; { 0 } :9 if is so, if S is cyclic for M, S is separating for M'. Conversely, if S is f M) . Conversely, f separating or if or so, - p')S ( l p ' ) S p t = P[MS M ' a n d t h a t separating 1 E M' and that (1 = n o t e that f o r M', M ' , note t h a t p' s e p a r a t i n gfor PluS l, = 1.) {OJ, p'' = l.) whence p { 0 ) , whence if and only if if (2.2.7) If 4>(x) == <xn,!l> <x4o> for x in M, M, then then 4l0 is tracial if Ilxoll = Ilx*oll for all x in 0n ir M. M.
ftiii'= iilsii'
Thus, functional 4l0 on M, faithful normal positive linear functional Thus, given aa faithful the GNS construction leads leads to aa realization of M as as aa von Neumann is aa cyclic 86, in which there algebra there is Hilbert space space lf4l' algebra of operators operators on aa Hilbert vector is and is automatically aa cyclic vector for f or n4l(M); n6(M)i this vector separating vector and separating and v e c t o r for f o r n4l(M)'. n6(M)'. s e p a r a t i n g vector a n d separating von c l a s s of o f von We w i t h an i m p o r t a n t class a n important t h i s section s e c t i o n with W e shall s h a l l conclude c o n c l u d e this and Neumann algebras with aa natural cyclic and come equipped equipped with algebras which come separating Neumann algebras algebras vector -- the so-called group-von the so-called separating vector Sroup-von Neumann groups. associated discrete groups. countable discrete associatedwith countable group, whose we shall shall Let G whose identity identity element element we discrete group, countable discrete G be be aa countable irreversibly ( t h e symbols denote h a v i n g already b e e n irreversibly a l r e a d y been a n d 1I having b y Ee (the s y m b o l s ee and d e n o t e by ! ' ( G ) denote p r o j e c t i o n sand identified L e t j2(G) denote i d e n t i t y operator). w i t h projections t h e identity o p e r a t o r ) . Let a n d the i d e n t i f i e d with the functions on on G: G: Hilbert space of square-summable square-summablefunctions spaceof the Hilbert
<-]. ! 2 ( G=) [ r ' o - o : I: IW)1 2
,r"li(r)12 where P(G), where There orthonormal basis G')of of j2(G), basisgt: There is is aa canonical canonical orthonormal {8,:tt Ee G}
Recall that a faithful normal tracial positive linear functional on a factor will, by restriction, yield a dimension function which assigns a finite value to 1. Hence, an infinite factor does not admit such a functional. However, every factor (operating on a separable Hilbert space) does admit several faithful normal states [cf. Example 2.2.3(c)]. Suppose, then, that ~ is a faithful normal state on a (not necessarily factorial) von Neumann algebra M. According to Ex. (2.2.7), ~ is tracial if and only if IIn~(x)n~1I = IIn~(x*)n~11 for all x in M. So, in order to study infinite factors, it might be instructive to examine (the lack of isometry of) the operator n~(x)n~ .... n~(x*)n~. The advisability of such an investigation is convincingly demonstrated by the celebrated Tomita-Takesaki theorem, which provides a powerful tool for the study of infinite factors (and
pue) srolceJ elrurJur 3o fpnls orll JoJ lool InJJa^\od e septrrord qclq^\ 'ruaroaql r{€se{BJ-BlruroI palerqoleJ oql ,(q pelerlsuoluep , t l E u r c u . r n u o c .s r u o g l e E l l s a n u l u e q c n s J o f l t l t q e s l a p e e q l '9u1*x;9u * 9u1x;9a rolerado eqt (3o frleruost Jo )toBI aql) oulruexa o l a ^ l l c n r l s u r a q l q t n u 1 1 ' s r o 1 c e 3a l l u l J u r , { p n l s o l r e p r o u l ' o S ' W 'Q'Z'() u r x I I B r o 3 ; 1 9 u ( * r ) Q u l= l 1 1 9 u 1 x 1 9 u3;r; i ( 1 u op u e J I I B I r B r l s \ q 'xE o l 6urpioccy 'yy' eqe?p uubunep uo,r (ler.lolce; flrressocou B uo el€ls lururou lnJr{lleJ e sl 0 teql 'ueql 'esoclclns lou)
'I(c)t'z'z
oldruexg 'Jcl salEls Ierurou InJrllr€J IBrelas llrup€ soop (aceds ' : a r r a m o 1 1' l e u o l l r u n J l r a q l l H a l q e r e d a sr u o E u r l e r a d o ) r o l c e g f r a r r a € qcns llurp€ lou soop rolJ€J ellulJul u? 'acueH 'l ol enlE^ ellulJ u suSrsseqJlq/rr uollcunJ uolsuorulp e plarf 'uo1lc1r1sar,{q 'lgn JolcBJ € uo I€uollcunJ r€aurl errlllsod I€lc€rl lerurou InJqll€J € leql ll€cau oq.;. 'eZ
2.3. The Tomita-Takesaki Theorem (For States) (sefefs rog) uerocql
rfssc{sl-Glluol
Tomita-Takesaki theorem, towards which landmark we shall now head.)
('peaq /t\ou IIBIIS e^\ {JErupu?l qCIq/t\ SpIB^\ol 'ruaroaql lIBseIeI-slIIuoJ eql Jo ecuanbasuocB s€ 'uoglcas lxeu eql ur eS.rauraIII/( slql i,,(g t l :rd1 = ,ou lBql assc aql 'lce3 ur 's1 11) 'palress€ sE "hl ro3 crlc{c s1
is cyclic for M', as asserted. (It is, in fact, the case that M' = (pt: t E G}"; this will emerge in the next section, as a consequence of the It is trivial to verify that each Pt commutes with each }.s' and hence {pt: t E G} f MI. Since ptn = ~ -1 for every t, it is clear that n t
'/,{rela rog r-1, = gld aculg ',Ni u leql realc sr 11 {g t l:rd1 3cueq pu€ "\ qc?o qlla salnuuoc ld qcea leql fgrran ol IBr^Irl sJ lI '@ G).
assertion established later, in Section 4.1.) Let n = ~E and observe that }.tn = ~t for each t, and, consequently, that n is cyclic for M. We shall verify that n is separating for M, by proving that n is cyclic for M ' [cf. Ex. (2.2.6)]. Analogous to the }.t'S, we may also construct the right-regular representation t ~ Pt, defined by (Pt~)(s) = ~(st) (or, equivalently, Pt~ = ~ 1 for all s,t in s st_ 1 8
u r , ' s I I e r o g I - ' l = ' l l d ' , { l l u a l u a r n b a ' r o ) ( 1 s ) != ( . r x l l d ) , { q p a u r g e p 'rd 's,11 I uoll€luasardar rulnEar-1qEu eql lJnJlsuoJ oslr feu en aqt ol snoEoluuy 'lO'Z'd 'xA 'Jcl thl roJ c11c,(csl U l€rlt Eurrr,ord f,q'n ro; Euylerudoss1 g leql fglroa lleqs e,,l[ 'n roJ cr1c,{csr U l€rl} ',(lluonbesuoc'pu€'t r l o e e r o 3 11 = Ul\ lerll a^rasqopue tl = U lo'I ( ' 1 ' p u o r l c e g u r ' r o 1 € 1p e q s l l q € l s ou o l l r o s s e 1e:eueE eroru € luorJ /AolIoJ III/'\ slql l- > ,l(l)xlDtl3 selJslles n * g : x a r a q / h ' l \ ( t ) r c ' t 3 = r S O T J Ol uSo E r e l u o c ] f 1 E u o r 1 s - o € J o u n s e q l se elqrsserdxa flanbrun sl .tg Jo luetualo {ra^e leql enrl 'lJ€J ur 's1 lI)
(It is, in fact, true that every element of M is uniquely expressible as the sum of a a-strongly* convergent series x = LtEGX(t».t' where x: G ~ [ satisfies LtEG lx(t)1 2 < "'; this will follow from a more general t
'acuaq :(tr =)t Eululeluoc 'ow ,r-1'1 = lr Jo ernsolc Euorls orll sl lg ecurs) erqeEle e s1 s,11 or{l Jo suolluulqruoJ r€ourl ellulJ lurofpe-g1os on les eqJ '@)*ll {q pelouep eq III,I\ pue g 3o erqoEle uu€runoN Jo uo,r dnorE eql pellec sI ,,{g r , :l\} = py etqeap uuBr,unaNuo^ orII '9
The von Neumann algebra M = {}.t: t E G}" is called the group von Neumann algebra of G and will be denoted by W*(G). The set M o of finite linear combinations of the }.t'S is a self-adjoint algebra (since }.t = ). -1) containing 1(= }.E); hence, M is the strong closure of Mo' G.
l'\ Jo uoll€luasardar relnEar lJoI pallec-oj aql sr 1r l(9)r1 ul (l\'\ = ''e'I) t\ - / deur oqj 'g u! s g IIe roJ Jo uolluluase:der ,{relrun e sl "l -'1'\',(lluolerr,rnba'ro'(sr-l)l = (sXllf) snql ll ^q uoll€Isu€rl lJeI o1 EulpuodsarJoc roleJedo frdlrun eql elouep t\ lel 'g il rlo€e ro{
For each t E G, let }.t denote the unitary operator corresponding to left translation by t; thus (}.t~)(s) = W- 1s), or, equivalently, }.t~s = ~ts for all s in G. The map t ~ }.t is a unitary representation of G (i.e., }.st = }.s}.t) in R2(G); it is the so-called left regular representation of if t
~ S
S I ' J I
if t = S
s=rJI
.:]
= 819= (s)r1
't'Z
2.3. The Tomita-Takesaki Theorem (For States)
45
(se1e15rog) tuaro0ql I{Es0{BI-ElltuoI
9V
oql
46 46
2.. T The Tomita-Takesaki Theory heory omita-Takesaki T 2 he T
of type III, III, in in particular). factors of Till further further notice, assume assume that M M is a von Neumann algebra algebra acting Till on lf}f and that O 0 is a cyclic and separating separating vector ffor or M.
2.3.1. Let So So and Fobe F 0 be the the conjugate-linear conjugate-linear operators,with operators, with Proposition L!-ldomains MQ MO and M' 0, respectively, respectively, defined (unambiguously) (unambiguously) by So(xo) So(xO) MtCl domains = x*$ x*o, Fo(x'a) F o(x' 0) = at*q x' *0. Then F 0 are are densely densely defined closable closable Then So So and Fo denoted by S Sand respectively, satisfy satisfy S = = operators; their closures, closures, denoted and F, respectively, operators: F S aand n d FF = SS** == SS6' 3. F** == FF6 0 is cyclic and separating separating for for M M as as well as as for for M' M' (cf (cf.. Proof. Since Since o (2.2.6», it it is clear that both So F 0 are densely densely and ^90 and Fo Ex. (2.2.6)), unambiguously defined. M', observe t, o hat a n d xx't e M b s e r v e tthat u nambiguously d e f i n e d . IIff x e M and 'O),xo> = <x' *o,xo> = <(\x 'xo> tO> = = <x*o,x < x * G x t'0> o> = = <So(xO),x' <So(xQ),x'o>' = 0>.
By t h i s says says o f a conjugate-linear c o n j u g a t e - l i n e a roperator, o p e r a t o r , this B y definition d e f i n i t i o n of o f the t h e adjoint a d j o i n t of = So; implies i.e., So ^to fg r.3. FSlMo = So; i.e., that MO F6 and that F61MO F6' This implies Ma f! dom Ffi the i s closable. I f F denotes d e n o t e sthe that 0 is i s densely hence Fo c l o s a b l e . If F 3 is d e n s e l y defined d e f i n e d and a n d hence t h a t F6 if S denotes denotes So So So is closable closable and if So. So closure F 0' then F* = Fl F6 ~ closure of Fs, I So' = f3 S. closure of So' Se, then F* = the F6"s.ippose ~ the closure : S. q#. f'fi and F6~ Ffi[' == ~#. For the reverse dom F6 ,eue.ie inclusion, suppose ~I ee dom w i t h domain M ' Q0 by Define d o m a i r t M' b y Qo(x' D e f i n e operators a n g Qt, b o t t r with operators Q Q o ( x t0) o ) == O 6 i , both Qo6 and i f x',y' M t , then then x' Notice i ' l #~#. . N b t i c e that t h a t if x ' , y ' e€ M', a n d Qt(x' x ' t ,~ and C f , { x ' 0) o ; == x' t a ) , y' y ' o0> > = <x' < x t ~,y' < O o ( x'0), o> = <~,x'*y'o> = = <~,Fo(Y'*x'O» = =
(#) ( b y definition (by d e f i n i t i o n of o f ~#)
= <x' o)>. <x'gOJ(y o,Qt(y ,' 0». is closable; closable;if Q Hence, Q and consequently consequently Hence,as as before, before, Q6 Q On Of, ~ OI and ) Qt o is verify It-is to verify trivial to denotes Qt. It is trivial si Qo' ihen Q* the closure closureof denotesthe Ot ~)Ot. Qo,then Q* == Q6 easy h e n c e ,by b y an a n easy a n d hence, M t , then x'Q o , and that Qo' i f x' t h e n Qox' t h a t if x t e€ M', Q o * ' ~2 x' passto and conclude conclude approximation we may to the the closure closureand may pass argument,we approximationargument, M. is affiliated affiliated to to M. that linear operator operatorQ denselydefined definedlinear that the the closed closeddensely O is
(sel€ls rog) ruo:oaql l{€sal€I-€llruol
LV
'E
Z
oslv 'u qcse roJ (lAl)lu'ol | =ua er?q^ n ? Jo uolllsodurocap reloct eql aq l)ln = O :c-l
Let Q = ulQI be the polar decomposition of Q. Then, u M where en = 1[O,nj(IQI) for each n. Also ,
E
E
M and en
n
111l)t"'olilAl)n=ua6 =lb
u(lQII [O,nj(lQI»
'A
qn = Qe n =
) n'uaql
E
47
'a pue n
agl
2.3. The Tomita-Takesaki Theorem (For States)
M
'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
(cf. Ex. (1.1.4». Note that en /' rp Q* and uenu* /' rp Q; since ~ = Qn and ~# = Q*n = Q+n, conclude that uenu*~ .. ~ and en~# .. ~ . On the other hand, an = plQln*nuan = TSlQlu an = glub *nu an = 7SQ*nu
= uenlQln = uenu*ulQln = uenu*Qn = uenu*~, '1
qnn
while SOn (q n) = q*n ~#' Conclude, # from the n = e nQ*n = e n definition of S, that ~ E dom S and S~ ~, thereby establishing that F~ f S, and hence, that S = F~ = F*. The dual statements, with F and S interchanged, are proved identically, with the roles of M and M' interchanged. 0
',{.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\ ary to uotltsodruocap tolod aLfi aq zltyf
'uarlJ 'g .toyotadopasop = S ta7 ?€Z uoplsodord
Proposition 2..3.2. Let S = J/l1/2 be the polar decomposition of the closed operator S. Then,
'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)
(a) J is a self-adjoint antiunitary operator and /l is a posItIve
self-adjoint operator which is invertible in the sense of unbounded operators (i.e., /l is 1-1); (b) F = J/l- 1/ 2 is the polar decomposition of F; further /l = FS and /l-1 = SF' (c) J/lJ: /l-1; more generally, if f is any (possibly unbounded but finite-valued) Borel function on [O,co), then Jf(/l)J = 1(/l-1); in particular J /lit J = /lit for all t E IR .
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
Proof. By definition, we have /l = (/l1/2)2 = S*S = FS and so /l is positive; also J is a conjugate linear partial isometry with initial space ran-J:' and final space rallS. Since Min = ran Fo fran F and Mn = ran So fran S, conclude that J is anti unitary (i.e., its initial and final spaces are both ~). Since So = SOl, a simple approximation argument shows that S is invertible and S = S-l; this implies that /l is invertible since ker /l1/2 = ker S = {OJ; the equation S = S-l also implies that J/l1/2 = S = S-l = /l-1/2J*. Hence J2/l1/2 = J/l- 1/ 2J*. Since J/l- 1/ 2J* is an invertible positive self-adjoin t opera tor, the uniqueness of the polar decomposition guarantees that J2 = 1 and /l1/2 = J/l- 1/ 2J*. As J is anti unitary, this yields J = ]"1 = J* and J/l- 1/ 2J = /l1/2. This completes the proof of (a) as well as of the decisive step in the proof of (c). The validity of (c) can be derived from J/l-1/2J = /l1/2 and a routine application of the spectral theory to /l1/2; the verification is purely routine in nature and may be safely left to the reader. As for (b), conjugate the equation S = J!l1/2 to get F = /l1/2J = J J /l1/2 J = J /l-1/2; this completes the proof. 0 = tclfl
.Joord aql selalduoc slq-l irlr_vt = tslyIt n = g nE ot = g uofwnba oql aleEnfuoc '(q) rog sy i6:-yr 'r3p€3r
4488
2. The The Tomita-Takesaki Tomita-Takesaki Theory Theory 2.
The stage stage is is now now set set for for the the Tomita-Takesaki Tomita-Takesaki theorem. theorem. The The proof proof The u i t e llong o w e r f u l aand o n g aand n d eelaborate. laborate. s qquite h e o r e m iis i f f i c u l t ttheorem n d ddifficult oof f tthis h i s ppowerful As we e sshall n l y ssupply upply h a l l oonly h i s cchapter, hapter, w n tthe n t r o d u c t i o n tto o tthis A s sstated h e iintroduction t a t e d iin the proof proof in in the the very very special special case case when when ,S S (or (or equivalently equivalently A) tJ.) is is the proof general the bounded. For a reasonably short proof of the general assertion, the assertion, of the reasonably short For a bounded. reader may may consult consult [BRI]. [BRI]. reader
Theorem L3-32.3.3_ With the notation established established in in Theoren propositions, the following following statements statements are valid: propositions,
the
preceding preceding
(a) AitMA-it tJ.itMtJ.- it = =M M lor for all all t in R; IA ; and M'. ((b) b ) JJMJ '. MJ = M are Suppose that ,S S is bounded. bounded. This means means that .S, S, A tJ. and F Fare Proof. Suppose everywhere defined defined bounded operators; operators; it it also also means means that A-r tJ.- 1 is an defined bounded operator, operator, since since a-l tJ.- 1 = JAJ. JtJ.J. everywhere defined For x,y and z in in M, note that that For
= zx*Q; = 1I to conclude (Sx^lXzQ) = 2*+q combined Apply with y./ = conclude that (SxS)(zn) Apply this with = (y(^Sx'S))(zn). from the (Sx'S)yXzo) Infer from with (1), this yields: (SxS)y)(zn) = (y(SxS»(zn). Infer yields: (l), with (Sx^S)y== y(SxS). y(SxS). Since arbitrary, cyclicity O that (SxS)y Since x and y were arbitrary, of n cyclicity of we MI. g e t : SMS t. w e get: S M S fC M (M',F) (M,S) and (M' roles of (M,S) An entirely analogous ,F) with the roles argument, with analogous argument, interchanged, yields: FMI F f M. y i e l d s : F M t F interchanged, 9M. results in above results A combination of inclusions established established above of the two inclusions 1 1 = = FS and tJ.!:J.MtJ.= SF); (since tJ. A-r ,SF);conclude conclude that A= AMA-Lf! M (since
(2) (2)
tJ.nMtJ.L n M An' nft MM
for 1 , 2 ,'". . . . f o r n = 1,2,
los \ for>. = g(tJ.). (0,-), and write tJ.z Ao = For g(I) = >.z \z = e"zZ log>' for \ Ee (0,"), C, let g(>.) For zz Ee cr, 8(A)' z i n cr. C A u f o r e a c h i s b o u n d e d , Since tJ. is invertible, it is clear that tJ.z is bounded, for each z in Since A is invertible, it is clear that ( c l e a r l y e n t i r e ) t t t h e Now, fix x E M, x' E M', t,n E :If and consider the (clearly entire) e a n d c o n s i d e r € M ) , e M , x t f i x Now, l,rt function by f u n c t i o n defined d e f i n e d by
l(z) = llall-'".ta"" a-o, xt)l,o>,z e (L A ab -- ba. ba. A by [a,b] where defined definedby the commutator commutator denotesthe where[a,b] la,bl== ab [a,b]denotes yields crude yields estimation crudeestimation
R If(z)1 II. z l lzall·IltJ." l l .zlll·llx l a -11·llx' " l l ' l l11·11 ' l l 'dl l."lin l' l n l ' l l 'r l l l l a l lRe - ' "Z" 211tJ. t f ( z,)
R"", n"zil"ll== 11tJ.112 z II2 = II
roJ rotccl Eulleredos pue c11c,{cs sv 0g uaql '1Qu'Qy'QU1 eyd1r19519 p a l e l c o s s €q \ / ^ n u o I E u o l l c u n J r E a u r l a n l l l s o d I € u r r o u I n J q l r € J e s r 0 leqt pue erqaEle uuuuneN uo.rrleraueE e sr.n leql ^\ou osodclns '79 3o rusrqdJoruolne-*.;o dnorE relouered-euo snonurluoJ *,(18uor1s-o € seurJep *_VrrrV e x leql eslcrexe etoq€ aql uoJJ s^\olloJ ll 'ruerooql r{Bsa{BJ-Blrruol or{l Jo Eurllas aql ol Eururnlag
Returning to the setting of the Tomita-Takesaki theorem, it follows from the above exercise that x .. t::,.itxt::,.-it defines a a-strongly* continuous one-parameter group of *-automorphism of M. Suppose now that M is a general von Neumann algebra and that ¢ is a faithful normal positive linear functional on M with associated GNS triple (:If¢, Tl¢,O¢). Then O¢ is a cyclic and separating vector for to prove (strong* and hence) strong continuity; this requires a standard E/3 argument.) 0
O ( ' l u e r u n E J eg / r p r e p u e l s € seJrnbtJ slql lfllnulluoc Euorls (ocueq pue *Euorls) orrord o1 s e c l J J n sl r ((p) (t'g'O) 'xg 'Jo) os 'pepunoq-rurou sl {UJ, I :(r)rn} 'l{ r x ' n u l x q c e a r o J ( * f 1 8 u o r 1 s - o )( r ) r n les arll JI :turH)
.. at(x) (a-strongly*) for each x in M. (Hint: if x E M, the set {at(x): t E IR) is norm-bounded, so (cL Ex. (0.3.4) (d» it suffices n
(*)uto € / - t/ leql esues eql ur snonulluoo *^lEuorls-o sl rlorrl/ir 'N n go swslqdrouolnE-* 3o dno:E relaurered-auo€ seulJep ']nxrn = (r)ln uorlenbo eql 't " ler{l ^\oqs ile roJ W = ? "' UJ > | inry'n lBql qcns ,{ uo srol€redo ,(relrun go dnor8 ralaur€rBd-euo snonurluoc ,(lEuor1s e sr Utlllnl esoddns pue ($h j 79 asoctcln5(q) (1'7'7 waneqJ Jo JI€rl puoooseql Jo Joord 'leruJou sr D '-n aql ol leeddy lBql epnlcuoJ :r_D soop sE ur ernlcnrls ropJo arll so^reserd r :1ur11; 'J1esll ol:uo n Jo ursrqdroruoauorl l€ea-o crrloruosr ue sr p ueql 'n lnv t p JI (e)
(a) If a E Aut M, then a is an isometric a-weak homeomorphism of M onto itself. (Hint: a preserves the order structure in M+, as does a-I; conclude that a is normal. Appeal to the proof of the second half of Theorem 2.2.1.) (b) Suppose M f :e(:If) and suppose {Ut}tEIR is a strongly continuous one-parameter group of unitary operators on :If such that utMUi = M for all t. Show that the equation at(x) = utXUi, t E IR, X E M, defines a one-parameter group of *-automorphisms of M which is a-strongly* continuous in the sense that t n .. t at (x)
*
'/t/
(2.3.4) Let Aut M denote the group of *-automorphisms of M. Jo srusrqdrourolnu-*3o dnorE aql olouop n $y
n1
(V'E'Z)
Exercises 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
Before proceeding with some of the consequences of the TomitaTakesaki Theorem, it would help to get some elementary statements out of the way. So, here come some more exercises.
0
E 'n3Itn.I
'(q) turqsrlqetsesnql
thus establishing (b).
= f61yYthl11yYt = ttWt
and
puB SWS = ta1yYN71{I
= Ilttt
= Jt::,.I/2Mt::,.-1/2J = SMS f
M'
tn3
JMJ
'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 = @){
if Re z ~ O. Thus, the entire function I is bounded (by 2 IIx II Ilx' II II d II 17 II> in the half-plane Re z ~ 0; also, by (2), I(n) = 0 for n = 0,1,2, .... It follows from Carlson's theorem (cL [Tit)) -- which is valid under a weaker condition on the growth of I (f(z) = 0(e k1zl ) for Re z ~ 0, with some k < n, to be brutally precise) -- that I(z) = 0 for all z in cr. Conclude that t::,.zMt::,.-z f M for all z in [; applying this to (-z) yields t::,.zMt::,.-z = M for all z in [; for z = it, this is precisely the contention of (a). It also follows that t::,.zM't::,.-z = M' for all z in
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
49
2.3. The Tomita-Takesaki Theorem (For States)
6'
2. The The Tomita-Takesaki Tomita-Takesaki Theory Theory 2.
5500
F6, J6 n~(M). Here Here and and in the the sequel, sequel, we we shall shall denote denote Uy by SO, S ~ F~, J ~ and and A ~
oft") 0t(x) = = n-q,(aiot n4/<~~t z6(x)A6it). n~(x)~4r). b) a a c t Athat that assertion Ex. 2.3.4 and . 3 . 4((b) n d tthe h e ffact rom E x. 2 ((The T h e ccontinuity o l l o w s ffrom s s e r t i o n ffollows ontinuity a r o u p ({at} o f ) is is o-weak homeomorphism.) The one-parameter group he o n e - p a r a m e t e rg omeomorphism.) T is a o -weak h zn~ 6 is o r ssimply, group off m modular automorphisms modular odular roup o he m i m p l y , tthe odular a u t o m o r p h i s m s ((or he g ccalled d t t e a tthe associated with with the positive functional functional 0. ~. 0 n group) associated
Exercises functional on M, (2.3.6) ~ be be a faithful (2.3.6) Let 0 faithful normal positive linear functional AO,SOFO, as above. above. "16 and lfg, lf~, nb n~ ~~, S~F~, J ~ and 46 ~~ be be as 04; F6oq o6 Ee dom S~ t1 dom F F6~ and s~n~ SO0O = F (a) Show 56 () (a) I/>nl/> = nl/>; Show that n~ = nl/>; = J" 1I/>nl/> ng; A g and A O a O= 6 0 6= e dom d o m lJ.1/> a n d ~I/>nl/> that'O conclude n~O E c o n c l u d e that aitn6(x)o6. nd(oP(n))ad (b) Show ot(x»n~ = ~~tn4!(x)n( (c) ( c ) Show a r e equivalent: equivalent: f o l l o w i n g conditions c o n d i t i o n d are t l i e folloWIng S h o w that t h a t the
(i) ( i ) I/> i s tracial; tracial; Q is (ii) I/> is (ii) S i s anti 56 a n t i uni u n i tary; tary; F ( i i i )S~ s o== J~ (iii) = F~; !lr'o = o ; (iv) ~I/> a6 =- llf~ (iv) ; (v) IR . ( " ) 0t(x) Y x Ee M, M ,tt Ee R o f ( x =) =x x Vx + (v) ( i v ) :} ( v ) are ( i i i ) #( 9 (iv) ( i i ) #< + (iii) e a s y ; use use (Hint: ( i ) #< + (ii) a r e easy; ( H i n t : The T h e implications i m p l i c a t i o n s (i) (b) ( v ) :} + (iv).) ( i v ) . ) 0n ( b ) for f o r (v) Tomitat h e TomitaLet w i l l illustrate i l l u s t r a t e the t h a t will l o o k at a t some s o m e examples e x a m p l e s that L e t us n o w look u s now Takesaki T a k e s a k i theorem. theorem.
Sf
where vv is is aa a", where Example oo(X,r,/L) and (a) Let and ~(f) Let M M == LL-(X,T,tt) 2.3.7.(a) Exanplc 2.3.7. Q(fl == It dv, g i v e n 1 f 5 == i s b y finite measure equivalent to /L. The GNS triple is given by lf~ T h e G N S t r i p l , i t o finite measure equivalent & = ' J~ "/d t h a t S~ ^ S d= ( d v / d 1 l r l 2 . It i s easy t o see s e ethat a5= l t is e a s y to LL 22(X,/L), = (dv/d/L)1/2. n;(f) = m mf r and a n d n~ ( X , y ) , nl/>(f) - - note = II -i! n o t e that t h a t I/>0 is 46 = == FFep6 is t h a t ~I/> ( X , t t ) and a n d that m a p f7 ...- 1 d n Lt z2(X,/L) i s the t l i e map 7 on = JMJ Mt = JMJ t h a t M' i n f < i r m sus u s that traclal T h u s , the t h e o r e m informs t h e theorem M is i s abelian. a b e l i a n . Thus, as M t r a c i a l as (0.4.5)(b). (b). Ex. (0.4.5) == M which is of Ex. (since Jm! is the the content content of nr;), JmrI == m M (since f ), which
'puu (r{h o t = Of(t o (a)f)Of ocuaH 'r Jo l€ql Jo al€Enluoc-xolduroc (esr,n,{:1ua)aql sl 1"1) srseq IErurouorllro oql .ol lJodso.l qll/r\ xrrleu s s o g / hr { u o r o l B J a d ol s g l s J x e r s q ^ \ ' r 6 I = Q r ( t . o r ; 9 1 l e q t s l e a ^ o r '(reou11 91 ^\ou u o l l e l n d u r o c i t s e o u y a l u S n f u o c s 1 l e r l l I J B J eql roJ "l -l = "1;91 (*l q l n q ' d s r u d l t J o q t i ( l l e r l u a s s as r s r q t ) o 1 e q 1q c n s Jot€rodo[rclrunlluu.lnblun aqr sl 94 pue ,_d o d = 0t '": o ulrrfou3 = \rU 'l @ f = (.y)vu 'll @ ll = 9J[ .alnlcrd clqdrouosr u€' uI
by"" $k( aj/p), a!).d conclude that, by self-adjoint ness, b. is this operator: (b.O(k) = ai/p~(k). (The details aJe spelt. out in Ex. (2.5.5) & (2.5.6.).) Since S = Jb. 1/ 2, conclude that J~~n) = ~~m) fQr all n~,n. .I t is easily deduced from the above formula for b. that af(x) = pltxp-It for all x E At, t E IR . In an isomorphic picture, :lert>. = :Ie @ :Ie, 7lrp(x) = x @ 1, 0rp = [na~/2~n @ ~n' b.rp = P @ p-1 and J d> is the unique antiunitary operator such that J rp( ~n @ ~m) = ~m @ ~n (this is essentially the flip map, but for the fact that Jrt> is conjugate linear). An easy computation now reveals that J rp(x @ I)Jrp = 1 @ X, where is that operator on :Ie whose matrix with respect to the orthonormal basis {~n} is the (entrywise) complex-conjugate of that of x. Hence J rp( :I:(:Ie) @ l)Jrp = 1 @ :I:(:Ie) and,
x IIe roJ
x
= (x)jo r€.qlv roJ BlnruroJe^oqe,u, u,orY
3":"#"i ,r-d"r,d ,(lrseesr li 'ttiiuge rd; 1-'il = ru]:I lcql apnlcuol ,r11vf= g oJurg z9 (S'S'Z)'x:r uI rho"iredddfe slrelepoq1) '(tr)i?y,]"= (U)(:v) ('('S'S'z) :rolerodosrql sr 'ssaulurofp€-Jlos ,(q 'l€ql epnlcuoJpte''1dr-{n){eaQ V r Jprr k
:Ie): [
: ( g : M ) r rr I E J12(lN;
k
~ IIpr(k) 11 2 < "'} a
t- "ll(z)3dll i 3
{r )
Do' Note that Do is a core for the posi ti ve self-adjoin t opera tor defined on
uo paurJap r o l e r a d o 1 u 1 o [ p e - g 1 aasa r l r s o d a r l l r o J e r o c B s 1 0 g l e q l e l o g ' 0 G x
ut ? II€ rog (1)!dr- p = (ZXIV) ltrql pue V ruop i og teql opnlouoJ uol ak-1P~(k)
"" 6
"" Conclude that Do f dom b. and that (b.O(k) n
for all ""~ in
'. (r u" )( J zf r r l g J= ( -u). l r
Fr~m) =
[:m f/2 r~).
It is not too hard to verify (do it!) that the linear span Do of (r~m): m,n E IN} is contained in dom F and that lBrll puB J urop ur poureluoc sg {g
r u'la
:1*jt) .lo og ueclsr e e u l l e g r r e q l ( i r l o p ) f S r r a r r o l p r E q o o l l o u s l l I
'(rf r * n l =(dlos ,,,15;J So~~m)
""
and thus
=
[a~ ]1/2 ""~~).
'nql Pue
m
^ 'rtJ r * p ' ) =ou(*1."i";eu ,r,LEJ
7lrp(x~m)*)orp =
[ : - ] 1/2
r~),
elrq/h'(ujl = 01r1,*jx10zlEr{l alou pu€
"
and note that llrp(x~m)orp = r~m), while
*:'ul,r/ilo
n
m
= t_j,
x(m) = a- 1 / 2t
~n'~m
ourJc.p'M ul u'tu rof ''l{*g = (Z)r-lt eroq/h'(g a u'ar:,-\1} i(q uanrE !l 911ro; srs?q lturouor{lro 1e-rnird} 'ulrlio = (r)Qu :(a)fx = ( u ) ( l ( x ) v s ) : ( A u r t p , { r l l e u l p r e cJ o l o s e , ( q p a c e l d e ia q l s n r u M ' l 6 u ;r l1efiorsuorulpellulJul sI 4 paunsse e^€rl e,r\) ("''Z't) = M aror{^\
where IN = {l,2,...} (we have assumed :Ie is infinite dimensio!).al; if ng.t, IN must be replaced by a set of cardinality dim :Ie); (llrp(xH)(n) = xj(n); 0rp(n) = a~/2~n' A""natural orthonormal basis for :lerp is given by (~~m): m,n E IN}, where ~~m)(k) = 5mk~n' For m,n in IN, define n
e l l (i l )?i tta * N 3 ) = ( s : s) r a =0 x1
{r::N
"':Ie: [
' {- >
:lerp = J12(lN ; :Ie) =
Ilr(n)1I 2 <
",},
where an > 0, [an < '" and {~n} is an orthonormal basis for:le. version of the GNS triple is given thus:
: s n q l u a r r r 8s 1 a 1 d t r 1S N O o q l J o u o l s r o ^ uo3'0 tn 'll toJ sls€q < eroqrn leurouoqlro ue sr {"11 pue o r r:,r1, ,o 3 = d
One
ouo
.
(b) Let At = :I:(:Ie) and rp(x) = tr px, where p is an injective positive trace-class operator, with canonical decomposition
u o l l l s o c h u o c a pI E c r u o u B cr l l r / r \ ' r o l e r a d o s s B l c - e c s J l a,rrlrsod err.rlcafurue s1 d ereq^\ 'xd tl = (x)0 pu€ (r{h = hl p1 (q)
(srluls rol) uroroeql r{sse{BJ,-ullruofeqJ. 't'z
51
2.3. The Tomita-Takesaki Theorem (For States)
I9
522 5
2.. T The Tomita-Takesaki Theory heorv 2 he T omita-Takesaki T
consequently, (l(13) (:f(lt) o @ l)' 1)' = I1 @ @ t(lf). :f(lt). consequently, it may be shown (see (see [Tak [Tak l] 1] for for details) using the More generally, it Tomita-Takesaki Neumann algebras M cf o r vvon on N eumann a h a t iif, f , ffor lgebras M T o m i t a - T a k e s a k i ttheorem h e o r e m tthat :f(lt) and N g f t( :f( K), K), we define define M @ @ N = = (x {x @ @ yi y: x e E M, yYEN}" :f(lt o @ l(18) e /{)" cf t(Xf = q u i t e M @ K K )) tthen @ n N)' M' @ N'. This fact is quite non-trivial and o n o n t r i v i a l M ' T h i s f a c t i s a n d N ' . h e n ((M r open problem for for a long time before Tomita Tomita resolved resolved it. remained an open We shall, shall, however, however, not go into a proof proof of of this here. here. We countable group and M = W*(G), (c) Let G be a countable W(G), the group von a s sshown Neumann algebra discussed att tthe end off Section 2.2. was . 2 . It It w nd o Section 2 hown N eumann a lgebra d i s c u s s e da he e and M.. Since O = l~E e c t o r for for M Since yclic a n d sseparating e p a r a t i n g vvector tthere h e r e tthat hat n e iiss a ccyclic _ r , it ( r ) r . " iiss rEe = i n e a r sspan r),:~E f(e = i t ffollows o l l o w s tthat h a t tthe h e llinear Pan D = \),r --1~E = (~ r -1' :Doo o off ((~t}tEG t
t
in dorn dom S and that contained in
(s(Xr)= i(r-r) all and ~ iin :Doo ---- rrecall i n e a r . Since e c a l l tthat o n j q g a t e llinear. Since ffor or a n G a n D h a t S iiss cconjugate l l t iin nd E preseruing this map is norm preserving and :Do Do is dense Iz(G), conclude conclude that dense in ,Q2(G), --:1 = I (cL = F = = , S and = W(cf. (S(Xt) = A = (SOU) I 2 ( G ) , that i n ,Q2(G), t h a t t> f o r all a l l ~( in that J = a n d that [ ( t - 1)) for (2.3.6)(c)). It Ex. (2.3.6)(c». J that J\rJ J),tJ == Pprl ; It follows from from the definition definition of ,I t = {p = JMJ hence M' = JMJ = G}"; thus, the commutant of the : t E of e G}"; thus, hence Mt {pr: t g e n e r a t e d by left-regular v o n Neumann N e u m a n n algebra by l e f t - r e g u l a r representation r e p r e s e n t a t i o nis i s the t h e von a l g e b r a generated the O t h e rright-regular i g h t - r e g u l a r rrepresentation. epresentation. 0
24. Hilbcrt Algebras Algcbras Yeights and and Generalized Gcncralizcd Hilbert 24. Weights (X,I) can A positive measure Borel space can be be viewed as as aa measure Ilp on aa Borel space (X,f) p o s i t i v e linear ( n e c e s s a r i l y faithful (necessarily l i n e a r functional f u n c t i o n a l on f a i t h f u l and n o r m a l ) positive on a n d normal) L oo(X,f,Il), via integration, only when it is measure. If If Ilp is is is aa finite finite measure. L'1X,T,1I1, - - typically n o n - z e r o constants infinite, f u n c t i o n s -t h e non-zero constants i n f i n i t e , some t y p i c a l l y the s o m e bounded b o u n d e d functions p-integrable. Worse be made made reasonable sense can be -- are Worse still, no reasonable sense can are not Il-integrable. = IE l F , where w h e r e E and a n d F are a r e disjoint disjoint i f , for f o r example, l E -- IF' of e x a m p l e , f| = o f ) f| d lIlt if, however, sets f u n c t i o n s , however, i n f i n i t e measure. m e a s u r e . With W i t h non-negative n o n - n e g a t i v e functions, s e t s of o f infinite there i n t e g r a l can a l w a y s be be t h e integral c a n always i s no n o such s u c h difficulty d i f f i c u l t y and a n d the t h e r e is " n u m b e r " in prompt h e u r i s t i c s prompt meaningfully i n [0,00]. T h e s e heuristics a s aa "number" m e a n i n g f u l l y defined d e f i n e d as [ 0 , - ] . These the f o l l o w i n g definition. definition. t h e following
J
is aa Definition algebra is Definition 2.4.1. 2.4-1. A weight on aa von Neumann M algebra = w h e n ever y ) I O ( x ) mapping M+ .... [0,00] such that 4>(),x + y) = ),4>(x) + 4>(y) whenever m a p p i n g 4>: M , ' s u c h t h a t 0 ( y ) @(\x Q: [0,-] o == 00 @and a n d ),.00 - - with \.x,y * 00 w i t h the c o n v e n t i o nthat), that I + M - and), t h e convention x , y Ee M+ a n d \ Ee [0,(0) [ 0 , - ) -- if - = = O. = 00 w e i g h t 4>Q is i s said = w h i l e 0. T h e weight s a i d to t o be be 0 . 00 0 . The 0 , while i f ),\ > 0, (i) (i) (ii) (ii)
faithful, i m p l y 4>(x) 0; i f x Ee M+ M - aand n d x i-l 00 imply f a i t h f u l , if 0 ( x ) >> 0; ( o - s t r o n g * ) limit limit normal, w h e n e v e r x is i s the t h e (a-strong*) n o r m a l , if i f 4>(x) s u p 4>(x) O ( x i ) whenever 0 ( x ) == sup net (x) it M+; M*: of aa monotone monotone increasing increasing net {x1} in (iii) aa trace, in'M. (iii) M. 0 if 4>(x*x) trace, if 0(xx*) for all x in 0(x*x) == 4>(xx*)
apply this with Tl n = ~n to get rllx~J2 = rllx*~J2; combine this with the previous equalitx applied to x*.) 2 (b) If x E :f(:If), define IIx = r Ilx ~n 11 , where {~n} is any orthonormal basis for Jf. Let C2(:If) = {x E :f(:If): Ilx 11 2 < co}; the members of C2(:If) are called Hilbert-Schmidt operators. Prove the following:
: E u 1 m o l 1 oagq l o^ord 'srolerado lplur{Js-lroqlrH palluc an (y)zg Jo sraqrueru : ( , t ) x , x ) = ( 1 1 )-z=J l e . 1 f i r o J s r s e q l e r u r o u o t l l r o eqr i(- t'll"ll " '(ah t x i(ue sl {ul) araqa,l 'rll"?x1;3 JI (q) lll"ll eulJep parlddb ,{tlyenba eql qll/t\
IIi
(1*x ol snorlard sr qteul quor :a llu t*xl=lzl 3 l " l x1 1ral 8o 1'l = ' r , q ll^\sr q l, ( t d cl€ = rl<-rr''lx>lff= r;1't"11f :zllu(r*xll3
r IIx ~J2 = rr I <x ~n,Tlm>12 = r Ilx*Tlm11 2;
:lulH) '1enbeeru seprso^u aql puu 'reqlo aql sl os "e'l -'allulJ sr sprs auo JI ,;;"rrxll3.= zll:.rrll3 l?ql a^orcl tt "e roJ seseqleurouorllro 3o rted ar't ('u) '{'l}'pue (r{h t x JI (s) (a) If x E :f(:If) and {~n}n {Tl n} are a pair of orthonormal bases for :If, prove that rllx~nll2 = rllxTlJ2 -- i.e., if one side is finite, so is the other, and the two sides are equal. (Hint:
0'v'z) (2.4.4)
Exercises saslcrexg
'sesrcrexe las Eur^\oIIoJ eql Jo u r e u o p s r s r q l i s r o l e r o c l ol p r u q c s - u a q l r H p e l l E c - o s e q l l n o q ? s l c € J clseq euos loolloc ol luapnrd oq lqttu ll '(n)X uo ac?rt IEsIuouBc 'uellaq? lou sl aql Eulssncslp ol Jolrd W eJeq^\ aldruexa u€ a a s o l p o o E e q I I I / { l l ' r e A e ^ \ o q' l 8 q l . e r o J e q ' p a q s l l q e l s ee q u o o s I I I ^ \ pu€ pll€^ sfe,nle sr i(1r1enbaslqt l+7g u 9W = 99 tu{1, 'as?c slql ul '((rt'!'X){1 'eloN '(TI'X)fl u (rt'X)-1 = 9y pue f!'X)rZ v (a'y)-7 = 9y 'ernsualu alIuIJuI t { pue 'a'B 0 < { :(rl:X)^1 ) {l = vC '3s€c s5tl uI ue s1 rl eroq^\ 'TIp II ='ArA pue (rf'4f)-'I = Iy fq uar18.s1 (at1u13 '[1rea13 lou sI qclq,$) 33Erl- IBruJOu InJIIIIEJ u 3o alduruxe euo
Clearly, one example of a faithful normal trace (which is not finite) is given by M = Lco(X,f,p,) and 4J(f) = dp" where p, is an infinite measure. In this case, D4J = (f E L co(X,p,): I ~ 0 a.e. and I E L 1(X,f,p,)}, N4J = Lco(X,p,) n L 2(X,p,) and M4J = Lco(X,p,) n L 1(X,p,). Note, in this case, that Dd> = Md> n M+; this equality is always valid and will soon be established. Before that, however, it will be good to see an example where M is not abelian. Prior to discussing the canonical trace on :f(:If), it might be prudent to collect some basic facts about the so-called Hilbert-Schmidt operators; this is done in the following set of exercises.
II
'^Ia{II suees 'pallFuo aq III/!\ p ldlrcsqns eql n uolsnJuoo ou pu€ 'uolssncslp rapun lqElo^\ euo ,(1uo sI eJeql uel{rl6
When there is only one weight under discussion, and no confusion seems likely, the subscript 4J will be omitted. 0
'aoloq ua,rr8ere suorlrulJapesoq^\i7g secedsqns 3o ur€lreo go slsfleue uE sr 0 lqElo,n e go [pn1s aqt ol leluauepunJ
Fundamental to the study of a weight 4J is an analysis of certain subs paces of M, whose definitions are given below.
+nW = luql qcns O f7g , O slslxa aroql (rrr) i- > (t)g E (gy)ierrulSsl O (I) :sdtllypuocEuptollog aqTJo ecuolu,rlnbeeql e^ord +n ur. x IIs roJ - > (x)0 JI allulJ oq ol ples sI 0 lqEIa/Y\Y G'V'C)
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(2.4.2) A weight 4J is said to be finite if 4J(x) < co for all x in M+. Prove the equivalence of $.he following conditiQ.lls: (i) 4J is finite; (it) 4J( I) < co; (iii) there exists 4J E such that 4J = 4JIM+. 0
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Exercises
'V'Z serqaElyuaqllH pezllerauogpue s1qE1a11
gs
2.4. Weights and Generalized Hilbert Algebras
53
5544
22..
heory The Tomita-Takesaki Theory T he T omita-Takesaki T
IIx II , Ilx 11 11.11
for xx eE C (Hint: ifif It is a unit unit vector, consider CzQt). ll"ll < llrll,2 for 2(:If). (Hint: = q t.)' ; a s i s ({t u c h tthat h a t qt 1' = r t h o n o r m a l bbasis aan n oorthonormal ' ( - )} ssuch (ii) ll. ll,2 ir is a norm norm on C'2(:If) with respect respect to which which - Cr{r) C2(:If) is a C'jttl with (ii) ( H i n t : ( i ) Hilbert space. (Hint: use (i) to locate the limit of Cauchy l i m i t a C auchy l o c a t e o f t o t h e u s e Hil6ert space. sequence in in Cr(Xf); C2(:If); the inducing inducing inner-product inner-product is given by by <x,y> <X,Y> sequence = Ir <xlrr,yE,r>, <xtn,yt n>, where (En) {tn} is any,orthono-rmal any orthonormal basis basis for for 1f') :If.) -llrll, (iii) x e E C 2(:If) x· e E iJ?i) C2(:If) ana and = ll"*llr. (Hint: see see hint hint to to (iii) c-i(h1-"+'x* 2 = 2. (Hint: (a).) (a).) (iv) 2(:If) is a two-sided ideal in in t(Xf): :e(:If); in in f.act, fact, if if #x fE .,Cr(Xf) C2(:If) and yY eE (iv) ' ' CC 2(1f) ri#j, ' llyll :e(:If), in.n then lly'll, lIyx 11 2 < lIy ll"ll, 11 2 un,i'llxyll, and IIxy 2 < Ily ll"llr." 11 2, lHint: (Hint: tt'e the ' llyll i i i ) , iitt iimplies easy, and, with i t h ((iii), m p l i e s tthe he ogether w n e q u a l i t y iiss e asy, a n d , ttogether ffirst i r s t iinequality second.) second.) 0. Then x eE C 2(:If) € x e E K(xf) K (:If) and x admits a (v) Let x )~ 0. Cr(lt) decomposition = Irant, 0,, Ir "a32 . '
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