Operator Algebras Generated by Commuting Projections: A Vector Measure Approach (Lecture Notes in Mathematics)

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1711

Springer Berlin Heidelberg New York Barcelona Ho ng Ko ng London Milan Paris Singapore Tokyo

Werner Ricker

Operator Algebras Generated by Commuting Projections: AVector Measure Approach

Springer

Author Werner Ricker School of Mathematics University of New South Wales Sydney, NSW, 2052 Australia e-mail: werner @maths.unsw.edu.au Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ricker, Werner: Operator algebras generated by commuting projections: a vector measure approach / Wemer Ricker. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1711) ISBN 3-540-66461-0

Mathematics Subject Classification (1991): 28B05, 06E15, 47B40, 47D30 ISSN 0075-8434 ISBN 3-540-66461-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10700204 41/3143-543210 - Printed on acid-free paper

For Margit, Simon and Sandra; without their constant encouragement notes would never have eventuated.

and support these

PREFACE In the summer semester of 1997 Professor Jim Cooper invited me to give an advanced set of lectures at the Honours/Masters level in the Mathematisches Institut of the Johannes Kepler Universitgt in Linz, Austria. He left the choice of topic up to me; his only request was that the topic should be of an interdisciplinary nature since the students already had a good background in such individual courses as algebra, linear algebra, real and complex analysis, functional analysis and measure theory, etc.. The content of this book is essentially an expanded version of the lectures given in Linz. The material was chosen in an attempt to illustrate to advanced students that it is indeed possible to present lecture courses within their mathematical reach which form a connecting bridge between many of the specialized courses that they have already had and such that it "all comes together." In addition to being able to absorb a body of mathematical knowledge (hopefully developed in a systematic and coherent way), students at this level should also become accustomed to the methodology of mathematical research. They should be able to go to libraries and consult research books and articles, extract from these the relevant information, do some independent thinking, come to the realization that not all problems have instant solutions, etc.. Accordingly, there are many references to the mathematical literature (which the reader is expected to follow up), both in the text and in the various exercises. The exercises are a mixture of fairly routine ones (indicated by [.]) to somewhat more challenging ones and form an integral part of the notes. This book is surely not a pure research book on the topic; for this we refer to the excellent monographs [13] and [15], for example. It is more of a hybrid and, for this reason, definitions and statements of results are carefully formulated and referenced, examples are included to illustrate various points, and many of the proofs are quite detailed as they are designed for the working student and future researcher, and not (necessarily) current experts. At the same time, several of the chapters contain a significant amount of material which may also interest current researchers in the area. Moreover, any reader who achieves a firm grasp of the material is well placed to begin serious research in the general area of operator theory alluded to in these notes, especially in some of the more recent directions. I have here in mind two general areas. Firstly, there is the extension of the theory to the setting of non-normable spaces, where genuine new phenomena arise which are not present in the Banach space setting. Even though much has already been achieved in this direction in the past 20-30 years (see the works of P.G. Dodds, C.K. Fong, J. Junggeburth, S. Okada, B. de Pagter, W.J. Ricker, A. Shuchat, W.V. Smith in the Appendix, and of C.

viii

PREFACE

Ionescu Tulcea, F.Y. Maeda, H.H. Schaefer, B.J. Walsh listed in the bibliography of [15]), there are still several important problems which remain unresolved. The other area is in the direction of harmonic analysis and differential operators in Euclidean//-spaces, which often generate families of commuting projections based on algebras or &rings of sets rather than a-algebras of sets; see the works of E. Albrecht, G.B. Folland, O. Oaudry, G.E. Huige, J. Locker, G. Mockenhaupt, M.A. Shobov, W.V. Smith, H.J. Sussmann, I.P. Syroid in the Appendix, and of V.E. Ljance, V.A. Mar~enko, M.A. Naimark, B.S. Pavlov, J.T. Schwartz listed in the bibliography of [15]. Such families of projections are typically not uniformly bounded and so will not lead to a Borel functional calculus of the type usually associated with a spectral operator. New techniques will be needed to analyze the operator algebras that such families of projections generate. Many of the results presented are classical so I have not attempted to record the source of every item. References are not always to the original source, but often to more recent works where further references can be found. The absence of a reference does not necessarily imply originality on my part. The reader is assumed to have a basic grasp of standard undergraduate courses in algebra, linear algebra, set theory (manipulation), topology, functional analysis, measure theory and integration. Since not all of the readers will have a common base of knowledge in this regard, and for the reason of self-containment, some of these basic notions and facts are included. This is especially true of those having a direct bearing on the subject matter. More specialized material (eg. measure algebras, vector measures and integration, Stone spaces, aspects of operator algebras, functional calculi, etc.) is developed along the way, but only to the extent required for these notes. In an Appendix at the end of the text I have attempted to form an extensive bibliography of research articles in the general area of spectral operators and Boolean algebras of projections which have appeared since 1979. For articles prior to 1979 we refer the interested reader to the excellent bibliographies in [13] and [15]. Some relevant papers prior to 1979 have also been included, provided they do not occur in [13] or [15]. The reason for this Appendix is two-fold. First, it is always useful for any student and/or researcher to have access to such extensive and up-to-date bibliographies. Second, and perhaps more important, I wish to illustrate to students and future researchers that this is an active area of modern research. This can be seen not only from the number of articles and their diversity, but also from the number of mathematicians who have contributed to the area. Special thanks go to my colleagues and friends J.B. Cooper, K. Kiener, E. Matou~kov~ and C. Stegall from Linz. Their encouragement, attention, assistance and above all, their patience, were remarkable. To all of my many colleagues and friends over the past years who have, at various stages, listened to my thoughts and ramblings on this topic (both directly and indirectly) and who have made helpful suggestions (both positive and negative), I especially wish to thank E. Albrecht, R.G. Bartte, I.D. Berg, E. Berkson, P.C. Dodds, I. Doust, D.H. Fremlin, T.A. Gillespie, D. Hadwin, B.R.F. Jefferies, I. Kluvgnek, H.P. Lotz, A. McIntosh,

IX S. Okada, M. Orhon, B. de Pagter, F. Rgbiger, P. Ressel, H.H. Schaefer, J.J. Uhr Jr., and A.I. Veksler. Finally I wish to thank Mrs J. Kos and Ms V. Pratto, both for their excellent typing and for their unlimited tolerance and understanding, and Dr P. Blennerhassett for his expert assistance in several of the finer points of I~TEX. Sydney; June, 1999

Contents PREFACE INTRODUCTION I

Vector measures a n d Banach spaces

II

A b s t r a c t Boolean algebras and Stone spaces

V

ix 1

25

I I I Boolean algebras of p r o j e c t i o n s and uniformly closed o p e r a t o r algebras 41 IV

Ranges of s p e c t r a l measures a n d Boolean algebras of p r o j e c t i o n s

57

V

I n t e g r a l r e p r e s e n t a t i o n of t h e strongly closed a l g e b r a g e n e r a t e d by a Boolean a l g e b r a of p r o j e c t i o n s 67

VI

B a d e functionals" an a p p l i c a t i o n to s c a l a r - t y p e s p e c t r a l o p e r a t o r s

91

V I I The reflexivity t h e o r e m and b i c o m m u t a n t algebras

105

Bibliography

121

Appendix

125

List of symbols

153

S u b j e c t index

156

INTRODUCTION One of the fundamental facts learnt in linear algebra courses is a basic structural result referred to as the Jordan decomposition theorem. Namely, in a finite dimensional vector space X every linear map T : X --+ X can be decomposed as T = S + N, where S is a diagonalizable operator (i.e. with respect to a suitable basis of X it is similar to a diagonal operator) and N is a nilpotent operator (i.e. the spectrum a ( N ) , of N, consists just of {0} or, equivalently, N k = 0 for some non-negative integer k) satisfying S N = NS. The operator S is called the scalar part of T and N is called the radical part of T. In particular, S has a representation of the form r

(1)

s : ~ A:Ej, j=l

where or(S) = or(T) = {"~j};:l consists of the distinct eigenvalues of S and {Ej};= 1 is a T E J = I (the identity operator on family of non-zero projections (i.e. Ey = E~) with Y]j=I X) and satisfying EjEk = 0 = EkEj whenever j # k. So, the study of such scalar operators S reduces to a study of the family of much simpler operators i j E j , for 1 _< j _< r. In fact, if r Xj = EyX is the range of Ej, then the family of vector subspaces {X j}j=l has the properties that Xj AXk = {0} for j # k, that X 1 0 . . . | XT = X and that S X j C_ Xj, for 1 _< j _< r. In particular, S restricted to X3 (which is the same as I j E j restricted to Xj) acts like l j I j in Xj, where Ij : X 5 ~ X~ is the identity operator. For an elegant and succinct account of this topic in terms of linear operators (rather than the usual matrix approach) we refer to [14; Chapter VII, Sections 1 & 2]. W h a t happens if X is infinite dimensional and the linear operator S is continuous? Consider first the case when X is a Hilbert space. If S is compact and normal (or selfadjoint), then the classical spectral theorem of D. Hilbert asserts that or(S) = {0} O {lj}j~__l is a countable set in C (or R) with l i m ~ ),~ = 0 (in the case when a(S) is infinite) and S has a representation of the form (compare with (1))

j=l

oo where the commuting family of non-zero, selfadjoint projections { E J}j=l is pairwise disjoint oo and satisfies ~ j = 0 Ej = I; here E0 is the orthogonal projection of X onto {x E X : Sx = 0}.

INTRODUCTION

XlV

The series (2) and the series ~ j : o E j = I both converge in the strong operator topology. Removing the compactness requirement on S has the effect that a(S) may no longer be discrete. Indeed, cr(S) can then be any compact subset of C (or R if S is selfadjoint). Moreover, to every Borel set A C_ C (the c~-algebra of all such sets is denoted by Bo(C)) there corresponds a selfadjoint projection E(A) such that E((0) = 0, E(C) = I and the projections in the range E(Bo(C)) of E satisfy

(3)

E(A)E(B) : E(A A B) = E(B)E(A),

A, B E Bo(C),

and

E(un=l fiE(A n )~),

(4)

=

oo

A

n=l

whenever {A~}~__I C_ Bo(C) are pairwise disjoint sets. Of course, the series (4) again converges in the strong operator topology. The condition (4) says that A ~-~ E(A) is a projectionvalued measure on Bo(C). What is the analogue of (2)? Adopting the naive approach that integrals usually replace sums (in the "limit") suggests that

(5)

s= s

s )`dE/)`),

where the operator-valued integral (5) needs to be suitably defined. This turns out to indeed be the case and (5) is a formulation of the classical spectral theorem for arbitrary normal (or selfadjoint) operators. The important features from the abstract point of view are that S is synthesized from a certain family of projections {E(A) : A r Bo(C)} via an integral formula of the type S = f c f()`) dE(),), where f()`) = )`, for )` E C. Moreover, the multiplicative property (3) of E implies that

s" = f )`ndE()`) = f f()`)~ de(X),

~=0,1,2,...,

and more generally, that 0

g(s) := J~ g()`) dE()`) for any Borel measurable function g : C ~ C which is bounded on a(S). Actually, ~(S) turns out to be the support of the measure E. So, all reasonable operators which are "functions of S", that is, operators of the form g(S) for suitable 9, are built up from the projections {E(A) : A r Bo(C)}. If we wish to stay within the realm of normal operators, then it is necessary to require {E(A) : E E Bo(C)} to be a selfadjoint family. However, the properties (3) and (4)

XV are independent of selfadjointness and so it is undesirable to require this condition from the outset. Moreover, removing this property is no great restriction. Indeed, the well known Mackey-Wermer theorem asserts that if the integral in (5) exists for an arbitrary projection-valued measure E (with respect to the strong operator topology), then there exists a selfadjoint isomorphism W : X ~ X such that the family of commuting projections { W E ( A ) W -1 : A E Bo(C)} consists entirely of selfadjoint projections. So, the infinite dimensional Hilbert space analogue of a scalar operator (still called a scalar operator) is any continuous linear operator S which is similar to a normal operator, in which case it has an integral representation of the form (5) for some projection-valued measure E defined on Bo(C). The analogue of a nilpotent operator N is still one which satisfies or(N) = {0}. However, in infinite dimensional spaces this becomes equivalent to l i m , ~ IINnll v~ = 0, rather than to some power of N being 0; such operators are called quasinilpotent. So, a natural class of continuous linear operators in an infinite dimensional Hilbert space which corresponds to the familiar class of all linear operators in a finite dimensional space, consists of those operators T which have a decomposition (6)

T= S + N= [

)~dE(),) + N,

Jo"(T) where S is a scalar operator and N is a quasinilpotent operator satisfying S N = NS. In this formulation we see that even the Hilbert space structure of X is no longer crucial; the definitions of a scalar operator and quasinilpotent operator make perfectly good sense in a general Banach space X. In this setting, operators T of the form (6) are called spectral operators. This important class of operators, initiated by N. Dunford in the late 1940's and early 1950's, has undergone intense research ever since. The aim of these notes is to concentrate on certain particular aspects of the theory of scalar operators, especially in the Banach space setting, where the results and methods differ significantly from those in the Hilbert space setting. As discussed above, the central notion is the family of projections B = {E(A) : A C Bo(C)}, the so called resolution of the identity, from which the scalar operator S is synthesized. However, to insist on indexing the projections in B by elements of Bo(C) is, from the theoretical and practical viewpoint, both unnecessary and unduly restrictive. So, the basic concept throughout will be that of a family of commuting projections B, assumed to form a Boolean algebra but otherwise not indexed in any particular way. Since we will be interested in those operators which can be "built up" from the elements of the Boolean algebra B, it is natural to require the linear span of/3 to be an algebra (not just a vector space) and, since some limiting procedures will have to be involved (to pass from sums to integrals, for example), it will also be necessary to take the closure of this linear span with respect to some suitable topology. Moreover, to have any hope of identifying elements which arise as some sort of limit from expressions of the form Y'~j=IPJ E J, where pj E C and EjEk = 0 = EkEy if j 7~ k, it is also a necessity to require sup{llEH : E ~/3} to be finite; this condition is automatic if/3 consists of selfadjoint projections in a Hilbert space, but not in general.

XVl

INTRODUCTION

So, we arrive at the following setting: given is a Banach space X and a commutative, unital subalgebra/A (of continuous linear operators on X) which is closed with respect to some topology and is generated by some Boolean algebra of projections B (assumed to be uniformly bounded). Our main purpose is to investigate, systematically and in detail, the theory of such operator algebras and to attempt to answer various natural questions. As a sample, we will consider the following problems. (i) Is it possible to give a concrete description of the elements of b/ in terms of those from B? The answer will depend on various factors; the properties of the underlying Banach space X, the topology used in b/, and on certain properties of B itself. This question is the central theme of Chapter III, where the uniform operator topology is considered, and of Chapter V, where the strong and weak operator topologies are relevant. (ii) Are the elements of b/all of the form 9(S) for suitable functions 9 and some scalar operator S? The important ingredients here turn out to be the "size" of B and certain properties of the Banach space X. One of the main results will be to show that the answer is affirmative if the Boolean algebra/3 is complete in a certain sense and if X is separable. This forms the core of Chapter VI and is a far reaching extension of the well known fact that every strongly closed Boolean algebra of selfadjoint projections in a separable Hilbert space is the resolution of the identity of some selfadjoint operator. (iii) Are there other descriptions of the elements of/~ with a more algebraic fiavour? For instance, if X is a Hilbert space, then a classical result due to J. von Neumann provides a positive answer in terms of the bicommutant of/3 (provided that/3 consists of selfadjoint projections). Other descriptions are known in terms of the lattice of closed, /3-invariant subspaces of X. A detailed discussion of this topic is presented in Chapter VII. Questions such as those above, and many more, were considered by N. Dunford and others. Several of the major results (but, certainly not all) concerning such operator algebras can be found in two penetrating papers by W.G. Bade [1,2]. These results, and others, are well documented in [13] and [15], for example. Anyone who spends time reading these monographs will realize immediately the beautiful combination of methods employed from a variety of areas within mathematics. From algebra we see the theory of partial orders, Boolean algebras and the representation results of M.H. Stone (as a sample), from functional analysis there is Banach algebra theory, functional calculi, Banach space geometry, weak and weak-star topologies, Alaoglu's theorem and so on, from measure theory we have the Riesz representation theorem, the Radon-Nikodym theorem, the Hahn decomposition theorem, operator-valued integrals and so on, from topology there occur various disconnected spaces, Urysohn's extension theorem, the Stone-Cech compactification, etc. etc.. So, there is no question that we axe dealing with an "interdisciplinary topic". In discussing commutative operator algebras which are uniform operator closed it is natural to employ Banaeh algebra techniques (as is the case in [15]). However, such methods are not always suitable to describe the strongly or weakly closed algebra generated by a Boolean algebra of projections. One of our main goals is to systematically employ the

• methods of vector measures and integration theory (developed in Chapter I to the extent needed for our purposes) to represent this algebra as an Ll-space of a spectral measure. Once this representation theory is available many of the results alluded to above are easy and natural consequences. In particular, our approach yields proofs of several of the well known theorems in the area which are quite different to the proofs given in [15]. That vector measure techniques can be employed at all relies on the fact that any Boolean algebra of projections/3 (with suitable completeness properties) can be realized as the range of a spectral measure defined on the Baire or Borel sets of the Stone space of/3. This subtle interplay between Boolean algebras of projections and spectral measures, which plays a crucial and unifying role throughout these notes, is carefully developed in Chapter IV. To fully appreciate this subtle connection it is necessary to first consider general Boolean algebras (i.e. not necessarily consisting of projections on some Banach space) and their representation via the closed-open subsets of some totally disconnected, compact Hausdorff space. It turns out that the a-algebra generated by these closed-open sets is precisely the family of Baire sets. Typically, the Baire sets form a proper sub-a-algebra of the a-algebra of all Borel sees. All of these features (and more) form the subject matter of Chapter II. In conclusion, I wish to make it clear that the material presented here forms a personal choice of topics taken from a rather extensive area of research. I have not even attempted to touch on the theory of spectral operators, unbounded operators of scalar type, multiplicity theory, sums and products of commuting spectral operators, and so on. For this I refer the interested reader to [13], [15] and to the vast research literature on these topics which has appeared since the publication of [13] and [15], most of which is recorded in the Appendix.

Chapter I Vector measures and Banach spaces The first half of this chapter recalls some important notions and basic facts from classical (scalar-valued) measure theory and functional analysis. The second half of the chapter introduces vector measures (with values in a Banach space) and develops the theory of integration with respect to such measures, but only to the extent required in the sequel. Special emphasis is given to the usual convergence theorems and the L1-space of a vector measure. The reader who is interested in more recent developments of such LLspaces should consult the works of G. Curbera [5, 6, 7]; these aspects of the theory will not be needed in these notes. Much of the basic theory of integration with respect to vector measures in real Banach spaces (and more general spaces) can be found in [27]. However, we wish to work in complex Banach spaces. Since the results for complex spaces do not always follow easily or directly from those for real spaces, we have decided to develop the theory for complex spaces directly. Many of these results can be found in [29] and others, such as the completeness of L I, are new. Throughout this chapter, and the remainder of the text, the symbols N, N and C will always denote the natural numbers {i, 2 ... }, the real numbers, and the complex numbers, respectively. So, let us begin. Let ft be a non-empty set. A family of subsets E of f~ is called an algebra (of sets) if (i) f~ E E and 0 E E (where 0 denotes the empty set), (ii) E c := ~2\E belongs to E whenever E C- E, and (iii) AjeTEj C E for every finite collection {Ej : 3' E )c} C_ E. If E is an algebra of sets with the additional property that N~=lEn C E for every sequence { E n}n=l _C E, then it is called a a-algebra. In this case the pair ([2, E) is called a measurable space. Let (f~, E) be a measurable space. A function t, : E ~ C is called a complex measure = 1E,,) = ~ n~= * u(En) whenever { E n}n=l oo if z~(tO~ C E is a sequence of pairwise disjoint sets, meaning that E~ A E m = 0 whenever n r We say that zJ is ~r-additive . In this oo oo case the triple (f~, E, u) is called simply a measure space . Since U~= 1E ~ = U~=IE~(~ ), for every bijection rr : N -+ N, it follows from the cr-additivity requirement that u(U~=IE~ ) = ~ = 1 u(E~(~)). Accordingly, the series ~ = 1 u(E~) is necessarily unconditionally convergent in C. Hence, it is actually absolutely convergent, meaning that ~ = ~ I~(E~)I is finite.

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H S P A C E S Whenever we write (f~, E, u) it is meant that E is a or-algebra of subsets of a non-empty set f~ and ~ : E ~ C is a complex measure. Example I. (a) Let Q -- [0, 1] and E denote the Borel sets in f~, by definition the smallest a-algebra containing all the open subsets of f~. Define ~ : E --+ C by

~,(E) ::

,~(,~ + IS

E c ~.

Then z~ is a complex measure. Here x s ( t ) := 1 if t 6 E and Xz(t) : : 0 if t ~ E. (b) Let (fL E) be as in (a) and define u : E ~ C by

. ( m :=

L ~,

dr,

E c <,.

Then u is a complex measure. (c) Let ft = [0, oo) and E denote the Borel subsets of f~. Then the usual Lebesgue measure u : E ----+ [0, oo] is not a complex measure as it takes the "value" oo ~ C. 9 E x e r c i s e 1.[*] (a) Let ft = N be the natural numbers and E be the family of all subsets of f~ which are finite or have finite complement. Show that E is an algebra (of sets) . Is it a a-algebra? Give reasons. (b) Let f~ = [0, 1] and E be the family of all subsets of f~ which are countable or have countable complement. Decide whether or not E is a a-algebra. 9 E x e r c i s e 2. Let E be a a-algebra of subsets of a set of f~ # 0 and let u : E ~ C be a complex, finitely additive measure, i.e. u(@[=lEn ) = Y-~[-1 u(E~) for all finite collections {E~}[_ 1 C_ E of pairwise disjoint s e t s . . (a) Show that u ( A \ B ) = u(A) - u(B) for all A, B E E with B C_ A. Here A\B :: {w C A: w r B}. (b) Show that t, is a-additive if and only if l i m ~ _ ~ u(A~) = 0 whenever {A~}~=, C_ E is a sequence of sets decreasing to 0 (i.e. A1 D A2 _D A a . . . and A~=IA n = 0). 9 D e f i n i t i o n 1.1 Let (f~, E, u) be a measure space. Define a function I~1 : z ~ [0, o~) by lul(E) = s u p E 7r

lu(A)l,

E ~ E,

AETr

where the supremum is taken over all finite partitions ~r = iF1,. 99 , F~} of E in N, that is, Fj 9 E, F j A F k = 0 i f j 7{ k, Uj~_I_Fj = E a n d n 9 N. The function ]ul : E ~ [0, oo) is called the variation measure of v. Note that if we allow lul to take the "value" oc, then lul is also defined for finitely additive measures u : E ~ C, even if E is merely an algebra of sets. 9 The following result summarizes some important features of complex measures and their variation; see [38; Chapter 6], [14; Chapter III], for example. T h e o r e m 1.1. Let (f2, E , u ) be a measure space, i.e. u is a-additive.

(b) The variation measure I~I : E ----+ [0, oo) is a a-additive, finite-valued measure. (c) I•(E)I < ]•](E), for all E E E. (d) / f # : E ~ [0, oo) is any finitely additive measure satisfying ]r,(E)] _< # ( E ) , for all E 6 E, then IL,I(E) _< #(E) for all E E E. (e) luI(E) _< l - l ( r ) , for all E , F E E with E C_ F. (f) s u P { I t , ( H ) I : E _ D H E E } _ < ) ~ I ( E ) < _ 4 s u P { I z ~ ( H ) I : E _ D H E E } ,

EEE.

The number I1~11 == I~l(f~) is called the total variation of ~. In particular, the range ~(E) := {u(E) : E E E} of any complex measure ~ : E ~ C is necessarily a bounded subset of C. For basic references to the theory of topological spaces we refer to [14], [24] or [37], for example. Suppose now that ft is a compact, topological Hausdorff space. Let C(f~) denote the set of all continuous, C-valued functions on ft. Given f, g E C(ft) and c~ E C we define functions f + g and c~f on ft by w H f ( w ) + g(w) and w ~-* cef(w), for w E f~, respectively. Then f + g and c~f are also elements of C(ft) and so C(ft) is a vector space . Define a function /l" I1~ = C(f~) ----+ [0, oo) by I l f l l ~ : = s u p { l f ( w ) l : w C f~},

fEC(ft).

It is routine to check that (i)

IIfll~ = 0 if and only if f = 0,

(ii) Ilc~flloo = ]c~I 9 Ilfll~, for all c~ E C and f E C(ft), and (iii) I / / + g / l ~

<-Ilfll~ + Ilgll~ for all

f , g E C(ft).

Any function II" [] : X ---+ [0, co) defined on a vector space X which has the properties (i) (iii) of I1' I1~ on C(~) listed above is called a norm on X. Recall that a sequence {x~}~_l in a normed space (X, II. II) is a Cauchy sequence if for every e > 0 there is a positive integer N~ such that IIx~-Xmll < e , for a l l m _ > N ~ a n d n _ > N ~ . If, for every Cauchy sequence {x~}~= 1 in X there exists a vector z E X such that l i m n + ~ x~ = x, then we say that (X, II" II) is complete and call it a Banach space. The space (C(f~), I1' Iloo) defined above is a well known example of a Banach space. E x a m p l e 2. (a) Every Hilbert space is a Banach space. (b) Let co denote the space of all sequences r - (r r ) with complex entries and satisfying l i m ~ r = 0. Then co is a vector space with respect to the co-ordinatewise operations ~b q2 r := (~bl + r r + ~b2,... and 0~r :-- (O~r , 0~r ) for each c~ E C and r ~ E co. A norm is defined in co by

I1r

:= s u p { l e v i

n E N},

r c co,

C H A P T E R I. VECTOR M E A S U R E S A N D B A N A C H SPACES with respect to which co becomes a Banach space. (e) Let Coo denote the collection of all elements r E co which have the property that there exists an integer ArC > 0 such that On = 0 for all n _> N e. Then Coo is a vector subspace of Co and so is a normed space with respect to the norm of co. The normed space (coo, II 9 II) is not a Banach space as it fails to be complete. 9 Exercise 3. Verify that co is complete, but coo fails to be complete. 9 Let (X, If" II) be a Banach space. A map ~ : X ~ C which satisfies r

+ ~y) = c~r

+/3r

c~, ~ e C and x, y C X,

is called a linear functional on X. If, in addition,

IIr

:= sup{lr

: x c X, HxH _< 1}

is finite, then r is called continuous (or bounded). We also use the notation (x, r := r for x c X. The space of all continuous linear functionals on X is called the (continuous) dual space of X and is denoted by X ~. E x a m p l e 3. (a) Let a = [0, 1] and X = C(fl) with norm II" H~- Define ~b: X ----+ C by f C X.

(f,@ = f(~),

Then ~b E X ' and I1~1[ = 1. (b) Let X = co with norm as defined in Example 2(b). Define ~ : X ~

C by

~r (r ~> :=

2~

,

r c x

n=l

Then ~ C X ~ and II~ll = 1. 9 Let r ~2 be continuous linear functionals on a Banach space X and a l , a2 E C. Then a1~1 + c~2r : X ~ C defined by <X, 0:~1~)1 -~ 0~2~2} : = O~I<X, l/)l> ~- O~2<X, @2),

X ~ X~

is a continuous linear functional on X as Ilct1r + c~2r _< I~ll 9 N~/JIHX) -}- 1(:1,219 II~/J2llx,. Hence, X ~ also becomes a vector space. It is a basic fact that X ~ is itself a Banach space for the norm II 9 IIx,; see [14; Chapter II, w for example. Elements of X ' are also traditionally denoted by x'. E x e r c i s e 4. (a) Let X = Co. Show if ~ E X ~, then there exists a sequence of complex numbers ~ = (~, ~2,.-. ) with 2~__1 I~1 < oc such that oo (2g, @> : ~ :En~n, a=l

Moreover, II~rlx, = En%l I~1,

X = (Xl, X2,... ) 9 X.

(b) Let Y = gl denote the space of all sequences of complex numbers ~ = (fl, {2,... ) for Oo which fl~lll := E j = I I~J[ < ~ . Verify that I1"II1 is a norm on Y and that (Y, I1' Ill) is a ganach space. Show if ~ 9 Y', then there exists a sequence of complex numbers p = (Pl, P2,. 99) with sup~ [p~[ < oc such that c~

<~, ~ / = ~ Moreover, IIr

=

SUPn

~p~,

~ = (~1, ~2,... ) 9 v.

Ip~l.

(c) The space consisting of all sequences p = (p~,p2,...), with entries p~ 9 C, which satisfy I[pll~ : = sup{lp~l : n 9 N} < oo is denoted by g~. Show that g~ is a Banach space with respect to II " [Io~ and that Co is a proper (i.e. co r g~o) closed subspace of g~. 9 Let f~ be a topological space. The smallest ~-algebra on ft containing all the open sets is called the Borel a-algebra on f~ and is denoted by Bo(f~). D e f i n i t i o n 1.2. A complex measure u : Bo(f~) ~ C, where ft is a compact Hausdorff space, is called regular if for each E 9 Bo(fl) and e > 0 there exists a compact set K C_ ft and an open set U C ~ such that K c E c_ U and I ~ l ( g \ / 0 < c. 9 T h e o r e m 1.2. Let ~2 be a compact, topological Hausdorff space and A : C(f~) ~ C be an element of the dual space of (C(~), II " II~). Then there exists a regular complex measure # : Bo(f2) ~ C such that (r

A>=

f Ja r d/z,

r 9 C(ft).

Moreover, # is unique in the sense that if ~ : Bo(f~) ~ measure such that P (r A) = ./o r du,

C is another regular complex

r 9 C(f~),

then # = u (i.e. #(E) = u(E) for all E 9 Bo(f~)). The above classical result is referred to as the Riesz representation theorem ; see [38; Chapter 6], for example. It is important to note that if A 9 C(f~) ~ is represented by the regular measure # : Bo(f~) ~ C (i.e. (r A) = f a r d#, for r 9 C(f~)), then its dual norm IIAll=sup{l
I[r

1}

equals the total variation [1#[[ := [#[(f0. We have seen if X is a Banach space, then so is its dual space X ' when equipped with the dual norm ]1" []x,. In Exercise 4 it is shown that (Co)' = s and (gl), = g~. The Riesz representation theorem shows that C(f~)' is the space of all regular Borel measures u: Bo(f~) ~ C equipped with the total variation norm ]['l[. D e f i n i t i o n 1.3. Let X be a Banach space. The weakest topology on X which makes each element r : X ---+ C of X ' continuous is called the weak topology of X. A typical open neighbourhood of x 9 X for the weak topology has the form

D9

1<x,r162

r149

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H SPACES for some ~ > 0 and some finite set 5 C X'. In particular, a net of elements {z~} C_ X converges to z E X for the weak topology if and only if

l~<x~, ~> = <x, ~>,

~'~ c x'.

The weak topology on X will be denoted by or(X, X'). 9 E x a m p l e 4. Let X = Co, in which case X ' = gl. Consider the sequence {e~}~~176 1 c_ X , where e. has 1 in the n-th co-ordinate and 0 elsewhere. Then {e~}~=l converges to 0 E X in the weak topology. Indeed, let ( = ((~, ~2. . . . ) C X ' in which case ~ _ _ ~ ][~l < oo. Then

since the individual terms of an absolutely convergent series in C converge to O. However, oe { e ~}~=1 does not converge to 0 with respect to the norm of X since [le~][o~ = 1, for every nEN. 9 E x e r c i s e 5.[*] Show that if a sequence { z ~ } ~ l in a Banach space X co~verges in norm to z E 32, then {zn}~=l also converges to z w i t h respect to the weak topology. 9 We record some basic facts about the weak topology. First, a subset A C X is called weakly bounded if IAI~ := s u p { l ( z , ~ ) l : z 9 A} < oo, for all lb E X'. It is called norm bounded if IIAll := sup{llzll : z e A} < oo. Recall that a subset A of a vector space is called convez if it has the property that Az + (1 - A)y E A whenever :c, y C A and A E [0, 1]. For the following basic facts about the weak topology we refer to [14; Chapter V]. T h e o r e m 1.3. Let X be a Banach space. (a) A subset of X is norm bounded if and only if it is weakly bounded . (b) I r A is any convex subset of X , then the closure of A with respect to the norm topology coincides with the closure of A with respect to the weak topology. (c) A linear functional A : X ~ C is continuous with respect to the topology or(X, X') on X if and only if A E X ~. E x e r c i s e 6.[,] Let X be a Banach space and Y be a vector subspace of X. Show that the closure of Y in X with respect to the weak topology or(X, X ~) is the same as its closure in X with respect to the norm topology. 9 D e f i n i t i o n 1.4. Let X be a Banach space and ~ = 1 z , be a series of elements x , E X. (i) The series is said to be unconditionally norm convergent if there exists x E X such that ~ - - 1 x~(~) = x, for all bijections Ir : N ~ N, that is, if limN~oo IIx - ~ = I N z~(~)II = 0, for all bijections 7r : N ~ N. (ii) The series is said to be weakly subseries convergent if each subseries ~ _ ~ z~ k converges (to some element of X ) in the weak topology. 9 The following remarkable result is due to W. Orlicz and B.J. Pettis, [8; Chapter 1, w T h e o r e m 1.4. Let X be a Banach space Then a series ~~1761 x,~ in X is unconditionally norm convergent whenever it is weakly subseries convergent. E x e r c i s e 7. A series }-~=1 x~ in a Banach space X is called absolutely convergent if

f i IIx.II < oo

(a) Show that every absolutely convergent series is unconditionally norm convergent. (b) Let X be the Hilbert space g2 of all sequences ~ = (~1, ~2,. 9 ), with ~j E C, which satisfy It'll2 : = (En~ I~nl2) 1/2 <~ OO. For each n E N, let x n E X be the vector with r~i in the n-th co-ordinate and 0 elsewhere. Show that ~,~--1 x~ is unconditionally norm convergent, but not absolutely convergent. 9 D e f i n i t i o n 1.5. The weakest topology on X ' which makes each of the linear functionals <x, .> : X ' - - ~ C, x E X, defined by ~ H <x, @, for ~ E X', continuous is called the weakstar topology of X ' and is denoted by ~(X', X). A typical open neighbourhood of ~ E X ' has the form

{teN':

I(x,~)-{z,r

x~7},

for some e > 0 and some finite set ~- C X. In particular, a net of elements {Ca} C_ X ' converges to r E X ' for the weak-star topology if and only if

lim<x,r

xEX.

9

Let X be a Banach space. Then X ' determines the norm of X in that (1)

Ilxll = sup{IKx, x'>l : z' C X', IIx'llx, _< 1}.

Now X ' is itself a Banach space with respect to the dual norm

Ilx'lIx ,=sup{l<x,x'>[: x E X ,

IIxll <

I}

and hence, X ' has again a continuous dual space (X')', denoted simply by X " , consisting of all the linear functionals p : X ' ~ C for which (2)

Ilpllx" := sup{l<x',p)l : x' E x ' ,

I[x'llx, < 1} < oe.

Clearly every element x E X defines an element ~ E X " by :x'H(x,z'>,

x'EX';

it is straight-forward to see from (1) and (2) that II~llx,, = Ilxllx. Hence, the map x ~ from X into X " is linear and isometric, meaning precisely that IIzllx = II~llx,, for all x E X. It is called the natural embeddin 9 of X into X". The space X " is also called the bidual of X. D e f i n i t i o n 1.6. Let X be a Banach space. If the natural embedding x ~ ~? of X into X " is surjective , that is, it maps X onto the bidual X", then X is called reflexive. 9 E x a m p l e 5. (a) Every Hilbert space is reflexive . (b) Let 1 < p < oc and gP denote the Banach space of all complex sequences ~ = (~1,~2,-..) with norm I/~llp = (E~_-~ I~[P) 1/; < oc. If 1 < q < oc is the unique number satisfying 1; + 1 q = 1, then the dual Banach space of gP is gq, where each p = (Pl, P2,... ) E gq acts on gP via the formula oo

<~, ~> := ~ ~ , n=l

~ = (~1, ~ , . . . ) c ep.

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H SPACES Clearly each space fP is reflexive. (c) The space Co is not reflexive , since (co)' = gl and (gl), = e ~ which contains Co as a proper closed subspace; see Exercise 4(c). 9 We now collect together a few basic facts needed later; parts (a) (e) can be found in [14; Chapter V]. Part (f) can be found in [14; Chapter II, w for example; it is a consequence of the Hahn-Banach theorem. T h e o r e m 1.5. Let X be a Banach space. (a) A subset A of X ' is norm bounded if and only if it is weak-star bounded, that is,

sup{l(x,x')l : z' e A} < ee,

z e X.

(b) A subset A of X ' is compact for the weak-star topology cr(X', X ) if and only if A is norm bounded and weak-star closed (AIaoglu's theorem). (c) If x ~-~ 2 is the natural embedding of X into X " , then ~2 = { 2 : x C X } is cr(X",X')dense in X " . (d) X is reflexive if and only i f { x : Ilxll <_ 1} is compact for the weak topology ~ ( X , X ' ) . (e) (Eberlein-Smulian) A subset A C_ X has the property that its closure in the cr(X,X') topology is compact for the weak topology if and only if every sequence of elements from A has a subsequence which is or(X, X')-convergent to some element of X . (f) Let Y be a closed subspace of X and y' G Y'. Then there exists x' E X ' such that IIx'llx ' = IlYql~" and (y,x'} = (y,y'} for a l l y 9 Y. E x e r c i s e 8. A Banach space X is called weakly sequentially complete if every sequence {x~}~__l C_ X which is Cauchy for the weak topology cr(X,X') converges to some x 9 X with respect to the weak topology. (a) Show that every reflexive Banach space is weakly sequentially complete. (b) Show that co is not weakly sequentially complete . (c) Note that ~1 is weakly sequentially complete , but not reflexive; see [14; Chapter IV, w for example. 9 For the remainder of this chapter we concentrate on vector measures and, more specifically, on the theory of integration with respect to such measures. D e f i n i t i o n 1.7. Let X be a Banach space and (f~, E) be a measurable space. A function m : E ---+ X is called a finitely additive vector measure if m(Ukn=lEn) = E~=lm(En ) for all finite collections {E~}~= k 1 C E of pairwise disjoint sets. If, in addition, m satisfies oo

n=l oo • E, where the series is convergent in the for all sequences of pairwise disjoint sets { E n}~=l norm topology of X, then m is simply called a vector measure. A similar remark as for complex measures applies to show that the series ~ = 1 m(En) is then necessarily unconditionally norm convergent in X. We also say that m is ~-additive on E. 9

Let m : E ---+ X be a finitely additive vector measure. For each x' C X ' we define a set function (m, x ' } : 2 ~ C by E ~ (re(E), x'}, for E e 2. P r o p o s i t i o n 1.1. Let X be a Banach space and (f~, E) be a measurable space. Let m : E ~ X be a finitely additive vector measure. Then m is a vector measure (i.e. m is a-additive ) if and only if (m, z'} : E ~ C is a complex measure, for each x' E X ' . P r o o f . If m is a-additive, then the continuity of x' and (3) imply that

(m,

:

:

n=l

:

n=l

n--I

whenever {mn}~= 1 C E is a sequence of pairwise disjoint sets. Hence, (m, x') is a-additive. Conversely, suppose that Ira, x'} is a-additive for each x' E X'. Let {E~}~%~ C_ E be a pairwise disjoint sequence of sets. If { k}k=l is any increasing sequence of elements from N, then the ~-additivity of (m, x'} implies that ( m ( U k ~ l E ~ ) , z') = }-~k~=l(m(E~k), x'}, for each c~ x' ~ X'. This shows that the subseries ~ kc=~ l m(E~k) is weakly convergent to m (Uk=lE~k). By Theorem 1.4 the series ~n~=l m(E~) converges unconditionally in norm to rn(U~:lE~). This is precisely the requirement for m to be a-additive in X. 9 C o r o l l a r y I . l . 1 . Let m : 2 ---~ X be a vector measure on a a-algebra ~. Then its range re(E) : : {re(E) : E 9 ~} is a norm bounded subset of X . P r o o f . By Theorem 1.3(a) it suffices to show that

(4)

sup{ I(m,x')(E)l: E 9 2} = sup{l(m(E),x')l: E 9 E} < oo,

x' 9 X'.

But, by Proposition I.l each (m, x') is a complex measure and so Theorem I.l implies that (4) is indeed finite, for each x' 9 X'. 9 It is important to note that Corollary 1.1.1 fails to hold for finitely additive vector measures (in general), even when E is a a-algebra and X = C. For example, let e~, n 9 N, be the standard unit vector in g~ with 1 at position n and 0 elsewhere. Extend {e~ : n 9 N} to a Hamel basis of ~ . For x 9 ~ , let f=(x) be the e~,-coordinate of x with respect to this Hamel basis. Then f~ : g~o ~ C is linear and {n : f~(x) # 0} is finite for each x 9 e ~. For E E E := 2N define m ( E ) = ~-~__~ f~0/E). Then m : E ~ C is a finitely additive measure with unbounded range m(Yl,) in C (since m({1, 2 , . . . , n}) = n for each n 9 N). In view of this example, a finitely additive vector measure m : E ----+ X defined on an algebra of sets E is called bounded if its range m ( E ) is a bounded subset of X. All finitely additive vector measures considered in these notes will be defined on either an algebra or a-algebra of sets. W i t h o u t any further quMification the phrase "m : E ----* X is a vector measure" will mean that E is a a-algebra and m is a-additive. We point out that for any bounded finitely additive measure ~ : E ~ C defined on an algebra of sets E, all the properties (a)-(f) of Theorem 1.1 remain valid (cf. [14; Chapter III]), except that lul in (b) is then only finitely additive. E x e r c i s e 9. (a) Let ~ = ~ x~ be a convergent series in a Banach space X. Show that l i m ~ _ ~ Ilx~ll = 0.

10

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H S P A C E S

(b) Let E be a a-algebra of sets of a non-empty set and m : E ~ X be finitely additive. Show that m is a-additive if and only if l i m ~ o o II-~(En)ll = 0 whenever {En}n~=l C_ E is a decreasing sequence with Qnc'~ = O. 9 E x a m p l e 6. (a) Let X := Co and E be the a-algebra of all subsets of f} = N. Define m : E - - - + X by re(E) = (XE(1), ~1 X E (2) , 51 X z ( 3 ) , - . . ) ,

E 9 E.

To see that m is a vector measure , let { = ( { 1 , { > . . . ) belong to X ' = ,71, in which case

n=l

n--1

where 5n(E) := xE(n), for E 9 E, is the Dirac point measure at n 9 fk It is clear from (5) that (m, ~} is a complex measure and so the conclusion follows from Proposition 1.1. (b) Let X = g~ and (f~,E) be as in part (a). Define re(E) = (XE(1),XE(2),...), an element of g~, for each E 9 E. Then m is finitely additive , but not a-additive . This follows from Exercise 9 applied to the sets E~ = {n, n + 1 , . . . }, for each n 9 N. 9 D e f i n i t i o n 1.8. Let X be a Banach space and m : E - - ~ X be a finitely additive vector measure defined on an algebra of sets E. The function Ilmll : < ---* [0, oo] defined by H m l l ( E D = s u p { l ( m , x ' } l ( E ) : Ilx'll < 1,

x' 9 X ' } ,

E 9 E,

is called the semivariation of m; here [<m, x')[ is the variation measure of the finitely additive complex measure (m, x'}. 9 The next result is one of the fundamental inequalities concerning vector measures. P r o p o s i t i o n 1.2. Let X be a Banach space and m : E ----* X be a bounded finitely additive vector measure defined on an algebra of sets E. Then (6)

sup{llm(H)ll : E ~ H E E} _< Ilmll(E) ~ 4sup{llm(H)H : E D H 9 E},

Proof.

FixEEE.

IfH_CEandHcE,

E e E.

then

]]m(H)N : sup{l(m(H),x')]: Ilx']]_< 1}_<sup{l(m,x')l(H):llz']l_< _< sup{l<.~,x'>l(E): IIx'll _< 1} : II-~II(E),

1}

where the first inequality follows from Theorem I.l(c) and the second from Theorem I.l(e); see the discussion after Corollary I. 1.1. This establishes the first inequality in (6). Now let x' E X ' satisfy IIx'll <_ 1 Corollary I. 1.1, we have that

Then, by Theorem I.l(f) and the discussion after

[<m,x')l(E) < 4sup{l(m(H),x'>l: E D_ H e E} < 4sup{Ilm(H)ll : E 2 H 6 E}

11

as tlm(H)ll = sup{l<m(H),S>l : z'E X', Ilz'll ~ 1}. Accordingly, (7)

sup{l(m,J)l(E) : IIz'll -< 1} _< 4sup{FIm(H)lF : E _~ H E Z}.

Since the left-hand-side of (7) is precisely Ilml[(E) the second inequality in (6) follows. 9 It is clear from Proposition 1.2 and Corollary I.l.1 that Ilmll (E) is finite, for each E C E. We now determine some further basic properties of the semivariation of a vector measure. L e m m a 1.1. Let X be a Banach space, m : E ~ X be a bounded finitely additive vector measure, Ilml] : E - ~ [0, oc] be its semivariation , and E, F 9 E.

(a) Ilmll(E) <_ Ilmll(F) whenever E c_ F (i.e. IIm[I is monotone).

(b) I]mll(E U F) _< Ilmll(E) + Ilmll(f), that is, Flml[ is subadditive. (c) [ llmll(E) - IImN(F)I <_ IImII(EAF), where E A F := ( E \ F ) t2 ( F \ E ) . P r o o f . Both (a) and (b) are clear from the definition of Ilmll. (c) We have to verify the two inequalities limll(E) and

<_ Ilmll(f) + IImlI(EAF)

Ilmll(F) _< Ilmll(E)+ [I-~II(EAF).

Since the roles of E and F are symmetric it suffices to establish the first inequality. Write E = (E A F) U ( E \ F ) and note that the two sets in the union are disjoint. Then

Ilmll(E) _< IImll(E n F) + Ilmll(E\f) _< Ilmll(F) + I/mII(EAF), as required, where the first inequality folIows from part (b) and the second from part (a) as (E n F) c F and E \ F c E A F . 9 ~ A sequence of sets { E ~}~=i is called c o n v e r g e n t if ,~ ,oo = 1 < r A ~k=n E k)~ : Fl~_l(tJ~_~Ek); this set is then denoted by limn E~. P r o p o s i t i o n 1.3. Let X be a Banach space, 'rn : E ----* X be a vector meas'a~e and {E~}~= 1 C_ E be a convergent sequence of sets. Then

Ilmll (1~ E.) = ~i2s Ilmll(E,~)P r o o f . We begin with a special case. Namely, suppose that the sequence {En}~_l is decreasing and Q~__lEn = 0. Assume that lim~_~o IlmlL(E~) r 0. Then there exists e > 0 such that Ilmll(E~) > e for all n C N. Let nl = 1. Then there must exist x' E X ' with IIz'li _< 1 such that I(m,x')l(mnl) > e. Since I(m,x')l(E~) ; O, there is n2 > nl such that [{m,x')l(En~) < 89 Then 4sup{llm(F)H:Fc

E, FC_ (E~\E~2)}_> I(rn, x'}l(Enl\E~2)

CHAPTER L VECTOR MEASURES AND BANACH SPACES

12

by Theorem

I.l(f) and Proposition 1.2. But, i

J(m,z')](Enl\E,~2) : J(m,z')J(E.~)- J(m, z'>I(E~2) > ~r since J(m,z')l(E~,) > s and J(rn, z')l(E,~2) < 89 Accordingly, there exists F1 C E with 1 FI C_ (E,~\E,~2) such that Ilm(F1)Jl > gE. Again, since JjmJJ(E~) > e, there exists z' e X ' with IIz'll _< 1 such that J(m,z')J(En=) > r Since [(m, z')l(En ) I 0, there exists n3 > n2 such that [(m, z')J(E~a) < 89 By the same estimates as above 1 4sup{Hm(F)J[ : F 9 E, F _c (E.~kE~3)} > i s 1 from which it follows that there exists F2 9 E with F2 C_ (E,2\En3) such that llm(F2)Jl > g~. Oo Continuing inductively gives an increasing sequence { 7~ k}k=1 C__ N and a sequence of sets 0r { F, k}k=1 C Z with Yk C (E.~\E~+,) such that ll'~(Fk)II > ~, for all k 9 N. Since the sets in {Fk}k~__1 are pairwise disjoint this contradicts the ~-additivity of m. So, the statement of the proposition is proved for decreasing sequences En i 0. oo Suppose now that { E ~}~=i C_ E is convergent with limit E. Since Ez~E~ C LJ~=~E/kEk, for all n 9 N, it follows from Lemma I.l that _

I Ilmll(E) -Ijmll(E~)l _< IImlI(EAE~) _< Ilmll(uk%,~EAZk) = oo eo e~ = II II([Uk:~EkEk] U [Ok:~Ek\E]) _< jlmlI(Uk:~EkkE) + llmll(Uk=~E\Ek) oo But, the sequence { [_]oo k=~E\Ek}~=l decreases to 0 since

n~:1(O~:nE\Ek ) = N~=I(E \ n~=~ Ek) = E\ 0~= I (N~=~Ek) = E\E = ~. The same is true of {Ok~:~Ek\E}~=l. So, by the special case proved above, both terms in the sum on the right-hand-side of the previous inequality converge to 0 as n -~ oo. It follows that ]]m]](E)= l i m . _ ~ ]{mII(S,). 9 Let L, : E ----+ C be a complex measure . Then a set E 9 E is said to be ~-null if lul(E) = 0. By Theorem I.l(c) and the definition of the variation measure lul this is equivalent to the statement that v ( F ) = 0 for every F 9 E with F C_ E. Suppose that # : E ~ [0, oe) is a non-negative measure. We say that ~ is absolutely continuous with respect to #, written as ~ << #, if every #-null set is also a ~-null set. It is known (combine Theorem I.l(f) with [38; Theorem 6.11]) that v << # is equivalent to the requirement that for every e > 0 there is a 6 > 0 such that I~I(E) < e, for all E 9 E satisfying #(E) < 6. The following result, known as the Radon-Nikodym theorem, is one of the most important facts in measure theory; see [38; Chapter 6], for example. Recall that a measure # : E ---~ [0, oe] is called or-finite if ~ is a union of countably many sets ft~ 9 E, for n 9 N, such that # ( f ~ ) < oo for each n 9 N.

13 Theorem

1.6.

Let (ft, E) be a measurable space . Let ~ : E ---* C be a complex measure

and p : E ---* [0, oo] be a non-negative, or-finite measure. unique element h E L l(p) such that

,(E) = /Ehdp,

Moreover, the variation measure

I f ~, << p, then there exists a

E E E.

I~1: E ---~ [0, oo) of ~ is given by

I.I(E) = f Ih] dp, JE

E ~ E.

Suppose now t h a t m : E ~ X is a v e c t o r measure . T h e n a s e t E E E i s said to be m - n u l l if r e ( F ) = 0 for every F E E with F C_ E. By P r o p o s i t i o n 1.2 this is equivalent to requiring Ilmll(E) = 0. Given a finite measure p : E ----+ [0, oo) we say t h a t m is absolutely continuous with respect to p, also denoted by m << p, if for every ~ > 0 there is a 5 > 0 such t h a t IImll(E) < c, for all E E E satisfying p ( E ) < 5. T h e o r e m 1.7. Let X be a B a n a c h space, m : E ~ X be a vector measure and p : E ----+ [0, oo) be a finite, non-negative measure. Then m << p i f and only i f re(E) = 0 whenever E E E satisfies # ( E ) = O. Since p ( E ) = 0 implies # ( F ) = 0 for every F E E with F _C E, it follows t h a t re(F) = 0 for all such sets F a n d hence, t h a t IImll(s = 0. Accordingly, m << p if a n d only if every p-null set is also m-null . T h e o r e m 1.7, due to B.J. Pettis , can b e found in [8; p.10]. Concerning t h e existence of measures p for which m << p, we record t h e following fund a m e n t a l result, due to R.G. Bartle , N. Dunford a n d J.T. Schwartz in 1955; see [8; p.14], for example. T h e o r e m 1.8. Let X be a Banach space and m : E ~ X be a vector measure . Then there exists a finite measure p : E ~ [0, oc) such that m << p. Moreover, p can be chosen to satisfy 0 <_ p ( E ) < Ilmil(E) , f o r each E E E. Let m : E ~ X be a b o u n d e d finitely additive vector measure defined on a n algebra r~ of sets E. If f = ~ j = l c~jXE(5) is a E-simple f u n c t i o n on ~, where c~j E C a n d E ( j ) , for 1 <_ j < n, are pairwise disjoint m e m b e r s from E, t h e n we define 7%

(8)

~ f d m := E

c~jm(E M E ( j ) ) ,

E E Z.

j=l

Note t h a t t h e s t a n d a r d a r g u m e n t used for finitely additive complex measures also applies to show t h a t t h e integral in (8) is well defined, t h a t is, it is i n d e p e n d e n t of t h e particular

CHAPTER I. VECTOR MEASURES AND B A N A C H SPACES

14

representation used for f. Let/9 : sup{If(w)l : w ~ t~} = max{l~jF : 1 < j < n}. Then

II ~ f d m l l

:

/9/Is

M E)I I

j=l

n

-- /9sup{l(}--~/9-1~jm(E(j)

nE),z'}l

: x' c x ' , II~'ll < 1}

j=l n

<

/gsup{~/9-'l~jl.

I<m(E(j)n E),x'>l : IIx'll-< 1}

j~l

< /gsup{~-'l<m(E(j)n E),x'>l: I[z'll _< 1},

as/9-'l~j[ _< 1,

<_ /gsup{~-'l<m,x'>l(E(j)nE): Ilx'll < 1},

by TheoremI.1,

j=l

= /gsup{r<m, x'>I(E n (uj~_IE(j)): II~'IP -< 1}, by disjointness of {E(j)}]~=I, _< /gsup{l<m,z'>l(E): IIx'll -< 1}, by Theorem 1.1, = /911roll(E), for each E E E, where the references to Theorem 1.1 are in combination w i t h the discussion

after Corollary I.l.1. So, for each E-simple function f : ~ ~ (9)

II ~ f d m l l

<- Ilmll(E) 9 II/11~,

C we have

E C Z.

Let B~~ denote the closure of the vector space of all E-simple functions, formed in the Banach space of all bounded functions f : f~ - - ~ C which are measurable with respect to the a-algebra generated by E and equipped with the norm ]lfll~ = s u p { I f ( w ) l : w c a}, [38; p.16]. Since the E-simple functions are clearly dense in B ~ ( E ) , it follows from (9) that the linear map f ~-+ fE f d m (for fixed E E E) can be extended to a continuous linear map (with norm at most Ilmll(E)) on all of B~176 this is a special case of Theorem 18 in [14; p.55]. We again write fE f d m for each f E B ~ ( E ) and note that (9) still holds for all f E B~176 Moreover, for each f C B~176 we have

This is immediate from (8) if f is a E-simple function. The case for a general function f E Boo(E) then follows by approximating f in B~176 by E-simple functions, applying (9) to the bounded finitely additive measure (m, cc'} in place of m, for a fixed x' c X', and using the fact that x ~ is linear and continuous. We now wish to present an elegant application of the Bartle -Dunford-Schwartz theorem (c.f. Theorem 1.8.), which is a significant refinement of Corollary 1.1.1 in non-reflexive spaces. P r o p o s i t i o n 1.4. Let X be a Banach space and m : E ~ X be a vector measure . Then its range re(E) is a relatively weakly compact subset of X. That is, the closure in X of re(E)

with respect to the weak topology a(X, X') is weakly compact.

15 P r o o f . By Theorem 1.8 there is a finite measure p : E ~ [0, oo) such that m << p. Recall that L ~ ( # ) is the space of (equivalence classes of) E-measurable functions f such that

IIfllL~ := inf{sup{lf(w)l:w E E } : E E E, p(E ~) = 0} < oc. So, given f E L ~ ( p ) there is a p-null set N(f) E E such that fxN

ECE.

It then follows from (10) that (11)

f e L~(p), x' E X '.

It is known that (Ll(p)) ' = L~176 [38; Theorem 6.16]. So, let {f~} C L~176 be a net which converges in the weak-star topology ~r(L~(p), Ll(p)) to some h ~_ L ~ ( p ) . Fix z' E X'. It follows from (11) that

li~n
16

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H S P A C E S

function f E L(m) it follows that both fxa(s)+ = f V 0 := f + and f - := ( - f ) + = ( - f ) x a ( s ) belong to L ( m ) , where a ( f ) + := {w 9 f~ : f(w) >_ 0} and a ( f ) - := {w 9 f~ : f(w) < 0}. Hence, also Ifl = f § + f - belongs to L(m). To deduce the same fact for C-valued functions f requires an additional result. Given f 9 L(m), define m S : E ~ X by m r ( E ) = fE f d m , for E 9 E. We call m I the indefinite integral of f with respect to m. L e m m a 1.2. Let X be a Banach space, m : E ~ X be a vector measure and f 9 L(m). Then m f : E ~ X is also a vector measure P r o o f . Let x' 9 X q Then

(m:(E),x') = I s f dm, x'l = SE f d(m,x'l,

E9Z

Since f 9 L l ( ( m , z')), it is known from standard scalar-valued measure theory that E fE f d(m, z'} is a-additive, that is, it is a complex measure . So, (m:, z') is a complex measure, for each x' 9 X', and the conclusion follows from Proposition 1.1. 9 L e m m a 1.3. Let X be a Banach space, m : E ~ X be a vector measure and f 9 L(m). Then also [fl 9 L ( m ) . P r o o f . Let E = f - l ( { 0 } ) = Ifl-l({0}), in which case E 9 E. Then define h := (~)X~o and note that h 9 B ~ ( E ) as IIhH~ _< 1. Clearly Ifl 9 n l ( ( m , x ' } ) for all x' 9 X', as LI(/]) has the property that r 9 Ll(v) implies Ir 9 LI(.) for any complex measure v. Since h 9 B~~ we have seen that h 9 L ( m f ) , as mf is a-additive by Lemma 1.2. So, let xF := fF h d m l , for F 9 E. Then, for x' 9 X ' , we have

(xF,x')= (Zihdm,,x')= ~ hdIms,-'>= s

= s Ifl d(m,x').

This shows that ]fl is m-integrable and fF f d m = XF, for each F 9 E. 9 A slight (but useful) extension of Lemma 1.3 is the following result. L e m m a 1.4. Let X be a Banach space, m : E ~ X be a vector measure and g >_ 0 be an m-integrable function. I f f : ~2 -----* C is a E-measurable function such that If(w)l g(w), for w E f~, then f 9 L ( m ) . P r o o f . Let E -- g-l({0}) and note that f - l ( { 0 } ) _D E. Define h := (f)x~o and note that h 9 B ~ ( E ) . With XF := fF h drag, for each F 9 E, the argument of the proof of Lemma 1.3 can be repeated. 9 The following result summarizes the previous few facts. P r o p o s i t i o n 1.5. Let ( ft, E) be a measurable space, X be a Banach space and m : E ~ X be a vector measure. Then L ( m ) is a vector space and a (complex) vector lattice in the sense that;

_<

(a) f 9 L ( m ) implies that Ill 9 L ( m ) . (b) 0 < g 9 L ( m ) and f is E-measurable with Ifl <- g implies that f 9 L ( m ) .

17

(e) B~(~,) C L(,~). If u : Z ---* C is a complex measure and f C Ll(u), then the measm'e u / : E H

Remark.

fE f & ' has variation measure given by lu/[(E) = fE If] diu[, for E C E; this follows from [38; Theorems 6.12 &: 6.13], for example. Suppose now that m : Z - - ~ X is a vector measure and f E L(m). T h e n we have (12)

I l m / l l ( E ) = Ilml/[ll(E),

E e 2.

Indeed, by the previous comment about complex measures and the definition of ms, we have IIm/ll(E)

=

sup{l(mf, x')l(E): I[z'll < 1}

=

sup{/ Ifldl<m,z')l : IIz'll-< 1} JE

= sup{l(mlfhZ'>l(E): IIx'll _< 1} = Ilml/lll(E), for each E C E. A similar calculation shows that if f and g are m-integrable functions satisfying 0 _< f _< g, then

Ilmfll(E) _< IImgll(E),

E e

E.

9

(13The)~ following result, known as the dominated convergence theorem for vector measures will be an important tool in the sequel. Theorem

1.9.

L(m). Let f~ : 9 (i) l i m ~

Let X be a Banach space, m : E ----* X be a vector measure and 0 <_ g E - - ~ C, for n C N, be a sequence of E-measurable functions such that

f~(w) = f ( w ) exists, for each w C f~, and

(ii) IAI -< g, for n = 1,2,...

Then f E L(m) and IIm(f_fn)il(f~) = Ilmf - m a l l ( 9 ) ~

0 as n --+ oc.

P r o o f . Fix e > 0 and define E~ = {w C 12: I f ( w ) - f ~ ( w ) l > e}, in which case Iim~ E~ = 0. Fix E 9 E. For each x' 9 X ' satisfying IIx'll _< 1 we know from the dominated convergence theorem for scalar measures that f 9 Ll((m, x')). Moreover, for each n 9 N,

I f E ( f -- f ~ ) d ( m , x ' ) '

<_ ~ ,f - fn, d](m,z'), =

fE

\E~

'f--fn]d'(m,x')'+

<_ eIImII(E\E~)+ fE

fE

nE~

,f-f~,d,(m,x'),

If - f~ldl(m,x')l . NEn

But, for w E E M E~ we have that I(f - f-)(w)I = lira IA(w) - A(w)I < 2g(w)

CHAPTER I. VECTOR MEASURES AND B A N A C H SPACES

18 and so, as IIx'll < 1, that

fE If-fnldl(m,d)l<_2s nE~

Hence, whenever

IIm'll

HEn

gdl(~,~/)l_<2/Im, ll(EOEn),

nCN.

_ 1, Lemma h i ( a ) applied to both m and mg yields

I ~ ( f - f~) d(m, x')l

_< d I - ~ I I ( E \ E - )

+

211%11(E ~ E~)

_< ellmll(~) + 211~.ll(E~),

n 9 N.

Take the supremum of the left-hand-side, with respect to IIz']l < 1, gives

II

SE(f -- fn) droll -< ~11"~11(O)+ 211%11(E,0,

~ ~ 1~.

It follows, for each E 9 E, that

_< 2 d l ~ l l ( O ) + 2 1 1 ~ . l l ( E n ) + m l ~ . l l ( m ~ ) ,

k, ~ 9 N.

Since lim~ E~ = 0 , we have from Proposition 1.3 applied to the vector measure mg that l i r n I I ~ . l l ( S . ) = IIm~ll(l~mE~) = I1%11(0) = 0. So, (14) shows (as {IIm, II(E,)}L~ is Cauchy in [0, co)) that {fE f~dm},~=l is Cauchy in X, uniformly with respect to E 9 E. By completeness of X there is ZE 9 X such that II I ~ f : d . ~

- x~ll --~

0 as ~

-~ oo.

By (i) and (ii), the dominated convergence theorem for scalar-valued measures, and the fact that 9 9 Ll((m,z')), (see Definition 1.9), we have f 9 Ll((m,m')), for all z' 9 X'. Moreover, defining fE f d m := ZE, gives

This shows that indeed i 9 L(m). Fix c > 0 again. We have seen that there is N > 0 such that

EEE

Fix F 9 E. Then for n >_ N we have (n fixed)

19 Accordingly,

supll f f d m - ; f , , d m l l < e / 4 ,

n>_N.

FEE J F Then, for every n _> N, we have by Proposition 1.2 that

IIms

I

_< 4sup I I f f d m FEE J F

f fndml] _< ~. JF

This shows that limf - mr, H(f~) ~ 0 as n -~ oc. 9 E x e r c i s e 10. Let X be a Banach space which is weakly sequentially complete . (a) Suppose ~ = i x~ is a (formal) series in X such that ~-~=1 I(x~, x')l < ec for each x' E X'. Show that there exists z E X such that ~ = 1 xn is unconditionally (norm) convergent to x. (b) Let X be as in part (a), m : E ~ X be a vector measure and f be a C-valued, E-measurable function such that fa Ifldl(m,x')l < oo, for every x' E X'. Show that f is m-integrable. 9 D e f i n i t i o n 1.10. Let m : E , X be a vector measure . A function f E L(m) is called m-null if my is the zero vector measure , that is, fE fdm = 0 for all E C 2. By Proposition 1.2 this is equivalent to IImfll(f~) = 0 . 9 Two functions f,g E L(m) are called m-equivalent if If - gl is m-null. This is an equivalence relation on L(m); see [37; Chapter 1, w for the definition of an equivalence relation. The equivalence class of f E L(m) is denoted by [f]m := {9 E L(m) : f is mequivalent to g}. The operations [f]~ + [9]m := If + g],~ and c~[f],~ := [af]m, for c~ E C and f, g ~ L(m), are well defined and the quotient space of L(m) with respect to this equivalence relation, denoted by Ll(m), is a vector space . For the definition of the quotient space we again refer to [37; Chapter 1, w for example. Moreover, we can define a norm in L ~(m) by H [/],~l]~ = Ilmfll(tE),

[f],~ E L l ( m ) ;

see [37; p.lS3], for example. Since there is no confusion likely, elements [f],~ E Ll(m) will also be denoted simply as f E Ll(m). E x e r c i s e 11. Let X = gl and E be the or-algebra of all subsets of N. Define m : E ---* X

by m(E)

= (Xs(1), 2-EXs(2), 3-2XE(3),... ),

E E E.

(a) Establish t h a t m is a vector measure . (b) Show that a function f : N --* C is m-integrable if and only if }-~'~=1n-Elf(n)I in which case

Efdm

= (:z.(1)f(1),2-2)/.(2)f(2),3

and II [f]mlll = E n ~ = l n - 2 1 f ( n ) l . (c) Exhibit an m-integrable function which is

2X.(3)f(3),...),

not bounded.

< oo

E E E,

9

20

CHAPTER L VECTOR MEASURES AND BANACH SPACES

To establish the crucial fact that Ll(m) is complete (i.e. is a Banach space) for any vector measure m : E ~ X we require two further results. The first result is from [29]. P r o p o s i t i o n 1.6. (a) Let (~2, E , # ) be a complex measure space and As : E ~ C be a sequence of complex measures such that A~ << tL ('i.e. It~I << ]#I), for each n E N. If l i m , ~ An(E) exists in C, for each E E E, then (15)

lira

Iffl(E)~0

I ~ ( E ) = 0,

uniformly with respect to n E N,

that is, for each c > 0 there is a 6 > 0 such that supn IAn(E)] < s, for all E C E satisfying

hi(E)

<

5.

(b) Let f : f~ ~ C be a function and {f~}~:l c_ LI(#) satisfy (i) fn ---+f pointwise on f~, and (ii) {rE f~ d#}~=] is Cauchy in C, for each E 9 E. Then f 9 LZ(t~) and IIf~ - fill ~ 0 as n ~ oo. P r o o f . (a) This is the Vitali-Hahn-Saks theorem, a classical result from measure theory; see [14; Chapter III, w for example. (b) For each n, define A~(E) := fE f~ d# for each E 9 E. Then ~ << #, for n 9 N, and limn_oo A . ( E ) exists in C, for each E 9 E; see assumption (ii). By part (a) we conclude that (15) holds. Fix s > 0. Define

E~ := {w 9 f~: IA(w) - f(w)l _> e}, By assumption (i) we have lim~ E~ = ~ and so I#I(E~) ~ (15) there is m0 9 N such that

n c N. 0 by the G-additivity of Ill. By

sup ( s u p , l ~ , ( E . ~ ) ) < s . rn>_mo \riCH

So, for m > m0, we have

alf fmI~I.I < s-Ip}(gt\E,~)+ liminf(f IA - fmld]#I) <- s(2+ H#]I). k J Em I

This inequality implies the desired conclusion. 9 Let m : E ~ X be a vector measure . The Bartle-Dunford-Schwartz theorem guarantees the existence of a finite measure p : E ---+ [0, oo) such that m << p. A substantial improvement of this result states that a more "special" p can be selected: this is the following remarkable fact known as Rybakov's theorem ; see [8; Chapter IX]. It should be noted that Rybakov's theorem came some 15 years after the Bartle-Dunford-Schwartz theorem. T h e o r e m 1.10. Let X be a Banach space and m : E ----* X be a vector measure. Then there exists x' C X ' with Ilx'll = 1 such that m << I(m,x')l. E x e r c i s e 12. Let ~2 be a c o m p a c t , topological Hausdorff space, X be a Banach space and m : Bo(f~) ~ X be a vector measure. Then m is called regular if for each E 9 Bo(~2) and e > 0 there exist a compact set K and an open set U with K C E C U and Ilmll(U\K) < s.

21 The measure m is called scalarly regular if each complex measure (m, x'}, for x' C X', is regular. Show that m is regular if and only if m is scalarly regular. Hint." Rybakov's theorem may be useful. 9 We now come to one of the most important properties of L l(m); the proof is adapted

from [36]. T h e o r e m 1.11. Let X be a Banach space and m : E ) X be a vector measure. Then Ll(m) is complete (i.e. it is a Banach space) and the integration map I.~ : Ll(m) ---+ X defined by Im([f]m) := ./o fdm,

[f],~ 9 Ll(m),

is linear and continuous. Moreover, the E-simple functions are dense in L l(m). P r o o f . By Theorem 1.10 there is x' 9 X ' with IIx'll = 1 such t h a t m << I(m,x'}]. Let {f~}n~__l c Ll(m) be a Cauchy sequence. Since

i

Jfl

I / n - f~i dl(m,x')l _< sup iJrl i / n - Dldl(m>z')l = El[f~]m - [D]~ii,, IIz'll
we see that {A}n%l is also Cauchy in Li((m, x')). By completeness of L~((m, x')) there is f 9 L l ( ( m , x ' ) ) with fn ---+ f in Ll((m,m'}). Hence, there is a subsequence {fnk}k~__l of {f~}~--1 such that f~k ----' f almost everywhere> briefly a.e., with respect to (m, x'), [38; Theorem 3.12], and so, also a.e. with respect to m (as m << I(m,x'}l). By redefining f and fnk, for k 9 N, on an m-null set (if necessary) we may assume that f~k ~ f pointwise everywhere on f~, as k ~ ~ . Fix y' 9 X'. Then, for each E 9 E and all k, g 9 N, we have

,s =

fnkd(m,y')-- ~ fned(m,y')]<

Ily'll 9

Ifn~ - f,~
dl(m,

)1 <

s IA~-f~l Ily'll"

IIf~

-

d](m,Y')l

I,~IIL~(m).

Hence, {fEf~kd(m,y')}~=l is Cauchy in C, for each E 9 E. So, Proposition 1.6(b) (with # = (m,y')) implies that f 9 nl((m,y')) and (16)

lim f~k = f,

in Ll((m,y')).

Fix E 9 E. Then the inequality

ii/y - AD droll _< IIA~- f~
k,e 9 N,

shows that {rE f~k dm}~~ is Cauchy in X and so, by completeness of X , there is XE 9 X such that fE f,~k dm ---+ xE in X, as k --+ oo. Then (16) implies t h a t

22

CHAPTER I. VECTOR MEASURES AND B A N A C H SPACES

This establishes that f E Li(m) and f E f d m = XE, for each E E E. Let E > 0. Then there exists N~ such that

(17)

II [f~],~ -[/~,]~111 _< ~/4,

k,~ _> N~.

Fix E E E. Then, for each k > N~, we have by (17) that

By Proposition 1.2 (applied for all k > Nr that

to

each vector

measure

E

~-+

]] [/Ira -[f~k]-~]h = ]]m(f-f,,k)]](~) ~-- 4sup [] f f d m JE EEE

IE(I - fn~) din, for k 9 N) we see,

- ffi~kdm]] < 4.c/4 = E. JE

This shows that [f~k]-~ ~ [fire in Ll(m), as k ---* no. O~ OO Using the fact that if { X ,~}~=i is a Cauchy sequence in a normed space X and {Xnk}k=l is a subseqnence with auk ~ x (as k ---* no), for some x 9 X , then also x~ --~ x as n -~ de, it follows that [fi,],~ ~ [f],~ in Li(m), as n --* oc. This shows that LI(m) is complete. Since ]/fagdm]] <- Hg]lL~(~), for each g E Ll(m), it is clear that the integration map I,~ : L l ( m ) --* X is continuous. Linearity of I,~ is obvious. To see that sire(E), the space of all E-simple functions, is dense in Ll(m) it suffices to consider [f]m 9 Li(m) with [fire _> 0. So, redefine f (if necessary) to be zero on the m-null set {w 9 ~ : f(w) ~_ [0, oo)}. Choose E-simple functions {s~}7:i such that 0 < s . T f pointwise everywhere on ~. By the dominated convergence theorem for vector measures (c.f. Theorem 1.9) we conclude that

tl [ ~ ] ~ - [ f ] ~ l l - ( ~ )

: liras - m~.ll(~) ---" 0,

~ --, oo,

and hence, [s~],~ -* [fire in Li(m), as n -* co. 9 We end this chapter with the vector-vMued Riesz representation theorem. Let f~ be a compact , topological Hausdorff space. Then we have seen that the Banach space dual of (C(f~), ]]. ]]o~) is the space M(Bo(f~)) of all regular complex measures y : Bo(~) --* C equipped with the total variation norm ]]~l] = I~](g~) 9 Hence, it makes sense to talk about the weak-star topology a(M(Bo(~)), C(~)) on M(Bo(fi)). If X and Y are Banach spaces, then Z:(X, Y) denotes the Banach space of all continuous linear operators T : X - - ~ Y equipped with the operator norm IITH : = s u p { i l T x i i r : x 9

[]xlix < 1}.

In the case when X = Y we denote LZ(X, Y) simply by LZ(X). The dual (or adjoint) operator T' : Y' ~ X ' is defined by

(Tx, y') = (x, T'y'),

x E X, y' E Y';

23 it satisfies IIT]I = IIT'II, [14; Chapter VI]. In particular, T' E Z](Y',X'). An operator T r s Y ) is called weakly compact if { T x : x e X , rlxll _< 1} is a r d a t i v e l y weakly compact subset of Y. The next two results can be found in [14; Chapter VI, w for example. T h e o r e m 1.12. Let ~ be a compact Hausdorff space, X be a Banach space and T r s X ) . Then there exists a unique function F : Bo(f~) ---+ X " satisfying the following properties. (a) (x', F(.)} e M(Bo(Q)), for each x' C X'. (b) The mapping x' ~ (x', F(.)} is continuous from X ' equipped with its weak-star topology cr(X', X ) into M ( B o ( ~ ) ) equipped with its weak-star topology a ( M ( B o ( ~ ) ) , C(~)). (c) (T f, x'} = f~ f ( w ) d(x', F(w)}, for f e C(~) and x' e X'. (d) I]TI] = sup I] }-~-j~l c~yF(Ej)]lx,,, where the supremum is taken over all finite collections of disjoint Borel sets { j}j=l in f~ and all finite sets of complex numbers cq,.. cg~ with [aj] _< 1, and n C N is arbitrary. Conversely, if F : Bo(s ~ X " satisfies (a) and (b), then the equation (c) defines a continuous linear operator T E s with norm ][T H given by (d) and such that T'x' = (x', F(-)}, for each x' C X'. The above result (see also [8; p.152]) shows that every T e s X) can be represented by a finitely additive vector measure F : Bo(f~) ---~ X " which is a-additive for the weak-star topology cr(X", X') on X". Hence, if X is reflexive (i.e. X" = X), then F : Bo(f~) ---, X is a genuine regular vector measure (c.f. Exercise 12) and (c) becomes (18)

Tf

=

s fdF,

f e C(f~),

after noting each f r C(f~) is bounded and Bo(~)-measurable and hence, is F-integrable. To get a similar X-valued representation result without requiring X to be reflexive requires a restriction on T. This is the following result, known as the Riesz representation theorem for vector measures ; see also [8; p.153]. T h e o r e m 1.13. Let ft be a compact Hausdorff space, X be a Banach space and T 9 s C( ~ ), X ) be weakly compact. Then there exists a regular vector measure m : Bo( ~ ) ~ X (necessarily unique) such that (a) (m, x'} 9 M ( B o ( ~ ) ) , for all x' e X', (b) T f = f ~ f d m ,

for each f 9

(c) IITII = Ilmll(n),

and

(d) T'x' = (m, x'}, for each x' 9 X'. Conversely, if m : Bo(~) ---+ X is a vector measure which satisfies the condition (a), then T : C(~) ----+ X defined by (b) is a weakly compact operator with norm given by (c) and whose dual operator T ~ is given by (d).

24

C H A P T E R I. V E C T O R M E A S U R E S A N D B A N A C H S P A C E S

Combining

Theorem

1.13 with the following result shows that (18) actually holds for

arbitrary operators T E ~(C(f~), X) in a class of Banach spaces X more general than the reflexive spaces. T h e o r e m 1.14.

Let X be a Banach space which does not have any closed subspace isomorp h i s m to co. Let t2 be a compact Hausdorff space. Then every continuous linear operator T : C(f~) ~ X is necessarily weakly compact . In particular, (18) holds f o r some unique regular vector measure m : Bo(f2) ~ X.

The above result can be found in [8; pp.159-160], for example. Exercise 8(b) shows that every weakly sequentially complete Banach space satisfies the hypothesis of Theorem 1.14 (for this class of spaces Theorem 1.14 can be found in [14; Chapter VI, w However, there exist Banach spaces which do not contain an isomorphic copy of Co and fail to be weakly sequentially complete; see [22; p.73], for example.

Chapter II Abstract Boolean algebras and Stone spaces The aim of this chapter is to develop in a systematic way the theory of (abstract) Boolean algebras, as far as is needed later in the text. Far more comprehensive discussions of this topic can be found in [16], [24], [26], [28], [40] and [41], for example. The fundamental result is the Stone representation theorem which states that a Boolean algebra B is isomorphic to the Boolean algebra Co(t2B) of all closed-open subsets of some (essentially unique) totally disconnected, compact Hausdorff space ft~. In topological spaces of the type f2B, called Stone spaces , the sets from Co(QB) form a base for the topology of f~B. Certain completeness properties of/3 (of an algebraic nature) manifest themselves in certain disconnectedness properties of t2B (of a topological kind). Two important or-algebras which arise are the Baire sets Ba(f~), which comprise the minimal or-algebra generated by all closed-open sets, and the Borel sets Bo(t2B), which are generated by all open sets. Examples are given to show that Ba(t2B) C Bo(t2~) is typically a strict inclusion. The Boolean algebra isomorphism Q : Co(t2~) --~ ]3 as given by Stone's representation theorem (which is always finitely additive) plays a fundamental role. Moreover, if ]3 is abstractly or-complete (resp. abstractly complete) , then Q has an extension to a Boolean algebra cr-homomorphism Q : Ba(f~B) ---~ ]3 (resp. Q: Bo(t2s) ~ ]3). Such extension theorems for Q, from the algebra of sets Co(f~t~) to the or-algebras Ba(t2B) and Bo(~t~), will play an important role in subsequent chapters where ]3 will be part of a vector space equipped with a topology and it will become important to decide whether or not the extensions Q and (~ are a-additive . In the case when they are a-additive, it will be possible to apply the methods and techniques of vector measures and integration theory as developed in Chapter I. We begin with some algebraic preliminaries. Definition II.l. A partially ordered set is a non-empty set A together with a relation _< satisfying the following properties; (i) a < a f o r a l l a E A , (ii) a _< b and b ~ a implies a = b, and

26

C H A P T E R II. A B S T R A C T

BOOLEAN ALGEBRAS

AND STONE SPACES

(iii) a < b and b _< e implies a < e.

9

Example 7. (a) A -- N with the usual order < of real numbers is a partially ordered set. (b) Let f~ be any non-empty set and A be the set of all subsets of ~. For each E, F C A define E < F if E C F. Then (A, <) is a partially ordered set. 9 If B is a subset of A, with (A, _<) partially ordered, then an element a E A is called an upper bound ofBifb_< a for allbC B. An upper bound a E A of B is said to be a least upper bound of B if every upper bound c of B satisfies a < c. In a similar fashion the terms lower bound and greatest lower bound of B are defined (if they exist). When it exists, the greatest lower (resp. least upper) bound of B is denoted by AB (resp. VB). D e f i n i t i o n I I . 2 . A partially ordered set (L, _~) is called a lattice if every pair x, y C L has a least upper bound and a greatest lower bound, denoted by x V y and x A y, respectively. The lattice L is said to have a unit if there exists 1 C L such that x _< 1 for all x ~ L. If 1" is another unit, then the conditions 1" < 1 and 1 _~ 1" together with the properties of a partial order imply that 1" = l, i.e. a unit is unique (when it exists). The lattice L is said to have a zero if there exists an element 0 E L (necessarily unique) such that 0 _~ x for all x E L. The lattice L is called distributive if x V (y A z) = (z V y) A (x V z) and z A (y V z) = (x A y) V (x A z), for all x , y , z E L. The lattice L is complemented if it has a unit and zero and if, for every x E L, there is an element x' C L (called the complement of x) such that x A x' = 0 and x V x' = 1. 9 E x e r c i s e 13. (a) Give an example of a partially ordered set which is not a lattice. (b) Give an example of.a partially ordered set for which x A y exists for each pair of elements x, y but such that not every pair of elements has a least upper bound . (c) Let L = {3, 4, 5 , . . . } be equipped with the usual partial order < inherited from R. Show that L has a zero element, but no unit. Show that the subset A = {4, 6, 8 , . . . } does not have an upper bound in L. (d) Give an example of a lattice which has a unit but no zero element. (e) Let L = {0, {1}, {2}, {3}, {1, 2, 3}} and partially order the elements of L by set inclusion. Show that L is a lattice with zero and unit, but that L is not distributive. (f) Let (L, _<) be a l a t t i c e . For x, y E L show the following statements are equivalent.

(i) x_
(ii)

xAy:x

(iii)

xVy=y.

9

E x e r c i s e 14. Let X be a Banach space and g C_ s be the family of all (continuous) projection operators on X. Define E < F to mean E F = E = F E . (a) Show that _< is a partial order such that 0 _< E < I for all E C E, where I denotes the identity operator on X. (b) Show that each E C g has a complement, namely (I - E) C g. (c) Let E, F C g satisfy E E : F E . Show that E F is a projection which is the greatest lower bound of E and F in g and that the range ( E F ) X of E F is the intersection of the ranges E X and F X .

27 (d) Let E, F C g satisfy E F = F E . Show that E + F - E F is a projection which is the least upper bound of E and F in $ and that its range (E + F - E F ) X is the linear subspace of X spanned by the set E X U F X . 9 D e f i n i t i o n I I . 3 . A lattice with zero and unit which is both complemented and distributive is called a B o o l e a n a l g e b r a , briefly B.a.. 9 E x a m p l e 8. (a) Let f~ be a non-empty set and let B = 2 a be the collection of all subsets of t2, partially ordered via set inclusion. Then the empty set (~ is the zero element, the whole set f~ is the unit, the B.a. complement A ~ of A C B is the set theoretic complement A c := f~\A, and the distributive law holds because of the set identity A N (B U C) = (A N B) U (A N C) and the fact t h a t A A B = A N B a n d A V B = A U B . Hence, B i s a B . a . . (b) Let (t2, E) be any measurable space. Then E is a B.a. with respect to the operations of part (a). The same is true if E is merely an algebra of sets. 9 Let B be a B.a.. Define multiplication and addition in B by (1)

xy:=xAy

and

z+y:=(xAy')V(x'Ay),

x, y C B .

It can be verified that with these operations B becomes a B o o l e a n r i n g with the unit 1 E B as its identity; see [14; Chapter I, w for the definition of a Boolean ring. On the other hand, if 7~ is a Boolean ring with identity e and we define a relation _< in ~ by x _< y if x y = x and complements by x ~ = e + x, then ~ becomes a B.a. with x V y = x + y - xy

and

x A y = zy,

x, y E ~ .

Of course, the B.a. unit is the element e. Concerning the B.a. of Example 8(@ we see that the Boolean ring operations are given by intersection for multiplication and symmetric difference for addition, that is, A B = A n B and A + B = A A B := (A N B C) U (A c n B), for each A, B. E x e r c i s e 15. Let B be a Boolean ring. S h o w t h a t x 2 = x a n d x + x = 0 , forxEB. 9 E x e r c i s e 16.[.] L e t / 3 be a Boolean algebra. (a) Show that (a')' = a, for all a E/3, and that (a V b)' = a' A bI and (a A b)' = a' V b', for all a, b C/3. Deduce that a _< b if and only if b~ <_ a'. (b) Let {b~} be a subset of/3 with a least upper b o u n d . For each b C/3, show that the subset {b A b~} also has a least upper bound and that b A (V~b~) = V~(b A b~). Show that {b'~} also has a greatest lower bound and that A~b~ = (V~b~)'. (c) Let {ca} be a subset o f / 3 with a greatest lower bound. For each c E /3, show that the subset {c V ca} also has a greatest lower bound and that c V (A~c~) = A~(e V ca). Show that { c'~} also has a least upper bound and that V~c~' = (A~c~)'. 9 D e f i n i t i o n I I . 4 . Let (B, _<) be a B.a.. T h e n / 3 is called abstractly c o m p l e t e (resp. abstractly or-complete) if every subset A of/3 (resp. every countable subset A of/3) has a greatest lower bound, denoted by AA, or equivalently, if every subset A (resp. every countable subset A) has a least upper bound, denoted by VA. 9 E x e r c i s e 17. (a) Let /3 be the collection of all subsets of N which are finite or have finite complement. Define A _< B if A c_ B. Show that (/3, _<) is a B.a. which is n o t abstractly a-complete.

28

CHAPTER II. A B S T R A C T BOOLEAN A L G E B R A S AND STONE SPACES

(b) Let B be the collection of all subsets of the interval [0, 1] which are countable or have countable complement. Define A < B if A c B. Show that (B, <) is a B.a. which is abstractly a-complete, but not abstractly complete. 9 Definition II.5. Let `4 and B be B.a. 's and 9 : .4 --+ B be a function. Then 9 is called a homomorphism if (i) (~(x))' = ~(x'), for x e `4, (ii) ~ ( x A y) = ~(x) A ~(y), for x , y 9 `4, and (iii) qS(x V y) = ~(x) V ~(y), for x, y 9 `4. If, in addition, 9 is injective and satisfies d)(`4) = B, then we call ~ an isomorphism.

9

Since x A 0`4 = 0`4 and x V 1.4 = 1`4, for all x 9 `4, it follows from (ii) and (iii) that ~5(0`4) = 0n and ~(1`4) = 1B whenever ~5 is a homomorphism. Moreover, x <_ y implies that ~5(x) _< q~(y), since x _< y if and only if x A y = z (c.f. Exercise 13). If ~5 is also injective , then ~(x) _< ~(y) implies that x <_ y. Indeed, ~(x) _< ~5(y) implies that ~5(x) A(P(y) = d2(x). But, by (ii), we also have that ~(x) A q)(y) = q~(x A y) and so ~ ( x A y) = ~(x). Then the injectivity of 9 yields x = x A y, that is, x _< y. E x e r c i s e 18.[*] Let `4 and B be Boolean algebras and 9 : `4 ) B be an isomorphism. (a) Suppose {x~} is a set in `4 such that V~x~ exists in `4. Show that the least upper bound V , ~ ( x , ) of {~(x~)} exists in B and V~q~(x~) = ~(V~x~). (b) Suppose {yz} is a set in ,4 such that Azyz exists in `4. Show that the greatest lower bound A ~ ( y z ) of {~P(yg)} exists in B and AZ~p(yz) = ~(A~y~). (c) Deduce that `4 is abstractly complete (resp. or-complete) if and only if B is abstractly complete (resp. a-complete). 9 We now t u r n our attention to some topological considerations. Suppose that f~ is a compact Hausdorff space. Denote by Co(f~) the collection of all subsets of f~ which are simultaneously open and closed (such sets are called clopen) . Note that always 0 9 Co(f~) and f~ 9 Co(f~). It is easy to exhibit examples where these are the only elements of Co(D). The collection of sets Co(f~) turns out to be a B.a. with respect to the operations given in Example 8; this follows from elementary properties of open and closed sets in a topological space. This particular B.a. Co(Q) will play a fundamental role in the sequel. We point out that Co(f}) is also an algebra of sets in the sense of measure theory; see Chapter I. D e f i n i t i o n I I . 6 . A compact topological Hausdorff space f~ is called totally disconnected if Co(f~) forms a base for the topology in f~. That is, every open set in f~ is the union of some subcollection from Co(f~). 9 E x e r c i s e 19. (a) Let 12 = {0} U {~ : n 9 N} be equipped with the relative topology from R. Describe all the clopen subsets of ~2. Show that f2 is totally disconnected. (b) Let f2 = [0, 1] with its usual topology. Using the fact that every open set in f~ is a countable union of pairwise disjoint, open intervals from f~ describe all the clopen subsets of fL Show that f~ is not totally disconnected . 9 Given a topological Hausdorff space f~, let sim(Co(gt)) denote the vector space of all

29 functions f : gt - - ~ C which are a finite linear combination of functions of the form XE, where E C Co(f~). Each f C sim(Co(ft)) has a unique expression (2)

f = ~ OrYXEs, j=l

called its standard representation, where the complex numbers {aJ}j~--1 are all distinct and {Ej}j~=I c Co(f~) is a finite family of non-empty, pairwise disjoint sets with Ujn=IEj = ~. L e m m a II.1. Let ft be a compact Hausdorff space. (a) The vector space sim(Co(f~)) is also a unital algebra of functions, meaning that f g E sim(Co(f~)) whenever f, g e sim(Co(f~)). The unit is the constant function 11 : w ~-~ 1, for weft. (b) The algebra sim(Co(ft)) is conjugate closed, that is, ] E sim(Co(~2)) whenever f C sim(Co(ft)), where ] denotes the function w ~ f ( w ) , for w E ~. (c) / f f e sim(Co(f~)) has standard representation given by (2), then ~ e sim(Co(f~)) if and only ifO f[ {c~y}]=l. Then ~ = ~ 2 = l ( ~ ) X s j is the standard representation of ~. (d) /f (2) is the standard representation of f C sim(Co(f~)) , then Iifil~=max{lc~Jl:

l_<j_
E x e r c i s e 20. Prove Lemma II.1. 9 Let A be a set and 5c be a family of C-valued functions on A. Then we say that 9r distinguishes the points of A if, for every pair of distinct points A1,A2 E A, there exists r E 5c such that r r r E x a m p l e 9. Let f~ be a compact Hausdorff space. A result from topology (called Urysohn's theorem) states for such spaces f~ that if A, B are disjoint, closed sets in f~, then there exists a continuous function f : f~ ~ C with f(f~) C_ [0, 1] such that f ( A ) = {0} and f ( B ) = {1}; see [14; Chapter I, w for example. Since singleton sets {w}, for w C f~, are closed sets it follows that C(f~) distinguishes the points of f~. 9 The following result shows that for certain topological spaces f~ the algebra C(ft) in Example 9 can be replaced by a smaller subalgebra. L a m i n a II.2. Let ~ be a compact Hausdorff space which is totally disconnected . Then sim(Co(f~)) distinguishes the points of [2. P r o o f . Let Wl and w2 be distinct points of f~. Since 12 is Hausdorff, there exist disjoint open sets U1 and U2 in f~ with wl c U1 and w2 C U2. But, Co(~2) forms a base for the topology of ~, and so there is a set W ~ Co(~) with w~ C W and W c_ U~. Then X~ c sim(Co(gt)) satisfies X~ (wi) = 1, but X~(w2) = 0. 9 We now recall a classical result concerning function algebras, namely the Stone- Weierstrass theorem ; see [14; Chapter IV, w for example. T h e o r e m II.1. Let 12 be a compact Hausdorff space and A be a norm closed subalgebra of C(f~) satisfying both (i) 11 E~4, and

CHAPTER II. ABSTRACT BOOLEAN ALGEBRAS AND STONE SPACES

30

(ii) 7 E A whenever f E A. Then M = C(~) if and only if M distinguishes the points of ~. As an immediate consequence we have the following useful result. P r o p o s i t i o n II.1. Let f~ be a compact Hausdo~ff space which is totally disconnected. Then sire(Co(a)) is dense in C ( a ) . P r o o f . By Lemma II.1 we know that 5 := sim(Co(f~)) is a conjugate closed, unital subalgebra of C(t2). Then A := 5c (the bar denotes closure) is a norm closed subalgebra of C(VI) with properties (i) and (ii) of Theorem II.1. Since 5 distinguishes the points of f~ (c.f. Lamina II.2) so does .4. Then the Stone-Weierstrass theorem implies that A = C(t2). 9 We can now formulate the fundamental representation theorem for Boolean algebras, the so called Stone representation theorem, due to M.H. Stone; see [14; Chapter I, w T h e o r e m II.2. Let B be a B.a.. Then there ezists a totally disconnected, compact Haus-

dorff space f~B, unique up to topological homeomorphism, such that B is isomorphic to the B.a. Co(~B). Remark.

(a) The uniqueness of Q~, up to homeomorphism, follows from the fact that

if ~1 and t22 are compact Hausdorff spaces such that C(~l) and C(f~2) are algebraically isomorphic, then fll and t22 are topologically homeomorphic , [14; p.279]. (b) The totally disconnected, compact Hausdorff space f ~ as given by Theorem II.2 is called the Stone space of B. Moreover, the B.a. isomorphism Q : Co(f~) ~ B given by Theorem II.2 is called the Stone map. (c) The Stone map Q is finitely additive. That is, if we turn B into a Boolean ring with multiplication and addition as given by (1), then (3)

n Q(Uj= 1A j) = Q(A1) +

...

+ Q(A~)

whenever {Aj}j~ 1 C_ Co(~B) is a finite collection of pairwise disjoint sets. We establish (3) when n = 2; the general case follows by induction. So, let A1,As E Co(~B) satisfy A1NAs = 0. Then it follows from (1) and the fact that Q is a homomorphism (c.f. Definition II.5), that

Q(A~) + Q(As) := [Q(A~) A Q(As)'] V [Q(A~)' A Q(As)] = Q((A~ A A~) V (Ai i &)). But, by definition of the operations A, V and complementation in Co(~B) we have that (A1 A A;) V (A'1 A As) = (A1 5/A~) U (A~ (7 As) = A1 U As, (where the last equality uses the disjointness of A~ and As) which implies that AI N A~ = A1 and A~ N As = As. Accordingly, Q(AI) + Q(As) = Q(A1 U As) as required. (d) Let t2 be a given totally disconnected , compact Hausdorff space. Define B to be the B.a. Co(t2). Then flz = f~; see [40; p.25]. 9 E x e r c i s e 21. Let B be a B.a. and consider B also as a Boolean ring with respect to addition and multiplication as given by (1). Show that x + y = x V y whenever x, y E B are disjoint, meaning that x A y = 0.

31

Hint: The identities z -- x A 1 = x A (y' V y) and y = 1 A y = (x' V x) A y may be useful. 9 For a given B.a. B it is not always easy to identify its Stone space f ~ . We proceed to give two examples of a quite general nature; further examples can be found in [16], [26], [28], [40]. But first we require some further concepts from topology. Recall that the discrete topology on a non-empty set A is simply the collection 7 := 2 A consisting of all subsets of A. Let (A, r) be any locally compact Hausdorff space, meaning that for each A E A there exists an open set Ua C 7 containing I such that its closure Ua is compact. Let w be any element not in A and define Am = A U {w}. Then the collection r ~ consisting of all subsets of A~o which are either open sets in A (i.e. belong to r) or the complement (in Am) of compact subsets of A, form a topology in Am. The topological space (A~, T~) is called the one-point compactification of (A, 7). E x e r c i s e 22. (a) Let A be a non-empty set equipped with the discrete topology . Show that A is both locally compact and Hausdorff. (b) Let (A, 7) be any locally compact Hausdorff space and (Aoo, Too) be its one-point compactification . Show that %0 is indeed a topology, that is,

(i) 0, A ~ C T~, (ii) U A V E T~ whenever U, V e T~, and (iii) tj~V~ C 7oo for every family of open sets {V~} C T~. Equip f~ := Aoo\{w} with tile relative topology from Aoo (of course, f~ = A as a set), meaning that a set U c_ t2 is defined to be open if it is of the form U = f~ • V for some V E Too. Show that the identity function from (A, T) onto g~ is a topological homeomorphism. Moreover, verify that (Aoo, too) is indeed a compact Hausdorff space. Show that A is dense in the space (A~, T~). 9 A topological space (A, r) is called completely regular if, given any closed set B C_ A and a point k E B ~, there exists a continuous function f : A ~ R such that f ( B ) = {0} and f(k) = 1. Every compact Hausdorff space is completely regular; this follows easily from Urysohn's theorem (c.f. Example 9).

A compactification of a topological space A is a compact topological space K together with a continuous, injective map r : A - - ~ K such that ~(A) is dense in K, that is, ~(A) = K. The one-point compactification of a locally compact, non-compact space is an example of a compactification; see Exercise 22(b). A more interesting compactification is that due to M.H. Stone and E. Cech ; see [24; Chapter 6, w or [41], for example. T h e o r e m II.3. Let A be a completely regular Hausdorff space. Then A has a compactification/3(A) with the property that every continuous map from A into any compact Hausdorff space f~ has a continuous extension from/3(A) into fL Furthermore, /3(A) is unique, in the following sense: if a compactification K of A also has the property that every continuous map from A into any compact Hausdorff space f~ has a continuous extension from K into ~2, then there exists a homeomorphism of/3(A) onto K that leaves A pointwise fixed.

32

C H A P T E R II. A B S T R A C T B O O L E A N A L G E B R A S A N D S T O N E SPACES

The compact Hausdorff space/3(A) is referred to as the Stone-Cech compactification of A. If A is already compact, then t3(A) is homeomorphic to A. Now to the two examples alluded to above. E x a m p l e 10. (a) Let A be a non-empty set and B := 2 A be the B.a. of all subsets of A (c.f. Example 8). If we equip A with the discrete topology, then it is routine to verify that A is a completely regular Hausdorff space. It turns out that f~u =/3(A); see [40; p.26]. (b) Let A be an infinite set and B denote the collection of all subsets of A which are either finite or the complement of a finite set. Then B is a B.a. (c.f. Exercise 17). If we equip A with its discrete topology, then it is a locally compact Hausdorff space (c.f. Exercise 22(a)). It turns out that f ~ = AM is the one-point compactification of A; see [40; p.26]. 9 The B.a. 's of projections that we will be dealing with in later chapters will often be complete, or at least a-complete; see Definition II.4. For such B.a. 's more can be said about their Stone spaces t h a n in general; see Proposition II.4 below. So, we now concentrate on properties of this more specialized class of B.a. 's, for which some topological notions are required. D e f i n i t i o n II.7. Let f~ be a compact Hausdorff space. (i) The space f~ is called basically disconnected in the restricted sense if the closure of every countable union of sets from Co(f~) is an open set. (ii) The space f~ is called extremely disconnected if the closure of every open set is again an open set. 9 E x a m p l e 11. (a) Let a = {0} U {88 : n C N} be the totally disconnected , compact Hausdorff space of Exercise 19(a). For each n E N, let U~ = { ~ } in which case U~ E Co(m). Then the closure U~=IUn equals {0} U {!2~: n E N} which is not an open set in ft. Indeed, its complement is the set E = {~(2~-1) : n E N}. Since 0 E E \ E we see that E is not closed and so its complement U~=IU~ is not open. Accordingly, f~ is not basically disconnected in the restricted sense. (b) Every extremely disconnected space is obviously basically disconnected in the restricted sense. 9 In relation to Example l l ( b ) the following result is useful in producing examples of compact extremely disconnected spaces. P r o p o s i t i o n II.2. Let A be a completely regular IIausdorff space. (a) An isolated point of A is also isolated in ~(A). (b) A is an open set in 13(A) if and only if A is locally compact . (c) I r A E Co(A), then the closure (in/3(1)) Of A and A \ A are open sets in ~(A) which are complements of one another (in/3(A)). In particular, they belong to Co(~(A)), are pairwise disjoint and have union equal to ~(A). (d) A is extremely disconnected (using the same formulation as in Definition II.7(ii)) if and only if ~(A) is extremely disconnected. (e) If A is compact, then A is extremely disconnected if and only if A =/3(Y) for every dense subspace Y of A.

33 For (a), (b) and (c) we refer to [24; p.90] and for (d) and (e) we refer to [24; p.96]. E x a m p l e 12. Let A be any non-empty set equipped with its discrete topology. Then A is clearly extremely disconnected and completely regular . So, Proposition II.2(d) shows that /3(A) is compact and extremely disconnected. 9 The question arises of whether there is an analogue of Proposition II.2(d) for basically disconnected spaces. We proceed to show that this is the case; see Proposition II.3 below. Let A be a topological Hausdorff space and Ca(A) denote the vector space of all continuous functions f : A ~ N. Any set of the form f - l ( ( 0 , oo)) o r / - 1 ( ( - o o , 0)) is called a co-zero set . In the following definition the topological space is not assumed to be compact (c.f. Definition II.6 and Definition II.7). D e f i n i t i o n II.8. (i) A topological Hausdorff space is called basically disconnected if the closure of every co-zero set is an open set. (ii) A completely regular Hausdorff space is called totally disconnected if its only connected subsets are those containing a single point. 9 R e m a r k . For a compact space A, Definition II.8(ii) is equivalent to the requirement that Co(A) forms a base for the topology in A, [24; p.247]. Accordingly, for compact Hausdorff spaces, Definition II.8(ii) agrees with Definition II.6. 9 P r o p o s i t i o n II.3. (a) Let A be a completely regular Hausdorff space. The space A is basically disconnected if and only if/3(A) is basically disconnected. (b) Suppose that A is also compact . Then A is basically disconnected, if and only if, A is both totally disconnected and basically disconnected in the restricted sense. (c) If A is the Stone space of some t?.a. , then A is basically disconnected if and only if it is basically disconnected in the restricted sense. P r o o f . (a) See [24; p.96], for example. (b) Suppose that A is basically disconnected. Let {U~}~=I be a sequence of clopen sets in A, in which case U~ = {~ E A : Xv~(),) > 0} for each n E N. By [24; 3N.4, p.52] the closure of tO~=lUn is again open. This shows that A is basically disconnected in the restricted sense. The Remark after Definition II.8 and [24; 4K.8, p.63] show that A is totally disconnected Now assume that A is totally disconnected and basically disconnected in the restricted sense. Let A = {~ E A : f(),) > 0} be a co-zero set, where f C c a ( a ) . Since A is open and Co(A) forms a base for the topology in A we have A = U~A~ for some sets A~ c Co(A). Since the closed (hence, compact) sets A (n) := f - l ( [ n , oc)), for n E N, satisfy A (n) C_ tO~A~ _ ~ there is a finite set of indices F(n) such that A (~) C U~eF(~)A ~. Then A := U~=l(tO~eF(~ )A ~) is a countable union of sets from Co(A) and so A is open (as A is basically disconnected in the restricted sense). This shows that A is basically disconnected . (c) This follows from part (b) and the fact that the Stone space of a B.a. is always a totally disconnected, compact Hausdorff space. 9 The following result, which will play a crucial role later, makes the connection between certain completeness properties of a B.a. with certain "disconnectedness" properties of its Stone space. We refer to [28; Theorem 7.21] or [41; p.47 & p.69], for example.

C H A P T E R II. A B S T R A C T B O O L E A N A L G E B R A S AND STONE SPACES

34

Let B be a B.a.. Then l~ is abstractly complete (resp. abstractly acomplete) if and only if its Stone space ~ is extremely (resp. basically) disconnected.

P r o p o s i t i o n II.4.

It is clear for compact

Hausdorff spaces that we have:

Extremely disconnected ~

Basically disconnected ~

Totally disconnected.

Indeed, the first implication is obvious and the second implication follows from Proposition II.3(b). Example l l ( a ) shows that the second implication is not an equivalence. To see that the first implication also fails to be an equivalence, let B be the B.a. of Exercise 17(b), in which case B is abstractly a-complete, but not abstractly complete. Then by Proposition II.4 the Stone space ~ of B is a compact , basically disconnected space which is not extremely disconnected . E x e r c i s e 23. Let A be an infinite set equipped with its discrete topology ~- and let (A~, ~-o~) be the one-point compactification of A. (a) Show U C_ Am is clopen if and only if U = F or U = Aoc\F for some finite set F C A. (b) Show that (Aoo, %r is totally disconnected. (c) Show that (Am, 7oo) is not basically disconnected . (d) Using part (c), deduce that the B.a.B of Example 10(b) cannot be abstractly acomplete.

Note:

If we let A = {~ : n E N}, then Am can be identified with the space ~ of Example l l ( a ) . Hence, the phenomenon in Example l l ( a ) is a special case of part (c) above. 9 Another important class of sets (for our purposes) will be the following one. D e f i n i t i o n II.9. Let f~ be a compact Hausdorff space. Then the a-algebra of Baire sets , denoted by Ba(~), is defined to be the smallest a-algebra with respect to which each function f E C(gt) is measurable. 9 Clearly Ba(f~) a_ Bo(f~) . Moreover, being a-algebras, both Ba(a) and Bo(f~) are abstractly a-complete as B.a.'s (with respect to the operations given in Example 8(b)), since the supremum (resp. infimum) of each countable subfamily is given by its union (resp. intersection). Also, the algebra of clopen sets satisfies Co(f2) c_ Ba([2), since sim(Co(f2)) C_ C(~). W i t h i n the class of Stone spaces the relationship between Co(f2) and Ba(f~) can be precisely described. Given any family ~ of subsets of f~ we let ~r(7-t) denote the a-algebra generated by 7-/. Proposition

II.5.

Let ft be a compact , totally disconnected Hausdorff space.

Then

Ba(f~) : a(Co(~)). Proof. Since Ba(U) is a a-algebra and Co(~) < B~(U) it is olear that a(Co(U)) < Ba(U). Let f E C(Q). Then there exists a sequence of functions {s.}~= 1 C_ sim(Co(~)) satisfying Hs~ - fIIoo ~ 0, as n --~ oo; see Proposition II.1. In particular, s~ --+ f pointwise on Q. Since each function s~ is a(Co(Q))-measurable, for n E N, the pointwise limit f is also a(Co(Q))-measurable. Since f E C ( 9 ) is arbitrary, it follows from Definition II.9 that

Ba(f~) c a(Co(~)).

9

The algebra of sets Co(~2), even if Co(f t) is abstractly a-complete or complete as a B.a., is typically not a a-algebra of sets. The reason is that if {E~}~_ I _a Co(f~), then the elements

35 o~ 1E ~ and A~= V~E~ and AmEn formed in the B.a. Co(f~) are generally not given by Un= ~ 1E 4, o e 1E ~ and A~Er~ is the interior of N~=IE~, zo respectively. Rather, V~E~ = U~= denoted by (C~n~= l E n / ~o9 The following example illustrates the point. E x a m p l e 13. Let N have the discrete topology in which case it is clearly extremely disconnected. By Proposition II.2(d) the space f~ = ~(N) is compact and extremely disconnected. Since ftu = ft (by Remark (d) after Theorem II.2), where B := Co(ft), it follows from Proposition II.4 that the B.a. Co(f~) is abstractly complete. Proposition II.2(a) shows that oo n each singleten set {n}, for n C N, is a clopen set in ft. Accordingly, N = O~=~{ } is certainly a Baire set in ft. If Uo(f~) was a a - a l g e b r a , then we would have N E Co(ft) and so N = N (the closure taken in ~(N)). But, N = ft (as 1 is always dense in /3(1)) and we have a contradiction. So, Co(f~) is not a or-algebra and the inclusion Co(ft) C_ Ba(f~) is strict. 9 It is always the case that Ba(f~) C_ Bo([~). Is it likely that this is an equality, with perhaps certain restrictions on ft? E x e r c i s e 24. Let ft = {0} U {88 : n E N} be the totally disconnected, compact Hausdorff space of Exercise 19(a). Using Proposition II.5 show that Ba(f~) = Bo(ft). 9 The underlying reason for equality in Exercise 24 is the general fact that Ba(f2) = Bo(ft) whenever f~ is a compact metric space, [37; pp.302-303], or if ft has a countable base for its topology , [26; p.100]. Unfortunately, Stone spaces are rarely metrizable. Let us show that Ba(f~) C_ Bo(f~) is strict for the example of Exercise 23. For this we will require the following useful fact; see [26; p.99], for example.

Let f~ be a totally disconnected, compact Hausdorff space. Then every open Baire set in f~ is the countable union of sets from Co(f~). E x a m p l e 14. Let A be an uncountable set equipped with its discrete topology 7. Let Lemma II.3.

f~ := A ~ be the one-point compactification of A, where AM = A U {w} with w ~ A. Then f~ is compact and totally disconnected ; see Exercise 23. The first claim is that Bo(f~) consists of all subsets of ~2, that is, Bo(~q) = 2 a. Indeed, since A E 7 _c Too, its complement {w} = f~\A is a closed set in ft. In particular, {w} E Bo(~2). Moreover, for each A E A the singleton set {),} C Bo(f~), since {A} C ~- C_ ~-~. Now let U C_ gt be arbitrary. I f w r U t h e n U i s o p e n i n f~, since U C 7- C_ 7%. Otherwise, U = {w} U V where V = U\{w} is ~--open in a and so ~-~-open in ft. T h a t is, V 9 Bo(ft). Since also {w} 9 Bo(f~) it follows that U 9 Bo(ft). So, Bo(gt) = 2 a. We show the compact set {w} r Ba(ft). Now, A is open in f~. If it were a Baire s e t , then Lemma H.3 would simply that a = U,~__IE~ for some sequence of sets {E~}~%~ C_ Co(f t). In particular, w r for every n 9 N. So, by Exercise 23(a) each set E~ C_ A is a finite set. Accordingly, A is countable which is a contradiction. T h e n A, and hence also {w}, is not a Baire set. 9 In the previous example the space f~ is totally disconnected , but not basically disconnected: can the same phenomenon occur if ft is basically or extremely disconnected? Example 15 below shows it can indeed occur in such spaces. First we require a further topological fact.

36

CHAPTER H. A B S T R A C T BOOLEAN A L G E B R A S AND STONE SPACES

A regular topological space A is called a Lindelof space if every open cover of A has a countable subcover. L e m m a II.4. (a) Let A be a compact Hausdo~ff space and Y C_ A be equipped with the relative topology. If there exists a countable family {Fn}~__l of closed subsets of A with the property that for every pair of points u, v with u C Y and v E A \ Y there is some n E N such that u E Fn and v ~ F~, then Y is a LindelSf space. (b) Let Q be a totally disconnected, compact Hausdorff space. Then every open Baire subset of ~ is Lindelbf. P r o o f . Part (a) can be found in [17; p.250, Ex.3.8F] (be careful of the misprint). (b) Let Y be an open Baire set in Q. By Lemma II.3 there is a sequence {Fn}~__l C_ Co(Q) with Y = Un~__IF~. Let u E Y and v E Q\Y. Then u E F~ for some n, and certainly v 9~ F~ as v ~ Y. Part (a) then implies that Q is a Lindel6f space. 9 Example 15. We give an example of an extremely disconnected, compact Hausdorff space Q for which the inclusion Ba(Q) C_ Bo(Q) is strict. Let A be an uncountable set equipped with its discrete topology . By Exercise 22(a) the space A is locally compact and Hausdorff, and it is clearly completely regular . Since A is obviously extremely disconnected, so is Q :=/~(A) by Proposition II.2(d). Moreover, Proposition II.2(b) implies that A is an open subset of Q. Since each point of A is isolated (in A), it follows from Proposition II.2(a) that the family of sets {{~} : ~ E A} is contained in Co(f~) and obviously forms an open cover of A in Q. Since this particular open cover of the open set A C Q clearly has no countable subcover, it follows from Lemma II.4(b) that A cannot be a Baire set in f~. But, A being an open set, it is immediate that A E Bo(f~). 9 The following exercise provides an example of a basically disconnected , compact Hausdorff space f~ which is not extremely disconnected and for which the inclusion Ba(Q) c_ Bo(Q) is strict. E x e r c i s e 25. Let Y be an uncountable set equipped with its discrete topology ~-. Let w be a point not in Y and let A = Y U {w}. Define a collection of sets p in A to consist of all sets from r together with all sets of the form A\C, where C is a countable subset of Y. (a) Verify that p is a topology on A. (b) Show that the topology p is Hausdorff. (c) Show that (a, p) is completely regular. (d) Verify that (A, p) is not locally compact. The space (A, p) is basically disconnected, but not extremely disconnected; see [24; p.64], for example. Let Q := ~(A). (e) Show that Q is a compact , basically disconnected Hausdorff space. (f) Show that a is not an open subset of f~ (Hint. Use part (d) and Proposition II.2(b)). (g) Show that each point of Y is an isolated point of (A, p) and hence, is also an isolated point of Q. Deduce that Y is an open subset of Q for which there exists an open cover with no countable subcover. (h) Using (g), prove that Y E Bo(Q) but Y ~ Ba(Q). 9

37

The Stone representation theorem makes the precise connection between B.a. 's and totally disconnected, compact Hausdorff spaces. Namely, a B.a./3 is isomorphic to Co(~2~), where [2B is the Stone space of B. It was observed that Co(f~B) is always an algebra of sets, in the sense of measure theory, and that its generated g-algebra a(Co(f~B)) is precisely the Baire g-algebra Ba(f~). A still larger g-algebra is the family Bo(f~) of Borel sets. For the remainder of this chapter we wish to discuss the problem of determining when the Stone map Q : Co(f~B) --~ 13 can be extended, as a B.a. homomorphism (still with values in/3), to either Ba(f~B) or Bo(f~). Recall that a subset U of a topological space f~ is called nowhere dense if it has empty interior, i.e. U ~ = 0. A subset V _c f2 is said to be of first category (or meager) if V is the union of at most countably many nowhere dense sets. An open subset U of f~ is called regular if it coincides with the interior of its closure , i.e. U = (U) ~ The collection of all regular open subsets of a topological space is an abstractly complete B.a. when partially ordered by set inclusion; see [16; Theorem 8.2]. Moreover, given a family {Us} of regular open sets we have V~U~ = (U~U~) o and AU~ = (ngs o. Exercise

26.

(a) Show

Let f~ be a compact

Hausdorff space.

that every element of Co(f~) is a regular open set.

(b) If, in addition, f~ is extremely disconnected, to

show that every regular open set belongs

Co(~).

(c) Let /3 be an abstractly complete B.a.. Show that /3 is isomorphic to the B.a. of all regular open subsets of some extremely disconnected, compact Hausdorff space. 9 Let ~4 and/3 be abstractly g-complete B.a. 's and ~5 : A ---+/3 be a B.a. homomorphism (c.f. Definition II.5). Then 9 is called a a-homomorphism if ~(V=an) = Vn~(a~) for every countable set {a~}~EN C_ ~4. Definition II.10. Let/3 be a Boolean algebra.

(i) A non-empty

collection .7- of elements from/3 is called an ideal if

(a) ~z V v E jr whenever u, v E jc and (b) u C jr whenever u E/3 and u _< v for some v E jr. (ii) Suppose that/3 is abstractly a-complete . Then an ideal jr in/3 is called a a-ideal if V~un E ~ for all countable sets {u~}~eN C_ jr. 9 E x a m p l e 16. Let f~ be a compact Hausdorff space. Let /3 := 2 a. Using the fact that the union of two sets of first category is again of first category and that a subset of a set of first category is again of first category, it is clear that the collection of all subsets of first oo 1A ~, for any category is an ideal in/3. Since/3 is a g-algebra it follows that V~A~ = U~= countable collection {A~}~e~ c_/3. It is routine to check that the countable union of sets of first category is also of first category and hence, the collection of all subsets of first category is a a-ideal in B. Let A,/B~(a) denote the collection of all elements from Ba(f~) which are of first category and AABo(a) denote the collection of all elements from Bo(f~) which are of first category. Since b o t h Ba(f~) and Bo(f~) are sub-g-algebras o f / 3 it is clear t h a t ~4B~(a) and A/IBo(a)

38

C H A P T E R II. A B S T R A C T B O O L E A N A L G E B R A S A N D S T O N E SPACES

are a-ideals in Ba(~) and Bo(f~), respectively. 9 Let 5c be an ideal in a B.a.B. We define an equivalence relation ~ in 13 by a ~ b if a + b E $'; see (1) for the definition of + in B. The coset of b C/3 is denoted by [b]. Then the set of all cosets B / 5 := {[b] : b E/3} becomes a B.a. with respect to the operations [a] V [b] := [a V b],

[a] A [b] := [a A b]

and

[a]' := [a'],

a, b E B .

The B.a. B/.P is called the quotient of B modulo 5 . The map h : B ----+ B / • defined by h(b) = [b], for b E /3, is a surjective B.a. homomorphism of B onto B/.F. On the other hand, given B.a. 's .4 and B and a surjective B.a. homomorphism h : .4 ~ B, its kernel ker(h) := {a C . 4 : h(a) = 0u} is an ideal in .4 and the map g : .4/ker(h) ~ B given by 9([a]) = h(a) is a B.a. isomorphism onto B. All of these notions and facts on quotient B.a. 's can be found in [40; w for example. The following important extension result concerning the Stone map is known as the Loomis-Sikorski theorem; see [16; Theorem 18.3], [26; Theorem 13] or [40; w for instance. T h e o r e m II.4. Let 13 be an abstractly or-complete B.a. , f~u be its Stone space and 3.4 B~(a~) be the a-ideal of all Baire sets of first category. Then the Stone map Q : Co(f~u) ~ B has a unique extension to a ~-homomorphism Q : Ba(12u) ~ 13 with kernel ker(Q) = A4B~(a~). Remark. (a) The Baire category theorem for compact spaces states that every set of first category in a compact Hausdorff space has empty interior , [16; Theorem 18.2]. It follows that if E, F C Co(f~u) satisfy E ~ F modulo 2t4B~(a~), then actually E = F. Accordingly, for each E C Ba(f~u) there is a unique set /~ E Co(f~z) such that the symmetric difference E A / ~ (which corresponds to E + / ~ in Ba(f2z)) belongs to AdBa(a~). The extension Q of Q is then given by Q ( E ) := Q(/~). (b) The uniqueness of the extension Q can be argued as follows. Let P : Ba(f~z) ~ B be another cr-homomorphism which coincides with Q on Co(12~). Define .4 := {E c Ba(f~u) : oo P(E) Q(E)}. By hypothesis Co(f~B) C_ .4. Let { ~ ~}~=1 _C .4 be increasing. Since Ba(f~u) is a or-algebra we have V~E~ = U~= ~ I E ~. Since P, Q are a-homomorphisms it follows that

Q(u,~=,E~) v,,~(E,,)

- -

oo

~

v,,P(E,d

z

--(u,,=IE,~). t~

oo

Accordingly, U~_IE n E ,4. A similar argument shows that M~=IF~ C ,4 for every decreasing family {Fn}~_l C_ .4. So, .4 is a monotone class of sets (c.f. [38; Chapter 7] for the definition) containing the algebra of sets Co(f2u), from which it follows that .4 = cr(Co(f~B)). Then Proposition II.5 shows that .4 = Ba(f~u) , that is, P = Q. 9 Under extra restrictions on B the map Q has a further extension. T h e o r e m II.5. Let B be an abstractly complete B.a., f~u be its Stone space and 3,teo(a~) be the a-ideal of all Borel sets of first category. Then the Stone map Q : Co(f~B) ~ B has an extension to a c~-homomorphism Q, : Bo(f~u) ~ B with the following properties. (a) ker(Q) = AdBo(a~). (b) V~s

= (~(U~V~) for every family {V~} of open sets in a s .

39 (c) For each open set E C_[~s we have ,~(E) = V { Q ( F ) : F

dosed,

F c

E}.

(d) If P : Bo(f~s) ---+ B is another a-homomorphism which coincides with Q on Co(f~s), then P and Q, agree on the open subsets of t~B and on the Baire a-algebra Ba(f~s). P r o o f . Since fts is extremely disconnected (c.f. Proposition II.4) it follows from Exercise 26 that the regular open subsets of ~B are precisely the elements of Co([2s). Hence the existence of Q, that it is a a-homomorphism, and property (a) are well known; see [26; Theorem 4] or [16; Theorem 21.7], for example. (b) Let V : O~V~. Since Q is surjective , there are sets U1, U2 E Co(gB) for which Q(UJ = V~4(V~) and Q(U2) = ~)(V). Since V~ c_ V for each a and ~) is a B.a. homomorphism, we have Q(U~) <_ Q(U~). Conversely, let {W~} be a subfamily of Co(gz) such that W~ is contained in some V~ and V = U~W~; this is possible as Co(~s) forms a base for the topology in f~s. Given any/3 we have, for some a, that Q(W~) <_ Q,(V~) <_ Q(U1) and so W~ C_ U 1 as Q is a B.a. isomorphism. Consequently, V = U~W~ C_ U~ and Q(U2) = Q(V) <_ Q(U1). Thus Q(U1) = Q(U2) which establishes (b). (c) Since Co(~s) forms a base for the topology in f~B there is a family of sets {Hg} C_ Co(~2B) whose union is E. By part (b) s

= v ~ e ( H p ) < V{t~(F) : F C_ E, F closed} < ~)(E).

This establishes (c). (d) The formula in (b) also hold with P in place of Q. So, if V c_ f~s is an open set, then the fact that Co([2s) forms a base for the topology in f~s implies that -P(V) = v { Q ( W )

: V _D W ~ Co(r~,,)} = O ( V ) .

That P and s also agree on Ba(f~s) can be argued along the lines indicated in Remark (b) after Theorem II.4. 9 Property (c) is a kind of regularity condition akin to that seen in measure theory. The question arises of whether (c) holds for arbitrary Borel sets E rather than just open scts? Unfortunately, this is not the case in general. Example 17. Let .A4Bo(~) denote the a-ideal in Bo(]R) consisting of all sets of first category 9 Then the quotient B.a. B := Bo(]R)/MBo(R) is abstractly complete , uncountable, atornless and has a countable dense set (in the sense of B.a. 's), [40; p.94]. So, its Stone space f~B is separable in the topological sense, meaning it has a countable dense set, say D = {d~ : n E N}. Since each singleton set {d~} is closed in f~, it must have empty interior . Otherwise, {dn} C Co([~s) in which case {d~} would be an atom of Co(gB) and hence, Q({d~}) would be an atom of B. So, D E Jt4Bo(a~) and hence (~(D) = 0B, by Theorem II.5(a). If (c) of Theorem II.5 was true for arbitrary sets E E Bo(~s) we would have

(4)

~)(D ~) = V{Q(F) : F closed, F C_ D~}.

40

CHAPTER H. ABSTRACT BOOLEAN ALGEBRAS AND STONE SPACES

But, if F is any closed set satisfying F C_ D c, then F c is an open set containing the dense set D and so (~(F c) = lB. Accordingly, (~(F) = 0B and so the right-hand-side of (4) equals 0B. However, the left-hand-side of (4) equals lB. So, (4) cannot be an equality. 9

Chapter III Boolean algebras of projections and uniformly closed operator algebras Let X be a Banach space and B C_ s be a B.a. of projections which is bounded. The aim of this chapter is to identify the closed subalgebra (B)~ generated by B with respect to the operator norm topology in E(X). Since (B}~ is a commutative, unital Banach algebra it is possible to apply the general methods of Banach algebras. Indeed, this is the approach adopted in [15; Chapter XVII, w However, our approach will be via the Stone space fib of/3 and the methods of B.a. 's as developed in the previous chapter; see also [13] and [33J for this approach. The reason for this approach is that it gives a consistency of treatment throughout the text, since B.a. methods and spectral measures must be used in the next chapters when considering the closed algebra generated by /3 with respect to other nonnormable topologies on s where Banach algebras no longer play an effective role. The basic idea in this chapter is to realize/3 as the range of a finitely additive spectral measure ; the boundedness of/3 enters to ensure that at least bounded measurable functions are "integrable" with respect to this finitely additive spectral measure (via a suitable extension process from the simple functions). It turns out that (B)~ is isomorphic to C(f~B) 9 Moreover, there exists a bounded B.a. of projections A in s indexed by the elements of Bo(f~), such that /3' := {F' : F r /3} c_ A and the isomorphism d) : C ( [ ~ ) ---* (/3)$ has the property that each dual operator ~(f)', for f C C(f~B), is given as the integral of f against an s ("almost") spectral measure taking its values in A. Without any further restrictions on X or/3 this is the best that can be expected. Let X be a Banach space. Then s is also a Banach space for the operator norm ]]T]]:=sup{HTr

zCX,

Uzl] < 1},

TcE(X).

This norm topology on s is called the uniform operator topology ; if we wish to stress this topology is being considered we will write s A subspace of s is called closed for the uniform operator topology if it is closed in s The inequality

IITSII < HTH. ]]S]I,

T, S c s

42

C H A P T E R III.

UNIFORMLY CLOSED OPERATOR ALGEBRAS

shows that /2(X) is a Banach algebra : it has the identity operator I as its unit. If X is at least 2-dimensional, then s is not commutative, that is, there exist operators S, T C s such that S T r T S . An operator T C s is called invertibIe if there exists S E s necessarily unique, such that S T = I = T S . We denote S by T 1. Let Inv(X) denote the set of all invertible operators in t;(X). L e m m a I I I . 1 . Let X be a Banach space. (a) I f T C s satisfies IrTIr < 1, then (I - T ) C Inv(X). (b) Inv(X) is an open subset o r g y ( X ) . P r o o f . (a) Since the geometric series ~--o Ilzll '~ < oo and E,~__0 I[T"II < E ~ _ 0 I[Trl~, it follows that the series ~ _ 0 T " is absolutely convergent in s to some operator S C s say. Since n

r~

(I - T ) . ~

TJ = Z - r ~+~ = ( Z r J ) . (I - r ) , j=o jr0

~ c N,

by letting n --* oo and noting that IIT'~+~II _< [IZll ~§ ~ 0 we see that (I - T ) S = [ = S(I T). Hence, (I - T) E Inv(X) with (I - T) -1 = S = ~ ~ 0 T ~. (b) Fix S C Inv(X). Let R C s satisfy [IR - SI[ < ~ . Then -

Ili - R s

111 = I I ( S - R)s-~I] _< lie - RII. lie-all < 1

and so, by part (a) applied to T = I - R S -~, we see that R S -1 = (I - T) C Inv(X). Hence, R = ( R S - 1 ) S is also invertible being the product of invertible operators. This shows that the open ball in s i with centre S and radius ~ is contained in Inv(X). Since S E Inv(X) is arbitrary we are done. 9 E x e r c i s e 27. Let X be a Banach space. Show that the map T ~ T -1 is a continuous map of Inv(X) onto Inv(X) with respect to the operator norm topology inherited from s 9 The following inequalities will be needed later. E x e r c i s e 28.[.] Let X be a Banaeh space with norm II" H. (a) Show that /Ilzll- Ilyll I _< I I ~ - y l l , for all z , y c X. (b) Deduce that IIz - yll > Ilzll - Ilyll, for all ~, y c x . 9 We now turn our attention to a special class of B.a. %. D e f i n i t i o n I I I . 1 . Let X be a Banach space. (a) A collection B C_ s of commuting projection operators is called a B.a. of projecfiords if 0, I C B and if it is a B.a. with respect to the partial order < defined by E _< F if EF = E = FE. (b) A B.a. of projections B C_ s is called bounded if

[1~11 := sup{llEll : E c B} < oo.

9

The notation IF,' for the B.a. complement in B of an element E E 13 (so, in particular, E' C L ( X ) ) should not be confused with the dual operator E' E ~ ( X ' ) . It will always be clear from the context which of the two distinct notions is meant by Es

43 E x e r c i s e 29.

Let X be a Banach space and B c s

be a B.a. of projections.

(a) For elements E, F E B show that E _< F if and only if E X c_ F X . (b) Show that the B.a. operations A, V and complementation in B are given by E A F = 9

E F and E V F = E + F - E F and E' = I - E (Mint: See Exercise 14).

(10/andS=(0 0)

E x e r c i s e 30.[*] (a) Let X = C 2 and consider the matrices A = 0 0 0 1 , interpreted as elements of s Show that both A and 8 are projections and satisfy A 8 = B A . Show that ( 8 - A) is not a projection. (b) Let X be a Banach space and B C_ s A , 8 E B satisfy A < 8 , then ( 8 - A) C g.

be a B.a. of projections.

Show that if 9

We will be dealing exclusively with bounded B.a. 's of projections. The following example shows that not all B.a. 's of projections are bounded. E x a m p l e 18. Let X = LP(R), for some p E (1, 2); see [38; Chapter 3]. For each t E R, define the translation operator Tt E s by T t f = fi, for f E X, where f i ( s ) = f ( s + t) for a.e. s E R. A projection E C s is called a p-multiplier projection if E T t = T t E , for all t E R. It is a known fact from harmonic analysis that the family Bp of all p-multiplier projections is a B.a. of projections for which sup{iiEi] : E E /3p} = oo. If p = 1, then /31 = {0, I} and if p = 2, then/32 consists of selfadjoint projections. So, both/31 and 132 are bounded B.a. 's of projections. 9 The following remarkable result, due to W.G. Bade [1], shows that a large class of B.a. 's of projections are always bounded. The proof given below is an expanded version of Bade's original proof. Theorem IliA. jections in s

Let X be a Banach space. Then every abstractly (z-complete B.a. of prois necessarily bounded .

P r o o f . Proceeding by contradiction, assume that there exists an abstractly ~-complete B.a. of projections /3 C s which is not bounded. Declare a projection E E 13 to have property (c~) if c~(E) := sup{llFll : F e/3, F _< E} -- oc. Suppose there exists E E /3 such that both E and E' := (I - E) E s do not have property (c~). Since every F E /3 satisfies F = F E + F E ' with F E <_ E and F E ' < E' it follows that IIFII _< IIFEII + IIFE'II _< ~(E) + ~(E'). Since F E/3 is arbitrary we contradict our assumption that /3 is not bounded. So, for every E C /3, at least one of E or E' must have property ((~). Suppose that E E /3 has property (c~) and F E B satisfies F _< E, in which case (E - F ) E /3 by Exercise 30(b). Suppose that both F and (E - F ) do not have property (c~). Since every H E/3 with H _< E satisfies H = H E = H F + H ( E - F), where H F < F and H ( E - F ) <_ ( E - F ) it follows that I]Hll _< I]HFII + IIH(E - F)I] _< c~(f) + c~(E - F). Accordingly, c~(E) < oc contrary to the choice of E. Hence, if E E /3 has property (a) and F _< E, with F E/3, then at least one of F or (E - F ) also has property (c~) . From the above discussion there exists some element E1 E /3 with property (a). Then,

44

C H A P T E R IlL

UNIFORMLY CLOSED OPERATOR ALGEBRAS

by definition of c~(E1) = o% there is E1 c / 3 with F1 _< E1 such that IIFlll >_ 2 + 2llElll, i.e. (1)

I[Flll - IIElll > 2 + IIEII[.

Let E~ be any element from {F1, (El - F1)} having property ( a ) . If E2 = F1, then certainly E2 _< El. On the other hand if E2 = (El - F1), then E2E1 = ( E l -- E l ) E 1 = E t - FIE1 = E1 - F1 - E2

and hence again E2 _< El. So, E2 C/3 satisfies E2 _< El. Moreover, if E2 = F1, then IIE2[[ = [IFI/I > 2 + 2]IEIII >_ 2 + IIE1[I. On the other hand if E~ = (El - F1), then (1) and Exercise 28(b) imply that

IIE2JI = IIE~ - F~ll _> JJF~II- JlE~II _> 2+ IIEIII. So, we have produced an element E2 E/3 satisfying both E2 _< E1 and I[E2[[ _> 2 + [[E~I[. Since E~ has property (a) there is F2 E /3 with F2 < E2 such that [IF2][ >_ 3 + 2[[E2[1. Let Ea be any element from {F2,(E2 - F2)} with property (a) . Then again Ea C /3 satisfies Es <_ E2 and a similar argument as above shows that [[Es[[ > 3 + []E2[[. Proceeding inductively produces a monotone decreasing sequence {E~}~_ 1 C_ B satisfying (2)

IlEal] _> r~+ llE~_~I[,

n _> 2.

Define G~ := ( E ~ - E ~ + I ) for n E N. Since E.~+I < E~ it is clear that each G~ E /3; see Exercise 30(b). Suppose that n > m, in which case E,,E~ = E , and E~+IE~ : E~+I and E,~+IE~+I = E~+I and E.,~E,~+I = E,~. Using these identities it follows, by expanding the right-hand-side of the expression G,~G,~ = (En - E,+~) (E~ - E~+l), that G~G,~ = 0. Hence, the projections {G,}~= 1 are pairwise disjoint in Z. Moreover, by (2) and Exercise 28(b), []anl[ = lien - E n + l l [ ~ liEn+l[ [ - l i e n ] ] ~ (Y~ -]- 1), that is, l i m ~ ]]anl] = e~. By selecting subsequences from {G~}~=I , a collection of mutually disjoint sequences of pairwise disjoint projections { H j,k}k=~, for each j E N, is obtained (all from B) such that (3)

lim []Hj,kl] = o~,

k~oc

j c r~.

Since B is abstractly or-complete the elements Py := Vk= 1H j,k exist in B, for each j E N. For m r n it follows from Exercise 16(b) that (4)

P.~P~ = P~ A P.~ = P~ A (Vk~__lH.~,k) = V~__~(P~ A H,~.k).

But, since H~,kHn,~ = 0, for all k, r E N and m r n, it follows that H~,k n P~ = Hm,k A (Vy=xH~,~)

=

Vr~176

A H~,~)

=

Vy=I(H~,~H~,~ )

for all k E N. It follows from (4) that P,~P,~ = 0 whenever m =~ n.

=

0,

45 Fix m and x r 0. Let Xm := PmX. Suppose that P,~x r O. Since H,~,n = H,~,nPm, for n E N, we have

IIH~,nzll. IlHm,.Pmzll _ IlP~xll HHm,~P.~xll Ilzl] Ilxll Ilxll I[Pmzll But, lIp~xrl <-- IIPmtl and II~m,=P~41 < HP~I[' II~.~,nlrL(x~). ItP~ll <-- IIHm,~llc(x~), and so ~ In the case when P ~ x = 0 we have

IIHm,~zll, IIHm,~P~zll - 0 <_ IIP~/I. IIH~,nlIL(xm). Ilxtl Ilzll So, we conclude that

IIHm,nll _< I[P~ll" IIH~,~II~(x,,,),

m, n ~ N.

It follows from this inequality and (3) that, for each fixed m E N, we have

lim IIH.~,~IILtx~) = oo.

n~oo

So, we can select an increasing sequence {nk}k~176C N and unit vectors xk E Xk such that IIHk,n~x~lt > k, for all k ~ N. net Q = Vk%lHk,~. Since Pkxk = xk (as xk E Xk) and QPt = He,n,, for all k,g E N we have IIQxkll = IIQPkxkll = IIHk,~xkll > k, for k E N. It follows that oo = sup{llQxkll : k ~ N} _< IIQ[I which is impossible since Q E s means that IIQII < oo. Hence, B must be bounded. 9 Let {A~ : a E A} be a family of subalgebras of L;(X), that is, each A~ is a vector subspace of L;(X) and T S E A~ whenever T, S C A~. Then the intersection A~A~ is also a subalgebra of L;(X). So, for any subset Ad c_ L;(X) there is a smallest subalgebra in L;(X), denoted by (Ad), which contains A4, namely the intersection of all subalgebras containing A4. We call (A4) the subalgebra of L;(X) generated by A4. The closed algebra generated by 31t is the smallest closed subalgebra of L;~(X) containing Ad; it is, of course, the closure of (A4) in L;~(X) and is denoted by (A4}~. We call (3A)~ the uniformly closed algebra generated by Ad. The following result gives a complete description of (B)g in the case when/3 is a bounded B.a. of projections. A proof based on the theory of Banach algebras can be found in [15; Chapter XVII, w We have decided to give a proof based on the theory of Stone spaces for B.a. 's as this is the underlying approach of the entire text; see also [13; Proposition 5.43]. The crucial point turns out to be the fact that every bounded finitely additive spectral measure P : E ----, L;(X) defined on an algebra of sets P. (see Definitions III.2 and III.6 below) yields a continuous homomorphism (via integration) of B~176 into L;~(X). T h e o r e m III.2. Let X be a Banach space and 13 C_ F~(X) be a bounded B.a. of projections. (a) (13)~ is inverse closed in s that is, T -~ E (B)~ whenever T C (13)~ M Inv(X). (b) (13)~ is isomorphic (as a commutative, unital Banach algebra ) to C(ftt~), where f~B is the Stone space of I3, via an isomorphism ~ : C ( f ~ ) ----4 (B}~ which satisfies

9 (x~) = Q(E),

E 9 Co(n~),

46

C H A P T E R III. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

where Q : Co(f~s) ~

B is the Stone map , and the inequalities Ilfll~ -< ]le(f)l[ -< 411/311 Ilfll~,

Proof.

/ ~ C(as).

(a) Note that (/3} consists of all operators of the form

je~where 2 C_ N is any finite set, {aj : j ~ S } is any set of distinct complex numbers and (6)

{Fj : j E 2 } C_ 13 are non-zero projections with FjFk = O, j 7~ k, and E Fj = I. jE~r

Let T E (B}~ N Inv(X). Choose {U~}~__a C (B) such that U~ --~ T in s Then by Lemma III.l(b) there is N such that Un c Inv(X) for all n _> N. Exercise 27 shows that Ug I ~ T -~ as n -+ ec. So, T i will belong to (B)j provided we know that U~-1 C (B) for all n >_ N. So, suppose that U E (/3) is given by (5) and (6) with U E Inv(X). We claim that c~j # 0 for all j E 2 . For, suppose that aj0 = 0 for some j0 C 2 . Since Fj0 # 0, there is x # 0 with FSc = x. Then (6) implies that jeJ: Hence, x E ker(U) contradicting U C Inv(X). So, aj # 0 for all j C .7- and hence U -1 = 1 / ~J, showing that U.- 1 C (/3). Y~jeT(~) (b) Let Q : Co(as) ~ / 3 be the Stone map in which case Q(0) = 0, Q(fi) = I and Q(AC~B) = Q(A)Q(B). Moreover, by Remark (c) after Theorem II.2 and the B.a. operations in/3 we see that Q is finitely additive (in the sense of measures) on the algebra of sets Co(as), n i.e. Q(Uj=IEj) = ~ j =n l Q ( J )j~ for all pairwise disjoint sets {Ej}j= t C Co(fie). So, Q has a unique extension to sim(Co(f~s)), denoted by 4~, defined by linearity and the property 9 (;g,) := Q(E) for E C Co(as). Suppose that f = ~ j =n ~ ajXzj E sim(Co(t2s)) has its standard representation, in which

case II/[l~ = max~j_<~l~jl. Then ~ ( f ) -- 2 j =n ~ c~jQ(Ej). Fix x ~ X with II~ll -< 1 and let rn := Qx be the X-valued, bounded finitely additive measure E ~-~ Q(E)x, for E ~ Co(f~s). By (9) of Chapter I and Proposition 1.2 we have (7)

1l~5(/)xH <- ll/lloo. II,~ll(as) <_ 4ll/lloosup{lIQ(E)xll : E ~ C o ( a s ) } ,

from which it follows (using IIQ(E)xil _< IIBII ]lzll and taking the supremum over all x ~ X satisfying Ilzll -< l) that Ile(f)ll _< 411/3[I. tl/llo~. Now fix j0 ~ 2 . Since Q(Ejo) r 0 we can choose Xjo ~ Q(Ejo)X with IIX~oll = 1. Then, using the fact that Q(Ej)Q(E~o ) = 0 whenever j r j0, we have

qS(f)x = ~ . ayQ(Ej)Q(Eyo)XJo = ayoQ(E~o)Xy o = ajoXjo j=l

47 and so II~(f)X3oll = laSol. Accordingly, Ild)(f)l] > II<~(f)xj0l/= I~jol. Since j0 is arbitrary we deduce that Ilfll~ = maxl_<j_<~I~jl < II~(fDl]. So, we have established that (8)

Ilfll~<- Ildg(f)ll-<41lBll'llflloo,

fcsim(Co(aB)). n

To see that ~5: sim(Co(a~)) ----+ {B) is also multiplicative note that if f = 2 j = ~ ajX.j has its standard representation and E E Co(f~B), then n

fXE = ~ ajXE~Ej j--1

is the standard representation of fx~ and so (using the fact Q is a B.a. isomorphism) n

e(/~) = ~ ( ~ o ~ ) = j=l

~/~J)= j-1

n

n

= ~ajQ(E)Q(Ej)

[~asQ(Ej)]Q(E ) =

j=l

=

j=l

~(f)~(x~).

This formula and the linearity of 9 then imply that (9)

r

= ~(f)de(g),

f, 9 e sim(Co(Qu)).

It is clear from (8) and (9) that 9 has a unique continuous extension, which is still linear and multiplicative, from the closure of sim(Co(Qu)), taken in C ( f ~ ) , into (B)~. Still denoting this extension by q~ it is clear that (10)

Ilfll~ <<-11~5(f)ll <- 411fll~lIBII,

f E C(aB),

after using Proposition II.1 to establish that sim(Co(f~B)) = C ( f ~ ) . Since (B) = r it is clear from (10) that r = (B}~-, i.e. r maps C(t2B) onto (B)~. Also (10) shows that r is injective on C(t2t~). Hence, q~ is a Banach algebra isomorphism . 9 E x e r c i s e 31. Let X be a Banach space and B c Z:(X) be a bounded B.a. of projections. Show that if P E (B)~ is a projection, then actually P E B. 9 Theorem III.2 shows, for any bounded B.a. of projections B, that {B}~ is isomorphic to C(glu) both as a Banach space and as an algebra. That is, there exists a linear and multiplicative isomorphism 9 : C(flB) ~ (B):. Changing the situation somewhat, suppose that A is a compact Hausdorff space and qJ : C(A) ~ s is a linear and multiplicative map which is injective , continuous and has closed range. W h a t can be said about the uniformly closed algebra ~ ( C ( A ) ) ? Is it of the form (B)~ for some bounded B.a. of projections B? And, if so, how are the operators ~ ( f ) E Z:(X), for f C C(A), related to B? The remainder of this chapter is devoted to a consideration of such questions.

CHAPTER III. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

48 Definition map

III.2.

P : E ---* s

Let E be an algebra of subsets of a set f~ and X is called a finitely additive spectral measure

be a Banach

space. A

if

(i) P(t2) = I and P(0) = 0, (ii) P(Uj"=IEj) = ~ = 1 P(Ej), for all finite collections {Ej}j~__I _C E of pairwise disjoint sets, and (iii) P ( E M F) = P ( E ) P ( F ) , for all E, F E E, that is, P is multiplicative. Suppose that F c-:_X ' is a vector space which distinguishes the points of X. If, in addition to (i) (iii) above, E is a ~r-algebra and P also satisfies the condition (iv) E H (P(E)x,x'}, for E E E, is a complex measure (i.e. is c>additive) for every xEX andx'EF, then P is called a spectral measure of class F . The complex measure in (iv) is denoted by (Px, x'}, for each x E X and x' E P. 9 E x a m p l e 19. (a) Let X = gv, 1 _< p < oc, and E denote the a-algebra of all subsets of ~2 = N. For each E E E define P(E) E s by

P ( E ) : x ~-+ (xlxs(1),x2XE(2),. .. ),

x = ( x l , x 2 , . . . ) E X.

Then P is a spectral measure of class F = X'. (b) Let X = g~ and (f~, E) be as in (a). Define P(E) E s by the same formula as in (a). Then P is a spectral measure of class F = gl C X ' but not of class P = X'. 9 We require some further topologies on spaces of linear operators. D e f i n i t i o n I I I . 3 . Let X be a Banach space. (i) The strong operator topology on s is defined by specifying a basic open neighbourhood of T E s by N's(T) := {R r s

I I ( R - T)xll < e,

x E be},

where e > 0 is arbitrary and bv is any finite subset of X. Hence, a net {T~} C_ s converges to T E s in the strong operator topology if and only if limT~x = Tx,

(11)

x E X,

c~

where the limit (11) exists in the norm topology of X. We also say that {T~} converges strongly to T. The space s equipped with the strong operator topology is denoted by

c~(x). (ii) The weak operator topology on Z2(X) is defined by specifying a basic open neighbourhood of T E s by

Hw(T) := {R c s

I((V- R)x,x')l < ~, x e 7 , ~' c ~ }

where e > 0 is arbitrary, and 5r C X and 7~ C X ~ are arbitrary finite sets. Hence, a net {T~} C s converges in the weak operator topology to T E s if and only if

limiT~x, x'} = (Tx, x'),

x r X, x' E X'.

49 The s p a c e / : ( X ) equipped with the weak operator topology is denoted by Z;~(X). 9 E x e r c i s e 32. Let X be a Banach space. (a) Suppose that {T~} c s is a net of operators converging strongly to T E s Show that {T~} also converges to T with respect to the weak operator topology. (b) Let X = g 2 a n d ~ C X b e a n y fixed non-zero vector. For e a c h n C N, let e~ E X have 1 in the n-th co-ordinate and 0 elsewhere. Define T~ : X + X by Tnx = (x, ~}en, for x E X, where (x, ~) ~ - - 1 x ~ . Show that T~ E / : ( X ) , for each n E N, and that the sequence {T=}n~=l converges to the zero operator in the weak operator topology, but {T~} does not converge strongly to the zero operator. 9 P r o p o s i t i o n I I I . 1 . Let X be a Banach space. (a) (Uniform Boundedness Principle) Let ,,4 C_ s equivalent: (i) sup{IlTl] : T C A} < oo.

Then the following statements are

(ii) sup{l]TxH : T E A} < oo for each x C X . (iii) s u p { l ( r x , z')l : r C A } < oo for each x 9 X and x' 9 X ' . (b) (Banach-Steinhaus theorem) Let {T~} C_ s be a net such that sups IIT~II < oo and T x := lima Tax exists in X , for each x 9 X . Then T G s (c) A linear functional on the vector space s is continuous for the strong operator topology if and only if it is continuous for the weak operator topology. (d) A convex subset of s has the same closure for the weak operator topology as it does for the strong operator topology. For (a) we refer to [14; p.66] and for (b) we refer to [14; p.55]. Parts (c) and (d) can be found in [14; Chapter VI, w for example. D e f i n i t i o n I I I . 4 . A finitely additive spectral measure P : E ~ s defined on a aalgebra E is simply called a spectral measure if it is a-additive with respect to the strong operator topology. T h a t is, whenever {E~}~__I C E is a decreasing sequence with M~=IEn = O, then l i m ~ o o P ( E , ) = 0 i n / : , ( X ) . 9 L e m m a I I I . 2 . Let X be a Banach space and P : E ~ s be a finitely additive spectral measure defined on a ~-algebra E. Then P is a spectral measure if and only if it is a spectral measure of class F = X ' , that is, if and only if (Px, x') is a complex measure for each x 9 X and x ~ 9 X t. P r o o f . Suppose that P is a spectral measure. Fix x E X and x' 9 X'. Let E~ $ ~ in E. Then l i m ~ P ( E ~ ) x = 0 in the norm of X. Hence, also (Px, x'}(E~) ~ 0 since I(P(E~)x, x')l < IIP(E~)xII . Ilx'll. Accordingly, (Px, x'} is a-additive. Conversely, suppose that P is a spectral measure of class F = X ~. Fix x 9 X. Then (Px, x ~} is a-additive, for each x ~ 9 X ~, and hence the X-valued set function P x : E P ( E ) x , for E 9 E, is (norm) a-additive; see Proposition 1.1. Exercise 9(b) then implies that P ( E ~ ) x ----* 0 in X whenever En ~ 0 in E. 9 E x e r c i s e 33. Let f~ -- N and E be the a-algebra of all subsets of f~. For each E 9 E define P(E) G s by P ( E ) x = (z~x~(1), x2X~(2),... ), for x 9 g~. Show that P : E ~ s

C H A P T E R IH. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

50

is a spectral measure of class F1 = 61, b u t P is not a spectral measure of class F2 = (6~176 '. T h a t is, verify t h e claims m a d e in Example 19(b). 9 D e f i n i t i o n I I I . 5 . Let X be a B a n a c h space, F C_ X ' distinguish the points of X , and P :E ~ s be a spectral measure of class F defined on a or-algebra E. A E-measurable function f : gt ----* C is called P-integrable if (i) f a Ifl dl(Pz, z'>l < oo, t h a t is, f E L l ( ( P z , x'}) for all x E X, x' C F, a n d (ii) for each E E E t h e r e exists a n element of s necessarily unique a n d denoted by

fE f d P , such t h a t ( ( ~ f dP)x,x') = ~ f d(Px, x'},

x e X, x' E F.

D e f i n i t i o n I I I . 6 . A finitely additive spectral measure P : P. ~ algebra of sets E is called bounded if IIP(E)H := sup{llr(EDll : E ~ E} < oo.

s

9 defined on a n

9

Let X be a Banach space and P : E ~ s be a spectral measure. Then P is necessarily bounded. Proof. It follows from L e m m a III.2 t h a t (Px, z'} is ~r-additive , for all x E X, z" E X ' and Lemma III.3.

hence, by T h e o r e m 1.1, t h a t

supl(P(E)x,x'}l
xEX,

x'EX'.

.

The conclusion follows from the Uniform Boundedness Principle ; see Proposition III.l(a).I E x e r c i s e 34.[*] Let X be a B a n a c h space a n d P : E ~ s be a spectral m e a s u r e . For each E E E let Q(E) = P ( E ) ' E s be the dual o p e r a t o r of P(E). Show t h a t the function Q : E ~ s so defined is a b o u n d e d spectral measure of class F = X C_ X " . 9 E x e r c i s e 35. A B a n a e h space X is said to contain a copy of 6 ~176 if t h e r e exists a closed subspace Y of X a n d a bicontinuous isomorphism of 6~ onto Y. T h e following result can be found in [8; p.23].

Let X be a Banach space which does not contain a copy of 6~ and let F C_ X ' be a subspace which distinguishes the points of X . Suppose that ~n~=~ x~ is a (formal) series in X such that every subseries is F-convergent in the sense that for each A C_ N there is XA E X satisfying Fact.

E(Xn,X') nEA

: <XA, X'>,

X ! e F,

Then ~ = 1 xn is (norm) unconditionally convergent. Let X be a B a n a c h space not containing a copy of 6~ a n d P : E ~

s be a spectral measure of class F _C X ' . Show t h a t P is actually a spectral measure (i.e. of class X ' ) . 9 I I I . 2 . Let X be a Banach space and P : E ----+ s be a spectral measure of class F C_ X'. I f P is bounded, then every bounded E-measurable function is P-integrable.

Proposition

51 P r o o f . Let ~b E B~176 Then certainly ~ C Ll(
=

Since [(T) := ( f x , x'), for T E / : ( X ) , is an element of (s for each x C X and x' C F, it is clear from (12) that (ii) of Definition III.5 is satisfied. Accordingly, r is P - i n t e g r a b l e . I We are now able to describe the nature of continuous linear and multiplicative homomorphisms from C(A) into s where A is a compact Hausdorff space, in terms of a B.a. of projections ; see [15; Chapter XVII, Theorem 2.4]. T h e o r e m I I I . 3 . Let X be a Banach space, A be a compact Hausdorff space and let S : C(A) ~ s X ) be a continuous linear map which is also a unitaI algebra homomorphism (i.e. S ( f g ) = S(f)S(g), for f, 9 9 C(A), and S(ll ) = I). Then there exists a unique bounded spectral measure R : Bo(A) - - + L(X') of class F = X c X " such that (i) (x, Rx') : Bo(A) ~ C is regular for each x 9 X and x' 9 X',

and (ii) S ( f ) ' = f a / d R , f 9 C(A). P r o o f . By assumption IIS]I := sup{i]S(f)l] : Ilflloo -< 1} < oo. Fixx 9 and define Sx : C(A) ~ X by Sx(f) = S ( f ) x , for f 9 C(A). By Theorem 1.12 there exists a unique finitely additive measure m~ : Bo(A) > X" such that (x', rex) is a regular G-additive measure for each x' 9 X ' and satisfies (13)

f , f d(x',mx) = (S~(f),x') = (S(f)x,x'),

f 9 C(A).

Let a l , a2 9 C and Xl, x2 9 X. Then S~1x1+~2~2 = c~1S~1+c~2S~ 2. Moreover, the mapping ~ X" satisfies <x', a l m ~ + (~2m~z) = aa(x', r n ~ ) + c~2<x',m~2), is regular and a-additive for each x' 9 X', and satisfies

alm~l + ~2m~2 : Bo(A)

fire/ d(x', oqmxl -4- a2m~2) = (S(f)(O~lX 1 Jr- o~2x2), Xl) ,

f 9 C(A).

By uniqueness we deduce that (14)

almzi + a2mx2

=

mc~1,~1+~2=2.

Fix E 9 Bo(A) . Then x ~ (x',m~(E)), for x 9 X, is linear (by (14)) and satisfies [(x',m~(E))[ < [[<x',m~)[[. Since C(A)' = M(Bo(A)), it follows from (13) that [[(x',m~)[[ =

=

[ffd(x',m~)]=

sup II/ll~<_] JA sup I G ( f ) x , x ' ) l Ilf[l~_
_< IlSll.llxll.llx'll,

CHAPTER III. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

52

for every x' E X'. This shows, for each fixed x' C X', that x H (x',m~(E)), for x E X, is a continuous linear functional on X. Hence, there is a unique vector R(E)x' C X ' such that (15)

(x, R(E)x') = (x', rex(E)).

Furthermore, we see from above that

IIR(E)z'I[ := sup I(z,R(E)x')I = sup r(z',m~(E))l <_ IISlI. IIx'll, Ilxll_
[Izll_
for all x' E X ' and E E Bo(A). Moreover, x' ~ R(E)x' is linear for each fixed E E Bo(A); ! ! this follows from (15) since, for each (h, c~2 E C and zl, z 2 E X', we have that

(x, R(E)(~Ix i + c~2z;)) =

(~ix~ + ~2x'2,rex(E))

:

~l(X'1,m=(S)) + c~2(x'2,mx(S))

:

~i(~,R(E)xi) + ~2(x,R(E)x;)

:

(x, OZlR(S)x i + c~2R(S)x;}.

Since this is true for all x E X, we conclude that R ( E ) ( ~ l x I + ~2x;) = ~ l B ( E ) x i

~R(E)x;.

So, x' ~ R(E)x' is indeed linear for each E E Bo(A). Moreover, R is bounded since sup{l[R(E)ll : E C Bo(a)} _< IISI[. Hence, R : Bo(A) ----+ s is a finitely additive measure with bounded range such that (x, Rx'} is a regular a-additive measure, for each x C X, x' 9 X', and satisfies

(S(I)x, x') = ~A f d(x, Rx'},

(16)

f 9 C(1).

We now show that R is a finitely additive spectral measure. P u t f = 11 in (16) and use

S(]I ) : I 9 s

gives (x, x') = (x, R(A)x'),

x 9 X, x' E X',

and so R(A) = I 9 s P u t f = 0 gives R(0) - 0. The finite additivity of R implies that R ( E ~) = I - R(E), for E 9 Bo(A). It remains to check the multiplicativity of R. Fix g 9 C(A) and x 9 X, x' C X'. Define a regular complex measure #~,~, by #~,~,(E) := fEgd(x, Rx'}, for E 9 Bo(a). Then, for each f E C(A) C_ Lt(#~,~,) we have (by (16) and the identity S(Ig) = S(f)S(g)) that

= (S(I)x, S(g)'x'} = [ f d(x, RS(g)'x'). JA

53 Since #x,~, and (x, R(.)S(g)'x') are both regular complex measures it follows that #x,~, = (x, R(.)S(g)'x') as an equality of measures, that is,

(17)

/~ g d<x, m:') : <x, n(E)S(g)'x'>,

g 9 C(A), E 9 Bo(A).

Suppose we know that S(g)'R(E) = R(E)S(g)' for all E 9 Bo(A) and g 9 C(A). Then it follows from (17), for E fixed now, that

leg d<x,Rx'> = <x,S(g)'R(E)x'>

<S(g)x, R(E)x'> : f, g ~<x,m,>,

where z' = R(E)x'. T h a t is,

/Agd(x, Rx'),~ = /Agd(x, Rz'),

g e C(a),

where (x, Rx')[ E denotes the restriction of the measure (x, Rx') to E. By uniqueness of regular measures we deduce that (x, Rx')[E : (x, Rz') as an equality of measures, that is,

(x, R(E A F)x'} = (x, R(F)z') = (x, R(F)R(E)x'},

F 9 Bo(A).

By also fixing F we conclude, since x 9 X and x' 9 X ' are arbitrary, that R(E n F) : R(F)R(E). So, it remains to verify that S(g)'R(E) = R(E)S(g)', for all g 9 C(A) and E 9 Bo(A). Accordingly, fix g and E. Choose Bo(A) -simple functions {gn}~n=~ such that IIg,~-gll~ ------+O. We have seen before that

II/AgndRIl<_4.11SIl.llgnlloo ,

heN,

and so there exists R(g) 9 s with fh gn dR -----* R(g) in s Since R(E) commutes with each operator fA g'~ dR, for n 9 N, it is clear that R(E)R(g) = R(g)R(E). So, it suffices to show that R(g) = S(g)'. By the dominated convergence theorem applied to (x, Rx'} we see (c.f. (16)) that

But, fA gn d(x, Rx') = (x, (fA g'~ dR)x') in s Hence, we deduce that

<x, s(g)'x')

~ (x, R(g)x'), as n --+ oo, since fA gn dR -----+R(g)

= <x, R(g)x'>,

which implies the desired equality S(g)' = R(g).

x 9 x , x' 9 x',

9

54

CHAPTER III. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

An unpleasant feature of Theorem III.3 is that the B.a. of projections {R(E) : E C Bo(A)} exists in s rather t h a n in s If the Banach space X in Theorem III.3 does not contain an isomorphic copy of e0, then more can be said. Indeed, the bounded linear maps Sx : C(A) ~ X given by S~ : f H S ( f ) x are then necessarily weakly compact ; see Theorem 1.14. Then applying the vector-valued Riesz representation theorem (see Theorem 1.13) in place of Theorem 1.12 it is possible to choose for each z E X a unique a-additive vector measure m~ : Bo(A) ~ X (rather t h a n X'-valued) which is regular and satisfies

S ( f ) z = / A f dm~,

f C C(A).

An analogues argument as in the proof of Theorem III.3 then establishes the following result. T h e o r e m I I I . 4 . Let A be a compact Hausdorff space, X be a Banach space not containing an isomorphic copy of co and S : C(A) ~ Z;~(X) be a continuous linear map which is also a unital homomorphism. Then there exists a unique spectral measure P : Bo(A) ~ s

such that (i) each vector measure Px : Bo(A) --~ X, for x 9 X , given by E ~ P ( E ) x is regular and (ii) S(I) = f A f alP, f 9 C(A). Let /3 be a bounded B.a. of projections in s and let (/3)~ be the uniformly closed algebra generated by B in s By Theorem III.2 there exists a finitely additive spectral measure Q : Co(f~s) ~ s where f s is the Stone space of/3, such that the map S : C(t2s) ,s given by (18)

S(f) = {

f dQ,

f 9 C(gs),

B

is a Banach algebra isomorphism of C(f~s) onto (/3)~. Recall that fa f dQ is defined by a continuous extension process from the dense subalgebra sim(Co(as)) to all of C ( f s ) . We note that Q is only defined on the algebra of sets C o ( f s ) . By Theorem III.3 there exists a unique regular, bounded spectral measure R : Bo(f~s) /2(X') of class F = X C X " (so some a-additivity is present) such that (19)

f dR,

S(f)'={

f E C(f~s),

J~2 B

where S is given by (18). The first point is that R is defined on the a-algebra B o ( f s ) and takes its values in s with X ' the dual space of X. If E 9 Co(fiB), then X~ 9 C ( f s ) and so we see from (18) and (19) that R(E) = Q(E)'. So, every projection in the B.a. {R(E) : E 9 Co(f~s)} c s is the dual operator of some projection from the B.a./3 = { Q ( E ) : E 9 Co(Us)} C_ s Fix x 9 X and x' 9 X'. Then the finitely additive measure (Qx, x') is actually a-additive on the algebra C o ( f s ) . Indeed, suppose that {E~}~=~ C_ Co(t2s) is a sequence of pairwise disjoint sets such that E := U~=IE,~ belongs to C o ( f s ) . Since E C Co(t2s), it is closed hence

55 compact.

Accordingly, { E

oc n}n=l is

an open cover of t h e c o m p a c t set E a n d so t h e r e exist

k finitely m a n y sets {E~j}~= I such t h a t E = Uj=IE~ j. Moreover, t h e disjointness property implies t h a t En = 0 for n ~ {nj}j= k 1. T h e n the finite additivity of Q yields

oo (Q(u~=I

E

~)x, x i ) =

~(Q(En)~, x'). n=l

This establishes the (7-additivity of Q on the algebra of sets Co(fB). A n e x a m i n a t i o n of Definition 1.1 (as ah'eady noted) shows t h a t t h e variation I~1 of any C defined on a n algebra of sets S is just as well defined as for a complex measure with domain a a - a l g e b r a of sets. D e f i n i t i o n I I I . 7 . A b o u n d e d finitely additive measure p : N - - - , C defined on a n algebra

bounded finitely additive function u : S ~

of sets S of some topological space A is said to be regular if for each E C $ a n d each e > 0 there is a set K E S with K C_ E a n d a set U E S with E C_ U ~ such t h a t I~t(U\K) < e. This is equivalent to [u(F)l < ~ for all F C S w i t h F C_ U \ K . 9 T h e following classical extension t h e o r e m for measures is due to A.D. Alexandroff ; see [14; p.138], for example. I I I . 5 . Let u : S ----* C be a bounded, regular, finitely additive measure defined on an algebra of sets 8 of some compact topological Hausdorff space A. Then t~ is a-additive on S and u has a unique extension to a regular, (7-additive measure on a(S), the (7-algebra generated by $. E x e r c i s e 36. Let .A be a n algebra of subsets of a set A a n d let E = a ( A ) be the (7Theorem

algebra g e n e r a t e d by .4. Let p : E ---* C and u : E ~

C be complex measures such t h a t

#(E) = u(E) for every E C A. Show t h a t #(E) = u(E) for every E E E. 9 After this short digression let us r e t u r n to the s i t u a t i o n of a bounded B.a. of projections B c_ s a n d its Stone m a p Q : Co(fu) ----* B, where f u is the Stone space of B. Fix x C X a n d x' E X'. It was n o t e d above t h a t (Qx, x') is a - a d d i t i v e on t h e algebra of sets Co(flu). Moreover, we have seen t h a t

sup{I(Q(E)z,z')I: E c C o ( ~ ) ) <_411Nl.llzll.llx'll,

z c X , x' ~X',

showing t h a t (Qx, x') is a bounded measure. It is a routine observation to note t h a t Definition III.7 is fulfilled with S = Co(fB) a n d so (Qx, x') is regular on Co(fB). Proposition 11.5 a n d T h e o r e m III.5 imply t h a t (Qx, x') has a unique, regular extension pz,~, : Ba(fB) ~ C which is also a-additive. By T h e o r e m III.3 t h e r e also exists a regular, a-additive measure (x, Rx') : Bo(fB) ~ C which coincides with (Qx, x'} on Co(f~). T h e n necessarily #~,~, = {x, Rx') on t h e a - a l g e b r a B a ( f s ) c_ B o ( f B ) ; see Exercise 36. Actually, {x, Rx') is t h e unique regular Borel extension of t h e Baire measure #~,~,; see [37; p.314]. P a r t of our aim in t h e next two chapters will b e to investigate more closely certain properties of the B.a. B which ensure t h a t the extension of t h e scalar measures {Qx, x'} from Co(f~) to B a ( f B ) or Bo(fB) is effected by members of B itself. T h a t is, for each E from B a ( f ~ ) or Bo(f~B) we require t h e existence of QE E B which satisfies

,~,~,(E) = (Q~x, x') : (x, r~(E)x'),

x e X , x' e X'.

56

C H A P T E R III. UNIFORMLY CLOSED O P E R A T O R A L G E B R A S

Equivalently, every projection from {R(E) : E ~ Ba(~Q~)} or {R(E) : E E Bo(t2B)}, rather than just those from {R(E) : E C Co(t2~)}, should be the dual operator of some projection from B. The appropriate conditions on B which ensure that this is the case will be formulated in the following chapters.

Chapter IV Ranges of spectral measures and Boolean algebras of projections At the end of Chapter II we saw that the Stone map Q : Co(~2~) ~ / 3 can be extended to a a-homomorphism Q (resp. s on Ba(f~B) (resp.Bo(f~)) provided that the B.a./3 is abstractly a-complete (resp. abstractly complete). In this generality not much more can be said. However, the more specialized case of a B.a. of projections 13 acting in a Banach space X has the additional feature that it is part of a vector space equipped with a topology , namely s or s Since both Ba(f~u) and Bo(f~u) are a-algebras of sets, it then has a meaning to ask whether Q a n d / o r s are a-additive, that is, whether they are spectral measures? The abstract a-completeness or abstract completeness of/3 by itself no longer suffices in this case; to see this combine Example 20(a) with Theorem IV.1 below. It turns out that this question has a precise (and positive) answer in terms of some very natural properties of B which intimately relate certain order properties of directed systems in /3 with some topological requirements. Such conditions, already known quite some time ago for B.a. 's of selfadjoint projections in a Hilbert space (see [18], [25], for example), were extended to the Banach space setting by W. Bade [1,2]. The purpose of this chapter is to make a detailed study of the class of those B.a. 's of projections which can be represented as the range of a a-additive spectral measure defined on a ~-algebra of sets. For such B.a. 's the powerful methods of vector measures and integration can be invoked; this will be exploited in the following chapter. Let us begin with various notions which connect the order properties of/3 with some topological requirements. D e f i n i t i o n I V . 1 . Let X be a Banaeh space a n d / 3 <_ s be a B.a. of projections. (i) B is called Bade complete (resp. Bade a-complete ) if it is abstractly complete (resp. abstractly a-complete) in the sense of Definition II.4 and if (A~B,~)X = A~B~X

and

(V~Bc~)X = sp{U~B~X}

whenever {B~} is a family (resp. countable family) of elements from/3.

58

CHAPTERIK

R A N G E S OF S P E C T R A L MEASURES

(ii) B has the monotone (resp. g-monotone) property if lim~ B~ exists for the weak operator topology and belongs t o / 3 whenever {B~} _C B is a monotone net (resp. sequence) with respect to the partial order of/3. (iii) /3 has the ordered (resp. or-ordered) convergence property if lim~ B~ exists for the strong operator topology and belongs to B whenever {B~} C /3 is a monotone net (resp. sequence) with respect to the partial order of 13. 9 Definition IV.l(i), without the term "Bade", was introduced by W. Bade in [1]. E x a m p l e 20. (a) Let X = g~ and, for each E E 2 ~, let P ( E ) E s be defined by

P ( E ) x = (xlxE(1),x2Xz(2),...),

x = (zl,x2 . . . . ) E goo.

Then B := { P ( E ) : E E 2 N} is an abstractly complete B.a. of projections which fails to be Bade a-complete. Indeed, let E , = {n}, for n E N, in which case V~P(En) = I. But, the linear hull sp{U~=IP(E~)X } is the subspace of goo consisting of all elements with only finitely many non-zero co-ordinates. Hence, sp{U~_I P( En)X } = co whereas ( V , P ( E~) ) X = g~. (b) Let X = g2([0, 1]), a nonseparable Hilbert space. Let E = Bo([0, 1]) and, for each E E E, let P ( E ) E s be the projection P(E) : ~ ~ Xsr (pointwise multiplication), for r E X. Then B := P ( E ) is Bade c~-complete. Let A C [0, 1] be a set which is not a Borel set. If ~C(A) is the family of all finite subsets of A, then { P ( E ) : E E .7-(A)} is a family of projections belonging to B which has no least upper bound in B. Hence, B is not abstractly complete and so also fails to be Bade complete. (c) Let X = g~ and E be the B.a. of all subsets of N which are finite or the complement of a finite set. For each E E.E define a projection BE E s by BEX

= (Xl)~E(1),Z2~E(2)

. . . . ),

X = (Xl,X2,.

. . ) E e 2.

T h e n / 3 := {BE : E E E} is a bounded B.a. of projections which is not abstractly or-complete. Indeed, if E , = {2, 4 , . . . , 2n}, for n E N, then {BE,,}~-I has no least upper bound in B. 9 Let X be a Banach space and P : E ~ s be a spectral measure defined on a a-algebra of sets E of some set ft # ~. A set E E E is called P-null if P ( E ) = 0. By the multiplicativity of P (e.f. Definition III.2) this is equivalent to P ( F ) = 0 for all F E E with F C E. Two sets E, F E E are called P-equivalent if their symmetric difference Ez~F is P-null. This is an equivalence relation in E; the equivalence class of E E E is denoted by [El and the family of all equivalence classes {[El: E C E} is denoted by E(P). It is routine to check that the operations [E] A [F] := [E A F]

and

[E] V [F] := [E V F]

and

[E]' := [E'],

with E A F = E M F and E V F = E U F and E' = E c being the B.a. operations in E, are well-defined in E ( P ) and t u r n E ( P ) into a B.a. Moreover, if B := P(E), then the induced map P : E ( P ) ~ B given by P([E]) := P ( E ) is a B.a. isomorphism of N(P) onto B. Note that the zero element [0] of E ( P ) consists of all the P-null sets in E. Part (a) of the following exercise shows that P is well defined.

59 E x e r c i s e 37. Let X be a Banach space and P : E ~ / : ( X ) be a spectral measure. (a) If E, F E E are P-equivalent (i.e. P ( E A F ) = 0) show that P(E) = P(F). (b) Let {t~j}jn=l C_ E be P-null sets. Show that Ujn=IE2 is also P-null. 9 Recall that a function p : Y x Y - - ~ [0, ec), where Y is any non-empty set, is called a pseudometric if it satisfies p(u, v) = p(v, u), for all u, v E Y, and p(u, v) < p(u, z) + p(z, v) for all u, v, z E Y; see [24; Chapter 15]. Let P : E ~ s be a spectral measure . For each x ~ X, define a pseudometric d~ on E ( P ) by

d~([E], [El) := IIPzlI(EAF),

[El, [F]E 2(P),

where IIPxlt is the semivariation of the vector measure P z : E ----* X given by E ~ P(E)z, for E E E. The triangle inequality for d~ follows from E A F C_ (E/\G) U ( G A F ) , valid for any E, F, G E E, together with the subadditivity of semivariation; see Lemma I.l(b). The topology and an/form structure on E(P) specified by the family of pseudometrics {d~ : z E X} (see [24; Chapter 15] for the definition of uniform structure) is denoted by 7~(P). E x e r c i s e 38. Let P : E - - ~ s be a spectral measure . Show that the family of pseudometrics {d~ : x E X} has the Hausdorff property, namely that [E] = [F] whenever [El, IF] C E ( P ) satisfy d~([E], IF]) - 0 for all z E X. 9 D e f i n i t i o n IV.2. A spectral measure P : E - - ~ ~2(X) is called a closed spectral measure if (E(P), Ts(P))is a complete uniform space. 9 The previous definition is a special case (for spectral measures) of the notion of a closed measure for arbitrary vector measures taking values in any locally convex Hausdorff space; see [27; Chapter IV]. For the definition and basic theory of locally convex Hausdorff spaces we refer to [14; Chapter V]. The main aim of this chapter is to establish the connection between the various notions of completeness for a B.a. of projections (as given in Definition IV.l) and spectral measures. T h e o r e m I V . 1 . Let X be a Banach space and B c_ s be a B.a. of projections. Then

the following statements are equivalent. (a) B has the or-monotone (rasp. monotone) property. (b) B has the a-ordered (rasp. ordered) convergence property. (c) /3 is Bade or-complete (rasp. Bade complete ). (d) B coincides with the range of some spectral (rasp. closed spectral) measure. The proof of Theorem IV. 1 will require a series of laminas and propositions. The extension of the Stone map Q from the algebra of sets Co(~2B) to the a-algebras Ba(f2B) and Bo(gtB) , as given by Theorem II.4 and Theorem II.5, will play a decisive role. We begin with a useful technical result. L e m m a I V . 1 . Let X be a Banach space and 13 C_ s be a B.a. of projections. (a) Let B have the monotone (reap. or-monotone) property. Then (i) 13 is abstractly complete (rasp. abstractly a-complete), and (ii) whenever {B~} a_ B is an increasing net (reap. sequence), then lim~ B~ = V~B~, whereas if {B~} is a decreasing net (reap. sequence), then lim~ B~ = A~B~, where the limits

60

CHAPTER IV. R A N G E S OF S P E C T R A L MEASURES

exist in s (b) Let B have the ordered (resp. a-ordered ) convergence property. Then (i) B is abstractly complete (resp. abstractly c~-complete) , and (ii) whenever {B~} is an increasing net (resp. sequence), then lima B~ = V~B~, whereas if {B~} is a decreasing net (resp. sequence), then lim~B~ = A~B~, where the limits exist in s . P r o o f . (a) (ii) Suppose that B has the monotone property. Let {B~}~E A C_ /3 be an increasing net. By definition of the monotone property B = lim~ B~ exists in s and B E/3. If a E A, then B~Bv = B~ for all -~ >_ c~ and so {BB~x, x'} = lim{B~B~x, x') = ( B j , x'),

x E X, x' E X',

which implies that BB~ = B~. That is, B~ _< B for all a E A. Suppose that D E B satisfies B~_
61 and Bo(~u), respectively. L e m m a I V . 2 . Let X be a Banach space and t3 C s be a B.a. of projections. Let ~u be the Stone space of B and Q : Co(~u) ~ B be the Stone map. (a) There exists a unique spectral measure P : Ba(~u) , s having the properties that 13 = P( Ba( ~u) ) and P coincides with Q on Co( Q~), if and only if, B has the a-monotone property. (b) Suppose that the B.a.B is abstractly complete . Then there exists a spectral measure P : Bo(~u) ~ s having the properties that B = P(Bo(~u)) and P coincides with Q on Co(~u), if and only if, 13 has the a-monotone property. P r o o f . (a) Suppose there exists a spectral measure P : B a ( ~ ) ----+ s such that P(Ba(~u) = B and P coincides with Q on Co(~u). Let { B ~}n=l be an increasing sequence in B in which case B~ = P(E~) for some sets E~ C Ba(~u). T h e n P(E~\E~+k) = P(E~) - P ( ~ n E~+k) = P(E~) - P(E~)P(E~+~) = B~ - B~B~+~ = O, for all n and k, since B~ _< B,+k means that B~B~+k = B~. Then the sets A~ = U~=IEJ are increasing in ~ s and P(An) = P([U~-~Ej]\En) + P(En) = P(U~-I(Ej\En)) + Bn = Bn,

n 9 N,

since if P(Fj) = 0, for 1 < j < r, then also P(O;=IFj) = 0; see Exercise 37(b). Since P is a-additive in s we have that c~

lira B . = lim P ( A ~ ) = P(Uk=IAk), n~co

n~oo

where P(Uk=IAk)~ 9 P(Ba(~u)) = B and the limit exists in s . If {B~} _C B is a decreasing sequence we consider the increasing sequence {I - B~}~= 1 C B to deduce again that l i m ~ o ~ B~ exists in s and belongs to B. Since the existence of a limit i n / : s ( X ) implies its existence in s we conclude that B has the ~-monotone property. Conversely, suppose that B has the a-monotone property. By Lemma IV.1 we know that B is then abstractly a-complete . Then Theorem II.4 guarantees that there exists a unique a-homomorphism P from Ba(~u) onto B such that P extends Q. Let {E~}~=~ C Ba(~z) decrease to ~. By Lemma IV.l(a), since (P(/~)}n~=~ is decreasing in B, it follows that lim P(E~) ~ A~=IP(E~ ) = P ( Q o=o I E ~ ) = 0, n~oo

where the limit exists in s

So, for x 9 X fixed, we deduce that

~i~flp(E~)~, x'~ = 0,

~' 9 x'.

This shows that the X-valued function m := P(.)x : E ~ X has the property that (m, x') is a-additive for all x ~ 9 X'. By Proposition 1.1 we conclude that m is a vector measure and so, in particular, P(E~)x = m(E~) ~ 0 with respect to the norm topology in X. Since x 9 X is arbitrary, it follows that P(E~) ---, 0 in s Hence, P is a-additive in s

62

C H A P T E R IV. R A N G E S OF S P E C T R A L M E A S U R E S

The uniqueness of P follows from the uniqueness statement in Theorem II.4 together with the fact that Ba(f)•) = cr(Co(f~B)). (b) Modify the proof of part (a) using Theorem II.5 in place of Theorem II.4. L e m m a IV.3. Let X be a Banaeh space and P : E - - ~ s the following statements are equivalent.

9

be a spectral measure. Then

(a) P is a closed spectral measure (c.f. Definition IV.2). (b) The B.a. E(P) is abstractly complete and, whenever {[E~]} c_ E(P) is a net downwards filtering to [0], it follows that lima P(E~) = 0 in s . (c) The B.a. E(P) is abstractly complete and, whenever {[E~]} _C E(P) is a net downwards filtering to [0], it follows that Iima P(E~) = 0 in s P r o o f . We first show that s is quasicomptete, meaning that every bounded Cauchy net {T~} c s (i.e. for which sup~ NT~zll < oc for each x e X) has a limit in s But, for fixed z C X, the definition of the topology in s implies that the net {Tax} is Cauchy in X and so has a limit, say Tx, in X (as X is complete). Then T : X ----* X defined by z ~ T z is an element of s and T~ ~ + T in s see Proposition III.l(b). The equivalence (a)<==:~(b) now follows (by choosing rn := P and Y := s from a general result about vector measures rn : E ----* Y, where Y is a quasicomplete locally convex Hausdorff space, which states: A vector measure m : E -----+ Y is closed if and only if E(m) (identified with {X~ : E E E} c L~(m)) is complete as an abstract B.a. and whenever {[E~]m} _C E(m) is downwards filtering to [0], it follows that rn(E~) ~ 0 in Y .

For the proof of this result and a more precise definition of the notation and concepts involved, we refer to [11; Proposition 1.1]. The equivalence (b)r follows from the fact that the locally convex Hausdorff space s equipped with its weak topology r163 (s is precisely the space s (this is essentially Proposition III. 1(c)), and a general fact about vector measures, [34; Proposition 2], which states: Let Y be a locally convex Hausdorff space and m : E ~ Y be a vector measure. Then m is a closed measure if and only if it is a closed measure for every locally convex topology on Y consistent with the duality between Y and Y'. 9

We can now formulate one of the main representation theorems for B.a. 's of projections (of a certain kind) in terms of spectral measures. T h e o r e m IV.2.

Let X be a Banach space and 13 C_ s X ) be a B.a. of projections.

(a) I f B has the a-monotone property, then there exists a unique spectral measure P: Ba(f~) , s X ) , where f~B is the Stone space of B, such that P ( Ba( fh3) ) = B and P restricted to Co(f~s) coincides with the Stone map Q : Co(f2~) ~ B. (b) I[ B has the monotone property, then there exists a unique closed spectral measure P : Bo(f2t3) ---* s where f2B is the Stone space of B, with the properties that P(Bo(f~B)) = B, the map P restricted to Co(F~B) coincides with the Stone map Q : Co(f~B) ~ B, and

63

the spectral measure P is regular in the sense that

(1)

P(E) =

lira P(V),

E C So(U~),

v~v(E)

where the limit of the net in (I) exists in f~s(X) and ])(E) is the downwards filteringsystem (directed by inclusion) of all open sets V C f~ such that E C_ V. Proof. (a) This is Lemma IV.2(a). (b) Lemma IV.l(a) implies that B is abstractly complete and so, by Lemma IV.2(b), there is a spectral measure P : Bo(f~u) --~ s which extends the Stone map Q from Co(f~s) onto Bo(~u) and satisfies P(Bo(~u)) = B. For simplicity of notation let E := Bo(~B). The quotient B.a. E(P) is abstractly complete because it is isomorphic to the abstractly complete B.a.B. To deduce the closedness of P we wish to use Lemma IV.3. So, let {[E~]} C_ E(P) be decreasing to [0]. Since the induced map P : ~ ( P ) - - ~ /3 given by iB([E]) = P ( E ) is a B.a. isomorphism it follows that {P(&~)} is decreasing in B to A~P(&~) = P(A~[E~]) = P([0]) = 0. By Lemma IV.l(a) it follows that P(E~) ~, 0 in E ~ ( X ) . Hence, Lemma IV.3 does indeed apply to show that P is a closed spectral measure. Let {P(F~)} c_ B be an increasing net. Since P : W,(p) ~ B is a B.a. isomorphism the net {[F~]} is increasing in ~ ( P ) . Since ~ ( P ) is abstractly complete there is IF] E W~(P) such that V~[F~] = [F] and hence, {[F\F~]} is downwards filtering to [0] in ~ ( P ) . By Lemma IV.3 we conclude that P ( F \ F ~ ) ~

0 in s

. Now, for each ~,

(2) P ( F ) - P(F~) = (P(F\F~) + P ( F A F~)) - P(F~) = P(F\F~) + ( P ( F 0 F~) - P(F~)). But P ( F A F~) - P(F~) = 0, since [F~] T [F] implies that [F] A [Fg] = [F~], and so

P ( F n F~) = P ( [ F A F~]) = P([F] A [F~]) = P([F~]) = P(Fa). It follows from (2) that P(F~) ~ P(F) in s Similarly, if {P(H~)} C_ B is a decreasing net, then {[Hv] } is decreasing in ~ ( P ) and so by abstract completeness of Z ( P ) again there is [HI E E ( P ) such that {[Hv\H]} ~ [0] in 2 ( P ) . Then Lemma IV.3 yields P(H.~) ~ P(H) in s . So, we have established that B has the ordered convergence property. Suppose for the moment that P is regular. Then the uniqueness of P follows immediately. Indeed, let R : Bo(f~u) ---+ s be another spectral measure which coincides with Q on Co(Qu), has range B and is regular. By Lemma IV.l(a) B is abstractly complete and so R and P coincide on the open sets in f~u; see Theorem II.5. Then (1) implies that R = P on

Bo(f~). So, it remains to establish the regularity of P. Fix z ~ X and x ~ ~ X ~, and let # be the real pm't of the complex measure (Pz, f ) , that is,

~(E) = ae((P(E)x, x'>),

E c Bo(a~).

CHAPTER IV. RANGES OF SPECTRAL MEASURES

64

Then # is a a-additive, R-valued measure and so by the Hahn decomposition theorem [14; p. 129] there are pairwise disjoint Borel sets E1 and E2 with ~ = El U E2 and #(Ey N E) _> 0, for all E E Bo(~B) and j C {1, 2}, such that #(E) = #(E n El) - #(E n E2),

E E Bo(~B).

Let 9 : Bo(gB) ----+ Co(gs) be the c~-homomorphism which maps each E E B o ( 9 s ) to the unique set 9 ( E ) E Co(fts) such that E A 9 ( E ) is of the first category . Now 9 is a B.a. homomorphism and so the sets Uj := 9(E0), j = 1, 2, are pairwise disjoint, belong to Co(~s) and satisfy 0-1 n 0"2 = F~B. Since E1 A U1 and E2 ~ 0"2 are both P-null we have

E 9 Bo(fttJ.

#(E) = # ( E N U1) - # ( E N U2), Define non-negative measures #1 and #2 by

#](E)=p(ENUj),

EEBo(as),

j E {1, 2}.

Fix an open set V C fls and let {V~}~cd be the family of all open sets in f~s such that V~ C V, for a E A. Ordering {Vo n U/}~ed by inclusion we deduce that {P(V~ N Uj)}~ed is increasing in B, as P is a homomorphism , and that P(V n Uj) = V~eAP(V~ N Ui) by Theorem II.5. Since B has the monotone property, it follows that P(V~ N U]) ~ P(V n Uj) in s see Lemma IV.l(a). Accordingly, (P(V~ U Uj)x, x'} - ~ (P(V N Uj)x, x'). Since z ~ Re(z) is a continuous function on C it follows that also p(V~ N Uj) ~ #(V N Uj), i.e. #j(V~) --% #j(V) for j = 1, 2. Accordingly, #/V)

=

sup#j(V~),

V c_ ~B, Vopen, j 9 {1,2}.

c~

That is, both Pl and p2 are r-additive measures; see [20] for this notion. It then follows from [20; Theorem 5.4] that both Pl and P2 are regular Borel measures and hence, p is also regular. By a similar argument the imaginary part of (Px, x'} is also a regular measure and hence, (Px, x') itself is regular. Accordingly,

(Px, x')(E) :

lim (Px, x')(V),

E 9 Bo(f~s).

veY(E)

Since x 9 X and x' 9 X ' are arbitrary we deduce that (3)

Iim P(V) = P(E)

V~V(E)

(ins

for each E 9 Bo(t2~). But, as already noted above, 13 has the ordered convergence property and so TE := limyev(E) P(V) exists in s . Then (3) implies that TE = P(E) and so (1) holds for every E 9 Bo(~28), that is, P is regular. 9 The notion of regularity used in the paper [20], which is cited in the proof of the previous theorem, appears different to that given in Definition 1.2. The following exercise shows that the two notions of regularity are actually the same.

65 E x e r c i s e 40. (a) Let E := Bo(f~) , where f~ is a compact Hausdorff space and v : E - - ~ C be a complex m e a s u r e . Let Ivl : E ----+ [0, oo) be the variation measure of v. Show, in the sense of Definition 1.2, that v is regular if and only if Ivl is regular. (b) Let # : Bo(f~) ~ [0, oo) be a finite, a-additive measure. For each subset E c ft define #{(E)

::

sup{p(Z) : Z c E, Z closed in f~}

and # J E ] "= infIu(U) " E C U, U open in f~}. Show that the following ~tateIhents af~quilz~en~. (i) # is regular. (ii) #(B) : p{(B), for every B E Bo(f~). (iii) p(B) = p~(B), for every B E Bo(a). Deduce that regularity as given in [20] coincides with that of Definition 1.2. 9 We are now ready to establish the main result of this chapter. P r o o f of T h e o r e m I V . 1 . ( b ) ~ ( a ) . This is clear as convergence of nets (and sequences) in f-.~(X) also implies their convergence i n / 2 ~ ( X ) and to the same limit. ( a ) ~ ( d ) . This is immediate from Theorem IV.2(a) if B has the a-monotone property and from Theorem IV.2(b) if B has the monotone property. ( c ) ~ ( b ) . Suppose that B is Bade complete. Let {Ba} C_ B be an increasing net and let B = V~B~. Fix x E X. Given e > 0, the fact that B x C B X = sp{U~B~X} means there is n a vector y = ~ j = l zy and indices a 5 such that Zy c B ~ j X (i.e. zj = B~hzj), for 1 < j _< n, and Ily - Bzll < e. F o r each c~ _> c~j (1 _< j _< n) we have B~y = y and B ~ B = B~ (as B~ T B), and hence IIS~x - Sxll

_<

IIS~x - yll + Ily - Szll = IIS~(Bx

_<

(ItB~II + 1)lly - Bxll _< (1 + I/BII)c,

- y)ll + Ily - Sx[I

where IIBIt < co (see Theorem III.1). This shows that lim~ B~x = B z , for the norm topology in X. Since x E X is arbitrary, we have shown that lima B~ = B in Z:~(X) . The dual statement for decreasing nets follows from the formula A~B~ = I - V ~ ( I - B a ) . So, B has the ordered convergence property. The proof when B is Bade a-complete is similar; just replace nets by sequences. ( b ) ~ ( c ) . Suppose that B has the ordered convergence property. By Lemma IV.l(b) B is abstractly complete as a B.a. Let A c_ B be a set and let {B~} be the increasing net consisting of the suprema of all finite subsets of `4, directed by the order induced from/3. Then an element of B is an upper bound for .4 if and only if it is an upper b o u n d for {B~}. Since (F1 V . . . V F ~ ) X = sp{U'~=~FjX} for any finite set { F #}j=~ ~ C .4, to construct a least upper b o u n d for .4 with the property required in the definition of Bade completeness (e.f. Definition IV.l(i)) it suffices to make the corresponding construction for {B~}. Now by Lemma IV.l(b) we have V~B~ = B, where B = lima B~ in Z:~(X). So, it remains to check that

(4)

Bx

= sp{u~B~x).

66

CHAPTER 1V. RANGES OF SPECTRAL MEASURES

Fix x C X. Since B~x -%+ Bx in X and B~x C sp{U~B~X}, for all ct, it follows that Bx c sp{U~B~X}. This shows that B X C_ sp{U~B~X}. On the other hand, since {B~} is increasing with V~B~ = B we have B~B = B~ i.e. B~X C_ B X , for all c~, and so sp{U~B~X} C_ B X follows. So, (4) is indeed satisfied. By considering the decreasing net {I - B~}, a greatest lower bound for A with the property required in Definition IV.l(i) can be constructed in a similar way as the least upper bound was constructed. Hence, /3 is Bade complete . The proof w h e n / 5 has the a-ordered convergence property is similar; just replace nets by sequences. (d)~(b). If 13 coincides with the range of a spectraI measure, t h e n / 3 has the a-ordered convergence property; this was established in the proof of Lemma IV.2(a). If /3 coincides with the range of a closed spectral measure , then /3 has the ordered convergence property; this was established in the proof of Theorem IV.2(b). 9 We conclude with an exercise showing that a finitely additive spectral measure can be a-additive in s only in trivial cases. E x e r c i s e 41. Let X be an infinite dimensional Banach space. (a) Let R E s be a non-zero projection. Show that IIRII > 1. (b) Let (f~, E) be a measurable space and P : E -----+s be a multiplicative set function which is a-additive in the Banach space s that is, l i m , ~ IIP(E~)H = 0 whenever {E~}~_ 1 c E is a sequence decreasing to 0; see Exercise 9(b). Show that there exists N E N such that P(E,~) = 0 for all n > N. (c) Let P : E - - + Z;(X) be as in part (b). Show that the range P ( E ) , of P, is a finite subset of s 9

Chapter V Integral representation of the strongly closed algebra generated by a Boolean algebra of projections Let B be a Bade complete B.a. of projections in a Banach space X. We have seen that there always exists some closed spectral measure P : E ---, s such that P(E) = B; for instance, E can always be taken to be Bo(f*~), where FIB is the Stone space of B. Associated with P is the space LI(p) of all (equivalence classes of) P-integrable functions. It will be shown that L I(P) can be topologized (via a non-normable topology) in such a way that L 1(P) is a complete locally convex Hausdorff space and the (linear) integration map f ~ ffl fdP is a bicontinuous topological and algebraic isomorphism of L I(P) onto the closed subalgebra (B}2, of s generated by/3. The closedness of P turns out to play an essential role in this respect. In particular, LI(P) itself turns out to be a commutative algebra of functions! In the first part of this chapter we develop the theory of integration with respect to spectral measures and investigate the space LI(p), but only as far as is needed to establish the integral representation theorem mentioned above. A fundamental result is the fact that the only P-integrable functions are the P-essentially bounded functions! Having established the representation theorem identifying (B}s as an Lz(P)-space, the remainder of the chapter concentrates on highlighting various non-trivial consequences of this theorem concerning Bade complete B.a. 's of projections B and the subalgebras
CHAPTER V. S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

68

or-algebra of subsets of some non-empty set f~. According to Definition III.4 a E-measurable function f : f2 ~ C is P-integrable if (i) f 9 LI((Px, x')), for each x 9 X and x' C X', and satisfying (ii) for each E 9 E there exists an operator fE f d P 9 s

s f
x 9 x, x' 9 x'.

If s is considered as a locally convex Hausdorff space, where the topology is specified by the family of seminorms {qx : x 9 X} given by

q~(T) := llTxll,

T 9s

for each x 9 X, then the above definition of P-integrability is a particular case of the definition of integration with respect to more genera] vector measures as developed in the book

[27]. Because of the multiplicativity of P it turns out that the definition of P-integrability can be somewhat simplified; see Proposition V.1 below. We begin with a technical result. L e m m a V . 1 . Let X be a Banach space and P : E ----+ s be a spectral measure . Let

f : 9 ----+ C be a E-measurable function such that f 9 LI((Px, x'}), for all x C X and x' E X'. If T 9 s satisfies (Tx, x') = ~ f d(Px, x'},

x 9 X, x' E X',

then T P ( E ) = P(E)T, for all E 9 E. Proof.

FixEEEandx 9

Then

, where y := P(E)x. Also, if ~ := P(E)'x' with P(E)' 9 s

the dual o p e r a t o r , then

(Py, x')(F) := (P(F)y, x') = (P(F)P(E)x, x') and ( P x , ( ) ( F ) := (P(F)x, ~) =
9

Let X be a Banach space and P : E ~ s be a spectral measure . Let f : f~ ~ C be a E-measurable function such that f 9 L~({Px, x')) for every x 9 X and x' 9 X'. Then f is P-integrable if and only if there exists T 9 s satisfying Proposition V.1.

(1)

(Tx, x') = f a f d(Px, x'},

x 9 X, x' e X'.

69

In this case (2)

s f dP = TP(E) = P(E)T,

E C E,

and, in particular, T = fa f dP. Suppose that an operator T c s exists satisfying (1). Define f E f d P := by Lemma V.1), for each E E E. Then, given x e X and x' E X', we have for fixed E C E that Proof.

TP(E)(= P(E)T

((~E f dP)x'x'l = (TP(E)x,x'I = ~ f d(Py, x'l where y :=

P(E)x.

But,

(Py, x')

is the complex measure on E given by

(Py, x')(F) := (P(F)y,x') = (P(F)P(E)x,x') = (P(E A F)x,x') ~ X~nFd(Px, x') = ~ XExFd(Px, x'} = ; x~d(Px, x'), for F e E. T h a t is,

(Py, x'}(F) = fFX~d(Px, x'}, f d(Py, x')

for F 9 E, and so

fu fXE d(Px, x ' ) = / E f d ( P x , x').

That is, for each E 9 E, we have

( ( ~ f dP)x,x'} = / E f d(Px, x'l,

x 9 X, x' E X'.

This means by definition that f is P-integrable . Conversely, if f is P-integrable, then the operator T := fa fdP 9 s satisfies (1) by definition of P-integrability. 9 Let X be a Banach space and P : E ~ s be a spectral measure . Then the vector space of all P-integrable functions is denoted by L(P). Given f 9 L(P), the operator fa fdP 9 C(X) is also denoted by P ( I ) ; it necessarily satisfies (1) with T = P(f) and commutes with each projection P ( E ) , for E 9 E. The next result clarifies the relationship between P-integrable functions and those functions which are integrable with respect to each X-valued vector measure Px, for x 9 X. This result is very useful in practice since it reduces the problem of determining whether a function is integrable with respect to an Es(X)-valued measure to verifying whether it is integrable with respect to a family of X-valued measures. P r o p o s i t i o n V . 2 . Let X be a Banach space and P : E ~s be a spectral measure

Let f : ~ ---* C be a E-measurable function. Then f is P-integrable if and only if f is Px-integrable for each x 9 X, that is, L(P) = N L(Px). xEX

70

CHAPTER V. STRONGLY CLOSED OPERATOR ALGEBRAS

P r o o f . Suppose that f is P-integrable. Fix z E X. Since f E Ll((Pu, u')) for all ~z E X and u' E X' it is clear that f C Ll((m,x')) for all x' C X', where m := Pz. For each E E P, define fE f dm := (rE fdP)oc. Then, using the fact that f is P-integrable, gives

for each z' E

X'.

So, f is Px-integrable and satisfies

Conversely, suppose that f E L ( P z ) , for each z e X. Given r~ E N, let E(n) := {w E f~ : < r~}. Then f~ := fXE(,~), for r~ E N, is a bounded g-measurable function and hence fn E L(P); see Lemma III.3 and Proposition III.2. Fix z C X. By the first part of the proof we know that f~ E L(Pz) and

If(w)l

~fnd(Pxa=(s

hEN.

Since lid _< I.fl with f E L(Pz) and f , ---+ f pointwise on f~ we can apply the dominated convergence theorem for the X-valued vector measure Px (see Theorem 1.9) to deduce that

Tx:= l i m ( ~ f~dP)x= 9

Iim n~cx~

[ f,d(Px)= s f d(Px) J~2

exists in the norm of X. In particular, sup~ II(fa L dP)xll < oe and so sup~ II fa f~ dPll < oo by Proposition III.l(a). Since x E X is arbitrary and {fa f~ dP}~=l C_ s it follows from the Banach-Steinhaus theorem (c.f. Proposition III.l(b)) that the so defined operator T E Z;(X). It is routine to verify that T satisfies (1) of Proposition V.1 and so, by that result, f is P-integrable . 9 E x e r c i s e 42. A Banach space X is said to have the BP-property if, whenever {z=}~=l is a sequence in X satisfying ~-~nC~1 [(Xn~Xt)] < 043 for each x' C X', then the series ~~176 x~ is convergent in norm to some element of X. It is a classical theorem that X has the BPproperty if and only if X does not have any closed subspace which is isomorphic (as a Banach space) to c0; see [3]. (a) Let X be a Banach space with the B P - p r o p e r t y and m : E ----+ X be any vector measure. Show that a E-measurable function f is m-integrable if and only if

(*)

falfldl(m,x')l < oc,

x' c X'.

(b) Give an example of a vector measure m : 2 - - ~ co, defined on some a-algebra g, and a g-measurable function f satisfying (.) of part (a) for every x' E (Co)' = gl, such that f is not m-integrable.

71 (c) Let X be a Banach space with the B P - p r o p e r t y and P : E ~ s be a spectral measure . Show that a E-measurable function f is P-integrable if and only if

s

1fldl(Px, x'}[ < o~,

x ~ X, x' ~ X'.

9

The next two results show that Proposition V.2 has some useful consequences. C o r o l l a r y V . 2 . 1 . Let X be a Banach space and P : E ~ s be a spectral measure . (a) If f C L(P), then also Ifl c L(P). (b) If 0 <_ g 9 L(P) and f : f~ ----, C is a measurable function such that Ifl <- g, then

f e L(P). P r o o f . (a) By Proposition V.2 we have that f C L(Px) for all x 9 X and hence, that ]fl e L(Px) for all x E X (see Lemma 1.3). Then If] is P-integrabie by Proposition. V.2

(again!). (b) Since g E L(Px) for all x E X (by Proposition V.2), it follows that f c L(Px) for all x E X (by Lemma 1.4) and so Proposition V.2 implies that f E L(P). 9 C o r o l l a r y V . 2 . 2 . (Dominated convergence theorem for spectral measures). Let X be a Banach space and P : E ~ s be a spectral measure. Let f , : f~ ~ C, for n c N, be a sequence of E-measurable functions such that f ( w ) := l i m , + ~ fn(w) exists pointwise on f~ and lfnl < g, for all n E N and some O < g ~ L(P). Then f c L(P) and P(fn) ~ P ( f ) in c~(x)

.

P r o o f . Fix x C X. Then g E L(Px); see Proposition V.2. It follows from the dominated convergence theorem for the X-valued measure P x (see Theorem 1.9) that f E L(Px) and fa fn d(Px) ~ fa f d(Px) in X. That is, (f~ f~ dP)x ----+ f~ f d(Px), where we have used Corollary V.2.1 to ensure that fn c L(P) for each n E N. Then Proposition V.2 again gives that f E L(P) and so

(/af~dP)x----~ s f d(Px) = ( f a f dP)x ,

n---* oo,

in the norm of X. Since x 9 X is arbitrary, we see that P ( f n ) - - ~ P ( f ) in s . 9 The next result shows that L(P) is an algebra with respect to pointwise multiplication of functions. It should be noted that this feature is special to spectral measures and is surely not typical of more general 0Perator-valued measures. The multiplicativity of P plays an essential role in this regard. P r o p o s i t i o n V.3. Let X be a Banach space and P : E ----* s be a spectral measure.

If f and g are P-integrable functions, then f g is also P-integrable . Moreover, (4)

/E(fg)dP = P(I)P(g)P(E) = P(g)P(I)P(E),

E E E.

P r o o f . It is a simple consequence of (1) and (2) that (4) is valid whenever 9 C sire(E) and f E L(P). Suppose that f C L(P) and g is bounded and E-measurable. Then ]fg] <_ ]]g]]~. If] co and so Corollary V.2.1 yields that f g E L(P). Select functions { 8 ~}~=1 C sim(E) with

72

CHAPTER V. STRONGLY CLOSED OPERATOR ALGEBRAS

Isnl <- Igl such that Sn ' g uniformly on ~. Since P(f)P(sn) = P ( f s ~ ) , for n c N, with Is,~l <_ Igl E L(P) and Ifs,~l <_ Ifl " Ilglloo C L(P) with fa,~ ---+ fg pointwise on ~, it follows from Corollary V.2.2 that both P(s~) ----, P(g) and P ( f s ~ ) ~ P(fg) in s . In particular, P(f)P(g) = P(fg) follows, that is, again (4) holds. Suppose now that both f,g E L(P) are arbitrary. Let E(n) := {w : tg(w)l < n}, for n E N, and define gn := gx~<~). Then fgn ---+ fg pointwise on ~ and

(5) for a l l n C N a n d m E X , x ' C X'. F i x E E E. Sincegxs~E{.) is bounded for e a c h n C N , it follows that Ifgx~o~=l < Ilgx~o~en~ll~o. Ill and so, by Corollary V.2.1, fgxs~u(~) 9 s with (6)

P(fgxE~u(~)) = P(f)P(gx~E(.~)),

n 9 N.

A routine calculation using (1) and (6) shows that N,

for each x 9 X, a:'9 X', where { := P(f)'x'. Sin~:e g 9 L(P) we know that g 9 L~((Px, {}). Moreover, since Igx.E(~)I <- Ig[ with gx~(~) ---+ g pointwise on ~, the dominated convergence theorem for the complex measure
lira [ n~c~

e<e ,e> = s ga<ex,e>.

J E

The existence of this limit and the identities (5) show that {rE fg'~ d
= L g d(Px, P(f)'x'}.

Since E ~ {P(E)x, P(f)'x'}, for E 9 E, coincides with the measure E ~ fE f d
(Tz, x') = s fg d(Px, x'),

x 9 X, x' 9 X'.

Proposition V.1 then implies that fg 9 L(P) and (4) is valid. 9 We now turn out attention to another algebra of P-integrable functions. D e f i n i t i o n V.1. Let X be a Banach space and P : E ---* f , ( X ) be a spectral measure . A Pqntegrable function f : f~ ---+ C is called P-null if P(f) := fu f d P = O. By Proposition V. 1 this is equivalent to

iEf

dP=O,

E 9 E.

9

73 Let P : E ~ s be a spectral measure. Two P-integrable functions f and g are called P-equivalent if If - 9 1 is P-null. This is the same as { w : f ( w ) r g(w)} being a P-null set; see the discussion prior to Exercise 37 (in Chapter IV). D e f i n i t i o n V . 2 . Let P : E ----+ s be a spectral measure. T h e n a E-measurable function f : f~ ~ C is said to be P-essentially bounded on ft if

lflP := inf{sup{If(w)l : w E E} : E E E, P ( E ) = I} < oc. E x e r c i s e 43. Let X be a Banach space and P : E ----+ s be a spectral measure. Let f : f~ ----+ C be a P-essentially bounded function. Show that there exists a set E0 E E such that P(Eo) = I and IflP. =,sup{lf(w)l : ~w,~ E0}.n .... (8) Let P : E ~ ~s be a spectral measure an~ f De ar-essen~lal• bounded function. By Exercise 43 there is a set Eo E E with P(Eo) = I such that (8) holds. Hence, there is

a bounded E-measurable function f0 on ft (e.g. fo = fx~o) such that { ~ : fo(~) r f ( ~ ) ) is P-null and If[P = Ilf0lI~. We define the equivalence class [f] of f to consist of all Emeasurable functions g on ft for which {w E f t : f ( w ) r g(w)} is a P-null set. The space of all equivalence classes of P-essentially bounded functions is denoted by L~176 it is a Banach algebra with respect to the P-essential supremum norm l" le- If If] E L~(P), then the integrals f~[f] dP are de~ned to be f~fodP, E ~ E, for any bounded E-measurable function f0 : ft ~ C such that f = f0, P-a.e.. By the usual abuse of notation we will write f rather t h a n [f] for elements of L~176 Note that Lemma III.3 and Proposition III.2 imply that fo is P - i n t e g r a b l e . E x e r c i s e 44. Let X be a Banach space and P : E ----+ s be a spectral m e a s u r e . (a) Show that L ~ ( P ) is complete with respect to the norm l" IP. (b) Show that the E-simple functions are dense in L ~ ( P ) . 9 T h e o r e m V . 1 . Let X be a Banaeh space and P : E ~ 12,(X) be a spectral measure .

Define ~ : L ~ ( P ) ~

C ( X ) by ~ ( f ) := f a fdP,

f E L~(P).

Then ~ is an isomorphism of the Banaeh algebra L ~ ( P ) onto an inverse closed Banaeh subalgebra of s and satisfies IflP <<-II~(f)ll < 41IP(E)II" IflP,

f E L~(P).

P r o o f . It is clear that 9 is linear and multiplicative (by Proposition V.3). Let f E L ~ ( P ) and choose f0 : ft ~ C bounded and E-measurable such that f = fo, P-a.e., and tfiP = IIf0II~ = sup{if0(w)I : w E f~}. Choose functions sn E sire(E) such that Is~I < If01 and s~ ~ f0 uniformly on ft. By the proof of Proposition III.2 we have, for each n E N, that (9)

II

fsndPll <_4I]P(E)]I'

IIs~IIo~ < 4IIP(E)II. Iif0H~ = 4]IP(E)II" IflP,

CHAPTER V. STRONGLY CLOSED OPERATOR ALGEBRAS

74

and ffl s~ dP ----* fa fo dP in Z:u(X). Hence, also n

---+ OO~

and it follows from (9) that Hw(f)ll < 411P(~)II. KIP. n n Let f = }-~j=l ajX% with EyNEk = 0 i f j r k and U3=IE j = g~. Then If[P = max{laJl : P(Ej) # 0} = Ic~j01 say (and assume IflP > 0). Since P(Ejo ) r 0 there is a unit vector x 6 X with P(Ejo):C = x. Then n

j=l and

so

I]~(f)l I ~ II~(f)x

j=l

= I~Jol = If[e.

Hence,

l i p ~/l~(f)ll,

(10)

f E sim(E).

By Exercise 44 the space slm(E) is dense in L~176 and so a continuous extension argument shows that (10) holds for all f E L~176 To complete the proof it remains to check that the range ~(L~~ is inverse closed in s So, let f C L~176 and suppose that ( ~ ( f ) ) - I exists in Z;(X). For each m E N let 61,... , 6 ~ be disjoint Borel sets (in C) of diameter less t h a n 1 / m whose union is the compact disc {z 6 C : ]z I <_ IflP}. Let Ej := f-l@j) and choose any point wj E Ej. Then the E-simple functions r~m

m6H, j=l

have the properties

If - fmIP <- ( l / m ) and Ilkt'(f) - g~(f~)l[ -< 41[P(E)II/m,

m e N.

Then r ---* tg(f) in Z:~(X). Since r is invertible in Z:(X), it follows from Lemma III.1 that for sufficiently large values of m each operator nm

'~(A) = ~_, A(wj)P(Ej) j=l

is invertible in s So, for all sufficiently large m we must have that f,~(wj) r 0 whenever P(Ej) r O, which means precisely that (1/fro) E L~176 and, of course, ~ ( 1 / f m ) = ( ~ ( f , 0 ) -1. By Exercise 27 I(1/fm)

-

(I/A)IP ~ I I ~ ( 1 / A )

- ~(I/A)II

= r l ( ~ ( f m ) ) -~ - ( q a ( A ) ) - l l l

~

o,

75 as k,m --+ co. Accordingly, {l/f,~},~__: is Cauchy in L ~ ( P ) and so converges (to 1If of course). Hence, ( l / f ) C L~176 and ( q ( f ) ) - : = li_m (~(f,~)) -1 = l i m ~ ( 1 / f , 0 = 9 ( 1 / f ) which shows that (9(f)) 1 belongs to the range of 9, as required. 9 The following technical result will needed for the proof of Proposition V.4 below. L e m m a V.2. Let X be a Banach space and P : E ---+ s be a spectral measure. If f is a P-integrable function, then

sup{ IIfagdPIl : g is E-measurable Proof.

and

Igl <- [fl,

P-a.e.} < oo.

Note that each such function g is P-integrab]e by Corollary V.2.1. Moreover, we

can write g = f . (g/f) where (g/f) G L ~ ( P ) satisfies I9/flP -< 1. Hence, by Proposition V.3 we have that

and so, by Theorem V.I we get

,,~gdP,,

<_ ,, f a ( g / f ) d P , , . , , ~ f d P , , < _ 4 , g / f , p . , l P ( E ) , ,

I'fafdP"

-< 41[P(E)[I" I I / f dPII. The following result shows that JCt~e only P-integrable functions are the P-essentially bounded functions! P r o p o s i t i o n V.4. Let X be a Banach space and P : E ----+ s be a spectral measure . Then a E-measurable function f : 9 ---* C is P-integrable if and only if it is P-essentially bounded. Proof. If f is P-essentially bounded, then f is bounded P-a.e. and so f is P-integrable; see the discussion after Definition V.2. Conversely, suppose that f E L(P). Define disjoint sets

6n(f):={we~:

nE<_lf(w)l<(n+:)2},

n= 0,1,2,...,

and E-simple functions Cn(f) :=

k Xsk(n,

n = 0, 1,2,...

k=0

Then Ir -< Ifl pointwise on [2, for all n _> 0. By Corollary V.2.1 the function P-integrable and so, by Lemma V.2, the sequence of operators

n_> 0, k=0

kfl

is

76

CHAPTER ~

S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

is uniformly bounded. Hence, also the consecutive differences

are uniformly bounded. So, there is fl > 0 such that

IIP(G(f))]I < 3 / n 2,

n ~ H.

Let E~ := Uk~=n~k(f), for n _> 0. Fix z E X. The a-additivity of P implies that P(En)x = oo ~-~k=n P(Sk(f))x and so, for n _> 1, we have

IlP(E,~)zll <_

IlP(~k(f))xll < Ilx[I, k=7~

rlP(&(f))ll <_ flllxll ~ ( 1 / k S ) k--n

e~

k=n

2

Accordingly, IIP(&)II -< f i G k = ~ ( 1 / k ), for ~ _> 1, showing that IIP(&)H - - ' 0 as n ~ c~. Since JIP(E~)I I >_ 1 whenever P(G) r 0 (see Exercise 41(a)), it follows that there exists N > 0 such that P(E~) = 0 for all n > N. In particular, P(EN) = 0 and so

EN=U~=NSk(f ) = { w C f ~ :

N 2_< If(w)l}

is P - n u l l , that is, If(w)l _< N 2 for P-a.e. w E f L Hence, f is P-essentially bounded. 9 Using the results of Chapter III we are now able to identify the compact space ~2 such that the Banach algebra L ~ ( P ) is isomorphic to C(f~). T h e o r e m V.2. Let X be a Banach space and P : E ----* s be a spectral measure . (a) The commutative, unital Banaeh algebra L ~ ( P ) is isomorphic to C(f~), where f~ is the Stone space of the Bade or-complete B.a. P ( E ) . In particular, f~ is a compact basically

disconnected Hausdorff space. (b) Suppose, in addition, that the spectral measure P is closed. Then L~176 is Banach algebra isomorphic to C(f~), where f~ is the Stone space of the Bade complete B.a. P ( E ) . In particular, f~ is a compact extremely disconnected Hausdo~ff space. P r o o f . (a) Let B := P(E), in which case B is a Bade a-complete B.a. (see Theorem IV.l). In particular, B is abstractly or-complete and so is uniformly bounded (see Theorem III.1). Let f~ := ft~ be the Stone space of B, in which case f~ is a compact basically disconnected Hausdorff space; see Proposition II.4. By Theorem III.2 the algebra (B}~ is isomorphic to C(f~). But, by Theorem V.1 the subalgebra ~ ( L ~ ( P ) ) of s is isomorphic to L ~ ( P ) , where ~ ( f ) = fa fdP, for f E L~(P). So, it remains to check that (B)~ = ~ ( L ~ ( P ) ) . Since ~(XE) = P(E), for each E C E, it is clear that 13 C_ qg(L~(P)) and hence, (13}~ C_ ~(L~176 Since sim(E) is dense in L~176 it follows that all operators which are finite sums of the form ~-~yeFa~P(Ej) with a j C C, Ej C E and F _C N (with F finite), are dense in tg(L~(P)). But, all such operators clearly belong to (13}~ since P(Ej) C 13 for all j E F. Accordingly, ~ ( L ~ ( P ) ) c_ (13}~. (b) The same proof as for (a) shows that L ~ ( P ) is isomorphic to C(~), where f~ is the Stone space of 13 := P(E). But, the closedness of P is equivalent to 13 being a compact extremely disconnected space; see Theorem IV.1 and Proposition II.4. 9

77 We now wish to present a refinement of Theorem III.2 in the case when the B.a. of projections B satisfies some additional properties. T h e o r e m V.3. Let X be a Banach space and B c_ s be a B.a. of projections. (a) Suppose that B is Bade a-complete . Then the closed subalgebra (B)~ of s is

isomorphic to C ( a s ) as a Banach algebra ('with f~s the Stone space orB) via an isomorphism

r

C(as) ---~ ~- given by g2(f) = fa fdQ,

f E C(f~s),

where-Q: B a ( a s ) ,s is a regular, a-additive spectral measure satisf'ying~(Ba(aS) ) = B and (I)(x~) : Q ( E ) = Q(E), for each E E Co(as), and Q : Co(as) ----* B is the Stone map . (b) Suppose that B is Bade complete. Then (B}~ is Banach algebra isomorphic to C(~s) (with i2s the Stone space orB) via an isomorphism 9 : C ( a s ) ~ {B)~ given by 9 (f) = ~ fdO,,

f

E

C(as),

where O, : Bo(az~) ~s is a regular, a-additive, closed spectral measure satisfying Q,(Bo(a~)) = B and q~(XE) = Q(E) = Q(E), for each E 9 Co(f~s), and Q: Co(f~s) ~ B is the Stone map. P r o o f . (a) By Theorem III.1 B is uniformly bounded and by Theorem III.2 the algebra (B}~ is isomorphic to C ( f ~ ) . By Theorem III.3 (with a s in place of A) there is a unique bounded spectral measure R : Bo(a~) ----* s of class F = X c X " such that (x, Rz') : Bo(f~s) ~ C is regular for all x 9 X and x' 9 X ' and, for each f 9 C ( a s ) , we have

(@(f)x, x') = [

(11)

f d(x, Rx'},

x

9

X, x' 9 X',

d ~2B

where (I) : C(f/s) ----+ (B)~ is the isomorphism of Theorem III.2. Let E denote the family of those Borel sets E C f~s for which R(E) = Q ( E ) ' for some projection Q ( E ) 9 B. Since B is a B.a. it follows that E is an algebra of subsets of aB (contained in Bo(~s)). To see that E is a a-algebra, let {E~}n~__l be an increasing sequence of sets from E. Since ~ ( E n + I ) ' ~ ( E ~ ) ' = R(E~+I)R(En) = R(En+~ n En) = R(En) = ~ ( E ~ ) '

we see that - - E n)Q(En+l)X, -(Q( x ')

=

- E n+l) ' -Q( - E n) 'x' } = {X, Q(En)tX'> (x, -Q(

=

(Q(En)x , 22'},

for all n E N, x 6 X, x' 6 X ' which implies that Q(E~)Q(En+I) = Q(En), for all n. Hence, {Q(En)}~__I is increasing in B. By Theorem IV.1 and L e m m a IV.1 it follows that

CHAPTER V. S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

78

B := l i m ~ o o Q(E~) exists in s by a-additivity of (x, Rz'), that oc ! <x, R(U~=IEn)x )

= ~lim ,-~

and B C U. Thus, for each x E X and z' E X ' we have,

(x, R(En)x')

= . .lim . . . ( -O-, ( E ~ ) x , x )' = ( s x ,

z') = (x, S ' x ' ) .

This shows that U~_IE~ C E and hence, that E is a a-algebra. The isomorphism (I) satisfies q)(XE) = Q(E), for each E E Co(f~s); see Theorem III.2. Since qS(XE) = Q(E) is a projection in s and satisfies Q(E)' = R(E) by the formula (11), it follows that Q ( E ) = Q(E) and Co(as) c E. Hence, also Ba(as) c E (see Proposition II.5). Since (Q(E)x, x'} = (x, R(E)x') for each z E X, z' E X ' and E E Ba(f2B) it is clear that Q : Ba(f~u) ~ s is a-additive, regular and multiplicative (as R is multiplicative). That is, Q is a regular, a-additive spectral measure. Since Q(Co(f~B)) = B it follows from the definition of E that Q(Ba(f~)) = 13. (b) If B is Bade complete, then the same argument establishes that E is a ~r-algebra. It remains to estabIish that E = Bo(f~B) . Since B has the monotone property (see Theorem IV.l) it follows from Theorem IV.2(b) that there exists a unique closed spectral measure Q, : Bo(au) ----* s such that O,(Bo(f~u)) = B and (~(E) = Q(E) for all E E Co(f~u), and that (~ is regulaT" (see also the proof of Theorem IV.2(b)). In particular, {Qz, z') is a regular complex measure on Bo(f~u), for each .T, E X and .T/ E X', and coincides with the regular measure {x, Rz') on the algebra Co(12u) and hence, also on the generated a-algebra a(Co(f~u)) = Ba(f~u); see Exercise 36. Since the regular extension to the Borel sets of any regular Baire measure on a compact Hausdorff space is unique (see [37; p.314]) it follows that {Qz, z'} = (z, Rz') as measures on Bo(f~u). In particular,

(:r, Q,(E)'z') = (O,(E)z, z') : (z,/{(E)z'),

E E Bo(f~u),

for each z E X and z' E X', from which it follows that ~)(E)' = / { ( E ) for all E E Bo(f~s) . Hence, Bo(Qu) C E and so Bo(Qu) = E. Since Q(Co(f~s)) = B it follows from the definition of F, that (~(Bo(ftu)) = B. 9 Remark. (a) Theorem V.3. should be compared with [15; Lemma 9, p.2202]. The proof given there is in the spirit of Banach algebra theory, whereas our proof is based on the theory of B.a. 's and Stone spaces . In particular, our proof that Bo(f~u) equals E (in part (b) of Theorem V.3) is quite different to the argument given in [15]. (b) It is clear from the proof of Theorem V.3 that the spectral measures Q and Q are precisely the more general a-homomorphisms of Theorem II.4 and Theorem II.5 specialized to the particular setting of B.a. 's of projections (which are Bade a-complete and Bade complete ). (c) Let B be a bounded B.a. of projections and Q : Co(f~u) ~ B be the Stone map Then we have seen (at the end of Chapter III) that each regular, bounded, a-additive measure (Qz, z'}, which is defined on the algebra of sets Co(f~u) for each z E X and z' E X', has an extension to a regular , a-additive measure #~,~, : Ba(f2u) ----+ C. By Theorem III.3 there also exists a regular, a-additive measure (x, Rz'} : Bo(f2u) ~ C which coincides with

79 #x,x' on Ba(f~B). The question raised at the end of Chapter III was to decide when each projection from the B.a. R(Ba(gt~)) or R(Bo(aB)), rather t h a n just those from R(Co(f~B)), is the dual operator of some projection from B? An examination of the proof of Theorem V.3 provides an answer for certain B. Namely, if/3 is Bade a-complete (resp. Bade complete), then each projection from R(Ba(t2B)) (resp. R(Bo(f~B))) is indeed of the form B' for some B E /3. The following exercise shows that this fails if the Bade a-completeness or Bade completeness assumption of/3 is reduced merely to its abstract a-completeness or abstract completeness . In particular, it fails for bounded B.a. 's of projections in general. 9 E x e r c i s e 45. Let X = t ~ and let S denote the a-algebra of all subsets of N. For each E r 8 define B(E) C s by

B ( E ) x : (x~(1)xl, x ~ ( 2 ) x ~ , . . . ),

x : (x~, x ~ , . . . ) c X.

It is shown in Example 20(a) that the B.a. B := { B ( E ) : E E $} is abstractly complete , but not Bade cT-complete . Observe that 11/311= sup{IIB(E)II : E E S} = 1. Let t2~ be the Stone space of B and 9 : C(t2s) ---* (/3)~ be the Banach algebra isomorphism of Theorem III.2. By Theorem III.3 there is a unique bounded spectral measure R : Bo(f~z3) ~ 12(X') of class F = X C X" such that (z, Rx') is a regular measure on Bo(flB), for each z C X and z' E X', and for each f E C(~2~) we have

(r

z'} = [

f d(x, Rx'},

z E X, x' E X'.

B

Let E denote the family of those sets E E Bo(f~z) for which R(E) = P(E)' for some projection P(E) E s (a) Show that E is an algebra of sets containing Co(t2B). (b) Show that E is not a a - a l g e b r a . 9 So far in this chapter we have seen that if P : E ~ Z;s(X) is a a-additive spectral measure , then the space L ( P ) of all P-integrable functions consists precisely of the Pessentially bounded functions . Moreover, if we equip (the space of equivalence classes of) L ( P ) with the norm I" IP, then we produce the Banach space (and algebra) L ~ ( P ) and the map f H fa fdP is a bicontinuous isomorphism of L ~ ( P ) onto (/3)~ -~ C(t2B), where /3 := P ( E ) . For the remainder of this chapter we wish to concentrate on identifying the closed algebra generated b y / 3 in ~:~(X) or, equivalently, in ~,,(X). For this purpose we will need to consider (the space of equivalence classes of) L ( P ) equipped with a locally convex Hausdorff topology which is non-normable and to investigate, in detail, the algebraic and continuity properties of the linear map f ~ fa fdP with respect to this locally convex topology (rather than the norm topology I" IP). The results on vector measures developed in Chapter I will play a crucial role in this respect. Given f E L ( P ) , where P : E ----* Z;~(X) is a spectral m e a s u r e , we define the equivalence class [f] = {g E L(P) : If-gl is P-null } and, as usual, let L I ( P ) = {If] : f E L(P)}. Then we have seen that L I ( p ) = L~(P) as a vector space and hence, LI(P) becomes a Banach

80

CHAPTER V. S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

algebra with respect to the norm I' [,. We now equip LI(P) with a different topology. By the usual abuse of notation we still denote elements of L I(P) by f , rather than If]. For each fixed x 9 X, define a seminorm qx(P) : LI(P) ~ [0, oo) by

qx(P)(f) :=

IINIIL,
IlCPx)sllCf~),

f 9 LI(p),

where we recall that Px : E ~ X is the vector measure E ~ P(E)x, for E E E, and (Px)f : 2 ----* X is the vector measure given by the indefinite integral of f with respect to Px, that is, by E ~ f E f d ( P x ) , for E 9 E. Of course, IICPz)slI(') denotes the semivariation of ( P x ) I ; see Chapter I. Observe that Proposition V.2 ensures f 9 LI(Px) whenever f 9 LI(p). The topology Ts(P) specified by the family of seminorms {qx(P) : x 9 X } turns LI(P) into a locally convex space. Moreover, the topology % (P) is Hausdorff meaning that if f 9 L 1(P) satisfies qx(P)(f) = 0 for a l l x 9 X , then f = 0 i n LI(P). To see this fix E 9 E. For each x 9 X, it follows from Proposition 1.2 applied to the vector measure m = (Px)f that fE f d P 9 f~(X) satisfies

II(s f dP)zll =

II ; f dCPx)ll = II(Pz)z(E)II-< HCPz)slICE)

_< II(rx)sll(a) = q~(P)(f) = O. Since x 9 X is arbitrary, it follows that fE f d P = 0 (in Z;(X)). Since this is true for every E 9 E, it follows that f = 0 in LI(P). So, we see that (LI(P),'r~(P)) is a locally convex Hausdorff space. Proposition V.3 shows that LI(P) is an algebra; the next result shows that L 1(P) is actually a locally convez algebra. This means that multiplication is separately continuous, that is, if {f~} _c LI(p) is a net such that f~ ~ 0 in LI(P), then for each fixed g 9 LI(P), also gfc, ~ 0 and fag - ~ 0 in L I ( p ) . Of course, since L~(P) is a commutative algebra it suffices to check that either one of gf~ ~ 0 or f~g - ~ 0 is satisfied. D e f i n i t i o n V . 3 . Let P : 2 ~ s be a spectral measure . Then the integration map IF: LI(P) ~ f.(X) is defined by

Ip(f) := P ( f ) = ~ fdP,

f 9 L~(P).

9

V . 4 . Let X be a Banaeh space and P : Y. - - ~ E~(X) be a spectral measure . Then (LI(P), ~-~(P)) is a locally convex (commutative) algebra with identity 11 and

Theorem

(12)

q~(Ip(f)) <_ q~(P)(f) <_ 411P(~)II. q~(IP(f)),

f 9 LI(P),

for each x 9 X , where q~ : s ) [0, oo) is the continuous seminorm defined by q~(T) := IlTxlI, for each T 9 f-.(X). In particular, the integration map IF is a bicontinuous (algebra) isomorphism of LI(P) onto its range Ip(LI(p)) equipped with the relative topology from s P r o o f . We have already seen that LI(P) is a commutative algebra with unit the constant function 11 on fL Let us establish (12). So, fix x 9 X. Then, by Proposition 1.2 applied to

81 m := (Px)f, we have

(13)

qx(I.(f)) = ]lP(f)xll = I] s

= ]](Pxb(~)]l _< ]b(Px)~l[(a)= q~(P)(f),

for each f E Lz(P). But, again by Proposition 1.2, for f E LI(P) we have

(14)

q~(P)(f) = II(Px)/l](~) _< 4sup{tl(Px)r(E)11: E c r~} = 4sup{il f E f d P x l l : E e E} = 4sup{llP(E)P(f)xi[: E 9 E},

where the last equality follows from the identity (3) and Proposition V.1. But, the inequality IIP(E)P(f)zll < iIP(E)ll. IlP(f)xn, valid for each E 9 E, implies that

sup{llP(E)P(f)xH: E 9 E} _< ]IP(E)I] 9 ]]P(f)xl] = IlP(E)l] 9qx(Ip(f)). Combining this inequality with (13) and (14) yields (12). Since Ip is injective (Proposition V.1 shows that fE f d P = 0 for all E 9 E, that is, f = 0 in LI(P), whenever P ( f ) = Ip(f) = 0), it follows from (12) that Ip is a bicontinuous isomorphism of LI(P) onto its range in s recalling that the seminorms {qx : x 9 X} determine the topology of s Finally, it follows from (12) that, for fixed x 9 X, we have

(4llP(E)ll)-Zqx(P)(fg)

<_ q~(Ip(fg))=q~(Ip(f)Ip(g))= IlZp(f)Ip(9)xll

-< IIIP(g)ll. IlZP(f)xll

= I]Ip(g)ll. q~(Ip(f)) <_ llZP(g)[I,

q~(P)(f),

for all f and g in LI(P), that is,

q~(P)(f g) <_411P(~)II. IlIP(g)ll- q~(P)(f). It follows if g 9 LI(P) is fixed and f~ > 0 in Lz(P), then also gf~ --~ 0 in LI(P). Hence (LI(P), %(P))is a locally convex algebra. E x e r c i s e 46. Let X be a Banach space and P : E ~ s be a spectral measure . (a) Show that the simple functions sim(E) are sequentially dense in L I ( p ) for the topology (b) Show that the identity function from the Banach space (L~(P), I" lP) into the locally convex Hausdorff space (Lz(P), T~(P))is continuous. 9 In view of Theorem IV. 1 it is clear that closed spectral measures will play an important role in determining the closed algebra in s generated by a Bade complete B.a. of projections. The following result, which will require Theorem V.4, gives some useful characterizations of closed spectral measures. T h e o r e m V.5. Let X be a Banach space and P : E ---* s be a spectral measure .

Then the following statements are equivalent. (a) The range P ( E ) := { P ( E ) : E 9 E} is a closed subset of E~(X).

82

CHAPTER ~

S T R O N G L Y CLOSED OPERATOR A L G E B R A S

(b) The range P(E) is a complete subset of the locally convex Hausdorff space s (c) The range P(E) is a Bade complete B.a. of projections. (d) P is a closed spectral measure, that is, the uniform space (E(P), 7s(P)) is complete9 P r o o f . ( b ) ~ ( a ) . This follows from the fact that any complete subset of a topological space is necessarily a closed set. (a)=~(b). Let {B~}~eA be a net from P(E) which is Cauchy in s Fix x C X. Then

and so {B~x}~eA is Cauchy in X. By the completeness of X there is Bx C X such that lim~/3~x = /3x. Since sup~ IlB~H _< lIP(E)][ < cxz (c.f. Lemma IH.3) it follows from the Banach-Steinhaus theorem (c.f. Proposition III.l(b)) that B C s Accordingly, /3~ ----+/3 in s Since P(E) is assumed to be a closed set, we deduce that /3 E P(E). Hence, P(E) is complete . (d)~-(c). This is part of Theorem IV.1. (c)~(d). By Definition IV.l(i) of Bade completeness the B.a. P(E) is abstractly complete 9 Since E(P) is B.a. isomorphic to P(E), also E(P) is abstractly complete. Suppose that {lEvi} C_ (E(P), %(P)) is a net which is downwards filtering to [0]. Then {P(E~)} C_ P(E) is a net which is decreasing to 0 (in the order of P(E)). Since P(E) has the ordered convergence property (c.f. Theorem IV.l) it follows from Lenmm IV.l.(b) that lim~ P(E~) = A~P(E~) = 0 in s Then Lemma IV.3 implies that P is a closed spectral measure. (b)=v(d). Suppose that P(E) is a complete subset of s Let {[E~]} be a Cauchy net in (E(P), %(P)). Fix :c e X . Then

(1,5)

q~(P(E~)-

P(&,))

= IIP~(E~)

- P~(E~,)IF = II f(X~o

- X~,,)dP~:ll

= q ~ ( I ~ ( x ~ - x ~ ) ) <_ q d P ) ( x ~ o - x ~ ) ,

for all c~ and /3, where the last inequality follows from (12). But q~(P)(Xz~ - XEe) = q~(P)(lXz~ - Xze]), by the Remark after Proposition 1.5 applied to rn := P z and f := Xz~ - XEe, and the identity ]XE - XF] - XzaF for all E, F C E. So, (15) implies that (16)

q~(P(E~) - P(Eg) ) < qx(P)(:g~aE,) = I r P x l I ( E J ~ E e ) ,

for all c~ and /3. By definition of the topology and uniform structure in E(P), as defined by the pseudometrics {d~ : z C X} specified after Exercise 37 in Chapter IV, (16) implies that {P(E~)} is a Cauchy net in P(P,). Then the topological completeness of P(E) in s ensures there is E C E such that P(E~) ~ P(E) in s So, for each c~ (with f denoting (X~ - Xz~)), we have by (12) that d~([E], [E~]) =

IIP~II(EAG)= II(Pz),,,ll(~)= II(Pz),ll(~)

= qx(P)(f) <_411P(P,)II 9q~(Ip(f)) = 41[P(P,)II" % ( P ( E ) - P(E~)).

83 Since P(E~) ----+ P(E) in Z:s(X) it follows that [E~] ~ %(P) complete , that is, P is closed spectral measure .

[El in Z(P). Accordingly, ~(P) is

(d)=~(b). Suppose that P is a closed spectral measure. Let {P(E~)}~eA be a net which is Cauchy in P(E) for the relative topology from Es(X). Fix z C X. We saw in the proof of ( b ) ~ ( d ) that

]]Px]](E~AEz~)<_4]]P(E)]].q~(P(E~)- P(E~)),

c~,/3E A,

and hence , {[E~]}~ea is Cauchy in (E(P),~-~(P)). So, there exists [El E 2 ( P ) such that [E~] ~ [E] in 2(P). The inequality

N~(P(E~) - P(E)) <_ IIPxlI(EAE~) = d~(fE],

[E~]),

c~ e A,

was also established in the proof of (b)=>(d) and hence, we see that P(E~) ~ P(E) C P(2) in s This shows that P(E) is topologically complete. 9 The following exercise shows that for B.a. 's of projections B (even abstractly complete ones) which are not the range of some spectral measure, the properties (a)-(d) of Theorem V.5 are no longer equivalent, in general. Exercise 47. Let X = g~ and E = 2N. For each E E E define B ( E ) C s by

B ( E ) z = (XE(1)xl,XE(2)X2,...),

X = (Zl,X2,...) e g~,

in which case B := {B(E) : E C E} is an abstractly complete B.a. of projections which is not Bade ~-complete ; see Example 20(a). So, B cannot be the range of any spectral measure; see Theorem IV.1. Nevertheless, show that B is a closed subset of s 9 The next result is a useful consequence of Theorem V.5. C o r o l l a r y V.5.1. Every Bade complete B.a. of projections in s with X a Banach

space, is a closed (= complete) subset of s Proof. If B c_ s is a Bade complete B.a., then Theorem IV.1 implies that B coincides with the range of some closed spectral measure . The desired conclusion then follows from Theorem V.5. 9 We are finally in the position where we can describe the closed subalgebra in s generated by Bade c~-complete and Bade complete B.a.'s of projections. The following result will be crucial in this regard. So, let B be a B.a. of projections from s Then the smallest closed subalgebra of E~(X) (resp. s generated by B is denoted by (B)2 (resp. (B};). Since the linear span of B in s is a convex set, it follows from Proposition III.l(d) that (g)~ = ( g ) ; as vector subspaces of s T h e o r e m V.6. Let X be a Banach space and P : E ~ s be a closed spectral measure. (a) The locally convex Hausdorff space (LI(P),%(P)) is complete. (b) The integration map Ip : LI(P) ~ E~(X) is a bieontinuous isomorphism of Ll(P)

onto (P(r,)h. Proof, It was noted in the proof of Lemma IV.3 that the locally convex Hausdorff space L;s(X ) is quasicomplete, Let Z denote the completion of fl-.s(X) and ~ : ~ ~ Z denote P

84

C H A P T E R V. S T R O N G L Y C L O S E D O P E R A T O R A L G E B R A S

considered as being Z-valued, in which case P is also or-additive. We note that the LCspace of a vector measure with values in a locally convex Hausdorff space is defined analogously as for Banach spaces; see [19], [27], for example. It is clear that L I ( p ) C_ L l ( p ) (as a vector space inclusion) and that r ( P ) induces the topology r(P) on s To see that actually L I ( p ) = LI(/~) it suffices to show that each P-integrable function f _> 0 is Pintegrable. Choose E-simple functions sn > 0, n C N, such that sn T f pointwise. Since fE s , d P C s for each E E E and n E N, it follows from the dominated convergence theorem for general vector measures (see [27; Chapter II] or [29; Theorem 2.2], for example) and the quasicompleteness of s that f ~ f d P = l i m ~ fE s,~dP belongs to s Hence, f is P-integrable with fE f d P = fE f d P for each E E E. Accordingly, L I ( p ) = L I ( P ) with equality as locally convex spaces. In particular, since 2 ( P ) (resp. E(_P)) is identified with the subset {X~ : E C E} of LI(P) (resp. L I ( P ) ) it follows that /5 is a closed measure because P is closed (by hypothesis). But, the completeness of Z then implies that L I ( P ) is a complete locally convex Hausdorff space, [27; Theorem 1, p.73], and hence, L I ( p ) is also complete (being equal to LI(_P)). This establishes part (a). By Theorem V.4 the integration map Ip is a bicontinuous isomorphism of LI(P) onto its range Ip(LI(P)) C_ s So, it remains to check that (P(E)} 2 = Ip(L~(P)). Since P ( E ) = Ip(X~) for each E C E, it is clear that P ( E ) _C Ip(L~(P)). Since Ip(L~(P)) is complete, it is also closed in s and so (P(E)}2 _c Ip(L~(p)). Conversely, it is clear that Ip(sim(E)) C (P(E));-. Since sim(E) is %(P) dense in LI(P) by Exercise 46, and the integration map Ip is continuous, it follows that Ip(LI(p)) C_ (P(E))~-. 9 Remark. If a Banach space X is infinite dimensional, then the quasicomplete space s is never complete, that is, there always exist Cauchy nets with no limit in s However, the closed subalgebra (P(E)) 2 C_ s is always complete in the case when P is a closed spectral measure. 9 We can now combine the results of this chapter with those of Chapter IV to deduce some interesting facts about Bade complete B.a. 's of projections. C o r o l l a r y V.6.1. Let X be a Banach space and B c_ s be a Bade complete B.a. of projections. If B C s is a projection such that B E (I3}~,, then necessarily B E B. P r o o f . Let f ~ denote the Stone space of/3 in which case we know that there exists a closed spectral measure P : Bo(f~z~) ---+ s such that P(Bo(f~s)) = 13; see Theorem IV.1 and Theorem IV.2. Since (B)~ = (B)~- and IF: LI(p) ---+ s is an isomorphism onto
I p ( f 2 - f) = [IF(f)] 2 - I p ( f ) = t? 2 - B = 0 and so f2 = f (as IF is injective) . That is, g = f 2 _ f is a P-null function and so E := {w C f~ : 9(w) # 0} is a P-null set . Since F I \ E is the disjoint union of the sets E0 := {w : f ( w ) = 0} and E1 := {w : f ( w ) = 1}, and E is P-null, it follows from the identity

B = P ( f ) = P ( f ) [ P ( E ) + P(Eo) + P(E1)]

85 that B = P(f)P(E,1) = rE1 f d P = P(E1). Hence, B E ~l 9 E x e r c i s e 48. Give an example of a Bade or-complete B.a. of projections /3 C_ s for which the conclusion of Corollary V.6.1 fails to hold. Hint: Consider Example 20(b). 9 Combining Theorem V.6 and the proof of Corollary V.6.1 yields the following important representation theorem. C o r o l l a r y V.6.2. Let X be a Banach space and B C_ s be a Bade complete B.a. of projections. Then (B>2 is isomorphic as a locally convex algebra to LI(P), where P : Bo(f~B) - - ~ B c_ s is a regular, closed spectral measure which extends the Stone map Q : Co(f~B) ~ 13 and f~B is the Stone space of B. A further consequence is the following interesting fact. C o r o l l a r y V.6.3. Let B c_ s be a Bade complete B.a. of projections. Then (/3)~ =
Moreover,
T E s

for each x E X, it is clear that [ c_ [. This establishes that [ = [ C_ 2 may be strict if/3 is only Bade a-complete. E x a m p l e 21. Let /3 be the Bade a-complete B.a. of Example 20(b). It was shown ([) in Exercise 48 that there exists a projection B E ~ c_ (B>~- is strict. 9 We make a slight digression. Let X be a Banach space and rn : E - - ~ X be a vector measure , where E is a a-algebra of subsets of some non-empty set. For each E E E, let [E] := { F E E : E & F is m-null}. Then the space E(m) of all such equivalence classes {[El : E E E} is an abstractly or-complete B.a. Considering E(rn) as a subset of Ll(m) we may consider the topology and uniform structure from Ll(m) relativized to E(rn). Then a net {[Eo]} c_ E(m) is Cauchy if for every e > 0 there is c~(e) such that IImlI(E~ZXE~) < c for all a , ~ > c~(e), where Ilmfl(') is the semivariation of m. It is known that E(m) is always a complete uniform space , that is, m is a closed measure, [27; Theorem 1, p.78]. Combining this fact with [11; Proposition 1.1] gives the following result. L e m m a V.3. Let X be a Banach space and m : E ----+ X be a vector measure . Then the B.a. E(rn) is abstractly complete and, whenever a net {[E~]} _C E(m) is downwards filtering to [~] in the order o r E ( m ) , then lim~ m(E~) = 0 in the norm of X .

CHAPTER ~

86

S T R O N G L Y CLOSED OPERATOR A L G E B R A S

Now, back to B.a. 's of projections. D e f i n i t i o n V . 4 . Let X be a Banach space and B C_ s be a B.a. of projections. (i) For each z E X the cyclic space generated by z with respect to B is the closed subspace B[z], of X, defined to be the closure of the linear span of { B z : B C B}. (ii) A vector z E X is said to be a cyclic vector for B if X = B[x]. 9 E x a m p l e 22. (a) Let X be any Banach space of dimension at least two and B := {0, I}. Given any non-zero vector x c X it is clear that B[x] is the 1-dimensional subspace spanned by z. Hence, B has no cyclic vectors. (b) L e t X = C

2 and B := {0, I, A, B}, where A = [ 10 00 ] a n d B =

[ 00 01 ] . T h e n B i s

a B.a. of projections with the property that any vector x = (:;) with xl 7! 0 and z2 r 0 is a cyclic vector for B. 9 It will be seen later that many proofs of results about B.a. 's of projections reduce the argument first to the case when a cyclic vector is present and then establish the result for that special case. Theorem V.? below turns out to be useful in this context. Let P : E ~ s be a spectral measure. Fix z C X. Since P ( E N F ) = P ( E ) P ( F ) for every E, F E E, it follows from Proposition h2 that P ( E ) z = 0 if and only if IIPzI[(E) = 0. Hence, the vector measure P z : E ~ X has the special property that E C E is a Pz-null set if and only if (Pz)(E) := P ( E ) z = 0. For a general vector measure rn : E ~ X we point out that r e ( E ) = 0 alone does not imply that E is an m-null set, even in the simplest case when X = C ! Indeed, let f~ = {1,2}, let E = 2 a and let m : E ~ C be defined by m := 52 - ~1, where d5 is the Dirac point measure at j E fL Then re(g1) = 0, but there certainly exists a set F E E With F C_ f~ such that r e ( F ) r 0. Hence, fi is not an m-null set. T h e o r e m V . 7 . Let X be a Banach space a~zd B c_ s be a Bade a-complete B.a. of projections with a cyclic vector. Then B is actually Bade complete . P r o o f . Let P : E ~ s be any spectral measure satisfying B = P ( E ) ; see Theorem IV.1. Let z C X be a cyclic vector for B, that is, X = B[z] = P(E)[z]. Suppose that E C E is a Pz-null set, that is, P ( E ) z = 0. Then also P ( E ) z = 0 for every z from the linear span of { B z : B E B}. Since P(E) is continuous and the set of all such vectors z is dense in X = B[z], it follows that P(E) = 0 in s Hence, the spectral measure P and the vector measure P z : E ~ X have the same null sets. Accordingly, E ( P ) and E ( P z ) are isomorphic as B.a. 's. Since E ( P z ) is abstractly complete (c.f. L e m m a V.3), so is E ( P ) . Now, let {lEvi} C_ E ( P ) be downwards filtering to [0], in which case {[E~]} is also downwards filtering to [~] in E ( P z ) . By Lemma V.3 we have that P(E~)z = Pz(E~) - - ~ 0 in the norm of X. Then also P(E~)z ----* 0 in X for every z in the linear span of { B z : B E B}. Since sups IIP(E~)II <_ IINI < oo, it follows from the density of such vectors z that P ( E ~ ) u ~ 0 for all u E X , that is, P(E~) ----* 0 in Z;~(X). Then Lemma IV.3 implies that P is a closed spectral measure and hence, B = P ( E ) is Bade complete ; see Theorem IV.1. The following technical result will prove to be quite useful. Lemma

V.4.

Let X be a Banach space and B G s

be a Bade a-complete B.a. of

87

projections. Suppose that Y is a closed subspace of X such that B Y C Y for every B E I3. Let By c_ s denote the collection of all restrictions By : Y ----* Y of elements B E 13. Then By is a Bade c~-complete B.a. of projections in s Proof. By Theorem IV.1 there exists a spectral measure P : E ---* s such that B = P(E). I f y E Y _c X, then for each E E E we have that (P(E)Iy)y = P(E)y. Hence, if ~ c_ E and E~ $ 0, then (P(En)ly)y --~ 0 in Y. This shows that Py : E ~ s {E n},,=l given by E ~ P(E)Iy is a-additive. Since it is routine to verify that Py(O) = 0, Py(f~) = Iy and P y ( E N F) = P y ( E ) P y ( F ) for each E, F E E, it follows that Py is a spectral measure. So, by Theorem IV.1 again we deduce that By =- Py(E) is a Bade a-complete B.a.

9

Exercise 49. Let X be a Banach space and {P~}~cA C s be a net of commuting projections. Suppose that sup{llPsH : ~ c A} < oo and that P = lim~ P~ exists in s Show that P is a projection and PP~ = PaP for all c~ E A. 9 Exercise 50. Let X be a Banach space and B C s Let g denote the closure of B in/28(X).

be a bounded B.a. of projections.

(a) Show that B is a bounded subset of s (b) Show that B is again a B.a. of projections.

9

Given a (general) B.a. B which is abstractly a-complete, it is not always easy to identify its abstract completion in a concrete manner in terms of B itself. However, if B is a Bade a-complete B.a. of projections, then the following result identifies the Bade completion of B in a very direct way in terms of B itself. T h e o r e m V.8. Let X be a Banach space and B c_ s projections. Let B denote the topological closure of B in s B.a. of projections .

be a Bade a-complete B.a. of Then -B is a Bade complete

Proof. Since B is bounded (c.f. Theorem III.1) so is B; see Exercise 50(@ By Exercise 50(b) B is again a B.a. of projections. Suppose that B is not Bade complete. By Theorem IV.1 and Lemma IV.1 there is a monotone increasing net {B~} C_ B and x E X such that {B~x} is not a convergent net in X. Let B[x] and B[x] be the cyclic spaces generated by x with respect to B and B, respectively. If B E B, then B = limAz in s for some net {A~} _C B and so Bx = lim~ A~x belongs to B[x], because each vector A~x E B[x] and B[x] is closed in X. Accordingly, {Bx : B E B} C_ B[z] and it follows that B[x] c_ B[x]. Since B C_ B, it is also clear that Nix] C B[x] and hence B[x] = B[x]. So, if Y := B[x], then it is routine to check that B Y = B(B[x]) C_ B[x] = Y, for each B E B. For each B E B, let By ~ s denote the restriction of B to Y. It can be verified that By := {By : B E B} is contained in the closure of {By : B E B} in s see Exercise 51 below. Since {B~x} is not convergent in X, it follows that {B~lyx} is not convergent in Y since z E Y and B~x = (B~ly)x, for all c~. That is, {B~IY} is not convergent in s and hence, by the previous paragraph, the closure of {By : B E B} in s contains the monotone increasing net {B~IY}which fails to have a limit in s By Theorem IV.1 the closure of {By : B c B} in s cannot be Bade complete . But, by Lemma V.4 it is Bade a-complete and so, by Theorem V.7 applied in Y = B[z], it is also Bade complete. This

88

C H A P T E R V. S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

contradiction shows the initial assumption that B is not Bade complete is false. Hence, B is Bade complete . 9 Exercise 51. In the notation of the proof of Theorem V.8 show that By is contained in the closure of {By : B E B} in s 9 As an immediate consequence of the previous theorem we have the following fact. C o r o l l a r y V.8.1. Let X be a Banach space and B C_ s be a Bade or-complete B.a. of projections. Then B is Bade complete if and only if B is a closed subset of s Proof. If B is Bade complete, then it is a closed subset of s by Corollary V.5.1. On the other hand, if B is closed in s then/3 = B and so B is Bade complete by Theorem V.8. 9 R e m a r k . In Exercise 47 it is shown that there exists an abstractly complete B.a. of projections/3 C s such that 13 is closed in s but/3 fails to be Bade complete. Hence, the assumption that /3 is Bade or-complete in Corollary V.8.1 cannot be replaced by the requirement that /3 is abstractly complete. The interested reader may wish to look at the paper [22] where it is shown that if X does not contain an isomorphic copy of e0, then every bounded B.a. of projections/3 C s which is a closed subset of s is necessarily Bade complete. 9 Exercise 52. Let X be a Banach space and /3 C_ s be a Bade (r-complete B.a. of projections. (a) Show that if A E s of/3 in s

is a projection such that A E
(b) Show that ~ c_ (/3)~ = (/3)j, is valid (compare with Corollary V.6.3). (c) Show that ~- = 2, where again B is the closure of B in s 9 Recall for a Hilbert space H that an operator T E s is called normal if TT* = T ' T , where T* denotes the (Hilbert space) adjoint operator. If T = T*, then T is called selfadjoint P r o p o s i t i o n V.5. (a) Let T E s A T = TA.

Let H be a Hilbert space. be a normal operator. Then AT* = T*A whenever A E s

satisfies

(b) Let 13 C_ s be a bounded B.a. of projections. Then there exists a selfadjoint operator S E s which is invertible in s such that the B.a. of projections { S B S -1 : B E/3} consists entirely of selfadjoint projections. Part (a) is the Fuglede theorem, [13; Theorem 7.21], and part (b) is the well known Mackey-Wermer theorem, [13; Proposition 8.2]. The following example shows that if B C_ s is a B.a. of selfadjoint projections, then the elements of ~- consists entirely n of normal operators. To see this observe that every operator of the form ~i=1 a j B j with aj E C and Ba E /3, for 1 _< j _< n, is clearly normal. Since every operator in
89 limit in s of a net of such operators it suffices to establish the following F a c t . Let {T~} C_ s be a net of pairwise commuting, normal operators such that T = lim~ T~ exists in s . Then T is normal. To establish this Fact fix an index c~. It is routine to check that TT~ = T~T and so, by Fuglede's theorem above, TT* = T2T. Since also T~ > T in s see Exercise 32(a)- it follows for each x, y C H that (17)

lim(T~z, Ty} = (Tz, Ty) = (T*Tx, y).

But, again by Puglede's theorem, (18)

=
r2y > =

.

Since T~ - - ~ T in s we have from (18) that (T~x, Ty) = (T~T*x,y) ~ (TT*x,y). It then follows from (17) that (r*Tz, y) = (TT*z, y>, for all z, y 9 H , and hence T*T = TT*. T h a t is, T is normal . 9 D e f i n i t i o n V . 5 . Let X be a Banach space. An operator T 9 s is called a scalar-type spectral operator if there exists some spectral measure P : E ~ s and a P-integrable function f such that T = fa f dP. 9 Given an operator T 9 s recall that the spectrum a(T), of T, consists of all numbers A 9 C such that (T - AI) is not invertible in s In the notation of Definition V.5 if T 9 s is a scalar-type spectral operator, then it turns out that (19)

. or(T) = A { f ( E ) : E 9 E, P(E) = I},

where f ( E ) is the closure (in C) of f ( E ) := {f(w) : w 9 E}. Indeed, if E 9 Z and P(E) : I , then we define, for each A r f ( E ) , the function

ha(w) := (A - f(w))-lXz(W),

w 9 a.

Observe that h~ 9 L~176 and satisfies

Accordingly, (AI - T) -~ = fa ha dP exists in / : ( X ) and so A belongs to the resolvent set p(T) := C\cr(T), of T. T h a t is, ~(T) C_ f ( E ) whenever P ( E ) = I and so A{I(E):

E9

P(E)=I}DD_cr(T).

Conversely, if X 9 p(T) = P(fa f d P ) , then fa(A - f ) d P = (AI - T) is invertible in s with ( A - f ) 9 LI(P) = L~176 (as vector spaces) . By Theorem V.1 the function ( A - f ) - I : w H 1/(A - f(w)), for w 9 f~, is an element of L~(P). Hence, there is a set Eo 9 E with P(Eo) = I such that ( 1 / ] A - / ( w ) ] ) < M, for w 9 E0, for some M > 0. Then A r f(Eo) and so A r A { f ( E ) : E 9 E, P(E) = I } , w h i c h shows that ~(T) _D A { / ( E ) : E 9 E, P(E) = I}.

90

CHAPTER ~

S T R O N G L Y CLOSED O P E R A T O R A L G E B R A S

Hence, (19) is indeed valid and is, moreover, independent of P and f in the sense that P and f only need to satisfy T = fa fdP. Define P T : Bo(C) ----+ s by PT(a) = P ( / - 1 ( 5 ) ) , for each 5 C Bo(C). It is routine to verify that P r is a spectral measure and satisfies PT(a) = 0 if a C_ p(T) ; see (19). Moreover, the identity function on C is Pr-essentially bounded (as a(T) is compact and PT(a(T)) = I) and satisfies

The spectral measure P r is called the resolution of the identity of T. The following result shows that in Hilbert spaces the scalar-type spectral operators are essentially the normal operators. P r o p o s i t i o n V.6. (a) Let H be a Hilbert space and T E s be a scalar-type spectral operator. Then there ezists a selfadjoint operator S E s which is invertible in s such that S T S -1 is a normal operator. (b) Let B C s be a Bade a-complete B.a. of projections. Then every clement of{B}; is a scalar-type spectral operator. P r o o f . (a) Let P : E ~ s be a spectral measure and f C L I ( p ) satisfy T = f a f d P . Then P ( E ) is a bounded B.a. of projections (see Lemma III.3) and so, by the MackeyWarmer theorem (c.f. Proposition V.5(b)), there exists an invertible, selfadjoint operator S Es such that { S P ( E ) S - ~ : E C E} consists entirely of selfadjoint projections. Then R : Z ~ Z;~(H) defined by R(E) = S P ( E ) S -1, for each E e E, is a selfadjoint-valued spectral measure satisfying

STS -1= S(~ fdP)S -1= ~ far C (R(~)} s. By putting B := R(E) in Example 23 it follows that S T S -1 is normal. (b) By Theorem V.8 the closure B, of/3, in s is a Bade complete B.a. of projections. By Theorem IV.1 there exists some closed spectral measure P : E ~ s such that N = P(E). Let T C (B)2. Then also T C (B)2; c.f. Exercise 52(@ So, Theorem V.6 implies that T = fn f d P for some f C L I ( p ) , i.e. T is a scalar-type spectral operator. 9 An examination of the proof of Proposition V.6(b) shows that it is irrelevant that the underlying space is a Hilbert space. Hence, the same proof establishes the final result of this chapter. P r o p o s i t i o n V.7. Let X be a Banach space and 13 c_ s be a Bade a-complete B.a. of projections. Then every element of (13)~ = (13};- is a scalar-type spectral operator. Recall that an operator S C s is called quasinilpotent if a(S) = {0} or, equivalently, if l i m ~ IlSnll 1In 0. Let T 9 L;(X) be a scalar-type spectral operator, say T = fa f d P in the notation of Definition V.5. It is clear from (19) that if T is also quasinilpotent, then f = 0, P-a.e. , and so T = 0. If then follows from Proposition V.7 that the closed subalgebra (13);- = (B)~ of s is semisimple (i.e. contains no non-zero quasinilpotent elements) whenever B C_ s is a Bade a-complete B.a. of projections. =

Chapter VI Bade functionals: an application to scalar-type spectral operators Let H be a Hilbert space and /3 c_ s be a B.a. of selfadjoint projections. Fix z C H. Using the fact that B 2 = B = B* and IIBzl] 2 = (Bx, Bx}, for each B E B, the calculation = (B~x, x> = = IIBxll ~ shows that (Bz, z} > 0, for B C B, and that B z - 0 whenever t3 E B satisfies (Bx, w/ = 0. Suppose now that B C_ s is an arbitrary bounded B.a. of projections. Let S E s be an invertible , selfadjoint operator such that J[ := { S B S -1 : B C 13} is a B.a. of selfadjoint projections; see Proposition V.5(b). Fix z E H and let z' := S2z. For each B C B let B # := S B S -1 c A, in which case (B#) 2 = B # = ( B # ) *. The calculation

:

<(B#)=Sx, Sx> : : = ItB#S~ll =,

valid for each B C B, shows that (Bz, x'} >_ O, for all B C B. Moreover, if B E B satisfies (Bz, z') = 0, then the previous calculation shows that 0 = B # S z : S B z . Since S is injective it follows that B z = 0. So, identifying H ' with H and interpreting z' = S2z C H' we have shown, for each z C H, that there exists z' C H ' with the properties (Bz, z'} >_ O, for all B E B, and B z = 0 whenever B E/3 satisfies (Bz, z') = O. W. Bade showed in [1] that, remarkably, this property of bounded B.a. 's of projections in a Hilbert space carries over to Bade d-complete B.a. 's of projections in Banach spaces. One of the aims of this chapter is to establish Bade's result. Our proof, based on Rybakov's theorem for vector measures, is quite different to Bade's original proof. For still another proof see [23]. Of course, Rybakov's theorem appeared some 15 years after Bade's proof! In the second part of the chapter we give a non-trivial application of Bade functionals to show that every Bade a-complete B.a. of projections in a separable Banach space coincides with the resolution of the identity of some scalar-type spectral operator with real s p e c t r u m , [31]. D e f i n i t i o n V I . 1 . Let X be a Banach space and 13 C_ s be a bounded B.a. of projections. Given z E X , any non-zero vector z' E X ' (if it exists) with the properties

CHAPTER VI. BADE FUNCTIONALS AND APPLICATIONS

92 (i)

(Bx, x') ~ O, for all B E/3,

and (ii) Bx = 0 whenever B r satisfies (Bx, x') = O, is called a Bade functional for x with respect to/3. 9 The existence of Bade functionals is guaranteed by the next result. T h e o r e m V I . 1 . Let X be a Banach space and 13 C L ( X ) be any Bade a-complete B.a. of

projections. Then every x C X has a Bade f~tnctional with respect to/3. P r o o f . Fix x C X. Let Y := ~3Ix] be the cyclic space generated by x with respect to /3. By Lemma V.4 the restricted B.a./3y c s o f / 3 to Y is also a Bade a-complete B.a. of projections. Since x r Y is a cyclic vector for /3z, it follows from Theorem V.7 and Theorem IV.1 that there exists a closed spectral measure P : E ~s satisfying P ( E ) = /3y. Then Rybakov's theorem (see Theorem 1.10) applied to the vector measure Px : E ~ Y guarantees the existence of a unit vector y' E Y' such that Px << I(Px, Y')I. Since (Px, y') << ](Px, y'} ], the Radon-Nikodym theorem for scalar measures (c.f. Theorem 1.6) guarantees the existence of a E-measurable function r for I(Px, y')l-a.e, w E ~, such that

(1)

I(Px, y')I(~) = f Cd(Px, y'),

~2 ----* C satisfying Ir

= 1,

5 e ~.

Define r to be zero on the I(Px, y')t-null set E for which Ir ~ 1. Since E is then also Pxnull and x is a cyclic vector for P ( E ) = / 3 z , it follows that E is also P-null . Accordingly, r e L ~ ( P ) . Define T := f a C d P e s Finally, let x' G X ' be any continuous linear functional on X which coincides with T~y~ r Y' on the closed subspace Y C_ X; see Theorem 1.5(0. Fix B E/3. Then Biy = P(5) for some 3 E F,. Since x c Y, we have

(Bx, x') = (P(~)~, ~') = ( P ( ~ ) x , T'y') where that last equality uses the fact that P(a)x c Y and x ' i r - T'y'. But,

Ip(~)x, T'y') = IP(~)~, (~ * dP)'y'l = IP(~)(~ ~ dP)~, Y'I

where the last equality follows from (1). Hence, (Bx, x') = [(Px, y')[(~) > 0 and so (i) of Definition VI. 1 is satisfied. Suppose that B e B satisfies (Bx, x') = 0. Then also (Byx, T'y') = 0, as B x C Y and x~iy =- T'y ~. Since By = P(Eo) for some E0 E E the same calculation as above shows that

o = (Bz, x') = I ( P x , r Since P x << I(Px, y')l, we deduce that B x = P(Eo)x = 0. Accordingly, (ii) of Definition VI. I is also satisfied. 9

93 We point out that Bade a-complete B.a. 's of projections are not the only ones having a Bade functional for each x E X. Example 24. Let X := ~oo and/3 C ZZ(X) be the B.a. consisting of all projections P(E) C s given by P(E)

:x ~

(x~(1)xl, x~(2)x~,...),

x = (x1,x~,...)

e x,

where E is an arbitrary subset of N. Then/3 is abstractly complete but not Bade or-complete 9 Given any non-zero vector x E X, let x ~ be the sequence defined by (x~)~ := ~n/2nlixlIoo, for each n C N. T h e n x ~ E gl C X t and (P(E)x, x ~) = ~=l(lxni2/2nilxiloo)xE(n ) for each set E C N. It is then clear from Definition VI.1 that x ~ is a Bade functional for x with respect to/3. 9 The existence of Bade functionals in Example 24 is due to the countability of the set N used to form the Banach space g~ = g~(N). Removing this condition provides an example of an abstractly complete B.a. of projections for which not every vector has a Bade functional. E x a m p l e 25. Let A be any uncountable set and E be the family of all subsets of A. Let X := g~(A) be the Banach space of all bounded fimctions f : A - - ~ C equipped with the norm II/11~ := sup{I/(A)l : A E A}. Then the dual Banach space X ' consists of all finitely additive set functions # : E -----* C for which sup{l~(E)l : E E ~} < oc; see [14; Ch.IV, w L e t / 3 be the B.a. consisting of all projections P ( E ) C s defined by P ( E ) Z = xEf, for f C X and each E E E. It is routine to verify that P : E ----*/3 is a B.a. isomorphism 9 Since E is abstractly complete the B.a./3 is also abstractly complete . Consider the particular element ll E X (i.e. the function constantly equal to 1 on A). Suppose that # C X ' is a Bade functional for 11 with respect t o / 3 , in which case

o < = f

n dp = p(m),

E c Z;

see Definition VIA(i). By Definition VI.l(ii) we see that ( P ( E ) l l , #) = 0 implies P(E)tl = Xz = 0 i n X (i.e. E = ~), from which it follows that # ( { l } ) > 0 for each A C A. That is, {~ E A : #({A}) > 0} = A. But, the finite additivity of # together with the property sup{Ip(E)l : E E E} < oe implies that E . := {)~ E A : #({~}) _> 88 is a finite set, for each n E N. Accordingly, the set U~=~E~ = {,~ E A : #({,~}) > 0} is countable. But, we saw above that this set equals A and we have a contradiction. So, 1l cannot have a Bade functional with respect to/3. It is interesting to note that 11 is a cyclic vector for/3. 9 Given a B.a. of projections B c_ s and a particular vector x c X , it is also of interest to be able to determine when x admits a Bade functional with respect to/3. For criteria in relation to this question we refer to [23]. As an immediate application of Theorem VI.1 we present an interesting result where weak operator convergence implies strong operator convergence; that this is not the case in general was noted in Exercise 32(b). P r o p o s i t i o n V I . 1 . Let X be a Banach space and/3 C_ s be a Bade ~-complete B.a. of projections. Let {B~}~cA C_ 13 be a net and B E s be a projection such that lim~ B~ = B in s . Then lim~ B,~ = B in s

CHAPTER VI. BADE FUNCTIONALS AND APPLICATIONS

94 m

m

P r o o f . Let 13 be the closure of B in Z:s(X), in which case 13 is a Bade complete B.a. ; see Theorem V.8. By Corollary V.6.1 the limit projection B c B. Observe that

(Bx, B'z') = (B2z, z') = ibm, z'),

m 9 X, z' e X',

showing that BoB -%~ B in s Since B~B <_ B we have, by Exercise 30(b), that (B - B~B) C 13 for all c~, and we just showed that (B - B,~B) ~ 0 in Z:~(X). A similar calculation shows that (I - B)B,, -%+ 0 in s since

((I

-

B)B~x, x') = ( B j ,

(I - B)'x') ~

(Bz, (I - B)'x')

=

((I

-

B)Bx, z') = O,

for all x E X and x / E X/. Moreover, for each a the identities

S-

S~ = ( S -

S~)[S + ( I - S)] = ( B - B ~ ) B + ( B - B ~ ) ( I - B ) ,

together with B ( I - B) = 0 show that

B-

So = ( S - S ~ S ) - ( I - S ) S ~ ,

~ e A,

is the difference of the nets { B - B ~ B } and { ( I - B ) B ~ } , both from B, which converge to 0 in ~(x). So, it sumces to show that if a net {D~} c_ B satisfies D~ ~ 0 i n / : ~ ( X ) , then also D~ ~-~ 0 i n / : ~ ( X ) . Suppose, by contradiction, that D~ - ~ 0 in Z;~(X). Then there exists x0 C X such that { D J 0 } does not converge to 0 in X. Let P : S ---* Z;~(X) be a closed spectral measure such that P(P.) = B. Then there exist sets E~ E ~ with P(E~) = D~, for each c~. Let x' E X ' be a Bade functional for x0 with respect to B. Then Pxo <<
95 bl, b2 E 13 are called disjoint if bl A b2 = 0 and a set E C / 3 is said to be disjoint if every pair of distinct elements of E is disjoint. 9 E x e r c i s e 54.[.] Let 2 N denote the B.a. of all subsets of N equipped with the partial order A _< B if and only if A c B. Show that 2 TM is countably decomposable. 9 P r o p o s i t i o n V I . 2 . (a) A (general) Boolean algebra I3 is countably decomposable if and

only if every set E c_ B has a countable subset D such that D and E have the same set of upper bounds , that is, if and only if {uEB:

d<_uforalldED}={vEB:

e
(b) A (general) Boolean algebra B which is countably decomposable and abstractly orcomplete is actually abstractly complete. P r o o f . (a) Assume first that B has the stated property. Let E C_ B be a disjoint set of non-zero elements. By hypothesis there is a countable set D C_ E such that D and E have the same set of upper b o u n d s . Suppose that E r D, in which case there exists an element e E E \ D . By disjointness of the elements in E it follows that e A d = 0 for all d E D. So, if e' is the complement of e in B, then e V e' = 1 and hence d = d A (e ~V e) = (d A e') V (d A e) = (d A e j) V 0 = d A e'. Accordingly, d <_ e' for all d E D (c.f. Exercise 13(f)) and so e' is an upper bound for D. But, e' is not an upper b o u n d for E since e A e' = 0 shows that e ;~ e' (as e r 0 and e < e' if and only if e A e' = e). This is a contradiction and so no such element e can exist, that is, E = D. In particular, E is a countable set. This shows that B is countably decomposable. Conversely, let B be countably decomposable. Let E be an arbitrary subset of B. Define A to be the family of all elements A in B such that A < V~()~e3= for some finite set { e l , . . . , ek(a)} _C E. Note that A is an ideal in B (see Definition II.10) and that E c_ A (as e _< e for any e E E). So, if v E B has the property that A _< v for all A E A, then also e <_ v for all e E E. On the other hand if u E B has the property that e _< u for all e E E, then V~=xej _< u for all finite subsets { e l , . . . , ek} c E and so A _< u for all A E A. This shows that E and A have the same set of upper b o u n d s . Let ~ca be the family of all subsets F of A such that F is disjoint and consists of non-zero elements. Partially order $-A by declaring F1 _< F2 (with Fj E $'A) if and only if F1 c_/'2. If {F~} is any totally ordered subset of 5cA, then it is easily verified that UoF~ E 5cA and so U~F~ is an upper b o u n d for {F~} in $cA. By Zorn's lemma, [14; p.6], 5cA has a maximal element, say F. The claim is that _P and A have the same set of upper bounds. Since F c_ A it is clear that any upper b o u n d of A is also an upper b o u n d of F. So, suppose that b E B has the property that f < b for all f E _P. If b is not an upper b o u n d of A, then there exists A E A such that A ;~ b. Consider w := A A b'. Then w r 0, for otherwise (i.e. if w = 0) AAb=(AAb)

Vw=

(AAb) V ( A A b ' ) = A A ( b V b ' ) = A A I = , ~ ,

which means that A < b; this is a contradiction. Also, if f E F, then

f A w = f Ab' A.~= f A b A b ' A A = O ,

96

C H A P T E R VI. B A D E F U N C T I O N A L S A N D A P P L I C A T I O N S

where the second equality uses the fact that f A b = f (as f _< b) and the last equality uses the fact that bAb ~ = 0. So, w is disjoint with every element in F. Finally, w ~ F. Otherwise, w := ~, A b' = f0 for some fo r F. Then b A (~ A b') = b A f0 from which it follows that 0 = f0 (as b A bt = 0 and b A f0 = f0)- But, all elements of/~ are non-zero and so this is impossible9 Moreover, since A is an ideal and ~ C A we have w =/k A b~ E A. But, F U {w} is then a set of non-zero, disjoint elements from A which contains F as a proper subset. This contradicts the maximality of F. Accordingly, b is an upper bound for A. Since/3 is countably decomposable , the set F is countable. Since each of the countably many elements of F is dominated by the suprenmm of some finite subset of E, it follows that the union, say D, of these finite sets is a countable subset of E with the same set of upper bounds . (b) Every countable supremum exists in 13 by the fact that /3 is abstractly a-complete. By part (a) the supremum of an arbitrary set exists as it coincides with some countable

supremum. 9 We have the following important consequence. First we require a definition. D e f i n i t i o n VI.3. Let X be a Banach space and /3 _C s be a B.a. of projections. A vector x E X is called separating for/3 if B = 0 whenever B E/3 satisfies B x = O. 9 P r o p o s i t i o n VI.3. Let X be a separable Banaeh space and /3 C_ s be a Bade acomplete B.a. of projections. Then, (a) /3 is countably decomposable, (b) /3 has a sepa~in9 vector, and (c) /3 is Bade complete. e~ P r o o f . (a) Let {B~}~eA be a disjoint set of non-zero elements from/3. Let {X ~}~=1 be a countable dense set in X. Consider {B~}~eA as being a subset of B (the closure of/3 in s in which case B is Bade complete). Then {B~}~eA is still disjoint and consists of nonzero elements in/3. By a Zorn's Iemma argument there is a maximal disjoint set of non-zero elements, say {BZ}Zev C_ B, containing {B~}~e A, Partially ordering the collection )c(V) of all finite subsets of the index set V by inclusion, it turns out that X~ZeF~/?Z -< Y~eF2/~ (with respect to the order in B) whenever F 5 r )c(V) satisfy F1 _ F2. So, {~ZeF/~Z: F G Jc(V)} is a monotone increasing net in B and hence, by the ordered convergence property of B, converges (in s to some B C B. If B ~r I, then ( I - B ) E B i s anon-zero element, disjoint with every BZ which contradicts maximality. Hence, ~Zev/3Z = B = I. Accordingly, z~ = EZ/~gx~ , for each n = 1, 2 , . . . , and so there is a countable subset V~ _C V such that /~x~ = 0 for all ~ r Vn. By density of {x~}~%1 in X it follows that /~p = 0 whenever /3 r U~=IV~. In particular, B~ = 0 for every a r A with a ff U~=IV~. But, {B~}~eA has no zero elements and so A C_ U~=IV~, i.e. A is a countable set. (c) Let {B~}~eA C_ /3 be an arbitrary set. Note by part (a) and Proposition Vh2 t h a t / 3 oo is abstractly complete 9 Also, by Proposition VI.2, there is a countable subset { / 5 ' ~}~=1 of {B~}~eA such that V~B~ = V~=IB~~. Then by the Bade a-completeness of/3, we have

(v~B~)X = (Vn%~B~o)X = sp{U~B~, X} C_ s p { U ~ B J } .

97 But, B ~ o X c_ (V~B~)X for each index a0 (as B~ 0 _< V~B~) and so U~B~ C_ (V~B~)X. Since (V~B~)X is a closed subspace of X it follows that sp(U~B~X) C_ (V~B~)X. This shows that (V~B~)X = sp(U~B~X). Similarly, oo

/~

oo

for some countable set of indices {/3k}k%~ C_ A; see Proposition VI.2. Then (A~B~)X = ( A OkO = I B ~ ) X = A ko o= ~ B ~ X (by the Bade ~-completeness of B) Accordingly, C ~ B ~ X c_ A ~ = I B ~ X = (A~B~)X. But, since A~B~ <_ B~ o for each index a0, also (A~B~)X C_ B ~ o X for every a0. That is, (A~B~)X _C M~B~X. This shows that (A~B~)X = A ~ B ~ X and we have established the Bade completeness of B. (b) Since/5 is Bade complete, the carrier projection C~ of x, which is defined to be the element of B given by C~ := A{B e g : B x = z } ,

exists for each x C X. By a Zorn's lemma argument there is a maximal, disjoint family {C~} of carrier projections in/3. Since B is countably decomposable it is of the form {C~(n)}n~_> where we can suppose that Hx(n)H = 1, for each n c N (since C~ = C ~ for all x E X and A ~ 0 in C). Let { := Y~=I 2-~x(n) 9 We claim that carrier projections always satisfy the following property. F a c t . C~x = x, f o r each x E X . To see this f i x x E X and let /3~ := {B E / 3 : B x = x } . For each finite set FC_ /3~let CF := A{B : B E F}. Given finite subsets _F1 and F2 of/3~ define F1 _< F2 if F1 C_ F2, in which case CF~ _< CF~ in/3. Then {CF : F C Y(/3~)}, where 5=(/3~) is the family of all finite subsets of/3~ directed by inclusion, is a decreasing net in/3 with A { C F : F E Y(B~)} = C~. Moreover, if F = {B1,... , B~} C_/3~, then B j x = x for each 1 _< j < n and so B 1 B 2 . . . B ~ x = B1B2... B n - l X . . . . .

B 1 B 2 x = B l X = x.

That is, C F x = (Aj~=IBj)x = B 1 B 2 . . . B ~ x = x. Since limF CF = Cx in Z:s(X) (see Theorem IV.1 and Lemma IV.l) we deduce that C~x = x. This establishes the Fact. C l a i m 1. C~(n)~ = 2-nx(n) = 2-~Cx(~)x(n), n C N. Indeed, fix n E N. Since C~(~)C~(k) = 0 if k ~ n, it follows from the continuity of oo --k C~(~) and the above Fact that C~(~)~ = }-~-k=~C~(~)2 x(k) = ~ = 1 2-kC~(~)C~(k)x(k) = 2-nc~(~)x(n) = 2-~x(n). This establishes Claim 1. C l a i m 2. V~=IC~(~) = ~n~=l C~(~) = I, where the series converges in Z:~(X). Observe that the disjointness of {C~(~)}~= 1 implies that V~=IC~(k) = Y~=I C~(~), for each n E N. Since the projections {V~=lC~(k)}n~_l are increasing in the order of B to the projection B := Vk~__lC~(k),it follows from Theorem IV.1 and Lemma IV.1 that r~

lim ~ B~OO

~

k=l

C~(k) = lim rt~oo

V~=lCz(k)

=

B,

in Z2~(X).

98

C H A P T E R VI. B A D E F U N C T I O N A L S A N D A P P L I C A T I O N S

In particular, B = ~ k = l C x(k) with the series converging in s Suppose that B r I, in which case (I - B) # 0. Since Cx(n)B = C~(~) (as Us(n) <_ B) for all n E N, it follows that (2)

Cx(n)(I - t3) = C~(,4B(I - B) = O,

n C N.

Choose any non-zero vector y E (I - 13)X, in which case (I - B ) y = y. So, by the definition of carrier projections we have Cy _< (I - B), that is, Cy(I - B) = Cy. It then follows from (2) that c~(,~)c~ = c~(,~)c~(l - 13) = o,

~ c N.

Accordingly, the carrier projection Cy ~& 0 is disjoint from {Cx(~)}~-1 which contradicts the maximality of {Cx(n)}noc_l. Hence, B = I and Claim 2 is established. C l a i m 3. The element ~ is a separating vector for B. To establish this suppose that B C 13 satisfies B~ = 0. By Claim 1 and the above Fact we deduce that

It follows that (I - B)C~(~)z(n)

=

(Z - B)z(n)

= z(n),

n C N,

and so, by the definition of carrier projections , we have that C~(~) _< (] - B)C~(~), that is, C~(~) <_ (C~(~)-BC~(~)). It follows that BC~(~) 0, for all n 6 N, and so B (}-~k=l C~(~)) 0, for alln E N. Let n--* oc and use Claim 2 to deduce that 0 = B(V~_IC~(~)) = BI = B, that is, B = 0. This shows that ~ is a separating vector and completes the proof of Claim 3 and of the proposition. 9 Exercise 55. Let X be a separable Banach space. Show that any spectral measure P :E ~ ~]~(X), with ~ any cT-algebra of sets , is necessarily a closed spectral measure. 9 Definition VI.4. Let X be a Banach space and B C s be a B.a. of projections . (i) A non-zero projection B E B is called an atom if, whenever D ~ ~ satisfies D _< B then either D = 0 or D = B. (ii) B is called atomic if there exists a family {B~}~ca of distinct atoms in B such that, whenever B E/3 there is a subset AB C_ A such that B = ~_AB B~, that is, B is the strong operator limit of the net of finite partial sums of {B~ : ct E AB}. If the index set A in (ii) is countable, then we say that B is countably atomic. 9 E x e r c i s e 56.[*] Let X be a Banach space and B c_ s be a B.a. of projections. Show that if B1 and B2 are distinct atoms in B, then B1 A B2 = 0 (i.e they are disjoint) . 9 E x e r c i s e 57. Let X be a Banach space and B C s be a Bade complete B.a. of projections which is atomic. Show that the set of atoms {B~}~eA which generates B in the sense of Definition VI.4 is maximal and pairwise disjoint . 9 L e m m a V I . 1 . Let X be a 13anach space and B C_ s be a countably atomic , Bade a-complete B.a. of projections. Then 13 is Bade complete. P r o o f . Let {B~}~= 1 be a countable family of atoms in B which generates B in the sense of Definition VIA. The map F ~ ~ , e F Bn, for each subset F _C N, is a B.a. isomorphism

99 of the B.a. 2N (of all subsets of N) onto B. Since 2TM is countably decomposable (see Exercise 54) so is B. Hence, B is Bade complete ; this follows from an examination of the proof of Proposition VI.3(c) which only used the countable decomposability of B and not the fact that X was separable . 9 D e f i n i t i o n VI.5. A (general) abstractly a-colaplete (resp. abstractly complete) B.a. B is called countably generated if there is a countable subset 7:) C_ B such that the smallest abstractly a-complete (resp. abstractly complete) B.a. jc _C B with 7? C_ jc is 5 = B. 9 R e m a r k . An examination of Lemma 4 in [14; p.167] and its proof, together with the fact that every abstract B.a. is isomorphic to a B.a. of subsets of some set (see Theorem II.2), shows that the countable set 7? in Definition VI.5 can always be chosen as a Boolean subalgebra of B, meaning that it contains the same 0 and 1 from B and is closed with respect to finite sups and infs and complements. 9 Exercise 58. Let X be the non-separable Hilbert space f2([0, 11) and, for each subset F C_ [0, 1], let P ( F ) E s denote the projection operator in X of multiplication by XF. Show that B := { P ( F ) : F C_ [0, 1]} is a Bade complete B.a. of projections in s which is

countably generated. 9 Let X be a Banach space and B C /2(X) be a Bade complete B.a. of projections with a separating vector, say x. (a) B is countably generated if and only if its restriction to the (B-invariant) cyclic space B[x] is countably generated. (b) If the cyclic space g[x] is separable, then B is countably generated. (c) If X is separable and x is a cyclic vector for B, then B is countably generated. L e m m a VI.2.

Proof. (a) Let Y := B[x]. Since I / i s invariant for each B E B it follows that the family By := {BIy : B c B} of restricted operators is Bade a-complete in /2(!/); see Lemma V.4. Since By has a cyclic vector in I/ (namely x), it follows from Theorem V.7 that By is actually Bade complete . Moreover, the map 9 : B ~ By given by ~(B) = By is a surjective B.a. homomorphism. Using the fact that x is a separating vector for B it is routine to check that 9 is also injective and hence, 9 is actually a B.a. isomorphism of B onto By. Accordingly, B is countably generated if and only if By is conntably generated. (b) This follows from part (a) and part (c) below after noting that By (with 1 / : = B[x]) is Bade complete; see the proof of part (a). (c) Let {D~x}~=l be a countable dense subset of the separable subset W := {Bx : B E B} z~ of X. Let Af (~) c_ / 2 ( X ) be the abstractly a-complete B.a. generated by {D n}~=l, in which case Af (~) _c B of course. Let {B ~}~=1 be a monotone sequence in Af (~ Since B is Bade ~ complete , it has the ordered convergence property and so l i m ~ B~ = B exists in/2,(X), with B = V~_~B~ if {B~} is increasing and B = A~B~ if {B~} is decreasing; see Theorem IV.1 and Lemma IV.1. But, since Af (~) is abstractly a-complete we have V~B~ and AnB~ both belong to Af (~). Hence, N "(~) has the a-ordered convergence property and so Af(~ is Bade a-complete by Theorem IV.1. Then Proposition VI.3 (together with the fact that any cyclic vector is necessarily a separating vector) shows that Af(~ is actually Bade complete .

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CHAPTER VI. BADE FUNCTIONALS AND APPLICATIONS

Fix B E/3. Since A/:x(~ := {Nx : N E A f (~ } is a dense subset of W there exist elements ~ C _ A/"w) such that Bx = l i m k ~ Nkx. It follows, for every vector z = ~y=~ m ajRjx { N k}k=l with a j E C and Ry E B, that l i m k ~ Nkz = Bz. Since the set of all such vectors z is dense in X and sup{i]NkII : k E N} < ee, we conclude that Nk ~ B in s as k --~ ~ . This shows that B belongs to the closure A/'W), of A; (~ in s By the Bade completeness of Af (~ and Corollary V.5.1 we deduce that A/'W) = AfW) and so B E A f w). Since B E /3 is arbitrary we conclude that Af w) = / 3 and so/3 is countably generated. 9 Remark. Proposition VI.3 and Lemma VI.2 imply that every Bade complete B.a. of projections in a separable Banach space X is necessarily countably g e n e r a t e d . In particular, in a separable Banach space X every Bade a-complete Boolean subalgebra of a Bade complete B.a./3 ( in which case/3 is necessarily countably generated) is itself countably generated. 9 D e f i n i t i o n V I . 6 . By a measure algebra we mean a (general) abstractly a-complete B.a./3 together with a function # :/3 ~ [0, ca) satisfying (i) # ( B ) = 0 if and only if B = 0, and (ii) #(V~= 1 ~) = }-]~=1 # ( , ~ ) whenever B~ A B,~ 0 for all m # n. In this case we call # a measure o n / 3 and denote the measure algebra by (/3, #). 9 E x a m p l e 26. Let E be a a-algebra of subsets of some non-empty set ft and u : E ~ [0, co) be a a-additive measure. Let Af := { E E E : u(E) = 0} and define B := E / A f to be the quotient B.a. of equivalence classes , where we define E ~ F if E A F E A f in which case [E]:={FEE:

EAFEN'},

[E] E/3.

If we define a function # : / 3 - - ~ [0, oc) by #([E]) := ~(E), then # is well defined and (/3, #) is a measure algebra. This is the classical example of a measure algebra. 9 Let (/3, #) be a measure algebra . For each pair bl, b2 E / 3 define

blab2 := (blAb;)V(biAb2), Then pg :/3 x / 3 - - ~ [0, ee) defined by p,(bl, b2) = #(blGb~) specifies a metric in/3. It turns out that (/3, p,) is a complete metric space and that the three mappings b ~-~ b', and (hi, b2) ~ hi V b2, and (bl, b2) ~ bt A b2,

are continuous on 13 and B x/3 with respect to pg and pg • Pw These facts can be found in [37; Chapter 15, Section 2], for example. Exercise 59. Let X be a Banach space and/3 C s be a Bade complete B.a. of projections which is countably generated by some countable Boolean subalgebra Z~ C_ /3. Suppose that x E X is a separating vector for /3 and that x ~ E X ~ is a Bade functional for x with respect to/3. Define p :/3 ~ [0, c~) by ~ ( s ) :=

(Sz, x'),

s E

s,

101 and let p , : 13 x 13 ~

[0, oc) be the associated metric given by p,(A, B ) : = # ( A A B ) ,

A, B 9 13,

where A A B :• (A A B') V (A' A B). (a) Show that the closure ~ , of I9, in the metric space (B, p,) is a Bade ~-complete B.a. of projections satisfying I9 C ~ C B. (b) Show, moreover, that actually ~ = B. 9 D e f i n i t i o n V I . 7 . Let (B, p) be a measure algebra . T h e n / 3 is called separable if (B, p,) is a separable metric space. 9 The next result gives an important procedure for generating measure algebras in the setting of B.a. 's of projections. P r o p o s i t i o n V I . 4 . Let X be a Banach space and B c_ s be a B.a. of projections with a separating vector x C X . (a) Suppose that 13 is Bade or-complete . For any Bade functional x' C X ' of x with respect to 13, define #(B) : : (Bx, x'), for B E 13. Then (13, #) is a measure algebra. (b) Suppose, in addition to (a), that X is separable. Then the measure algebra (13,#) of part (a) is separable. P r o o f . (a) It is clear from the definition of Bade functionals that # _> 0 is a finite-valued function on 13. Moreover, if B E 13 satisfies #(B) = 0 (i.e. (Bx, x') = 0), then B x = 0 (see Definition VI.1) and hence B = 0 (as x is a separating vector for 13). Suppose that {B~}n~=~ ~ n IB kI~=1 leo is increasing in 13. Since 13 has the is a pairwise disjoint sequence in 13. Then r~vk= or-monotone property we know that oo n Vn=IB~ = lim Vk=lBk,

in ~ ( x ) ,

r*~oo

and hence, that

p(V~=IB~ )

((V~= 1 ~)z,x')

lim ((Vk:lBk)x, x').

But, by the disjointness of {B~}~__I it follows that n n <(vk:lB~)x, x!) : ~(S~x,

x ! >,

nED,

k:l

and so

n oo #(V~=I

B

n)

=

n--~lim E k:l

oo

#(Bk) = E #(Bn). n=l

This shows that (B, #) is a measure algebra. (b) Let us suppose now, in addition to (a), that X is separable. Proposition VI.3 shows t h a t / 3 is Bade complete and hence, by the Remark after Lemma VI.2 we see that B is countably generated via some countable Boolean subalgebra l) of B (see the Remark after Definition

CHAPTER VI. BADE FUNCTIONALS AND APPLICATIONS

102

VL5), It follows, since the closure of D in (/3, p~) is a Bade a-complete B.a. containing 2? (see Exercise 59), that this closure equals B and hence, that D is dense in/3. Accordingly, (/3, #) is separable. 9 We now come to another main result of this chapter. T h e o r e m V I . 2 . Let X be a separable Banach space and/3 C_ s

be a Bade a-complete B.a. of projections. Then there exists a scalar-type spectral operatorT E s with a(T) C R whose resolution of the identity Pr : Bo(R) ~ s satisfies/3 = {PT(a) : 5 ~ Bo(R)}. P r o o f . By Proposition V1.3 B is Bade complete. Hence, if {B~} is a maximal disjoint family of atoms in/3, then the series P,~B~ is summable in s to the element B= = V~Bo of B. Let x be a separating vector for/3 (see Proposition VI.3). Then E~B~x = B~x, with the series being unconditionally norm summable in X. Accordingly, at most countably many of the vectors {B~z} can be non-zero. Since x is a separating vector it follows that B has at most countably many atoms, say {B~}~=I, and hence B~ = V~=IB~. Let X~ := B=X, in which case X~ is also a separable Banach space. Then the restricted B.a./3~ := {Blxo : B C 13} is a Bade complete , countably atomic B.a. in X~; see Lemma V.4 and Lemma VI.1. Let X~ := (I - B~)X, in which case Xr is also a separable Banach space and we have the direct sum X = X~ G X,. Let /3r := {BIx ~ : B C /3}. Suppose, for the moment, that there exists a scalar-type spectral operator T~ E s with a(Tc) C IR, such that the range of its resolution of the identity Pc : Bo(a(T~)) ~ s coincides with B~. Then we choose an interval [a,b] C R having a(T~) in its interior. Hence, Ta := ~ =oo 1 ( b + I~R(~) nJ ~ (where B(~~) is the restriction of B~ to X~ and the series converges in s is a scalar-type spectral operator in X~ with a(T~) = {b} U {b + 88 contained in ]R and disjoint from a(T~), and whose resolution of the identity P~ : Bo(a(T~)) ~ s has range coinciding with/3=. By construction A := a(T~) U a(T~) is a disjoint union in R. Moreover, every Borel set G C Bo(A) has a unique decomposition G = G~ U Gc into disjoint Borel sets G~ C Bo(a(T=)) and G~ ~ Bo(a(T~)). Define Pc(G) c s by Pr(G) = P~(G=) ~ P~(Gr Then the set function PT : Bo(A) ----+ s so defined is a spectral measure . Moreover, if T := T~ | in which case a(T) = A, then T = fo(T)AdPr(k) is a scalar-type spectral operator with or(T) C R which satisfies s = {Pr(a)

: a e Bo(~)} = {Pr(a)

:

a r SoW(T))}.

So, it remains to exhibit an operator Tc C s with the properties required above. Now, Bc is a Bade complete B.a. in the separable Banach space Xc (hence is countably generated)

, has no atoms

and

has z := (I - Bo,)x as a separating

vector.

Choose

a Bade

functional z' E X~' such that (z, z') = 1. By Proposition VIA with #(D) = (Dz, z'), for D E B~, we see that (Be, #) is a separable measure algebra with range p(/3c) C_ [0, 1], satisfies p(I) = 1 and has no atoms. By a classical result of C. Carath6odory, [37; p.321], there is a B.a. isomorphism ~ of/3c onto the measure algebra gt generated by Lebesgue measure in [0, 1], which preserves countable suprema and infima. If Af denotes the Lebesgue null sets of Bo([0, 1]) , then A is B.a. isomorphic to the quotient Bo([0, 1])/AA Let p : Bo([0, 1])

103 Bo([0, 1])/N" be the quotient map. Then Pc : Bo([0, 1]) ~s given by Pc(G) = q)-l(p(G)), for G c Bo([0, 1]), is a spectral measure whose range is precisely Be. Indeed, it is clear that P~(Bo([O,1])) = Be, that Pc is multiplicative, and that Pc satisfies Pc(0) = 0 and Pc(J0, 1]) = I. To verify the a-additivity of Pc let Gn $ 0 in Bo([0, 1]). Since p is a surjective B.a. homomorphism, there exists G C Bo([0, 1]) such that p(C~) I p(G) in Bo([0, 1])/A/. Let A denote Lebesgue measure, in which case A(Gn) ; 0. Observe that A induces a well defined action in the quotient B.a. Bo([0, 1])/N" via the formula A(p(E)) - A(E), for every E c Bo([0, 1]). Since p(Gn) I p(G), we have p(G) = p(G)p(G~) = p(G C~G~) for all n C N, and so A(G) = A(G C~G~) _< A(G~) ~ 0. This shows that a E N" and so p(G~) ,[ 0 in the B.a. Bo([0, 1])/N'. Since 9 is a B.a. isomorphism it follows that P~(G~) = ~-~(p(Gn)) i 0 in the order of Bc. But, B~ is Bade complete and so P~(G=) , 0 in Z;~(X~); see the equivalence (b)e=a(c) in Theorem IV.1. This establishes the a-additivity of Pc. Hence, T~ := f[0,1] a dPc(k) is a scalar-type spectral operator with the desired properties as cr(T~) = supp(P~) C_ [0, 1]. 9 We end this chapter with an example to show that Theorem VI.2 is no longer valid without the separability assumption (even in Hilbert spaces). Let X be a Banach space, P : E ~ t;~(X) be a closed spectral measure and Ip : LI(P) ~ s be its associated integration map f ~ fafdP, for f C LI(P). If the underlying a-algebra E is countably 9enerated as a a-algebra (i.e. there is a countable collection of sets { A ~ } ~ I C E such that the a-algebra generated by {A~}~%1 coincides with E), then the locally convex Hausdorff space (LI(p), r~(P)) is necessarily separable ; see [35; Proposition 2]. Accordingly, its isomorphic image

Ip(LI(p)) = { ~ IdP : f E L~(P)} must be a separable subspace of s for the relative topology. Let T = f~(T) ~ dPT(),) be a scalar-type spectral operator (with resolution of the identity PT: Bo(a(T)) --~ s such that a(T) C R. Since the a-algebra Bo(a(T)) is countably generated it follows from the above remarks that IPr(LI(PT)) is a separable subspace of s and hence, so is its closure IPr(LI(PT)) in s Since IPT(LI(PT)) = (B}s, where B := Pr(Bo(a(T))), we have established the following result. P r o p o s i t i o n V I . 5 . Let X be a Banach space and 13 c_ s be a Bade complete B.a. of

projections. If 13 coincides with the closure (in s of the resolution of the identity of some scalar-type spectral operator with real spectrum, then (B}2 is necessarily a separable subspace of s E x a m p l e 27. Let (f~, E, #) be a finite, positive measure space such that # is a non-separable measure (eg. # can be taken to be Haar measure on the Bohr compactification ft of the locally compact abelian group R). Let X denote the non-separable Hilbert space L2(#). For each E C E, let P(E) E s be the operator in L2(#) of multiplication by X~. Then it is routine to verify that P : E ~ g~(X) so defined is a (selfadjoint) spectral measure and hence, its range B := P ( E ) is a Bade a-complete B.a. Since the E-simple functions are dense in L2(#)

104

CHAPTER VI. BADE FUNCTIONALS AND APPLICATIONS

it is easily verified that 11 (the constant function 1 on f~) is a cyclic vector for B. Then Theorem V.7 shows that B is actually Bade complete . Suppose that T = fo(T) )' dPT(),) is a scalar-type spectral operator with a(T) C_ R and satisfying PT(BO(cr(T))) = 13. The evaluation map k~: S ~ $11 from (B}2 into X is linear, continuous and its range Y := ~ ( ( B ) 2 ) is dense in X (as 11 is a cyclic vector for B). Since (B)2 is a separable subspace of s see Proposition VI.5 - i t follows that Y is separable in X and hence, that X = Y is separable. This contradiction shows that no such operator T can exist. Can there exist a resolution of the identity PT : Bo(cr(T)) ~ s of some scalar-type spectral operator T E s with or(T) _C R such that B = PT(BO(~(T))), where the closure is taken in s If this were the case, then (PT(Bo(~(T))))2 = (B)2 and so again (B)2 would be separable, which is not the case. Hence, B cannot coincide with the closure (in Z;s(X)) of a resolution of the identity either. 9

Chapter VII The reflexivity theorem and bicommutant algebras The purpose of this final chapter is to establish a beautiful result, called the Bade reflexivity theorem, which states that the strongly closed operator algebra generated by a Bade ~complete B.a. of projections B c_ s consists of all continuous linear operators in X which leave invariant every closed subspace of X which is invariant under every member of B. If B is actually Bade complete , then this algebra also coincides with the uniformly closed operator algebra generated by B. The proof of this theorem is reduced to a consideration of the restricted B.a. of projections Bu[~] to the cyclic spaces B[x], for each x E X. For this special case of cyclic spaces , the proof becomes a routine consequence of an integral representation theorem characterizing the cyclic spaces B[x] as spaces of the type Li(m), for certain kinds of X-valued vector measures m. This integral representation theorem is established in the first part of the chapter. The rest of the chapter is then devoted to verifying the reflexivity theorem. As an elegant application we deduce the classical von Neumann bicommutant theorem for a bounded selfadjoint operator in a separable Hilbert space. So, let us begin with a representation theorem of the cyclic spaces B[x], with B a Bade complete B.a. of projections, as spaces of the type Ll(rn) for vector measures m : E ~ X of the kind rn = P z (with P a spectral measure satisfying P(E) = B). For this purpose the following useful result will be needed. P r o p o s i t i o n VII.1. Let X be a Banach space and P : E ~ s be a spectral measure.

Fix x E X . Then a E-measurable function f : f~ ---* C is Px-integrable if and only if f E LI((Px, x'}), for each x' E X', and there exists a vector z~ E X satisfying (1)

(xa, x'} = / ~ f d(Px, x'},

x' E X'.

In this case, fE f d(Px) = P ( E ) x a for each E E E. In particular, f is Px-null if and only if fa f d(Px) = O. Proof. If f is Px-integrable, then by definition f E Ll(
CHAPTER VII. REFLEXIVITY AND BICOMMUTANT ALGEBRAS

106

Conversely, suppose that f C LI((Px, x')), for all x' C X', and that there exists an element xa c X satisfying (1). It suffices to show that P(E)xa satisfies

L f d(Px, x') = (P(E)x~, x'),

(2)

x' ~ X',

for each E E E. So, fix E E E. For each x' E X', the formula (1) implies that

(P(E)xa, x') = (xa, P(E)'x') = L f d(Px, P(E)'x').

(3)

I/(w)l _< n}, for each n - 1, 2 , . . . , and f~ : - fXE~. Then each function f~ belongs to L(P) and so

Let E~ := {w E a :

Since [f~I -< Ifl and f~ ---~ f pointwise on f~, the dominated convergence theorem for the complex measures (Px, x') and (Pz, P(E)'x'} yields (via (3) and (4)) that

(P(E)xe, z') = f f d(Px, P(E)'x') = lim f f~ d(Px, P(E)'z'} n~oo j[~ J~

which is (2). Hence, f E L(Px). Finally, if xn = fa Zd(Pci) = 0, then fE f d(Px) = P(E)za = 0 for all E C E and so f is Px-null. 9 E x a m p l e 28. Proposition VII.1 shows that any vector measure of the form m = Px, for some spectral measure P : E ---+ s evaluated at a vector x C X , is rather special in that a E-measurable function f : ~ ---+ C is m-integrable if and only if (i) f C L l ( ( m , x ' ) ) , for each x' E X', and (ii) there exists xa C X satisfying (an, x'} = fa f d(m, x'), for all x' e X'. Of course, if the Banach space X does not contain an isomorphic copy of co, then (i) alone suffices for f to be m-integrable ; see Exercise 42. However, for an arbitrary Banach space X and an arbitrary vector measure m : E ----+ X conditions (i) and (ii) do not suffice for f to be ra-integrable. To see this let X := co. Let f~ := N and E := 2 a. Define m : E ~ X by

re(E) = ((6~ - 62)(E), 3-~(6a - 64)(E), 5-1(65 - 6 6 ) ( E ) , . . . ),

E E E,

where ~ is the Dirac point measure at n E f L If ( = ((1, ~2,... ) belongs to X ' = g l then it is routine to verify that

(m, [)(E) = ~1" (61 - 6z)(E) + 3-1r 9 (6z - 64)(E) + 5-1r 9 (65 - 66)(E) + . . . ,

E C Y?,,

107 from which it is clear that (m, ~) is a complex measure . So, Proposition 1.1 implies that m is a c0-valued vector measure. Now, the function f : f~ ~ C defined by f(n) = n = f ( n + 1), for each n E {1, 3, 5 , . . . }, clearly satisfies (i) since, for each ~ E gi, we have

(*) fEfd(m,~}:~l.(61-62)(E)+~2.(63-64)(E)+~a.(65-66)(E)+...

,

EEl.

In particular, fa Ifldl(m,~)l <- 21l~ll~ < ec. Moreover, the element xa := 0 of X clearly satisfies (ii). However, for an arbitrary set E E E it can be seen from (*) that only the vector X E : = ((61 -- 6 2 ) ( E ) ,

(63 - 6 4 ) ( E ) ,

(65 - 6 6 ) ( E ) , . . .

)

satisfies (XE, ~} = fE f d(m, ~) for all ~ E X ' = gl. But, xE is typically only an element of the bidual X" = g~ (unless E is finite or cofinite) and not an element of X = co itself. Accordingly, f is not m-integrable (in X). 9 We can now establish the following important representation theorem. T h e o r e m V I I . 1 . Let X be a Banach space and 13 C_ s be a B.a. of projections which

coincides with the range of some closed spectral measure P : E ~ z E X , the integration map Ipx : LI(Px) ~ X given by [px : f H f a f d(Px),

s

Then, for each

f C LI(Px),

is a Banach space isomorphism from L~(Px) onto the cyclic space 13[x]. P r o o f . Proposition VII.1 implies that Ipx is injective . Fix z E X. Suppose that f >_ 0 is Pz-integrable . Choose functions Sn E sim(E) , for n C N, such that 0 < s~ T f pointwise on f~. By the dominated convergence theorem for the X-valued vector measure Px (see Theorem 1.9) we have that fa s~ d(Px) ~ f a f d(Px) = Ip~(f) in X. Since clearly fa s~ d(Px) E 13[x], for each n E N, and 13[x] is closed, it follows that Ip~(f) E 13[x] for all non-negative f E LI(Px) and hence, for arbitrary f E LI(Px). So, Ip~ takes all of its values in 13[x]. If f is Px-integrable, then we know from Proposition h2 applied to m := Px that

IlIp~(f)ll = II [ f d(Px)ll <-II(Px)/ll(a) = J~

IlfllLl(p~)

and also that

Ilfh~(P~)

<

f 4sup{II/ fd(Px)ll:

EEE}=4sup{IIP(E

)/ofd(Px)li:EEr}

-< 4IIBII. II fnfd(Px)I] = 4IIBll" IIIpx(f)l[. That is, for each f E LI(Px), we have (5)

IlIPx(f)ll ~ I[fllLl(Px) ~_ 4111311" IlIPx(f)ll,

which shows that Ipx is an isomorphism of LI(Px) onto its range Ipx(L1(Px)) C_ 13[@

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CHAPTER VII. R E F L E X I V I T Y AND BICOMMUTANT ALGEBRAS

So, it remains to check that Ip~ is onto 13[x]. Let z E B[x]. Then z = l i m ~ z~ with each z,~ being some finite linear combination of elements from {P(E)x : E E E}. It is then possible to write z~ = } - ~ 1 ~j ~(~)''~(~)~ ~(~) E C and the sets t~a /~(~)lk~ r~j ~, with c~j J j : l C E pairwise disjoint, - -

so that z~ = 1p~(s~) with s~ := }-]~"--1@~)XE]"/ C sim(E) . Hence, z = l i m ~ o o Ip~(s~). It is clear from (5) that {Sn}~~ is then a Cauchy sequence in LI(Px). Since LI(Px) is complete (see Theorem 1.11), there is f E LI(Px) such that s~ ~ f in LI(Px). By (5) again it follows that l i m , ~ [px(Sn) = Ip~(f). Accordingly, z = Ipz(f). 9 The following fact is a useful consequence of Theorem VII.1. E x e r c i s e 60. Let X be a Banach space and B C_ s be a Bade c~-complete B.a. of projections with a cyclic vector z E X (i.e. X = B[z]). Let x' E X' be a Bade functional for x with respect to B and let Z C_ X ~ be the linear span of {B'z' : B E B}. Show that Z is dense in X ' for the weak-star topology cr(X', X ) I 9 We now turn towards the main topic of this chapter. Let A _c s be a commutin 9 family of operators. By Lat(A) we denote the family of all closed subspaces of X which are invariant for every operator A E A. We then define AlgLat(A):={TEE(X)

: T Y c_ Y for a l l Y E L a t ( A ) } .

E x e r c i s e 61. Let X be a Banach space and 13 C_ s be a bounded B.a. of projections. Let B denote the closure of g in s (a) Show that Lat(B) = Lat(B). (b) Show that T E s belongs to AlgLat(B) if and only if TI3[x] C_ B[z], for each x E X, where B[x] is the cyclic space generated by x with respect to B. (c) Show that AlgLat(B) is a subalgebra of s which is closed in each of the spaces f-.w(X), ~,s(X) and s , and that (B}2 C_ AlgLat(B). (d) Show that Lat(B) = Lat((B}y ). (e) Show that AlgLat(g) = AlgLat((B>Z ). 9 Let A c_ s be a family of operators. Then the commutant A c, of A, is defined by A~:={TEs

TA=ATforallACA}.

The family of operators (Ac) ~, denoted simply by ,4 ~, is called the bicommutant of A. Of course, we have that

~4~ c : { T E s

TS:ST

for alISCAC}.

E x e r c i s e 62. Let X be a Banach space and B C s be a B.a. of projections. (a) Show that B c is a subalgebra of s and that it is closed in each of the spaces s s and s Show that each of the subalgebras (B>~ and (13}Z = (B}~ is contained in B c. (b) Show that B c~ is a commutative subalgebra of s which contains B and that it is closed in each of the spaces s s and s

109 (c) Give an example of a Banach space X and a B . a . B C_ s such that B c is not commutative. (d) Let g be bounded and B be the closure of B in s Show that B c = (~)c and B c c = (~)cc. 9 The main aim of this chapter is to establish the Bade reflexivity theorem. We begin with the following weakened form of this result. P r o p o s i t i o n V I I . 2 . Let X be a Banach space and I3 C_ s be a Bade or-complete B.a. of

projections with a cyclic vector. Then

(6)

(BL

: 13c.

P r o o f . It is routine to verify that (13}~ C_ 13c; see Exercise 62(a). To prove the converse, let T E 13c. Let P : E >s be a spectral measure such that P ( E ) = 13. Since 13 is Bade complete (c.f. Theorem V.7) it follows t h a t P is a closed spectral measure (see Theorem IV.l) and hence, by Theorem VII.l, that the integration map Ip~: f H fa f d ( P x ) is a bicontinuous isomorphism of LI(Px) onto X = 13[x]; here x E X is any cyclic vector for B. Let f E LI(Px) satisfy Tx = fa f d(Px) and define sets E~ := {w E f~: If(w)l _< n}, for n = 1 , 2 , . . . Then f x ~ , C L~176 and so T~ := fafx~ dP E (13)~ for each n = 1, 2 , . . . ; see the proof of Theorem V.2(a). It follows from Proposition VII.1 that

(7)

P(E,OTx= ~ fxE d(Px) = (fnfxE dP)z= T,~x,

hEN.

Using (7) and the fact that T P ( E ) = P(E)T, for all E E E, we deduce that P(En)Tz = T,~z, for each n E N a n d every z E s p { P ( E ) x : E E E}. Since the space s p { P ( E ) x : E E E} is dense in X (as x is a cyclic vector), it follows that P ( E n ) T = T~, for each n E N. Noting that /~, t ~ it follows from the G-additivity of P that l i m ~ o o T~ = T in/2~(X) and hence, T E (B);- = (P(E))~. Since ( B ) , = (B)g (c.f. Corollary V.6.3), we see that T E (B)g. 9 E x e r c i s e 63. Let X be a Banach space and 13 C_ s be a Bade a-complete B.a. of projections with a cyclic vector . Show that B c is necessarily commutative. 9 B.a.'s of projections 13 which satisfy (6) are rather special. There are many known criteria on 13, other than those imposed by the hypotheses of Proposition VII.2, which also imply (6). We will only discuss one such set of criteria arising in connection with a notion of completeness for B.a. 's of projections (due to C.Rall, [33]) which is closely related to the existence of carrier projections and to Bade completeness. D e f i n i t i o n V I I . 1 . Let X be a Banach space. A bounded B.a. of projections 13 C_ s is called "r-complete if, for each z E X, there exists a smallest projection P E 13 (depending on z) satisfying P x = z. T h a t is, whenever Q E 13 satisfies Qz = z, then necessarily P < Q. 9 E x a m p l e 29. Any Bade complete B.a. of projections 13 C_ s is T-complete. Indeed, for each z E X the carrier projection C~ := A{B E 13 : B z = z} exists in 13 (by abstract completeness) and it necessarily satisfies C~z = z (see the Fact in the proof of Proposition VI. 3(b)). 9

110

CHAPTER VII. R E F L E X I V I T Y AND B I C O M M U T A N T A L G E B R A S

In view of Example 29 it is natural to ask whether every abstractly complete B.a. of projections is r-complete ? As shown by the following example this is false in general, even if B has a cyclic vector ! Example 30. Let N be equipped with its discrete topology and let Q :- ~(N) . Equip the subset A := Q\N with the relative topology from Q and consider the Banaeh space X := X1 e X 2 := {(fl, f2) : fj C X3} equipped with the norm II(f~, f2)lE = max{llflll~, IIf211~}, where X1 := { f C C ( a ) : fXA = 0} and X2 := Cb(a) is the Banaeh space of all bounded continuous functions on A. We note that X1 is Banach space isomorphic to Co and that X can be identified with a subspace of g~(Q). Since A A E E Co(A) whenever E E Co(Q), we can define a projection P ( E ) E s by

P(E) (fl, f2) = (xEfl, XsnAf2),

(/1, f2) E X,

for each E C Co(Q). If B := { P ( E ) : E C Co(Q)}, then it is routine to verify that 111311 = 1 and that P : Co(Q) ~ 13 is a B.a. homomorphism of Co(Q) onto 13. To see P is injective suppose that P(E) = 0 for some E C Co(Q). Then, for each fixed n E N, we have P(E)(x{~},O) = (0,0), that is, XE~{n} = 0 in X~. So, ~(s(n) = 0 for each n C N. But, Xs is continuous on Q and N is dense in Q from which it follows that Xs = 0 in C(Q), i.e. E = 0. Hence, P is actually a B.a. isomorphism of Co(Q) onto 13. Since Q is compact and extremely disconnected, it follows that the B.a. Co(Q) is abstractly complete (c.f. Example 13). Hence, its isomorphic image 13 is also abstractly complete (e.f. Exercise 18). Let el C X1 be the function on Q given by el := }-2~=1 ~X{,dl and e2 := XA C X2, in which case e := (e~, e2) C X. Then the cyclic space 13[e] C X is generated by

{P(E)e: EECo(Q)}:{(XseI,XS~A) : E6Co(Q)}. By considering the sets {n} C Co(Q), for each n C N, it follows that {(X{~},0) : n E hence that X , O {0} C_ 13[@ Given any E C Co(Q), the element P(E)e = (Xsel,0) | (0, Xs~ae2) belongs to 13[@ Since also x s e , C X1 (i.e. ( x s e , , 0 ) c 13[@, it follows that (0, Xu~ae2) = (0, Xsoa) E 13[e]. Since E C Co(Q) is arbitrary it follows that {0} 9 X2 _C 13[e]. Hence, B[e] = X and so e is a cyclic vector for 13. To see that 13 is notr-complete, let E~ := Q \ { n } and z := (0, e2), and note that P(E~)z = z for all n C N. The claim is that A~P(E~) = 0s. To establish this it suffices to show (by taking complements and using Exercise 16(b)) that V~P({n}) = I . So, let F c Co(9) be any set such that P({n}) < P(F) for all n E N. Since P is a B.a. isomorphism, it follows that {n} c F for all n E N, i.e. N c_ F. But, F is closed and so also H = Q C F. Hence, F = f~ and so P(F) = I or, equivalently, A,,P(E~) = 0z~. In particular, (A~P(E~))z = 0 and so (A~P(E,,))z # z. Accordingly, there is no smallest projection P in 13 satisfying P z = z. That is, 13 is not T-complete . 9 In contrast to Example 30 we now record some positive facts about r-completeness in relation to Bade completeness ; all of the following facts (and the terminology used below) can be found in or follow immediately from the results in [33]. In particular, (ii) of part (b) of the following theorem is a version of (6).

N} c_ 13[e] and

111 T h e o r e m V I I . 2 . (a) Let X be a Banaeh lattice with a-order continuous norrn (i.e. X is Dedekind a-complete) and B C_ s be the (necessarily bounded) B.a. of all band projections in X . Then B is r-complete. (b) Let X be a Banach space and B c s be a bounded B.a. of projections which is T-complete and has a cyclic vector. Then, (i) B is abstractly a-complete, and (ii) (B}: = (B)2 = AlgLat(B) = B c. (c) Let X be a Banach space and B c s be a bounded B.a. of projections with a cyclic vector. Then the following statements are equivalent. (i) B is Bade a-complete . (ii) B is Bade complete . (iii) B is r-complete and Lat(B) = { B X : B 9 B}. Remark. (1) Part (a) of Theorem VII.2 is false in general if the Banach lattice X is not Dedekind a-complete. Let X := c be the closed subspace of goo consisting of all elements x = (xl, x 2 , . . . ) for which limn~oo xn exists. Then X is a Banach lattice (with respect to the order defined co-ordinatewise) which is not Dedekind a-complete. Let E denote the algebra of subsets of N which are finite or have finite complement and, for each E 9 E, define a projection P(E) 9 s by P ( E ) x = (XE (1)xl, XE (2)X2,...),

X

=

(Xl, X2

. . . .

) 9 X.

Then B := { P ( E ) : E 9 E} is the B.a. of all band projections in X. To see that B is not T-complete, let x = (1, 0, 5, * 0, g, 1 0 , . . . ). If E 9 E satisfies P ( E ) x = x, that is, 1 1 101 (X~(1),0, sXE(3),0, g x A s ) , 0 , . . . ) = (1,0, 5' '5- . .0,. .

)

then we see that n 9 E for all odd integers n from N. In particular, E is then an infinite set and so E c must be finite (as E E E). Hence, there must exist some even integer no 9 E in which case E \ { n 0 } 9 E and so P(E\{no}) 9 B. But, P ( E \ { n o } ) x = x and P(E\{no}) <_ P(E) with P ( E \ { n o } ) ~ P(E). So, there can be no smallest projection P(E) e B satisfying P ( E ) x = x. In particular, B is not T-complete . It may be interesting to note that e := (1, 1 , . . . ) 9 X is a cyclic vector for/3. It is also not difficult to verify that B is a closed subset of s but B is not abstractly a-complete. (2) Let X be a Dedekind a-complete Banach lattice and B c s be the B.a. of all band projections. If B has a cyclic vector, then it follows from parts (a) and (c) of Theorem VII.2 t h a t / 3 is Bade complete if and only if Lat(/3) = { B X : B 9 In particular, if/3 is known not to be Bade complete (or Bade a-complete ), then the inclusion { B X : B 9 C Lat(/3) is necessarily strict. 9 We now return to the main result of this chapter; its proof is an elaboration of that given in [15; Theorem 16, p.2209].

112

C H A P T E R VII. R E F L E X I V I T Y AND B I C O M M U T A N T A L G E B R A S

Theorem VII.3. projections. Then

Let X be a Banach space and 13 c s

be a Bade complete B.a. of

(13)~ = AlgLat(13). Proof. The inclusion (13)$ C_ AlgLat(B) is clear; see Exercise 60(c). For the converse, let f~u be the Stone space of B. By Theorem V.3(b) there is a bieontinuous isomorphism #p : C(f~s) ~ (B}$ given by fdP,

4p(f) := s

f E C(~u),

B

with P : Bo(g~u) ---+ s a regular, closed, or-additive spectral measure satisfying P(Bo(Ftu)) = /3 and P ( E ) = Q(E), for E E Co(f~s), where Q : Co(f~u) ---+ 13 is the Stone map. Thus, each projection B E B determines a characteristic function X~(m, for some clopen set 5(B) E Co([~u) (i.e. X,(s) E C(~qu)), such that S = r = P(d(B)). Since Qu is totally disconnected (actually, extremely disconnected) the sets from Co(~u) form a base for the topology of ~u. Since B is Bade complete , the carrier projections (8)

C~:=A{B~B:

Sx=x},

x~X,

exist in /3; the corresponding clopen set ($(C~) will be denoted simply by 6x. Moreover, by the Fact in the proof of Proposition vI.a(b), we have (9)

z = C:~x,

z E X.

The next claim is (for a fixed x E X) that: (10)

f E C(f~u) with both fxn~\,x = 0 and q)(f)x = 0 implies that f = 0.

To see this, suppose that f r 0 (under the given hypotheses). Since Co(f~B) forms a base for the topology in ~28 and f - l ( C \ { 0 } ) is an open set which is non-empty, there exists B r 0 in B such that f ( w ) r 0 for every w E 5(B). Since 6(B) is clopen it is compact and so 9 := (1/f)'xa(m E C(gZ~). By the fact that ~5 is a homomorphism we deduce that q)(g)~(f) = q~(X,(m) = B. Since fxa~\a= = 0, we have f = IXa= and so r = r ) = ~(I)C~. Hence, multiplying by ~(g) and using ~ ( g ) ~ ( f ) = B gives B = B C , , that is, B _< C~. So, (C~ - B) is a projection in B; see Exercise 30. Now, B z = ~ ' ( g ) ~ ( f ) z = 0 (as ~5(f)x = O) and so (9) yields that (C~ - B)oc = z. By (8) we then have that Cx _< (C~ - B). Hence, C~(C~ - B) = C~ which yields C~ - C~B = C,, that is, C~B = 0. But, we saw above that B = B C , ( = C~B) and so we conclude that B = 0 which is a contradiction to the choice of B. This establishes (10). Let T E AlgLat(B). If B E /3, then the range B X E Lat(B) and the range (I - B ) X E Lat(/3) and so T B X C_ B X and T ( I - B ) X C_ (I - B ) X . Hence, if x E X, then T B x E B X

113 and so B ( T B x ) = T B x . Also T ( I - B ) x E ( I - - B ) X and so B [ T ( I - B)x] = 0 (as B ( I - B) = 0). Since x E X is arbitrary, we deduce that B T B = T B and B T ( I - B) = O, from which it follows that

TB = BTB = BTB + BT(I-

B) = BT.

So, whenever T E AlgLat(B) we have established that

T B = BT,

(11)

13 E B.

Fix now T E AlgLat(B). By Exercise 61(b) we have that TB[x] C_ B[x], for each x E X. Fix x E X. Then the restriction TI~[~] of T to the cyclic space B[x] commutes with the restricted spectral measure Ps[~] : E ~ P(E)lu[x]; see (11). An examination of the proof of Proposition VII.2 (applied in the space B[x] to PBN(E) C_ Z:(B[x])) shows that TIB[~1 = fa g~ dPB[xI for some g~ E L~(P~[~]). Choose an everywhere defined, bounded Bo(f~B)-measurable function hx such that sup{Ih~(w)l : w E f~B} = Ig~lP~i~l and h~ = g~, P~[~]-a.e., in which case TIB[~] = [ h~ dPB[~I. Jn

Then Theorem V.1 shows that

sup Ih~(~)q = Ig~]P~M ~ IITIBMII ~ IITII.

wEf~B

If we define Tx E s

by T~ := C ~ ( s

(12)

hx dP),

x E X,

B

then clearly Tz E (B)~, as Cx E /3 = P ( B o ( f ~ ) ) and h~ E L~~ Since C,:x = x and (f~s h,: d P ) x = [fnB h,: dP~[xl]x = (Tl~[x])x = T x we also have that Txx = x. It is immediate from (12) that CxT~ = T~C,: = T~. Moreover, (13)

h~x~ dPll <_ 41IBll. Ih~x~xIp <_ 4IIBll. Ihxlp

ItT~II = I I f d~2 B

<

411NI sup [h~(w)l < 4111311. IITll. wEf~B

Since T~ E (B); there is a unique function f~ E C(fIB) such that T~ = ~5(f~). Then q~(f~)x = T,:x = x. Moreover, (I - C~)q~(f~) = (I - C,~)T~ = 0 as C~T~ = T,~. However, we also have that Since d) is injective, it follows that Xa~\~=f= = 0. Hence, the function f~ has the properties (14)

f~ E C(t2B) with both f=xa~\& = 0 and qh(f~)x = T z .

114

CHAPTER

VII.

REFLEXIVITY

AND BICOMMUTANT

ALGEBRAS

Moreover, if g is another function with the properties (14), then it follows from (10) that g = f~ in C ( f ~ ) . Hence, the properties (14) characterize fxThe next claim is, for each x e X, that

(15)

dB~ = 5(B) n 5~,

B E B.

To see this fix x E X and B E B. Let z := B z and note that C'~Bz = C x B x = B C ~ x = B x = z, so that CB~ <_ C x B . However, if CBx < C~B, then the non-zero projection G C B defined by G := ( C x B - C B x ) satisfies G < C ~ B and GCB~ = 0. Moreover, G B x = G C B x B x = 0 and G ( I - B ) = 0 (since G < C x B implies that G C ~ B = G and so G ( I - B ) = G C x B ( I - B ) = 0). Thus Gx = GBx + G(I - B)x = 0 + 0 = 0 and so (I - G ) x = x. By (8) we see that C~ < (I - G), i.e. G _< (I - C~). Hence, G <_ C z B and G < (I - C~) and so C < ( C ~ B ) A ( I - Cx) = C ~ B ( I - C~) = O,

that is, G = 0 which is a contradiction. So, we must have CBx = C ~ B . T h a t is, ~5(Xe~~) = (I)0l,,)q)(X,(m) = ~P(X,.~(m ). Then the injectivity of (I) establishes (15). For each z E X, we now establish the identity (16)

fs~ = f~x,(~),

B c B.

Let g := AX,( m. If w E f ~ s \ 5 ( B ) , then clearly g(w) = 0 (as X,(m(w) = 0) and if w E Q~\6~, then f ~ ( w ) = 0 by (14) and so again g ( w ) = 0. Accordingly, 9X~\{~,~(B)) = 0 and so by (15) we also have that gXa~\~Bx = 0. Moreover, by (11) and (14) we see that ( P ( g ) B x = ( P ( I ~ x e { m ) B z = e2(L)~5(Xa(m)Bx = r

= B~(I~)z

= BTx = TBx.

So, with f~ replaced by g and z replaced by B z in (14) we see from the uniqueness statement after (14) that g = fB~. This establishes (16). The next step is to verify the following property: (17) For each x, y E X we have that f , and fy coincide on 6x M 6y. The first observation is to note that we may assume 5~ = 6y. For, if z := C y x and w := C~y , then putting B = Cy into (15) yields 5~ = @ A 6~ and putting B = C~ and replacing x by y in (15) yields 5~ = 5, (-1 5~. Hence, 5, = 5~ C~5~ = f~. Similar substitutions into (16) show that f~ = f~xe~ and f~ = fyxe~. Thus, if f z and f~ coincide on 5z = 6~, then (17) will hold. So, we may assume that 5~ = 5y. Since T ( x - y) = T x - T y it follows from (14) that d2(f~ y ) ( x - y) = T ( x - y) = T x - T y = e ) ( f , ) x - cp(fy)y

which, upon rearrangement, yields (18)

e2(f~_y - f,~)x = (P(f~_y - f y ) y .

115 To complete the proof of (17) we proceed by contradiction. So, suppose that fx # fy on 5~:(= by). Choose a point )~0 E 5:: such that fx(~0) ~ fv(~o). Then also f~-v(~o) - fx(/~0) f : : - ~ ( ~ o ) - fA~0) and hence, at least one of L-~(~0) - f::(~o) or f x - , ( ~ o ) - f~(~o) is non-zero. So, one of the functions fx or fy, say f::, differs from both fy and f~_~ at some point Ao E 5x. Since C o ( f ~ ) forms a base for the topology of f~s and {P(5(B)) : B E B} =/3, there exists some element B C B and E > 0 such that 6(B) C 6~ = 5> with t0 E 5(B), and ]f:~_y(w) - fx(w)] > g, Since 5(B) 9 Co(f~B), the function g := Xar h := g(f~-v - fv)" Then it follows from (18) that

w 9 5(B). - f,:) belongs to C(f~8). Let

9 (h)y = (p(g)~a(f~_y - fv)Y = ah(g)(p(f~_v - fi:)x = (P(g(f~_y - f~))x = O~(X,(B))x = Bx, and so by (14) (i.e. ~,(fi:)x = T x ) and (11) (i.e. r 9 (f~)B~

It follows that

-= T#P(h)) we have that

=

Bq~(fx)X = BTx

=

~(h)~2(fy)y = ~(fy)ah(h)y = ~ ( f y ) B x .

= TB;

= T|

(P(fvx,(m)Bx - rP(fy)~(X,(m)Bx - ~ ( f y ) B B x

= ~(h)Ty

= (P(fv)Bx = T B x ,

and that fv (w)x , r ) = 0 for all w r 5y N 5(B) = 5:: N 5(B) = 5B:: (using (15)). Since the equations (14) determine f~ uniquely, we have that fyxa(m = fB~. On the other hand, (16) shows that also fs~ = f~x,(m and hence f~ = fy on 5(B), contradicting the fact that Ao 9 5(B) and f~(Ao) # fy(Ao). This contradiction yields (17). Because of (17) the function

~, f~(w), f O,

r

w 9 5~ w r U~exS~,

is well defined and continuous on the open set UxexS~. Since

IIf~ll~ = II'~-~(T~)II <- II~-lll " IIT~II <- 411~'-111 9 IIBII " IITII,

x ~ X,

(see (13)) we see that r is a bounded Borel function. Define To := s r dP.

(19) Then, for each x 9 X, we have

Tox

=

ToC,:x = T o P ( 5 x ) x = /a C d P x =

=

e(5~)q~(fx),

= q,(f~)P(5~),

fa f ~ d P x =

= ~(f~)c~

P(5~) s

= e(f~)~ = r~.

Accordingly, T = To and it follows from (19) that To 9 (P (B o(f ~u)}: = (13):.

9

C H A P T E R VII. R E F L E X I V I T Y A N D B I C O M M U T A N T A L G E B R A S

t16

The following consequence is usually referred to as the Bade reflexivity theorem. C o r o l l a r y V I I . 3 . 1 . Let X be a Banach space and B c_ s be a Bade a-complete B.a. of projections. Then (/3); = (/3}; = AlgLat(B). P r o o f . Let ~ be the closure of/3 in s Then ~ is a Bade complete B.a. (c.f. Theorem V.8) and so, by Corollary V.6.3, we have that (B}~ = {B}~- = (B}~. Combining this observation with (B>~ = AlgLat(B) (c.f. Theorem VII.3) and the identities (B}.7 = (B)~(c.f. Exercise 52(c)) and AlgLat(B) = AlgLat(B) (which follows from Exercise 61(a)) yields the desired conclusion. 9 E x e r c i s e 64. It was shown in the proof of Theorem VII.3 that AlgLat(/~) C_ B~; an examination of that part of the proof shows this is the case for any B.a. of projections I3 C_ s Give an example to show that, in general, this inclusion is strict. 9 E x e r c i s e 65. Let X be a Banach space and g C_ s be a Bade ~-complete B.a. of projections with the property that, for each x E X, there exists a projection P r B c such that P X = B[x]. Show that (13}2 = 13co. 9 A single operator T E s is called reflezive if the closed algebra in s generated by {I, T} is reflexive. Of course, this closed algebra consists of the closure, in s of {p(T) : p a complex polynomial}. D. Sarason showed in [39] that every bounded normal operator in a Hilbert space is reflexive. T.A. Gillespie extended this result to the Banach space setting by showing that every scalartype spectral operator is a reflexive operator; see [21]. We end this chapter with an application of the Bade reflexivity theorem to deduce a classical result of J. von Neumann , usually referred to as the bicommutant theorem. T h e o r e m V I I . 4 . Let X be a separable Hilbert space and T E s be a selfadjoint operator with resolution of the identity P r : Bo(R) ~ s so that T = f~)~dPr(A) = f~(r) A dPT(A). Then the following seven algebras are the same. (a) { f e r

: r

1R ~

C a bounded Borel function}.

(b) The closed subalgebra of s (c) The bicommutant {T} co, of T. (d) (PT(BO(N)))~.

(e) (PT(Bo(X))): = (P~(Bo(R)))g. (f) AlgLat(PT(Bo(R))). (g) AlgLat({T}).

generated by {I, T}.

117 P r o o f . Let `41,... , `47 denote the algebras defined by (a), . . . , (g), respectively. T h a t A2 = A7 is precisely the result of D. Sarason referred to above. Let B := PT(BO(R)). Then the separability of X implies that B is Bade complete (c.f. Proposition VI.3) and hence, .44 = .45 by Corollary V.6.3. The equality .44 = .46 is precisely the Bade reflexivity theorem (see Corollary VII.3.1). Theorem V.1 and an examination of the proof of Theorem V.2 show that .41 = .44. Since T = f~ r dPT, where r := tX~(r)(t), for t C R, is an element of L ~176 (PT) ~ L I(PT), and PT is a closed spectral measure, it follows from Theorem V.6 that T E .As. Since As is closed in Z:~(X) and is a subalgebra, it follows that `42 C_ As. To prove that `45 C `42 it suffices to show that PT((5) E A2, for all ~ c Bo(R) . Since `42 is an algebra of operators, the family E of all Borel sets ~ C N for which PT(~) E A2 is an algebra of sets. Let {6n}~=1 C_ E be a monotone sequence with limit 6. By the countable additivity of PT in the weak operator topology it follows that P(~n) -----+P(6) in s . Since {P(o~,)}~=l C_ `42 and `42 is closed in Z;~(X) we see that P(~) E .4e. Accordingly, ~ ~ E. So, E is a a-algebra o/sets. Let [a,b] be a bounded closed interval with a(T) c_ (a, b). Let [u, v] c_ (a, b) and r be the continuous, piecewise affine function such that r = 11 on [u, v] and r = 0 on ( - c o , u - e] U Iv + e, oc), where e > 0 satisfies a < ( u - e ) and ( v + e ) < b. Then there exists a sequence of polynomials {%}n~=1 such that % ----+ r in C([a, hi). By the dominated convergence theorem for PT (see Corollary V.2.2), after noting that supp(PT) = a ( T ) C In, b] and

IqnlPr <- supte[a,bll%(t)l <- M,

n C N,

for some constant M > 0, we have that q~(T) = ~ (T) qndPT

= ~,b] q~dPr ----~ /[a,b]r

:= r

n--+ oo,

where the convergence is in f;~(X). Hence, r C .42 for every such e. Let e~ ; 0, in which case Cs, ~ X[~..I pointwise on [a, b], and use the dominated convergence theorem again yields lim C e , ( T ) = lim .

.

.

.

.

.

f J[a,b]

r162

lim n~z

f X[~IdPT=PT([U,V]), j[a,bl '

where the convergence is again in s Hence, PT([u, v]) C .4~ and so [u, v] 9 E. We note that u = v is allowed and so also {u} E E, for each u E (a, b). Since E is a a-algebra it follows that E contains all subintervals of [a, b]. But, PT(R\[a, b]) -- 0 and so it follows, for an arbitrary interval J C_ R, that

PT(J) = PT(J\[a, b]) + PT(J (-1[a, hi) = PT(J N [a, b]) C .42. Hence, E contains all intervals from R and so E = Bo(R) . This establishes that A2 = .45. It remains to show that .42 = .43. The inclusion .42 C .43 is clear. For the converse, let Y E Lat(B) and Q be the orthogonal projection of X onto Y. Fix x C X, in which case

118

C H A P T E R VII. R E F L E X I V I T Y A N D B I C O M M U T A N T A L G E B R A S

Q z E Y. I f B c B, t h e n B Y C_ Y and so B Q z C Y. Hence, Q ( B Q z ) = B Q z . S i n c e z i s arbitrary it follows that Q B Q = BQ. Take adjoints of this identity gives Q B Q = Q B (using B* = B and Q* = Q). So, we see that BQ = QB for all B C B, that is,

(20)

PT(5)Q = QP~(5),

~ ~ Bo(•).

Since T = a dP~(a) is the limit (in s of operators of the form f~ s dPT, where s is a Borel-simple function, it is clear from (20) that T Q = QT. Given R E A3 it follows that RQ = QR. Since Q is the orthogonal projection onto Y it follows that R Y C_ Y. So, we have shown that every operator from A3 belongs to AlgLat(B). That is, A3 _c A6 (= A2 as already shown) and so A3 _C A2. Hence, .A2 = A3. 9 If X is a separable Banach space and we use the fact that A2 = A7 (by the paper of T.A. Gillespie mentioned above in place of D. Sarason's paper), then an examination of the proof of Theorem VII.4 shows, for T a scalar-type spectral operator from s with a(T) C_ R and resolution of the identity PT, that

(21)

AI = A2 = A4 = A5 = A6 = A7 C A3.

We remark that the last inclusion may be strict, even in a separable, reflexive Banach space. An example can be found in an elegant paper of J. Dieudonn~, [9] . For some positive results, where the inclusion in (21) is actually an equality, we refer to [32]. In particular, if the range of PT is atomic, then this is the case. Hence, for any compact scalar-type spectral operator T satisfying or(T) _CN the inclusion in (21) is an equality. More generally, if X is an arbitrary Banach space and B C_ s is any Bade or-complete B.a. of projections (not necessarily the resolution of the identity of some scalar-type spectral operator) one may ask when is it the case that (22)

(B); = / 3 ~

?

It is shown in [32; Proposition 1] that (22) holds whenever B is atomic. This is of some interest since there exist classes of Banach spaces X in which every Bade a-complete B.a. of projections is automatically atomic. For instance, if X is a Grothendieck space with the Dunford-Pettis property, or a hereditarily indecomposable space, or a space with the Schur property, or a complemented subspace of an s176176 then this is the case; see [32; Proposition 2]. We note that there also exist other classes of Banach spaces X (besides Hilbert spaces) for which (22) holds for all Bade a-complete B.a. 's of projections 13 C_ s and such that non-atomic B.a. 's B exist in s For example, this is known to be the case if X is any complemented subspace of an s [30; Corollary 8], since any Bade complete B.a. of projections B in such a space X necessarily satisfies the hypotheses of Exercise 64; see [30; Theorem 7]. For the definition, examples and theory of s we refer to [4] and the references there in. We conclude with the remark that (22) also holds in any Banach space X for any bounded B.a. of projections B C_ s which is T-complete and has a cyclic vector, even in the stronger form

119

This is an easy consequence of Theorem VII.2(b). As noted before, every Bade complete B.a. of projections is ~--complete.

Bibliography [1] Bade, W.G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345-360. [2] Bade, W.G., A multiplicity theory for Boolean algebras of projections in Banach spaces, Trans. Amer. Math. Soc. 92 (1959), 508-530. [3] Bessaga, C. and Petczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. [4] Bourgain, J., New Classes ofgP-spaces, Lecture Notes in Mathematics No. 889, SpringerVerlag, Berlin-Heidelberg-New York, 1981. [5] Curbera, G.P., Operators into L 1 of a vector measure and applications to Banach lattices, Math. Ann. 292 (1992), 317-330. [6] Curbera, G.P., When L 1 of a vector measure is an AL-space, Pacific J. Math. 162 (1994), 287-303. [7] Curbera, G.P., Banach space properties of L I of a vector measure, Proc. Amer. Math. Soe. 123 (1995), 3797-3806. [8] Diestel, J. and Uhl, J.J. Jr., Vector Measures, Amer. Math. Soc., Providence, 1977. [9] Dieudonn~, J., Sur la bicommutante (1955), 35-38.

d'une alg~bre d'op~rateurs, Portugaliae Math. 14

[10] Dixmier, J., Von Neumann Algebras, North Holland, Amsterdam, 1981. [11] Dodds, P.G. and Ricker, W.J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), 136-163. [12] Dodds, P.G., de Pagter, B. and Ricker, W.J., Reflexivity and order properties of scalartype spectral operators in locally convex spaces, Trans. Amer. Math. Soc. 293 (1986), 355-380. [13] Dowson, H.R., Spectral Theory of Linear Operators, Academic Press, London, 1978. [14] Dunford, N. and Schwartz, J.T., Linear Operators I: General Theory, Wiley Interseience Publ. (2nd Printing), New York, 1964.

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BIBLIOGRAPHY

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and s

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Appendix The following collection of references is (hopefully) a comprehensive list of all papers on the general topic of spectral operators and Boolean algebras of projections which have appeared since 1979 or are not recorded in the monograph Linear Operators III: Spectral Operators (by N. Dunford and J.T. Schwartz, Wiley-Interscience, New York, 1971) or in the monograph Spectral Theory of Linear Operators (by H.R. Dowson, Academic Press, London, 1978). Any omissions are unintentional.

Abramovich, Y.A., Arenson, E.L., and Kitover, A.K. 1. Operators in Banach C(K)-modules and their spectral properties. (Russian) Dokl. Akad. Nauk SSSR 301 (1988), no.3, 525-528; translation in Soviet Math. Dokl. 38 (1989), 93-97. MR90a:47091. 2. Banach C(K)-modules and operators preserving disjointness, Pitman Research Notes in Mathematics, 277, Longman Scientific and Technical, Harlow, 1992. MR94d:47027. Akhmedov, A.M. 1. Perturbations of compact spectral operators. (Russian) Izv. Akad. Nauk Azerba{dzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk 5 (1984), 22-27. MR86e:47041. 2. A perturbation of compact spectral operators. (Russian) Akad. Nauk Azerba{dzhan. SSR Dok. 40 (1984), 10-13. MR86g:47043. 3. Spectrality of perturbations of compact operators. (Russian) Linear operators and their applications (Russian), 3-14, Azerba{dzhan. Gos. Univ., Baku, 1986. MR90j:47039. Albrecht, E. 1. On some classes of generalized spectral operators. Arch. Math. (Basel) 30 (1978), 297-303. MR57~10486.

Glasgow Math.

J. 23

1. Local spectral properties of constant coefficient differential operators in D~ Operator Theory 24 (1990), 85-103. MR92b:47048.

J.

2. A characterization of spectral operators on Hilbert spaces. (1982), 91-95. MR83a:47040. Albrecht, E., and Ricker, W.J.

126 2. Local spectral properties of certain matrix differential operators in D~ ator Theory 35 (1996), 3-37. MR98b:47060.

APPENDIX J. Oper-

A1-Khezi, S. 1. Analytic functions of a prespeetral operator. Glasgow Math. J. 23 (1982), 171-175. MR83h:47022. A1-Khezi, S., and Dowson, H.R. 1. Quasispectral operators, Proc. Roy. IrishAead. Sect. A 81 (1981), 25-28. MR82m:47026. Amrein, W. (see Jauch, J.M.) Anderson, J., and Foia~, C. 1. Properties which normal operators share with normal derivations and related operators. Pacific Y. Math. 61 (1975), 313-325. MR54#1010. Andruchow, E., Recht, L., and Stojanoff, D. 1. The space of spectral measures is a homogeneous reductive space. Integral Equations Operator Theory 16 (1993), 1-14. MR93j:46078. Applebaum, D. 1. Spectral families of quantum stochastic integrals. Comm. Math. Phys. 170 (1995), 607-628. MR96f:81061. Apostol, C. 1. Invariant subspaces for subquasiscalar operators, J. Operator Theory. 3 (1980), 159164. MR83a:47005a. 2. The spectral flavour of Scott Brown's techniques, J. Operator Theory. 6 (1981), 3- 12. MR83a:47005b. Arenson, E.L. (see Abramovich, Y.A.) Azoff, E.A. 1. A note on direct integrals of spectral operations. Michigan Math. Y. 23 (1976), 6569. MR55#13280. Azzouni, A. 1. Opdrateurs spectraux g points critiques. (French) [Spectral operators with critical points] Bull. Math. Soc. Sci. Math. Roumania (N.S.) 35-(83) (1991), 3-14. MR95k:47051. Bacalu, I.

127 1. Residual spectral measures. MR53#3777.

(Romanian) Stud.

Cerc. Math.

27 (1975), 377-379.

2. Restrictions and quotients of spectral systems. (Romanian) Stud. (1980), 113-118. MR81h:47031.

Cerc. Mat. 32

Bai, F.D. (see Li, Y.S.) Battle, R.G. 1. Selfadjoint operators and some generalizations. Operator theory and functional analysis (Papers, Summer Meeting, Amer. Math. Soc., Providence, R.I., 1978), pp36-50, Res. Notes in Math., 38, Pitman, Boston, Mass.-London, 1979. MR81f:47029. Baskakov, A.G. 1. Methods of abstract harmonic analysis in the theory of perturbations of linear operators (Russian), Sibirsk. Mat. Zh. 24 (1983), no.l, 21-39, 191. MR85j:47010. 2. The method of similar operators and formulas for regularized traces. (Russian), [zv. Vyssh. Uchebn. Zaved. Mat. 1984, no.3, 3-12. MR86a:47009. 3. Regularized trace formulas for powers of perturbed spectral operators. (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1985, 68-71, 86. MR87i:47017. 4. Spectral analysis with respect to finite-dimensional perturbations of spectral operators. (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1991, 3-11. MR92m:47023. 5. Spectral analysis of perturbed non-quasi-analytic and spectral operators. (Russian) Izv. Ross. Akad. Nauk Set. Mat. 58 (1994), 3-32; translation in Russian Acad. Sci. Izv. Math. 45 (1995), 1-31. MR96d:47004. Bauer, G., and Mennicken, R. i. St6rungstheorie fiir diskrete Spektraloperatoren und Anwendungen auf im Parameter nichtlineare Eigenwertprobleme. (German) [Perturbation theory for discrete spectral operators and applications to eigenvalue problems that are nonlinear in the parameter] Regensburger Mathematische Schriften [Regensburg Mathematical Publications], 10. Universitdt Regensburg, Fachbereich Mathematik, Regensburg, 1985. vii+104pp. MR87k:47028. Berezanski, Yn. M. 1. The projection spectral theorem, Uspekhi Mat. Nauk 39:4(1984), 3-52; Russian Math. Surveys 39:4(1984), 1-62. MR86e:47029. 2. Projection spectral theorem and its applications to the infinite-dimensional harmonic analysis. Gaussian random fields (Nagoya, 1990), 114-128, Set. Probab. Statist., 1, World Sei. Publishing, River Edge, N J, 1991. MR93b:47032. Berezanski, Yu. M., Zhernakov, N.W., and Us, G.F.

128

APPENDIX

1. A spectral approach to quantum stochastic integrals. Rap. Math. Phys. 28 (1989), 347-360. MR93:47142. Bernstein, A.R. 1. The spectral theorem-a nonstandard approach. Z. Math. Logik Grundlagen Math. 18 (1972), 419-434. MR47~4048. Birman, M. Sh., and Solomyak, M.Z. 1. Operator integration, perturbations and commutators. (Russian) Zap. Nauchn. Sam. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), Issled. Linein. Oper. Teorii g~nktsii. 17, 34-66, 321. MR91b:47086. 2. Tensor product of a finite number of spectral measures is always a spectral measure. Integral Equations Operator Theory 24 (1996), 179-187. MR96m:47038. Birman, M. Sh., Vershik, A.M., and Solomyak, M.Z. 1. The product of commuting spectral measures may fail to be countably additive. (Russian) Funktsional. Anal. i Prilozhen. 13 (1979), 61-62. MR80h:28010. Blashchak, V.A. 1. On a differential operator of the second order on the whole axis with spectral singularities, Dopovidi Akad. Nauk Ukrain. RSR (1966), 38-41 (Ukrainian). Burnap, C., and Zweifel, P.F. 1. A note on the spectral theorem. Integral Equations Operator Theory 9 (1986), 305324. MR87h:47078. Byrne, C.M. 1. Banach function spaces and spectral measures, PhD Thesis, University of Edinburgh, 1982. Byrne, C.M., and Gillespie, T.A. 1. The representation of spectral measures. Proc. Roy. Irish Acad. Sect. A 85 (1985), 31-42. MR87e:47040. Cao, X.D. 1. Some results on spectral systems of commuting operators. (Chinese) Acta Sci. Natur. Univ. Jilin. 1993, 27-33. MR96c:47045. Chabauty, R. 1. Op~rateurs semi-simples. (French) [Semisimple operators] Nonlinear analysis, 1982/1983, Exp. No.10, 40pp., Publ. Math. Fac. Sci. Besan~on, 7, Univ. Franche-Comtg, Besangon, 1983. MR86h:47025.

129 Cheng, Qingpeng I. Well-bounded operators on general Banach spaces. PhD 1999.

Thesis, Murdoch

University,

Chourasia, N.N. i. Decomposable MR82g:47025.

and spectral operators. Indian Y. Pure Appl. Math. 12 (1981), 73- 80.

2. On Weyl's theorem for spectral operators and essential spectra of direct sum. Pure Appl. Math. Sci. 15 (1982), 39-45. MR84b:47039. Cleaver, C.E. I. A characterization of spectral operators of finite type. Compositio Math. 26 (1973), 95-99. MR47#9336. Conway, J.B., and Gillespie, T.A. i. Is a selfadjoint operator determined by its invariant subspace lattice? J. Funct. Anal. 64 (1985), 178-189. MR87h:47041. Curtain, R.F. I. Spectral systems. Internat. J. Control 39 (1984), 657-666. MR85h:93039. Cu~o, G.I. (see Ljance, V.I~.) Davidson, K.R. I. Essentially spectral operators. MR84i:47048.

Proc.

London

Math.

Soc.

(3) 46 (1983), 547-560.

Dayanithy, K. I. Interpolation of spectral operators. Math. Z. 159 (1978), i-2. MR80a:47048. Delanghe, R., and Van hamme, J. 1. Generalised spectral measures. Proc. Roy. Irish Acad. Sect. A 87 (1987), 17-26. MR89b:47050. Dinescu, G. 1. On semiscalar operators. Rev. Roumaine Math. Pures Appl. 28 (1983), 359-380. MR85k:47055. Dodds, P.G. 1. Boolean algebras of projections in locally convex spaces. Miniconference on operator theory and partial differential equations (Canberra, 1983), 67-76, Proc. Centre Math. Anal. Austral. Nat. Univ., 5, Austral. Nat. Univ., Canberra, 1984. MR86a:47044.

130

APPENDIX

2. Scalar type spectral opeators in locally convex spaces. Miniconference on linear analys~s and function spaces (Canberra, 198~), 185-193, Proc. Centre Math. Anal. Austral. Nat. Univ., 9, Austral. Nat. Univ., Canberra, 1985. MR825524. Dodds, P.G., and de Pagter, B. 1. Orthomorphisms and Boolean algebras of projections, Math. Z. 187 (1984), 361- 381. MR86a:47045. 2. Algebras of unbounded scalar-type spectral operators. Pacific J. Math. 130 (1987), 41-74. MR89e:47070. Dodds, P.G., and Ricker, W.J. 1. Spectral measures and the Bade reflexivity theorem. 136-163. MR86i:47042.

J. Funct.

Anal.

61 (1985),

Dodds, P.G., de Pagter, B., and Ricker, W.J. 1. Reflexivity and order properties of scalar-type spectral operators in locally convex spaces. Trans. Amcr. Math. Soc. 293 (1986), 355-380. MR87d:47046. Doust, I. 1. Contractive projections on Banach spaces. Miniconference on Fucntional Analysis Optimization, Proc. Centre Math. Anal., ANU, 20 (1988), 50-58. MR90i:46020. 2. Well-bounded and scalar-type spectral operators on spaces not containing co. Proc. Amer. Math. Soc. 105 (1989), 367-370. MR89f:47048. 3. Well-bounded and scalar-type spectral operators on LP-spaces. J. London Math. Soc. (2) 39 (1989), 525-534. MR90f:47043. 4. An example in the theory of spectral and well-bounded operators. Miniconference on Operators in Analysis (Sydney, 1989), 83-90, Proc. Centre Math. Anal. Austral. Nat. Univ., 24, Austral. Nat. Univ., Canberra, 1990. MR91k:47071. 5. Interpolation and extrapolation of well-bounded operators. (1992), 229-250. MR95i:47064.

J. Operator Theory 28

6. A weaker condition for normality. Glasgow Math. g. 36 (1994), 249-253. MR95f:47041. Doust, I., and de Laubenfels, R. 1. Functional calculus, integral representations, and Banach space geometry. Quaestiones Math. 17 (1994), 161-171. MR95e:47028. Doust, I., and Ricker, W.J. 1. Spectral properties for Hermitian operators. Linear Algebra Appl. 175 (1992), 75- 96. MR93g:47024. Dowson, H.R. (see also A1-Khezi, S.)

131

1. Spectral theory of linear operators. London Mathematical Society Monographs, 12. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. xii+422 pp.ISBN:0-12-220950-8. MR80c:47022. 2. Restrictions of scalar-type spectral operators. Bull. London Math. Soc. 10 (1978), 305-309. MR80a:47050. 3. Solution to a problem of Lior Tzafriri. Bull. London Math. Soc. 23 (1991), 285- 292. MR92j :47083. Dowson, H.R., and Gillespie, T.A. 1. A representation theorem for a complete Boolean algebra of projections. Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 225-237. MRS0m:47039. Drewnowski, L., Florencio, M., and Patil, P. 1. The space of Pettis integrable functions is barrelled. Proc. Amer. Math. Soc. 114 (1992), 687-694. MR92f:46045. 2. Uniform boundedness of operators and barrelledness in spaces with Boolean algebras of projections, Atti. Sere. Mat. Fis. Univ. Modena 41 (1993), 317-329. MR94m:46004. 3. Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections, Glasgow Math. J. 36 (1994), 57-69. MR94m:46005. Dubrovskii, V.V. and SadovniSii, V.A. 1. Unbounded perturbations of spectral operators. (Russian) Problems in mathematical physics and numerical mathematics (Russian), pp. 137-144, 325. Nauka, Moscow, 1977. MR58~30463. Duncan, R. 1. Weak convergence of spectral mesaures, Ann. Univ. Mariae Curie-Sklodowska Sect. A 51 (1997), 35-39. MR99a:47004. Dudley, R.M. 1. A note on products of spectral measures. Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972), pp. 125-126. Academic Press, New York, 1973. MR49#1193. Dunford, N. 1. An expansion theorem. Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. 61-73, Contemp. Math., 2, Amer. Math. Soc., Providence, R.I., 1980. MR82i:47048. 2. Spectral theory in topological vector spaces. Functions, series, operators, Vol.I, H (Budapest, 1980), 391-422, Colloq. Math. Soc. J~nos Bolyai, 35, North-Holland, Amsterdam-New York, 1983. MR86a:47029.

132

APPENDIX

Emamirad, H. (see de Laubenfels, R.) Erdos, J.A. 1. On Boolean algebras of projections. Glasgow Math. J. 18 (1977), 69-72. MR55~1131. Farwig, R., and Marschall, E. 1. On the type of spectral operators and the nonspectrality of several differential operators on Lp. Integral Equations Operator Theory 4 (I981), 206-214. MR82b:47037. Faulkner, G.D., and Huneycutt, J.E., Jr. 1. The canonical form of a scalar operator on a Banach space. Proc. Amer. Math. Soc. 71 (1978), 81-84. MR58#7196. Fialkow, L. 1. A note on quasisimiiarity. II. Pacific J. Math. 70 (1977), 151-162. MR57~17341. Fixman, U., and Tzafriri, L. 1. The full algebra generated by a spectral operator. J. London Math. Soc. (2) 4 (1971), 39-45. MR45#943. Fleming, R.J., and Jamison, J.E. 1. Classes of operators on vector valued integration spaces. J. Austral. Math. Soc. Set. A 24 (1977), 129-138. MR585r Florencio, M. (see Drewnowski, L.) Foals, C. (see Anderson, J.) Folland, G.B. 1. Spectral analysis of a singular non-selfadjoint boundary value problem. J. Differential Equations. 37 (1980), 206-224. MR81m:34022. 2. Spectral analysis of a non-selfadjoint differential operator, J. Differential Equations. 39 (1981), 151-185. MR82m:34021. Fong, C.K., and Lam, L. 1. On spectral theory and convexity. MR82c:46061.

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APPENDIX

146

19. A concrete realization of the dual space of Ll-spaces of certain vector and operatorvalued measures. J. Austral. Math. Soc. Set. A 42 (1987), 265-279. MR88m:46037. 20. Functional calculi for the Laplace operator in LP(R). Miniconference on harmonic analysis and operator alyebras (CanberT~a, 1987), 242-254, Proc. Centre Math. Anal. Austral. Nat. Univ., 15, Austral. Nat. Univ., Canberra, 1987. MR90b:47086. 21. Spectral projections and the commutant of the Laplace operator in LP(R). Rend. Sere. Mat. ['is. Milano 57 (1987), 519-528 (1989). MRY0j:47034. 22. "Spectral subsets" of R TM associated with commuting families of linear operators. Proceedinys of the analysis conference, Sinyapore 1986, 243-247, North-Holland Math. Stud., 150, North-Holland, Amsterdam-New York, 1988. MR89e:47007. 23. Vector measures with values in the compact operators, Proc. Amer. Math. Soc. 102 (1988), 441-442. MR90f:28007. 24. Spectral properties of the Laplace operator in LP(R), Osaka J. Math. 25 (1988), 399410. MR89h:47071. 25. Bade's theorem on the uniformly closed algebra generated by a Boolean algebra, Comment. Math. Univ. Carolin. 29 (1988), 597-600. MR90a:47112. 26. An Ll-type functional calculus for the Laplace operator in LP(R). J. Operator Theory 21 (1989), 41-67. MR91b:47109. 27. Operator algebras generated by Boolean algebras of projections in Montel spaces. Inegral Equations Operator Theory, 12 (1989), 143-145. MR90a:47113. 28. Analogues of von Neumann's bicommutant theorem. Semesterbericht Funktionalanalysis Tiibingen, Wintersemester, 15 (1989), 213-221. 29. Spectral measures and integration: counterexamples. Semesterbericht Funktionalanalysis Tiibingen, Sommersemester, 16 (1989), 123-129. 30. Boolean algebras of projections of uniform multiplicity one. Miniconference on Operators in Analysis (Sydney, 1989), 206-212, Proc. Centre Math. Anal. Austral. Nat. Univ., 24 Austral. Nat. Univ., Canberra, 1990. MR91d:47031. 31. Boolean algebras of projections and spectral measures in dual spaces. Linear operators in function spaces (Timi~oara, 1988), 289-300, Oper. Theory Adv. Appl., 43, BirkhSuser, Basel, 1990. MR92e:47059. 32. Scalar-type spectral operators and C(t2)-operational calculi. Integral Equations Operator Theory 13 (1990), 901-905. MR92k:47069. 33. Non-spectrality of generators of some classical analytic semigroups. (N.S.) 1 (1990), 95-103. MR91b:47089.

Indag. Math.

Hokkaido Math.

J. 20 (1991),

34. Kakutani's example on product spectral measures. 593-600. MR92h:47049.

147 35. Spectral-like multipliers in L'(R). Arch. MR92j :42009.

Math.

(Basel) 57 (1991), 395- 401.

36. Integral transforms of vector measures on semigroups with applications to spectral operators. Semigroup Forum 45 (1992), 342-363. MR93h:28017. 37. Weak compactness of the integration map associated with a spectral measure. Indag. Math. (N.S.) 5 (1994), 353-364. MR95j:47026. 38. Well-bounded operators of type (B) in H.I. Spaces. Acta Sci. Math. (1994), 475-488. MR96j:47027.

(Szeged) 59

39. Weak compactness of integration maps associated with indefinite integrals of spectral measures. Indag. Math. (N.S.) 6 (1995), 495-503. MR97e:47047. 40. Spectrality for matrices of Fourier multiplier operators acting in //-spaces over lca groups. Quaestiones Math. 19 (1996), 237-257. MR97e:47048. 41. Existence of Bade functionals for complete Boolean algebras of projections in Fr~chet spaces. Proc. Amer. Math. Soc. 125 (1997), 2401-2407. MR97j:47050. 42. The sequential closedness of or-complete Boolean algebras of projections. Anal. Appl. 208 (1997), 364-371. MR97m:47022.

J. Math.

43. The strong closure of or-complete Boolean algebras of projections, Arch. Math. (Basel), 72 (1999), 282-288. 44. Lipschitz continuity of spectral measures, Bull. Austral. Math. Soc., 59 (1999), 369373. 45. The spectral theorem: a historical viewpoint. Ulmer Seminare: Funktionalanalysis und Differentialgleichungen. To appear in vol.4, 1999. Ricker, W.J., and Schaefer, H.H. 1. The uniformly closed algebra generated by a complete Boolean algebra of projections. Math. Z. 201 (1989), 429-439. MR91b:47081. Rosenberg, M. (see Masani, P.) Rosenthal, P., and Sourour, A.R. (see also Radjavi, H.) 1. On operator algebras containing cyclic Boolean algebras. Pacific J. Math. 70 (1977), 243-252. MR58~17876a. 2. On operator algebras containing cyclic Boolean algebras. II. J. London Math. Soc. (2) 16 (1977), 501-506. MR58•17876b. Rybkin, A.V. 1. The Stieltjes B-integral and the summation of spectral decompositions of some classes of contraction operators. (Russian) Dokl. Akad. Nauk SSSR 319 (1991), 562-566; translation in Soviet Math. Dokl. 44 (1992), 166-170. MR92k:47024.

148

APPENDIX

Sadovni[ii, V.A. (see Dubrovskii, V.V.) Saeks, R., and Goldstein, R.A. 1. Cauchy integrals and spectral measures. Indiana Univ. Math. J. 22 (1972/73), 367378. MR474p435. Schaefer, H.H. (see also Ricker, W.J.) 1. Banach lattices and positive operators, Springer'-Verlag, New York-Heidelberg-Berlin, 1974, xi +376pp. ISBN:0-387-06936-4. MR54~11023. Sen, D.K. 1. Some results on Hermitian operators on Banach spaces. Bull. Calcutta Math. Soc. 77 (1985), 287-290. MR87i:47029. Schonbek, T.P. 1. On a calculus for a generalized scalar operator. Y. Math. Anal. Appl. 58 (1977), 527-540. MR58~7155. Shobov, M.A. 1. Spectral operators generated by damped hyperbolic equations. Integral Equations Operator Theory 28 (1997), 358-372. Shuchat, A. 1. Vector measures and the spectral theorem. In: Vector and operator valued measures and applications (Proc. Sympos., Alta, Utah, 1972), pp.339-341. Academic Press, New York, 1973. MR49#3589. 2. Vector measures and scalar operators in locally convex spaces. Michigan Math. J. 24 (1977), 303-310. MR58#12497. 3. Spectralmeasures and homomorphisms. Rev. Roumaine Math. Pures Appl. 23 (1978), 939-945. MR80a:47053. 4. Infrabarreled spaces, linear operators and vector measures. Bull. Soc. Roy. Sci. Liege, 48 (1979), 153-157. MR81d:46002. Smith, W.V. 1. The Kluv~nek Kantorovitz characterization of scalar operators in locally convex spaces. Internat. J. Math. Math. Sci. 5 (1982), 345-349. MR83g:47038. 2. Differential operators in Banach spaces with densely defined spectral measures, Houston J. Math. 8 (1982), 429-448. MR84d:34019. 3. Spectral measures. I. General theory in a topological vector space. Period. Math. Hungar. 13 (1982), 205-227. MR84d:46063.

149 4. Spectral measures. II. Characterization of scalar operators. Period. Math. Hungar. 13 (1982), 273-287. MR84g:47031. 5. Spectral measures. III. Densely defined spectral measures. Period. Math. Hungar. 15 (1984), 189-203. MR86d:47039. Solomyak, M.Z. (see Birman, M. Sh.) Sourour, A.R. (see also Rosenthal, P.) 1. On groups and semigroups of spectral operators on a Banach space. Acta Sci. Math. (Szeged) 36 (1974), 291-294. MR55:/r 2. Semigroups of scalar type operators on Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 207-232. MR51#1481. 3. Unbounded operators generated by a given spectral measure. (1978), 16-22. MR81h:47020.

Y. Funct. Anal. 29

Spain, P.G. 1. Boolean algebras of projections. 287-289. MR52~r

Proc.

Edinburgh Math.

Soc.

(2) 19 (1974/75),

Stochel, J. 1. The Bochner-Kolmogorov extension theorem for semispectral measures. Colloq. Math. 54 (1987), 83-94. MR89b:47053. Stojanoff, D. (see Andruchow, E.) Stroescu, E. 1. Dunford subspectral systems of operators on a Banach space. (Romanian) Stud. Cerc. Mat. 35 (1983), 399-409. MR86a:47030. 2. A-subspectral systems of operators on a Banach space. (Romanian) Stud. Cerc. Mat. 35 (1983), 518-528. MR86a:47031. 3. A Stone theorem for subscalar operators on Banach space. Rev. Roumaine Math. Pures Appl. 30 (1985), 571-578. MR87f:47061. Sul'man, V.S. 1. Linear operator equations with generalized scalar coefficients. (Russian) Dokl. Akad. Nauk SSSR 225 (1975), 56-58=(English translation) Soviet Math. Dokl. 16 (1975), 1465-1468. MR58Jr Sun, S.L. 1. Decomposability for a class of nonspectral second-order ordinary differential operators. (Chinese) Acta Sci. Natur. Univ. Jilin. 1982, no.4, 9-16. MR85b:34032.

150

APPENDfX

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Univ.

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Chinese Ann.

Math.

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MR82j :47055. Weibang, G. 1. Scalar-type operators that are commutators of compact operators. An. cure~ti Mat. 32 (1983), 89-90. MR84m:47045.

Univ. Bu-

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(Chinese) Dongbei Shida Xuebao

152

APPENDIX

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List of symbols T h e page reference indicates t h e page in

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Index absolute continuity of measures, 12, 13, 15, 20, 92, 94 absolute convergence, 1, 6, 42 abstractly complete B.a., 25, 27, 28, 34, 35, 37-39, 57, 60 63, 79, 82, 83, 85, 86, 88, 93, 95, 96, 99, 110 abstractly a-complete B.a., 25, 27, 28, 34, 37, 38, 43, 44, 57, 59-61, 76, 79, 85, 87, 95, 96, 99, 100, 111 a-additive, 1, 9, 10, 25, 49, 50, 52, 54, 55, 76, 109, 112 Alaoglu theorem, 8, 15 Alexandroff (extension of measures), 55 algebra of functions or equivalence class of functions, 29, 47, 71, 73, 80 algebra of operators, 45 algebra of sets, 1, 2, 28, 34, 38, 46, 54, 55, 59, 77, 79, 111, 117 a-algebra of sets, 1, 2, 10, 25, 34, 35, 37, 54, 55, 58, 59, 78, 79, 98, 100, 103, 117 a-algebra generated by a family of sets, 14, 34, 37, 38, 55, 78, 103, 117 atom of a B.a. of projections, 98, 102 atomic B.a. of projections, 98, 118 Bade complete B.a. of projections, 57, 59, 65, 66, 76-78, 81 88, 90, 94, 96 102, 104, 105, 109 112, 116, 117, 119 Bade a-complete B.a. of projections, 57, 59, 65, 76-79, 83, 85 88, 90 94, 96, 98-103, 105, 108, 109, 111, 116, 118 Bade functional, 92 94, 100 102, 108 Bade reflexivity theorem, 116, 117 Baire category theorem, 38 Baire set, 25, 34-39, 55, 61, 62, 77, 78

Banach algebra, 41, 42, 45, 54, 73, 76-80 Banach lattice, 111 Banach space (definition), 3 Banach-Steinhaus theorem, 49, 70, 82 band projection, 111 Bartle-Dunford-Schwartz theorem, 13, 14, 20 base for a topology, 28, 33, 35, 39, 112, 115 basically disconnected, 33, 34, 36, 76 basically disconnected in restricted sense, 32, 33 bicommutant of a family of operators, 108, 116, 118 bidual X" of a Banach space X, 7, 8, 23, 107 Boolean algebra (B.a.), 25, 27 Boolean algebra of projections, 42, 51, 54, 98, 108 Boolean ring, 27, 30 Boolean subalgebra of B.a., 99 101 BP-property for a Banach space, 70, 71 Borel set, 2, 5, 20, 22-25, 34 39, 51 57, 59, 61-65, 67, 77-79, 85, 90, 102-104, 112, 113, 116-118 bounded (finitely additive) vector measure, 9-11, 13, 46, 51 bounded B.a. of projections, 42, 43, 45, 47, 54, 55, 58, 78, 87, 88, 90, 91, 108, 109, 111 bounded (finitely additive) spectral measure, 45, 50, 54, 55, 77, 79 Carath~odory, C., 102 carrier projection, 97, 98, 112-115 Cauchy sequences and nets, 3, 8, 20 22, 72, 75, 82, 83, 85, 108 clopen set, 28, 33-35, 112 closed set and closure of a set, 6, 8, 14,

157

INDEX

28, 29, 31-33, 35, 37, 45, 47, 54, 55, 81-83, 86-89, 94, 96, 100-104, 108, 109, 116 closed spectral measure, 59, 62, 63, 66, 67, 76, 78, 81 86, 90, 92, 94, 98, 103, 107, 109, 112, 117 closed vector measure, 59, 85 commutant of a family of operators, 108, 109, 111, 116 compact (Hausdorff) space, 3, 5, 8, 14, 15, 20, 22-25, 28-38, 47, 51, 54, 55, 65, 76, 78 compact operator, 118 compactification of a topological space, 31 complement (in a lattice), 26, 42 complete (topological), 3, 8, 19-21, 24, 59, 67, 73, 82-85, 100, 108 T-complete for a B.a. of projections, 109111 completely regular topological space, 31 33, 36 completion, 83 complex measure, 1-3, 5, 9, 10, 12, 13, 15 17, 20 22, 48, 49, 52, 53, 55, 65, 69, 72, 78, 107 conjugate closed, 29, 30 connected set, 33 continuous (or bounded) linear functional, 4 continuous linear operator, 22 convergent sequence of sets, 11 convex set, 6, 49, 83 countably atomic B.a. of projections, 98, 102 countably decomposable B.a., 94-97, 99 countably generated or-algebra, 103 countably generated B.a., 99, 100, 102 co-zero set, 33 cyclic space, 86, 87, 92, 99, 105, 107, 108, 110, 113 cyclic vector, 86, 92, 93, 99, 104, 108-111 Dedekind a-complete, 111 dense set, 31 Dieudonn6, J., 118 Dirac point measure, 10, 86, 106 directed set, 60

discrete topology, 31-36 disjoint elements in a B.a., 44, 94-98, 100 102 distinguish points of a set, 29, 48, 50 distributive lattice, 26, 27 dominated convergence, 17, 22, 53, 70-72, 106, 107, 117 dual Banach space X t, 4, 5, 22, 54 dual operator, 22, 23, 42, 50, 51, 54, 68, 77, 79 Dunford-Pettis property, 118 Eberlein-Smulian theorem, 8 equivalence relation (and classes), 19, 38, 58, 59, 73, 79, 85, 100 P-essentially bounded function, 67, 73, 75, 76, 79, 90 extremely disconnected, 32, 34 37, 39, 76, 112 finitely additive measure, 2, 9, 10, 23, 25, 30, 41, 46, 48-50, 52, 55, 66, 93 finitely additive vector measure, 8-10 a-finite measure, 12 first category (or meager) set, 37-39, 64 ~glede theorem, 88, 89 Gillespie, T.A., 116, 118 Grothendieck space, 118 Hahn decomposition (signed measures), 64 Hahn-Banach theorem, 8 Hamel basis, 9 hereditarily indecomposable Banach space, 118 Hilbert space, 3, 7, 57, 58, 88, 90, 91, 94, 103, 105, 116 homomorphism (for B.a. 's), 28, 37, 38, 64, 99 homomorphism (for operator algebras), 51, 54, 112 c~-homomorphism (for B.a. 's), 25, 37-39, 57, 61, 64, 78 ideal (in a B.a. ), 37, 95, 96 identity operator, 42, 46, 48, 73, 81, 89, 97, 110 a-ideal (in a B.a. ), 37-39

158

indefinite integral, 16, 80 injective, 28, 31, 47, 81, 84, 91, 99, 107, 110, 113, 114 integrable function (for a vector measure), 15-17, 19, 70, 105 107 P-integrable function (for a spectral measure), 50, 67 73, 75, 79, 89 integration map, 21, 22, 67, 80, 83 85, 103, 107, 109 interior of a set, 35, 37 39, 55, 102 inverse closed (algebra of operators), 45, 73, 74, 85 invertible (operator), 42, 74, 88-91 isolated point, 32, 36 isomorphism (of B.a. 's), 28, 30, 37, 38, 47, 58, 63, 82, 86, 99, 102 isomorphism (of operator algebras), 28, 45, 47, 54, 67, 73, 76, 77, 79 81, 83-85, 103, 112 kernel, 38, 46 lattice, 26 Lindel6f space, 36 linear functional, 4 locally compact space, 31, 32, 36 locally convex algebra, 80, 81, 85 locally convex Hausdorff space, 59, 67, 68, 79-81, 103 Loomis-Sikorski theorem, 38 lower (and greatest lower) bound, 26-28 g~-space, 118 Ll-space, 118 Mackey-Wermer theorem, 88, 90 measurable space, 1, 8, 9, 13, 16, 27, 66 measure algebra, 100-102 measure space, 1, 2 monotone class of sets, 38 monotone property (for B.a. of projections), 58-60, 62, 64, 78 monotone set function, 11 a-monotone property (for B.a. of projections), 58-62, 101 multiplier projection, 43 natural embedding of X into X ' , 7, 8 non-separable measure, 103

I ND EX

norm, 3, 4, 19, 41, 42, 79, 80 norm bounded set, 6, 8, 9, 49 normal operator, 88-90, 116 normed space and topology, 3, 4, 61, 65, 85, 94 nowhere dense set, 37 null function (for a spectral measure), 72, 84, 90 null function (for a vector measure), 19, 105, 106 null set (for a complex measure), 12, 15 null set (for a spectral measure), 58, 59, 64, 73, 76, 84, 92 null set (for a vector measure), 13, 21, 85, 86, 92 one-point compactification, 31, 34, 35 operator norm, 22, 41 a-order continuous norm, 111 ordered convergence property (of a B.a. of projections), 58 60, 65, 66, 82, 99 Orlicz-Pettis theorem, 6 a-ordered convergence property (of a B.a. of projections), 58 60, 66, 99 partial order, 25-27, 42, 58, 95 partition, 2 Pettis theorem, 13 property (c~) (for a projection in a B.a.), 43, 44 pseudometric, 59, 82 quasicomplete, 62, 83 quasinilpotent operator, 90 quotient space, 19, 38, 39, 63, 100, 102 Radon-Nikodym theorem, 12, 15, 92 reflexive Banach space, 7, 8, 23, 118 reflexive operator, 116 regular extension (of a measure), 55, 78 regular open set, 37, 39 regularity (of a complex measure), 5, 20, 22, 51, 52, 55, 64, 65, 77-79 regularity (of a vector measure), 23, 24, 54, 77, 78 relative topology, 31, 36, 80, 85, 103 resolution of the identity, 90, 91, 94, 102104, 116, 118

159

INDEX

resolvent set of an operator, 89, 90 Riesz representation theorem, 5, 22, 23, 54 Rybakov theorem, 20, 91, 92

surjective, 7, 38, 39, 99 symmetric difference of sets, 11, 27, 38, 58, 64, 82, 85, i00

Sarason, D., 116, 118 scalar-type spectral operator, 89-91, 94, 102104, 116, 118 Schur property, 118 selfadjoint operator, 57, 88, 90, 91, 94, 105, 116 seminorm, 68, 80, 81 senfisimple algebra, 90 semivariation of a vector measure, 10, 11, 13, 17, 19, 23, 59, 80 separable measure algebra, 101, 102 separable space (topologically), 39, 91, 94, 96, 98 103, 116-118 separating vector, 96, 98 102 simple function, 13, 14, 21, 22, 28, 29, 34, 46, 53, 54, 71, 73-76, 81, 84, 85, 103, 107, 108, 118 spectral measure, 41, 48-52, 54, 57 59, 6163, 66 69, 71-73, 75-81, 83 87, 89, 90, 98, 102, 103, 105-107, 109 spectral measure of class F, 48 51, 77, 79 spectrum of an operator, 89 91, 94, 102104, 118 standard representation of a simple function, 29, 46, 47 Stone map, 30, 37, 38, 46, 55, 59, 60, 62, 63, 77, 78, 85, 112 Stone representation theorem for B.a. 's, 25, 30, 37 Stone space (of a B.a. ), 25, 30, 34, 35, 38, 41, 45, 54 57, 61, 62, 67, 76 79, 84, 85, 112 Stone-r compactification, 31, 32, 35, 36, 110 Stone-Weierstrass theorem, 29 strong operator topology, 48, 58, 60-65, 71, 72, 87, 89, 90, 93, 94, 98 strongly closed algebra of operators, 67, 83-85, 88, 103, 104, 108, 109, 116118 subadditivity of a vector measure, 11, 59 subalgebra generated by a family of operators, 45

~--additive measure, 64 topology and topological space, 3, 28 31, 35-37, 55, 57, 59, 85 total variation, 3, 5, 22 totally disconnected space, 28 30, 32-36, 112 totally ordered set, 95 unconditional convergence, I, 6 9, 19, 50, 102 Uniform boundedness principle, 49, 50 uniform operator topology, 41, 42 uniform structure (and space), 59, 82, 85 uniformly closed subalgebra of operators, 41, 45, 47, 54, 76, 77, 79, 85, 88, 108, 109, 111, 112, 116 upper (and least upper) bound, 26 28, 58, 65, 66, 95, 96 Urysohn theorem, 29 variation measure, 2, 3, 12, 13, 17, 55, vector lattice, 16 vector measure, 9-11, 13 20, 23, 24, 57, 59, 61, 67 71, 79, 80, 85, 92, 105, 106 vector space, 3, 6, 15, 16, 19, 28, 29, 79, 83, 89 Vitali-Hahn Saks theorem, 20 yon Neumann (bicommutant) theorem, 116

65 54, 86, 69,

105,

weak operator topology, 48, 49, 58, 60 62, 64, 89, 93, 117 weak topology, 5, 8, 14, 15 weak-star topology, 7, 8, 15, 22, 23, 108 weakly bounded set, 6 weakly closed subalgebra of operators, 8385, 88, 116, 117 weakly compact operator, 23, 24, 54 weakly sequentially complete, 8, 19, 24 weakly subseries convergent, 6 Zorn's lemma, 95-97

Druck: Verarbeitung:

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