An Invitation to von Neumann Algebras (Universitext)

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F.W. Gehring P.R. Halmos

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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

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An Invitation to von Neumann Algebras

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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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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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CONTENTS

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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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Appendix: Topological Topological GrouPs Groups Appcndir

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Notcs Notes

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Bibliography Bibliography

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Indcx Index

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122 122 132 132 1148 48 1155 55

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LIST OF SYMBOLS

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The accompanying number usually refers to the page on which the symbol first occurs, or is explained. The symbols are arranged in the order of their appearance in the text.

vr'*n

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P(M), 14

vt'O[)d

gg')q

9l'lav'la|

gg 'suJ

Ve i , l\ei' 16

M4>' 53

V 4>' N4>' hE' 56

fns, 58

19'qA

M.,14

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LI

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MX, M4>, 70

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6I

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U(M), 85 M(G),93

9 6 'Q

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Symbols LList i s t ooff S ymbols

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eneral G f o r ggeneral MOo: G), @ dG ((for ), M

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r(M), 107 f(M), r07

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a,, 1150 a 50

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1149 49

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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

Jlssg

O

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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 = '

{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 == <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

' 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 >


'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 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

(0.2.4). If x

E

l(X), there exists a unique

*o-' ?"',':[y".],"l,l"Jt*,,f uBseurrep fftt#rT","="iltti; 4>x(tt

TI) = <x~,TI>

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== 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

\xd, srrluopu,Jr;ffij;'l':i {:rk",ilj'fl:li;l','iJri'":h (e) (r's'o) (0.3.1).

Exercises

soslcraxe

'sarEoloclol 'tuorls-o aql .f .sartolodol Jeqlo *Euorls-o pue *-8uor1s leru?u aarql Jo suolllurJep aql erB sosrJrexe aql ur popnlcur osle lsasrcroxo eql uI polsll 0JB serEolodol eseql Jo soJnlBeJ {relueruelo otuog

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 r l < * & ' ' r 'tl = u l l ' t lil L 1< ~n,17m>12
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 ; 112 :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):

:fu1)guo sel8olodol luaJaJJrp 0rll u3e,{l3q suollEler ,,uorsnlcur,,Eur.no11o3er{l e^oJd (q)

'2r.5 r.l e11r^ ' p l o q s u o l l l p u o c e s e q l ueq71;pesolc-zt sl tes pesolc-rt f:ene (rrr) : X ' q u 1 . r p u ? Xul (!x) leu fue rog '(rr)x - tr € (zr)r - !r (g)

(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 ,w (q) tg uo sroleractoJo erqe8leruioJpe-31es alureuaEap-uouB sl ,t|^ (e) fnlf d.Jrre,rol ssalur€d^la^rleler sl lI

' 0 0

tt

= W

" o o

0 0

[^

' 0 t )

't'0

0.4. The Double Commutant Theorem

13

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'.

'erqaEle uueruneN uol ? ol sEuoleq roleredo uB ueq^\ tulururralep JoJ uorJelrJc InJasn Eu1,no11og eqt sp1a1f ,(tW = t, qll^\ pcrlddu) €ruruol o^oqu eql qll^\ paldnoc uaq,r 'ruaroeql lu€lnruuoc elqnop orII

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

k[ ~ ++ ikT), 4[ ~,T)] == rt ii\q ikn1. ikn, ~q,++ ikT)). 4[!,,n] k=O k=0

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, :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).)

'Jc -- ueroeql Ierlceds aql uroJJ flrpear s^\olloJ uollrasss lsEI slql tf;u oo =

Do = n~1 ran I [o,n](H);

:1.g;tu'ollur,

'"

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 (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 / ) 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

] +

],


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 "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 ^ ("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

€€

334 4

(In) (I-) ((10)) I) ((Ill) IIl) (110)) (II-) (III) (III)

l1. .

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 ;; (n (n == 1,2, 1.2....) .'.) (0, ..., nZ), , 1 , 2 , . . o>}. . , * )w , h e r e00