Fundamentals of the Theory of Operator Algebras: Elementary Theory [Vol. 1, 1st ed.] 9780080874166, 9780123933010, 9780821808191, 0123933013, 0821808192

Table of contents :
Fundamentals of the Theory of Operator Algebras......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
Contents of Volume II......Page 14
1.1. Algebraic results......Page 18
1.2. Linear topological spaces......Page 29
1.3. Weak topologies......Page 45
1.4. Extreme points......Page 48
1.5. Normed spaces......Page 52
1.6. Linear functionals on normed spaces......Page 60
1.7. Some examples of Banach spaces......Page 65
1.8. Linear operators acting on Banach spaces......Page 76
1.9. Exercises......Page 82
2.1. Inner products on linear spaces......Page 92
2.2. Orthogonality......Page 102
2.3. The weak topology......Page 114
2.4. Linear operators......Page 116
2.5. The lattice of projections......Page 126
2.6. Constructions with Hilbert spaces......Page 137
2.7. Unbounded linear operators......Page 171
2.8. Exercises......Page 178
3.1. Basics......Page 190
3.2. The spectrum......Page 195
3.3. The holomorphic function calculus......Page 219
3.4. The Banach algebra C(X)......Page 227
3.5. Exercises......Page 240
4.1. Basics......Page 253
4.2. Order structure......Page 261
4.3. Positive linear functionals......Page 272
4.4. Abelian algebras......Page 286
4.5. States and representations......Page 292
4.6. Exercises......Page 302
5.1. The weak- and strong-operator topologies......Page 321
5.2. Spectral theory for bounded operators......Page 326
5.3. Two fundamental approximation theorems......Page 342
5.4. Irreducible algebras—an application......Page 347
5.5. Projection techniques and constructs......Page 349
5.6. Unbounded operators and abelian von Neumann algebras......Page 357
5.7. Exercises......Page 387
Bibliography......Page 401
Index of Notation......Page 404
Index......Page 408
Pure and Applied Mathematics......Page 416

Citation preview

FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS VOLUMEI Elementary Theory

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.

FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS VOLUMEI Elementary Theory Richard V. Kadison

John R. Ringrose

Department of Mathematics University of Pennsylvania Philadelphia, Pennsylvania

School of Marhematics University of Newcastle Newcastle upon Tyne, England

1983

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco Silo Paulo Sydney Tokyo Toronto

COPYRIGHT @ 1983, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAQE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITINQ FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON)LTD.

24/28 Oval Road, London NWl

IDX

Library of Congress Cataloging in Publication Data Kadison, Richard V., date Fundamentals of the theory of operator algebras. (Pure and applied mathematics) Includes index. 1. Operator algebras. I. Ringrose, John R. 111. Series: Pure and applied mathematics 11. Title. (Academic Press) QA3.P8 [QA326] 512’.55 82-13768 ISBN 0-12-393301-3 (v. 1)

PRINTED IN THE UNITED STATES OF AMERICA a3 a4 a5 86

9

a

7 6 5 4 3 2 1

CONTENTS

vii xiii

Preface Contents of Volume I1

Chapter 1 . Linear Spaces 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9.

Algebraic results Linear topological spaces Weak topologies Extreme points Norrned spaces Linear functionals on normed spaces Some examples of Banach spaces Linear operators acting on Banach spaces Exercises

1 12 28 31 35 43 48 59 65

Chapter 2. Basics of Hilbert Space and Linear Operators 2.1. 2.2. 2.3. 2.4.

2.5. 2.6.

2.7. 2.8.

Inner products on linear spaces Orthogonality The weak topology Linear operators General theory Classes of operators The lattice of projections Constructions with Hilbert spaces Subspaces Direct sums Tensor products and the Hilbert-Schmidt class Matrix representations Unbounded linear operators Exercises V

75 85 97

99 100 103 109 120 120 121 125 147 154 161

C0N TEN TS

vi Chapter 3. Banach Algebras

3.1. Basics 3.2. The spectrum The Banach algebra L,(W) and Fourier analysis 3.3. The holomorphic function calculus Holomorphic functions The holomorphic function calculus 3.4. The Banach algebra C(X) 3.5. Exercises

173 178 187 202 202 205 210 223

Chapter 4. Elementary C*-Algebra Theory 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

Basics Order structure Positive linear functionals Abelian algebras States and representations Exercises

236 244 255 269 275 285

Chapter 5. Elementary von Neumann Algebra Theory 5.1. 5.2. 5.3. 5.4.

The weak- and strong-operator topologies Spectral theory for bounded operators Two fundamental approximation theorems Irreducible algebras-an application 5.5. Projection techniques and constructs Central carriers Some constructions Cyclicity, separation, and countable decomposability 5.6. Unbounded operators and abelian von Neumann algebras 5.7. Exercises

304 309 325 330 332 332 334 336 340 370

Bibliography

384

Index of Notation Index

387 391

PREFACE

These volumes deal with a subject, introduced half a century ago, that has become increasingly important and popular in recent years. While they cover the fundamental aspects of this subject, they make no attempt to be encyclopaedic. Their primary goal is to teach the subject and lead the reader to the point where the vast recent research literature, both in the subject proper and in its many applications, becomes accessible. Although we have put major emphasis on making the material presented clear and understandable, the subject is not easy; no account, however lucid, can make it so. If it is possible to browse in this subject and acquire a significant amount of information, we hope that these volumes present that opportunity-but they have been written primarily for the reader, either starting at the beginning or with enough preparation to enter at some intermediate stage, who works through the text systematically. The study of this material is best approached with equal measures of patience and persistence. Our starting point in Chapter 1 is finite-dimensional linear algebra. We assume that the reader is familiar with theresults of that subject and begin by proving the infinite-dimensional algebraic results that we need from time to time. These volumes deal almost exclusively with infinitedimensional phenomena. Much of the intuition that the reader may have developed from contact with finite-dimensional algebra and geometry must be abandoned in this study. It will mislead as often as it guides. In its place, a new intuition about infinite-dimensional constructs must be cultivated. Results that are apparent in finite dimensions may be false, or may be difficult and important principles whose application yields great rewards, in the infinite-dimensional case. Almost as much as the subject matter of these volumes is infinite dimensional, it is non-commutative real analysis. Despite this description, the reader will find a very large number of references to the “abelian” or “commutative” case-an important part of this first volume is an analysis of the abelian case. This case, parallel to function theory and measure theory, provides us with a major tool and an important guide to our vii

viii

PREFACE

intuition. A good part of what we know comes from extending to the noncommutative case results that are known in the commutative case. The “extension” process is usually dimcult. The main techniques include elaborate interlacing of “abelian” segments. The reference to “real analysis’’ involves the fact that while we consider complex-valued functions and, non-commutatively, non-self-adjoint operators, the structures we study make simultaneously available to us the complex conjugates of those functions and, non-commutatively, the adjoints of those operators. In essence, we are studying the algebraic interrelations of systems of real functions and, non-commutatively, systems of self-adjoint operators. At its most primitive level, the non-commutativity makes itself visible in the fact that the product of a function and its conjugate is the same in either order while this is not in general true of the product of an operator and its adjoint. In the sense that we consider an operator and its adjoint on the same footing, the subject matter we treat is referred to as the “self-adjoint theory.” There is an emerging and important development of non-selfadjoint operator algebras that serves as a non-commutative analogue of complex function theory-algebras of holomorphic functions. This area is not treated in these volumes. Many important developments in the selfadjoint theory-both past and current-are not treated. The type I C*algebras and C*-algebra K-theory are examples of important subjects not dealt with. The aim of teaching the basics and preparing the reader for individual work in research areas seems best served by a close adherence to the “classical” fundamentals of the subject. For this same reason, we have not included material on the important application of the subject to the mathematical foundation of theoretical quantum physics. With one exception, applications to the theory of representations of topological groups are omitted. Accounts of these vast research areas, within the scope of this treatise, would be necessarily superficial. We have preferred instead to devote space to clear and leisurely expositions of the fundamentals. For several important topics, two approaches are included. Our emphasis on instruction rather than comprehensive coverage has led us to settle on a very brief bibliography. We cite just three textbooks (listed as [HI, [K], and [R]) for background information on general topology and measure theory, and for this first volume, include only 25 items from the literature of our subject. Several extensive and excellent bibliographies are available (see, for example, [2,24,25]), and there would be little purpose in reproducing a modified version of one of the existing lists, We have included in our references items specifically referred to in the text and others that might provide profitable additional reading. As a consequence, we have made no attempt, either in the text or in the exer-

PREFACE

ix

cises, to credit sources on which we have drawn or to trace the historical background of the ideas and results that have gone into the development of the subject. Each of the chapters of this first volume has a final section devoted to a substantial list of exercises, arranged roughly in the order of the appearance of topics in the chapter. They were designed to serve two purposes: to illustrate and extend the results and examples of the earlier sections of the chapter, and to help the reader to develop working technique and facility with the subject matter of the chapter. For the reader interested in acquiring an ability to work with the subject, a certain amount of exercise solving is indispensable. We do not recommend a rigid adherence to ordergach exercise being solved in sequence and no new material attempted until all the exercises of the preceding chapter are solved. Somewhere between that approach and total disregard of the exercises a line must be drawn congenial to the individual reader’s needs and circumstances. In general, we do recommend that the greater proportion of the reader’s time be spent on a thorough understanding of the main text than on the exercises. In any event, all the exercises have been designed to be solved. Most exercises are separated into several parts with each of the parts manageable and some of them provided with hints. Some are routine, requiring nothing more than a clear understanding of a definition or result for their solutions. Other exercises (and groups of exercises) constitute small (guided) research projects. On a first reading, as an introduction to the subject, certain sections may well be left unread and consulted on a few occasions as needed. Section 2.6, Tensor products and the Hilbert-Schmidt class (this “subsection” is the largest part of Section 2.6) will not be needed seriously until Chapter 11 (in Volume 11). All the material on unbounded operators (and the material related to Stone’s theorem) will not be needed until Chapter 9 (in Volume 11). Thus Section 2.7, Section 3.2, The Banach algebra L,(R) and Fourier analysis, the last few pages of Chapter 4 (including Theorem 4.5.9), and Section 5.6, can be deferred to a later reading. Some readers, more or less familiar with the elements of functional analysis, may want to enter the text after Chapter 1 with occasional back references for notation or precise definitions and statements of results. The reader with a good general knowledge of basic functional analysis may consider beginning at Section 3.4 or perhaps with Chapter 4. The various possible styles of reading this volume, related to the levels of preparation of the reader, suggest several styles and levels of courses for which it can be used. For all of these, a good working knowledge of point-set (general) topology, such as may be found in [K],is assumed. Somewhat less vital, but useful, is a knowledge of general measure the-

X

PREFACE

ory, such as may be found in [HI and parts of [R]. Of course, full command of the fundamentals of real and complex analysis (we refer to [R]for these) is needed; and, as noted earlier, the elements of finite-dimensional linear algebra are used. The first three chapters form the basis of a course in elementary functional analysis with a slant toward operator algebras and its allied fields of group representations, harmonic analysis, and mathematical (quantum) physics. These chapters provide material for a brisk one-semester course at the first- or second-year graduate level or for a more leisurely one-year course at the advanced undergraduate or beginning graduate level. Chapters 3, 4, and 5 provide an introduction to the theory of operator algebras and have material that would serve as a onesemester graduate course at the second- or third-year level (especially if Section 5.6 is omitted). In any event, the book has been designed for individual study as well as for courses, so that the problem of a wide spread of preparation in a class can be dealt with by encouraging the better prepared students to proceed at their own paces. Seminar and reading-course possibilities are also available. When several (good) terms for a mathematical construct are in common use, we have made no effort to choose one and then to use that one term consistently. On the contrary, we have used such terms interchangeably after introducing them simultaneously. This seems the best preparation for further reading in the research literature. Some examples of such terms are weaker, coarser (for topologies on a space), unitary transformation, and Hilbert space isomorphism (for structure-preserving mappings between Hilbert spaces). In cases where there is conflicting use of a term in the research literature (for example, “purely infinite” in connection with von Neumann algebras), we have avoided all use of the term and employed accepted terminology for each of the constructs involved. Since the symbol * is used to denote the adjoint operations on operators and on sets of operators, we have preferred to use a different symbol in the context of Banach dual spaces. We denote the dual space of a Banach space 3 by 3”.However,,we felt compelled1by usage to retain the terminology “weak *” for the topology induced by elements of 3E (as linear functionals on 3’ ), Results in the body of the text are italicized, titled Theorem, Proposition, Lemma, and Corollary (in decreasing order of “importance”though, as usual, the “heart of the matter’’ may be dealt with in a lemma and its most usable aspect may appear in a corollary). In addition, there are Remarks and Examples that extend and illuminate the material of a section, and of course there are the (formal) Definitions. None of these items is italicized, though a crucial phrase or word frequently is. Each of these segments of the text is preceded by a number, the first digit of which

PREFACE

xi

indicates the chapter, the second the section, and the last one- or twodigit number the position of the item in the section. Thus, “Proposition 5.5.18” refers to the eighteenth numbered item in the fifth section of the fifth chapter. A back or forward reference to such an item will include the title (“Theorem,” “Remark,” etc.), though the number alone would serve to locate it. Occasionally a displayed equation, formula, inequality, etc., is assigned a number in parentheses at the left of the display-for example, the “convolution formula” of Fourier transform theory appears as the display numbered (4) in the proof of Theorem 3.2.26. In its own section, it is referred to as (4) and elsewhere as 3.2(4). The lack of illustrative examples in much of Chapter 1 results from our wish to bring the reader more rapidly to the subject of operator algebras rather than to dwell on the basics of general functional analysis. As compensation for their lack, the exercises supply much of the illustrative material for this chapter. Although the tensor product development in Section 2.6 may appear somewhat formal and forbidding at first, it turns out that the trouble and care taken at that point simplify subsequent application. The same can be said (perhaps more strongly) about Section 5.6. The material on unbounded operators (their spectral theory and function calculus) is so vital when needed and so susceptible to incorrect and incomplete application that it seemed well worth a careful and thorough treatment. We have chosen a powerful approach that permits such a treatment, much in the spirit of the theory of operator algebras. Another (general) aspect of the organization of material in a text is the way the material of the text proper relates to the exercises. As a matter of specific policy, we have not relegated to the exercises whole arguments or parts of arguments. Reference is occasionally made to an exercise as an illustration of some point-for example, the fact that the statement resulting from the omission of some hypothesis from a theorem is false. During the course of the preparation of these volumes, we have enjoyed, jointly and separately, the hospitality and facilities of several universities, aside from our home institutions. Notable among these are the Mathematics Institutes of the Universities of Aarhus and Copenhagen and the Theoretical Physics Institute of Marseille-Luminy. The subject matter of these volumes and its style of development is inextricably interwoven with the individual research of the authors. As a consequence, the support of that research by the National Science Foundation (U.S.A.) and the Science Research Council (U.K.) has had an oblique but vital influence on the formation of these volumes. It is the authors’ pleasure to express their gratitude for this support and for the hospitality of the host institutions noted.

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CONTENTS OF VOLUME 11 Advanced Theory

Chapter 6. Comparison Theory of Projections 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9.

Polar decomposition and equivalence Ordering Finite and infinite projections Abelian projections Type decomposition Type I algebras Examples Ideals Exercises

Chapter 7. Normal States and Unitary Equivalence of von Neumann Algebras 7.1. 7.2. 7.3. 7.4. 7.5. 7.6.

Completely additive states Vector states and unitary implementation A second approach to normal states The predual Normal weights on von Neumann algebras Exercises

Chapter 8. The Trace 8.1. Traces

The trace in finite algebras The Dixmier approximation theorem The dimension function Tracial weights on factors Further examples of factors An operator-theoretic construction Measure-theoretic examples 8.7. Exercises 8.2. 8.3. 8.4. 8.5. 8.6.

xiii

xiv

CONTENTS OF VOLUME I1

Chapter 9. Algebra and Commutant 9.1. The type of the commutant 9.2. Modular theory A first approach to modular theory Tomita's theorem-a second approach A further extension of modular theory 9.3. Unitary equivalence of type I algebras 9.4. Abelian von Neumann algebras 9.5. Spectral multiplicity 9.6. Exercises

Chapter 10. Special Representations of C*-Algebras 10.1. The universal representation 10.2. Irreducible representations 10.3. Disjoint representations 10.4. Examples Abelian C*-algebras Compact operators !a(,#)and the Cakin algebra Uniformly matricial algebras 10.5. Exercises

Chapter 11. Tensor Products 11.1. Tensor products of represented C*-algebras

11.2. Tensor products of von Neumann algebras Elementary properties The commutation theorem The type of tensor products Tensor products of unbounded operators 1I .3. Tensor products of abstract C*-algebras The spatial tensor product C*-norms on 'LI 0 d Nuclear C*-algebras 11.4. lnfinite tensor products of C*-algebras 113. Exercises

Chapter 12. Approximation by Matrix Algebras 12.1. 12.2. 12.3. 12.4.

Isomorphism of uniformly matricial algebras The finite matricial factor States and representations of matricial C*-algebras Exercises

CONTENTS OF VOLUME I1

Chapter 13. Crossed Products 13.1. 13.2. 13.3. 13.4.

Discrete crossed products Continuous crossed products Crossed products by modular automorphism groups Exercises

Chapter 14. Direct Integrals and Decompositions 14.1. 14.2. 14.3. 14.4.

Direct integrals Decompositions relative to abelian algebras Appendix-Bore1 mappings and analytic sets Exercises

Bibliography

xv

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CHAPTER I

LINEAR SPACES

This chapter contains an account of those basic aspects of linear functional analysis that are needed, later in the book, in the study of operator algebras. The main topics - continuous linear operators, continuous linear functionals, weak topologies, convexity - are studied first in the context of linear topological spaces, then in the more restricted setting of normed spaces and Banach spaces. In preparation for this, some related material is treated in the purely algebraic situation (that is, without topological considerations). 1.1. Algebraic results

In this section we shall consider linear spaces (that is, vector spaces) over a field K, and it will be assumed throughout that K is either the real field R or the complex field C. We sometimes distinguish between these two cases by referring to real vector spaces or complex vector spaces. Our main concern is with linear functionals, convex sets, and the separation of convex sets by hyperplanes. Suppose that $ . is a linear space with scalar field K. If X and Y are nonempty subsets of $; and U E K, we define further subsets a X , X f Y by ax= (as:sEXJ,

x+

Y = {s+y:xEX,yEY},

and

x-

Y = X + ( - 1)Y.

When X consists of a single element s,we write x f Y in place of X f Y. To avoid ambiguity in the use of the symbol-, the set theoretic difference { ~ E A : ~ $oftwosetsGandBwillbedenotedbyA\B. BJ Avectoroftheform a 1 x , + . . . + a,x,, where x I ,. . . ,x, E X and a l ,. . . ,a, E 06, is called a (finite) linear combination of elements of X . The zero vector is always of this form (in a . . . ,x,) an arbitrary finite subset of X , and aj = 0 for trivial way), with {s,, each j . If it can be expressed as a non-triuiul linear combination of elements of X (that is, with x l , .. . ,x, distinct, and at least one aj non-zero), then Xis said to be linearly dependent; otherwise X is linearly independent. The set of all

2

1. LINEAR SPACES

finite linear combinations of elements of X is a linear subspace of the smallest containing X ;we refer to it as the linear subspace generated by A'. If 9; is a linear subspace of % we denote by Y-/Y,j the set of all cosets x + Yo(XE Y ) in the additive group $C Of course, Y/%, is a group, with addition defined by (x + Yo)+ ( y + Yo)= (x + y ) + 9';. If a € H, and x1 + %$= x2 + G, we have axl - axz = a(xl - x 2 ) ~ V o ,so axl + $; = ax2 + 6. From this it follows easily that $'-/Yobecomes a linear space over 06, the quotient of Y'. by Y;, when multiplication by scalars is defined (unambiguously) by a(x + G) = ax + Yo.If V / V i has finite dimension n, we say that YG has finite codimension n in 9'.' Suppose that Y and W are linear spaces over K. By a linear operator (or linear transformation) from Y into .Iyc we mean a mapping T : Y + W such that T(ax + by) = aTx

+ bTy

whenever x, Y E V and a, b~ K (the notation T : Y + W indicates that T is defined on Y and takes values in $6'"; it can be read " T , from Y into W " ) .If Y; is a linear subspace of K the equation Q x = x + Yo defines a linear operator Q from Y onto V - / P ; ,the quotient mapping. When T : Y + Y#" is a linear operator, the nullspace of T i s the linear subspace { X E Y :T x = 0 ) of Y, and the image (or range) T ( Y ) = { T X : X E V } is a linear subspace of %KIf T( Pi) = { O } , the condition x + = y + 6 entails x - Y E U;, and hence T X - Ty = 0 ; moreover, if 6 is the null space of T, Tx = 0 entails X E Yo. From this, the equation T,(x + Yo)= T x defines (unambiguously) a linear operator To from V / V ; onto T ( Y )( E W"),when T(Yo)= ( 0 ); and To is oneto-one if $;' is the null space of T. Note that T = T o e , a fact sometimes described by saying that Tfactors through Y/Yowhen T ( 6 ) = { O } . Given any linear operators S, T : Y- + W" and scalars a, 6, the equation (aS + b T ) x = aSx + bTx ( X E V ) defines another such operator aS + bT, and in this way, the set of all linear operators from 9- into W becomes a linear space over K. By a linearfunctionalon Y we mean a linear operator p : *Y' + H (of course, K is a one-dimensional linear space over W). The set of all linear functionals on 3' is itself a linear space over K, the algebraic dualspace of Y': When p is a nonzero linear functional on V (that is, p does not vanish identically on U')the image p ( V )is K. 1.1.1. PROPOSITION. If p is a linearfunctionalon a linear space $ ;then every linear functional on Y' that vanishes on the null space Yoof p is a scalar multiple of p . I f ' p # 0, Y; has codimension 1 in $C Conversely each linear subspace of codimension 1 in V - is the null space of a non-zero linearfunctional. U p ,, . . . ,pn are linearfunctionals on %< then every linearfunctional on 9' that vanishes on the intersection ojthe null spaces of p l , . . . ,p . is a linear combination of p l , . . . ,pn.

3

1 . 1 . ALGEBRAIC RESULTS

Proof We may suppose that p # 0. The equation p o ( x + Yo)= p ( x ) defines a one-to-one linear operator p o from *Y /YG onto the one-dimensional linear space K ;so 9 ‘/Yois one dimensional. Of course, p o is a non-zero linear functional on $ ’/$i; and in the same way, if a linear functional B on 3’vanishes on T i , there is a linear functional go on V/Yo,defined by o0(x + r’;) = cr(x). Since $//Yo is one dimensional, o0 = upo for some scalar a, and B = o0Q = apoQ = up, where Q is the quotient mapping from Y onto Y/Y&

If 9 ; is a linear subspace with codimension 1 in 0 ( j = 1, . . . ,n) and 2 a, = 1. It makes no difference in this definition if the condition a, > 0 is relaxed to a, b 0 (because zero terms can be deleted), but strict inequality is slightly more convenient for our present purposes. We say that Y is conres if b , y , + h2y2E Y whenever y , , y 2 E Y and b , , 6, are positive real numbers with sum 1 (that is, Y contains each convex combination ofjust two elements of Y ;geometrically, this means that each line segment with endpoints in Y lies wholly in Y ) .A simple proof, by induction on n, shows that a convex set Y contains every convex combination a l x l + * . . + a,x, of elements s1, . . . , x, of Y ; the “inductive step up,” from n - 1 to n, depends on the observation that a , x , + . . . + a,x, = b , y , b,y,, where b , = a , , b2 = a2 + . * . + a n r y l= sI, andy, is theconvexcombination b; l(U2.Y2 + . . . + a,x,) of x 2 , . . . ,x,. It is sometimes useful to note that a subset Y of Y is convex if and only if a, Y a, Y = ( a , + a 2 )Y whenever a, and a, are non-negative scalars; for this

+

+

4

I . LINEAR SPACES

isequivalenttn(ai + u 2 ) - l ( u ,Y + a 2 Y )= Y, whena, anda2arenon-ncgativc scalars, not both 0. When X G $',.we denote by co X the set of all finite convex combinations of elements of X . A straightforward calculation shows that if y , , . . . ,y . E C O X, then every convex combination of y , ,. . . ,y . lies in co X. Thus co Xis a convex set, the smallest one containing X; it is called the convex hullof X . By an internal point of X we mean a vector x in Xwith the following property: given any y in Y'; there is a positive real number c such that x + ayE X whenever 0 6 a < c. Our next result is concerned with real vector spaces. By a hyperplane, in a linear space V over R, we mean a set of the form xo + Yo,where xoE V and U,j is a linear subspace with codimension 1 in K From Proposition 1.1.1, a subset H of V ' is a hyperplane if and only if it can be expressed in the form

H

=

{ x E Y : ~ (=xk)} ,

where p is a non-zero linear functional on Y and k E R; of course, p and k are not uniquely determined by H, but the only possible variation is to replace them by ap and ak, respectively, where a is a non-zero real number. With the hyperplane H we can associate the two closed half-spaces, { X E f " : . p ( x )2 k } and { X E V : p ( x )< k } , and the two open half-spaces, which are defined similarly but with strict inequalities. We say that H separates two subsets Y and Z of YT if Y is contained in one of the closed half-spaces determined by Hand Z is contained in the other; strict separation is defined similarly in terms of the open half-spaces. If the hyperplane is described in terms of ap and ak, the property of separation remains unchanged (although the two half-spaces are interchanged if a < 0). 1.1.2. THEOREM.I f Y and Zare non-empty disjoint convex subsets of a real rector space V;at least one of which has an internalpoint, they are separated by a hyperplane H in 9.:I f either Y or Z consists entirely of internal points, it is contained in one of the open half-spaces determined by H . Ifboth Y and Z consist entirely of internal points, they are strictly separated by H . Proof: We may suppose that Y has an internal point, and denote by Yi the set of all internal points of Y. It is easily verified that Yi is a convex subset of Y, and that (1 - a)yl + a y e Yi whenever y , E Yi, Y E Y , and 0 < a < 1 . We assert that every point of Yi is an internal point of Yi, and that a hyperplane which separates Yi and Z also separates Y and Z . For this, suppose that y , E Yi and X E K Since y , is an internal point of Y , y , + C X E Y for some positive scalar c. From the preceding paragraph,

+ a c x = ( I a)yl + a ( y l + C X ) E Yi when 0 G a < 1 ;so y, + bx E Yi whenever 0 G b < c, and thus y , is an internal y1

-

point of Yi. If H is a hyperplane separating Yi and Z , there is a non-zero linear

functional p on Y and a scalar k such that H p(yi) 2 k k p(=)

(4'1 E

U)P(Yl)

+ w(i)= P(( 1

and

Y,, ZEZ).

Given y in Y , choose any y , in Y ,. Since ( 1 - a)y have -

f : p ( ~= ) k}

=

'

(1

5

ALGEBRAIC RESULTS

1.1.

, + ay

- Q)Yl

E

Y, when 0 < a < I , we

+ ay) 2 k ;

and when a + I , we obtain p ( y ) 2 k . Thus H separates Y and Z . Upon replacing Y by Y , , it now suffices to prove the theorem under the additional assumption that each point of Y is an internal point of Y. In this case, Y - Z is a convex subset of I . consisting entirely of internal points and not containing 0. Let @? be the family of all convex subsets C of Y for which 0 4 C, Y - Z c C, and each point of C is an internal point of C. Then W is partially ordered by the inclusion relation G . If Wo is a totally ordered subfamily of @?, let C1 be the union of all the sets in go.It is apparent that O $ C , , Y - Z c C , , and C, consists entirely of internal points. Given u and u in C , ,there is a single set Coin 'Z0containing both u and u (because V 0is totally ordered by G ) . Thus Co (and hence, also C,) contains every convex combination of u and u, and so C , is convex. Accordingly, C1EV, and C, is an upper bound for % o. It now follows from Zorn's lemma that there is an element C of % that is maximal with respect to inclusion. It is immediately verified that the set { a u : u ~ Ca ,> 0 )

is an element of $5 and contains C. By maximality, it coincides with C ; so U U E C whenever U E C and a > 0. From this, and since C is convex and 0 4 C , it now follows that C n - C = 0 (the empty set), and U U E C , au + bilEC,

bwE T \ C

whenever u, LIEC, W E 9-\C, a > 0, and b 2 0. We assert next that au + be E Y -\C whenever u, I ) E .Y\C and a, b 2 0. For this, suppose the contrary, so that au + be E C for some a, b, u , u satisfying the stated conditions. From the preceding paragraph, au, bue Y \ C ; so, upon replacing u by au and c by bc, we may suppose that u, u E Y \ C and u + c E C. When r 2 0, we have 2rr~"Y'\C,

2rc

= r(r

+ u) + r(u - u)

and L' + U E C ; so, again from the preceding paragraph, Accordingly, if C, = {x

then 0 4 C, , C,

2

C (3Y

-

T(L'

- u ) $ C.

+ r(u - u ) : x ~ Cr ,k 0 ) .

Z ) , and C1 is convex and consists entirely of

6

I . LINEAR SPACES

internal points. Thus C , EV, by maximality C1 = C, and hence

2u = ( u + u )

+ ( u - U ) E C 1 = c,

a contradiction (since U U E Y‘\C for all a 2 0). This proves our assertion that au + ~ V Y’\C E whenever u, L ~ E Y \ and C a, b 2 0. It follows that the set VO = { x E ” / ” : x, - X E V . \ C } = Y-\(cu- C )

is a linear subspace of V (and Von C = 0, whence Vo# V - ) . We prove next that has codimension 1 in V:To this end, we have to show that any two non-zero elements of Y‘/Y; are linearly dependent; that is, if u, L‘ E Y T \ %$, then au + bu E Yo for suitable non-zero scalars a, b. Since $/\YO = C u - C, we may suppose (upon replacing u by - u, or L] by - t i , if necessary) that u E C and L’ E - C. Since C consists entirely of internal points, the same is true of - C. From this, the disjoint subsets

+ s(u - U ) E C } , 1]:u + s(u - U)E - C }

So = {SE[O, I]:u S1 =

{SE[O,

of the real interval [0, 13 are both open; indeed, if u + so(u - U ) E C (or - C), then u - u ) E C (or - C) for all s sufficiently close to so, since both the points u s0(tl - u) f t(u - u ) lie in C (or - C) for all sufficiently small non-negative 1. Since 0 E So, 1 E S1,and [0,1] is connected, So u S1is not the whole of [0, I]; so there is a real number s such that 0 < s < 1 and

+ +

s(tl

( I - s)u

+ su = u + s(v-

u)EY‘\(Cu - C ) =

%O.

This completes the proof that V; has codimension 1 in %’: Let p be a (non-zero) linear functional on Y‘ whose null space is %$.Since C is convex and Y 0 n C = 0, the subset p(C)of R is convex and does not contain 0 ; so either p(C) G ( 0 , ~ or ) p(C) G ( - o 0 , O ) . Upon replacing p by - p if necessary, we may suppose that p(u) > 0 for all u in C. Since Y - Z E C , it follows that p ( y ) > p(z) whenever y e Yand Z E Z From . this, the subset p ( Z ) of R is bounded above, and its least upper bound k satisfies P(Y) 2 k 2 p ( 4

( Y E Y,

zEZ )

Thus the hyperplane { X E Y . :p ( x ) = k} ( = H ) separates Y and Z . From the assumption that Y consists entirely of internal points, we now deduce that it is contained in the open half-space { x E Y - : p ( x ) > k ) . For this, suppose that Y E Y, and choose xo in V such that p(x,) > 0. Then y - axo E Y (and therefore p(y) - ap(xo)2 k ) for all sufficiently small positive scalars a, and thus p ( y ) > k. If Z (as well as Y) consists entirely of internal points, a similar argument shows that p(z) < k for each z in Z ; so in this case Y and Z are strictly separated by H .

7

1 . 1 . ALGEBRAIC RESULTS

Theorem 1.1.2 is our first example from a group of related results, described loosely as Hahn-Banach theorems. These results, which occur both in the present algebraic setting and also in the context of linear topological spaces, can be divided broadly into two main types. The first group (separation theorems) is concerned with separation of convex sets; closely related to this, there is a second group (extension theorems for linear functionals). A complex vector space Y can be viewed also as a real vector space simply by restricting attention to real scalars. Occasionally, for emphasis, we shall denote the real vector space so obtained by K-. A linear functional on % is described as a real-linearfunctional on V,and linear subspaces of %are called real-linear subspaces of .Y: A set X ( E V )is convex if and only if it is convex and the internal points of Xare the same in both when viewed as a subset of cases, since the concepts of “convex set” and “internal point” depend only on real scalars. In proving Hahn-Banach theorems for complex vector spaces, we shall require the following simple result.

k ( Y E Y ) if Y consists entirely of internal points, and k > Re p(z) ( z E Z ) if Z consists entirely of internal points. Proof. By considering Y and 2 as subsets of the real vector space obtained from K i t follows from Theorem 1.1.2 that there is a non-zero reallinear functional a on 9" and a real number k such that a(y) > k 3 ~ ( z ) whenever y E Y and z E Z. Moreover, if either Y or Zconsists entirely of internal points, the corresponding one of the inequalities 2 can be replaced by > . By Lemma 1.1.3, there is a linear functional p on Y such that a(x) = Re p ( x ) for each x in % H Let 3' be a linear space with scalar field H ( = R or C). By a sublinear functional on V - we mean a function p : Y -,R such that

+ Y ) G P(X) + P(Y),

p(x

p(ax)= d x )

whenever x , y ~ V "and a is a non-negative real number. If, further, (x E K

p ( a 4 = lalp(x)

a E H),

p is described as a semi-norm on % If p is a semi-norm, then P(X) 2 09

Indeed, 2p(x) = p ( x )

I&)

- p(y)l

G A x - v)

(X,YE %

-1.

+ p ( - x) > p ( x - x) = 0; while

P(X) = P ( ( X

- Y ) + v)d p ( x - y ) + P(Y)?

whence p ( x ) - p ( y ) d p ( x - y ) , and similarly P ( Y ) - P(4

< p ( y - x) = p ( x - y ) .

By a norm on V i we mean a semi-norm p such that p ( x ) > 0 whenever X E $1 x # 0.

As an example note that if H is R or C and n is a positive integer, the set H" consists of all ordered n-tuples ( a l , . . . ,an)of elements of 06, and is an ndimensional vector space when the algebraic structure is defined by a(a1,. . . ,a,)

+ b(b1,. . . ,b,) =

(a01

+ bbl, . . . ,UU, + bb,).

The equations ~~((a1~...~an))=Ia11 + ...

+Ian13

p,((al?...,a,)) = max{la,l,. . . la,l> 9

define norms, p1 and p a , on K". In particular, the modulus function is a norm on K. A subset Y of V is said to be balanced if ay E Y whenever y E Y , a E H,and la1 < 1. Ifp is a sublinear functional on Kit is immediately verified that the set V,, = {XE V : p ( x )< I } is convex, contains 0, and consists entirely of internal points; V p is balanced if p is a semi-norm.

9

ALGEBRAIC RESULTS

1.1.

1.1.5. PROPOSITION. Suppose that V is a convex subset of a linear space ooer K ( = R or C ) , and 0 is an internal point of V. Then the equation

v

p ( x ) = inf{c:cER, c > 0, X E C V }

(xEV) dejines a sublinearjunctionalp on 9; I f V consists entirely of internalpoints, then V = {.YE ^I : p ( x ) < 1 }. If V is balanced, p i s a semi-norm.

Proof: Given .Y in %; c- ‘ X E V (and thus X E C V )for all sufficiently large positive scalars c, since 0 is an internal point of V ; so p ( x ) , as defined in the proposition, is a non-negative real number. Suppose that x , y ~ Vand a > 0. Since X E C Vif and only if axEacV, it follows that p ( a x ) = ap(x) (and this remains true when a = 0, since it is apparent that p(0) = 0). Given any positive real number E , we can choose real numbers b and c so that

0 < b I , a n d O , c - ’ z ~ V , i t now follows that z E V . Accordingly, V = { x E V : p ( x )< 1 }. The sublinear functional p occurring in Proposition 1.1.5 is called the support ,functional of V. Our next two results are Hahn-Banach theorems of the “extension” type. 1.1.6. THEOREM. I f p is a sublinear .functional on a real vector space V; while po is a linear,functionalon a linear subspace of Y< and

Po(Y) G P O )

(YE

m r

there is a linear functional p on P- such that p ( s ) G p ( x ) (sE

f’T

p ( y ) = po(y)

( y E PO).

Proof: The product set R x V ” becomes a real vector space when addition and scalar multiplication are defined by (r,x)

+ b y ) = (r + s,.Y + y).

a(r,x)

= (ar,ax),

10

I . LINEAR SPACES

for x , y in Y and a, r, s in R. From the defining properties of sublinear functionals, it is immediately verified that the set V = { ( r , x ) E R x P . : r > p ( x ) } (ER x 3 . )

is non-empty, convex, and consists entirely of internal points. The set

w = {(Po(Y)7Y ): Y E VO} is a linear subspace of R x Y (and is therefore convex), and V n W = $3. From Theorem 1.1.2, there is a linear functional c o n R x $‘and a real number k such that a(u) > k b a ( w )

If

W E W , then

(U€

V,

W).

WE

awE W , and thus aa(w) = a(aw) < k , for every scalar a ; so a(w) = 0

(W€

W),

and k 3 0. From this, and since (1,O) E V , it follows that a(( 1,O)) > k 2 0 ; upon replacing a by a suitable positive multiple of a, we may assume that a((1,O)) = 1. The equation p ( x ) = - a ( ( 0 , x ) ) defines a linear functional p on f ; and a(@, x ) ) = a(r( 1,O)

+ (0,x)) = r - p ( x )

(r E R, x E $ ‘).

Given any x in V; we have ( r , x ) E V , and therefore r - p ( x ) = a ( ( r , x ) )> k 2 0 , whenever r > p ( x ) ; so p ( x ) < p ( x ) . When

YE

9$, ( p 0 ( y ) , y ) € W , and thus

Po(Y) - P(Y) = a((po(r),y)) = 0. H

I . 1.7. THEOREM. I f p is a semi-norm on a linear space P over K ( = R or C),while p o is a linear functional on a linear subspace 3 0 of’ 3 ;and IPo(Y)l

< P(Y)

( Y E YO),

there is a linear functional p on Y‘ such that IP(4l

0 and q l , . . . ,qmE r2.Since qjo T is continuous ( j = 1 , . . . ,m), the set W = {,re% :qj(Tx) < E ( j = 1,. . . , m ) } is a neighborhood of 0 in

K , and it is apparent that

T ( W )E V(0:q1,...,ym;&) G v. Thus T is continuous (at 0, and therefore throughout %). (iii) The scalar field H is a locally convex space, its (usual) topology being obtained from a single norm, the modulus function; and p : % + I6 is a linear operator. Thus (iii) is a special case of (ii). H

Our next two results are Hahn-Banach (separation) theorems. They are formulated so as to apply to both real and complex linear topological spaces, the notation Re being redundant in the real case. 1.2.9. THEOREM.r f Y and Z are disjoint non-empty convex subsets of a linear topologicalspace and Y is open, there is a continuous linearfunctional p on V and a real number k such that

e

Rep(y) > k 2 Rep@)

(.YE

Y, ~ € 2 ) .

%further, Z is open, then k > Rep(z) for each z in Z . Proof. In view of the fact that an open set consists entirely of internal points, the assumptions of Theorem 1.1.2 are fulfilled in the real case, while those of Theorem 1.1.4obtain in the complex case. From those theorems, there is a linear functional p on V that satisfies the stated inequalities; and by applying Lemma 1.2.4(i),with - p and Yin place of p and G , it follows that p is continuous. H

1.2.10. THEOREM.If Y and Z are disjoint non-empty closed convex subsets of a locally convex space K at least one of which is compact, there are real numbers a, b and a continuous linear functional p on .lr such that Rep(y) 2 a > b 2 Rep(z)

(YE

Y, Z E Z ) .

Prooj,f: We may suppose that Y is compact. For each y in Y, there is a balanced convex neighborhood V y of 0 such that ( y + V,,) n Z = 0, since Z is closed and y $ Z . The open covering { y + i V , , : y Y~ } of Y has a finite subcovering { y ( j )+ : V y c n f = j I , . . . ,m } , and the set V = $Vycjr is a balanced convex neighborhood of 0. The convex sets Y + V and Z + V are open since, for example, Y + V = U { y + V : Y EY ) ; we assert also that they are disjoint. For this,

ny=

I.?.

21

LINEAR TOPOLOGICAL SPACES

suppose the contrary, and choose y in Y, i‘ in Z , and cl, u2 in V , so that J’ + 1’1 = + “ 2 . For Some j ( 1 6 j < nz), y ~ y ( j ) :vycj,; moreover 1.1 ,- 1.2 E V E j VyI,). Thus

+

7

=y

+ v1 -

1’2

= y(.j)

+ ( y - y ( j ) ) + c 1 - u2 +

contradicting our assumption that (y(.j) Vylj,)n Z = 0. This proves our assertion that Y + V , Z + V are disjoint. From Theorem 1.2.9, we can choose a continuous linear functional p on ’I‘ and a real number k such that Rep(y) > k > Re&)

(JJE Y

+ V,

+

ZEZ V);

in particular, these inequalities are satisfied when y E Y and z E Z. Since Y is compact, the continuous function y : Y + R,defined by&) = Rep(y), attains its lower bound at a point y o of Y. Hence Re p(y) 3 Re p ( v d > k > Re p(z) and it suffices to take a

=

Rep(yo) and b

(vE Y ,

z EZ),

= k.

1.2.11. C O R O L L A R Y . I f ’ x is a non-zero rector in a locally convex space V,’ there is a continuous linear functional p on $ such that p ( x ) # 0. ’

Proq/: This follows from Theorem 1.2.10, with Y = {x} and 2 = {O}. 1.2.12. COROLLARY. IfZ is a closed convex subset of a locally convex space ,-and},E $-\Z, there is a continuous linearfunctional p on Y” and a real number b W C / J that Re p(.v) > b, Re p(z) 6 b ( z E Z ) . f

Proof. This follows from Theorem 1.2.10, with Y = { y } . 1.2.13. C O R O L L A R Y . If2 is a closed subspace of a locally convex space Y i andyE $ \Z, there is a continuous linearfunctional p on V- such that p ( y ) # 0, p(z) = 0 ( z E Z ) .

Proof. From Corollary 1.2.12 there is a continuous linear functional p on a real number b such that

f and

Re y ( y ) > b,

Re p ( z ) 6 b

(z E Z ) .

The latter inequality implies that the range of values assumed by the linear functional p 1 Z on Z is not the whole of the scalar field. Hence p I Z = 0, b 2 0, Re p(y) > b 2 0, and thus p(y) # 0. We now prove a Hahn-Banach extension theorem.

22

I . LINEAR SPACES

1.2.14. THEOREM.I f p o is a continuous linear functional on a subspace $ 0 of a locally concex space there is a continuous linear functional p on Y such that p ) 3 ; = po. Proof. Let r be a family of semi-norms that gives rise, as in Theorem 1.2.6, to the topology on Y: By restricting each member of to f 6,we obtain a family of semi-norms that defines the relative topology on 9;. Since po is continuous, it follows from Proposition I .2.8(iii) that there is a positive real number C and a finite set p l , . . . ,pmof elements of f such that

IPO(Y)~

6 Cmax{ P I ( Y ) ,. . . tPm(Y))

(YE

$0).

By Theorem 1.1.7, p o extends to a linear functional p on f ; such that Ip(x)l

< Cmax{ P l ( X ) ,

*

. .,p,(x))

(x E Y, I

since the equation

AX) = Cmax{p,(x), . . . ,Pm(-K)J defines a semi-norm p on K A further application of Proposition 1.2.8(iii) shows that p is continuous. If 3'. is a locally convex space and X E we denote by [ X I the closure of the set of all finite linear combinations of elements of X. It is apparent that [ X I is the smallest closed subspace of V that contains X ; we describe it as the closed subspace generated by X. When X is a finite set, {xl,. . . ,x,,},we write [ x ,,..., x,] in place of [XI. In fact, [xl ,..., x,] is the set of all linear combinations of xl, . . . ,x,, since it results from Theorem 1.2.17 that this set is closed in -K We now consider some properties of finite-dimensional subspaces of locally convex spaces. In fact, the main results obtained remain valid in all linear topological spaces, without the restriction of local convexity. However, we shall not need that degree of generality, and have preferred to derive these results as simple consequences of the Hahn-Banach theorems in the locally convex case. We have already noted that, when Dd is R or C and n is a positive integer, the finite-dimensional vector space K", with its usual (product) topology, is a locally convex space. Each linear functional p on Dd" is continuous, since it is given by a formula p ( ( a l , .. . ,a,)) = alcl

+ . . . + a,c,,

where cl, . . . ,c,, are fixed elements of Dd. From elementary linear algebra, if ./11 is a proper subspace of K", there is a non-zero linear functional on K" that vanishes on ,A.

23

1.2, LINEAR TOPOLOGICAL SPACES

1.2.15. LEMMA.Suppose that $ is a locally convex space with scalarfield x,) is a basis of afinite-dimensionalsubspace ?; o f f Then there are continuous 1inear.functionalsp I ,. . . ,pn on Y such that pj(xj) = 1 and p,(.yk) = 0 when j # k . The equation '

K ( = R or C)and { xl, . . .

+

T.y = (PI(-y), . . . ,pfl(+y)) (x E V) defines a continuous linear operator T : P- -+ K", and the restriction TI% is a oneto-one hicontinuous linear operator from $;' onto K". Proof: The set IT'of all continuous linear functionals on V is a linear subspace of the algebraic dual space of V.'The equation s p = (P(XI),

. . ., p(xfl))

(PE

YU)

defines a linear operator S : Y' + 06". If the subspace S ( P ) is not the whole of K", there is a non-zero linear functional on K" that vanishes on S ( V ' ) ) ;in other words, we can choose c l , . . . ,c, in H,not all 0, so that ClP(X1)

+ . . . + c,p(x,)

=0

( p € V').

Thus every continuous linear functional p on Y vanishes at the non-zero vector c l x l + . . . + c,x, contradicting the conclusion of Corollary 1.2.11. It follows that S( V ' ) = 06". Accordingly, we can find p l , . . . ,pn in V'such that Spj = (0,. . . ,0, 1,0,. . . ,0) (with the 1 in thejth place); that is, p l , . . . ,pn are continuous linear functionals on and pj(xk) is 1 or 0 according a s j = k or j#k. I t is apparent that T, as defined in the lemma, is a continuous linear operator from V into K". Moreover, since pj(clxl

if follows that

+ . . + c.x,,) *

T ( c , x ~+ . . .

( j = 1,. . . ,n),

= cj

+ cp,) =

. . ,c,,).

( ~ 1 , .

Thus T carries V; onto K", and the restriction TlVo has a continuous inverse mapping (c1,

..., c n ) 4 c 1 x l + . . . +c,x,:

K"+YO.

H

I f f is a finite-dimensional linear space, with scalar 1.2.16. PROPOSITION. field K (= R or C),there is a unique locally convex topology on Y..

Proof. Let {x . . ,x,} be a basis of -Y; and define a one-to-one linear mapping T, from Y* onto K", by T(C,X,

+ . . . + c,x,)

= (c1,.

. . ,C").

There is a unique topology T on Y; which makes T a homeomorphism; locally convex, since K" is a locally convex space.

T

is

24

I . LINEAR SPACES

If Y has another locally convex topology q,,we can apply Lemma 1.2.15 (with 9'; = 9'-). Since the mapping T just defined is the same as the one occurring in the lemma, ro makes T a homeomorphism, and thus coincides with t. W 1.2.17. THEOREM.If Yois afinite-dimensional subspace of a locally convex space $; then Yois closed in *I", moreover, there is a closedsubspace % of Y such thaf Y ; and % are complementary subspaces of Y and the projection from Y onto $0 parallel to % is continuous. Proof. Let { x l , . . .,xn} be a basis of Yo.By Lemma 1.2.15, there are continuous linear functionals p, ,. . . ,pn on Y such that pj(xk) is 1 or 0 according as j = k or j # k. Each element y of Yois a linear combination clxl + . . . + cnxn.and p j ( y )= c j ; so n

Y

=

1 Pj(Y)xj

YE^;).

j= 1

The equation n

EX =

C pj(x)xj

(XE

Y)

j= 1

defines a continuous linear operator E : Y -+ f l From the preceding paragraph, Ey = y when y E $$,and it is apparent that E ( Y ) s Yo.It follows easily that

v; = {x E v :Ex = x}, that E Z x = Ex for each x in K and hence that E 2 = E. B y Theorem 1.1.8, 9; has a complementary subspace, "y; = { x E Y : E x = O } ,


1, and define a neighborhood G, of y by G, = {.YE V :Ip,(x)l > 1). The open covering { G , : ~ EV - \ V } of V - \ V has a finite subcovering; and if the linear functionals p,, corresponding to the sets G, in this subcovering, are enumerated as p l , . . . ,pn, then for each x in V - \ V there is at least one integerj such that 1 < j < n and Ipj(x)l > 1. The equation Tx = (pr(x), . . . ,p,(x)) defines a linear mapping T from Y into K". In order to prove that Y is finitedimensional, it suffices to show that T is one-to-one, and this follows easily from the two preceding paragraphs. Indeed, if x, E ,Y\(O},then ax, E V - \ Vfor some scalar a, and Ipj(ax,)l > 1 for some integerj with 1 < j < n ; so that pj(x,) # 0, and hence Tx, # 0. H We conclude this section with a discussion of nets and (unordered) infinite sums in a linear topological space K Suppose that v E K and ( u j , j € J , 3 ) (or, more briefly, { v j } ) is a net in f: the index set J being directed by the binary relation 2 . Then { i i j } converges to v if and only if, given any neighborhood V of 0, there is an index j , such that uj E v + V(equivalently, tij - E V ) whenever j >j,. From the definition of the uniform structure on C as set out in the discussion preceding Proposition 1.2.1, { u j } is a Cauchy net if and only if the following condition is satisfied: given any neighborhood V of 0, there is an indexj, such that uj - uk E V wheneverj, k 2 j o. If Y is a locally convex space and r is a separating family of semi-norms that determines its topology, each neighborhood V of 0 contains a basic neighborhood

n m

~ ( 0,,..., : ~p m ; E ) =

{xE~'-:pj(X 0 and x l , . . . ,x, E 9: From Theorem 1.3.1 (with Y replaced by V', and 9 = Y = +.), the weak* continuous linear functionals on Y' are precisely the elements of 9; so we have the following result. 1.3.5. PROPOSITION. A linearfunctionalo, on the continuous dualspace Y"' of a locally convex space -V; is weak* continuous ifand only ifthere is an element x of $'such that w(p) = p ( x ) for each p in V'. 1.4. Extreme points

Suppose that Y is a locally convex space. By the closed convex hull of a subset Y of V" we mean the closure CO Y of the convex hull co Y ; it is clear that this is the smallest closed convex set that contains Y. An element xo of a convex set Xin ./''is described as an extremepoint of Xif the only way in which it can be expressed as a convex combination xo = ( 1 - a ) x l + a x 2 , with 0 < a < 1 and x l , x2 in X , is by taking x1 = x2 = x0. We shall prove (Theorem 1.4.3) that every compact convex subset of P-has extreme points and is the closed convex hull of the set of all its extreme points. In the locally convex space R2 (that is, in the plane), a closed triangle is a convex set that has just three extreme points, its vertices. For a closed disk in R2, the extreme points are precisely the boundary points. In each of these examples, it is apparent that the set specified is the convex hull of its extreme points. In the case of a triangle, one might expect that the sides, as well as the vertices, have some significance in terms of convexity structure; in fact, each side is a "face," in a sense now to be defined.

32

I . LINEAR SPACES

By a face of a convex set X in Y we mean a non-empty convex subset F of X,such that the conditions O O . Proof. It suffices to show that x 1 E F if a, > 0. If a, = 1, then a2 = a 3 = . * . = a , = O and x1 = a l x l + . . * + a , x , ~ F . If 0 < a , < 1, let a = 1 - a , and let y be the convex combination a - l ( a Z x 2+ . * * + a,,.~,,)of x 2 ,..., x , , . T h e n x l , y ~ X , O < a < l , a n d

( I - a)xl + ay = a l x l

+ a2xz + . . . + U,X,E

F

Since F is a face of X , it follows that x , E F. I The results that follow are formulated so as to apply to both real and complex locally convex spaces, the notation Re being redundant in the real case. 1.4.2. LEMMA.If X is a non-empty compact convex set in a locally convex space ^y; p is a continuous linear functional on K and c = sup{Rep(x):xEX}, then the set F = { X E X: Re p(x) = c ) is a compact face of X . Proof. Since a continuous real-valued function on a compact set attains its supremum, Fis not empty; and it is evident that Fis compact and convex. If x , , x 2 e X , 0 < a < 1, and (1 - a)xl axzEF, we have R e p ( x , ) < c, Re p(xJ < c, and

+

(1 - a) Re p(xl)

+ a Re p(x2) = Rep(( 1 - a ) x l + a x z ) = c;

so Re p ( x l ) = Re p(x2) = c, and x, ,x2E F. Thus F is a face of X. 1.4.3. THEOREM(Krein-Milman). If X is a non-empty compact convex set in a locally convex space K then X has an exrremepoinr. Moreover, X = E6 E, where E is the set of all extreme points of X .

1.4. EXTREME POINTS

33

Proof: The family 9 of all compact faces of Xis non-empty since XE 8 and is partially ordered by the inclusion relation G . Let 9, be a subfamily of 9 that is totally ordered by inclusion. It is evident that 9,, has the finiteintersection property, so by compactness the set Fo = n { F : F E ~ , is} nonempty. Thus Fo is a compact face of X , and is a lower bound of 9,, in $? Since every totally ordered subset of 9 has a lower bound in it follows from Zorn’s lemma that 9 has an element F that is minimal with respect to inclusion. We shall show that F consists of a single point x, and since F is a (compact) face of X , it then follows that x is an extreme point of X . To this end, suppose the contrary, and let x,, x 2 be distinct elements of F. By the Hahn-Banach theorem, there is a continuous linear functional p on Y such that Re p(.ul) # Re p ( x 2 ) . From Lemma 1.4.2 we can choose a real number c so that the set

Fo = {xEF:Rep(x)= c)

is a compact face of F. Accordingly, Fois a compact face of X ; that is, F, E 9? Since Rep(xl) # Rep(x2), at least one of .xl, x2 lies outside F,; so F, is a proper subset of F, contrary to our minimality assumption. Hence Fconsists of a single point. So far, we have shown that each non-empty compact convex subset of V has an extreme point. If E denotes the set of all extreme points of X , it is clear that= E c X , and we have to show that equality occurs. Suppose thecontrary, and let xo E x\cO E ; we shall obtain a contradiction. From the Hahn-Banach theorem, we can find a continuous linear functional p on Y and a real number a such that Rep(x,) > a >, Rep(y)

(1)

If

c 1 = sup{Rep(x):xEX},then

(~ECOE).

cl > a , and the set

F, = { x E X :Re@)

= cl}

is a compact face of X by Lemma 1.4.2. In particular, F1 is a non-empty compact convex subset of -U; and so has an extreme point xl.Since x1 is an extreme point of a face of X , it is an extreme point of X ; that is, x, E E. However, Rep(x,) = c1 > a, contradicting (1). 1.4.4. COROLLARY. If X is a non-empty compact convex set in a locally concex space and p is a continuous linear junctional on Vi there is an extreme point so o j X such that Re p(x) < Re p(xo)for each x in X . Proof: Let c = sup{Rep(.x):.xEX}.

By Lemma 1.4.2, the set { x E X :Rep(x) = c} is a compact face of X . In

34

I . LINEAR SPACES

particular, it is a non-empty compact convex set in and so has an extreme point xo. Since xo is an extreme point of a face of X , it is an extreme point of X ; and Re p(x,)

=c

2 Re p ( x )

( x X~) .

1.4.5. THEOREM.IfXisanon-empty compact convexset in a locally convex space V and Y is a closed subset of X such that CO Y = X , then Y contains the extreme points of X . Proof. Suppose that xo is an extreme point of X . In order to show that it suffices to prove that xo + V meets Y whenever V is a balanced convex neighborhood of 0 in .Y ; for Y is closed, and sets of the form xo + V constitute a base of neighborhoods of xo. Given V as above, the family { y + + V :Y E Y } is an open covering of the compact set Y , and so has a finite subcovering { y j + : V : j = 1,. . . ,n } , with y , , . . . ,y , in Y. Let V - denote the closure of V , and f o r j = 1 , . . , ,n, let X j be the non-empty compact convex set ( y j i V - ) n X . Then X ~ Y, E

+

Y= YnXc

u

1+

(yj+iV) nXG

[j:,

u xj.

j:,

Let S be the set of all vectors of the form a l x l . . . + anxn,where xi€ Xi . . , n ) and the coefficients a , , . . . , a , are non-negative real numbers with sum 1. Then Scontains each X j , and so contains Y ; from the convexity of X I , . . . ,X,, it is readily verified that S is convex. We assert also that S is compact. To prove this, let A be the compact subset ( j = 1,.

{ ( a l , .. . , a , ) : a l 2 0 , . . . , a , 2 0, a ,

+ . . . + a, = 1 )

of R", and write a for the element ( a , , .. . ,a,) of A . By Tychonoff's theorem, the set A x X1 x * ' . x X , is compact in the product topology; and hence S is compact, since it is the image of the product set under the continuous mapping ( a , x l ,. . . ,x,)

+alxl

+ . . . + a,x,.

From the preceding paragraph, S is a compact convex set containing Y . Thus S contains the closed convex hull X of Y ; in particular, X,ES. Accordingly, xocan be expressed as a convex combination a l x l + . * . + anxn, where xi€ X j ( s X ) . Since xo is an extreme point of X , xo = x j for somej in { l , . . . , n } (take a n y j with a j > 0 ) , and xo = x j E

xj G yj + iv-.

:

Thus .xo - y j lies in theclosure of i V ; so the neighborhood xo - y j + Vmeets i V . Since V is balanced and convex, it now follows that y j e x o + V , whence xo + V meets Y.

35

1.5. NORMED SPACES

1.5. Normed spaces

In the discussion preceding Proposition 1.1.5, we introduced the concepts of “semi-norm” and “norm” on a (real or complex) linear space. In Theorem 1.2.6, we showed how a separating family f of semi-norms on such a space gives rise to a locally convex topology. The present section is concerned with the case in which f consists of a single norm. By a normedspace we mean a pair (X,p)in which X is a linear space whose scalar field K is either R or C andp is a norm on X. When x E X,we usually write llxll rather than p ( x ) , and refer to llxll as “the norm of x.” With this notation, the defining properties of a norm can be set out as follows: whenever x, y e X and U E K, (a) llxll 2 0, with equality only when x = 0; (b) llaxll = la1 Ilxll; (c) IIx + yl( Q llxll + llyll (the triangle inequality). We recall also another property, lllxll - llylll d IIx - yll, which is an easy consequence of the triangle inequality. Suppose that (X, 11 11) is a normed space. From the properties (a), (b), and (c), just noted, it is apparent that the equation

YE X) d(x,y) = IIx - YII defines a metric d on X. Also, X has a locally convex topology, the norm topology, derived as in Theorem 1.2.6 from the family f consisting of the single norm 11 11 on X; and this topology gives rise to a uniform structure on X, described in the discussion preceding Proposition 1.2.1. In the norm topology, each xo in X has a base of neighborhoods consisting of the sets V(xo: 11 )I;E ) ( E > 0), where V(x0:II ( ( ; E ) = { x E X : ( I X - X ~ ~ ~ < E } = { X E X : ~ ( ~ , X ~ ) < E } ,

the “open ball” with center xo and radius structure has a base, consisting of the sets {(X,y)EX x X:.u - y E V ( O : I l

I[;&)}

E.

The corresponding uniform

= {(?C,Y)EX x X:llx-yll = {(X,Y)€X x

0, we denote by (X), the closed ball {xE X: JJxJJ < r } ;we refer to (X), as the unit ball of X. Since (X), is convex and is closed in the initial (that is, the norm) topology, it is weakly closed, by Theorem 1.3.4. A subset 9 of X is bounded if %? c_ (X), for some positive real number r. A linear mapping T from a normed space X into another such space Y is said to be norm preserving if I)Txll = llxll for each x in X. Such a mapping is necessarily isometric (that is, distance preserving) and hence one-to-one, since

d(Tx1, Tx2) = llTx1 - Txzll = IIT(X1 - x2)ll = 11x1 - x2Il

= d(x,,x2);

and conversely, an isometric linear mapping is norm preserving. A normpreserving linear mapping from a normed space X onto another such space 9is sometimes described as an isometric isomorphism from X onto Y. A subspace X of a normed space Y is itself a normed space since the norm on OY restricts to a norm on X.The following theorem shows that every normed space can be viewed as an everywhere-dense subspace of an (essentially unique) Banach space. 1.5.1. THEOREM.If X is a normed space, there is a Banach space OY that contains X as an eiierywhere-dense subspace (and such that the norm on X is the restriction of the norm on YY).IfYl is another Banach space with theseproperties, the identity mapping on X extends to an isometric isomorphismfrom 9 onto W l . Proof. Let d be the metric on X derived from the norm; let 2 be the completion of the metric space X so obtained, and let adenote the metric on .%. Thus .% is a complete metric space, X is an everywhere-dense subset of 3, and

J(u, U ) = d(u,V ) = IIu

-

uII

(u,U E3).

We shall show that P can be made into a Banach space, with addition, scalar multiplication, and norm, extending those of X. When u, u’, 11, u’ E X and a E K (the scalar field), Il(U

+

2))

+ 0’)ll = I((u - u’) + (u - d)((< ((u- U’((+ ( ( u - cy,

- (U’

llau - au’ll = la1 IIU -

4

1

7

I llull - IIU’III < Ilu - 41.

Accordingly, the equations

f(u, 4 = u + 0,

g,(u) = au,

h(u) = IIuII

37

1.5. NORMED SPACES

define uniformly continuous mappings

f:X x X + X ( d ) ,

g,:X+X(sf),

h:X+R.

Since%. and R are complete, and X is everywhere dense in 2 (so that X x X is everywhere dense in f x A), it follows that go, h extend by continuity to uniformly continuous mappings

f :%.

x

f -+ .%, &: .% -+ .%, h: R + R,

respectively. The addition and scalar multiplication, already defined for elements of X, can now be extended to f by the equations u+v=3(u,c),

au=@,(u)

(U,UE.%,

UEK);

and we assert that, in this way, 2 becomes a linear space over K. To prove this, it suffices to verify the relations

+ 1‘ = L’ + u, u + ( - l ) =~ 0, u

+ ( 0 + w ) = (u + 0 ) + U’, u + 0 = u, (a + b ) = ~ au + bu, U ( U + U) = uu + U U , u

lu = u,

a(bu) = (ab)u,

for all u, P, M’in 2 and a, b in K. Of course, all these relations are satisfied when u, c, W E X, and simple continuity arguments show that they remain valid for elements of 2. For example, the relation a(u + ti) = au + au can be rewritten in the form

@,(f(4 11)) - f ( @ h h

40([3))

= 0,

and is satisfied when u, C E X. The left-hand side of this last equation is a continuous function of (u, 11) on .% x .%, which vanishes on the everywheredense subset X x X, and so vanishes throughout %. x $ (as required). Similar arguments establish the other relations; so $ is a linear space over 06. We prove next that h is a norm on .%.The relations h(u - 1’) - h(u, 0) = 0,

h(au) = la)h(u)

( a €K)

are satisfied when u, u E X, since h extends the norm on X. By continuity, they remain valid for all u, ti in .%.Accordingly,

h(u + 1‘) = h(u + 0,O) < h(u + 11, D )

+ &,O)

= h(u)

+ h(U),

and h(u) = d(u, 0) 3 0 (with equality only when u = 0), when u, u E 2. It now follows that h is a norm on f , which extends the norm on X,and gives rise to the complete metric a on 2. Hence $ is a Banach space and contains X as an everywhere-dense subspace. Finally, suppose that ?/ and OTY, are Banach spaces, each containing X as an everywhere-dense subspace and each with its norm extending the norm on 3.

38

I . LINEAR SPACES

The identity mapping on X can be viewed as a continuous linear operator, from an everywhere-dense subspace of the Banach space into the Banach space By Corollary 1.2.3, it extends to a continuous linear operator T from Y into ?V1. The equation f ( u ) = IITull - llull defines a continuous mapping f:9 -,R, and f vanishes on the everywhere-dense subset X of 9.Hence f vanishes throughout 9, and T is norm preserving, and therefore isometric. Since 4Y is complete and Tis isometric, the range TCY) of Tis complete, and is therefore closed in @YI. Thus T ( g )= g1,since T ( Y )contains the everywheredense subset X of Y l ; and T is an isometric isomorphism from !Y onto ?Yl, which extends the identity mapping on X. The Banach space ?2/ occurring in the statement of Theorem 1.5.1 is called the completion of the normed space X. The following lemma provides a useful criterion for the completeness of a normed space. It is couched in terms of the convergence of certain infinite series of vectors in the space; by the convergence of such a series, 1; x,, we mean convergence of the sequence of partial sums s, = x1 + . . . + x,. The result could easily be reformulated in terms of unordered sums of the type considered at the end of section 1.2. 1S . 2 . LEMMA. equioalent ;

if

X is a normed space, the ,following

two

conditions are

(i) X is a Banach space. (ii) /fxI,x2,. . .~XaandCIIx,,lI< oo,theseriesEx,conoeryes, in themetric of 3, to an element of X. Proof. Suppose first that X is a Banach space. Let {x,) be a sequence of elements of X, such that Cllx,ll < a,and write s, for the partial sum x 1 + . . . + x,. Given any positive E , there is a positive integer N such that ))x,)J< E (and hence Its, - s,)) = ))I;+ x,)) < E ) whenever m > n 2 N . Hence {sn} is a Cauchy sequence, in the complete space X, and so converges (that is, Ex,, converges). Conversely, suppose that condition (ii) is satisfied. If {y,} is a Cauchy sequence in X,there is a strictly increasing sequence {n(I), n(2),. . .) of positive integers such that

xr+

- Yflll < r k (m> n 2 n ( W . In particular, IIy,(k+ - yn(k)l(< 2 - k , and therefore llYm

0s

Ikn(1,II

+ 1 IIYn(k+l)

-Yn(k)ll


n(k), lbn

- Yll

6

lbn

- yn(k)ll

+ l b n ( k ) - yll < 2 - k + 11Yn(k) - yll;

and since the right-hand side tends to 0 when k + 00, {y,} converges to y. Hence X is a Banach space. 1.5.3. THEOREM.If J is a closed suhspace of a normed space X, the equation 1I.x

defines a norm 1) quotien t mapping

+ "Yllo = inf{llx + y ( 1 : y ~ J Y )

(~€3)

[lo on the quotient space XlQ. With this norm on X/"Y, the Q : x - + x + ? V : .X+X,+?/

isa continuous linear operator, and llQxllo 6 llxll; and X/.Y is a Banach space i f X is a Banach space. Proqt

Suppose that x, xl, x2E X and a is a scalar. Since

+

inf{((x y ( I : y ~ u Y=) inf{l)u)l:U E X

+ @Y},

the definition of IIx + Yllo is unambiguous (that is, it dependsonly on thecoset .Y + 9, not on the choice of x within that coset); moreover, IIx + 9YlIo 2 0. For all y , y l , and y 2 in J , 11x1

+ .y2 + Y l + Y2ll G I b 1 + Y l l l + 11x2 + Y21L

llax

+ ayll = la1 llx + Yll.

Thus

+ x2 + +Y?'l(Io < llxl + 9(lo+ JIx2+ Yyllo, lax + tVl0 = la1 I(x + ??/l o. If I(x + 9110= 0, there is a sequence {y,} in ??/ such that IIx + y,JI < l / n ( n = 1,2,. . .); since 9Y is closed, while - y , E I and lim( - y,) = x, it follows that ~ € 9 whence , x + "Y is the zero vector in X/J.Hence )I (lo is a norm on (Ixl

X1.Y. The quotient mapping Q is a linear operator, and is continuous since

IIQ-YI - QxzIlo = 11x1 - x2 = inf{llxl

+ Jllo

- x2 + r l l : y ~ t V 6 } 11x1 - x ~ l l

for all x, and x2 in X. Suppose that X is a Banach space. If {x, + tV) is a sequence of elements of X / I , and CIlx,, + ?Vllo < 00, we can choose y , , y 2 , .. . in 9Y so that ((x, + y,(( < ((x, 9lIo 2 - " . Thus C((x, + y,(( < 00, and by Lemma 1.5.2, the series C(x, + y,) converges to an element z of X. Since Q is a continuous

+

+

40

1. LINEAR SPACES

linear operator, and Q(x,, + y,,) = x,

n= 1

+ y , + ?Y = x, + Y, it follows that \n=

1

as rn + 00 ;that is, C(x, + "Y) converges to Qz (E X/?/). Again by Lemma 1.5.2, X/?q is a Banach space. 1

V Y and 3 are subspaces of a normedspace 3, with Y 1.5.4. COROLLARY. closed and 2Tfinite-dimensiona1, then (?7 + 2'is closed in X. Proof. The quotient mapping Q : X + X/oY is a continuous linear operator; the subspace Q ( 3 )o f X/"Y is finite-dimensional and is therefore closed in Xkq (Theorem 1.2.17); and (?7 + 2T is the inverse image Q - l ( Q ( 9 ) ) . 1 We now consider some elementary properties of linear operators acting on normed spaces. 1.5.5. THEOREM.I f X and Y are normed spaces and T :X + 3Y is a linear operator, the following four conditions are equivalent.

(i) T is continuous. (ii) There is a non-negative real number C such that IITxIJ< Cllxll for each x in X. (iii) sup{)lTxll/llxll:x~X, x # 0) < co. (iv) sup{llTxll:x~X, llxll = I } < co. When these conditions are satisfied, the suprema occurring in (iii) and (iv) are both equal to the smallest real number C with the property set out in (ii). Proof. The equivalence of (i) and (ii) is a special case of Proposition 1.2.8(ii), with both rl and Tz consisting of a single norm. A real number C has the property set out in (ii) if and only if IITxll/llxll < C ( X E X, zc # 0). Upon taking x, = [lxIl-'x, it follows that this last condition is satisfied if and only if IlTXlll < c (x1 E X , IlXlll = 1). This proves the equivalence of (ii), (iii), and (iv), and shows that the suprema in (iii) and (iv) coincide with the smallest possible value for C in (ii). 1 When X,? arel normed spaces and T :X + Y is a linear operator, we denote by llTll the (equal, and possibly infinite) suprema occurring in parts (iii) and (iv) of Theorem 1.5.5; we refer to IlTll as the (operator) bound of T. Thus T is continuous if and only if llTll < co; then IITxll

IlTll llxll

(XEX),

41

1.5. NORMED SPACES

and llTll is the smallest real number with this property. It is clear that ((TI1= 0 only when T = 0. Since continuity is equivalent to the existence of a finite bound, continuous linear operators between normed spaces are often described as bounded linear operators. The set d(X,Y) of all bounded linear operators from X into "Y is a linear space (with the same scalar field I6 as X and c!V). If S, TE&?(X,"Y)and a € 06, Il(S +

m-11= llSx + Txll 6 IlS4l + IITxll d IlSll + llTll9 Il(aT)-4l = Il4Tx)ll = la1 IITXII

whenever S E X and llxll = 1. By taking suprema, as x varies subject to the conditions just stated, we obtain

11s + TI1

IISII + IlTll?

IlaTIl = la1 IlTll.

Since, also, llTll 2 0, with equality only when T = 0, it follows that &(X, 9) is a normed space, with the operator bound as norm. When Y = 3, we usually write B(X)rather than a(X,X). If X, Y, Y are normed spaces, and S : 9 + I ,T :X-+ "Y are continuous linear operators, then S T : X -+ Y is continuous. Since IlST.UIl 6 IISII IITXll d IlSll IlTll

( x E x

llxll =

it follows that IlSTll 6 IlSll IlTll. This applies, in particular, when X = 9 = Y. The set d ( X ) is an associative linear algebra with a unit element I(the identity mapping on 3 ) ;it is also a norrned space, and its norm (the operator bound) satisfies 11111 = 1, JISTII< IlSll IlTll. These properties of d ( X ) are characteristic of Banach algebras, which are studied in Chapter 3.

I .5.6. THEOREM.If X is a normed space and tV is a Banach space, both hariny the same scalar field K, then the set d?(X,"Y) of all bounded linear operators from X into 'Y is a Banach space oijer K,with the operator bound as norm. Proof. We have already seen that d ( X , 'Y) is a normed space, so it remains to prove that it is complete. Let { T,,)be a Cauchy sequence in 9 ( X , "Y). Given any positive E , there is a positive integer N(E)such that IlT,,, - T1.1 d

E

(m > n 2 N(E)).

When X E X , (1)

(m > n 2 N(E)).

IIT,,,.u - T,,.xll d /IT,,, - T1.1 IIxlI 6 cllxll

Thus { Tnx)is a Cauchy sequence in the Banach space c!Y, and so converges to an element of d.Accordingly, the equation

Tx = lim T,,x n-

T

(XE

X)

42

I . LINEAR SPACES

defines a mapping T: X + '9, and it is apparent that T is a linear operator. Upon taking limits as m + c;o in (I), we obtain llTx - T,,xll < ~llxll

( n 2 N(E), XEX).

This shows that T - T, is a bounded linear operator (whence, so is T )and that IIT - T1.1 < E whenever n N(E).It follows that { T,) converges to the element T of .%?(X,Yq), and hence B(X,Y) is complete. is a Banach space, both I f ' X is a normed space and 1 S . 7 . THEOREM. having the same scalar field, then every bounded linear operator T: X + 9 extends uniquely to a bounded linear operator p: 2 -+ 9,where %. is the completion of X. The mapping T + $is an isometric isomorphismfrom g ( X , #) onto a(%, Y). Proof. By Corollary 1.2.3, each continuous linear operator T: X -+ 9 extends uniquely to a continuous linear operator T : .% -+ 9. The inequality IlTll llxll - IIFxll 2 0 is satisfied when X E X , and by continuity it remains valid for all x in 2. Thus 11T11 < IlTll; the reverse inequality is evident since Textends T, SO ( ( ~ 1 1= IITII. It is apparent that the norm-preserving mapping T + F : g ( X , Y) -+ B(.%,3 ) is linear. Its range is the whole of B(2,fq) since, when S 0 € g ( . % ,Y),we have So = Po, where To (EB(X,?Y)) is the restriction SolX. H 1 S . 8 . THEOREM. Suppose that X and ?iY are normed spaces, T: X + 'Y is a bounded linear operator, Xo is a closed subspace of X such that T(Xo) = { 0 } ,and Q: X -+ X/Xo is the quotient mapping. Then there is a bounded linear operator To:X/X0-+ 9such that T = Toe; moreover, llToll = JJTJI, and To is one-to-one if Xo is the null space of T.

Proof. From purely algebraic considerations T has a factorization TOP, where To:X/Xo + 9 is a linear operator, and is one-to-one if .X0 is the null space of T. When X E X, llQxll < llxll by Theorem 1.5.3; so llQll < 1. Moreover, IIToQxll = IITxll = IIVx + xo)ll

IlTlllI~+ xoll

(XOEXOL

so

IIToQxll < llTllinf{llx+ x o l l : x o ~ X o )= IlTll IlQxll. Since Q(X)

= X/Xo,

it now follows that To is bounded, and

IlToll < IlTll = IlToQll < IlToll IlQll so IlTOll = IlTll.

< IlToll;

1.6. LINEAR FUNCTIONALS O N N O R M E D SPACES

43

1.5.9. LEMMA. IfX and 9/ are normed spaces, S : X -+ 'Y and T :9 -+ X are linear operators such that S T = I (the identity operator on "Y), and a = inf{llSxll:xE T(JY), llxll = I},

then IlTll = a - ' (where0-' is to he interpretedas m).Inparticular, Tishounded i f and only i f a > 0. Proof. Since S T = I, T can be viewed as a one-to-one linear mapping from ,Y onto T(!Y),and as such, it has an inverse mapping, the restriction SI T(iY). Also, when 0 # Z E T(CiY),11z11-'z is a unit vector x in T(9/). Thus

=

sup{

: Z E T(?Y),z # 0) IlSZll

1.5.10. COROLLARY. Suppose that X and 9 are normed spaces, S is a oneto-one linear operator from X onto 'I, T :9/ + X is its inuerse operator, and

a

= inf{IJSxll:xEX, llxll =

I]

(i) IlTll = a-' (where 0 - ' is to he interpreted as 03); in particular, T is bounded i f and only i f a > 0. (ii) I f X is a Banach space, S is hounded, and a > 0, then 9 is a Banach space. Proof: (i) This follows from Lemma 1.5.9, since T ( g )= X. (ii) Since a > 0, T (as well as S) is bounded; so both S : X -+ 9/ and its inverse T : I -+ X are uniformly continuous. Hence the completeness of X entails completeness of 'Y. 1.6. Linear functionals on normed spaces

In this section we shall be concerned with the continuous dual space X uof a normed space X and with the properties of the weak topology o(X, Xu)on X and the weak* topology o(X',X) on X'. I t turns out that, in a natural way, X u becomes a Banach space and X is isometrically isomorphic to a subspace of the second dual space Xu' ( = (3')').We describe a necessary and sufficient condition for this subspace to be the whole of X". In Section 1.5 we considered linear operators from a normed space X into another such space ,!Y and introduced the normed space @(X, 3 )ofcontinuous linear operators. By taking, for 0), where .yo E X\’Y,

d=inf{(lx, + y ( ( : y ~ q } , the distance from xo to +Y. Proof. The quotient mapping Q : X + X/.Y is a bounded linear operator, and d = IIQxoll > 0. By Corollary 1.6.2, there is a bounded linear functional p o on X/.Y such that llpoll = 1 and po(Qxo) = d. The equation p ( x ) = po(Qx) defines a bounded linear functional p on X; p ( x 0 ) = d, and p ( y ) = 0 for each y

1.6. LINEAR FUNCTlONALS ON NORMED SPACES

45

in #. Since p has the factorization p0Q through XIY, it follows from Theorem 1.5.8 that 1 1 ~ ) )= ) ) ~ o = ) l 1. H 1.6.4. THEOREM.tf X is a normed space and x E X, the equation 2(P) = p ( x )

(PEX')

defines a bounded linearfunctional 2 on the Banach dual space Xu. The mapping x + i is an isometric isomorphismfrom X onto the subspace f = {a:x E X} of the second dual space .Xu'. Proof. It is evident that 2, as defined in the theorem, is a linear functional on X', and that the mapping x + 2 is a linear operator from X into the algebraic dual space of X'. When x0eX, I-$o(P)I = IP(X0)l

< IlPll llxoll

( P E - 0

When p is chosen as in Corollary I .6.2, M P ) I = ll-yoll = llpll IIxoll.

Thus ,tois a bounded linear functional, and Il2,,ll = ~ ~ x so o /the ~ ;mapping x is an isometric isomorphism from X onto a subspace f of 3". H

+2

When X is a normed space, the mapping x -+ ?, occurring in Theorem 1.6.4 is called the natural isometric isomorphismfrom X into X", and 2 is described as the natural image of X in X". The weak* topology o(X', X) (as defined, in the discussion preceding Proposition 1.3.5, for locally convex spaces) is the weak topology induced on X' by .%. If .% = X", the normed space X is said to be reflexive. A reflexive normed space X is necessarily a Banach space since it is isometrically isomorphic to the Banach dual space 3". However, many Banach spaces are not reflexive (see, for example, Exercise 1.9.24). Part (i) of the following result is known as the Alaoglu-Bourbaki theorem. 1.6.5. THEOREM.Suppose that X is a normed space and f is the natural image ofX in X". (i) (ii)

The unit ball (X')), is compact in the weak* topology g(X', 3 ) on X'. The weak* closure in Xu' of the unit ball (f), off is the unit ball (X"), of

XU'. Proof. (i) For each x in X, let D, denote the compact subset { a :la1 < Ilxll} of the scalar field K. The product topological space

P= n D x r;EX

is compact, by Tychonoff's theorem. It consists of all functionsp: X + tt6 such

46

1. LINEAR SPACES

that P ( X ) E D, (xEX); each element p o of P has a base of neighborhoods consisting of all sets of the form { p P :~Ip(xj) - po(xj)l < E ( j = 1 , . . . , m ) } ,

where&> 0 and x l , . . . , x , E X . For each x i n X, the mappingp + p ( x ) : P - t K is continuous. The unit ball (X‘)),consists of all linear mappings p : X + K such that P ( X ) E D, (that is, Ip(x)l < Ilxll) for each x in X. Thus

(X’)),= { p ~ P : p ( a x + b y ) - a p ( x ) -b p ( y ) = O ( x , y E X ; a , b E K ) } . From the final sentence of the preceding paragraph, it now follows that (X’)), is a closed subset of P and is therefore compact in the relative topology. In view of the form of the basic neighborhoods of points in P (as described above) and the definition and discussion of the weak* topology (preceding as a subset of P Proposition 1 . 3 3 , it is clear that the relative topology on (X’)), coincides with its relative weak* topology as a subset of Xu. Thus is compact in the latter topology. (ii) From (i), (X”)), is compact in the weak* topology a(X’’, X’), and so contains the weak* closure %? of its convex subset (g)),; we have to show that % = (fig*)),.Suppose the contrary, and choose a. in (X”)),\%?;we shall obtain a contradiction. By the Hahn-Banach theorem, there is a weak* continuous linear functional wo on X” and a real number a such that Rew,(a,) > a,

Rew,(o)

a, contradicting the previous inequality.

1.6.6. COROLLARY. If9 is a bounded weak* closed subset of the Banach dual space Xuof a normedspace X, then .Yis weak* compact. in addition, Y is contiex, it is the weak* closed convex hull of its extreme points. Proof. For some positive r, 9 is a (weak* closed) subset of the ball (X’)), ( = r(X‘)l),and this ball is weak* compact by Theorem 1.6.5(i); so Y is weak*

compact. The final assertion in the corollary now follows from the Krein-Milman theorem (1.4.3), since Xu,with the weak* topology, is a locally convex space.

47

1.6. LINEAR FUNCTIONALS ON NORMED SPACES

1.6.7. THEOREM. A normed space X is reflexive ifand only if its unit ball (X), is compact in the weak topology. Proof. It is clear that X is reflexive (that is, .% = 3") if and only if (Xs%)l.By Theorem 1.6.5, (X"), is weak* compact, and is the weak* closure of (f)l.Thus (R), = (X"), if and only if (g), is weak* compact. The natural isometric isomorph is^ x -+ 2 : X + f { c 3'') carries (X), onto When x 0 E X, y , ,. . . ,pmE XP,and c > 0, it carries the basic neighborhood (.%)l =

( x € X : ( p j ( x ) - p j ( s , , ) l < & ( j =1, ..., m)]

of x,, (in the weak topology on X) onto the basic neighborhood

{a&:

[a(pj) - a,(pj)l < & ( j = 1,.

. . ,m)}

(in the relative weak* topology on f , as a subset of 3"). It is therefore a of io homeomorphism between X and f , with the topologies just mentioned. Thus (.%),is weak* compact if and only if (X), is weakly compact. Suppose that X and 9are normed spaces and T :X .+ 9is a bounded linear operator. If p is a continuous linear functional on 9, the composite mapping p~ T is a continuous linear functional on X. Accordingly, we can define a -+ X' by mapping T': 9%

T'p = p o T

(p~%*).

We assert that T' is a bounded linear operator and that [[T'ljl= Il7'll. The linearity of TPfollows from the fact that (alp,

+ a2p2)

T = a l b l a 7-1 + az(P2 O T ) ,

when p , , p2 E 9'and a t , a2 are scalars. For each p in g',

IIPII 1 1 ~ ~ s1 1 IIPII IlTlI IIXII (XEX), and thus IIT'plI < llTll IIpII; so T s is bounded, and ((?"'I[< IITJI.To prove the reverse inequality, it suffices to show that IlTxll < IIT'II JIxIJfor each x in 3. ltT'p)(x)/ = lp(Tx)l d

Given such x, it follows from Corollary 1.6.2that we can choose p in @so that llpll = I and p(Tx) = /l7'xll; and

IITxll = Ip(Tx)l = I(z-'p)(-4 d IITPPllIlxll < lITPllMI IlXJl = IIT'll IIXlL as required. When T , , T2 : X -3 $9 are bounded linear operators and p&@, it results from the linearity of p that (UlT,

+ a2T2Yp = P " h T 1 + a2T2) =U ~ ( P O Ti)

+ az(po T2) = a1T:p +

~ 2 7 ' : ~

48

I . LINEAR SPACES

for all scalars a , , a 2 . Thus (al T I

+ a2T2)' = al T : + a2T i ,

and the mapping T -+ T' is a norm-preserving linear operator from g(X,?Y) into g(@,3'). If X, g,3 are normed spaces, and S E ~ ( . Y , Y ) ,T E ~ ( X , then ~ ) ,

( S T ) $ = ~ ~ ( S T ) = ~ . ( S =. T( )~ . s ) . T = ~'(s'p) for each p in 2'; so (ST)' = T'S'. The operator T': g' + X' is called the (Banach) adjoint of the bounded linear operator T : X -+ @. When X and Y are Hilbert spaces, there is another (and, in that case, more important) adjoint operator, the (Hilbert) adjoint T* :??/ X, which will be described in Section 2.4. -+

If T is a bounded linear operator from a normed space 1.6.8. PROPOSITION. X into another such space Y, then T' is continuous relative to the weak* topologies on 9'and X'. Proof. We use the criterion set out in Proposition 1.3.2. The weak* topology on X' is cr(X',,), where 2 is the natural image of X in 3". Accordingly, it suffices to show that the linear functionals i o T' ( x e X) on [Y' are weak* continuous. Suppose XEX, and let y = T x ; for each p in g', (2" T')(p)= 2(T'p) = (T'p)(x) =

P(W

= P(Y) = Bb).

Thus 20T' = BE @, and therefore 2 0T' is continuous in the weak* topology f?(a',,).

1.7. Some examples of Banach spaces In this section we describe some of the Banach spaces that will be used in later chapters. In some cases, we shall first indicate a general process by which, from a given Banach space X, another such space can be constructed; we then obtain specific examples by taking X = DB or @. Whenf, g are mappings from a set A into a Banach space X, and c is a scalar, f + g and cf will always denote the mappings defined by

( f + s ) ( a )= f ( a ) + g(a),

(cf)(a)= c f ( 4

A).

1.7.1. EXAMPLE.1, spaces. If A is a set and X is a Banach space with scalar field K, we denote by I,(& 3 ) the set of all functionsf: A X such that sup(I1f ( a ) l l : a E A }< co. Given two such functions, f and g, and a scalar c, -+

1.7. SOME EXAMPLES OF BANACH SPACES

49

f + g and d a r e functions of the same type. Thus I x ( A ,X) is a linear space over H,and it has a norm defined by llfll = suP{llf(a)ll:aE A}. We shall show that, with this norm, Im(A,X) is a Banach space. To this end, suppose that Ifn} is a Cauchy sequence in Im(A,3 ) ;we have to show that it converges to some elementfof I , (A,X).Given any positive E , there is a positive integer N(E)such that llfm -fn" Q E whenever m > n 2 N ( E ) . Hence

Ilfm(a) Am11Q l l f m -hll Q & ( a E 4 m > n 2 W E ) ) . Accordingly, for each a in A, { f , ( a ) }is a Cauchy sequence in the Banach space X, and so converges; we can define a mapping f : A + X by fTa) = limf,,(a). When m + x, in (l), we obtain (1)

(2)

I f l a ) -f(a)Il

6

A, n 2 ME)).

6

+

Thus Ilf(a)ll 6 E Ilfn(a)ll < c + llhll, when U E A and n 3 N ( E ) ; so sup{llfla)ll:aEA} < oo,andf€I,(A,X).Sincel)f-f,,ll 6 ~ w h e n n2 N ( ~ ) , b y (2), {fn} converges tof; and Im(A,X) is a Banach space. If a sequence (or net) {fn) converges in I x ( A ,X ) , with limit f , then supllfia) -.L(a)Il a6

=

Ilf-fnll

0;

A

that is, f n ( a )+ f l u ) uniformly on A. Convergence in the Banach space I , (A,X) is uniform convergence on A. By taking X to be R or @, the construction just described gives rise to Banach spaces I , (A, R) and I, (A, @); the latter is usually denoted by In@).

1.7.2. EXAMPLE.Spaces of continuous functions. If S is a topological space and X is a Banach space with scalar field H,we denote by C(S,X) the set of all continuous functions f : S + X such that sup{llf(s)ll:s~S}< a.It is apparent that C(S,X) is a subspace of the Banach space la(& X) defined in Example 1.7.1; we shall show that it is a closed subspace. From this it follows that C(S,X) becomes a Banach space when the norm is defined by

Ilf'll

= suP{llf(s)ll:s~Sl.

Suppose, then, thatf'E I=(S,X). andfis the limit of a net {f,,} of elements of the subspace C(S,3 ) .Thenji(s) +As) uniformly on S , and sincefis a uniform limit of continuous functions, it too is continuous on S. ThusfE C(S,X), and C ( S ,X) is closed in 3). We shall usually be interested in the case in which Sis a compact Hausdorff space. In this case, C(S,X) consists of all continuous functions f:S + X.

50

I . LINEAR SPACES

Indeed, given any suchJ the mappings + Ilf(s)ll: S 4 R is continuous, and is therefore bounded on S ; so sup{IIf(s)ll:s~S}< co. By taking X = R and C,we obtain Banach spaces C(S, R) and C(S,C);the latter is usually denoted by C ( S ) .

In discussing the next example, we shall make use of Minkowski’s inequality [R: p. 62, Theorem 3.51: if 1 < p < co, and x1, . . . , xn,yI,...,yneCrthen (3)

{

Ixj j= 1

+ yjl.)“p < { i

j = 1 lxjlp}l’p

+

{i

j = 1 lyjlp}l’p.

The inequality extends at once to infinite sums; iff and g are complex-valued functions defined on a set A and the sums 2 If(u)lP, C Ig(a)lP converge, then so does 1 IAu) g(u)lP,and

+

For this, it suffices to observe that (by Minkowski’s inequality for finite sums), the net of finite subsums of Ifla) + g(u)lp is bounded above by

1.7.3. EXAMPLE.1, spaces. If A is a set, X is a Banach space with scalar field [M, and 1 < p < 00, we denote by Ip(A,X) the set of all functionsf: A 4 X such that CaEAIlf(~)llP < co.Given two such functions,fand g, it follows from Minkowski’s inequality that

so

f+ g

E

lp(A,X).

Also, c f ~Ip(A,X), and

{c aE

A

llc~u)ll~}l’p= ICI

{ c llAu~llp}l~p a€A

for each scalar c. Accordingly, /JA, X) is a linear space over K, and has a norm

1.7. SOME EXAMPLES OF BANACH SPACES

51

defined by

We shall show that, with this norm, I,(& X) is a Banach space. To this end, suppose that if,)is a Cauchy sequence in I,(& 3). Given any positive E , there is a positive integer N(E)such that Ilfm -frill < E whenever rn > n 2 N ( E )that ; is,

1 IIfmta)

(5)


n 3 N(E)).

QEA

It follows that Ilfm(a) - fn(a)ll < E when m > n 2 N(E)and a~ A;so, for each a in A, { f , ( a ) }is a Cauchy sequence in the Banach space X, and therefore converges. Accordingly, we can define a mappingf: A + X byf(a) = limfn(a). For each finite subset iF of A,it results from ( 5 ) that

C IIfm(a)

< cp

-fn(a)IIP

(m > n 2 N(E))*

asli

When rn 4 co, we obtain

1 IMa) -fn(a)IIp

tp

tn 2 N ( E ) ) ;

aeF

and since this last inequality is satisfied for every finite subset iF of A,

1 IlA4 -fn(a)llP

(6)

G EP

( n k N(4).

ae A

Thus

f-.L

E

lp(&X)

when n 2 N(E),and thereforef= (f-fn) + f n ~ l ~ ( A , X By ) . (6), !If-All < E whenever n 3 N(E),so { f n } converges toJ This proves that IJA, 3)is a Banach space. The construction just described gives rise, in particular, to Banach spaces Ip(A,R) and l J A , 42);the latter is usually denoted by Ip(A). H By taking for A the set { 1,2,. . . ,n ) in Examples 1.7.I and 1.7.3, it follows that there are norms I / ( I p ( I < p < 00) on the linear space 116” (where 06 is 88 or C),defined by the equations

Il(r1,. lI(c1

. . ,(.n)(lp = ElCll”

,+..,CJIlx

+

*

. . + Icnlp]l’p

= max(lc,I,. .

(1 G p < a),

. ,ICnl).

(In the case of 11 /Il and // [ I x , this has already been noted near the beginning of Section 1.5, the two norms being denoted there by p I , p a .) It is easily verified

52

I . LINEAR SPACES

that each of these norms gives rise to the usual product topology on H" (necessarily so, by Proposition 1.2.16). Our next two examples are drawn from measure theory, and we refer to [H, R] as standard sources of information on this subject. For the sake of simplicity we confine attention to a-finite measures, which suffice for our purposes. Accordingly, we shall assume throughout the remainder of this section that m is a a-finite measure defined on a a-algebra Y of subsets of a set S. m ) ) the set 1.7.4. E X A M P L E . L , spaces. We denote by L , (= L,(S, 9, of all measurable complex-valued functions f on S that are essentially bounded in the following sense: there is a positive real number c such that If(s)l Q c for almost all s in S. It is evident that L , is a complex vector space,cdntaining as a subspace the set N of all null functions (those that vanish almost everywhere) on S.We shall observe that, with a suitable norm (defined in (7) below), the quotient space L,/N is a Banach space. It is easily verified that, when f E L , , there is a smallest constant c such that 1fls)l < c almost everywhere on S [R: p. 641. It is denoted by ess sup{(f ( s ) l :SES},the essential supremum of I f [ ; and it is 0 only when f is a null function. There is an equivalence relation on L , , in which f g if and only iffls) = g(s) for almost all s in S;and the equivalence class [fl off is the coset f + N . The equation

-

-

(7) IlCflll = esssupIlAs)l:s~Sl ( f E L , ) defines a norm on L,/N. It is a straightforward result in measure theory that, with this norm, L,/N is complete, and is therefore a Banach space [R : p. 66, Theorem 3.1 I]. There is a common convention, which we shall usually follow, that the Banach space just defined is denoted by L , (rather than L , / N ) , and that its elements are described as functions (although, strictly speaking, they are equivalence classes of functions, modulo null functions). This convention is convenient, and should not lead to confusion, provided it is remembered that two essentially bounded functions must be regarded as the same member of L , if they are equal almost everywhere. It is not difficult to verify that a sequence {h}in L , converges, in the norm topology, if and only if there is a null set Z such that the functions { h ( s ) } converge uniformly on S\Z [R: p. 671. If m is a regular Bore1 measure on a compact Hausdorff space S andfE L , , there is a sequence {h}of continuous functions on S , such that f(s) = limf,(s) almost everywhere on S and If,(s)l < llf'll for all s in S and all n = I , 2 , . . . [R: p. 54, Corollary]. H 1.7.5.

EXAMPLE.

L, spaces. Suppose that 1 Q p
0 and g is a measurable complex-valued function on S. for every measurable set X ( E S ) of finite measure, g is integrable over X and

u,

then Ig(s)( < c for almost all s in S . Proof. We have to show that the set Y = {sES:Jg(s)l> c} is null. Let {z, ,z2,. . .} be a countable everywhere-dense subset of the unit circle

{ z € @ :IZI = l}, and note that n

y=

u Yjk, j.k = 1

where Yjk = {sES:Rezjg(s) 2 c + l/k}. Thus it suffices to show that each of the sets Y j k is null. If Yjk is not null, it has a measurable subset X such that 0 < m(X)< 00 (since m is a-finite). Then,

a contradiction; so each Yjk is a null set. H

1.7. SOME EXAMPLES OF BANACH SPACES

55

1.7.8. THEOREM. Suppose that m is a a-finite measure defined on a oalgebra Y of subsets of a set S. For each g in L , , the equalion

P,W

=

i

(fg L 1 )

f(s)g(s) dm(4

defines a bounded linearfunctional pe on the Banach space L1, and the mapping g -+ pg is an isometric isomorphismfrom L , onto the Banach dual space (L,)'. Proof. The usual norms on LI and L , will be denoted by 11 Ill and 11 Ilm respectively. When f~L1 and g E L,, the function f g is measurable, and If(s)g(s)l G l l g l l * l ~ ~ ) l for almost all s in S. ThusfgE L1 and

From this, it follows easily that p , , as defined in the theorem, is a bounded linear functional on L , ,with llpgll < llgllou.When X ( c S ) is a measurable set of finite measure, its characteristic function xX is an element of L 1 ,and

G

lP,II

IIXXlll

= IIPellm(X).

From Lemma 1.7.7, I&)[ G IIp,II for almost all s in S ; so Ilgll, ,< llp,ll, and therefore IlPglI = lI91lm. The mapping g pe is linear, and from the preceding paragraph it is an isometric isomorphism from L , onto a subspace of (Ll)'. It remains to prove that its range is the whole of (L,)'. Suppose that p E ( L l ) ' ;we shall show in due course that p = p e for some g in L , . Observe first that, if { Y,) is an increasing sequence of measurable sets whose union Y has finite measure, then --f

IIXY - XY"ll1 = m(Y\ Y")-b 0; and, since p is continuous, it follows that

P(XY)

=

lim

dXYJ

n-r co

We now choose (and, for the moment, fix) a measurable set X of finite measure, and we define a complex-valued function 1.1 on Y by (9) P( r) = P f X X n Y ) ( Y E 9). It is apparent that p is a finitely additive set function, and (10)

p(Y)=p(YnX),

p(Z)=O

( Y , Z E Y , m(Z)=O).

56

I . LINEAR SPACES

Moreover, if { Y,,} is an increasing sequence of measurable sets with union Y , it results from the preceding paragraph that P ( Y ) = P ( X X n Y ) = lim n-

P(XX,Y,)

=

lim P( Y n ) . n- ou

00

Thus p is a complex measure on 9. Since it vanishes on the null sets of m , it follows from the Radon-Nikodym theorem that there is an element h of L , such that h(s)dm(s)

(11)

( Y E9).

Now

G llpll I l X X n Y I I I = IIpllm(Xn Y ) G IIpllm(Y)

(YE9).

By Lemma 1.7.7, Ih(s)l < llpll for almost all s i n S ; moreover, jYh(s)dm(s)= 0 whenever Y is a measurable subset of S\X, and thus h(s) = 0 almost everywhere on S\X. When the values of h are suitably adjusted on a null set, (1 1) remains valid, and in addition (12)

hEh,

Ih(s)l < llpll

(SES),

h(t) = 0 ( f E S \ X ) .

Since m is a-finite, S is the disjoint union of a sequence {XI, ,'A ,. . .} of measurable sets of finite measure. For each X,, we can use the process described in the preceding paragraph to obtain a complex measure p,,and the corresponding Radon-Nikodym derivative h,, ; and X,,, h,, p, satisfy conditions analogous to (9), ( 1 l), and (12). Each s in S lies in exactly one of the sets X,, ; so the sequence {h,,(s)}has at most one non-zero term, and

The equation a

g(s) =

c hflW

n= 1

defines a bounded measurable function g on S (so g E Lou). We shall prove that pe = p. To this end, it suffices to show that p , ( f ) = p ( f ) whenever f is the characteristic function of a measurable set Y (cS ) of finite measure; for p and pe are continuous linear functionals, and linear combinations of such functionsfform an everywhere-dense subset of LI

1.7. SOME EXAMPLES OF BANACH SPACES

(the integrable simple functions). Now Y =

u

57

Y,, where ( n = 1,2 ,...).

Yn= Y n ( X , u X 2 u . . . u X n )

Thus P ( x ~= ) lim n-

P(xY,)

=

n

=

=

x1

i j =

XYnX,

(J:1

)

T.

C

lim n-

2

lim P n- Ix

D

P(XYnX,)

=

I

C PJ(Y) j =

]rhj(s)dm(s)=

I

I

JY

g(s)dm(s);

the last step follows from the dominated convergence theorem, since x.Hence p ( x Y ) = pg(xy),as required. W

m( Y )
r. Since each Yr is a countable subset of lp(A),so is the set Y = Y,. If ,YEI,(A) and E > 0, we can choose first a positive integer r and then rational complex numbers c1, . . . ,c, so that

u

r

C

lx(an)- cnIp < it”. , n= 1 If y(an)= cn ( 1 < n < r ) and y(an)= 0 (n r), then y e Yr ( G Y ) and 1I.y - y11 < c. Hence Y is an everywhere-dense countable subset of lp(A). Conversely, if IJA) is separable, the same is true of its subset X = {,xu: U E A},where xu denotes the function whose value is 1 at a and 0 elsewhere on A.Thus Xhas an everywhere-dense countable subset, which must

C Is(a,,)lP


=-

58

1. LINEAR SPACES

be the whole of X , since Ilx, - xbll = 2l’, when a # 6 . In other words, X (and therefore, also, A) is countable. Suppose that 1 < p < co and m is a a-jinnite measure 1.7.10. PROPOSITION. m) defined on a a-algebra Y of subsets of a set S. Then the Banach space L,(S, 9, is separable i f and only i f there is a sequence { X , ,X 2 , . , .} of measurable sets of finite measure with the following property: given any measurable set X offinite measure and any positioe E , there is an integer j for which

rn((Xj\X) u (fix,))< E . Proof: Let .4”0 be the family of all measurable sets of finite measure. When

X , Y E Y ~the , characteristic functions x x , xy lie in L,, and Ilxx - xvllP = m ( ( x \ Y ) u ( Y \ X ) ) .

Accordingly, the existence of a sequence { X , ,X 2 , . . .} with the properties set out in the proposition is equivalent to the existence of a countable everywheredense subset of the set C = { x x : X € Y 0 } ( G L,). If L, is separable, so is C. Conversely, suppose that C has a countable everywhere-dense subset { g , , g 2 , . . .}. Let R be the countable subset of L, that consists of all finite linear combinations z l g , + . * . + zngn,in which the coeflicients z l , .. . ,z, are rational complex numbers. Given anyfin L, and any positive E , there is a simple functionf, in L, such that )If - f,(1 < $8. Sincef, is a finite linear combination (with complex coefficients) of elements of C, there is a finite linear combination f 2 (with rational complex coefficients) of elements of the everywhere-dense subset { g l , g 2 . . . .} of C such that [Ifl - j i l l < ic. Then f2E R and 1) f - f i l l < E ; so R is a countable everywhere-dense subset of L,. 1.7.1I . REMARK.It follows from Proposition

1

1.7.10 that, when

< p < GO, the L, space, associated with Lebesgue measure on a measurable

subset E of R”, is separable. To prove this, we consider the set of all “rational cells” {(x,, . . . ,x,): a j < x j

< b j ( j = 1,. . . , n ) }

in R“, where the aj’s and bj’s are specified rational numbers. We can list, as a sequence { Y , , Y 2 , .. .}, the set of all finite unions of rational cells, and take X j = E n Y j .The sequence { X j } then has the properties set out in Proposition 1.7.10. We shall show, in Exercise 2.8.7, that certain 0-finite measures give rise to inseparable L2 spaces. We shall see later that C ( S )is separable when S is a compact metric space (see Remark 3.4.15), while infinite-dimensional I,, and L,, spaces are not separable (Exercises I .9.31, 1.9.32).

1.8. LINEAR OPERATORS ACTING ON BANACH SPACES

59

1.8. Linear operators acting on Banach spaces In this section we prove four basic results concerning linear operators acting on Banach spaces, namely, the open mapping theorem, the Banach inversion theorem, the closed graph theorem, and the principle of uniform boundedness. The first three are so closely related as to be more or less equivalent, and the fourth is easily deduced from the third. We recall that a mapping cp from a topological space X into another such space Y is said to be open if q(G)is open in Y whenever G is an open subset of X. 1.8.1. PROPOSITION. Suppose that X, CY are normedspaces and T :X -+ 9 is a linear operator. Then Tis open ifandonly ifthe image { T x :x E (X),} of the unit ball (X), contains the ball (9),, for some r ( > 0). If T is open, T(X) = ”Y. Proof: If T is open, it carries the open unit ball { x E X : llxll < l } onto an open subset of Y that contains 0 and so contains (Y), for some r ( > 0). Thus (Y),

G

{ T x :X E X , llxll < 1)

G

T((X)I).

Moreover, the subspace T(X) of 3 contains (”Y),, and is therefore the whole of “Y. Conversely, suppose that r > 0 and (CY), G T((X)l).We have to show that T(G)is open in Y when G is an open subset of X. If xoE G, then G contains a ball xo + c(X), (where c > 0), and

T ( G ) 2 TxO

+ c T ( ( 3 ) l )2 TXO+

C(9’)r.

Hence each point Txo of T(G) is an interior point, and T ( G ) is open, as required. W 1.8.2. PROPOSITION. If T is a bounded linear operatorfrom a Banach space

X inlo a normed space CY, and the closure C - of the set C = { T x :X E ( X ) ~ } contains the ball (“Y),, for some r ( > 0), then T is open. Proof. From Proposition 1.8.1, it suffices to show that Ccontains the ball (Y)r,2.To thisend, suppose t h a t y e g a n d llyll < ir. Since2yE(”Y), G C - , we can choose y1 in C so that IVY - Y l l l < 9.

Since 22y - 2y, ~ ( G 9C)- ,~ we can choose y 2 in C so that

1P2Y - 9,- Y211 < ir. Since z 3 y - 2 2 y , - 2y2e(g), E C - , we can choose y 3 in C so that 1 1 2-~ 22y, ~ - 2y2 - Y,I

< tr.

60

I . LINEAR SPACES

By continuing in this way, we obtain a sequence {yl , y 2 , .. .} in C such that

11Yy - 2"-'y1

-T

( n = I , 2 , . . .),

2 y , - . . . - y,l( < +r

Thus n

1Iy-

C 2-jyjll 0, let B(y, r ) denote the closed ball y + ("Y),. It suffices to show that, for some n, the closure C,- of C, contains such a ball. We assume the contrary, and in due course obtain a contradiction. (There is a short cut available here for the reader who is familiar with the following result, known as the Baire category theorem: a complete metric space X cannot be expressed as the union of a sequence { X , ) of subsets, each of which is nowhere dense in X . Indeed,=;J( C,, is the complete metric space Y since T has range "Y, and the category theorem implies the required conclusion that at least one of the sets C, is not nowhere dense in Y (that is, for at least one value of n, the closure C,- has non-empty interior [ K :p. 2011).The argument that follows is in fact a proof of the category theorem, within the particular context now under consideration, but in a form easily adapted so as to apply to the general case.) From our assumption, C ; does not contain the unit ball ("Y), ; so we can choose y , (EY) and rl ( > 0) so that Yl€("Y)l\c;,

B ( y 1 , r l ) n C l = 0.

1.8. LINEAR OPERATORS ACTING ON BANACH SPACES

61

Since C; does not contain the ball B ( y l ,+rl),we can choose y , ( ~ g and ) r2 so that

( > 0)

yzEB(yl.:rl)\C;, B ( y 2 , r 2 ) n C 2= 0, r2 < $rl. Since C j does not contain the ball B ( y 2 ,$r2),we can choose y 3 (E@?) and r3 ( > 0) so that . v ~ E B ( ~ z , $ ~ ~ \ CB;( ,y 3 , r 3 ) n C 3= 0, r3 < $r2. By continuing in this way, we obtain sequences {y,,}in OY and {r,} in R such that, for n = 2 , 3 , . . ., ynEB(yn-l,:r,-l)\Cn-, These conditions imply that

BO?,,r,)nC,

=0, O
- ( Y o -

E

c-.

C - , and T is open by Proposition 1.8.2. W

1 . 8 . 5 . THEOREM (Banach inversion theorem). If T is a one-to-one hounded linear operator from a Banach spuce X onto a Banach space Y, the inverse T - I : 9 -+X is a bounded linear operator.

62

1. LINEAR SPACES

Proof. It is apparent that T - ' is a linear operator. We have to show that it is continuous; that is, we must prove that the inverse image, under T - I , of any open subset G of X, is open in Y. Since this inverse image is T(G),the result follows at once from Theorem 1.8.4. H

Suppose that X and Y are normed spaces with the same scalar field K. The product set X x Y is a normed space when the algebraic structure and norm are defined as follows: (x1 l Y l )

+ (XZ9YZ) = (x1 + XZ7YI + YZ)?

a(x9y) = (ax?aY),

Il(X~Y)II=

llxll + Ilvll.

When X and Y are both Banach spaces, so is X x 9. If T :X Y is a linear operator, the graph of T is the subspace Y ( T ) of X x 9 defined by %(T)= {(x, T x ) : x E X } . As a linear subspace of a normed space, %(T) is itself a normed space. Note that Y(T) is closed (in X x Y) if and only if the following condition is satisfied: if a sequence {x,} in X converges to an element x of X, while { Tx,} converges to an element y of Y, then Tx = y. It is clear that a bounded linear

operator has a closed graph; for operators acting on Banach spaces, there is a converse. 1.8.6. THEOREM(Closed graph theorem). If'X and 9 are Banach spaces, and T is a linear operator from X into g,then the graph of T is closed if and only if T is bounded. Proof. In view of the preceding discussion, it suffices to show that T is bounded when its graph % (T )is closed. In this case, Y( T ) is complete, and is therefore a Banach space, since it is a closed subset of the complete metric space X x ?Y. The equation H(x, Tx) = x (X E X) defines a one-to-one linear operator H from 9(T ) onto X; and H is bounded with IlHll 6 1, since

IIWx, Tx)ll = llxll 6 llxll + lIT4l = IKX, Tx)ll. By the Banach inversion theorem, H - is bounded; and the same is true of T since IITxll 6 Il(x,Tx)ll = IIH-'xll 6 l l ~ - ' l l l l ~ l l ( X E X ) . If E is an idempotent bounded linear operator acting on a normed space X, the corresponding complementary subspaces

Y={X€

x : Ex = x},

z = { X E X : Ex = 0)

1.8. LINEAR OPERATORS ACTING ON BANACH SPACES

63

of X are closed, being the null spaces of the continuous linear operators I - E and E, respectively. For Banach spaces, there is a converse. 1.8.7. THEOREM. If Y and 2 are closed complementary subspaces of a Banach space X, then the projection E from X onto Y parallel to Z is bounded,

Proof. The graph of E can be expressed in the form {(X,Y)€X x X:y€ Y , x

-y€z}

and is therefore closed in X x X; so the result follows from the closed graph theorem. W 1.8.8. COROLLARY. If Y and Z are closed subspaces of a Banach space X and Y n Z = { 0 } ,then Y Z is closed in X ifand only ifthere is a positive real number C such that

+

llvll G CllY + zll

(vE y ,

z E Z).

Proof. Suppose that there is such a constant C. If a sequence {x,} in Y + Z converges to an element x of X, let x, = y , + z,, where y , E Y and z, E Z . Since IlYm - Ynll G Cll(Ym - Yn)

+ ( z m - z,)ll

= C l b m - xnll,

{y,} is a Cauchy sequence in the closed (and hence complete) subspace Y of X, and so converges to an element y of Y. Since 2 is closed, and x

-y

= lim(xn - y,) = lim z,,

it follows that x-y

E

z,

x=y+(x-y)

E

Y+Z.

Hence Y + Z is closed in X. Conversely, if Y + Z is closed in X,it is a Banach space Xocontaining Y and Z as complementary subspaces. From Theorem 1.8.7, the projection E from Xo onto Y parallel to 2 is bounded; and the stated condition is satisfied, with c = IIEJJ.W If y and z are unit vectors in a Banach space X, it is reasonable to consider the “angle” between y and z to be large, or small, according as Jly- zll is large or small. Accordingly, when Y and Z are closed subspaces of X, the lower bound

inf{lly - z l l : y ~Y , Z E Z ,llyll = llzll = 11 can be regarded as indicating the (minimum) angle between Y and Z . It is not difficult to show that this lower bound is strictly positive if and only if there is a constant C with the property set out in Corollary 1.8.8 (see Exercise 1.9.5).

64

I . LINEAR SPACES

Thus the corollary can be interpreted in the following geometrical form: if Y n Z = {O}, then Y + Z is closed if and only if the angle between Y and Z is strictly positive. We conclude this section with various forms of the principle of uniform boundedness. 1.8.9. THEOREM.Suppose that { T,: aE A} is a family of bounded linear operators from a Banach space X into a normed space 9 and sup{llT,xll:aEA} < cc for each x in X. Then sup{llT.II:a~A)< m. Proof. Since T, can be viewed as a bounded linear operator from X into the completion @ of 9, we may suppose that UY is a Banach space. For each x in X, the equation (Sx)(a)= Tax

( a € A)

defines a mapping Sx: A -+ ??/,and S x is an element of the Banach space I z ( A , 9 )(Example 1.7.1). It is evident that the mapping x-+ S x is a linear operator S from X into Im(A,CiY). We assert that the graph of S is closed. For this, suppose that a sequence {x,} in X converges to x (EX), while {Sx,} converges to an element f of l % ( AY); , we have to show that S x =f. For each a in A,

Ilm - (Sxfl)(a)lld I l f - Sxnll

-+

0.

From this, and since T, is continuous, f ( a ) = lim(Sx,)(a)

so f

= lim

Tax, = Tax = ( S x ) ( a ) ;

= Sx, and the graph of S is closed. From the closed graph theorem, S is bounded. For each x in X and a in A,

IITaxll = II(Sx)(a)ll d llsxll Q SO

IlTall

llsll

llsll llxll;

A). R

1.8.10. THEOREM.Suppose that { p a :a € A} is a family of bounded linear functionals on a Banach space X and sup{lp,(x)( : a E A) < co for each x in X. Then sup{llp,ll:aEA} < m.

Proof. This follows from Theorem 1.8.9, with +Y the scalar field. R 1.8.1 1. COROLLARY. Suppose that {x,: a E A} is a family of elements of a normedspace X andsup{ lp(x,,)I :a E A} < mfor each bounded linearjunctionalp on X. Then sup{(lx,ll:aEA}< 00.

65

1.9. EXERCISES

Proof. For each p in the Banach dual space X’,

sup{li,(p)l: a € A} = sup{Ip(x,)l: a € A} < cc (where x 4 iis the natural isometric isomorphism from X into X”). From Theorem 1.8.10 (with X‘ in place of X and i, in place of pa), and since ~ ~= iIIx,,II, J we have sup{llx,ll: a E A} < ca. 1.8.12. THEOREM.Suppose that {To:a € A} is a family of bounded linear operators from a Banach space X into a normed space t!Y and sup{(p(T,x)l: a € A } < m for each x in X and p in CV‘. Then sup{11 Tall: aE A} < cc. Proof. By Corollary 1 . 8 . 1 1 , s u p { ~ J T , x J ~ : a 0. [See Corollary 1.8.8 and the discussion following it.]

1.9.6. Show that, if X is a real Banach space, then X x X becomes a complex Banach space X, when its linear structure and norm are defined by (x, y ) + (u, 0) = (x (a

+ 4y + u),

+ ib)(x,y ) = (ax - by, bx + ay), Il(x,y)ll = sup{ll(cos0)x + (sin 0)yll:0 < U d 27c}

for all x , y , u, u in X and a, b in R. Prove also that the set {(x,O): X E X} is a closed real-linear subspace X, of X,, that X, = { h ik: h , k E X R } , and that the mapping x -+ (x,O) is an isometric isomorphism from X onto (the real Banach space) XR.

+

1.9.7. Suppose that X is an infinite-dimensional normed space, and V i s a neighborhood of 0 in the weak topology on X. Show that V contains a closed subspace of finite codimension in X. Deduce that the weak topology on X is strictly coarser than the norm topology.

67

I .9. EXERCISES

1.9.8. Show that, if X is a separable Banach space, each bounded sequence X' has a subsequence that is weak* convergent to an element of X'.

{ p , } in

1.9.9. Prove that, if X and f9are normed spaces, X # {0}, and W(X, tY) is complete, then 9 is complete. 1.9.10. Suppose that Xo is a closed subspace of a Banach space X, Q :X + X/Xois the quotient mapping, and X t (c3') is the closed subspace consisting of all bounded linear functionals on X that vanish on Xo. (i) Show that the (Banach) adjoint operator Q* is an isometric linear mapping from (X/Xo)' onto Xt. (ii) Show that the mapping T : p + X,l+ p I X o is an isometric isomorphism from X*/X,I onto Xi.

1.9.11. Show that, if Xo is a closed subspace of a Banach space X,then X is reflexive if and only if both Xo and X/Xoare reflexive. 1.9.12. Show that a Banach space X is reflexive if and only if its dual space 3' is reflexive. 1.9.13. Suppose that X is a Banach space with the following property : given any positive real number E , there is a positive real number 8 ( ~such ) that IIx - y(I < E whenever x and y lie in the unit ball (X),and 1 1 % ~ + y)ll 1 -6(~) (such a Banach space is said to be uniformly convex). Prove that X is reflexive. [Hint.Suppose that sZo E X" and llsZoll = 1. Choose po in X' so that ( ( p o (=( 1 and (sZo(po)- 11 < B(E); and let

=-

x

= {.?:xE(X)1, Ipo(x) -

11 < 8 ( E ) } ,

where I + ,i is the natural isometric isomorphism from X into 3". Prove that sZo lies in the weak* closure of X, and that 112 - jll < E whenever 2 , j e . X Deduce that JJQ0- i l l d E for each .? in X.] 1.9.14. Suppose that X is a normed space, .Iis a linear subspace of the dual space X', and .,M separates the points of X. Let p be a linear functional on X.

(i) Suppose that the restriction p((X), of p to the unit ball (X)] is continuous in the weak topology a(X, A!).Show that p e 3'. Prove also that, if E > 0, there is a finite subset {q,. . . , m n } of . X such that n

lp(x)J < c

whenever ~ E ( X ) and ~

loj(x)l j= 1

< 1.

68

1. LINEAR SPACES

Deduce that n

IP(x)I 6

EIIxII + IbII C

IWj(x)I

(~€3).

j= 1

From this inequality, together with the result of Exercise 1.9.2, deduce that p has the form p1 + p 2 , where p1 EX’, llplll 6 E , and p2 EM. (ii) Prove that pI(X), is o(X,d)-continuous if and only if p lies in the norm closure .I= of A in X’. 1.9.15. Suppose that X is a Banach space, x + .t is the natural isometric isomorphism from X into X“, and W E X“. Show that, if the restrictiod wl(.X’), of w to the unit ball of Xgis continuous in the weak* topology o(X‘, X), then Q = 1 for some x in X. [Hint. Use the result of Exercise 1.9.14(ii).] 1.9.16. Suppose that X and 3 are Banach spaces, and S E ~ ( O ~ ’ , X ’ ) . (i) Prove that, if S is continuous relative to the weak* topologies on gg and X’, then S = T’ for some T i n g ( X , 3 ) . (ii) By using the result of Exercise 1.9.15, show that (i) remains valid when the weak* continuity of S is replaced by weak* continuity of S[(@)l.

1.9.17. Suppose that X, Oy are Banach spaces and T E ~ ( X 9). , Prove that (i) the set { T ’ p : p ~ ( O y ‘ is ) ~weak* } compact, and hence norm closed in 3’; (ii) if X is reflexive, the set { T X : X E ( X is ) ~weakly } compact, and hence norm closed in 9. 1.9.18. Suppose that X and Oy are Banach spaces, TEYB(.X,1v), and the image T ( 9 )( = { Tx : X E X})is a closed subspace of ‘3.Prove that T‘(9’) is the closed subspace of X’ consisting of all bounded linear functionals on X. that vanish on the null space of T. [Hint.If p E X‘ and p vanishes on the null space of T, the equation wO(Tx) = p(x) defines (unambiguously) a linear functional w0 on T(X).By applying the open mapping theorem to T, as an operator from 9 onto T(X),prove that w0 is bounded. Deduce that p = T’u, for some w in ?Yo.] 1.9.19. Let I, denote the Banach space I,( N, C) of Example 1.7.I , where N is the set of positive integers; an element of I, is a bounded complex sequence{xl,x2,...), and Il{x,,}II= sup{lx,,l:n~N). Let candc,, bethelinear subspaces of I, defined by

i

I

c = {x,,}€1, : lim x, exists , n-r

69

1.9. EXERCISES

Prove that (i) (ii) I,, and (iii)

c and co are closed subspaces of I , ; the sequence { 1, I , I , . . .} is an extreme point in theclosed unit ball of also in the closed unit ball of c; the closed unit ball of co has no extreme point.

Deduce that co is isometrically isomorphic neither to c, nor to any Banach dual space. 1.9.20. With the notation of Exercise 1.9.19, let U be the element { 1, 1 , 1 , . . .) of c; and, for k = 1,2,. . . , let Ek(in co) be the sequence that has 1

in the kth position and zeros elsewhere.

(i) Provethat,ifX = {x,} E C ~thenX , =Z =; I SkEk, theseriesconverging -,0 as m --* cc). in the norm topology of co (that is, JIX- I:=, x,, then X - X U E Cand ~ (ii) Prove that, if X = {x,} E C and x = X = .xu + C;= (xk - x)&, the series converging in the norm topology of c. 1.9.21. Adopt the notation of Exercises 1.9.I9 and I .9.20; in addition, let

Il denote the Banach space ll(N, C ) of Example 1.7.3, so that an element of f l is a complex sequence Y = {yl, y 2 , .. .) such that (11 Y ( (=) C."= (y,l < m . (i) Show that, if Y = { y l ,y,, . . .} € I I , the equation x

c

P(X) =

( X = { x n ) E co)

Yn-Y,

n= 1

defines a bounded linear functional p on co, and llpll = IIYII. Prove also that Y n = P(E,).

(ii) Show that each bounded linear functional on co arises, as in (i), from an element Y = {yl, y 2 , .. .) of Il . (iii) Deduce that the Banach dual space c$ is isometrically isomorphic to I , . 1.9.22. Let c and lI be the Banach spaces defined in Exercises 1.9.19 and 1.9.21. (i)

Show that, if { y o , y , , y 2.,. . } is a complex sequence such that

C,"=ol.vnl < x,the equation a

p ( X ) = y o lim x, fl+

7c

+ C ynxn

( X = {x,}

EC)

n= 1

defines a bounded linear functional p on c, and llpll = Iy,,l. (ii) Prove that every bounded linear functional on c arises, as in (i), from such a sequence {yo,yl, y 2 , .. .}. [Hint. Use the results of Exercises 1.9.2O(ii) and 1.9.21(ii).]

70

I . LINEAR SPACES

(iii) Deduce that the Banach dual space c' is isometrically isomorphic to

11.

(iv) Deduce that ch and c' are isometrically isomorphic, while c,, and c are not. 1.9.23. Let I, and l1 be the Banach spaces defined in Exercises 1.9.19 and 1.9.21, and, for each positive integer k,let e, (in 11)be the sequence that has 1 in the kth position and zeros elsewhere. Without using Theorem 1.7.8: (i) Prove that, if Y = { y l , y z , .. .} € I I , then Y = C ; = l y k e k ,the series converging in the norm topology on II . (ii) Show that, if X = {xl, x z , . . .} el,, the equation

c OU

P(Y)

=

( Y = Lhl}€11)

XnYn

n= 1

defines a bounded linear functional p on 11,and llpll = IlXll. (iii) Prove that each bounded linear functional on II arises, as in (ii), from an element X of I,. (iv) Deduce that the Banach dual space 1; is isometrically isomorphic to I,. 1.9.24. By using the results of Exercises 1.9.21, 1.9.22, and 1.9.23, show that neither of the Banach spaces co, c is reflexive. Deduce that neither of the Banach spaces II , I, is reflexive. 1.9.25. Give a second proof that none of the Banach spaces co, c, I, is reflexive, by using the results of Exercises 1.9.19 and 1.9.11. : = 1,2,. . .} is a double 1.9.26. (i) Suppose that E > 0 and { x ~ , ~m,n sequence of complex numbers that satisfies the conditions CC

4E
~ i.Show that X is the closed linear span of

{-%}.I (ii) Show that a reflexive Banach space is separable if and only if its dual space is separable. (iii) Give an example of a Banach space that is not reflexive but has a separable dual space (and is, therefore, separable).

1.9.36. Show that a bounded sequence {x,,} of elements of a reflexive Banach space X has a subsequence that is weakly convergent to an element of .X. [Hinr. Show that it is sufficient to consider the case in which X is separable, by replacing X by the closed linear span of {x,). In the separable case show, by use of Exercise 1.9.35(ii),that the required result can be deduced from Exercise 1.9.8.1

74

I . LINEAR SPACES

1.9.37. Suppose X is a separable normed space and {x,,:n E N} is a (norm-)dense subset of (X), . Define m

d(P,P’) =

c 2-“KP

- P’)(Xn)l

(P9P’EX’).

n= 1

(i) Show that d is a metric on X‘. (ii) Show that the metric topology induced by d o n (X’)),is the weak* topology on (X’)), . (iii) Use the fact that each sequence in a compact metric space has a convergent subsequence to solve Exercise 1.9.8 (and 1.9.36) again.

1.9.38. Use the Baire category theorem (see the proof of Lemma 1.8.3) to give another proof (direct) of the uniform boundedness principle (Theorem 1.8.9). 1.9.39. If { T,,} is a sequence of bounded linear transformations of one Banach space X into another Banach space Y and { T,,x}converges for each x in X, show that T, defined by Tx = limn T,,x,is a bounded linear transformation of X into Y. 1.9.40. Suppose X and $Y are Banach spaces and T is a linear transformation of 3 into Y. With q in the algebraic dual of Y, let ( T ‘ q ) ( x )be q( Tx) for each x in X; and suppose that T’pE X’ for each p in OY‘. Show that Tis bounded and that T‘lY’ = T’. [Hint. Consider the graph of T.]

CHAPTER 2 BASICS OF HILBERT SPACE AND LINEAR OPERATORS

This chapter deals with the elementary geometry of Hilbert spaces and with the simplest properties of Hilbert space operators. Section 2.1 is concerned with inner products, and the corresponding norms, on linear spaces with complex (or, occasionally, real) scalars. It introduces the concept of Hilbert space and provides a number of examples. Section 2.2 is devoted to the notion of orthogonality in a Hilbert space. In it we deal with orthogonal complements of closed subspaces, orthogonal sets, orthonormal bases, dimension, and the classification of Hilbert spaces up to isomorphism. This is followed, in Section 2.3, by Riesz’s representation theorem concerning the form of bounded linear functionals on a Hilbert space, and some corollaries concerning the weak topology of such a space. Section 2.4 is devoted to bounded linear operators acting on Hilbert spaces, with primary emphasis on elementary properties of the “Hilbert adjoint” of such an operator. Special classes of operators (normal, self-adjoint, positive, unitary) are considered briefly, and illustrative examples are given. Section 2.5 is concerned with orthogonal projections, corresponding to the decomposition of a Hilbert space as the direct sum of a closed subspace and its orthogonal complement. It includes an account of the order structure of projections, and its relation to the strong-operator topology. In Section 2.6 we deal with elementary constructions with Hilbert spaces, such as direct sums and tensor products, together with related aspects of operator theory. Section 2.7 is concerned with unbounded linear operators on Hilbert spaces. 2.1. Inner products on linear spaces

By an inner product on a complex vector space a?, we mean a mapping ( x , y ) + ( s , ~ )from , x‘ x X into the scalar field @, such that

+

z> = a ( x , z > + &Y, z > , (i) ( a x by,-~ (ii) ( Y . X > = (x,Y>, (iii) (x,x> 2 0, whenever x, y , z E a? and a, b E @. If, in addition, (iv) (x,x> = 0 only when x = 0, 15

76

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

the inner product is said to be definite (sometimes, positioe definite is used). In (ii) we adopt the convention that 2 denotes the complex conjugate of an element c of @. From (i) and (ii), an inner product satisfies the further condition (conjugate linearity in its second variable)

+

+

(v) ( z , a x by) = i i ( z , x ) 6(z,y). When ( , ) is an inner product on a complex vector space X, the pair ( X ,( , )) is called a (complex) inner product space, and we refer to the complex number ( x , y ) as the inner product of the vectors x and y in 2. For real vector spaces, the definition of inner products is the same as the one given above, except that scalars and the values ( x , y ) are required to be real, so that the "bars" denoting complex conjugation no longer appear in (ii) and (v). As regards elementary geometrical properties of inner product spaces, there is very little difference between the real and complex cases. In the main, we shall restrict attention to the complex case, making only 'occasional comments on the modifications needed to deal with real spaces. For the theory of linear operators on inner product spaces and algebras of such operators, the complex case has significant advantage over the real one. The finite-dimensional linear spaces C" and R" provide the simplest examples of inner product spaces, with the inner product defined by

+ . . + a,,&

( ( a l , .. .,an),( b l , .. . ,b,,)) = a161

*

(complex conjugation being redundant in the case of UP).Just as the real space R"can be viewed as a real-linear subspace of the complex space @", it can be shown that every real vector (or normed, or inner product) space can naturally be imbedded in a complex space of the same type (see Exercises 1.9.6 and 2.8.3). 2.1.1. PROPOSITION. Suppose that ( , ) is an inner product on a complex vector space 2. (0 I(x,y>12 < ( x , x ) ( y , y ) , f o r a11 x a n d y in 2. (ii) The set 9 = { z E ~( z: , z ) = 0) is a linear subspace of .X, and the equation (x

+9,y +9>1 =

defines a definite inner product ( ,

(x9.Y)

(X,YE*)

on the quotient space 219.

Proof. (i) When x, YE X and a, b E @, (ax

+ by,ax + b y ) = a ( x , a x + by) + b ( y , a x + b y ) = aii(x, x ) + a 6 ( x , y ) + bii(y, x) + b 6 ( y , y ) ,

so (1)

(ax

+ by,ax + b y ) = laI2(x,x) + 2 Re a 6 ( x , y ) + Ib12(y,y).

77

2.1. INNER PRODUCTS ON LINEAR SPACES

By taking a = t( y, x), where t is real, and b = 1, we obtain 0 < (ax

+ by,ax + b y )

= mx,Y)12(x,x)

+ 2tl(x,y)I2 + ( y , y >

( t E W

If ( x , x ) = 0, it follows, by considering large negative t, that I(x,y)l = 0, so I(x,y)I* = ( x , x ) ( y , y ) = 0 in this case. If ( x , x ) > 0, we can take t = - l / ( x , x ) , to obtain 0 d I(xtY>I21(x1x>- 21(x,Y)l2/(x,x) + (YtY) =(%Y)

- I(xlv>l2I(x,x),

whence I(-w9l2d ( X , ~ > ( Y , Y > . (ii) Let

Yl = {zEX: (z,y) = 0 for each y in X ) . It is evident that Y l is a linear subspace of X, contained in the set Y defined in the proposition, and that

PI= ( z E X :( y , t ) = 0 for each y in X ) . With z in 9, it follows from (i) that l(Z,V>l2 < ( z , z ) ( y , y ) = 0, (Z,Y> = 0 (YE-% so z E Yl. Hence Y = Yl, and Y is a linear subspace of A?. If x , y ~ %and z l , z z ~ (= P PI), we have ( x + Z1,Y + z t ) = ( X , Y > It follows that the equation (x

+ (x,zt) + ( Z l t Y ) + ( Z l l Z 2 )

+ P , Y + -2% = ( X , Y >

= (x,v>.

(X,YEX)

defines (unambiguously) a mapping (u, u ) ( u , u ) ~from ( X / Y )x (X/2’) into C. It isclear that ( , inherits from ( ,) the threedefining properties of an inner product. If 0 = ( x + 9,x + Y)l ( = ( x , x)), then X E 9,and x + 9 is the zero element of X/Y.Thus ( , is a definite inner product on X / Y . H The inequality stated in Proposition 2.1.1ti) is known as the CauchySchwarz inequality.

If( , ) is an inner product on a complex vector space 2.1.2. PROPOSITION. X, the equation

(2)

llxll = (x,x)”2

( X E X )

defines a semi-norm 11 11 on X. Ifthe innerproduct is definite, 11 11 is a norm on X. Proof. With 11 11 defined by (2), it is apparent that llxl} 2 0 and Ilaxl} = la1 lixil whenever x E X and a E @. Moreover, if the inner product is definite,

78

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

llxll = 0 only when x = 0. The Cauchy-Schwarz inequalitycan be written in the form I(x,y>l < ( x , x ) 1 ’ z ( Y 9 Y ) 1 ’ = 2 llxll llvll

(X9YEm.

From (l), with a = b = 1, IIX

+ Y1I2 = llXllZ + 2 Re(x,y) + llYllZ < llxl12 + 2ll + llY1I2 < 11x112 + 2llxll llvll + llY1I2 = (Ilxll + llY11)2.

Hence JIx+ yll

< llxll + llyll for each x and y in X .

W

When referring to the norm on a (definite) inner product space, it is understood, in the absence of an explicit statement to the contrary, that the norm intended is the one constructed as in Proposition 2.1.2 from the inner product. For such spaces, we have proved the triangle inequality IIX

+ Yll < llxll + IlYll

and the Cauchy-Schwarz inequality I(X,Y>l

< llxll Ilvll.

For each of these results, we now determine the conditions under which equality occurs. 2.1.3. PROPOS~T~ON. r f ( , ) is a definite inner product on a complex vector space 2 and x, y E 2,the following three conditions are equivalent:

0) IIX + YII = llxll + llvll; (ii) = IIXII Ilvll; (iii) one of x and y is a non-negative scalar multiple of the other. (x9.Y)

Proof.

(3)

For any scalars a and 6, it follows from (1) that llaX

+ byl12 = la1211X112+ 2ReaRx9y) + 1~1211y11z.

Thus

(Ilxll + l l ~ 1 1 )-~ Ilx + y1I2 = 2(llxll Ilyll - Re(x,y)). If (i) is satisfied, the last equation and the Cauchy-Schwan inequality give Re(x,v) = llxll llvll 2 I(X,Y)I9 and therefore ( x , y ) = Re(x,y) = llxll Ilyll. Thus (i) implies (ii). If (ii) is satisfied and a, b are real, (3) gives

llax + bY1l2 = (allxll + 41ull)z.

2.1. INNER PRODUCTS ON LINEAR SPACES

79

With a = IlyJJand b = - Ilxll, it follows that llyllx - llxlly = 0. Hence either x = 0 ( = 0 . y ) or y = IIxII-lllyJIx,and so, (ii) implies (iii). If (iii) is satisfied, we may suppose that x = ay, where a 2 0. Then

so (iii) implies (i). W 2.1.4. COROLLARY. If( , ) is a definite inner product on a complex vector space I x and x, y c Ix, then I(x,y)I = IIx((J(yI(ifand only ifx and y are linearly dependent.

Proof. If ( ( x , y ) l = I/xII Ilyll, we can choose a scalar a so that la1 = 1 and a ( x , y ) = llxll Ilyll; that is, ( a x , y ) = I(axlI Ily((.By Proposition 2.1.3,one of ax

and y is a non-negative scalar multiple of the other; so x and y are linearly dependent. Conversely, suppose that x and y are linearly dependent. We may assume that x = ay for some scalar a, and then I(X,Y>I

= la(Y,Y)l

= la1 llYIl* =

llxll Ilvll.

A complex normed space &' is said to be apre-Hilbert space if its norm 11 11 can be obtained, as in Proposition 2.1.2, from a (necessarily) definite inner product on Z.If, in addition, &' is complete relative to 11 (I, then X is described as a Hilbert space (of course, one can consider real Hilbert spaces - complete real inner product spaces). Accordingly, Hilbert spaces form a particular class of Banach spaces, and the theory developed in Chapter 1 for linear topological spaces, normed spaces, and Banach spaces is available in the case of Hilbert spaces. The geometry of Hilbert spaces is in many respects analogous to elementary euclidean geometry, and is simpler and more extensive than any corresponding theory for general Banach spaces. In consequence, the analysis of Hilbert space operators is more fully developed than its Banach space counterpart. The main objects studied in this book are certain algebras of linear operators acting on Hilbert spaces.

The inner product on a pre-Hilbert space &' is a 2.1.5. PROPOSITION. continuous mapping from &' x X into @. Proof. When x, y, xo,yo E X,

+ (x - X 0 ) l Y O + (u - Y o ) ) = (X0,Yo) + (X0,Y -Yo> + (x - X0,YO) + ( x - X O - Y -Yo>.

(X,Y) =

(xo

80

2. BASICS OF HILBERT SPACE A N D LlNEAR OPERATORS

From this and the Cauchy-Schwarz inequality, (4)

l(x9.Y) - l

6 llxoll IIY - Yoll + IIX - xoll llyoll + IIX - xoll IIY - YOlL and the right-hand side is small when x is close to xo and y is close to y o . W

In Theorem 1.5.1, we showed that a normed space X can be embedded, essentially uniquely, as an everywhere-dense subspace of a Banach space f, the completion of X. We now consider the case in which 3E is a pre-Hilbert space.

If 2 is a pre-Hilbert space, its completion 2.1.6. PROPOSITION. Hilbert space.

2 is a

Proof. For n = 1,2,. . . ,let

s,,= {(x,y):x,yEx, $n=

llxll < n, llvll < n>, { ( x , y ) : x , y ~ * *IMI < n , IIYII< n > ,

and note that S,, is everywhere dense in (41,

s,,in the topology on 2 x 2.From

+ nlly - Yo11 + Ilx - xoll IIY - Y O I L when ( x ,y), (xo,y o )E S,,. From this, the mapping fn : S,, + @, defined by l(x,y) - (xo,yo>l < nllx - xoll

(5)

f . ( x , y ) = (X,Y>,

is uniformly continuous on S,,, and so extends uniquely to a continuous mappingx: + C. When m 2 n, the restrictionfmIsn is another continuous extension offn, SOT,I $,, =A. It follows that there is a mappingf: 2 x 2 + 43 such that fl,$ = A (n = 1,2,. . .). We assert that fis an inner product on 2, and gives rise to its norm, whence 2 is a Hilbert space. For this, we have to show that

s,,

f ( a x + by, 4 = a f k f(Y9

4 + b f ( y ,4,

x) = f ( x ,Y ) ,

f(x,x) = llX1l2,

whenever x , y, z E 2 and a, b E @. We can choose an integer n that exceeds the norm of each of the vectors x, y , z , ax + by, and by continuity off 1 ( =I,,), it then suffices to prove the three required equations under the additional assumption that x, y , z E X. However, in this case, these three equations follow at once from (9,since f l S,, = f..

s,,

Next, we prove two identities, both of which are frequently useful, concerning vectors in a pre-Hilbert space. The first of these is known as the parallelogram law.

2.1. INNER PRODUCT’S ON LINEAR SPACES

81

2.1.7. PROPOSITION. Ifu, u, x, and y are uectors in a pre-Hilbert space, IIx

+ Yl12 + IIx - Y1I2 = 211x1I2 + 211Y1I2,

and

+ u, x + y ) - ( u - u, x - y ) + i(u + iu,x + i y ) - i ( u - iu,x - iy).

4(u, y ) = ( u

Proof. The first identity is an immediate consequence of the equations IIx f Yl12 = llx1I2f 2 Re(x,y)

+ lly1l2,

which are particular cases of (3). For the second identity, note first that ( u f u,x f y ) = ( u , x )

+ ( 4 y ) f((u,y> + (u,x))

(with the same choice of the ambiguous sign throughout). Thus (u+u,x+y) -(u-u,x-y)=2(u,y)

+2(u,x).

Upon replacing u by iu and y by iy, we obtain (u

+ i u , x + i y ) - ( u - i v , x - i y ) = - 2 i ( u , y ) + 2i(u,x).

From the last two equations, (u

+ u , x + y ) - ( u - u , x - y ) + i ( u + iu,x + i y ) - i ( u - iu,x - i y ) =4(u,y).

We illustrate the use of the two identities just established, in obtaining the following characterization of pre-Hilbert spaces within the class of normed spaces. 2.1.8. PROPOSITION. A complex normed space 2 is a pre-Hilbert space f and only i f

IIx + y1I2 + IIx - yl12 = 211xIl2 + 211y1I2 (X,YEW. (6) When this condition is satisfied, there isa unique inner product on 2 that defines its norm, and this is given by

(7)

4 ( x , y ) = 11x

+ y1I2 - I(x - y1I2 + illx + iy1I2 - illx - iyI12.

Proof: When JV is a pre-Hilbert space, it follows from Proposition 2.1.7, with u = x and L’ = y , that the norm satisfies (6) and the inner product is determined by (7). Conversely, suppose that a? is a normed space whose norm satisfies (6); and when x , y e X , define a scalar ( x , y ) by (7). Then

+

(.u,x) = i(112~11~ ill(1

+ i)xl12

-

ill(l - +[I2)

+ ill + iI2 - ill - i I 2 ) = IIxI12,

= +11~11~(4

82

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

and

= .

In order to complete the proof that ( , ) is an inner product on X, which defines the norm on X , it remains only to show that, for each fixed y in i#,the equation (8)

+

f ( x ) = IIX y1I2 - IIx - yIl2 + illx + iy1l2 - illx - iy1I2

defines a linear functional f on 2.We begin by proving that

fW = is(x),

(9)

f(x1

+ x2) =f(x1) + f ( X Z ) ,

for all x, x l ,x2 in X. For this, note that

+ yII2 - llix - yllz + illi(x + y)IIz - illi(x - y)I12 = i(- illi(x - iy)1I2 + illi(x + iy)llz + IIx + yllz - IIx - y1I2)

f ( i x ) = llix

= if(x).

Since the norm satisfies the parallelogram law (6), 11x1

+ x2 + 2YllZ+11.:

+ Y1I2 + 11x2 + YIIZ = 1:1. = 21l:(x1

- X21l2

+ x2) + Y1I2 +

11 .:

- X21I2.

From this and the three similar equations obtained when y is replaced by - y, iy, - iy, it follows that (10)

f(x1) + f ( x 2 ) = 2f(t(Xl With x1 = x and x2 = 0, (10) gives

f(x) = 2f(:x)

+ x2)).

(x E A?),

since it is apparent from (8) thatf(0) = 0. Hence (10) can be rewritten in the form f(x1)

+fM

=Ax1

+ xz),

and (9) is proved. It now remains to show that f ( a x ) = af(x) whenever x E i# and a E @. Equivalently, we must prove that IF = C where IF = { a ~ @ : f ( a x=)af(x) for each x in A?}.

From (8),fis continuous, so IF is closed. It is evident that 1 E IF, and that ab, E IF whenever a, b, c ( # O ) E IF. From (9), ie IF, and a + b e ff whenever

c-

2.1. INNER PRODUCTS ON LINEAR SPACES

a, b g IF. The properties just listed imply that s + it€ rational, and so IF = @, since IF is closed.

[F

83

whenever s and t are

2.1.9. REMARK.Most of the theory developed above for complex inner product spaces is valid also in the real case. For real inner product spaces, the second relation in Proposition 2.1.7 is omitted. In Proposition 2.1.8, the relation (7) between the inner product and norm is modified by the deletion of the last two terms on the right-hand side and is easily proved by direct computation. The remaining proofs require only minor alterations. H 2.1.10. REMARK.Equation (7) gives an immediate alternative proof of the continuity of the inner product on a pre-Hilbert space. Moreover, if 2 is a normed space that satisfies the parallelogram law (6), it follows by continuity that the completion $ has the same property. This, together with Proposition 2.1.8, provides an alternative proof that the completion of a pre-Hilbert space is a Hilbert space. 2.1.1 1. EXAMPLE. With n a positive integer, the complex vector space @", consisting of all n-tuples x = ( x l , , . . ,x,,), y = (yl,. . . ,yn) of complex numbers, has a definite inner product defined by

(x, y)

= XlVl

+ . . . + x&.

The associated norm is given by

llxll = (IXl12

+ ... +

IX"12)"2.

Since @" is complete, relative to the metric d(x, y) = IIx - yll, it is a Hilbert space. In this example, the Cauchy-Schwarz and triangle inequalities reduce t o

for all complex numbers xl,. . . ,x,,,yl,. . . ,y,. In the same way, the equation (x, Y) = X l Y l

+ ... +

XnYn

(x,Y E R")

defines a definite inner product on the real vector space R". The equation

(X9Y)l

(X,YE@") defines an inner product on C"; when n 2 1, ( , ) l is not definite, for 9 (see = XlVl

Proposition 2.1.1) consists of all vectors whose first component is zero. In this case, @"/Y is a one-dimensional Hilbert space (isomorphic to C).

84

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2.1.12. EXAMPLE.Given a set A,the Banach space 12(A)described in Example 1.7.3 consists of all complex-valued functions x on A for which the (unordered) sum CaEAIx(u)I2 is finite, and its norm is given by

When x, y E /,(A), the sum CEAx(u)y(a)converges, since

Ix(a)y(a)I G tClx(4l’ + lr(4I2),

(Ix(4l’

+ IY(4l2) < a).

acA

From this, it follows easily that /,(A) has a definite inner product, defined by aeA

which gives rise to the norm in (11). Hence /,(A) is a Hilbert space. In this example, the Cauchy-Schwarz and triangle inequalities assert that

for all x and y in 12(A). When A = { 1,2,. . . ,n}, l,(A) is the Hilbert space @”considered in the preceding example. When A is the set { 1,2,3,. . .} of all positive integers, we write 1, in place of /,(A), and sometimes denote an element of this space as a sequence {x”}. W 2.1.13. EXAMPLE.Let 1:’) be the class of all complex-valued functions defined on the set A = { 1,2,3,. . .} that take non-zero values at only finitely many points of A.Thus 17)is a linear subspace of I,, and so inherits from 12, by restriction, a definite inner product and the associated norm. Hence I:’) is a preHilbert space; we assert that it is not a Hilbert space, that is, it is incomplete. For this, we show that 1:’) is everywhere dense in 12, from which it follows (since l2 # ’”I\> that I\’) is not closed in l,, and therefore not complete. With x in 12, define x1, x 2 , x 3 , .. . in 1;’) by xj(k) =

Then

{;(k)

if k ~j if k > j .

85

2.2. ORTHOGONALITY

a s j + a .This shows that each element of f 2 is the limit of a sequence in I\’), so proves our assertion that fy)is everywhere dense in 12.

and

2.1.14. EXAMPLE.Suppose that m is a a-finite measure defined on a a-algebra Y of subsets of a set S. The Banach space L2 ( = L,(S,.Y:m)), described in Example 1.7.5, consists of all (equivalence classes modulo null functions of) complex-valued measurable functions x on S for which

with the norm defined by

~ the function x(s)y(s) is integrable, since it is measurable and its When x , y L2, absolute value is dominated by the integrable function ~lx(s>12 Iy(s)12). From this, it follows easily that L2 has a definite inner product, defined by

+

(X9Y)

= lsx(slyodm(s)7

which gives rise to the norm in (12). Hence L2 is a Hilbert space. The Cauchy-Schwarz and triangle inequalities reduce, in this example, to

for all x and y in L 2 . 2.2. Orthogonality

The theory of Hilbert spaces and Hilbert space operators is more tractable than its Banach space counterpart, largely because the presence of an inner product permits the introduction of a satisfactory concept of orthogonality. In the present section we study this concept, after obtaining some preliminary results. We show first that, in a Hilbert space, the minimal distance from a point to a closed convex set is attained.

86

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

2.2.1. PROPOSITION. lf Y is a closed convex subset of a Hilbert space X, and a unique element yo of Y such that

x ~ E ,there ~ , is

11x0 - Yoll G 11x0 - Yll

(1)

(YE

Y).

Moreover, R e ( y o , x o - Y O > 2 Re(y,xo - YO>

(2)

(YE

Y).

Proof. With d = inf{llxo - y l l : y ~ Y}, there is a sequence {y,} of elements of Y such that llxo - ynll + d. By the parallelogram law, 211x0 - ymI12

+ 211x0 - ynI12 = 112x0 - y m - ynI12 + IIyn - ymI12

for all positive integers m and n. Since :(ym + y,) 112x0 - y m - ynII = 211x0 - + + t211Y - Yol12.

Hence

- 2 W y - yo,xo - y o ) and this gives (2) when t

-+

0.

+ tlly - yO1l23 0

(0< t < 11,

87

2.2. ORTHOGONALITY

2.2.2. REMARK.We have proved, in Theorem 1.3.4, that a closed convex subset Y of a locally convex space is closed also in the weak topology. For a Hilbert space X, Proposition 2.2.1 permits an alternative proof of this result. For this, suppose that xoE X \ Y , and let yo be the element of Y that satisfies (1) and (2). Then llxo - yell > 0, and , -YO) W y , xo - yo> < R ~ ( Y oxo = Re(x0, xo

- yo)

- (xo - Yo, xo - yo)

= Re(x0,xo - yo) -

11x0

- YO1l2

for each y in Y. Thus X \ Y 2 V, where V = {xEX Re(x,xo : - y o ) > Re(xo,xo - y o ) - llxo - yO1l2}.

The equation p(x) = (x, xo - y o ) defines a linear functional p on 2 ;and from continuity of the inner product, p is bounded, and is therefore weakly continuous. Since V = { x E XRep(x) : > Rep(xo) - llxo - y o J I z } ,

it follows that V ( c X \ Y ) is a neighborhood of xo in the weak topology on X. Hence X \ Y is weakly open, and Y is weakly closed. Suppose that X is a Hilbert space, u, u E X, and X,Yare subsets of X. We say that u is orthogonal to u if ( u , v ) = 0, that u is orthogonal to Y if (u, y) = 0 for each y in Y , and that X is orthogonal to Y if (x, y ) = 0 whenever x E X and y E Y . The set of all vectors, in X and orthogonal to Y, is denoted by Y '. When Y is a closed subspace of X, we sometimes write X 0 Y in place of Y I. If u is orthogonal to u, then also u is orthogonal to u, and by expanding the inner product (u + u,u + u) we obtain IIU

+ uIl2 = llU1l2 + lIuIIZ.

From continuity and linearity of the inner product in its first variable, Y' is a closed subspace of H. It is apparent that X c Y if and only if Y G X I , and that X' c Y' if X 2 Y . If U E Y', then Y G {u}'. Moreover, since {u}' is a closed subspace containing Y, it contains the closed subspace [ Y] generated by Y , and so u E [ Y ]I.This shows that Y I c .[ Y ] * , and the reverse inclusion is apparent since Y E [ Y ] ; so

Y' = [ Y ] ' . If Y E Y n Y', then ( y , ~ = ) 0, whence y = 0; so Y n Y' = (0).

In particular, X' = X n X' = (0).

88

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

The following theorem includes the assertion that, if Y is a closed subspace of a Hilbert space X, then Y and Y l are complementary subspaces in the sense discussed in Section 1.1 (preceding Theorem 1.1.8). For this reason, Y ' is called the orthogonal complement of Y.

2.2.3. THEOREM.If Y is a closed subspace of a Hilbert space X, each element xo of X can be expressed uniquely in the form yo + zo, with yo in Y and zo in Y l. Moreover, yo is the unique point in Y that is closest to xo. Proof. Since Y is a closed convex subset of X, we can choose yo as in Proposition 2.2.1, and define zo = xo - yo. From (1) and (2), yo is the (unique) point in Y that is closest to xo, and Re(y, zo) ,< Re(yo, zo) for each y in Y. By writing ay in place of y , we obtain

< Re(yo,zo)

Rea(y,zo)

(YE

Y, a€@).

Hence ( y , zo) = 0 for each y in Y, and zo E Y l. This proves the existence of a decomposition xo = yo zo, with yo in Y and zo in Y l . If, also, xo = y , + zl, with y , in Y and z1 in Yl, then

+

yo

+ zo = y1 + z , ,

yo - y1 = z1 - Z

~ Y E

nYl

=

{O};

and so yo = y , , zo = zl. H

2.2.4. COROLLARY. If Y is a closed subspace of a Hilbert space H' and

X _c X, then (Y1)l = Y,

(Xl)' = [XI

Moreover Y = X ifand only if Y l = (0). Proof. Since [ X I is a closed subspace of 2, and (Xl)' = ( [ X I ' ) ' , it suffices to prove only the results concerning Y. If y E Y, then y is orthogonal to each element of Y I,and so y E ( Y l)l. This shows that Y E (Yl)', and we have to prove the reverse inclusion. With xo in ( Y ')', we can choose yo in Y and zo in Y so that xo = yo + zo, by Theorem 2.2.3. Then xo E ( Yl)', yo E Y G ( Y l)l, and therefore zo = xo - yo E ( Y.')l Hence

z o ~ y l n ( Y 1 ) ' = {O}, and xo = yo E Y. This gives the required inclusion ( Y *)l E Y , so ( Y ')' = Y. If Y = 2,then Y' = #1 = {0};conversely, if Y ' = { O j , then Y = (Y ')' = {O}'= X. A subset Y of a Hilbert space X is described as an orthogonalset if any two distinct elements of Yare mutually orthogonal. By an orthonormal set we mean an orthogonal set of unit vectors. An orthonormal set Y is linearly independent

2.2. ORTHOGONALITY

89

(by which we mean that every finite subset of Y is linearly independent); for if y , ,. . . ,yn are distinct elements of Y , a , , . . . ,an E @, and I;=, ajyj = 0, then n

ak = ( C a j y j , y k )= 0

(k = I ,..., n).

j= 1

In developing the theory of orthogonal expansions in a Hilbert space, we make use of the concept of unordered summation introduced in Section 1.2 (following Theorem 1.2.18). I f Y is an orthogonal set in a Hilbert space X, the sum 2.2.5. PROPOSITION. CWYy converges ifand only ifCyEylly112 < 00. When this condition is satisfied,

(3) Proof. With F a finite subset of Y , expansion of the inner product y1(~ expression for l l , ~ ~ ~gives (4)

From this, and the Cauchy criterion for unordered sums, it follows that convergence of either of the sums in (3) impliesconvergence of the other. When these sumsconverge, they are limits of the finite subsums occurring in (4); since the norm is continuous, (3) is an immediate consequence of (4). W I f Y is an orthonormal set in a Hilbert space X, andf is 2.2.6. COROLLARY. a complex-valuedfunction defined on Y , the sum cWy f ( y ) y converges ifand only ifzFyIf(y)I2 < 00. When this condition is satisfied,

II Cf(Y)Yl12= C lf(Y)I2. YEY

FEY

Proof. It suffices to apply Proposition 2.2.5 to the orthogonal set { f ( Y ) Y : Y EY ) . 2.2.1. PROPOSITION. If Y is an orthonormal set in a Hilbert space X and ME&,

then

(i) CyeY I(U9Y>l2 6 Ilul12; (ii) the sum CBtY( u ,y ) y converges, and

u-

(U,Y)YE YE

w

Ib - C y E Y (U7Y)YIl2

y*;

r

= 1 1 ~ 1 1 2 - CWY I(UtY>l2.

90

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

Proof. With F a finite subset of Y and p(y) = ( u , y ) for each y in Y,

IIU -

(U9Y)Y1I2

= (u

-

YSF

c P(Y)Y,U - c

YEF

c P(Y) llUllZ - c IP(Y)l2 c ll~IlZ c

= (4u> -

YGF

=

-

YEF

=

ZEF

P(Z)Z)

reF

-

1pO(u7 z> + 1 P ( Y ) p o ( Y ?z> ZEF

lP(Z)I2

YJEF

+ c lP(Z)IZ zeF

I(U,Y>IZ.

-

Y ~ F

Hence (5)

1

l(UJJ>l2 =

llul12 - IIU -

1

(U9Y)Y1I2

G llu1I2

Y EF

YEF

for each finite subset F of Y . This proves (i), and the convergence of IFy ( u , y ) y now follows from Corollary 2.2.6. For each y o in Y , continuity and linearity of the inner product in its first variable entail (U,Y)Y,YO) = (U9Yo) -

(u YEY

1 (U~Y>(Y~YO> YEY

= (u,ro> - ( U , Y O > = 0,

so u - CyeY( u , y ) y E Y I.This proves (ii), and (iii) is an immediate consequence of ( 5 ) since the norm is continuous. The inequality in Proposition 2.2.7(i) is usually known as Bessel's inequality. 2.2.8. COROLLARY.Y is an orthonormal set in a Hilbert space JP and u E X, then CFY ( u , y ) y is the unique vector closest to u in the closed subspace [ Y ] generated by Y. Moreover, the following three conditions are equivalent:

(0 u E [ y l ; (ii) u = CyeY (u,Y>Y; (iii) lu112 = CyeY I(4Y>I2. Proof. With v ' = (u, y ) y , it is evident that U E [ Y], and Proposition 2.2.7(ii) asserts that u - U E Y l = [ Y ] l .Since, also, u = u + (u - v ) , Theorem 2.2.3 now implies that u is the unique point closest to u in [ Y ] . From the last statement, it follows that v = u if U E [ Y ] , so (i) implies (ii). The reverse implication is apparent, and the equivalence of (ii) and (iii) follows from Proposition 2.2.7(iii). W

2.2. ORTHOGONALITY

91

2.2.9. THEOREM.If Y is an orthonormal set in a Hilbert space 2, the following six conditions are equivalent:

(i) (ii) (iii) (iv) (v) (vi)

for each u in 2,u = CYEY ( u ,y ) y ; for each u and v in 2, (u, 0) = CyEY ( u ,y ) ( y , 0); for each u in 2, llu112 = CYEY I(u, y)12; Y is not contained in any strictly larger orthonormal set ;'A Y 1 = {O}; [ Y ] = S.

Proof. By the linearity and continuity of the inner product in its first variable, (i) implies (ii). It is apparent, by taking v = u, that (ii) implies (iii). If Y is contained in a strictly larger orthonormal set,'A (iii) fails since, when XEX\Y.

1 I(X,Y>I2= 0 # 1 = llX1l2. YEY

It follows that (iii) implies (iv). If Y l has a non-zero element x, Y is contained in a strictly larger orthonormal set Y u {llxll- ' x }; so (iv) implies (v). By Corollary 2.2.4, and since Y 1 = [ Y ] l ,(v) implies (vi). It follows, from the equivalence of the first two conditions stated in Corollary 2.2.8, that (vi) implies (i). H An orthonormal set Y in a Hilbert space X that satisfies (any one, and hence all six, of) the equivalent conditions set out in Theorem 2.2.9 is called an orthonormal basis of 2. When Y is an orthonormal basis, the equation in condition (ii) is known as Parseval's equation. 2.2.10. THEOREM.Each Hilbert space 2 has an orthonormal basis, and every orthonormal set in X is contained in an orthonormal basis. Moreover, all orthonormal bases of X have the same cardinality. Proof. The class of all orthonormal sets in 2 is partially ordered by inclusion. If a family { Y,} of orthonormal sets is totally ordered by inclusion, then U Y, is an orthonormal set that contains each Yo;for any two distinct elements y , z of U Y, are contained in the union Yb u Y, of two sets in the family, Ybu Y, coincides with Yb or Y, and is therefore orthonormal, and so ( y , z ) = 0. In view of this, it follows from Zorn's lemma that there is a maximal orthonormal set Y o ;since Yo is not contained in a strictly larger orthonormal set, it is an orthonormal basis. If Y is a given orthonormal set, we can repeat the above argument, restricting attention throughout to orthonormal sets containing Y . In this way, we prove that there is an orthonormal basis containing Y .

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2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

Suppose that X and Y are two orthonormal bases of X, and that the cardinal numbers corresponding to these sets are m and n, respectively. In proving that m = n, we consider separately two cases. If X is finite dimensional, the (linearly independent) sets X and Y are necessarily finite. Since both satisfy condition (i) in Theorem 2.2.9, each has linear span X, and is therefore a basis of X in the elementary algebraic sense. The basis theorem for finite-dimensional vector spaces now implies that m = n. We note also an alternative proof, that m = n when m and n are known to be finite, which is more akin to the argument needed in the infinite-dimensional case. By condition (iii) in Theorem 2.2.9

m =

c Ilx1l2

=

xcx

1c

I(X9Y>12

xsx ysY

yoY xox

YeY

If X is infinite dimensional, both X and Y are infihite sets, since [ X I = [ Y] = X. If m # n, we may assume that m c n ; we show, in due course, that this assumption leads to a contradiction. For each x in X ,

1

I(X9Y>12 =

llX1l2 = 1,

YeY

so the set Y, if y~ Y,

= {YE Y : ( x, y>

# 0} iscountable. Moreover, Y =

uxcx Y,; for,

1 I(x3r>l2= llYl12 = 1, xcx

whence ( x , y ) # 0 and Y E Y,, for at least one element x of X. The remainder of the argument consists of elementary cardinal arithmetic, amounting, essentially, to the observation that n 6 mKo = m (where, as usual, KOdenotes the cardinal of the set of natural numbers). In order to prove that n < m, and so obtain the desired contradiction, we need only show that there is a mapping from Xonto Y. For this, it sufices to prove that Xcan be expressed as a disjoint union UXEX X , of a family (indexed by X) of countably infinite subsets; for then each X , can be mapped onto the corresponding Y,, and X ( = U X,) can be mapped onto Y (= U Y,). To prove the existence of such a family (XJxeX, it is enough to show that X x Z has the same cardinality as X , when Z is a countably infinite set. Since Z x Z is countably infinite, it now suffices to prove that X has the same cardinality as A x Z, for some set A. This, in turn, amounts to showing that Xcan be expressed as the disjoint union UOsa Z, of a family (with arbitrary index set A) of countably infinite subsets of X.For this, observe that a simple argument using Zorn’s lemma proves the existence of a maximal disjoint family { Z o }of countably infinite subsets of X. The maximality of this family implies that X \ U Z , has only a finite number of

93

2.2. ORTHOGONALITY

elements; by adding these to any one Z,, we obtain the required partition x=uz,. W By the dimension of a Hilbert space .F we mean the cardinal number dim H corresponding to an orthonormal basis Y of X. From the preceding theorem, this does not depend on the choice of Y; moreover, it coincides with the elementary algebraic concept of dimension when X is finite dimensional. Suppose that ,X,and 2,are Hilbert spaces, and U is a linear mapping from 2,onto H,. By expressing inner products in terms of norms, as in Proposition 2.1.8, it follows that U preserves inner products if and only if it preserves norms. Accordingly, the concept of isomorphism, from XI onto X,, is the same whether 2,and 2,are regarded as Banach spaces or as Hilbert spaces. It is evident that isomorphic Hilbert spaces have the same dimension. 2.2.11. EXAMPLE.With A any set, the Hilbert space I,(A) has an orthonormal set Y = { y , :a E A}, in which y, is the function taking the value 1 at a and 0 elsewhere on A. Since ( x ,y , ) = x(a)for each x in I,(A) and a in A, it follows that YL = (0).Hence Y is an orthonormal basis of I2(A),and the dimension of I,(A) is the cardinal number corresponding to the set A. A necessary and sufficient condition for two spaces I,(A) and I,(B) to be isomorphic is that the sets A and B have the same cardinality. The condition is necessary because isomorphism preserves dimension; it is sufficient since, iff is a one-to-one mapping from A onto B, the equation (Ux)(a)= x(f ( a ) ) defines an isomorphism U from I,@) onto I2(A). W 2.2.12. THEOREM.Two Hilbert spaces are isomorphic fi and only have the same dimension.

if they

Proof. In view of Example 2.2.1 1, it suffices to prove that a Hilbert space 2 with an orthonormal basis Y is isomorphic to I,( Y). For each x in 2,we can define a complex-valued function Ux on Y by ( V x ) ( y )= ( x , y ) . From condition (iii) in Theorem 2.2.9,

llX1l2 =

1 I(X,Y)l2 = 1 I ( ~ X ) ( Y ) I 2 , YeY

YGY

so U is a norm-preserving linear mapping from X into I,( Y). With f in f,( Y), it follows from Corollary 2.2.6 that the sum CFrJIY)y converges to an element x of X Moreover, for each yo in Y , ( W ( Y 0 ) = (X,YO) = C f ( Y ) ( Y , Y O ) w so U x = J Hence U is an isomorphism from 2 onto I,( Y). =f(Yo)9

2.2.1 3. COROLLARY. Every Hilbert space is isomorphic to one of the form I2(A).A Hilbert space with finite dimension n is isomorphic to @”.

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2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2 . 2 . 1 4 . REMARK. We assert that a Hilbert space X is separable if and only ifdim X 6 KO.In consequence, all separable infinite-dimensional Hilbert spaces have dimension KO,and are therefore isomorphic. In particular, 1, spaces for countably infinite sets, and L2 spaces for Lebesgue measure on measurable subsets of R", are all isomorphic. To prove the above assertion, let Y be an orthonormal basis in X . If Y is countable, X has a countable everywhere-dense subset, which consists of those finite linear combinations of elements of Y in which each coefficient has rational real and imaginary parts. If Y is uncountable, the open balls with radius +$ and centers in Y form an uncountable disjoint family, since llYl - Y2112 = IlYI1l2 + llY21I2 = 2

when y , and y , are distinct elements of Y. An everywhere-dense subset of ,# meets each of these balls, and is therefore uncountable. In proving the following result, we describe the GramSchmidt orthogonalization process, by which a linearly independent sequence of Hilbert space vectors gives rise to an orthonormal sequence. The linearly independent sequence may be finite or (countably) infinite, and the orthonormal sequence has the same number of terms. We recall that the (necessarily closed) subspace generated by a finite set x ] , . . . ,x,, of vectors is denoted by [ x ] , . . . ,x,,].

2 . 2 . 1 5 . PROPOSITION. If (xl, x2,x3, . ..) is a linearly independent sequence of vectors in a Hilbert space X, there is an orthonormal sequence (y, ,y2,y,, . . .) such that [xl,. . . ,x,] = [yl,. . . ,y,] for each n = 1 , 2 , 3 , . . . . Proof. We construct yl, y,, y,, . . . inductively, and start the process by x l . suppose that we have produced an defining y1 to be ~ ~ x l ~ ~ - lNow orthonormal set {yl,. . . ,yr- ]}, with the property that (1

[yl,...,ynl = [ x ~ , . . . , x , , I

Yj j= 1

is non-zero, and [YI,* -

* 9.Y-

1

9

~

= 1 [XI9

1

xr-

1,

xrl.

Moreover, for k = 1,. . . , r - I, r- 1

= IIzII > O . j= 1

It is not difticult to verify that the conditions

determine the orthonormal sequence { y,} uniquely. We conclude this section with a brief discussion of certain orthogonal families of functions in L z spaces, which are encountered in classical analysis and its applications. The best known examples arise in connection with the theory of Fourier series, for functions in Lz( - A, n). With Z the set {0, k 1, f 2 , . . .} of all integers, we can define functions x, (n E Z) in L2( - n,a) by x,(s) = exp(ins)

(- n

< s < n).

By evaluating the appropriate integrals, we obtain

ItxnII =

fi,

- 1, we obtain the Jacobi polynomials P:+"(s). The Laguerre polynomials L:(s) ( v > - 1) arise when E = [0, to) and w(s) = s'exp( -s). The Hermite polynomiais correspond to the choice E = R, w(s) = exp( - s2). These are the main three classical sequencesof polynomials, from which the others can be derived; for example, the Legendre polynomials are a particular case (v = p = 0 ) of the Jacobi polynomials.

2.3. The weak topology In this section we prove Riesz's representation theorem (Theorem 2.3.1), which describes the general continuous linear functional on a Hilbert space X.

98

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

By means of this result, we establish certain properties of the weak topology on 2 (see Section 1.3).

2.3.1. THEOREM.If X is a Hilbert space and

YE%,

the equation

cpy(x)= ( x , y ) ( ~ €defines 2 ) a continuous linear functional cpy on X, and llcpyll = Ilyll. Each continuous linear functional on X arises, in this way, from a unique element y of X.

Proof. For each y in 2, Icpy(x)l = I ( X 9 Y ) I

< llxll llYll

(XEWr

with equality when x = y. Thus cpy is a continuous linear functional on X, and IIcpyll = IlYll.

If cp is a non-zero continuous linear functional on X, the closed subspace Y = cp- '(0) is not the whole of X, so Y l # ( 0 ) .Let u be a unit vector in Y l , and note that cp(cp(u)x - cp(x)u) = cp(u)cp(x) - cp(x)cp(u) = 0

for each x in 2.It follows that cp(u)x - cp(x)u E Y , and since u E Y '-,we have 0 = = cp(u)

- cp(x).

Hence cp(x) = cp(u)(x,u> = < X , Y >

( X E W ,

~

where y = cp(u)u. This shows that cp has the form cpy for some y in X (and the same conclusion is apparent when cp = 0). If, also, cp = cpz with z in X, then IlY - zll = Ilcpy-ZII = IIcpy - cpzll = IIcp - cpII = 0,

whence y = z; so there is only one y in X for which cpy = cp.

2.3.2. COROLLARY. If2 is a Hilbert space, the equation (JY)(X) = < X , Y >

( X , Y E W

defines a conjugate-linear norm-preserving mapping J from X onto the Banach dual space 2'. Proof. Since Jy is the continuous linear functional cpy occurring in Theorem 2.3.1, J is a norm-preserving mapping from 2 onto 2',and it is evident from the conjugate linearity of the inner product in its second variable that J also is conjugate linear.

2.3.3. COROLLARY. Every Hilbert space is reflexive. Proof. Suppose that X is a Hilbert space and @ is a continuous linear functional on its Banach dual space Xp. With J defined as in Corollary 2.3.2,

99

2.4. LINEAR OPERATORS

the equation P(Y) = @(JY) (VEX) defines a bounded linear functional cp on M.By Theorem 2.3.1, we can choose z in Ad so that ~ ( y=) ( y , z ) for each y in H. Every element of &” has the form Jy, with y in X , and

+

- -

Y ) = cp(Y) = ( Y - 2 ) = (Z.Y> = ( J Y ) ( Z ) = +(a. Since each bounded linear functional @ on %“ arises in this way from an element z of .X. it follows that X is reflexive. H @(+) = W

2.3.4. COROLLARY. Suppose that family of all sets of the form

3y; is

a Hilbert space and x o e X The

1,..., n)}, with y , , . . . ,y. ( E X ) and E ( > 0 )preassigned, is a base of neighborhoods of xo in the nveak topology on X. A net { x u }of elements of JV converges weakly to xo f and only if(xa,y ) (xo, y ) f o r each y in A? The closed unit ballof X is weakly compact . { x ~ X : J ( x - . x ~ , y< ~E ()Jl=

Proof. Since X is reflexive, its unit ball is weakly compact by Theorem 1.6.7. The remaining assertions in the corollary are simply reinterpretations of the appropriate Banach space definitions, taking into account the information in Theorem 2.3.1 concerning the form of continuous linear functionals on .?F.

2.3.5. PROPOSITION. v a net {xu}of rectors in a Hilbert space X converges weakly to an element x of X, and Ilxall

Ilxll?

then { x u }converges to x in the norm topology. Prooj: Since (xav

x> 7(-v, -v> = IIxII’,

we have llxu - XI(’ = I(X,((* - 2 Re(xa,x)

+ llxll’ y 0.

H

2.4. Linear operators We recall from Theorem 1.5.5 that a linear operator T, from a normed space 3E into another such space ?iY, is continuous if and only if it is bounded, in

100

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

the sense that there is a real number c such that IITxll 6 cllxll for each x in 3. The set .%?(3E,%) of all such bounded operators is itself a normed space, when the norm of an element T is defined to be the least such constant c; equivalently, IlTll = sup{IITxl/:xEX, llxll Q 1).

By Theorem 1.5.6, 99(3,?4’)is a Banach space when 9is a Banach space. In this section we obtain more detailed information concerning bounded linear operators acting on Hilbert spaces. General theory. Suppose that X and X are Hilbert spaces. By a conjugate-bilinear functionalon X x X, we mean a complex-valued function b on H x X that is linear in the first variable and conjugate-linear in the second. We say that such a functional b is bounded if there is a real number c such that Ib(x,y))6 c1)x11))yllfor all x in JV and y in X. When this is so, we denote by llbll the least possible value of c, which is given by

llbll = ~ ~ P I l b ( x , y ) l : x ~ ~ , y ~ 6 ~1,llvll , l l xQ l l 1). When X = Jr, we refer to a conjugate-bilinear functional “on X,” rather than “on .X x X.” 2.4.1. THEOREM. If Jr, X are Hilbert spaces and T E B ( . # ” X ) ,the equation

(1)

b,(X,y) = ( T x , ~ )

(XE

X, Y E X )

defines a bounded conjugate-bilinearfunctional bT on 2 x X, and llbTll = 11 TI[. Each bounded conjugate-bilinear functional on 2 x X arises in this way from a X). unique element of 99( - (T(x -Y ) , X - Y> + (i T ( x + iy), x + iy) - (i T(x - iy),x - i y ) .

4(TX,Y) = ( T ( x + Y ) , X

This relation is called the polarization identity. It is a particular case of the second relation stated in Proposition 2.1.7 (and is essentially equivalent to it). When T = Z, it reduces to the expression of inner products in terms of norms, already noted in Proposition 2.1.8. 2.4.3. PROPOSITION. If S and T are bounded linear operators acting on a Hilbert space 2 and (Sx, x> = (Tx, x) for each x in S, then S = T. Proof. Since ( S X , ~ )= ( T x , ~for ) each vector x , it follows by polarization (that is, by means of the polarization identity (3)) that (Sx, y ) = ( T x , y ) for all x and y in X. Hence S and T give rise to the same conjugate-

103

2.4. LINEAR OPERATORS

bilinear functional on 2,and from the uniqueness clause in Theorem 2.4.1, S=T. 2.4.4. REMARK.For linear operators acting on real inner product spaces, the analogue of Proposition 2.4.3 is false. For example, the equation T ( x l , x 2 )= ( x 2 ,- x l ) defines a non-zero operator T acting on R2, and ( T x , ~=) 0 for each x in R2. 2.4.5. PROPOSITION. If X and X are Hilbert spaces and T E B ( X X , ), then T is an isomorphism from X onto X if and only if it is invertible, with T - ‘ = T*.

Proof. The operator Tis an isomorphism from X onto X if and only if it is both invertible and norm preserving. Accordingly, we may suppose that T has an inverse, and it suffices to show that T preserves norms if and only if T*T = I (which is equivalent to T - = T* when T is invertible). Since

( T * T x , x ) - ( x , x ) = ( T x , T x ) - ( x , x ) = IITX112 - I x I ’ , for each x in X, the required result follows from Proposition 2.4.3.

Classes of operators. A bounded linear operator T, acting on a Hilbert space 2,is said to be self-adjoint if T* = T, and unitary if TT* = T * T = I. Both these conditions imply that T is normal, by which we mean that TT* = T* T. We say that T is positive if ( T x , x) 2 0 for each x in X. For every Tin $#(X),

(T*Tx,x)=(Tx,Tx)>O

(xEX),

so T * T is positive; we shall see later (Theorem 4.2.6(iii)) that each positive operator arises in this way. A conjugate-bilinear functional b on X is said to be symmetric if b(y,x ) = b(x,y ) for all x and y in X, and positive if b(x, x ) 2 0 for each x. With T in &I(%), and bT the conjugate-bilinear functional defined by bT(x,y ) = ( T x ,y ) , it is evident that T is positive if and only if bT is positive; moreover, ~~

bT*(X,y)= ( T * x , y ) = ( x , TY) = (Ty, x> = bT(y, x), so bT = bT*(equivalently, T is self-adjoint) if and only if bT is symmetric. Suppose that Tis a bounded linear operator acting on a 2.4.6. PROPOSITION. Hilbert space 2. (i) T is self-adjoint if and only if ( T x , x > is real for each x in X. In particular, positive operators are self-adjoint. (ii) T is unitary if and only fi T is a norm-preserving (equivalently, inner product-preserving) mapping from X onto 2. (iii) T is normal ifand only if IITxll = IIT*x(lfor each x in X.

104

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

Proof. (i) Since

( T x , x ) - (T*x, x ) = ( T x , x ) - ( x , T x ) = 2iIm( Tx, x ) , it follows from Proposition 2.4.3 that T = T* if and only if ( T x , x ) is real for each vector x. (ii) An element T of A?(%) is unitary if and only if it is invertible, with inverse T * ; so the assertion (ii) is a special case of Proposition 2.4.5. (iii) Since ( T * T x , x ) - ( T T * x , x ) = ( T x , T x ) - ( T * x , T * x ) = IITxll’

-

((T*x(~’,

if follows from Proposition 2.4.3 that T*T = TT* if and only if [(TxlI= IIT*x(l for each vector x. From part (ii) of the above proposition, a “unitary operator” acting on a Hilbert space X is simply an isomorphism from X onto itself. We shall sometimes describe isomorphisms between different Hilbert spaces as unitary operators (or unitary transformations). 2.4.7. REMARK.When X is an infinite-dimensional Hilbert space, a norm-preserving linear operator Tacting on X is not necessarily unitary, since its range may fail to be the whole of X . For an example in which this occurs, consider the operator that acts on the sequence space l2 (see Example 2.1.12), and maps the vector ( x 1 , x 2 , x 3 ., ..) onto ( 0 , x 1 , x 2 ,...). H 2.4.8. LEMMA. r f T is a bounded normal operator on the Hilbert space 2 and 0 < inf{llTxll:xEX, llxll = 1) ( = a ) ,

then T has a bounded, two-sided inverse, and 11 T - ‘ ( 1

= a - l.

Proof. By Corollary 1.5.10, T is a bicontinuous linear mapping from 2 onto the range W(T ) of T, and the inverse mapping T - : W(T ) -,X satisfies llT-lll = a - l ; moreover, W(T)is complete, and is therefore closed in X. It remains to prove that W(T)= X. If W(T)# 2,there is a unit vector x in W(T)*; by Proposition 2.4.6(iii),

0 = ( x , TT*x) = ( T*x, T*x) =

IIT*xll’ = IITx11’ 2 a’,

a contradiction. Thus W ( T ) = S. The simple properties of the adjoint operation * on a(%), as set out in (i), . . . ,(iv) in the discussion preceding Proposition 2.4.3, in some respects resemble those of the process of complex conjugation for elements of the scalar field @. The analogy can usefully be pressed a good deal further. The self-

105

2.4. LINEAR OPERATORS

adjoint elements of g ( X ) (those for which T = T*) correspond to real numbers (the scalars for which a = a). Parallel to the expression of a complex number in terms of its real and imaginary parts, each Tin B ( X )can be written (uniquely) in the form H + iK, with H a n d K self-adjoint operators acting on X ; moreover

H = i ( T + T*),

K = $i(T* - T).

The operators H a n d K are sometimes called the “real” and “imaginary” parts of T, and denoted by Re T and Im T, respectively. It is easily verified that T is normal if and only if H and K commute. The classes g9, Jf and g ( i f ) + of all self-adjoint, unitary, normal, and positive operators (respectively) on X are norm-closed subsets of g(&). Moreover, 9is a multiplicative group, while .Y is a real-linear subspace (that is, aH + bKEY whenever H, K E Y and a, b E IF!). From Proposition 2.4.6(i), a(.%)+ is a subset of 9; it is apparent that aH b K E B ( H ) + whenever H , K E B ( X ) +and a, b are non-negative real numbers. If both HE^?(#)+ and - HEg ( X ) ’ , then (Hx, x) = 0 for each vector x, and H = 0 by Proposition 2.4.3; so B ( X ) +n - g(#)+= (0). In view of the properties of a(#)+just stated, there is a partial order relation < on Y: in which H < K if and only if K - HEB ( X ) +As . in the case of real numbers, one can add such inequalities between self-adjoint operators and multiply throughout by non-negative scalars. Multiplication by negative real numbers reverses the inequalities. Moreover, if H, K e g T E ~ ( % ) ,and H < K, then T*HT < T*KT; for K - HEB(.%)+,and therefore

+

(T*(K - H)Tx,x) = ( ( K - H)Tx, Tx) 2 0 ( x ~ i f ) , whence T*(K - H ) T E ~ ( # ) + .For each H in Y: the operators llHllZ f H a r e positive, since IIHll(x,x) f (Hx,x) 2

llHll llxll’

-

WxlI IIxII 2 0

(xEJ~‘).

It follows that -

IIHIII < H ,< llHlll

( H = H* E B ( X ) ) ,

and that each self-adjoint operator H can be expressed as the difference of positive operators llHllZand IIHIJI- H. Each Tin a(#)has the form H + iK, with Hand K self-adjoint, and is therefore a linear combination of at most four elements of B ( X ) + . We shall see later, in Theorem 4.1.7, Theorem 6.1.2, and Proposition 4.2.3, that each element of a(.%) is a linear combination of at most four unitary operators, and has a “polar decomposition” analogous to the expression of a complex number in terms of its modulus and argument; moreover, there is an “optimal” way of expressing a self-adjoint operator as a difference of positive operators.

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2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2.4.9. REMARK. For the study of bounded linear operators, a Hilbert space is much more convenient than a real inner product space. This is due, in part, to the fact that, in contrast with the complex case, a non-self-adjoint operator acting on a real inner product space cannot be expressed as a linear combination (necessarily with real coefficients) of self-adjoint operators. 2.4.10. EXAMPLE.Suppose that Y is an orthonormal basis in a Hilbert space X, g is a bounded complex-valued function on Y, and

k = suP{lg(Y)l:YE y > . For each x in 2,

1 Ig(Y)(x,Y)lz 6 kZ 1 I(X,Y>lZ YCY

= k211X112;

YCY

so the equation (4) defines a vector Tx in X, and

It is apparent that Tx depends linearly on x, so T is a bounded linear operator on 2, with IlTll 6 k. From (4),

TY = d Y l Y

(6) From this,

(YE

Y).

IlTll 2 suP{llTY/l:YE Y ) = suP{lg(Y)l:YE Y ) = k, and so (7)

IlTll = suP{lg(Y)l:YE y>. By the same process, the complex-valued function defined by g ( y ) = g(y),gives rise to a bounded linear operator S on X. For all u and u in

a,

3,

(SU, u>

=

(

1 s(v)(u,v>r, u>

YSY

107

2.4. LINEAR OPERATORS

Hence S = T* and, in parallel with (4), (9,and (6), we have

(9)

IIT*xll =

(1la(Y)(x,Y)l2)

1/2 3

YEY

(10) T*y= g(ylY ( Y E r). From (5) and (9), IITxll = IIT*xJI for each vector x, so T is normal; alternatively, this can be proved by a simple direct calculation, which shows that (1 1)

TT*x = T*Tx =

1 Ig(v)12(x,y)y. YEY

Similar calculations show that sums and products of bounded complex-valued functions correspond to sums and products of the associated operators; in particular, all such operators commute. If Tis self-adjoint,it results from (6) and (lo) that g(y) = g(y)for each y in Y; the reverse implication follows from (4) and (8). Thus Tis self-adjoint if and only if g is a real-valued function. From (4) (Tx, x> =

1 g(Y)(X,Y)(Y? x) = c s(Y)l(x,v>12; YEY

YEY

in particular, (Ty,y) = g(y) (YE Y). Hence Tis positive if and only if g takes non-negative real values throughout Y. If Tisunitary, it follows from (6) that Ig(y)l = IITyll = llyll = 1 for each y in Y. Conversely, if Ig(y)I = 1 ( Y E Y), we deduce from (1 1) that TT*x = T*Tx =

1 (x,y)y

=x

(~€2).

YEY

Hence T is unitary if and only if Ig(y)l = 1 for each y in Y. 2.4.11. EXAMPLE. Suppose that m is a a-finite measure defined on a o-algebra Y of subsets of a set S, gE L , (= L,(S, Xm)),and k is the essential supremum of 191. For each x in the Hilbert space S = L2(S,,Y;’rn), the equation

(M,x)(s) = g ( M 4 (SE S) defines a measurable function M,x on S, and I(M,x)(s)l ,< klx(s)l almost everywhere. Accordingly, M , x E(= ~ L 2 ) and IIMgxll < kllxll since (12)

108

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

It is apparent that M,x depends linearly on x; so M B is a bounded linear operator on X, and llM,ll 6 k . If 0 < a < k, the measurable set { s E S : Ig(s)l > a } has positive measure, and so has a measurable subset Y such that 0 c m( Y ) < 00, since m is rr-finite. The characteristic function y of Y is a non-zero vector in 2, I(Ma)(s)l 2 aly(s)l for each s in S ; hence llM,gll 2 allyll, and so llMJ 2 a. From this, llM,ll = k ; that is,

where (1 1(, is the usual norm on L,. With g defined by g(s) = g(s) (sE S ) , we have g E L , and

I-

therefore

M,* = M i .

(14) It is apparent that (15)

M,,+~, = a ~ +,b ~ , , M ,, = M , M ,

um,,

a,b~c).

From this, M , and M , commute for all f and g in L,; in particular, M g commutes with its adjoint Ms, and is therefore normal. Since llMg

- M,*ll= llMg-ull= esssup Ids) sss

-

go(,

M Bis self-adjoint if and only if g(s) is real for almost all s in S. Moreover, =

s,

s(s)lx(s)tz dm(s)

(x E 2 1,

from which it follows that MBis positive if and only if g(s) 2 0 for almost all s. With u in L , defined by u(s) = 1 (SE S),M u= I and

Thus M gis unitary if and only if Ig(s)l = 1 for almost all s in S . It is well known that a linear operator T, acting on a finite-dimensional complex vector space, has at least one eigenvector x, with corresponding

2.5.

THE LATTICE OF PROJECTIONS

109

eigenvalue A (that is, x # 0 and Tx = Ax). As indicated in (6), the operator considered in the preceding example has eigenvectors forming an orthonormal basis Y. In contrast, the present example permits the construction of selfadjoint and unitary operators that have no eigenvalue. For this purpose, note that the equation MBx= cx (with g in L,, x a non-zero vector in X (= L z ) , and c in C ) implies that g(s) = c almost everywhere on the measurable set {SE S: X ( S ) # 0 } , which has positive measure. Accordingly, MBhas no eigenvalue if g assumes each of its values only on a null set. When m is Lebesgue measure on the o-ring Y of Bore1 subsets of the interval S = [0,1] and

f ( s ) = s,

g(s) = exp(is)

(sE S),

M J is positive, M , is unitary, and neither has an eigenvalue. H 2.5. The lattice of projections

If Y is a closed subspace of a Hilbert space X, Theorem 2.2.3 asserts that each vector in Z can be expressed uniquely in the formy + z, with y in Y and z in Y'. From Theorem 1.1.8, the equation (1)

E(y

+

Z)

=y

( Y E Y,

ZE

Yl)

defines a linear operator E acting on Z,the projection onto Y, parallel to Y *. Moreover, E 2 = E, (2)

Y = { E x : x E X }=

EX: Ey = y } ,

and Y ' = { zE Z : Ez = 0).We call E the (orthogonal)projectionfrom Z onto Y. Note that I - E is the orthogonal projection from X onto Y l , because (Y')' = Y, and

(I-E)(z+y)=z

(zEY', ~ E Y ) .

Since (y, z ) = 0, when y E Y and z E Y ', we have

IIm + 4112= llYIl2 G llY1I2 + 1 1 ~ 1 1 2 = IlY + z1IZ, (E(Y + Z),Y

+ z> = ( y , y + z> = lIy1lZ2 0.

It follows that E is bounded, with IlEll ,< 1, and is positive (hence, also, selfadjoint). Since Ey = y ( y e Y), IlEll = 1 except in the case in which Y = {O} and E = 0. Moreover (3)

Y = { x € x :((Ex((= ((x((}.

Conversely, suppose that E E ~ ( Xand ) E Z = E = E*. From Theorem 1.1.8, E is the projection from X onto the closed subspace Y defined by (2),

110

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

parallel to the closed subspace 2 = {z E X :Ez = 03.Since Z = {zEX:(Ez,x)=OforeachxinX} = {zEX: ( z , E x ) = 0 for each x in X } ,

it follows from (2) that Z = Y l . Hence E is the projection onto Y, parallel to Y l ; that is, the orthogonal projection from X onto Y. In the context of Hilbert space theory, it is understood that the term “projection” refers to an orthogonal projection unless there is an explicit statement to the contrary. The following proposition summarizes the results of the preceding discussion. 2.5.1. PROPOSITION. Relations (1) and (2) establish a one-to-one correspondence between closed subspaces Y of a Hilbert space X andprojections E acting on X. A projection E is apositive operator, and IlEll = 1 unless E = 0. The projections are precisely the self-adjoint idempotents in B ( X ) . The projections acting on a Hilbert space X inherit from the set of all selfadjoint operators the partial order relation < described in the discussion preceding Remark 2.4.9. 2.5.2. PROPOSITION. I f E and Fare theprojections from a Hilbert space 2 onto closed subspaces Y and 2, respectively, the following conditions are equivalent : (i) (ii) (iii) (iv) (v)

Y E 2; FE = E ; EF = E ; IlEXlI < IlFXll E < F.

( X E Z ) ;

Proof. If Y G Z, then, for each x in X, E X EY c Z , and therefore FEx = Ex; so (i) implies (ii). If FE = E, then EF = (FE)* = E* = E, whence (ii) implies (iii). If EF = E, then llExll = llEFxlI < llFxll for each x in 2,since IlEll < 1 ; so (iii) implies (iv). Since

( E x , x ) = ( E ’ x , ~ )= ( E x , E x ) = I ~ E X I I ~ , and similarly ( F x , ~ = ) (JFx~)’, it is apparent that (iv) implies (v). If E < F, then, for each y in Y, llY11’ = (EY,Y) i(FY9Y) = llFYllZ < llYI1’;

whence IIFyII = Ilyll, and Y E Zby (3). Hence, (v) implies (i). When the five equivalent conditions in Proposition 2.5.2 are satisfied, we describe E as a subprojection of F.

111

2.5. THE LA'ITICE OF PROJECTIONS

From the equivalence of conditions (i) and (v) in Proposition 2.5.2, it follows that the partial ordering of projections (as self-adjoint operators) corresponds to the partial ordering of closed subspaces by the inclusion relation E . Given any family { Y,} of closed subspaces of a Hilbert space X , there is a greatest closed subspace A Y, that is contained in each Y, and a smallest closed subspace V Y, that contains each Y,. Specifically, A Y, is n Y,, while V Y, is the closed subspace [U Y,] generated by U Y,. From this it follows that each family { E,} of projections acting on A? has a greatest lower bound A E, and a least upper bound V E, within the set of projections (ordered as selfadjoint operators). Of course, the projections A E, and V E, correspond to the closed subspaces A E , ( X ) and V E , ( X ) , respectively. We write E A F and E v F for the lower and upper bounds (often called the intersection and union) of two projections E and F. Since the mapping E -,I - E reverses the ordering of projections, we have

V ( I - E,) = Z - A E,, A(Z - E,) = I - V E, (4) for each family {E,,} of projections. This gives corresponding relations A Y ; = ( V Y,)' V Y t = ( A Yo)', (which can easily be verified independently) for each family { Y,} of closed subspaces of X. Our next few results are concerned with commuting sets of projections.

If E and F are commuting projections acting on a 2.5.3. PROPOSITION. Hilbert space X, corresponding to closed subspaces Y and Z , respectively, then EvF=E+F-EF,

EAF=EF,

YvZ=Y+Z.

In particular, the linear subspace Y + Z of X is closed. Proof. When U E Y A Z , Eu = Fu = u, so EFu = u. For each x in Y l , Ex = 0, so EFx = FEx = 0 ; similarly, EFx = 0 for each x in Z'. Since ( Y A Z)' = Y' v Z', it follows from the linearity and continuity of EF that EFv = 0 whenever u E ( Y A Z ) l . We have now shown that EF(u + v ) = u

(UE

Y A Z , V E ( YA Z)'),

whence EF is the projection from X onto Y A 2. By applying the same result to the commuting projections I - E and I - F, we have

( I - E ) A (I - F) = ( I - E)(I - F), and (4) gives

E vF= I

- (I - E ) A

( I - F) = I

- (I - E)(I - F ) = E

For each x in Y v 2,

x = ( E v F ) x = ( E + F - EF)x = y

+ Z,

+F -

EF.

112

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

where y = ( E - EF)x€ Y and z = F x E Z . Thus Y v Z G Y + Z , and the reverse inclusion is apparent. H 2.5.4. COROLLARY. Suppose that Eand Fare theprojections from a Hilbert space X onto closed subspaces Y and Z , respectively. Then EF = 0 if and only i f Y is orthogonal to Z , and when this is so, YvZ=Y+Z.

EvF=E+F,

Proof. Since Y = E ( X ) , Z = F ( X ) , and (EFu, v ) = (Fu, E v ) for all u and v in X, it is evident that Y is orthogonal to Z if and only if EF = 0. When this is so, FE = (EF)* = 0 (= EF), and it follows from Proposition 2.5.3 that

EvF=E+F,

YvZ=Y+Z.

2.5.5. COROLLARY. I f E and F are the projections from a Hilbert space A? onto closed subspaces Y and Z , respectively, and E < F, then F - E is the projection from X onto Z A Y l . Proof. By Proposition 2.5.2, EF = FE = E, so the projections F and I - E commute. From Proposition 2.5.3, F(Z - E ) (= F - E ) is the projection F A (I- E ) from X onto Z A Y l . H

Projections E and F, from X onto closed subspaces Y and Z , commute if and only if Y A ( Y A Z)* and Z A ( Y A Z)* are orthogonal (loosely, if and only if the spaces Y and 2 are “perpendicular”); for these spaces are orthogonal if and only if 0 =(E- E and EF = E

A

A

F)(F- E

A

F) = E F - E

A

F,

F if and only if (see Proposition 2.5.3) EF=EAF=(EAF)*=FE.

2.5.6. PROPOSITION. Zf{E,,} is an increasing net ofprojections acting on a Hilbert space X, and if E = V E,,, then Ex = lim, Eaxfor each x in X.

is an increasing net of closed subspaces of u,, and isSince a linear subspace of and has norm closure Suppose > Since we can choose an element in one of the

Proof. {E,,(X)} EJX) X E(X). XEX E 0. Ex E E ( X ) , y subspaces E , , ( X ) so that IJEx- yll < E . When b 2 a, we have En 6 Eb

< E,

y E E J X ) G &(A?)

G

E(X),

and thus llEx - Ebxll = IIE(Ex - Y ) - ‘%(Ex - Y)II

< IIE - Ebll IJEx- YII < &.

A?,

2.5. THE LATTICE OF PROJECTIONS

113

2.5.7. COROLLARY. I f {E,} is a decreasing net of projections acting on a Hilbert space X, and if E = A E,, then Ex = lim, E,xfor each x in 2. ProoJ In view of (4), it sufices to apply Proposition 2.5.6 to the increasing net { I - E,}.

By an orthogonal family of projections we mean a family (Ea)aeAof projections such that E,E, = 0 (equivalently, E , ( 2 ) is orthogonal to &(A?)) whenever a and b are distinct elements of A. 2.5.8. PROPOSITION. If(Ea)aEEn is an orthogonalfamily ofprojections acting on a Hilbert space X, E = V E,, and x E X, then Ex = 1E,x; the sum converges in the norm topology on X. Proof. When A is a finite set, it follows from Corollary 2.5.4, together with a straightforward argument by induction on the number of elements in A, that E = CaEaE,. When A is an infinite set, let 9denote the class of all finite subsets of A ;for each ff in Edefine G, = CEFE,. By the preceding paragraph, G , = VaEF E,, so (GF,F E E 2 ) is an increasing net of projections, and VG, = V{ FE3F

v E,: ad

IFEF

1v =

E, = E.

aEA

By Proposition 2.5.6, Ex is the limit, in norm, of the net (G,x,F E 4 s ) ;that is (since G,x = CEFE,x), CasAE,x converges in norm to Ex. H When H is a Hilbert space and x E X, the equation p,( 7') = I ITxll defines a semi-normp, on B ( X ) .The family of all such semi-norms separates the points of B ( X ) ,in the sense of Theorem 1.2.6, and so gives rise to a locally convex topology on B ( X ) ,the strong-operator topology. In this topology, an element To of B ( 2 )has a base of neighborhoods consisting of all sets of the type V ( T o : x ,..., l x , ; E ) = { T E B ( X ) : I I ( T -To)xjII c ~ ( j 1=,..., m)},

where xl,. . . ,X,EXand E > 0. In fact, it suffices (and is sometimes convenient) to take E = 1, since

...)X , ; & )

V(T0:Xl)

= V ( T O : E - l X I ) . . .E, - l X m ; l ) .

In a similar way, one can introduce the strong-operator topology on B(X,X ) ; the semi-norms and basic neighborhoods are defined as above, but the vectors Tx, ( T - To)xjlie in X . The strong-operator topology can be described as the restriction to W ( X ) of the point-open topology on mappings from X into X, with X in its norm topology. It is apparent that V ( T o : x , ,. . .,x,,,;E)contains the open ball with center To and radius 6 , provided that bllxjll < E for each j = 1,. . ., m .

114

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

Accordingly, the strong-operator topology is coarser than the norm topology in fact, it is strictly coarser when % is infinite-dimensional (see on a(&'); Exercise 2.8.32). In working with the strong-operator topology, it is often possible (and useful) to confine attention to those basic neighborhoods V(To:x l , . . . ,x , ;E ) in which x l , . . .,x,,, are drawn from a suitable preassigned subset 9 'of %. We mention two cases in which this occurs. First, when 9'has (algebraic) linear span %, the sets V(To:y l , . . . ,y , ;S), where 6 > 0 and yl ,. . . ,y, E 9, already form a base of neighborhoods of To. Second, if Y has closed linear span 2, while 93 is a bounded subset of 93(&') and ToE 93, the sets @>O,

Wo:y1,...,yn;6)ng

yl ,...,y , E Y )

form a base of neighborhoods of To in the (relative) strong-operator topology on 8 .We shall prove the second of these assertions (the first follows from a similar, but simpler, argument). Wemay assume that IJTJI < Mfor each T i n a . Given any positive E , and x l , . . . ,x , in S,each xi can be approximated within c/4M by a finite linear combination of elements of 9 Hence we can choose y , ,. . . ,y , in Y and scalars a j k (some of which may be 0) such that n

Let 6 be a positive real number such that 26(G= lajkl) c

j = 1,. . . ,m. It now suffices to show that

E

for each

V( To:yl ,. . . ,y, ;6 ) n 9 G V(To:xl, .. . ,x,,,;E ) ;

and this results from the fact that, for J = 1,. . . ,m, n

n

-

TO>xjll


ll

+ 11

n

< IIT -

llxj

-

1 k= 1

ajk(T -

TO)ykll

k=l

k= 1

n

ajkykll

+

bjkl

ll(T - TO)ykll

k= 1

when T (as well as To)lies in &and I ll(T - To)ykll < 6 (k = 1,. . . ,n). We can summarize the results of the preceding paragraph as follows: a set of vectors with algebraic linear span % suffices to determine the strongoperator topology on O ( X ) ; a set of vectors with closed linear span % suffices to determine the strong-operator topology on bounded subsets of

gw ).

2.5. THE LATTICE OF PROJECTIONS

115

2.5.9. REMARK.From the discussion following Theorem 1.2.18, a net { T j } of elements of A?(&) is strong-operator convergent to To ( E W ( Xif) and only if, given any x in X and any positive E , there is an indexjo such that

l

< 0,

Accordingly, the family {x, + yo}is in 10Ha;it followseasily that 10H a i sa pre-Hilbert space when the algebraic structure, inner product, and norm are defined by

+ {Y,}

{x,}

=

{x.

+Yo},

({x,}, { Y J > = C(Xo,Ya)r

C{XJ =

{cx,},

IHxJIl = [Cllxall211/2.

The element {x,} of 10X, is sometimes denoted by 10 x,. We assert that I@ X, is complete, relative to the norm just defined, and is therefore a Hilbert space. For this, suppose that (x("))is a Cauchy sequence in X,, so that each x(")is a family {xr)} of the type considered above. Given ) that any positive real number E , there is a positive integer n ( ~ such IIx(") - x(")II< E whenever m,n 2 n ( ~ )that ; is,

c@ (1)

C Ilxr) - xF)l12< E'

(m, n

n(~)).

aE A

From this, Ilx:") - xr'll < E (m,n2 n ( ~ ) ,U E A); so, for each fixed a, {xr):n = 1,2,. . .} is a Cauchy sequence in X,, and therefore converges to an element x, of A?,.With [F a finite subset of A, it follows from (1) that

C Ilx:"'

- x,(n) 11 2

< E'

(m,n 2 44).

< E2

(n 2 n(E)),

a€F

When m + 00, we obtain

C 11%

ad

- xb"'l12

124

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

and since the last inequality is satisfied for every finite subset IF of A, we have

This shows that the family { x , - x:)}, as well as {x:)}, is in 10i f a when n 2 n ( ~ )Accordingly, . {x,} (= { x , - x:)} + {x:)}) is an element x of Xu, and (2) asserts that JIx- x'")(I< E whenever n 2 n(e). Thus (x("))converges to x; so I@ ifais complete, and is therefore a Hilbert space. With b in A, %b is isomorphic to the closed subspace XL of COX, consisting of those families {x,} such that x, = 0 whenever a # b. We obtain a unitary transformation ub, from z b onto if;,by taking for ubx the family {x,} in which xb is x and x, = 0 (a # b). The subspaces X i (aE A) are pairwise 2,. orthogonal, and V 2;= If { X,} is a family of mutually orthogonal subspaces of a Hilbert space X, and V X, = if,the corresponding projections form an orthogonal family {E,} with (strong-operator convergent) sum I. Just as in the case of finite direct sums, the equation U x = {E,x} defines an isomorphism U from 2 onto X,, and we consider X as an "internal" direct sum of the family (2,). Moreover, U - ' { x , } = E x , when {x,} EX@ X,; the sum converges since { x , } is an orthogonal set in X, and 111xaI12< co. Suppose next that X,, X, are Hilbert spaces, and T, E a(#,,X,) for each a in A. If

c@

c@

c@

s~P{llTaIl:aEAl< a,

c@

the equation T { x a } = { Tax,} defines a bounded linear operator Tfrom Za into X,. We call T the direct sum 10 T, of the family { T,}. Just as in the case of finite direct sums, we have

c@

and R , E ~ ( X , , ~ ~ ) . when S,, T,E~J(#,,X,) When 2,is the one-dimensional Hilbert space @, for each a in A, c0 8, reduces to the Hilbert space /,(A) of Example 2.1.12.

2.6. CONSTRUCTIONS WITH HILBERT SPACES

125

Tensor products and the Hilbert-Schmidt class. The material in this subsection will be used in an essential way in later parts of the book (from Chapter 1 1 onward), but has a relatively minor role until that point. The reader who wishes may bypass it until that stage, and will have only very occasional need to refer back in the meantime. There are several ways o f defining the (Hilbert) tensor product X of two Hilbert spaces 2l and ifz,each method having advantages in particular circumstances. Our approach, set out below, emphasizes the “universal” property of the tensor product. The Hilbert space X is characterized (up to isomorphism) by the existence of a bilinear mapping p, from the Cartesian product Xl x X2 into X, with the following property: each “suitable” bilinear mapping L from $‘1 x X ; into a Hilbert space X has a unique factorization L = Tp, with T a bounded linear operator from %‘ into X. Before starting the formal development of the theory, we indicate some of the intuitive ideas that underly it. When x, E Xl and xz E X z ,we shall want to view the elementp(xl, x2)of 2 as a “product,” x, 8 x2, of x, and x2.It turns out that linear combinations of such products form an everywhere-dense subspace of X. The bilinearity of p implies that these products satisfy certain linear relations; for example, as the product notation suggests, (x1

+ Y l ) 0 (x2 + vz) - x1Ox2 - XI OYZ - Y l 0 x2 - Y , OYZ = 0,

whenever xl, y , E XI and x2, y 2 E SZ. In fact, all the linear relations satisfied by product vectors can be deduced by (possibly repeated) use of the bilinearity o f p . It turns out that the inner product on X satisfies (and is determined by) the condition (x1 O X Z 9 Y l

0 Y Z ) = (xl,Yl)(X2,Yz);

in particular, llxl @ x211= llxlII Ilxzll.There are various constructions leading to a Hilbert space X with the required properties. In the method we shall use, the elements of 2 are certain complex-valued functions defined on the product Xl x S2and conjugate-linearin both variables (and by introducing a concept of “conjugate Hilbert space,” these functions are viewed as bilinear functionals). When u1 E X ,and uz e X 2 , u1 @ uz is the function that assigns the value ( u , , x l ) ( u 2 , x z ) to the element (x1,x2) of Z1x Xz. In the formal development of the theory, we first introduce the class of bilinear mappings used in formulating the universal property mentioned above. The tensor product is then defined, and identified (up to isomorphism) with certain specific Hilbert spaces, such as the completion of the algebraic tensor product and the class of “Hilbert-Schmidt operators” from the conjugate Hilbert space into X 2 .We conclude this subsection with a discussion of tensor products of bounded linear operators. It is convenient in the initial stages to consider the tensor product of a finite family of n Hilbert spaces, specializing later to the case in which n = 2.

126

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

Suppose that Sl,. . . ,Xnare Hilbert spaces and cp is a mapping from the Cartesian product XI x . . x Xninto the scalar field C. We describe cp as a bounded multilinear functional on Xl x * . x X,, if cp is linear in each of its variables (while the other variables remain fixed), and there is a real number c such that

-

IV( X I

7 * *

9

xn)l

< cllxlll .

* *

.

(XIE XI9 .

llxnll

xn E %)*

When this is so, the least such constant c is denoted by llcpll. Then, cp is a continuous mapping from Xl x . . x X, into C, relative to the product of the norm topologies on the Hilbert spaces; the estimates required to prove this are much the same as those needed in showing that the mapping ( a l ,..., a n ) + c a l . * * a , ,C: x

x C+C

is continuous, so we omit the details. In the following proposition, we consider certain sums of positive terms, which may converge or diverge, and a divergent sum is to be interpreted as co. In part (ii) of the proposition, inequalities involving co are to be understood in the obvious sense, and we adopt the convention that 0 00 = 0. Whether or not the sums considered converge, the manipulations required in the proof are easily justified, in view of the final paragraph of Section 1.2.

+

1

2.6.1. PROPOSITION. Suppose that Xl ,. .. ,Xnare Hilbert spaces and cp is a bounded multilinear functional on X I x * . x X,,.

-

(i) The sum

1

(3)

1 IdYl,...,Y,)I2

* * *

YneY,

YlEYl

has the same uinite or infinite) valuefor all orthonormal bases Y1 of XI,. . . , Y, of 2,. (ii) If.%,, . . .,X, are Hilbert spaces, A , E @(X,,X,) (m = 1, . . . ,n), I) is a bounded multilinear functional on Xl x . . x X,, and ~p(xl, *.

.y x n )

= +(A1x1,.

*

,Anxn)

(XI

EX^, . . * , x n ~ X n ) ,

then

1 YlEYl

* *

.

1

I ~ Y I*

* . ,Yn)Iz

< IIA1112 . .

*

IIAnII'

YneYn

1 ZlEZl

..

*

1 I+(zl,

when Y, and Z , are orthonormal bases of .X, and X,, (m = 1,. . . ,n).

1

C Y&Y"

Icp(Y1,*..,yn)Iz


is a Hilbert-Schmidt functional on Xl x * x Xn; (ii) there is a real number d such that llLul12< dllull for each u in X. When these conditions are satisfied, the least possible value of the constant din (ii) is denoted by IILl12. H As in the case of multilinear functionals, a bounded multilinear mapping L: Xl x . . . x X,,+ X is (jointly) continuous relative to the norm topologies on the Hilbert spaces. Condition (ii) is in fact redundant, since it follows from (i), by an application of the closed graph theorem to the mapping u + L,:X + X Y F (Exercise 2.8.36). We shall not make use of this implication, and have incorporated (ii) in the definition for convenience.

132

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

2.6.4. THEOREM. Suppose that Xl,. . . ,X,, are Hilbert spaces. (i) There is a Hilbert space & and a weak Hilbert-Schmidt mapping p : HI x x X,,+ X with the following property: given any weak Hilbertinto a Hilbert space X, there is a Schmidt mapping L from Xl x * . . x Z,, unique bounded linear mapping Tfrom 2 into X, such that L = Tp ; moreover, IITII = llLl12. (ii) If&' andp' have the properties attributed in (i) to X andp, there is a unitary transformation U from &Y onto Z'such that p' = Up. (iii) I f u,, w, E X, and Y, is an orthonormal basis of 2 ,(m = 1 , , . . ,n), then

( P ( U ~ * uAP(w1,* . ., wn)> = 9 . .

9

wl> . . *

( ~ 1 3

(unr

wn>,

the set { p ( y l, . . . ,y,,):y , E Y , , . . .,y,, E Y,,} is an orthonormal basis of X, and llPll2 =

1.

Proof. With 2, the conjugate Hilbert space of 2,,let X be the set of all with the Hilbert space Hilbert-Schmidt functionals on P1x . . . x 2,, structure described in Proposition 2.6.2. When v( 1) E XI,. . . ,v(n)E X,,, let p(u(l),. . . ,v(n)) be the Hilbert-Schmidt functional (pvcl,,...,",) defined on

2,x . . '

x

2,,

by (~"(1)

XI

7

. , x n ) = ( X I ~ ( 1 ) ) .- . * ( x n ,v(n)>9

=

< v ( l ) , x l ) * . ( W ,xn>.

Since Y j is an orthonormal basis of X j ( j = 1 , . . . ,n), it follows from Proposition 2.6.2 that the set { p ( y l , .. . ,y,,):y l E Y1,. . . ,y , , Y,,} ~ is an orthonormal basis of A?,and that (P(~I,...,U~),P(WI,...,W~)> = ( ~ 1 y u 1 ) .- . * ( W n , u n > -

= (01 IlP(vl9..

-

9

9

un)112 = llulll

Wl> *

. . (4l, wn), *

. . IIvnll.

From the preceding paragraph, p : XI x . * x X,,+ .# is a bounded multilinear mapping: we prove next that it is a weak Hilbert-Schmidt mapping. For this, suppose that cp E X, and consider the bounded multilinear functional p v : X1x . * 'x X,, + C defined by 1

P J ~ I ., xn) = 9 . .

3

*

., -4, ~p>.

With y(1) in Y1,. . . ,y(n) in Y,, orthonormality of the bases implies that (pvC1, ,.__, ),,, takes the value 1 at (y(1),..., y(n)) and 0 elsewhere on

2.6. CONSTRUCTIONS WITH HILBERT SPACES

= cp(Y(l), * .

1

1

...

Y(~)EYI

133

. 3 Y W 7

lP,(Y(l), . . . ,An))I2 = l cpll;.

y(n)EYn

From this, pq is a Hilbert-Schmidt functional on Xl x x Xn and llpIll~= llrpllz; sop: Xl x . . . x Xn+ X is a weak Hilbert-Schmidt mapping with llpllz = I . Suppose next that L is a weak Hilbert-Schmidt mapping from Xl x . * . x Xninto another Hilbert space X. If u E X and Luis the HilbertSchmidt functional occurring in Definition 2.6.3, while cp E X and [F is a finite subset of Y1 x * . . x Yn, we have

I(

1

W 1 3 . .

*

,Yn)UYl,

* * *

,Yn),

u>I

( Y r . . .. . A H

6

1

(Yl.. . , .Y"kF

I d y l9 *

* *

,Yn)I ILu(Y19.. * ,Yn)I

134

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

converges to anelement T q of X, and IlTqll < llL11211q112.Thus Tis a bounded linear operator from &' into X, and IlTll < IILl12. When y ( 1 ) ~Y1, . . . ,y(n)E Y,,, we have

. . . ,y(n)) = Tqy(1),....y(n) TP(Y(~),

C

=

C

* * *

YlEYI

qy(1),...,y(ndy1, * * * ,yn)L(ylY. . . , Y n )

Y"EY.

= L(y(l), *

,v(n)).

*

Since L and Tp are both bounded and multilinear and Y,,,has closed linear span

Xm(rn = 1,. . . ,n),it follows that L = Tp.

The condition Tp = L uniquely determines the bounded linear operator T, x Y,,) of 2. because the range ofp contains the orthonormal basisp( Y1 x For each u in X, Parseval's equation gives

I I L ~ I I ~=

1

+

*

.

=

C C

a

*

1

Y&Y"

.

YlEYI

=

I(L(y1, * * ,yn),u)12

1 I(TP(yl,***,yn),u)12

*

YlEYl

=

C Y.EY"

YlEYl

I(p(y1,.

*

*.

,yn), T*u)12

Y+Y.

llT*uIl2< lIT11211u112;

so llLll2 < IlTll, and thus llLll2 = IlTll. It remains to prove part (ii) of the theorem. For this, suppose that X ' and p': Xl x . . x &',, -+ &''(as well as &' andp) have the properties set out in (i). When X is &" and L is p', the equation L = Tp' is satisfied when T is the identity operator on X ' , and also when T is the projection from 2" onto the closed subspace [p'(X1 x x Xn)]generated by the range p'(Xl x . . . x &',,) of p'. From the uniqueness of T, [pyx1 x

* .

. x &', )I

= 2';

moreover, llP'll2 = l l U 2 = IlTll = 1141= 1. With the same choice, X = &" and L = p', it follows, from the properties of X andp set out in (i), that there is a bounded linear operator U from &' into X ' such that p' = Up and

IlUll = IILII2 = IIP'll2 = 1. The roles of X, p and A?',p' can be reversed in this argument, so there is a bounded linear operator U' from X ' into &'such thatp = U'p' and IlU'll = 1. Since U'Up(x1,. . . , X n ) = UlP'(x1,..., X n ) = p(x1,. . . ,x,,),

2.6. CONSTRUCTIONS WITH HILBERT SPACES

for all x1 in Zl, . . . ,x, in

135

x , while ..

[p(X1 x

*

x Zn)] = X,

it follows that U‘U is the identity operator on S‘; and similarly, UU‘ is the identity operator on Hi.Finally,

llxll = IIU‘Wl G IIUxll G llxll

(XEZ);

so I(Ux((= ( ( ~ ( 1 , and U is an isomorphism from X onto X ’ . By part (ii) of Theorem 2.6.4, the Hilbert space X appearing in that theorem, together with the multilinear mapping p: Xl x * x Xn-P 2, is uniquely determined (up to isomorphism) by the “universal” property set out in (i). We describe Z as the (Hilbert)tensorproduct of Zl, .. .,S,,, denoted by Zl @ . * Q X,,, and refer to p as the canonical (product) mapping from Xl x . * x #, into Xl 0 * . 6Xn. The vector p ( x l ,. . . ,x,) in Xl @ . . Q X,, is usually denoted by x1 @I . . 6x,. Finite linear combinations of these “simple tensors” form an everywhere-dense subspace of XI @ . ’ * Q Z, ; indeed, if Y, is an orthonormal basis of # , (rn = 1,. . . ,n), then +

{y l

6 . 6yn :y1 E Yl . . . ,Y n E Yfll *

9

is an orthonormal basis of X I dim(Zl O X 2 @

1

.

@ Xn.Thus

*

@ Z , ) = d i m Z l d i m X 2 ...dim#,.

-

-

As the notation suggests, the vector x1 @ . . 6x,, behaves in some respects like a formal product of xl,.. . , x n ; for example, it results from the multilinearity of p , and from Theorem 2.6.4(iii), that (7)

Oxrn-l6(ux:,+bx~)Ox,+lO

XlQ...

Qx,,,-~ @X;@X,+~

=a(x1 Q

+b(xl6 (8)

(XI@

. * *

(9)

a

*

*

OXn,y16 11x1

@

6xn * * *

OX,)

6xm-l6x~Oxm+lO...6xn), . . . O y n ) = ( ~ l , ~ l ) . . . ( x n , ~ n > ,

6 . . 0 xnll = IlXlll ’ . . IlXnll. a

In studying tensor products of Hilbert spaces, the properties just listed are usually more important than the detailed constructions employed in the proof of Theorem 2.6.4. Many of the arguments involve two stages; the first stage deals with the linear span Soof the simple tensors, and is based on the identities (7)-(9), while the second employs “extension by continuity” from Xo to its closure Xl 6 * . . 6 X,,. Since

0 ~2 Q . . 6xn) = (uxl)6x2 6 * * * 6xn, Xo consists of all finite sums of simple tensors. In dealing with X o ,it is 4x1

136

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

important to bear in mind that the simple tensors are not linearly independent. Relation (7) can be viewed as the assertion that a certain linear combination of three simple tensors is zero, and repeated application of (7) yields more complicated identities of this type. We shall look at this question in more detail in Proposition 2.6.6. In the meantime, we establish the “associativity” of the tensor product. 2.6.5. PROPOSITION. I f X l , .. . ,Xm+, are Hilbert spaces, there is a unique unitary transformation U from Xl 6 . ’ 6 X m +onto ,

( 2 1 6 . * .O - % m ) 6 ( X m + 1 6 . . . 6 X m + n ) such that (10) U ( x 1 O * * * 6 x m + n ) = (XI 6 * O x , ) 6 ( x m + 1 O Ox,+,) whenever x j €Xj ( j = 1, . . .,m + n). Proof. Since the set of all simple tensors in Xl 6 . * 6 Xm+, ( = X ) has linear span everywhere dense in X, there is at most one unitary operator U with the stated property; so it suffices to prove the existence of such an isomorphism. For this, let 1

.

.

X ‘ = (XI6 . * * 6 S m ) 6 ( X m + 1 O . . . 6 x m + n ) T and when x j ~ X( j = 1,. ..,m + n), define ~ ( ~ 1 , . . . , x m + n ) = ~ 1 6 . ” 6 ~ m + n(

P‘(XI

9 * * * 9

x m + n ) = (XI

EX)

6 * * * 6 xm) @ ( x m + 1 O * + * 6 x m + n ) ( E X ’ ) .

The ranges of p and p‘ contain orthonormal bases of X and X ’ , respectively, and so generate everywhere-dense subspaces X ( G X ) and X’ (cX ’ ) . I f x j , y j ~ X( j = 1 , . . . , m + n), we have

(14x1

, x m + n ) , p ( Y l ~ . * * ,Ym+n))

7 * *

=(XI,YI)

(Xm,Ym)(Xm+l,Ym+l)

*

a

*

(xm+n,Ym+n)

= ( ~ 1 6 * * * 6 x m , Y 1* 6* * Q Y m ) ( ~ m + l O. . * O X m + n , Y m + l O . . . @ Y m + n ) = (P’(XI

9

*

*

rXm+n),P’(Yl,

-

*

9

ym+n))*

From this,

k= 1

k=l

1=1

wheneverakE@andx)k)EX,(j=1,..., m + n ; k =

1 ,..., q).

137

2.6. CONSTRUCTIONS WITH HILBERT SPACES

The remainder of the argument is of frequently recurring type. The equation

\k=

1

1

k=l

defines a norm-preserving linear mapping Uo from Xonto X’. The definition is unambiguous since, given two expressions Ca,p(xy), . . . and Cb,p(yy),...,y$)+,,) for a vector x in X, it follows (upon replacing Cakp(x‘f),. .. ,x:!+,) by Cakp(xy),.. . ,x:’+,,) - 1blp(yy),. . . in the last chain of equations) that the two corresponding expressions

,XI’+,) ,YE)+,,)

1a,p’(xY),. . . , m +

X(k) n)

and

14 P’(Y:“

1

. . Y:)+ n ) *

9

for Uox are equal. By continuity, Uo extends to an isomorphism U from X onto X ’ , and

By use of the “associativity” established in the preceding proposition, questionsconcerning the n-fold tensor product of Hilbert spaces can usually be reduced to the particular case n = 2. Our next few results are directed toward this case. We consider first the question of linear dependence of simple tensors. 2.6.6. PROPOSITION. Suppose that Xl and X2 are Hilbert spaces, X = S10 X 2 ,and X o is the everywhere-dense subspace of 2‘generated by the simple tensors.

cjn= x j 0 y j

(i) U x , , . . . ,x, E Xl ,y , ,.. . ,ynE X 2 ,then there is an n x n complex matrix [cjk] such that

= 0 ifand only

if

n

1 cjkxj=o

( k = 1, ..., n),

j= 1

(ii) If L is a bilinear mapping from Xl @ X2 into a complex vector space X, there is a (unique) linear mapping T from Xo into X such that L(x,y ) = T(x 0 y ) for each x in Xl and y in X 2 .

138

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

Proof. (i) If there is a matrix [ c j k ] with the stated properties, bilinearity of the mapping (x,y) + x @I y implies that n

n

/

\

n

n

n

Conversely, suppose that Cj”= xi 6y j = 0. I f u l , . . . ,ur is an orthonormal basis of the linear subspace of X2 generated by y l , . . . , y n , we can choose an n x r matrix A = [ a j k ] and an r x n matrix B = [ b j k ] such that r

C

y j =

ajpl

( j = 1, ..., n),

b1kyk

(f

1=1 n 01

C

=

= 1,.

. .,r).

k= 1

With

the n x n matrix AB, we have

[cjk]

Yj =

i

ajl( 1=1

i

=

blkyk)

k=l

( j = 1,.

Cjkrk

. .> n),

k= 1

and n

0=

n

1

x j

C

Oy j =

j= 1

x j

O

j= 1

ajiul (1:1

)

=

,Il

~1

o 01,

where n

( I = 1,.

aj,xj

u1 =

. . ,r).

j= 1

F o r e a c h m = 1, ..., r, r

r

0=

(ui O ui, urn O urn> = 1=1

C (ui,urn)(Vi, urn> = IIurnII’. 1=1

Thus u1 = u2 = . . . = ur = 0, and n

C j= 1

n CjkXj

=

r

r

11U j l b r k X j = 1b l & j=ll=l

=0

(k = 1,. . . ,n).

I= 1

(ii) Suppose that L is a bilinear mapping from XI x 2P2 into X . If . . , x , E X ~, y l , . .. ,yn€ 2 2 , and Cj”= x j O y j = 0, we can choose a matrix

XI,.

2.6. CONSTRUCTIONS WITH HILBERT SPACES [cjk]

139

as in (i). The bilinearity of L then entails

j= 1 n

n

c

=

1

cjkl(xj,yk)

= k= 1

k=l j=l

Suppose next that x l , . . . ,x n ,u l , . . . ,um E X l ,yl ,. . .,y,, u l , . . . ,vmE X 2 , and x j 0 y j = Cj”= uj 0 vj. Then

c

m

n

x j o y j +

j= 1

c ( - u j ) o V j = o ; j= 1

the preceding paragraph shows that m

n

and therefore

j= 1

j= 1

From this, it follows that the equation /

n

\

n

2.6.7. REMARK.The first part of Proposition 2.6.6 asserts, in effect, that the only finite families of simple tensors that have sum zero are those that are “forced” to have zero sum by the bilinearity of the mappingp: ( x ,y) + x 0 y. From this, APo can be identified with the algebraic tensor product of Xl and X 2 ,which was defined, traditionally, as the quotient of the linear space of all formal finite sums of simple tensors by the subspace consisting of those finite sums that must vanish ifp is to be bilinear. The second part of the proposition shows that Xo has the “universal” property that characterizes the algebraic tensor product. We can identify X with the completion of its everywhere-dense subspace Xo.Accordingly, the Hilbert tensor product Xl 0 X 2 can be viewed as the completion of the algebraic tensor product Xo,relative to the unique inner product on Sothat satisfies (x1

@ Y l , X 2 0 Y 2 )

=(X1,X2)(Yl,Y2)

(Xl,X?EJEoll

Y19Y2EX2).

rn

140

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

2.6.8. REMARK.We show that the tensor product of Hilbert spaces A? and X can be viewed as the n-fold direct sum of A? with itself (that is, the direct sum of n copies of A?),where n is the (finite or infinite) dimension of X. For this purpose, let { Y b : b E B} be an orthonormal basis of X, and for each b in B let A?b be X. The mapping w b :x x @ Y b : A?b A? @ is a norm-preserving linear operator from A?b onto a (necessarily closed) subspace A?; of A? Q X. Since c

0 Y a ~ @~Y b2>

f

= (XI,X2)(Ya,Yb)

=0

for all x 1 and x 2 in A?,when a and b are distinct elements of B, it follows that the subspaces {A?::be B} are pairwise orthogonal. With {za} an orthonormal basis of A?,the closed subspace V A?; of A? @ X contains the orthonormal basis {za @ ' y b } , so VA?; = A? Q X. Accordingly, A? @ X is the internal , we have isomorphisms W direct sum of its subspaces A?; ( ~ E B ) and (= I@ w b , from 10A?b onto Z@ A?:) and V (from 10A?: onto A? @ X ) , defined by Thus VW is an isomorphism U , from

Z@A?b onto A? Q X, and

u(c@x b ) = c x b @ Y b * (1 1) From this, each element of A? @ X can be expressed (uniquely, once the orthonormal basis { y b } is specified) in the form 1 x b @ Y b , where x b E A?(bE 5 ) and 1llxbl12 < co.When .fis finite dimensional, the elements of A? @ X are finite sums of simple tensors; the same is true when A? is finite dimensional, since the roles of A? and X are interchangeable. W We show next that the tensor product of Hilbert spaces X and X can be represented as a certain linear space of operators from the conjugate Hilbert space 2 into X. For this, note first that the equation YEX) X ) onto the set of all defines a one-to-one linear mapping T -+ bT from a(%, bounded bilinear functionals on A? x 2 (since these are, precisely, the bounded conjugate-bilinear functionals on A? x X ) .With T i n a(*,X ) , it follows by applying Proposition 2.6.1 to bT that the (finite or infinite) sum b,(X,Y)

= (%Y)

( X E S ,

has the same value, for all orthonormal bases X of A? and Y of X. From Parseval's equation, this sum can be written also in the alternative forms

2.6. CONSTRUCTIONS WITH HILBERT SPACES

141

We describe T as a Hilbert-Schmidt operator if the value of the sums is finite; equivalently, T is a Hilbert-Schmidt operator if and only if bT is a Hilbert-Schmidt functional on X x 2. With X Y 9 the linear space of all Hilbert-Schmidt functionals on S x 9, the Hilbert-Schmidt operators from Y? into X form a linear su bspace XYU = {TEB(X,X):bTEY?Y%} of B(2,X ) .By means of the mapping T + bT, the Hilbert space structure on as described in Proposition 2.6.2, can be transferred to XYO. Accordingly, XYU is a Hilbert space, when the inner product and norm are defined by ( S , T) =

c (SX,Y)(Y, Tx), [ c c I(TX,Y)l’] XEX YEY

1/2

lIT112 =

9

x e x yeY

these being independent of the choice of the orthonormal bases X of X and Y of X , Of course, the mapping T bT is an isomorphism from Z Y O onto X Y R The equality of the four sums appearing in (12) and (13) implies that there are three other, equivalent, expressions for I(T(12 ; and similarly, the inner product ( S , T) can be expressed in the alternative forms

c c (T*Y,XXX,S*Y),

(SX,

Tx),

X€X

YEY X E X

c (T*Y,S*Y). YEY

If X,,, X,, are Hilbert spaces, A E ~ ( X X,,), ,, EEB(Y?,,, X ) , and T is a Hilbert-Schmidt operator from A? into X, then ATE is a Hilbert-Schmidt operatorfromS,, into X,, with IIATEl12 G IlAll llTl12IlEll. For this, let X , bean orthonormal basis of So,and observe that

1 IITW2 =

lIB*T*Yl12, YEY

XEXO

since these sums are the analogues, for TE, of the ones in (13). The stated result now follows from the inequalities

c IIATBxl12 G llA1I2 c IITBxl12 xexo

xexo

= llAl12

IIB*T*Y1I2

YEY

G

11~11211~*112

c IIT*Y1l2 YEY

=

I I A II211 TllzZllBll 2 .

This result can be proved also by means of Proposition 2.6.1.

142

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

The identification of X 6 X with the Hilbert space of all Hilbert-Schmidt operators from 2 into X is described in the proposition that follows. r f X and X are Hilbert spaces, then,for each x in % 2.6.9. PROPOSITION. and y in X, the equation

(~€2)

T x , y=~( u , x ) - ~= ( x , u ) ~

defines a Hilbert-Schmidt operator Tx,yfrom 2 into X. With 29 0 the Hilbert space of all Hilbert-Schmidt operators from 2 into X, there is a unitary transformation U from X Q X onto 2 9 0 ,such that w x €3 Y ) = T x , y

YE.%?-).

(XEX,

Proof. As constructed during the proof of Theorem 2.6.4, X €3 X is the Hilbert space X Y 9 of all Hilbert-Schmidt functionals on R x 9. Moreover, when x E X and y E X, x 6y ( = p(x,y ) ) is the bilinear functional ( P ~defined, , ~ throughout 2 x 9,by (PX,Y(U,

0) =

0,U ) ( Y ,

v).

The discussion preceding Proposition 2.6.9 shows that there is an isomorphism U from X Y 9 onto &‘YO that associates with each Hilbert-Schmidt functional on R x 9 the corresponding Hilbert-Schmidt operator from 2 into X. It is apparent that Tx,y,as defined in the proposition, is the bounded linear operator from 2 into X that corresponds to the bilinear functional qx,y. Since ( P ~E, X ~ Y 9 , it follows that Tx,y E 2 9 0 ,and

u x 6 Y ) = Uqx,, = T X J .

rn

2.6.10. EXAMPLE.With A and B arbitrary sets, we can associate with each x in /,(A) and y in 12(B) a complex-valued function px,y defined throughout AxBby

Px,y(a,b) = x(aly(b).

We shall show that there is a (unique) unitary transformation U from [,(A) Q I,@) onto 12(A x B) such that

w @ v)=

Px,y

(XE12(A),

For this, note first that p , , , ~ l ~ ( xA B) since

c

YEMB)).

2.6. CONSTRUCTIONS WITH HILBERT SPACES

143

Moreover, (P x , y , P”,”)=

c

(o,b)oA x B

(the series manipulations being justified by absolute convergence), when x , u E /,(A) and y , u E 12(B). By expressing norms in terms of inner products, it

now follows that, for any finite linear combination of elements px,y,

We sketch the remainder of the argument, which follows the same pattern as the second paragraph of the proof of Proposition 2.6.5. The linear span Xo of { P ~ ,x ~~ :l ~ ( ~A E) I, ~ ( Bis) everywhere } dense in &(A x B) since it contains the usual orthonormal basis (consisting of functions with value 1 at a single point of A x B and 0 elsewhere); and the linear span Soof the simple tensors is everywhere dense in ],(A) @ 12(B). From (14), there is a norm-preserving linear mapping Uo from Xo onto Z0 such that UOpx,,= x @ y ; and this mapping extends by continuity to an isomorphism U from I,(A x B) onto M A ) @ MB). 2.6.1 1. EXAMPLE.We now consider the tensor product of the L2 spaces m) and (S’, Y ‘ ,m’).We show associated with 0-finite measure spaces (S,9, that this can be identified with the L2 space of the product measure space (S x S ‘ ,Y x Y ‘ ,m x m’),in such a way that x @ y corresponds to the function px,ydefined throughout S x S’ by PX,Y(S,

s? = x(s).Y(s’).

For this, note first that p x , yis a complex-valued measurable function when x E L 2 ( S , Y , m )(= 2)and yEL2(S‘,Y’,m’)(= 2’); moreover, px,yE L2(S x S’,Y x Y ‘ ,m x m‘) (= x ) ,

144

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

since

Also,

=

(

x(s)u(s) dm(s)) S

(1

y(s’)uo)dm’(s’))

S‘

= (x, u > ( y ,v> = (x

6y , u 0 v>,

whenever x, U E A ? and y , V E A?‘.From this, we have n

n

II 1 cjPxj,yjII = II 1 cjxj 6YjII, j= 1

j= 1

for every finite linear combination of elements p x , y . Accordingly (by the argument already used in the preceding example and in the proof of Proposition 2.6.5), there is a norm-preserving linear mapping Uo, from the linear span Xo of { P ~ ,x ~~: 2 , y ~ Xonto ’ ) the linear span Xo of the simple tensors in A? 6 X ‘ ,such that U o p , , = x @ y . Now X0is everywhere dense in X 6X ’ , and Xois everywhere dense in X since it contains the characteristic extends by function of every measurable rectangle of finite measure. Thus 17, continuity to an isomorphism U from X onto A? 6W . H We now introduce tensor products of bounded linear operators. r f %1, . . . ,X n ,Xl,. . . ,Xnare Hilbert spaces and 2.6.12. PROPOSITION. A,,, E~(X,,,, X,) ( m = 1, . . . ,n), there is a uniquq bounded linear operator A from Xl 6 . . * 0 A?n into Xl 6 6Xnsuch /hat

-

A(x1 8 . . . 6x,) = A i x l

6

* *

(XI EX i ,

6Anxn

-

. . . ,X, E 2,).

Proof. The canonical mapping p: X, x x X, -+ Xl @ * * * 6 Xn (= X ) is a weak Hilbert-Schmidt mapping, with llpllz = 1. With u in Y,andp,

145

2.6. CONSTRUCTIONS WITH HILBERT SPACES

defined by (p(z1 . . ,Zn), u>, pu is a Hilbert-Schmidt functional on Xl x . . * x X,, and IIpu112< ((u(1.The equation ( p ( X 1 , . . .,xn) = P ( A l X l t . . . ,Anx,)

P&,

9 .

. ., z,)

=

9 .

defines a bounded multilinear mapping cp: Xl x . . . x X,, + X, and cpU(Xl,.. x,) = (44x1 9.. - 3

=

., xfl),u>

-

(p(A1x1,. . ,Anx,),

4

1x1,. . A,x,). It now follows from Proposition 2.6.1(ii) that cpu is a Hilbert-Schmidt functional on a?, x . . . x H,, with = p,(A

a ,

-

llcpullz G 11Alll . * IlAnll IIPUIIZ G IlAlll * * IlAnll IlullAccordingly, tp: Xl x . . x X, + X is a weak Hilbert-Schmidt mapping, with (Icp((2 6 llAlll . . * llA,,ll. By the universal property of the tensor product (see Theorem 2.6.4(i)), there is a unique bounded linear operator A , from X1Q . . . Q &',, into X, such that cp = Ap', wherep' is the canonical mapping Moreover, from X', x . . . x Z,into H I Q * . . Q a?,,.

IlAll = I I d l z 6 IlAlll . . *

11~4.11.

Also, A(x1 Q ' . . Q x,) = Ap'(x1,.. . ,x,)

. . .,x,) =p(Alxl, ...,A,x,)= A l X , Q ... @A,x,, when x1€a?,,...,X,E a?,,. = cp(x,,

The operator A described in Proposition 2.6.12 is called the tensor product of A l , . . . , A , and denoted by A l Q . . . Q A,. It is apparent that A , Q . . . Q A , depends linearly on each A,,, and that ( A , Q . . . Q A,)(B, 0 - .. Q B,) = A l l ? , Q . . Q A,B,. Since

((A1 0 * . . 0 M x 1 0 . * * 0 X,),Yl 6 . . . QY,> = (AlXl

0 . . . 0 A,x,,y, 0 . . * 8 Y,>

= (AlX1,Yl)

. . * (A,xn,yfl)

* . . (Xfl,An*Yn> = (xl Q . . . Q x,,A:y, Q . . . Q A,*y,)

= (X1,ATYl)

= (X1Q

. . . Qx,,(A:O ' . . QA,*)(y1Q

* * .

Byfl)>,

146

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

it follows by linearity and continuity that ( ( A , 0 . * . @A,)u,u) = ( u , ( A : @

6A,*)u)

. * *

for all vectors u and u in the appropriate tensor product spaces. Thus

(15)

(A,@

* * .

@ A , ) * = A T @ * . *@A,*.

We assert also that llA1 @ . . . @ All = IlAlll * * * IlAnll. (16) Indeed, given by any unit vectors x1 in Sl, . . . ,x, in X,,, we have llA1 6 . * * @ An11 = 11-41 @

04 111x1 @ . . . @ XnII

* *

3 ll(A1 @

0 An)(x1 @

* * *

= llAlx1 6

*

+

*

+

@ XJll

@ Anxnll = IlAlxlll . . * IIAnXnll.

*

Upon taking the supremum of the right-hand side, as the unit vectors . . ,xnvary, we obtain

xl,.

llA1 0 . . . @An11 2 IlAlll .

9

*

IlAnll;

the reverse inequality was noted during the proof of Proposition 2.6.12. XI,. . . ,Xm+,, are Hilbert spaces and Suppose next that X , , .. .,Sm+,,, that Aje99(Xj,X j )( j = 1,. ..,m + n). We can construct isomorphisms u:X1@

@ ~ m + n ~ ( ~ , @ . . . @ ~ m ) 6 ( ~ m + *1. .@@ X m + n ) ,

v:

6X m + n

@

* * *

+

(XI@ . . 0 X m ) @ ( X m + l @ . . . 0 X m + n ) ,

as in Proposition 2.6.5, and it is at once verified that

v(A,O * . . OAm+n)U-’

=(A1

O

*

*

a

OAm)O(Am+I O

1

.

.

OAm+n)*

This proves the “associativity” of the tensor product of bounded linear operators on Hilbert spaces. With X and X Hilbert spaces, the linear mapping A

+A

6I:a(%)-+ 99(S@ X )

preserves operator products, adjoints, and norms; from this last, it is norm continuous. We consider next its continuity properties relative to the strongoperator topology. With v a simple tensor x @ y in % @ X,

ll(A 0 0 0 - (A0 0 Oull = ll(A - Ao)x OYll = ll(A - Ao)xlI Ilvll, for each A and A . in a(&).From this, it follows that, if ol,. . . ,v,,, are simple tensors in 2 @ X and c > 0, then the set { A E B ( X )ll(A : @ Z ) u j - ( A , @I)ujll

-=

E

( j = 1 , . ..,m)}

is a strong-operator neighborhood of A . in g(S).Since the simple tensors

2.6. CONSTRUCTIONS WITH HILBERT SPACES

147

have closed linear span X‘ Q Y, they suffice to determine the strong-operator topology on bounded subsets of X‘ 0 X ; so the preceding sentence implies that the mapping A + A Q I is strong-operator continuous on bounded subsets of a(*).When one or other of X‘ and X is finite dimensional, the simple tensors have algebraic linear span X ’ Q X (Remark 2.6.8), and therefore suffice to determine the strong-operator topology on (the whole of) Q X ) ; so, in this case, the mapping A + A @ I is strong-operator continuous on a(%). Suppose, finally, that { y b : b E B} is an orthonormal basis of X. As noted in Remark 2.6.8, the equation defines an isomorphism U from CbBQ3 X‘b onto X‘ With A in a(X),

6X when each x b is X.

( A 8 I)u(c@ x b ) = ( A 8 I > ( cx b 8 y b ) = x ( A @ I)(xb =Z A X b

8y b )

@ Y b = u(c@ AXb),

so U-’(AQI)LI= C @ A

(17)

(AEW(X‘)).

b€B

Matrix representations. We conclude this section with an account of the matrix representations of operators acting on a direct sum CbB@ %b, where each X‘b is the same Hilbert space 3. Before embarking on this program, we consider the numerical matrices of operators relative to orthonormal bases. Suppose that { Y b : b E B} is an orthonormal basis of a Hilbert space X. With S in a ( X ) ,each vector s y b has an expansion

(18)

syb

=

c

( b E B),

SabYu

aEB

in which the coefficients are given by sub = < S Y b , Y a ) * (19) In this way, we associate with each S in 9 ( X ) a complex matrix [Sabla&B, relative to the orthonormal basis { y b } .When the index set B is finite, every complex matrix [ S a b l u , & ~corresponds, as in (18), to some element S of B ( X ) . When 5 is infinite, however, boundedness of S imposes certain restrictions on its matrix. For example, Parseval’s equation gives babl’ a&

=

1 asB

I(SYb,ya>l’

=

llSyb11’

< lls112,

148

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

so the “columns” and “rows” of the matrix [ s o b ] form (bounded sets of) vectors in &(B). The algebraic relations between operators and matrices follow the pattern familiar in the finite-dimensional case. From (19) and (18), the matrix elements S,b depend linearly on S, and [ s o b ] is the zero matrix only when S = 0. Since (s*Yb,Yo)

= ( Y b , s Y a > = ( s Y o , Y b ) = sba,

the matrix of S* has &,a in the (a,b) position. If two elements S and T of g ( X ) have matrices [ s u b ] and [ f o b ] , respectively, and R = ST, then (RYb,Ya)

=(sTYb,Ya) =(TYb,s*Ya),

and Parseval’s equation gives (RYb,Ya)

=

(TYb,Yc)(yc,S*ya) CEB

1( = 1 =

S Y c 9 Y a > ( TYb 9 Y c )

CEB

Sactcb.

ceB

Accordingly, the matrix

[rgb]

of R (= S T ) is given by rob

=

1

Soctcb.

CEB

The results of the preceding paragraph can be summarized in the assertion that the matrices, corresponding (through a fixed orthonormal basis) to bounded operators on X, form an algebra relative to the usual concepts of sum, product, and scalar multiple of matrices. Moreover, the mapping from bounded operators to the corresponding matrices is an isomorphism. 0 x b , with each x b We now consider operators acting on a direct sum CbeB the same Hilbert space X. For this purpose, we introduce a closed subspace 3Pb of &, and bounded linear operators

c@

ua :

Z@x b ,

v g

:10 x

b

,?%c

for each a in B,as follows. When X E Y and u = { x b } EX@ x b , V,u = x, and U,x is the family { z b } in which z, = x and all other z b are 0;A?; is the range of U,,and so consists of all elements { z b } of I@ x b in which z b = 0 when b f a. Observe that V,U, is the identity operator on 2 and U,V, is the projection E, from x b onto 2;. Since the subspaces 2;(a E B) are pairwise orthogonal, and V X ; = 10 %b, it follows that the sum LEBE,is strong-operator convergent to I . Note also that U, = V,*, since

z@

( uax, { x b } ) = ( x , xa> = ( x , Va{xb})

whenever x E

~

and

{xb}E

c0 X b .

149

2.6. CONSTRUCTIONS WITH HILBERT SPACES

With each bounded h e a r operator T acting on I@ s matrix [TablaqbB, with entries Tabin g(#) defined by (20) If u

=

Tab

= {xb}

EX@

s b ,

Ya =

vaTU

VaT

c ) EbU

we associate a

vaTub.

then Tu is an element { y b } of =

b ,

=

I@ #b,

and

1v a T u b v b U =

( b

b

T&b. b

Thus (21)

T(c@ xb)

=

10y b

where

=

ya

c

Tabxb

(a E 8).

k B

The usual rules of matrix algebra have natural analogues in this situation. From (20), the matrix elements Tabdepend linearly on T. Since vaT*ub

= UtT*

vt = ( V b T U a ) *

= (Tba)*,

the matrix of T* has ( Tba)*in the (a, b) position. If S and Tare bounded linear operators acting on s b , and R = ST, then

c@

Rub

=

vaRub

=

VaSTUb

=

VaSEcTUb C€B

=

c

VasucVcTub

CE EJ

=

c

SocTcb,

CEB

the sum converging in the strong-operator topology if the index set B is infinite. In this way, we establish a one-to-one correspondence between elements of with entries Tabin B(&‘).When @ #b) and certain matrices the index set B is finite, each such matrix corresponds to some bounded operator Tacting on I@ s b ;indeed, Tis defined by (21), and its boundedness follows at once from the relations II{yb}112

=

1lly011~= 1 1 1c a

G

a

b

TabXb112

G

cc a ( b

IITabll l b b l l

>’

~a ( b~ ~ ~ T a b ~ ~ z )=(( ~~a ~b ~~ ~x Tb a ~ b ~~ 2~ )2 ) ~ ~ { x b } ~ ~ 2 .

When the index set B is infinite, it is apparent that some matrices with entries in

a(#)do not arise in the above manner from bounded operators. In formal matrix calculations, it is necessary to ensure that no such “unbounded” matrices appear at any stage. While there is no simple general procedure for determining whether or not a given matrix corresponds to a bounded operator, a criterion that is sometimes useful is set out in Proposition 2.6.1 3 below. In the meantime, we describe certain special types of “bounded” matrices that arise frequently in applications.

150

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

In the first place, a matrix [Tab]gives rise to a bounded operator if it has only a finite number of non-zero entries Tab.Indeed, the proof used above, for the case in which B is finite, applies also in the circumstances just described. More generally, the same argument shows that the matrix corresponds to a bounded operator whenever the sum c ~ c ~ J J isT finite, ~ ~ J since J ~ it is easily verified in this case that the series for y,, in (21), is absolutely convergent. However, there are bounded operators on I@ X b (for example, I ) whose matrices do not satisfy these conditions. Second, suppose that n is a permutation of B, and { Tb:b E El} is a bounded family of elements of g ( H ) .Since

defines a bounded linear operator Ton the equation T(C@ xb) = I@ Tbxn(b) C @ X b . The matrix [Tab]of T is given by Tab= da(a),bTa,where d a b is the Kronecker symbol ( d a b is 1 when a = b, 0 when a # b). If each Tbis unitary, the same is true of T . When n is the identity mapping on B, T is the direct sum 10 Tb, and corresponds to the diagonal matrix [dabTa]. Finally, we consider the matrix representation of certain tensor products of operators. Suppose that {yb : b E EB} is an orthonormal basis in a Hilbert space X, and Uis the isomorphism from C@ x b onto H @ X (where each Z b is H ) , definedby U(C@Xb) = CXb@yb. W h e n A € g ( X ) , U - ' ( A @ I)Uisthedirect sum CkB@ A , and has matrix [ & , A ] This . characterizes the operators To of the form A @ I acting on 2 @ X as those for which U - ' ToU has a diagonal matrix with the same element of @ ( X in ) each diagonal position. We assert also that an element To of a(&'@ X ) can be expressed as I @ S, with S in .@(X), if and only if the matrix of U - ToU has the form [ & I ] , with each Sob a scalar. For this, suppose first that S E ~ ( X so ) , that S has a complex matrix satisfying (18). With x in H and b in B, UUbx = x @ yb, so

'

va(

u- ' ( I @ s ) u ) u b x = vau- ' ( I @ s ) ( x @ yb) =

v a u - ' ( x @ s y b ) = VaU-'

=

@va: (

LB

)

cscbx@yc

scbx) = Sabx.

Hence v,(U-'(Z@ s)u)&= &blr and U - ' ( I @ S)U has matrix [&(,I]. Conversely, suppose that ToE a(# @ X ) and the matrix of U - ToU (= 7') has the form [ & I ] . With x a unit vector in A? and f an element of lZ(5), u = C@f(b)x is a vector in C@ %b and Tu = xb, where

'

c@

xa =

c sabf(b)x*

as

2.6. CONSTRUCTIONS WITH HILBERT SPACES

151

Hence the sum CkBsabf(b)converges, and

so

Accordingly, the equation

defines a bounded linear operator S on X. Since S has matrix [sob], U - ' ( I 0 S ) U has the same matrix [SabI] as does U - ToU, so To = I 0 S. We now establish a criterion for determining whether or not a matrix [Tab] &B, with entries Tabin i?d(X), corresponds to a bounded linear operator acting on CkBQ #b, where each X b is X. As noted above, each such matrix gives rise to a bounded operator when the index set B is finite.

'

2.6.13. PROPOSITION. Suppose that X is a Hilbert space and [ Tabla,bs~ is a matrix, with entries Tabin a(*). For eachfinite subset ff of B,let T(F)be the bounded linear operator, corresponding to the matrix [Tab] a,kF, that acts on the Hilbert space ChF0 X b (where each s b is 2 ) . 0) IlW1)ll 6 IlT(mll i f E 1 F2. a,bB corresponds to a bounded linear operator T acting (ii) The matrix [Tab] on CkBQ ixg ifand only ifthe set { 11 T(IF)II : IF afinite subset of B} of realnumbers is bounded above. When this is so,

IlTll = sup{l(T(IF)II:ff afinite subset of B}, Proof. Let 9denote the class of all finite subsets of B, and when ff E let F E and ~ u is an element

*(IF) be the Hilbert space ChEF O X b .Note that if I CkF0 xb of ,)FI(% then

llu112

=

c

t€F

llXb112,

llT(F)u112 =

11

c TabXb112.

hb

(i) Suppose that IF1, IF, €9 and ff E IF2. With u an element ChF,Q X b of 2(ff'), let w be the element CkFl0 xb of #(IF2) obtained by taking X b to be 0

152

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

when bEIFz\IF1. Then

= I I ~ ( ~ z ) w 1G I 2llm2)l1211w11z = l

l ~ ~ ~ 2 ~ l l z l l ~ l l z ~

and therefore IIT(!F1)ll< IIT([Fz)ll. (ii) Suppose first that the set { llqlF)ll: IF E F}has finite supremum k. In view of the discussion preceding the proof of (i), it follows that

whenever IFE 9and x b E %b ( bE IF). Let u be an element CkB0 x b of Eke @ %b. Our proof that [Tab]is the matrix of a bounded linear operator is now divided into two stages. First, we show by the Cauchy criterion that, for each c in B, the sum CkB T c b X b converges to an element yc of X. For this, suppose E > 0, and choose IF, in 9 so that

1

lIxb(12

< 2/k2

whenever IF €9 and

F nF, = @.

kF

By enlarging IF, if necessary, we may suppose that C E [F,. When IF E 9 and ff n IF, = 125, let IFo = IF u {c}, and define xi = X b ( bE IF), xi = 0. From (22),

11

TcbXbIIZ

= 11

bsf

1

TcbXi112

kF0

6

c 11 c

aeFo

Tabxb1Iz

&F,

< kZ 1 IlxJl’ = kZ C llxbllZ < E kFrJ

~ .

k F

Thus IICkFTcbXbll < E whenever F E and~ IF n [F, = 0,the Cauchy criterion is satisfied, and CbEB T c b X b converges to an element yc of X. Second, we prove that

153

2.6. CONSTRUCTIONS WITH HILBERT SPACES

For this, suppose that ff ff, E% let ff = [F1 u [F2, and define xb = x b ( b IF,) ~ xb = 0 ( bE ff\F2). From (22),

When IF2 increases to B, llCbFlT,,bXbl( from the preceding inequalities that

-,IlypII;and since IF1

C IIYaII’ < k2

is finite, it results

IIxbII’. b B

ad,

Since the preceding relation has been proved for each

1 llYa1I2 < k2 1 ll€B

[F1

in

g

IIXb1l2.

b B

From the two assertions just proved, there is a bounded linear operator T, acting on CbB0 Z b , with

T(C@x b ) = 10Y b

where

Ya =

C TabXb

(a E B),

bsB

Since this is a restatement of (21), T has matrix [ T a b ] . Conversely, suppose that [ T a b ] is the matrix of a bounded linear operator T. With IF in 9and u an element CbEF 0 x b of Z(Q, let u be the vector Eke 0 x b in CkB0 x b , obtained by defining x b = 0 when b E B\F. Then

Ilr(F)ul12

=

111 1 a€F

b€F

TabXbl12

154

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

Thus IIT([F)II < IlTll, and the set {Ilr([F)II:[ F supremum at most IlTll. This, with (23), gives

E ~ } is

bounded above, with

IlTll = suP{llT(~)Il:F E 5 ; ) . In the circumstances described in Proposition 2.6.13, we regard the operator T(F)as the “finite diagonal block” of the matrix [Tab]corresponding to the finite subset [F of B. The main result of the proposition is the assertion that [Tab]is the matrix of a bounded linear operator if and only if the set of norms of all the finite diagonal blocks is bounded above.

2.7. Unbounded linear operators

In Section 1.5, we discussed linear transformations from one normed space into another. We noted in Theorem 1.5.5 that the continuity of such a transformation is equivalent to its boundedness (on the unit ball), so that we speak, interchangeably, of “continuous” and “bounded” linear transformations. In Section 2.4, we specialized the discussion of bounded linear transformations to Hilbert space. In this section, we take up the study of discontinuous (and, necessarily, unbounded) linear transformations between Hilbert spaces. We have only to think of the process of differentiation to be convinced that unbounded linear operators arise in the most natural way and that they are important. Without proceeding carefully, let 9 be the linear manifold of allfin L,(R) (relative to Lebesgue measure) almost everywhere differentiable with Then D is a linear transformation and derivativef’ in L2(R);and let D ( f ) kf’. D is not bounded. (If fk(t) = exp(- kltl), with k a positive integer, then IlDfklllllfkll = k.) Although D is defined on a dense submanifold of L,(R) (as follows from classical approximation results), it is certainly not defined on all of L,(R). We must expect, then, in dealing with unbounded linear operators, to specify a domain of definition 9(T)for our operator T (and, thereafter, to exercise care not to apply T, in a formal way, to each element that suits our convenience). Not only is the subject of unbounded linear operators natural and important, but the literature devoted to it is vast (almost as a consequence). Not to divert ourselves from the purpose at hand, we restrict the examples and results described in this section to a bare minimum. Unbounded operators will appear again in Section 5.6, when we extend to unbounded self-adjoint operators the spectral theory developed there for bounded self-adjoint operators. A “polar decomposition” for (closed) unbounded operators appears in Section 6.1, and a formulation of Theorem 7.2.1 in terms of unbounded operators appears in Section 7.2; but the essential use of

2.7. UNBOUNDED LINEAR OPERATORS

155

unbounded operator theory occurs, for us, in the presentation in Section 9.2 of modular theory. For the most part, the naturally arising unbounded operators retain some vestiges of orderly “limit properties” - notably, the possibility of extending them to operators with closed graphs. While this assumption may seem like a negligible replacement for continuity, it can be turned to remarkable advantage, as we shall see. In Section 1.8 we associated a graph B(T) with a linear transformation T, where B(T ) = { (x, Tx) : x E 9(T)}.The closed graph theorem (1 3 . 6 ) tells us that if Tis defined on all of A? (mapping into the Hilbert space X ) and B(T )is closed, then T is bounded. (Conversely, if T is bounded and everywheredefined, B(T )is closed.) This provides us with the possibility of an assumption intermediate between continuity and the totally unrestricted linear operator. Let T be a linear mapping, with domain 9 ( T ) a linear submanifold (not necessarily closed), of the Hilbert space A? into the Hilbert space X. We say that T is closed when B(T) is closed. The unbounded operators T we consider will usually be densely defined, that is, 9(T)is dense in S. Whatever T we consider, it has a graph ’3(T),and the closure B(T)- of B(T ) will be a linear subspace of 2 0 X. It may be the case that Y(T)- is the graph of a linear transformation T, but it need not be. If it is, T “extends” T and is closed. We say that To extends (or is an extension of) T, and write T G To, when 9 ( T ) E 9(To) and Tox = Tx for each x in 9(T).If B(T)- is the graph of a linear transformation T, clearly T is the “smallest” (“minimal”) closed extension of T. In this case, we say that T ispreclosed (the term closable is also used) and refer to Tas the closure of T. If ’3(T)- contains elements ( x , y ) and (x,y’) such that y # y’ (equivalently, -since B(T)- is a linear space, if (O,Z)EB(T)-with z not 0), then B(T)- is not the graph of a (single-valued) mapping and Tis not preclosed. This is, of course, the only way in which Tcan fail to have a closure (for, otherwise, the mapping that sends the first to the second coordinate of Y(T)- defines T). Interpreting B(T)- as the closure of B(T) in limit terms, we see that T is preclosed if and only if convergence of the sequence {x,} in 9 ( T ) to 0 and {Tx,} to z implies that z = 0. From the point of view of calculations with an unbounded operator T, it is often much easier to study its restriction Tlgo to a dense linear manifold goin itsdomain 9(T)than to study Titself. If Tisclosed andB(TIQ0)- c= B(T),the information obtained in this way is much more applicable to T. In this case, we say that gois a core for T. Each dense linear manifold in B(7‘)corresponds to a core for T. 2.7.1. EXAMPLE. With the notation of Example 2.4.10, remove the restriction that g be bounded and let Tbe defined as in that example for those x in 3Ep such that CyEY Ig(y)(x,y)12 is finite (so that 9 ( T ) consists of such vectors x). Of course 9(T) contains the submanifold go of all finite linear

156

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

combinations of the basis elements in Y, from which 9 ( T ) is dense in X. At the same time, with u and u in 9(T),from the Cauchy-Schwarz inequality,

so that B(T) is closed. The submanifold 9,,is easily seen to be a core for T.

2.7.2. EXAMPLE.With the notation of Example 2.4.11, once again remove the restriction that g be bounded (requiring only that g be measurable and finite almost everywhere) and let MBbe defined as in that example for those x in %' such that jsI(Mex)(s)12dm(s) is finite (so that 9 ( T ) consists of such x). The present example extends the preceding example to the case of non-discrete (a-finite) measure spaces (so that the important case in which Y is denumerable is included). Again 9 ( T ) is dense since it contains the submanifold g o of measurable functions on S with support in a set of finite measure on which g is essentially bounded. The Cauchy-Schwarz inequality assures us once again that 9(7')is a linear manifold. A more general measure-theoretic argument of the character of that appearing in the preceding example establishes that M Bis closed and g ois a core for it. (See the comments following Theorem 5.2.4.) W 2.7.3. EXAMPLE.IS%' is a separable Hilbert space with orthonormal basis {y,,}n=1,2,... and TyP= ny,, then T extends linearly to the (dense) linear

manifold g oof finite linear combinations of basis vectors y n .If we denote this extension by T again (so that 9(T)= go),then T is densely defined, unbounded, and not preclosed. To see this, it suffices to note that n-'y, -+ 0 while T(n-'y,,) = y1 + y l . H

2.7. UNBOUNDED LINEAR OPERATORS

157

We define the operations of addition and multiplication for unbounded operators so that the domains of the resulting operators consist precisely of those vectors on which the indicated operations can be performed. Thus 9 ( A + B) = 9 ( A ) n 9 ( B ) and ( A B)x = Ax Bx for x in 9 ( A + B). Assuming that 9 ( B ) E X and 9 ( A ) c X , where B has its range in X, A B is defined as the linear transformation, with { x :x ~ 9 ( Band ) B x E ~ ( A )as} its domain, assigning A(Bx) to x. Of course 9 ( a A ) = 9 ( A ) and (aA)x = a(Ax). More care is needed in defining the adjoint of an unbounded operator.

+

+

2.7.4. DEFINITION. If Tis a linear transformation with 9(T)dense in the Hilbert space X and range contained in the Hilbert space X, we define a mapping T*, the adjoint of T, as follows. Its domain consists of those vectors y in X such that, for some vector z in Z, ( x , z ) = ( T x ,y ) for all x in 9(T).For such y , T*y is z. If T = T*, we say that T is self-adjoint. In connection with this definition, we must note that there is at most one z (for a given y ) since x can assume values in the dense set 9(T);so T* is well defined. Note, too, that the existence of z is equivalent to the boundedness of the linear functional x + ( Tx,y ) on 9(T )(for, given that it is bounded, it has a unique bounded extension from 9 ( T ) to X, and Riesz's representation theorem (2.3.1) provides us with z). The formal relation ( T x , y ) = ( x , T * y ) , familiar from the case of bounded operators, remains valid in the present context only when x E 9(T ) and y e 9(T*).

2.7.5. REMARK.If To is densely defined and Tis an extension of To,then TO*is an extension of T*. To see this, suppose that y ~ 9 ( T *and ) uc9(T0). Then

so that y ~ 9 ( T g *and ) Tgy = T*y.

2.7.6. REMARK.If Tis densely defined, T* is aclosed linear operator; for, with u and u in 9 ( T * ) and x in 9 ( T ) ,

+ T*u) = (ax, T*u) + ( x , T*u) = ( T x , au) + ( T x , u ) = ( T x , au + u ) , so that au + U Eg ( T * )and T*(uu + u) = aT*u + T*u. Thus 9 ( T * )is a linear (x,aT*u

manifold and T* is a linear operator. If {u,} is a sequence in 9 ( T * )converging to u such that {T*un}converges to u', then, with u in 9(T),(u, T*u,) = (Tu, u,); and { (u, T*u,)} converges to (Tu, u ) . Thus (Tu, u ) = ( u , u ' ) for each u in 9(T);so u ~ 9 ( T * )and , T*u = u'. It follows that T* is closed.

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2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2.7.7. REMARK.There are several ways in which we can use the hypothesis that T (that is, % ( T ) )is closed. The mapping P taking (u, u) to u is a bounded linear transformation of the Hilbert space %(T)into S. Thus P has a bounded adjoint P* mapping 2 into %(T).Since ( O , z ) € % ( T ) only when z = 0, P has null space (0). From Proposition 2.5.13, the range of P* is dense in Y(T). Thus, with u in 2,if P*(u) = (w, w') (in %(T)),then Tw = w' and, for each u in 9(T), (u, U )

= (P*(u),(u, Tu)) =

(w, U )

+ (w',Tu).

Hence ( Tu, w') = (u, u - w ) and w' E 9(T*). Moreover, T*w' (= T*Tw) = u

- W,

so that (T*T + I)w = u. While it is not clear, apriori, that B ( T * T )consists of more than the vector 0, our brief computation, relying on the information that %(T) is closed, allows us to conclude that B ( T * T )contains (and, therefore, is) a core for T (namely, the first coordinates of the range of P*). We learn, at the same time, that T*T + Z has range 2 (for D was an arbitrary element of 2).

Making use of the preceding remarks, there is no difficulty in proving the main theorem of this section.

2.7.8. THEOREM.If T is a densely defined transformationfrom the Hilbert space S to the Hilbert space X, then

(n*

(i) if T is preclosed, = T* ; (ii) T is preclosed if and only 9(T*) is dense in A!-; (iii) if T is preclosed, T** = T ; (iv) if T is closed, T* T + I is one-to-one with range 2 andpositiue inverse of bound not exceeding 1 ; (v) T* T is self-aa'joint when T is closed.

(n*

Proof. (i) Since T E T ; from Remark 2.7.5, s T*. Suppose y ~ g ( T * )For . each x in 9(T),there is a sequence {x,,} of vectors in 9(T) converging to x such that {Tx,,} converges to Tx. Thus ( T x , ~=) lim( Tx,,y) = lim(x,,, T * y ) = ( x , T*y),

so that y ~ g ( ( n * and ) (T)*y = T*y. Hence (T)* = T*. (ii) If T is preclosed, from Remark 2.7.7, %(T)contains a dense linear manifold (the range of P*) consisting of pairs (x,Tx) with Tx in 9 ( T * ) ( = 9(T*)).If y is orthogonal to the range of T, then 0 = ( T x , y ) = ( Tx, y) for each xin 9(T);and y is in 9(T*)(y is annihilated by T*).Thus 9(T*)contains a dense subset of the range of T as well as the orthogonal complement of this range. Since 9 ( T * )is a linear manifold, it is dense in X.

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2.7. UNBOUNDED LINEAR OPERATORS

Suppose, now, that 9(T*) is dense in X and {u,} is a sequence in 9 ( T ) converging to 0 such that {Tu,} converges to u. With y in 9(T*), (Tu,,,~)= (u,,, T*y); so (u,,, T*y) converges both to 0 and to ( u , y ) . Since 9(T*) is dense in X, u = 0 and T is preclosed. (iii) If Tis preclosed, 9(T*) is dense, from (ii), and T* has an adjoint T**. If y ~ 9 ( T * and ) x ~ 9 ( T ) then , (T*y,x) = (y, Tx), so that x ~ 9 ( T * * and ) T**x = Tx. Thus T** is a closed (from Remark 2.7.6) extension of T, and T E T**. From Remark 2.7.5, T*** c T* = T*. Since T* is closed, we have, as well, T* c (T*)**. Thus T* = T***. As noted T c T** (equivalently, %(T)E %(T**)).If ( x , T**x), in Q(T**), then (x, u ) + (T**x, Tu) = 0 for each u in 9( This is orthogonal to %( holds, in particular, when T U E ~ ( T *(= ) g(T***)); and, for such u,

n,

n.

0 = (x,(T*T+ I ) u ) .

But, from Remark 2.7.7, (T*T + I)u takes on all values in 2.Thus x = 0, %(T**)= and T** = T. (iv) We noted in Remark 2.7.7 that the domain of T*T (and, hence, of T*T + I)is a core for T when T is closed and densely defined. We noted, too, that T*T + I has 2 as its range. If X E ~ ( T *+TI>, then

%(n,

llxll’

< (x,x) + (Tx, Tx)

= ((T*T+ I)x,x)

< II(T*T + 0x11 IIxII.

+

Thus T* T I has (0) as null space, is one-to-one, and has a bounded inverse H of bound not exceeding 1. From this same computation, and since each z in 2 has the form (T* T + I)x, it follows that ( z , H z ) is (( T*T + I ) x , x ) , which is real and non-negative. Thus H is positive. (v) As noted in (iv), 9(T*T) is a core for T; hence it is dense in 2. Since

+(XJ) when x E 9(T* T), we see that (T* T)* and (T*T + I)*have the same domain and that (T*T)* + I = (T*T + I)*. With y in 9(T*T), ((T*T+ I ) x , y ) = (T*Tx,y)

(T+Tx,y) = (x, T*TY), so that T*T G (T*T)* and T*T + I c (T*T + I)*. Itfollowsthat(T*T+ I)* has 2 as its range. If (T*T + Z)*y = 0, then, for each x in 9(T*T),

0 = ((T*T+ I ) * ~ , x )= (y,(T*T+ I ) x ) . Since T*T + I has range 2, y = 0. Thus (T*T + I)*is one-to-one, extends T*T + I, and has the same range as T*T + I. It follows that T*T

+ I = (T*T + I)*= (T*T)* + I,

so that T*T = (T*T)*. H

160

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

The statement that Tis self-adjoint (T = T*) contains information about the domain of T as well as the formal information that (Tx, y ) = (x, Ty) for all x and y in 9(T).When 9(T ) is dense in 2 and (Tx, y ) = (x, Ty) for all x and y in 9(T), we say that T is symmetric. Equivalently, T is symmetric when T c T*. Since T* is closed and 9(7') c 9(T*), in this case, Tis preclosed if it is symmetric. If Tis self-adjoint, Tis both symmetric and closed. The operation of differentiation on an appropriate domain provides an example of a closed symmetric operator that is not self-adjoint. In Proposition 2.7.10 we describe conditions that guarantee that a given closed symmetric operator is selfadjoint. If A c T with A self-adjoint and T symmetric, then A c T c T*, so that (T** c ) T * E A* = A G T G T* and A = T. It follows that A has no proper symmetric extension. That is, a selfadjoint operator is maximal symmetric. 2.7.9. LEMMA. Zf Tis closedandsymmetric, T f ilhave closedranges. Zf T is closed and 0 < (Tz, z ) for z in 9(T), then T I has a closed range.

+

Proof. Suppose {x,} is a sequence in 9 ( T ) such that {( T f il)x,} tends to y . Note that, with z in 9(T), (Tz, z ) is real, so that

< ( ( T z , ~ )+~( z , ~ ) ' ) ' ' ~= I((Tf il)z,z)l < ll(Tf il)zlJ11z11. Thus Ilx, - xmll< ll(T f il)(x, - xm)lland {x,} isconvergent. Suppose x, llzll'

-+

x.

Since {Tx,} converges to T ix + y and T is closed, x ~ 9 ( T and ) Tx = f ix + y . Thus y = ( T f il)x, and T f i l have closed ranges. Suppose, now, that T is closed and 0 < (Tz, z ) for each z in 9(T). Then

llzl12 < (Z,Z> + (Tz,z) for z in 9 ( T ) , and, as above, T

< ll(T+ Ozll 11z11,

+ I has closed range.

W

2.7.10. PROPOSITION. If T is a closed symmetric operator on the Hilbert space 2, the following assertions are equivalent; (i) (ii) (iii) (iv)

T is self-adjoint; T* f i l have (0) as null space; T f il have 2 as range; T f iI have ranges dense in 2.

Pro05 (i) -+ (ii). If T = T*, for each x in 9(T),(Tx, x) (Tx,x) is real. Thus

= (x,Tx) ; and

f i l ) x , x ) = ((T f i l ) x , x ) = (Tx,x) f illxl12 = 0 only if x = 0. Hence T* f i l have (0) as null space. ((T*

2.8. EXERCISES

161

(ii) + (iii). From Lemma 2.7.9, T k il have closed ranges. Thus, it suffices to show that these ranges are dense in X . If ( ( T f il)x,y ) = 0 for all x in 9(T),then ( T x , ~=) f i ( x , y ) , so that y ~ 9 ( T *and ) T * y = f iy. Since T* t- il have (0) as null space, y = 0. Hence T f il have dense ranges. (iii) -(iv). This follows from the preceding discussion. (iii) + (i). Since T is closed and symmetric, T G T* and Y ( T )is a closed subspace of the closed space Q(T*).If ( y , T*y)in Q( T*)is orthogonal to Y( T ) , then (y,X>

+ ( T*Y,T x ) = 0

for each x in 9(T).Since T f i l have range X, there is an x in 9(T)such that ( T x e 9 ( T ) ,and) y = ( T + il)(T - il)x ( = ( T 2 + Z)x). For this x, (Y,Y> = ( y , ( T 2 + O x ) = ( Y , x > + ( T * y , T x ) = 0.

Thus ( y , T*y) = (O,O), Y ( T ) = Y( T*),T = T * , and T is self-adjoint. 2.7.11. REMARK.If T is self-adjoint, it follows from (iii) of Proposition 2.7.10 and the inequality at the beginning of the proof of Lemma 2.7.9 that T f il have everywhere-defined, bounded inverses with bound not exceeding 1 . rn 2.8. Exercises

2.8.1. Show that a finite set { x l , .. . ,xn}of n vectors in a Hilbert space X is linearly independent if and only if the n x n matrix that has ( x j ,x k ) in the ( j ,k ) position is non-singular. 2.8.2. Show that a Hilbert space is uniformly convex (in the sense defined in Exercise 1.9.13). 2.8.3. Show that, if X is a real Hilbert space, then X x X becomes a (complex) Hilbert space Xc when its linear structure, inner product, and norm, are defined by ( x ,y ) + (u, u ) = ( x + 4 y (a

+ 01,

+ ib)(x,y ) = (ax - by, bx + uy), + ( y , u > + i - 0,u>, l l ( X ~ Y ) I l 2= llX1l2 + llY1I2,

( ( x , Y ) (u, , ~ 1= ) ( x , u>

for all x , y , u, u in X and a, b in R.

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2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

Prove also that the set {(x, 0) :x E X } is a closed real-linear subspace XRof Zc,that Xc = { h + i k : h , k ~ X ~and } , that the mapping x + (x,O) is an isometric isomorphism from X onto (the real Hilbert space) XR. 2.8.4. Suppose that {xl, x2,x3,.. .} is an orthonormal basis in a Hilbert space X and that

Prove that Y is a bounded closed convex set that has no element with greatest norm. 2.8.5. Suppose that Y is a closed convex set in a Hilbert space X, 4!y is the set of all unitary operators U acting on X for which U ( Y ) = Y, and Yo = {YE Y : Uy = y for each U in

aY}.

(i) Prove that Yo is not empty. [Hint. Use Proposition 2.2.1.1 (ii) Show that, if Yo consists of a single non-zero vector yo, then Y is a subset of the hyperplane

EX: Re(x - y o , y o ) = 0). 2.8.6. Prove that a bounded sequence of vectors in a Hilbert space has a weakly convergent subsequence. 2.8.7. Suppose that A is an uncountable set and, for each a in A, ma is Lebesgue measure on the a-algebra of Bore1 subsets of the interval [0,1] (= S,). Show that, if (S,g m ) is the corresponding infinite-product measure space (see [H: p. 158]), then L 2 ( S , g m )is non-separable. 2.8.8. Suppose that X is a Hilbert space in which the inner product is denoted by ( , ) and that K e g ( % ) + . Show that the equation (X,Y)l

= (KX9.Y)

(X,YEX)

defines an inner product ( , on X. By means of the Cauchy-Schwarz inequality for ( , ) 1 , prove that

IIKIJ= min{a:aER, K G a l } . 2.8.9. Let &' be a Hilbert space in which the inner product and norm are denoted by ( , ) and 11 1 , respectively. Suppose that ( , )1 is another definite inner product on Z and the corresponding norm 11 Ill satisfies llxlll < llxll for each x in X. Prove that there is a positive self-adjoint operator K, acting on X,

163

2.8. EXERCISES

2.8.10. Suppose that X is a Hilbert space in which the inner product and norm are denoted by ( , ) and 11 I), respectively. Let K be a positive element of a(%'), and define an inner product ( , )1 on &' by (X,Y)l = (KX,Y) Let IIxIll

=

(XlYEX).

[(x,x)~I~'~.

(i) Prove that 11 [I1 is a norm on X if and only if K has null space (0). (ii) Show that, if K has null space {0},then the norms 11 11 and 11 [I1 give rise to the same topology on 2 if and only if K has an inverse in a(&'). (iii) Suppose that K has an inverse in a(%). If A* denotes the adjoint of an element A of a(X)relative to the inner product ( ,),find a formula for the adjoint of A relative to the inner product ( , 2.8.1 1. Suppose that T is a bounded self-adjoint operator acting on a Hilbert space X and k is a positive real number such that - kZ < T < kZ. By using the identity 4 Re(Tx9.Y) = (T(x

+ Y ) , x + r>- (T(x - Y ) ,x - Y>,

show that IRe(Tx9Y)l d ;kk(llxl12 + llY1l21 for all x and y in X. Deduce that llTll < k and that

IlTll =min{a:aER, - a Z < T < a Z } = sup{l(Tx,x)l : X € X : llxll =

l}.

2.8.12. A bounded linear operator A , acting on a Hilbert space X, is said to attain its boundif llAxll = llAll for some unit vector x in 2.Give examples of (a) a bounded self-adjoint operator with an orthonormal basis of eigenvectors, (b) a bounded self-adjoint operator with no eigenvector, neither of which attains its bound. 2.8.13. Let X be a Hilbert space. (i) Prove that each unit vector x in X is an extreme point of the unit ball (X)l. (ii) Prove that each isometric linear operator V from X into X is an extreme point of the unit ball (a(#)), .

164

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2.8.14. Show that the projection E from a Hilbert space &' onto a closed of all positive operators in subspace X is an extreme point of the set the unit ball of a(&'). 2.8.15. Determine a necessary and suficient condition for the operator MB,defined in Example 2.4.1 1, to have a bounded inverse. ) each a in A, 2.8.16. Suppose that T = CacA@ T,, where T , E ~ ( & ' ,for and sup{11 Tall:a E A} < co. Show that T has a bounded inverse if and only if the following two conditions are satisfied :

(i) each T, has a bounded inverse, (ii) sup{llT,-'Il:a~A} < 00. 2.8.17. Let 9 denote the set of all projections from a Hilbert space &' onto its closed subspaces, and suppose that F E ~0 # , F # I. Prove that the mappings E+EAF,

E-EvF

are not continuous, from 9 with the norm topology into 9 with the strongoperator topology. 2.8.18. Let 9'denote the set of all bounded self-adjoint operators acting on a Hilbert space X. If A, B, CE% we say that Cis a lower bound of { A , B} if C < A, C < B. We say that Cis the greatest lower boundof { A , B } if it is a lower bound of {A, B}, and D < C whenever D is a lower bound of {A, B}. (i) Show that, if A , B E % then {A, B} has a lower bound in 9 (ii) Suppose that A, Bare non-zero elements of a(&')'. Show that there is a vector xo such that ( A x o ,xo) > 0 and ( E x , , xo) > 0. Prove that, if Po is the projection onto the one-dimensional subspace containing xo,a and b are suitable positive real numbers, and T = aPo - b(I - Po),then T < A, T < B, T 6 0. Deduce that 0 is not the greatest lower bound of {A, B}. (iii) Suppose that A, B E % and {A, B} has a greatest lower bound Cin 9 By applying the result of (ii) to {A - C, B - C}, show that either A < B or B < A. 2.8.19. Suppose that A and B are mappings from a Hilbert space X into itself, and ( A x , y ) = ( x , B y ) for all x and y in 2.Prove that A and B are bounded linear operators; and A = B*. 2.8.20. Suppose that &' is a Hilbert space and A €a(&'). Prove that the following five conditions are equivalent.

165

2.8. EXERCISES

(i) A is continuous as a mapping from the unit ball (%), (with the weak topology) into % (with the norm topology). (ii) If x, x l , x 2 , . . . € 2and {x,) is weakly convergent to x, then { A x , } is norm convergent to Ax. ( 5 ) Every bounded sequence {x,} in A? has a subsequence {x,(k)} such that {A+..} is norm convergent. (iv) The set { A x : x E ( # ) , } is relatively compact in the norm topology of x. (v) The set { A x : x ~ ( # ) ~is} compact in the norm topology of S. [An element of a(%)that has any (and, hence, all) of the above properties is described as a compacf linear operator.] 2.8.21. Prove that the identity operator, acting on an infinite-dimensional Hilbert space, is not compact (in the sense of Exercise 2.8.20). 2.8.22. Suppose that % is a Hilbert space and

9 = { A €&?(A?): A has finite-dimensional range}. (i) Prove that, if A €9and {yl,. . . ,y,} is an orthonormal basis of the range of A, there exist vectors x l r . .. ,x, in % such that n

Ax =

C ( x , x,)yn

(xEH).

j= 1

(ii) Prove that 9 is a two-sided ideal in a(%)and that every non-zero two-sided ideal in g(%) contains A (iii) Prove that an element A of a(&)lies in 9 if and only if A is continuous as a mapping from % (with the weak topology) into 2' (with the norm topology). (iv) Prove that the elements of 9 are compact linear operators (in the sense of Exercise 2.8.20). 2.8.23. Suppose that {A,} is a sequence of compact linear operators acting on a Hilbert space 2, A E&?(%), and ( [ A ,- ,411 -,0. Prove that A is compact. [Hint. Use condition (i) of Exercise 2.8.20 as the defining property of a compact linear operator.] 2.8.24. Suppose that A is a compact linear operator acting on a Hilbert space A? (see Exercise 2.8.20).

(i) Prove that the (closed) range space [A(%)] is separable. (ii) Suppose that [A(%)] is infinite-dimensional, and let {yl , y z ,y,, ..-} be an orthonormal basis of [A(%)]. For each positive integer n, let P, be the

166

2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS

projection from 2 onto the subspace spanned by yl , . . . , y n . Prove that $4 - P,,A(I + 0 as n 3 00. (iii) Deduce that A lies in the norm closure of the ideal f (see Exercise 2.8.22), and A* is compact. 2.8.25. Let X denote the set of all compact linear operators acting on a Hilbert space X. By using the results of the three preceding exercises, show that:

(i) Z is the norm closure of the ideal 9 in a(%); (ii) X is a norm closed two-sided ideal in g ( 2 ) ; (iii) each non-zero norm closed two-sided ideal in @ ( X contains ) X. 2.8.26. Suppose that {yl,y2,y3, .. .} is an orthonormal system in a Hilbert space S, and { A1,A 2 , A,, . . .} is a sequence of real numbers such that 11112 IAzI 2 2 . * Suppose also that the sequences {y,,} and {A,,} are either both finite and of the same length or both infinite, with {A,,} converging to 0. Show that the equation a .

Ax

=

C An(X, yn)yn

(X E 2 )

n

defines acompact self-adjoint operator A on 2,with IlAll = /All.[The result of Exercise 2.8.29 below shows that every compact self-adjoint operator has the form just described.] 2.8.27. Suppose that A is the compact self-adjoint operator constructed in Exercise 2.8.26 from an orthonormal system { y , ,y 2 ,y,, . ..} in a Hilbert space X and a real sequence {A,, A 2 , A,, . . .} that satisfy the conditions set out in that exercise. Extend the orthonormal system to an orthonormal basis {YI r ~ ~ , * ~ 3u ,{ Z a } ,

(i) Prove that, if 3, is a non-zero scalar that does not appear in the sequence {An}, the operator A - AI has an inverse in a(%'), and

[Hint. Consider the matrix of A.] (ii) Show that, if A is a non-zero scalar that appears in the sequence {A,,} and x E X, the equation ( A - AI)z = x

167

2.8. EXERCISES

has a solution z in X if and only if ( x , y k )= 0 for each integer k satisfying = 1. What is the most general solution z , when this condition is satisfied?

,Ik

2.8.28. Let A be a bounded self-adjoint operator acting on a Hilbert space X. (i) Show that each eigenvalue of A is real. (ii) Show that eigenvectors corresponding to distinct eigenvalues of A are orthogonal. 2.8.29. Let A be a compact self-adjoint operator acting on a Hilbert space X. (i) By using the result of Exercise 2.8.11, show that there exist unit vectors xl, x2,x3, . . . in 4 such that the real sequence { ( A x , , ,x,,)} converges, Show that IIAx,, - px,,ll-, 0 as n + 00. with limit p equal to IlAll or - JIAJJ. (ii) Prove that A x = px for some unit vector x in 2 (so that a non-zero compact self-adjoint operator has a non-zero eigenvalue). (iii) Show that, if A is a non-zero eigenvalue of A , then the null space of A - LZ has finite dimension. (We call this finite dimension the multiplicity of Iz as an eigenvalue of A . ) (iv) Prove that, if E is a positive real number, there are only a finite number of different eigenvalues p of A such that 1p( > E. Deduce that the distinct non-zero eigenvalues of A either form a finite set or form a sequence converging to 0. (v) Let {pl ,p 2 , p 3 , . . .} be the (finite or infinite) sequence of all distinct and non-zero eigenvalues of A , arranged so that lpl(2 Ip21 2 Ip31 2 . suppose that p,, has multiplicity m(n). Let {Al, 12,,I3,. ..} be the real sequence consisting of p1 (m(1) times), followed by p2 (m(2) times), followed by p3 (m(3) times), and so on. Let {yl ,y 2 ,y,, . . .} be a sequence of unit vectors consisting of an orthonormal basis of the null space of A - p u l l , followed by an orthonormal basis of the null space of A - p21, followed by an orthonormal basis of the null space of A - p3Z, and so on. Show that {y,,}is an orthonormal system, and Ay,, = Any, for each n. Prove that, if A . is the compact self-adjoint operator defined by

-

Aox = CAn(x,yn)yn

a ,

(XEW

n

(see Exercise 2.8.26), then A - A . has no non-zero eigenvalue. Deduce that A = Ao. (vi) Show that A 0 if and only if A,, 2 0 for all n. Deduce that, in this case, A has a (compact) “positive square root” A l (that is, A: = A and A l 2 0) such that IJA11(2= ( [ A [ ( .

168

2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS

2.8.30. Let A be a self-adjoint operator acting on a Hilbert space X, and = JJAJJ. let x be a unit vector in X such that JJAxJI (i) Show that x is an eigenvector for A' corresponding to the eigenvalue

llA1I2. (ii) Show that either Ax = llAllx or Az = - IJAllzfor some unit vector z.

[Hint. Consider the vector JlAlJx- Ax.] (iii) Under the added assumption that A 3 0, show that x is an eigenvector for A corresponding to the eigenvalue llAJ1. 2.8.31. Let E and F be projections acting on a Hilbert space X.

(*I

(i) Show that, if E and F commute, EvFinf{llAB-(C1B+AC2 - C1Cz)(l:C1,C2E9}

2 inf{llAB - CII: C E ~=}ll(A %/Sis a Banach algebra in its quotient norm.

+ 9 ) ( B + $)I[,

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3. BANACH ALGEBRAS

3.2. The spectrum Our Banach algebras are intended to provide the general framework for the study of algebras of (linear) operators on a Hilbert space. In the case of a finitedimensional space (and when they are present in the infinite-dimensional case), the eigenvectors and their associated eigenvalues play an important role in the analysis of the individual operator. The concept of spectrum, which we study in this section, is devised as the replacement, in the general setting of Banach algebras, for the set of eigenvalues in the finite-dimensional case. Henceforth our Banach algebras are assumed to be complex. 3.2.1. DEFINITION. If A is an element of the Banach algebra 2l,we say that a complex number A is a spectral value for A (relative to a) when A - AZdoes not have a two-sided inverse in N. The set of spectral values of A is called the spectrum of A and is denoted by spW(A). When there is no danger of confusion, we write sp(A) in place of sp,(A). Before beginning the general study of the spectrum, let us note that it serves the purpose for which it is designed. If 3’ is a finite-dimensional (Banach) space and A is a linear transformation of 3’ into itself, then A - AZ will fail to have an inverse (in a(#),the family of all linear transformations of # into itself) if and only if it annihilates some (unit) vector xo in 2 -that is, if and only if A has some unit eigenvector xocorresponding to the eigenvalue A. At the same time, let us note that while an eigenvalue is always in the spectrum, the reverse need not, in general, be the case. 3.2.2. EXAMPLE.Let 2 be the Hilbert space of complex-valued squareintegrable functions on [0,1] relative to Lebesgue measure, and let A be multiplication by the identity transform on [0,1] (so that (Af)(x) = xf(x)). Then A has no eigenvalues; for if Af= AJfmust be 0 at all points of [0,1] other than A. Hencefis 0 almost everywhere, andfis the element 0 in X. Nonetheless, sp,,,,(A) = [0, 11. To see this, let y, be the characteristic function of [A - 1/2n,A+ 1/2n], and let x, be nl”y,. The obvious modification is made when A is 0 or 1 ;and, for 1 in (0, l), the sequence {x,) has as first element xno, where no is so large that [A - 1/2n, A + 1/2n] E [O, 11, when no 6 n. Each x, is a unit vector, and ll(A - AZ)x,ll = >/6n (so that {x,} is a sequence of “approximate” eigenvectors for A corresponding to the eigenvalue A). If B is a left inverse to A - AZ, then

179

3.2. THE SPECTRUM

A - IZdoes not have a two-sided inverse in A?(%). (In essence this is part of the argument of Corollary 1.5.10, which could be applied here.) It follows that 11 SPB,,,(A)If J.$[O, 11, then A defined as (x - I)-' for x in [0, I], is continuous in [0, I]. Multiplication by f o n &' is a bounded operator, which is the two-sided inverse (in A?(%)) to A - IZ. Thus I $ s ~ ~ ( ~ , (and A )sp,(,,(A) , = [O, 11. r o 9

We shall see presently (see Lemma 3.2.13) that for normal operators (see Section 2.4) the situation of Example 3.2.2 holds generally: spectral values correspond to (sequences of) approximate unit eigenvectors. 3.2.3. THEOREM. I f A is an element ofthe Banach algebra 2l then sp,(A) is a non-empty closed subset of the closed disk in C with center 0 and radius llAll.

Proof. If I $ s ~ p l ( A )then, , by Proposition 3.1.6, A - I'Zis invertible for all 1'in a small open disk with center 1.Let p be a continuous linear functional on 8. Since p ( ( A - I ' Z ) - l ) - p ( ( A - I Z ) - 1 ) - p((I' - A)(A - I'Z)-l(A - I z ) - ' )

A' - I

I' - I

= p ( ( A - A'Z)-'(A -+

p((A - u

- IZ)-')

- 2 )

as I' -+ I , by continuity of inversion (see Proposition 3.1.6) on the set of invertible elements of 8, and the continuity of p , the function I -+ p ( ( A - IZ)- l ) is holomorphic on C \ sp,(A). Note, too, that p((A - u as l,ll-+

00; for

- 1 )

= I-lp((I-1A - Z ) - l ) + O

A-'A - I is invertible when (IA(I< 111,and

(I-lA - I ) - '

-+

-I

as 121 -+ 00.We see, at the same time, that I ( I - ' A - Z) (= A - AZ) is invertible when llAll < 111;so that sp,(A) is a subset of the closed disk in C with center 0 and radius IlAll. If sp,(A) were empty, the function I -+ p((A - AZ)-') would be an entire function that vanishes at a.By Liouville's theorem, this function would vanish everywhere on C. In particular, we would have p ( A - l ) = 0, for each continuous linear functional p on a. From the Hahn-Banach theorem (see Corollary 1.2.1l), it would follow that A - = 0, a contradiction. Thus sp,(A) is not empty. We observed, during this argument, that C \ sp,(A) is open, so that sp,(A) is a non-empty closed subset of the disk C with center 0 and radius llAll. H Despite the fact that the spectrum of A is not empty, it may consist ofjust 0. If { e l ,e 2 }is a basis for two-dimensional Hilbert space and A is the operator on

180

3. BANACH ALGEBRAS

this space that maps e l to e z and e2 to 0, then sp,,#,(A) consists ofjust 0. Note, too, from this example, that A may be non-zero and have just 0 in its spectrum. If each element of 2l other than 0 has an inverse in 2I (so that ‘21 is a division algebra), then, if A E spm(A),A - IIl must be 0 (being a singular element of ‘91). Thus, in this case, 2I consists of just scalar multiples of I. Since M is, then, isomorphic to @, we say, loosely, that 2l is @. A (complex) Banach division algebra (or field) is @. 3.2.4. COROLLARY.

We noted, in Proposition 3.1.8,that a maximal ideal .M in a Banach algebra 2I is closed. Since .A is maximal, if 2I is commutative, %/.Ais a field -and a Banach algebra. From the preceding corollary, ‘%/.Ais C,and the quotient mapping is a continuous multiplicative linear functional on ‘i!l(that is, a homomorphism of 2I onto C). Conversely, if p is a homomorphism of 2I onto @, its kernel .A is a maximal two-sided ideal in 2I (since %/A! is the field C). Hence A! isclosed and, from Corollary 1.2.5,p iscontinuous. In general, if ’21is not commutative and .Ais a maximal two-sided ideal in 2I, we cannot conclude that ‘%/A is a field; so that no multiplicative linear functional need be associated with .I.

3.2.5. COROLLARY. If ’21 is a commutative (complex) Banach algebra and .Iis a maximal two-sided ideal in %, then %/Ais C and the quotient mapping from ‘91 to 2lIYXis a (continuous) multiplicative linear functional on a.If2l is an arbitrary (complex) Banach algebra and p is a multiplicative linearfunctional on PI, then p is continuous with kernel A! a maximal two-sided ideal in CU such that Vlldt is @. We saw that sp,(A) is contained in the disk in C with center 0 and radius (IA((.The radius of the “smallest” disk containing the spectrum will appear in our considerations.

3.2.6. DEFINITION. The spectral radius r,(A) of an element A of a Banach algebra 2I is sup{(k(:AESPyl(A)}.

w

3.2.7. REMARK.When no confusion can arise, we write r(A) in place of r4((A).As noted in Theorem 3.2.3, r(A) < l1All. It is apparent from the definition that r(A) is the radius of the smallest disk in C with center 0 containing sp(A). W 3.2.8. PROPOSITION. I f A and B are elements of a Banach algebra PI, then sp(AB) u ( 0 ) = sp(BA) u {0),and r(AB) = r(BA).

181

3.2. THE SPECTRUM

Proof If2 # O a n d l ~ s p ( A B )then , A B - Aland,hence,(A-'A)B - l a r e not invertible. On the other hand, if l#sp(BA), then BA - A/ and, hence, B(i.- ' A ) - I are invertible. Our task, then, is to show that I - A B is invertible in V l if and only if I - BA is invertible in 2l, for arbitrary elements A and B of

PI. Arguing formally, for the moment, (I

-

A B )- ' =

c (AB)" x

=

I

+ A B + ABAB +

n=O

and

B(I- AB)-'A

=

BA

+ BABA + BABABA + . . * = ( I -

BA)-' - I .

Thus if I - A B has an inverse, we may hope that B(I - A B ) - ' A inverse to I - BA. Multiplying, we have ( I - BA)[B(I - A B ) - ' A =

B(I - A B ) - ' A

+ I is an

+I]

+ I - BAB(I - A B ) - ' A

-

BA

=B[(I-AB)-' -AB(I-AB)-']A + I - B A = I ,

and similarly for right multiplication by I - BA. W

3.2.9. R E M A R K . It is apparent that sp(A + I ) = { l + a : a E s p ( A ) ) .We shall prove the more general result concerning the relation between sp( p ( A ) ) and sp(A), for an arbitrary polynomial p , in the proposition that follows. (We prove the full spectral mapping theorem (Theorem 3.3.6) in Section 3.3.) Combining the simple initial observation with the preceding proposition yields the fact that the unit element I of a Banach algebra 2l is not the commutator A B - BA of two elements A and B of 2l. (If I = A B - BA, then sp(AB) = 1 + sp(BA), which is not consistent with sp(AB) u (0)= sp(BA) u { O } . ) This fact is familiar in quantum theory where it takes the form the commutation relations are not representable in terms of bounded operators. However, there are unbounded operators whose commutator is I restricted to a dense linear manifold. (See Exercise 2.8.49.) 3.2.10. PROPOSITION. I f A is an element of the Banach algebra 91 andp is a polynomial in a single cariable, then

SP(P(A)) = M A ) : 1 E SP(A)J ( = P(SP(A))).

182

3. BANACH ALGEBRAS

If A is int:ertible, then sp(A-') = {1-':1Esp(A)) ( = (sp(A))-'). r f A and B are elements of the commutative Banach algebra 'u, then

sp(AB) G sp(A)sp(B), r(AB) < r(A)r(B),

sp(A r(A

+ B) G sp(A) + sp(B),

+ B) d r(A) + r(B).

Prooj: If 1~ sp(A), then A - 11 does not have a two-sided inverse in PI. Thus one of ( A - 11)BI or ' u ( A - 11) is a proper ideal .a in PI. If p ( x ) = a,x" + . . . + ao, then p ( A ) - p ( i L ) l= an(A"- 1")

+ . . . + a,(A - 11).

Noting that

+ lAk-' + ' . + Jk-'f) = (Ak-1 + A A k - 2 + . . . + Ak- ' I ) ( A - I I ) ,

Ak - IkI = ( A - l l ) ( A k - '

we conclude that p ( A ) - p ( 1 ) 1 ~ 9so, that p ( A ) - p ( 1 ) l does not have a twosided inverse in %, and p(1) E sp(p(A)). If y~ sp(p(A)) and I1,. . . ,A, are the n roots of p ( 1 ) - y, then p(A) - y l = ( A - 1'1) *

. . ( A - 1J),

so that at least one of A - 1'1,.. . , A - &I is not invertible. If A - AjI is not invertible, then l jE sp(A) and y = p(Aj)~p(sp(A)).Thus sp(p(A)) = p(sp(A)). Suppose A is invertible in 'u (equivalently, 0$spw(A)). If ?, f 0, then A - ' - 1 - ' 1 = 0.1- ,4)(2A)-', so that L-'Esp(A-')ifand onlyifAEsp(A). Thus sp(A-') = sp(A)-'. Suppose, now, that A and B are elements of the commutative Banach algebra 'u. If 1 E sp(AB), then AE - 21 lies in a proper ideal (necessarily, twosided) 9 of %. Since 2l has an identity, Zorn's lemma, applied to the set of proper ideals in 'u containing 9,shows that 9 is contained in a maximal ideal .X of 'u. From Corollary 3.2.5, .I is the kernel of a multiplicative linear functional p on %. Thus 1 = p(AB) = p(A)p(B). Since A - p ( A ) I and B - p(B)I are in the kernel JZ of p, p(A)~sp(A)and p(B)~sp(B).Thus 1 E sp(A)sp(B) and sp(AB) E sp(A)sp(B). Again, if 1 E sp(A + B), there is a multiplicative linear functional p on our commutative 'u such that p(A + E ) = 1. As p(A) E sp(A), p(B) E sp(B), and 1 = p ( A ) p(B); l.~sp(A) sp(B), and sp(A + B) 5 sp(A) + sp(B). The inequalities for the spectral radius are immediate consequences of the corresponding relations for the spectra. m

+

+

3.2.11. R E M A R K . We make special note of the fact established at the end of the proof of Proposition 3.2.10. If A is an element of a commutative Banach

3.2. THE SPECTRUM

I83

algebra CLI and E. E sp,(A), then there is a multiplicative linear functional p on such that p ( A ) = E.. Conversely, if p is a (non-zero) multiplicative linear functional on ‘21 (not necessarily commutative), then ~ ( A )sp%(A) E for each A in ‘$1. For this last assertion, note that A - p ( A ) I is in the kernel of p, a proper two-sided ideal in 91. The examples that follow illustrate the concepts of spectrum and spectral radius in the Banach algebra g ( X ) of bounded operators on the Hilbert space .X. 3.2.12. EXAMPLE.Let { e n } be an orthonormal basis for a separable Hilbert space X . Recalling Example 2.4.10, we have a bounded operator A on d such that Ae, = %,,en,where {A,,} is an arbitrary bounded (denumerable) subset of @.We saw that llAll = sup{lA,,l}, that A is normal, in general, and selfadjoint exactly when all A,, are real, unitary when all A,, have modulus 1, and positive when all A,, are real and non-negative. Since each A,,is an eigenvalue (with eigenvector en),{All} E sp,,,,(A). From Theorem 3.2.3, sp(A) is closed, so that {A,,} -,the closure of {A,,}, is contained in sp(A). If 1 is not in this closure, then inf{li - in[} > 0, and {(A,, - A)-’} is a bounded subset of @. Thus there is a bounded operator B on A? such that Be,, = (A,,- 1)-‘en.Since ( A - Al)e, = (All - ;,)en,we have B(A - AI)e,, = e,,and ( A - 1I)Be,, = e,,for all n. Thus Bis a two-sided inverse in g ( H )to A and A#sp,,,,(A). Hence {A,,}- = sp,,,,(A). If {EL,,) is an enumeration of the rationals in [0,1], then sp(A) = [0,1]. In Example 3.2.2 we considered an operator with spectrum [0,1] but no eigenvectors. Although the present example and Example 3.2.2 exhibit selfadjoint operators with the same spectrum, and, in a sense still to be made precise, both of these operators have spectra without “multiplicity”; these operators are quite different structurally. One has an orthonormal basis of eigenvectors, while the other has not a single eigenvector. In the finitedimensional case, self-adjoint operators having the same spectrum, each without multiplicity, have identical structure (are “unitarily equivalent”).

With the aid of an extension of the “approximate eigenvector” technique encountered in Example 3.2.2, we shall be able to extend to the spectra of selfadjoint, positive, and unitary operators, the information we have about the eigenvalues for the corresponding operators with an orthonormal basis of eigenvect or s. 3.2.13. LEMMA.’I .x‘ is a Hilbert space and A is a normal operator in g ( X ) ,then I E sp(A) ifand only ifthere is a sequence {x,,} of unit veclors in *x‘ such that “ ( A - ,IJ)x,,ll + 0 as n -+ 00.

Proof. Since A is normal, A - 1I is a normal operator in LA?(#). From Lemma 2.4.8, A - AI fails to have a bounded two-sided inverse (that is,

184

3. BANACH ALGEBRAS

i ~ s p ( A )if) and only if inf{ll(A - Al)xll: llxll = 1 , X E X=}0. Thus I E sp(A) if and only if there is a sequence {x,} of unit vectors in X such that l((A - Il)x,ll+ 0 as n + co. 3.2.14. THEOREM.If2 is a Hilbert space and T E ~ ( . @then ),

(i) sp( T ) consists of real numbers if T is self-adjoint ; (ii) sp( T ) consists of non-negative real numbers if T is a positive operator; (iii) sp(T) E { O , I } if T is a projection; (iv) sp(T) consists of complex numbers of modulus 1 if T is a unitary operator ; (v) sp(T*) consists of the complex conjugates of numbers in sp(T). Proof. Suppose Tis self-adjoint and 1E sp( T). From Lemma 3.2.13, there is a sequence of unit vectors x, such that ll(T - AZ)x,ll + 0 as n + 00. Then (( T - Al)x,,,x,) + 0 as n + 00. Since (Ax,, x,) = A, ( Tx,, x,) tends to 1.But (Tx,, x,) is real. Thus A is real, and (i) follows. In the same way, if T 2 0, then ( T x , , ~ , 2 ) 0 and I b 0. Thus (ii) is established. If T is a projection, then T 2 = T, so that

T(T - Al)x, = (1 - A)Tx, + 0. Thus (1 - A)Ax, + 0. But l(x,,ll = 1 . Hence ( 1 - l)A = 0, and A is either 0 or I , so that (iii) is established. If T is unitary, then 1 = (x.,x,) = ( T x , , Tx,). Since (Axn,1xn) = Ill2, and ( T x , , Tx,) - (Ax,, Ax,) -+ 0 as n + 00, 1 = 1AI2 and (iv) follows. From the properties of the adjoint operation on W ( 2 )(see Theorem 2.4.2), B* is a bounded inverse to T* - 11if and only if B is a bounded inverse to T - AZ, and (v) follows. With the added assumption that T is normal, the converses to (ik(iv) of Theorem 3.2.14 are valid. It is more convenient to establish these after the spectral theory of normal operators has been developed (see Theorem 4.4.5). 3.2.15. PROPOSITION. If 2 is a Hilbert space and A is a self-adjoint operator in @(#), then at least one of llAll or - IlAll is in sp(A). Proof. By working with IIAII-lA in place of A , we may assume that (IA((= 1. In this case, there is a sequence {x,} of unit vectors such that llAx,ll + 1 as n -+ co. Thus

- A2)x,1I2 =

+ llA2~,112- 2 Re(A2x,,x,) < 2 - 211A~,()~ -+0

as n + 00. From Lemma 3.2.13, 1 esp(A2); and by Proposition 3.2.10, 1 ~ ( s p ( A ) )Thus ~ . 1 or - 1 is in sp(A).

185

3.2. T H E SPECTRUM

It follows from the preceding proposition that r(A) = llAll when A is selfadjoint. More generally, r( T ) = IlTll when Tis normal. These facts will follow directly from a general formula for the spectral radius that will be developed in Section 3.3 (see Theorem 3.3.3 and Proposition 4.1 .l(i)). 3.2.16. EXAMPLE.There is no difficulty extending the construction used in Example 3.2.2 to identify the spectrum of more general “multiplication operators” (see Example 2.4.1 I). If ( X , p) is a o-finite measure space and fis an essentially bounded measurable function on X , then MJ(g) = f g defines a bounded operator M , on L 2 ( X ,p ) . The essential range s p ( n offis the set of complex numbers i. such that p ( f - ’ ( O ) ) > 0 for each open subset 0 of C containing L. Suppose L ~ s p ( f ) .For each positive integer n, let yn be the characteristic function of a measurable subset of f - ’ ( O , ) of finite positive p-measure a,, where 8, is the open disk in C with center 1 and radius n-l. Then {x,} is a sequence of unit vectors, where x, = ~ , - ” ~ y , ,and

Thus ll(MJ - A I ) x , , ~+~ 0 as n + SO, and IESP(M/). Conversely, if L $ sp(f) there is a disk 0, of radius n - with center i, such that p ( f - ’ ( O , ) ) = 0. Then l/(f- A) is a measurable function g with an essential bound n, and M, is a two-sided bounded inverse to MJ - At. Thus i$sp(M,). It follows that sp(M,) = s p ( n (and that r(M,) = IlfllX, = IlM~ll). H 3.2.17. EXAMPLE.Let ,P be a separable Hilbert space and {e,:n = 0, 1, & 2 , . . .} an orthonormal basis for 2.The transformation U on 2 such that Ue, = en+ is a unitary operator. From Theorem 3.2.14, sp(U) is a subset of C1, the complex numbers of modulus 1. If 1 E C1, then, in a formal sense,

If= - ].-,en is an “eigenvector” for U corresponding to the “eigenvalue” 1. Although this sum is not a “genuine” element of ,X; its partial sums, multiplied by suitable normalizing factors, provide us with a sequence of approximating eigenvectors. Specifically, let x, be (2n + x i = -nA-kek.Then Ilx,ll = 1 and ‘1

I~(U i l ) x , J ~= (2n +

n

1)-~”11

C

n

A - k e k + l-

k= - n

= (2n

+ I)-’i2IIA-”e,,+

=21’2(2n as n

+

+ 1)-”2+0

x.Thus i ~ s p ( ( i ) and , sp(U) = C l .

k-“-l)ek1I k= -n

- An+ le-,ll

186

3. BANACH ALGEBRAS

Let Y?’ be L2(Cl) relative to Lebesgue measure on C1 normalized so that the total measure on C,is 1. Then {z”:n = 0 , f 1 , . . .) (where z denotes the identity transform on C1) is an orthonormal basis for 8’ [the Weierstrass approximation theorem (see Remark 3.4.15) is used to show that this system generates X ‘ ] . There is a unitary transformation V of ,X’ onto .X such that V(z”)= en. The multiplication operator M , on H ’has spectrum C1, from Example 3.2.16. Note that VM,V-’ = U . From this “unitary equivalence” of the “two-sided shift” operator with “multiplication by z,” we can deduce the spectral properties of one from those of the other. W 3.2.18. EXAMPLE.With .K a separable Hilbert space and { e , : n = 0,1, 2 , . . .) an orthonormal basis for 2,let W be the bounded operator on X such that We, = en, In this case, W , the “one-sided shift” operator, is not a unitary operator on .X, since eo is not in its range (although W is a unitary transformation of .X onto the range of W ) . Note that if 111 = 1, then U , C ; = ~l - k e k( = x,) is a unit vector in ,X, where a, = (n + 1)-1’2,and n

Il(W - II)x,ll

= anll

C A-kekt

k=O

n

-

C j.-(k-llekll k=O

as n + cx;, since a, + 0. Thus ).ESP( W ) . If 111 < I , then W * x = A x , where x = X;=oIkek, for ( W * x , e k )= (x, We,) = ,Ik+’. Thus A€sp(W*) and ~ T Esp( W ) (see Theorem 3.2.14(v)). Since 11 WII = 1 , sp( W ) is contained in the closed disk of radius 1 with center 0 in C (see Theorem 3.2.3).Thus sp( W )is this closed disk. W 3.2.19. EXAMPLE.Returning to H and U of Example 3.2.17, we let CLI be the Banach subalgebra of B ( H )consisting of the norm closure of the algebra of polynomials of a single variable in U (and I ) . If U has an inverse in $11,then that inverse must be U * . Each polynomial in U , and hence each element of CLI, however, maps the closed subspace generated by {en:n = 1 , 2 , . . .) into itself, whereas U* does not map this space into itself ( U*el = eo).Thus U* 4 CU and U has no inverse in ’11. Stated in terms of spectrum, we have O E S P ~ ( Ubut ) 0 4 SP,B,,I(U). Of course, as we pass from a Banach algebra to a Banach subalgebra, an element of the subalgebra may “lose its inverse.” Thus, in theory, the spectrum may “grow” on passage to a subalgebra. The present example indicates that this increase of spectrum can occur in practice. We shall note (Proposition 4.1 .5) that no change occurs in the spectrum when passing from a C*-algebra to a C*-subalgebra. This fact plays a crucial role in the application of spectral theory to C*-algebras. W

3.2. T H E SPECTRUM

187

3.2.20. PROPOSITION. The non-zero multiplicative linear functionals on a Bunach algebra '$S form a weak* compact subset of the unit ball of a'. Proof: From Remark 3.2.1 1, if p is a non-zero multiplicative linear functional on VI, then p(A)~sp,(A), for each A in '$1. Thus, from Theorem 3.2.3, Ip(A)I ,< llAll, and p lies in the unit ball of a'. such that p ( A B ) - p(A)p(B) = 0 is weak* The set of elements p in closed. The intersection of these sets (as A and Brange through 2l)is the weak* closed set of multiplicative linear functionals in a'. The further condition, p ( l ) = I , singles out the weak* closed subset consisting of non-zero multiplicative linear functionals on VI -a subset of the unit ball of a'. From Theorem 1.6.5(i), the unit ball of rU' is weak* compact, as is this closed subset. H The Banach alyebra L , ( R ) and Fourier analysis. In this subsection, we study the maximal ideals of the special Banach algebra Ll(R) provided with convolution multiplication. By letting L , ( R ) act on L,(R) as a convolutionmultiplication algebra, we define an algebra , d , ( R ) of operators acting on the Hilbert space L,(R). We adjoin I to d , ( R ) and take the (operator) norm closure to obtain another algebra VS0(R) of operators on L,(R). The algebra !&(R) is an example of a class of operator algebras, abelian C*-algebras, whose general properties will be studied intensively in Chapter 4. For the present, we identify the maximal ideals of PSo(R) and use this information to develop some of the basic theory of Fourier transforms. The Banach algebra L , ( R ) and its ideal structure is the general framework for this theory. We shall have occasion to use Fourier transforms in Sections 9.2, 13.2, and 13.3. The results we obtain here on the ideal theory of L , ( R ) will play an important role in the analysis of the (continuous) homomorphisms of the additive group R into the group of unitary operators on a Hilbert space (see Stone's theorem (5.6.36)). The various algebras we define may be viewed as generalizations of the complex group algebra of a finite group to the case of the group R. The methods we describe apply to more general (locally compact) topological groups and can be extended without great difficulty to such abelian groups.

3.2.21. DEFINITION. With f and y measurable functions on R, the conrdution of fand y is the function f * y whose domain consists of those real numbers s for which the integral f ' ( t ) g ( s- t ) dt converges and whose value at s is this integral. H

Sw

Since Lebesgue measure on R is invariant under the transformations -+ t + s, for each real s, we have, for each h in Ll(R),

t -, - t and t

h(s + t ) d t =

h(s - t ) d t = J R

188

3. BANACH ALGEBRAS

3.2.22. PROPOSITION.

f*Y

=

(i) I f f and g are measurable functions on R, then

s*f.

(ii) I f f € L1(R) and g E LJR) (where 1 < p), then f * g E L J R ) und

Ilf*sll p G llflll . Ilsllp.

(1)

(iii) I f f € L , ( R ) , g,hELp(R), and a E C , then both f * ( a . g a . j * g + f * h are in L,(R) and f * ( a .g

+ h) = a . f * g

+ h ) and

+f*h.

(iv) I ~ J ~ E L ~andhELp(R), ( R ) then both (f*g)*h a n d f * ( g * h ) are in L p ( R ) and

( f * g ) * h=f*(g*h). (v) Provided with the mappings ( J g ) + f * g and f - Ilflll, Ll(R) is a commutative Banach algebra.

Proof. (i)

(f*g)(s) = J =

flt)g(s - t ) dr = R

lR

s ( t ) f ( s - 4 dt

JR

= (9

f(s

+ r)s(- t ) dt

*ncs>.

(ii) If h,(t) = g(s - t ) and p(S) = Js If(t)l dt for each measurable subset S of R, then h S ~ L p ( Rand ) p is a finite measure on R. Applying the Holder inequality [R: p. 62, Theorem 3.51 to h, and the constant function 1 relative to p, we have

and

Now (s,t ) + Ig(s - t)lPf(t) is in L , ( R x R) since

3.2. T H E SPECTRUM

189

Thus, using Fubini’s theorem,

from which (1) follows. (iii) From (ii), f * (a . g

and u

+ h) and a .f*g +f* hare in L,(R). In particular

[flI)co . g(s - t )

+ h(s - t ) ] dt

s

s

f l t ) g ( s - t ) dt

+

f(t)h(s - t ) dt

converge for almost all s, and (iii) follows. (iv) From (ii), (f*g) * h and f*(g * h) are in L,(R). Now

and

Since If1 and IgI are in Ll(R), and Ihl E L,(R); whenf, g, and h are replaced by their respective absolute values, the last two integrals converge for almost all s. Fubini’s theorem applies, and, for almost every s,

Since

s

g ( r - r)h(s - r)dr =

( f * 9 ) * h = f*(Y * h).

s

g(r)h(s - t - r)dr,

(v) From (i)-(iv), the Ll-norm and convolution multiplication provide L , ( R ) with the structure of a commutative Banach algebra (without unit).

I90

3. BANACH ALGEBRAS

If we define LJy) to kf*y, wherefg Ll(R) and g E L,(R), (ii) and (iii) of Proposition 3.2.22 tell us that L, is a bounded linear operator on Lp(R) and llLs 11 < IIJ'II1. From (iv) of that proposition (and right-distributivity of convolution multiplication - proved as in (iii)], we have that the mapping f --,L , is a homomorphism of the algebra Ll(R) into B(L,,(R)).In particular, if p = 2, the image d l ( R ) of Ll(Ew) under this homomorphism is an algebra of operators on the Hilbert space L2(R).We denote by %,(R) the norm closure of v d l ( W

3.2.23. PROPOSITION. For each f in t l ( R ) , $ G llLf - 111,so that 14 91 The linear space %o(Ew) generated by I and 911(R)is a norm-closedcommuiatitie algebra of operators on L2(Ew)and %,(R) is a (proper)maximal ideal in &,(R).

Proof. Let tlnft)be (n/2)1i2 for tin [ - n - I , n - J and 0 for other values o f t . If f i n L,(R) is such that llLr - ZII < i, then

=

11"

r

lj*in f(s - t)dr - 1 ds, 2 - I/. - lin for all positive integers n. Since feL,(R), we can choose n so large that ft';/,If(t)\dt < $. With s in [ - n - ' , n - $ 3 ,

so that

+ < If!':,,f(s

Is":,.

I

f(s - i)dr g

- r)di

If(t)l dr

1

- ~and ) that the set of such z is open and contains all (small) z such that IIzAIl < I . Thus, for small z,fis defined and is represented by the power A'z". From Theorem 3.3.1 (and the comment preceding it on the series uniqueness of series representation), this series represents f on the largest open disk with center 0 on whichfis defined. On the other hand, Theorem 3.3.2 informs us that this series failsto converge for z of modulus exceeding ~ ' i nis, an a such that a' < la1 for (lim IIA"Illi")-'.Thus, if 0 < a' < lim ~ ~ A n ~there which I - a - ' A and, hence, A - a l fail to have inverses in 2l. Therefore a E sp(A) and a' < r(A). Since a' is an arbitrary non-negative number less than lim IIA"llli",the inequality (7) follows. H In case lim llAnlllin= 0, the inverse of limIIA"II"" is interpreted, as is customary, as co.When this occurs, r(A) = 0 and sp(A) consists of 0 alone. We discussed an instance of this in the comment following Theorem 3.2.3. An element A in 2I for which r(A) = 0 is said to be a generalized nilpotent in 2I. If A" = 0 for some positive integer n, we say that A is nilpotent (so that a nilpotent element in N is, in particular, a generalized nilpotent). One notes from Theorem 3.3.3 that r,(A) = r & 4 ) when A lies in the Banach subalgebra 98 of %. If A and Bare commuting elements of 2l, applying Proposition 3.2.10 to the commutative Banach subalgebra of N that A and B generate, together with this observation, we have the following result. 3.3.4. COROLLARY. If A and B are commuting elements in the Banach algebra 2I, then r(AB) 6 r(A)r(B)and r(A B) < r(A) r(B). The holomorphic function calculus. Turning now to the case where 2l is a Banach algebra, we note that

+

+

'S

'

A" = - z"(zZ - A ) - dz, (8) 271i where n is a positive integer, A EN,and C is a smooth closed curve whose interior contains sp(A). To see this, observe that z + ( z l - A ) - is holomorphic on C\sp(A) (as proved in Theorem 3.2.3). Employing the Cauchy theorem (see (4)) in the case of the 2l-valued function z + z"(zl - A ) - we may replace C by the (circular) perimeter of a disk with center 0 and large radius. Assuming C is this circle and z is on C,

'

',

(9)

z"-l(l-

~ ~ - 1 ) -= 1 zn-l

m

a3

k=O

k=O

1 Akz-k = C Akf-k-1

9

206

3. BANACH ALGEBRAS

where convergence is in the norm topology (and uniform on C). It follows from (3) and this convergence that term-by-term integration of (9) is justified. Now

s,

AkZn-k-l

dz=( ~czn-k-ldz)Ak=O~Ak=O

unless k = n, in which case the integral is 27tiA“.This proves (8). It follows that

‘S

f(A) = 7

2x1 c

f(z)(zZ - A ) - dz

for each polynomialf, when C is as in (8). With the foregoing in mind, we take (10) as the definition of f ( A ) for holomorphic functionsf. More precisely, whenfis holomorphic in an open set containing sp(A), we can choose a smaller open set 8 containing sp(A) whose boundary consists of a finite number of closed piecewise linear curves C1,.. . ,C,,. If C denotes the collection of these curves oriented in the customary way in complex function theory, then (10) definesf(A). To find the smaller open set with boundary as described, an argument involving a square grid in the plane (with squares of diameter less than the distance from sp(A) to the boundary of the initial open set) will suffice. Since the integral in (10) converges in norm, from our discussion of line integrals, it represents an element f(A) in 2l. From (4) (Cauchy’s theorem) f ( A ) is independent of the curve C (consisting of a finite number of smooth closed curves constituting the boundary of an open set in which f is holomorphic). Let %(A) be the set of functions holomorphic in some open set containing sp(A) (the open set may vary with the function). The following two results constitute a “calculus” of such functions - the holomorphicfunction calculus.

3.3.5. THEOREM. The mapping f 3 f ( A ) is a homomorphism from # ( A ) into 2lfor each A in the Banach algebra 2l. I f f is represented by the power series I,,“=anznthroughout an open set containing sp(A), then

,,

m

n=O

Proof. Since

the mappingf+f(A) is linear. The proof thatf(A) . g(A) = (f. y)(A) requires more effort. Let 0 be an open set, containing sp(A), on which bothfand g are holomorphic. We can choose open sets O1 and O2 such that sp(A) E 01,

01 v C1 G 02,

02 v C2 E 8 ,

207

3.3. THE HOLOMORPHIC FUNCTION CALCULUS

where C1and C2, the boundaries of O1 and 0,, consist of a finite number of smooth closed curves. Then

f(z)g(w)(zl- A)-'(wl- A ) - ' dzdw

[ ( z l - A)-' - ( w l - A ) - ' ] w-z

dz dw

- ( ~2ni ) ' S , , g ( w ) ( w l - A ) - ' ( J c1W-z E d z ) d w 1 2ni

==(

f(z)g(z)(zl- A ) - ' dz-

g(w)(wl-A)- ' ( 0 )dw

c1

f . g)(A)*

To prove the last assertion of the theorem, we may assume that f is defined on the disk of convergence of the series. Let C be a circle with center at 0 containing sp(A) in its interior and contained in an open set on which f is holomorphic and represented by I,"=a,z". Then this series converges uniformly on C, so that, from (8),

c a, (& c a,A". m

=

n=O

Ic

z"(zl - A )- dz

m

=

n=O

In our next result, we identify the spectrum off@). The special case wheref is a polynomial has been treated in Proposition 3.2.10.

3.3.6. THEOREM(Spectral mapping theorem). r f A is an element of a Banach algebra 2l and f is holomorphic on an open neighborhood of sp(A), then (12) SP(f(A)) = { f ( a ) :a E SP(41 ( =f(sp(AN). Proof. Suppose aEsp(A), so that either %(A - al) or ( A - al)% is a proper (left or right) ideal in 2l. Say, %(A - a l ) is a proper (left) ideal in 2l.

208

3. BANACH ALGEBRAS

Then 2 n i [ f ( A ) -f(a)Z] = =

f(z)[(zZ

- A)-'

- (z - a)-'Z]dz

(lC

)

[ ( z l - A)(z - a)]- ' f ( z )dz ( A - aZ) E %(A - a l )

and f ( a )E sp(f(A)). Thus (13) f(SP(A)) c S P ( f ( 4 ) . If b$f(sp(A)), then (f- b ) - l (= g ) is holomorphic on an open neighborhood of sp(A). From Theorem 3.3.5, g(A) is a two-sided inverse to f ( A ) - b l in % (since g . (f- b) is 1 on an open neighborhood of sp(A)). Hence b $ sp(f(A)), and

(14) SP(f(A)) s f(SP(4). Combining (13) and (14), we have (12).

An interesting and simple corollary of the holomorphic function calculus and the spectral mapping theorem asserts that if the Banach algebra 2l has an element A whose spectrum is not connected, then 9l has an idempotent E different from 0 and I. To see this, suppose that sp(A) = S1u S2 ,where S1and S, are disjoint closed sets. Since sp(A) is compact (see Theorem 3.2.3), both S1 and S, are compact. It follows that there are disjoint open sets O 1 and 0,such that S1 E O1 and S, E 0,.The functionftaking the values 1 on O1 and 0 on 0, is holomorphic on O1 u O,, an open neighborhood of sp(A). Thenf(A) is an idempotent E since E = f ( A ) = ( f Z ) ( A = ) f ( A ) f ( A ) = E Z , and sp(E) = (0, l}. Thus E is neither 0 nor I. +

3.3.7. COROLLARY. r f A is in the Banach algebra 2l and sp(A) is not connected, then 2I contains an idempotent different from 0 and 1. The composite-function result that follows is an important addition to the function calculus.

3.3.8. THEOREM. IfAisanelementof the Banachalgebra%,gEX(A), and f E W S ( A ) ) ,then f 0 9 E %(A) and ( f o g ) W =f ( g ( 4 ) . Proof. By assumption, f is holomorphic on an open set O1 containing sp(g(A)) and g is holomorphic on an open set 0, containing sp(A). From Theorem 3.3.6, g(sp(4) = sp(g(A)) s

017

so that sp(A) E g - l ( O 1 ) . By continuity of g , g-'(O,) n 0, is an open set O (containing sp(A)) on which f a g is holomorphic. Thus f o g E X(A).

3.3. THE HOLOMORPHIC FUNCTION CALCULUS

209

Choose open sets @ and 6&l with boundaries C and C1consisting of a finite number of smooth closed curves such that sp(A) c 42 E 9 u C G 8 and sp(g(A)) c g(% u C ) c 9

1

e 421 u c1 E 8,.

By continuity of g on 8, there is an open set 9' such that 9u C c 9'E 0 and g ( 4 ' ) 5 +21.Then, for each w on C1, h, is holomorphic on W ,where h,(z) = [ w - g ( z ) ] - ' . From Theorem 3.3.5, h,(A) = [ w l - g ( A ) ] - ' for each w on C1. It follows now that .

P

We conclude this section with a result that allows us to treat convergence in the holomorphic function calculus. 3.3.9. PROPOSITION. ZfS is an open set containing sp(A), where A is an element of the Banach algebra a, and { f n }is a sequence of functions holomorphic on 0 and converging uniformly tof on compact subsets of 0, then f E X( A)and Ilf"(4 -f(A)II 0 as n a* +

+

Proof. Choose an open set 9 with boundary C consisting of a finite number of smooth closed curves such that

sp(A) 5 %! E 42u C c 8.

Since {fn} converges uniformly to f on compact subsets of 0,f is holomorphic on 0.Thus fe.%?(A) and {f,}converges to f uniformly on C. It follows that

Ic

2nllf,(A) - f ( M = II

Cf,(Z)

- f ( z ) l ( z l - A)-'dzll

G klC1 . llfn - f

IIC

+

0 9

210

3. BANACH ALGEBRAS

3.4. The Banach algebra C(X)

In Example 3.1.4 we introduced the algebra C ( X ) of continuous complexvalued functions on the compact Hausdorff space X together with its (supremum) norm and established that it is a Banach algebra. From the point of view of C*-algebras, C ( X ) is, by far, the most important example of a commutative Banach algebra. We shall see in Section 4.4 that C ( X ) is the example of a commutative C*-algebra. In its function algebra form it provides the basis for the spectral theory and “function calculus” of a self-adjoint (or normal) operator. Our purpose in this section is to study C ( X )both with respect to its Banachalgebra structure and with respect to its order structure. We begin by identifying the closed ideals (and, hence, the maximal ideals in C(X)).

rfS is a closed ideal in C(X),there is a closed subset S of 3.4.1. THEOREM. Xsuch that $ is the set of allfunctions vanishing on S. U S is a closed subset of X , the set of allfunctions vanishing on S is a closed ideal in C ( X ) .The maximal ideals in C(X)are those closed idealsfor which the corresponding closed subset of X (on which all the functions of the ideal vanish) consists of a single point. Proof. Since the set of points at which a continuous function vanishes is closed and the set of points S at which all the functions of 9 vanish is an intersection of such sets, Sis a closed subset of X . We use the assumption that $ is closed to show that a functionf that vanishes on S is in 9.Note that this has the implication that 9 = C ( X ) if S is null. Suppose, then, that f E C ( X )andf vanishes on S. Given any positive E , let F, be the set of all pointsp at which If(p)l 2 E , so that F, is compact, and does not meet S. We shall construct an element g, of 9 such that 0 < g , ( p ) < 1 for all p in X , while g , is 1 throughout F,. Once this is done, we have fg,EY (since g c E 9 ) ; moreover, 1) f - fg,ll < E , because 11 - g,(p)l never exceeds 1, and is zero at all points where If(p)I 2 E . Since 9 is closed, we can conclude thatf E A We now construct g , with the properties set out above. By definition of S, and since F, does not meet S,for each point p of F,, there is an f, in 9 such that f,(p) # 0. Thus3, .fp is (strictly) positive on some open neighborhood ofp. A finite number of such neighborhoods cover the compact set F,. Ifp, ,. . . ,pnare thecorresponding points, then l&J2 + . . . + 1f P J 2 is a function h,, in Y,which is non-negative on X and positive throughout F,. From the compactness of F,, h, has infimum c (> 0) on F,. The equation k,(p) = max(h,(p), c ) defines an element k, of C ( X ) , and

k,(P) > 09

k,(P) 2 h&(P)2 0

3.4. THE BANACH ALGEBRA C ( X )

21 1

for all p in X , while k, coincides with h, on F,. Since k; ' E C ( X ) and heE 9, it now suffices to take h,k; ' for g,. The set of functions vanishing on an arbitrary subset of X will be a closed ideal f in C(X),but the set of points at which all the functions of 9 vanish will be a closed subset containing that set, its closure. To establish this Galois-like correspondence between closed subsets of X and closed ideals in C(X), note that if S is a closed subset of X and f is the closed ideal of functions in C ( X ) vanishing on S, the Tietze extension theorem tells us that, given a pointp not in S, there is a continuous functionfon X that is 0 on Sand 1 atp. ThusfE 9 andp is not in the closed set corresponding to 9.Hence the closed set corresponding to Y is S. Of course, now, the maximal ideals in C ( X )are those whose corresponding closed set consists of a point. 3.4.2. COROLLARY. Each non-zero multiplicative linear functional p on C ( X ) corresponds to a point po in X ; and p ( f ) =f ( p o )for each f in C ( X ) .

Proof. The kernel A of p is a proper ideal (since p # 0) and is a maximal ideal of C ( X ) .From Theorem 3.4.1, A is the set of functions in C ( X )vanishing at some point po of X . Since ~ ( 1 ' ) = ~ ( 1 ) '= p(1) # 0, p(1) = 1 ; and f- p(f)l E Afor each f in C ( X ) .Thus f ( p o )= p ( f ) for each f in C ( X ) .

We say that the functional p in Corollary 3.4.2 is evaluation at p o . The closed subset corresponding to an ideal (or just the common zeros of any set of functions) is referred to as the kernel of that ideal (or that set of functions). The (closed) ideal of functions vanishing on a set of points in Xis referred to as the hull of that set. Corollary 3.4.2 (or Theorem 3.4.1) provides us with a means of recapturing the topological space X from the algebraic structure of C(X). 3.4.3. THEOREM.A mapping cp of C ( X ) onto C( Y), with X and Y compact Hausdorff spaces, is an algebraic isomorphism if and only if there is a homeomorphism r] of Y onto X such that q ( f )= f r] for each f in C(X). 0

Proof. If r] is a homeomorphism of Y onto X and q(f)= f0 r], then for each f in C ( X ) , f or] E C( Y ) , and cp is an algebraic isomorphism of C ( X ) onto C( Y ) . Suppose, now, that cp is an algebraic isomorphism of C(X)onto C( Y ) .If .A is a maximal ideal in C( Y ) ,then q - ' ( A is ) a maximal ideal in C(X).Ifp is the point in Y corresponding to A,denote by r](p)the point in Xcorresponding to q - ' ( A )Since . each maximal ideal A. in C ( X )has the form cp-'(cp(Ao)), with cp(d0) a maximal ideal in C( Y ) , q is a one-to-one mapping of Y onto X . If f~C ( X ) , f - f(r](p))lvanishes at r](p)in Xfor eachp in Y. Thus, by definition of r], q(f - f(r](p ) )1) vanishes at p . Hence cp( f)( p ) = f0 r]( p ) for all p in Y, and cp(f) = f o r]. Since X is a compact Hausdorff space, it is completely regular.

212

3. BANACH ALGEBRAS

Hence sets of the f o m f - l(O), with 0 an open subset of C andf in C ( X ) ,form a subbasefortheopen setsinX. Now q-l(f-l(O)) = cp(f)-’(Co);and cp(f)-’(O) is an open set in Y, since q(f) is a continuous function on Y . As the inverse images of subbasic open sets in X, under q, are open in Y, q is continuous. Symmetrically, q - is continuous; and q is a homeomorphism of Y onto Xsuch that cp(f) = f o q for eachfin C(X).

’

3.4.4. R E M A R K . It follows from Theorem 3.4.3 that cp mapsreal functions in C ( X ) onto real functions and positive functions onto positive functions. It follows, too, that cp is an isometry. Thus the assumption that cp is an algebraic isomorphism entails very strict response from cp in terms of other structure that C ( X ) possesses (in particular, the norm and order structure on C(X)).These consequences of the assumption that cp is an algebraic isomorphism become more apparent when we note that cp “preserves” spectrum, that is sp,-,,,(f) = spc0,(cp(f)), and that the spectrum offrelative to C(XJ is the range of the functionf(see Example 3.2.16, in this connection, where the spectrum of M , is the essential range off). That cp preserves spectrum is a consequence of the fact that spectrum is defined in terms of inverses and cp preserves inverses. Of course, f - I 1 fails to have an inverse in C ( X ) if and only if it vanishes at some point of X , that is, if and only if 1is in the range off: The property of being real or positive forfand the norm offare all determined by the range off: We see, at the same time, that cp preserves the operation of complex conjugation on C ( X ) (that is, cp(f) = cp(f))- for cp maps real functions onto real functions. Note the formal similarity between the operation of complex conjugation of functions on C ( X ) and the adjoint operation on g ( X )(see Theorem 2.4.2). Both are conjugate-linear, involutory, (anti-)automorphisms on their respective algebras. With Theorem 4.4.3, this similarity becomes more than “formal.” When Theorem 4.4.3 has been established, the view of a C*-algebra as a non-commutative generalization of C ( X ) (that is, as a “non-commutative function algebra”) will be quite plausible. Up to this point, we have been studying the Banach-algebra structure of C ( X ) . In application to noncommutative C*-algebras, extending the order properties of C ( X ) ,rather than its Banach-algebra structure, proves to be the more fruitful procedure. The order structure of C ( X ) is a natural partial ordering of its real-linear subspace C(X, R), the continuous real-valued functions on X . It is introduced by means of the “cone” 8 of positive functions, which has the properties (of a cone) :

(i) iff and - f a r e in 8, then f = 0; (ii) if a is a positive scalar and f E 8, then af E 9; (iii) f+ g e 9 iffand g are in 9

3.4. THE BANACH ALGEBRA C ( X )

213

A real vector space V with such a cone is said to be a partially ordered vector space. Defining (as is usual in C(X, R)), f < g when g -f E fl induces a partial ordering on .Y:An element I in V (the constant function 1 in C(X,R)) is said to be an order unit when, given any f in "y; we have -aI < f < aZ for a suitable positive scalar a (depending onf -in the case of C(X,R), we may choose a to be

I l f 11). 3.4.5. DEFINITION.If V is a partially ordered vector space with order unit I , a linear functional p on Y is said to bepositive when p ( f ) 2 0 iff 2 0. If, in addition, p ( l ) = 1, p is said to be a state of % If p is an extreme point of the (convex) family Y ( Y )of states of "y; we say that p is a pure state of %

Note that the set of positive linear functionals on Y is a cone relative to which the dual space of Y becomes a partially ordered vector space (generally without an order unit). A simple modification of the condition for a state to be pure proves useful to us. 3.4.6. LEMMA.If V is apartially ordered vector space with order unit I , a state p of Y is pure ifand only ifeachpositivefunctional z on Y such that z < p is a scalar multiple of p . Proof. If the stated condition holds for p and p = upl + (1 - a)p2,with 0 < a < 1 and p1 and p 2 states of "y; then 0 < upl < p , so that u p , = bp. Since pl(l) = p ( I ) = 1, a = b and p1 = p . Similarly p 2 = p , and p is pure. On the other hand, if p is pure and 0 < < p , then 0 < z ( I ) < p ( l ) = 1. If z(Z) = 0, then, for each f in K 0 = z( - aZ) < z(f) < z(aI) = 0 for some scalar a, and t ( f ) = 0; so 7 = 0 (= 0 * p ) . If T ( I ) = 1 (= p ( l ) ) , a similar argument shows that the positive functional p - z is 0 (and z = 1 . p ) . Finally, if 0 < z ( I ) < 1, we have p = (1 - b)pl bp,, where b = z ( I ) and p1,p2are the states defined by p1 = (1 - b ) - ' ( p - z), p 2 = b-%. Since p is pure, p 2 = p , and T = bp. In each case, z is a multiple of p .

+

By expressing elements of C ( X )in terms of their real and imaginary parts, it is apparent that each (real-)linear functional on C(X,)!FI extends uniquely to a linear functional on C(X).A linear functional on C ( X )is said to be positive (or a state, or a pure state) if its restriction to C(X, R) is positive (or a state, or a pure state). Since the states of C(X,R) form a convex set with the pure states as extreme points, the same is true of C ( X ) . The significance of states and pure states for C*-algebras will appear in Sections 4.3 and 4.5. For the present, we establish that the pure states of C ( X ) are precisely the (non-zero) multiplicative linear functionals on C ( X ) . 3.4.7. THEOREM.A non-zerofunctional p on C ( X )is apure state of C ( X )i f and only if it is multiplicative.

214

3. BANACH ALGEBRAS

Proof. If p is multiplicative, from Corollary 3.4.2, there is a point po in X such that p c f ) =f(po) for all f in C ( X ) . Since each continuous function vanishing at po is a linear combination of positive continuous functions vanishing at po; if 0 < z < p for some linear functional t on C ( X ) , then z ( f ) = 0 whenf(po) = 0. Thus z and p are linear functionals on C ( X )with the same (maximal linear) null space and z is a scalar multiple of p. From Lemma 3.4.6, p is a pure state of C(X). Suppose p is a pure state of C(X).If 0 < f < 1 and z(g) = pug),then z is a linear functional on C ( X ) such that 0 < z < p . Thus z = up. If p(h) = 0, then p(fh) = z(h) = ap(h) = 0. Since each function g in C ( X ) is a linear combination of functions between 0 and 1, p(gh) = Ofor all g in C ( X ) .Thus the null space of p is an ideal -clearly maximal since the null space of p is a maximal linear subspace. As p(Z) = 1, p is multiplicative. W The linear order structure in C ( X )is strong enough to characterize it. We say that a mapping cp between two partially ordered vector spaces (or two C ( X ) spaces) that is a linear isomorphism of one onto the other is a linear order isomorphism when q ( f )> 0 if and only iff 2 0.

3.4.8. COROLLARY. A linear order isomorphism cp of C(X)onto C( Y) such that cp(1) = 1 is an algebraic isomorphism. Proof. If po is a pure state of C( Y) corresponding to the point po of Y, by Theorem 3.4.7, then po 0 cp is a pure state of C ( X ) corresponding to a point q(po) of X. Now, q is a one-to-one mapping of Y onto X , and f(q(po)) = po(cp(f)) = cp(f)(po) for all f i n C ( X ) and po in Y. Thus cp(f) =f 0 q. As in Theorem 3.4.3, q is a homeomorphism of Y onto X; and cp is an algebraic isomorphism. W

3.4.9. REMARK.The partial ordering on C ( X )induces a lattice structure on the set of real-valued functions. Iffand g are two such functions, we define (fv g ) ( p ) to be max{f(p), d p ) } and (fA g)(p) to be min{f(p), g(p)} for each p in X . Then fvs=%-+g)+:lf-gL

fAs=t(f+g)-:lf-gI;

so that f v g and f A g are in C(X).Clearly, f v g is the smallest function greater than bothfand g, andf A g is the largest function less than bothfand g. Moreover,f = f - f-,wheref + = f v 0 and f- = - (fA 0); so thatfis the difference of two positive functions in C(X) (with disjoint supports). The lattice structure assures us that C ( X ) has the Riesz decomposition property: iff < g1 g2, wherefl gl, and g2are positive functions in C(X),then f = f l +f2, where 0 I= 0

(fc L ~ ( um)). ~~

m

3.5.40. Let Sf be f (as defined in Exercise 3.5.39) for f in L2(U1, m ) ( G Ll(Ul, m)),and let T be the unitary transformation of Iz(Z)onto L2(Ul,rn)

described in Exercise 3.5.36(ii). (i) Show that S is a unitary transformation of Lz(Ul,m) onto 12(Z). (ii) Let U be the (self-adjoint) unitary operator on Lz(Ul,rn) that maps 5, onto for each t in Z. Show that T W f ) =f

(YE L2(U19 m))

sum)= 9

(9EW)).

and (iii) Deduce that T* = T-’

and

= S(J

3.5. EXERCISES

233

3.5.41. Let L,(R) denote the Banach algebra formed by providing Ll(R) with convolution multiplication. Withfin L,(R), denote byfi the “translate” off by the real number r (fi(t) =At - r)). Let 9 be a norm-closed linear subspace of Ll(R). (i) Show that if 9 is an ideal in L,(R), then 9 is invariant under translations (that is,f;E 9 whenfE9). [Hint. Use the approximate identity of Lemma 3.2.24 and (5) of Section 3.2.1 (ii) Show that if 9 is invariant under translations, it is a (norm-closed) ideal in Ll(R). [Hint. Use the Hahn-Banach theorem to support the “view” of y *fas Sg(t)f;dt and recall the identification of the dual of L1.] (iii) Show that if the span of the translates of a functionfin L,(R) is norm dense in L,(R),fvanishes nowhere on R. (With its converse, this is one of the main theorems in a body of work known as “Wiener’s Tauberian theorems.”) 3.5.42. Let Ll(Ul ,m)be the Banach algebra described in Exercise 3.5.37. denote byfw the “translate” offby w ( f w ( z )=f(zW)). Let Withfin Ll(Ul ,m), 9 be a norm-closed linear subspace of L1(T1,m). (i) Show that the sequence {u,} satisfies

as n + 00, where u, = 2n(u, tj- l ) , {u,} is the “approximate identity” described in Lemma 3.2.24 and tj is defined by q(s) = exp is. (ii) Show that if 9 is an ideal in Ll(Ul, m),then 9 is invariant under translation. (iii) Show that if 9 is invariant under translation, it is a (norm-closed) ideal in Ll(Ul ,m). (iv) Show that if the span of the translates of a functionfin L1(T1,m)is norm dense in Ll(Ul,m), thenf((t) # 0 for each t in Z. 0

3.5.43. Let 9I be a Banach algebra and A be an element of ‘2I such that sp,(A) E @ \ R - , where R - = { z : z E @ z, = - lzl}. (i) Show that there is an element A . in ‘2I such that (Ao)’ = A . (The element A . is said to be a square root of A in ’u.) (ii) Deduce that if B E 9I and 111 - BII < 1, then B has a square root in a. 3.5.44. Let A be an element of the Banach algebra 9I. (i) Show that if sp,(A) E {A: A E R, 0 < i}, then A has a squareroot in ’u with positive spectrum. (ii) If the hypothesis of (i) is weakened to

sp,(A) E {A:AER, 0 < A} (= R+),

234

3. BANACH ALGEBRAS

does A still have a positive square root in %? [Hint. Consider

in the Banach algebra of complex 2 x 2 matrices.] (iii) Show that if sp,(A) E R’ and IlBll = r,(B) for each Bin the Banach subalgebra 910 of 9l generated by A and I, then A has a square root in 910 with spectrum (relative to 910) in R + .

[Hint.Study the square roots of A applied to 9t0.]

+ n- ‘ I as n + co and use Remark 3.2.11

3.5.45. Let p be a hermitian functional on C(X),where X is a compact Hausdorff space. Suppose llpll = p(1). (i) Use Proposition 3.4.1 1 to show that p is a positive linear functional on C ( X ) . (ii) Show that p is a positive linear functional on C ( X ) without using Proposition 3.4.1 1. 3.5.46. Let Xbe acompact Hausdorff space and d be aclosed subalgebra of C(x> containing the constant functions and containing f when f~d. (i) Show that iffis a real-valued function in d,I € spd(f), and I is not real, then there is a real-valued function g in d such that iEspd(g). ( - ig)”/n! (= exp( - ig)) is an element (ii) With g as in (i), show that of d of norm 1. (iii) Use a multiplicative linear functional on d to show that e E sp&dexp( - i d l ’ (iv) Conclude that sp,(f) s R for each real-valued function f i n d. (v) Conclude that spd(f) E R, again, with the aid of Exercise 3.5.28(v). 3.5.47. Let C ( X ) and d be as in Exercise 3.5.46 and let the partial ordering of the (real) algebra d ,of real-valued functions in d be that induced by the partial ordering of C(X, R). (i) Show that each (non-zero) multiplicative linear functional p on d has an extension p’ to C(X)such that p’ is a state of C(X). (ii) Show that p is a pure state of d. (iii) Show that the set d of state extensions of p to C(X) is convex and weak* compact. (iv) Show that each extreme point of 6 is a pure state of C(X).

235

3.5. EXERCISES

(v) Conclude that p has an extension to C ( X ) that is a (non-zero) multiplicative linear functional on C ( X ) .

3.5.48. Let C ( X ) and d be as in Exercise 3.5.46. (i) Show that sp,(fl = sp,,,,(fl for eachfin d. (ii) Use Exercise 3.5.44to show that iffis a positive function in d,then the (unique) positive square root off in C ( X ) lies in d. (iii) Conclude that If1 ~d when f ~ d .

3.5.49. Let X be a compact Hausdorff space, and let 9 be a subset of C(X, R) such that, withfand g in 9,Y containsf v g andf

A

g (in this case,

9is said to be a sublattice of C(X, R)). Suppose that for each pair r, s of real numbers and each pairp, q of distinct elements of Xthere is anfin 9such that f ( p ) = r andf(q) = s. Show that 9 is norm dense in C(X, R). 3.5.50. Combine the results of Exercises 3.5.48and 3.5.49to give another proof of the StoneWeierstrass theorem (3.4.14).

CHAPTER 4 ELEMENTARY C*-ALGEBRA THEORY

In this chapter we study a special class of Banach algebras, termed C*algebras, the ones that have an involution with properties parallel to those of the adjoint operation on Hilbert space operators. With X a compact Hausdorff space and X a Hilbert space, C(X) and B ( 2 )are examples of C*-algebras, and so is each norm-closed subalgebra of B ( S )that contains the adjoint of each of its members. Two basic representation theorems (4.4.3 and 4.5.6) assert that, up to isomorphism, these are the only examples; every C*-algebra can be viewed as a normed-closed self-adjoint subalgebra of .93(2),for an appropriate choice of S, and every abelian C*-algebra is isomorphic to one of the form C(X). Earlier sections of the chapter are devoted to studying the spectral theoretic properties of certain special elements in C*-algebras and the order structure in such algebras and in their Banach dual spaces. These are basic tools, both for the proofs of the representation theorems just cited, and also for all the subsequent theory. 4.1. Basics

By an involution on a complex Banach algebra 'u, we mean a mapping

A

-,A*, from 2I into 'u, such that (i) (aS + bT)* = dS* (ii) (ST)* = T*S*, (5) (T*)* = T,

+ 6T*,

whenever S, T E'u and a, b E C and a, bdenote the conjugate complex numbers. A C*-algebra is a complex Banach algebra (with a unit element I ) with an involution that satisfies the additional condition (iv) IIT*TII = llTl12

(Tea).

This last condition ensures that the involution in a C*-algebra preserves norm (and is therefore continuous); for

IITIIZ= IlT*TII llT*11llTll7 236

4.1. BASICS

237

whence IlTll < IIT*ll, and we obtain the reverse inequality upon replacing T by T*. We have already encountered several examples of C*-algebras. If S is a Hilbert space, a(%)is a C*-algebra, with the adjoint operation as its involution; indeed, the defining conditions (i)-(iv) above are abstracted from the properties of adjoints of Hilbert space operators, as set out in Theorem 2.4.2 and the discussion that follows it. Several Banach algebras of complexvalued functions are C*-algebras, with an involution that assigns to an element f the conjugate complex function f,defined by Ax) =f(x). In this way, the Banach algebra C ( X ) , of all continuous functions on a compact Hausdorff space A’, becomes a C*-algebra. The same applies to the Banach algebra l,(X) of all bounded functions on an arbitrary set X (with pointwise algebraic operations and supremum norm), and to the Banach algebra L,(S, g m ) of all essentially bounded measurable functions (with pointwise algebraic operations and essential supremum norm), associated with a measure space (S,g m ) . We now introduce some terminology, concerning elements of a Banach algebra % with involution, and note certain immediate consequences of conditions (ib(iv) above. Motivated by the example of the algebra g ( Z ) ,we refer to A* as the adjoint of A (EB), and describe A as self-adjoint if A = A*, normal if A commutes with A*, unitary if A*A = AA* = I. With S = I* and T = I , it follows from (ii) and (iii) that I* = I ; so the unit element Iis both selfadjoint and unitary. The set of all self-adjoint elements of % is a real vector space, while the unitary elements form a multiplicative group, the unitary group of %. Each A in % can be expressed (uniquely) in the form H + iK, where H (= $(A + A * ) ) and K (= $i(A* - A ) ) are self-adjoint elements of %, the “real” and “imaginary” parts of A ;moreover, A is normal if and only if H a n d K commute. From (ii), A is invertible if and only if A* is invertible, and then ( A - ’)* = ( A * ) - ’ . By applying this result, withal - A and its adjoint iil - A* in place of A and A*, it follows that the spectra of A and A* satisfy sp(A*) = { a : U€SP(A)}. Accordingly, these elements have the same spectral radius, r(A*) = r(A). If ‘$ and I93 are Banach algebras with involutions, a mapping cp from B into is described as a * homomorphism if it is a homomorphism (that is, it is linear, multiplicative, and carries the unit of B onto that of 98)with the additional property that cp(A*) = cp(A)* for each A in a.If, further, cp is one-to-one, it is described as a * isomorphism. Although we impose no continuity condition in these definitions, we shall see later (Theorem 4.1.8) that * homomorphisms do not increase norm and * isomorphisms are norm preserving, when B and 9?are C*-algebras. If is a Banach algebra with involution, a subset 9of % is said to be selfadjoint if it contains the adjoint of each of its members. A self-adjoint subalgebra of % is termed a * subalgebra. If the involution is continuous (in

238

4. ELEMENTARY C*-ALGEBRA THEORY

particular, if 2€ is a C*-algebra), the closure of a * subalgebra is again a * subalgebra. It is clear that a closed * subalgebra 28 of '$ that Icontains the unit of CLI is itself a Banach algebra with involution ;if, further, 2l is a C*-algebra, then so is a. In this last case, we describe 9 as a C*-subalgebra of 2€. The proposition that follows extends, to appropriate elements of a C*algebra, the information concerning Hilbert space operators contained in Theorem 3.2.14, together with Proposition 3.2.15 and the comments following it. Suppose that A is an element of a C*-algebra 2€. 4.1.1. PROPOSITION.

(i) I f A is normal, r(A) = IlAll. (ii) If A is self-adjoint, sp(A) is a compact subset of the real line R,and contains at least one of the two real numbers f 11A11. (iii) I f A is unitary, IlAll = 1 andsp(A) is a compact subset of the unit circle { a E C : la1 = I}. Proof. (i) With H self-adjoint in 2l and n a positive integer, llH2"11= ll(H")*H"ll = llH"112.By induction on m, llHqll = llH114 when q has the form 2* (m = 1,2, . . .); so, by Theorem 3.3.3,

r ( H ) = lim

llHq11"q

= llHll.

4-

With A normal and H the self-adjoint element A*A, it follows from the preceding argument, together with Corollary 3.3.4 and the C* property of the norm, that llAllZ= IIA*AII = @*A)

< r(A*)r(A)= r(A)'

6 IIAl12;

so r ( A ) = IlAll. (ii) With A self-adjoint in 2l, sp(A) is compact (Theorem 3.2.3) and so contains a scalar with absolute value r(A);and r(A) = 114,11, from part (i) of the present proposition. Consequently, if suffices to prove that sp(A) G 02. For this, suppose that c ~ s p ( A ) ,where c = a + ib. For each integer n, let B,, = A - a I + inbl, and observe that i(n

+ 1)b

=a

+ ib - a + inb E sp(B,,).

= li(n

+ l)b12 < [r(B,,)]' < 11B,,112

Accordingly

+ + 1)b'

(nZ 2n

= IIB,*Bnll= " ( A - a l - inbI)(A - a l + inbI)ll = ll(A - a1)'

+ n2b2111< 11-4 - alllZ + n2bz.

Thus (2n + l)b2< IIA - aIllZ(n= 1,2,. . .); so b = 0 , and c = ~ E R .

239

4.1. BASICS

(iii) With A unitary in a,it results from the C* property of the norm that IlAll' = llA*All = 1 14 = 1,

so llAll = 1. With a in sp(A), we have a-'Esp(A-')

= sp(A*);

hence la1 < IlAll = 1,

1aI-l

< IIA*II = I ,

and thus Jal= 1. 4.1.2. COROLLARY. I f A isa normalelement of a C*-algebra a,andAk = 0 for some positive integer k , then A = 0. llAll

Proof. Since A" = 0 when n 2 k , it results from Proposition 4.1 .I(i) that = r(A) = lim (IA"((""= 0.

Suppose that A is a self-adjoint element of a C*-algebra a,and denote by C(sp(A)) the C*-algebra of all continuous complex-valued functions on the spectrum sp(A). We now introduce the (continuous) function calculus for A , a mapping that associates with each f in C(sp(A)) an element f ( A ) of a.The existence and properties of this mapping are the subject of Theorems 4.1.3, 4.1.6, and 4.1.8(ii) and Propositions 4.1.4 and 4.2.3(i) below. At a later stage (Theorem 4.4.5) we shall construct a similar function calculus for a normal element of a C*-algebra. For self-adjoint elements, the two methods lead to the same function calculus; Remark 4.4.6, Theorem 4.4.8, and Example 4.4.9 provide information that, even in the self-adjoint case, is not contained in the other results just cited. 4.1.3. THEOREM.I f A is a self-adjoint element of a C*-algebra a,there is a unique continuous mapping f -,f ( A ): C(sp(A)) -, such that (i) f ( A ) has its elementary meaning when f is a polynomial. Moreover, when f,g E C(sp(A)) and a, b E @, (ii) Ilf(A)II = llfll; (iii) (af bg)(A) = a f ( A ) bg(A); (iv) ( f g ) ( A )= f (A)g(A); (v) f(A) = [ f ( A ) ] * ,wherefdenotes the conjugate complex function; in particular, f ( A )is self-adjoint ifand only i f f takes real values throughout sp(A); (vi) f ( A ) is normal; (vii) f ( A ) B = Bf(A) whenever B E % and A B = BA.

+

+

Proof. By Proposition 4.1.1(ii), sp(A) is a compact subset of the real line. The Weierstrass approximation theorem shows that the set P of all

240

4. ELEMENTARY C'-ALGEBRA THEORY

polynomials with complex coefficients, considered as a subset of C(sp(A)), is everywhere dense. If p is such a polynomial, say p ( t ) = a,

+ a,t + a2t2+ . . + ant", *

then p ( A ) = a , l + a,A [ p ( A ) ] * = rioz

+ a2A2+ . . . + a , # ' ,

+ &A +

&A2

+ . + ii,A". * '

Hencep(A) and [ p ( A ) ] * commute; that is, p ( A ) is normal. From the spectral mapping theorem for polynomials (see Proposition 3.2. lo), together with Proposition 4.1. I(i),

IIP(A)ll = r ( p ( 4 ) = max{lsl:sEsP(P(A))) = max{Ip(t)l: tESP(41 = IIPII

(the norm o f p as an element of C(sp(A))). If two distinct polynomialsp and q are identically equal on sp(A), we can replacep byp - q in the above argument, and deduce thatp(A) = q ( A ) ;of course, this question arises only when sp(A) is finite. The preceding argument shows that the linear mappingp + p ( A ) : P + % is well defined and continuous (in fact, isometric). Since % is complete and P is everywhere dense in C(sp(A)), there is a unique extension to a continuous mapping f +f(A) : C(sp(A)) -+ 2I. We have now proved the existence of a unique continuous mapping f +f(A) satisfying condition (i) in the theorem. In view of the above argument, each of the remaining properties (ii)-(vii) is easily verified when f and g are polynomials, and by continuity remains valid for all f and g in C(sp(A)). H Clauses (i)-(v) of Theorem 4.1.3 amount to the assertion that the function calculusf+f(A): C(sp(A)) + 2I is an isometric * isomorphism that carries the identity mapping on sp(A) to the self-adjoint element A of %. Since C(sp(A)) is a complete metric space, the same is true of its image {f(A) :fE C(sp(A))j in N, so this set is an abelian C*-subalgebra %(A) of %, containing I and A. Since polynomials form an everywhere-dense subset of C(sp(A)), each element of %(A) is the limit of a sequence of polynomials in A. A closed subalgebra W of % that contains I and A necessarily contains all polynomials in A and therefore contains %(A). We have now proved the following result. 4.1.4. PROPOSITION. ZfA is a selfadjoint element of a C*-algebra 2I, the set { f( A ) :fE C(sp(A))} is an abelian C*-subalgebra%(A) of 2I, and is the smallest closed subalgebra of 'ill that contains Iand A . Each element of U ( A )is the limit of' a sequence of polynomials in A .

4.1. BASICS

24 1

Suppose that 2I is a complex Banach algebra, W is a closed subalgebra that contains the unit I of %, and BE99.If a E spa(B), then a l - B has no inverse in 2I; accordingly, it has no inverse in a,so aEspg(B). Hence spa(B) s spa(@, and Example 3.2.19 shows that strict inclusion can occur. By use of the function calculus described in Theorem 4.1.3, we now show that the two spectra coincide when 2I and $3 are C*-algebras. 4.1.5. PROPOSITION. r f % is a C*-algebra, W is a C*-subalgebra of 2I, and Bc.99, then spa(B) = spa(B). Proof. As noted above, spa(B) c spa(@. In order to establish the reverse inclusion, it suffices to prove the following result : if A E W,and A has an inverse A - ' in 2I, then A - ' E ~ . We consider first the case in which A is self-adjoint. Since O#spa(A), the equationf(t) = t - defines a continuous function on spa@). By means of the function calculus for A relative to a, we obtain an element f(A) of 2I, and deduce from Proposition 4.1.4 that f ( A ) e W . Since t f ( t ) = 1 for each t in spa(A), it follows from Theorem 4.1.3(i) and (iv) that Af(A) = I; so A-' = f ( A ) E B . Consider next a (not necessarily self-adjoint) element A of W that has an inverse Cin 2I. Then A* lies in W and has inverse C* in 9. Since A*A is a selfadjoint element of W,with inverse CC* in %, it follows from the preceding paragraph that CC* E B. Accordingly, A - = C = (CC*)A* E 3. In the circumstances considered in Proposition 4.1.5, we can now omit the suffices 2I and W,and denote by sp(B) the spectrum of B relative to either algebra. From the preceding proposition, if A is an invertible self-adjoint element of a C*-algebra a, then A - ' is the norm limit ofp,(A), with eachp, a polynomial (for this, in the proposition take for W the C*-subalgebra generated by A and I).This can be strengthened as follows. If A is self-adjoint and has inverse A - ' in 'u, then there is a sequence {p,} of polynomials without constant terms such that IIpn(A)- A - ' " -,0. To see this, extend t - ' on sp(A) to a continuous function f on an interval containing sp(A) and 0, so that f ( 0 ) = 0. From the Weierstrass approximation theorem, f is the norm (uniform) limit of polynomials q, on this interval. Of course, q,(O) +f(O) = 0. Thus the polynomials pn obtained from qnby omitting the constant terms (q"(0))tend tofin norm on this interval (hence on sp(A)). From (i) and (ii) of Theorem 4.1.3, IIp,(A) -f(A)II -,0, and from (i) and (iv) of that theorem,f(A) = A - ' . Our next result is another spectral mapping theorem (compare Theorem 3.3.6).

'

'

4.1.6. THEOREM.If A is a self-adjoint element of a C*-algebra 2I, and f E C(sp(A)), then SP(f(4) = { f ( t ) :tESP(4).

242

4. ELEMENTARY C*-ALGEBRA THEORY

Proof. In view of Propositions 4.1.4 and 4.1.5, we can interpret sp(f (A)) as the spectrum of f ( A ) relative to the C*-subalgebra %(A) = M A ) :9 E C(SP(A))}

of 9l. Since the function calculus g + g ( A ) : C(sp(A)) --f 9l is a * isomorphism from C(sp(A)) onto %(A), it follows that sp(f (A)) coincides with the spectrum off as an element of C(sp(A)); that is, by Remark 3.4.4, SP(f(4) = I f

(0:t E SP(A)}.

We conclude this section with some further applications of function calculus. 4.1.7. THEOREM.Each element A of a C*-algebra 2l is a finite linear combination of unitary elements of 9l.

Proof. It is sufficient to consider the case in which A is self-adjoint and IlAll < 1. In these circumstances, sp(A) is a subset of the interval [ - 1,1], and Since we can define f in C(sp(A)) by f (t) = t id=.

+

trf(0 +m1,

mf

f(t>T(t)= (0 = 1, for each t in sp(A), it follows that the element f(A) (= U ) of 9l satisfies t=

A=t(U+U*),

UU*=U*U=Z.

H

Suppose that 9l is a C*-algebra, A = A* ~ 9 l and , f is a continuous complex-valued function whose domain includes sp(A). We denote byf (A) the element of % that, in the function calculus for A, corresponds to the restriction flsp(A). This convention is used in Theorem 4.1.8(ii), where we refer tof (cp(A)) although the domain off may be strictly larger than sp(cp(A)). 4.1.8. THEOREM.Suppose that 9l and 9 are (?-algebras homomorphism from 9l into a.

and cp is a

*

(i) For each A in 9l, sp(cp(A)) E sp(A) and llcp(A)ll < IlAll; inparticular, cp is continuous. (ii) If A is a selfladjoint element of 9l a n d f e C(sp(A)), then cp( f ( A ) )= f (cp(A))* (iii) Zfcp is a * isomorphism, then llcp(A)ll = IlAll and sp(cp(A)) = sp(A)for each A in 9l, and cp(%) is a C*-subalgebra of 93.

Proof. (i) The unit elements of 9l and 9will both be denoted by Z,since the context in each case indicates which one is intended. With A in 9l, we prove first that sp(cp(A)) E sp(A). For this, if a $ sp(A), a1 - A has an inverse S i n %; since cp(Z) = Z, aZ- q ( A ) has inverse cp(S) in 9,so a$sp(cp(A)); hence SP(cp(4) E SPU).

243

4.1. BASICS

With A in 2l, it results from Proposition 4.1.1(i) that (IA1I2= IIA*AII

(1)

llcp(41I2

= r(A*A),

= Ilrp(A)*cp(A)II = lIcp(A*A)lI = r(cp(A*A))*

Since sp(cp(A*A)) E sp(A*A), we have r(cp(A*A))< r(A*A), and therefore IIcp(A)II < IlAll. (ii) If { p n }is a sequence of polynomials tending to f uniformly on sp(A) (hence, from (i), on s ~ ( d A ) ) ) ,then d P n ( A ) ) d f( A ) ) and p n ( d A ) ) f(cp(A)).Since cp(p,(A)) = p,(cp(A)) for each n (because cp is a homomorphism), (ii) follows. (iii) Suppose now that cp is a * isomorphism. With B self-adjoint in %, it follows from (i) that sp(cp(B)) G sp(B). If strict inclusion occurs, there is a nonzero element f of C(sp(B)), whose restriction to sp(cp(B)) is identically zero. From part (ii) of the theorem, we have +

f (4f 09

cp(f

+

(4) = f (cp(B))= 0,

contrary to the assumption that cp is one-to-one. Hence sp(cp(B)) = sp(B), and so r(cp(B))= r(B), for each self-adjoint B in 2l. With A in 2l and B = A*A, it follows from the preceding paragraph and (1) that IIAIIZ = r(A*A) = r(cp(A*A))= llcp(A)1I29

Ilcp(4II = IlAll.

Since 2l is a complete metric space and cp: % + W is an isometry, the * subalgebra cp(2l) of W is closed, contains I, and is therefore a C*-subalgebra of W. By Proposition 4.1.5, the spectrum of cp(A) in W is the same as its spectrum in cp(%); and sp(A) = spV($)(cp(A)),since cp is an isomorphism from 2l onto

cp(W The following result strengthens the final conclusion of Theorem 4.1.8(iii). 4.1.9. THEOREM.If% and W are C*-algebras and cp is a * homomorphism from 2l into W,then cp(2l) is a C*-subalgebra of W. Proof. Since cp(2l) is a * subalgebra of W (containing I),it suffices to show that cp(2l) is closed in W. Accordingly we must prove that, if B E B and IIB - cp(A,)II + 0 for some sequence { A n }of elements of 2l, then B~cp(%). By expressing B, A ,A 2 , . . . in terms of their real and imaginary parts, we reduce to the case in which B and the An's are self-adjoint. Upon passing to a subsequence of { A n } ,we may suppose also that llq(An+l)

- cp(An)II < 2-"

(n = 1,2,**.).

Let fn be a continuous function on R, with values in the real interval [-2-",2-"], such thatf,(t) = t when It1 < 2-". From Theorem 4.1.8(ii), and

244

4. ELEMENTARY C*-ALGEBRA THEORY

sincef, restricts to the identity mapping on sp(cp(A,+,) - &I,, we )) have , d A n t 1)

- d A n ) =fn(q(An+

1

- An)) = dfn(An+ 1 - An)).

+

SinceIlfn(An+l- An)ll < 2-", theseriesA, C,"=lfn(An+l - AJconverges to an element A of a; and by continuity of cp (Theorem 4.1.8(i)),

{

m- 1

c lim { d A d + c

c p ( ~ ) = lim

m- m

V ( A1)

+

p(fn(An+1 - An))}

n= 1

m- 1

=

m-r m

Ccp(An+l)

- r(A.)l}

n= 1

= lim cp(A,) = B. m-

00

Thus BE(P(%). H At later stages (Corollary 4.2.10 and Theorem 10.1.7) we shall show that a closed two-sided ideal X in a C*-algebra is automatically self-adjoint, and that the quotient algebra % / Xis a C*-algebra. With the aid of the latter result, Theorem 4.1.9 becomes a simple consequence of Theorem 4.1.8(iii) (see the proof of Corollary 10.1.8).

4.2. Order structure In Section 2.4 we introduced a concept of positivity for operators acting on a Hilbert space, and a partial order relation on the set of self-adjoint operators. In this section we study the corresponding notions for elements of a C*algebra. We recall that a bounded linear operator A, acting on a Hilbert space X, is said to be positive if (Ax, x) 2 0 for each x in X. By Proposition 2.4.6(i) and Theorem 3.2.l4(ii), a positive operator A is self-adjoint, and its spectrum is a subset of the non-negative half-line R + = { Z ER: t 2 0). Conversely, if A is a self-adjointelement o f g ( X ) , andsp(A) E R+, theequationf(2) = t l ' Z defines a real-valued continuous function f on sp(A). By means of the function calculus, for A as a member of the C*-algebra B ( X ) ,we obtain a self-adjoint operator H ( = f ( A ) ) such that H Z= A. Since ( A x , x ) = ( H x ,H x ) 2 0

(X€X),

A is a positive operator. Accordingly, the positive operators are precisely those that are self-adjoint and have spectrum contained in R+. Motivated by these considerations, we describe an element A of a C*algebra 2l aspositioe if A is self-adjoint and sp(A) E R+ ;we denote by at the set of all positive elements of a. From the preceding discussion, this definition

245

4.2. ORDER STRUCTURE

is consistent with our earlier conventions when 'u = 9-3(%). If 9-3 is a C*subalgebra of 'u, a self-adjoint element B of 9-3 is positive relative to 9-3 if and only if it is positive relative to 'u (that is, 9?+ = 9n a+),since it has the same spectrum in 9-3 as in 2l. If cp is a * homomorphism from 'u into a C*-algebra V and A €a+,then c p ( A ) ~ % ? +for ; cp(A) is self-adjoint, and sp(cp(A)) c sp(A) G IW', by Theorem 4.1.8(i). From Proposition 4.1.1(ii), IIAIIES~(A> when A€%+. With X a compact Hausdorff space, andfin the C*-algebra C(X),fis selfadjoint if and only iffis real valued throughout X ; moreover, sp(s) = { f ( x ) : x € X } . Accordingly,fis positive (in the C*-algebra sense just defined) if and only if f ( x ) 3 0 for each x in X . 4.2.1. LEMMA. I f A is a self-adjoint element of a C*-algebra 'u, a E R, and a 2 llAll, then A € % + ifand only ifllA - all( < a. Proof. Since sp(A) c [ - a, a ] , and

IIA - all( = r(A - a l ) = sup It - a1 = sup ( a - t), fesp(A)

tesp(A1

it is apparent that IIA - all1 6 a if and only if sp(A) E R+. Clauses (ii), (iii), and (v) of the following theorem tell us that 'u+ is a (positive) cone (in the sense explained in the discussion preceding Definition 3 . 4 3 , in the real vector space of all self-adjoint elements of %. 4.2.2.

(i) (ii) (iii) (iv) (v)

THEOREM.

Suppose that 'u is a C*-algebra.

2l+ is closed in 'u. aAE'u+ $ A € % + and a E R + . A + B E % +i f A , B ~ ' u + . ABg'u' i f A , B ~ ' u +and A B = BA. I f A ~ 2 l +and - A € % + , then A = 0.

Proof. (i) From Lemma 4.2.1, 'u+ = { A € ' u : A = A* and [ [ A- llAlllll G 11A11},

whence '$isI+ closed (since the norm is continuous on 2l). (ii) If A E 'u+ and aE R+, then aA is self-adjoint, and sp(aA) = {at:tEsp(A)} G R+. (iii) If A , B E % ' , it results from Lemma 4.2.1 that IIA - llAll4l G llAll9

IIB - IlBlllll G

IlBll.

246

4. ELEMENTARY C'-ALGEBRA THEORY

Thus IIA

+ B - (IlAll + IIBll>Zll < ll4l + 11J41;

and from the same lemma (with a = IlAII + ( ( B (2( ((A + Bll), it follows that A +BE%+. (iv) With A, Bcommuting self-adjoint elements of %+, AB is self-adjoint since (AB)* = BA = AB. Since each of A, B, AB has the same spectrum in 2l as in the commutative C*-subalgebra generated by { I , A, B ) , it follows from Proposition 3.2.10 that sp(AB) E (st:s~sp(A),t~sp(B)} E R+. =

(v) If A, - A € % + , then A is self-adjoint and sp(A) E R+ n - R t ( 0 ) ;so JJAll= r(A) = 0.

4.2.3. PROPOSITION. Suppose that A is a self-adjoint element of a C*algebra %, and fE C(sp(A)).

(i) f(A) E %+ if and only i f f ( 1 ) 2 0 for each t in sp(A). (ii) ((AllZf A E %+. (iii) A can be expressed in the form A + - A -, where A', A - E %' and A + A - = A - A + = 0. These conditions determine A + and A - uniquely, and IlAll = max(llA+II,IP-II). Proof. (i) By Theorem 4.1.6, f(A) has spectrum { f ( t ) :t E sp(A)} ; so f takes non-negative values throughout sp(A) if f(A) E a'. Conversely, if f(t) 2 0 for each t in sp(A), then f(A) is self-adjoint (sincefis real valued) and has spectrum a subset of R+. (ii) Withfin C(sp(A)) defined byf(t) = J(Allk t, f takes non-negative values throughout sp(A). By (i), f(A) E %+ ; that is, IlAllZ k A E a+. (iii) With u, u + , u- the continuous real-valued functions defined, for all real t, by u(t) = I,

u ' ( t ) = max{t,O},

u - ( t ) = maxi-

t,O},

we have u = u+

- u-,

u+u- = u - u + = 0.

Since u(A) = A, we have A = A + - A-,

A + A - = A-A'

= 0,

where A + = u+(A) and A - = u-(A); moreover, A + , A - c % + , by (i). The supremum norms of u, u + , and u - , as elements of C(sp(A)), satisfy

llull = max{llu+ll,Ilu-lll~ so IIAII = max{llA+lI,IlA-113.

247

4.2. ORDER STRUCTURE

To prove the uniqueness clause of (iii), suppose that A = B - C , where B, C E N + and BC = CB = 0. Then A" = B"

+ (-

( n = 1,2,3,...),

C)"

and therefore p ( A ) = p(B) + p( - C) whenever p is a polynomial with zero constant term. There is a sequence {p,,} of such polynomials that converges to U + uniformly on sp(A) u sp(B) u sp( - C); and u + ( A ) = limp,(A) = lim[p,(B)

+ p,( - C ) ] = u+(B) + u + ( - C ) .

Since u + ( s )= s (sEsP(B)),

u+(t) = 0

(tEsp(- C)),

we have u+(B) = B , u + (- C) = 0. Thus B = u + ( A )= A + ,

C= B- A

=A+

- A =A-.

4.2.4. COROLLARY. Each element A of a C*-algebra combination of at most four members of N+.

is a linear

Proof. From Proposition 4.2.3(ii) or (iii), the real and imaginary parts of A can each be expressed as a difference of elements of a+.

Our next objective, achieved in Theorem 4.2.6, is to give a number of conditions equivalent to positivity for elements of a C*-algebra. For this purpose, we require the following preliminary result. 4.2.5. LEMMA.If% isa C*-algebra,A ~ N , a n d- A*AE%I+,then A = 0. Proof. Let A = H + iK, with Hand K self-adjoint in a. Since sp(H) c [w and sp(H2) = {Iz: tEsp(H)} G R+, it follows that H2 (and similarly P)is positive. Since AA* is self-adjoint, and sp( - AA*) E sp( - A*A) u ( 0 ) c R + by Proposition 3.2.8, - AA* is positive. Now A*A

+ AA* = ( H - iK)(H + iK) + ( H + iK)(H - iK) = 2H2 + 2 K 2 , A*A = 2H2 + 2K2 + (- AA*).

Since all three terms on the right-hand side of the last equation are positive, A*A (as well as - A*A) is positive and so A*A = 0, by Theorem 4.2.2(iii) and (v). Thus I(A1(2= JIA*AII= 0, and A = 0. 4.2.6. THEOREM.If% is a C*-algebra and A E a, the following conditions are equivalent: (i) A € % + . (ii) A = H 2 ,for some H in a+. (iii) A = B*B, for some B in a.

248

4. ELEMENTARY C*-ALGEBRA THEORY

When these conditions are satisfied, the element H occurring in (ii) is unique. If X is a Hilbert space and 2l is a C*-subalgebra of a(%)), the preceding three conditions are equivalent to

(iv) ( A x , x ) 2 0, for each x in X. Proof. If A E%+, the equation f ( t ) = t 1 / 2 defines a non-negative realvalued continuous function f on sp(A) ( E R+). With Hdefined asf(A), HE%+ and H2 = A. This shows that (i) implies (ii), and it is apparent that (ii) implies (iii). Suppose next that A = B*B, for some B in 3 .Since A is self-adjoint, it has the decomposition A + - A - described in Proposition 4.2.3(iii). With C defined as B A - , C*C = A - B * B A - = A - ( A + - A - ) A - = - ( A - ) 3 . Since A - E%+ and ( A - ) 3 has spectrum { t 3 :tEsp(A-)}, it follows that - C * C = (A-)3~21+.Fr~mLemma4.2.5,C= O ; S O ( A - )=~ Oand,sinceAis self-adjoint, A - = 0 by Corollary 4.1.2. Thus A = A E 2l+, and (iii) implies (0. Having proved the equivalence of (i)-(iii), we show next that, when A E a+, the element H in (ii) is unique. For this, suppose that K is any element of 2l+ satisfying K 2 = A, while (as above) H =f(A), wheref ( t ) = t1l2( t E sp(A)). Let { p n }be a sequence of polynomials converging tofuniformly on sp(A), and let qn(t)= pn(t2).Since sp(K) c R + and +

sp(A) = sp(K2) = { t2 :t E sp(K)}, it follows that lim qn(t)= lirn pn(t2)= f ( t 2 ) = t , n+ m

n+m

uniformly for t in sp(K). Hence

K = lim qn(K)= lim p n ( K 2 ) n-. m

n+ m

=

lim p,(A) =f ( A ) = H, n+ m

and the uniqueness assertion is proved. With 2l a C*-subalgebra of a(%), %+ = 2l n a(%)+. Accordingly, in proving the equivalence of (i) and (iv) in this situation, it suffices to consider the case in which 2l = a(&). The required result then amounts to the following assertion, already proved in the introductory discussion of the present section : if A E ~ ( % ) ,then ( A x , x ) 2 0 for each x in X if and only if A = A* and sp(A)s R+. When A E 2l+, the element H occurring in condition (ii) of Theorem 4.2.6 is A similar procedure called thepositive square root of A , and is denoted by

4.2. ORDER STRUCTURE

249

can be used to introduce an element A" of a+,for other real values of a. Withf, defined byf,(t) = t", is a continuous non-negative real-valued function on sp(A) when a > 0 (for all real a, if A is invertible). Note thatfa(t)fa(i) = f a + B ( t ) , fl(t) = t , andfo(t) = 1 when A is invertible, for all t in sp(A). With A" defined as fa(A), we have A"€%', A"AB= A"'B, A ' = A, and A' = Zif A is invertible. It follows easily that this definition of A" agrees with the elementary one when a is an integer; in particular, if A is invertible, its inverse is the positive element A (=f-I(A))of 2I. 4.2.7. COROLLARY. If 2I is a C*-algebra, A€%', B* A B E 2I'.

and BE%, then

Proof. This follows from Theorem 4.2.6, since B*AB = (A1/2B)*A1/2B.

rn Suppose that a,, is the real linear space consisting of all self-adjoint elements of a C*-algebra CU. Since the adjoint operation is norm continuous, %,,is closed in 2l, and is therefore a real Banach space. From Theorem 4.2.2, it is a partially ordered vector space with a closed positive cone a+.In the partial ordering on a,,, A < B if and only if B - A E 2l' ; and, of course, %+ = { A E ( U h : A 2 0 ) .

From Proposition 4.2.3(ii), - llAlll< A < IIAl(Iforeach A in a,,;in particular, therefore, I is an order unit for ah. Moreover

2 0, - a1 < A < aZ}, since in its function calculus A corresponds to the identity mapping i on sp(A) (c R), while I corresponds to the constant function 1, whence llAll

= inf{a: a

llAll = 1 1 ~ 1 1= sup{ldt)l: t€SP(A)J =inf{a: - a . 1 < z < a . 1).

Just as in the case of real-number inequalities, one can add inequalities between self-adjoint elements of 2I (because sums of positive elements are positive), multiply through by positive scalars (because positive multiples of positive elements are positive), and take limits (because %' is closed in a), while multiplication by a negative scalar reverses inequalities. Since a product of commuting positive elements is positive, it follows that AC < BC whenever A < B, C E ~ I ' and , C commutes with both A and B. This last condition is essential, since without it, AC and BC are not self-adjoint. The corresponding non-commutative result, that C*AC < C*BC whenever A < Band C E2I, is a consequence of Corollary 4.2.7. An element A of 2I is invertible if and only if A 2 alfor some positive real number a. Indeed, A 2 a1 if and only if A - ale a ', and since sp(A - a l ) = { t - a : t E sp(A)}, this occurs if and only if sp(A) E [a, m). Since sp(A) is a +

250

4. ELEMENTARY C'-ALGEBRA THEORY

compact subset of R + , sp(A) G [a, co) for some positive a if and only if 0 $ sp(A) (equivalently A is invertible). Suppose that A and B are self-adjoint elements of a 4.2.8. PROPOSITION, C*-algebra %.

(i) r f - B < A < B, then llAll < IlBll. (ii) ZfO < A < B, then A'/' < B'". (iii) If0 < A < B and A is invertible, then B is invertible and B - ' < A Proof.

'.

(i) Since -

lIB((Z< - B < A

< B < IIBllZ,

(i) follows. (ii) and (iii) Suppose that 0 < A < Band A is invertible. Then A >, a l , for some positive real number a ; hence B >, aZ, and so B is invertible. Moreover 0 B - 1/2AB- 1 / 2 < B - 1/2gg- 112 = 1, and IJB-'/ZAB-'/211< 1 by (i). Thus (1) IIA '/2B- 11 = II(A1/2B- 1/2)*A1/2B-1/2111/2 = JIB-1/ZAB- 112 IIl / z

< 1.

From this, JIAI/ZB- 1A1/211 = IIA1/2B-I/2(A1/2B-1/2)Il.1, *
IIp[I - E . Thus [lpll < sup{p(H): H = H * E A ,llHll s l}, and the reverse inequality is evident. A linear functional p on A is said to be positive if p ( A ) 2 0, for each A in A + ;if, further, p(Z) = 1, p is described as a state of A. A positive linear k A) 2 0 since functional p is hermitian; for if A = A* E A, then p(((A((Z IJAllZk A E +,and p ( A ) is real because p ( A ) = :Cp(llAIII

+ A ) - p(llAlll- 41.

The real vector space Ah,consisting of all self-adjoint elements of A, is a partially ordered vector space, with positive cone A and order unit I. A linear functional p on A is hermitian if and only if its restriction p I A h is a +

256

4. ELEMENTARY C'-ALGEBRA THEORY

linear functional (of course, real-valued) on A,,;and each linear functional on Ahextends, uniquely, to a hermitian linear functional on A. Moreover, p is positive (or a state of A)if and only if p l A his positive (or a state of A,,)in the sense of Definition 3.4.5. The positive linear functionals on A form a cone P, in the real vector space consisting of all hermitian linear functionals on 4 (9n - 9 = {O},because A is the linear span of A +).Hence there is a partial order relation on the hermitian linear functionals; p1 < p 2 if and only if p 2 - p1 is positive. With X a Hilbert space and x in 2,the equation w x w = (Ax,x )

( A € -%XN

defines a linear functional w, on a(%). In view of the equivalence of two concepts of positivity for Hilbert space operators (conditions (i) and (iv) in Theorem 4.2.6, with 8 = a(X)), w,(A) 2 0 whenever A E W ( X ) + .Since, also, w,(I) = 11x112,it follows that w, is a positive linear functional on a(%), and is a state if llxll = 1. If % is a C*-subalgebra of a(#),and (as usual) JZ is a selfadjoint subspace of % that contains I, the restriction w x l Ais a positive linear functional on A.The states of A that arise in this way, from unit vectors in 2, are termed vector states of A. 4.3.1. then

PROPOSITION.

I f p is apositive linear functional on a C*-algebra %,

Ip(B*A)I2 < p(A*A)p(B*B)

( A , B E 8).

Proof. With A in 8,we have A*A E 8 +and , therefore p ( A * A ) b 0. From this, and since p is hermitian, the equation ( A , B ) = p(B*A)

( A ,B E % )

defines an inner product ( , ) on 8, and we have the Cauchy-Schwarz inequality

I ( 4 B>I2< ( A , A ) ( & B ) ; that is, )p(B*A)I2< p(A*A)p(B*B). W We refer to the inequality occurring in Proposition 4.3.1 as the Cauchy-Schwarz inequality for p. This inequality appeared, in the case of the C*-algebra C(X), at the end of Remark 3.4.9 and, again, in Remark 3.4.12. 4.3.2. THEOREM. If A is a selj-aa'joint subspace of a C*-algebra % and contains the unit I of 8,a linear functional p on A is positive if and only if p is bounded and llpll = p ( l ) . Proof. Suppose first that p is positive (and therefore hermitian). With A in A,let a be a scalar of modulus 1 such that ap(A) 2 0, and let H be the real part

257

4.3. POSITIVE LINEAR FUNCTIONALS

b(4I= p ( a 4 =

= p(dA*)

+ dA*)) = p ( H ) 6 p(l)llAll. This shows that p is bounded, with (IpIJ< p ( l ) ; and the reverse inequality is = p(f(aA

evident. Conversely, suppose that p is bounded and llpll = p(Z); it suffices to consider the case in which l(pll = p ( I ) = 1. With A in & +,let p ( A ) = a + ib, where a and b are real. In order to prove that p is positive, we have to show that a 2 0 and b = 0. For small positive s, sp(Z - sA) = {l - s t : tESP(A)} E [ O , 11, since sp(A) E Iw ; so (II - sAll = r(I - sA) < 1. Hence +

1 - su < 11 - S(U

+ ib)( = ( p ( I - sA)I < 1,

and therefore a 2 0. With B, in A! defined as A - a I + inbl, for each positive integer n,

11Bn,112= IlS,*B,((= ll(A - al)'

+ n2b2Z((< (IA - a1(I2+ n2b2.

Hence (n2

+ 2n + l)b2 = Ip(Bn)12,< JIA - aIllZ+ n2b2

(n = 1 , 2 , . . .),

and thus b = 0. H From Theorem 4.3.2, each state p of A is a bounded linear functional on A!, with llp[l = 1. Accordingly, the set Y ( M )of all states of A! is contained in the surface of the unit ball in the Banach dual space A'. It is convex and weak* closed, since Y ( A )= { p E & " : ( l )

= 1 , p(A) 2 0 (A€&+)},

and is therefore weak* compact, by Corollary 1.6.6. It follows that Y(A!),with the weak* topology, is a compact Hausdorff space, the state space of A!. 4.3.3. PROPOSITION. rf'u is a C*-algebra with unit I, A is a self-adjoint subspace of Cu containing I , A E .A',anda E sp(A), then there is a state p of A such rhar p ( A ) = a . Proof. For all complex numbers b and c, ab + cEsp(bA + c l ) , and therefore lab c ( < ((bA clll. Accordingly, the equation po(bA + c l ) = ab + c defines (unambiguously) a linear functional p o on the subspace

+

+

258

4. ELEMENTARY C*-ALGEBRA THEORY

+

{bA cI: b, C E C } of A, and p o ( A ) = a , p o ( I ) = 1 , llpoll = 1. By the Hahn-Banach theorem, po extends to a bounded linear functional p on A, with llpll = 1 (= p ( I ) ) . From Theorem 4.3.2, p is positive (and is therefore a state); and p ( A ) = a.

4.3.4. THEOREM.Suppose that W is a C*-algebra with unit I, A? is a sevadjoint subspace of '9l containing I, and A E A . (i) I f p ( A ) = 0, for each state p of A, then A = 0. (ii) I f p ( A ) is real, for each state p of A, then A is self-adjoint. (iii) If p(A) 2 0, for each state p of A, then A E A + . (iv) I f A is normal, there is a state p of &'such that Ip(A)I = IlAll. Proof.

(i) Suppose first that A is self-adjoint and p ( A ) = 0 for each state

p of A. From Proposition 4.3.3, sp(A) = {0}, so IlAll = r(A) = 0, A = 0.

Next, let A = H + iK, with Hand Kself-adjoint in A. If p ( A ) = 0, for each state p of A, then p(H) = p(K) = 0, since p ( A ) = p(H) + ip(K)and p(H) and p ( K ) are real. From the preceding paragraph, H = K = 0, whence A = 0. (ii) If p ( A ) is real, for each state p of A, then

p ( A - A*) = p ( A ) - p ( A ) = 0, and A - A* = 0 by (i). (iii) If p ( A ) > 0, for each state p of A, then A is self-adjoint by (ii), sp(A) E R + by Proposition 4.3.3, and so A E A + . (iv) If A is normal, r(A) = 11A11, so sp(A) contains a scalar a such that la1 = IlAll. By Proposition 4.3.3, a = p(A), for some state p of A, and then Ip(A)I

=

IlAll.

Our next objective is to prove a non-commutative analogue of the Hahn-Jordan decomposition for linear functionals on C(X) (Proposition 3.4.11 and Remark 3.4.12). We show in Theorem 4.3.6 that every hermitian functional on A can be expressed (in a unique optimal manner, when A is the whole of '9l) as a difference of positive linear functionals. For this purpose, we require the following lemma, which will be needed again later when we characterize those subsets of the state space Y ( A )that retain some of the properties of Y ( A )set out in Theorem 4.3.4. With X a subset of the Banach dual space A', we write E ( X )for the weak* closed convex hull of X.

4.3.5. LEMMA.Suppose that W is a C*-algebra with unit I, A? is a seuadjoint subspace of W'containing I , and % is a set of states of A. If IlHll = sup{lp(H)I: P E % } ? for each self-adjoint H in A, then =(Yov - Yo)is the set of all hermitian functionals in the unit ball of An.

259

4.3. POSITIVE LINEAR FUNCTIONALS

Proof. The set of all hermitian functionals in the unit ball (An)l is convex and weak* closed, and contains You - Yo;so it contains Co(.u?, u - Yo). We have to show that the two sets coincide. Suppose the contrary, and let po be a hermitian functional on A, such that llpoll < 1, po#C5(% u - %). By the Hahn-Banach theorem (Corollary 1.2.12), and since the weak* continuous linear functionals on A narise from elements of Jt (Proposition 1.3.5), there is an A in .I, and a real number a, such that

Re p ( A ) < a

Re p o ( A ) > a,

( p ~ E 5 ( %u - %)).

With H the real part of A, p(H) = : C P W

+ P(A*)l

=

Re P ( 4 9

for every hermitian functional p on A ; so p(H) < a

p o w > a,

(PE=(%

u - %)).

Thus Ip(H)I ,< a ( p E %), and a

a contradiction.

-= p 0 ( W < IlHll = sup{lp(H)I: p€%I < a, H

4.3.6. THEOREM.If% is a C*-algebra with unit Z and A is a seljladjoint subspace of CU containing I , each bounded hermitian functional p on A can be expressed in the form p - p - , where p + and p - are positive linearfunctionals = )Jp+JI llp-11. If A is the whole of %, these conditions on Jll and JJp1I determine p + and p - uniquely. +

+

Proof. We may assume that llpll = 1. With Y the state space of A,

l l 4 = suPo7(4l: .r E 9 1 for each self-adjoint A in A, from Theorem 4.3.4(iv) and since 11711 = 1 when ~ €By9 Lemma 4.3.5, p ~ E 5 ( 9 u- 9). A straightforward calculation shows that the subset { a o - b 7 : o , t ~ Y : a , b ~ R + , a + b1)=

ofC5(Y u - 9) is convex. It contains Y u - Y: and is weak* compact since it is the range of the continuous mapping (o,7 , a ) -,ao - (1 - 4 7 :

9 x 9'x [O,l]

+ An.

Accordingly, it is the whole of C5(Y u - 9). From this, and the preceding paragraph, p has the form ao - b7, with p and 7 in Y: a and b in R and a + b = 1. With p + and p - the positive linear functionals ao and b7, respectively, p = p + - p - and +

IIP+II + IIP-ll

=a

+ b = 1 = llpll.

260

4. ELEMENTARY C*-ALGEBRA THEORY

Suppose now that A = a. To prove the uniqueness of the decomposition of p, we assume that p = p - v = p‘ - v‘, where p, p’, v, v’ are positive linear functionals on 9l and

llpll + llvll = 11P:Il + llv’ll

=

llpll = 1.

Given E (> 0), choose a self-adjoint H in the unit ball of ‘illfor which p ( H ) > llpll - $ E ~ ,and let K = :(I - H).Then 0 < K < I, p(I)

+ v(l)

= 11pIl

+ llvll = IIPII < p(H) + y

p(I - H )

= p(H)

- v(H) + fEZ,

p ( K ) + v(I - K ) < iEi.

+ v ( l + H ) < $&Z,

Since K , l - K Ea+,while p and v are positive linear functionals, 0 d v(I - K ) < $&? 0 < p ( K ) < $&2, With A in a, the Cauchy-Schwarz inequality gives

Ip(KA)12 = Ip(K’l2 K’/2A)IZ< p(K)p(A*KA) < $ E ~ ( I A ~ ( ~ , 9

Iv((l- K)A)12 < v ( I - K)V(A*(I - K ) A ) < $&211AJI2. From this, and a similar argument for p’ and v‘, we have

< tEllAll, I W - K)4I < t&IIAll.

< ;&ll~ll, Iv((l - KM)I < ;&ll4l,

IP’(K4

lP(KA)I

Since p - p‘ = v

- v’,

p ( A ) - $(A) = p(KA) - p’(KA)

+ v((Z - K ) A ) - v’((I - K ) A ) ,

and so Ip(A) - $(A)[ < 2&llAIJ.Since the last inequality has been proved for each positive E , it follows that p = p’, whence v = v’. 4.3.7. COROLLARY. Zf9l is a C*-algebra with unit I and A is a self-adjoint subspace of N containing I, each bounded linear functional on A is a linear combination of at most four states of A. Proof. Each bounded linear functional z on A has the form p (T bounded hermitian functionals. Hence

+ io, with

p and

z = p+ - p-

+ ia+ - ia-,

and each term on the right-hand side is a scalar multiple of a state of A. We shall later give an alternative proof of the existence of the decomposition p = p + - p - , by reduction to the case in which rU is abelian, and appeal to Proposition 3.4.11 (see Remark 4.3.12). The uniqueness clause of

26 1

4.3. POSITIVE LINEAR FUNCTIONALS

Theorem 4.3.6 fails, in general, if one deletes the assumption that A = % (see Exercise 4.6.22). Since the state space 9(A)of is convex and weak* compact, it has extreme points; indeed, by the Krein-Milman theorem, Y(&)is the weak* closed convex hull cO(9(A)) of the set 9(A)of its extreme points. Elements of 9(A)are termed pure states of A,and the weak* closure 9(A)is called the pure state space of A. In general, 9(A)is not a closed subset of An, and the pure state space then has elements that are not pure states. When X is a compact Hausdorff space, the pure states of the C*-algebra C ( X ) are precisely the non-zero multiplicative linear functionals (Theorem 3.4.7); and the set 9 of pure states is therefore weak* compact (Proposition 3.2.20). Thus 9 coincides with the pure state space, in this case. A linear functional p on A is a pure state if and only if its restriction PI&, , to the partially ordered vector space Ahconsisting of all self-adjoint elements of A, is a pure state of A,,in the sense of Definition 3.4.5. Indeed, this follows from similar assertions, concerning hermitian linear functionals and states, occurring in the discussion preceding Proposition 4.3.1. 4.3.8. THEOREM. Suppose that % is a C*-algebra with unit I , A is a sewadjoint subspace of % that contains I, and A E A . (i) I f p ( A ) = 0, for each pure state p of A, then A = 0. (ii) I f p ( A ) is real, for each pure state p of A, then A is seif-adjoint. (iii) If p(A) 2 0, for each pure state p of A, then A E A . (iv) I f A is normal, there is apure state po of A such that Ipo(A)I = IlAll. +

Proof. If p ( A ) = 0 (or p ( A )is real, or p ( A ) 2 0) for all p in 9(A), then the same is true for all p in 9(A),since every state is a weak* limit of convex combinations of pure states. In view of this, the first three parts of the theorem follow, at once, from the corresponding assertions in Theorem 4.3.4. Suppose now that A is normal. By Theorem 4.3.4(iv), there is a scalar c and a state t of A such that 4 4 )= c, (c( = 11A11. Let be the (weak* continuous) linear functional on Asthat takes the value p ( T ) at p, and let a be a complex number of modulus 1 such that t ( a A ) = IcI = IIAJJ.From Corollary 1.4.4,there is a po in 9(A)such that IlAll 2 Ipo(A)I 2 Re ape>

2 sup{Re a ( p ) : ~ E Y ( A ) } 2 Re

a(t)Re t ( a A ) =

=

IlAll.

4.3.9. THEOREM. If% is a C*-algebra with unit I , A is a self-adjoint ' is a subset of the state space 9(A),the subspace of % thal contains I, and 9 following four conditions are equivalent:

262 (i) (ii) (iii) (iv)

4. ELEMENTARY C*-ALGEBRA THEORY

I ~ A E and A p(A) > 0 for each p in 9& then A E A ' . IlHll = sup{lp(H)I: p € Y 0 } ,for each self-adjoint H in A. cO(Yo)= Y ( A ) . 9(A)E (%)- the weak* closure of % in A'.

Proof. With H self-adjoint in A, define a (< IlHll) by

a = sup{lp(WI : ~ € 3 3 , and note that

p(aZ k H ) = a

kp(H)2 0

(p~g).

If (i) is satisfied, then aZ k H E & + , - aI < H < aZ, hence IlHll < a, and so IlHll = a. Thus (i) implies (ii). Suppose next that (ii) is satisfied. With Y; defined a s s ( % ) , and (A')1the unit ball in A', we have YoE Y; c_ Y ( 4 )c_ (A')l,so (ii) remains true when Yois replaced by From Lemma 4.3.5, cO(Y; u - .yI) is the set of all hermitian functionals in (A')l;in particular, Y(&) E E(Y;u - Y;). The set

x.

{ao - b z : o , z ~ Y ; ,a , b E R + , a

+ b = 1)

contains Y; u - Y ; , inherits convexity and weak* compactness from Y;, and so coincides with cO(Y; u - 3 )(compare this with the proof of Theorem 4.3.6). Accordingly, each state p of 4has the form ao - br, with o and z in Y;, a and b in R +,and a + b = 1. Since 1 = p(Z) = ao(Z) - bz(Z) = u - b = 1 - 2b, we have b = 0, a = 1 and p = o E Y;. Hence Y ( A )E Y;, and since the reverse inclusion has already been noted, Y ( A )= Y; = =(%Po). Thus (ii) implies (iii). If =(Yo)= 9'(A), it follows from Theorem 1.4.5 that 9(A)E (Yo)-;so (iii) implies (iv). E (sP,)-. If A E 4 and p ( A ) 2 0 for each p in Finally, suppose that 9(4) Yo,the same is true for each p in (Yo)-(in particular, for each p in 9(A)), by weak* continuity of the mapping p -+ p(A). By Theorem 4.3.8(iii), A E A +,so (iv) implies (i). W 4.3.10. COROLLARY. I f H is a self-adjoint operator acting on a Hilbert space X, then IlHll = suP(l(Hx,x)l: XEX, llxll = 11.

If& is a self-adjoint subspace of a(#)containing Zand Yois the set of all vector states of A, then 9(A)c (Yo)-and Y ( A )= cO(Yo). Proof. If A E A and p(A) 2 0 for each p in Yo,then l: XE3E4 llxll = I > ,

for each self-adjoint H in A. Since the last conclusion applies, in particular, when A = a(%), the corollary is proved. W The formula for IlHll, in Corollary 4.3.10, can also be proved without reference to C*-algebra theory, by combining Lemma 3.2.13with Proposition 3.2.15. By a function representation of A on a compact Hausdorff space X , we mean a linear mapping (P: A -,qAfrom A into the C*-algebra C ( X ) ,such that qr is the constant function with value 1 throughout X , and qAE C ( X ) +if and only if A E A + . If, in addition, given any two distinct points x and y in X , there is an element A of A such that qA(x)# ( P ~ ( Y we ) , describe (P as a separating function representation. Two function representations, (P: A + C ( X ) and t,b: A -,C( Y ) ,are said to be equivalent if there is a homeomorphismffrom X onto Y , such that ( P ~ ( x= ) t,bA( f ( x ) ) , for each A in A and x in X . If (P: A?+ C ( X ) is a function representation of A, and XEX, the evaluation mapping p x : A + (P~(x) is a state of A, since it is a positive linear functional and p x ( l ) = q I ( x )= 1. Hence I(PA(x)l = lpx(A)I 6 llAll

( A E A ,

XEX)

and ll(PAl1

= SUP I(PA(x)l 6

llAll

( AE d ) *

*EX

For each self-adjoint Hin A,q H is a real-valued function, since ( P ~ ( x=) px(H) and px is hermitian. With c defined as II(PHII (PclfH(4

=c

* (PH(X)2 0

(XEX);

so qEI f H E C ( X ) +; hence c l 2 H E A +,- c l 6 H 6 cl, and therefore

IlHlI 6 c = ll(PHll 6 l l H l l . Thus (P maps self-adjoint elements of A, isometrically, onto real-valued functions. By expressing an element A of A in the form H + iK, with Hand K self-adjoint in A, it follows that (P preserves adjoints; moreover, since (PA

= (PH + I'(PK,

llAll 6

llHll + llKll

= ll(PHll

+ II(PKI1 6 211(PAll*

Accordingly,

:ll4 6 II(PAll 6 IlAll,

II(PHII =

IlHll

( A E A , H = H*E.m.

The set Jf = ((P, :A E A} is a self-adjoint subspace of the C*-algebra C(X), and contains its unit. Since (P is a one-to-one bicontinuous linear

264

4. ELEMENTARY C*-ALGEBRA THEORY

mapping from the normed space A onto the normed space Jy; its Banach adjoint cp': 7 + cp," is a bicontinuous linear mapping from N' onto A'. Since cpA is self-adjoint, or positive, if and only if A has the same property, while cp:(A) = 7(cpA) ( A E A , T E N 'it) ,follows that cpf is hermitian, or positive, if and only if the same is true of z. Since cp is isometric on self-adjoint elements of A,while the norm of a hermitian functional is unchanged by restriction to selfadjoint elements, cp' is isometric on the hermitian functionals in N'. We now exhibit some canonical function representations associated with a C*-algebra. For each A in A', the equation =

( P E Y(&)

defines acontinuous complex-valuedfunction A on the state space Y ( A )If. SP, is a closed subset of Y(&) and contains the pure state space P ( A ) - ,it is apparent that the restriction A 1% takes non-negative values throughout % if A E A +,and the converse assertion follows from Theorem 4.3.9. Since p(aA we have

n

+ bB) = ap(A) + bp(B),

( a A + b B ) = a2

+ bB

( A , B E A , a,bE@).

If p1 and p2 aredistinct elements of Yo,then we have p l ( A ) # p 2 ( A ) for some A in A ; that is, &pl) # A ( p 2 ) . Moreover, j ( p ) = p ( l ) = 1, for each p in SP,. Accordingly, the mapping A+A(SP,: A+C(%) The following theorem is a separating function representation of A on goPo. shows that every separating function representation of 4 is equivalent to one that arises in this way, by appropriate choice of %. The most important function representations of A are the two "extreme" ones, obtained from the above construction when Yois either the state space Y ( M )or the pure state space &'(A)-. 4.3.11. THEOREM. r f '2.l is a C*-algebra with unit I, A is a self-adjoint subspace of '2.l containing I, Xis a compact Hausdorffspace, and cp :Jf + C(X) is a separatingfunction representation of A, then there is a unique closed subset Yo of Y ( A ) ,such that &'(A)c SP, and cp is equivalent to the function representat ion

A - * A l S : A+C(%). Proof. For each x in X,the equation p,(A) = cpA(x)defines a state pr of A.Given two distinct points x and y of X , there is an element A of J2 for which cpA(x) # cpA(y), whence p,(A) # py(A) and so px # p y . If { x a } is a convergent net of elements of X , with limit x, it follows from the continuity of the function

4.3. POSlTlVE LINEAR FUNCTIONALS (PA

265

that

so pxm+ px in the weak* topology. From the preceding paragraph, the mapping

f: x + p x :

X+Y(A)

is one-to-one and continuous. Since Xis compact, the same is true of its range, X onto Yo.If A E A and p ( A ) 2 0 for each p in z,then

Yo= { px : x E X), and f is a homeomorphism from

(XEX); 20 so cpA€ C ( X ) + ,and therefore A E A ' . Since Yois closed, it now follows from Theorem 4.3.9 that P ( A ) - c Yo.Finally, (PAW

=p x ( 4

(XE X, A E 4, W ( x , >= 4 P x ) = P x ( 4 = c p A ( 4 and therefore cp is equivalent to the function representation

A-+C(%).

A+ai$:

To prove the uniqueness clause of the theorem, suppose that cp is equivalent also to the function representation A+&q

A+C(%),

hhere 3 is a closed subset of Y(&) containing 9(A)-.Let g : x + g x be a homeomorphism from X onto g,such that (PA(x) = a = (PA@) = P X W

Hence g x = p x for each x in X,and

y; = { g x :X E X }

=

{ p x :X E X }

=

z. rn

4.3.12. REMARK.We illustrate the use of function representations by giving an alternative proof of the existence of a decomposition of a bounded hermitian functional as a difference of positive linear functionals (Theorem 4.3.6), by reduction to the abelian case (Proposition 3.4.11). With cp : A + cpA a function representation of A on a compact Hausdorff space X, and Af the subspace { q A :A E A ) of C(X), we recall that cp has a Banach adjoint operator cpI: 7 -+ cp: from N aonto .A!'; that cp: is hermitian, or positive, if and only if 7 has the same property; and that cp' is isometric on hermitian elements of N 8 . In order to show that a bounded hermitian

266

4. ELEMENTARY C*-ALGEBRA THEORY

functional p on A can be expressed as p + - p - , where p + and p - are positive linear functionals and JIp+II+ IIp-ll = llpll, it now suffices to prove the corresponding statement for J1/: With p a bounded hermitian functional on N, p extends without increase of norm to a bounded linear functional 7 on C(X). We can suppose that 7 = 71 iz2, where 71, 7 2 are hermitian, and I(zjJId llzll ( j = 1,2). Since p and the restrictions 71 IJV and .rZ(JV are hermitian, while Upon replacing z by 71, we may p = (ti + i.t2)IXit follows that p = 71 assume that 7 is hermitian. By Proposition 3.4.1 1, 7 can be expressed as 7 + - 7 - , where 7 + and 7 - are positive linear functionals on C ( X ) , and Il~+ll Ilr-ll = IITII. With p+ and p - defined as 7 + 1N . and 7 - 1.V;respectively, p + and p - are positive linear functionals on Jv; and p = p + - p - . Moreover,

+

+

IIP+II + IIP-ll d 117+11 + 117-11 = 11711 = IlPll = IIP+ - P-ll SO

d IIP+II + IlP-II,

IIP+II + lip-11 = llpll.

We conclude this section with-some further results concerning states and pure states. 4.3.13. THEOREM. Suppose that 'u is a C*-algebra with unit I , JZ is a selfadjoint subspace of % that contains I, and p is a state of 4. For each self-adjoint H in 'u, defne I H = S U ~ { ~ ( B ) : B = B * E AB ', < H } , uH = inf{p(B) : B = B* E A, B > / H }.

(i) - IIHII d 1" d U H < IIHll ( H = H * E a). (ii) p extends to a state 7 of 'u. I f c is a real number and H is a self-adjoint element of 'u, the extended state 7 can be chosen so that 7 ( H ) = c if and only iJ 1H

< c < uH.

(iii) p extends uniquely to a state 7 of 'u ifand only $1, = uHfor each selfaa'joint H in %. (iv) I f p is apure state of A, then p extends to apure state of 2I. I f , further, p has only one extension as a pure state of 'u, then p has only one extension as a state of a. Proof. (i) With H self-adjoint in 'u,

l l H l I I ~ 4 , - IIHIII d H d IIHIII. If B = B * E A and B < H , then B d ((HI(Zand therefore p(B) = IlHll. Hence the set

< llHllp(I)

{p(B): B = B * E A , B < H } is bounded above by IlHll and (with B = - IIHllZ) contains - IlHll.

267

4.3. POSITIVE LINEAR FUNCTIONALS

Accordingly, its supremum lH satisfies - IlHll 6 lH 6 11Hl1; and a similar argument shows that (IHll< uH 6 IIHII. If B1 and B2 are self-adjoint elements of & and B1 6 H 6 B2, then p ( B , ) < p(B2). By allowing B1 and B2 to vary, subject only to the conditions just stated, it follows that I , 6 u H . (ii) With H self-adjoint in 'Lz, 1, 6 uH by (i), so we can choose a real number c such that 1, 6 c 6 uH. We shall prove that p has an extension to a state t of 8 such that t ( H ) = c. As a first step, we show that the equation

-

to(aH + A ) = uc + p ( A )

(1)

defines a positive linear functional A? = {uH

t oon

(a€@, A E A )

the self-adjoint subspace

+ A : U E C ,A E A }

generated by Hand A.This is evident when HE4, since in this case Jlr = A, 1, = uH = p(H), hence c = p(H); and to,as defined by (l), is p . We assume henceforth that H$ A, whence each element T of Jlr is uniquely expressible as aH A , with a in C and A in A.Accordingly, (1) defines a linear functional t o on J,'to(H) = c and 70 extends p ; in particular, zo(l) = p(Z) = 1. In order to show that t o is positive, suppose that T E N ' , and let T = aH + A , as above. Since

+

0 = T* - T = (U - a ) H

+ A* - A ,

(ii - a)H = A - A*€&!, and it follows that a is real. If a = 0, then T = A E A ' and t O ( T )= p ( A ) b 0. If a > 0, then H + a - ' A = a - ' T a 0, so - u - ' A E A ,- a - ' A 6 H ; from the definition of lH, - u-'p(A) 6 lH

( 6 c), and therefore

t,(T) = a[c + a-'p(A)] 2 0.

If a

-= 0, then - H - a - ' A = (-

a)-'T 2 0, so

- a-'A 2 H; -a-'AEA, from the definition of uH, - a - ' p ( A ) 3 uH ( 2 c), and therefore to(T) = - a [ - c - a - ' p ( A ) ] 2 0. The preceding enumeration of cases shows that zo(T)2 0 whenever T EN + , so t ois a positive linear functional on M. From Theorem 4.3.2, t o is bounded, and I(tOll= to(Z) = 1. By the Hahn-Banach theorem, t o extends without change of norm to a bounded linear functional t on 8 ;moreover, t is positive, again by Theorem 4.3.2, since T(l) = fo(1) = 1 = [ ( T o ( ]=

Accordingly, t is a state of 8, t(H)

= to(H) = c,

Il'Tll. and

t

extends p .

268

4. ELEMENTARY C'-ALGEBRA THEORY

Conversely,suppose that t l is any state of % that extends p. If B1and B2 are self-adjoint elements of A, for which B1< H < B2, then tI(Bl) < t l ( H ) < t l ( B 2 ) ;that is, p(B,) < tl(H) < p(B2). By allowing B1 and B2 to vary, subject only to the restrictions just stated, it follows that 1, < t l ( H ) < uH. (iii) This is an immediate consequence of (ii). (iv) Suppose now that p is a pure state of A, and define Yoto be the set {tEY(%)

: T(B) = p ( B ) ( B E d ) }

of all states of 2I that extend p. Then % is a closed convex subset of Y(U),is therefore weak* compact, and is non-empty by (ii). From the Krein-Milman theorem, = =(Po),the closed convex hull of the set Poof all extreme points . Pois not empty, and consists of a single element if and only if Yo of 9,Hence has just one element. It now suffices to show that each T in Po is a pure state of a. For this, suppose that z = a t l + (1 - a)z2, where tl, t 2E Y(%) and 0 c a c 1. Since the restrictions z1 IA, t 2! Aare states of A, while p is a pure state of d and p=TlA=a(Tll~@

+(I

-U)(TzId),

+

it follows that z l l A = z 2 l A = p . Thus t l , t 2 ~ Yand 0 ; since t ( = a t , (1 - a ) t 2 )is an extreme point of Yo,t 1= t 2 = T . Hence T is an extreme point of Y(9l);that is, T is a pure state of a. The results on extensions of states and pure states, set out in Theorem 4.3.13, remain valid in the context of a partially ordered vector space V with order unit I, and a subspace of V that contains I (see Exercise 4.6.49).

If% is a C*-algebra with center V and p is a pure 4.3.14. PROPOSITION. state of a, then p ( A C ) = p ( A ) p ( C )for all A in % and C in V. Moreover, the restriction plV is a pure state of V. Proof. In order to show that p ( A C ) = p ( A ) p ( C )when A E % and CEV, it suffices (by linearity) to consider the case in which 0 < C < I. In this case, for each H in %+, we have 0 < HC < H , and thus 0 < p ( H C ) < p ( H ) , since H commutes with C (see the discussion following Corollary 4.2.7). Hence the equation po(A) = p ( A C ) defines a positive linear functional p o on %, and p o < p. Since p is a pure state, so is its restriction PI%, to the partially ordered vector space a,,of all self-adjoint elements of (see the paragraph preceding Theorem 4.3.8). Since polah < PI%,,, it follows from Lemma 3.4.6 that pol%h= a(pl2I,J,for some scalar a. Hence p o = u p ; and P(AC) = P O W

for each A in

a.

= a p ( 4 = w ( O p ( A ) = PO(I)P('4 = p(C)p(A),

269

4.4. ABELIAN ALGEBRAS

From the preceding paragraph it follows, in particular, that the non-zero linear functional ~ 1 % ' is multiplicative on V; so the final assertion of the proposition follows from Proposition 4.4.1 oust below). H 4.4. Abelian algebras With X a compact Hausdorff space, C ( X ) is an abelian C*-algebra. Our main purpose in this section is to show that every abelian C*-algebra 2I is * isomorphic to one of the form C ( X ) . As a first step, we prove that the pure states of 2I are precisely the multiplicative linear functionals on 2I, a fact already noted in Theorem 3.4.7 for the algebra C ( X ) . 4.4.1. PROPOSITION. A non-zero linear functional p on an abelian C*algebra 2I is apure state ifand only i f p ( A B ) = p ( A ) p ( B )for all A and B in 2l.

Proof. The first assertion of Proposition 4.3.14 includes, as a special case, the fact that pure states of an abelian C*-algebra are multiplicative. Conversely, suppose that p is a multiplicative linear functional on 2I. By Proposition 3.2.20, p is bounded and llpll = p(1) = 1, so p is a state of 2l. In order to prove that pis pure, suppose that p = apl bp2,where pl, pz E 9(2I), a > 0, b > 0, and a + b = 1. With C self-adjoint in 2I,

+

Cpj(C)12= Cpj(IC)IZG pj(Opj(C2)= pj(C2) by the Cauchy-Schwarz inequality. Accordingly,

(i= 1,2),

0 = P(C2)- ")I2

+ bpz(C2)- Cap1(C) + bPZ(C)l2 >, a(a + b>CP1(C>I2 + b(a + b)CP2(C)l2- Cap1(C)+ bPZ(C)l2 = ap1(C2)

= abCp1(C) - PZ(C)l2. From this, p l ( C )= p2(C)for each self-adjoint Cin 2I;so p1 = p z , whence p is a pure state. H

The argument just given shows that a multiplicative linear functional on a (not necessarily abelian) C*-algebra is a pure state. This can also be proved by reduction to the case of algebras of the form C ( X ) and an application of Theorem 3.4.7. Indeed, if p = (1 - a)pl + ap2, we can show that p ( A ) = p l ( A ) = p 2 ( A ) ,for a given self-adjoint A in 2l,by restricting p, p l , and p2 to the C*-subalgebra generated by I and A , and identifying this algebra with C(SP(A)). 4.4.2. COROLLARY. The set 9(%) ofpurestates of an abelian C*-algebra 2I is a closed subset of the state space Y(2I).

270

4. ELEMENTARY C*-ALGEBRA THEORY

Proof. By Proposition 4.4.1,

9(%) = { p E Y ( % ) : p ( A B )= p(A)p(B) ( A , B E % ) } .

m

4.4.3. T H E O R E M . Suppose that % is an abelian C*-algebra, P('9l)is the set of allpure states of 'u, andfor each A in %, a complex-valuedfunction A is defined by &p) = p ( A ) . Then 9(%) is a compact Hausdorfs space, throughout 9(%) relative to the weak* topology, and the mapping A 4A is a * isomorphismfrom % onto the C*-algebra C(P(%)). Proof. A substantial part of the argument required to prove this theorem is already contained in the discussion of function representations of (not necessarily abelian) C*-algebras, preceding Theorem 4.3.1 1. Two new features that are special to the abelian case, those set out in Proposition 4.4.1 and Corollary 4.4.2, suffice to complete the proof. For the sake of clarity, the argument is presented below in unified form, even though this involves some repetition of the earlier discussion. From Corollary 4.4.2, P(%)is weak* compact. With A in %, it is apparent from the definition of the weak* topology that A is a continuous complexvalued function on P(%).For all A and B in %, a and b in @, and p in 9(%) n (aA + b B ) ( p ) = p(aA bB) = ap(A) + bp(B) = a&) + b&),

+

I&)l

= IP(A)I G

- -

IlAll,

A*(P)

= P(A*) = p

(4

=

m;

and ( m P ) = d A B ) = P(A)P(B)= since p is multiplicative, by Proposition 4.4.1. Since % is abelian, A is normal, so by Theorem 4.3.8(iv) there is a pure state p o of % such that Ipo(A)I = llAll. From this, &mP)9

IlAll = I 4 P o ) l


=

r r

=

{ f (f)(

/ r

U A Y )df,

where the Fubini theorem applies since (Igl* If l)lhl E Ll(R). Bibliography: [4, 191 4.6. Exercises Sz, T1,T2,and A are elements of a C*-algebra 2I, 4.6.1. Suppose that S1, and 0 < S 2 < T2. O < S1 < Ti,

Prove that llS:’2All G llT:’2AlI,

and deduce that llS;’ZAS;’211 < IIT;’2AT;’21(.

4.6.2. Suppose that 2I and 9are C*-algebras and cp is a * homomorphism , B self-adjoint and K positive, and from 2I onto L’d.Suppose that B, K E ~with

286

4. ELEMENTARY C*-ALGEBRA THEORY

let V be the exponential unitary expiB. Show that there exist A, H, U (= exp iA) in W, with A self-adjoint and H positive, such that cp(A) = B,

cp(H) = K ,

q(U)= V.

[See also Exercises 4.6.3 and 4.6.59.1

4.6.3. Suppose that D is the unit disk { Z E a3 :1zI < l}, U is its boundary {ZEC: JzI = l}, and S is the union {exp itl :tlE R, n/4 < 101 ,< 3 4 4 ) of two closed arcs in U. Consider the C*-algebras C(D), C(B), C ( S ) and the * homomorphisms cp (from C(D) onto C(U)) and 9 (from C(T) onto C ( S ) ) defined by restriction; that is,

0=flT

9(d = glS

(fEC @ ) , 9 E C(W*

Find

(i) a unitary element u of C(T) that is not of the form ~ ( ffor ) any invertible element f o f C(D); [Hint. Use the fact (from elementary algebraic topology) that there is no continuous mapping of D onto B that leaves each point of U fixed-that is, T is not a retract of D.] (ii) a projection q in C ( S )that is not of the form $ ( p ) for any projection p in C(U). [See also Exercise 4.6.59.1

4.6.4. Determine whether the following assertion is true or false : if 2I and

a are abelian C*-algebras, cp is a * homomorphism from 'illonto W,and B is an invertible self-adjoint element of 99, there is an invertible self-adjoint element A of 2I such that q ( A ) = B.

4.6.5. Use the results of Exercises 4.6.2and 4.6.3(i)to provide an example of an abelian C*-algebra 99 and a unitary element Vof W that is not of the form expiB for any self-adjoint B in 99. 4.6.6. Let 42 be the (multiplicative) group of all unitary elements in a C*algebra a. (i) Show that, if U ~ 4 and 2 111 - Lrll < 2, then U = exp iH for some selfadjoint H in 2I. (ii) Show that, if V, WE% and IIV- WII < 2, then V = WexpiH for some self-adjoint H in 2I. (iii) Let ( E %) be the set of all products of the form (exp iH,)(exp iH2) * . * (exp iHk), where { H I ,H 2 , .. .,H k } is a finite set of self-adjoint elements of 21. Show that

287

4.6. EXERCISES

is open, closed, and arcwise connected, in the (relative) norm topology on 9. 4.6.7. Suppose that 9I is an abelian C*-algebra, 4 is its unitary group (considered as a topological group with the norm topology), 4, is the connected component of 9 that contains the identity I, and U E ~Use . the results of Exercise 4.6.6(iii) to show that the following three conditions are equivalent.

(i) U = expiA for some self-adjoint A in 2l. (ii) U is connected to I by a continuous arc in 4 . (iii) U E ~ , . 4.6.8. Show that, in both of the following cases, each unitary element in the C*-algebra 2l has the form expiA for some self-adjoint A in a.

(i) 2I is the C*-algebra L , associated with a o-finite measure space. (ii) 2l is the C*-algebra C(X), where the compact Hausdorff space X is contractible (that is, there is a point xo in X and a continuous mapping f : X x [0,1] + X such thatf(x,O) = x, f ( x , 1) = xo for each x in X).[Hint. Use the result of Exercise 4.6.7.1 4.6.9. (i) Show that, if a, b are complex numbers and P, Q are projections with sum I in a C*-algebra, then the function calculus for the normal element UP+ bQ (= N ) is given by

f ( N ) =f ( 4 P + f(b)Q. (ii) Suppose that U is the unit circle {ZEC: IzI = 1) and A is the algebra of all 2 x 2 complex matrices, so that & becomes a C*-algebra when identified in the usual way with the set of all linear operators acting on the twodimensional Hilbert space @'. The Banach space C(U,&) (see Example 1.7.2) becomes a C*-algebra when products and adjoints (as well as the linear structure) are defined pointwise. Let E and F be the projections in 2l given by E(eie)=

[i ]

1 -cos8 -sin0

,

-sin8 1 +cos8

1

(0 < 8 Q 2x), and let U be the unitary element (exp ixE)(exp ixF) of 2l. Show that U(e")

+ e-"Q

= eioP

(0 Q 8 Q 2x),

where P and Q are the projections in & given by p=!['2 i

-:I,

a=![2

-i

1i].

288

4. ELEMENTARY C*-ALGEBRA THEORY

Deduce that U , a product of two exponential unitary elements of a, is not itself an exponential unitary. 4.6.10. Let 'u be a C*-algebra and 9 be a norm-closed (though not necessarily self-adjoint) subalgebra of 2l. Let A be a self-adjoint element of 2l in 39. Show that spd(A) = sppI(A).[Hint. See Exercise 3.5.28.1 4.6.11. Suppose that A is a positive element of a C*-algebra a, E and F are orthogonal projections in 2l, and EAE = 0. Show that EAF = 0. 4.6.12. Suppose that a C*-algebra 2l has a maximal abelian * subalgebra XI that is finite dimensional.

(i) Show that d is the linear span of a finite orthogonal family { E l ,. . . ,En} of projections in 2l with sum I (the identity of 2l). (ii) By considering the family { E , ,. . . , En, E j A E j } ,where A = A* E 2l, show that Ej21Ej = {aEj :U E C} for j = 1,. . . ,n. (iii) Suppose that j , k E { 1,. . . ,n} and j # k. Given A and B in 55, let a,b, c, d be the scalars determined by EjA*EkAEj= aEj, EjA*EkBEj= bEj, EjB*EkBE, = CEj, EkBEjA*Ek = dEk. Prove that a 2 0, c 2 0. By considering suitable expressions for bdE,, bdEk, and acEj, show that b = d and ac = JbI2. Deduce that (SEkAEj

+ tEkBEj)*(SEkAEj + tEkBEj) = 0

for suitable scalars s and t (not both 0). Deduce that Ek21Ej is at most one dimensional. (iv) Prove that 2l is finite dimensional. 4.6.13. Suppose that 2l is an infinite-dimensional C*-algebra. By using the result of Exercise 4.6.12, show that there is an infinite sequence { A , ,A 2 , .. .} of non-zero elements of a+such that AjAk = 0 when j # k. 4.6.14. By using the result of Exercise 4.6.12, show that, in an infinitedimensional C*-algebra, there is a positive element with infinite spectrum. 4.6.15. Let p be a state of a C*-algebra 2l. We say that p isfaithfulif A = 0 when A E W and p(A) = 0. (i) Show that Y,,,the left kernel of p , is (0) when pis a faithful state of 2l. Deduce that Xp is the completion of 2l relative to the inner product ( A , B) .+ p(B*A) ( = ( A , B)), and that np is faithful. (ii) Let p be a state of 2l such that n,,is faithful. Must p be faithful?

4.6. EXERCISES

289

4.6.16. Let A be a self-adjoint element in a C*-algebra 2I. Let p be a state of 2I such that p(A2) = P ( A ) ~(We . say that p is definite on A in this case.) Show that p(AB) = p(BA) = p(A)p(B) for each B in A. 4.6.17. Suppose that '$I, d,and { E l ,. . . ,En}are as in Exercise 4.6.12. Let pj be a state of 2I that extends the state of d assigning 1 to Ej and 0 to Ekwhen k # j. Let p be n-' pj.

c;=l

(i) Show that p is a faithful state of 2I. (ii) Choose Ajk ill 'u such that (EjAjk&)*(EjAjkEk) = nEk for those k a n d j for which Ej21Ek is one dimensional. Show that the set of EjAjkEkforms an orthonormal basis for Xp. 4.6.18. Let X be a Hilbert space of finite dimension n, { e l , .. . , e n ] be an orthonormal basis for H,Ejkbe the element of g ( X )that maps ek to ej and ek, to 0 when k' # k,and p be an element of the Banach dual space @(X)'.Define A to be the element of W ( X ) with matrix representation [ajk] relative to { e l , .. . , e n } , where ajk = p(Ekj), and t to be the normalized trace ( t ( B ) = n-' Cj"= bjj = n-' tr(B), where B has matrix [bjk] relative to { e l , .. . ,e,,]). Show that: (i) z is a faithful state of a(%); (ii) p(B) = tr(A B) for each Bin g ( X ) ,and p is a state if and only if 0 < A and tr(A) = 1; (iii) A is independent of the basis { e l , .. .,en] ; (iv) p is a pure state of W ( X ) if and only if A is a projection with onedimensional range (in which case p = ox,where x is a unit vector in the range of A); (v) p is a faithful state of g ( 2 )if and only if tr(A) = 1 and A 2 aZ for some positive scalar a. 4.6.19. With the notation of Exercise 4.6.18, suppose p is a state and E is the projection in W ( X )with range the range of A. Show that the left kernel Y of p is ,%?(X)(Z - E). 4.6.20. Suppose that phism from 2I onto 58.

'$I and

B are C*-algebras, and cp is a

* homomor-

(i) Let B 1 and B2 be elements of 9' such that B I B l = 0. By considering B1 - B 2 ,show that there exist elements A l and A 2 of 2I' such that A l A 2 = 0, cp(A1) = B1, and cp('42) = B2. (ii) Suppose that { B 1 ,B 2 ,B 3 , . . . .} is a sequence of elements of 2 ' such that BjBk = 0 whenj # k. Prove that there is a sequence {A,, A 2 , A 3 , . . .} of elements of '$I' such that AjAk = 0 when j # k, and cp(Aj)= Bj for each

290

4. ELEMENTARY C*-ALGEBRA THEORY

j = 1,2,3,. . . . [Hint. Upon replacing Bj by bjBj for a suitable positive scalar

b j , we may assume that the series C Bj and C B ; / 2 converge to elements of a. Prove, by induction on n, the following statement: there exist elements A 1 , A 2 , . . . , A , , A', of 2l+ such that AjAk = A j X , = 0 for allj, k = 1,. . ,n with j # k, and v(A 1) = BI . . ., V(An) = Bn V(Xn) = Bn + 1 9

9

*

-1

are C*-algebras, cp is a * homomorphism

4.6.21. Suppose that 2l and from 2l onto 9, and AEN',

+ Bn + 2 +

BE^,

0 6 B 6 cp(A).

(i) Show that there is a self-adjoint element R of 2l such that R 6 A and cp(R)= B. Deduce that there exist positive elements S and T of 2I such that

cp(S) = B, (ii) For n

=

cp(T)= 0,

S6T

+ A.

1,2, . . . ,let

U, = S"'(n-'Z

+ T + A ) - ' ( T + A)1/2A1/2.

Show that U,*U,,< A ,

cp(U,) = B'I2[n-'Z

+ cp(A)]-'cp(A).

(iii) By using the result of Exercise 4.6.1, show that

llu,,,- u , ~ ~ < -~~~l () ~( t- ~1l 1

,-1

T+A)-~(T+A)~/~II

z~ ++ ~ ) - 1 ( n - 1 1 +

1/2

I , llB1l2- cp(U,,)ll 6 Iln-1cp(A)1'2[n-11+ cp(A)]-'II f} = sup{&)

: A E K d >) defines an inner product ( ,)o on Y o and , the corresponding semi-norm 11 on Yois given by ({%}?

Iu,}>o

= P(I(Un3

Il{un}llo = P({IIunll}).

(iii) Let Nobe the subspace {{u,} €Yo : II{u,}llo = 0) of Y o With . TI the quotient space Yo/No, let ( , be the definite inner product on Yl , and 11 the corresponding norm, derived (as in Proposition 2.1 .l) from ( , )o. Let 9be the completion of the pre-Hilbert space Yo/No so obtained, and use the same symbols, ( , and 11 I l l , for its inner product and norm. Show that, for each T in a(%), the equation no(T>{un>= {Tun}

(Iun}EzO)

defines a linear operator no(T)acting on Yo,and Il.o(T){un}llo

G IITII Il{un>llo.

Deduce that the mapping { u,)

+ Mo

+

{Tun]

+ Jlro :

2%

+9 1

is well defined and extends uniquely to a bounded linear operator n(7')acting on 9 with lln(T)ll < IITI(. (iv) Show that the mapping 71: T-,

n(T) :

a(%) a(9) -b

is a representation of @(H). (v) Show that the kernel of z is the ideal 3? consisting of all compact linear operators acting on H.[Hint. Use condition (ii) in Exercise 2.8.20 as the defining property of a compact linear operator.] is the 4.6.58. Suppose that Jf is a separable Hilbert space, X (ca(%)) ideal consisting of all compact linear operators acting on %, and {e, : q E Q} is an orthonormal basis of % indexed by the (countable) set Q of all rational numbers. Let cp be a representation of g(%)that has kernel X (see Exercise 4.6.57). For each real number t , choose a sequence { q ( l ) ,q(2), . . .} of rational numbers (with no repetitions) that converges to t, and let El be the projection from YF onto the subspace generated by {e,(l),e,,2,, . . .}. Show that: (i) {El : t E W} is a commuting family of projections such that El 4 X, EsElE X ,whenever s, t E W and s # t ;

300

4. ELEMENTARY C*-ALGEBRA THEORY

(ii) the Hilbert space on which c p ( W ( 2 ) )acts is not separable; (iii) the result of Exercise 4.6.20(ii) cannot be extended to the case in is which { B , ,B 2 , .. .} is replaced by an uncountable family, even when abelian. 4.6.59. Suppose that {eo,el, e 2 ,. . .} is an orthonormal basis in a separable Hilbert space X , and Wis the isometric linear operator on X defined (as in Example 3.2.18) by Wej = ej+ ( j = 0,1,2,. . .). Let X be the ideal in g ( X ) that consists of all compact linear operators acting on X, and let 7c be a representation of g ( X )that has kernel X (see Exercise 4.6.57). .

(i) Show that W* W = I and WW* = I - E o ,where Eo is the projection from 2 onto the one-dimensional subspace containing eo . (ii) Show that n(W) is a unitary operator U . (iii) Show that there is no invertible operator T in g ( X ) such that n(T) = U. [Hint. If K EX and T = W + K, use the relation T* W = I + K* W and the result of Exercise 3.5.18(iv) to show that T is not invertible.] (iv) Show that there is no normal operator N in B ( H ) such that n ( N ) = U . [Hint. Suppose that K E X and W + K is a normal operator N . Let A?be the null space of N, so that .At is also the null space of N * (Proposition 2.4.6(iii)). Use the relation W*N = I + W*K and the properties of compact linear operators to show that .& is finite dimensional. Prove also that N + E (= W + K + E ) is normal and one-to-one, where E is the projection from X' onto A. Hence reduce to the case in which Jt = {O}. In this case, use the relation N* W = I + K* W and the properties of compact linear operators to show that N is invertible in W ( X ) , contradicting the conclusion of (iii).] 4.6.60. Suppose that 9 is a proper closed two-sided ideal in a C*-algebra 9I, ( V L } is an increasing two-sided approximate identity for 3, and cp: 9I ---i %/9is the quotient mapping from 9I onto the Banach algebra %/Y (Proposition 3.1.8). Prove that: (i) 9I/9has an involution defined by cp(A)* = cp(A*) ( A EJU); (ii) the usual quotient norm on 2l/Y satisfies ((cp(A)((= lim ( ( A- A VL(l= lim ( ( A- VLAJ( 1

( A €91);

1

(iii) with the quotient norm and the involution defined in (i), %/Y is a C*algebra. [This important result will be proved by another method in Theorem 10.1.7.1 4.6.61. Show that, if 9 is a proper closed two-sided ideal in a C*-algebra 2l, then there is a representation of % that has kernel 9.

30 1

4.6. EXERCISES

4.6.62. Suppose that 2I and 93 are C*-algebras, cp is a from CU into a, and 4 is a closed two-sided ideal in CU.

* homomorphism

(i) By adapting the proof of Theorem 4.1.9, show that cp(Y) is closed in &l. (ii) Let % be the C*-subalgebra { c I + S : CEC,SEY}of %, and let X be the kernel of the * homomorphism cp I V: % -+ &l. By considering the induced * isomorphism $ from the C*-algebra %/Xinto B, give a second proof that ~(9 is closed ) in B. 4.6.63. Suppose that 4 and 9are closed two-sided ideals in a C*-algebra %. By using the results of Exercises 4.6.6O(iii) and 4.6.62, show that the ideal

Y

+ 9 is closed in 2I.

4.6.64. Suppose that 9 and 9are closed two-sided ideals in a C*-algebra = B + CE%+,where B E Y and C E ~ .

a, and A

(i) Show that A = S + Tfor suitably chosen self-adjoint elements S of 4 and T of f . (ii) Suppose that E > 0, and define H = IS1 + IT1 + EI, D = A”2H-”2, Ti

S1 = DISID*,

= DITID*,

where IS1 and IT1 denote the positive square roots of S 2and T2, respectively. Prove that D*D Q I, and deduce that A - EIQ S1

+ Ti < A .

Prove also that S1 E

Y+,

TIE$+,

IlSlll Q IlAll,

IlTlll Q IlAll,

and 0 Q A l 6 EI,where A,

= A - S1

- Ti = ( S - S , ) + ( T - T 1 ) ~ ( +9 9)’.

(iii) By repeated application of the result of (ii), show that A can be expressed in the form X + Y , with X in 4 and Y in 9+ . [This exercise shows that (Y+ 2)’= Y 4 when Y and 9 are closed two-sided ideals in a C *-algebra.] +

+

+

+

4.6.65. Suppose that CU is a C*-algebra and 6: 2I+ 2I is a linear mapping such that 6(AB) = Ah(B)

+ 6(A)B

( A , BE%).

(Such a mapping 6 is called a derivation uf2I.) Let Y be the set of all elements A

302

4. ELEMENTARY C*-ALGEBRA THEORY

in '% for which the linear mapping

T+d(AT) :

%4%

is continuous. (i) Show that if A € % , then A €9if and only if the linear mapping

T + A 6 ( T ) : %4% is continuous. (ii) Show that 9 is a closed two-sided ideal in %. (iii) Show that the restriction 819 is continuous. [Hint.If d l 3 is discontinuous, there is a sequence { A l , A 2 , .. .} in 9 such that C llAjllZ 1 and Ild(Aj)ll -+ co.Use the result of Exercise 4.6.40 to obtain a contradiction.] (iv) Show that the quotient C*-algebra %/9 is finite dimensional. [Hint. Suppose the contrary, and deduce from Exercises 4.6.13 and 4.6.20(ii) that the unit ball of % contains a sequence {Sl,S 2 , .. .} of positive elements not in 9 such that SjSk = 0 whenj # k. Prove that there is a sequence { T1, T 2 , .. .} in % such that ( j = 1,2,. . .). IId(S;Tj)II a j IIS(Sj)II IITjII < 2-j,

-=

+

Obtain a contradiction by considering Sj6(C),where C = 1SjTj.] (v) Deduce that 6 :'% + % is continuous. 4.6.66. Suppose that % is a C*-algebra and 3 is a Banach space. We describe 3 as a Banach %-module if there are bounded bilinear mappings (A,x)+Ax,

(A,x)+xA : % x 34%

such that Zx = x l = x for each x in %, and the associative law holds for each type of triple product A l A 2 x , A l x A 2 ,x A I A z .By a derivation from % into a Banach %-module 3, we mean a linear mapping 6: % + 3 such that

6(AB) = A6(B)

+ d(A)B

( A ,B E % ) .

Adapt the program set out in Exercise 4.6.65 to prove that every derivation from a C*-algebra '% into a Banach '%-module is continuous. 4.6.67. Show that the set B of pure states of B ( 2 )is weak* closed when

4 is finite dimensional. 4.6.68. Let J? be a Hilbert space. Show that each vector state oxof g ( 2 ) is pure. 4.6.69. Suppose 2 is an infinite-dimensional Hilbert space, X i s the ideal ofcompact operators in g ( X ) ,P i s the set of pure states of a(#),and Yois the set of vector states of W ( 2 ) .

4.6. EXERCISES

303

(i) Use Exercises 4.6.57 and 4.6.23 to show that there is a pure state p of g(%)that is 0 on X. (ii) With a in [0,1], x a unit vector in &‘,and p in 8- and 0 on X, let w be the state am, + (1 - a)p of B(&‘).Show that w is in 8 - ,the pure state space of 9(2), and that o is in 9’;[Hint. . Use Corollary 4.3.10 to approximate p by a vector state w,,, of B ( X ) .With A l ,. . . ,A , self-adjoint operators in ( B ( J V ) ) ~ and E the projection with range [x, A l x , . . . ,A,x], estimate I(w - wz)(Aj)l where z = a1I2x + (1 - u)li2yand y = ll(I - E)y’ll- ‘ ( I - E)y’.] (iii) Conclude that 8 is not weak* closed (that is, 9 # 8 - )when &‘ is infinite dimensional. 4.6.70. Let CU be a C*-algebra and 3 be a self-adjoint subalgebra of containing I. Suppose that for each pair p l , p2 of distinct states of ‘$ there I is a B in B such that p l ( B ) # p2(B)(that is, B separates the states of a).Show that B is norm dense in CU.

CHAPTER 5 ELEMENTARY VON NEUMANN ALGEBRA THEORY

Those C*-algebras (von Neumann algebras) that are strong-operator closed in their action on some Hilbert space play a fundamental role in the subject. Historically they were the first class of such operator algebras introduced. Their study will occupy us in this and the following four chapters. In the present chapter we develop the elements of the subject. The strengthened closure assumption on the algebra entails significant structural changes. On the technical level, the strong-operator closed algebras abound in projections; while the general C*-algebra may contain no projections other than 0 and I. In a less technical (and deeper) sense, the passage from the general to the strong-operator closed C*-algebra corresponds to the passage from the algebra of continuous functions to the algebra of (bounded) measurable functions. This correspondence can be made precise in the commutative case and lends force to the interpretation of the theory of von Neumann algebras as “non-commutative measure theory.”

5.1. The weak- and strong-operator topologies Recall that the strong-operator topology on B ( 2 ) has a base of neighborhoods of an operator To consisting of sets of the type V(To:x,) . . . )x m ; &= ) { T € a ? ( X ) : l l ( T -T0)Xjll < & ( j = 1,..., m)},

where xl,. . .,x, are in 2 and E is positive. Thus the net {Ti} is strong-operator convergent to To if and only if {[I( Tj - To)xll}converges to 0 for each x in X, that is, if and only if the net { Tjx}of vectors in &‘converges to Toxfor each x in 2. (See the discussion following Proposition 2.5.8 and the comments in Remark 2.5.9.) Another topology on a?(*) will be important for us. 5.1.1. DEFINITION. The weak-operator topology on A?(&) is the weak topology on B(#)(in the sense described in Section 1.3) induced by the family 304

5.1. THE WEAK- AND STRONG-OPERATOR TOPOLOGIES

305

flW of linear functionals w , , ~ :W ( X )-+ C defined by the equation

w,,,(A) = ( A 4 . Y )

(X,YEXO, A E2wf)).

rn

If o,,,(A) = 0 for all x and y in X, then A = 0, whence & is a separating family of linear functionals for B(X).It follows that the weak-operator topology on 3?(X)is a locally convex topology determined by semi-norms (w,,,(A)I.The family of sets of the form V(TiJ: w,,,,,

Y

.

*

3

Wxm,ym

;4

= { T e B ( X ) : I ( ( T -T o ) x j , y j ) J < ~ (= j1,.

. . ,m)},

where E is positive and x , , . . . ,x,, y,, . . .,y, are in X, constitutes a base of convex (open) neighborhoods of To in the weak-operator topology. Since ( ( ( T- To)x,y ) ( < E when Il(T - To)xll < ~ ( 1 ~ ~ y ~ each ~ ) - open l , set relative to the weak-operator topology is open relative to the strong-operator topology. Hence the weak-operator topology is weaker (coarser) than the strong-operator topology. (See Exercise 5.7.2 where it is noted that this relation is “strict.”) As a consequence, the requirement that a subset of B ( X ) be strong-operator closed is less stringent than the requirement that it be weakoperator closed. An important exception to this occurs in the class of convex sets of operators.

+

5.1.2. THEOREM.The weak- and strong-operator closures of a convex subset X of W ( X ) coincide.

Proof. An operator in the strong-operator closure of X is in the weakoperator closure of X. Suppose A, in the weak-operator closure of X, and vectors x l r . .. ,x, in X are given. Let 9 be the direct sum X @. . . @ Xof X with itself n times. For Tin g ( X ) ,let T(y,, . . . ,y,) be ( T y l , .. . ,Ty,) (that is, T = T g . . @ T). Then { T : Tin X } is a convex subset 2 of a(*); and 2iis a convex subset of 2,where 1= ( x ~ , . ,.x,). As A’ is in the weak-operator closure of .$,A ’ i is in the weak closure of 21(in 9).From Theorem 1.3.4,21 is in the norm closure of 2 2 (in 2). Thus for some Kin X, llKxj - Axjll is small for eachj in { 1,. . . ,n}. It follows that A is in the strong-operator closure of X and that the weak- and strong-operator closures of X coincide. W By polarization (see 2.4(3)) the span of the functionals wx,x (= w,) coincides with the span of Fw, so that the semi-norms defined by ( ( A x , x ) l determine the weak-operator topology on W ( X ) . In fact, restricted to (.B(X)), ,the unit ball in g ( X )(or, equally, any bounded subset of &?(if)the ), weak-operator topology is determined by the semi-norms ( ( A x j ,xk)l where ( x j )spans a dense linear manifold in X. For this, note that AX, x ) ( is small (with A in (g(#)),) provided ( ( A y ,y)l is small with y (in the span of ( x j ) ) sufficiently near x .

306

5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

Since ( B A x , y ) = ( A x , B*y), the mappings A .+ BA and A AB ( B e g ( # ) ) of B ( 2 )into g ( H )are weak-operator continuous. That is, left and right multiplication by B are weak-operator continuous. We note, too, from Theorem 1.3.1, that each weak-operator continuous linear functional on g ( H ) lies in the linear span of Fw. Another useful aspect of the weak-operator topology resides in a special compactness property it possesses.

5.1.3. THEOREM.The unit bail compact.

of B ( 2 ) is weak-operator

Proof. Let D I , be the closed disk of radius IIxJJ* llyll in the plane C of complex numbers. The mapping which assigns to each Tin ( W ( 2 ) ) , the point { ( T x , y) : x, y in 2')of IDx,, is a homeomorphism of with the weak-operator topology, onto its image Xin the topology induced on X by the product topology on ID,,, (from the very definition of these topologies). As D, is a compact Hausdorff space in the product topology (Tychonoff's theorem), Xis compact if it is closed. If b is a point in the closure of X and x l , y l , x z , y z are elements of 2, then, for each positive number E , there is a Tin (a(&)), such that each of

n,,, n,,,

nx,,

la ' b(xj,yk) - a(Txj,yk)l,

Ib(ax1 + xz,yj) - (T(ax1 + x A y j ) l , is less than

E,

Ib(xj,yk) - (Txj,yk)l, Ib(xj,ayl

+ YZ) - ( T x j , a y , + ~ 2 ) l

wherej, k = 1, 2. It follows that Ib(ax1

+ X2rY1) - a . &l,Yl)

- b(X2rYl)I

< 3&

and

I&,

yay1

+Y2) - a

*

K X l ,Yl>

- 4 x 1 9YdI c 3 E .

Thus b(ax1 + xz,y1) = a * b ( x 1 , y A + 4 X 2 , Y l ) and b ( x , , a y , + y z ) = a . b ( x , , y , ) b(xl,yz). In addition lb(x,y)l < llxll . Ilyll, since b ( x , y ) I~D,,,. Hence b is a conjugate-bilinear functional on H bounded by 1. From the Riesz representation (Theorem 2.4.1) of such bilinear functionals, there is an operator Toin (B(H))lsuch that b(x,y ) = (Tax, y) for all x and y in H.Thus b E X,X is closed, Xis compact, and (a(&')), is weak-operator compact.

+

The weak-operator topology and the Riesz representation of bounded conjugate-bilinear functionals on a Hilbert space appear once again in establishing a key order-topological property of g ( 2 ) . It concerns nets { I f a , A, < } of self-adjoint operators If, for which the operator-ordering and the partial ordering of the directed index set A agree (that is, Ha < Ha, if a < a'). We say that such nets are monotone increasing (decreasing if the operator-ordering reverses the ordering of A). Although it will prove useful to

5.1. THE WEAK- AND STRONG-OPERATOR

TOPOLOGIES

307

have the result that follows for nets, for present purposes the simpler circumstances of sequences would suffice. 5.1.4. LEMMA.If { H a } is a monotone increasing net of self-adjoint operators on the Hilbert space A? and Ha 6 kf for all a, then {Ha} is strongoperator convergent to a self-adjoint operator H , and H is the least upper bound of {Ha).

Proof. Since the convergence of { H a } and that of { H a , a2 ao} are equivalent, we may assume that { H a } is bounded below (by Hao)as well as above. Thus - ~ ~ H a o ~Ha~ < I A}, so that el is the greatest lower bound of this set in C ( X ) . It follows that E, is the greatest lower bound of { EA,: A' > A} in d . From Proposition 5.1.8, A , > E,, E d . From Corollary 2.5.7, A,,>, EL. is the greatest lower bound in a(*) of {Ex:A'> A}. Thus EA = AA*,AEAP. To prove (v), choose A. less than - l.411 and let {Ao, Al,. . .,A,} be a partition of [Ao, llAll] (so that A, = IlAll). If [Aj- 1 , Aj] n sp(A) # 0, let A; be a point of this intersection-otherwise, let A; be Aj-l. If this intersection is empty, f - ' ( [ A j - l , Aj]) = 0, since sp(A) is the range off. Thus f-l((Aj-17

a))= f - l ( ( A j , a))

and e A J -= , e A J It . follows that Cj"= A; (el, - e,,-,) (= h) is a linear combination of mutually "orthogonal" characteristic functions e,. - e, with coeficients in sp(A). Now each p in X lies in exactly one set Xnj\ (= Yj), j = l , . . . ,n, since X,, = 0 and X," = X. If P E Yj, then h ( p ) = A; and Aj-' < f ( p ) < A j . Hence Ilf- hll < maxj{lAj - Lj-'1}, and (v) follows.

A family {E,} of projections indexed by R, satisfying (i')

A l s l w E ,= 0 and V L E a E = , I

and (ii), (iii) of Theorem 5.2.2is said to be a resolution ofthe identity. Since (i) of Theorem 5.2.2 guarantees (i') above, the family {E,} determined in the argument of Theorem 5.2.2 is a resolution of the identity. If there is a constant a such that El = 0 when A < - a and E, = Zwhen a c A (as there is in the case of Theorem 5.2.2), we say that {E,} is a bounded resolution of the identityotherwise we say that { E,} is an unbounded resolution of the identity. At this point, we have a resolution of the identity for A in each abelian von Neumann algebra containing A . In Theorem 5.2.3, we show that a resolution of the

312

5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

identity satisfying either (iv) or (v) of Theorem 5.2.2 is the resolution of the identity for A in the abelian von Neumann algebra generated by A and I; so that we may speak of the resolution of the identity for A (or the spectral resolution of A ) . 5.2.3. THEOREM. I f { F , } is a resolution of the identity and A is a (bounded) self-adjoint operator such thaf AF, < AF, and A(I - FA)< A(I - Fa)for each A, or i f A = AdF, for each a exceeding someao, then { F A }is the resolution of the identity for A in d o ,the abelian von Neumann algebra generated by A and I.

r-,,

Proof. The fact that AF, is self-adjoint is implicit in the assumption that AFa < AF,. Thus A commuteswith each F A .Since Fa < FA,when A < A’, { F A }is an abelian family. Let d be an abelian von Neumann algebra that contains A and {F,} and X be the extremely disconnected compact Hausdorff space such f in C ( X ) that d E C ( X ) . If {E,} is the resolution of the identity for A in d, correspondsto A , and el in C ( X )corresponds to E,, then enis the characteristic function of X,,the largest clopen set in Xon whichftakes values not exceeding A. Iffn in C ( X )corresponds to FA,then f,is the characteristic function of a clopen set Y , on which f takes values not exceeding A, sincef fa < A,,. Thus Ya c X,. As F, = A ,,>, FA., Y, is the largest clopen set in X contained in fl,, > a Y,. . Now A’ G A P )if p E X\Y,. , since A‘(I - FA.)< A(I - Far),so that X\YapE f-’((A, 00))- when 1’ > A. Thus Xa c Y,. when A‘ > A ; and X, is a YA,.Since Y , is the largest such clopen set, clopen set contained in n,. X, E Ya. Hence X, = Y, and En = F A .The resolution of the identity for A in d osatisfies (iv) of Theorem 5.2.2 and d oE d . From what we have just proved, that resolution coincides with {E,} (and {F,}). Suppose, now, that A = A dF, for each a exceeding some a,. If, for such an a, A E [ - a, a] and {A,, . . . ,A,} is a partition of [ - a, a ] , with A as some A k , such that ( B =) Cj”= A;(F,, - FA,-J is close (in norm) to A ; then IIAF, - BFJ is small and

p-.

k

k

C A;(F,, - FA,-^) < C &(FA, - F A , - , )= A(F, - F - J < AF,. j= 1 Thus AF, < AF,. At the same time, IIA(Z - FA)- B(I - FJl is small and BF,

=

1

j=

n

B(I - FA)=

1 j=k+ 1

n

AJ(FA,-

- 1)

2

C

Ak(F2., - FA,- 1) = &Fa

- Fa).

j=k+ 1

+

Thus A(I - FA)3 A(Fa - FA)for each a greater than ao. Letting a tend to co, Fa tends to I in the strong-operator topology, so that A(Z - FA)3 A(I - Fa). From the first part of this proof, F, = E, for each A.

In Theorem 5.2.4 we start with a bounded resolution of the identity and construct a bounded self-adjoint operator whose spectral resolution is the given resolution of the identity.

5.2. SPECTRAL THEORY FOR BOUNDED OPERATORS

313

5.2.4. THEOREM.If{ Ed}is a bounded resolution of the identity on a Hilbert l d E Aconverges to a self-adjoint operator A on Z such that space 2,then Stla ( [ A [< ( a and for which { E d }is the spectral resolution, where Ed = 0 $1 < - a and EL = I $ a < 1. Proof. If { I , , . . . ,I,} ( = 9’) and {p,, . . . ,p,,,} (= 9) are partitions of [ - a , a ] , 19’1 and 191 are the lengths of their largest subintervals, and {yo,.

. . ,y r } is their common refinement, then r

n

II

c qJ%,

1 YXEh - En-JI < 191’

- EdJ-J-

j= 1

k= 1

and r

m

II

c PJ(EPj

- EPJ-,)-

c Y;(Eh

-

E7k-l)ll

< 191,

k= 1

j= 1

so that n

II

c 1pnj-

j= 1

m

-

p p & - J%!-l)ll

< 191+ PI.

k= 1

Thus the family of approximating Riemann sums to Jta 1dEA,indexed by their corresponding partition of [ - a, a] and the set of these partitions partially ordered (and directed) by refinement, forms a Cauchy net in the norm topology on 3?(2). Since B ( Z )is complete in its norm topology, this net converges in riorm to a bounded self-adjoint operator on H.From Theorem 5.2.3, { E d }is the spectral resolution of A . Passing to C ( X ) , where d z C ( X ) and d is an abelian von Neumann algebra containing A , we see that the conditions, EA= 0 if I < - a and Ed = I if a < I , imply that the function in C(X)representing A has range in [ - a , a ] . Thus IlAll ,< a. We studied unitary operators in C*-algebras in Section 4.4, and noted, there, that exp iH is a unitary element in each C*-algebra containing the selfadjoint element H . We remarked, in the discussion preceding Proposition 4.4.10, that not each unitary element of a C*-algebra has this form. In essence, the possibility of finding “log U” in the C*-algebra generated by U (an algebra of continuous functions) may be blocked by topological (homotopy) considerations. This is not the case in the von Neumann algebra generated by U - where the topological obstructions vanish before the (essentially measuretheoretic) constructions available in von Neumann algebras. We prove this von Neumann algebra analogue to Proposition 4.4.10 in the theorem that follows. 5.2.5. THEOREM.r f U is a unitary operator acting on the Hilbert space Z and .dis the (abelian) von Neumann algebra generated by U and U*, there is a

314

5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

positive operator H in d such that IlHll < 2n and U = exp iH. In addition, U is the norm limit offnite linear combinations of mutually orthogonal projections in d with coefficients in sp(U). Proof. From Theorem 5.2.1, d z C ( X ) with X an extremely disconnected compact Hausdorff space. If u in C ( X ) corresponds to U , then ii corresponds to U * ; and lu12 = 1. Let X , be the complement of the closure of the set of points at which the values of u do not lie in {expiI': I' in [0, I ] } (= C,), for I in [0,2n). Arguing as in the proof of Theorem 5.2.2, X , is the largest clopen set on which u takes values in C,. Let e, be 0 if I 0, 1 if 2n < I ; and let e, be the characteristic function of X , for I in [0, 2n). Then el is the greatest lower bound of {el. : I < A'} if I < 0 or I > 2n. As el < e,, when I < A', el is a lower bound of {el. : I < A'} for all I . To see that e, is the greatest lower bound when I E [0,2n), note that each clopen subset 0 of n,. , X,. is contained in X , (for u takes values on 0 in each C,., with I' exceeding I , so that u takes values on 0 in C,). As in Theorem 5.2.2, the projections Ed in d corresponding to e, give rise to a (bounded) resolution of the identity { E d } . From Theorem 5.2.4, I dE, converges (in norm) to a self-adjoint operator H in d . Let h be the function in C ( X ) corresponding to H . Letting X , be @ when I < 0 and X when I 2 271, X , is the largest clopen set on which h takes values not exceeding I . The range of h is contained in [0,2n] so that H is positive and IlHll < 2n. Note, too, that h cannot take the value 2n at each point of a non-null clopen set 0,; for otherwise 0, is disjoint from U, < 2n X,, . But then u(po) # 1 for somep, in 0,(otherwise 0, G X,). By continuity of u, there is a clopen subset O1 of Oo containingp, and there is a I , in ( 0 , 2 n )such that u(q)E C,,, for each q in O1. Thus O1 E X,, contrary to the choice of 0, disjoint from XA0.With this information, we can now see that X , is the largest clopen set on which exp ih takes values in C , ( I E [ 0 , 2 x ) )- whence exp ih = u, and exp iH = U. Indeed, if 0 is a clopen set such that exp ih(p) E C, for each p in 0, then either h( p ) E [0, I ] or h( p ) = 271for each p in 0. If h( p ) = 2n for some p in 0, then, by continuity of h, there is a clopen subset of 0 containingp on which h takes values near 2n -in particular, not in [0, A ] , since A 271. By choice of 0, then, h takes the value 2n on this entire clopen subset-contrary to what we have just proved. Thus h ( p ) E [0, I ] for all p in 0, and 0 c X,. If Ij(E,, - Ed,_ is close to H in norm, then

-=

,

-=

c;=

n

1 (exp iI;)(Ea, - Edj- J j= 1

is close to exp iH (= U ) in norm. From Theorem 5.2.2, we can choose I: in sp(H) if Ed, # E a j - l .With this choice, expiIJEsp(U) if Eaj # E a j - l . In Example 5.1.6 we noted that the multiplication algebra d of a o-finite measure space ( S ,9, m)is an abelian von Neumann algebra. Iff is a real-valued

5.2. SPECTRAL THEORY FOR BOUNDED OPERATORS

315

essentially bounded measurable function on S, M , is a (bounded) self-adjoint operator on L 2 .With El the projection corresponding to multiplication by the characteristic function of the set Sdon which f takes values not exceeding 1, { E d }is the spectral resolution of M,. The key observation needed for this is the fact that n,, , Sap= Sd(and, thus, A El. = El). In Theorem 5.2.6 we describe a simultaneous spectral resolution for a commuting family of operators forming an abelian C*-algebra. In this case, the (joint) spectrum of the family is the pure state space. Somewhat more precisely, we describe the spectral resolution of a representation of an abelian C*-algebra. 5.2.6. THEOREM. I f X is a compact Hausdor-space, 2 is a Hilbert space, then, to each Borel subset S of X there and (p is a representation of C ( X ) on S, corresponds a projection E(S) such that (i) E ( S )E d,the strong-operator closure of (p(C(X)); (ii) E ( S ) = A ( E ( 0 ) :S E 0, 0 open}; (iii) E(U,"= S,,) = =C : E(Sn)for each countable family {S,,} of mutually disjoint Borel subsets of X, in particular, E(Sn)E(S,,,)= 0 if n # m , and E ( 0 ) = 0; (iv) E(Xo) = I,for X o a Borel subset of X , ifthe span of the ranges of those such that fE C ( X ) and f vanishes on X\ X o is dense in X ; (v) for each x in 2,S -P ( E ( S ) x ,x ) is a regular Borel measure, px,and,for f i n C(X),

(p(n

(cp(f)x,x) =

s

f l P ) dPx(P)*

X

Proof. If 0 is an open subset of X and f in C ( X ) has range in [0,1] and vanishes on X\0, then 0 ,< q(f) < I. Thus (p(F(0))has a least upper bound E ( 0 )in the abelian von Neumann algebra d,where F(0) is the set (directed by its natural order) of such functionsf. As { f ' : f E F ( 0 ) ) = F(0), E(0) is a projection. With S a Borel subset of X , let E ( S )be A{ E(0):S G 0 , O open}. I f x is a unit vector in S, f -,(tp(f)x, x) is a state of C(X). From the Riesz representation of such functionals (see the discussion preceding Lemma 1.7.7), there is a regular Borel measure px on Xsuch that (cp(f)x, x ) = J f ( p )dpx(p). By (inner) regularity of p x , given an open set 0, there is a compact subset X of 0 such that px(-X) is close to px(0). Since X is a normal space, there is a continuous function f on X with range in [0,1], vanishing outside 0, and 1 on X. Then Px(.x) G

s

f(P)dPx(P)= ( P ( f ) X , X >G ( E ( 0 k X ) -

316

5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

It follows that p x ( 0 ) 6 ( E ( O ) x , x ) . From the definition of E(0), ( E ( O ) x , x ) 6 p x ( 0 ) ,so that (E(O)x,x) = px(0). From (outer) regularity of p x , we have that px(S)= inf{px(0):S c 0, 0 open} = inf{(E(O)x,x):

S G 0, 0 open} = ( E ( S ) x , x )

for each Borel set S. If, now, {S,} is a family of disjoint Borel subsets of X,then m

m

1 (E(Sn)x,x)*

px(Sn) =

n= 1

In particular, if x is a unit vector in the range of one of the E(S,,),say, E(S,), then m

12 ( E ( \n=]

Sn)x,x) = /

1 (E(S,,)x,x)2 ( E ( S l ) x , x ) = 1. n= 1

Since 0 6 (E(Sn)x,x) for all n, (E(S,)x, x) = 0 unless n = 1. It follows that E(S,)E(S,,,) = 0 if n # m and that E(U,"= S,,)= E(S,,). With X, as in (iv), if 0 is an open set containing X, and f (real-valued) in C ( X )vanishes on X\0, then the range projection of cp( f o )is a subprojection of E(0),wheref, = )If ll-llfl. But cp( f o ) and ~ ( fhave ) the same range projection (for dlfl)= c p ( f + ) + df-), df) = 4 u + ) - cpu-1, and f + f - = O-see Remark 3.4.9). Thus E ( 0 ) contains the range projection of the image of each function in C ( X ) vanishing on X \ X o ; and, by assumption, E(0)= I. Hence E(x,) = I. We apply this theorem to the important special case of R-essential representations of C({R, co}) (that is, representations essential on the ideal of functions in C({R, a})vanishing at co, where, as usual, {R, co} denotes the one-point compactification of R). 5.2.1. COROLLARY. Each R-essential representation cpo of C({R, co}) corresponds to a (possibly unbounded) resolution of the identity { E L }such that, for each f in C({R, co}) with Am) = 0, (cpo(f)x,x> =

1

A4W

A X , x>.

R

Proof. From Theorem 5.2.6, there is a projection-valued measure S + E(S) on { R, co} such that (q0(f ) x , x) = JR A I ) dp,.(I), where feC({R, co}),f(co) = 0, and px(S) = ( E ( S ) x , x ) . If EA= I - E((1, a)),then { E l } is a (possibly unbounded) spectral resolution of the identity. To see this, note that E((I, a))< E((A', co)), when I' < I , so that EA,< E A ,in this case. Since cp, is R-essential, from (iv) of Theorem 5.2.6, we have that E(R) = I. As

317

5.2. SPECTRAL THEORY FOR BOUNDED OPERATORS

(m, a)= U n B m ( n , n = 0. It follows that

+ 11, E((m,a))= I,"=,E((n,n + 11). and

I = I - A E((A,C O ) ) = V(I- E((1, 00)))

= YEA. I

1

1

A m E((m,00))

At the same time, m

m

C

I = E(R)=

E((n,n

+ 11) = V

E((m,CO)),

m=-w

n=-m

so that oc

02

0=I-

V m= - w

OD

A

E((m,00))=

A

( I - E ( ( ~ , c o )= ))

m=

m = -a:

Em

-a,

and m

O,on X, where X = { (w, 00} if A is self-adjoint and X = {C, a}, the one-point compactification of C, if A is normal. From Theorem 5.2.6, there is a projection-valued measure assigning a projection E ( S ) on X to a Borel subset Sof Xand such that, for eachfin C(X),

j

( f ( A ) x , x )=

f(P)&x(P),

X

where pX(S)= ( E ( S ) x ,x). If 0 is an open set disjoint from sp(A) andfin C ( X ) is in 9 ( 0 )(see the notation of Theorem 5.2.6), thenAA) = 0, so that E ( 0 ) = 0 and p x ( 0 ) = 0 for each x in X. We write ( A A ) ~x> , =

J

~

pdcLx(p), )

sp(A)

and speak of S + E(S) as the spectral measure for A. In case A is self-adjoint, Corollary 5.2.7 shows us how to pass from this spectral measure for A to a spectral resolution {E,}. From the proof of that corollary, we have that E, = Z - &(A, a)).As just noted, E((A, a))= 0 if (A, 00) is disjoint from sp(A). Thus E, = I if A > IlAll. At the same time, I - E((1,co))= E(X\(A, 00)) = 0 if A .c - IlAll, so that E, = 0 when A - llAll. Thus { E n }is a bounded resolution of the identity in this case, and

-=

(2)

(f(& x) =

J

IlAll

f(4 d(E,x, x> -llAll

(at first, for eachfin C ( X ) vanishing at a,but then for eachfcontinuous on sp(A) since each such agrees on sp(A) with some function vanishing at 00). In particular

s

IlAll

( A X ,X)

=

A d(E,x, x)

- IlAll

for each x in X, so that AE, < AE, and A(Z - EL)< A(I - El). It follows (from Theorem 5.2.3) that {E,} is the spectral resolution of A. Since px(S) = ( E ( S ) x ,x) for each Borel subset S of X and x in X and (f(A)x,x)=

1

AP)dPx(P),

X

polarization of ( E ( S ) x , y ) allows us to define a complex (Radon) measure on X as a linear combination of positive measures pz and (3)

( f ( A ) x , y )=

1

~(P)~P,,(P)

X

5.2. SPECTRAL THEORY FOR BOUNDED OPERATORS

319

for eachfin C(X) and all x, y in X. If A is self-adjoint, (3) amounts to the formula,

j

IlAll

( f ( A ) x , y )=

f(4~(J%x,JJ),

-llAll

which “polarizes” to (2). If A is normal, (3) provides us with the possibility of defining g(A) when g is a bounded Borel function on @. Note that ISg(P)dPx,,(P)I G llxl1 IIYII suPlg(P)l (this is apparent from (3) when g E C ( X ) , and extends by a measure-theoretic argument to the case where g is a bounded Borel function). Thus ( X J )

-+

jxg(P)dPx,y(P)

is a bounded conjugate-bilinear functional on & with bound not exceeding sup)g(p)Jand, so, corresponds to an operator g ( A ) satisfying

= (g(Aly9 x> = ( g ( A ) * x , y )

for all x and y in %. Thus (5)

&A)

=g

W*

for each bounded Borel function g on C and each normal A . Since A and A* commute with a,g(A) and g(A) commute with and, hence, with each other. Thus g ( A ) is normal. We proceed, now, to establish the other properties of a (bounded) Borel function calculus for (bounded) normal operators. If g and h are bounded Borel functions on C (or R if A is self-adjoint), then, for each x in a?, ( ( a s + h)(A)x,x> =

Ix

( W ( P ) + h(PN dPx(P) = ((ag(A) + h(A))x,x),

so that (6)

(ag

+ h)(A) = ug(A) + h(A).

320

5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

If g is the characteristic function of a Borel subset S of X,then, for each x in X, (g(A)x,x > =

J

g(p)dlcx(P)= PX(s) = (E(s)x, x > . X

Thus g(A) = E ( S ) ; and, in particular g2(A)= g ( A ) = g(A)2. If h is the characteristic function of a Borel set disjoint from S, then, from Theorem 5.2.6(iii), 0 = g(A)h(A) = ( g h)(A). It follows, now, that g2(A) = g(A)2, so that ( g + h)'(A) = [g(A) + h(A)I2 and (9 . h)(A) = g ( 4 . h ( 4 ,

(7)

when g and h are finite linear combinations of characteristic functions of disjoint Borel subsets of X ("step functions"). Since each bounded Borel function is a uniform (norm) limit of such step functions (and Ilg(A)II < llgll for each bounded Borel function g), we have (7) for arbitrary bounded Borel functions g and h. The identities (5), (6), and (7) that we have established thus for assure us that the rule (9 h)(A) = g(h(4)

(8)

O

holds when g is a polynomial (in z and 5) and h is an arbitrary bounded Borel function. Using the Stone-Weierstrass theorem (3.4.14) to approximate a continuous function uniformly on a closed disk in C containing the range of h by a polynomial (in z and 5), it follows that (8) is valid for each continuous function g and each bounded Borel function h. Since g(h(A)) = g(h(A)) = g(h(A))* if g is real-valued and g(h(A)) 2 0 if g 2 0, we have that {g,,(h(A))}is an increasing sequence of self-adjoint operators when {g,,} is an increasing sequence of bounded Borel functions. If each g,, is continuous and tends pointwise to a bounded Borel function g, then g,,(h(A))= (g,, h)(A) and {gn h } is an increasing sequence tending pointwise to g h. It will be useful for us to note that if {f,} is an increasing sequence of bounded Borel functions tending pointwise to the bounded Borel functionf, then { f , ( A ) }is an increasing sequence of self-adjoint operators with least upper bound f ( A ) (and a similar conclusion holds for decreasing sequences). We say that the mappingf-Ad) with this monotone sequential convergence property is m-normal. To prove this, choose x in X and note that 0

0

0

( f , ( A ) x , x )=

1

X

f"(P)dPX(P)

+

j

f(P)dPx(P) = (f(A)x,x), X

from the monotone convergence theorem. Thus, in the case of the continuous the characteristic function of an open set 0 in C (and, so, for a closed set as well). To construct the

g,,, we have (8) for their limit g . In particular, we have ( 8 ) for g

5.2. SPECTRAL THEORY FOR BOUNDED OPERATORS

321

sequence {g,,} for g , express 0 as a countable union of open disks Oj(with radius r,). Letfj" be a continuous function on a3 with range in [0,13, vanishing outside Lo,, and 1 on the closed disk with the same center as 0,and radius ( n - l)rj/n. Then f i n v fin v * . v f,,,, will serve as g,,. Let S be the family of Borel sets S whose characteristic function g satisfies (8) for all bounded Borel functions h. We have just seen that 9 contains all open and all closed sets. From the properties we have established for the mapping f + A A ) (in particular, a-normality) we see that 9 is a a-algebra. Hence S is the family of all Borel sets; and (8) holds for all bounded Borel functions h, when g is the characteristic function of a Borel set. As a consequence, (8) is valid for each step function g and then, by passing to (norm) limits, (8) follows for each pair of bounded Borel functions g and h. We summarize this discussion in the theorem that follows.

5.2.8. THEOREM. If A is a (bounded) normal operator on the complex Hilbert space X the * homomorphism f +f(A) of C ( X )into the C*-algebra % generated by A, A*, and I, where X is the one-point compactijkation of @, extends to a a-normal * homomorphism g -+ g(A) of the algebra W of bounded Borel functions g on C into the abelian von Neumann algebra d consisting of operators commuting with each operator commuting with %. Ifg in W vanishes on sp(A), then g(A) = 0. With g and h in W,g ( A ) = g(A)* and ( g 0 h)(A) = g(h(A)). Letting S be a Borelsubset of X , g be its characteristicfunction, and E(S)be g(A), the mapping S + E(S) is a projection-valued measure on X . Moreover Ilf(A)II G sup{If(a)l: and

=

j X

f ( P ) dPx(P) =

SP(41

1

f(P)dPx(P)

sp(A)

for each f in 93,where px(S)= ( E ( S ) x , x ) . If A is self-adjoint its spectral resolution is { E A } ,where EA= I - E((I, 00)).

I f f is the ideal of functions in 9 vanishing on sp(A), then 9 is the kernel of the homomorphism of W onto W(sp(A)) obtained by restricting a function in W to sp(A). Thus &?/Yz W(sp(A)). As noted in Theorem 5.2.8, the kernel of the (a-normal) homomorphism, g -,g(A), of W into d, contains 3 Thus the mapping, g + 9 -,g ( A ) , gives rise to a homomorphism of W(sp(A)) into d.If g E W(sp(A))and we define $jto be g on sp(A) and 0 on the complement of sp(A), then 8 E 9and g ( A ) is the image of g under the homomorphism described. We write g ( A ) for this image. From this same observation, we see that g + g(A)is a a-normal homomorphism of W(sp(A)) into d. In the theorem that follows, we prove that our bounded Borel function calculus is unique. (Compare Theorem 4.4.5 and Remark 4.4.6.) The

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uniqueness is stated in terms of B(sp(A)), although the preceding paragraph applies to any Borel subset of C containing sp(A) and the following result (and argument) apply to each bounded Borel subset of C containing sp(A).

5.2.9.THEOREM.I f A is a (bounded)normal operator on a complex Hilbert space 2, B(sp(A)) is the algebra of (complex-valued)bounded Borelfunctions on sp(A), cp is a a-normal homomorphism of B(sp(A)) into an abelian von Neumann algebra d, cp( I) = Z, and cp( z) = A , where z(a) = a for each a in sp(A), then cp maps W(sp(A)) into d o the , abelian von Neumann algebra generated by A, A*, and Z, and cp(g) = g(A) for each g in W(sp(A)). Proof. With complex conjugation as involution (* operation) and sup{lg(a)l : a E sp(A)} as Ilgll, g(sp(A)) is a C*-algebra; for 11s .g1( = 11g112.If g is real-valued and aEC\R, then g - a1 has an inverse h in B(sp(A)). Since Z = [cp(g) - aZl . cp(h), a#sp(cp(g)). The elements of d are normal, so that cp(g) is self-adjoint. Thus cp is a * homomorphism of the C*-algebra B(sp(A)) into d. From Theorem 4.1.8(i), cp is order preserving and does not increase norm. Applying Theorem 4.4.5 to cp restricted to C(sp(A)), we have that q ( f )= A A ) for each f in C(sp(A)) a n d A A ) E d o . Since cp and the mapping g + g ( A ) are a-normal, we have, as in the argument proving (8), that cp(g) = g ( A ) ( € d o when ) g is the characteristic function of an open set. If 4 is the family of Borel sets whose characteristic function g satisfy cp(g) = g(A), then, again, by a-normality, 4 contains the union of each countable subfamily. Moreover, since cp( 1) = Z,9contains the complement of each set in 3F Thus 4 coincides with the family of all Borel subsets of sp(A); and cp(g) = g(A) (E d o for ) the characteristic function g of such a set. Since cp is linear and norm continuous, Ilg(A)II 6 Ilgll, and the step functions are norm dense in ) each g in &?(sp(A)). 4 B(sp(A)), it follows, now, that cp(g) = g(A) (E d o for Lemma 5.2.10 provides the foundation for developing the Borel function calculus in purely topological-function-theoretic terms. The approach of Theorem 5.2.8, stemming from Theorem 5.2.6, is essentially measuretheoretic, while the construction of the spectral resolution { E A }in Theorem 5.2.2is topological and function-theoretic. We shall take the latter path, and refer to the following results, when we treat the function calculus for unbounded normal operators (Section 5.6). Recall that a subset of a topological space Xis nowhere dense (in X ) if its closure has empty interior and that a subset is meager (or of thefirst category) in X if it is a countable union of sets nowhere dense in X . (See [K: p. 2011.)A subset of a nowhere-dense set is nowhere dense, so that a subset of a meager set is meager. A countable union of meager sets is meager. 5.2.10.LEMMA.If X is an extremely disconnected compact Hausdorff space, each Borel subset of X differsfrom a (unique) clopen set by a meager set.

5.2.

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323

Each bounded Borelfunction g on X differsfrom a (unique)continuousfunction f on a meager set. The mapping that assignsf to g is a (conjugation-preserving,0normal> homomorphism of B(X),the algebra of bounded Borel functions, onto C ( X ) with kernel consisting of those functions vanishing outside a meager set. Proof. Let 9be the family of subsets of Xthat differ from a clopen set by a meager set. If S Eand~ X o is a clopen set such that (S\Xo)u(Xo\S)is meager, then X\S and X\X, differ by this same set. As X\X, is clopen, X\Se$? Each open set 0 lies in % since 0- is clopen and 0-\0is nowhere dense. If S j E 9for j = 1,2, . . . and X j is aclopen set such that (Sj\Xj) u (Xj\Sj) (= M j ) is meager, then

,

; S j E $?Hence 4” contains the 0As UT= M j is meager and UT= X j is open,=U algebra generated by the open subsets of X ; that is, 9contains the Borel subsets of X . The Baire category theorem [K: p. 200, Theorem 341 assures us that the complement of a meager set is dense in X , so that two continuous functions agree on the complement of a meager set only if they are equal. Thus there is at most one continuous function agreeing with a given bounded Borel function on the complement of a meager set. If S is a Borel subset of X , g is its characteristic function, X, is a clopen subset of X such that (X,\S)v (S\Xo) (= M ) is meager and f is the characteristic function of X o , then f is continuous and g - f is 0 on X\M. We see, from this and the preceding “uniqueness” remark, that there is at most one clopen set differing from S by a meager set. At the same time, we see that a finite linear combination of characteristic functions of (disjoint) Borel subsets of X(step functions) agrees with a (unique) continuous function on the complement of a meager set. Since the step functions are (supremum-)norm dense in B ( X ) ;if g is in B(X),there is a sequence {g,,} of step functions such that 119 - grill + 0. Let {fn} be a sequence of continuous functions such that fn and gn agree on the complement of a meager set M,,. Then [If,-f,ll < 119. - gml(,sincefn - f , and g,, - gm agree on the complement of M,, u M,, a dense set (so that I( f n -f,)(p)l < 119. - gml(for eachp in this dense set). Thus {f,}is a Cauchy sequence and converges in norm to somef in C(X).As { g n }tends to g and {fn} tends tof pointwise, f and g agree on the complement of UTZl M j , a meager set. If g1 and g2 in B(X)differ fromf l andf 2 in C ( X )on the meager sets M1 and M 2 , then g l ,ag, g2 and g l g 2 differ fromyl, afl +f 2 and f i f i on a subset of M I v M 2 . Thus the assignment to g in B(X)of the unique f in C ( X )differing from g on a meager set is a (conjugate-preserving) homomorphism of B(X) onto C ( X ) .Of course g corresponds to 0 if and only if g vanishes outside a meager set. If {g,,} is a monotone increasing sequence of bounded Borel

+

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functions on X tending pointwise to the bounded Borel function g and fn in C ( X ) differs from g,, on the meager set M,,, then fn(p) is dense in E ( H ) , {FA’~:A’EW’}is dense in F E ( 2 ) (= F ( X ) ) .Thus F is cyclic in B and Fx is a generating vector for F. Suppose E is an arbitrary projection in 9. If Eis 0, then E is cyclic in W with generating vector 0. If E # 0 and x is some non-zero vector in its range, then [W’x] is the range of a cyclic projection Eo. Since the ranges of Eo and E are Eo < E and Eo E 9‘‘ = W. Thus E has a stable under the self-adjoint family 9‘; non-zero cyclic subprojection if E # 0. The set of orthogonal families of nonzero cyclic subprojections of E is non-empty, and the union of each totally ordered subset is an upper bound for that subset under inclusion ordering. Zorn’s lemma guarantees the existence of a maximal orthogonal family { E,} of non-zero cyclic subprojections of E. If E - V, E, is not 0, it contains a non-zero cyclic subprojection Eo. Adjoining Eo to {E,} contradicts the maximality of { E , } . Thus E is the union of the orthogonal family {E,} of non-zero cyclic projections. The simple Zorn’s-lemma argument employed at the end of the preceding proof will be needed (with minor modifications) frequently. For the most part, all that will appear will be the imperative, “Let { E,} be a maximal orthogonal family of cyclic subprojections.”

5.5.10. PROPOSITION. If {Q,,} is a countable, orthogonal family of central projections in a von Neumann algebra W and {En}is a family of cyclic projections in W such that E,, < Q,,, then 1En is cyclic in W. Proof. If x,, is a generating unit vector for En under W‘, then {x,} is an orthonormal set, so that C n - l x , , converges to a vector x. As [w’x,] = [B’Q,,x] E [w’x], the range of 1Enis contained in [W’x]. Since the range of C E,, is stable under w’ and x is in that range, it coincides with [W’x]. That is, x is a generating vector for 1En, and C En is cyclic in 9.

5.5.1 1 . PROPOSITION. If9 is a von Neumann algebra acting on the Hilbert space 2,a subset Y of # is generating for W ifand only ifit is separatingfor 9’. Proof. Suppose Y is generating for W,A ‘ E W ’ ,and A‘y = 0 for ally in Y. Then 0 = AA‘y = A’Ay for all A in W and y in Y. As { A y :Y E Y, A E W}spans 2,A’ = 0. Thus Y is separating for 9’. If Y is not generating for 9, then [WY] is the range of a projection E‘ in w’, different from I . Thus I - E‘ # 0, and ( I - E’)y = 0 for all y in Y (since y is in the range of El). Hence Y is not separating for w’ in this case.

The special case of the preceding result in which Yconsists of a single vector is the one used most frequently.

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A vector is generating for a von Neumann algebra i f 5.5.12. COROLLARY. and only i f it is separating for the commutant. I f 9 is a von Neumann algebra and E and E‘ are 5.5.13. PROPOSITION. projections in Wand 9‘with ranges [W‘x] and [Wx],respectively, then CE= CE.. Proof. From Proposition 5.5.2, CE and CE. have ranges [Ww’x] and [W’Wx], respectively. Since 9and w’commute, these subspaces coincide; and

cE=cE‘.

5.5.14. DEFINITION. A projection E in a von Neumann algebra W is said to be countably decomposable relative to 9 when each orthogonal family of non-zero subprojections of E in W is countable. When I is countably we say that W is countably decomposable. H decomposable relative to 9, The term “c-finite” is often used in place of “countably decomposable.” The von Neumann algebra W relative to which countable decomposability is asserted for a projection Eis important, as can be seen by taking E to be Iand W to be, first, 9(Z)with X non-separable, then to be { a ] } .The (minimal) projections corresponding to an orthonormal basis for X‘ form an uncountable orthogonal family of subprojections of I, so that I is not countably decomposable relative to B ( 2 ); but I is countably decomposable relative to { a I } . Despite the necessity for caution when a projection is claimed to be countably decomposable, reference to W will be omitted when no confusion can arise. If &? is a separable Hilbert space, each orthogonal family of projections is countable, so that a(#)and each von Neumann algebra on A? are countably decomposable. 5.5.15. PROPOSITION. If E is a cyclic projection in a von Neumann algebra 9,then E is countably decomposable. Proof. If x is a unit generating vector for E under 9‘and {E,} is an orthogonal family of non-zero subprojections of E in 9, then, from Proposition 5.5.9, E,x is a generating vector for E,. Since E, # 0, E,x # 0. From Bessel’s inequality (Remark 2.5.17), CaIIEaxllz< llxllz = 1. If the set of indices a is uncountable, there is a positive integer n such that l / n < IIE,xll for is not finite. Thus {Ea} is an infinite set of indices a. In this case, C,llE,~11~ countable. It follows that E is countably decomposable. H

5.5.16. PROPOSITION. A centralprojection P in a von Neumann algebra W is the central carrier of a cyclic projection in 9 if and only i f P is countably decomposable relative to the center W of 9.

5.5. PROJECTION TECHNIQUES AND CONSTRUCTS

339

Proof. Suppose P = C, with E a cyclic projection in 9, and x is a unit generating vector for E. If { P a } is an orthogonal family of central subprojections of P , then CallPaxll* 6 llxll* = 1. Thus (as in the proof of Proposition 5.5.15) Pax = 0 for all but a countable set of indices. If Pax = 0, then 0 = A'P,x = P,A'x for each A' in 9'. As [W'x] is the range of E, P,E = 0. From Theorem 5.5.4, 0 = PaC, = P a p = Pa. Thus P is countably decomposable relative to W. Suppose now that Pis countably decomposable relative to $9. Let { E,} be a family of non-zero projections cyclic in Wmaximal with respect to the property that their central carriers form an orthogonal family of subprojections of P . By hypothesis { CEa}and consequently { E,} are countable. From Proposition 5.5.10, C E, is cyclic in 9; and, from Proposition 5.5.3, its central carrier is V, CEa.If P - V, C,. # 0, it contains a non-zero projection Eo cyclic in 9, Since C,, is orthogonal to each CEO,adjoining Eo to {E,} contradicts the maximality of { E , } . Thus P = V, C,. ;and 1E, is a cyclic projection in W with central carrier P. H

An important consequence of the preceding result deals with the case where

W is abelian. 5.5.17. COROLLARY. If d is a countably decomposable abelian von Neumann algebra acting on the Hilbert space X, d has a separating vector. Ifd is maximal abelian, in addition, the separating vector is generating for d . Proof. Since d is its own center and Zis countably decomposable relative to d ,Z is (the central carrier of) a cyclic projection in d .With x a generating vector for d ' ,x is separating for d (= d"),from Corollary 5.5.12. If d is as well. W maximal abelian, x is generating for d (= d')

A partial converse to the preceding result states that if an abelian von Neumann algebra has a generating vector then it is maximal abelian. While an ad-hoc argument could be given to establish that converse, at this point it is best to defer further discussion to Section 7.2 (see Corollary 7.2.16), where suitable techniques are developed. An illustration of the situation discussed in Corollary 5.5.17 is provided by the example (see Example 5.1.6) of the multiplication algebra of the unit interval under Lebesgue measure. In this case the algebra is maximal abelian; and the constant function 1 is a separating (and generating) vector for it. 5.5.18. PROPOSITION. If W is a countably decomposable von Neumann algebra acting on the Hilbert space 2,there is a centralprojection cyclic in W whose orthogonal complement is cyclic in 9'. Proof. Let {x,,} be a set of unit vectors in X maximal with respect to the property that {En}and { E i } are orthogonal families of projections, where E,,

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has range [W’x,,] and Ei has range [Wx,,]. Since { E n is } an orthogonal family of projections in 9,it is countable. We may assume that the index n is a positive integer. As {x,} is an orthonormal set, E n - l x , converges to a vector x in S. If E is 1En and E‘ is 1E ; , then E and E‘ are cyclic projections in 9 and W’, respectively;and xis a generating vector for each. Of course, x is in the range of both Eand E’. In addition [wlx,] = [a’E:x] G [W‘x].Thus [ a ’ x ] is the range of E. Symmetrically, [ a x ] is the range of E‘. If (I- E)(I - E’) # 0, a unit vector x o in the range of (I - E ) ( I - E‘) will generate cyclic projections Eo in W and Eb in W‘orthogonal to { E n }and { EA}, respectively. Adjoining xo to {x,,} contradicts the maximality of {x,,}.Thus (I - E)(I - E’) = 0; and, from Theorem 5.5.4, C,-EC,-E,= 0. Since I - C,-E < E, I - C,-E is cyclic in W (from Proposition 5.5.9). Similarly I - C,-E, is cyclic in 9’.As Cj-E < I - Cj-E’,C j - Eis cyclic in 9’. Thus I - C , - E is a central projection cyclic in 9 whose orthogonal complement, Cj-E, is cyclic in 9’. H 5.5.19. PROPOSITION. I f E is the union of a countable family { E n }of cyclic then E is countably decomposable in W. projections in a von Neumann algebra 9, Proof. Let x,, be a unit generating vector for En under 9‘. Let {Fa: a E A} be an orthogonal family of non-zero subprojections of E in 9; and let % be { a :Fax, # O } . If a is not in %, Fax, = 0; and (0) = [&?’Fax,,]= [FaW’x,]. Since Fa < V, En,Fa does not annihilate the range [W’x,,] of each En.Thus U % = A. On the other hand, CacAllFaxnllZ < Ilx,,llz = 1, so that Fax, # 0 for at most a countable number of elements a of A -that is, % is countable. If K is the cardinal number of A, K < KO * KO = KO(see the final paragraph of the proof of Theorem 2.2.10), since A = U %. H

5.6. Unbounded operators and abelian von Neumann algebras In this section we study the spectral theory of unbounded self-adjoint and normal operators. We associate an (unbounded) spectral resolution with each (unbounded) self-adjoint operator (compare Theorem 5.2.2 and the discussion following it). We extend the function calculus (both continuous and bounded Borel) to (unbounded) normal operators (compare Theorem 5.2.8). We begin with a discussion (really, a continuation of Example 5.1.6) that details the relation between unbounded self-adjoint (and normal) operators and the multiplication algebra of a measure space. If we are prepared to confine attention to, say, the case of separable Hilbert space, this discussion combined with some of the general theory of abelian von Neumann algebras to be developed in Chapter 6 contains all that we need in dealing with unbounded self-adjoint (and normal) operators.

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341

If g is a (complex) measurable function (finite almost everywhere) on S now, without the restriction that it be essentially bounded -multiplication byg will not yield an everywhere-defined operator on L,(S), for many of the products will not lie in L2(S).Enough functionsfwill have productfg in L2(S), however, to form a dense linear submanifold 9 of L 2 ( S ) and constitute a (dense) domain for an (unbounded) multiplication operator M,. To see this, let En be the (bounded) multiplication operator corresponding to the characteristic function of the (measurable) set on which 191 < n. Since g is finite almost everywhere, {En}is an increasing sequence of projections with union I. The union goof the ranges of the En is a dense linear submanifold of L,(S) contained in 9.A measure-theoretic argument shows that M , is closed with go as a core. In fact, if { f , } is a sequence in 9 converging in L,(S) tofand {gf,} converges in L 2 ( S )to h, then, passing to subsequences, we may assume that { f , } and {d,}converge almost everywhere tofand h, respectively. But, then, { a f , } converges almost everywhere to gf, so that gfand h are equal almost 9 ,= M,(f), and M , is closed. Withf, in 9, everywhere. Thus gj"~L 2 ( S ) , f ~ h { E n f o }converges tofo and {M,Enfo} = {E,,M,fo} converges to M,fo. Now E,f0 e g o ,so that gois a core for M,. Note that M,E,, is bounded and that its bound does not exceed n. Using the lemma that follows, we see that M Bis an (unbounded) self-adjoint operator when g is real-valued, since M,E,, is a bounded self-adjoint operator in that case.

5.6.1. LEMMA.Ij" {En} is an increasing sequence of projections on the ;l En(#) Hilbert space 2 and A . is a linear operator with dense domain U= (= go)such that AoEn is a bounded self-adjoint operator on Z,then A . is preclosed and its closure is self-adjoint. If A is closed with core goand AE,, is a bounded self-adjoint operator, A is self-adjoint. Proof. With x and y in go,there is an m such that

( A o x , y ) = (AOEnlX,Y) = (X,AoEmY) = (X,AoY). Thus y ~ g ( A ; and ) A,* is densely defined. From Theorem 2.7.8(ii), A . is preclosed. With A the closure of Ao, it remains to prove the last assertion of this lemma. With x and y in 9 ( A ) , we can choose sequences {x,,} and {y,,} in go converging to x and y , respectively, such that {Ax,,} and { Ay,} converge to A x and A y . Then (AXfl,Yn) = (AEmxfl,yfl)= (Xfl,AEmYn) = (xn,Ayn). Now ( A x , , , y n )tends to ( A x , y ) and (x,,, Ay,,) tends to ( x , A y ) as n tends to infinity. Thus ( A x , y ) = ( x , A y ) ; and A is symmetric. Note that ( A iI)E,, has range En(&') since AE, is bounded and self-adjoint (so that AE,, f iE,,has a bounded inverse on E , , ( 2 ) ) , Thus A _+ iI has a dense range. From Lemma 2.7.9, this range is 2 ;and, from Proposition 2.7.10, A is self-adjoint. H

*

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If Ma is unbounded, we cannot expect it to belong to the multiplication algebra d of the measure space (S,Y: m).Nonetheless, there are various ways in which M gbehaves as if it were in d -for example, M gis unchanged when it is “transformed” by a unitary operator U commuting with R In this case (see Example 5.1.6), U E ~so, that U = M u where u is a bounded measurable function on S with modulus 1 almost everywhere (see Example 2.4.1 1). Withf in 9 ( M g ) ,gujc L 2 ( S ) ;while, if guh E L2(S), then gh E L,(S) and h E 9 ( M a ) . Thus U transforms 9 ( M g )onto itself. Moreover

( U * M g U ) ( f )= Uguf= lu12gj-=

trf:

Thus U*MaU = M g . The fact that Ma “commutes” with all unitary operators commuting with d in conjunction with Theorem 4.1.7 and the double commutant theorem (5.3.1) (from which it follows that a bounded operator having this property lies in d)provides us with an indication of the extent to which Ma “belongs” to R We formalize this property in the definition that follows. 5.6.2. DEFINITION. We say that a closed densely defined operator T is affiliated with a von Neumann algebra W (and write T q B ) when U*TU = T for each unitary operator U commuting with 9.

Note that the equality, U*TU = T, of the preceding definition is to be understood in the strict sense that U*TU and T have the same domain and (formal) equality holds for the transforms of vectors in that domain. As far as the domains are concerned, the effect is that U transforms 9 ( T ) onto itself. 5.6.3. REMARK.If Tis a closed densely defined operator with core goand U* TUX = Tx for each x in goand each unitary operator U commuting with a von Neumann algebra 9, then T q 9. To see this, note that, with y in 9(T), there is a sequence { y,,} in gosuch that y,, + y and Ty, + Ty (since g ois a core for T).Now Uy, -+ U y and TUy, = UTy,, + UTy. Since Tis closed, U y E 9(T) and TUy = UTy. Thus g(T)G V*(g(T)).Applied to U*, we have 9 ( T )s U(g(T)),so that U ( 9 ( T ) )= a ( T ) . Hence 9(U*TU) = 9(T)and U* TUy = Ty for each y in 9(T). Our discussion to this point establishes that M gis a closed operator (selfadjoint, when g is real-valued) affiliated with the multiplication algebra d. Conversely, if A is a closed (unbounded) operator affiliated with d,then A has the form Ma (and g is real-valued when A is self-adjoint). We prove this in the discussion that follows. Since each Bo in d is a linear combination of four unitary operators in d (see Theorem 4.1.7) and UA E A U for each such unitary operator (as d c_ d‘ and A q d), we have BOAc ABo. In particular, if E is the projection

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343

corresponding to multiplication by the characteristic function of a measurable subset So of S , EA c AE, so that E f e g ( A ) iffeQ(A). Let Q 1 be the set of essentially bounded functions in Q(A). Iff€ Q(A) and E, is multiplication by the characteristic function of { x :1fix)l < n } , { E n }is an ascending sequence of projections in .d tending to I in the strong-operator topology (sincefis finite almost everywhere). From the foregoing Enf e g1, {E,f)tends to f and AE, f = E,Af+ Afas n + co. Thus g1is a core for A . Iffand g are in g1,then f A g = M f A g = AMJg = A(f g ) = A M , f = MgAf = gAf. (1) Let { S , } be a family of mutually disjoint sets of finite positive measure with union S. If x, is the characteristic function of S,, there is a sequence {f,,}of elements of g1tending to x,. If

S,O = {s :S E S,, f,,(s) = 0 for all j } and x is the characteristic function of S,O, then 0 = M X f n+ j Mxxn= x , so that S,O has measure 0. Let g(s) be [ ( A f , , ) ( ~ ) ] [ f , , ( s ) ]where - ~ , seS,\S,O and j is the least integer such thatf,,(s) # 0. Then g is a measurable function on S defined almost everywhere. For eachfin g1,f,,(s)(Af)(s)=fis)(Af,,)(s) for all n a n d j except on a set of measure 0, from (1). Thus ( A f ) ( s )= g(s)fis) almost everywhere. We have noted that M gis closed and affiliated with d.Since M gis an extension of the restriction of the closed operator A to the core g1for A , we have that A c Mg.As we have noted earlier, the set of functions h in L2 vanishing on the complement of sets {s:s E S, 1g(s)l < m } forms a core for M g. With h such a function, let {f,}be a sequence of functions in B1tending to h. If x is the characteristic function of the set of points at which h does not vanish, we may replace f, by x f , . Changing notation, we may assume that each f, vanishes when h does. In this case, since M g M , is bounded, Af, = M g M x f ,+ MgMxh= Mgh.

As A is closed, h E 9 ( A ) and Ah = M,h. It follows that A = M g . If A is self-adjoint, Mgxis a bounded self-adjoint operator, so that gx is realvalued almost everywhere. Hence g is real-valued almost everywhere. Again, as with bounded multiplication operators, the projections E d , corresponding to multiplication by the characteristic function of the set where g does not exceed 2, form a spectral resolution {Ed}of A (see the discussion following Theorem 5.2.5). In this case, if A is, in fact, unbounded, there will be no non-negative real number a (replacing llAll when A is bounded) such that EA= 0 if A, c - a and Ed = I if a < A, ;and we speak of {Ed}as an unbounded resolution of the identity (see the discussion preceding Theorem 5.2.3). We summarize the foregoing conclusions in the theorem that follows. 5.6.4. THEOREM. r f ( S , x m ) is a o-finite measure space and d is its multiplication algebra acting on L2(S), then A is a closed densely defined

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5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

operator affiliated with d ifand only i f A = M Bfor some measurable function g finite almost everywhere on S. In this case, A is self-adjoint ifand only i f g is realvalued almost everywhere.

An unbounded self-adjoint operator A can be associated (“affiliated”) with an abelian von Neumann algebra. As in the case of a bounded self-adjoint operator (see Theorem 5.2.2),we can use Theorem 5.2.1 to locate a function on an extremely disconnected compact Hausdorff space that “represents” A. As might be expected, this function is neither everywhere defined nor bounded. We will be able to use this representing function (as in Theorem 5.2.2)to find a resolution of the identity (unbounded) for A. (See the discussion of resolutions following Theorem 5.2.2.) In preparation for this analysis, in the definition that follows we describe the functions that appear. 5.6.5. DEFINITION. If X is an extremely disconnected compact Hausdorff space a normalfunction on Xis a continuous complex-valued functionfdefined on an open dense subset X\Z of Xsuch that limq+plf(q)l= 00 for eachp in Z (where q E X\ Z ) . A self-adjoint function on Xis a real-valued normal function on X. Iff is self-adjoint and defined on X\ Z , we denote by Z, those pointsp of Z such that lim,,,f(q) = +00 and by Z - those p in Z such that lim,,,f(q) = - 00. We denote by N ( X )and Y ( X )the sets of normal and selfadjoint functions on X. H It is one of the many surprising properties of extremely disconnected compact Hausdorff spaces (and their associated constructs) that Z = Z, u Z - (that is, there are no points near which an f i n Y ( X ) takes arbitrarily large positive and negative values). This follows from the fundamental property of such spaces that disjoint open sets O1 and have disjoint closures (X\Ol is closed and, thus, contains 0; ,a clopen set, so that X\0; is closed and contains 0;).To see this, note that the sets

are disjoint open sets (sincefis continuous on X\ Z and X\ Z is open in X), and that a point near which f takes arbitrarily large positive and negative values would have to lie in both of their closures. Clearly If1 is normal on X iff is normal. The following simple lemma will prove useful to us.

5.6.6. LEMMA.Iff and g are normal functions on X defined on X\Z and X \ Z , respectively, and f ( q ) = g(q) for each q in some dense subset S of X \ ( Z u Z’), then Z = Z ’ and f = g . Proof. Since X\(Z u 2’)is dense in X,Sis dense in X. Ifp E Z‘ and q in Sis near p, then Ig(q)l (= If(q)1) is large so that P E Z . Thus Z’ c Z and,

5.6. UNBOUNDED OPERATORS AND ABELIAN VON NEUMANN ALGEBRAS

345

symmetrically, Z G Z'. Hence f - g is defined and continuous on X\Z and 0 on the dense subset S. It follows t h a t f = g.

5.6.7. LEMMA. ZfA is a self-adjoint operator acting on a Hilbert space X, A is affiliaied with some abelian von Neumann algebra. Zf A q d and d is isomorphic to C(X ) , with X an extremely disconnected compact Hausdorffspace, there is a unique self-adjoint function h on X such that h'e is in C ( X ) and represents A E when E is aprojection in d such that A E is a bounded everywheredefined operator, where e in C ( X ) corresponds to E, and (h :e)(p) is h ( p ) fi e( p ) = 1 and 0 otherwise. There is a resolution of the identity { E,} in d such that UF= Fn(H)is a core for A , where Fn = En - E-,,, and A x = J:, A dE, x for each x in F,,(X) and all n, in the sense of norm convergence of approximating Riemann sums. Proof. From Proposition 2.7.10 and Remark 2.7.11, A + il and A - il have range X, null space (0), and inverses T + and T - that are everywhere defined with bound not exceeding 1. Note that (T,(A

+ iI)x,( A - i l l y ) = ( x , ( A - i l l y ) = ( ( A + i l ) x , y ) = ( ( A + il)x, T - ( A - illy),

when x and y are in 9 ( A ) ,since A is self-adjoint. Thus T - = TT . (Recall that A f iZ have range X, so that ( A + i l ) x and ( A - illy represent arbitrary vectors in X.)Again, since A il have range X, we can represent an arbitrary vector as ( A - il)(A + il)x, where x E 9 ( A ) and A x E 9 ( A ) . In this case,

*

( A - il)(A + iI)x = ( A 2 + Z)x = ( A + il)(A - il)x, and T +T - = T - T , . Since T- = T: , T , is normal. Let d be an abelian von Neumann algebra containing I, T , , and T - . If U is a unitary operator in d', for each x in 9 ( A ) , U x = U T + ( A+ i l ) x = T+U(A+ i l ) x so that ( A + iI)Ux = U(A + i l ) x ; and U - ' ( A + iI)U = A + il. Thus U - ' A U = A and A q d . In particular A q d O , where d ois the (abelian) von Neumann algebra generated by I, T , , and T - . From Theorem 5.2.1, d 1C ( X ) , where X is an extremely disconnected compact Hausdorff space. Let g + and g- be the functions in C ( X )corresponding to T , and T - . Let h + and h- be the functions defined as the reciprocals of g + and g - , respectively, at those points where g+ and g - do not vanish. Then h , and h - are continuous where they are defined on X , as is g h + + h - ) ( = h). In a formal sense, h is the function that corresponds to A. We shall see that h is real-valued and find the spectral resolution of A by subjecting h to the same spectral analysis as was performed on the functions of C ( X ) in proving Theorem 5.2.2. Since T , and T - are adjoints of one another, g+ and g- are complex conjugates of one another (in particular, they vanish at the same points of X ) .

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5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

Thus h , and h- are complex conjugates of one another; and h is real-valued. The set Z on which g+ (and g-) vanishes is closed (since g+ is continuous) and nowhere dense; for if it contains a non-null open set it contains its closure, a non-null clopen set. The projection corresponding to this non-null clopen set would have product with T , (and T - ) equal to 0-contradicting the fact that T , (and T - ) have null space (0). Thus each point p in Z is a limit of points q in X\Z (at which h is defined). For each y in 2, A T , T - y = ( A + iZ - iZ)T, T - y = T - y - iT, T-y,

so that A T , T (2)

=

T - - iT, T - . Similarly AT- T , = T ,

+ iT, T - . Hence

2iT+T- = T - - T ,

and (3)

A T + T - = i ( T ++ T - ) .

It follows from (2) that (h(q) + i)-’ = g+(q) (and that (h(q) - i ) - l = g-(q)) forqinX\Z. Hence,foreachqinX\Znearp(inZ),g+(q)isnearOand Ih(q)lis large. Thus h is a self-adjoint function on X . Let 0,be U , u Z, ,where U , is the set of points of X \ Z a t which h exceeds I and Z, has the meaning explained in Definition 5.6.5. We show that 0, is open. Since h is continuous on X\Z and X\Z is open in X , U , is open in X . If p E Z , , there is an open set 0 containing p such that if q E 0 n ( X \ Z ) , then h(q) > 0 -and since Ih(q)Jis large for q (in X\ Z ) near p , we may choose 0 such that h(q) > I for q in 0 n ( X \ Z ) . In this case, 0 n (X\Z) E U , E 0,.If there were ap’ in 0 n Zwithp’ in Z- ,then, from the definition of Z - ,there would be a q in 0 with h(q) strictly negative-contradicting the choice of 0. Thus 0 n Z E Z, ; 0 E 0,;and 0,is open, as asserted. Let X , be X\0;. Again, as in the proof of Theorem 5.2.2, X , contains eachclopen set Y such that h ( p ) ,< I for allp in Y. (For pointsp of Z - , where “ h ( p ) = - co,” we write “ h ( p ) ,< I” as well, so that Y may contain points of Z- .) Indeed, if q E O,, then q E X \ Y, a closed set, so that 0, c X\ Y and Y E X\0; = X , . At the same time, A’, is such a clopen set, for if PE X , n (X\Z), then, since pq! U,, h ( p ) < A. If p E X , n Z , then p E Z\ Z, (= 2 - )and h ( p ) < I (in the extended sense). Thus X, is the largest clopen set on which h ( p ) < I ; and X , n Z = Z - . We proceed now as in the proof of Theorem 5.2.2. Let e, be the characteristic function of X , and Ed be the projection in d corresponding to e,. In this case, { E d }satisfies El < Ed, if A < 1’ and Ed = A a < L , EA..Since Z is nowhere dense, V, e, = 1 and A e, = 0, so that V, Ed = I and A Ed = 0. That is, we have constructed a resolution of the identity { E d } (and this resolution is unbounded if h $ C(X)). Let F be Eb - E,, where a < b. Then eb - e, (=A, the characteristic function of Xb\X,, corresponds to F. Since X b n Z = X , n Z (= Z - ) , we have that &\Xu c X \ Z ; and g + ( p ) g - ( p ) # 0

,

,

5.6. UNBOUNDED OPERATORS AND ABELIAN VON NEUMANN ALGEBRAS

347

when f(p) = 1. For p in X\ 2,

(4) by choice of h. Moreover, there is a positive function k in C ( X ) such that kg+g- =fand kf = k (since g + g - is continuous and vanishes nowhere on the clopen set &\X,). If K in d corresponds to k , then (5)

K T + T - = F.

From our information about X l , if p E Xb\X,, then p 4 2 and a Thus from (4), w+g-f< & J +

+g-)f
n ( j - 1)) such that J U , , ~ , J< j - 2 f o r j in N. Let x' be x ; = l j - l x n ( j ) .Show that for each A in the strong-operator closure 2I- of a, Ax0 # (iii) Conclude that a' is finite dimensional if there is a vector xoin 2 such that { A x , : A E%-} = 2. XI.

5.7.41. Let 2I be a C*-algebra that acts topologically irreducibly on a and let {x,, . . . ,x,} and { y ,,. . . ,y,} be sets of vectors in 8. Hilbert space 2, (i) Let H be a self-adjoint operator in g ( 2 )such that H x j = y j f o r j in { 1,. . . ,n } . Show that there is a self-adjoint operator Kin % such that Kxj = y j for j in { 1,. .. ,n} and llKll < llHll. [Hint. Use a diagonalizing orthonormal basis for EHErestricted to [xl ,. . . ,x,, y , ,. . . ,y,], where Eis the projection in g ( 2 )with this subspace as range, and apply Remark 5.6.32.1 (ii) Let B be an operator in a(2)such that Bxj = y j f o r j in { 1 , . . . ,n } . Show that there is an operator A in '3 such that A x j = y j f o r j in { 1,. . .,n} and

[ [ A1) < IlBll. [Hint. With E as in the hint to (i), use the fact that EBE 1 E ( 2 ) has the form VH with V a unitary operator and H a positive operator on E(X).] 5.7.42. Let .X0be a Hilbert space and 2 be the direct sum 1,@ 2,of countably many copies 2,of X 0 .

(i) With the notation of Subsection 2.6, Matrix representations, let W be the subalgebra of g ( 8 )consisting of operators whose matrix has the same element of a(*,) at each diagonal entry and 0 at each off-diagonal entry. (With 2 viewed as a tensor product 2,0 X of 8o with a separable, infinitedimensional Hilbert space X, W is { T Q I : TEg(iW0)}.) Show that 9' consists of those operators whose matrix representations have scalar multiples of I at each entry. (In tensor product form, w' = { I @ S : S e g ( X ) } . ) Show that 9"= 9 and conclude that W is a von Neumann algebra (as well as 9'). Call this sequence 12(ii) Let xl, x2,. . . be a sequence of vectors in 2,. independent when I ;= llxj112 < co and IT=ajxj = 0 for a sequence { a j } in 12(N,C) only if aj = 0 for allj. Note that an 12-independentsequence is linearly independent and find a linearly independent sequence that is not 12independent when 8, is infinite dimensional. (iii) Show that {xl, x2,. . .} is a generating vector for W if and only if xl, x 2 , .. . is 12-independent.

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5. ELEMENTARY VON NEUMANN ALGEBRA THEORY

(iv) Show that { x l , x 2 , . .} is separating for 9 if and only if [x1,x2,

...I = s o .

5.7.43. Let X be L2([0,13) relative to Lebesgue measure. With d the multiplication algebra of 2 and ‘illthe C*-algebra of multiplications by continuous functions, find a unit vector u that is separating for ‘ill(if A E ‘illand Au = 0, then A = 0) but not for d.[Hint. Use the “Cantor process” to find an open dense subset of [0,1] that has measure $1 5.7.44. Let X‘ be L2([0,13) relative to Lebesgue measure and d be the multiplication algebra of X.

(i) Describe the vectors in X‘ that are generating for d. (ii) Show that the set of generating vectors for d is dense in X (iii) Deduce that a norm limit of generating vectors need not be a generating vector. 5.7.45. Let W be a von Neumann algebra and { E l ,E 2 , .. .} be a countable family of countably decomposable projections in 9. Show that V=; En is a countably decomposable projection in 9. 5.7.46. Let 9 be a countably decomposablevon Neumann algebra acting on a Hilbert space X. Define a metric on W with the property that its associated metric topology coincides with the strong-operator topology on bounded subsets of 9. 5.7.47. Let (S, Z m )be a a-finite measure space, X‘ be L2(S,m),d be the multiplication algebra, andfand g be measurable functions on S finite almost everywhere. Show that:

(i) (ii) (iii) (iv)

M f = M , if and only i f f = g almost everywhere; Mar +, = a M f 4 M , for each scalar a ; M,., = M , M , ; M f 2 0 if and only i f f 2 0 almost everywhere.

5.7.48. Let (S, Z m )be a a-finite measure space, X‘ be L,(S, m),d be the multiplication algebra, g be a measurable function on S finite almost everywhere, and f be a Bore1 function on sp(M,).

(i) Define a concept of “essential range” sp(g) analogous to that of Example 3.2.16 and show that sp(g) = sp(M,). (ii) Show that there is a measurable go on S equal to g almost everywhere such that the range of go is contained in sp(g). (iii) With the notation of (ii), show thatf(M,) = MJeao.

38 1

5.7. EXERCISES

5.7.49. Let ,Y?' be L2(R) relative to Lebesgue measure and A be the (unbounded) multiplication operator corresponding to the identity transform I (the function t + r ) on iw with domain 9 consisting of thosefin L2(iw)such that I .f€L,(R).

(i) Note that A is self-adjoint and that the spectral resolution for A is { E l } ,where El is the multiplication operator corresponding to the characteristic function of (- co,A]. (ii) Let gobe the set of continuously differentiable functions on iw that vanish outside a finite interval. Note that gois a dense linear submanifold of X. Let Do be the operator with domain gothat assigns if' (= idfldt) t o 5 With T the unitary operator defined in Theorem 3.2.31, show that T - l A T f = Dof for each f in go. (iii) Conclude that T - ' A T (= D)with domain T - l ( 9 ) is a self-adjoint extension of Do. (iv) Show that exp(itD) is the unitary operator V , , where ( U , f ) ( p )= f ( p - t ) . How does this relate to Stone's theorem? 5.7.50. Let f be a jointly continuous function of two complex variables defined on S1 x S 2 , where S1 and S2 are subsets of @. Let A be a normal operator such that sp A G S2acting on a Hilbert space X. For each z1 in S1, the mapping z +f(zl, z) is a continuous (hence, Borel) function defined on sp A so that f(zl,A ) is a normal operator on #. Define an operator-valued function g on S1 by g(z) = f(z,A ) .

(i) Suppose A is bounded and So is a compact subset of S1. Show that the restriction of g to So is norm continuous (that is, the mapping z -,g(z) is continuous from So to A?(%) with its norm topology). (ii) Suppose S1is closed and f is bounded (but no longer that A is bounded). Show that g is strong-operator continuous. (iii) Let H be a positive operator on X. Show that exp( - izH) (= V,) is defined for each z in the closed lower half plane C - (= {z: I m z < 0}), that llUzll d 1, and that the mapping z -,V , is strong-operator continuous. 5.7.51. (i) With the notation and hypotheses of Exercise 5.7.50(ii), make the following additional assumptions about f:

(1) for each z2 in S2,z +f(z, z2)is differentiable at each point zo of the interior S(: of S1 with derivative fi(zo, z2), (2) given zo in Sy ,a bounded subset S ; of S2,and a positive E , there are a positive 6, aclosed disk D with center zo in S y , and a positive C , such that for all z 2 in S ;

ICf(Z922) -f(zo,z2)1(z - Zo)-l -fi(zo,z2)l
0; HxO = 0 ; xo is generating for 9.

W and each t in

[w;

(i) Define U, as in Exercise 5.7.5O(iii) when Z E @ - , and show that Uzxo= xo for each z in C- . With A and A’ self-adjoint operators in W and #,respectively,definef(z) to be ( U,Axo, A’xo) for z in C - . (ii) Show thatf(t) is a real number for real t , and thatfis continuous on C - , analytic on C! , and bounded on Q= - . (iii) Show that U,EW for each real t. 5.7.53. Let H be a self-adjoint operator acting on a Hilbert space 2,d be the von Neumann algebra generated by H , and dobe the von Neumann algebra generated by { U, : t E W}, where U, = exp( - itH). and a0 (i) Assume H is bounded and show that the C*-algebras generated by H (and I ) and by { U, : t E W}, respectively, coincide. (ii) Show that d = do. (iii) With the notation and assumptions of Exercise 5.7.52, show that HVW.

S.7. EXERCISES

383

5.7.54. Let A be a normal operator acting on a Hilbert space 2.

(i) Show that exp iA = I if sp A G (2nn : n E Z} and only if A is selfadjoint and sp A c {2nn :n E Z}. (ii) Show that expitA = I for all t in R if and only if A = 0. (iii) Let A and B be self-adjoint operators on A? such that expitB = expitA for each real t . Show that A = B. (iv) Let t + U, be a one-parameter unitary group acting on A?.Show that there is a unique self-adjoint operator H on 2 such that U, = exp itH for all real t.

5.7.55. With the notation of Exercise 5.7.52, assume conditions (l), (2) and (3), and in place of (4) assume that x, is separating for the center V of W. Show that there is a positive self-adjoint operator K on A? such that K q 3 and W,AW-, = U,AU-, for each A in W and all real t , where W, = exp(- itK) (EW), and such that Kx, = 0. [Hint.Consider the projection E' with range [ a x , ] and the von Neumann algebra a E ' acting on E ' ( X ) . ]

BIBLIOGRAPHY General references [HI P. R. Halmos, “Measure Theory.” D. Van Nostrand, Princeton, New Jersey, 1950; reprinted, Springer-Verlag, New York, 1974. [K] J. L. Kelley, “General Topology.” D. Van Nostrand, Princeton, New Jersey, 1955; reprinted, Sppnger-Verlag, New York, 1975. [Rl W. Rudin, “Real and Complex Analysis,” 2nd ed. McGraw-Hill, New York, 1974.

References [I] W. Ambrose, Spectral resolution of groups of unitary operators, Duke Math. J. 11 (1944), 589-595. [2] J. Dixmier, “Les C*-Algkbres et Leurs Representations.” Gauthier-Villars. Paris, 1964. [English translation: “C*-Algebras.” North-Holland Mathematical Library, Vol. 15. North-Holland Publ., Amsterdam, 1977.1 [3] J. M. G . Fell and J. L. Kelley, An algebra of unbounded operators, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 592-598. [4] I. M. Gelfand and M. A. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sb. I2 (1943), 197-213. [5] J. G. Glimm and R. V. Kadison, Unitary operators in C*-algebras, Pacific J. Math. 10 (1960), 547-556. [6] F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and Lowner’s theorem. Marh. Ann. 258 (1982), 229-241. [7] D. Hilbert, Grundziige einer allgemeinen Theorie der linearen Integralgleichungen IV, Nachr. Akad. Wiss. Gotringen Math.-Phys. K1. 1904, 49-91. [81 R. V. Kadison, Irreducible operator algebras, Proc. Nar. Acad. Sci. U.S.A.43 (1957), 273-276. [91 I. Kaplansky, A theorem on rings of operators, Pacific J. Math. 1 (1951), 227-232. [lo] J. L. Kelley, Commutative operator algebras, Proc. Nat. Acad. Sci. U.S.A.38 (1952), 598-605. [Ill J. von Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102 (1930), 370-427. [ 121 J. von Neumann, AIlgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Mark Ann. 102 (1930), 49-131. [13] J. von Neumann, Uber Funktionen von Funktionaloperatoren, Ann. of Math. 32 (1931), 191-226. [I41 J. von Neumann, Uber adjungierte Funktionaloperatoren, Ann. of Marh. 33 (1932), 294-3 10. 384

BIBLIOGRAPHY

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[IS] M. Neumark, Positive definite operator functions on a commutative group (in Russian, English summary), Bull. Acad. Sci. URSS Str. Math. [Iz. Akad. Nauk SSSR Ser. Mat.] 7 (1!N3), 237-244. [16] G. K. Pedersen, “C*-Algebras and Their Automorphism Groups,” London Mathematical Society Monographs, Vol. 14. Academic Press, London, 1979. [I71 F. Riesz, “Les Systbmes d’kquations Lineaires 9 une InfinitC d’hconnues.” Gauthier-Villars, Paris, 1913. [18] F. Riesz, Uber die linearen Transformationen des komplexen Hilbertschen Raumes, Acta Sci. Math. (Szeged)5 (1930-1932), 23-54. [19] I. E. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Sac. 53 (1947), 73-88. [20] M. H. Stone, On one-parameter unitary groups in Hilbert space, Ann. ofMath. 33 (1932), 643-648. [21] M. H. Stone, “Linear Transformations in Hilbert Space and Their Applications to Analysis”. American Mathematical Society Colloquium Publications, Vol. 15. Amer. Math. SOC.,New York, 1932. [221 M. H. Stone, The generalized Weierstrass approximation theorem, Math. Mag. 21 (1948), 167-183, 237-254. [231 M. H. Stone, Boundedness properties in function-lattices; Canad. J. Math. 1 (1949), 176- 186. 1241 S. Stratila and L. Zsidb, “Lectures on von Neumann Algebras.” Abacus Press, Tunbridge Wells, 1979. [251 M. Takesaki, “Theory of Operator Algebras I.” Springer-Verlag, Heidelberg, 1979.

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INDEX OF NOTATION

Algebras and related matters

closed sum of operators, 352 closed product of operators, 352 algebra of convolution operators, 190 positive cone in a;244 set of self-adjoint elements of a, 249 weak-operator closure of a, 328 norm closure of a, 328 norm closure of d l ( R ) , 190 'UI(R) with unit adjoined, 190 algebra of Borel functions on C, 359 algebra of Borel functions on X,358 central carrier, 333 reduced von Neumann algebra, 336 reduced von Neumann algebra, 335 { A X : A E . P } 276 , { A x :A E S " , X E X } 276 , commutant, 325 double commutant, 326 { A * : A E ~ }326 , set of holomorphic functions, 206 unit element, identity operator, 41 isomorphism between algebras, 310 left kernel of the state p , 278 operator, on L2,of convolution byf, 190 positive cone in A, 255 set of self-adjoint elements of A, 255 set of multiplicative linear functionals on a,@), 197 195 algebra of operators affiliated with d , 352 algebra of normal functions on X, 344 set of pure states of d,261 pure states space of A, 261 GNS constructs, 278 dual group of R, 192 spectral radius, 180 spectral radius, 180 restricted von Neumann algebra, 334 restricted von Neumann algebra, 336 spectrum of A , 178, 357 - a Q ) \ { P a J 9

387

388

INDEX OF NOTATION

spectrum of A in 8,178 essential range of j , 185, 380 set of self-adjoint affiliated operators, 349 state space of J?, 257 state space of V, 213 set of self-adjoint functions on X , 344 dual group of TI, 231 T is affiliated with 9, 342 vector state, 256 vector functional, 305 dual group of Z, 230 Direct sums direct direct direct direct direct direct direct direct

sum of Hilbert spaces, 121 sum of Hilbert spaces, 121 sum of Hilbert spaces, 123 sum of vectors, 123 sum of operators, 122 sum of operators, 124 sum of representations, 281 sum of von Neurnann algebras, 336

Inner products and norms