Saks spaces and applications to functional analysis

SAKS SPACES AND APPLICATIONS TO FUNCTIONAL ANALYSIS

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NORTH-HOLLAND MATHEMATICS STUD I ES

28

Notas de Matemhtica (64) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Saks Spaces and Applications to Functional Analysis JAMES BELL COOPER Johannes-Kepler Universitat. Linz, Austria

1978

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

@ North-HollandPublishing Company - 1978 All rights reserved. No part of this publication may be reproduced,stored in a retrievalsystem. or transmitted, in any form or by any means, electronic, mechanica1,photocopying. recording or otherwise, without the prior permission of the copyright owner.

North-Holland ISBN: 0 444 8.51 00 3

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM*NEW YORK*OXFORD SOLE DISTRIBUTORS FOR THE U.S.A.ANDCANADA

ELSEVIER / NORTH HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Cooper, James Bell. Saks spaces and applications to functional analysis. (Notas de matemhtica ; 64) (North-Holland mathematics studies) Bibliography: p. Includes index. 1. Saks spaces. 2. Functional analysis. I. Title. 11. Series. @l.N86 no. 64 [@I3221 510'.8s [515'.73] 78-1921 ISBN o - w A - ~ ~ ~ o o - ~

PRINTED I N THE NETHERLANDS

PREFACE

This monograph is concerned with two streams in functional analysis

-

the theories of mixed topologies and of strict topo-

logies. The first deals with mathematical objects consisting of a vector space together with a norm and a locally convex topology which are in some sense compatible. Their theory can be regarded as a generalisation of that of Banach spaces, complementary to the theory of locally convex spaces. Although closely related to locally convex spaces, the theory of mixed spaces (or Saks spaces as we propose to call them) has its own flavour and requires its own special techniques. The second theory is concerned with a number of special topologies on spaces of functions and operators which possess the common property that they are substitutes for natural norm topologies which are not suitable for certain applications. The best known example of the latter is the strict topology on the space of bounded, continuous, complex-valued functions on a locally compact space, which was introduced by Buck. The connection between these two topics is provided by the fact that the important strict topologies can be regarded in a natural way as Saks spaces and this fact allows a simple and unified approach to their theory.

The author feels that the theory of Saks spaces is sufficiently well-developed and useful to justify an attempt at a first synthesis of the theory and its applications. The present monograph is the consequence of this conviction.

V

vi

PREFACE

The book is divided into five chapters, devoted successively to the general theory of Saks spaces and to important spaces of functions or operators with strict topologies (spaces of bounded cont nuous functions, bounded measurable functions, operators on Hilbert spaces and bounded holomorphic functions respectively

. An

appendix contains a category-theoretical

approach to Saks spaces. Each chapter is divided into sections and Propositions, Definitions etc. are numbered accordingly. Hence a reference to 1.1.1 is to the first definition (for example) in

9 1 of Chapter

abbreviated to 1.1.

I. Within Chapter I this would be

At the end of the book, indexes of notations

and terms have been added.

An attempt has been made to make the first five chapters as self-contained as possible so that they will be useful and accessible to non-specialists. However, some results have been given without proof, either because they are standard or because they seemed to the author to provide useful insight on the subject but the proofs were unsuitable for inclusion. Some of the latter are relegated to Remarks. In the first Chapter, only familiarity with the basic concepts of Banach spaces and locally convex spaces is assumed. In Chapter I1 GILLMAN and JERISON has been applied

-

-

V the principle of

namely to make them com-

prehensible to a reader who understands the words in their respective titles. The monograph should therefore be suitable say for a graduate course or seminar. However, by providing each Chapter with notes (brief historical remarks and references to

vii

PREFACE

related research articles) and a list of references, an attempt has been made to produce a work which could also be useful to research workers in this and related fields. In addition to those references which we have used directly in the preparation of this book, we have tried to include a complete bibliography of papers which relate to Saks spaces and their applications, including some where the connection may seem rather remote.

A s a warning to the reader, we mention that all topological

spaces (and hence also locally convex spaces, uniform spaces etc.) are tacitly assumed to be Hausdorff.

It goes without saying that the author gratefully acknowledges the contributions to this book of all those mathematicians whose published works, preprints, correspondence or conversation with the author have been used in its preparation. Special thanks are due to Frxulein G. Jahn who made a beautiful job of typing a manuscript in a foreign language under rather trying circumstances.

J.B. Cooper Edramsberg, October, 1977

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LIST OF CONTENTS

Preface

V

I Mixed topologies :

1 4

1

Basic theory;

2

Examples;

20

3

Saks spaces;

25

4

Special results;

37

5

Notes;

63

References for Chapter I.

I1

111

68

Spaces of bounded, continuous functions:

75

1

The strict topologies;

77

2

Algebras o € bounded, continuous functions:

89

3

Duality theory;

101

4

Vector-valued functions;

112

5

Generalised strict topologies:

119

6

Notes;

136

References for Chapter 11.

14 1

Spaces of bounded, measurable functions:

155

1

The mixed topologies;

156

2

Linear operators and vector measures;

172

3

Vector-valued measurable functions;

176

4

Notes;

190

References for Chapter 111.

ix

192

X

IV

LIST OF CONTENTS 195

Von Neumann a l g e b r a s :

1

The a l g e b r a of o p e r a t o r s i n H i l b e r t s p a c e s ;

197

2

Von Neumann a l q e b r a s :

203

3

Notes:

22 1

References f o r Chapter I V .

V

S p a c e s o f bounded h o l o m o r p h i c f u n c t i o n s :

e:

223

225 226

1

Mixed t o p o l o g i e s on

2

The mixed t o p o l o g i e s on H m ( U ) :

230

3

The a l g e b r a Hm:

24 0

4

The H m - f u n c t i o n a l c a l c u l u s f o r c o m p l e t e l y non

5

unitary contractions:

24 9

Notes:

2 60

R e f e r e n c e s f o r C h a p t e r V.

267

Appendix :

.

263

C a t e g o r i e s of S a k s s p a c e s :

268

A.2.

D u a l i t y f o r c o m p a c t o l o g i c a l and uniform s p a c e s :

272

A.3.

Some f u n c t o r s :

282

A.4,

D u a l i t y for s e m i q r o u p s and g r o u p s :

2 84

A.5.

E x t e n s i o n s of c a t e g o r i e s :

295

A.6.

Notes:

305

A. 1

R e f e r e n c e s f o r Appendix.

308

Conclusion

31 3

Index of n o t a t i o n

319

Index of t e r m s .

323

CHAPTER I

-

MIXED TOPOLOGIES

Introduction: The objects of study in this chapter are Banach spaces with a supplementary structure in the form of an additional locally convex topology. The motivation lies in the interplay between certain mathematical objects (topological spaces, measure spaces etc.) and suitable spaces of (complex-valued) functions on them. These often have a natural Banach space structure. However, by passing over from the original spaces I

to the associated Banach spaces, one frequently loses crucial information on the underlying space. A good example (which will be the subject of our most important application of mixed topologies) is the Banach space Cm(S) of bounded, continuous, compl x-valued functions on a locally compact space it is impossible to recover

of cm

S)

S

S

where

from the Banach space structure

(in contrast to the case of compact spaces

S).

As we shall see in Chapter 11, this situation can be saved by enriching the structure of Cm(S) with the topology convergence on the compact subsets of

S.

' I ~ of

uniform

The class of spaces

that we consider can be regarded as a generalisation of the class of Banach spaces (we can "enrich" a Banach space in a trivial way, namely by adding its own topology). In fact these spaces can be regarded as projective limits of certain spectra of Banach spaces with contractive linking mappings (just as one can regard (complete) locally convex spaces as projective limits of arbitrary spectra of Banach spaces) and we shall lay particular emphasis on this fact for two reasons: for

1

2

I. MIXED TOPOLOGIES

purely technical reasons and secondly because, in applications to function spaces, we shall constantly use the fact that our function spaces are constructed out of simpler blocks which correspond exactly to the members of a representing spectrum of Banach spaces. As an example, dual to the fact that one can consider a locally compact space as being built up from its compact subspaces, we find that one can construct the space Cm (S) from the spectrum defined by the spaces {C ( K )1 as K runs through these subsets.

One of our main tools in the study of our enriched Banach spaces will be a natural locally convex topology

-

the mixed

topology of the title of this chapter. It turns out that this can be regarded as a generalised type of inductive limit. The latter were systematically studied by GARLING in his dissertation. For this reason, we begin with a treatment of this theory in the generality suitable to our purposes.

For the convenience of the reader, we now give a brief summary of Chapter I. In the first section, we give a basic treatment

of generalised inductive limits. Essentially, we consider a vector space with two locally convex topologies which satisfy suitable compatibility conditions. We then introduce in a natural way a "mixed topology" and this section is devoted to relating its properties to those of the original topologies However, a closer examination of the definitions and results shows that, for one of the topologies, only the bounded sets

INTRODUCTION

3

are relevant. We have taken the consequences of this observation by replacing this topology by a "bornology", that is a suitable collection of sets which satisfy properties which one would expect of a family of bounded sets. We really only use the language (and not the theory) of bornologies and introduce explicitly all the terms that we use. In section 2, we give a list of examples of spaces with mixed topologies. Some of these will be studied in detail (and in more generality) in the following chapters. Others are introduced to supply counter-examples. All are used to illustrate the ideas of the first section. In section three, we define the class of enriched Banach spaces mentioned, restate the results of sections 1 in the form that we shall require them for applications and describe the usual methods for constructing new spaces (subspaces, products, tensor products etc.). It is perhaps not inappropriate to mention here that one of the main reasons for our emphasis on spaces with two structures (a norm and a locally convex topology) rather than on locally convex spaces of a rather curious type is the fact that it is important that these constructions be so carried out that this double structure is preserved and not in the sense of locally convex spaces. The fourth section is devoted to attempts to extend the classical results on Banach spaces to enriched Banach spaces (e.g. Banach-Steinhaus theorem, closed graph theorem). The results obtained are perhaps rather unsatisfactory since they involve special hypotheses but, as we shall see later, they can often be applied to important function

I. MIXED TOPOLOGIES

4

spaces. In any case, there are simple counter-examples bhich show that such results cannot hbld without rather special restrictions.

I.1. BASIC THEORY

As announced in the Introduction to this chapter, it is convenient for us to use the language of bornologies. We begin with their definition:

1.1.

Definition: Let E be a vector space. A

ball in E is an

absolutely convex subset of E which does not contain a nontrivial subspace. If B is a ball in E l we write EB for the 03

linear span U nB n=l

11

of B in E. Then

/IB

:

x

is a normon E. If (EB,II

+ inf

llB)

(X > 0 : x

E

XB)

is a Banach space, B is a

Banach ball.

Note that any absolutely convex, bounded subset of a locally convex space is a ball. The following Lemma qives a sufficient (but not necessary) condition for it to be a Banach ball.

1.2.

Lemma: Let B be a bounded ball in a locally convex space

(E,T). Then if B is sequentially.complete for lar if it is T-complete), B is a Banach ball.

T

(and in particu-

Proof: Let (x,)

be a Cauchy sequence in ( E B l

since B is bounded, (x,) so

that xn

If

E

>

-+

5

BASIC THEORY

I. 1

x for

Then,

is T-Cauchy. Hence there is an x

(xm -

We show that IIxn

T.

]IB).

11

0, there is an N E N

so that

B

E

0.

xllB-

xn) belongs to EB

for m,n 2 N. Since B (and so also E B ) is sequentially complete and so sequentially closed, we can take the limit over n to deduce that xm

-

x

belongs to EB for m 2 N.

Definition: If E is a vector space, a (convex) bornology

1.3.

on E is a family B of balls in E so that (a) E =

U

(b) if B

E

B;

> 0, then

8, X

XB E 8 ;

(c) €3 is directed on the right by inclusion (i.e. if B,C (d) if B A

B then there exists D E B with B

E E

u C

2 D);

B and C is a ball contained in B , then C

E

B

.

subset B of E is B-bounded if it is contained in some ball

in 8 . A

basis for B is a subfamily B 1 of B so that each B

a subset of some B1

E

E

B is

B1.

is complete if B has a basis consisting of Banach balls.

(E,B)

B is of countable type if B has a countable basis.

If

(E,T)

is a locally convex space, then B,,

the family of

all T-bounded, absolutely convex subsets of E is a bornology on E

-

the von Neumann bornology. In many of our applications B

will be the

von Neumann bornology of a normed space ( E l l [

This is of countable type (the family {nB is the unit ball of E is a basis).

II

I b E

where B

11 ) .

II II

I. MIXED TOPOLOGIES

6

Now we consider a vector space E with a locally convex topology -r and a bornology B of CounKable type which are compatible in the followinq sense: and 8 has a basis of -r-closed sets. Then we can

8 18,

choose a basis (B ) for B with the following properties: n for each n; (a) Bn + B n L Bn+l (b) each Bn is -r-closed; (if (C,)

-

is a countable basis for 8 , we can define (Bn)

inductively as follows: take B1

=

C1. Once B1,...,Bn have

been chosen, we can find a T-closed ball in B which contains Bn + Bn + Cn+l. This is our Bn+l). In future, we shall tacitly assume that a given basis (B,)

has the above properties.

Definition: We define the mixed locally convex structure

1.4.

y = Y[E,TI

Let U

=

as follows: m

(Un)n=l be a sequence of absolutely convex T-neighbour-

hoods of zero and write m

.....

U (U, n B1 + + Un fl Bn). n= 1 Then the set of all such y ( U ) forms a base of neighbourhoods

y(u)

:=

of zero for a locally convex structure on E and we denote it by

y [ 8 , ~ ] (or simply by y if no confusion is possible).

In the case that 8 is the bornology defined by a norm on E, we write y [ l l

11 , T I

for the structure y [ E , r ] .

The following Proposition gives a natural characterisation of y .

I. 1

1.5. Proposition: (i) y is finer than (ii) y and

7

BASIC THEORY T:

coincide on the sets of B ;

T

(iii) y is the finest linear topology on E which coincides with

T

on the sets of 8.

Proof: (i) if U is a T-neighbourhood of zero, then U 3 y (Un) where Un := 2-"U. (ii) if B B

-

B

C_

8 , we can choose a positive integer r so that

E

Br. A typical neighbourhood of the point xo

E

B

for the topology induced by y on B has the form B fl (xo + y (Un)). Then

Ur n (B-B) z Ur

n B r L y((Un)) and

so

(xo + Ur 1 fl B i (xo + y((Un)) n B. (iii) let T~ be a linear topoloqy on E which coincides with

T

on the sets of B. We show that y is finer than 'r1. Let W be a neighbourhood of zero for

T,

and choose neighbourhoods Wn

.

of zero so that Wo = W and Wn + Wn C_ Wn-l (n 1 1 ) There are T-neighbourhoods (Un) of zero so that Un n B n G Wn. Then, for any n

(u, n

B ~ ) +

..... + (unnB

~ C_ )

w

and so y((Un)) C_ W.

1.6. Corollary: (i) y is independent of the choice of basis (Bn); (ii) if

T

(i.e. if

and T

' I ~are

suitable locally convex topologies on E

and T~ are compatible with Blthen y [ B , ~ l = y [ B , T l l

if and only if

T

and

T~

coincide on the sets of B.

I. MIXED TOPOLOGIES

8

The localisation property of y expressed in 1.1.5 implies that the continuity of linear mappings is determined by their behaviour on the bounded sets of E.

1.7. Corollary: Let H be a family of linear mappings from E into a topological vector space F. Then H is y-equicontinuous if and only if HIB is T-equicontinuous for each B

E

B. In

particular, a linear mapping T from E into F is continuous if and only if TiB is T-continuous for each B

E

8.

Proof: We must show that if W is a neighbourhood of zero in F then

n

T-' (W) is a y-neighbourhood of zero in E. TE H But ( T 1 ~ 1 -(W) l = T-' (w) f7 B and so we can apply T EH TEH 1.5. (iii).

n

n

In view of the above property, the following Lemma of GROTHENDIECK will be useful:

1.8.

Lemma: Let ( E , T ) be a locally convex space, B an absolute-

ly convex subset of E and T a linear mapping from B into a locally convex space F. Then TiB is rB-continuous if and only if it is TB-continuous at zero.

Proof: Let V be an absolutely convex neiqhbourhood of zero in

F, x a point of B. We must find a neighbourhood E so that T((x

+

U)

(-l B ) C Tx

+

U

of zero in

V. We choose U (absolutely

1.1

BASIC THEORY

9

convex) so that T(B n (U/2)) C_ V/2. Then if y x-y

E

B

-

B

=

E

((x+U) n B)

2B and the result follows from the inclusion

T(2B n U) C V.

As we shall see later, the topology y is, in the interesting cases, never metrisable (or even bornological)

. However,

it does, sometimes, have one useful property in common with such spaces.

1.9.

Proposition: Suppose that 8 has a basis of T-metrisable

sets. Then a linear mapping from E into a topological vector space F is continuous if and only if it is sequentially continuous.

Proof: For any B

E

B , TIB is sequentially continuous and so

continuous. The result then follows from 1.7.

In the following Propositions, we characterise certain properties (boundedness, compactness, convergence) with respect to y directly in terms of 8 and

1.10.

Proposition: A sequence (x,)

T.

in E converges to x in

(E,y) if and only if Exn) is 8-bounded and xn

Proof: We can suppose, that x = 0. By show that if xn

-+

0 in (E,T), then

-+

x in

(ErT).

1.5 it suffices to

{xn) is 8-bounded. If this

were false, we could find a subsequence (x ) so that x "k "k

4

Bk.

I. MIXED TOPOLOGIES

10

Since Bk is -r-closed,we can choose a T-neighbourhood of

4 Bk + 2Uk and we can suppose that zero Uk so that x "k (k > 1). Then for each k > 1 k' + k' 'k-1 m

y((un)) =

u

n= 1

..... + un n B ~ )

( u l n B~ +

m

c u (B1 + p= 1

..... +

Bk-l

+

Uk

.....

+

+

'k+p)

c Bk + 2Uk. Hence xnk

4

that (x,)

is a y-null-sequence.

1.11.

y ( (Un)) for each k which contradicts the fact

Proposition: A subset B of E is y-bounded if and only

if it is B-bounded.

Proof: Suppose that B is B-bounded. Then B E Br for some r. Let (Un) be a sequence of absolutely convex neighbourhood of zero. Then there is a K > 1 so that B L KU,.

Hence

B S K(Urn Br) E K y ((Un 1 and so B is y-bounded.

Now suppose that B is y-bounded. If B were not B-bounded, we could find a sequence (x,) positive integer n. Now n-'x n B-bounded - contradiction.

in B so that xn -+

4

nBn €or each

0 in ( E , y ) and so {n

-1

xn) is

1.12. Proposition: A subset A of E is y-compact (precompact, relatively compact) if and only if it is B-bounded and r-compact (precompact, relatively compact).

1.1

BASIC THEORY

11

Proof: This follows immediately from 1.11 and 1.5.(ii).

We recall that a locally convex space is semi-Monte1 if its bounded sets are relatively compact. It is Montel if, in addition, it is barrelled. In the next Proposition, we characterise semi-Monte1 mixed spaces. As we shall see below, non-trivial mixed topologies are never barrelled

-

and so

never Montel.

1.13. Proposition: (E,y) is semi-Monte1 if and only if B has a basis of .r-compact sets.

Proof: This is a direct consequence of 1.12 and the definition.

We now consider the completeness of (E,y). It is customary to use here a completeness theorem of RAIKOV which generalises KBTHE'S completeness theorem. However, this result is rather inaccessible so

we give a direct proof, using a method of DE WILDE and HOUET.

1.14. Proposition: (E,y) is complete if and only if B has a basis of .r-complete sets.

Proof: The necessity of the given condition follows from 1.5. (ii) and 1.11. Now suppose that we have a basis (B,)

(i,;)be 5, choose xo 4

for B consisting of

.r-canplete sets. Let

the locally convex completion

of E. If E

\ E. Then for each n, xo

#

4

2Bn

I. MIXED TOPOLOGIES

12

and so by the Hahn-Banach theorem ([561, p.39, Folg. 2 1 , there is an fn

E

(i) I

only finitely many

with fn

E

0 Bn and fn(xo) = 2. Since

fn are non-zero on a given B ,

Ifn) is

A

equicontinuous on E (1.7) and hence also on E. Then by the theorem of ALAOGLU-BOURBAKI ([311 , p.250, 5 10.10(4) 1 Efn) A

is u (El ,E) compact and so has a a(E' ,G)-limit point f. Then f vanishes on each Bm and so on E. Thus f = 0 which contradicts the fact that f(xo)

=

2.

We now make some remarks on the case where 8 is the bornology associated with a norm on E. Then there are three natural locally convex topologies

T,Y

and

II III

the norm topology,

on E and we discuss their distinctness. We first note that the equality

'I

= y means essentially that T is already a mixed

topology i.e. we have gained nothing by mixing. On the other hand the condition y = T 11

11

means that we are in the trivial

situation (trivial from the point of view of mixed topologies)

of a normed space "enriched" by its own topology (since y[ll

II,'I]

=

y [ l l I I , T ~ ( 111).

The following result shows that if

y belongs to the traditional classes of well-behaved locally

convex.spaces, then we have this trivial situation.

1.15. Proposition: If y is bornological (in particular, metrisable) or barrelled, then T =

T

II II.

Proof: If y is bornological, then the identity mapping from (E,y) into ( E , I I

11) , being

bounded (1.11)

is continuous

I. 1 and so y 1 'I

II II

. The

being a barrel in ( E , y )

11

13

inverse inequality is obvious.

If y is barrelled, then B

are assuming that B I I

BASIC THEORY

,

II I l l

the unit ball of (El11

11) ,

is a y-neighbourhood of zero (we

is 'I-closed - strictly speaking, this

need not to be the case. However, it follows easily from the compatibility conditions that we can find an equivalent norm

so that this condition is satisfied).

The essential property of y is that given in 1.5.(iii). The neighbourhood basis used in the definition was chosen so that this would hold. However, in applications, we shall frequently require a much less obvious description of y-neighbourhoods of zero. Let U be as in 1.4 (except that it is now convenient to index from zero to infinity) and put

-Y ( U )

m

:=

u0 n 0 (un + n= 1

B~).

Then the family of such sets forms a neighbourhood basis for a locally convex topology on E which we denote by y[B,'11.

Proof: Firstly, 7 is coarser than 'I on each set Bk. k -., ( B +~ un) n B~ and this is a For y(U) n B~ = (u0 n n=1 T -neighbourhood of zero. Hence by 1.5. (iii), y is finer I Bk than y.

n

Now we show that

y is coarser than

T.

Let y((Un)) be a typical

y-neighbourhood of zero. There exists a decreasing sequence

I. MIXED TOPOLOGIES

14 m

of -c-neighbourhoods of zero so that

(Vn)n=O

(n

1 + Vn-l E Untl -'yn- ( (Vn) 1 y ( (U,) 1 .

Choose x

C_

..) . We

0,l ,2,.

=

shall show that

y( (Vn)) .

E

Then x

for each n, x has a decomposition y, + zn zn

E

x,+. and

x1

Vn. Define

Zn-l

. .+ xn+ zn

:= y1 , xn := y,

y l + (y2 -y 1 ) +.

=

-

E

Vo and,

where y,

E

Bn,

Yn-1 (n > 1). Then

. .+ (yn-yn-l1 +zn

- y, + zn

=

x

- xn + zn.

We have

xn

and

xn - yn

-

= z n-1

-

-

zn

Y,-1

Vn-1

+

E

Bn

Bn-1 E Bn+l*

Hence xn E Un+l Bn+l * If no is so chosen that x On the other hand, Zno

E

vn E

E

B

E

+

"0

Vn-1

+

Vn-1

then zno = x-yno

Un+1

E

Bno+l.

V n o s Uno+2.

Hence we have x = x

1

+...+

xn+zn

E

U2 n B2+ ...+Uno+l

n Bno+l

un0+2

n Bno+2

1.17. Corollary: y has a basis consisting of -r-closed s e t s .

Proof: If Un is an absolutely convex -c-neighbourhood of zero, then

(2-lU )

n

+

Brit

(2-'Un)

+ B n C Un + B,

03

and so y((2-'Un))

E Uo n

n= 1 and this implies the result.

((2-'Un)

+ Bn) c_y((Un))

1.1

15

B A S I C THEORY

We now consider duality for ( E , y ) . E has three dual spaces:

-

E; E'

Y

Ei

the dual of the locally convex space ( E , . r ) ; the dual of the locally convex space ( E , y ) ; the dual of the space ( E , B ) , that is, the space of linear forms on E which are bounded i.e. bounded on the sets of 8.

Then E; E E ' E Ei

Y

and we regard each of these spaces a s a

locally convex space with the topology of uniform convergence on the .r-bounded sets (resp. the y-bounded sets, resp. the sets of B). Since B is of countable type, E ' is metrisable and it is also clearly complete (since the uniform limit of bounded functions is bounded). Hence it is a Frgchet space. Our next results characterise E ' and its equicontinuous sub-

Y

sets in terms of E;

and E i .

Proposition: (i) E'; is a locally convex subspace of Ei

1.18.

(ii) E7; is the closure of E:

;

in E i and so is a Frgchet space.

Proof: (i) follows directly from 1 . 1 1 . (ii) E' is closed in Ei since the limit of a sequence in E' Y Y is continuous on the sets of B and so is in E ' by 1 . 7 .

Y

We show that E;

B and

E

is dense in E'. Let B be a .r-closed ball in Y be a positive number. If f E E ' then there is an

Y

absolutely convex .r-neighbourhood U of zero so that If(x)l if x

E

B n U i.e. f belongs to E ( B n UIo (polar in E",

IE

the

algebraic dual of E). Now the polar of B n U is the closure

I. MIXED TOPOLOGIES

16

of

1 / 2 (Bo

+ Uo) in u(E",E). But this set is closed since

Uo is u (E",E)-compact by the theorem of ALAOGLU-BOURBAKI and so

(B

n u)Oc

+

Hence f belongs to E(UO 0

to EU if x

E

C_

+ u0.

BO

Bo) and so there is a g belonging

ElT such that f-g belongs to EB'

i.e. If (XI -g(x) I


0 . It is sufficient

to find a T-neighbourhood V of zero so that if x f E H, then

I f (x)I S E. We choose V so that H c (E/~)BO+ (E/~)v'G E(V n

E

B n V,

B)O

(for the last inclusion, cf. the proof of 1.18.(ii)). Necessity: suppose that H is y-equicontinuous and U is a strong neighbourhood of zero in El

Y'

We can suppose that U

=

0 Bk

for some positive integer k . Then there is a y-neighbourhood of zero y ( (Un)) so that

. Then

1.23. Corollary: Let (xn) be a null-sequence in E'

Y

Ixn) is y-equicontinuous.

Proof: If U is a strong neighbourhood of zero in E;,

then all

but finitely many of the elements of the sequence lie in U. Hence we can apply 1.22.

1.24. Corollary: Let A be a compact subset of E'

Y'

y-equicontinuous.

Then A is

I. MIXED TOPOLOGIES

18

Proof: Since E' is a Fri?chet space, A is contained in the

Y

closed, absolutely convex hull of a null sequence ( 1 5 6 1 , Ch.7,

9

2 , Lemma 2). By 1.23 the range of this sequence is equi-

continuous and hence s o is its a(E',E)-closed absolutely

Y

convex hull and this set contains A .

If ( F , T ~ )is a locally convex space and E is a subspace, there is a natural vector space isomorphism between F'/Eo and El induced by the restriction mapping from F' onto E'. Since the latter mapping is continuous for the strong topologies, the map from F'/Eo onto E' is continuous. In general, it is not a locally convex isomorphism. This is the case, however, when E has a mixed topology.

1.25. Proposition: Let (E,y) be a locally convex subspace of ( F , . r l ) . Then the strong topology on F'/Eo as the dual of (E,y) coincides with the quotient of the strong topology on F'.

Proof: We show that the natural mapping from E' onto F'/Eo is

Y

continuous. Since El is a Fr6chet space, it suffices to show

Y

that every sequence which converges to zero in E' is bounded Y in F'/Eo. But such a sequence is y-equicontinuous (1.23) and

so is the restriction of an equicontinuous set in F' (HahnBanach theorem). The image of such a set in F'/Eo is bounded.

1.26.

Corollary: If, in addition, E is dense in F, then every

bounded set in F is contained in the closure of a bounded set in E.

I. 1 Proof: By 1 . 2 5 ,

BASIC THEORY

19

the natural vector space isomorphism between

the duals of E and F is an isomorphism for the strong topologies and this is equivalent to the statement of the Corollary (bipolar theorem).

1 . 2 6 can be used to give an alternative proof of 1.14.

1.27.

Remark: It follows from the results of this paragraph

that the spaces of type (E,y[B,'C]) can be internally characterised as the class of locally convex spaces (E,T) with the following properties: a) BT is of countable type: b) an absolutely convex subset V of E is a neighbourhood of zero if and only if V n B is a T-neighbourhood of zero for each B

E

8,.

Hence they form a generalisation of the class of (DF)-spaces of GROTHENDIECK. They have been studied from this point of view by NOUREDDINE who calls then Db-spaces (see [411). NOUREDDINE has shown that they possess many of the properties of (DF)-spaces. We mention without proof the following: I. if (Vn) is a sequence of neighbourhoods of zero in

E, there is a sequence (an) of positive scalars so that nanVn is a neighbourhood of zero. 11. every continuous linear mapping from E into a metrisable locally convex space F is bounded (i.e. it carries some neighbourhood of zero in E into a bounded set in F).

20

I. M I X E D TOPOLOGIES 111. t h e c l a s s of Db-spaces i s c l o s e d under t h e

f o r m a t i o n o f q u o t i e n t s , c o u n t a b l e l i m i t s , s u b s p a c e s of f i n i t e codimension and even s u b s p a c e s of c o u n t a b l e codimension when t h e s p a c e i s complete. IV.

a Db-space E h a s t h e p r o p e r t y (B) of PIETSCH and

so e v e r y a b s o l u t e l y summable sequence i s t o t a l l y summable

( i . e . i s a b s o l u t e l y summable i n t h e normed s p a c e (EB,II f o r some B

E

B.,).

\IB)

As a consequence, t h e theorem o f DVORETSKY-

ROGERS h o l d s i n a s i m p l e Db-space

( i . e . a Db-space whose von

Neumann bornology i s g e n e r a t e d by a norm): i f E i s i n f i n i t e d i m e n s i o n a l , t h e n E p o s s e s s e s a sequence which is summable b u t n o t a b s o l u t e l y summable.

1.2.

EXAMPLES

A. L e t F be a F r g c h e t s p a c e and d e n o t e i t s d u a l by E . On E

we c o n s i d e r t h e s t r u c t u r e s : 8 := t h e c o l l e c t i o n of a b s o l u t e l y convex, e q u i c o n t i n u o u s s u b s e t s of E ( e q u i v a l e n t l y , t h e bounded sets o f E , p r o v i d e d w i t h t h e s t r o n g t o p o l o g y as t h e d u a l o f F ) ; U(E,F)

-

t h e weak t o p o l o g y d e f i n e d on E by F.

Then t h e t r i p l e

( E , 8 , a ( E , F ) ) s a t i s f i e s t h e c o n d i t i o n s of

1 . 4 and so w e can form t h e a s s o c i a t e d mixed t o p o l o g y . W e

i d e n t i f y t h i s t o p o l o g y . F i r s t l y , t h e d u a l of

(E, a(E,F)) is,

of c o u r s e , F and on F t h e t o p o l o g y of uniform convergence on B i s t h e o r i g i n a l t o p o l o g y . Hence, by 1 . 1 7 , t h e d u a l of

EXAMPLES

1.2

21

(E,y) is F. Now the u(E,F)-equicontinuous subsets of F are the bounded, finite dimensional subsets. Hence, by 1 . 2 2 and a standard characterisation of precompact subsets of a locally convex space (1561, p.58), the y-equicontinuous subsets of F are precisely the precompact subsets i.e. y is the topology

T

C

(E,F) of uniform convergence on the compact

subsets of F.

B. If S is a set and p

E

Lp(S) denotes the space of

[l,m],

complex functions x on S so that llxll < P

llxllm (lp(S)I T~

11

[Ip)

:=

sup (Ix(t)l : t

E

m

where

s). consider the topology

is a Banach space and we

of pointwise convergence on S. Then (lp(S),I[ Ilp,rs) satis-

fies the conditions of 1 . 4 and we can define the corresponding mixed topology. For p > 1 , lp(S) is the dual of l/p

+ l/q

= 1 (p

>

unit ball of lp(S) ness !

)

1 ) and T~

q = 1 (p = m)

eq (S) where

. Hence, since on the

coincides with u (lp(S),lq(S)) (compact-

we are in the situation of A and so y = -rC(lp,lq).

Similarly, for p co(S) := {x

= 1,

l1 (S) is the dual of

E lm(S)

: for each

E

> 0, there is a finite

subset J of S so that Ix(t)l < for t and so y is rC(l I ,c0).

4

J)

E

I . M I X E D TOPOLOGIES

22 C.

If p




(c) : suppose x

E

E with llxll > 1

i.e. x

4

BII

We need only find a continuous seminorm p on E so that p and p(x) > 1. By the Hahn-Banach theorem, there is an f

so that Ilfll seminorm.

I

1 on B

II I1

E

and f ( x ) > 1. Then IlflI is such a

II. I 11 11 (E,T)'

28

I. MIXED TOPOLOGIES Definition: A Saks space is a triple (E,ll

3.2.

E is a vector space, and

11 11

T

11,~)

where

is a locally convex topology on E

is a norm on E so that B

II II’

the unit ball of (E,11

11) ,

is .r-bounded and satisfies one of the conditions of 3 . 1 .

11,~)

If (E,II

and ( F , l l

111,~1)

are Saks spaces, a morphism from

E into F is a linear norm contraction from E into F

TiBll

II

if B I I

continuous.

is

II

A

Saks space ( E , I I

11)

is .r-c:omplete. Then ( E , l l

11,~)

so that

is complete

is a Banach space (1.2).

In constructing Saks spaces, one occasionally produces triples (E,II

11,~)

of B I I

where all but the last condition (on the closure

1 1 ) is

satisfied. This forces us to take the following

-

precaution: we define B1

:= B I I

11 Ill of 1 1 1 , ~ ) is a

11

(closure in

T)

. Then the

(as defined in 1 . I ) is a norm

Minkowski functional

B1

on E so that ( E l l [

Saks space. The following Lemma

ensures that we do not lose any morphisms in this process :

3 . 3 . Lemma: Let T be a linear mapping from E into a locally

convex space F so that TI Then TIB

B I I II

is T-continuous.

is T-continuous.

1

Proof: By 1 . 8 , it suffices to show that T ~ is B continuous ~ at zero. Let U be an absolutely convex neighbourhood of zero in F and choose an open neighbourhood V of zero in E so that T(V n B 11

11) c

For if x

E

1 / 2 U. Then

T(V n B 1 ) C U.

V n B1 and we choose X E ]0,1[

then we can, by the continuity of T in B

so that Ax

II II‘

find y

E B E

II II’ II II

V n B

1.3

so that T(Ax) Tx

=

-

T(Ay)

A-’(T(Ax)

SAKS SPACES

29

X/2 U. Then

E

-

T(Xy)) + Ty

E

U.

11,~) I\,T] -

3.4. The associated topoloqy: If ( E , \ l

is a Saks Space,

we can form the mixed topology

it is called the

y[ll

assuciated locally convex topology of E. Then a morphism between two Saks spaces is continuous for the associated topologies (1.5. (ii) and 1.7) and a Saks space is complete if and only if (E,y) is a complete locally convex space (1.14). We repeat that, despite these facts (and others to follow), the relevant structure is that of a Saks space and not a locally convex space (this is one of the reasons that we have been careful not to forget the norms in the definition of a morphism

-

thus a y-continuous linear mapping need not be a morphism although it is, of course, a scalar multiple of one). Hence we shall stubbornly persist in defining notions like completeness, compactness etc. in terms of the structure as a Saks space even when these can be expressed in terms of y (using the theory of

5

3.5.

1.1).

Subspaces and quotient spaces: Let ( E l11

space, F a vector subspace of E . Then if

11,~)

11 I I F , ~ F

be a Saks denote the

norm (resp. the locally convex topology) induced on F , ( F , ( I is a Saks space. We shall see later that coincide with

y[

11

ll,T]

y[ll

IIFfTF)

I I F f ~ F 1 need not

IF.

If F is a y-closed subspace, then we denote by

F I I 11

and F~ the

structures induces on the quotient space E / F . The triple

I. MIXED TOPOLOGIES

30

(E/F,FII

ll,F~)

need not be a Saks space since it can happen

11)

that the unit ball of (E/F,FII

is not FT-closed. However,

by the process described before Lemma 3 . 3 , we can obtain a Saks space which we shall call the quotient Saks space.

I] 1 1 , ~ )

3 . 6 . Completions: Let (E,

ET

n

the completion of the locally convex space (E,T). We write B

for the closure of B 11 in

be a Saks space and denote by

ElT.

Then if

11 ]IA

11

in

ET

and $ for the linear span of

is the Minkowski functional of

locally convex structure induced on

6

from

sT, (;,]I

and

6 is the is a

complete Saks space. We call it the (Saks space) completion of E. It has the following universal property: for every morphism T from (Ell[1 1 , ~ ) into a complete Saks space (F is a unique morphism

from

(E,II /In,;)

into

extends T (for we can extend T firstly to

11 Ill

, T ~ ) ,there

F,II

lll,Tl)which

y uniform continui-

ty and then to $ by linearity), As

an amusing example, consider the Saks space (E,ll II,IJ(E,E'))

where (E,ll

11)

is a normed space. The completion of (E,a(E,E'))

is (Em)+: the algebraic dual of E' (ROBERTSON and ROBERTSON [ 5 6 1 , Sat2 18, p.71). The closure of B in (El)?: is its bipolar, i.e. the unit ball of E", the bidual of (E,11

11).

Hence the completion

of (E,ll II,u(E,E')) is El', with the Saks space structure described in 2 . 1

(as the dual of El). Thus we can regard the bidual as

a completion.

1.3

3 . 7 . Products and projective limits: Let

be a family of Saks spaces. We can give normed space product of {E,},

and so is .c,-closed.

T

~

IL,T,)

IT E,. C~EA If {

(E,,

,T,)

the

the , product of { T ~ ) .

TI B clEA 1111, For the same reason, (Em,II I l m , ~ m ) is

II ,tI

of Em is the product

complete if and only if each (E,,II (E,,II

11 1, (Ern, 1 1 Ic) , {

a Saks space structure by

considering on Em the topology Then the unit ball B

31

SAKS SPACES

IL,.cm) is.

the Saks space product of {E,}

We call

and denote it by

S

:~ EB~d E,,

T

c1 I f?

, a ,@

system of Saks spaces ( s o that the

E

A)

is a projective

' s are Saks space 6, morphisms), we can define its (Saks space) projective limit

(E,ll

11,~)

T

as follows: as in the first paragraph of this section,

we consider the space E of threads as a subspace of give it the induced structure in the sense of 3.5.

ll E, and ,€A It can easi-

ly be checked that the unit ball of E is .c,-closed

in

S

S

TI E,

and so E is complete if each E, is. We denote this projective limit by

S-lim {E,,IT~,}.

Sums

and inductive limits of Saks

spaces can be defined without difficulty but we shall not require them. We recall that each Banach space can be regarded as a Saks space

-

namely the Saks space (E,II lI,.cll

11).

The following

result shows that the Banach spaces are in a certain sense dense in the Saks spaces and corresponds to the fact that complete locally convex spaces are projective limits of Banach spaces.

I. MIXED TOPOLOGIES

32

3 . 8 . Proposition: A Saks space (E,ll

11 ,T

is complete if

)

and only if it is the Saks space projective limit of a system of Banach spaces.

Proof: The sufficiency follows from the remarks above. Necessity: denote by S the family of T-continuous seminorms p on E which are majorised by

11 11.

Then S is a directed set

with the natural (pointwise) ordering and If p

E

S we denote by

E

P

P

p

:= (x E E : p(x) =

I q,

=

sup S ( 3 . 1

P

.

where

01, with the norm induced by p). If 6

A

let T~~ denote the natural contraction from E

then

)

the Banach space associated with p

(i.e. the completion of the space E / N N

11 1 1

into E P' q is a projective system of

A

p I q) PI Banach spaces and it is not difficult to show that (E,ll

11,~)

is its projective limit.

We remark that if (E,II

11,~)

is not complete, then the above

construction produces its completion in the sense of 3.6. As an example of a Saks space product, consider a family {Ta)acA of locally compact spaces and let S,

be a family of subsets

of T as in 9 I.2.D. Denote by T the topological sum of the

u S, (T, is regarded as a ~ E A subspace of T) Then the underlying vector space of S TI Cm (T,) CX€A can be naturally identified with C"(T) and this induces a Saks spaces (T,)

and by S the family

.

space isomorphism between (C" (TI,11

~ L , T and ~)

S II C" (Ta). This ,€A example displays the suitability of a Saks space product in a situation where any locally convex product is hopelessly inadequate.

1.3

33

SAKS SPACES

If T is a locally compact space, then I P K l ,K : C(K1)

+ C(K)

K,K1

E

K(T)

K C K1I

r

(where C ( K ) denotes the space of continuous, complex-valued functions on K and p

K1 , K

is the restriction operator) is a

projective spectrum of Banach spaces and its Saks space projective limit is. CC" ( T I I

II [I,, T ~ .)

3.9. Duality: The dual of ( E , l l

11,~)

is defined to be the

linear span of the set of morphisms from E into a: i.e. it is the dual of the locally convex space ( E , y

-

[

11 11 ,T 1) -

we denote

it by El. It is a Banach space. Suppose now that E is the Saks

Y

space projective limit of the spectrum (

1

:~ E ~

B

~

Eci, ci I B ,

ci

,B

E

A}

of Saks spaces. We assume, in addition, that rci(B

II I t )

T

ci

-dense in B

II IIcl

for each

ci

is

(rciis the natural morphism from

E into E 1 . This condition is satisfied, for example, by the ci

canonical representation of E ( 3 . 8 ) . Then each ( E a ) regarded as a Banach subspace of ( E , injection i

aB

of

71

from (E

'

U Y

11 11)

into ( E B ) ;

I

can be

and the natural

( a 5 B ) is the transpose

Ba *

3.10. Proposition: (a) E;

is the Banach space inductive limit

of the spectrum { iciB

: (E,);

d (ED);,

(b) a subset H of (E,11

11)

I

c1

I

B 1 a,B

E

A).

is y-equicontinuous if and only if

there is a sequence (an) with values in A and, for each n

E

N,

I. MIXED TOPOLOGIES

34

-equicontinuous; n (ii) c Sup{IIfll : f E Hn} < m ; n (iii) H 5 C Hn (i.e. if f E H, f (i) Hn is

'ca

W

=

C

fn where fn

n= 1

E

Hn).

Proof: (a) By a standard result on the duals of locally convex

5 IV.4.41, the dual of

projective limits (see SCHAEFER [ 6 1 1 , (E,T) is the subspace

U (Ea,~cl) I of ( E l l ] C~EA

11)

I .

Hence by

1.18(ii) and the remarks at the beginning of this section, El is the inductive limit of the Banach spaces Y

(E ) '1. a Y

(b) We note firstly that if H' is a T-equicontinuous subset of E' then there is an

c1 E A

so that H' E Ed, and H' is

T

c1

-equi-

continuous. The sufficiency of the condition follows then from 1.22. On the other hand, if H is y-equicontinuous then there are a 0 , ~ , in A ( a o < a,) and Ho Y

HI

aO

-equicontinuous in E' (resp.

~,,-equicontinuous in El) so that H

(E

T

G

Ho

+

E/2B, H C HI

> 0, B the unit ball of

Then if h ho

+

E

+

E/2B

El)

H, it has a representations €/2bo;

h, + ~ / 2 b , (ho

2 and Then h = ho + (hl-ho) + ~ / bo

E

Ho, h,

11 hl -

hoII

We define HI to be the set of all such (ho

-

E

H I , bo,bl

E

B)

S E.

h,) required in

the representations of the elements of H. Continuing inductively, we can construct a sequence (H,) ties.

with the required proper-

1.3

35

SAKS S P A C E S

Spaces of linear mappings: Let ( E l l \

3.11.

IlrT)

and ( F r l l

lll,Tl)

be Saks spaces and denote by L(E,F) the space of y-continuous linear mappings from E into F. Let C be a saturated family of norm-bounded sets in E. Then on L ( E , F ) we consider the following structures:

11 11

-

the uniform norm;

T~

-

the topology of uniform convergence (for T ~ ) on the sets of C .

Then

(L(ErF)

11

r

IlrTE)

is a Saks space. It is complete if

is and if C is the family of all bounded sets in E.

(FrII l l l , T l )

It is a Banach space if F is Banach.

3.12.

Tensor products: Let ( E J I

spaces. We denote by

1Ir~)

and ( F , "

be Saks

lllr~l)

the algebraic tensor product of

E 8 F

E and F. On E 0 F we consider the following structures:

/I He

SUP

: X-

{I

(f 8

4 )( X I [

: f E

g

E

Eir

IIflI

F;!r

IlglI 5 1 )

S 1

(here we are using the norm of f (resp. g ) in E;' (resp. in F;')) T

8 8

'cl

: the projective tensor product

T~

:

of

T

and T,;

A

T

Then ( E 0 F,II

the inductive tensor product of T and IhrT

6

Tl)

and ( E

Q F,II

spaces (this follows from 3.1 since If T

G

(E

TI

i,

2

and Frll

8 TI). llrT

6

TI)

I~,T6 Q

T ~ .

' 1 ~ 1 are Saks

gl is continuous for

We denote their completions by and ( E

6,

F,ll

1 1 , ~ Qf

T ~ ) resp.

Using the

representation of the completion by projective limits, we can

.

I. MIXED TOPOLOGIES

36

4

give another construction of the tensor product E By F : Let

L.

A

be the canonical representations of E and I?.

Then

is a projective spectrum of Banach spaces and its Saks space

11,~

projective limit is naturally isomorphic to (E By F,ll

.2 @

T~).

We finish this section with some brief comments on spaces which generalise the class of Banach algebras (resp. C"-algebras) exactly as the Saks spaces generalise the class of Banach spaces.

3.13. Definition: Let A be an algebra with unit e. A submultiplicative seminorm on A is a seminorm p with the

properties

p(xy)

I p(x)p(y)

If A has an involution x -++

(x,y

E

A)

and

p(e)

=

1;

xgr,p is a C"-seminorm if, in

addition, it satisfies the condition p(xgrx)= {p(x)I2 A

Saks algebra is a triple ( A f l l I l f . r ) where

(Afll

11)

(x E

A).

is a Banach

algebra with unit and the locally convex topology can be defined by a family A

S

of submultiplicative seminorms so that

11 11

= sup S.

Saks C":-algebra is defined in exactly the same way with the

additional requirements that A have an involution and the seminorms of S be C*':-seminorms. It follows from this condition that (A,]]

11)

is then a @-algebra.

1.4

31

SPECIAL RESULTS

If p is a submultiplicative seminorm, then

(as defined P in the proof of 3.8) has a natural Banach algebra structure. Hence a Saks algebra A has a canonical representation as a projective limit of a spectrum {IT

qP

:

where each

ii

iiq +iP'

P

p I q, p , q

E

S}

is a Banach algebra and the linking mappings

are unit-preserving homomorphisms. Similary, a Saks @-algebra has a representation with @-algebras

as components and C"-

algebra homomorphisms as linking mappings. 3.14. The spectrum: If (A,]]1 1 , ~ )

by

My(A)

is a Saks algebra, we denote

the set of y-continuous multiplicative functionals

f from A into a: with

f(e)

= 1.

My(A) is called the spectrum

of A. We regard M (A) as a topological space with the weak

Y

topology induced from

(A', a(A' ,A)1 . Then My ( A )

,

as a subspace

of a locally convex space, is completely regular. T ( A ) is a (topological) subspace of the spectrum of the Banach algebra (A,II

II)

1.4. SPECIAL RESULTS

In 1.5 we saw that the mixed topology could be characterised as the finest linear topology (and hence also as the finest locally convex topology) which agrees with T on the sets of €3.

I. MIXED TOPOLOGIES

38

In general, it is not the finest topology which satisfies this condition. In the Corollary to the following Proposition, we give sufficient conditions for this to be the case. Proposition: Let (E,g,-r)be as in 1.4. Then the following 4.1. are equivalent: (a) B has a basis of -r-compact sets; (b) there is a Frhchet space F so that E

=

F', with the

structure described in 2.A.

Proof: (b)

(a) follows from the BOURBAKI-ALAOGLU theorem.

(b): since (E,y) is semi-reflexive (1 . 1 3 ) , we can

(a)=7

identify E with the dual of the Fri!chet space E'. Then 8 is

Y

the equicontinuous bornology of E and

'I

= a(E,E') on the sets

of 8 by compactness.

4.2. Corollary: If (a) or (b) is satisfied then: (c) y[8,'11 with

T

is the finest topology on E which agrees

on the sets of 8;

(d) a subset A of E is y-closed (resp. open) if and only if

A rl B

is 'IIB-closed (resp. open) for each B

E 8.

Proof: In view of 4. 1,

this is a restatement of the Banach-

Dieudonni! theorem " 3 1 1

, p.274).

1.4

39

SPECIAL RESULTS

W e remark t h a t a famous counter-example of GROTHENDIECK

( [ 2 6 ] , pp.98-99)

which w e s h a l l n o t r e p r o d u c e h e r e shows

t h a t 4 . 2 ( c ) d o e s n o t hold i n g e n e r a l . See a l s o PERROT [471.

4.3. C o r o l l a r y : L e t ( E , l l

11,~)

be a Saks s p a c e i n which B

i s T-compact and l e t A be a s u b s e t of E so t h a t

(A

>

II II

XA = A

0) ( i n p a r t i c u l a r , i f A i s a s u b s p a c e ) . Then A i s y - c l o s e d

i f and o n l y i f

A Cl B I I

11

i s T-closed.

P r o o f : T o check t h a t A i s y - c l o s e d , A

n

nBll

A

n

nB

I~

II II

is

closed f o r e a c h n

= n(A

E

v e c t o r subspace of E . Then B and T~

N. But

BII 1 1 ) -

Now suppose t h a t w e have a s p a c e

a topology

w e need o n l y v e r i f y t h a t

(E,B,T) T

and t h a t F i s a

i n d u c e a bornology B F and

on F ( B F i s g e n e r a t e d by t h e b a l l s of B which

are c o n t a i n e d i n F ) . An i m p o r t a n t q u e s t i o n i s t h e f o l l o w i n g : do w e have t h e e q u a l i t y

W e c l e a r l y have

n

n

However, t h e r e i s a counter-example which shows t h a t e q u a l i t y d o e s n o t h o l d i n g e n e r a l (ALEXIEWICZ and SEMADENI [ 8 1 , p.133) b u t t h e following Proposition g i v e s s u f f i c i e n t c o n d i t i o n s f o r equality:

I. MIXED TOPOLOGIES

40

4.4. Proposition: Suppose that (E,8, T ) is such that (E,y)

is semi-reflexive and let F be a y-closed subspace. Then Y[B

IT]

IF

=

Y[BF"IF]

-

Proof: Consider the Frkchet space

G := E;.

Then

E

=

G'.

Since F is a(E,G)-closed, F is the dual of the quotient space G/Fo (Fo is the polar of F in G). Hence it suffices to show that every y-equicontinuous set H in G/Fo is the image, under IT,of

the quotient mapping

a y-equicontinuous subset of G

(note that G/Fo is the dual of F under both of the topologies under consideration). Now we can choose a neighbourhood basis (Un) of 0 in G so that n(Un) is a neighbourhood basis of zero in G/Fo (since both are Frgchet spaces). Then by 1.22 there is, for each n, a T-equicontinuous subset Hn of G/Fo so that H C Hn We can

-H H

:= _C

+

r(Un).

extend Hn to a T-equicontinuous subset

gn of

G. Then

n (%n + Un) is a y-equicontinuous subset of F and IT@).For IT

and so

-1

H

(HI E

IT-'

=

n

IT

E

fl

(En+

-1

= IT (IT

~

(

-1

n ( H +~ n (un)) ( H +~ IT(U,)

(HI

Un

+

C _ IT( n

Fo)

(in+ un +

Fo))

( ( Hn +~ .rr(un))).

We have defined the mixed topology in 1.4 by displaying a neighbourhood basis of zero. However, it is often convenient

1.4

41

SPECIAL RESULTS

to characterise a locally convex topology by its continuous seminorms. It is not too difficult to show that the topology y can be defined by the following seminorms: choose for each

n

N a r-continuous seminorm pn and define, for x

E

p(x) := inf

I

C

E

E

Pk(xk))

where the infimum is taken over all representations of x in the form

x,

... +

+

xn

where xk

E

Bk. Then the set of all

such seminorms determines y (cf. DE WILDE 1741 , where a corresponding representation for seminorms on inductive limits of locally convex spaces is given). This characterisation is not very useful for applications and we shall now show that in certain cases a much simpler representation for the y continuous seminorms on a Saks space can be given. Let (E,l[ 1 1 , ~ ) be a Saks space and let

S

be a defining family of seminorms

for r which is closed for finite suprema and is such that

11 I ]

= sup S.

(where pn

An

t

E S

Then for each pair (p,) and (1,)

and (An) of sequences

is a sequence of positive numbers with

m), Y

p : x

+sup

"1

pn(x)

is a seminorm on E. The family of all such seminorms defines

-

a locally convex topology

Y[ll

4.5. Proposition: Let (E,II

11,~)

11,~1 on E.

be a Saks space and suppose

that either (a) for every x x,y

E

E so that

x

E

E,

E

> 0, p

E S,

= y+z, p(z) = 0 and

there are elements [IyII I p(x) +

E:

I. MIXED TOPOLOGIES

42

Proof: We verify firstly the inclusion

Y [/I

IIrTl

7 [/I

2

IIrTl

which is valid without assumptions (a) or (b). We show that

-

7 [I/ ]],TI

A 7-neighbourhood of zero in m

where (p,)

and (An)

Xn > 1

that

(this suffices by 1 . 7 ) .

is coarser than T on B

for

It I1 has B II I1

the form

are as above. But if N

E

is chosen so

N

n 1 N, then N

m

and the latter is a T-neighbourhood of zero in

€3

I1 11-

We now consider the reverse inclusion under assumption (a). Every y-neighbourhood of zero contains a set of the form W

U, n

where

Un

:=

P

(un +

n= 1 Ix : pn(x) I

nB

E ~ I

I1 It) (

E

We can assume, in addition, that Put with

A,

:=

p(x)

min ( ~ ~ , 1 / 2 ) X,, 5

1

-p

where

and

p n ( z ) = 0. Then

z

E

Un

p,

1;’

(n

2 1).

y

E

S) (1.16).

for each n. Choose x

E

E

Then for any n there

p,.

of x where and

E

I pn+l

:= n / 2

:= s;p

is a decomposition x = y+z

>~ 0, pn

((~(1

nB

II II

I

pn(x)

+

so that x

(n/2) E

U

+

nB

1.4

Now we assume that B

II II

SPECIAL RESULTS

is .r-compact. Let U be a y-open

neighbourhood of zero. Then there is a

11

so that U n B 1 1

-7 {x : p0(x)

Suppose that we can find

fi

ix

k= 1

where

Xo

:

= E,

of a

E

n+l

S

pk(x)

I

€1

pl,...,p,

I 1 ),

n

nB

po

n

B

E

S

II II c:

E

S and an

E

>

0

I1 II. so that

" nB

Xn = (n-1) (n > 1). We shall prove the existence so that

{x : pk(x)

(71 k=

43

I

x,)n (n+l)B II

I I -c u

n (n+l)B

Then we can construct, by induction, a sequence (p,)

II II. in S so that

for each n and so 03

iX : pk(x) I k= 1 which proves the result. we argue by contradiction.

To prove the existence of

If no such seminorm exists, then

has non-empty intersection with the .r-compact set for each

q E S. Hence, by the finite intersection property,

there is an

x

0

(n+l)BII11 \ U

E

xo

E

U n nB

(ntl)~

II I I

I1 I1

\

cq.

u n qEs

E U n (n+l)B

which is a contradiction.

I1 I1

Hence

q(xo) s n

I. M I X E D TOPOLOGIES

44

Using this result, we can obtain new sufficient conditions for the equality y[

11

llFrTF]

=

Y[

11

]ItT]

IF

to hold for a subspace F of a Saks space ( E l l ]

11,~)

(cf. 4.4).

4.6. Proposition: Let F be a subspace of a Saks space ( E l l \

and suppose that (F,ll

I]F,~F)

11,~)

satisfies (a) or (b) of 4.5.

Then y[

11

IIF"F1

= y[

11

i11'1

IF

Proof: It is sufficient to show that a y [ I I I]F,~F]-continuous -1

seminorm of the form p := m i A n of a y [

11

pn

on F is the restriction

I~,T] -continuous seminorm -p on E .

But if

UPn

is an

extension of pn to a T-continuous seminorm on E then

5

:= sup A n '

nEN

zn has the required property. 11,~)

4.7. Corollary: Let ( E , l l

be a Saks space, F a subspace

so that the unit ball of (F,ll ),1

is TF-compact.

Then the restriction mapping induces an isometry from ( E , y ) '/Fo onto (F,y)'.

Proof: The restriction mapping from ( E , y )

'

into (F,y) I is

obviously a norm-contraction and it is surjective by 4.6. We must show that if

f

E

then there is an extension with ]!TI/5 1 + ~ . Since B

II

(F,y)' with Ilfll

7 IIF

= 1

and if

E

of f to an element of (E,y) ' is TF-comp&ct, there is an

> 0,

45

SPECIAL RESULTS

1.4

a b s o l u t e l y convex y-neighbourhood U of z e r o so t h a t

w

:=

1 (1+~ BII llF) +

U

c

{x

F :

E

I f(x)l

I

1)

W e can e x t e n d t h e Minkowski f u n c t i o n a l of W t o a y - c o n t i n u o u s seminorm

on E s o t h a t

1 BII

11 C

I X

-

: P(X)

1)

S

-

The r e s u l t t h e n f o l l o w s by a p p l y i n g t h e Hahn-Banach theorem t o f t o o b t a i n a l i n e a r form

?

on E which e x t e n d s f and i s

N

m a j o r i s e d by p.

Now w e d i s c u s s a theorem of Banach-Steinhaus t y p e f o r mixed t o p o l o g i e s . S i n c e such s p a c e s are n o t , i n g e n e r a l , b a r r e l l e d t h e c l a s s i c a l approach c a n n o t be employed. I n d e e d , i f one c o n s i d e r s t h e i d e n t i t y mapping from ( L 1

-

which i s n o t y - c o n t i n u o u s ,

lll,~J

1

into ( R

, 11 I)

b u t i s t h e p o i n t w i s e l i m i t of t h e

p r o j e c t i o n mappings pn : (x,)

(~l,***r~~rOrOr---)

which a r e c o n t i n u o u s , i t i s c l e a r t h a t a Banach-Steinhaus theorem can o n l y hold under r a t h e r s p e c i a l r e s t r i c t i o n s .

4.8. D e f i n i t i o n : L e t E be a l o c a l l y convex s p a c e . Then E h a s t h e Banach-Steinhaus p r o p e r t y i f f o r e v e r y sequence ( T n ) of c o n t i n u o u s l i n e a r mappings from E i n t o a l o c a l l y convex s p a c e F such t h a t t h e p o i n t w i s e l i m i t

T : x M l i m (Tnx)

(x

E

E)

e x i s t s , t h i s l i m i t i s c o n t i n u o u s . The f o l l o w i n g remarks are evident:

I. MIXED TOPOLOGIES

46

a)

in this definition, one can assume that F is a

Banach space: b)

inductive limits and quotients of spaces with the

Banach-Steinhaus property also have this property: c)

the Banach-Steinhaus property is possessed by a-barrelled

spaces (and in particular by barrelled spaces) (recall that a locally convex space is a-barrelled

if every barrel which

is the intersection of a countable family of neighbourhoods

of zero is itself a neighbourhood of zero): d)

if E is a Mackey space, it is sufficient to verify

the condition for linear forms (i.e.that El is a(E',E) sequentially complete).

4.9.

Definition: Let ( E , B , ' I )

be as in 1 . 4 . We say that E satis-

fies condition C 1 if, for every n

E

N every xo

E

Bn, and every

T-neighbourhood of zero U there is a -r-neighbourhood V of zero so

that

v

n B~ c_ ((x0 + U I n B ~ )- ((x0 + u) n B ~ ) .

The importance of this property is that it allows us to deduce 'IIB-continuity of an operator on some B

E

B from its

'I

I B-'On-

tinuity at a single point.

4.10. Lemma: Let (E,B,T) satisfy condition

C,

and let F be a

family of linear mappings from E into a locally convex space F. Suppose that the following condition is satisfied:

SPECIAL RESULTS

1.4

for each B

8 and each absolutely convex neighbourhood

E

W of 0 in F there is an xo so that

T( (xo

47

+

E

B and a T-neighbourhood U of 0

Txo

U ) fl B ) C

+ W. Then

F is y-equicontinuous.

Proof: By Grothendieck's Lemna ( 1 . 8 ) we need only show that FIB

is

T I B-equicontinuous

at 0. Let W be an absolutely convex

neighbourhood of zero in F. Then there is a T-neighbourhood U of zero so that

T ( ( x ~+ for each T

E

T(V

u) n

B) EL T X + ~ w/2

F. Choose V as in 4.9..

n B) E. T ( ( x ~+ u ) n (Txo + W/2) =

4.11.

B

-

Then for each T

-

(x0 +

u) n

E

F,

B)

(Txo + W/2)

w.

Proposition: Suppose

a)

that (E,B,T) satisfies condition 1,;

b)

B has a basis B1 so that ( B , T I ~ )is a Baire space for

each B

E

B1.

Then (E,y) has the Banach-Steinhaus property.

Proof: Let (T,)

be a pointwise convergent sequence of y-con-

tinuous linear mappings from E into a Banach space F. Let B be a TIB-Baire space. We shall find a point xo

E

E

8

B and a

T-neighbourhood U of zero so that Tn( (xo + U) n B)

C_

Tn(xo) + EBII1lF

for each n.

Then (Tn) will be y-equicontinuous by 4 . 1 0 and this will Suffice

I . M I X E D TOPOLOGIES

48

t o prove t h e r e s u l t . L e t := i x

E

B :

IIT~x - T~xII

N

E

0

so t h a t An

0

B =

0

(xo + U) flB

a T-neighbourhood U of 0 s o t h a t

II TnoX -

TnoXo II 5 ~ / 3 if

x

+ u)

11 T

E

(x0 ~ X -

4.12.

T"0

XI]

f o r p,q 2 nl

U An. Hence t h e r e i s an nEN c o n t a i n s an i n t e r i o r p o i n t x NOW choose

d Then An i s ~ I ~ - c l o s eand n

I ~ / 3

x E (xo

+

C_

.

A

"0

and

U ) n B. Then i f

n B I

IIT~x -

TnoXII +

C o r o l l a r y : Suppose

IIT~X -~

~

a) that (E,B,T)

b ) B h a s a b a s i s of TIB-compact o r

TI

B

+

II TnoXo ~ -1 T ~ x 1~ I I I

s a t i s f i e s C,

and

- m e t r i s a b l e and complete

s e t s . Then (E,y) h a s t h e Banach-Steinhaus p r o p e r t y .

One problem which h a s proved t o be c e n t r a l i n t h e t h e o r y of f u n c t i o n s p a c e s w i t h s t r i c t t o p o l o g i e s i s t h a t of d e t e r m i n i n g whether t h e g i v e n s p a c e i s a Mackey s p a c e ( i . e . i f it h a s t h e f i n e s t l o c a l l y convex topology c o m p a t i b l e w i t h i t s d u a l ) . Most f a m i l i a r Mackey s p a c e s have t h i s p r o p e r t y by v i r t u e o f . some s t r o n g e r p r o p e r t y ( e . g . t h a t of b e i n g b o r n o l o g i c a l ) . O n c e a g a i n , w e must seek a n o t h e r approach f o r mixed t o p o l o g i e s . O u r f i r s t r e s u l t g i v e s a p o s i t i v e answer f o r Saks s p a c e pro-

d u c t s of Banach s p a c e s . We r e q u i r e t h e f o l l o w i n g p r e p a r a t o r y result.

E

I. 4 4.13.

SPECIAL RESULTS

AEa),l

Proposition: Let { (Ea,II

49

be a family of Banach

spaces. Then ( S n Ea, B C Ed,)

1)

-

is a dual pair under the bilinear

mapping ((x,), (fa))

fa(xa).

C

~ E A

2)

Under this duality, B C Ed,, with its Banach space struc-

ture is identifiable with the y-dual of 3)

a subset C of

B C Ed, is y-equicontinuous if and only

if it is norm-bounded and, for each subset J of A so that

Proof: We can regard

SII Ea;

C Ilf,lI ~ E AJ\ SnEa

E

IE

> 0, there is a finite

for each (fa)

E

C.

as the projective limit of the

spectrum defined by the spaces

{B

ll

E, : J

UEJ

J(A))

E

where

J(A) denoted the family of finite subsets of A. Now there is a natural isometry from

( B a!J

E,)'

onto

B

C

~ E EJ J

and the

result follows then from 3.10.

We shall require the following version of Schur's theorem:

4.14.

Proposition: A subset C of

1

(S) is relatively u (eel,em)-

compact if and only if it is norm-bounded and for each there is a finite subset J of S so that each (x,)

E

C

a€S\J

IIxaII

IE

E

> 0 for

C.

In particular, there is a countable subset S1 of S so that

I. MIXED TOPOLOGIES

50

Proposition: Let {E,]

4.15.

be as in 4 . 1 3 with

Then (E,y) is a Mackey space (i.e. y

=

E := S

IT Ea.

T(E,E')). Y

Proof: It is sufficient to show that if C is a weakly compact

B C E;,

subset of

then C satisfies condition 4 . 1 3 . 3 ) .

We

first show that the support A1 of C is countable. (A1 is the

C exists with f B For each B E A1 we choose an x E E so that IIx 11 = 1 B B B B 0 for some (fa) E C. Then the mapping f (x ) B B set of those f3

B .c E;

from

E

A for which an (fa)

into L 1 ( A ) is

0

0).

E

(B c E;,

and

B IT E ~ )- u ( t l ( A ) , L ~ ( A ) )

continuous. For this is equivalent to the fact that for each

(Aa)

on

E

Lm(A)

is defined by an element of

B C EA

clearly is

the linear form

-

the element (1 x

cla

)

. The

B ll Ea

-

which it

image of C under this

mapping is thus weakly compact and so has countable support (4.14).

But, by construction, the support of this set is A1.

Thus we can consider the case where the indexing set is N. If C does not satisfy the condition of 4 . 1 3 . 3 ) ,

there is a

strictly increasing sequence of positive integers (nk ) and a sequence (f(k)) in positive and : f

E.

c

"k+ 1

c Ilfnkll 2 E for some n=nk+ I For each n, there is an xn E En so that IIxnII so that

11

/2

(n, < n

I

nk + l

'

= 1

SPECIAL RESULTS

1.4

and so

{(fn(xn)) : f

E

51

C ) is not weakly compact (4.14)

-

contradiction.

4.16. Proposition: Let

spaces and let (E, 11

11,~)

c ( E a r 11 ]la) IaEA

be a family of Banach

be the product

S

n

,€A

E,.

Then (E,y)

has the Banach-Steinhaus property.

Proof: By 4.15 and 4.8.d), it is sufficient to show that the pointwise limit of a sequence (f,)

of continuous linear forms

is continuous. We use the isometry

E' y

B C Ela. Let ~ E A

=

a

fn = ( f n ) .

a

Then the sequence (fn)aEA is pointwise convergent to a continuous linear form f" on E, for each

c1 &

of uniform boundedness, sup{IIfnII

E

sup {

c

aEA

IIfnII

:

n

E:

N)


E

I

8' 0

i.e. that and

is coarser

A := It

E

S : $(t) t € 1

IEsu l@(t)Il-' we have

tER the other hand, if V is a 8-neighbourhood of zero,

then, by 1.4.5, it contains a set of the form {x

E

C"(X) : PAn(X) I An)

where (An) is an increasing sequence in S and (A,)

is a strictly

increasing sequence of positive numbers which converges to infinity. Then if

11.1

A;’ +:t0 @

83

THE STRICT TOPOLOGIES (t

E

A,)

(t

E

An \ An-l)

(t

E

X \

U

An)

+

is in LS and V contains the unit ball of p

+ *

We now give a Stone-WeierstraB theorem for B S . For convenience, m

we consider the space CIR (X) of real valued functions on X as a real vector space. The results can be extended to complexvalued functions using standard methods. Since the sets of S are not necessarily compact, we need a refinement of the classical Stone-WeierstraR theorem due to NEL. Recall that a subset of X is a zero-set if it has the form x-’(O) for some x

m

E

.

CIR (X) A subset M of C i (X) separates disjoint zero-sets

if for each pair A,B of disjoint zero sets in X, there is an x

E

-

-

M so that x(A) and x(B) are disjoint. The following Lemma

follows from the fact that the points of the Stone-zech compactification f 3 X of X are limits of z-ultrafilters in X (cf. GILLMAN and JERISON [551, Ch. 6). Its proof can be found in NEL [ 1081

. m

1.12. Lemma: Let M be a subset of CIR (X) which separates disjoint zero sets in X. Then M, regarded as a subset of C i (8 XI, separates the points of B X . m

1.13. Proposition: Suppose that M is a subalgebra of Cn (X) so that for each A

E

S, MA, the restriction of M to A , separates

11. BOUNDED CONTINUOUS FUNCTIONS

84

disjoint zero-sets in A and contains a function which is m

bounded away from zero on A. Then M is 6s -dense in Cn (X)

.

Proof: We can assume that M is BS-closed. Then it is norm closed and so is a lattice under the pointwise ordering. We show that if x E C L (XI I 0 < is a y

E

E

< 1

M so that IIylloos 11x11, + 1

and if A and

E

S I then there

pA(x-y)

IE.

MA, regarded as an algebra of functions on BA, satisfies the conditions of the classical Stone-WeierstraR theorem (for MA seperates the points of f3A by 1.12). Hence MA is norm-dense m

in Cn (A) and so there is a y 1

E

M with

pA(x-yl) 5 E. Then

y := sup{ inf ( y l l llxllm + 1 1 , - ( I l x l ~+ 1))

is the required function.

1.14. Corollary: Let M'be a subalgebra of C i (X) so that M

separates the points of X and for each t k X there is an x E M with

m

x(t) f 0. Then M is BK-dense in Cn (X).

We now consider the problem of characterising those spaces for which (Cm(X),B~)is separable. Since the Banach space Cm(A) ( A

E

S) can only be separable if A is compact (this is a

classical result of M. KREIN and S. KREIN and follows from the fact that the Stone-Eech compactification of a non-compact space is never metrisable Cm(A) is

-

see GILLMAN and JERISON 1551, 5 9.6) and

at least when X is S - S normal, a continuous image

of (C"(X , 8 , ) ,

it is natural to impose the condition that S C K

i.e. the sets of S are compact.

11.1 1.15. Proposition: If

THE STRICT TOPOLOGIES

F C

S C K

then

(C"

85

(X),Bs)

is separable

if and only if there is a weaker separable, metrisable topology on X.

Proof: Since we shall use the Stone-WeierstraR theorem, it is convenient to restrict attention to C,

m

(XI. m

If M is a countable BS-dense subset of C , ( X ) ,

then the weak

topology defined by M satisfies the given conditions. Sufficiency: let

T

be a suitable separable, metrisable topology

on X. Then by Urysohn's metrisation theorem (WILLARD (X,T)

[

can be embedded in a compact, rnetrisable space Y

571

,

23.

For

each positive integer n, we can find a finite open covering (Un) of Y so that maxi diam U : U

E

Un) S l/n

(diam (U) is the

diameter of U). Let (Pn be a partition of unity of Y subordinate to Un and denote by M the subalgebra of C G (XI generated by the restrictions of the elements of m

by 1 .I4 and so

(C, ( X I ,Bs)

;Qn

to X. Then M is BS-dense

is separable.

1.16. Remark: It follows from SMIRNOV's metrisation theorem (see WILLARD [1571, 23.G.3) that if X is locally compact and paracompact and S possesses a weaker metrisable topology, then X is metrisable. Hence if X is locally compact, paracompact

and ( C m ( X ) ,Bs)

is separable, then X is metrisable (cf. SUMMERS

11391, Th. 2.5).

)

11. BOUNDED CONTINUOUS FUNCTIONS

86 1.17.

Proposition: If X is discrete, then (C"(X),Bs) is

separable if and only if

card (X) I card (IR)

.

Proof: The necessity follows from 1 . 1 5 and the fact that the cardinality of a separable metrisable space is at most card (XI). On the other hand, XI metrisable topology

1.18.

-

(and hence any subset) has a separable, the natural topology.

Remark: A more intricate argument shows that the same

result holds for metrisable spaces X (see SUMMERS [ 1 3 9 1 , Th. 3.2).

In the third section of this chapter we shall consider duality for Coo(XI , with strict topologies. However, using the theory of Chapter I we can already provide some information on this duality, without specifically calculating the dual space

-

in

particular, we can give sufficient conditions for B K to be the Mackey topology and we can characterise the relatively weakly compact subsets of C"(X).

1.19.

Partitions of unity for COD(X): Let X be a locally compact,

paracompact space. Then there exists a partition of unity on X (see BOURBAKI [ 1 5 1 , IX.4.3) so that Now (C"(X) , I ] system

1 1 , ~ ~ ) is

{C(K) : K

E

: K E K )

supp $ I K G K .

the Saks space projective limit of the

K)

of Banach spaces and if we define the

mappings TK :

{$IK

x

-

(x

11.1

THE STRICT TOPOLOGIES

from C(K) into C-.(X) where (x $IK)

A

a7

denotes t h e extension of

x $IK to a function on X obtained by setting it equal to zero off K, then (TK) is a partition of unity in the sense of 1.4.17. Hence we have, by 1.4.19 and 1.4.24:

1.20. Proposition: Let X be locally compact and paracompact. Then (C" (X), B K ) is a Mackey space and has the Banach-Steinhaus property. Also a linear mapping from Cm(X) into a separable Frgchet space is BK-continuous if and only if its graph is closed.

In the next results, the phrase "weak topology on C" (X) will I'

be used to denote the weak topology defined by the dual of (C" (XI ,BK).

1.21. Proposition: A sequence (x,)

in C" (XI converges weakly

to x if and only if {xnl is uniformly bounded and the functions xn converge pointwise to x. 1.22. Proposition: A bounded subset B of C" (XI is weakly precompact if and only if it is precompact for the topology of pointwise convergence on X. Hence if X is K-complete, then B is relatively weakly compact if and only if it is relatively compact for the topology of pointwise convergence on X.

Proof: Using 1.1.20, we can reduce 1.21 and 1.22 to the case where X is compact (see, for example, GROTHENDIECK [62], pp. 12 and

209

for this case).

11. BOUNDED CONTINUOUS FUNCTIONS

88

1.23. Remark: Using results from GROTHENDIECK 1601 , one can

strengthen 1.22 as follows: suppose that X is K-complete and has a dense subset which is the union of countably many compact sets. Then the following conditions on a bounded subset B of a)

c”(x) are equivalent: B is relatively countable compact for

T

P

(resp. for

the weak topology); b)

B is relatively sequentially compact for

T

P

(resp.

for the weak topology); c) topology 1 Here

T

P

B is relatively compact for

.

T

P

(resp. for the weak

denotes the topology of pointwise convergence on X.

1.24. Remark: One of the main themes of this Chapter will be

that of relating the topological properties of X with the linear (or algebraic) and topological properties of C“ (X) with various mixed topologies (cf. 1.8, 1.9, 1.15 for example). We list here some examples without proofs: 1)

X is hemi-compact (i.e. K is of countable type) if

and only if *)

BII

BII II,

11, is

BK-metrisable;

is BK-separable and metrisable if and only if

X is hemi-compact and each K 3)

E

K is metrisable;

if X is locally compact, then B

I1 II,

is BK-separable

and metrisable if and only if X is separable and metrisable (alternatively if X is the countable union of compact, metrisable sets);

11.2 4)

ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

89

the following conditions are equivalent: a) B II11 is BK-compact (i.e. (C" (X),BK) b) (C" (X),BK)

is semi-Montel);

is semi-reflexive;

m

c) (C (X),BK) is a Schwartz space; d) X is discrete. 5)

m

(C (X),BK) is nuclear if and only if X is finite;

B K = ' I ~on C"(X) if and only if the union of countably many compact subsets of X is relatively compact. If this is the 6)

case, then CCm (XI , B K ) is a (DF)-space.

1.25. Remark: A number of results given in this section for Cm (X) with the topology B K (e.g. those of 1.24) can be extended

to B,

with the natural changes. We leave the task of carrying

out such extension to the interested reader, mentioning only that 1.20 can be extended to the topology B B by replacing the assumption of local compactness by local boundedness (obvious definition ! ) and that SCHMETS and ZAFARANI have studied the topology B p in I: 1 2 4 1 .

11.2. ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

In the first part of this section, we work exclusively with the strict topology defined by the family K of compact subsets of X. To simplify the notation, we denote it by 8 . First we note that (Cm(X),11

1 1 , ~ ~ ) is

a pre-Saks algebra (that is, its

completion is a Saks algebra).

11. BOUNDED CONTINUOUS FUNCTIONS

90

2.1

. Proposition: Multiplication is continuous on

(C" (X),B )

Proof: We use the representation of $ given in 1 . 1 1 . does JI :=

so

If @

.

E

+

LK,

and we have the following inequality

In general, inversion is not continuous on C"(X) and so (Cm(X),B) is not a locally multiplicatively convex algebra in the sense of MICHAEL [ 9 9 1 .

If t

E

X then

6,

:

x

x(t)

.

is an element of the spectrum M (C"(X)) of C"(X) We have thus Y (Cm(X)). 6 t from X into constructed a mapping 6 : t

5

We call it the generalised Dirac transformation. It is injective since

2.2.

c"(x)

separates X. m

proposition: 6 is a homeomorphism from X onto Mv(C (X)).

Proof: Since the topologies on X and M (Cm(X)) are the weak

Y

topologies defined by Cm(X), it is sufficient to show that 6 is surjective. Let f be a $-continuous multiplicative functional on C"(X) m cR

x

E

and denote by M the kernel of the restriction of f to

(X). Then there is a to E X so that

M (for otherwise M would satisfy the conditions of 1 . 1 4 and m

SO

x(to) = 0 for each

would be $-dense in CR (X) i.e. € would be zero).

11.2

91

ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

-

Note that M separates X. For otherwise there would be points s1,s2 in X so that

x(sl) = x(s2)

in the kernel of the linear form

for x x

M. Then M would lie

E

x(sl)

.

so would have codimension at least two

-

f = 6

If X,X1 are completely regular spaces, $ : X d XI tinuous, then C"($)

:

x

x

0

and

Then M = {x : x(to) = 0 )

(for both these sets have codimension one) and so

-

x(s2)

tocon-

$ m

m

is a @-continuous star homomorphism from C (X,) into C (XI. In fact, every such homomorphism has this form as the following result shows:

2.3. Proposition: If 0 is a 8-continuous homomorphism from Cm (XI) into

Cm (X)

then

has the form Cm 9 ) for some continuous

mapping $ from X into X I . Proof: If t -

E

X then 6t

Q, is a @-continuous multiplicative

form on Cm(Xl) and so is defined by a unique element of X I

-

we denote this element by $(t). By the construction of this mapping I$ we have @(x) = x

for each x

E

0

9

Cm(Xl). Hence for each x

E

m

C (XI),x o I$

E

Cm(X) and

this property characterises continuity for mappings between completely regular spaces.

11. BOUNDED CONTINUOUS FUNCTIONS

92

2.4. Corollary: X and X, are homeomorphic if and only if

C"(X) and C"(X1) are isomorphic as pre-Saks algebras.

We remark that the following version of the Banach-Stone theorem for non-compact spaces can be deduced from 2.4: if there is an isometry from C"(X) onto C"(X,) which is also 8-bicontinuous, then X and XI are homeomorphic.

If (A,lI if x

11,~)

-

is a commutative pre-Saks algebra with unit and

A, then the mapping

E

2

:

f

f(x)

from M ( A ) into C is an element of Cm(M ( A ) ) . Thus we have con-

Y

Y

structed an algebra homomorphism from A into Cm(M (A)). We call

Y

it the generalised Gelfand-Naimark transform. Note that we can regard My (A) as a subspace of the spectrum M(A) of the normed algebra (A,11

11).

The generalised Gelfand-Naimark transform is

then the composition of the Gelfand-Naimark transform for A and the restriction operator from C(M(A)) into Cm(My(A) )

. In

particular, if A is a pre-Saks C"-algebra, then the generalised Gelfand-Naimark transform is a star-homomorphism (this also follows directly from 2.3).

2.5. Proposition: If (A,II

11,~)

is a commutative Saks (?-algebra

then the generalised Gelfand-Naimark transform is an algebra isomorphism from A onto Cm(M (A)). Y

93

ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

11.2

P r o o f : We f i r s t n o t e t h a t t h e image of A i n C m ( M y ( A ) ) i s a self-adjoint,

s e p a r h t i n g s u b a l g e b r a which c o n t a i n s t h e c o n s t a n t s

and so i s @ - d e n s e by t h e complex v e r s i o n o f 1 . 1 4 . Now l e t P be

a f a m i l y of C"-seminorms on A which d e f i n e For each p

E

P I w e d e n o t e by A

'I

(as i n 1 . 3 . 1 3 ) .

t h e a s s o c i a t e d @-algebra

P

and

by M ( A ) i t s s p e c t r u m . W e c a n r e g a r d M(A ) as a (compact) s u b s e t P P o f M ( A ) and w e show t h a t M y ( A ) = U M ( A 1 . I f f E My ( A ) , t h e n , Y PEP P by 1 . 3 . 1 0 , t h e r e i s an i n c r e a s i n g s e q u e n c e (p,) i n P and a n fn

E

Ap,l

so t h a t C f n is a b s o l u t e l y summable t o f . W e c a n a l s o

suppose t h a t

11 C

frill

Choose no so t h a t enough

E.


no

For i f

+


0. Then { s : x(s)

>

0)

is a relatively compact neighbourhood of t.

Using the generalised Gelfand-Naimark transform, it is easy to see that if A is a perfect, commutative Saks C"r-algebra,

11. BOUNDED CONTINUOUS FUNCTIONS

96

then M (A) is locally compact. The reverse implication is Y not true.

Let I be an ideal in C"(X) and write ~ ( 1 )= where

Z(x) := x-'(O)

n

XE I

z(~)

is the zero-set of x. Then we put

I(z(I)) := {y

E

c"(x)

:

y

=

o

on

Then I(Z(1)) is obviously a 8-closed ideal and We shall now show that

I = I(Z(1) )

~(1)). I C I(Z(1)).

if and only if I is 8 -

closed. This result is well-known for compact X and we shall use it for the proof of the result in the general case. We sketch briefly how it can be proved. Suppose that x E I(Z(1)). For each

E

>

0 we can find an open neighbourhood U of Z(1) and

a function xE in C(X) so that xE vanishes on U and We shall show that xE

E

IIx

-

xEI)I E.

I which will finish the proof. By a

...,

compactness argument, there exist x,, xn in I so that n U 2 n Z(xi). Then U contains the zero-set of the element i=l y := I xi[ of I. But then the zero-set of xE is a neigh-

*

bourhood of the zero-set of y and so xE is a multiple of y.

2.7. Proposition: Let I be a 8-closed ideal of C"(X)

Then

.

I = I(Z(1)).

Proof: I is a norm-closed ideal in C(gX) and so there is a closed set KO in BX so that

I = (x E C(f3X) : x = 0

on KO).

11.2

97

ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

It is obviously sufficient to show that KO = clf3XZ(1) (closure in 8x1 for then if a function vanishes on Z(1) its extension to f3X vanishes on KO and so is in I. If this were not the case, there would be a with yo(to) = 1

to

E

and

We now show that yo

KO \ clgxZ(I). Then there is a yo

y = 0 on a neighbourhood of

-

yKIE)

C(f3X)

clgxZ(I).

I which gives a contradiction. To do this,

E

we show that for each K that PK(Y,

E

E

E

K,

E

>

0, there is a yK

IE

in I so

and IIYK,EII 5 IIYoII. Then (YK,€) is a

net in I which is BK-convergent to yo. To construct yK

I E

we

proceed as follows: let IK denote the projection of I in C(K). Then IK is an ideal in C(K) and so

TKIits

space C(K) I is a closed ideal in C(K) {y

E

C(K) : y = 0 on

closure in the Banach

. Hence

Z(1) n K ) . Hence

yIK

it has the form E

yK and so Tietze’s

theorem implies the existence of the required yK,€.

The results of this section can be used to give a natural construction of the Stone-Eech compactification and the real-compactification of a completely regular space. We describe this briefly, firstly to display the connection between mixed topologies and the theory of topological extensions and secondly because it will allow us to give a significant generalisation of 2.2. If X is a completely regular space, (C”(X) ,[I

11)

is a Banach

algebra. Its spectrum M(Cm (XI 1 is a compact space which we denote by BX. The Dirac transformation can be regarded as a (topo-

98

11. BOUNDED CONTINUOUS FUNCTIONS

logical) embedding of X into BX. It has the following universal property: if 4 is a continuous mapping from X into a compact space K, then there is a unique continuous extension

7

of I$ to a continuous mapping from BX into K. For consider the operator Cm(I$) : C(K) d C"(X) = C(BX) ,which is

11

1l-B continuous and so (by the closed graph theorem

or, more elementarily, by 1.1.11) by 2.3, C"(@)

11 11 - 11 11

continuous. Hence,

(regarded as a mapping from C(K) into C(BX), has

7:

the form C"(7) for some

BX

___j

K.

7

has the required

property.

Now we denote by I(X) the set of those functions in C"(X) which have no zeros in X (i.e. are invertible in the algebra C(X)). Every x

E

Cm(X) has a unique extension to a function in C(@X)

which we shall continue to denote by x. Then we put

Ux where C (x) = Cs BX in BX.

:=

E

x

n E

I (XI

cBx(x)

gX : x(s)

0 )

is the co-zero set of x

UX is the real-compactification of X and X is real-compact if UX = X. The above rather unfamiliar definition is the natural one from the point of view of strict topologies. The following equivalent forms are better known:

11.2

ALGEBRAS OF BOUNDED, CONTINUOUS FUNCTIONS

ux

2.8. Proposition: 1 ) where

2

fl

=

x

E

99

x-I (1R)

C,(X)

denotes the extension of x

E

to a function from

C,(X)

BX into the 2-point compactification of El. 2)

uX is the completion of X with respect to the C(X)-uniformity

on X.

Proof: 1) follows from the simple fact that if x l/lxl

E

C,(X)

E

I(X), then

and the zeros of x in gX are precisely the points

where (l/lxl)- is infinite in value. For 2) see GILLMAN and JERISON [551,

5

15.13.

2.9. Corollary: For a completely regular space X I the following

are equivalent: 1)

X is complete for the C(X) -uniformity;

2)

X is real-compact;

3)

for each s

E

BX \ X, there is an x

E

I(X) with x(s) = 0.

Now it is clear that the bounded subsets of X are precisely those subsets of X which are precompact in the C(X)-uniformity or, by the above result, relatively compact in uX. This remark makes it natural to generalise the definitions of 1.1

to include saturated families S of subsets of UX (for reasons

which will be clear later, it is convenient to drop the assumption that the sets be closed). Hence if S is such a family, we can define the strict topology B s on C"(X).

We shall always assume

11. BOUNDED CONTINUOUS FUNCTIONS

100

that the subsets of S are relatively compact in uX. The following result is a significant generalisation of 2 . 2 .

2.10.

Proposition: The spectrum of the topological algebra

(c" (XI,

B ~ )

is

u

BES

cluX(~) (closure in UX)

Proof: It is clear that any point in

B;

.

cluX(B) defines

an element in the required spectrum. On the other hand, any point in the spectrum is defined by a member of uX (apply 2 . 2 to the space uX). Hence it is sufficient to show that if

4

is not BS-continuous. But this t0 follows easily from the fact that for each B E S there is an to

cluX(B) then 6

BII

II,

so that x(to) = 1

and

x = 0 on B.

(C"(X) ,BB)

is

u

2.11.

Corollary: The spectrum of

2.12.

Remark: The space of 2 . 1 1 has been introduced by BUCH-

BEB

clUX(B).

WALTER [ 2 2 ] who denoted it by X" (because of a certain formal analogy with the bidual of a locally convex space) in connection with the concept of a u-space i.e. a completely regular space in which

B

= K (cf. the concept of semi-Monte1 locally convex

space). Every space has a "u-ification" pX which is obtained as the limit of the transfinite series X,X", (X")",... and X is a p-space if and only if

X = ~ J X(or alternatively if X = X").

It is now clear that X is a p-space if and only if B K = B g C"(X).

on

11.3

101

DUALITY THEORY

11.3. DUALITY THEORY

A classical result of BUCK for the space C ” ( X )

(X locally

compact) is that the dual of (C” (X), B ) is the space of bounded Radon measures on X. In this section, we shall extend this result to completely regular spaces. We shall take this opportunity to discuss various equivalent definitions of Radon measures on completely regular spaces.

3.1. Definition: A premeasure on X is a member of the (vector space) projective limit of the system : M(K1)

,K

“K1

------+

M(K);

K Z K1

, K,K1

E

K (XI)

In other words, a premeasure is a system p = { p K ) of Radon measures which satisfies the compatibility relations u K1IK - ’K ( K C K 1 ) . If LI = {p,) is a premeasure on X, I L I ~ ~ ” denotes the outer measure on K defined by Thus Il-1~1’‘

I uKI

(see BOURBAKI

is defined as follows: if U E K is open,

is defined to be

sup { J f dl uKI 1

I uKl*

IuKI” (u) : u

open in K , A E

Now if C is a subset of X we define

xu.

IuKI”’ (U) For general

Iul”

u).

(C) to be

sup { l u K \ ” ( C

fl

K) : K

A premeasure

u

on X is said to be tight if for each

E

IV. 1 . 4 )

(A) is defined to be

inf {

is a K

5

where f ranges over the family

of positive, continuous functions on K with f I A 5 K,

131

K ( X ) s o that

E

K(X)).

lu]*

(X

\ K)
0 there

An equivalent condition

11. BOUNDED CONTINUOUS FUNCTIONS

102

is the existence of an increasing sequence (K,) that I pI" (X \ Kn) ---+

in K(X) so

0. We denote by Mt (X) the space of

tight measures on X. It is clearly a vector space. If x E C"(X) p E Mt(X)

, then

the limit

lim n-

XI

n

dp

Kn

independent of the particular choice of (K,).

exists and is We write

j x dp

for this limit.

If K

E

K (X) and

p

E

M(K) , then l.~ defines a tight measure on X

in a natural way: if K1 E K(X) and K1 E K we define p

K1

to be

to be the K1 measure induced on K, by p (for example, as a linear form on the restriction of

to K1. If K, 2 K we define p

-

C(K1 1 , VK1 is the mapping

x Then

:= (pK

1

xlK,du

).

1 is a tight measure on X and IFI"(X \ K) = 0.

Hence we can (and do) identify the space of measures on K with the subspace of Mt(X) consisting of those p for which

I PI''

u M(K) K E K(X) the space of measures with compact support. Then p E Mo(X) if (X \ K ) = 0. We denote by Mo(X) the subspace

and only if there is a K

E

K (X) so that

-

(X \ K) = 0 .

3.2. Proposition: The dual of (c" (x)8't.K) is naturally isomorphic

to Mo(X) under the bilinear form

Proof: (C" (X), T ~ )is a dense subspace of the locally convex

11.3

103

DUALITY THEORY

-

projective limit of the system

{ P K ~,K : C(Kl)

C(K); K c K 1 , K,K1

E

K(X)}

Now by a standard result on the duals of projective limits (see SCHAEFER, Ch.1 [61] §IV.4.4), the dual of the latter space is the union of the spaces {M(K)}KE K(X) i.e. Mo(X) under the above identification.

-

3.3. Proposition: The dual of (Cm(X), B ) is isomorphic to the space Mt(X) under the bilinear form (xru)

Proof: Each 1-1

E

j x d1.I*

Mt(X) defines a linear form on Cm(X) and we

show that it is TK-continuous on the unit ball of Cm(X). If E

>

0, choose K

x

E

Cm(X) with llxll

and 11.11'" (X)




6/2

-

l/m > 0

which gives a contradiction.

A similar argument, applied to a uniformly tight set C in VMt(X) produces a uniformly tight set in Mt(X) whose image is C. Hence the sufficiency follows from 3.15.

11.3

DUALITY THEORY

111

As an application of the theory developed in this section, we give a functional analytic proof of PROHOROV's theorem

on the existence of projective limits of measures. We suppose that

{ @ B a: X

F Xa, a I 8 , a,@

o

E

A}

is

a projective spectrum of completely regular spaces and that X is a completely regular space with continuous mappings

6,

X------+

:

Xa

so that

(a

$ B a $~B =

I

(thus the

8)

system { $ a } corresponds to a continuous mapping from X into the projective limit of the system {Xal). Suppose that : c1

{pa

E

A)

is a compatible system of bounded Radon measures

Mt(Xa) and M ( 4 ) ( p 1 = p a for a < B ) . We t Ba B seek necessary and sufficient conditions for the existence of

on {Xa) (i.e. 1-1,

a p

E

E

Mt(X) so that Mt($a) (11) = p a for each a .

3.16. Proposition: Such a ~.r exists if and only if 1)

SUP ~llPallMt(xa) : a ~ A ) < - i

2)

for each

E

lua] (X \

$,(K))

>

0 there exists a K


0, there is a K

( P I (Y \ K)
0 so that for each 6 > 0 there is a measurable set A and an x

E

H with P(A) < 6

Ixldp 2 E . We construct inductively a sequence (B,) of A measurable sets and a sequence (x,) in H as follows: choose x1 and

I

and B1 so that

IxlIdp 2 E. Now suppose that

B1

B1l...lBn are chosen. There is 6, and p (A) < 6,

>

x1 I

. .

.,xn

0 so that if A is measurable

we have

Then we can choose Bn+l and x ~ so + that ~ p(Bn+,) < 6n

I

IXn+1ldP 2 Bn Now if we put

and

and

E.

'I

Bkt one can verify that the An are k>n Ixnldp 1 ~ / 2 .The result then follows An by a standard argument (reduce to the real-valued case and then

An := Bn disjoint. Also we have:

consider positive and negative parts).

111.

164

BOUNDED MEASURABLE FUNCTIONS

1.8. Proposition: Let H be a subset of L 1 (M:p). Then the following conditions are equivalent: 1)

H is B1-equicontinuous:

2)

H is relatively o ( L ,Lm) ~ -compact:

3)

H is norm-bounded and I-equi-integrable.

Proof: 1)-

2)

follows from the ALAOGLU-BOURBAKI theorem.

3 ) : it is clear that H is norm-bounded. If it were not

2)-

1-equi-integrable, then we could choose (xn) and (A,)

as in 1.7.

Now consider the mapping T

xdpInEN An which is continuous from L1 (M:p) into L 1 (N) Then the sequence (Tx,) is not relatively weakly compact in L 1 (N) by 1.4.14 :

x -

((

.

contradiction. 3 ) W 1 ) : let H be a norm-bounded, equi-integrable subset of I?.

We use the criterium of 1.1.22 to show that H is Bl-equicontinuous.

>

0 we choose K

For

E

x

H. Choose n

E

for each x

E

E

E

K (MI so that

H where

Then p ( K ' ) s N1n-l :=

x

-

[ xi dp

M\K N so that p(A) I N1.n-1 N1 := sup{[lxllI: x

K' := It

x2

(

E

K : Ix(t)l

and so, putting

x1

I~

/ 2for each

implies

A

lxldp

E

HI. Then if x

2

n}.

:=

X'X~\Kl

E

and

x l , we have x1

E

HI := {y

E

L1 : supp y C K

and

I l y l k I n}

s ~ / 2

H I put

THE MIXED TOPOLOGIES

111.1

Hence

H E EBII

1,

165

' -equicontinuous. + HI and-HI is T loc

Corollary: (Lm,gl)is a Mackey space.

1.9.

1.10.

Corollary: Let T be a continuous linear mapping from 1

(Lm,a(Lm,L ) ) into a locally convex space F with its weak topology o(F,F')

. Then T

is B1-continuous and so transforms bounded

sequences in Lm which converge in mean on compact sets into strongly convergent sequences in F.

As an application of 1.10 consider a weakly summable mapping Q

-

from M into a locally convex space F. That is, for each f t

f (@(t) 1

is in L 1 (M;p). Then for each x

E

F',

Lm, the

E

integral

/

M

-

f (@(t))x(t)dp

exists and the mapping

f

of the algebraic dual (F')''

x

E

/

M

f ( @(t))x (t)dp

is an element

of F'. The mapping which takes

Lm into this form is then u(Lm,L 1 )-u"F')''',F')

Suppose that the range of this mapping lies in F

continuous. (@

is then said

to be strictly weakly summable or Pettis integrable). Then it is gl-continuous by 1.10.

We now consider Banach-Steinhaus theorems and closed graph theorems for Lm. We first show that (Lm,ll IL,T:~~) satisfies condition

C 1 of 1.4.9.

111.

166 1.11.

BOUNDED MEASURABLE FUNCTIONS

Lemma: The Saks space (Lml11

dition

lLI~iOc) satisfies con-

C1.

Proof: It is convenient to consider real-valued functions. Suppose that xo be an element of B

B I I IL' ll II,

composition x = x 1 +x2 and

1 1 xo-x2 11

i

E.

E

> 0 and K

with 11x11:

E

K(M) are given. Let x We

5 E/2.

of x with x1,x2

E

BII

shall find a de-

II, and

IIxo-xlII

5 E

Denote by M, the set where xo and x have the

same sign. Then we define x1 by x. := I

and

l[ xX++ xo(l xo(l + x )

on M 1 on M \ M I

x 2 := x - x l .

Then a simple calculation shows that x1,x2 IIXo-X1(I 5 E I

IIXo-X211 5

BII II, and

E.

In the next results we assume, for simplicity, that M is a-compact ( s o that B

II II,

is f31-metrisable). In fact, the results hold with-

out this assumption (see 1.14).

1.12. Proposition: If M is a-compact, then (Lm,B1) has the Banach-

Steinhaus property.

Proof: 1.11

and 1.4.11.

111.1

THE MIXED TOPOLOGIES

167

In particular, this implies the well-known result that L 1 (M;p) is weakly sequentially complete.

1.13. Proposition: Let M be a-compact. Then a linear mapping from (Lm,B1) into a separable Frgchet space is continuous if and only if it has a closed graph.

1.14. Remark: We sketch briefly the method of extending 1.12 (and hence 1.13) to general M. Since (Lm,B1) is a Mackey space, it suffices to show that it has the Banach-Steinhaus property 1 for functionals. Now each functional is represented by an L

-

function and this has a-compact support. Hence if (f,)

is a

pointwise convergent sequence of functionals on LO3, then there is a a-compact subset Mo of M so that the corresponding L 1 functions vanish outside of Mo. We can then deduce that the pointwise limit is continuous by applying 1.12 to Lm(Mo;p ) . MO 1.15. Remark: We note that it follows from 1.4.33.11

that (Lm,B1)

is nuclear if and only if Lm is finite dimensional (i.e. if and only if p has finite support). If p is atomic (so that (Lm,B1) has the form (Sm(S),B) for some set S, then (Lm,B1) is semiMonte1 and even a Schwartz space. The converse is also true (as DAZORD and JOURLIN have remarked) since if p is not atomic then, by a result of GROTHENDIECK, L 1 possesses a weakly convergent sequence which is not convergent in the norm. (Sw,B) is even a universal Schwartz space in the sense that every Schwartz space is a subspace of a product of (L",B)-space.

168

111.

BOUNDED MEASURABLE FUNCTIONS

1.16. Remark: It is perhaps interesting to point out that various ORLICZ-PETTIS theorems on unconditionally summable series (cf, THOMAS [29] ,

5

0) can be obtained from the closed

graph theorem (1.13). The point is that if E is a complete locally convex space, there is a bijective correspondence between the set of B1-continuous linear mappings T from t " ( l N ) into E and the unconditionally summable sequences in E, obtained by mapping T into (Ten) where (en) is the standard basis in k?"(N) (see GROTHENDIECK, Ref. [62] to Chapter 11, p.97).

As far as we know, the topology

B1 was first studied (implicitly)

by GROTHENDIECK [12] in his investigation of the DUNFORD-PETTIS property, in particular, to show that C(K) has this property. Since we now have sufficient machinery at our disposal to reproduce Grothendieck's proof we do so here. We recall the definition.

1.17. Definition: A locally convex space E has the strict DunfordPettis property (abbreviated to SDPP) if every a(E,E')-Cauchy sequence is Cauchy for the topology of uniform convergence on the equicontinuous, ~~(E',E")-compact discs on E' (this is equivalent to the fact that every weakly compact, continuous linear operator from E into a Banach space maps weakly Cauchy sequences into strongly Cauchy sequences). If E is a Banach space, this implies the Dunford-Pettis property for E (i.e. every weakly compact operator from E into a Banach space maps weakly compact sets into relatiuely compact sets).

111.1

THE MIXED TOPOLOGIES

169

1.18. Proposition: Let K be compact. Then the Banach space C(K) has the SDPP.

Proof: Let (xn) be a weak Cauchy sequence in C(K)

. Then

(x,)

is

uniformly bounded and pointwise Cauchy (cf. 11.1.21) and so has a bounded, pointwise limit x (which is then measurable for any Radon measure on K). We must show that

(

(K

xn dp) converges

uniformly on weakly compact sets of Radon measures. If this were not the case, we could find a relatively weakly compact sequence (pk) of Radon measures on which convergence is not uniform. By

a standard construction, there is a positive Radon measure p on K so that each pk is absolutely continuous with respect to p (take 11 := C 2-kl pkl 1 . Then we can regard {pk) as a subset of k L 1 ( K ; p ) and it is relatively weakly compact there. Now by Egoroff's theorem, the sequence (x,) in Lm(K;p)) converges to x in

B1

(regarded as a sequence

and so uniformly on weakly com-

pact subsets of L' ( K i p ) by 1.9. This gives a contradiction.

1.19. Remark: I.

Since a Banach space predua1,of a space with

SDPP has SDPP and the dual of L1 is a C ( K ) , it can be deduced from this result that every L1-space has the SDPP. 11.

The same proof shows that ( C m ( S ) , B )

has the SDPP f o r any

locally compact space S.

1.20. Remark: We have chosen to develop the theory of Lm -spaces in the context of Radon measures on locally compact spaces.

111.

170

BOUNDED MEASURABLE FUNCTIONS

In fact, the theory given here can be developed in the more general setting of abstract measure theory and we indicate briefly how this can be done. Let (M,C) be a pair where

C

is a

a-algebra on the set M. A positive regular measure is a uadditive function p from C into

IR'U

{m)

which satisfies the

(regularity) condition:

for each A

E

C.

Then we can develop as usual a theory of integration with respect to p for complex-valued functions on M and, in particular, we can define the spaces L 1 (M;p) and L 1 (M;p) of absolutely integrable functions respectively of equivalence classes of such functions. L1 (M;u) is a Banach space under the norm

1 1 Ill

: x

+---+

(MIxldp

. Note that the regularity of 1.1

can be

1

expressed in functional analytic terms as follows: L (M;u) is the Banach space inductive limit (cf. the introductory remarks to 5 1.3) of the spectrum {L' (A;vA) : A

E

Co 1

where Co denotes the family of sets of C with finite U-measure.

For the time being, we suppose that LOD

u

is finite (i.e. p(M)
-

be

and which

-Z

is

Q-linear and satisfies the condition

We can extend p by continuation to obtain the required lifting.

3.11

Proposition: Let E be a Banach space so that El is separable

Then the mapping constructed in 3.6 is surjective (i.e. the dual of L (M;E) is L ~ ( M ; E ~ ) ) .

111.3

VECTOR-VALUED MEASURABLE FUNCTIONS

185

Proof: Let 9 be an element of L(M;E)'. Then for each x

E

E,

the mapping

is an element of the dual of L 1 (M;u) and so is represented by a function in Lm(M;p). Hence we can regard T

9

as a mapping from

I(@II.

E into Lm(M;p). It is clearly linear and has norm 5

Now

the range of T is a separable subspace of Lm(M;p) (since E is 9 separable) and so there is a lifting p on it.

Now if t

E

M I we define f(t) to be the composition

-

where the last arrow is evaluation at t. Then f(t) norm is f : M

I

E

E' and its

II$II. Hence we have constructed a bounded function El

and for any x

E

E, < x , f > is measurable.

Hence we can apply 3.8 to deduce that f is Lusin measurable (note that any countable dense subset of E is norm-determining as a subset of the dual of El). To complete the proof, we must show that for any x

E

L 1 (M;E) we

have $(XI =

1 <x,f >dp.

It suffices to show this for the case where x has the form (y

E

L1 (M;p), z

E

El. Then we can calculate:

y

@

z

111.

186

BOUNDED MEASURABLE FUNCTIONS

Using this result, we can obtain immediately from 1.1.18 the following result:

3.12. Proposition: Let F be a Banach space whose dual is sepal rable. Then the natural duality bewteen Lm(M;F) and L (M;F1) establishes an isometry from L 1 (M;F') onto the p1-dual of L~(M;F).

Before stating the next result, we remark that the definition of equi-integrability (given before 1.2) can be extended to functions with values in a Banach space simply by replacing absolute values by norms.

3.13. Proposition: When F is as in 3.12 and H is a subset of L1 (M;E

,

)

then the following are equivalent:

1)

H is Bl-equicontinuous;

2)

H is relatively U(L 1 (M;FI),L~(M;F) )-compact;

3)

H is norm-bounded and I-equi-integrable;

Proof: 1)

2)

and

3)

1 ) can be proved exactly as in

the scalar case. 2)-

3): H is clearly norm-bounded. If H were not l-equi-inte-

grable, then by 1.7 (applied to the set (IlflI : f

E

H}

in L1 (M;p))

there would be a disjoint sequence (An) of measurable sets, a sequence (f,)

in H and an

E

> 0 so that

111.3 For every n by An'

E

VECTOR-VALUED MEASURABLE FUNCTIONS

N choose yn

IIYnIIm~1

E

187

Lm(M;F) such that yn is supported

and

(this can be done by approximating fn by simple functions). Now consider the mapping

from L1 (M;F') into 11 (IN) weakly compact

-

. Then T

is continuous but ET fn 1 is not

contradiction.

3.14. Corollary: Under the assumption of 3.12 , (Lm(M;E),B1 ) is a Mackey space.

3.15. Remarks on the Radon Nikodym property: If we compare 2.4 with the scalar result, we are led naturally to pose the following question: when can we identify the space Mv (C;E) of bv v-continuous measures of bounded variation with the space L ' (M;E) i . e . when does every E-valued v-continuous measure of bounded

variation have a v-derivative? The following example shows that this is not always the case: Example: Our measure space is [0,11 with Lebesgue measure. We define a vector-valued measure p by defining p(A) to be the sequence

((A

sin(2mt)du(t))i=l. Then p takes its values in co

(RIEMANN-LEBESGUE Lemma) and it is routine to show that it is a-additive and v-continuous. Now if 1.1 has a v-derivate, then it must be the function t

I----

--G

.

(sin(21~nt) ) However, this

188

BOUNDED MEASURABLE FUNCTIONS

111.

function does not take values in co. We formalise the above fact in the following definition: Definition: A Banach space E has the Radon Nikodym property (abbreviated RNP) if for each measure space (M;v) the canonical injection L1 (M;E)

qV(C;E)

is surjective. We remark that it is sufficient to check this condition for the canonical measure space [0,11. The above example shows that co does not have the RNP. As a positive result we have : Let E be a Banach space with a boundedly complete basis (x,). Then E has the RNP. We sketch the proof (as we shall see later, this result follows from 3.12

. It is convenient to renorm E so that n

m

NOW let p be an E-valued measure which is v-continuous. Then the induced measures Ifn

o

p)

(where (f,)

is the associated biortho-

gonal sequence) are v-continuous and so there are measurable functions yn so that fn

o

p(A) =

1

A

Yndp

(A

E

1).

Now it follows from the fact that the basis is boundedly complete that the E valued series Cynxn converges p-almost everywhere to a measurable function (since the simple estimate

111.3

VECTOR-VALUED MEASURABLE FUNCTIONS

189

bounded almost everywhere). 11-11 is the variation of F (i.e. 11-11 (A) = Sup CIIF(An)

11

;

the supremum being taken over the finite

partitions of A). The limit is a v-derivative for

u.

For a detailed study of the Radon-Nikodym property, we refer to DIESTEL and UHL [ 8 1 and BUCHWALTER and BUCCHIONI [3]. For the reader's orientation, we quote the following results: I.

Any reflexive space and any separable (or even weakly

compactly generated) dual space has the RNP. In fact, the dual of a separable Banach space has RNP iff it is separable. 11.

A Banach space has the RNP if and only if every separable

subspace has the RNP. 111. The spaces L1 (M;p), C (K) and Lm(M;p) do not have the RNP (except for the trivial exceptions and 1-1 of finite support resp. )

-

p atomic, K finite

.

In fact, Proposition 3.12 is a special case of the following result:

the space L'(M';E') is naturally isometric to the 6 , dual

of LOD(M;E) if and only if E' has the Radon-Nikodym property. Hence we can interpret 3.12 as stating that a separable dual space has the RNP (and this result contains the result that a space with a boundedly complete basis has the RNP of course). In the general case, one can identify the dual of (Lm(M;E),B1) with the lower star space LA (M;E')

introduced by SCHWARTZ

(roughly speaking, this is the space of those functions x which are weak-star measurable and are such that the upper integral

190

111.

BOUNDED MEASURABLE FUNCTIONS

of the norm function llxll is finite). Since a reflexive space has the RNP, 3.12 and its Corollaries hold for such spaces.

111.4. NOTES

The topology 6, has never been studied before (of course, 1.1.1) identifies it with a familiar topology of the dual pair ( L " , L ' ) ) . The study of B 1 was suggested by GROTHENDIECK's paper [I21 where the version of the DUNFORD-PETTIS theorem proved here (1.8) can be found. 1.4 is due to GODEMENT. Many of the results of

5

1,

in particular 1.9, were obtained independently by DAZORD and JOURLIN 171. The result that ( L m l B 1 )is a Mackey space has also been obtained by STROYAN (for bounded measures). The remark that (e",B,) is a universal Schwartz space is due to JARCHOW [15].

SENTILLES [271 has also introduced a strict topology for LEOspaces in his study of a measure free approach to L'-spaces based on the Boolean algebra of measurable sets. For a complete &

account of the integration theory used in 1.20 and in particular, for the definition of Xm(M;C) see the lecture notes of BUCHWALTER and BUCCHIONI [3].

Our basic reference for

9

2 was DIESTEL and UHL ( [81, Ch. 1)

-

in

particular, 2.2 and 2.3 can be found there.

As mentioned in the introduction,

9

3 is devoted to the elementary

beginnings of a theory of measurable functions with values in a

111.4

NOTES

19 1

Saks space. We refer the reader to SCHWARTZ [25] for a dis-

cussion of the problems involved in the duality theory for Banach space valued LP-functions.

The final form of this chapter owes a great deal to the assistance of W.SCHACHEFMAYER. In addition to simplifying and improving some proofs he pointed out the important references [31 and [81 to me. Paragraphs 1.20. 3.6

-

3.13 are based on [231and [241.

192

BOUNDED MEASURABLE FUNCTIONS

111.

REFERENCES FOR CHAPTER 111.

[I] T. AND0

Weakly compact sets in Orlicz spaces, Can. J. Math. 14 (1962) 170-176.

[21

N. BOURBAKI

Elbments de Mathgmatique: Intggration, Chs. 1-4 (Paris, 1965).

[31

H. BUCHWALTER, D. BUCCHIONI Intggration vectorielle et th&or$me de Radon-Nikodym (Dgpartement de Mathematiques, Lyon, 1975).

[41

A.K. CHILANA

The space of bounded sequences with mixed topology, Pac. J. Math. 48 (1973) 29-33.

[51

H . S . COLLINS

On the space L"(S) , with the strict topology, Math. 2. 106 (1968) 361-373.

161

J.B. CONWAY

Subspaces of (C(S), B ) , the space ( L " , B ) and (Hm,B) Bull. Amer. Math. SOC. 72 (1966) 79-81.

171

J. DAZORD, M. JOURLIN Une topologie mixte sur l'espace LOD, Publ. DBp. Math. Lyon 11-2 (1974) 1-18.

[81

J. DIESTEL, J.J. UHL, Jr. The theory of vector measures (to appear in the American Mathematical Society's surveys).

[91

J. DIXMIER

Les algebres d'opgrateurs dans l'espace hilbertien (Paris, 1969)

.

[lo] D.H. FREMLIN

Pointwise compact sets of measurable functions, Man. Math. 15 (1975) 219-242.

[I11

A. GOLDMAN

L'espace des fonctions localement Lp sur un espace compactologique st (Preprint).

[I23

A. GROTHENDIECK

Sur les applications linbaires faiblement compactes d'espaces du type C(K), Can. J. Math. 5 (1953) 129-173.

REFERENCES

193

[131

E.A. HEARD

Kahane's construction and the weak sequential completeness of L I , Proc. Amer. Math. SOC. 44 (1974) 96-100.

1141

A. IONESCU TULCEA

On pointwise convergence and equicontinuity in the lifting topology I;II (Z. Wahrsch. u. verw. Gebiete 26 (1973) 197-205 and Adv. in Math. 12 (1974) 171-177.

1151

H. JARCHOW

Die Universalitst des Raumes co. Math. Ann.

[161

J.W. JENKINS

On the characterization of abelian W"algebras, Proc. Amer. Math. SOC. 35 (1972)

203 (1973) 211-214.

436-438.

[171

S.S. KHURANA

A vector form of Phillip's Lemma, J. Math. Anal. Appl. 48 (1974) 666-668. Weakly convergent sequences in Lm(Preprint).

[181 [191

V.L. LEVIN

Lebesgue decomposition for functionals on the space LT of vector-valued functions, Func. Appl. 8 (1974) 314-317.

[20]

D.R. LEWIS

Conditional weak compactness in certain inductive tensor products, Math. Ann. 201 (1973) 201-209.

[21]

S. SAKAI

C2'-algebras and W"-algebras (Berlin, 1971

[22]

S. SAKS

On some functionals, Trans. Amer. Math. SOC. 35 (1933) 549-556.

1231

W. SCHACHERMAYER Mixed topologies on Lm for abstract measures (written communication).

[241

Mixed topologies in spaces of Banach valued functions (written communication).

.

111.

194 1251

L. SCHWARTZ

[261

2.

[27]

F.D. SENTILLES

SEMADENI

BOUNDED MEASURABLE FUNCTIONS Fonctions mgsurables et fc-scalairement mgsurables, mesures banachiques majorges, martingales banachiques et proprigtb de Radon Nikodym, Sgm. Maurey-Schwartz 1 9 7 4 / 7 5 , EXp. 4 . Banach spaces of continuous functions I, (Warsaw, 1 9 7 1 )

.

1

L -space for Boolean algebras and semi-reflexivity of spaces L ~ ( M , c , ~ ) (Preprint)

An

.

[281

K.D. STROYAN

A characterisation of the Mackey uniformity m(Lm,L1) for finite measures, Pac. J. Math. 49 ( 1 9 7 3 ) 223-228.

[291

E. THOMAS

1301

J. VESTERSTRgM, W. WILS On point realization of Lm-endomorphisms, Math. Scand. 2 5 ( 1 9 6 9 ) 178-180.

1311

C.R. WARNER

The Lebesgue-Nikodym Theorem for vector valued Radon measures, Memoirs of the American Math. SOC. 1 3 9 ( 1 9 7 4 ) .

Weak Jc dense subspaces of Lm(lR) , Math. Ann. 1 9 7 ( 1 9 7 2 ) 180-181.

CHAPTER IV

-

VON NEUMANN ALGEBRAS

Introduction: In this chapter, we study algebras of operators on Hilbert space from the point of view of mixed topologies. On the algebra L(H) (H a Hilbert space) there are three classical topologies

-

corresponding to uniform, strong and

weak convergence of operators. For most applications (in particular, to spectral theory), the uniform topology is too strong. From a theoretical point of view, the weak and strong topologies are very unsatisfactory. VON NEUMANN introduced rather complicated topologies topologies

-

-

the ultraweak and ultrastrong

to overcome these difficulties. We introduce here

three natural mixed topologies on L(H) which are related to (but distinct from) the ultraweak and ultrastrong topologies and have several advantages with respect to these topologies.

There are two distinct approaches to algebras of operators: the direct approach by defining them as self-adjoint, ultraweakly closed subalgebras of L(H) (von Neumann algebras) or

.

the axiomatic approach (W" -algebras) Comprehensive treatments can be found in the books of DIXMIER [ 8 1 and SAKAI [I41 respectively. The latter approach is certainly the most elegant. However we have chosen the former since it allows us to give a more elementary treatment. In fact, with the exception of one rather deep theorem of SAKAI, we require only acquaintance with spectral theory of self-adjoint operators and (what is

195

IV.

196

VON NEUMANN ALGEBRAS

essentially the same) the representation theory of commutative von Neumann algebras.

In

9

1 we consider two mixed structures on L(H) and identify

its dual as the space of nuclear operators on H. We then present the corresponding Saks spaces as a projective limit of spaces of finite dimensional operators and deduce that L(H) has the approximation property.

In the second section, we consider three mixed topologies

B,,

Bs and

Bs2b

on a von Neumann algebra. For commutative von

. . Lm-spaces) these coincide with the

Neumann algebras (i e

topologies considered in Ch. I11 (in this case, Bs

-

BsgC

=

8,).

After giving routine properties of these topologies, we use them to give a short proof of KAPLANSKY's density theorem. The main result is that Bsn is a Mackey topology and this has several important consequences. The section ends with some brief remarks on the relation between the mixed topologies and classical topologies on von Neumann and W*-algebras.

VI. 1 IV.l.

197

THE ALGEBRA OF OPERATORS

THE ALGEBRA O F OPERATORS I N HILBERT SPACES

L e t L ( H ) d e n o t e t h e a l g e b r a of bounded, l i n e a r o p e r a t o r s

on a H i l b e r t s p a c e H . On L ( H ) w e c o n s i d e r t h e f o l l o w i n g structures:

11 11 'lS -

t h e uniform norm: t h e topology of p o i n t w i s e convergence on H w i t h r e s p e c t t o t h e norm ( t h e s t r o n g o p e r a t o r t o p o l o g y ) ;

T

U

-

t h e topology of p o i n t w i s e convergence on H w i t h r e s p e c t t o t h e weak topology u ( H , H ' )

( t h e weak

o p e r a t o r topology).

Note t h a t w e can c o n s i d e r L ( H ) as a v e c t o r subspace of C"(B11

I I ; H ) and i t i s a l o c a l l y convex s u b s p a c e f o r t h e

following p a i r s of s t r u c t u r e s :

1 . 1 . P r o p o s i t i o n : The u n i t b a l l of L ( H ) i s complete f o r and compact f o r

'la.

Hence w e have two complete Saks s p a c e s a v a i l a b l e : (L(H),

11 1 1 , ~ ~ )

and

(L(H)

11 1 1 , ~ ~ ) .

'lS

IV.

198

VON NEUMANN ALGEBRAS

We denote the corresponding mixed topologies by B s and B,. Note that the above spaces are Saks space subspaces of

I L , T ~ ) and

IL,T~)

(C"(B11 ~ ~ : H ) , ~ ~ and

(C"(Bl1 ~ ~ : H , , ) , ~ ~

is a locally convex subspace of

(L(H)f B , )

that

( C m ( B I I II;H) f f 3 K )

(1.4.4 and 1.1).

We shall consider the properties of the above locally convex spaces in more detail in the next section. In the following we consider duality for L(H). We require two preliminary results:

1.2.

Lemma: Let E and F be locally convex spaces and let

(L(E;F)

f~

P

)

be the space of continuous linear operators from

E into F with the topology of pointwise convergence on E. Then the dual of (L(E;F)

P

f~

)

under the bilinear form n (T, C fi w xi) i=1 Proof: We regard (L(E;F),T (C(E;F),T~).Then if f

P

is,identifiable with n I-----+

C

i=1 )

E

b

F'

fi(Txi).

as a topological subspace of

(L(E;F),T~)'there is an extension

E

-

of f to an element of the dual of C(E;F). Such an element has the form T

n C fi(Txi) i=l

...,fn)

for some finite subsets {x,~...,x~)of E and {fl,

of F'.

IV. 1

THE ALGEBRA OF OPERATORS

1.3. Lemma: An operator A

E

L(H) is compact if and Only if

there exist orthogonal sequences (x,)

-

a positive element (A,) A :x (i.e. A

=

199

in H, (f,)

in H ’ and

of co(N) so that C

n

Xnfn(x)xn

c Anf, a x,). n

Proof: We shall prove that every compact operator has this form. The operator A”:A : H d H

is positive and compact.

Hence, by the spectral theorem for compact operators, there is an orthogonal sequence (xn) in H and a positive sequence 2 in co(N) so that (1,) A“A

:

x w c n

x ~ 2( x I x ~ ) x ~ .

Let M be the orthogonal complement of A”A(H). Then AIM = 0 since if y

E

M

= ( A ~ I A=~ ( )~ \ A $ C= A ~0. ) 11~y11~

-

Then a simple calculation shows that the sequence (f,) is orthonormal (where fn : x A = C XnXn

8

fn.

in H ’

(Xi’xlAxn)) and that

1.4. Definition: The above proof shows that the positive sequence (An)

is an invariant of A (i.e. independent of the

partiaular representation of A

-

up to permutations of COUrSe)l

since it is the sequence of the square roots of the eigenvalues of A*A. Hence we can make the following definition:

IV.

200 A

VON N E U M A " ALGEBRAS

compact operator A : H

---+

H

We denote the latter sum then by

is nuclear if

1 1 All

1.5. Proposition: If A = C Anfn o xn

C

An
0;

A n rBM

is T a -

if M is of countable type, a subset A of M is $ a -closed if and only if it is sequentially closed; 6)

7)

multiplication is jointly B,+$-continuous on BM;

8)

8, and

‘cS

(resp. BS,.,

and

‘cS9$)

have the same convergent

sequences and compact sets; 9)

B,

-

is defined by the seminorms -1 T sup A n lTr(AnT) 1 ncIN

where (A,)

runs through the sequences in the unit ball of N(H)

and (An) runs through the family of sequences of positive numbers which increase to infinity; 10)

if MI is a von Neumann algebra with MI E M then 8 , induces

8, on M I .

206

IV.

VON NEUMANN ALGEBRAS

I f M i s a von Neumann a l g e b r a , a l i n e a r o p e r a t o r from M i n t o a l o c a l l y convex s p a c e i s normal i f and o n l y i f i t i s 6,-continuous

as w e s h a l l see

( i t i s t h e n Bs9:-continuous

b e l o w ) . Note t h a t M9< i s t h e s p a c e of normal mappings from M i n t o C. A n - a l g e b r a homomorphism from M i n t o a von Neumann a l g e b r a M,

i s normal i f i t i s B - c o n t i n u o u s U

a u t o m a t i c a l l y a norm c o n t r a c t i o n

-

(note t h a t it is

see D I X M I E R [8], p. 8 ) .

Once a g a i n , it i s e q u i v a l e n t t o demand Bs9:-continuity. 1.4.32,

By

it s u f f i c e s t h a t t h e homomorphism have a 6,-closed

g r a p h . Two von Neumann a l q e b r a s are isomorphic i f t h e r e i s a b i j e c t i v e normal 9:-algebra homomorphism from M o n t o M,

(its

i n v e r s e i s t h e n normal).

W e r e c a l l now t h a t i f

(M;u) i s a measure s p a c e t h e n L m ( M ; p )

can be r e g a r d e d ( v i a m u l t i p l i c a t i o n ) a s a s u b a l g e b r a of L ( H ) where H i s t h e H i l b e r t s p a c e L L ( M ; u ) and so as a (commutative) von Neumann a l g e b r a . Then t h e t o p o l o g y B, c o i n c i d e s w i t h t h e 8,

of

5

introduced here

1 1 1 . 1 and

Bs = Bsn = B, on L".

The r e p r e s e n t a t i o n theorem f o r commutative von Neumann

a l g e b r a s s t a t e s t h a t e v e r y such a l g e b r a i s isomorphic t o an a l g e b r a of t h e above form.

The s p e c t r a l theorem s t a t e s t h a t i f T

E

L(H) is self-adjoint

t h e n t h e r e i s a p a r t i t i o n of u n i t y { E ( X ) ) on a ( T ) so t h a t

IV.2 Then if

h

E

Lm(u(T))

h

The mapping

we can define

-

h(T) :=

1

u (TI

h(X)dE(X).

h(T)

from Lm(u(T)) into L(H)

E

M

-

is a normal $

+

E

0 so that

I[h

- ElL

so that if S

E

5

~ / 3 .There is a Bs-neighbour-

V then K(T)

-

x(S)

E

U/3.

IV.

208 2.5.

VON NEUMANN ALGEBRAS

C o r o l l a r y (KAPLANSKY's d e n s i t y t h e o r e m ) : L e t A be a

Bs*-dense

9:

. Then

- s u b a l g e b r a o f t h e von Neumann a l g e b r a M

BA , t h e u n i t b a l l o f A

,

i s Bs,-dense

P r o o f : W e f i r s t show t h a t B

i n BM '

i s Bs-dense

assume t h a t A i s norm-closed.

W e can

in B

AS

MS'

If T

E

B

MS

there is a net

i n A w i t h Ta d T i n 8 , . W e can assume, by (Ta a E A 1 . 1 . 1 4 , t h a t (Ta) i s bounded. L e t h b e a c o n t i n u o u s f u n c t i o n on IR so t h a t 2.4,

h(Ta)

llhlL

---+

= 1

h(T)

and

h = Idn f o r 8.,

in B

MS

on [ - 1 , 1 ] .

Then. by

Now h ( T a ) i s t h e u n i f o r m

l i m i t o f p o l y n o m i a l s i n Ta and so i s i n A .

o p e r a t o r s , we use t h e following

To d e a l w i t h n o n - s e l f - a d j o i n t trick: let

b e a H i l b e r t s p a c e d i r e c t sum

fy operators

5

on

?? w i t h 2

[z :"I

t h e c o n d i t i o n s o f t h e C o r o l l a r y . Now i f T then

-T is i n

so t h a t i n B,n.

-

ns and Ta

IIFII

:=

5 1 . Hence t h e r e

-T

in

and i d e n t i -

( T ) where e a c h i j be t h e s e t of o p e r a t o r s ( T i j )

E L(H) L e t x (resp. ij whose components are i n A ( r e s p . i n M ) . Then

T

H

x 2 matrices

a)

.

H

Bs.

Then

E

is a n e t a T2,

E

and

satisfy

M with

IlTll s 1 ,

BA

T')=

and

T'.) il a T21-->

W e now p r e s e n t one o f t h e most i m p o r t a n t r e s u l t s o f t h i s

C h a p t e r : t h e r e s u l t t h a t (M,Bs,t)

i s a Mackey s p a c e .

i n BAS

T

IV.2

VON NEUMANN ALGEBRAS

209

We require the following characterisation of normal forms on a von Neumann algebra. A fairly accessible proof can be found in RINGROSE [13].

2.6. Proposition: Let M be a von Neumann algebra, f a form in

Mft. Then f is Bu-continuous (i.e. f

E

M9:) if and only if for

each orthogonal family {E } of projections in M Q

f( C EQ) = C f(E,). a

Q

2.7. Corollary: f

E

Ms': is BU-continuous if and only if f

I MI

is Ba-continuous for each commutative von Neumann subalgebra

M I of M. 2.8. Proposition: Let M be a von Neumann algebra, K a subset of Mi.;. Then the following are equivalent:

K

1 MI

1)

K is relatively u (Mf:,M) -compact:

2)

for each commutative von Neumann subalgebra M, of M,

is u(M1g:,M1)-compact.

Proof: 1) = > 2) 2)

>

1):

7

is trivial.

Firstly, if K satisfies 2) then K is pointwise

bounded on MS and so pointwise bounded on M. Hence, by the principle of uniform boundedness, K is norm-bounded and

0

relatively weakly compact in M*. Thus it will suffice to show that the u(M",M)-closure of K lies in Mg. Suppose that f lies in this closure. Then f 1

lies in the weak closure of K

M1

IV.

210

VON NEUMANN ALGEBRAS

for each commutative von Neumann subalgebra MI of M . Thus f is normal ( 2 . 7 ) .

Corollary: T ( M , M + ~ ) is the finest locally convex topology

2.9.

~ on each commutative von on M which is coarser than T ( M ,MI+ 0 there is a sequence (En) of pro-

in 8 , .

jections in M so that En-

in 8 , and llTnEnll I 6

1

for

each n.

Proof: Let

En

:=

X[-S,SI

(T ) . Then since n

2 1 6 2 (I - En) 2 0 Tn

I-En+

2.11.

0 in B.,

Also llTnEnl]5 6

Proposition: Let M be a von Neumann algebra of countable

type. Then

B,s

= T(M,M~).

Proof: We shall work in the real vector space MS and show that B s = T(M',M:)

there. Since M is of countable type,

BMs

is B s -

metrisable and so it will suffice to show that if (Tn) is a sequence in B

MS

so that Tn+

0

in B s then

Tn-

0

uniformly on each weakly compact subset K of M Z . If this were not the case, there would be a relatively compact sequence (f,), E

>

0 and a subsequence of (T )

n

(which we assume, for simplicity

an

IV.2

VON NEUMANN ALGEBRAS

211

of notation, to be (Tn) itself) so that

I fn(Tn)l >

each n. Now by EBERLEIN's theorem, (f,)

has a weakly convergent

for

E

subsequence and s o , by passing to a subsequence and by translating, we can assume that {En] so that

En+

f

n

d 0 weakly. Choose projections for some 6 > 0

in 8 , and [[EnTnllI 6

1

to be chosen later (2.10). Then Ifj(Ti)l

for each i,j Now

f> -.

I

[f.(EiTi)l 3 + lf.((I 3

5

2M6

+ lf.((I 3

N where

E

-

-

Ei)Ti)l

Ei)Ti)l

M := supCIIfll : f

K).

E

0 pointwise on the u (MS,Mf)-compact set B

3

MS

so, by Baire's theorem, there is a point To

E

B

and

so that (f.) 7

is equicontinuous at To i.e. there is a o(MS,M.:)-neighbourhood U of zero in M S so that Ifj(T) for each T

E

-

fj(To)l < 6

V := (To + U) n BMs. Choose J

E

N so that

if j 2 J. Then

Ifj(To)l < 6

If.(T)I < 36 7

for j 2 J, T

V. Now if

E

Si := E.T i oE i

then Si

E

B

and

+

(I

-

Ei)Ti

To. Hence there is an I

Si+

E

N

MS

that Si and EiToEi are in V for i 2 I. Then Ifj((I-Ei)Til for i 1 I, j

1

J. Then

I

If.(Si)l 3 + Ifj(EiToEi)l

5 66

so

IV.

212

VON NEUMANN ALGEBRAS

for i 2 I, j 2 J and for small 6 this contradicts the inequality

2.12.

I fn(Tn) I >

E.

Remark: It follows from a characterisation of relatively

weakly compact subsets of MB due to AKEMANN (see C33 or AARNES C 2 l ) that 2 . 1 1 holds without the assumption that M be of countable type. However, the extension to general von Neumann algebras requires some more technical results. Nevertheless, in view of this fact, we shall drop the assumption of countable type in the formulation of the following consequences of 2 . 1 1

2.13.

.

Proposition: Let M be a von Neumann algebra, F a locally

convex space. Then a linear mapping @ from M into F is

BSt.t-

continuous if and only if its restriction to each commutative von Neumann subalgebra of M is Bs-continuous.

Proof: Since ( M , B ~ $ : ) is a Mackey space, we can reduce to the case when

F =

Q:

and this is precisely 2 . 7 .

2 . 1 3 is equivalent to the fact that

Bsn is the finest locally

convex topology on M which is coarser than $,

on each commutative

von Neumann subalgebra MI of M (cf. 2 . 9 ) .

2.14.

Proposition: If M is a von Neumann algebra, then (M,Bs,+c)

has the Banach-Steinhaus property.

IV.2

VON NEUMANN ALGEBRAS

213

Proof: By 2 . 1 3 we can reduce this to the case where M is commutative and this is 1 1 1 . 1 . 1 2 .

2.15.

Corollary: If M is a von Neumann algebra then Mf: is

u (M%,:,M) -sequentially complete.

2.16.

Proposition: Let

M and

M 1 be von Neumann algebras, Q a

linear mapping from M into M I .

Then 0 is Bu-continuous if and

only if it is Bsft-.continuous.If , in addition, 0 is a

9:-

morphism, then these conditions are equivalent to its continuity for 6.,

Proof: If

@

is Bu-continuous, then it is continuous for the

Mackey topologies T ( M , M ~ : ) and ‘r(M1 ,MI*)

i.e. for the B,+:-topo-

logies. On the other hand, if it is Mackey continuous, then its adjoint (which maps

M1h

into M,:)

is weakly continuous and so

norm continuous. Hence it maps norm compact sets into norm compact sets and so the result follows from 2 . 3 . 5 ) .

If @ is Bs-continuous, then it is #3,gC-continuous (proof as for Bu-continuity). On the other hand, if @ ( M S ) C MY

and

8,

tinuous, then it is

@

- B S * on MS and M:.

continuous on MS

is star-preserving, then Hence if @ is Bsr-COnand the result follows

by considering real and imaginary parts.

214

IV.

VON NEUMANN ALGEBRAS

2.17. Proposition: If M is a von Neumann algebra, then a linear mapping 0 from (M,Bs>ps)

into a separable Frgchet space

is continuous if and only if it has a closed graph.

Proof: By 2.13, it is sufficient to show that the restriction of @ to each commutative von Neumann subalgebra B,-continuous. But 0

MI of M is

also has a closed graph and so is

I M1

continuous by 111.1.13.

As mentioned in the introduction, von Neumann defined two locally topologies, the ultraweak and ultrastrong topologies,

on L(H) (and so on any von Neumann algebra) as follows: if

*,...)

X := (x,,x

and

elements in H so that

Y := (y,,y2,.. .) are sequences of < m and C 11 ynll 2 < m then

I I x ~ ~ ~ ~

C

are seminorms on L(H) and the family of all such seminorms (p,)

(resp.

defines a locally convex topology on L(H)

-

the ultrastronq (resp. ultraweak) topology. We denote these

us

and

uu. Then

C us

and

us = T s

topologies by Similary,

'cS

T,

E uu and uu

= T~ on B

L(H) on BL (HI (see DIXMIER CS] ,

p. 34).

One can also define a symmetric form the seminorms T

c--j

pX(T") ) and

on the unit ball. Thus B,,

us*

of

us"' 2

~~g~

us

(by adding

with equality

6 , and Bsfs are stronger than

uu, us

IV.2 and

usB

VON NEUMANN ALGEBRAS

215

respectively. In general, they are strictly stronger

as the following Proposition shows:

2.18. Proposition: Let M be an infinite dimensional von Neumann algebra. Then B,,

8, and

u u , us and u s B are strictly weaker than

respectively.

Bsiy:

Proof: The topology uu on M is exactly o ( M I M A ) and 8,

is the

topology of uniform convergence on the norm compact subsets of Mfe. Hence

uu = 8,

if and only if the norm compact subsets

of Mi,: are finite dimensional and this is the case precisely when

Mg:

(and hence also M ) is finite dimensional.

For the second part, it suffices to show that 8, is strictly finer than

on M S . Since M is infinite-dimensional, there

us

is a mutually orthogonal sequence (En) of non-zero projections in M S . Let

{n’/2En}. Then we can claim that 0 lies in

S :=

the us-closure of S since for each X I px(n 1 /2En) 2 1 For if this were not the case, there is an px(n1/2En) > 1

for some n.

X = (xi) so that

i.e. pX(En) > n-1’2. But this contradicts the

additivity of the form fx in Mi) defined by X (i.e. in the notation of DIXMIER [ E l ) since then

CuXiryi C fX(En) 2 C l/n.

We shall show that 0 does not lie in the Bs-closure. For each n let xn be a unit vector in H so that

-

Enxn = xn

and

define fn to be the form

T Then fn

E

Me and

n- 1 /2 (TX~~X,).

K := {fnl is relatively compact in

(MS,II

11).

IV.

216

VON NEUMANN ALGEBRAS

Hence U := {T

E

MS : I f n ( T ) I


0 then there is an absolutely continuous

measure (i.e. absolutely continuous with respect to Lebesgue measure) v in M(G) so that

u-v (where u

and

- v means that l.~

IIVII 5

-

IIuII

+ E

v vanishes on Hm).

In fact, the result is true with the E-term omitted (cf. RUBEL and SHIELDS [31] , Prop. 4 . 1 )

.

V.l

MIXED TOPOLOGIES ON Hm

229

1.3. Definition: Let S be a subset of G. S is dominating if,

x

for each

E

Hm(G) ,

i.e. if the restriction mapping

:

ps

x

isometry from H ~ ( G )into ~ " ( s ) .

1.4.

-

xis

is an

Lemma: If S is a countable, dominating set then p, (Hm)

is B-closed in L m ( S ) .

Proof: By 1.4.3

and the fact that the unit ball of L " ( S ) is

@-metrisable, it is sufficient to show that ps(Hm) is sequentially closed. Let (xn) be a sequence in Hm so that ps(xnl is B-convergent in Lm(S) to y. Then (xn) contains a @-convergent subsequence .(x ) and it is clear that y = ps(lim xnk). "k It follows from this Lemma that p,

is an isomorphism from

(Hm(G), B ) onto a closed subspace of (L"(S1

1.5.

Proposition: Let S be a subset of

G.

,@)

(cf. 1.4.32).

Then the following

are equivalent: 1)

S is dominating;

2)

for every 1-1

(i.e. 1-1

-

E

G \

M(G)

there is a v in M(S) so that 1-1

-v

m

v vanishes on H 1.

Proof: 2 ) ho

E

1 ) : if S were not dominating, we could find a S

and

x

E

Hm(G)

so that

BOUNDED HOLOMORPHIC FUNCTIONS

V.

230

x(Xo)

for X

Ix(X)I I k

= 1,

If S satisfies condition 2) there is a v v

- &Ao

1

__7

I

I

xn

61

0

=

S (k < 1).

M(S) so that

(Dirac measure at lo). Now for each n 1 =

and so 1

E

E

E

IN,

1 xndv

k"Ilvll which is impossible.

2): by going over to a countable dense subset, we can

suppose that S is countable. The result then follows immediately from 1 . 4 and duality.

V.2. THE MIXED TOPOLOGIES ON Hm(U)

In this section, we specialise to function spaces on U, the open unit disc

{A

:

1x1

< 1).

and by aU the boundary

We denote by

the closed unit disc

\ U of U. We recall some notation and

results on Hardy spaces -see HOFFMAN 1211, Ch. 3 , 4 for a detailed discussion.

We denote by Hp(LJ) (or simply by Hp) (1

I

p


x

is an iso-

metry from Hp(U) onto a closed subspace of LP(aU). The range of this mapping is the space of those 27

J

0

y(e ie )einede = 0

y

E

Lp(aU)

such that

(n = 1,2,...)

i.e. the non-negative Fourier coefficient of y vanish. If

y

x

Hp(U)

E

2.1.

E

LP(aU)

satisfies these conditions, then

y

=

where

can be recovered from y via the Poisson integral

Proposition: 1 )

(Hm(U), @ ) is a topological subspace of

(Lm(au), B J ; 2) 1

(Hm(U),@)' is naturally isomorphic to the quotient space

L (BU)/HA

where HA denotes the space of functions in H 1 (U)

which vanish at the origin.

Proof: 1 ) follows from 1 . 4 . 4 and the fact that on the unit ball of Hm, the topology a(Lm(au) ,L1 (au)1.

T

P

agrees with the weak topology

2) Since the dual of (Lm(aU),@,) is L 1 (aU), it suffices to identify the polar of Hm in L 1 (aU) with Ho1 and this is clear.

V.

232

BOUNDED HOLOMORPHIC FUNCTIONS

For the next Proposition, we recall the following definition:

(xn)z in a topological vector space E is a Cesaro basis for E if for each x E E there is a unique sequence a sequence

(6,) of scalars so that the partial sums n

sn :=

C

k=O

5,

Xk

... + Sn) = XI.

1 converge to x in Cesaro mean (i.e. lim n+ 1 (so + 2.2.

(Hm,B)

2)

m

The sequence (znlo is a Cesaro basis for

Proposition: 1 )

where zn is the function

( H ~ , B ) is separable; m

Proof: 1) If x

E

H I let

Then if

n :=

s

C

k=O

Ek

C 0

cnX n

be its Taylor expansion.

... +

1

an := n+l (so +

zkI

Sn)

s dx

in T~ and so anx . On the other n hand, we have the following kernel representation of an:

we have

where Kn is the Fejer kernel t -

1 [I

-

1

cos nt cos t

(cf. HOFFMAN [ 2 1 ] , pp.16,17 and Ch. 3 ) . Then Ilanll I llxll (since Kn is positive and

l I T

2.rr

-IT

Kn = 1 ) .

MIXED TOPOLOGIES ON Hm(U)

V.2

Hence

a

n

d x

233

in B by 1.1.10.

2 ) follows immediately from 1 ) .

It is well known that if E is a locally convex space with a basis (xn) such that the corresponding projection operators n

m

are equicontinuous, then E has the approximation property. The following Lemma is proved similarly.

2 . 3 . Lemma: Let E be a locally convex space with a Cesaro basis

(xn) so that the projection mappings (Pn) are equicontinuous, where Pn maps C Skxk

into the Cesaro mean an. Then E has the

approximation property.

2 . 4 . Proposition: (H- ,B 1 has the approximation property.

Proof: By 2 . 3 and 1.1.7 jections (P,)

it is sufficient to show that the pro-

are equicontinuous for the topology defined by

the seminorms {pr : 0 < r Pr : x-




(q,)

... t s,). is a complete Saks space and

X

Proof: We have already seen that

in 5,.

a

n

d x

in

suffices to show that (an) is bounded i.e. that

T

E’ Hence it

(j

nEN

u

XEU

un(X)

V.

238

BOUNDED HOLOMORPHIC FUNCTIONS

is bounded in E. By the uniform boundedness theorem, it suffices to show that for each

f

E

E'

u

I!

f

Y' n e N X ~ U

o

an(X)

is bounded in C i.e. we can reduce to the scalar case which we have already treated in 2.2.

2.10.

Corollary: If we identify the algebraic tensor product

Hm(U)

8

E

with a subspace of Hm(U;E) under the natural (vector

space) isomorphism

then Hm(U) ca E

2.11.

is BE-dense in Hm(U;E).

Proposition: There is a natural (Saks space) isomorphism

H~(u;E)= H ~ ( u )

Y

E.

Proof: Having identified Hm(U) e E

with a dense subspace of

Hm(U;E), we need only show that the Saks space structure induced on

Hm(U)

8

E

from Hm(U;E) coincides with the tensor

product structure. This is standard.

We finish this section by quoting two important Propositions. The first states that the dual of (Hm(U), @ ) is sequentially complete and is due to MOONEY who thus settled a long-standing conjecture. The second is a characterisation of the weakly compact subsets of the dual of Hm(U) and is due to CHAUMAT. It is equivalent to the fact that B 1 is the Mackey topology on

V.2

MIXED TOPOLOGIES ON Hm(U)

239

Hm(U). Unfortunately, the only existing proofs are too long and technical to be reproduced here.

2.12. Proposition: Let (fn) be a sequence in (Hm(U), 8 )

I

which

converges pointwise to a functional f on Hm. Then f is 8-continuous.

2.13. Proposition: Let K be a bounded subset of the dual L1/(Hm)O of (Hm(U), 8 , ) .

Then the following are equivalent:

1) K is weakly relatively compact;

ITXII

: x E CBHm) = 0. lim sup (inf Ilf C fcK (n. is the projection from L' (aU) onto the dual of H" (U)).

2)

2.14. Corollary: (Hm( U ),B1) is a Mackey space.

Proof: The condition 2) means that for

E

>

0 there is a C

> 0

so that K C C B 1 + cB2 where B1 is the unit ball of the dual of H 1 (U) and B2 is the unit ball of L 1 (aU)/ (HOD)'. Hence the result follows from 1.1.22.

Note that 2.12 and 2.14 imply that (Hm(U),B,) satisfies a closed graph theorem with a separable Fr&chet space as range space.

V.

240

BOUNDED HOLOMORPHIC FUNCTIONS

V.3. THE ALGEBRA H m

As we have already remarked, (Hm(U),f3) is a topological algebra. In this section, we describe the closed ideals of

.

Hm (U) It turns out that these have an especially simple

-

form

in fact, they are all principal ideals. This classifi-

cation is, of course, of intrinsic interest but it also has a number of useful applications. Following RUBEL and SHIELDS, we use it to give a description of the invariant subspaces of a class of Hardy spaces. In the next section, we give an application of the ideal theory to operators in Hilbert space.

We begin by describing the f3-closed maximal ideals of Hm. Of course, Hm(U) is a Saks algebra and can be represented as the projective limit of the spectrum {A(K) : K compact in U) of Banach algebras where A(K) denotes the set of continuous

-

functions on K which are holomorphic in the interior of K. If 1

E

u, :

-

x

x(X)

is an element of M (Hm). Thus we have constructed an injection T :

x

Y

from

4,

u

3.1. Lemma: Suppose that A

Then

x

:=

:

X

E

A}

into M (H~). Y U is not relatively compact in U.

is not f3-equicontinuous.

V. 3 Proof: If

were equicontinuous, there would be an r

and an

E

>

x

E

BII

11

with 1x1

n

E

N so that

I

0

0 so that for each f

x : X

Then

(x)I

241

THE ALGEBRA Hm

-

There is a Xo

E.

An

2 1/2

-

If(x)I

on the set (X :

I E

rn 5

x,

E

1x1 E

I


1

then every c.n.u. contraction

of class Co on H has a non-trivial invariant subspace.

The proof of 4.12 is obvious (in fact, one can show that Ho is hyperinvariant, that is invariant for every operator which commutes with TI. The Corollary follows immediately if mT has a non-trivial factorisation. Otherwise, it has the form

for some la1 T = XoI

= 1,

X0

E

U. An easy calculation shows then that

and so has non-trivial invariant subspaces.

V.

260

BOUNDED HOLOMORPHIC FUNCTIONS

V.5. NOTES

As mentioned in the introduction to this chapter, the topology 6 on Hm was first introduced by BUCK

[lo]. It was systematically

studied by RUBEL and SHIELDS 1 3 1 1 and RUBEL and RYFF [30]. The results of section 1 can be found in these papers (usually with different proofs).

The results and methods of

5

2 are mostly new

-

in particular,

the topology P 1 seems to be new. The fact that f3 is not the Mackey topology is due to CONWAY [13]. This result has been generalised to more general domains by RUBEL and RYFF. BIERSTEDT [6] and [7] has obtained tensor product representations for

Hm(Ux U) and considers the approximation property using different methods. BIRTEL [8] and BIRTEL, DUBINSKY [9] have considered Banach space tensor products of Hm-spaces. MOONEY's theorem (2.12) seems to have been conjectured originally by A.E. TAYLOR

(see PIRANIAN, SHIELDS, WELLS [26]). For proofs see

AMAR 1 1 1 and MOONEY [241. BARBEY 121 and 131 has given a version

of this theorem in the setting of spaces of abstract analytic functions. CHAUMAT's theorem is in [12]. The topology 8, can be defined for domains G more general than U. It is necessary that the boundary of G be rectifiable and regular enough that Hm-functions have (non-tangential) boundary values and be recoverable from then. A nice discussion of this problem can be found in ZALCMAN [371.

V.5

8

NOTES

261

3 is based on the work of RUBEL and SHIELDS. Concerning the

main result (3.9) (which is originally due to SRINAVASAN), they write: "we prove the result here by using Beurling's characteri2 It would be sation of the closed invariant subspaces of H

....

good to find a direct intrinsic proof". The proof given here is adapted from the proof of the corresponding result for the space of functions in Ha which have continuous extension to 3 (a result due to BEURLING and RUDIN

-

see HOFFMAN [211, Ch. 6). The theorem (3.4) of KAKUTANI was published in [221. we have taken the proof from HOFFMAN 1211, p. 144. a

U, the y-spectrum of H (U), is naturally embedded in the spectrum of a Banach slgebra (Ha,11

11) .

The famous problem whether U is

a dense subspace (the "Corona problem") was solved positively by CARLESON in [ I l l .

In

9

4 we have given a brief introduction to part of the theory

of completely non-unitary contractions in Hilbert space. The theory is developed in detail by SZ.-NAGY and FOIAF in [251. We have used some ideas of TON-THAT-LONG 1361.

Further references on (Ha,B) are BARTELT (141 and [51), and SHAPIRO (1331, [341 and [351).

There are three recent survey articles on spaces of bounded analytic functions

-

GAMELIN [ 1 7 1 , RUBEL [291 and SARASON 1321.

V.

262

BOUNDED HOLOMORPHIC FUNCTIONS

The paper of ZALCMAN [ 3 7 1 also contains an excellent account of current work on algebras of analytic functions.

In

$5

2 and 3 we have considered only functions on the open

unit disc in order to simplify the presentation. However, many of the results can be extended to more general regions by simple methods. For example, 2 . 1 1 holds when U is replaced by a region which is the finite union of disjoint, simply connected domains (as BIERSTEDT has shown). The question of whether these results hold for general regions G can lead to complicated analytic questions. For example, RUBEL and RYFF give a fairly detailed account of extensions of Proposition 3.2. RUBEL and SHIELDS have introduced the notion of inner and outer functions for general regions (even in higher dimensions) but little seems to be known about them.

REFERENCES

263

REFERENCES FOR CHAPTER V.

[I]

E. AMAR

Sur un theoreme de Mooney relatif aux fonctions analytiques bornges, Pac. J. Math. 49 (1973) 311-314.

[2]

K. BARBEY

Ein Satz tiber abstrakte analytische Funktionen, Arch. der Math. 26 (1975) 521 -527. Z u m Satz von Mooney fur abstrakte analytische Funktionen (Preprint)

I. 31

[ 41

.

M. BARTELT

Approximation in operator algebras on bounded analytic functions, Trans. Amer. Math. SOC. 170 (1972) 71-83.

[ 51

161

Multipliers and operator algebras on bounded analytic functions, Pac. J. Math. 37 (1971) 575-584.

K.D. BIERSTEDT

Injektive Tensorprodukte und Slice-Produkte gewichteter R h n e stetiger Funktionen, Jour. reine angew. Math. 266 (1974) 121-131. Gewichtete Raume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt 11, Jour. f. reine angew. Math. 260 (1973) 133-146.

171

[81

F. BIRTEL

191

F. BIRTEL, E. DUBINSKY Bounded analytic functions in two complex variables, Math. 2 . 93 (1966) 299-310.

[I01

R.C. BUCK

Slice algebras of bounded analytic functions, Math. Scand. 21 (1967) 54-60.

Algebraic properties of classes of analytic functions, Seminars on analytic functions 11, 175-188 (Princeton, 1957).

264

V.

111

L. CARLESON

Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962) 542-559.

[I21

J. CHAUMAT

Une g6nsralisation d'un thborsme de Dunford-Pettis (Sem. Analyse Harmonique d'Orsay No. 85 1974).

[

BOUNDED HOLOMORPHIC FUNCTIONS

-

[ 131

J.B. CONWAY

Subspages of (C(S),B , the space (Loo,@ and (H , B ) , Bull. Amer. Math. SOC. 72 (1966) 79-81.

[ 141

J.B. COOPER

The Hm-functional calculus for completely non-unitary contractions, Math. Balkanica 4.15 (1975) 89-90.

[ 151

F. DELBAEN

Weakly compact sets in H1 , Pac. J. Math. 63 (1976) 367-369.

1161

T.W. GAMELIN

Uniform algebras (New Jersey, 1969)

.

The algebra of bounded analytic functions, Bull. Amer. Math. SOC. 79 (1973) 1095-1 108.

1171

[ 181

A. GROTHENDIECK

Sur certains espaces de fonctions holomorphes, I,II, J. reine angew. Math. 192 (1953) 35-64, 77-95.

1191

V.P. HAVIN

Weak completeness of the space L 1 /HA (Russian), Vestnik Leningrad Univ. 13 (1973) 77-81.

[20]

H. HELSON

Lectures on invariant subspaces (New York, 1964).

[21]

K. HOFFMAN

Banach spaces of analytic functions (New Jersey, 1962).

[22]

S. KAKUTANI

Rings of analytic functions, Lectures on functions of a complex variable, 71-83 (AnnArbor, 1955).

REFERENCES

265

[ 231

A. KERR-LAWSON

A f i l t e r d e s c r i p t i o n of homomorphisms o f H m , Can. J . Math. 17 (1965) 734-757.

1241

M . MOONEY

A theorem on bounded a n a l y t i c f u n c t i o n s , Pac. J. Math. 43 (1972) 457-463.

[251

B.

[261

G. P I R A N I A N , A.L.

1271

M . von RENTELN

SZ.-Nagy,

C.

Harmonic a n a l y s i s of o p e r a t o r s on H i l b e r t s p a c e (Amsterdam, 1970).

FOIAF

SHIELDS, J . H . WELLS Bounded a n a l y t i c f u n c t i o n s and a b s o l u t e l y c o n t i n u o u s m e a s u r e s , P r o c . Arner. Math. SOC. 18 ( 1967) 81 8-826. F i n i t e l y s e n e r a t e d i d e a l s i n t h e Banach a l g e b r a H", C o l l e c t . Math. 26 (1975)

115-1 25.

1281

J.R. ROSAY

[291

L.A.

RUBEL

[301

L.A.

RUBEL, J.V. RYFF The bounded w e a k - s t a r t o p o l o g y and t h e bounded a n a l y t i c f u n c t i o n s , J . F u n c t i o n a l Anal. 5 (1970) 167-183.

[311

L.A.

RUBEL, A.L.

Une e q u i v a l e n c e du Corona p r o b l s m e e t un probldme d ' i d e a l s dans H m ( D ) , J o u r . Func. A n a l . 7 (1971) 71-84. Bounded c o n v e r g e n c e o f a n a l y t i c f u n c t i o n s , B u l l . Arner. Math. SOC. 77 (1971) 13-24.

SHIELDS The s p a c e o f bounded a n a l y t i c f u n c t i o n s i n a r e g i o n , Ann. I n s t . F o u r i e r

16 (1966) 235-277.

1321

D.

[ 331

J.H.

SARASON

SHAPIRO

Algebras of f u n c t i o n s on t h e u n i t circles, B u l l . Amer. Math. SOC. 79 (1973) 286-299. Weak t o p o l o g i e s on s u b s p a c e s of C (S), T r a n s . Amer. Math. SOC. 157 (1971)

471 -479.

266 [34]

V. J.H. SHAPIRO

BOUNDED HOLOMORPHIC FUNCTIONS The bounded weak star topology and the general strict topology, J. Func. Anal. 8 ( 1 9 7 1 ) 275-286.

Noncoincidence of the strict and strong operator topologies, Proc. h e r . Math. SOC. 35 ( 1 9 7 2 ) 81-81.

1351

[361

TON-THAT-LONG

Sur le calcul zonctionnel d'une contraction completement non unitaire, Proc. Func. Anal. Week, 26-36 (Aarhus, 1 9 6 9 ) .

[371

L. ZALCMAN

Bounded analytic functions on domains of infinite connectivity, Trans. h e r . Math. SOC. 1 4 4 ( 1 9 6 9 ) 241-269.

APPENDIX

Introduction: In this appendix we assemble some results on Saks spaces from a category-theoretical point of view. It is our opinion that this approach puts a number of the results of the previous Chapters in their proper perspective. The leitmotiv is the establishment of a duality theory for various important categories.

In the first section, we define formally the category of Saks spaces, list some of its simple property and identify its dual category. We then use these categories to establish in

5

2 a

duality theory for uniform spaces and for compactologies. The latter are a generalisation of the compact spaces and allow us to give a symmetrical form of the duality theory for completely regular spaces established in Chapter 11. In

5

3 we consider

some important functors and bifunctors on categories of Saks spaces and their duals.

In the fourth section, we use the duality theory of

5

2 to give

a duality theory for certain classes of semigroups and groups.

In the last section we describe an extension process forcategories which formalises the transition from Banach spaces to Saks spaces and demonstrate thereby that this is a special exapmle of a construction which includes, for example, the

267

268

APPENDIX

generalisation from Banach spaces to locally convex spaces or to convex bornological spaces.

A.1. CATEGORIES OF SAKS SPACES

1.1.

The categories PSS, SS and CSS: We introduce three cate-

gories

PSS, SS and CSS

PSS

-

with the following objects:

triples ( E , l l 11,~)where ( E , I I

11)

is a normed space,

r is a locally convex topology on E and B unit ball of ( E , l l

-

SS

triples ( E , l l

11,~)

assumption that B CSS

-

11)

II II'

the

is T-bounded;

as above with the additional

II It

is r-closed and

r

=

v[ll 1 1 , ~ l ;

as for SS with the additional assumption that (E,T) be complete.

In each case, if (E,11 II,r), (F,1 1

Ill

,rl) are objects of one of

the above categories, the morphisms from E into F are the linear contractions T : E

F

so that TiB

I I II

is r-rl-continuous.

1.2. Proposition: SS and CSS are full, reflective subcategories of PSS.

Proof: The reflection from PSS into SS is obtained by first employing the procedure described after 1.3.2 BII

II by

(i.e. replacing

its r-closure) and then replacing r by y [ l l 11,rI. The

reflection from PSS into CSS is obtained by composing this functor with the completion functor (1.3.6).

A.l

CATEGORIES OF SAKS SPACES

269

We remark that all of the above categories are complete and cocomplete. Products and equalisers in PSS are formed as described in 1.3.5 and 1.3.7. The construction of sums and co-equalisers is a little more complicated and we shall not describe it explicitely here. The various limits in SS and CSS are constructed by first forming them in PSS and then reflecting down into SS and CSS respectively.

1.3. The categories PMW and MW: An object of PMW is a triple

(E,1 1 I1,B) where (E, 11 E so that each B

E

11)

B is

is a normed space, B is a bornology on

11 11-

bounded and, in addition, if B

there is a locally convex topology

‘ I ~ on

normed space EB so that

B)

TB-compact. If

(EB,

11 llB,

E

8,

the corresponding

is a Saks space with B

(E,ll ] 1 , B ) , (E1,II I I l , B 1 )

are objects of PMW,

a PMW-morphism from E into E l is a linear norm-contraction T from E into El which satisfies the following condition: For each B T

If

-B

B , there is a B1

E

B1 so that T(B)G B, and

is a Saks space morphism from EB into ( E

“B

(€,I1

E

11,?3) :=

B1

.

is an object of PMW, we put

{C C ! E : for each

E

>

0 there is a B

E

B so that

C C B + E B MW denotes the full subcategory of PMW whose objects satisfy the conditions B =

% and

(E,II

11)

II I?

(€,[I

l1,B)

is a Banach space.

In addition, we require the existence of a (separated) locally

270

APPENDIX

convex topology on E which coincides with object of MW is called a

1.4.

Examples: I.

on each B. An

T~

CoSaks space.

If (Ell/1 1 , ~ )

is a Saks space, then its dual

space E' has a natural PMW-structure ( 1 1 l l , B ) where 11 11 is the Y Y dual norm and B is the family of absolutely convex, weakly

Y

closed y-equicontinuous subsets of El

Y'

If B

E

8

Y

then

T~

is

defined to be the weak topology a(E',E). Y Then (E;,ll l l , B is even an object of MW by 1.1.22. In particu-

Y

lar, if

S

is a completely regular space, then the spaces Mt (S),

MT (S) and M,(S) 11.

of measures on S, are CoSaks spaces.

Let S be a completely regular space. Cm(S) has a natural

) where 11 11 is the supremum norm and 8 equ equ is the set of uniformly bounded, pointwise closed equicontinuous

MW-structure

(11

Il,B

subsets of Cm(S). For B

E

Bequ, the topology

T~

is that of point-

wise convergence on S. 111.

We can generalise I1 as follows: let S be a uniform space

and denote by Um(S) the vector space of bounded, uniformly continuous functions from S into C. Then Um(S) has a natural CoSaks space structure defined as above (replacing "equicon-

.

tinuous" by "uniformly equicontinuous") If S is a completely regular space and we regard it as a uniform space with the

fine

uniformity (i.e. the finest uniform structure compatible with its topology), then the CoSaks structures defined in this section and in I1 coincide.

CATEGORIES OF SAKS SPACES

A.l IV.

If (E,II

11)

271

is a Banch space, we can regard E as a CoSaks

space by defining 8 to be the family of all absolutely convex, compact subsets of (E,"

11).

This allows us to regard the cate-

gory of Banach spaces as a full subcategory of MW.

-

1.5. The dual of a CoSaks space: If ( E , I I 11,B) is an object of PMW, we define its dual E ' to be the space of all linear functionals f : E

restrictions to each B

which are norm-bounded and whose

Q:

E

B are TB-continuous. Then E ' has a

( 1 1 1 1 , ~ ~ ) where 11 11

natural Saks space structure norm and

'cB

is the dual

is the topology of uniform convergence on the sets

of B .

1.6. Lemma: Let f be a linear functional on the CoSaks space (E,"

1 1 , B ) . Then f is norm-bounded (and so an element of

provided flB is TB-continuous for each B

Proof: If f were not bounded on B

(x,)

in B

II II

so that

If (x,)

I

II II'

2 2 n

Proposition: If ( E , l l 11,8)

is an object of CSS.

E 8.

we could find a sequence

. Now

in 8 and hence f is bounded on this set

1.7.

E l )

{xn/n) is in

-

ff and so

contradiction.

is a CoSaks space, then ( E ' , l l

11,~~)

APPENDIX

272

are obviously functorial (if we map morphisms into their adjoints) and so we have defined duality functors P and D where

0 : CSS

1.9.

___j

MW

and

D : MW

j . CSS.

Proposition: CSS and MW are quasidual under the functors

D and D. Proof: The proof that D and D are mutually adjoint on the left is standard. It then suffices to show that the natural transformation from D

o

P

to the identity functor is equivalent to

the identity or, speaking loosely, that

DPE = E

for each

complete Saks space. This follows from Grothendieck's completeness theorem.

A.2. DUALITY FOR COMPACTOLOGICAL AND UNIFORM SPACES

In 5 11.2 we established a form of Gelfand-Naimark duality for W

completely regular spaces. Because the space (C (S), B K )

is not,

in general, complete, this duality is not perfect and in this section we show how a complete duality can be obtained by replacing the category of completely regular spaces by that of compactological spaces. We also develop a duality theory for uniform spaces using CoSaks algebras.

A.2

DUALITY

273

Definition: A compactology on a set S is a family K of

2.1.

subsets of S, together with a compact (Hausdorff) topology T

K on each K

K so that

E

1)

K is closed under finite unions and covers S;

2)

if K 1 C K where K

E

K then K 1

is TK-closed. If this is the case

T

K~ -

A compactological space is a pair ( S , K )

-

K if and only if K 1

E

T

~

)* ~

l

where K is a compacto-

logy on S. The class of compactological spaces becomes a category when we define morphisms from (S,K) f : S that

S1

f (K) E; K 1

so that for each K and f l K is

into (S1,K1) E

to be mappings

K there is a K1

E

K1

so

T ~ - T-continuous. ~

1

If S is a Hausdorff topological space, S has a natural compactology K ( S ) (define T~ to be the induced topology). This construction defines a forgetful functor from the category of Hausdorff topological spaces into the category of compactological spaces.

If

(S,K)

is a compactological space, we denote by

Cm(S)

the space

of bounded morphisms from S into 6! (with its natural compactologyl

. Then

(Cm(S), 11

mum norm and

T~

1 1 , ~ ~ ) is

a Saks space where

11 11

is the supre-

is the topology of uniform convergence on the

sets of K. In fact, (C"(S),ll

1 1 , ~ ~ ) is

even a Saks C"-algebra

(and, in particular, complete).

S is said to be regular if C m ( S ) separated S; of countable type if K possesses a countable

APPENDIX

274

basis K1 (i.e. every K

E

K is subset of some K1

K1).

E

We denote by RCPTOL the full subcategory of regular compactological spaces.

2.2. Proposition: Let (S,K) be a compactological space. Then

the following statements are equivalent: regular;

1)

( S , K ) is

2)

there is a completely regular topology on S so that K

is the induced compactology; the restriction operator

3)

jective for each K

E

K.

Cm(S)

-

C(K)

is sur-

Each of these conditions is satisfied if K is of countable type.

Proof: 1 )

+2 ) :

the weak topology on S induced by Cm (S) satis-

fies the required condition. 2) d 3 ) follows from Tietze's theorem. 3)

1 ) : let s,t be distinct points of S. Then {s,t}

E

K.

Let x be the function t-1

s-0 ,

from {s,t} into

ct.

Then an extension ';i of x in Cm(S) separates

s and t.

Now let S be of countable type

-

suppose that (K,)

is an in-

creasing basis (i.e. KnC_ Kn+l for each n). We verify condition 3 ) .

A.2 Choose K

E

K , x1

suppose that K

=

E

DUALITY

C(K). It is no loss of generality to IIxl1 1

K1 and

5 1.

tively a sequence (xn) where xn IlxnlI I 1

275

E

Then we can choose inducC(K,),

xnflIKn = xn and

for each n. Then the xn define an extension of xl.

The equivalence of 1 ) and 2) of 2.2 might suggest the conclusion that there is no difference between completely regular spaces and regular compactologies. This is not the case, the point being that if

and S1 are completely regular spaces, there are

S

more compactological morphisms from S into S1 as there are topological ones.

2.3. The spectrum of a Saks algebra: If (A,1 1

11,~)

is a commuta-

tive Saks algebra with unit, then M (A), the y-spectrum of A,

Y

has a natural compactology: the sets of K are y-equicontinuous subsets and, for each K topology a(A;,A).

E

K , T~ is the restriction of the weak

Then M (A) is even regular. It is obvious

-

Y

that the correspondences S

Cm(S)

and

A

My(A)

are functorial i.e. we have constructed functors Cm (from the category of compactological spaces into the category CSC" of commutative, Saks Cfc-algebraswith unit) and M

Y

(from the cate-

gory of commutative Saks algebras with unit into RCPTOL).

2.4.

Proposition: CSC" and RCPTOL are quasi-dual under the pair

of functors

COD

and M Y'

APPENDIX

276

Note that 2.4 means that

COD

and M

Y

are right and left adjoint

to each other and that the corresponding unit and co-unit (i.e. the generalised Gelfand-Naimark transform and generalised Dirac transformation) are isomorphisms. The proof is exactly that of 11.2.2 and 11.2.5 with the suitable changes.

We now establish a duality theory for uniform'space. The dual object of a uniform space S will be the CoSaks space UOD(S) described in 1.4.11. We repeat the details. We denote by D the family of all bounded, uniformly continuous pseudometrics on S. Um(S) denotes the vector space of all bounded, uniformly continuous complex-valued functions on

s.

11)

(U"(S),ll

is a Banach

space (even a Cfr-algebra). H denotes the family of all uniformly

.

bounded, uniformly equicontinuous subsets of Urn(S) We remark that a set B is in H if and only if there is a

B is majorised by d (i.e. Ix(s) s,t

E

-

x(t) I

I

d

d(s,t)

E

D

so that

for x

E

B,

S). The sufficiency of this condition is clear. On the

other hand, if B

E

H, then

is a suitable majorant in U.

If we supply each B

E

H with the topology of pointwise (or

compact) convergence, then ( U m ( S ) ,11 II,H)

is a CoSaks space.

In addition, it is a commutative algebra with unit and so we can define the CoSaks spectrum MC(Urn(S)) to be the subset of the dual of U m ( S ) consisting of the unit-preserving multipli-

A.2

DUALITY

277

cative functionals. We regard MC(Um(S)) as a uniform space with the structure of uniform convergence on the sets of H. We can define, in the obvious way, a mapping 6 from S into Mc(Um(S) )

-

the generalised Dirac transform.

2.5. Proposition: MC(Um(S)) is a complete uniform space and 6 is a uniform isomorphism from

S

into MC(Um(S)).

Proof: The statement about the completeness of MC(Um(S) 1 is clear (for example, MC(Um(S) 1 is closed in the dual of the CoSaks space Um(S) and this is a complete locally convex space).

The 6-mapping is injective since Um(S) separated S . Now, by its very definition, the uniform structure of MC(Um(S) ) is generated by the pseudometrics

where B runs through H. Now

aB induces the pseudometric

on S and the uniform structure of S is generated by

{d, : B

E

BI.

For, as we have already seen, each dB is in V and, on the other hand, if d

is in H and

E

V then

d = dB.

APPEND IX

278

We shall now show that S is dense in MC(Um(S)) so that we can identify MC (Um(S)1 with the uniform completion of S.

2.6. Lemma: Let

s

E

f

E

MC(Um(S)). Then

a)

f is positive:

b)

if x 2 0, f(x) = 0, then

c)

for each

S so that

x,,

. . .,xn

sup /xi( s ) i=l , ,n

.

0

E

5);

in Urn(S) and

-

f (xi)I


E

0 there is an

E.

Proof: a) This is standard (for example, we can regard f as an element of the spectrum of the C"-algebra Urn(S)1. b) If 1x1 2 c) Put

x

for some

E

:=

C

f(x) = Cf(lxi for each

E

E

> 0, then If(x)l

2 f(E) =

[xi - f(xi) 1 . Then we have: f(x)

-

f(xi)I) = C If(xi)

> 0, there is an

s

E

-

E.

= 0

(for

f(xi)l = 0). Hence, by b),

S so that

x(s)