Topological algebras

TOPOLOGICAL ALGEBRAS

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NORTH-HOLLAND MATHEMATICS STUDIES

24

Notas de Matematica ( 6 0 ) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Topological Algebras EDWARD BECKE NSTE I N St. John's University, Notre Dame College, Staten island, New York

LAWRENCE N A R l C l St. John's University, Jamaica, N e w York

CHARLES SUFFEL Stevens Institute of Technology, Hoboken, N e w Jersey

1977

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

@ North-Holland Publishing Company - 1977 All rights reserved. No parf of this publication may be reproduced, stored in a retrieval sysrem, or transmitted, in any form or by any means, electronic, mechanical, phofocopying, recording

or otherwise, without the prior permission of the copyright owner.

North-Holland ISBN: 0 7204 0724 9

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM, NEW YORK, OXFORD SOLE DISTRIBUTORS FOR THE U S A . A N D C A N A D A :

ELSEVIER / NORTH HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017 Library of Coogrens Cataloglog la Pmblicatleo Data

Beckenstein, Edward, 1940To~~2~gica.l algebras. (Not- de m t d t i c 8 ; 60) (Norhh-Houana m a t h a tics studies ; 24) Includes index. 1. TqpologicaJ algebras. I. Nmici, I.&wPence, joint author. 11. S u f f a , Charles, joint author. 111. Title. IV. Series.

PRINTED IN THE NETHERLANDS

Kilimanjaro is a snow-covered mountain 19,710 feet high, and is said to be the highest mountain in Africa.

Its western summit is called by the

Masai "Ngaje Ngai," the House of God.

Close to the western summit there is

the dried and frozen carcass of a leopard.

No one has explained what the

leopard was seeking at that altitude. From "The Snows of Kilimanjaro," by Ernest Hemingway

For Paul and Maria, Dori and Chuck, and Marshall and Cheryl.

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PREFACE Let T be a completely regular Hausdorff space and let real numbers R or the complex numbers

C

F

stand for the

without specifying either.

Three main subjects are dealt with in this book:

(1) general

topological algebras; ( 2 ) the space C(T,F) of continuous functions mapping T into

F

as an algebra only (with pointwise operations); and ( 3 ) C(T,F)

endowed with compact-open topology as a topological algebra C(T,F,c).

We

wish to characterize the maximal ideals and homomorphisms of C(T,F) and the closed maximal ideals and continuous homomorphisms in general and C(T,F,c) in particular.

of topological algebras

In addition a considerable inroad

is made into the properties of C(T,F,c) as a topological vector space in Chapter 2.

Naturally enough, many of the results about C(T,F,c) serve to

illustrate and motivate results about general topological algebras. Attention is restricted to the algebra

C(T,R) of real-valued

continuous functions in Chapter 1 and to the pursuit of the maximal ideals and real-valued homomorphisms

of such algebras.

The clue to their identity

and capture is found in the case when T is compact, The collection M

of t functions x 6 C(T,R) which vanish at the point t is a maximal ideal whether

T is compact or not, but when T is compact, every maximal ideal of C(T,g) is of this type.

For noncompact TI the maximal ideals of C(T,R) are tied V

to the points of the Stone-Cech compactification BT of T in a very similar way.

When T is compact, all homomorphisms

of C(T,@

are evaluation maps,

maps t* taking functions x 6 C(T,R) into their values x(t) at t. that the kernel of t* is Mt.)

(Note

By a quirk of nature, this remains

essentially true even for noncompact T, but the whole story is a little more complicated. Gnerally the homomorphisms

of C(T,F) are evaluation

maps but associated with the points of the repletion UT of T, a certain subspace of BT6 rather than just T. homomorphisms

The quirk mentioned above by which the

of C(T,R) are usually given by just the points of T is that

UT = T for most spaces.

These things and others are discussed in Chapter 1.

Rather than deal

with them from the z-ultrafilter point of view however, as Gillman and Jerison do for example, we have used uniform spaces as the habitat for the development of the theory.

The idea that such an environment provides a

felicitous setting for the development of the theory of rings of continuous functions is due to Nachbin and Chapter 1 owes a great deal to the way in which Warner carried such a development through in a set of lectures given vii

viii

PREFACE

at Reed College.

Some background results on uniform spaces are given

without proofs in Chapter 0 with references given for the details. As

mentioned above, the Stone-zech compactification BT and repletion

UT of T plays an important role in the development of the algebraic properties of C(T,R).

Thus, significance attaches to obtaining them and

viewing T as a uniform space enables a simple and direct realization of each. (=

The theory of uniform spaces provides that every Hausdorff uniform

completely regular) space T has a unique completion. With no further

fuss this fact produces one form of the Stone-Eech compactification of T I as the completion of T with respect to the weakest uniformity with respect to which each bounded continuous function is uniformly continuous. (Mercurial entity that it is, BT emerges as a space of measures in Section 1.7 and as a space of homomorphisms of a Banach algebra in The repletion is obtained similarly - a s a completion of T

Section 4.10.)

with respect to a different uniformity. An

interest that is always present when studying C(T,F) is the

correlation of algebraic properties of C(T,F) with purely topological properties of T.

For examples:

(1) If consideration is restricted to

compact spaces, S and T say, then C(S,@

and C(T,R) are isomorphic as

algebras if and only if S is homeomorphic to T; ( 2 ) T is connected if and only if 0 and 1 are the only idempotents in C(T,F).

When C(T,F) takes on

the compact-open topology to become C(T,F,c), the scope for possible interactions broadens as one now seeks interplay between topological properties of T and topologico-algebraic properties of C(T,F,c).

In this spirit, in

Chapter 2, the famous theorems of Nachbin and Shirota are presented which settled Dieudonfie's question: bornological?

Must a barreled topological vector space be

Nachbin and Shirota independently obtained negessary and

sufficient conditions on T for C(T,g,c) to be barreled and for it to be bornological.

The condition for bornologicity is especially simple:

C(T,R,c) is bornological if and only if T = uT.

Referring to the necessary

and sufficient conditions on T which makes C(T,R,c) barreled as "condition NS", one can investigate the question:

Is there a T which satisfies

condition NS but for which T # uT? 1.e. is there a T for which C(T,R,c) is barreled but for which T # uT? There are such spaces (of ordinals, predictably enough) and so bornologicity is not implied by barreledness. All of Chapter 2 is devoted to correlating topological properties of T with topological vector space properties of C(T,F,c).

In particular, in

PREFACE

ix

addition to barreledness and bornologicity, conditions which guarantee or characterize metrizability, completeness, and separability of C(T,F,c) are obtained. Another compactification, the Wallman compactification, plays an important role in characterizing the maximal ideals of certain topological algebras (Chapter 5 ) .

To develop the Wallman compactification however some

knowledge of lattice theory is required.

What is needed, together with the

Wallman compactification itself, is presented in Chapter 3 .

In Chapter 4

the general subject of commutative topological algebras (with identity) is introduced and developed.

To be more accurate, it is the theory of com-

mutative locally m-convex algebras that is developed there. . For just as topological vector spaces display an almost disappointing similarity to topological groups without the added assumption

04 local

convexity, general

topological algebras are similar to the point of disinterest to topological rings without “local m-convexity“. With this added property, scalar multiplication plays an important role. In Chapter 6 a special type of algebra is dealt with which we call an LB-algebra-

The reason for the “LB” is that they are essentially just

inductive Limits of Banach algebras. In dealing with algebras of continuous functions as algebras we have stuck to real-valued functions in the text.

As pointed out in Section 1.4

tho, most of what appears in Chapter 1 remains true for algebras of complex-valued functions.

In dealing with topological algebras, where

whether the underlying field is

or

C can make a significant difference,

we have tried to treat the real and complex cases on an equal footing wherever possible, and pointed out where it is not possible.

In excursions

in the exercises we consider algebras of K-valued functions where 5 is a topological field or, more specially, a nonarchimedean valued field. What sort of background should one have to read the book?

Basically

some algebra (one should certainly know what prime and maximal ideals are), some topology (having taken a year course in it somewhere along the line should suffice), and some functional analysis including some things about Banach algebras.

A s to the functional analysis, the elementary properties

of locally convex spaces plus their duality theory should do; as for Banach algebras, not much is required per se, but the generalization of Banach algebra results to locally m-convex algebras will be more meaningful if something about Banach algebras is known.

Essentially it is the same

argument that would be given as to the desirability of some acquaintance

PREFACE

X

w i t h metric s p a c e s b e f o r e s t u d y i n g g e n e r a l t o p o l o g i c a l spaces: absolutely necessary, b u t i t ' s

It's not

n i c e t o have.

Our n o t a t i o n a l c o n v e n t i o n s do n o t r e q u i r e a n y s p e c i a l comment.

The

o n l y one t h a t i s u n u s u a l i n any way i s t h e u s e o f V t o mark t h e end o f a p r o o f , b u t t h e r e a s o n f o r t h a t c h o i c e had b e s t remain a mystery.

Except

f o r C h a p t e r 6 , e a c h c h a p t e r h a s a l a r g e number o f e x e r c i s e s a t t a c h e d t o it. The e a r l i e s t o n e s are r o u t i n e and meant f o r p r a c t i c e as w e l l as informat i o n ; t h e r e v e r s e i s t r u e o f t h e l a t e r ones.

R e f e r e n c e s are g i v e n as w e l l

as e x t e n s i v e h i n t s , many of which are r e a l l y p r o o f s w r i t t e n i n t e l e g r a p h i c These l a t e r e x e r c i s e s are meant mainly t o p r o v i d e a d d i t i o n a l

style.

i n f o r m a t i o n a b o u t o r i n f o r m a t i o n t a n g e n t i a l t o t o p i c s developed i n t h e

text.

A s simply " e x e r c i s e s " ,

t h e y s h o u l d b e approached w i t h extreme

caution. Before commending o u r l u c u b r a t i o n s t o you, t h e r e are many p e o p l e w e would l i k e t o t h a n k f o r many d i f f e r e n t r e a s o n s :

St. John's University f o r

p r o v i d i n g f a c i l i t i e s and a r e d u c e d t e a c h i n g l o a d t o N a r i c f : t h e N a t i o n a l S c i e n c e Foundation f o r p r o v i d i n g a summer g r a n t t o S u f f e l : E l i z a b e t h S u f f e l f o r d o i n g some o n e r o u s l y d i f f i c u l t t y p i n g : t o many o t h e r f r i e n d s , some f o r having made comments a b o u t t h e book which were d i r e c t l y h e l p f u l t o i t , o t h e r s f o r j u s t being there.

As f o r errors t h a t might a p p e a r i n t h e t e x t ,

w e s h u d d e r , a p o l o g i z e now, and v i g o r o u s l y s t i p u l a t e t h a t any t h a t might remain are a l l t h e f a u l t o f t h e f i r s t - n a m e d a u t h o r .

CONTENTS

i

Chapter 0 Fundamentals

0.1 0.2

Topologies defined by families of functions Uniformities defined by families of functions

Chapter 1 Algebras of Continuous Functions 1.1 1.2

1.3 1.4 1.5 1.6 1.7 1.8

The Stone-Cech compactification Zero sets Maximal ideals and z-filters Maximal ideals and the Stone-Cech compactification Replete spaces Characters and J T 0 - 1 measures, BT, and Ulam cardinals Shirota's theorem on repleteness

Chapter 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Metrizability of C(T,E,c) and hemicompactness Completeness and k -spaces k-spaces, k -spaceg and pseudofinite spaces Continuous 8ual of C(T,E,c) and support Barreledness of C(T,_F,c) Bbrnologicity of C(T,T,c) Separability of C(T,F,c) The bornology of C(T,;,c)

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6

Lattices and Wallman Compactifications

Lattices Lattices and associated compactifications Wallman compactifications of topological spaces BT and Wallman compactifications A class of Wallman-type compactifications Equivalent Wallman spaces

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4 . 11 4.12 4 . 13

Topological Vector Spaces of Continuous Functions

10

13

17 20

23 34 44

61 62

63 65 73 93 99 107 110 135 136 14 2 14 7 152 155 16 1 175

Topological Algebras

Topological algebras Multiplicative sets and multiplicative seminorms Locally m-convex algebras Final topologies and quotients The factor algebras Complete M C H algebras and projective limits The spectrum Q-algebras and algebras with continuous inverse Topological division algebras and the Gelfand-Mazur theorem Maximal ideals and homomorphisms The radical and derivations Some elements of Gelfand theory Continuity of homomorphisms

xi

176 18 1 184 18 9 192 197 201 204 2 10

220 236 241 268

xii

CONTENTS

Chapter 5

5.1 5.2 5.3 5.4 5.5 5.6

Hull-kernel topologies R e g u l a r a l g e b r a s and n o r m a l i t y c o n d i t i o n s C o n d i t i o n hH f l as a Wallman c o m p a c t i f i c a t i o n o f The X - r e p l e t i o n o f M c Frechet algebras

M

Chapter 6

6.1 6.2 6.3

299

Hull-Kernel Topologies

LB-Algebras

LB-algebras Some p r o p e r t i e s of LB-algebras Complete LMC LB-algebras

300 302 306 308 311 3 12 329 329 332 340

References

349

Index of Symbols

363

Index

365

ZERO Fundamentals THIS SHORT CHAPTER c o n t a i n s some t h i n g s w h i c h a r e b a s i c f o r what f o l l o w s

a n d makes c e r t a i n t h i n g s e x p l i c i t , s u c h a s " c o m p l e t e l y r e g u l a r " n o t i n c l u d i n g "Hausdorff."

M a i n l y i t d e a l s w i t h t o p o l o g i e s and u n i f o r m i t i e s

d e t e r m i n e d by f a m i l i e s o f f u n c t i o n s and how t h e two a r e r e l a t e d ;

a few

f a c t s a b o u t uniform s p a c e s a r e l i s t e d , w i t h r e f e r e n c e s t o Bourbaki f o r p r o o f s , i n Sec. 0.2.

These l a t t e r f a c t s a r e p u t t o u s e r i g h t away i n S e c .

1.1 where t h e Stone-Cech c o m p a c t i f i c a t i o n BT o f a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e T i s o b t a i n e d a s a u n i f o r m s p a c e c o m p l e t i o n o f T w i t h res p e c t t o t h e u n i f o r m i t y i n d u c e d by t h e s p a c e C(T,g) o u s f u n c t i o n s o n T.

of real-valued continu-

I n S e c . 1 . 5 t h e r e p l e t i o n ( r e a l c o m p a c t i f i c a t i o n ) uT

of T is o b t a i n e d s i m i l a r l y . 0.1

T o p o l o g i e s d e f i n e d by f a m i l i e s o f f u n c t i o n s .

We assume f a m i l i a r i t y

w i t h t o p o l o g y a n d the t h e o r y of u n i f o r m s p a c e s and c h o o s e B o u r b a k i ' s G e n e r a l Topolo=,

P a r t s 1 and 2 , h e r e i n a f t e r r e f e r r e d t o a s B o u r b a k i 1966a

a n d 19bbb r e s p e c t i v e l y ,

a s o u r s t a n d a r d r e f e r e n c e on t h e s e s u b j e c t s .

In d e a l i n g with t o p o l o g i c a l s p a c e s , Hausdorff s e p a r a t i o n is n o t includ ed i n any i n s t a n c e u n l e s s s p e c i f i c a l l y i n h i c a t e d .

By s a y i n g t h a t two s u b -

s e t s A a n d B o f a t o p o l o g i c a l s p a c e T a r e S e p a r a t e d by open s e t s , w e mean

t h a t d i s j o i n t o p e n s e t s U and V e x i s t c o n t a i n i n g A and B r e s p e c t i v e l y . a continuous function x:T

-

If

[ 0 , 1 ] e x i s t s which maps A i n t o { O ] and B i n t o

{ l ] , w e s a y t h a t A and B a r e s e p a r a t e d

5 continuous function.

Thus a

c o m p l e t e l y r e g u l a r s p a c e i s one i n which e a c h p o i n t t and t h e complement o f

a n y n e i g h b o r h o o d o f t may b e s e p a r a t e d by a c o n t i n u o u s f u n c t i o n .

Occasion-

a l l y "Tihonov s p a c e " i s u s e d a s a synonym f o r " c o m p l e t e l y r e g u l a r H a u s d o r f f space.

I'

U n l i k e B o u r b a k i , H a u s d o r f f s e p a r a t i o n i s n o t i n c l u d e d i n "compact.

'I

By " l o c a l l y compact" w e mean t h a t e a c h p o i n t i n t h e s p a c e p o s s e s s e s a n e i g h b o r h o o d whose c l o s u r e i s compact.

A s p a c e is a-compact i f i t i s a

c o u n t a b l e u n i o n o f compact s u b s e t s , L i n d e l B f i f e v e r y open c o v e r c o n t a i n s a countable subcover. C a l l a s e t c l o p e n i f i t i s c l o s e d and open and a t o p o l o g i c a l s p a c e z e r o - d i m e n s i o n a l i f i t p o s s e s s e s a b a s e of c l o p e n s e t s .

An example o f a

zero-dimensional space follows.

- 13

Example 0.1-1 (I:K

VALUED FIELDS

A f i e l d K t o g e t h e r w i t h a r e a l - v a l u e d map

s u c h t h a t f o r a l l a , b€K 1

2

0 . 1 TOPOLOGIES BY FUNCTIONS

12

(a)

la

(b)

lab

0 and =O i f f a=O;

I=

1p1 la 1 +

(a

I

(c) Jb la+bI I i s c a l l e d a v a l u e d f i e l d ; t h e map

(1

I I

i s c a l l e d a v a l u a t i o n on K .

s a t i s f i e s ( c ' ) below i n s t e a d o f ( c ) , t h e n

I I

If i s a nonarchimedeaq v a l u a t i o n

and K a nonarchimedean v a l u e d f i e l d : (c')

la+bI

I max(

la

1,

(bI.

I n e i t h e r c a s e d ( a , b ) = ( a - b I i s a m e t r i c on K and when K c a r r i e s t h e m e t r i c I f t h e v a l u a t i o n i s nonarchimedean, i t

topology, K i s a t o p o l o g i c a l f i e l d .

i s s t r a i g h t f o r w a r d t o v e r i f y t h a t s p h e r e s , open o r c l o s e d , i n K ,

1

{aCK la] < r o r 9 r?, r > 0 , a r e c l o p e n i n K .

Thus any nonarchimedean

valued f i e l d i s a zero-dimensional topological space.

E

NOTATIONS

and

;s t a n d

f o r t h e r e a l and complex numbers r e s p e c t i v e l y

E

carrying t h e i r usual topologies. either.

FT,

5,

and

9

denotes

E

or;

d e n o t e t h e n a t u r a l numbers,

without specifying i n t e g e r s , and r a t i o n a l s

respectively. I f S and T a r e t o p o l o g i c a l s p a c e s , C(S,T) c o n t i n u o u s maps o f S i n t o T .

s t a n d s f o r t h e s e t of a l l

I f t h e r e i s a n o t i o n of "bounded" s e t i n T ,

i f T was a t o p o l o g i c a l v e c t o r s p a c e , f o r example, t h e n Cb(S,T) d e n o t e s t h e c o l l e c t i o n o f a l l bounded c o n t i n u o u s maps from S i n t o T , i . e .

ous maps whose r a n g e i s a bounded s u b s e t o f T.

a l l continu-

I f T is a topological f i e l d

t h e n C(S,T) and Cb(S,T) a r e e a c h T - a l g e b r a s w i t h r e s p e c t t o t h e p o i n t w i s e operations:

(x+y)(t)=x(t)+y(t),

( x y ) ( t ) = x ( t ) y ( t ) , and ( a x ) ( t ) = a x ( t ) f o r

x and y i n C(S,T) o r Cb(S,T) and aFT.

I t i s always t o t h e s e o p e r a t i o n s

t h a t we r e f e r when we speak o f s p a c e s o f c o n t i n u o u s f u n c t i o n s a s a l g e b r a s , rings, o r l i n e a r spaces. lo.1-1)

(Bourbaki 1966a, p. 30, Prop. 4 ) .

INITIAL TOPOLOGIES

s e t , ((Tp,

3

))

Let T be a

a f a m i l y o f t o p o l o g i c a l s p a c e s , a n d , f o r e a c h pcM, x

I.L (Im map from T i n t o T . The t o p o l o g y 3 g e n e r a t e d by t h e s e t s

u

-1

a LL

pcMxp ( p) ' CL t h e t o p o l o g y h a v i n g IJ -'(gP) a s a s u b b a s e , i s t h e c o a r s e s t t o p o l WMXP ogy f o r T w i t h r e s p e c t t o which e a c h of t h e maps x i s c o n t i n u o u s . 3 i s P c a l l e d t h e i n i t i a l t o p o l o g y d e t e r m i n e d by t h e maps (x ) and a b a s e f o r

i.e.

IJ. WE1 i t i s g i v e n by f i n i t e i n t e r s e c t i o n s o f s e t s of t h e form X (G)

where CLEM P and G C a LL I n i t i a l t o p o l o g i e s a r e a means t o t r a n s p o r t t o p o l o g i e s i n r a n g e s of

.

f u n c t i o n s back t o t h e domain.

F i n a l topologies, discussed next, a r e a

vehicle f o r the reverse direction. 10.1-2)

FINAL TOPOLOGIES

(Bourbaki 1966a, p. 32, Prop.

6).

If T i s set,

0. FUNDAMENTALS

3

a f a m i l y of t o p o l o g i c a l s p a c e s and x a map from T i n t o T f o r e a c h ( 'b) M M )I c1 b € M t h e n t h e r e i s a f i n e s t t o p o l o g y 3 f o r T w i t h r e s p e c t t o which e a c h x 12 i s c o n t i n u o u s . 3 i s c a l l e d t h e f i n a l t o p o l o g y f o r T ( d e t e r m i n e d by t h e -1 maps (x ) ) and 3 c o n s i s t s of t h o s e s u b s e t s U o f T s u c h t h a t x (U) i s open P !J i n T f o r each p a . P

I f ( T , d ) i s a t o p o l o g i c a l s p a c e and C(T,R) t h e c l a s s o f a l l c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on T t h e n c e r t a i n l y t h e i n i t i a l t o p o l o g y mined by C(T,B)

on T i s c o a r s e r t h a n

d.

3,

deter-

The same i s t r u e o f t h e i n i t i a l

abC

d e t e r m i n e d by C (T,E) on T. Moreover gcC d. When T b i s a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e , however, the t h r e e t o p o l o g i e s topology

3b

coincide ((0.2-5)). Example 0.1-2

COMPACT-OPEN AND POINT-OPEN TOPOLOGIES ON C(T,F)

I f K is a

compact s u b s e t o f t h e t o p o l o g i c a l s p a c e T t h e n t h e map pK:C(T,F) x

+F +

(=$ o r C )-

I ()I

sup x K

i s a seminorm on t h e l i n e a r s p a c e C(T,E). pK i s a m u l t i p l i c a t i v e seminorm on C ( T , F )

Viewing C(T,I?) a s a n a l g e b r a , i n the sense t h a t

5 p K ( x ) p K ( y ) . The i n i t i a l t o p o l o g y d e t e r m i n e d by t h e maps pK a s K r u n s t h r o u g h t h e compact s u b s e t s o f T i s t h e compact-open t o p o l o g y f o r

pK(xy)

C(T,E). C(T,F,c).

When C(T,E) c a r r i e s t h e compact-open t o p o l o g y i t i s d e n o t e d by The compact-open t o p o l o g y i s a l o c a l l y convex H a u s d o r f f t o p o l o g y

f o r t h e l i n e a r s p a c e C ( T , F ) a n d a l o c a l l y m-convex H a u s d o r f f t o p o l o g y f o r t h e a l g e b r a C(T,F) ( s e e ( 4 . 3 - 2 ) ) .

A n e i g h b o r h o o d b a s e a t 0 i n C(T,,F,c) i s

g i v e n by t h e c o l l e c t i o n o f a l l p o s i t i v e m u l t i p l e s o f s e t s o f t h e form

a s K r u n s t h r o u g h t h e c l a s s o f compact s u b s e t s of T.

I f T i s compact, t h e n

a b a s e a t 0 i s g i v e n by p o s i t i v e m u l t i p l e s o f j u s t V C(T,E,c)

and i n t h i s c a s e P i s a Banach a l g e b r a , i t s norm b e i n g s i m p l y t x e s u p norm.

A n o t h e r name f o r t h e compact-open t o p o l o g y i s t h e t o p o l o g y o f u n i f o r m

2compact convergence 0

sets, t h e

compact-open t o p o l o g y i f f

-

r e a s o n b e i n g t h a t a n e t ~1

-, x i n t h e

x u n i f o r m l y on e a c h compact s u b s e t of T.

I f , i n s t e a d of t a k i n g t h e c l a s s of a l l compact s u b s e t s o f T, w e t a k e

t h e c o l l e c t i o n o f a l l s i n g l e t o n s , t h e e n s u i n g weaker l o c a l l y m-convex H a u s d o r f f t o p o l o g y f o r C(T,E) i s t h e p o i n t - o p e n t o p o l o g y ( t o p o l o g y E f e o i n t w i s e c o n v e r g e n c e or s i m p l e c o n v e r g e n c e ) .

C(T,E+,p) d e n o t e s C ( T , F )

0.1

4

TOPOLOGIES BY FUNCTIONS

endowed w i t h t h e p o i n t - o p e n t o p o l o g y . Example 0.1-3

u(X.X') AND TVS CONVENTIONS

when d e s i r e d w e w i l l s a y H a u s d o r f f TVS

(TVS) d o e s n o t i n c l u d e H a u s d o r f f ; (HTVS)

.

"Topological vector space"

" L o c a l l y convex t o p o l o g i c a 1 v e c t o r s p a c e ' ' and " l o c a l l y convex

Hausdorff t o p o l o g i c a l v e c t o r space" w i l l o c c a s i o n a l l y be a b b r e v i a t e d t o

A l l TVS's a r e assumed t o h a v e ,R o r

LCS and LCHS r e s p e c t i v e l y .

underlying f i e l d unless otherwise specified.

l i n e a r s p a c e o f a l l c o n t i n u o u s l i n e a r f u n c t i o n a l s on X. weakened) t o p o l o g y a ( X , X ' )

C

as their

I f X i s a TVS, X ' d e n o t e s t h e The

&

(or

f o r X i s t h e i n i t i a l t o p o l o g y g e n e r a t e d by t h e

maps P X 3 'F x a s x ' runs through X ' .

I

-t

< x,x' >

I

It i s c l e a r l y the c o a r s e s t topology f o r X with

r e s p e c t t o which e a c h x ' f X '

i s continuous, i . e .

g e n e r a t e d by t h e f a m i l y o f maps X ' o n X . d e t e r m i n e d on X ' by t h e maps

X I

+

I < x,xI

a l s o t h e i n i t i a l topology

a(X',X) i s t h e i n i t i a l topology 7

1

a s x r u n s t h r o u g h X.

These

weak t o p o l o g i e s , b e i n g d e t e r m i n e d by f a m i l i e s of seminorms, a r e c l e a r l y l o c a l l y convex t o p o l o g i e s . The p o l a r S o o f a s u b s e t S o f a TVS X i s t h e c o l l e c t i o n of a l l x ' f X ' such t h a t sup Ix'(S)

I

5 1; t h e d u a l c o n s i d e r a t i o n a p p l i e s t o p o l a r s o f

subsets of X ' . Example 0.1-4

=-TOPOLOGIES.

o f u(X,X')-bounded So o f s e t s

T(X,X') AND g ( X , X ' )

If 6 i s a collection

s u b s e t s o f t h e LCHS X , t h e n t h e c o l l e c t i o n

6" o f p o l a r s

ScG i s a s e t o f a b s o r b e n t b a l a n c e d convex s u b s e t s o f

XI.

Hence

t h e c o l l e c t i o n o f p o s i t i v e m u l t i p l e s o f f i n i t e i n t e r s e c t i o n s o f s e t s from

6 '

forms a n e i g h b o r h o o d b a s e a t 0 f o r a l o c a l l y convex t o p o l o g y f o r X '

c a l l e d the G-topology.

Dual c o n s i d e r a t i o n s a p p l y t o G - t o p o l o g i e s f o r X .

A n o t h e r way t o v i e w 6 - t o p o l o g i e s i s a s f o l l o w s . o f u(X',X)-bounded f o r e a c h Sc@

s u b s e t s o f X I , t h e n < x , S > i s a bounded s e t of s c a l a r s

and e a c h XCX.

The maps ps:x x

a r e seminorms ( p

S

If @ i s a c o l l e c t i o n

-1 +

sup) < x,s

i s a c t u a l l y t h e gauge o f So)

d e t e r m i n e d by t h e seminorms ( p )

s SCG

>

I

on X .

The ( i n i t i a l ) t o p o l o g y

i s t h e G - t o p o l o g y m e n t i o n e d above.

T a k i n g G t o b e t h e c l a s s o f a l l b a l a n c e d convex o(X',X)-compact s e t s of

X I ,

o r a l l a(X',X)-bounded

subsets of X I ,

the =topologies

sub-

FUNDAMENTALS

0.

5

g e n e r a t e d a r e r e s p e c t i v e l y t h e Mackey t o p o l o g y T ( X , X ' )

and t h e s t r o n g

topology B(X,X').

0.2

U n i f o r m i t i e s d e f i n e d by f a m i l i e s of f u n c t i o n s

10.2-1)

INITIAL UNIFORMITIES

17p, P r o p . 4 ) .

( B o u r b a k i 1 9 6 6 a , p.

Let T

a f a m i l y o f u n i f o r m s p a c e s , a n d , f o r e a c h WM, x a map U (TP)pfM from T i n t o T Then t h e r e i s a c o a r s e s t u n i f o r m i t y f o r T, c a l l e d t h e P i n i t i a l u n i f o r m i t y d e t e r m i n e d by t h e maps (x ) w i t h r e s p e c t t o which

be a s e t ,

u

.

each x

P

I-I

is uniformly continuous.

PCM'

A fundamental system of entourages f o r

u

i s g i v e n by s e t s o f t h e form

where (k,

,..., P,]

i s a f i n i t e s u b s e t o f M and e a c h V . i s a n e n t o u r a g e i n

.

u),

I f y i s a map f r o m a u n i f o r m s p a c e S i n t o t h e u n i f o r m s p a c e (T, Pi then y is uniformly continuous i f f x y:S 4 T i s u n i f o r m l y c o n t i n u o u s f o r IJ. P each ~ c M . T

.

I f each T

i s a H a u s d o r f f u n i f o r m s p a c e and i f t h e f a m i l y (x )

p uFM s e p a r a t e s p o i n t s i n T ( i . e . f o r e a c h p a i r ( t , t ' ) o f p o i n t s from T , i f t # t o !J

t h e n t h e r e i s some ~ c Ms u c h t h a t x ( t ) # x ( t ' ) ) t h e n !J P

u is

a Hausdorff

uniform s t r u c t u r e . As s t a t e d i n ( 0 . 2 - 2 )

below, t h e t o p o l o g y d e t e r m i n e d by

u is

j u s t the

i n i t i a l t o p o l o g y d e t e r m i n e d by t h e maps (x ) (0.2-2)

p.

INITIAL UNIFORMITIES VS.

177, C o r o l l a r y ) .

P

maps

P

(Bourbaki 1966a,

The t o p o l o g y d e t e r m i n e d by a n i n i t i a l u n i f o r m i t y

d e t e r m i n e d by maps (x ) each x

P pCMM' INITIAL TOPOLOGIES

i s continuous,

lLCM

i s t h e c o a r s e s t t o p o l o g y w i t h r e s p e c t t o which

i . e . i t i s t h e i n i t i a l t o p o l o g y d e t e r m i n e d by t h e

(XJPCMM.

A uniformity

u is

compatible w i t h a topology

mined by u i s j u s t 3 . i b l e uniformity

u exists

3i f

t h e topology d e t e r -

A t o p o l o g i c a l s p a c e T i s u n i f o r m i z a a i f a compat-

on T.

The c e n t r a l c h a r a c t e r i z a t i o n o f u n i f o r m i z a -

b i l i t y is:

(0.2-3)

UNIFORMIZABILITY

( B o u r b a k i 1966b, p . 144, Theorem 2 ) .

A topologi-

cal space T is uniformizable i f f i t i s completely regular. 10.2-4)

p.

THE UNIFORMITY OF A COMPACT HAUSDOFSF SPACE

199, Theorem 1).

(Bourbaki 1966a,

O n a compact H a u s d o r f f s p a c e T t h e r e i s e x a c t l y one

u n i f o r m i t y c o m p a t i b l e w i t h t h e t o p o l o g y o f T.

The e n t o u r a g e s o f t h i s

u n i f o r m i t y a r e a l l n e i g h b o r h o o d s o f t h e d i a g o n a l i n TxT.

0 -2 UNIFORMITIES BY FUNCTIONS

6

TWO u n i f o r m i t i e s o f s p e c i a l i n t e r e s t a r e t h e i n i t i a l u n i f o r m i t i e s

d e t e r m i n e d by t h e r e a l - v a l u e d c o n t i n u o u s f u n c t i o n s C(T,P_) and C (T,,@ on a b t o p o l o g i c a l s p a c e d e n o t e d by &(T,R) and Gb(T,&) r e s p e c t i v e l y and some times shortened t o simply

and

?$

sb.I f

T i s c o m p l e t e l y r e g u l a r and H a u s d o r f f ,

and C (T,F.) e a c h s e p a r a t e p o i n t s i n T and s o d e t e r m i n e H a u s d o r f f b u n i f o r m s t r u c t u r e s f o r T. L e t t i n g and Jb d e n o t e t h e t o p o l o g i e s C(T,&)

.yc

d e t e r m i n e d by $ and

b

clear that JbCJcCJ.

be T ' s o r i g i n a l topology, i t is

on T a n d l e t t i n g J

When T i s c o m p l e t e l y r e g u l a r , t h e t h r e e t o p o l o g i e s

coincide. 10.2-5)

COMPLETE REGULARITY AND THE INITIAL TOPOLOGY DETERMINED BY (T,J ) i s co mp let ely r e g u l a r i f f

c(T,&) ON T

SCT.

T h e r e i s some xcCb(T,R) s u c h t h a t x ( s ) = O a n d x(CV)={l],

I < 1/21 is a

{tCTIIx(t) by

J=J-~=J'.

Suppose t h a t T i s c o m p l e t e l y r e g u l a r and l e t V be a n e i g h b o r h o o d o f

Proof

Gb

b a s i c neighborhood of s i n t h e topology determined

It follows thatg'='Tb.

t h a t is contained i n V.

.$ =Jb,

Conversely, i f

and

then T is uniformizable, hence completely

r e g u l a r by ( 0 . 2 - 3 ) . V C o n c e r n i n g c o m p l e t i o n s o f u n i f o r m s p a c e s , we n e e d the f o l l o w i n g res u l t s , t h e u p s h o t o f which i s t h a t H a u s d o r f f u n i f o r m s p a c e s a r e d e n s e l y embedded i n a n e s s e n t i a l l y u n i q u e c o m p l e t e H a u s d o r f f u n i f o r m s p a c e . 10.2-b)

COMPLETION

( B o u r b a k i l g b b a , p.

1 9 1 , Theorem 3 ) .

For any uniform s p a c e T t h e r e i s a complete Hausdorff uniform

(a)

space

^T,

c a l l e d t h e H a u s d o r f f c o m p l e t i o n o f T , and a u n i f o r m l y

c o n t i n u o u s map i : T (P)

+

^T

which t h e p r o p e r t y :

G i v e n a n y u n i f o r m l y c o n t i n u o u s map f o f T i n t o a c o m p l e t e Hausdorff uniform space S , t h e r e i s a unique uniformly continuous map g:T

-+

S such t h a t f = g * i.

T ) is a n o t h e r p a i r c o n s i s t i n g of a complete Hausdorff uniform 1' 1 and a u n i f o r m l y c o n t i n u o u s map i l : T -, T h a v i n g p r o p e r t y ( P ) t h e n space T 1 1 t h e r e is a unique u n i f o r m s p a c e isomorphism h:T T such t h a t i =h- i. If (i

-

(b)

s p a c e t h e n t h e c a n o n i c a l map i : T

-$

phism o f T o n t o a d e n s e s u b s p a c e o f t h e completion of T.

1

1

I f T i s a Hausdorff uniform

( B o u r b a k i 1 9 6 6 a , p . 194, C o r o l l a r y ) .

i s a uniform s p a c e isomor-

^T.

In t h i s case,

?

is called

ONE

Algebras o f Continuous Functions

A S P. SAMUEL has remarked, there a r e two p r i n c i p a l methods of

i n v e s t i g a t i o n i n p o i n t s e t topology.

The f i r s t

refers t o t h e t o p o l o g i c a l space alone.

- the

-

" i n t e r n a l " method

Separation axioms, compactness, and

connectedness, f o r example, a r e u s u a l l y expressed s o l e l y i n terms of t h e t o p o l o g i c a l space.

-

The second method

numbers as an a n a l y t i c t o o l .

t h e "external" - u s e s

the r e a l

Here they u s u a l l y appear v i a t h e channel of

real-valued continuous f u n c t i o n s on t h e topological space T.

A t times t h e

e n t i r e c l a s s of such functions i s c a l l e d upon, a s i n t h e d e f i n i t i o n of complete r e g u l a r i t y ; i n other i n s t a n c e s a subclass such a s t h e continuous pseudometrics i s s i n g l e d o u t , a s w i t h uniform spaces.

From t h e e x t e r n a l

s t a n d p o i n t , i f one wants t h e t o p o l o g i c a l space T described a c c u r a t e l y by

i t s continuous f u n c t i o n s , C ( T , R ) , one must have enough of them.

And t o

guarantee a good supply of n o n t r i v i a l continuous f u n c t i o n s , T i s assumed t o be completely r e g u l a r and Hausdorff throughout. A goal of t h i s chapter i s t o develop an a l g e b r a i c e x t e r n a l method

-

i n t e r p l a y between t h e t o p o l o g i c a l s t r u c t u r e of T and t h e a l g e b r a i c s t r u c t u r e

of t h e algebra C ( T , R ) .

I n Chapter 2 t h e more complex i n t e r a c t i o n s between

t h e t o p o l o g i c a l space T and t h e topological vector space (or algebra) C(T,R,c), C ( T , g ) with compact-open topology,are i n v e s t i g a t e d .

But here con-

s i d e r a t i o n i s r e s t r i c t e d t o topologicoalgebraic i n t e r a c t i o n s .

Generally

what can be s a i d of t h e a l g e b r a i c s t r u c t u r e of a l g e b r a s of continuous funct i o n s C(T,R)? How s e n s i t i v e i s C ( T , R ) t o T?

1.e. can e s s e n t i a l l y d i f f e r e n t

T ' s determine t h e same space of continuous f u n c t i o n s ?

is clearly t r i v i a l : tinuous f u n c t i o n s . Section 1.6. along t h e way.

The converse question

homeomorphic spaces do have t h e same space of conThe former question i s answered (negatively) i n

T h e machinery t o answer it, and o t h e r s l i k e it, i s developed

To answer t h a t p a r t i c u l a r question, t h e idea of a

" r e p l e t i o n " uT, a c e r t a i n type of completion of a completely r e g u l a r Hausdorff space T , i s needed. spaces.

This l e a d s t o t h e r e l a t e d i s s u e of " r e p l e t e "

( I f r e p l e t e spaces produce t h e same a l g e b r a of continuous

f u n c t i o n s , they must be homeomorphic.) The Stone-zech compactification f3T of T i s a l s o c u l t i v a t e d a s a completion (Sec. 1.1) and l a t e r , i n Section 1 . 7 , a s a space of f i n i t e l y a d d i t i v e 0 - 1 measures.

The kinship between BT and UT, both being

7

8

1.

ALGEBRAS OF CONTINUOUS FUNCTIONS

completions, emerges in the measure-theoretic setting with UT appearing as the countably additive members of BT. Because of our interest in replete spaces, for reasons which will become apparent as the chapter develops, questions concerning which particular spaces are replete

(R

is, for example) or the broader issue of what

classes of spaces are replete are unavoidable.

In trying to answer the

latter question we are led to certain fundamental set-theoretic questions

-

a prospect apt at first to strike terror in the hearts of many mathematicians (us, in particular)

- such as:

Can Ulam cardinals exist within the

framework of Zermelo-Fraenkel set theory?

Some discussion of these matters

is given in Section 1.7 before going on to Shirota's theorem in Section 1.8 governing repleteness of complete uniform spaces. 1.1

The Stone-Zech Compactification Very early on, with respect to the definition of uniform spaces, Weil

recognized that every compact Hausdorff space -hence every subspace of a compact Hausdorff space - could be viewed as a uniform space. He did this (Weil 1937, p. 2 4 ) using only internal methods. converse statement

- namely

In order to prove a

that every uniform space could be viewed as a

subspace of a compact Hausdorff space

- he had

to use the real numbers.

An

internal construction of this fact, similar to the way in which the Wallman compactification is obtained in Chapter 3 as a space of ultrafilters, was given by Samuel (1948). That every uniform (= completely regular) Hausdorff space T possesses a compactification" to which every bounded continuous function on the space may be extended is the subject of this section.

The reason for our

interest in it lies in the close connection between the maximal ideals of C(T,€J) and the points of the compactification, the Stone-zech compactification, as elaborated on further in Section 1.4.

The construction of the

Stone-zech compactification given here, as a uniform space completion

-

hence as a space of ultrafilters - is due to L. Nachbin. Existence and uniqueness of the Stone-Eech compactification BT were first proved by Stone (1937), using the methods of Boolean rings. (1941) has a construction using Banach lattices.

Kakutani

zech (1937) simplified

Stone's original proof while Wallman-type compactifications (cf. Chapter 4

*

A compactification S of a topological space T is a compact space containing a dense homeomorphic image of T. A Hausdorff compactification is a compactification which is a Hausdorff space.

1.1 STONE-CECH COMPACTIFICATION

and Wallman 1938) y i e l d BT i f T i s normal.

9

An a p p r o a c h u s i n g Banach

a l g e b r a t e c h n i q u e s , due t o G e l f a n d and S i l o v , i s d i s c u s s e d i n S e c t i o n 4 . 1 2

A r e a l i z a t i o n a s a s p a c e of measures due t o V a r a d a r a j a n i s g i v e n i n S e c t i o n 1 . 7 w h i l e s t i l l o t h e r avenues t o BT a p p e a r i n t h e e x e r c i s e s . I n t h i s s e c t i o n T i s a c o m p l e t e l y r e g u l a r Hausdorff space, C ( T , R ) i s t h e a l g e b r a ( w i t h p o i n t w i s e o p e r a t i o n s on the f u n c t i o n s o f C ( T , R ) ) o f cont i n u o u s f u n c t i o n s t a k i n g T i n t o R, and C ( T , R ) w i l l b e t h e s u b a l g e b r a o f b C ( T , R ) c o n s i s t i n g o f a l l bounded f u n c t i o n s . The p r o t e a n Stone-Cech c o m p a c t i f i c a t i o n o f T a p p e a r s as t h e maximal i d e a l s o f C ( T , R ) ( a p p r o p r i a t e l y t o p o l o g i z e d ) , as a s u b s p a c e o f a p r o d u c t b o f c l o s e d u n i t i n t e r v a l s , as t h e z - u l t r a f i l t e r s o f T ( a p p r o p r i a t e l y t o p o l o g i z e d ) , a s p a c e o f measures ( S e c . 1 . 7 ) and as t h e c o m p l e t i o n of T w i t h respect t o a c e r t a i n uniform s t r u c t u r e .

W e examine t h e l a s t c o n s t r u c -

t i o n and show (Theorem 1.1-1) t h a t i n s o m e s e n s e BT i s u n i q u e .

{

L e t x E C ( T , ? ) and c o n s i d e r t h e s e t s V(X,E) = b ( s , t )E TxTl I x ( s ) x ( t ) < E } f o r E > 0. Then t h e c o l l e c t i o n o f e n t o u r a g e s

-

(V(X,E)

Ix E C b ( T , R )

I

,E

> 0 ) form a s u b b a s e f o r a u n i f o r m s t r u c t u r e

c,

on T

c o m p a t i b l e w i t h t h e t o p o l o g y o f T. D e f i n i t i o n 1.1-1.

THE STONE-CECH COMPACTIFICATION.

The c o m p l e t i o n BT Of

t h e c o m p l e t e l y r e g u l a r Hausdorff s p a c e T w i t h r e s p e c t t o the u n i f o r m s t r u c -

C

i s c a l l e d t h e Stone-Cech c o m p a c t i f i c a t i o n of T. b Theorem 1.1-1. ELEMENTARY PROPERTIES OF BT. L e t T be a c o m p l e t e l y r e g u l a r ture

Hausdorff s p a c e .

Then:

(a)

BT i s a compact Hausdorff s p a c e .

(b)

Each x E Cb(T,R) c a n b e ( u n i q u e l y ) * e x t e n d e d t o

2 (c)

E C(BT,g)

= Cb(BT,9.

I f BT is a c o m p a c t i f i c a t i o n of T which i s a Hausdorff s p a c e and e a c h x E Cb(T,R) c a n be ( u n i q u e l y ) * e x t e n d e d t o

2 E

C(ET,R) , t h e n

BT and ET are e q u i v a l e n t c o m p a c t i f i c a t i o n s o f T ( i . e . BT and BT

a r e homeomorphic under a mapping which e x t e n d s the " i d e n t i t y " on T ) .

( a ) To show t h a t BT i s compact, it i s s u f f i c i e n t t o show t h a t T

Proof

wi t h t h e uniform s t r u c t u r e

cb

i s t o t a l l y bounded f o r t h e n BT i s complete

and t o t a l l y bounded and (Bourb. 1966a, p.

2 0 2 ) t h e r e f o r e compact.

x E Cb(T,R), t h e r e e x i s t s a c l o s e d i n t e r v a l [ a , b ] C R such t h a t

*

As

5

i s a Hausdorff s p a c e , t h e e x t e n s i o n

must be u n i q u e .

For e a c h

1.

10

consider a f i n i t e s e t of p o i n t s s

x(T) C [ a , b l . such t h a t I s

- si I i+l

# B, there exists t

x ( t . 1 E . [ s i , ~ i + l l and it f o l l o w s t h a t x [ t i ] = ( t € T I ( t i , t )€ V ( x , f ) ).

V(X,E)

x

-1

= a < s1


(4).

Hence

S

to define x

(s).

z-ultrafilter F S

c clS(z(x)nv

!J

1.

# $3.

u

s €

clS(z(x)nz(y)).

TO prove ( 4 ) we must extend an arbitrary x € Cb(T,R) up to

so that the resulting function xs is continuous.

S

s

# 0 and, as Vs is an arbitrary zero set

neighborhood of s , it follows that (3)

# $3.

and therefore

For each

s € S

we wish

To accomplish this we show that there is a unique on T which converges to

x ( s ) as the limit of a z-filter on

s

and then unambiguously define

derived from

F

.

To that end let

be any z-ultrafilter on T containing the z-filter base (V

U

)

where

Fs

($) is

16

1. ALGEBRAS O F CONTINUOUS FUNCTIONS

in S and V = VSnT for each p. V P it follows that the filterbase F + s.

a base of zero set neighborhoods of Since (V

u

) C

F

s

F

If F is another z-ultrafilter convergent to s and distinct from may choose z(w)

€

F and z(y)

F

C

such that z(w)

n z(y)

more fundamentally by ( 2 ) , cl z(w) fl cl z(y) = $.

s

S

s is an adherence point of F

-

s

B

=

9.

we

By ( 3 ) , or

- i.e.

Since s F cl z(y) S

cl z(w) and s is not an adherence point S

of F thus contravening the convergence of

F

to

s.

s

Now, to obtain the definition of x ( s ) , let [a,b] be any closed interval containing x(T) and 8 be the class of all closed subsets E of -1 a z-filter of subsets of [a,b] is [a,bl such that x (E) € fs. Clearly 8 S'

S

a filterbase and, as [a,b] is compact, there is an adherence point x ( s ) of B s . To see that B

actually converges to

s

we show first that

BS is prime,

i.e. if A and B are closed subsets of [a,b] whose union belongs to 8 one or the other of A and B belongs to 8 -1 x (A) U x-l(B) = x-l(AUB) C Fs. But -1 that x - l ( A ) or x (B) belongs to Fs and To show that 8 S

of x ( s ) in [a,bl. U

-+

fore

Indeed if A U B C Bs then

is a z-ultrafilter and it follows is seen to be prime.

S

x (s), let V be an arbitrary zero set neighborhood

As [a,b] is completely regular there is a zero set

c [a,bl such that xS ( s )

since 8

.

F

[a,b] - u c V .

Now U U V = [a,bl F

is prime, either U or V belongs to 8

u k Bs.

then

Hence V 4 8

.

and it follows that Bs

Bs and,

S

But x ( s ) k U and thereS

-+

x

(s).

to show that xs is continuous. To this end S let V be any zero set neighborhood of x ( s ) . The job now is to exhibit a S neighborhood W of s € S such that x (W) C V. By the complete regularity of

All that now remains'is

S

[a,bl there exists a zero set V ' C [a,b] such that x ( s ) F [a,bl -V'. Thus -1 -1 -1 -1 V U V ' = [a,b] and x (V) U x (V') = T, so clS x (V) U clsx (V') = S. -1 -1 We contend that s cl x (V'). If it did, if s Q clS x (V'), then, s -1 as s adheres to each set in F cl x (V') fl clsz(w) # @ for each Z(W) c Fs. s'

s

-1 BY (3) it now follows that x (v') fl z(w) # 9 for all z(w) f Fs. Since Fs -1 -1 is a z-ultrafilter and x (V') is a zero set, x (V') F Fs. By the S S V' E Bs. On the other hand Bs -+ x ( s ) and x ( s ) t! V' - a definition 8 SI

contradiction. Thus s

&' cl x -'(Y'). S The foregoing argument shows that s belongs to the open set -1 -1 If p F w C clsx (v) then as W is a s - c l x (v') = W C61 x-'(V). S S -1 neighborhood of p, every neighborhood of p will intersect x (V) and

1.3

therefore x

s x (p)€

-1

V.

Fp.

(V)c

MAXIMAL IDEALS AND z-FILTERS

17

S

Thus V E 8

-f

x (p) s o , as V is closed in [a,b],

s p This proves that x (W) c V. V

1.4 Maximal Ideals and the Stone-Cech Compactification It is clear that if t is a point of the completely regular Hausdorff space T and M = [xf C(T,R) Ix(t) = O l , then Mt is a maximal ideal of C(T,R). t In fact it is the kernel of the nontrivial homeomorphism x + x(t) from C(T,R) onto

5.

As is verified in (1.4-2) below, if T is compact, then

these "fixed" ideals M constitute the set of all maximal ideals of C(T,R). t Even if T is not compact tho, each of the maximal ideals of C(T,R) is "fixed" on a point p of the Stone-Cech compactification BT of T, a fact which is the essential content of Theorem 1.4-1, the main result of the section. h

Definition 1.4-1. Hausdorff space.

FIXED MAXIMAL IDEALS. A

Let T be a completely regular

maximal ideal M of C(T,R) is fixed if there is some

Otherwise M is called free. T such that M = M = (xfC(T,R) Ix(t) = O } . t z ( x ) $ 0. (1.4-1) FIXED IDEALS. The maximal ideal M is fixed iff xfM

t

€

na(x) = {tl for some t in the completely regular Hausdorff x €M

In this case space T.

then surely t f n z(x). If s # t then, since T is t x €M completely regular, there is some x € C(T,g) such that x(t) = 0 -hence

Proof

If M = M

n z(x). Thus x CM Conversely suppose that M is maximal and

x F M~ -while

X(S)

= 1 SO S{

n Z(X) = {tj. x €M n z(x) # $. If

xcM

n , z(x), then x(t) = 0 for all x C M and M is seen to be a subset of x FM v The maximality of M then implies that M = M t' Mt. For any T, completely regular and Hausdorff as usual, C(T,FJ) possesses t

C

fixed maximal ideals - namely those of the form M

As = {xlx(t) = O ] . t (1.4-2) shows, if T is compact, then all maximal ideals of C(T,R) are of

this type.

Conversely if T is not compact, C(T,R) always possesses free

ideals. V (1.4-2)

MAXIMAL IDEALS ALL FIXED WHEN T IS COMPACT.

If T is a Compact

Hausdorff space then the maximal ideals of C(T,R) are all fixed. Proof

We show that there are no free ideals in C(T,R) when T is compact:

we show that if M is free then M contains a unit. If M is free then for each t xt(t) # 0.

F

T there is some xt

6

M such that

Since each xt is continuous there must be open neighborhoods Ut

ALGEBRAS OF CONTINUOUS FUNCTIONS

1.

18

for each t on which x does not vanish. Since T is compact, finitely many t ,...,Ut say, must cover T. It is now easy to see that the n function

y

=

x2 +

+ x2 tn

***

tl

<M

never vanishes on T and is therefore invertible. V As a consequence of (1.4-2) we may determine the form of the maximal

ideals of C (T,R) whether T is compact or not. b ~ p :Cb(T,P)

+

First consider the map

C(BTrR)

X’X

where

i

denotes the unique extension of x to BT.

Through the algebra

isomorphism @the maximal ideals of C (T,?) are seen to be in 1-1 correb spondence with those of C(BT,R). Since BT is compact, the maximal ideals of C(BT,R) are all of the form

ii

P

{;tc C(BT,R) I;C(p) = o ]

=

pc

BT.

corresponds (1.e. cp(M ) = h ) to the maximal ideal M in Cb(T,R) P P P P consisting of those x € C (T,?) such that j(p) = 0. b In the case when T i s not necessarily compact, Theorem 1.4-1 below

And

characterizes the maximal ideals of C(T,?).

Before proving it, note that

(1.4-2) may evidently be recast as follows: If T is comuact then all maximal ideals of C(T,R) are of the form M

P

Theorem 1.4-1.

=

(x€C(T,R)

IPF

clBz(x)]

BT AND MAXIMAL IDEALS. $T

P

+

+

p €BT.

(Gelfand-Kolmogorov) The mapping

M

{xcC(T,R) Ipc cl z(x)) = M B P

establishes a 1-1 correspondence between the class M of maximal ideals of C(T,R) and the points p of BT. Proof y

€

.First we show that M

C(T,R), then z(x)

P

c z(xy),

is an ideal in C(T,R).

If x E M and P For x,y c Mp,

so p C clBz(x) C cl z(xy1.

B clBz(x) f l clBz(y) = cl [z(x)n z(y)J c c1 z(x+y), the equality B B concerning closures of zero sets following from Theorem 1.2-2. Thus M

p

F

P

an ideal.

is

1.4

MAXIMAL I D E A L S AND STONE-CECH C O M P A C T I F I C A T I O N

19

To prove t h a t M i s maximal we f i r s t show t h a t z(M ) i s a P P -1 z - u l t r a f i l t e r and then t h a t z ( z(Mp)) = Mp (Theorem 1.3-1). L e t y be such that z(y)

n z(x)

# @ for all x

M P that p C c l z(y).

h

i.e. P' of p , we claim t h a t x =

that y C

f

.

B

We show t h a t z ( y ) c z(M ) by showing P I f z ( & ) i s any z e r o set neighborhood

.

belongs t o M To s e e t h i s l e t p F i n t ~ ( 2 ) . P B n i n t z(f;) i s a nonempty neighborhood

21

T Then f o r each neighborhood V of p , V

of p .

nT

meets z ( x ) = z ( ; )

B

i n t z(f;) f l T i s a l s o nonempty.

A s T i s dense i n BT, V

B

Thus V

f o r each neighborhood V of p and it f o l l o w s t h a t P€

.

We have shown t h a t every member of a b a s i c family of Hence x C M P neighborhoods of p ( i . e . t h e z e r o s e t neighborhoods; s e e (1.2-4))meets

.

I t only remains t o show t h a t z ( y ) , so p c c l g z (y) and y 6 M P ) ) = Mp: To s e e t h i s l e t W F z - ~ z(Mp)). ( As such t h e r e m u s t be P some m € M such t h a t z(w) = z(m) Hence p C c l z(m) = c l z ( w ) and B B P therefore w € M The d e s i r e d e q u a l i t y now f o l l o w s . P We wish t o show t h a t M = M Now l e t M be a maximal i d e a l i n C(T,€J). P f o r some p c BT. To prove t h i s we show t h a t MC M f o r some p C BT. P I f no M c o n t a i n s M then f o r each p C BT t h e r e i s some x c M such P P t h a t p Q' c l z ( x ) . Thus BT = C(c1 z ( x ) ) . Since BT i s compact, B P p€BT

z-'(z(M

.

.

u

n BT =

f o r some p l * - . . l P n

c

BT.

U C(c1 z ( x )) B - Pi i=l

Thus

n

. n

\

s i n c e c l o s u r e and i n t e r s e c t i o n may be interchanged f o r z e r o s e t s by n it f o l l o w s t h a t n z ( x ) = @. But x , . . . I ~ F M and Theorem 1.2-2; Pi P1 Pn n I t now f o l l o w s t h e r e f o r e n z(xpi) = @ i s impossible by Theorem 1.3-1. i= 1 t h a t t h e r e i s some p C BT f o r which M = M P I f p and q a r e d i s t i n c t , F i n a l l y we show t h a t i f p # q then M # M q' P t h e r e a r e d i s j o i n t z e r o s e t neighborhoods fr and 9 of p and q r e s p e c t i v e l y

.

i n BT and f u n c t i o n s x , y c C ( T , R ) z(y) =

nT

= V.

maximal i d e a l .

v

As U

nV

=

@I

such t h a t z ( x ) =

ir n T

= U and

x and y cannot belong t o t h e same

20

1.

ALGEBRAS OF CONTINUOUS F U N C T I O N S

Theorem 1.4-1 enables us to characterize compactness externally, namely:

The completely regular Hausdorff space T is compact if and only if

each maximal ideal in C(T,R) is fixed.

To see the sufficiency of the

condition, we need only note that if T is not compact, then for any p

C

BT-T, M

P

is not fixed.

The algebra C(T,C) of complex-valued continuous functions does not differ markedly from C(T,R) if one takes preservation of the results of this chapter as the datum. the C(T,C)-completion.

The C(T,R)-completion of T yields BT; so does

The maximal ideals of C ( T , C ) are in 1-1 corre-

spondence with the maximal ideals of C(T,R) under the mapping M

+

re M + i re M where re M denotes the collection of real parts re x of

functions x in M.

The inverse of this map is the map sending the maximal

ideal J of C(T,R) into J+iJ. the point p

C

The Gelfand-Kolmogorov theorem then pairs

BT with the maximal ideal M +iM

P

P

of C(T,C) where

By similar considerations, a marked M = {x Q C(T,R) Ip Q clBz(x)]. P propinquity is exhibited between the subalgebras of bounded continuous functions, whether real- or complex-valued. Even if

5

is replaced by the

e

of quaternions with euclidean (i.e. 54 ) topology, and "ideal" by "two-sided ideal," the similarities persist: topological division ring

The

collection H (TI!) of horn omorphisms of C ( T , ! ) onto endowed with the b b again yields $T, for example, so the initial topology generated by C (TI!) b maximal two-sided ideals of C(T,H) are in 1-1 correspondence with the maximal ideals of C(T,g).

It is worth noting here that 8, C and

are the

only connected lecally compact topological fields; if a locally compact field is not connected, then it must be totally disconnected (Bourbaki 1964, Ch. 6, 59, no. 3 , Cor. 2). Some discussion is devoted to algebras of continuous functions taking values in a topological field or a topological ring in the exercises at the end of this chapter and scattered in a few other places throughout the book. 1.5

Replete Spaces The Stone-Cech compactification $T of T enabled us to characterize the

maximal ideals of C(T,g) in a certain way (Theorem 1.4-1, the GelfandKolmogorov theorem):

Each p M

P

When is M

P

=

Q

BT corresponded to the maximal ideal

{xCC(T,R)

IPQ

clBz(x)]

.

the kernel of a horn omorphism of C(T,R) into

R?

A

sufficient

condition is that p c T for then M is the kernel of the hom omorphism 6 P sending x into x(p). AS we shall see later ( (1.6-1)and (Theorem

1.5

21

REPLETE SPACES

1.5-1), t h e "super a d h e r e n c e p o i n t s " o f T - t h o s e p o i n t s ~ c B Tsuch t h a t f o r a l l c o u n t a b l e f a m i l i e s (V ) o f neighborhoods o f p i n B T , n V

must meet T

d e t e r m i n e a l l maximal i d e a l s which are k e r n e l s o f homomorphisms.

-

The

c o l l e c t i o n uT o f a l l s u c h p o i n t s i s t h e r e p l e t i o n o f T and it i s t o t h i s c o l l e c t i o n t h a t t h i s s e c t i o n i s devoted.

Once t h e maximal i d e a l s which are

have been c h a r a c t e r i z e d a s above, w e a l s o know

k e r n e l s o f homomorphisms

which s p a c e s T have e v a l u a t i o n maps as t h e o n l y homomorphisms namely t h o s e and o n l y t h o s e f o r which UT = T.

of C ( T , R ) :

Spaces T w i t h t h i s l a t t e r

p r o p e r t y are c a l l e d r e p l e t e and c o n s t i t u t e t h e o n l y s p a c e s f o r which C ( T , F , c ) - C ( T , g ) w i t h compact-open t o p o l o g y - i s b o r n o l o g i c a l , as i s shown A s u r p r i s i n g f a c t about r e p l e t e spaces i s t h a t n o n r e p l e t e

i n C h a p t e r 2.

s p a c e s a r e so h a r d t o come by. f i r s t uncountable o r d i n a l of

En i s

The s p a c e [ O , Q ) o f o r d i n a l s l e s s t h a n t h e

i s n o t r e p l e t e (Example 1.5-1)

b u t any s u b s p a c e

(Theorem 1 . 5 - 3 ) , as i s any complete u n i f o r m s p a c e T , p r o v i d e d o n l y

t h a t T i s o f non-Ulam

(nonmeasurable) c a r d i n a l (Theorem 1.8-1), t o name

j u s t t w o broad c l a s s e s o f r e p l e t e spaces. Before s t a r t i n g t h e d e f i n i t i o n of r e p l e t i o n w e recall t h e following r e s u l t on e x t e n s i o n by c o n t i n u i t y ; f o r a p r o o f see K e l l e y 1955, page 153 o r Dugundji 1966, page 243. (1.5-1)

L e t T b e a c o m p l e t e l y r e g u l a r Hausdorff

EXTENSION BY C O N T I N U I T Y .

s p a c e and

g U

x C C(T,R)

{-I

be t h e one-point

c o m p a c t i f i c a t i o n o f g.

c a n b e u n i q u e l y e x t e n d e d t o a f u n c t i o n ;€

Any f u n c t i o n

C(BT,EU{-])

Using t h e e x t e n s i o n s whose e x i s t e n c e were j u s t n o t e d , w e now d e f i n e t h a t s u b s p a c e of BT c a l l e d t h e r e p l e t i o n o f T. D e f i n i t i o n 1.5-1.

REPLETION.

The r e p l e t i o n ( r e a l c o m p a c t i f i c a t i o n ) U T o f

t h e c o m p l e t e l y r e g u l a r Hausdorff s p a c e T c o n s i s t s o f t h o s e p € BT f o r which e a c h G(p) i s f i n i t e , i . e . G(p) #

-,

f o r every x E C(T,R).

I f T = UT, t h e n

T is called replete.

S i n c e T i s d e n s e i n BT, t h e o n l y way t o e x t e n d a f u n c t i o n x t o a continuous real-valued continuity.

C C(T,R)

f u n c t i o n on any s u b s p a c e S , T C S C BT, i s by

Thus UT i s t h e l a r g e s t s u b s p a c e o f BT t o which e a c h x

possesses a real-valued

continuous extension.


0 then by n=l AS p

n

'

m

choosing m sufficiently large we may guarantee that 00

I n ; + l

I

xn( 5

2-n < E on T. Hence Ix-'(tll) > 1 / ~for each index LI and, by the n=m+l (x-') (p) 1 . 1 / ~ .The desired result that continuity of (x-') (x-1) 6 (p) = m now follows.

',I

To prove (a)

that z(y) c z ( M

I

* (b) it remains to show that z ( x )

).

P

previous argument,

Then z(y) n z(xn)

€

z (M 1 .

P

Suppose

z(M ) for each n and so, by the

P

1.7

MEASURES, BT AND UL4M CARDINALS

As z(x) meets each set in the z-ultrafilter z(M

P

Last we show that (c) such that x B (p) = n E n 6

1.7

3. y.

a.

it follows by (1.3-2) (b)

),

P

.

that z (x) € z(M )

* (a).

If p

By (1.2-1) (f), x

-1

A s xB is continuous at p and x

k

33

U T then there is some x

([n,m))

< C(T,R)

is a zero set for each -1 cl x ([n,m)) for each

B

(p) = m, p € B -1 -1 Thus each x ([n,m)) E z(M ) andn x (In,-)) = @.

P

v

0-1 Measures, BT, and Ulam Cardinals

There is a 1-1 correspondence between z-ultrafilters on T and points of BT, as well as with maximal ideals of C(T,R).

Associated with each

z-ultrafilter F is a 0-1 measure (on the algebra A the zero sets

z)

defined by taking m(E) = 1 if E

Moreover associated with each t

of sets determined by

c F , 0 otherwise.

T is the 0-1 measure m

concentrated at t t (mt(EC A ) = 1 if and only if tE E), thus providing a natural embedding of

T in the space M

0

f

of 0-1 measures on T. By suitably topologizing M

0

, the

map t + m actually embeds T homeomorphically in M , as shown in 0 t Theorem 1.7-1. So topologized, not only is M compact, but functions

x

6 C ( T , R ) may be continuously extended from T to

b which this is done is defining the extension

2

Mo.

The vehicle by

to be /xdm at m, i.e.

G:Mo-t? m + /xdm This is what Varadarajan (1965) did and this result is presented here. Converting the z-ultrafilters into a space of measures - a topological vector space of measures in fact -has the advantage of making a host of results from measure theory available. A difficulty which this leads us into is confrontation with some

problems in set theory.

The measures m concentrated at points of T are t countably additive set functions which may be defined on the class P(T) of all subsets of T, with m (T) = 1. But are these the only such set t functions? Could there be a countably additive {0,1]-valued set function p defined on P(T) such that p(T) = 1 but u({t]) question of whether such a function

- an

=

0 for each t C T?

The

Ulam measure -can exist on a set

T has antecedents as far back as 1904 when Lebesgue asked: exist a measure m defined on P([O,ll) such that m([0,11)

=

Does there 1 and such that

34

1.

ALGEBRAS OF CONTINUOUS FUNCTIONS

Since all one-point sets must have

congruent sets* have the same measure?

the same measure under these conditions, each singleton must have measure The question was answered in the negative in 1 9 0 5 by Vitali: measure could exist.

0.

No such

Moreover, as shown by Banach and Kuratowski in 1929,

even if the congruence requirement is dropped, no such measure can exist on [0,1] if the continuum hypothesis is assumed.

If the generalized continuum

hypothesis (GCH) is assumed, then no such measure can exist on any set E , Calling a cardinal IT1 an Ulam cardinal if an

as Banach showed in 1930.

Are

Ulam measure can be defined on P(T), the question may be phrased:

there sets of Ulam cardinal? Within the framework of Zermelo-Fraenkel (ZF) set theory, the existence of such cardinals cannot be proved - at least if one assumes ZF to be consistent, for their nonexistence has been shown to be consistent with ZF.

Their nonexistence is even consistent with

ZF+Axiom of Choice (ZFC) (Shepherdson 1952, Ulam 1 9 3 0 ) .

In ZF+GCH, as

previously mentioned, their nonexistence can be demonstrated.

Possibly

their nonexistence can even be proved in ZF or ZFC. For us a finitely additive measure on T is a finitely additive nonnegative real-valued function m defined on the algebra

A

of sets

of the completely regular Hausdorff space T

generated by the zero sets

for which the following "regularity" condition holds: For each A € AZ, (Thus knowledge of m on

z

m(A) = sup{m(Z)IZc

z,

ZCA]

.

is sufficient to determine its behavior on

A

.)

The difference of two finitely additive measures on T is called a finitely additive signed measure on T.

In the event that the set function is

countably additive, 1.e. m ( u E ) = C m ( E ) whenever (E ) is a pairwise n n n rxtj with union in A , it is called either a measure disjoint collection from A A 0 - 1 measure has {0,1] as its

or signed measure, as the case may be. range.

The collection of all finitely additive 0 - 1 measures is denoted by

+ , and

the finitely additive measures by M measures by M.

+ (A)

where m A m

*

A=. -+

=

For each m

C

M,

sup{m(B) IBCA, BC

the finitely additive signed

+ there are m , m

AZ}

+

c M such Fhat m

=

+

m -m

-

and m-(A) = -inf{m(B) IBCA,BC Azl for

+ (A)

The total variatidn of m is Iml (A) = m

im-(A).

- The function

llrnll = Iml (T) defines a norm on the linear space M with pointwise

x and Y are congruent if there is some r c [O,ll such that for all y there is some x C X such that y = x + r (mod l), and for each x is some y € Y such that y = x + r (mod 1).

C

6 Y X there

1.7

35

MEASURES, BT AND ULAM CARDINALS

operations with respect to which M is a Banach space.

It is not difficult

to see that n Iml (A) = sup{ C Im(Ei) lEi €AZ, EiC A, (Ei) pairwise disjoint 1 i=l

I

(see Dunford and Schwartz 1958, p. 137).

Moreover a bounded finitely

additive real-valued set function m is a finitely additive signed measure if and only if for each A F

A

there are zero sets Z1, Z2 such that

E s o that the elements of M are just the Z C A C CZ2 and Iml (CZ -Z 1 2 1 "regular" finitely additive bounded real-valued functions on T, "regular"


0, there is a closed set C C A

such that Iml (G-C)
0.

initial topology determined by the maps { Ixd. Ix( Cb(T,R) from M converges to m

€

3

on M.

net (m

A

M in the vague topology if and only if Jxdm

u

u

+

lxdm

In our first result we present an alternate T , ! ) . b( characterization of "vague" convergence to be used in the sequel. for each x

€ C

(1.7-1) VAGUE CONVERGENCE.

A net (m ) from

u

M+ converges to m

F

+

M

in the

vague topology if and only if (1) m (T) + m(T) and F! (2) lim sup m (2) 5 m(z) for each zero set

u u

2.

Condition ( 2 ) may be replaced by (2')

lim inf m (CZ) > m(CZ) for each zero set Z.

u u

+

Thus a net converges in the vague topology of M whenever it converges in

*

The proof of (2.4-5) Theorem 1.7-1.

of course does not depend on anything following

36

1.

ALGEBRAS OF CONTINUOUS FUNCTIONS

+

the product topology of M

(a collection of functions mapping A

into

+

5)

so

that the product topology of M is at least as strong as the vague topology. Proof

First suppose that (m

the vague topology and let

!J Z C

)

+

?JCL

is a net from M

2 , 2 denoting the zero sets of the

completely regular Hausdorff space T. there exists Z' F

z

+ in .

converging to m C M

By the regularity of m, given

such that Z C CZ' and m(CZ') < m(Z)

Theorem 1.2-1 (c), we may choose x x(Z) = {I] and x(Z') = { O ] .

5

> 0,

By

+E.

C (T,:) such that 0 b For this x and each !J 6 L C

E

x

5 1,

On the other hand

Since m

!J

As E

-+

m, lxdm -+ lxdm so that !J

is arbitrary, lim sup m (Z) 5 m(Z).

It is clear that m (T) -+ m(T).

!J?J

?J

Conversely, suppose that the conditions hold. x 6 Cb(T,R), lim sup

lxdm

?J

< Jxdm.

?J-

We claim that for each

Furthermore it suffices to prove this

inequality for those x's for which 0 < x < 1, for by appropriately choosing scalars a > 0 and b we can always force a x + b to satisfy the condition 0 < a x + b < 1. Assuming the result to hold for 0 < y < 1 then yields

lim sup ?J

But

I

(ax+b)dm

< 11-

I

(ax+b)dm.

(ax+b)dm = a lxdm + bm(T) and (ax+b)dm = a*lim sup

u

u

+ b-lim m

(T) !J

.

Since b-lim m (T) = bm(T), it follows that !J

To prove the inequality for 0 < x < 1, note first that for each k T = U x-1([(i-l)/k,i/k)).

i=l

Q

N,

1.7

Letting

Z.

MEASURES, BT AND ULAM CARDINAIS

37

-1 he the zero set x ([i/k,m)), this becomes k T = U (Zi-l-Zi) i=l

x defined at t

Now the simple functions 5 and x(t)

=

(i-l)/k

satisfy the inequality 5

and

5x5

x.

. - Zi by

Q Zi-l

x(t) = i/k

(i=l,. ..,k)

Thus

and

k

c

l/k

m(Zi) =

i=l As lim sup m ( 2 . )

5

m(Z.) for each i, it follows that, taking superior k k limits in (1) and replacing (l/k) C lim sup m (Zi) by (l/k) C m(Zi), P U i=l i=l and using (2).

u u

1

lim sup Ixdm

u

for each k €

y.

Letting k

+

< l/k

u-

+

Ixdm

-, establishes the desired inequality.

Applying the inequality for -x yields

and we may conclude that lxdm + lxdm for each x

Cb(T,R).

u

v

Now the measures m concentrated at the points t in T, defined to be t 1 at the sets A € AZ to which t belongs and 0 otherwise, evidently

constitute an injective image of T in M via the map

(P:T-+M0 C t+m Indeed if t #

s

and x

C C(T,R)

Mi

t

maps t into 0 and

s

into 1 then m (z(x)) = 1 t r~ (T) are

while m ( z (x)) = 0. It is equally evident that the elements of countably additive.

38

ALGEBRAS OF CONTINUOUS FUNCTIONS

1.

Theorem 1.7-1.

BT =

Mo

Let M denote the collection of all finitely

additive 0 - 1 measures on the completely regular Hausdorff space T.

Then:

(a) Mo with vague topology is the Stone-Cech compactification of TI and (b) the repletion UT of T is the collection of all countably additive elements of Mo, i.e. the measures in M Proof

.

(a) First we shall show that the 1-1 mapping tn given above is a

homeomorphism. Using the fact that there is a base of zero sets at each point t € T, one may show that for each x c Cb(TIF$ and t /xdmt

=

x(t).

€

T that

Since lxdm = x(t), t

=

M o n v(m

,x,E)

and cp is seen to be a homeomorphism. Next it is shown that(P(T) is dense in M

0

m F Mo,

by showing that, given

there is a net from $T) which converges to m (in the vague

topology) Let

zm

denote the collection of all zero sets Z for which m ( Z ) = 1.

m

becomes a directed set with respect to the ordering

f z m is nonempty, we may choose some element tZ from Z. We contend that the net converges to m in the product topology,

Since each Z

therefore also in the vague topology by (1.7-1). Suppose that A F that m ( 2

0

) =

1.

AZ.

If

m(A) =

Thus for each Z F

1 then there is a zero set

zm'

if Z

2 Z (i.e. Z C 2

)

2

C

A such

then

(A) = 1 so that m (A) + m(A). If m(A) = 0, then m(CA) = 1 and some tZ tZ zero set Z0C CA must exist whose measure is 1. Hence each Z F zm greater

m

than or equal to Z

, being a subset of Z0 ' fails to meet

and m ( A ) = 0. tZ m(A) in both cases and dT) is seen to be dense in M . 0

Thus m

(A) + tZ The next step in verifying that M

0

show that M

0

A

0

is a compactification of T is to

is compact and Hausdorff in its vague topology.

separation, let m1 and m 2 be distinct points of M

.

As for the

Then m l - m 2 is a

1.7

MEASURES, PT AND UIAM CARDINALS

39

nontrivial finitely additive regular set function. Clearly the total variation Iml - m2

I

xi I (m1-m2) I (A. ) 1

=

of m -m2 is either 1 or 2. 1 Choose pairwise disjoint sets A1, ...,An 6 A z with the properties that

I

Iml - m2 (T) and ml (A1) # m2 (A1)

.

Assuming that

m ( A ) = 1 and m (A ) = 0, choose zero sets Z and F such that Z and F are 1 1 2 1 each subsets of A1, m,(Z) = 1, and m (F) = 0. Then for the zero set 2 Z1 = A U F we have Z,F C Z1 Z A and it follows that m (Z ) = m ( A ) and 1 1 1 1 1 such that m 2 (Z1) = m2(A1). For each i, 2 5 i 5 n, choose zero sets Z1. C A. 1 (m -m ) (Z.)= (m -m ) (A.) in the same manner as Z. was chosen. As the Zi's 1 2 1 1 2 1 are pairwise disjoint, so are their closures in BT. Thus there is an xB

such that x B (cl Z.) =

€ C(T,F$

11 and x B (

$ 1

x denote x B

1 T'

i=2

cl (Zi)) = B

0

.

Letting

it follows that

I

xd(m -m ) = (m -m ) 1 2 1 2

Finally then [ m

F

Mo

I I lxd (m-m,) 1 < 1/2

disjoint neighborhoods of ml and m

2

= 1

(Z,)

add {m 6 Mo

-

I 1 lxd (m-m2) I < 1/23

are

in the vague topology.

Since a typical continuous linear functional x' on C (T,?) with the b uniform norm is of the form x' (x) = lxdm for some unique finitely additive signed measure m 6 M by (2.4-5), the vague topology of M is in fact the weak-topology a(C (T,R)',Cb(T,R)). Thus, by Alaoglu's theorem, a subset b of M is compact in the vague topology whenever it is closed in the vague topology and norm-bounded. As the norm, i.e. the total variation, of each element of M in M.

equals 1, it only remains to show that M

is (vague) closed

To this end let (m ) be a net from M convergent to m < M. lJ UCL 0 First we claim that m € Mf. If not, then there is some A 6 A such

that m(A) < 0. Let d = -m(A)/3 > 0 and choose Z, IZ, C I

L

such that

Z1 C.A C CZ2 and Im((CZ -Z ) < d. It follows that m(Z ) and m(CZ ) are 2 1 1 2 both less than -2d. Since Z, and Z, are disjoint, so are their closures in B < 1, x (cl z ) = { O ] , BT and we may choose xB f C(;T,R) s k h that 0 5 xB 8 2 Let x denote the restriction of xB to T and consider and xB (cl Z ) = {l.]. 8 1 r

Jxdm-

r

J

xdm =2

+I

'

xdm+f

Jcz2-z1

xdm=f Z1

xdm+

Jcz2-z1

1.

40

ALGEBRAS OF CONTINUOUS F U N C T I O N S

< -2d and

As m(2,)

it follows that ixdm < - d. On the other hand, for each p L, [xdm is P nonnegative since it is the integral of a nonnegative function with respect

+

to a finitely additive measure.

Thus lxdm lxdm which contravenes the P + choice of (m ) and m. We conclude that m C M . Furthermore as each m and lJ P m belong to M+ and m -+ m, the net m (T) + m(T) by (1-7-1) so that lJ P 0 2 m(A) 5 m ( T ) = 1 for each A E A Z . To see that m C M we make the contrary assumption that 0 < m(A) < 1 for some

A

A €

.

Then there must be

zero sets Z and Z such that Z C A C CZ2 and 0 < m(Zl) 5 m(CZ ) < 1. 1 2 1 2 Once again, by the fact that the 8-closures of disjoint sets are disjoint, we may choose x8 c C(BT,R) such that x8 (cl 2 ) = YO] and x B (cl Z ) = f1). 8 1 8 2 Now let x denote the restriction of x 6 to T, and choose zero sets Z* Z**

(by (1.2-3)) such that CZ* = It€ TI Ix(t)

1/43

and

By (1.7-1) lim sup m (Z**) 5

It follows that Z1 C CZ* C Z** C CZ2. m(Z**) < 1, so that an index P

lJ

c L exists such that n (Z**) < 1 for each P

.

1. p0. As each mP c M0, mP (Z**) = 0 for each l~ p 0 Using condition ( 2 ' ) of (1.7-1) and a similar argument it follows that an index p1 F L exists P

Choosing l~ 2 pO,pl we see that such that m (CZ*) = 1 for each p 2 u P 1' m (Z**) = 0 and m (CZ*) = 1 even though CZ*c *Z**,a contradiction. Hence P P m 6 Mo, Mo is closed, and therefore compact in the vague topology. Having shown that M

0

is a compactification of T the one thing

remaining to do is to prove that each x c C (T,R) has a continuous b

extension to M

0

(Theorem 1.1-1 (c)).

We effect the extension of x

c

Cb(T,g)

to M by taking G(m) = Clearly

;c

I

xdm

(mc Mo)

.

is continuous on M by the very definition of the vague topology,

but does

2 extend x? Since there is a base of zero set neighborhoods at

each t

T, we may conclude that G(mt) = lxdmt = x(t) at each t F T, and it

C

1.7

is seen that

41

MEASURES, BT AND ULAM CARDINALS

is an extension of x.

It now follows that

with

vague topology is the Stone-Cech compactification of T. (b) Recall (Theorem 1.6-1) that a point m in M

BT) belongs to the

(=

0

repletion UT of T iff (1) its associated maximal ideal Mm

=

{ x E C(T,R) Im fcl zfx)) has the property that z(M

B

m

)

is stable under

countable intersections. [Or (2) the codimension of M is 1, i.e. m C(T,€J)/M is isomorphic to g.1 m Prior to showing that each m C U T is countably additive via statement (1) above, we establish the following technical fact:

zm

is clearly a z-filter and z(M ) is a z-ultrafilter, it is only m necessary to show that z(M ) C to establish the equality of the two sets. m m A point in z(M ) is of the form z(x), x E C(T,R), where m E cl ( z ( x ) ) . m B Thus there must be a net (t ) from z f x ) such that m + m. Therefore, by t !J As

z

lJ

(1.7-1),

lim sup m

t

(z(x))

5 m(z(x)).

By the way the m

u

m (z(x)) t

=

t

are defined, each

!J

1 however.

Hence m(z(x))

=

1, and the desired inclusion is

u

established. Suppose now that m

€

uT,

that

so

zm

is stable under countable

intersections, and that m is not countably additive. there must be a sequence (A ) of sets from A each m(A m(F

) =

) =

n 1.

1.

This being the case,

decreasing to $ such that

And we may choose zero sets F C A such that each n n

n

Now each of the zero sets Z

n

=

Fm c

zm

nZn = n F n C

and

n A n = $.

rn=l

This brings us to the contradictory conclusion that m ( n 2 ) = 0, i.e. n

nzn B

zm.

Conversely, suppose that m c

z

M0

is countably additive.

To show that

m F UT we show that z (M ) = [Z f Im(2) = 11 is stable under countable m intersections. To this end, let (Zn)be a countable family of sets from z(Mm).

Thus, for each n c

N,

m(Z

) =

1 and, since a countable intersection

of zero sets is a zero set by (1.2-1) (e), m(c

n

zn))

= m(

n€CJ

u cz n ) 5 c

nc 3

nE N

m(cZ

n

)

= 0 .

v

TO view Part (b) of the above theorem in a somewhat different light, note that the correspondence

42

ALGEBRAS O F CONTINUOUS F U N C T I O N S

1.

!do

=

BT

-+

Zu

m + z(M ) m (see Theorem 1.3-1) between 6T and the z-ultrafilters on T pairs the elements of UT with the 6-z2ultrafiltersr those z-ultrafilters stable under the formation of countable intersections: In the theorem it was established that z (M,)

= { z (x)lm(z (XI ) = 11

.

Thus, by Part (b) of the

theorem, there is a 1-1 correspondence between the measures (i.e. countably additive members) of M and the 6-z-ultrafilters. 0

In proving that UT consists of the measures in M we made no mention of how an x

€

C(T,R) is extended up to a continuous function on uT. We did

see, however, how to continuously extend bounded functions to BT: For x c Cb(T,R), take

Thus it is not unreasonab-5 to suspect that this same met -2d might be used Indeed if to extend the functions of C(T,R) up to elements of C ( U T , R ) . -1 x € C(T,R), then x ([-n,n]) is an increasing sequence of zero sets -1 converging to T. Hence for any 0-1 measure m, m(x ([-n,n])) = 1 for some n.

as I x

I

Thus

-1 is bounded by n on x ( [-n,nl)

.

To see that the real-valued

function

-

x:UT+R

m + /xdm is continuous on UT, let (m vague topology.

)

be a net from UT converging to m C UT in the

-1

As x ((-n,n)) is the complement of a zero set Zn (see -1 (1.2-3)) and (x ((-n,n))) increases to T, there is an index N such that

) = 0. Thus, by (1.7-1), lim sup m ( Z ) 5 m ( Z ) = 0 and m (ZN) + 0. N V U N N !J Consequently there must be an index p such that m ( Z ) = 0 for all p 1. po.

m(2

Setting x = min(x,N) , it follows that N

U

N

1.7

f o r each p

2

p 0

.

43

MEASURES, BT AND U I A M CARDINAIS

Hence lxdm

v

+

lxdm and

A t t h i s j u n c t u r e w e examine P a r t

c l a s s of d i s c r e t e s p a c e s .

2

is seen t o be continuous.

( b ) of Theorem 1.7-1 above f o r t h e

Our p u r p o s e i s t o p r o v i d e a f o u n d a t i o n f o r

S h i r o t a ' s r e s u l t presented i n t h e next section.

There it i s e s s e n t i a l l y

shown t h a t a c o m p l e t e l y r e g u l a r Hausdorff s p a c e endowed w i t h a complete c o m p a t i b l e uniform s t r u c t u r e is r e p l e t e i f and o n l y i f e a c h c l o s e d d i s c r e t e subspace i s r e p l e t e . If T i s d i s c r e t e t h e n i t i s e v i d e n t t h a t

c o l l e c t i o n P(T) o f a l l s u b s e t s o f T .

and A

coincide with t h e

Thus a d i s c r e t e s p a c e T is replete i f

and o n l y i f e a c h ( c o u n t a b l y a d d i t i v e ) 0-1 measure on P(T) i s c o n c e n t r a t e d a t a p o i n t o f T. D e f i n i t i o n 1.7-1.

ULAM CARDINALS.

A

( c o u n t a b l y a d d i t i v e ) 0-1 measure

d e f i n e d o n t h e c o l l e c t i o n o f a l l s u b s e t s o f a s e t T which i s n o t c o n c e n t r a t e d a t a p o i n t o f T i s c a l l e d a n U l a m measure.

Since t h e

e x i s t e n c e o f s u c h a measure i s c l e a r l y a p r o p e r t y of t h e e q u i p o t e n c e c l a s s o f T r a t h e r t h a n j u s t T , t h e c a r d i n a l numbers IT1 f o r which U l a m measures e x i s t are c a l l e d U l a m ( m e a s u r a b l e ) c a r d i n a l s . I n t h i s t e r m i n o l o g y P a r t ( b ) o f Theorem 1 . 7 - 1 y i e l d s t h e r e s u l t t h a t : A d i s c r e t e s p a c e i s r e p l e t e i f and o n l y i f it i s n o t of U l a m c a r d i n a l .

A complete u n i f o r m s p a c e (T,

Uf provided r a g e s which are c l o s e d i n g w k

UW )

Uf

a f i n e r uniformity

g w

remains complete when equipped w i t h

p o s s e s s e s a fundamental system of e n t o u (Bourbaki 1966a, p.

185, P r o p . 7 ) .

Be-

c a u s e of t h i s and t h e f a c t s t h a t p r o d u c t s and c l o s e d s u b s p a c e s of r e p l e t e s p a c e s a r e r e p l e t e , we h a v e :

(1.7-2)

NON-ULAM CARDINALITY I S HEREDITARY AND PRODUCTIVE.

n o t of U l a m c a r d i n a l i t y t h e n n e i t h e r i s any subspace of T .

(a)

If T is

(b) I f (Tk)

is

a f a m i l y o f sets none of which i s of U l a m c a r d i n a l , t h e n nT i s n o t of U l a m P cardinality either. Furthermore

on

(I)t h e n

m({n])=l. w i t h (1.7-2)

IBI

i s n o t an Ulam c a r d i n a l .

l=m(E)=CncN m ( { n ) )

Hence m=mn,

iye.

Indeed i f m i s a 0-1 measure

s o t h a t a n i n t e g e r n e x i s t s f o r which

m i s c o n c e n t r a t e d a t n.

This f a c t t o g e t h e r

y i e l d s t h e c o n c l u s i o n t h a t e v e r y c a r d i n a l less t h a n o r equal

t o a c a r d i n a l of t h e form **

k1

44

ALGEBRAS OF CONTINUOUS F U N C T I O N S

1.

is not an Ulam cardinal, a fact which indicates that if Ulam cardinals exist at all, they must be very large. 1.8 Complete Implies Replete

The fact that broad classes of spaces are replete was mentioned in Section 1.5.

In Theorem 1;5-3, for example, it was shown that each

Lindelgf space is replete.

In the principal result of this section another

broad class of such spaces is established:

each space whose cardinality is

not Ulam and whose topology is given by a complete uniform structure. For the sake of that demonstration it is helpful to single out the following notion of "discreteness." Definition 1.8-1. d-DISCRGTE. set T.

A family (T

lJ

d(T ,T )

=

u x

)

inf{d(s,t)

Let d be a pseudometric defined on the

of subsets of T is d-discrete of gauge p > 0 if

I (s,t)E

T XT

1b

A. A set S C T is d-discrete if

A,u.

lJ

p for all distinct pairs of indices

([s])

SCS

is d-discrete.

The basic properties of d-discreteness follow. d-DISCRETENESS.

(1.8-1)

Let d be a continuous pseudometric on the

completely regular Hausdorff space T.

Then

(a) Any d-closed set A C T is a zero set (viz. the map t

-f

d(A,t));

(b) the union of any d-discrete family (T ) of d-closed (i.e. closed Fr with respect to the topology induced by d) sets is d-closed; (c) any d-discrete set

S

is closed;

(d) any d-discrete set

S

is discrete.

Proof

(a) Clear. (b) Suppose that t is a d-adherence point of U T

lJ

set [ s € Tld(s,t) < E) meets j u s t one of the sets T

U

(c) By (b) the set

5=

;

U cld{s] is d-closed.

.

For each

E

p the

it follows that

If t is an adherence

S€S

it is also a d-adherence point of S so that t e cl { s ] for some d. s and if V is a neighborhood of t in the t Hausdorff space T excluding s , then the neighborhood (in T), point of s E

S.

S

If we assume that t #

V fl {re T/d(r,t) 0 such that d(s,t) < 2p =C (s,t) €U. well-ordering 5 on T.

By the well-ordering principle there is a

For each n E

we now define by transfinite

induction a d-discrete family (Z ) 0-f u-small zero sets as follows: ns sCT

# @ for each n € N; any other Z If u is the first element of T then Z ns nu may be empty. The next stage of the proof consists of showing that some nonempty Z ns ). To do this we introduce the sets

is a (U-small) member of z(M

P

and show that some Z € z(M ) . We first contend thatU Z = T. n P n n For t € T let s be the first element of the set { c Tld(r,t) < p] choose n c rJ such that d(s,t) 5 p-p/n.

If r
n which c o n t r a and i t f o l l o w s t h a t 0 i s n o t a l i m i t p o i n t of

Q nnLa.v Even i f T i s n o t a k - s p a c e , topology

3k

t h e compact s u b s e t s o f T "generate"

which r e n d e r s T a k - s p a c e .

a

Indeed t h e c o l l e c t i o n o f a l l s u b -

sets U C T t h a t meet each compact s u b s e t K o f T i n a n open s u b s e t of K i s a topology c a l l e d t h e k - e x t e n s i o n t o p o l o g y o f T.

-

-

I t i s t h e s t r o n g e s t topology

on T f o r which a l l t h e i n j e c t i o n maps i :K T, t t , K compact, a r e conK tinuous. Clearly the k-extension topology is a t l e a s t a s f i n e a s the

-

T h i s , combined w i t h t h e f a c t t h a t e a c h i :K ( T g ) is K ' k c o n t i n u o u s , i m p l i e s t h a t t h e o r i g i n a l topology and t h e k - e x t e n s i o n topology

o r i g i n a l topology.

have t h e same compact s e t s .

Thus T i s a k-space i n t h e k - e x t e n s i o n t o p o l -

I n a d d i t i o n w e make n o t e t h a t b o t h t h e o r i g i n a l and k - e x t e n s i o n

ogy.

t o p o l o g i e s i n d u c e t h e same s u b s p a c e topology on e a c h compact s u b s e t o f T. With t h e n o t i o n of t h e k - e x t e n s i o n topology a t our d i s p o s a l we a r e i n a p o s i t i o n t o d e s c r i b e t h e c i r c u m s t a n c e s under which a kR-space i s a k space.

(2.3-1)

WHEN I S A kR-SPACE A k-SPACE?

w i t h t h e k - e x t e n s i o n topology.

L e t T d e n o t e t h e s p a c e T equipped k A k - s p a c e T i s a k - s p a c e i f f Tk is comR

pletely regular.

Proof.

S i n c e t h e k - e x t e n s i o n topology c o i n c i d e s w i t h t h e o r i g i n a l topology

when T i s a k - s p a c e ,

T =T i s c o m p l e t e l y r e g u l a r . Conversely suppose t h a t k i s c o m p l e t e l y r e g u l a r . A s t h e k - e x t e n s i o n topology i s a t l e a s t a s f i n e

T k a s t h e o r i g i n a l t o p o l o g y , C(T,EJcC(Tk,E).

On t h e o t h e r hand, i f xfC(Tk,E),

S i n c e T and T have t h e then x i s c o n t i n u o u s on each compact s u b s e t of Tk. k same compact s e t s , i n d u c e i d e n t i c a l subspace t o p o l o g i e s on t h e compact sub-

s e t s , and T i s a k - s p a c e , t h e n x f C(T,F) and C ( T , g ) = C ( T k , g . Now, s i n c e R T and T a r e c o m p l e t e l y r e g u l a r , t h e n T=T and i t f o l l o w s t h a t T i s a k k k space. v Though i t i s n o t g e n e r a l l y t r u e t h a t a c o m p l e t e l y r e g u l a r Hausdorff s p a c e T i s a k-space when C(T,F,c) s p a c e s f o r which i t i s t r u e

-

i s c o m p l e t e , t h e r e a r e c a t e g o r i e s of

two such a r e t h e hemicompact s p a c e s and t h e

p s e u o f i n i t e s p a c e s ( d e f i n e d below).

2.3

D e f i n i t i o n 2.3-1.

k-SPACES

PSEUDOFINITE SPACES.

71

A completely r e g u l a r Hausdorff

s p a c e T i s p s e u d o f i n i t e i f e a c h compact s u b s e t K C T i s f i n i t e . Certainly every d i s c r e t e space i s pseudofinite.

Every P - s p a c e ( i . e .

t h o s e c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e s T f o r which e a c h p r i m e i d e a l i n C(T,g) i s maximal) i s p s e u d o f i n i t e a n d t h e r e i s a p l e n t i f u l s u p p l y of nond i s c r e t e P - s p a c e s ( G i l l m a n and H e n r i k s e n , 1 9 5 4 ) .

Most o f t h e r e m a i n d e r o f

t h i s s e c t i o n i s d e v o t e d t o c h a r a c t e r i z i n g c o m p l e t e n e s s o f C(T,,F.,c) p s e u d o f i n i t e s p a c e s and hemicompact s p a c e s .

for

F i r s t w e c h a r a c t e r i z e pseudo-

f i n i t e spaces. (2.3-2)

CHARACTERIZATIONS OF PSEUDOFINITE

The f o l l o w i n g a r e e q u i v a l e n t

f o r a completely r e g u l a r Hausdorff space T.

T is p s e u d o f i n i t e .

(a) (b)

The compact-open and p o i n t - o p e n t o p o l o g i e s c o i n c i d e on C(T,X).

(c)

Tk i s d i s c r e t e .

Proof.

S i n c e t h e compact s u b s e t s of T a r e f i n i t e , t h e i m p l i c a t i o n ( a )

is c l e a r .

To o b t a i n t h e c o n v e r s e , i . e . t h e i m p l i c a t i b n ( b )

b e a compact s u b s e t of T.

-

-

etc..

4

-

(b)

(a), let K

By ( b ) t h e r e i s a f i n i t e s e t F C T s u c h t h a t

Now s u p p o s e t h a t K$F,

i.e.

t h a t t h e r e i s a p o i n t t c K-F.

Since T

i s c o m p l e t e l y r e g u l a r t h e r e i s a n xFC(T,E) s u c h t h a t x ( t ) = 2 w h i l e x(F)=(O]

.

!

Thus K C F a n d K i s f i n i t e . and x d 7 pF. A s Tk ( a s i n ( 2 . 3 - 1 ) f k a n d T have t h e same compact s e t s , t h e i m p l i c a t i o n

so that x €

(c)

(a) i s c l e a r .

--t

On t h e o t h e r hand i f T i s p s e u d o f i n i t e , e a c h compact

(hence f i n i t e ) s u b s e t of T i s d i s c r e t e .

Therefore any s u b s e t U of T meets

e a c h compact s e t K i n a r e l a t i v e l y open s e t a n d Tk must b e

It i s now c l e a r by ( a ) --. ( c ) of ( 2 . 3 - 2 )

and ( 2 . 3 - 1 )

discrete.^

t h a t any pseudo-

f i n i t e k -space i s a k-space. However, a s w e s h a l l p r e s e n t l y s e e , t h e o n l y R p s e u d o f i n i t e k -spaces a r e t h e d i s c r e t e ones. R (2.3-3) PSEUDOFINITE kR-SPACES ARE DISCRETE I f T i s p s e u d o f i n i t e , t h e n the following statements a r e equivalent. (a)

C(T,F,c)

(b)

Tk i s d i s c r e t e .

(c)

T i s a k-space.

Proof.

i s c o m p l e t e ( i . e . T i s a k - s p a c e , by Theorem 2 . 2 - 1 ) . R

S i n c e T is p s e u d o f i n i t e , by ( 2 . 3 - 2 ) ,

e q u i v a l e n c e o f ( b ) and ( c ) i s c l e a r .

Tk i s d i s c r e t e .

Thus t h e

To see t h a t the i m p l i c a t i o n ( a )

-, (b)

72

2.

SPACES OF CONTINUOUS FUNCTIONS

h o l d s , i t i s enough t o n o t e t h a t F i s dense i n t h e p r o d u c t s p a c e T-

-

by ( 2 . 3 - 2 ) ( b ) and t h e f a c t t h a t C(T,,F) F C(T,F)=T-. F i n a l l y ( c ) -t ( a ) f o l l o w s

'

by Theorem 2 . 2 - 1 and t h e f a c t t h a t k - s p a c e s a r e k

R-spaces' A s w e mentioned e a r l i e r t h e c l a s s of hemicompact s p a c e s i s a l s o a c l a s s

of s p a c e s i n which k - s p a c e s a r e k - s p a c e s . R ( 2 . 3 - 4 ) HEMICOMPACT k -SPACES L e t t h e c o m p l e t e l y r e g u l a r Hausdorff s p a c e -R T be hemicompact. Then C(T,F,c) i s complete ( o r , e q u i v a l e n t l y , T i s a k -space) i f f T i s a k-space. R Proof. The i m p l i c a t i o n k -. k

i s o b v i o u s . C o n v e r s e l y , suppose t h a t T i s R a k -space. S i n c e T i s a k - s p a c e whenever T i s c o m p l e t e l y r e g u l a r by R k ( 2 . 3 - l ) , i t s u f f i c e s t o prove t h a t i f T i s hemicompact, t h e n Tk i s normal.

Furthermore Tk must be hemicompact i f T i s , because Tk h a s t h e same compact s e t s a s T. regular.

Thus T i s LindelDf s o i t i s o n l y n e c e s s a r y t o show t h a t Tk i s k To t h i s end l e t t f Tk and U be a n open neighborhood o f t. A s T

i s hemicompact, T may be w r i t t e n a s t h e union of an i n c r e a s i n g sequence of compact set s Kn h a v i n g t h e p r o p e r t y t h a t e a c h compact s u b s e t of T i s cont a i n e d i n some K

.

We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t t F K

I'

Our f i r s t c o n t e n t i o n i s t h a t t h e r e e x i s t s an i n c r e a s i n g sequence (W ) of c l o s e d neighborhoods of t i n Kn such t h a t ( i ) W n C U 0 K n

WnnKm=Wm f o r a l l m int

5

f o r each n , ( i i )

n , and ( i i i ) ( i n t n W n ) n Km = i n tmWm f o r a l l m

denotes t h e i n t e r i o r taken i n K

.

5 n where

S i n c e K1 i s a compact Hausdorff

s p a c e and t h e r e f o r e r e g u l a r t h e r e i s a c l o s e d neighborhood W of t i n K contained i n U n K

1'

1

L e t us assume t h a t c l o s e d neighborhoods W l , . . . y W n - l

of t e x i s t s a t i s f y i n g t h e t h r e e p r o p e r t i e s s t a t e d above.

A s Wn-l

is a

c l o s e d s u b s e t of U n K n i n t h e normal s p a c e Kn t h e r e i s a n open s u b s e t V

K

such t h a t W n - l c

V n c c l V n C UnK,.

S i n c e W n-1 i s a neighborhood i n K n-1'

a n open s e t U n C K n e x i s t s which meets Kn-l c l UnnKn-l=cl(intn-lWn-l)CWn-l

i n t h e s e t intn-lWn-l.

Hence

and i t f o l l o w s t h a t t h e c l o s e d n e i g h b o r -

hood o f t i n Kn, Wn=(cl U n n c l Vn)UWn-l meets Kn - 1 i n t h e s e t Wnml a s u b s e t of U n K , .

of

Furthermore t h e r e l a t i o n s i n t n W n 3 V n n U n , K n - l f l

and i s

un

=

nKn-13

i n t n - 1W n-1' and V n nKn 3 intn-lWn-l imply t h a t i n t nWn i n t n-1Wn-1' The r e v e r s e i n c l u s i o n i s obvious and t h e c o n t e n t i o n h a s been e s t a b l i s h e d .

i s a c l o s e d neighborhood o f t i n T Next we c l a i m t h a t W= k UfWn ncIt i s c l e a r t h a t WCU. t a i n e d i n U, t h e r e b y p r o v i n g t h a t Tk i s r e g u l a r . That W i s c l o s e d i n t h e hemicompact k - s p a c e Tk f o l l o w s immediately from

ga

t h e r e l a t i o n W n K 'kl, f o r e a c h n. intnWn. Since Certainly W 3 n UnintnW,nKm=intmWm f o r e a c h m y u n i n t n W n i s an open s u b s e t of W i n Tk

2.4

containing t.

73

CONTINUOUS DUAL OF C ( T , F , c )

Thus W i s a n e i g h b o r h o o d o f t a n d t h e p r o o f i s c o m p 1 e t e . V

Thus i t i s n a t u r a l t o i n q u i r e when C ( T , F , c )

i s f u l l y complete.

In

t h e e v e n t t h a t t h e c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e T i s hemicompact o r pseudofinite,

C(T,,F,c)

i s f u l l y complete i f f T i s a k-space.

i s a hemicompact k - s p a c e ,

Indeed i f T

C ( T , F , c ) i s a F r e c h e t s p a c e by Theorem 2 . 1 - 1 and

A s any F r e c h e t s p a c e i s f u l l y c o m p l e t e ( H u s a i n , 1 9 6 5 , 4 . 1 , P r o p .

(2.3-4).

3) t h e d e s i r e d r e s u l t fo ll o w s .

The c o n v e r s e i s t r i v i a l .

A s for the case T and

of p s e u d o f i n i t e s p a c e s , we saw i n t h e p r o o f o f ( 2 . 3 - 3 ) - t h a t C(T,,F)=C

t h e compact-open and p r o d u c t t o p o l o g i e s a g r e e whenever T i s a p s e u d o f i n i t e k-space.

T h i s , combined w i t h t h e f a c t t h a t a n a r b i t r a r y p r o d u c t o f r e a l

l i n e s i s f u l l y complete i n t h e product topology (Husain, 19b5, 5.5, Prop. 1 4 ) , l e a d s t o t h e c o n c l u s i o n t h a t C(T,&,c) i s f u l l y c o m p l e t e .

U n f o r t u n a t e l y i t i s n o t g e n e r a l l y t h e c a s e t h e C(T,X,c) p l e t e when T i s a k - s p a c e , f o r C(T,F,c)

i s f u l l y com-

and n e c e s s a r y and s u f f i c i e n t c o n d i t i o n ..s on T

t o b e f u l l y c o m p l e t e a r e n o t known.

t h a t T i s a k - s p a c e whenever C ( T , F , c )

P t a k (1953) h a s shown

i s f u l l y c o m p l e t e and h a s g i v e n a

c o u n t e r example f o r t h e co n v ers e ( s e e E x e r c i s e 2 . 3 ( b ) ) . Some o f t h e i n t e r r e l a t i o n s h i p s between k - s p a c e s and k R - s p a c e s , e t c . , a r e summarized i n T a b l e 1 below.

T denotes a completely r e g u l a r Hausdorff

s p a c e a n d K , w i t h o r w i t h o u t s u b s c r i p t , a compact s u b s e t o f T. 2.4

The C o n t i n u o u s Dual o f C ( T , F , c )

and t h e S u p p o r t

A m a t t e r o f c o n s i d e r a b l e i m p o r t a n c e i n t h e t h e o r y o f l o c a l l y convex spaces X i s c h a r a c t e r i z i n g the continuous dual X ' , l i n e a r f u n c t i o n a l s on X .

t h e s p a c e of cont i nuous

The c o n s i d e r a t i o n s o f t h i s s e c t i o n h a v e p r i m a r i l y

t o do w i t h t h e c o n t i n u o u s d u a l C ( T , F , c ) ' o f C(T,F,c)where T i s a completely r e g u l a r Hausdorff space.

C(T,F,c)'

i s c h a r a c t e r i z e d h e r e i n t w o w a y s , one

i n v o l v i n g t h e e v a l u a t i o n maps T" = {tJ'lt(T1

(t'(x)=x(t)

f o r x f C ( T , F ) ) and

t h e o t h e r , more i m p o r t a n t f o r o u r p u r p o s e s , r e p r e s e n t s t h e e l e m e n t s o f C(T,F,c)'

a s i n t e g r a l s w i t h r e s p e c t t o c e r t a i n s e t f u n c t i o n s d e f i n e d on t h e

B o r e l s u b s e t s o f T.

T h i s r e s u l t h a s p r e d e c e s s o r s a s f a r b a c k a s 1909 when

R i e s z f i r s t c h a r a c t e r i z e d C([O, 1 ] , R )

Y

grals.

'

i n terms of R i e m a n n - S t i e l t j e s i n t e -

S u b s e q u e n t g e n e r a l i z a t i o n s were made which c u l m i n a t e d i n t h e w e l l -

known c h a r a c t e r i z a t i o n o f C ( T , F , c ) '

f o r compact H a u s d o r f f s p a c e s T i n term

o f i n t e g r a l s w i t h r e s p e c t t o r e g u l a r s e t f u n c t i o n s d e f i n e d on t h e B o r e l s u b s e t s of T ( s e e Theorem 2 . 4 - 1 ) .

A s t h e e l e m e n t s o f C(T,F)

are a l l

bounded when T i s c o m p a c t , i t was n a t u r a l f o r o t h e r s t o c o n s i d e r the s p a c e

C (T,R) b

( w i t h s u p norm) a n d i n t e g r a l r e p r e s e n t a t i o n s o f t h e e l e m e n t s of

74

2.

SPACES OF CONTINUOUS FUNCTIONS

definition

property hemi compact

(Def. 2.1-1) (K,)

A countable family

e x i s t s such t h a t each K

Kn

i m p l i e d by

equivalent t o

l o c . compact

C(T,E,c)

+

metrizable

0-compact

f o r some n .

k -space

( S e c . 2.1)

R

For x:T -,R

i f xIK

i s continuous.

seq. conti-

C(T,F,c)

n u i t y -, con-

complete

tinuity or

k-s pa c e k - s pa c e

(Sec. 2.3) open i n K

For each K , G n K

-

G open i n T.

k +pseudo-

R f i n i t e or k

+

R

hemicompac t

o-compa c t -\ loc

+

C(T,F,c) me t r i za b l e

\

compact hemicompact

kR

* C(T,E,c)

+hemicompact

o r pseudofinite

k

Table 1

2.4

Cb(T,_R)

75

CONTINUOUS DUAL OF C(T,F,C)

were o b t a i n e d f o r T s a t i s f y i n g v a r i o u s n o r m a l i t y c o n d i t i o n s ( s e e ,

f o r e x a m p l e , Dunford a n d S c h w a r t z , 1958 and A l e x a n d r o v ( 1 9 4 0 , 1941, 1 9 4 3 ) ) . By a b a n d o n i n g t h e s p a c e o f r e g u l a r s e t f u n c t i o n s on t h e B o r e l s u b s e t s o f a t o p o l o g i c a l s p a c e T f o r the l a r g e r c l a s s of r e g u l a r s e t f u n c t i o n s on t h e B a i r e s u b s e t s o f T i t was p o s s i b l e t o e s t a b l i s h i n t e g r a l r e p r e s e n t a t i o n s of t h e e l e m e n t s o f C ( T , F , c ) ' .

W e follow t h i s approach i n o b t a i n i n g a rep-

r e s e n t a t i o n of C b ( T , x ) '

(where C ( T , F ) c a r r i e s t h e s u p norm) f o r c o m p l e t e l y b r e g u l a r Hausdorff spaces T, b u t then take an a l t e r n a t e r o u t e f o r C ( T , F , c ) ' -

one which u t i l i z e s knowledge o f t h e d u a l o f C(T,E,c)

f o r compact T.

In so

d o i n g w e o b t a i n i n t e g r a l r e p r e s e n t a t i o n s i n terms o f s e t f u n c t i o n s on t h e Borel s u b s e t s .

These r e p r e s e n t a t i o n s a r e t h e n u s e d t o p r o v e t h e e x i s t e n c e

of the "support" of an element x ' c C ( T , F , c ) ' ,

a m i n i m a l compact s u b s e t S o f

T w i t h t h e p r o p e r t y t h a t i f y c C ( T , F ) v a n i s h e s on S , t h e n x' must v a n i s h on

y.

The n o t i o n o f s u p p o r t i s p u t t o u s e i n t h e n e x t s e c t i o n where n e c e s s a r y

a n d s u f f i c i e n t c o n d i t i o n s on T a r e o b t a i n e d f o r C ( T , F , c )

t o be b a r r e l e d a n d

bornological respectively. The f o l l o w i n g r e s u l t i s u s e f u l l a t e r on. (2.4-1)

T* AND THE DUAL OF C ( T S &

L e t K d e n o t e t h e f a m i l y o f compact

s u b s e t s o f t h e c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e T , K* = [t;':';ltCK],

and

t h e b a l a n c e d convex h u l l o f K . Then C ( T , F , c ) ' = [ U KC)< c lu(x',X> ( W b c ' J = Kbc H w h e r e X = C ( T , F , c ) , X* t h e a l g e b r a i c d u a l o f X o f all l i n e a r f u n c t i o n a l s o n

X , and t h e s q u a r e b r a c k e t s d e n o t e l i n e a r s p a n . C ( T , F , C ) ' = [ c l (X',

Proof

X) (T*)bc

S i n c e t*cH

Mackey-Arens

I f T i s compact, t h e n

3.

f o r e a c h t c T , t h e n (X,H) i s a d u a l p a i r .

By t h e

t h e o r e m w e need o n l y show t h a t t h e compact-open t o p o l o g y i s a

t o p o l o g y of u n i f o r m c o n v e r g e n c e on a c o l l e c t i o n o f b a l a n c e d convex a(H,X) compact s u b s e t s o f H t o p r o v e t h a t H i s t h e d u a l o f X .

To d o t h i s w e show

f i r s t t h a t each s e t E = c l compact.

(K$k)bc i s a b s o l u t e l y convex and o(H,X)u (X>t,x) And t o d o t h i s i t i s s u f f i c i e n t t o show t h a t E i s u(X*,X)-bounded

f o r i t i s a l r e a d y a(X*,X)-closed, s e t i s a b s o l u t e l y convex;

ing t h a t f o r each x and

bl,.

. ., u

CE

C

and t h e c l o s u r e of a n a b s o l u t e l y convex

o(X*,X)-boundedness,

X , s u p \ < x,E

be such t h a t C (pi

>I

im.

I 5 1.

i n t u r n , i s shown b y show-

To t h i s end l e t t 1$

Then f o r a n y x F X

* * *

3

tn 0 t h e r e a r e Z , Z ' c Z Z C A C C Z ' and

1u.l

(CZ'-Z) < C.

S i m i l a r l y i f p i s d e f i n e d on

is r e g u l a r i f such t h a t

a

o r 8 p is

2.4

77

CONTINUOUS DUAL OF C ( T , z , C )

r e g u l a r whenever B i s i n t h e domain o f d e f i n i t i o n o f c l o s e d and open s e t s C and U s u c h t h a t C C A C U and

u. and

ILL((U-C)

> 0 there are

C

< F.

Our f i r s t g o a l i s t o c h a r a c t e r i z e C ( T , Z ) ' a n d , a s might b e e x p e c t e d , b A s i t happens t h e e l e m e n t s o f Cb(T,,R)'

we b e g i n w i t h t h e r e a l - v a l u e d c a s e .

c a n be decomposed i n t o a d i f f e r e n c e of " p o s i t i v e " components.

More p r e -

are positive where x ' and x t h e n x ' = x I-x P n P n l i n e a r f u n c t i o n a l s , i . e . x ' ( x ) and x '(x) a r e 2 0 whenever x 5 0. P r i o r P t o e s t a b l i s h i n g such a decomposition w e v e r i f y t h a t p o s i t i v e l i n e a r func-

c i s e l y , i f xkCb(T,R)'

t i o n a l s a r e always continuous. 12.4-2)

POSITIVE

CONTINUOUS Any p o s i t i v e l i n e a r f u n c t i o n a l d e f i n e d on

Cb(T,x) where T is a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e i s c o n t i n u o u s . P r o o f To see t h a t a p o s i t i v e l i n e a r f u n c t i o n a l h d e f i n e d on C (T,F) i s b c o n t i n u o u s when C (T,F) carries t h e sup norm t o p o l o g y , l e t x c C b ( T , z ) . b Clearly then - PT(X)l I x 5 P , W 1 where l ( t ) = l f o r e a c h t C T .

and i t f o l l o w s t h a t lows t h a t (2.4-3)

Ih(x)

Thus, s i n c e h i s p o s i t i v e ,

PT(X)h(U

I5

5

htx)

(h(1) \pT(x).

5

PT(X)h(l)

I f x i s complex-valued, i t f o l -

J h ( x ) \ 5 2 h ( l ) p (x) so t h a t h i s continuous.V T

I f T is a c o m p l e t e l y

Cb(T,_R)' AND POSITIVE LINEAR FUNCTIONALS

r e g u l a r H a u s d o r f f s p a c e and C (T,R) c a r r i e s t h e t o p o l o g y i n d u c e d by t h e s u p b norm, t h e n c o r r e s p o n d i n g t o e a c h x'FCb(T,R)' t h e r e a r e p o s i t i v e l i n e a r

' and x ' such t h a t x ' = x ' - x ' . P P " P r o o f F i r s t we d e f i n e x ' on t h e n o n n e g a t i v e e l e m e n t s o f C,@,,): P x _> 0 t h e n functionals x

x

P

' (x)

= SupCx' (Y) I Y F Cb(T,R),

0

5

Y

5

if

x]:

C l e a r l y x ' ( a x ) = a x ' ( x ) f o r a 2 0 and x 2 0 . Next w e c l a i m t h a t x ' ( x + y ) = P P P x ' ( x ) + x p ' ( y ) whenever x , y 2 0. To see t h i s s u p p o s e t h a t 0 5 w 5 x+y. P C l e a r l y " 0 5 W A X 5 x and 0 5 w - ( w h x ) 5 y . Thus x'(w)

= x'(wAx)

+ x ' ( w - ( W A X ) ) 5 x P ' ( x ) + xP ' ( y ) ,

a n d , t a k i n g t h e supremum o v e r a l l s u c h w , w e o b t a i n

*

-

-

The meet xA y and join x V y o f r e a l - v a l u e d f u n c t i o n s x a n d y d e f i n e d on a s e t T a r e the functions t m i n ( x ( t ) , y ( t ) ) and t --t m a x ( x ( t ) , y ( t ) ) .

7a

2.

SPACES OF CONTINUOUS FUNCTIONS

x

5

(X+Y)

P

xp'(x>

+

x p I (Y).

+w

On t h e o t h e r h a n d , i f 0 5 v 1=_ x a n d 0 0;

P

I

i s l i n e a r on

P

(x v o+y

+ yV0) -

v 0) -x

P

I

( (-x)

((-x)VO

+

v O+(

-y )

v 0) .

(-y)VO) and ( x v o )

+

( y V O ) , (-x)V 0

+

thus x

P

(x)

+ xp

(y) = x p ' (x+y>

.

w e n o t e f i r s t t h a t x ' ( a x ) = ax ' ( x ) P P P f o l l o w s d i r e c t l y from t h e f a c t t h a t i t i s v a l i d

To e s t a b l i s h t h e h o m o g e n e i t y o f x for a

2

0 and x 0. On t h e o t h e r hand f o r X C C (T,&) we h a v e b

x l(-x) P

= x

P

'((-x)+

-

(-x)-)

- +1%P ' ( x - )+xp'(-x +)=x P ' ( x - ) - x p f ( x +)=-xp'(*).

= x P' (x -x Hence x

P ing t h a t x

is a positive linear functional. I

= x

I

P

-

X I

We c o m p l e t e t h e p r o o f by o b s e r v -

v

is a p o s i t i v e l i n e a r functional.

Now how a r e p o s i t i v e l i n e a r f u n c t i o n a l s r e p r e s e n t e d ? (2.4-4)

POSITIVE LINEAR FUNCTIONALS ARE "GENERATED" BY REGULAR ADDITIVE NONNEGATIVE SET FUNCTIONS I f T i s a n o r m a l ( c o m p l e t e l y r e g u l a r )

H a u s d o r f f s p a c e and h i s a p o s i t i v e l i n e a r f u n c t i o n a l o n C (T,_R), t h e n b t h e r e e x i s t s a f i n i t e nonnegative r e g u l a r a d d i t i v e set function u defined on t h e a l g e b r a

a

(a,) generated

o f T s u c h t h a t h = J dp

(s

by t h e c l o s e d s u b s e t s

6

(zero sets

2)

du i s t a k e n h e r e i n the same s e n s e a s i n S e c . 1.7).

2.4

Proof lar

-

Both s i t u a t i o n s

-

79

CONTINUOUS DUAL OF C ( T , i , c )

where T i s normal and where T i s c o m p l e t e l y r e g u -

a r e dealt with similarly.

I n f a c t i f t h e t e r m s " z e r o set" and

"complement of z e r o s e t " a r e s u b s t i t u t e d f o r " c l o s e d "

and "open"

respec-

t i v e l y i n t h e argument g i v e n below f o r t h e c a s e where T i s n o r m a l , we o b t a i n t h e proof f o r completely r e g u l a r T.

We b e g i n by d e f i n i n g a r e a l - v a l u e d s e t f u n c t i o n on t h e c l a s s o f a l l

We s u b s e q u e n t l y show t h a t

s u b s e t s o f T which we p r o v e t o be s u b a d d i t i v e .

t h i s s e t f u n c t i o n when r e s t r i c t e d t o t h e a p p r o p r i a t e a l g e b r a i s , i n f a c t , r e g u l a r and a d d i t i v e and i t i s f i n a l l y e s t a b l i s h e d t h a t h and t h e r e g u l a r a d d i t i v e s e t f u n c t i o n p a r e r e l a t e d by t h e formula g i v e n a b o v e . Suppose t h a t T i s a normal H a u s d o r f f s p a c e . la)

Definition of

o n p m .

I f U i s a n open s u b s e t of T we d e f i n e p(U)

t o be t h e supremum of t h e v a l u e s h ( x ) where x c Cb (T,R) and 0 5 x 5 kU' kU d e n o t i n g t h e c h a r a c t e r i s t i c f u n c t i o n o f U. I f A i s a n a r b i t r a r y s u b s e t of

-

T , P(A) i s d e f i n e d t o b e t h e infimum o v e r a l l open U 3 A of t h e v a l u e s p(U).

C l e a r l y p i s a n o n n e g a t i v e monotone s e t f u n c t i o n and ~ ( 8 ) = 0 . F u r t h e r more i f U i s open and 0 t h a t p(U)
p ( U U V ) - € where

0
0 and t h e proof i s complete. 0

I t i s c l e a r now t h a t e a c h e l e m e n t o f C b ( T , E ) ' i s r e p r e s e n t a b l e i n terms o f a n i n t e g r a l w i t h r e s p e c t t o a bounded r e g u l a r a d d i t i v e r e a l -

aZ

v a l u e d s e t f u n c t i o n on

i f T i s c o m p l e t e l y r e g u l a r and H a u s d o r f f o r a

Qc i f T i s a n o r m a l

bounded r e g u l a r a d d i t i v e r e a l - v a l u e d s e t f u n c t i o n on Hausdorff space.

The r e s u l t i s e q u a l l y c l e a r f o r t h e complex c a s e .

C e r t a i n l y i f p i s a g i v e n bounded r e a l - o r c o m p l e x - v a l u e d r e g u l a r a d d i t i v e s e t function then

1.r

xdw

SIX

I5

( i j

I 6 I I:< 1 1

111

u.

so t h a t a l i n e a r correspondence

.+

x'

1s I(T)

i s e s t a b l i s h e d b e t w e e n t h e bounded

F-valued r e g u l a r a d d i t i v e set f u n c t i o n s and C ( T , E ) ' . b It follows from t h e i n e q u a l i t y above t h a t [XI

u

11

I

5

To o b t a i n

IpI(T).

...,An€ac (or I 'I \ p \ ( T ) - € . A s & is r e g u l a r ,

t h e r e v e r s e i n e q u a l i t y , l e t f > 0 be g i v e n and l e t A l , be p a i r w i s e d i s j o i n t s e t s s u c h t h a t Ci(&(Ai)

c l o s e d s e t s ( z e r o s e t s ) C . and open s e t s (complements o f z e r o s e t s ) U . e x i s t s u c h t h a t C i c A i c Ui

IIL ((Ui-Ci)

and

€ I n , f o r i = l , . .. , n .

Now C l ,

.. . , C

a r e p a i r w i s e d i s j o i n t s o t h a t p a i r w i s e d i s j o i n t open s e t s (complements o f z s r o sets) V. e x i s t such t h a t C . C V . : 1

1

This i s c l e a r i f T i s normal.

T i s c o i n p l o t e l y r e g u l a r and p .is d e f i n e d on e n c e o f t h e V. f o r t h e c a s e n = 2 ;

a,we

When

demonstrate the e x i s t -

t h e g e n e r a l c a s e f o l l o w s w i t h t h e a i d of

the observation t h a t a f i n i t e union of z e r o sets i s a z s r o set [ ( 1 . 2 - 1 ) ( ~ ) ] . By Theorem 1 . 2 - 2 , function

c l Cl and c l C2 a r e d i s j o i n t s o t h a t t h e r e i s a c o n t i n u o u s

B B xB, 0 < X B < 1, d e f i n e d on

BT s u c h t h a t x P ( c 1 C,)={O]

8 2

by ( 1 . 2 - 3 ) t h a t V =x-'(-m,1/4) 1

1

i'

W

i

i s o p e n when T i s n o r m a l a n d i s t h e complement

o f a z s r o s e t when T i s c o m p l e t e l y r e g u l a r .

pr

while

a n d V = x - l ( 3 / 4 , m ) s e r v e the d e s i r e d p u r p o s e . 2

1

Now l e t W.=U.nV:

llp((Wi-Ci)

B

T h u s , d e n o t i n g t h e r e s t r i c t i o n o f x @ t o T by x , i t f o l l o w s

x P ( c 1 C )=[1].

...,n .

C € / n f o r i=l,

As

( p ( i s monotone,

By t h e n o r m a l i t y o f T ( o r the n o r m a l i t y of

when + i s d e f i n e d on Q ) c o n t i n u o u s f u n c t i o n s 0

e x i s t such t h a t xi(Ci)={l}

Z

and xi(CWi)={O].

5

Letting a

x.

< 1 d e f i n e d on T

1 -

i

be such t h a t

a4

2.

aip(Ai)=Ip(Ai)l

1s

SPACES OF CONTINUOUS FUNCTIONS

a n d s e t t i n g x=C.a . x .

1 1 1'

x d ! ~- Ci I!dAi)

I I=

x.dw wi

lCiaiJ

= IZihi,J

-

= ICiai"4Ci>


-

3c.

AND REGULAR ADDITIVE ____~-

i s o m e t r i c a l l y i s o m o r p h i c t o t h e l i n e a r s p a c e o f a l l bounded

f i n i t e l y a d d i t i v e r e g u l a r F - v a l u e d s e t f u n c t i o n s d e f i n e d on (When

g=,$t h e s e

0 z ( ac)

a r e the elements

f i n i t e l y a d d i t i v e measures of Sec. 1.7.)

Certainly both r e s u l t s of (2.4-5) situation.

-

I f T i s c o m p l e t e l y r e g u l a r ( n o r m a l ) and H a u s d o r f f ,

e q u i p p e d w i t h t h e t o t a l v a r i a t i o n norm.

m , the

Ik)(T)

and w e may s t a t e :

SET FUNCTIONS

of

Wi-Cy&I

-I- xiaiJ-

C 2F.

CONTINUOUS LINEAR FUNCTIONALS ON C,(T,F)

t h e n Cb(T?g)-!,is

3

+ CiaiJ W i - C i ~ i d ~

Since the W ' s a r e pairwise d i s j o i n t , l l x l l Thus

1 p(Ai)l I

-

a p p l y t o t h e compact H a u s d o r f f

I n d e e d i n t h i s c a s e w e may s a y e v e n more.

A s a consequence of

o u r n e x t r e s u l t t h e s e t f u n c t i o n s may b e c o n s i d e r e d t o b e c o u n t a b l y a d d i t i v e s e t f u n c t i o n s d e f i n e d on @ a o r (2.4-6)

(Alexandrov)

@

A BOUNDED REGULAR ADDI'CIVE SST FUNCTION

COUNTABLY A D D I T I V E

closed subsets

t

1 s

I f p i s a bounded r e g u l a r , r e a l - or complex-

v a l u e d , a d d i t i v e s e t f u n c t i o n d e f i n e d on

a d d i t i v e on

a s t h e c a s e may b e .

a

C'

t h e a l g e b r a g e n e r a t e d by t h e

+

of a compact H a u s d o r f f s p a c e T , t h e n

a,, i.e.

is countably

p((JAn)=% +(An) whenever (A ) i s a c o u n t a b l e f a m i l y n of p a i r w i s e d i s j o i n t s e t s from w i t h union i n Moreover p h a s a

ac

ac.

unique r e g u l a r countably a d d i t i v e extension t o t h e a-algebra s u b s e t s of T, i . e . Remark --

the closed s e t s

6

aat h e 0 - a l g e b r a Proof

a,.

t h e 0 - a l g e b r a g e n e r a t e d by

As i n previous r e s u l t s , e.g.

(2.4-4),

@

o f Bore1

6. the t h e o r e m r e m a i n s v a l i d i f

2 , a cby

a r e replaced by the z e r o sets

a, and

@

by

g e n e r a t e d b y 2 (sometimes c a l l e d t h e B a i r e s e t s ) .

F i r s t w e show t h a t t h e t o t a l v a r i a t i o n 111

I

is c o u n t a b l y a d d i t i v r on

L e t 6 > 0 b e g i v e n a n d l e t (A ) b e a s i n t h e s t a t e m e n t o f t h e theorem.

Choose E

€ 6 such

t h a t E C A and I p l (A-E)

a n open s u b s e t o f T c o n t a i n i n g A compact, t h e r e a r e Ul,

...,Un


CnCNp(Akn) N

-

k c/2

.

) from

S

k k

ac s u c h

and

kn Now S C U k U n A k n so t h a t

f

To s e e t h a t i t

>

3 is given.

that S k i

Unc_N4k n and

2.

86

SPACES OF CONTINUOUS FUNCTIONS

5sXn

$(S)

ii.(Akn)-

On t h e o t h e r hand

*

xk i ~ . (Sk) > %Cn p(Akn) 4

5

and i t f o l l o w s t h a t p ( S )

/L*(S~~).

-

E

>'

T o o b t a i n a c l a s s o f s e t s o n w h i c h i ~ . i s c o u n t a b l y a d d i t i v e , we s i n g l e Jr

A s u b s e t R C T i s a y -set i f ( t h i s c o n d i t i o n

out t h e family of b*-sets:

sometimes b e i n g r e f e r r e d t o a s t h e C a r a t h e o d o r y c o n d i t i o n )

9
0 , and c h o o s e a s e q u e n c e (A,) €

Hence, by t h e d e f i n i t i o n o f

U''

+

p"(S)

from

>

zn

a

a

a

4-

is a p"-set,

s u c h t h a t S C U n A n and

p(An).

and i t s s u b a d d i t i v i t y

let

2.

88

SPACES OF CONTINUOUS FUNCTIONS

>

P(sn9) +

pL*(sn,cA) 2 l L * ( ~ ) . k

As Q i s a r b i t r a r y , i t follows t h a t A i s a p -set. Having d e m o n s t r a t e d t h e e x i s t e n c e o f a n o n n e g a t i v e c o u n t a b l y a d d i t i v e

@ w e now p r o v e € 0 ,a n d (A ) b e a n y

e x t e n s i o n of p t o

Let 1 be another such

i t s uniqueness.

a,

&,.

such t h a t B c U s e q u e n c e from n "b S i n c e h i s c o u n t a b l y a d d i t i v e a n d n o n n e g a t i v e , i t i s monotone and c o u n t a b l y

extension, B

subadditive so t h a t

A(B)

5 X(UnAn> 6)that t h e r e i s a c o u n t a b l y K a d d i t i v e r e g u l a r F - v a l u e d s e t f u n c t i o n p d e f i n e d on t h e Bore1 s u b s e t s 63 I< K o f K s u c h t h a t f ( * ) = j * d p K . It o n l y remains t o show t h a t a c o u n t a b l y a d d i K t i v e r e g u l a r 2 - v a l u e d s e t f u n c t i o n IJ. can be d e f i n e d on t h e B o r e l s u b s e t s

(Bra)

of T by t h e e q u a t i o n P(B)=P ( B n K ) K

f o r then i t follows t h a t p has

compact s u p p o r t and x'

Let

a

= f K ( x IK) = jKx IKdPK =

1 xdw

F i r s t i t i s n e c e s s a r y t o show t h a t B n K

f o r e a c h x(C(T,E).

B €@.

(XI

~qf o r

each

d e n o t e t h e c o l l e c t i o n o f a l l s u b s e t s of T o f t h e form

EU(B-K) where E

€aKand B €a.It

i s a 0-algebra.

F u r t h e r m o r e i f C i s c l o s e d i n T t h e n CnK

a

is straightforward t o verify t h a t

€ 0K

and

:: =

(CnK)U(C-K)€a.

Hence, a s a c o n t a i n s a l l t h e c l o s e d s u b s e t s of T , B 2

Thus d b n K = { D n K I D

€ 8 )3

contains ously

)L

@nK and

@flK=(BnKIB

€63.But WnKK=aK so that

t h e s e t f u n c t i o n LL h a s a m e a n i n g f u l d e f i n i t i o n .

i s c o u n t a b l y a d d i t i v e , b u t what a b o u t r e g u l a r i t y ?

€a

@.

@K

Obvi-

Let B €@.

As

g i v e n any € > 0 t h e r e i s a c l o s e d s e t C C K and a n open s u b s e t U K of K s u c h t h a t C C B n K C U and \ ( U - C ) < C. L e t ? b e any open s u b s e t o f

BnK

1.

T s u c h t h a t C 2 C K and

i?nK=U.

Y

I f B1,

...,Bk

s u b s e t s of T w i t h u n i o n c o n t a i n e d i n ??-C CIIJ.(Bi)

I = C I PK(Bin

a r e pairwise d i s j o i n t Borel

then K)

U(BinK)CU-C.

Thus

I 0 t h e r e i s a p o s i t i v e d such t h a t

s u p B"(y) < a whenever s u p \ y ( s u p p ( B ) ) ( < d .

By t h e d e f i n i t i o n o f B" i t f o l -

lows t h a t s u p l B ( x ) [ < a whenever s u p / x ( s u p p ( B ) ) 1 < d a n d B i s e q u i c o n t i n u o u s . V I n p r o v i n g t h a t t h e weakly compact s e t B i s e q u i c o n t i n u o u s whenever supp(B) i s c o m p a c t , t h e weak c o m p a c t n e s s o f B i s u s e d o n l y t o g u a r a n t e e t h a t B(x) i s bounded f o r e a c h xfC(T,;).

Consequently t h e implication i s a l s o

v a l i d f o r w e a k l y bounded B. (2.5-3)

SUPPORT AND E Q U I C O N T I N U I T l

A w e a k l y bounded s u b s e t B o f C(T,,F,c)',

T c o m p l e t e l y r e g u l a r a n d H a u s d o r f f , i s e q u i c o n t i n u o u s whenever s u p p ( B ) i s compact.

98

2.

SPACES OF CONTINUOUS FUNCTIONS

W e now make u s e o f (2.5-3)

c o n d i t i o n f o r C(T,X,c)

i n e s t a b l i s h i n g a n e c e s s a r y and s u f f i c i e n t

t o b e i n f r a b a r r e l e d , a c o n d i t i o n analogous t o t h e

one g i v e n i n Theorem 2.5-1. (2.5-4)

INFRABARRELEDNESS OF C ( T , m

Hausdorff s p a c e .

Let T be a c o m p l e t e l y r e g u l a r

is infrabarreled (i.e.

C(T,l',c)

every bornivorous b a r r e l

i s a neighborhood of 0 ) i f f f o r e a c h c l o s e d noncompact s u b s e t S of T t h e r e i s a n o n n e g a t i v e lower s e m i c o n t i n u o u s f u n c t i o n y d e f i n e d on T which i s unbounded on S and bounded on e a c h compact s u b s e t of T. Proof

R e c a l l t h a t a l o c a l l y convex H a u s d o r f f s p a c e i s i n f r a b a r r e l e d i f f

e a c h s t r o n g l y bounded s u b s e t o f t h e c o n t i n u o u s d u a l i s e q u i c o n t i n u o u s (Horvath 1966, p. 217, Prop. 6 ) . Now, t o p r o v e n e c e s s i t y , l e t S i t be t h e homeomorphicimage i n t h e weakly topologized space C(T,Z,c)'

of t h e c l o s e d non-compact s e t S,

o f e v a l u a t i o n maps t J x as t r u n s t h r o u g h S .

j::

C(T,X,c)

is n o t

i s i n f r a b a r r e e d , S;k

Prop. 6 ) .

q

r

i s n o t s t r o n g l y bounded (Horvath 1966, p. 2 1 7 , C(T,,F,c) e x i s t s s u c h t h a t A(S)=S*(A)

Consider t h e nonnegative f u n c t i o n t

i s weakly bounded, t*(A)=A(t) Moreover e a c h f u n c t i o n t

-

i s n o t weakly compact,

i ic w i t i i i i ~ o r r s ; moreover s i n c e t h e s p a c e

Thus a weakly bounded s e t A

i s unbounded.

t

~

S i n c e Sf:

i.e. t h e s e t

-

As A

y(t)=suplA(t)l.

i s bounded f o r e a c h t < T a n d y i s r e a l - v a l u e d .

i x ( t ) l i s c o n t i n u o u s on T s o t h e map

-, s u p x F A l x ( t ) l i s lower s e m i c o n t i n u o u s (Dieudonng 1970,

F i n a l l y s u p i y ( S ) l =sup 1A(S)I =

p. 2 5 ) .

(12.7.6),

w h i l e f o r e a c h compact K C T , sup]y(K)[ =pK(A)<m

s i n c e A i s bounded and y i s t h e d e s i r e d f u n c t i o n . Conversely suppose t h a t t h e c o n d i t i o n h o l d s . s t r o n g l y bounded s u b s e t H C C ( T , L , c ) ' bounded i t s u f f i c e s by (2.5-3)

We must p r o v e t h a t e a c h

is equicontinuous.

S i n c e H i s weakly

t o show t h a t supp(H) i s compact.

Assuming

t h a t supp(H) i s n o t compact, w e u s e t h e c o n d i t i o n t o c o n t r a d i c t t h e s t r o n g boundedness o f H.

I n d e e d i f t h e c l o s e d s e t supp(H) i s n o t compact t h e r e

i s a r e a l - v a l u e d n o n n e g a t i v e lower s e m i c o n t i n u o u s f u n c t i o n y d e f i n e d on T

-1 and unbounded on supp(H). Thus y (n,m) meets supp(H) f o r e a c h n. -1 s u p p ( x ' ) ) then, f o r each y (n,m) i s open and s u p p ( H ) = c l ( U f'CH

i s a n x ' F H whose s u p p o r t m e e t s y-

(n,m).

As

"€2, t h e r e

Next w e c o n s t r u c t a weakly

bounded s u b s e t { y n l o f c ( T , F , c ) on which H i s unbounded, t h e r e b y e s t a b l i s h -1 (-m,n] so ing t h e d e s i r e d c o n t r a d i c t i o n . F i r s t note t h a t supp(x ' ) e y n t h a t by ( 2 . 4 - 8 ) ( b ) a f u n c t i o n xnFC(T,,F) e x i s t s s u c h t h a t xn v a n i s h e s on

y-'(-@,n] yn=Ck,lak%

and x n ' ( x n ) = l .

We d e f i n e a weakly bounded s e q u e n c e o f f u n c t i o n s

s u c h t h a t x n ' ( y )=n f o r each n. n h e n c e weakly bounded

bounded i n C(T,E,c)

-

To s e e t h a t t h e s e t {y,]

-

is

l e t K be a n y compact s u b s e t

2.6

S i n c e y i s bounded on K , y

o f T.

-1( n , m ) n k = @ f o r a l l n l a r g e r than o r

Thus f o r e a c h t f K c y - l ( - m , n f ,

e q u a l t o some m.

yn(K)=ym(K) f o r a l l n

99

INFRABARRELEDNESS OF C(T,F,C) N

2 m.

xn(t)=O f o r n

2

m

SO

that

Hence w i t h p ( w ) = s u p t C K l w ( t ) l , {pK(y,)ln6E] i s K ] i s bounded. A s f o r t h e s p e c i f i c v a l u e s

a bounded s e t f o r e a c h K and ( y

t h e y a r e e a s i l y e s t a b l i s h e d by t h e e q u a t i o n s C a x '(x ) = n ( n = l , n' k=l k n k 2 , . . ) . This e s t a b l i s h e s t h e c o n t r a d i c t i o n and completes t h e p r o o f . V of a

.

2.6

A l i n e a r map b e t w e e n normed s p a c e s i s c o n -

B o r n o l o g i c i t y of C ( T , g , c )

t i n u o u s i f f i t maps bounded s e t s i n t o bounded s e t s . s p a c e s f o r which boundedness'implies f i r s t d e f i n e d by Mackey ( 1 9 4 6 ) .

A wi der c l a s s of

continuity is the bornological spaces,

The e x t e r n a l d e s c r i p t i o n o f s u c h s p a c e s i s

t h a t a l o c a l l y convex s p a c e X i s b o r n o l o g i c a l i f , f o r any l o c a l l y convex space Y,

a l i n e a r map A : X

must b e c o n t i n u o u s .

-, Y

which maps bounded s e t s i n t o bounded s e t s

An e q u i v a l e n t i n t e r n a l d e s c r i p t i o n i s t h a t a l o c a l l y

convex s p a c e X i s b o r n o l o g i c a l i f e a c h b a l a n c e d convex b o r n i v o r e i n X i s a n e i g h b o r h o o d o f 0. T h e r e a r e i n c o m p l e t e normed s p a c e s w h i c h a r e b o r n o l o g i c a l ( s i n c e t h e y a r e metrizable) but not barreled.

*

I n t h e c o n v e r s e d i r e c t i o n , e a c h com-

p l e t e b o r n o l o g i c a l s p a c e must be b a r r e l e d a n d , a s w i l l b e shown i n t h i s s e c t i o n (('2.b-l)),

i f C(T,F,c)

i s b o r n o l o g i c a l , i t must be b a r r e l e d .

Quite

e a r l y on Dieudonne (1953) r a i s e d t h e q u e s t i o n o f w h e t h e r t h e r e a r e b a r r e l e d s p a c e s which a r e n o t b o r n o l o g i c a l .

S h o r t l y a f t e r w a r d s t h e q u e s t i o n was

a n s w e r e d i n t h e n e g a t i v e by N a c h b i n (1954) a n d S h i r o t a (1954) who d e t e r m i n e d n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s o n T f o r C(T,F.,c) r e m 2.5-1:

C(T,L,c)

c a l (Theorem 2 . 6 - 1 ) : being r e p l e t e .

t o b e b a r r e l e d (Theo-

i s b a r r e l e d i f f T i s a n NS-space) and t o b e b o r n o l o g i The b o r n o l o g i c i t y of C(T,&,c) i s e q u i v a l e n t t o T

Thus t o e x h i b i t a b a r r e l e d s p a c e which i s n o t b o r n o l o g i c a l ,

i t s u f f i c e s t o p r o d u c e a n NS-space which i s n o t r e p l e t e , and t h i s i s done

i n Example 2 . 6 - 1 .

-2-

''

The s u b s p a c e X o f the normed s p a c e

a2

w h i c h a r e e v e n t u a l l y 0 , b e i n g d e n s e Fn

c o n s i s t i n g of the sequences (p )

a2, h a s a2

Since X i s metrizable, X i s bornological.

a s i t s c o n t i n u o u s d u a l X'.

B u t t h e r e a r e w e a k l y bounded s e t s

\ C n , n€E} f o r example - which a r e n o t s t r o n g l y ( = i n n t h e norm h e r e ) bounded, hence n o t e q u i c o n t i n u o u s . Hence t h e B a n a c h - S t e i n -

i n X'

-

{(pn)c.t2

h a u s t h e o r e m d o e s n o t h o l d and X i s n o t b a r r e l e d .

100

2.

SPACES OF CONTINUOUS FUNCTIONS

C ( T , F , c ) I S BORNOLOGICAL I F F T I S REPLETE

Theorem 2 . 6 - 1

For C(T,E,c) t o

be b o r n o l o g i c a l i t i s n e c e s s a r y and s u f f i c i e n t t h a t t h e c o m p l e t e l y r e g u l a r Hausdorff space T be r e p l e t e , i.e. Proof

uFT.

Suppose t h a t T i s n o t r e p l e t e , s o t h a t t h e r e i s some

Necessity:

&

t coT-T.

I t f o l l o w s t h a t t h e map t

phism of C ( T , F ) .

phisms of C ( T , E , c ) ,

-

x U ( t o ) , x u denoting the unique

t o u T , i s a n o n t r i v i a l F - v a l u e d homomor-

c o n t i n u o u s e x t e n s i o n o f xcC(T,g) Only t h o s e

", x

from T c a n p r o d u c e c o n t i n u o u s homomor-

t " I s

s o tk must b e d i s c o n t i n u o u s . t To c o n c l u d e t h a t C(T,L,

c ) i s n o t b o r n o l o g i c a l , we need o n l y show t h a t t* maps bounded s e t s i n t o bounded s e t s .

I f t" d o e s n o t h a v e t h i s p r o p e r t y t h e n t h e r e i s a bounded

set B C C ( T , z , c )

-

t:(xn)

a.

:qd

o ( t )= n o x n Y j t o ) -13.

a s e q u e n c e (x ) o f p o i n t s o f B s u c h t h a t x

Now c o n s i d e r t h e open s e t s V ={tqpT

tocur,

A s t h e V ' s a r e n e i g h b o r h o o d s of

I x nU ( t )

>

t h e r e i s a p o i n t s c ( n V n ) n T by

But s c f l V n i m p l i e s t h a t s"'(x )=x " ( s ) which c o n t r a n n (which must be c o n t i n u o u s s i n c e s c T ) must t a k e B

Theorem 1 . 5 - 1 ( b ) .

d i c t s t h e f a c t t h a t s"

i n t o a bounded s u b s e t of F .

Hence t*(B)

i s bounded a n d t h e n e c e s s i t y o f

t h e c o n d i t i o n h as been d emo n s t rat ed . Sufficiency:

I n p r o v i n g t h a t C ( T , F , c ) was b a r r e l e d i n Theorem 2 . 5 - 1 ,

an

a r b i t r a r y b a r r e l V was shown t o b e a n e i g h b o r h o o d o f 0 i n t h r e e s t e p s . F i r s t th'e e x i s t e n c e o f a d > 0 was e s t a b l i s h e d s u c h t h a t dV c V . It PT was proved n e x t t h a t f o r a n y s u b s e t S o f T w i t h t h e p r o p e r t y t h a t xcV

-

whenever x(S)={O'j,

t h e r e was a n a

> 0 s u c h t h a t aV

C V. It then only PS remained t o p r o d u c e a compact s u b s e t K o f T w i t h t h e a b o v e p r o p e r t y . A s i m i l a r approach i s used h e r e .

For C(T,E,c)

t o be b o r n o l o g i c a l , i t i s

n e c e s s a r y a n d s u f f i c i e n t f o r e a c h a b s o l u t e l y convex s e t t h a t a b s o r b s a l l bounded s e t s t o be a n e i g h b o r h o o d o f 0. condition actually holds, i.e.

We s h a l l p r o v e t h a t a s t r o n g e r

t h a t e v e r y a b s o l u t e l y convex s e t V t h a t

a b s o r b s a l l bounded s e t s o f a c e r t a i n t y p e ( o r d e r s e g m e n t s ) i s a n e i g h borhood o f 0.

F i r s t w e d e f i n e a n o r d e r segment and p r o v e i t t o be

I f x and y a r e r e a l v a l u e d c o n t i n u o u s f u n c t i o n s on T s u c h t h a t

bounded. x

5

y , t h e n t h e o r d e r seRment [ x , y ] c o n s i s t s o f a l l wcC(T,&) s u c h t h a t

x

5

w

5

wc[x,y]

I f K i s compact t h e n p (w) K s o t h a t [ x , y ] i s bounded.

y.

5

max ( p ( x ) , p ( y ) ) f o r e a c h K K

+An a p o l o g y i s p e r h a p s d u e t h e r e a d e r h e r e f o r t h i s r e s u l t i s proved i n Example 4 . 1 0 - 2 .

Our d e s i r e t o p l a c e Theorem 2 . 6 - 1 n e a r t o c l o s e r r e l a -

t i v e s m o t i v a t e d us t o l o c a t e i t h e r e , r a t h e r t h a n i n Chap. 4.

2.6

101

INFRABARRELEDNESS OF C(T,F,C)

P r o c e e d i n g i n t h e f a s h i o n o u t l i n e d a b o v e w e show t h a t a d

> 0 exists

C V. S i n c e V a b s o r b s a l l o r d e r segments t h e r e i s a b > 0 PT such t h a t [-bl,bl] C V. Now c h o o s e d = b / 2 and s u p p o s e t h a t p ( x ) 5 d . . I f T x=B, i t f o l l o w s t h a t x c [ - b l , b l ] C V . I n t h e e v e n t t h a t F=C w e see t h a t 2x

s u c h t h a t !d

N

and 2 x . b e l o n g t o [ - b l , b l ] of x r e s p e c t i v e l y . (i/2)(2xi)cV.

where x

U

r

a n d x . a r e t h e r e a l and i m a g i n a r y p a r t s

Thus, by t h e a b s o l u t e c o n v e x i t y o f V , x = ( 1 / 2 ) ( 2 x r ) +

Now, a s was shown i n t h e p r o o f o f t h e p r e v i o u s t h e o r e m , f o r

a n y s u b s e t S o f T w i t h t h e p r o p e r t y t h a t x b e l o n g s t o t h e a b s o l u t e l y convex

-

s e t V whenever x(S)={O},

t h e r e e x i s t s a p o s i t i v e number a s u c h t h a t aV c V . PS Thus i t i s j u s t a m a t t e r o f p r o d u c i n g a compact S w i t h t h e a b o v e p r o p e r t y . To d o t h i s w e b e g i n by d e f i n i n g t h e n o t i o n o f a s u p p o r t s e t o f V . c l o s e d s u b s e t K o f BT i s a s u p p o r t

set o f

A

V i f t h e c o n t i n u o u s f u n c c i o n xfV

b b e n e v e r x@(K)=O ( h e r e xB d e n o t e s Lile u n i q u e e x t e n d e d r e a l - v a l u e d e x t e n s i o n t o BT which e x i s t s by ( 1 . 5 - 1 ) ) .

o f xcC(T,;) itself.

An example o f s u c h a s e t i s 81

The i n t e r s e c t i o n o f a l l s u c h s u p p o r t s e t s i s c a l l e d t h e s u p p o r t

V a n d i s d e n o t e d by K ( V ) .

A f t e r showing t h a t K(V)

i s a s u p p o r t s e t of V w e

s h a l l c o m p l e t e t h e p r o o f by showing t h a t K ( V ) C T . The f a c t t h a t K(V)

i s a s u p p o r t s e t w i l l b e e s t a b l i s h e d w i t h t h e a i d of

two f a c t s :

(1) A c l o s e d s u b s e t K o f PT i s a s u p p o r t s e t o f V i f f xcV whenever xB v a n i s h e s on a n e i g h b o r h o o d o f K ( i . e .

a s u p e r s e t o f K i n BT whose i n t e r i o r

contains K).

(2)

Any f i n i t e i n t e r s e c t i o n o f s u p p o r t s e t s i s a s u p p o r t s e t o f V

One h a l f o f (1) i s t ; . i J i a l . condition holds, i.e.

To o b t a i n t h e o t h e r h a l f , s u p p o s e t h a t t h e

t h a t xCV w h e n e v e r xB v a n i s h e s on a n e i g h b o r h o o d o f K.

To see t h a t K i s a s u p p o r t s e t of V , s u p p o s e t h a t x @ v a n i s h e s on K .

I t re-

mains t o show t h a t x @ v a n i s h e s on some n e i g h b o r h o o d o f K f o r t h e n x w i l l belong t o V.

-

s u c h t h a t dV

F i r s t suppose t h a t

E=&

y=max(x,d/Z)+min(x,-d/Z)

and c l a i m t h a t ( 2 y ) B v a n i s h e s on G .

n e s s o f t h e e x t e n s i o n w --. min(xp,(-d/Z)l).

WB,

T

be [ t € @ T I x r @ ( t )

Hence (2y)B must v a n i s h on G and 2ycV.

,

xi@(t) < d/4],

yi=max(xi, ( d / 4 ) l ) + m i n x ( x i , ( - d / 4 ) 1 ) .

-

4 ( x . - y . ) b e l o n g t o dVp C V , 1

belong t o V.

By t h e u n i q u e -

i t now f o l l o w s t h a t yp=max(xB,(d/2)1)+

C V and t h e r e f o r e x=(1/2)(2y+2(x-y))CV.

2(x-y)cd!p

< d/2 f o r d Next w e d e f i n e

C V ] . G i s c l e a r l y a n o p e n n e i g h b o r h o o d of K.

PT

1

and l e t G={t - ( l ] and

B

S i n c e WUW2 i s open

C l e a r l y 2x2 v a n i s h e s on (WUW ) n T . 2 B i n BT and T i s d e n s e i n BT,

z P ( c 1 W,)=(O].

WUW,

= (WUW2)ncl T C c l ((WUW2)nT).

B

B

Thus by t h e c o n t i n u i t y of ( Z X Z ) ~ ,(2xz)B v a n i s h e s on c l ( W U W 2 ) n T ) a n d

B

t h e r e f o r e a l s o on WUW2. o f V and WUW 2xzcV.

2

Now w e c a n u s e t h e f a c t s t h a t Y i s a s u p p o r t s e t

i s a neighborhood o f Y t o g e t h e r w i t h (1) t o conclude t h a t

I n t h e same way t h e e x t e n s i o n ( 2 x ( l - z ) ) v a n i s h e s on W1 a n d , t h e r e -

f o r e , 2x(l-z)EV.

F i n a l l y x=(1/2)(2xz)+(l/2)(2x(l-z))~V

and i t f o l l o w s by

(1) t h a t K i s a s u p p o r t s e t of V. Having e s t a b l i s h e d (1) and ( 2 ) , t h e s u p p o r t of V,

w e a r e now r e a d y t o p r o v e t h a t K ( V ) ,

i s a s u p p o r t s e t of V.

To t h i s end s u p p o s e t h a t x B v a n -

i s h e s on a n open n e i g h b o r h o o d W of K(V) i n

8:.

S i n c e BT i s a 'compact

H a u s d o r f f s p a c e a n d K(V) i s t h e i n t e r s e c t i o n o f a l l s u p p o r t s e t s of V ,

...,Kn

t h e r e a r - support sets Kl,

s u c h t h a t nKiCW.

nKi

By ( 2 ) ,

is a

s u p p o r t s e t o f V s o t h a t x ii.ust b e l o n g t o V s i n c e xB v a n i s h e s on t h e open neighborhood W of

nKi.

Thus (1) may b e i n v o k e d a g a i n t o c o n c l u d e h a t

K(V) i s a s u p p o r t s e t o f V. The f i n a l t h i n g t o b e shown i s t h a t K ( V ) C T . p r o v e t h a t t k K(V).

L e t t C ET-T;

we s h a l l

By Theorem 1 . 5 - 1 ( b ) a n d t h e r e p l e t e n e s s o f T , there i s

a d e c r e a s i n g s e q u e n c e (W ) o f c l o s e d n e i g h b o r h o o d s o f t s u c h t h a t

=0. V.

We c l a i m t h a t a t l e a s t one o f t h e s e t s BT

tc

As

int W

i t follows

x

f V.

-

i s a s u p p o r t s e t of V.

t h a t tkK(V). Then, f o r

n

L e t y=supnnlx 1 .

v a n i s h e s on T-W IT-W,,,

i s a - s u p p o r t s e t of

t h e r e i s a n e l e m e n t x n < C ( T , g , c ) s u c h t h a t x @ ( @ T - i n tW ) = 0 and To see t h a t y i s c o n t i n u o u s on T , f i x a p o s i t i v e

i n t e g e r m and c o n s i d e r any n x

int W

- a f t e r establishing the claim

n' Suppos.:. t h a t none o f t h e s e t s pT-int W

e a c h nc;,

-

(nnW n ) n T

. m

> m.

C l e a r l y T-WmC BT-int Wn a n d , t h e r e f o r e ,

Thus y=max(lx 1 . 2 1 ~I , . . . , m i x

i s c o n t i n u o u s on T-W

m

.

1

2

m

I ) on T-W,

so that

A s (T-W ) i s a n i n c r e a s i n g s e q u e n c e o f open m

s e t s , whose u n i o n i s T , w e see t h a t y i s c o n t i n u o u s on T.

Since V absorbs

a l l o r d e r s e g m e n t s , t h e r e i s a p o s i t i v e number k s u c h t h a t [ - y , y ] C k V .

2.6

p,g

Now i f

then f o r each n , nxnc[-y,y]CkV

o t h e r hand, i f

K=C,

103

INFRABARREJXDNESS OF C(T,X,c)

and xncV f o r a l l n

2

On t h e

k.

I < y f o r e a c h n and i t f o l l o w s t h a t t h e r e a l

then 2nlx

2n b e l o n g t o V f o r a l l n 2 k . Hence, by t h e a b s o 2n CV f o r a l l n 2 k . Thus i n b o t h c a s e s , t h e c o n -

and i m a g i n a r y p a r t s o i 2x

l u t e c o n v e x i t y of V , x 2n t r a d i c t i o n t h a t x EV f o r some p @ h a s becn e s t a b l i s h e d .

P

i s a s u p p o r t s e t o f V f r o m which i t f o l l o w s t h a t

one o f t h e s e t s BT-int W th(V)

We c o n c l u d e t h a t

+

I n summary w e h a v e shown t h a t K(V) i s a s u p p o r t s e t o f V , c o n t a i n e d i n T s o , f o r some a

R e c a l l (Sec.

-

> 0 , aV

C V and t h e p r o o f i s c o m p l e t e . V 'K(V> 1.5) t h a t a s u b s e t E of t h e completely r e g u l a r Hausdorff

s p a c e T i s r e l a t i v e l y pseudocompact i f t h e r e s t r i c t i o n x

of e a c h x C) t r e q u a l t o t h e s m a l l e s t e l e m e n t s o f A and B r e s p e c t v e l y l a r g e r t h a n

e l e m e n t s of [O,a) s u c h t h a t b

sup[F([O,c))nA]

A s F(b) i s

and s u p [F ([O , c))n B].

5

sup(F([O,c))nA)

or

i t follows t h a t ' F ( c ) i s l a r g e r than F(b).

sup(F([O,c))nB),

i s a n o r d e r - i s o m o r p h i s m , aCP, and P=[O,n]. the d e f i n i t i o n of h

Hence F IC0.4 By t r a n s f i n i t e i n d u c t i o n and

i t follows t h a t F i s an order-isomorphism.

Thus, by

( i i ) , sup F(CO,~))CT. Next w e c l a i m t h a t s u p F ( [ O , n ) ) E A n B . directed set

[O,n)

and s u p F([O,h2))

S i n c e F i s a n e t d e f i n e d on the

is i t s l i m i t ,

i t i s only necessary t o

show t h a t t h e n e t i s f r e q u e n t l y i n e a c h o f t h e s e t s A and B.

L 6 t aC[O,n).

I f F ( a ) t A , t h e n F ( a ) i s t h e s m a l l e s t e l e m e n t o f B which i s l a r g e r t h a n t h e f i r s t element of A l a r g e r t h an s u p ( F ( [ O , a ) ) n A ) . q u e n t l y i n A , c o n s i d e r any a € [ O , n ) . a'C[a,62) F(a+l)€A.

To see t h a t F i s f r e -

We w i s h t o show t h a t t h e r e i s some

I f F(a)kA, t h e n i t w i l l b e shown t h a t

such t h a t F(a')cA.

I f F(a)jA, then sup(F([O,a))n A)=sup(F([O,a+l))n A ) .

t h e s m a l l e s t element of A est element of A

2

2

sup(F([O,a))flA),

sup(F([O,a))nA).

coincides with x

Moreover s i n c e F ( a ) ( B ,

Thus x

a+l' the small-

a' then F(a) is

o f {b€B (b 2 s u p ( F ( [ O , a + l ) ) f l B ) ] . Hence b a+l= a+l x =x a n d , by t h e d e f i n i t i o n o f ha+l, F ( a + l ) c A . Thus F i s f r e a a+l q u e n t l y i n A and t h e same i s t r u e o f B . the s m a l l e s t element b

F(a)

>

R e t u r n i n g t o t h e t a s k o f showing t h a t e a c h f u n c t i o n xCCb(T,&) i s c o n s t a n t on a t a i l o f T , w e n o t e t h a t f o r e a c h t C T t h e s e t c l { x ( a ) la z t ] i s a nonempty compact s e t .

Consequently

e l e m e n t r c a n b e e x t r a c t e d from i t .

i s c l o s e d and c o f i n a l i n T f o r each

n t c T c l ( x ( a ) la

> t}

i s nonempty and a n

It f o l l o w s t h a t G n = { a € T l ; x ( a ) - r ; i l / n }

ncE.

S i n c e t h e s e t F ={acT(!x(a)-rt>l/n} n f o r each n c l , 2n Fn a n d c h o o s e aCT s u c h t h a t a > s u p a

i s c l o s e d and d i s j o i n t from t h e c l o s e d c o f i n a l s e t s G must b e bounded by ( i i i ) .

Let a =sup F

Certainly then x ( t ) = r f o r each t

n'

2 a ( o t h e r w i s e tCFn f o r some n ) .

A p p e a l i n g t o t h e d i s c u s s i o n a t t h e b e g i n n i n g o f ( b ) , w e may c o n c l u d e t h a t PCBT-T.

To see t h a t wCuT, i t i s enough t o show t h a t ( n V n ) n T # O f o r

any s e q u e n c e (V ) o f n e i g h b o r h o o d s of p i n BT. s e q u e n c e f r o m T s u c h t h a t (b , p ] C V n f l S . {b,}

is not cofinal i n T,

p l e t e s the proof. V

L e t ( b ) be a n i n c r e a s i n g

S i n c e T i s c o f i n c . 1 i n [O,p)

( s u p b n , p ] nT#0 and ( s u p b n , p ]

c nnVn.

and

T h i s com-

2.7 -

107

SEPARABILITY OF C(T,,F,c)

2.7

M . and S . K r e i n ( 1 9 4 0 ) were t h e f i r s t t o

S e p a r a b i l i t y o f C(T,F,c)

mention t h e c h a r a c t e r i z a t i o n o f s e p a r a b i l i t y o f C(T,F,c)

f o r compact T

Warner (1958) g e n e r a l i z e d t h i s t o t h e g e n e r a l

t h a t appears i n (2.7-1).

c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e T and t h i s r e s u l t , a s w e l l a s some of i t s c o n s e q u e n c e s , a p p e a r s below. (2.7-1)

SEPARABILITY OF C ( T , F , c ) WHEN T I S COMPACT

Hausdorff s p a c e , then C(T,g,c) Proof

is s e p a r a b l e so t h a t a countable dense s u b s e t

Suppose t h a t C(T,F',c)

(xnIn&

I f T i s a compact

is separable i f f T i s metrizable.

e x i s t s i n C(T,E,c).

Let U(y,a)={(s,t) Ily(s)-y(t)(

< a ] be a

t y p i c a l s u b b a s i c e n t o u r a g e i n t h e u n i f o r m i t y C ( T , E ) g e n e r a t e d by t h e c o n t i n u o u s 2 - v a l u e d f u n c t i o n s on T. form pT(xn-y)


0, f o r t h e unique 1 '***'Xkn N ( s i n c e T i s compact) u n i f o r m i t y G(TJ). S i n c e T i s H a u s d o r f f , t h e u n i -

a b l e b a s e o f e n t o u r a g e s (U(x

IX

formity is separating, i . e . given (t,s)cTxT, b a s i c entourbges f a i l s t o contain ( t , s ) .

t # s , a t l e a s t one of t h e

T h u s , f o r some nc;,

and some n n 1

0 /lylcrN(x)j=

Now by p a r t s ( c ) and ( d ) i t i s

-

s u f f i c i e n t t o show t h a t g i v e n any s e q u e n c e ( x ) from X w i t h xm 0 , y= m X' C l e a r l y t h e s e r i e s c o n v e r g e s p o i n t w i s e t o y and we show 2-mx E C ( T , F ) .

'm<E

m

2.

116

t h a t y(C(T,F)

SPACES OF CONTINUOUS FUNCTIONS

For

a s follows.

c > 0 , choose a neighborhood V o f

t FT such

t h a t y i s bounded o n V , and a p o s i t i v e i n t e g e r M s u c h t h a t

2

-m

suptCv m I' 2M of t such t h a t

( x m ( t ) - x m ( t o ) ) 1 5 c / 2 . Now t h e r e e x i s t s a n e i g h b o r h o o d V' M 2-m(xm(t)-xm(to) Thus s u p ly(t)V'cV and s u p t c v l (CmXl

I.

y ( t o ) l s C and y < C ( T , L ) .

Now i t r e m a i n s t o show t h a t yCX

I f T i s r e p l e t e , t h e n C(T,E,c)

(f)

c l e a r t h a t C(T,F)= U x

X

and a s X

t EV

.

is ultrabornological.

i s a Banach s p a c e w e need o n l y show

t h a t a n a b s o l u t e l y convex a b s o r b i n g s e t

U C C(T,X,c)

0 i f and o n l y i f UnXx i s a n e i g h b o r h o o d o f O i n X

X

i s a neighborhood o f

f o r a l l n o n n e g a t i v e func-

Using ( a ) , t h i s f o l l o w s i f f U i s b o r n i v o r o u s .

t i o n s xcC(T,L).

It is

Hint:

But C ( T , I , c )

i s b o r n o l o g i c a l whenever T i s r e p l e t e s o t h a t t h e b o r n i v o r e U must b e a

n e i g h b o r h o o d of 0 . 2.2

A t o p o l o g i c a l s p a c e T i s a k - s p a c e i f a s e t U C T i s open

k-Spaces

whenever U n G i s open i n G f o r e a c h compact s u b s e t G o f T .

C l e a r l y any

f u n c t i o n x mapping a k - s p a c e T i n t o a t o p o l o g i c a l s p a c e Z i s c o n t i n u o u s whenever e a c h o f i t s r e s t r i c t i o n s t o a compact s u b s e t o f T i s c o n t i n u o u s .

A s p a c e T h a v i n g t h e p r o p e r t y t h a t a Z - v a l u e d f u n c t i o n d e f i n e d on T i s cont i n u o u s whenever e a c h r e s t r i c t i o n t o a compact s u b s e t i s c o n t i n u o u s i s r e f e r r e d t o a s a k - s p a c e ( c f . t h e d e f i n i t i o n of a k - s p a c e i n S e c . 2 . 2 ) . -Z R Thus i f T i s a k - s p a c e t h e n i t i s a k - s p a c e f o r e a c h t o p o l o g i c a l s p a c e Z . Z The c o n v e r s e i s a l s o t r u e : (a)

k

M

FOR ALL Z

k

A t o p o l o g i c a l space T is a k-space i f f it is a

L.

k -space f o r a l l t o p o l o g i c a l spaces Z . Hint f o r sufficiency: Consider the Z f u n c t i o n I: (T,T) 4 (T,Tk) where -7 i s t h e o r i g i n a l t o p o l o g y o f T and -Yk i s the k-extension topology (defined before (2.3-1)). (b)

LOCAL COMPACTNESS OR 1 s t COUNTABILITY

-.

k

s p a c e s and a l l 1 s t c o u n t a b l e s p a c e s a r e k - s p a c e s .

A l l l o c a l l y compact A s there a r e locally

compact s p a c e s which a r e n o t 1st c o u n t a b l e and 1st c o u n t a b l e s p a c e s which a r e n o t l o c a l l y compact t h e c l a s s o f k - s p a c e s

i s s t r i c t l y smaller than

e i t h e r t h e c l a s s of l o c a l l y compact s p a c e s o r t h e c l a s s o f 1st c o u n t a b l e spaces. Though a k - s p a c e n e e d n o t be l o c a l l y compact i t must have a l o c a l l y corn pact "ancestor": (c)

A k-SPACE I S A QUOTIENT OF A LOCALLY COMPACT SPACE

A topological

s p a c e i s a k - s p a c e i f f i t i s a q u o t i e n t of a l o c a l l y compact s p a c e .

Hint.

N e c e s s i t y : L e t 3 d e n o t e t h e c l a s s o f a l l compact s u b s e t s o f T , G ' = {(t,G)

ItcG] f o r e a c h GEa, and T ' =

(GI I G f d ] .

t o p o l o g y g e n e r a t e d by t h e i n j e c t i o n maps

I f equipped w i t h t h e f i n a l

117

EXERCISES 2

T'

iG:G t

+

(GtY),

(t,G)

TI i s r e f e r r e d t o a s t h e f r e e u n i o n o f

0,

i.e.

i f t h e topology of G i s

t r a n s f e r r e d t o G ' i n t h e n a t u r a l way a s u b s e t o f T ' i s open i f f i t s i n t e r s e c t i o n w i t h e a c h G ' i s open i n G ' and s o T' i s c l e a r l y l o c a l l y compact. The r e l a t i o n

-

d e f i n e d on T' b y : (t,G)

-

(s,H) i f f t = s ,

i s a n e q u i v a l e n c e r e l a t i o n on TI and t h e mapping h:T t

is a bijection.

-+

+

T'fR h ( t ) = [(t,G)

(tcG1

I t r e m a i n s t o show t h a t h i s a homeomorphism when T ' / R t h e f i n a l t o p o l o g y g e n e r a t e d by t h e map

c a r r i e s the q u o t i e n t topology, i.e. p : ~ ' +T ' I R (t,G)

h ( c ) = ( ( t , H ) ItfH].

+

The r e s u l t f o l l o w s from the o b s e r v a t i o n t h a t f o r e a c h U C T , p

u[U'nG' I G € g ] , each G E d

-1

(h(U) =

and t h e f a c t t h a t U i s open i n T i f f U n G i s open i n G f o r

.

Sufficiency:

I f t h e r e e x i s t s a l o c a l l y compact s p a c e S , a n d e q u i v a l e n c e

r e l a t i o n R and a homeomorphism h t a k i n g S / R o n t o T t h e n t h e mapping f:S

s

-

-

S/R Rs

+

4

T

h(Rs)

h a s t h e p r o p e r t y t h a t a s e t U C T i s o p e n i f f f-'(U) p r o v e t h a t T i s a k - s p a c e i t s u f f i c e s t o show t h a t f

UnG i s open i n G f o r compact G C T .

i s open i n S . -1

Thus t o

(U) i s open whenever

S i n c e S i s l o c a l l y compact s o t h a t

t h e r e e x i s t s a c o v e r i n g o f S by r e l a t i v e l y compact open s e t s { V

a

the

&A'

d e s i r e d c o n c l u s i o n f o l l o w s from t h e sequence of o b s e r v a t i o n s : (1)

U n f ( c 1 V ) i s o p e n i n t h e compact s e t f ( c 1 V ) ,

(2)

t h e r e e x i s t s a n open s e t W C T s u c h t h a t U n f ( c 1 V ) = w n f ( c l V ) ,

(3)

f-l(u)nv;f-'(w)nvCy

(d)

SUBSPACES

a

CI

.

a

A subspace o f a k-space need n o t be a k - s p a c e .

cy

Indeed

i f T i s a n y c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e which i s n o t a k - s p a c e , e . g . A where A i s u n c o u n t a b l e ( d i s c u s s e d i n Example 2 . 3 - l ) , t h e n T i s c e r t a i n -

T=W

l y a s u b s p a c e o f the k - s p a c e BT. They a r e a l s o k - s p a c e s .

What a b o u t c l o s e d s u b s p a c e s o f k - s p a c e s ?

2.

118 (e) Hint:

QUOTIENTS

SPACES OF CONTINUOUS FUNCTIONS

A q u o t i e n t s p ace of a k-space i s a k-space.

Use ( a ) .

(f)

( B a g l e y and Young 1 9 6 6 ) .

PRODUCTS

W e a l r e a d y know by Example

2 . 3 - 1 t h a t a n i n f i n i t e p r o d u c t o f k - s p a c e s n e e d n ' t be a k - s p a c e . about f i n i t e products? k

R

What

U n f o r t u n a t e l y a p r o d u c t of two k - s p a c e s need n o t be

e v e n i f one o f t h e s p a c e s i s m e t r i z a b l e . L e t T b e a c o m p l e t e l y r e g u l a r H a u s d o r f f hemicompact k - s p a c e which

(fl)

i s n o t l o c a l l y compact. w h i l e C(T,$,c)X Hint:

Then C(T,R.,c)

i s m e t r i z a b l e (and a k - s p a c e by ( b ) )

T i s n o t a kR-space.

I t s u f f i c e s t o show t h a t t h e e v a l u a t i o n map e : C(T,g,c) X T (x,t>

4

-

,R x(t)

i s c o n t i n u o u s on e a c h compact s u b s e t o f C ( T , R , c ) x T b u t i s n o t c o n t i n u o u s on a l l o f C ( T , R , c ) X

T.

To p r o v e t h e f i r s t a s s e r t i o n i t i s enough t o con-

s i d e r e on compact s u b s e t s o f t h e form F x K where F i s compact i n C ( T , S , c ) and K i s compact i n T. s u b s e t o f C(K,&c) neighborhood U of t

0'

t o ) € Fx K.

i n T such t h a t

It follows t h a t

a l l x€F.

x u ) n FX

F i x (x

A s F IK={x

and t h e r e b y e q u i c o n t i n u o u s , Ix(t)-x(to)

Ix(t)-xo(to)

I


0 there e x i s t s a

1 < €/2

f o r a l l t C U n K and )

€ whenever ( x , t ) € ( ( x o + f / 2 V

PK

K.

To e s t a b l i s h d i s c o n t i n u i t y o f e w e p r o d u c e a n e t ( x , t ) o!

i n C(T,E,c)

cy o!€A

x T c o n v e r g e n t t o ( 0 , t ) (where 0 d e n o t e s t h e f u n c t i o n on T which s e n d s each tCT i n t o

o€&

and t

O=O(t ) .

i s f i x e d i n T) s u c h t h a t x ( t ) = l

As T

acY

i s n o t l o c a l l y compact t h e r e e x i s t s t g g t>g g n c o v e r e d by f i n i t e l y many W ‘ s , . s a y W W , U =.nU serves the degn t 1=1 t , g i 8 81 s i r e d end.

,...,

(f5)

If

^x:

T

4

C(S,Z,c)

i s c o n t i n u o u s and S i s l o c a l l y compact and

Hausdorff then t h e a s s o c i a t e d f u n c t i o n x ( d e f i n e d above) i s a l s o c o n t i n u -

ous.

. t -i n t :

I.

Suppose ( t

01’

s )

a

cy€A

i s a n e t i n TxS c o n v e r g e n t t o a p o i n t ( t , s ) C TxS

and V i s a neighborhood of x ( t , s ) i n Z . eventually i n V.

We s h a l l p r o v e t h a t x ( t , s ) i s

(Yo

A s $ ( t ) € C ( S , Z ) and S i s l o c a l l y compact H a u s d o r f f t h e r e

e x i s t s a r e l a t i v e l y compact n e i g h b o r h o o d W Thus ; ( t ) c

[clWs,V].

Now

2

t

.All

5 B,

(clWs)CV.

i s c o n t i n u o u s s o t h e r e e x i s t s a neighborhood U

o f t i n T s u c h t h a t x”(U ) C [ c l W , , V ] . QFA s u c h t h a t f o r cy

of s such t h a t :(t)

S i n c e ( t ,s )

-t

(t,s)

t there e x i s t s an

l Y ( Y

(t,,s

B 2 a.

B

)C

UsxWs.

Hence x ( t ,s )=;(t

B P

R e t u r n i n g t o t h e p r o o f o f ( f 3 ) we c l a i m t h a t x:TxS

-

B

)(s

B

)cV f o r

Z i s continuous

i s c o n t i n u o u s f o r e a c h compact s e t K C T .

To see t h i s s u p IKxS pose t h a t x i s continuous. Then i s a l s o c o n t i n u o u s by ( i ) . A s T IKXS K i s a k - s p a c e i t f o l l o w s t h a t ^x i s c o n t i n u o u s a n d s o by ( i i ) x i s c o n t i n u o u s .

whenever x

^XI

Thus i t r e m a i n s t o show t h a t x o u s f o r a l l compact G C S .

IK~G

i s c o n t i n u o u s whenevs: x is continulKxS But t h i s f a c t f o l l o w s by e x a c t l y t h e same a r g u -

ment s i n c e K i s l o c a l l y compact H a u s d o r f f and S i s a k - s p a c e (g)

(by p a r t ( b ) ) .

(Noble 1 9 6 7 ) T A k-SPACE 4 u T A k-SPACE I n Example 2 . 2 - 1 i t was e s t a b A , t h e c o l l e c t i o n of a l l non-negative integer-valued functions

lished that

W

d e f i n e d on t h e u n c o u n t a b l e s e t A ,

i s not a k-space.

t h e s e t S c o n s i s t i n g o f a l l e l e m e n t s of

FA which

Moreover w e saw t h a t

v a n i s h a t a l l b u t a t most

2.

120

SPACES OF CONTINUOUS FUNCTIONS

a c o u n t a b l e number o f e l e m e n t s from A i s a k - s p a c e w i t h G S = p . R c l a i m : S i s a k - s p a c e whose r e p l e t i o n i s n o t a k - s p a c e .

H-i n t :

Here w e

To see t h a t S i s a k - s p a c e l e t s b e f i x e d i n S a n d U b e a s e t c o n -

t a i n i n g s which i s n o t a n e i g h b o r h o o d o f s .

K such t h a t C U n K i s n o t clo s ed i n K.

We s h a l l e x h i b i t a compact s e t

S i n c e U i s n o t a neighborhood o f s

t h e r e e x i s t s a n el-ement s €S s u c h t h a t s d U . A s s l , s f S t h e y d i f f e r on a 1 1 lj€,N) ( i f A1 i s f i n i t e a =a f o r some J and a l l c o u n t a b l e s u b s e t A =[a 1 lj Ij 1J t h e n t h e n e i g h b o r h o o d { t F S I t ( a l l ) = s ( a l l ) ) ~ U s o we j 1 J ) . L e t F 1= ( a l l ] ; L e t A2={a (jcx] d e n o t e t h e e l e can f i n d s IU s u c h t h a t s 2 ( a l l ) = s ( a ll). 2 2j Then ments o f A on which s and s d i f f e r and s e t L.',,={aij 11 < i , j 5 21. 2 t h e n e i g h b o r h o o d { t € S I t ( a ) = s ( a ) f o r e a c h aCF2]& U s o t h e r e e x i s t s s I U 3 C o n t i n u i n g by i n d u c t i o n we o b t a i n a s e q u e n c e ( s ) s u c h t h a t s = t on F2. 3 from S a l l of whose e l e m e n t s l i e i n C U a n d , d e n o t i n g t h e c o l l e c t i o n o f e l e L

and s d i f f e r by A ={a . l j c l ] , s =s on F = n nJ n+ 1 i , j 5 n). S e t C = U F n . I f akC t h e n s ( a ) = s ( a ) f o r e a c h n s o t h a t ( a , . 11 1J I f a€C t h e r e e x i s t s m > 0 s u c h t h a t a € F f o r e a c h n 2 m. sn(a) + s(a).

ments of A o n which s

-

( a ) = s ( a ) whenever n 2 m and s ( a ) 4 s ( a ) . T h e r e f o r e s s i n S, n+ 1 K={s,\n€Z]fl{s) i s compact and C U f l K = { s IncbJ] i s n o t c l o s e d i n K a s s k C U n K . Thus s

ASCOLI THEOREMS

(h)

( K e l l e y 1 9 5 5 , pp. 223-249;

B a g l e y & Young 19bb)

The p o i n t o f t h i s e x e r c i s e i s t o o b t a i n g e n e r a l i z a t i o n s

[ ( i i i ) and ( i v )

below] o f A s c o l i ' s Theorem a s p r e s e n t e d i n K e l l e y 1955 (Theorem 2 1 , p . 2 3 b ) , a s w e l l a s t h e v e r s i o n i n K e l l e y ' s Theorem 7 . 1 7

(p. 233).

I f F i s a f a m i l y o f maps from a t o p o l o g i c a l s p a c e S i n t o a t o p o l o g i c a l s p a c e T , t h e n any t o p o l o d y . 7 f o r F which makes tFic e v a l u a t i o n map e s e n d i n g (x,s)

i n t o x ( s ) f r o m FxS i n t o T is c a l l e d j o i n t l y c o n t i n i c j g s .

Two r e s u l t s

on j o i n t c o c t i n u i t y a r e needed f o r t h e A s c o l i Theorems. (i)

I f S and T a r e H a u s d o r f f s p a c e s , F C C ( S , T ) , z

which i s f i n e r t h a n t h e compact-open t o p o l o g y a k-space,

then

n

d

-7 and

is a topology f o r F

which makes (F,$)xS

i s j o i n t l y continuous.

L e t C b e a c l o s e ( : Yubset o f T , K a compact s u b s e t o f ( F J ) x S , -1 and ( x , s ) b e a p o i n t o u t s i d e of M=Kfle ( C ) . I f ( x , s ) !?K t h e n o b v i o u s l y -1 ( x , s ) t! c l M. Suppose (x,s)CK and ( x , s ) d e ( C ) . L e r U=T-C a n d l e t K (Hint:

S

be t h e p r o j e c t i o n o f K i n t o S .

relative to K

T h e r e i s a compact n e i g h b o r h o o d N o f S

s u c h c h a t x ( N ) C U and e([N,U]xN)C U w h e r e [ N , U ] = -1 {y€F Iy(N)CU)€JCC-J. Thus([N,U]xNfle (C)=0. It f o l l o w s t h a t ( x , s ) i s S

n o t i n t h e c l o s u r e of M r e l a t i v e t o ( F , d ) x K S . follows t h a t ( x , s ) k c l M. proof is complete.)

But s i n c e K - K C F x K it S' -1 S i n c e FxS i s a k - s p a c e , e (c) is and t h e

EXERCISES 2

i s l o c a l l y compact, S a Hausdorff k- space,

I f (F,.Tc)

(ii)

12 1

and T a

i s j o i n t l y continuous.

Hausdorff space then

Use ( f 3 ) and ( i ) a b o v e . )

(Hint:

Using ( i i )

,

t h e g e n e r a l i z a t i o n s o f K e l l e y ' s A s c o l i Theorems m e n t i o n e u

a b o v e ( K e l l e y ' s Theorem 7 . 1 7 and 7 . 2 1 r e s p e c t i v e l y ) may now b e o b t a i n e d u s i n g t h e same p r o o f s t h a t a r e i n K e l l e y . ASCOLI THEOREM

(iii)

Let S be a Hausdorff k-space,

T a Hausdorff uni-

f o r m s p a c e a n d FC-C(S,T).

Then ( F Z ) i s compact i f f ( I ) ( F , d ) i s c l o s e d

i n C(S,T,c) where C(S,T,c)

d e n o t e s C(S,T) w i t h compact-open t o p o l o g y , ( 2 )

F ( s ) h a s compact c l o s u r e f o r e a c h s c S and ( 3 ) F i s e q u i c o n t i n u o u s . (iv)

ASCOLI THEOREM

d o r f f s p a c t and F C C ( S , T ) . C(S,T,c),

L e t S be a H a u s d o r f f k - s p a c e , T a r e g u l a r Haus( F , Z c ) i s compacr i f f (1) (F,$)

is closed i n

( 2 ) F ( s ) h a s compact c l o s u r , : f o r e a c h S C S and (3) F i s e v e n l y

continuous.

( T o s a y t h a t F i s e v e n l y c o n t i n u o u s means t h a t f o r e a c h s € S ,

e a c h t € T , and e a c h n e i g h b o r h o o d U o f t t h e r e i s a n e i g h b o r h o o d V o f s and a n e i g h b o r h o o d Y of t s u c h t h a t x ( V ) c U whenever x(s)(W.)

2.3

k-SPACES AND FULL COMPLETENESS OF C(T,J,,c)

ment f o l l o w i n g (4.12-8).) C(T,E,c)

( P t a k 1953;

s e e a l s o corn-

L e t T be a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e .

i s c o m p l e t e i f f T i s a k - s p a c e , whenever T i s a hemicompact s p a c e T h i s e q u i v a l e n c e i s n o t g e n e r a l f o r , a s we h a v e s e e n i n Example

((2.3-4)).

t h e r e a r e c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e s which a r e n o t k - s p a c e s

2.3-1,

b u t f o r w h i c h C(T,X,c)

i s complete.

w i l l f o r c e T t o be a k-space?

What c o n d i t i o n

on t h e LCHS C(T,_F,c)

P t a k (1953) h a s shown t h a t T i s a k - s p a c e

whenever C ( T , I , c ) i s f u l l y c o m p l e t e .

Unfortunately t h e converse i s n o t

R e c a l l t h a t a LCHS X i s f u l l y c o m p l e t e i f e v e r y c o n t i n u o u s l i n e a r

true.

t r a n s f o r m a t i o n A t a k i n g X o n t o a LCHS Y w h i c h i s a l m o s t o p e n ( i . e .

c l A(V)

i s a n e i g h b o r h o o d o f 0 i n Y f o r e a c h n e i g h b o r h o o d V o f 0 i n X) i s a n o p e n map. (a)

I f C(T,F,c) (i)

i s f u l l y complete then T i s a k-space

The image o f a f u l l y c o m p l e t e s p a c e u n d e r a t o p o l o g i c a l homomor-

phism ( i . e .

a c o n t i n u o u s and open l i n e a r map) i s f u l l y c o m p l e t e .

(Suppose

t h e t o p o l o g i c a l homomorphism A t a k e s t h e f u l l y c o m p l e t e LCHS X o n t o t h e LCHS Y and l e t B be a n a l m o s t open c o n t i n u o u s l i n e a r map from Y o n t o t h e

i s a neighborhood o f

LCHS Z.

I f U i s a n e i g h b o r h o o d o f 0 i n Y t h e n A-'(U)

0 i n X.

T h e r e f o r e , b e c a u s e X i s f u l l y c o m p l e t e and B - A i s a l m o s t open,

B(u)=(B.A)(A-'(u)) (ii)

i s o p e n i n 2.)

I f S i s a c l o s e d s u b s e t o f T and C ( ' J , F , c )

s o i s C(S,F,c).

F u r t h e r m o r e i f C(T,,F,c)

i s f u l l y complete, then

i s f u l l y c o m p l e t e , t h e n T i s normal.

122

2.

SPACES OF CONTINUOUS FUNCTIONS

C o n s i d e r t h e r e s t r i c t i o n map R: C ( T , E , c )

-

C(S,&c)

: XIS

X

R is clearly linear.

Indeed,

C(S,x,c).

Next we c l a i m t h a t t h e r a n g e o f R i s d e n s e i n

i f ycC(S,T) and K C S i s compact t h e n y l K h a s a c o n t i n u -

d e f i n e d o n a l l o f T.

ous e x t e n s i o n

n e i g h b o r h o o d o f y, s a y y+€V e l e m e n t o f t h e r a n g e o f R.

Thus 9(,-y

IK

and s o i n a n y b a s i c

p K ( z ) < F],

={y+zIzFC(S,i),

t h e r e e x i s t s an

pK

It o n l y r e m a i n s t o show t h a t R i s a t o p o l o g i c a l homomorphism f o r t h e n ,

by ( i ) , t h e r a n g e o f R i s

As R i s c e r t a i n l y continuous

f u l l y complete.

o u r c o n c e r n i s t o e s t a b l i s h o p e n n e s s o f R.

But C ( T , F , c ) i s f u l l y c o m p l e t e

s o i t s u f f i c e s t o show t h a t R i s a l m o s t open.

L e t K b e compact i n T and

c o n s i d e r t h e s e t c l R(V ). I n t h e e v e n t t h a t K n S # @ w e c o n t e n d t h a t PK c l R(Vp )2CycC(S,x) \ p K n s ( y ) 5 1/21 so t h a t c l R(V ) i s a n e i g h b o r h o o d o f Suppose t h a t y € C ( S , g ) s u c h t h a t p p K ( y ) 5 1/2 and H C S i s K nS Now by w h a t h a s a l r e a d y b e e n shown, t h e r e e x i s t s ucC(T,E) s u c h

0 i n C(E,E,c).

compact.

t h a t u a g r e e s w i t h y on ( K n S ) U H . j o i n t from H. S e t x I = z u and p o s e t h a t tcW.

Choose z f C ( T , E ) s u c h t h a t 0 XI

Is=x.

z(K)-{O],

-


5 1, z(H)={l],

l x ' ( t ) !=O 5 1.

13 i s d i s -

and z(W)={O].

I f t C K then there are two possibilities.

Then z ( t ) = O and

I I z ( t ) I < 1.

lu(t)

yCc1 R(V ). pK I f KnS-g

Thus t h e s e t W-EtCK

F i r s t sup-

I f tfW t h e n I x ' ( t )

A s x a g r e e s w i t h y o n H ( s i n c e z(H)-{l]

I=

and u=y o n H),

t h e n we may c h o o s e a f u n c t i o n zfC(T,_F) s u c h t h a t 0 5 2 5 1,

and z ( S ) - { l ] .

Now i f y i s any e l e m e n t o f C ( S , x ) and H C S i s com-

and p a c t t h e r e e x i s t s uFC(T,g) w i t h t h e p r o p e r t y t h a t u I =y t h e n X ' = Z U F V H PK X-X' a g r e e s w i t h y o n H. Thus c l R(V )=C(S,_F) i s a n e i g h b o r h o o d o f 0 i n pK C(S,_F). Hence R i s a t o p o l o g i c a l homomorphism o n t o a f u l l y c o m p l e t e d e n s e sub-

1,

s p a c e o f C(S,E) s o t h a t R i s , i n f a c t , s u r j e c t i v e , and C(S,_F) i s f u l l y complete.

Moreover, b y t h e o n t o n e s s o f R, e a c h c o n t i n u o u s r e a l - v a l u e d

t i o n d e f i n e d o n S h a s a c o n t i n u o u s e x t e n s i o n t o a l l o f T. i s normal b y t h e T i e t z e e x t e n s i o n (iii)

func-

Consequently T

theorem.

I f C(T,X,c) i s f u l l y c o m p l e t e t h e n T i s a k - s p a c e

E C T and E n K i s c l o s e d f o r e a c h compact K C T .

Suppose t h a t

Show t h a t SIC1 EkE.

To

t h i s end c o n s i d e r t h e t o p o l o g y o n C ( S , i ) g e n e r a t e d by t h e c o l l e c t i o n o f seminorms p K n E ( . ) = s u p I ( . ) ( K n E )

I

a s K r u n s t h r o u g h t h e compact s u b s e t s o f

EXERCISES 2

123

S f o r w h i c h K n E f f l and l e t C(S,_F,c ) d e n o t e C ( S , z ) w i t h t h i s t o p o l o g y .

see t h a t t h i s s p a c e i s H a u s d o r f f , l e t x ~ c ( S , s ) ,x#o.

To

Then t h e r e e x i s t s s C S

s u c h t h a t x ( s ) # O and s o t h e r e e x i s t s a n e i g h b o r h o o d V s o f s i n S i n w h i c h Now E i s d e n s e i n S s o E f l V s # @ and p r t ] ( x ) # 0 f o r a n y

x never vanishes.

trEnVs. A s t h e compact-open t o p o l o g y i s f i n e r t h a n t h e c

t o p o l o g y o f C(S,E)

t h e b i j e c t i v e mapping

I : C (S

,F, c )+C

( S ,F, co )

-.,

x-x

i s continuous.

-

Since C(S,F,c)

i s f u l l y c o m p l e t e ( b y ( i i ) ) i t i s o n l y nec-

e s s a r y t o show t h a t I i s a l m o s t o p e n i n o r d e r t o p r o v e t h a t t h e compacto p e n and c

topologies coincide.

To see t h i s show t h a t c l V

(where c l

O P

d e n o t e s cl:sure

i n the c

e a c h compact K C S .

topology) i s a c -neighborhood o f

tk

I n d e e d , i f K n E # @ we c l a i m t h a t 1 / 2 V

o r i g i n fz-r

C C l V

.

P K ~ E O PK and s u p p o s e t h a t H C S i s compact and H n E # f l . Using t h e

L e t x( 1 / 2 ~ P p r o c e d u r e o u l f l $ e d below, show t h a t t h e r e e x i s t s a f u n c t i o n x'cC(S,X)

such

t h a t x ' a g r e e s w i t h x o n H n E and p ( x ' ) c 1 t o e s t a b l i s h t h e d e s i r e d i n K clusion. L e t W = { t F K I l x ( t ) l > 11; n o t e t h a t W i s compact a n d , s i n c e s u p l x ( K n E ) l < 1/2, H n E i s a l s o compact s o t h e r e e x i s t s zqC(S,x) s u c h t h a r

0 5 z 5 1, z ( H n E ) = { l ) and z(W)=(O]. so x ' ( t ) = x ( t )

S e t x'=x.z.

and x ' and x a g r e e o n H n E .

t h e n z ( t ) = O and t h e r e f o r e I x ' ( t ) l = O < 1. I z ( t ) ( < l , l x ' ( t ) l < 1.

I f t c H n E then z ( t ) = l

Next suppose t h a t t f K .

I f tFW

I f t # W t h e n I x ( t ) l S 1 and, s i n c e

i s a c -neighborhood o f 0 p r o v i d e d K n E K ' What i f K n E = @ ? Then by u s i n g a n a r g u m e n t s i m i l a r t h e one a b o v e show

#@.

Thus c l V

i s c - n e i g h b o r h o o d o f 0. PK 0 Thus, s i n c e C ( S , x , c ) i s f u l l y c o m p l e t e , I i s a t o p o l o g i c a l i s o m o r p h i s m and that cl V

'K

=C(S,x) so t h a t i n any e v e n t c l V 0

t h e compact-open and c - t o p o l o g i e s a g r e e o n C ( S , x ) . Now show t h a t E=S.

I f tFS-E t h e n t h e compact-open n e i g h b o r h o o d V

b e i n g a l s o a c -neighborhood o f t h e o r i g i n c o n t a i n s a neighborhood

V Pr t i

,

p ~E n w h e r e K i s compact i n T and KnE#fl. S i n c e K n E i s c l o s e d and t # K n E t h e r e e x i s t s yfC(S,_F) s u c h t h a t 0 w h i l e y#V

YCV

k- s 'Kfl p a c e .E

c

-

a contradiction.

Hence

I t f o l l o w s t h a t S=E and T i s a

T h e r e a r e ( c o m p l e t e l y r e g u l a r H a u s d o r f f ) k - s p a c e s T f o r which

(b)

w

PM

5 y 5 1, y ( t ) = l , and y ( K f l E ) = ( O ] .

) i s n o t f u l l y complete.

Hint:

C o n s i d e r t h e o r d i n a l s p a c e s [O,tu]

and [0,0]

i n f i n i t e o r d i n a l and 61 t h e f i r s t u n c o u n t a b l e o r d i n a l .

where w i s t h e f i r s t Then

[o,'J]x[o,d

is

124

2.

SPACES OF CONTINUOUS FUNCTIONS

-

a compact Hausdorff s p a c e and t h e open s u b s e t T=[O,w]X[O,n] l o c a l l y compact c o m p l e t e l y r e g u l a r Haiisdorff space.

is a

On t h e o t h e r hand T i s n o t normal ( s e e Dugundji 1966,

by E x e r c i s e 2.2(b). p.

{(w,")]

Hence T i s a k-space

145, Ex. 4 ) s o , by ( i i ) o f p a r t ( a ) , C(T,F,,c) i s n o t f u l l y complete. N

2.4

C(T,R)' AND BAIRE MEASURES

Throughout t h i s e x e r c i s e we t a k e T t o be

a c o m p l e t e l y r e g u l a r Hausdorff space.

We know from (2.4-4)

i t i v e l i n e a r f u n c t i o n h on C (T,E) h a s t h e form h ( . ) = J ( . ) b r e g u l a r a d d i t i v e n o n - n e g a t i v e s e t f u n c t i o n d e f i n e d on

t h a t each pos-

dp where

aZ,t h e

g e n e r a t e d by any

Z , the

R e c a l l how p was d e f i n e d :

z e r o s e t s o f T.

0 5 x 5 k cz

262, p(Z)=sup{h(x) Ixc Cb(T,F.),

IJ

is a

algebra for

] and f o r a r b i t r a r y A F R T ) ,

I n ( a ) below we show t h a t p i s , i n f a c t ,

P(A)=inf[p(CZ) I Z C Z , A C C 21.

c o u n t a b l y a d d i t i v e on t h e a - a l g e b r a

d3

generated b y Z ( t h e Baire s e t s )

and t h a t , i f h i s o b t a i n e d a s t h e r e s t r i c t i o n o f a p o s i t i v e l i n e a r funct i o n a l on C(T,E) t h e r e p r e s e n t a t i o n a s an i n t e g r a l g i v e n above h o l d s f o r a l l xCC(T,B). (a)

POSITIVE LINEAR FUNCTIONS ON C(T,fi.) AND BAIRE MEASURES

Let h be a

p o s i t i v e l i n e a r f u n c t i o n on C(T,g) and P be t h e s e t f u n c t i o n d e f i n e d above. Then

P i s a r e g u l a r measure on @ and

-

h(x)

x dk

For each xcC(T,&) t h e r e e x i s t s E C @

(xCC(T,S)).

such t h a t x i s bounded on E and P(CE)

4.

L e t P d e n o t e t h e c o l l e c t i o n o f complements o f z e r o s e t s .

Sketch o f Proof F i r s t show t h a t (i)

f o r each

I.L(G)-q c b ( H ) :

G c P and

C

>

0 t h e r e e x i s t s H q P such t h a t c l H C G and

such t h a t 0 5 w 5 k

Choose w€Cb(T,P,)

Now H = ~ - ~ ( C / 2 , 1 ] d pby (1.2-3) = h(w-(c/2)1)

h(w)-(/2

and c l H C G .

G

and h(w) > w(G)-C/2.

I f W(H) 5 ~ ( G ) - c t h e n

5 h((w-(C/2)1)V 0 ) _< P(H)

5 k(G1-c c h ( w ) - r / 2 -a contradiction. To e s t a b l i s h c o u n t a b l e s u b a d d i t i v i t y on

i s a sequence from

P such

P prove:

f o r a l l n and n+l Gn=O t h e n l i m n p(Gn)=O: Consider f i r s t t h e s i t u a t i o n where r\ c l ( G )=@. n As p i s monotone on (p(Gn)) i s a d e c r e a s i n g sequence o f r e a l numbers; ( i i ) i f (G,)

t h a t Gn>G

n

aZ,

a s such i t p o s s e s s e s a l i m i t a

2

0.

I f we assume a t o be p o s i t i v e t h e n f o r

e a c h n > 0, t h e r e e x i s t s x CC (T,E) such t h a t 0 5 x 5 kG and h ( x n ) > a/2. n b Since 4, each t f T c a n belong t o o n l y a n f i n i t e number of n

Let x=C x n n

.

nG

EXERCISES 2

125

s o t h a t x i s a r e a l - v a l u e d n o n - n e g a t i v e f u n c t i o n on T. Moren’ B u t Un o v e r , s i n c e n c l Gn=@, e a c h t c T b e l o n g s t o U = C ( c l G ) f o r some n. n i s open and o n l y a f i n i t e number o f t h e x k ’ s a r e n o n - z e r o on U s o x i s the sets G

Applying h t o x we o b t a i n t h e c o n t r a d i c t i o n :

continuous a t t for each t€T.

Next c o n s i d e r a s e q u e n c e (G ) f o r w h i c h

nG 4.

For any g i v e n c: > 0, by

( i ) , t h e r e e x i s t s V Fk’ s u c h t h a t c l ( V ) C G and ;L(G ) - f / 2 ” < u ( V n ) f o r n n n n > 0. Thus i f we s e t H =V and i n d u c t i v e l y d e f i n e H n = V n n Hn-l i t f o l l o w s

1

1

t h a t c l H n C G n and H 3 H n + l n the observation:

f o r each n

Gn-HnC

u is

and t h e f a c t t h a t

/

0.

Furthermore, w i t h t h e a i d of

( n > 0)

(Gn-Vn)lJ(G,-l-Hn-l)

s u b a d d i t i v e on

a

( s i n c e i t i s a d d i t i v e and mono-

k

1 / 2 < p(Hn) n H n C nGn=@, s o n t h a t lim,u(k;j=O, i t fol-

t o n e on a z ) , i t i s r e a d i l y e s t a b l i s h e d t h a t u(G ) c u(H )+C

As

+C f o r e a c h p o s i t i v e n.

ncl

Hence l i m P(G )=O. lows t h a t l i m p(G ) 5 C. n n n n ( i i i ) li. i s c o u n t a b l y s u b a d d i t i v e o n P , i . e . t h e n P ( U G n ) 5 rnG(Gn):

if G

cp

F i r s t we claim t h a t f o r each G

for each n

0

t h e r e e x i s t s a se-

.

) o f z e r o s e t s such t h a t C = 2 I n d e e d , by ( 1 . 2 - 1 ( d ) t h e r e nm n m nm -1 (0,1]. Then c h o o s i n g e x i s t s a 0 _< y < 1 from C(T,R) s u c h t h a t G =y ry n n -1 n [l/m,l] - a z e r o s e t b y ( 1 . 2 - 3 ) - t h e c l a i m f o l l o w s . Now G = U n G n Zn,m=yn W where e a c h W e q u a l s some Z Set M = Wm. S i n c e G F p b y m m nm’ m z k ( 1 . 2 - 1 ( e ) ) and e a c h % xCz by ( 1 . 2 - 1 ( ~ ) ) t h e s e q u e n c e (GnCMk) s a t i s f i e s t h e

quence (2

u

=u

hypothesis of ( i i ) . )

;(GoC%

< F.

Thus f o r a n v p o s i t i v e

But

y”,

f

there exists k

i s c o n t a i n e d i n some f i n i t e u n i o n

0

that

u1G) 5 (iv)

U.

u(Un ~

NGn) -+ @ ( G n C M k )

+

c.

L e t A=CJ A and c h o o s e n n G and n o t e t h a t S e t G= f o r e a c h n > 0. n n

u

Thus w(A) 5 u(G)

I n summary t h e n ,(@)=O;

I.L(G,)

G so L l n i N n

i s c o u n t a b l y s u b a d d i t i v e onb7(T):

GnX An s u c h k(Gn) < b(An)+C/Zn G3A.

in-

0

such t h a t

i.e.

-.Zn w(An)

+ c.

w i s n o n - n e g a t i v e , monotone, c o u n t a b l y s u b a d d i t i v e and

+ i s a n o u t e r m e a s u r e on P ( T ) .

measure s e t s , i.e. each A € P ( T ) ,

1” ?(Gn)

t h e s e t s EO’(T)

Hence t h e c o l l e c t i o n o f

u-

f o r w h i c h u(A) L u ( A n E ) + U ( A n C E ) f o r

form a a - a l g e b r a of s u b s e t s o f T on w h i c h u i s c o u n t a b l y

126

2.

SPACES OF CONTINUOUS FUNCTIONS

a d d i t i v e ( s e e Dunford and S c h w a r t z 1958, 111 5 . 4 , p. 1 3 4 ) .

u i s a r e g u l a r measure on 6

(v)

a measure o n ACT, U

C

a

-

To see t h a t u i s

the Baire s e t s :

i t i s enough t o show t h a t e a c h GC?

i s u,-measurable.

> 0, and c h o o s e HcP s u c h t h a t H 3 A and w(A) > ,,,(H)-c.

Let

Then, s i n c e

i s a d d i t i v e on Q, and monotone on R T ) , p(A)

+C

9

p(H)

-

and t h e r e s u l t f o l l o w s .

u(HnG)

+

p ( H n C G ) _" IL(AI)G)

+ u(AnCG)

A s f o r r e g u l a r i t y , given

and H Z C A s u c h t h a t u,(G)-U(A) < C/2 and p(H)-!J(CA)

there e x i s t s GJA

< C/2.

But H-CA=A-CH

so t h a t

-

W(A)

u(CH) = b(A-CH) = u(H-CA)

m

w(H)

-

k(CA)


0. Suppose t h a t Cl(x-l[a b])=p(T): -1 It f o l l o w s t h e n t h a t a sequence o f r e a l ~ ( x[ O , r ] ) < p ( T ) f o r a l l r > 0. numbers O < a

1

< b

1

< a

2

< b

2

0 w e d e f i n e gcC(&,&) a s f o l l o w s : n n

12 7

EXERCISES 2

f o r e a c h n 2 0.

This c o n t r a d i c t i o n e s t a b l i s h e s the r e s u l t .

A s a consequence of ( v i ) w e s e e t h a t t h e f u n c t i o n a l x d^h(x) =

J"

i s r e a l - v a l u e d on C(T,E).

>

(vi), there exists r xrimin(x,r). h(x)

2

x dp Furthermore suppose t h a t x

0 such t h a t u (x

C l e a r l y 'k(x)=%(xr).

-1

[O,r])=u,(T)

2 0.

Then, a g a i n b y

A df 6 j t

s o t h a t h ( x ) = xdpr

Let

On t h e o t h e r hand x r F C b ( T , l l ~ - ~ 6

h

h(xr)=J" x r d u A ( x r ) = h ( x ) .

We may t h e r e f o r e c o n c l u d e t h a t h-$

p o s i t i v e l i n e a r f u n c t i o n a l on C(T,E) w h i c h v a n i s h e s on C (T,R). b f i n i s h t h e p r o o f o f ( a ) we need o n l y p r o v e t h a t (vii)

a positive linear functional x

Cb(T,R) v a n i s h e s i d e n t i c a l l y o n C(T,&):

>k

is a

Thus t o

on C(T,&) which v a n i s h e s o n

S i n c e e a c h xcC(T,E) may b e ex-

p r e s s e d i n t h e form x = ( x V O ) - ( ( - x ) V 0 ) i t i s enough t o p r o v e t h a t x*(x)=O whenever x > 0.

I f xCC (T,&) t h e r e i s n o t h i n g t o show, s o l e t u s assume

b

t h a t x i s unbounded o n T.

Choose a n i n c r e a s i n g s e q u e n c e ( a ) of r e a l n n 2 1 numbers from x ( T ) and l e t t C T b e s u c h t h a t x ( t )=a f o r e a c h n z 0. Next n n d e f i n e t h e bounded f u n c t i o n s

w h e r e a PO and a

0

3

then \(t)=a

"€3.

Each s u c h f u n c t i o n i s c o n t i n u o u s on T.

k k-1 Thus, f o r any s u c h t , x ( t ) = $

xk(t).

and x ( t ) = O f o r k > n. k I f x ( t ) = O t h e n x,(t)=O f o r a l l k s o

t h a t once a g a i n x ( t ) = F x ( t ) . Therefore k k T. Next s e t y =a x f o r e a c h p o s i t i v e n.

n

n

t h e open s e t xnl[O,al) x

-1

[O,al).

I f x(t)€(an-l,

f o r k < n, x ( t ) = x ( t ) - a n - l ,

-a

xk c o n v e r s e s p o i n t w i s e t o x on C l e a r l y En y n ( s ) = O f o r a l l s i n

s o t h a t y ( s ) = z n y n ( s ) e x i s t s and i s c o n t i n u o u s on

For s c x - l (an-l,an+l)

f o r some n 2 1 w e s e e t h a t

128

2.

%

%Yk(')

=

%

+

ak(ak-ak-l)

-1

+ a n ( x ( t ) - an-l)

ak(ak-ak-l)

n-l

I'

SPACES OF CONTINUOUS FUNCTIONS

an+l(x(t)

f o r scx

-1

-

an) for sex

s o t h a t y ( s ) = s y k ( s ) e x i s t s and i s c o n t i n u o u s o n x

- %5

n 'k

Ck

a

ck

> n yk

>

-1

(an-

1 9

an]

[an,an+,)

(an- 1' an++

n a k xk2% > n a n Xk

Now

an(X

-

cklnyk)

a n d , a s x" v a n i s h e s o n C ( T , B ) , b $c

x ( y ) = x"(Y

-

-

>

- 3

a n x9'(x

-L

xk) = an x"(x)

~

Thus x''(x)=O.

f o r e a c h p o s i t i v e n. I n (2.4-2)

y,)

and ( 2 . 4 - 3 )

we saw t h a t a l i n e a r f u n c t i o n a l on C (T,&) i s b c o n t i n u o u s i n t h e u n i f o r m norm i f f i t i s t h e d i f f e r e n c e o f two p o s i t i v e linear functionals.

C l o s e s c r u t i n y o f t h a t r e s u l t r e v e a l s t h a t we a c t u a l l y

proved t h a t a l i n e a r f u n c t i o n a l i s order-bounded

(i.e.

i t sends sets of the

x ] = f x Ixl 5 x 5 x ] i n t o bounded s e t s o f numbers) i f f i t i s a d i f 1' 2 2 f e r e n c e o f two p o s i t i v e l i n e a r f u n c t i o n a l s . The same p r o o f may b e u s e d t o

form [x

o b t a i n t h e c o r r e s p o n d i n g r e s u l t f o r l i n e a r f u n c t i o n a l s on C(T,&). (b)

A l i n e a r f u n c t i o n a l x* o n C(T,$)

ORDER-BOUNDED LINEAR FUNCTIONALS 9:

9


F

0, x

+C

is continuous.

Of t h e homomorphisms o f C(T,R)

ItcT] a r e c o n t i n u o u s i n t h e compact-open

Any homomorphism f i x e d t o a p o i n t o f uT-T, i . e . 9
JcGK xK du. =l. On t h e o t h e r hand i f P PP t h e compact s u b s e t s o f T a r e o r d e r e d by s e t i n c l u s i o n , ( x ~ i)s ~a n e t i n C(T,&,c) c l e a r l y converges t o t h e z e r o f u n c t i o n , t h e r e b y c o n t r a d i c t i n g t h e c o n t i n u i t y of x'

P'

L e t t i n g K=K UKn i t i s s e e n t h a t ~1 (K)=p (T) and un(K)= P P P

Wn(T). A t t h i s point i t i s c l e a r t h a t it s u f f i c e s t o consider p o s i t i v e

XI,

i.e

.

I and P=p The n e x t t h i n g t o do i s t o t r a n s f e r x ' o v e r t o C(K;&). P P Each xcC(K,F.) h a s a c o n t i n u o u s e x t e n s i o n t o T ( e x t e n d x t o @T and t h e n re-

x'rx

s t r i c t i t t o T) s o t h a t i t i s n a t u r a l t o a t t e m p t t o d e f i n e k ( x ) = x ' ( y ) where y i s some c o n t i n u o u s e x t e n s i o n o f x up t o T.

To s e e t h a t t h i s d e f i -

n i t i o n i s meaningful we need o n l y show t h a t x f ( z ) 5 0 f o r any zfC(T,E) which v a n i s h e s on K.

I f z i s such a f u n c t i o n t h e n G=z

Thus p(T-G )=O,

p(G )=u(K) and t h e r e f o r e

c

f o r each F

-1

(-c,c)cp

and c o n t a i n s K.

F

> 0.

Now r(' i s c l e a r l y p o s i t i v e s o t h e r e e x i s t s a n o u t e r measure

L4


such t h a t 0 an

€1.

S i n c e c l CH and F a r e d i s j o i n t i n BT t h e r e e x i s t s uhC(T,E) s u c h t h a t

P 0 5 u _< 1, u(F)={1] and u(CH)={O]. t o T, x"=u

y'

and $ IK=x.

-

uK(G)

f

any c o n t i n u o u s e x t e n s i o n o f y

It f o l l o w s t h a t y 5 x

_< x K ' ( y ) 5 x K ' ( X = x'(2)

< As € i s a r b i t r a r y c1 ( G )

K such t h a t J f l K = G .

JEP

9 be

Let

p(H)

_< k ( G ) .

+

+

= XK'(x)

C)

+c

+

c and 0 5 x 5 k

+

H'

Thus

F PK(K)

UK(X)

E rq s u c h thaL x 5 y i m p l i e s $ ( x ) homeomorphic

5

$ ( y ) ) t h e n T and S a r e

(Ka>lansky 1947).

S i n c e t h e d i s t r i b u t i v e f u n c t i o n l a t t i c e s o f t h e p r e v i o u s example d i f f e r s o d r a s t i c a l l y i n a p p e a r a n c e f r o m t h e s e t l a t t i c e s g i v e n i n Examples 3.1-1 a n d 3.1-2

i t i s p e r h a p s s u r p i i s i n g t o f i n d t h a t any l a t t i c e c a n b e " r e p r e -

s e n t e d " by a s e t l a t t i c e .

More p r e c i s e l y , i f L i s a l a t t i c e t h e n t h e r e

e x i s t s a s e t T a n d a c o l l e c t i o n o f i t s s u b s e t s which,when s u p p l i - d w i t h t h e o r d e r i n g d e f i n e d by s e t i n c l u s i o n , i s o r d e r - i s o m o r p h i c t o L. t a k i n g T'L

I n d e e d by

a n d 1: t o be t h e c o l l e c t i o n o f a l l s u b s e t s o f T o f t h e f o r m

La = {b c L i b 5 a 3

( a F L)

140

3.

WALIMAN COMPACTIFICATIONS

t h e mapping a -, La i s s e e n t o b e a n orc!c:r-isomorphism b e t w e e n L and

e.

Even more c a n b e s a i d - i f L i s a d i s t r i b u t i v e l a t t i c e t h e n L i s o r d e r i s o m o r p h i c t o a s e t l a t t i c e i n w h i c h t h e m e e t a n d j o i n of a p a i r of s e t s a r e given by p.

n

and U

r e s p e c t i v e l y ( N o b e l i n g 1 9 5 4 , p . 2 8 , a n d Hermes 1955,

lob).

The l a t t i c e o f t h e f o l l o w i n g example i s q u i t e s i m i l a r i n form t o t h e l a t t i c e of c l o s e d s e t s m e n t i o n e d i n Example 3 . 1 - 2 .

After linking l a t t i c e s

and c o m p a c t i f i c a t i o n s , i t w i ' l b e s e e n (Example 3 . 3 - 3 )

,.hat t h e l a t t i c e of

s e t s l e r d s t o t h e Wallman c o m p a c t i f i c a t i o n .

S u b l a t t i c e s of the

a l l closed

l a t t i c e of a l l closed sets y i e l d o t h e r compactifications. z e r o s e t s of

2

The l a t t i c e o f

c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e T, f o r e x a m p l e , p r o d u c e s

BT ( ( 5 . 4 - 3 ) ) .

-Example

3.1-4

THE ZERO SET LATTICE

Let T

be

a completely r e g u l a r Hausdorff

s p a c e a n d 2 t h e c o l l e c t i o n o f a l l z e r o s e t s of f u n c t i o n s i n C(T,R).

By t h e

i s a d i s t r i b u t i v e l a t t i c e w i t h 0 and 1 w i t h r e s p e c t

r e s u l t s of Sec. 1 . 2 , 2 t o U and fl :

z(x) U

n

z(.)

=

Z(Y)

z(x

2

+ y

2

>cz

Z(Y) = ~ ( X Y ) F Z

The n o t i o n s t o b e c o n s i d e r e d n e x t a r e a b s t r a c t i o n s o f c o n c e p t s p a r t i c u l a r t o t h e power s e t l a t t i c e . D e f i n i t i o n 3.1-1

F i r s t we h a v e a g e n e r a l i z a t i o n o f " f i l t e r " .

LATTICE FILTERS AND ULTRAFILTERS

A s u b s e t F of a d i s t r i b -

u t i v e l a t t i c e L w i t h 0 and 1 i s a l a t t i c e f i l t e r ( o r s i m p l y " f i l t e r " )

(1)

0 c! F,

(2)

F i s s t a b l e under f i n i t e products ( i . e .

(3)

( t h e " u p p e r bound" c o n d i t i o n ) i f a c F a n d b

if

i f a , bCF t h e n a b c F ) and

2

a then bcF.

A l a t t i c e f i l t e r F is a l a t t i c e u l t r a f i l t e r (or j u s t " u l t r a f i l t e r " ) i f no other f i l t e r contains F as

E

proper subset.

A n a l o g o u s t o t h e power s e t s i t u a t i o n , u l t r a f i l iers c a n b e c h a r a c t e r i z e d i n terms of a " p r i m a l i t y " c o n d i t i o n .

(3.1-2)

ULTRAFILTERS

t i c e L w i t h 0 and 1. bcF.

Let F be a l a t t i c e f i l t e r , i n the d i s t r i b u t i v e l a t Then i f F i s a n u l t r a f i l t e r and a+bcF, e i t h e r aFF o r

C o n v e r s e l y , i f L i s a B o o l e a n l a t t i c e and a + b c F - a

c

F o r b c F

(a, b

c

L)

then F i s an u l t r a f i l t e r . Proof

Suppose t h a t F i s a n u l t r a f i l : e r ,

a+bcF, and bkF.

H=[ccLlc+bcF?, i t f o l l o w s t h a t H i s a f i l t e r :

Then, l e t t i n g

I n d e e d OdH a s b d F .

I f c,d€H

3.1

t h e n , s i n c e cd+b=(c+b)(d+b)(F,

14 1

LATTICES

cdcH.

2 c

M o r e o v e r , i f c € H and d

d+b 2 c+b, i t f o l l o w s t h a t d+bcF and d€H.

then, as

C l e a r l y a€H and F C H s o t h e

m a x i m a l i t y o f F i m p l i e s t h a t aCH=F. C o n v e r s e l y , i f F . s a f i l t e r i n t h e Boolean l a t t i c e L and F c o n t a i n s a o r b whenever a+bcF, s u p p o s e t h a t H 3 F and aCH w h i l e akF.

A s I f F , akF,

a n d l = a + a ' where a ' i s t h e complement o f a , i t f o l l o w s t h a t a ' F F . b o t h a a n d a ' b e l o n g t o H s o t h a t O=aa'€H F is an u l t r a f i l t e r .

-

a contradiction.

and

v

LATTICE FILTERS VS. SET FILTERS

Example 3.1-5

But t h e n

Thus HF'

I n a power s e t l a t t i c e ?(T)

a c o l l e c t i o n of subsets i s a l a t t i c e f i l t e r i f f it is a f i l t e r i n the o r d i nary set

-

respect t o

I n a l a t t i c e 1: o f s u b s e t s o f a s e t T ( w i t h

theoretrc sense.

u

a n d fl ) a l a t t i c e f i l t e r i s n o t , i n g e n e r a l , a s e t f i l t e r ,

a s i t need n o t c o n t a i n belonging t o

all

s u p e r s e t s of elem ent s of t h e f i l t e r

-

j u s t those

C.

The p r i m a l i t y c o n d i t i o n g i v e n i n ( 3 . 1 - 2 ) f i l t e r s i n Boolean l a t t i c e s .

completely c h a r a c t e r i z e s ultra-

Below w e o b t a i n a c h a r a c t e r i z a t i o n o f them i n

a n y d i s t r i b u t i v e l a t t i c e w i t h 0 and 1 (Theorem 3.1-1) by e s s e n t i a l l y s t r e n g t h e n i n g c o n d i t i o n (3) o f t h e d e f i n i t i o n o f a l a t t i c e f i l t e r .

First,

however, w e d e f i n e a n o t i o n w h i c h i s somewhat weaker t h a n t h a t o f a l a t t i c e filter. D e f i n i t i o n 3.1-2

LATTICE FILTER SUBBASE

A subset S of a d i s t r i b u t i v e l a t -

t i c e L w i t h 0 a n d 1 i s a l a t t i c e f i l t e r s u b b a s e i f i t h a s the f i n i t e i n t e r section property, i.e.

a l l f i n i t e produccs of el em ent s of S a r e non-zero.

Every l a t t i c e f i l t e r subbase S i s c o n t a i n e d i n a l a t t i c e f i l t e r f o r c l e a r l y the c o l l e c t i o n F(S) o f a l l e l e m e n t s b o f L w h i c h a r e g r e a t e r t h a n o r e q u a l t o some f i n i t e p r o d u c t o f e l e m e n t s f r o m S i s a l a t t i c e f i l t e r a n d , in fact,

i s t h e s m a l l e s t l a t t i c e f i l t e r of L containing S.

As promised, w e

now show t h a t by s t r e n g t h e n i n g t h e t h i r d c o n d i t i o n o f D e f i n i t i o n 3.1-1, a c h a r a c t e r i z a t i o n of u l t r a f i l t e r s i s o b t a i n e d . Theorem 3.1-1

ULTRAFILTERS AND THE '!MEET" CONDITION

t i v e l a t t i c e L w i t h 0 a n d 1.

L e t L be a d i s t r i b u -

Then F C L i s a l a t t i c e u l t r a f i l t e r i f f

(1)

F i s a l a t t i c e f i l t e r subbase,

(2)

F i s s t a b l e u n d e r f i n i t e p r o d u c t s , and

(3)

( t h e " m e e t " c o n d i t i o n ) f o r a n y CCL, i f ca#O f o r e a c h a € F t h e n cCF.

Remark

C l e a r l y c o n d i t i o n s (1) and (2) a r e e q u i v a l e n t t o c o n d i t i o n s (1) a n d

( 2 ) o f t h e d e f i n i t i o n of a l a t t i c e f i l t e r ( D e f i n i t i o n 3 . 1 - 1 ) .

Moreover t h e

meet c o n d i t i o n i s s t r o n g e r t h a n t h e u p p e r bound c o n d i t i o n of D e f i n i t i o n 3.1-1:

Whenever b

2

a f o r some a € F t h e n , f o r e a c h dEF, bd#O

-

indeed i f

142

3 WALLMAN COMPACTIFICATIONS

bd=O t h e n ad=bad=a(bd)=O

-

Hence, by t h e meet c o n d i t i o n

a contradicrion.

t h e u p p e r bound c o n d i t i o n o f D e f i n i t i o n 3.1-1 i s f u l f i l l e d : LCF. C l e a r l y (1) and ( 2 ) a r e n e c e s s a r y f o r F t o be a l a t t i c e u l t r a f i l t e r .

Proof

A s f o r t h e meet c o n d i - i o n s u p p o s e t h a t ca#O f o r e a c h a c F .

A s the smallest

l a t t i c e f i l t e r H g e n e r a t e d by t h e l a t t i c e f i l t e r s u b b a s e F U f c } c o n t a i n s t h e u l t r a f i l t e r F, H=F a n d , t h e r e f o r e ,

cCF.

C o n v e r s e l y , a s was n o t e d i n t h e remark a b o v e , c o n d i t i o n s (l),

( 3 ) f o r c e F t o be a l a t t i c e f i l t e r . t h e n ca#O f o r e a c h a c F . ultrafilter.

(2) and

I f H i s a l a t t i c e f i l t e r 3 F and CCH

Thus cFF by ( 3 ) , H=F, and F i s t h e r e f o r e a l a t t i c e

V

The c o l l e c t i o n o f a l l l a t t i c e f i l t e r s u b b a s e s o f a g i v e n d i s t r i b u t i v e l a t t i c e L w i t h 0 and 1 i s p a r t i a l l y o r d e r e d by s e t i n c l u s i o n a n d , v i a t h e Z o r n ' s lemma a r g u m e n t , l a t t i c e f i l t e r s u b b a s e s which a r e maximal w i t h res p e c t t o s e t i n c l u s i o n c a n be shown t o e x i s t . t o a s maximal l a t t i c e f i l t e r s u b b a s e s .

Such o b j e c t s a r e r e f e r r e d

I t i s p e r h a p s s u r p r i s i n g t h a t max-

imal lattice f i l t e r subbases are i n f a c t l a t t i c e u l t r a f i l t e r s v i z .

13.1-3)

MAXIMAL FILTER SUBBASE "=" ULTRAFILTER

:

In a d i s t r i b u t i v e l a t t i c e

L w i t h 0 a n d 1 a l a t t i c e f i l i e r s u b b a s e i s maximal i f f i t i s a l a t t i c e ultrafilter. I f H i s a l a t t i c e u l t r a f i l t e r and S i s a l a t t i c e f i l t e r s u b b a s e con-

Proof

: a i n i n g H t h e n t h e s m a l l e s t l a t t i c e f i l t e r F(S) c o n t a i n i n g S c o n t a i n s H. Thus by t h e m a x i m a l i t y of H , H C S C F (S)

= H

and S=H t h e r e b y p r o v i n g t h a t H i s a maximal l a t t i c e f i l t e r s u b b a s e . C o n v e r s e l y , s u p p o s e t h a t S i s a maximal l a t t i c e f i l t e r s u b b a s e . a s S'

-

t h e c o l l e c t i o n of a l l f i n i t e products of S

-

Then,

is a lattice filter

s u b b a s e c o n t a i n i n g S , S'=S.

By Theorem 3.1-1,

whenever ca#O f o r e a c h aCS.

But t h i s i s c l e a r l y t h e c a s e a s S U { c ] = S ' U {c]

i t r e m a i n s t o show t h a t c f S

is a l a t t i c e f i l t e r subbase containing S. V S e c t i o n 3.2

LATTICES AND ASSOCIATED -COMPACTIFICATIONS

In this section w e

show t h a t g i v e n a d i s t r i b u t i v e l a t t i c e L w i t h 0 and 1 t h e r e i s a n a s s o c i a t e d compact s p a c e wL:

WL c o n s i s t s of a l l l a t t i c e u l t r a f i l t e r s .

A s ob-

served i n Sec. 3.1 given a t o p o l o g i c a l space T t h e r e a r e c e r t a i n l a t t i c e s which one n a t u r a l l y t e n d s t o c o n s i d e r , e . g .

the l a t t i c e

6

of a l l c l o s e d

s u b s e t s o f T a n d , f o r T c o m p l e t e l y r e g u l a r and H a u s d o r f f , t h e l a t t i c e o f a l l z e r o s e t s of T.

D e t e r m i n i n g t h e p r o p e r t i e s o f WL i n t h e s e s p e c i a l

c a s e s i s what t h i s s e c t i o n i s d i r e c t e d t o w a r d .

3.2

14 3

LATTICES AND COMPACTIFICATIONS

To b e g i n w i t h we s i n g l e o u t a c o l l e c t i o n o f s u b s e t s o f t h e f a m i l y o f a l l u l t r a f i l t e r s of a d i s t r i b u t i v e l a t t i c e L w i t h 0 and 1 t o s e r v e a s a For e a c h aEL w e d e f i n e t h e b a s i c

ba se of closed s u b s e t s f o r a topology.

set

6

g e n e r a t e d by a t o be t h e c o l l e c t i o n o f a l l u l t r a f i l t e r s t o which a

belongs.

A s n o u l t r a f i l t e r c a n c o n t a i n 0,

aO=O.Thus

f o r the b a s i c sets

t o be a b a s e r a t h e r t h a n a s u b b a s e o f c l o s e d s e t s f o r a t o p o l o g y i t s u f f i c e s t h a t they be s t a b l e w i t h r e s p e c t t o t h e f or m at i on of f i n i t e uni ons. I n d e e d , a s w e now show,

Bb=da+bf o r

@,U

t h e n , s i n c e f o r a n y b 0 t h e r e e x i s t s e t s A1, n CAi a n d , u s i n g " 0 " f o r " o s c i l l a t i o n " , T= i=l o ( x , CAi) = s u p { l x ( s ) - x ( t ) l Is, t € C A i ] < c

...,Anc2

such t h a t

u

...,n.

f o r each i=l,

C e r t a i n l y a n y 2 - u n i f o r m c o n t i n u o u s f u n c t i o n i s bounded.

Furthermore,

t h e c o l l e c t i o n o f a l l s u c h f u n c t i o n s f o r m s a c l o s e d s u b a l g e b r a o f C (T,W) b i n t h e u n i f o r m norm. I n o u r n e x t r e s u l t w e show t h a t a n y xqCb(T,&) t h a t p u l l s c l o s e d s u b s e t s of

8

b a c k i n t o members o f 1: must b e 2 - u n i f o r m l y con-

tinuous.

A CLASS OF 2-UNIFORMLY CONTINUOUS FUNCTIONS

(3.4-2)

r e g u l a r H a u s d o r f f s p a c e and 2 a n o r m a l b a s e f o r T. x-'(C)C2 Proof

f o r each closed C C J

Let



0 be given.

number o f r e a l numbers a l , .

Let T be a completely

I f x

-

x(t)l =

3s , 2 t E vaj,

since

WALIMAN COMPACTIFICATIONS

- xw(J-tt)15 c

l X W G S )

and x i s X-uniformly c o n t i n u o u s .

Conversely, suppose t h a t x i s 2 - u n i f o r m l y c o n t i n u o u s and l e t g € w ( T , l ) . As

3

i s a f i l t e r b a s e on T , x($)={x(A)

I A C Z ] i s a f i l t e r b a s e on

thermore, a s x(T) i s bounded, x ( 2 ) h a s a a d h e r e n c e p o i n t i n ;

~"(3). We

n o t e by

s h a l l show t h a t x @ )

+

~"(3) by p r o v i n g

E.

Fur-

which w e d e -

x u 3 t o be a

Cauchy f i l t e r b a s e w i t h t h e a i d of t h e 2-uniform c o n t i n u i t y o f X . Indeed n f o r a g i v e n c > 0 t h e r e e x i s t s A l,. ,A cf, s u c h t h a t T = u CA. and i=1 1 " n o(x,CA.) 6 c f o r e a c h 1 s i 5 n . S i n c e n A . = @ t h e r e e x i s t s 1 s j 5 n i=1 By t h e maximality of 8 t h e r e e x i s t s A € g such t h a t such t h a t A . f 2 J Thus f o r any p a i r t , S E A , I x ( t ) - x ( s ) 1cc A . n A = @ or, equivalently, A C C A . . J J and i t f o l l o w s t h a t xu? i s Cauchy.

..

.

To complete t h e proof we must d o two t h i n g s .

We must show t h a t x

a s d e f i n e d above, e x t e n d s x and i s c o n t i n u o u s on w ( T , L ) .

-

e x t e n d s x , l e t tCT and c o n s i d e r ~ ( 2 , )

W

-9t .

x ( 2 t ) there e x i s t s A c 9 ,

W

,

To s e e t h a t xw

We c l a i m t h a t x W ( s t ) = x ( t ) .

As

s u c h t h a t f o r e a c h SEA, I x ( ~ ) - x ~ ( ~ ~ ) l c € .

Thus, s i n c e t c A , Ix(s>

-

x ( t > 15 I x ( s ) - x w ( 2 t l I + I x ( t )

-, x ( t ) .

w -

-

x (St)

I
=0.

V

a,fluH=@

165

EXERCISES 3

Exercises 3

3.1

Let L b e a d i s t r i b u t i v e l a t t i c e w i t h 0 a n d 1.

SEPARATION I N IATTICES

L i s a d i s j o i n t e d l a t t i c e i f a , b s L , afO, b#O, t h e n t h e r e e x i s t s cgL s u c h t h a t ac#O a n d bc=O o r bcf.0 and ac=O.

If L i s a d i s j o i n t e d l a t t i c e t h e n a c 3 f o r all 3 < w L

(a)

(Sec. 3.2)

i f f a=l. The l a t t i c e of a l l c l o s e d s u b s e t s o f a t o p o l o g i c a l T

(b)

space T i s a

disjointed lattice i f f T is a T

1 space* I f T i s a t o p o l o g i c a l s p a c e and L t h e l a t t i c e o f a l l c l o s e d s u b -

(c)

sets of T, then i f L is an a - l a t t i c e , L i s a d i s j o i n t e d l a t t i c e .

Conversely

i f L i s a d i s j o i n t e d 8 - l a t t i c e , then L i s an a - l a t t i c e . 3.2

BOOLEAN LATTICES;

THE STONE REPRESENTATION THEOREM

In t h i s e x e r c i s e

some r e s u l t s a r e p r e s e n t e d a b o u t B o o l e a n l a t t i c e s (complemented d i s t r i b u t i v e l a t t i c e s w i t h 0 a n d 1) i n c l u d i n g t h e S t o n e r e p r e s e n t a t i o n t h e o r e m ( f ) which c a n b e g e n e r a l i z e d t o d i s t r i b u t i v e l a t t i c e s ( S t o n e ,

1936), a s can a

number o f t h e p a r t s o f t h i s e x e r c i s e . A l l l a t t i c e s a r e assumed t o c o n t a i n 0 and 1; z e r o e l e m e n t a of a l a t t i c e L i s a n

eif

a ' denotes 1-a.

when bcL and b

5

A non-

a t h e n b=O o r

b=a. I f L i s a f i n i t e l a t t i c e t h e n f o r e v e r y CCL t h e r e e x i s t s a n atom

(a)

aCL s u c h t h a t a

5

c.

If L i s a l a t t i c e and a a n a t o m i n L , t h e n f o r e v e r y b€L, ab=O o r

(b) ab=a.

L e t acL where L i s a l a t t i c e a n d l e t R(a) b e t h e s e t o f a l l atoms

(c) that are

5

a.

Then R ( a b ) = R ( a ) q R ( b ) , R ( a ' ) = R ( l ) - R ( a ) .

t h e n R(a)=R(b) if and o n l y i f a = b . a l g e b r a L.

Then R(max.a . ) = ( a l , . 1 1

Let al,

...,a k

If L is finite,

be atoms i n a Boolean

. .,an}.

Two B o o l e a n a l g e b r a s Ll a n d L a r e s a i d t o b e i s o m o r p h i c i f t h e r e e x 2 i s t s a 1-1 c o r r e s p o n d e n c e f b e t w e e n them s u c h t h a t f ( a ' ) = f ( a ) ' f o r a l l a c L a n d f ( a l A a ) = f ( a ) A f ( a ) f o r a l l a l , a €L 2 1 2 2 1' (d) I f L i s a f i n i t e Boolean l a t t i c e , t h e n L i s isomorphic t o t h e B o o l e a n l a t t i c e o f a l l s u b s e t s o f t h e s e t o f atoms o f L.

Hint:

L e t X=[a l , . . . , a

} b e t h e atoms o f L a n d c o n s i d e r f:L a

-P(X) +R(a)

1

166

3.

WALLMAN COMPACTIFICATIONS

( e ) Two f i n i t e B o o l e a n l a t t i c e s w i t h t h e same number of e l e m e n t s a r e isomorphic. S t a t e m e n t (d) i s t r u e f o r i n f i n i t e B o o l e a n l a t t i c e s a s w e l l . i n p r o v i n g t h i s , u s e i s made o f Z o r n ' s lemma.

However,

As is often the case, the

lemma i s u s e d i n showing t h a t e v e r y n o n z e r o aqL c a n b e embedded i n a n u l t r a f i l ter (f)

.

Every Boolean l a t t i c e L i s i s o m o r p h i c t o a Boolean l a t t i c e o f s u b -

s e t s of a set X.

Hint:

L e t X be t h e s e t o f u l t r a f i l t e r s o f L.

Let

T:L --cp(X) a

-15%

x

lac23

for a # O

a n d T(O)=0. A Boolean l a t t i c e i s c a l l e d c o m p l e t e i f f o r e v e r y WCL, s u p W e x i s t s .

L is c a l l e d atomic i f every element i n L i s

(g)

2

A Boolean l a t t i c e L i s isomorphic t o

some a t o m ,

P(X)

u n d e r t h e mapping T o f

( f ) i f and o n l y i f L is c o m p l e t e and a t o m i c .

3.3

ZERO-DIMENSIONALITY OF w ( T , G ) AND ULTRANORMALITY

In t h i s e x e r c i s e T

I t i s shown i n ( a ) t h a t w ( T , G ) i s z e r o - d i m e n s i o n a l i f f T

i s a T1 s p a c e . is ultranormal.

(a)

Let

6 be

t h e l a t t i c e of a l l c l o s e d s u b s e t s of T.

Show t h a t

w(T,G

I s zero-dimensional i f f T is ultranormal.

Hint:

If w(T,C)

i s u l t r a r e g u l a r t h e n , b e i n g compact, w ( T , C ) is u l t r a -

a3

and @ a r e d i s j o i n t b a s i c c l o s e d s e t s i n w ( T , G ) , Hence i f F K (where T - U i € & f o r t h e r e e x i s t s (see Example 3.2-1) a c l o p e n s e t i€I i E I ) s u c h t h a t BFC and =0. Thus w i t h 'p a s i n S e c . 1EI 1 iFI i

normal.

u )(,.

BKr)u V u

uv,.

C o n v e r s e l y , i f T is u l t r a n o r m a l , t h e c l o p e n s u b s e t s o f T s e p a r a t e t h e closed sets.

(\/,IUfG} i s

Letting

d e n o t e t h e c l o p e n s u b s e t s o f T, by Example 3 . 2 - l ( b ) ,

a b a s e f o r t h e t o p o l o g y of w ( T , & ) .

i t follows t h a t

(b)

6

vU

A s each

UcG i s c l o p e n ,

f o r each U < G i s clopen.

A n o r m a l T1 s p a c e T i s u l t r a n o r m a l i f f BT i s u l t r a r e g u l a r and i n

t h i s c a s e @T=w(T,6).

Hint:

I f BT i s u l t r a r e g u l a r t h e n show t h a t BT i s u l t r a n o r m a l .

Utilizing

n o r m a l i t y o f T, show t h a t t h e d i s j o i n t c l o s e d s e t s i n T have d i s j o i n t c l o s u r e s i n BT.

167

EXERCISES 3

L e t S C U where U i s

C o n v e r s e l y , s u p p o s e t h a t T i s u l t r a n o r m a l and SQBT.

open i n 8T and t a k e xBcC(BT,R) s u c h t h a t z ( x B ) C U w h e r e z ( x p ) i s a n e i g h borhood o f s .

A s T i s u l t r a n o r m a l , l e t V be c l o p e n i n T s u c h t h a t z ( x ) = O b s e r v i n g t h a t V a n d CV a r e z e r o s e t s i n T show t h a t

z(x8)flTTCVCUOT.

c l V and c l CV a r e d i s j o i n t c l o p e n s e t s i n fiT w i t h s C c l V C U .

8

B

B

L e t T b e a n u l t r a r e g u l a r s p a c e which i s n o t u l t r a n o r m a l .

(c)

Show

t h a t {vulUpq i s n o t a base f o r the topology of w ( T , G ) . The s e t s

Hint:

'/v

a r e c l o p e n i n w ( T , G ) and w ( T , G ) i s n o t u l t r a r e g u l a r .

AN ULTRAREGULAR SPACE WHICH I S NOT ULTRANORMAL

(d)

u n c o u n t a b l e o r d i n a l and

UJ

the f i r s t i n f l n i t e cardinal.

Let

R

be t h e f i r s t

L e t S=[O,u]x[O,n]

Show t h a t S and T ( c a r r y i n g t h e p r o d u c t of t h e o r d e r

a n d T=S-{(w,O)].

t o p o l o g y on t h e o r d i n a l numbers) a r e b o t h u l t r a r e g u l a r b u t t h a t T i s n o t u l t r a n o r m a 1.

T i s n o t n o r m a l a s t h e c l o s e d s e t s K=((cy,u) 10

Hint:

{ ( n , n ) 10

5

1 9 6 6 , p.

145).

n


0. The set of

nomials which converges to may assume that

(f,)

x-p

therefore, has a positive distance from Cn' and as a result for each m € 2 , l/fm is expressible as a zeros of each

fm,

power series whose radius of convergence is over since follows that

(l/fm)

converges to

l/(x-v)

l/(x-p)

> l-l/n.

More-

uniformly on Cn, it

is expressible as a uniform limit of

.

polynomials on Cn Hence if in P (Cn)I i.e. point of x 1 n'

p ,.( p , (

0

x(Cn),

(x

Ic

then

p

is a regular

1 which establishes the n

4 . COlfllUTATIVE TOPOLOGICAL A L G E B M S

204

equality. Now f o r any

x E H , a ( x ) = cpr(Cn) = x ( S I ( 0 ) ) .

Thus, as

w a s t h e case i n t h e p r e c e e d i n g example, t h e spectrum of an element i s j u s t t h e s e t o f v a l u e s i t a s s u m e s . The r e s u l t s of t h e p r e c e e d i n g two examples make it c l e a r t h a t t h e s p e c t r u m of a n element i n a complete LMCH a l g e b r a need b e n e i t h e r c l o s e d n o r bounded.

trast,

a(x)

I n a Banach a l g e b r a , by con-

may b e e x p r e s s e d a s t h e c o n t i n u o u s image of a

compact set and s o i s c l o s e d and bounded ( c f . S e c t i o n 4 . 8 ) ;

i s always c o n t a i n e d i n t h e d i s c o f r a d i u s

Ilx~l a b o u t 0 .

it

Since

t h e a l g e b r a H i s a F r e c h e t a l g e b r a , m e t r i z a b i l i t y is s e e n n o t t o be the critical ingredient. 4.8

Q - a l g e b r a s and a l g e b r a s w i t h c o n t i n u o u s i n v e r s e . A s is w e l l known maximal i d e a l s i n Banach a l g e b r a s a r e

closed sets. a l case.

gebra

T h i s f e a t u r e i s u n f o r t u n a t e l y l o s t i n t h e gener-

I t i s r e c o v e r e d , however, i n t h e s p e c i a l t y p e of a l -

w e d e f i n e below a l t h o u g h w e d e f e r p r o v i n g t h i s u n t i l

(4.10-1).

D e f i n i t i o n 4.8-1

Q-ALGEBRAS.

A topological algebra

in

which

t h e s e t of u n i t s i s open i s c a l l e d a Q-algebra. Any Banach a l g e b r a i s a Q - a l g e b r a ( 4 . 8 - 2 ) ; o f i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s on proved a f t e r ( 4 . 8 - 1 )

below.

so i s t h e s p a c e [a,b]

as i s

An example of a t o p o l o g i c a l a l g e -

b r a which i s n o t a Q - a l g e b r a i s g i v e n i n Example 4.8-1 Example 4.8-1 C(T,E,c)

C(T,J,c)

I S NOT GENERALLY A Q-ALGEBRA.

now. Let

b e t h e t o p o l o g i c a l a l g e b r a of c o n t i n u o u s s c a l a r - v a l u -

ed f u n c t i o n s on t h e non-compact

s p a c e T of Example 4.5-1,

c o m p l e t e l y r e g u l a r Hausdorff

w i t h compact-open

topology.

To show

4 . 8 Q-ALGEBRAS ANQ ALGEBRAS W.TH CONTINUOUS INVERSE

that

is not a Q-algebra, we show that each neighbor-

C(T,Z,c)

hood of the identity e contains a non unit.

To this end con-

sider a typical basic neighborhood of the identity {x E C(T,E,c) ~supttG~x(t)-ll< subset of T and

E

> 0.

t ,( G, y(t)

member of

e+W(G,E)

=

where G is a proper compact

E),

By the Tietze extension theorem there

is a continuous function y on T such that some fixed

205

=

0.

y(G) = (1)

and, for

The non-unit y then is clearly a

e+W(G,€).

(4.8-1) Q-ALGEBRA IFF INT Q # $3.

The topological algebra X is

a Q-algebra iff its set Q of units has nonempty interior. Proof.

Clearly only the sufficiency of the condition need be

To this end, let x E int Q and let y be any unit. Consider the map w+yx -1w. As remarked in Section 4.1 such a

proved.

map must be a homeomorphism which maps Q onto Q. V E V(x)

such that

V c Q, yx-lV

Hence for

is a neighborhood of y which

lies in Q.V As an application of (4.8-1)we now show that the space of infinitely differentiable functions on

a

[arb] with the top-

pn(x) = sup (t)I t€[a,bl (first discussed in Example 4.3-2) is a Q-algebra.

ology generated by the seminorms (n=O,l,. . . )

(4.8-1) we need only show that the neighborhood

By

and to this end consider any and

I x(t)-1 1

function [a,b]

< 1

l/x(t)

for each

CQ

PO

. Since e(t) = 1 PO t E [arb], it follows that the x E e

+

V

is defined and infinitely differentiable on

thereby implying that x is invertible.

(4.8-2)

BANACH ALGEBRAS ARE Q-ALGEBRAS.

gebra then it is a Q-algebra. *-

e+V

is a unit.

If X is a Banach al-

In particular if IIe-xll < l then

4 . COlINUTATIVE TOPOLIGICAL ALGEBRAS

206

L e t X b e a Banach a l g e b r a w i t h s e t of u n i t s Q.

Proof.

t h a t Q i s open i t s u f f i c e s by ( 4 . 8 - 1 ) To accomplish t h i s w e n o t e t h a t

t o show t h a t

To show

Sl(e) c Q .

IIe-xll < 1 i m p l i e s t h a t t h e

03

sequence o f p a r t i a l sums o f

C

n= 0

i s Cauchy.

Ile-xf

Hence t h e

m

C (e-x)" i s Cauchy i n X and t h e n= 0 series t h e r e f o r e converges t o y s a y . T o see t h a t xy=e, w r i t e

sequence o f p a r t i a l sums of

x as

and c o n s i d e r (e- (e-x) ) (e+(e-x) +.

e- (e-x)

e- (e-x) n+l As

..+ (e-x) n )

=

from which t h e d e s i r e d r e s u l t fol1ows.V

w a s mentioned i n t h e p r e v i o u s s e c t i o n a p l e a s i n g f e a f a c t t h a t a l l e l e m e n t s have com-

t u r e of Banach a l g e b r a s - - t h e

l o s t i n t h e g e n e r a l case.

pact spectra--is

Our n e x t two re-

s u l t s combine t o r e c o v e r t h i s p r o p e r t y i n Q - a l g e b r a s . I N Q-ALGEBRAS ALL ELEMENTS HAVE BOUNDED SPECTRA.

(4.8-3)

any t o p o l o g i c a l a l g e b r a X , ( t h e s e t of a l l x w i t h (b)

(a)

X i s a Q-algebra i f f

For

U(0 )

r0 (x) 5 1) h a s non-empty i n t e r i o r ;

I f X i s a Q - a l g e b r a t h e n t h e spectrum of each ele-

ment i s bounded.

(a)

Proof.

U

(0 )

,

F i r s t w e assume t h a t x i s a n i n t e r i o r p o i n t of

o r e q u i v a l e n t l y , t h a t t h e r e i s a neighborhood

such t h a t

x+W c U ( 0 ) .

(1/2)W so f o r each s u c h y ,

e

+

(1/2)x ( 1 / 2 ) W

r0 ( y ) 5 1 / 2 f o r each

Now

e+y

is invertible.

W E V(0)

y E (1/2)x + Thus

and i t f o l l o w s , by (4.8-1), t h a t Q i s

C Q

open. Conversely i f Q i s open, choose a b a l a n c e d neighborhood W o f 0 such t h a t

with cQ

Ihl

2 1,

so t h a t -A

e + W c Q.

Then f o r each

i t i s clear t h a t

1 a(x).

an i n t e r i o r p o i n t of

Thus

U(U).

Ae

+ x

=

r a ( x ) 5 1,

x E W

and

+

A-lx)

A (e

W c U(U),

A E F, E A (e+W) and 0 i s

20 7

4 . 8 0-ALGEBPAS AND ALGEBRAS WITH CONTINUOUS INVERSE

S i n c e t h e r e i s a neighborhood

(b)

,

W c U(U)

it follows t h a t f o r each

such t h a t

W E V(0)

x E X

there is a

w i t h s u f f i c i e n t l y s m a l l a b s o l u t e v a l u e , such t h a t U(a).

Thus s i n c e

r,(Ax)

=

IAlru(x)

ru(x)
0

F

u

Thus

X

+

-+

C l e a r l y f o r any ae-x

x E X

E Q

W E V(0)

t h e map

i s c o n t i n u o u s and it f o l l o w s t h a t

such t h a t

+

S E (Ale-x c he-x

W c Q.

SE(A) c p(x).V

S i n c e Banach a l g e b r a s are Q - a l g e b r a s [ ( 4 . 8 - 2 ) ] , e a c h element i n a Banach a l g e b r a X h a s compact spectrum. = I l x i i / ~ u ~< 1

11 e-(e-x/u)ll

since

s o f o l l o w s by ( 4 . 8 - 2 ) { A € FI

1x1

whenever

t h a t each such

Moreover,

1 ~ >1

llxll,

1~ E p ( ~ ) . Thus

u(x) c

5 Ilxll).

Y e t a n o t h e r p r o p e r t y of Banach a l g e b r a s - - c o n t i n u i t y

map

it a l -

x+x-’--

i s l o s t i n t h e g e n e r a l case.

of t h e

I n f a c t it i s n o t

g e n e r a l l y t h e case t h a t i n v e r s i o n i s c o n t i n u o u s i n Q - a l g e b r a s

(see ( E x e r c i s e 4 . 7 ( c ) ) a l t h o u g h LMC a l g e b r a s do p o s s e s s t h e property 4.8-1,

( w e prove t h i s i n ( 4 . 8 - 6 ) ) .

A s mentioned i n Example

i s n o t a Q-algebra b u t

C(R,&c)

x+x

-1

i s continuous

anyway by ( 4 . 8 - 6 ) . D e f i n i t i o n 4.8-2

CONTINUOUS INVERSE.

A topological algebra X

4 . COHtllJTATIVE TOPOLIGICAL ALGEBRAS

268

-1 x+x

i n which t h e map

i s c o n t i n u o u s a t e i s an a l g e b r a w i t h

W e a l s o say t h a t X h a s continuous i n v e r s e .

continuous i n v e r s e .

THE RESOLVENT MAP I S ANALYTIC.

(4.8-5)

continuous i n v e r s e :

( a ) t h e map

x+x

where on Q , t h e s e t of u n i t s of X ; x E X, i f

b r a t h e n f o r any

-1

i s continuous every-

( b ) i f X i s a complex a l g e -

i s an open s e t ( e . g . i f X i s

p(x)

a Q - a l g e b r a ) t h e r e s o l v e n t map

I n an a l g e b r a X w i t h

r :p(x)+X X

,

i s an-

A-.(x-Ae)-l

a l y t i c on p ( x ) * .

(a) L e t

Proof.

show t h a t @

-’

% !

be a f i l t e r b a s e i n Q c o n v e r g e n t t o

= {B-ll B

E

a} converges

to x

-1

observed p r e v i o u s l y , i f y i s a u n i t t h e map morphism o f X o n t o X , any

x E Q,y+x

-1

Q converges t o x ,

t h e map

w+w

-’

then

x-b+e.

rx(h)

-

-1

.

rx(h)

which ( * )

follows.

X+x-Ae+ (x-Ae)

X-tr

X

(A)

is t h a t

( X r P € P ( x ) ) . TO see t h i s

l r X( u ) = (x-Ae)r X 0.i) = (x-pe+ve-Ae)rx(p)

consider

-1

from which i t

x6’+eI

The key o b s e r v a t i o n i n prov-

r X (11) = ( A - u ) r x ( h ) r x ( v )

-

Thus, f o r

By t h e assumed c o n t i n u i t y of

i n g t h e a n a l y t i c i t y of t h e r e s o l v e n t map

(“1

i s a homeo-

- l+x- 1

rx(X) = (x-Xe)

Let

x+xy

I f the filterbase 6 i n

a t e, it follows t h a t

r e a d i l y f o l l o w s t h a t (a (b)

A s h a s been

f o r any t o p o l o g i c a l a l g e b r a X.

i s a homeomorphism.

y

.

x E Q. W e

from

By p a r t ( a ) , t h e r e s o l v e n t map p ( x ) - Q + Q ,

i s continuous.

Thus t h e a n a l y t i c i t y o f r x ( A )

on t h e open s e t p ( x ) now f o l l o w s from ( * ) . V

*

The s e t p ( x ) i s open i n any Q - a l g e b r a ( ( 4 . 8 - 4 ( a ) ) ) .

however, t h a t i t need n o t b e c o n n e c t e d .

W e note,

For example, c o n s i d e r

t h e f u n c t i o n x ( t ) = t i n C ( T , C , c ) where T i s a c l o s e d a n n u l u s i n

.% C.

C l e a r l y a ( x ) = T and p ( x ) , b e i n g t h e complement of T , i s

n o t connected.

209

4.8 Q-ALGLBRAS AND ALGEBRAS NITH CONTINUOUS I N V E R S E

A t t h e o u t s e t of t h e d i s c u s s i o n on c o n t i n u i t y of i n v e r s i o n

w e mentioned t h a t n o t a l l t o p o l o g i c a l a l g e b r a s have c o n t i n u o u s I t i s t h e case, however, t h a t a v e r y l a r g e c l a s s o f

inverse.

t o p o l o g i c a l a l g e b r a s c o n t a i n i n g t h e normed a l g e b r a s , namely t h e LMC a l g e b r a s , p o s s e s s t h e p r o p e r t y .

(4.8-6)

LMC ALGEBRAS ARE ALGEBRAS W I T H CONTINUOUS INVERSE. I f

X i s ( a ) a normed a l g e b r a , o r , m o r e g e n e r a l l y ,

( b ) an LMC a l -

gebra, then X has continuous i n v e r s e .

.x+x-1

( a ) To see t h a t t h e map

Proof.

i s continuous a f e ,

suppose ( x ) i s a sequence of u n i t s c o n v e r g e n t t o e. To show n -1 1 ) Ile-xnllr t h a t xn l + e , w e need o n l y n o t e t h a t Ilxn-l-ellgxn t h e boundedness of

(tixn-lll )

being apparent f r o m the r e l a t i o n m

[ i n t h e c o m p l e t i o n of X I (see ( 4 . 8 - 2 ) ) (b) prove t h a t

Let

xn

c (e-xn) k f o r l l x n - e l l ~r < 1

=

k=O

and t h e convergence o f

/a

@-'

(x,)

t o e.

be a f i l t e r b a s e of u n i t s c o n v e r g e n t t o e . = {B

-1

We

converges t o e.

IB

L e t P b e a f a m i l y of m u l t i p l i c a t i v e seminorms g e n e r a t i n g

-1 (0). Since P /R+e and t h e map x+x+N from X i n t o t h e f a c t o r a l g e b r a X i s P P continuous, it follows t h a t +N + e + N in X But X is a P P P' P Banach a l g e b r a , s o i n v e r s i o n i s c o n t i n u o u s and i t f o l l o w s t h a t t h e t o p o l o g y on X and f o r each

p E P,

-'+eiN

I n o t h e r words P P' Since p i s a r b i t r a r y , it follows t h a t

%-l+N

= (g+Np)

let

Since t h e algebr a

C(T,E,c),

N =p

$ (a l - e i N p )

=p (B-l-e) + O f

ld-'+e.V

T non-compact,

completely

r e g u l a r , and H a u s d o r f f , of Example 4.8-1

i s an LMC a l g e b r a , it

follows t h a t i n v e r s i o n i s continuous i n

C(T,E,c);

n o t a Q - a l g e b r a however, as i s p o i n t e d o u t t h e r e .

C(T,_F,c)

is

4 . COMMUTATIVE TOPOLIGICAL ALGEBRAS

210

4 . 9 T o p o l o g i c a l D i v i s i o n A l g e b r a s and t h e Gelfand-Mazur Theorem A d i v i s i o n a l g e b r a i s a n o t n e c e s s a r i l y commutative a l g e -

b r a i n which each nonzero element h a s a n i n v e r s e . Gelfand theorem,Theorem 4 . 9 - 1 ,

The Mazur-

f o r a l g e b r a s w i t h continuous i n -

v e r s e shows t h a t t h e r e i s e s s e n t i a l l y o n l y one complex l o c a l l y convex Hausdorff verse.

(LCH) d i v i s i o n a l g e b r a w i t h c o n t i n u o u s i n -

Both p a r t s o f t h e theorem remain t r u e when o u r g e n e r a l

assumption of commutativity i s l i f t e d and one needs o n l y t o c a r e f u l l y p r e s e r v e t h e o r d e r i n which t h i n g s are w r i t t e n down t o see t h a t t h i s i s so.

Something which is e s s e n t i a l t o t h e

v a l i d i t y o f t h e theorem, however i s t h e f a c t t h a t t h e a l g e b r a s The r e a l LCH t o p o l o g i c a l d i v i s i o n a l g e b r a

b e complex.

i s c e r t a i n l y n o t t o p o l o g i c a l l y isomorphic t o

u s u a l topology f o r example.

2,

The q u a t e r n i o n s t o o c o n s t i t u t e a r e a l t o p o l o g i -

cal d i v i s i o n algebra--in

f a c t a Banach a l g e b r a , hence a topo-

l o g i c a l a l g e b r a w i t h c o n t i n u o u s i n v e r s e by (4.8-aa)--which d i s t i n c t from

i n its

is

C.

I t w a s f i r s t e s t a b l i s h e d by F r o b e n i u s t h a t any f i n i t e - d i m e n t i o n a l d i v i s i o n a l g e b r a o v e r t h e r e a l numbers i s ( t o p o l o g i c a l l y ) i s o m o r p h i c - t o E,C, o r t h e q u a t e r n i o n s . e r a l i z a t i o n s of t h i s r e s u l t w e r e made by

s.

Subsequent gen-

Mazur (19381, G .

S i l o v ( 1 9 4 0 ) , and I . G e l f a n d , f o r normed a l g e b r a s .

In addition

t o e s t a b l i s h i n g t h e r e s u l t f o r complex LCH a l g e b r a s (Theorem 4 . 9 - 1 ( b ))

,

Arens ( 1 9 4 7 a ) a l s o proved t h a t e s s e n t i a l l y t h e o n l y

r e a l LCH t o p o l o g i c a l d i v i s i o n a l g e b r a s w i t h c o n t i n u o u s i n v e r s e a r e II,&, and t h e q u a t e r n i o n s , p a r t of which i s e s t a b l i s h e d i n Theorem 4 . 9 - 2 . The c l a s s i c a l L i o u v i l l e theorem s t a t e s t h a t a bounded en-

4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELPAND-MAZUR THEOREM

tire function must be constant.

211

With s u i t a b l e a n a l o g s f o r

"bounded" and " e n t i r e " w e prove a v e r s i o n of L i o u v i l l e ' s theo-

r e m f o r e n t i r e vector-valued ((4.9-1))

functions

x:C_+X.

This version

i s needed t o prove o u r main r e s u l t , Theorem 4.9-1.

I f X i s a t o p o l o g i c a l v e c t o r s p a c e and G a s u b s e t of then

i s bounded i f x(G) i s a bounded s u b s e t of X I

x:G+X

'bounded' s u b s e t o f X i n t h e s e n s e t h a t it i s a b s o r b e d by any neighborhood o f 0 . Definition 4.9-1

ANALYTICITY.

L e t G b e an open s u b s e t of t h e

complex p l a n e and X b e a t o p o l o g i c a l v e c t o r s p a c e . x:G+X i s a n a l y t i c i n G i f t h e l i m i t x'

(u0)

e x i s t s a t each

lim

The map

x(')-x(uo)

-

u-uo

P+U0

p EG. 0

I n t h e v e r s i o n of L i o u v i l l e ' s theorem t o follow, it i s i m p o r t a n t t h a t t h e v e c t o r s p a c e p o s s e s s s u f f i c i e n t l y many c o n t i n uous l i n e a r f u n c t i o n a l s .

I n p a r t i c u l a r w e want t h e t o p o l o g i c a l

v e c t o r s p a c e X t o have enough c o n t i n u o u s l i n e a r f u n c t i o n a l s so t h a t t h e i n f o r m a t i o n t h a t each c o n t i n u o u s l i n e a r f u n c t i o n a l v a n i s h e s on a c e r t a i n v e c t o r i s enough t o g u a r a n t e e t h a t t h e v e c t o r i s 0.

Whenever a subspace S o f l i n e a r f u n c t i o n a l s on a

v e c t o r space has t h e p r o p e r t y t h a t

f (x)=O

for a l l

p l i e s x=O, t h e n t h e subspace i s c a l l e d t o t a l . the set X'

fES

im-

In particular

of a l l c o n t i n u o u s l i n e a r f u n c t i o n a l s on a l o c a l l y

convex Hausdorff s p a c e i s always t o t a l , as shown by t h e HahnBanach theorem.

(4.9-1) L I O U V I L L E ' S THEOREM.

L e t X be a topological vector

s p a c e and suppose t h a t i t s d u a l X ' i s t o t a l .

If

x:s+X

is

e n t i r e and bounded, t h e n x m u s t b e c o n s t a n t . Proof.

S i n c e t h e c o n t i n u o u s l i n e a r image of a bounded s e t i s

4 . COMMUTATIVE 'POPOLOGICAL ALGEBRAS

212

bounded, if x is bounded, then so is fx for any

fEX'.

Thus

the standard Liouville theorem applies to each of the entire functions fx and we conclude that fx is constant for each fcX'. Hence for any

p,AFC,

f(x(p))

=

f(x(A)),

f(x(u)-x(A))=O.

so

Holding u and A fixed and noting that the last equality holds for every f in a total set of linear functionals, we conclude that

x(p)

=

Since p and A are arbitrary, the constancy

x(A).

of x is pr0ved.V Theorem 4.9-1 COMPLEX LCH DIVISION ALGEBRAS WITH CONTINUOUS IN(Gelfand-Mazur) Let X be a complex LCH algebra with

VERSE.

continuous inverse.

Then (a) for any x E X, u(x) # pI and (b)

if each non-zero element in X has an inverse, X is topologically isomorphic to

5.

x E X, suppose a(x) = B. Then for each ,C, -1 rx(A) = (x-Ae) , the map A+r X (A) is seen to be en-

Proof.

(a) For

letting

tire by part (b) of ( 4 . 8 - 5 ) .

We show that

rx(A)

is bounded.

T o this end, consider the filterbase formed by the sets

{A EC2I I h l 2 n}, n=1,2,

... .

For any seminorm

p E P,

-

Bn -

where

P generates X's topology, A E Bn, and any x E X, p(A-'x) = -1 -1 I A1 p(x) 5 n p(x). Thus, letting la-' = (Bn-'), it follows -1 that (since p is arbitrary) @-'x+O. Hence e-I, x+e. Since -1 X has continuous inverse and since e - A x must be regular for any plies

A # 0,

p((e-B-'x)-l)+p(e)

Thus, for (*I

it follows that

E

(e-*'x)-'+e

for any

-1

=e which im-

p E P.

> 0, n sufficiently large and

p((x-he)-')=~~~-lp((e-x/~)-')

the upshot of it all being that sufficiently large.

1AI

2 n,

< l~~-lp(e)+E 5 (l/n)p(e)+E,

p(rx(A) )

is bounded for

Since rX is a continuous function,

1 XI

4 . 9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM

p(rx(A))

IAl 5 n.

i s c e r t a i n l y bounded f o r

bounded s u b s e t of t h e LCHS X.

since p is

NOW,

a r b i t r a r y , i t f o l l o w s t h a t rx i s bounded, i . e .

213

rx(C)

is a

That i s rx i s bounded e n t i r e

f u n c t i o n , so i t m u s t b e c o n s t a n t by L i o u v i l l e ' s theorem (4.9-1). Certainly and ( * )

p-rx

implies t h a t

i s a bounded f u n c t i o n t o o f o r any p ( r x ( A ) ) = O f o r each

s.

Since p i s

rx(A)=O

for

A E

a r b i t r a r y and X i s H a u s d o r f f , it f o l l o w s t h a t

p E P

each A which i s a b s u r d f o r how can 0 be t h e i n v e r s e of anything?

Thus t h e assumption

o(x)=g has led t o a contradiction

( a ) , follows.

and t h e d e s i r e d r e s u l t ,

( b ) According t o t h e r e s u l t j u s t proved i n ( a ) , g i v e n any

x E X , x-Ae

i s s i n g u l a r f o r some A .

ment i s r e g u l a r , i t m u s t b e t h a t x=Ae

f o r some A .

Thus

X=se

But i f each non-zero e l e -

x-Ae=O

f o r some A ,

and t h e map

x=Ae+A

an a l g e b r a isomorphism from X 0 n t o . S . meomorphism as w e l l ,

t h e above

A s f o r i t s b e i n g a ho-

Since P i s s a t u r a t e d , b a s i c

neighborhoods of 0 a r e of t h e form a V x = h e € aV iff P t r i v i a l seminorms p ,

is clearly

suppose P i s a s a t u r a t e d f a m i l y of s e m i -

norms which g e n e r a t e X ' s t o p o l o g y .

Now

that is,

P

where

p(Ae)=IAlp(e) < a.

a > 0

and p~ P .

T h a t i s , f o r non-

aV =CAel 1 x 1 < a / p ( e ) } and t h e f a c t t h a t P map i s a homeomorphism i s now c1ear.V

A s a consequence of Theorem 4 . 9 - 1 LMC a l g e b r a h a s c o n t i n u o u s i n v e r s e

LMCH d i v i s i o n a l g e b r a i s

s.

and t h e f a c t t h a t any

[(4.8-6)],

W e can go s l i g h t l y f u r t h e r and s a y

t h a t t h e o n l y complex LMC d i v i s i o n a l g e b r a i s it s u f f i c e s t o show t h a t

t h e o n l y complex

a(x)#$

s.

To see t h i s ,

f o r any x i n a complex LMC

division algebra X (with nontrivial topology).

A s t h e topology

on X i s n o t t h e t r i v i a l t o p o l o g y t h e r e e x i s t s a p r o p e r b a l a n c e d

4 . CONMUTATIVE TOPOLOGICAL ALGEBRAS

2 14

m-convex neighborhood of 0 i n X.

Consequently t h e r e e x i s t s a

n o n t r i v i a l m u l t i p l i c a t i v e seminorm p on X ,

and an a s s o c i a t e d

S i n c e X i s a Banach a l g e P' P x E X by t h e p r e v i o u s theorem.

f a c t o r a l g e b r a (See Sec. 4 . 5 )

X

o(x+N # 0 f o r any P i s a n o n t r i v i a l homomorphism from X i n t o S i n c e t h e map x+x+N P X t h e n p E o(x+Np) i m p l i e s 1-1 E a ( x ) PI For a t i m e i t seemed t h a t t h e r e w e r e p r o b a b l y no complex

bra,

.

t o p o l o g i c a l d i v i s i o n a l g e b r a s ( c f . Kaplansky, 1948, p. 8 1 1 )

2.

o t h e r than

Williamson (1954) however showed t h a t t h i s was

n o t t h e c a s e by e x h i b i t i n g a topology f o r t h e a l g e b r a q u o t i e n t f i e l d of t h e polynomial a l g e b r a

s[t]of

c ( t ) ,t h e

polynomials i n

t w i t h complex c o e f f i c i e n t s , which i s c o m p a t i b l e w i t h t h e a l g e b r a i c operations.

W e p r e s e n t t h i s c o n s t r u c t i o n i n o u r n e x t ex-

ample. Example 4 . 9 - 1 FROM

2.

Let

A COMPLEX TOPOLOGICAL DIVISION ALGEBRA DISTINCT

M(0,l)

b e t h e s p a c e of a l l Lebesgue measurable

f u n c t i o n s on (0,l) t h a t assume f i n i t e complex v a l u e s a l m o s t everywhere.

The q u o t i e n t f i e l d o f t h e a l g e b r a of polynomials

may b e i d e n t i f i e d w i t h t h e class o f f u n c t i o n s of i n t , _C(t), M(0,l)

c o n s i s t i n g of r a t i o s of polynomials w i t h complex co-

efficients.

B(k,&)

=

L e t @ b e t h e f i l t e r b a s e of sets of t h e form

i f € M ( 0 , l ) ( m ( I ( l f l 2 k))
0

and

m ( I ( ( r 1 > k)

=

E

0.

> 0.

r(t) = Then f o r any

Thus m ( C t E (0,l)1

m

r ( t ) # 0 ) ) 5 . Z m ( I ( l r 1 > l / i ) )= 0 . Hence p ( t ) = 0 a l m o s t 1=1 everywhere on (0,l) so by t h e c o n t i n u i t y of p ( t ) on ( O , l ) , p ( t )

i s i d e n t i c a l l y zero.

*

The topology of convergence i n measure, however i s n o t l o c a l l y convex.

To prove t h i s it i s s u f f i c i e n t t o prove t h a t

i n v e r s i o n is continuous a t e ( t h e function i d e n t i c a l l y e q u a l t o 1 on ( 0 , l ) ) f o r i f w e assume t h a t

g(t)

i s l o c a l l y convex t h e n

t h e a b s u r d c o n c l u s i o n f o l l o w s , by Theorem 4.9-l(b) t h a t C_(t)i s isomorphic t o

5.

To show t h a t

*On a l l of M ( O , l ) ,

g ( t ) has

continuous i n v e r s e l e t

t h e topology of convergence i n m e a s u r e i s

n o t a Hausdorff t o p o l o g y , because t h e r e a r e non-zero measurable f u n c t i o n s t h a t a r e z e r o a l m o s t everywhere. T o a v o i d t h i s probl e m one u s u a l l y d e f i n e s M ( 0 , l )

t o be t h e c l a s s of a l l equiva-

l e n c e classes of measurable f u n c t i o n s on (0,l) g e n e r a t e d by t h e e q u i v a l e n c e r e l a t i o n of e q u a l i t y a l m o s t everywhere. T h e equival e n c e classes c o n t a i n i n g a r a t i o n a l f u n c t i o n c o n t a i n e x a c t l y one r a t i o n a l f u n c t i o n a n d , s ( t ) may s t i l l be i d e n t i f i e d w i t h a s u b a l g e b r a of M ( 0 , l ) .

Furthermore, i f f E B ( k , E ) t h e n s o d o e s

e v e r y o t h e r f u n c t i o n i n t h e e q u i v a l e n c e c l a s s g e n e r a t e d by f .

4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM

B(k,E)

€a;we

shall exhibit k' > 0

+ B(k',E).

B (k,E ) whenever

f E e

k' satisfies

(t E (0,1)If(t) #

(*

such that

217

l/f E e +

This will be accomplished if

f3)n I(ll/f-el>- k)

c

I(lf-el 1 k').

Treating k' as an unknown and assuming that

f(t) # 0 while

If(t) - 11 < k'

1 (l/f (tl) (1-f(t ) I

it follows that

ll/f(t)//f(t)- 11 < /l/f(t)/k'

=

sure that the above inclusion holds we set Since have

Il/f(t)Ik' 5 k k' < k/(l+k).

Il/f(t)

k' 5 If(t)Ik

iff

I l/f (t)Ik'

and

Any such k' satisfies

.

-

I/=

To in< k. -

If(t)I 2 1 - k ' we

(*).

One might wonder: could there exist locally convex complex division algebras other than C?

There can, and this is dis-

cussed in Exercise 4.6. For multiplication to be continuous in CJt)

endowed with a

linear topology3,Jcan be neither too coarse nor too fine: is discussed in Exercise 4.7,

if

As

is locally convex, it cannot

be a weak topology--i.e. there can be no linear space Y such that

(g(t),Y) is a dual pair for which-7 = o(C(t) ,Y)--and

multiplication is discontinuous when g(t) carries the finest locally convex topology. In addition to Theorem 4.9-1 being interesting in its own right it plays an important role in the development of "Gelfand theory", roughly the consequences of topologizing the set of maximal ideals of an algebra in a certain way (see Sec. 4.10 and Sec. 4.12)).

*Can't:

Thus the sets of

63

may be considered as being com-

posed of equivalence classes rather than just functions, thereby making M ( 0 , l )

a Hausdorff topological algebra in the topo-

logy of convergence in measure.

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

218

The analog of Theorem 4.9-1 for real algebras is presented next.

As usual, consideration is restricted to commutative al-

gebras.

If the (real) algebra is non-commutative however, it

must be the quaternions, as discussed in Exercise 4.8. Theorem 4.9-2

REAL LCH FIELDS WITH CONTINUOUS INVERSE.

Every

real LCH algebra X with continuous inverse in which each nonzero element has an inverse is topologically isomorphic to either ,R or &. iff for all

Furthermore X is topologically isomorphic to g

x,y

E

is called formally Proof.

X, x2+y2=0 implies

x=y=O; in this case

X

&.

We consider separately the cases when X is formally

real and when it is not.

When X is formally real, we introduce

operations to

XxX

(identical in form to the operations one

introduces to

gxg

to form

c) which make

it a complex LCH al-

gebra with continuous inverse and then apply Theorem 4.9-1 to conclude that

XxX

is topologically isomorphic to

striction of this topological isomorphism to to be topologically isomorphic to

E.

X x {O)

Suppose that X is formally real and let and

a+ib

(a+ib)(x,y)=(ax-by,ay+bx)

E

2 and

re-

shows X

If X is not formally real

then X is shown to be topologically isomorphic to

(x,y), (w,z)EY

s; the

we define

2.

Y = XXX.

For

(x,y)+(w,z)=(x+w,y+z)

(x,y)(w,z)=(x~-yz,yw+xz). With

these operations and the product topology Y is easily seen to be a complex LCH algebra.

(To verify that complex scalar mul-

tiplication is continuous, it is helpful to observe that the map

(x,y)+(y,-x)

is continuous.)

To see that Y is a field,

note first that (e,O) is the multiplicative identity of Y and 2 2 suppose that (x,y)#(O,O). Then x +y # 0, (x2+y2)-l exists in

4.9 TOPOLOGICAL D I V I S I O N ALGEBRAS AND THE GELFAND-MAZUR THEOREM

219

2 2 -1 2 2 -1 X and (x,y)[(X,-Y)((x +y ) , O ) l = [(x,Y) (x,-y)l ((x +Y 1 , o ) 2 2 = (x +y , O ) ((x2+y2)-',O) = (e,O) so that (x,y) is invertible and Y is seen to be a field. In order to be able to call upon Theorem 4.9-1 to conclude that Y is topologically isomorphic to show that Y has continuous inverse.

G I it only remains to

To this end suppose that

(xply,,)) is a net convergent to (e,O). Then x +e and y +O. P lJ -1 2 2 -1 As (X,,iYP) = (xyi-Y,,) ((x +y,, ) ,O) for each 1-1 and X has (

continuous inverse, it follows that

-

(xPlyP)'+(e,O)

as well.

Thus Y has continuous inverse and is topologically isomorphic to

2 as

established by the map

Theorem 4.9-1. Xx{O)--it

a+ib+(a+ib)(e,O) = (ae,be) of

By restricting the map to X--more precisely to

follows that X is topologically isomorphic to

What if X is not formally real? X,YEX

such that

x2+y2-0.

E.

i.e. there are nonzero

If so, let

j=xy-'

and extend mul-

tiplication to multiplication by complex scalars as follows: (a+ib)z

=

az + b(jz)

( z c X).

is now a complex LCH algebra.

It is easily verified that X Theorem 4.9-1 may now be applied

and it follows that X is topologically isomorphic to

2.V

Following the proof of Theorem 4.9-1 we remarked that the only complex LMC division algebra with nontrivial topology is C.

I

An analogous statement can be made for real algebras: the

only real LMC fields with nontrivial topology are

E and 2.

Just as in the complex case, the fact that the topology on the real LMC field X is nontrivial guarantees the existence of a proper balanced m-convex neighborhood of 0 in X.

Thus there is

a nontrivial multiplicative seminorm p on X and an associated factor algebra X P'

Now

X/N =z(e+N 1 P P

where

s=E

or C, and the

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

220

mapping

v ( e + N p ) +u

t h a t t h e mapping

i a l homomorphism. verses,

h

-1

i s a t o p o l o g i c a l isomorphism.

I t follows

,

x+x+N = p ( e + N 1 - q ~ i s a n o n t r i v P P S i n c e a l l nonzero e l e m e n t s o f X have i n -

h:X+X/Np+x

(O)={O)

and h i s an isomorphism.

That h i s con-

t i n u o u s i s c l e a r ; t h a t i t i s a l s o open f o l l o w s from t h e openn e s s of t h e c a n o n i c a l homomorphism

x+x+N P' Maximal i d e a l s and Homomorphisms.

4.10

A p r o p e r i d e a l which i s n o t p r o p e r l y c o n t a i n e d i n any

o t h e r p r o p e r i d e a l i s a maximal i d e a l .

I f I i s a proper i d e a l ,

a s t r a i g h t f o r w a r d Z o r n ' s lemma argument shows t h a t t h e r e exi s t s a maximal i d e a l c o n t a i n i n g I. s i n g u l a r element then

In particular, i f x is a

(x)=xX i s an i d e a l c o n t a i n i n g x c a l l e d

t h e p r i n c i p a l i d e a l g e n e r a t e d by x.

Thus any s i n g u l a r e l e m e n t

I n t h i s s e c t i o n w e p r o v e two

i s c o n t a i n e d i n a maximal i d e a l .

b a s i c t o p o l o g i c a l r e s u l t s a b o u t i d e a l s i n c e r t a i n t y p e s of topo l o g i c a l a l g e b r a s and t h e n i n v e s t i g a t e t h e c o n n e c t i o n s between

maximal i d e a l s and homomorphisms.

W e a l s o look a t some exam-

ples. (4.10-1)

I f X i s a Q-algebra then ( a

IDEALS I N Q-ALGEBRAS.

t h e c l o s u r e of a p r o p e r i d e a l i s a p r o p e r i d e a l ;

( b ) maxima

i d e a l s are c l o s e d . Proof. X,

C l e a r l y w e o n l y need t o prove ( a ) .

x , y E c l I and z E X ,

I f I i s an i d e a l i n

it i s e a s y t o show by c o n s i d e r i n g

f i l t e r b a s e s on I c o n v e r g e n t t o x and y t h a t

x+y

The i m p o r t a n t p a r t i s t o show t h a t

S i n c e X i s a Q-

a l g e b r a , however, t h e r e e x i s t s a

sists e n t i r e l y of u n i t s .

In the algebra

Thus

C(€?,z,c)

cl IfX. V E V(e)

v n I=g

and

and xzE c l I.

such t h a t V con-

e k cl 1.7

as i n Example 4.8-1,

t h e set I

221

4.10 MAXIMAL IDEALS AND HOMOMORPHISMS

of continuous functions x which vanish outside some compact set G

X

constitutes a proper ideal which is clearly dense in

C(E,z,c).

Thus no maximal ideal containing I can be closed.

In the result which follows, we consider (maximal) ideals I of an LMC algebra X such that the quotient topology on X/I is

If I is a closed ideal (as is the

not the trivial topology.

case for maximal ideals in Q-algebras) , then the quotient topology on X/I is Hausdorff, so the quotient topology is certainly not trivial in this case. A weaker sufficient condition for nontriviality of the

quotient topology when X is LMC is that I not be dense in X. More generally, if H is a linear subspace of the locally convex space X and

cl HfX,

then the quotient topology on X/H is

not trivial and we now outline a proof of this fact.

Indeed if

H is not dense in X, then there is a convex neighborhood of the

origin V and an element x E X

such that

(x+V)

the quotient topology on X/H is trivial, then -x-V+H

E H

If

= (a.

(x+H) + V = X =

(since (x+V)+H is a neighborhood of x+H in X/H).

follows that there exist elements v,w E V w/2

nH

such that

It

x+v/2+

n (x+V), and this is a contradiction. As was mention-

ed after (4.10-1),

C(s,;,c)

is an algebra containing dense

ideals. (4.10-2) QUOTIENTS OF MAXIMAL IDEALS IN LMC ALGEBRAS.

In the

event that X is a (commutative) real LMC algebra and M is a maximal ideal in X such that the quotient topology on X/M is not trivial then X/M is topologically isomorphic to either C.

z

or

In particular if X is a Banach algebra then X/M is topolo-

gically isomorphic to f3. or

if X is a real algebra or just

5

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

222

i f X i s a complex a l g e b r a .

Furthermore t h e q u o t i e n t topology

on X/M i s induced by t h e i n f norm: IIx+M[I = infmcM(ix+m[l. A s r e g a r d s t h e main a s s e r t i o n a l l w e need do i s a p p l y

Proof.

t h e remarks f o l l o w i n g Theorem 4 . 9 - 1

and Theorem 4.9-2

t o the

I f X i s a Banach a l g e b r a t h e n it i s

LMC d i v i s i o n a l g e b r a X/M.

LMC and, b e i n g a l s o a Q - a l g e b r a [ ( 4 . 8 - 2 ) ] , t h e maximal i d e a l M

i s c l o s e d i n X from which it f o l l o w s t h a t t h e q u o t i e n t topology on X/M i s H a u s d o r f f , hence n o n t r i v i a l .

I t i s easy t o v e r i f y

t h a t t h e q u o t i e n t t o p o l o g y i s induced by t h e i n f n0rm.v A s mentioned b e f o r e

(4.10-2)

i t s u f f i c e s f o r t h e maximal

i d e a l M t o be c l o s e d t o g u a r a n t e e t h a t t h e q u o t i e n t t o p o l o g y b e nontrivial.

I n a Banach a l g e b r a ,

all

maximal i d e a l s a r e

c l o s e d b u t t h i s i s n o t g e n e r a l l y t r u e f o r LMC a l g e b r a s .

There

may b e non-closed maximal i d e a l s even i n F r e c h e t a l g e b r a s , as shown by t h e d i s c u s s i o n of property--existence

C(E,g,c)

a f t e r (4.10-1).

This

of non-closed maximal i d e a l s - - c o n s t i t u t e s ,

t h e r e f o r e , a major d i f f e r e n c e between LMC a l g e b r a s and Banach algebras. Notation.

"Homomorphism" h e r e means "complex o r r e a l homomor-

phism", depending on whether t h e a l g e b r a i s r e a l o r complex,fl ( o r M ( X ) ) d e n o t e s t h e maximal i d e a l s of an a l g e b r a

NC ( X ) )

x, Ncr

(or

d e n o t e s t h e c l o s e d m a x i m a l i d e a l s of a t o p o l o g i c a l a l -

g e b r a X and Xh d e n o t e s t h e n o n t r i v i a l c o n t i n u o u s homomorphisms of t h e t o p o l o g i c a l a l g e b r a X .

Note t h a t Xh

c X'

,

t h e continu-

ous d u a l of t h e t o p o l o g i c a l v e c t o r s p a c e X . Our f i r s t r e s u l t c o n n e c t i n g t h e n o t i o n s o f maximal i d e a l and homomorphism i s obvious. (4.10-3)

MAXIMAL I D E A L S AND HOMOMORPHISMS.

The k e r n e l of any

4.10 MAXIMAL IDEALS AND HOMOMORPHISMS

223

non-trivial homomorphism of any algebra X is a maximal ideal. Clearly the kernel of any non-trivial continuous homomorphism is a closed maximal ideal in any topological algebra.

It

follows immediately from (4.10-2) and the remark following (4.10-2) that any closed maximal ideal in an LMC algebra is the kernel of a continuous homomorphism.

We state these facts in

our next result. (4.10-4)

MC

IN LMC ALGEBRAS.

then a maximal ideal

McX

If X is a complex LMC algebra

is closed iff M is the kernel of

some continuous complex-valued homomorphism.

If X is a real

LMC algebra then M is closed iff M is the kernel of a real-valed homomorphism or a complex-valued homomorphism.

Thus for

complex LMC algebras there is a 1-1 correspondence betweenNc, h the closed maximal ideals, and X , the continuous homomorphisms, namely that established by pairing M with the homomorphism

x+x+M.

Frequently it will be convenient to identify when X is a complex LMC algebra. of

M c "with

a(X',X)-topology".

Mc and

h X

Thus, for example, we speak An examination of the basic

neighborhoods of 0 shows that this is the weakest topology on with respect to which each of the Gelfand maps h x:M,+S, M+x+M, (or x -+€, f+f (x)) continuous; the topology h induced by o ( X ' ,X) on X (or A,) is called the Gelfand top-

A

ology. In Q-algebras all maximal ideals are closed [(4.10-1)1, in complex LMC Q-algebras each non-trivial homomorphism continuous by (4.10-4).

SO

is

Furthermore, if X is a Banach algebra,

real or complex, and f is a non-trivial homomorphism (real-val-

4. COMMUTATIVE TOPOLOGICAL ALGEBRAS

224

ued if X is real, complex-valued if X is complex) on X I then for any

xc X

f(x)E u(x).

Ib 11

we have

f(x-f(x)e) = 0

which implies that

Thus, by the remark following, (4.8-41, If(x)I 5

for every x and it follows that

IJf11 _c 1.

We summarize

these facts in our next result. (4.10-5) HOMOMORPHISMS OF LMC Q-ALGEBRAS ARE CONTINUOUS.

is an LMC Q-algebra, all homomorphisms are continuous.

If X

Fur-

thermore, each homomorphism on a Banach algebra is not only continuous, it has norm less than or equal to one. Questions concerning the continuity of homomorphisms of topological algebras into topological algebras are treated in Section 4.13.

In particular a result similar to (4.10-5) is

established in Theorem 4.13-1 where it is shown that any homomorphism of a complete barreled LMCH Q-algebra into a strongly semisimple fully complete LMCH algebra is continuous. Unfortunately (4.10-5) does not remain true for an arbitrary LMC algebra.

Our next result is of critical importance

in Example 4.10-1 where we construct a whole class of LMCH algebras on which discontinuous homomorphisms exist. (4.10-6) DISTINCT HOMOMORPHISMS ARE LINEARLY INDEPENDENT. Any

collection of distinct non-trivial homomorphisms of an algebra X is linearly independent.

Proof.

Suppose

{fl,...,fn}

is a linearly dependent set of

distinct non-trivial homomorphisms of X into

which is mini-

mal in the sense that the removal of any one homomorphism from the set leaves a linearly independent set. n#l.

Suppose that

there exists y e X

C%fi=O

where no

such that

It is clear that

cii=O.

fn(y) # fl(y).

Since

fnffl Holding y fixed

225

4.10 MAXIMAL IDEALS AND HOMOMORPHISMS

and permitting x to be any vector in X we see that Cn a.f.(xy) = C n a.f.(y)fi(x) = 0, and fn(y)C:=laifi(x) = i=l 1 1 i=l 1 1 n Ci=laifn(y)fi(x) = 0. Subtracting the second equation from the n first, we have Ci=lai(fi(y)-fn(y))fi(x) = 0 for all x~ X. Since

fl(y)-fn(y) # 0

the last equation implies that

{fl,...,fn-l}

is linearly dependent which condradicts the

minimality of

Ifl,

...,fn).V

Example 4.10-1 DISCONTINUOUS HOMOMORPHISMS.

Let X be an alge-

bra and H be a family of non-trivial homomorphisms on X with the property: {O},

where

(*)

there exists

Ho=H-(fo).

foE H

such that

In this case f0

o u s homomorphism when X carries

p

ker f = Ho will be a discontinu-

u(X,Ho).

There are many algebras which satisfy ( * ) .

In particular

if X is the complex algebra of continuous functions on C([a,bl,c,c)

[arb],

with sup norm topology, then let H be the collec-

tion of homomorphisms determined by the evaluation maps t*:C [a,bl +E that

(*)

(tE [a,bl)

,

x+x(t)

and fo to be to*.

To see

is satisfied it is enough to note that a continuous

function x which vanishes for all ish at to.

tc [a,b]-{to) must also van-

Since the essential feature of to here is that it

is not an isolated point of

[a,b] it is easy to see that,

more generally, we can take x to be any B-* algebra, i.e. an algebra of continuous complex-valued functions C(T,S,c)

on a

compact space T with sup norm topology, and to to be any point

of T which is not an isolated point. Letting follows by

(*)

[Ho] denote the linear span of Ho in X*, that

(X,[Ho])

is a dual pair.

it

Furthermore,

since the seminorms pf defined at each xc X by pf(x)=lf(x) 1 ,

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

226

fc Hor (X,U(Xr

[Hol

which generate [Hol))

a(Xr

[Hal),

are clearly multiplicative,

is an LMCH algebra.

In view of that fact that

( x , u ( x , [Hal)) , it fol-

is the continuous linear dual of

lows that any non-trivial continuous homomorphism g of X must belong to Ho U { g }

[Ho].

g ,( Ho

If we assume that

itself then

is linearly independent by (4.10-6) and, therefore,

[Ho] is a proper subset of [Ho u { g } ] . But this is ridiculous, S O g € Ho. Thus Xh= H 0 and, consequently, ( o f Xh

.

As it happens the topological algebras of Example 4.10-1 are rarely, if ever, complete (see Exercise 4.5).

An example

of a complete LMCH algebra on which discontinuous homomorphisms exist is afforded by

C(T,E,c)

T=[O,Q) and f2 is the

where

first uncountable ordinal.

T is not replete (Example 1.5-1) so

there are homomorphisms of

C(T,E,c)

which are not evaluation

maps by (1.6-1). But a l l the continuous homomorphisms of C(T,R,c) are evaluation maps as shown in Example 4.10-2, so discontinuous homomorphisms exist on open subset of the compact space

C(T,E,c).

[O,Q],

it is locally compact

and therefore also a k-space (Exercise 2.2(b)). rem 2.2-1,

C(T,_R,c) is complete.

As T is an

Hence by Theo-

Other examples of complete

LMCH algebras on which discontinuous homomorphisms exist are given in Exercise 2.2(g). relationship between

uM=uAC

Nevertheless there is still a close

M and

(see (4.10-9) and

Mc

in complex LMCH algebras; e.g.

r$t=nMc

(see (4.11-1)).

Our next result is another example of how information about an LMC algebra can be gleaned from the factor algebras. It asserts that the continuous homomorphisms of an LMCH algebra can be "obtained" from the (continuous) homomorphisms of a

227

4.10 MAXIMAL IDEALS AND HOMOMORPHISMS

set of factor algebras.

First we state some conventions about

certain things in topological vector spaces. If X is a topological vector space then the continuous dual X' is the linear space of all continuous linear function-

als on X.

The polar Bo of a subset B of X is the set of x'EX'

such that

supI 1

Then the contradictory facts follow that

is unbounded while

1

for each

XE

v,

xnc

b,

and

for all 0 fc Xh n V,

nc

E.

.

(c) It is easy to see that Xh is closed in the continuous dual X' of X so that the

u(X' ,X)-compactness of

[the polar

lJ

of a neighborhood of 0 in any topological vector space X is always

D(X',X)-compact

by Alaoglu's theorem] and part (a) combine to show that KIJ'(X,h ) is o(X',X)-compact. As Xuh is Hausdorff in its Gelfand topology it suffices to show that K

,'(f,)+f,,

is well-defined to conclude that it is a homeomor-

phism (it is clearly injective). K

lJ

' (glJ)

SO

for each

that

x+N

CI

f

f (x+N

X/N

P

. u

)=(K

?J

'

Thus suppose that

U

'(f

,

P

) =

(f 1 ) (x)=(K,' (g

1 ) (x)=g (x+N 1

is dense in X

f =g

lJ

I J l J

As X/N

K

,*

)-I

,

1.I

and the

lJ

proof is comp1ete.V As an application of (4.10-7), in our next example we determine all the continuous homomorphisms of C(T,E,c) is a completely regular Hausdorff space.

when T

Note that the result

is trivial if T is replete since in that case all the homomorphisms of

C(T,_F)

into

X

are evaluation maps by (1.6-1) and

evaluation maps are always continuous when

C(T,E)

carries the

compact-open topology. Example 4.10-2 C ( T , E , c ) h=T* FOR COMPLETELY REGULAR HAUSDORFF T. Since T is a completely regular Hausdorff space, a set of fac-

4.10 MAXIMAL IDEALS AND HOMOMORPHISMS

tor algebras for C(G,E,c),

is the set of Banach algebras

where G is a compact subset of T [see Example 4.5-11

We claim that can show that this let

C(T,E,c)

229

C(T,g,c) h=T* h C(G,F,c) =G*

follows from (4.10-7) provided we for each compact set G.

To see

f € C(T,E,cIh; then by (4.10-7) and the statement

"C(G,Z,c)h=G* and a point

for each compact C" , there is a compact set G c T

tc G

such that for any

x~ C(T,lJ,c)

f(x) =

(KG'(t*)(X)=t*(KG(X) )=t* (Xl,)=X(t). To show that C(G,X,c) h=G*, when G is compact, it suffices to note that G* constitutes all the homomorphisms, for all homomorphisms on a Banach algebra are continuous, [(4.10-5)]. That

G*

constitutes all the homomorphisms follows immediately

from (1.6-1). V

Example 4.10-3 Hh =D*.

Consider the LMCH algebra H of analytic

functions on the open unit disc D of the complex plane carrying the compact-open topology (See Examples 4.5-2 and 4.7-2).

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

230

We have already seen (Example 4.5-2) that H is a Frechet algebra with a set of factor algebras

(P(Cn))

where P(Cn) con-

sists of all uniform limits of polynomials on l-l/n}

endowed with sup norm topology.

cn={tEgl It[_
0

in X.

P

where

If

XEI

PEP

such that

VP=ix6Xlp(x) < 1). and

YEN^

then

I n (e+EVp)=g

Consider the ideal

p (x+y-e)=p (x-el2 E.

(I+N ) n (e+eV ) = # and J=I+N is not dense. Now deP P P noting the factor algebra (Def. 4.5-1) associated with p by X P‘ it is clear that clX ( K ( J )) is an ideal in X (where K is the P P canonical homomorphism of X into X 1. Moreover we claim that Hence

P

it is proper.

Indeed if we make the assumption to the contrary

then

e+N E clX ( K ( J ) ) . Thus, with E as above, there must be P P elements X ~ E I and YEN such that (; (xE+y+Np)-(e+N ) ) < E. P P However p (xE-e)=p ( (xE+N ) - (e+N ) =fi ( (xE+y+N ) - (e+N 1 ) < E and P P P P this is a contradiction. Since clX ( K (J)) is a closed proper P ideal in the Banach algebra X there is a (perforce closed) P -1 maximal ideal M 3 clX ( K ( J) ) We contend that M=K (Mp) is

.

p

P

the desired maximal ideal.

Certainly M is a closed ideal con-

taining I; it only remains to show that it is maximal. h be the surjective complex homomorphism of X

P

Letting

determined by

it immediately follows that M=~-’(h-’(0)) and M is the MP kernel of the nontrivial homomorphism h-K. Thus M is maximal and the proof is comp1ete.V As mentioned earlier, an immediate consequence of (4.10-9) is that any proper principal ideal in a complete complex LMCH algebra X can be embedded in a closed maximal ideal.

It is

natural to inquire if this is true for any finitely generated ideal

(zl,

...,zn)={Cx.z. ]xi€ X,i=l,...,n}. 1 1

In other words,

4.10 ItAXIMAL IDEALS AND HOMOMORPHISMS

when is

(zl,..

. ,zn)

proper?

233

With the added condition of me-

trizability--i.e. if X is a complex Frechet algebra--then this is so, as is proved in (4.10-12).

First, however, we must es-

tablish the following technical fact. (4.10-11). Let h be a homomorphism between the topological al-

gebras X and Y such that h (X) is dense in Y, and h (e)=e. Given

z1 ,

e1ements

...,zn E X,

exist such that

.

yl,.. ,ynE Y

(y5=h(ui)

there exist

yi+Ui

for

Proof. 1

for example) such that

yl,...,yn

and neighborhoods

vl,...,vnE X

x

such that

Cy.h(zi)=e. 1

ul, ".,Un

Cv.z.=e and 1 1

of

h(vi)E

i=l,...,n.

Suppose that

ul,...,u

1 1

i=l ,...,n,

3 3

1

n E X. Then setting it follows that Cv.z.=e

Cu.z.=e for

v.=w.+u.(e-Cw.z.1 for 1

.

ul,.. run€

Cu.z.=e. It follows that there exist 1 1

Then for any such 0,

suppose that elements

1 1

By the hypothesis

regardless of the choice of the w's. then, choosing Y1'

-.. Yn

such that

Cyih (zi)=el (a): h (vi)-yi=

(h(wi)-yi)+h(ui)h(e-C.w.z.) for each i. 3 3 3

Now let V and W be

balanced neighborhoods of the origin such that

V + V c Ui

for

Cn W c V. Note that h(u. )h (e-C .w.z. ) = i=l 1 3 3 3 h(ui) (e-C .h(w.)h(z.)). Since e=Cyih(zi) , the right-hand side

each i and 3

becomes

C

3

3

.h(ui)h ( z . ) (y .-h(w. )

3

3

3

3

)

.

Thus, as multiplication is

...,wn

continuous and h(X) is dense in Y we can choose wl, such that

h(ui)h(z.) (y.-h(w.))E W 3

.

i,j=l,.. ,n.

3

3

and

h(wi)-yiE V,

for

By (a) and the choice of V and W it follows that

h (vi)-yi E Ui. V (4.10-12).

EMBEDDING FINITELY GENERATED IDEALS IN CLOSED MAXI-

MAL IDEALS.

Let X be a complex Frechet algebra with topology

generated by the saturated family

(p,)

of seminorms (where it

2 34

4 . COPMJTATIVE TOPOLOGICAL ALGEBRAS

is assumed without l o s s of generality that pkl pk+l for each k) and with associated factor algebras (X ) . Then (a) k (Zlr rzn) is proper in X iff ( K ~ z ~ , . . . ~ K z ) is proper in k n Xk €or some k; (b) any proper finitely generated i d e a l can be

...

embedded in a closed maximal ideal. Proof.

(a) If

X ~ ~ . . . ~ EX

n

(KkZIP.*

.,K

..

( ' ~ ~ 2 ~

X.

(zl,...,zn)=X Hence

then

Cx.z.=e for some 1 1

( x . ) (z.)=e+Nk ~ and it follows that k i k i for each k. Conversely suppose that

k z n)=Xk K ' . z ) =Xk k n

CK

for each k. By an induction process we con-

struct a Cauchy sequence in Xk for each fixed i, 1 5 i 5 n, convergent to an element

(ui(k)) k,

ui (k) such that

element of the projective limit of the Xk's and e+Nk

for each

k l 0:

C

,

C.U. (k)Kk ( zi) = i i

We conclude the proof by invoking Theo-

rem 4.6-1 to obtain elements for each i and k as

is an

x =u i i

U.EX such that 1

K ~ ( u ~ ) = u(k) . 1

turns out to be a solution of

x . z . =e.

1 1 1

We remind the reader that

hrs(sl r)

sion by continuity to XS of the mapping rem 4.6-1) and

K~

choose y10,...ryn

denotes the exten-

x+NS +x+Nr

(see Theo-

the canonical homomorphism of X into Xr. 0 E

Xo

s u c h that

Ciyi0K (zi)=e+N 0

We

Pro-

0'

m m ceeding inductively we find that elements y1 r...ryn m' m exist by (4.10-11) such that Ciyi ~,(z.)=e+N and 1 m m m-1 fim-l(hm-l,m(~i)-yi ) < l/Zm (i=lr...rn). If m> k then,

Set in

m u. (krm)=hkm(yi ) 1 (*)

we obtain

for

m> k

and

1 5 i 5 n.

Substituting

4.10 PWXIMAL IDEALS AND HOMOMORPHISMS

fik(ui(k,m)-ui(k,m-l)1 < 1/2m

for

m> k.

235

We now claim that

(ui(k,m)1m> k is a Cauchy sequence in X k' Indeed if m> j > k we have pk(ui(k,m)-ui(k, j ) )< Cmt=j+lpk (ui(k,t)-ui(k,t-l)) 5 ':=j+l

ui(k)

2-t.

Since Xk is complete there exists an element m such that ui(k,m) + ui(k). To see that the

Xk

E

"tuple"

is in the projective limit of the Xk's

(ui(k)) k >

for each i we observe that for m m hk,k+l (hk+l,m(~i 1 ) = hkm(yi 1 the continuity of ui(k)

=

hk,k+l(ui(k+l,m))

ui(k,m).

hk,k+ll we see that

for each i so

Thus as

=

m-,

hk,k+l (u. 1 (k+l))

using =

( ~ ~ ( k ) ) ~ ?is indeed an element of the

projective limit for each i. e+Nk

m> k

The fact that

Ciui(k)Kk(zi)

=

for each k also follows by taking a limit with respect to

m in the equation (recall that hkrn-Km='ckand the way in which m m that yi were chosen) Ciui(k,m)K (z )=h (C.Y. K ( 2 . ) = k i km 1 1 m 1 hkm(e+Nm)=e+Nk. Finally elements ulI...,u E X exist by n Theorem 4.6-l(d) such that ~ ~ ( u ~ ) = u ~ for ( k )each k i 0, so pk(Ciuizi-e)=+

(1.u.( k ) ~ (z.)-e+N ))=O for each k i 0 and the i i k i k concluding statement--C.u z =e-- follows since X is Hausdorff. i i i (b) If ( ~ ~ r - - zn) . r is proper in X then, by (a), there is a

k? 0

k

such that

( K ~ Z ~ , . . .k,zKn)

is proper in Xk.

Since Xk is

a Banach algebra there exists a closed maximal ideal

..

M k ( ~ K ~ z ~.,K , z 1. k n X containing ( z , , . .

Clearly

.,'n) .

M=Kk-I(Mk)

is a closed ideal in

Since Xk is a complex Banach alge-

bra there is a complex homomorphism h of Xk such that kernel of h. of M fol1ows.V

Thus M is the kernel of ~

%

is the

~ and - the h maximality

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

236

4 . 1 1 The R a d i c a l and D e r i v a t i o n s

The r a d i c a l of a n a l g e b r a is an ideal of g r e a t u t i l i t y i n t h e s t u d y of t h e s t r u c t u r e o f t o p o l o g i c a l a l g e b r a s .

Especially

s h a r p s t a t e m e n t s can b e made when t h e r a d i c a l i s t h e z e r o i d e a l For example, i f t h e complex a l g e b r a X h a s

t o ) as i t s r a d i c a l ,

t h e r e i s a t most one t o p o l o g y on X w i t h r e s p e c t t o which X i s a Banach a l g e b r a . I n t h i s s e c t i o n w e u s e t h e p r o p e r t i e s of t h e r a d i c a l and d e r i v a t i o n s t o produce an example of a F r e c h e t LMC Q - a l g e b r a which i s n o t a Banach a l g e b r a . Definition 4.11-1

THE RADICAL AND SEMISIMPLICITY.

The r a d i c a l ,

Rad X, of an a l g e b r a X i s t h e i n t e r s e c t i o n of a l l maximal If

i d e a l s i n X.

Rad X = { 0 l f

X i s r e f e r r e d t o as semisimple

w h i l e a t o p o l o g i c a l a l g e b r a i s c a l l e d s t r o n g l y semisimple whenever

nMc

=

to}.

Thus i n a s t r o n g l y semisimple complex LMC a l g e b r a where t h e r e i s a 1-1 c o r r e s p o n d e n c e between

Mc

and Xh , i f e v e r y

c o n t i n u o u s homomorphism v a n i s h e s on an element x , t h e n x must be 0.

Moreover i n s u c h a l g e b r a s , s i n c e t h e c o n t i n u o u s homo-

morphisms seperate t h e p o i n t s of X ( i . e . i f x f 0 , t h e r e i s some f E Xh

f ( x ) # 0 ) t h e topology on X must b e Hausdorff

such t h a t

Thus " s t r o n g l y semisimple LMC" is t h e same a s " s t r o n g l y s e m i s i m p l e LMCH," Example 4 . 1 1 - 1 algebra

f o r complex a l g e b r a s . A STRONGLY SEMISIMPLE ALGEBRA.

C(T,Ffc),

where T i s a c o m p l e t e l y r e g u l a r Hausdorff

s p a c e o f Example 4.5-1. morphisms of

Consider t h e

C(T,g,c)

S i n c e t h e c o n t i n u o u s n o n t r i v i a l homo-

a r e j u s t t h e e v a l u a t i o n maps on t h e

p o i n t s of T (See Example 4 . 1 0 - 2 )

if

xe

nMc

then x ( t ) = 0 f o r

237

4.11 THE RADICAL AND DERIVATIONS

all

tc T

and

x=O.

Clearly X is semisimple whenever it is strongly semisimple.

We now show that in complete complex LMCH algebras, semi-

simplicity implies strong semisimplicity. (4.11-1)

x

RAD

n&

=

IN COMPLETE COMPLEX LMCH ALGEBRAS.

is a complete complex LMCH algebra then

Rad X

nMc.

=

If

x

Thus a

complete complex LMCH algebra X is semisimple iff X is strongly semisimple. Proof. Certainly f (x)=O

for all

so by (4.10-8)

nMc.

Rad X c

Suppose that

f E Xh, f (e-xy)=l for each

for some

M

f E Xh

is invertible in X for each

e-xy

suppose that (,x

nMc.

XE

MEMl

and y e X.

As y f X, If we

then since, X is a commu-

tative ring with identity, X/M is a field and there exists y~ X such that

(x+M)(y4-M)

invertibility of

=

e-xy.

e+M. Thus

Thus

e-xyc M

xc Rad X

contradicting the

nMcc Rad

and

X.v

The notion of "derivation" defined below is purely algebraic.

For the sake of the definition X needn't be commutative

or possess an identity. Definition 4.11-2 map xDy

D:X+X

+

DERIVATIONS.

Let X be an algebra. A linear

is a derivation on X if for all

x,yC X, D(xy)

=

(Dx)y. Clearly the trivial linear transformation is a derivation

and so is the differentiation operator on spaces of infinitely differentiable functions.

Let X be a linear space and &(X,X)

the noncommutative algebra of all linear maps taking X into X [where (AB)x for any

=

A(Bx)].

AEX(X,X).

Fix

BfZ(X,X)

and define

The transformation AB-BA

DB(A)=AB-BA

is called the

commutator of A and B and the derivation DB is referred to as a

4 . COMNUTATIVE TOPOLOGICAL ALGEBRAS

238

commutator operator.

Finally, let E be a field and X the alge-

bra of all formal power series in a single variable t with com

z ~1 tnlanE El where addin=O n tion and scalar multiplication are performed componentwise

efficients from E .

That is,

X

=

while multiplication is taken to be the Cauchy product.* m

Then

m

1 a tn

n=O n

+

is a derivation on X.

nnntn-l n=O

(4.11-3) CONTINUOUS DERIVATION MAPS A COMPLEX BANACH ALGEBRA INTO ITS RADICAL.

If X is a (not necessarily commutative) com-

plex Banach algebra and D a continuous derivation on X, then D(X) c Rad X. Proof.

As usual

L(X,X)

denotes the complex Banach space of

continuous (=bounded) linear transformations on X.

Since D is m

bounded and

L(X,X)

is a Banach space, the series Y, (an/n!)Dn n=O is absolutely convergent in L(X,X) for any E 2 . We denote ,D the sum of this series by e Let f be any nontrivial homo, D morphism of X and define f, to be f - e Since f must be con-

.

.

tinuous (see the discussion preceeding (4.10-5)) , f, is a continuous linear functional on X.

To see that fa is in fact a

homomorphism we first note that since D is a derivation, a "Leibniz rule" holds:

for any positive integer n,

Thus, since f is continuous,

m

=

* Yn

c

n=0

anz

The Cauchy product of =

i+j=na iB j*

i+j=n

i!

zantn

and

XBntn

is

zYntn

where

239

4.11 THE RADICAL AND DERIVATIONS

and t h e two

s e r i e s above converge a b s o l u t e l y , t h e p r o d u c t of t h e s e two series e q u a l s t h e Cauchy p r o d u c t ( * ) and Since

I[f ii 5 1 (see t h e d i s c u s s i o n p r e c e e d i n g ( 4 .10-5))

and D i s bounded t h e series s o l u t e l y f o r each f i x e d a+fa(x)

f,(x)

xc X

(Dnx) n!

= C

and any a.

c o n v e r g e s ab-

Thus t h e mapping

( w i t h x h e l d f i x e d ) i s a n e n t i r e f u n c t i o n of a .

w a s the case f o r (Ix /i

f a ( x y ) = f,(x) f a ( y ) .

f o r each

a€

f,

s.

ii

5 1,

Hence

and it f o l l o w s t h a t fa(x)

As

llf,(x)

11

5

i s a bounded e n t i r e func-

t i o n of a and t h e r e f o r e i s a c o n s t a n t by L i o u v i l l e ' s theorem

[(4.9-1) 1.

I t f o l l o w s from t h e i d e n t i t y theorem f o r power

series t h a t f(Dx)=O.

f(Dnx)=O

for a l l

n i l ;

thus, i n particular,

S i n c e f w a s any homomorphism w e conclude t h a t

Dx E

Rad X . v Next w e s t a t e an immediate c o r o l l a r y o f t h e p r e c e e d i n g p r o p o s i t i o n f o r ease of r e f e r e n c e . (4.11-4)

CONTINUOUS DERIVATIONS ON COMPLEX SEMISIMPLE BANACH

ALGEBRAS ARE T R I V I A L .

I f X i s a semisimple complex Banach a l -

g e b r a t h e n t h e o n l y c o n t i n u o u s d e r i v a t i o n on X i s t h e t r i v i a l one. The f a c t t h a t t h e commutator of a p a i r of bounded operat o r s on a complex Banach s p a c e i s n e v e r t h e i d e n t i t y t r a n s f o r mation w a s f i r s t proved by H .

W i e l a n d t (1949-50).

An indepen-

d e n t proof c a n b e g i v e n f o r matrices u s i n g t h e e l e m e n t a r y prop e r t i e s of t h e t r a c e f u n c t i o n on matrices. (4.11-5)

t

COMMUTATORS AND THE I D E N T I T Y . ?

I f X i s a complex

The v a l i d i t y of t h e proof below depends on Theorem 4.6-1, (4.10-7), ( 4 . 1 0 - 8 ) and ( 4 . 1 1 - 3 ) , a l l of which remain t r u e w i t h o u t t h e assumption of c o m m u t a t i v i t y a s t h e r e a d e r may v e r i f y .

4 . COM4UTATIVE TOPOLOGICAL ALGEBRAS

240

Banach s p a c e and

then

ArBE L(X,X)

c a n n o t b e t h e iden-

AB-BA

t i t y map x+x on X. Proof.

consider the derivation

I t i s r e a d i l y s e e n t h a t DB i s bounded

Since

= DB(A)

( IpB(H)

H+HB-BH.

lk 2 Ip 11 IF 11).

i s a complex Banach a l g e b r a , it f o l l o w s by

L(X,X)

(4.11-3) t h a t AB-BA

,

DB:L(X,X)+L(X,X)

DB(L(X,X)

c Rad L ( X , X ) .

1k Rad L ( X , X )

But

,

so

# l.V

The r e s u l t s j u s t o b t a i n e d now e n a b l e u s t o e x h i b i t a comp l e t e b a r r e l e d semisimple Q - a l g e b r a which i s n o t a Banach a l g e bra. Example 4 . 1 1 - 1

A BARRELED SEMISIMPLE FRECHET Q-ALGEBRA

NOT A BANACH ALGEBRA.

Consider t h e a l g e b r a

l y d i f f e r e n t i a b l e f u n c t i o n s on

[a,b]

.8

[arb]

I X ( n ) ( t )I

ample 4 . 3 - 2 , s

(n=O, 1,

. . .) .

of a l l i n f i n i t e -

of Example 4.3-2

t h e LMCH t o p o l o g y g e n e r a t e d b y t h e seminorms

suptc

W H I C H IS

with

pn(xl =

A s was e s t a b l i s h e d i n Ex-

is a Frechet algebra.

S i n c e any complete m e t r i c

s p a c e i s a Baire s p a c e any LCS which i s a B a i r e s p a c e i s b a r r e l e d (see Horvath, 1 9 6 6 , pp. 213-214), I n a remark f o l l o w i n g ( 4 . 8 - 1 )

reled.

a

element maps

t*

w e proved t h a t

x c Rad X

:a+g,

was a Q-

V

must v a n i s h on

[arb]

as t h e e v a l u a t i o n

x+x ( t ) are c o n t i n u o u s homomorphisms on

To show t h a t

a is

Clearly the differentiation operator

D:8+a,

t r i v i a l d e r i v a t i o n and it i s c o n t i n u o u s s i n c e

nz 0 .

.

n o t a Banach a l g e b r a w e have o n l y t o

e x h i b i t a n o n t r i v i a l c o n t i n u o u s d e r i v a t i o n on

f o r each

i s bar-

= { x c a I p o ( x ) < 13c Q. The f a c t PO i s semisimple i s a p p a r e n t from t h e o b s e r v a t i o n t h a t any

a l g e b r a by showing t h a t that

a

the algebra

a by x+x' -1 D

(4.11-4)

.

i s a non( V )=V Pn Pn+1

4.12 SOME ELEMENTS OF GELFAND THEORY W e can a l s o o b s e r v e t h a t ( 4 . 1 1 - 4 )

241

i s no l o n g e r t r u e i f

t h e c o n d i t i o n t h a t X b e a Banach a l g e b r a i s r e l a x e d :

It is not

even t r u e f o r b a r r e l e d F r e c h e t semisimple Q - a l g e b r a s a s t h i s example i l l u s t r a t e s . F o l l o w i n g t h e proof of

( S i n g e r and W e r m e r (1955))

(4.11-3)

t h e s u s p i c i o n grew t h a t p e r h a p s c o n t i n u i t y of t h e d e r i v a t i o n

w a s n o t a n e c e s s a r y i n g r e d i e n t i n t h e h y p o t h e s i s of t h a t r e s u l t and t h i s s u s p i c i o n h a s been somewhat b o r n e o u t . proved t h a t e v e r y d e r i v a t i o n on a r e g u l a r

*

Curtis

(1961)

commutative s e m i -

s i m p l e Banach a l g e b r a w i t h i d e n t i t y i s c o n t i n u o u s .

This w a s

s u b s e q u e n t l y g e n e r a l i z e d t o semisimple F r e c h e t a l g e b r a s by Rosenfeld ( 1 9 6 6 ) .

B.E.

Johnson (1969)

proved t h a t e v e r y

d e r i v a t i o n on a semisimple commutative Banach a l g e b r a i s cont i n u o u s and hence t r i v i a l by ( 4 . 1 1 - 4 ) . M i l l e r (1970) and Gulick ( 1 9 7 0 ) have c o n s i d e r e d h i g h e r

o r d e r d e r i v a t i o n s and some of G u l i c k ' s r e s u l t s subsume Rosenf e l d ' s g e n e r a l i z a t i o n o f C u r t i s ' theorem. A number of t h e r e s u l t s j u s t mentioned may b e found i n

t h e E x e r c i s e s f o r Chapter

5.

4 . 1 2 SOME ELEMENTS OF GELFAND THEORY

-

THE T O P O L O G I Z I N G O F Xh

AND THE M A P P I N G I.

I n t h i s s e c t i o n w e c o n s i d e r a mapping u s e f u l i n a n a l y z i n g

*

L e t X b e a commutative Banach a l g e b r a a n d M i t s s p a c e of max-

i m a l i d e a l s . I f xc X , d e f i n e f ; $ + C_ by t h e formula $ ( M ) = h ( x ) , where M=ker h , and p r o v i d e M w i t h t h e w e a k e s t t o p o l o g y w i t h res p e c t t o which each

2

i s continuous.

I f the functions

s e p a r a t e p o i n t s and c l o s e d s e t s i n M , i . e . and

MEM

t h e r e i s an x such t h a t ; ( M ) = l

called regular.

2,

x~ X ,

f o r each c l o s e d F c M

and $ ( F ) = { O } , t h e n X i s

For f u r t h e r d i s c u s s i o n s e e Sec. 5.2.

242

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

mapping Y s e n d i n g X E X i n t o

t h e s t r u c t u r e of LMCH a l g e b r a s - - t h e

t h e c o r r e s p o n d i n g e v a l u a t i o n map on Xh t a k i n g a t o p o l o g i c a l a l gebra X i n t o

h

X

C(Xh,_F)

t h e a l g e b r a of c o n t i n u o u s f u n c t i o n s on

equipped w i t h t h e r e l a t i v e

4.12-1).

u(X',X)-topology

(see D e f i n i t i o n

As w e have s e e n it i s f r e q u e n t l y p o s s i b l e t o r e d u c e

q u e s t i o n s p e r t a i n i n g t o t h e s t r u c t u r e of LMCH a l g e b r a s t o rel a t e d q u e s t i o n s about t h e f a c t o r a l g e b r a s with t h e a i d of t h e c a n o n i c a l homomorphisms and some of t h e i r p r o p e r t i e s . s e c t i o n w e s h a l l see t h a t i n t h e e v e n t Xh i s

In t h i s

u(X',X)-compact,

Y can a t t i m e s b e e x p e c t e d t o behave i n a s i m i l a r way i n red u c i n g q u e s t i o n s c o n c e r n i n g t h e LMCH a l g e b r a X t o r e l a t e d quest i o n s a b o u t t h e Banach a l g e b r a

h

C(X

,E,c).

I n order forY t o

b e u s e f u l i n t r a n s f o r m i n g q u e s t i o n s from X i n t o

i s sometimes d e s i r a b l e f o r Y t o be continuous--our d e a l s with t h i s .

h

C ( X ,,F,c)

it

third result

W e c o n c l u d e t h e s e c t i o n by e s t a b l i s h i n g cer-

t a i n c o n d i t i o n s under which a t o p o l o g i c a l a l g e b r a i s a Banach a l g e b r a , by p r o v i n g t h a t Y i s a t o p o l o g i c a l isomorphism. Michael (1952) h a s r e f e r r e d t o LMCH a l g e b r a s X f o r which

I i s an isomorphism o n t o 4.12-1).

h

C(X

,F)

as

full a l g e b r a s

(Definition

I n Michael (1952) t h e q u e s t i o n w a s r a i s e d :

Is Y a

t o p o l o g i c a l isomorphism when X i s a f u l l F r e c h e t a l g e b r a ? Warner (1958) answered t h e q u e s t i o n i n t h e a f f i r m a t i v e and h i s proof a p p e a r s i n ( 4 . 1 2 - 8 ) .

( I n f a c t t h e analogous s t a t e m e n t

a l s o h o l d s f o r many t o p o l o g i c a l a l g e b r a s o v e r nonarchimedean valued f i e l d s . )

Thus Y can a l s o b e used t o i d e n t i f y c e r t a i n

topological algebras as function algebras. D e f i n i t i o n 4.12-1.

THE M A P P I N G Y .

L e t X b e a complex t o p o l o -

g i c a l a l g e b r a o r a r e a l t o p o l o g i c a l a l g e b r a f o r which

Xh

# @

243

4.12 SOME ELEMENTS OF GELFAND THEORY

h and Xh carry the Gelfand topology, i.e. topology induced on X by

o(X',X) (see the discussion following (4.10-4)*). Then we A h define the mapping Y:X+C (X ,E) , x+x by the rule: $(f)=f (x) for each

f E Xh.

As Xh carries the induced weak-* topology,

{fc Xhl IYx(f) - Yx(fo) I
0, h C(X , g ) . Although

E)

is open in Xh for any

and, therefore, Yx is indeed an element of Y

is seen to be a homomorphism, it is not

generally 1-1 or onto.

In the event that X is a LMCH algebra

and Y is 1-1 and onto, X is referred to as a

full algebra.

In order for Y not to be 1-1, there must exist a nonzero XEX such that Yx=O--in other words an x for which f ( x ) = O for Thus Y is 1-1 iff X is strongly semisimple (Def. each fEXh

.

In case X is a complete complex L X H algebra so

4.11-1). that

Rad X =

of some

fEXh

nMc

((4.11-1)) and each

ME.N,

is the kernel

((4.10-4)) , it follows that Y is 1-1 iff X is

semisimple. We record these findings in our first result. (4.12-1)

SEMISIMPLICITY-

Y 1-1.

If an LMCH algebra is (a)

strongly semisimple or (b) semisimple, complete, and complex, then Y is 1-1. As a consequence of (4.12-3) we see that Y takes any complete complex LMCH Q-algebra into a Banach algebra: that for such algebras Xh is weakly compact.

We show

The following

technicality, (4.12-2) , is a convenience. (4.12-2) Proof.

In any complete complex LMCH algebra (Xh)O =

U (a).

h In a complete complex LMCH algebra X, a(x)={h(x) IhE X 1

for each

X E X by (4.10-8).

Thus

* If H is any collection of nontrivial homomorphisms on X then the topology induced on H by a(X*,X) is also called the Gelfand topology (See Exercise 4.4).

244

4 . COMIUTATIVE TOPOLOGICAL ALGEBRAS

{ x c Xlsup hlh(x) 15 11 = {xlro(x) 5 1 3 = u(a).V he x (4.12-3) Q-ALGEBRAS AND COMPACTNESS OF Xh. Let X be a complete

(XhIo

=

complex LMCH algebra.

If X is a Q-algebra, then

neighborhood of 0 in X and Xh is if X is barreled and Xh is

(Xh)O

u(X',X)-compact.

is a

Conversely,

u (X',XI -compact, then X is a Q-al-

gebra. Proof. set

Suppose that X is a Q-algebra.

U ( a ) = {xc Xlru(x)

O c int(U(o)) (4.12-2)

X.

Now

5 11

Then, by (4.8-31, the

has non-empty interior; in fact,

(see the proof of(4.8-3) and

Ufu) =

Xhc

(Xh)O; hence

c

V(0).

By

is a neighborhood of 0 in

so we can utilize the fact that the polar

(xh)O0,

of a neighborhood of 0 in X is the fact that Xh is

(Xhto

U(u)

a(X',X)-compact together with

a(X',X)-closed in X' to complete this part

of the argument. Conversely, suppose that X is barreled and Xh is a(X',X)compact. X.

Then Xh is

o(X',X)-bounded so

(Xh)O

is a barrel in

Since X is complete and barreled, it follows that U(a)

(Xh)O

=

Thus X is a Q-algebra by

is a neighborhood of 0 in X.

(4.8-3) .V

(4.12-3) can be used to obtain still another realization of the Stone-Cech compactification BT of a completely regular Hausdorff space T, namely as the space of continuous nontrivial homomorphisms of the Banach algebra

Cb(T,crc) of bounded con-

tinuous s-valued functions on T with sup norm. Cb(T,g,c)h

The space

of continuous homomorphisms is compact in its Gel-

fand topology by (4.12-3). of evaluation maps on

So,

identifying T with the space T*

Cb(T,g,c),

a possibility afforded by

the complete regularity of the Hausdorff space T, we see that,

245

4.12 SOME ELEMENTS OF GELFAND THEORY i n some s e n s e , t a i n i n g T.

If

( a ) T and T* can b e i d e n t i f i e d as t o p o l o g i c a l

( b ) T i s dense i n

spaces,

i s a compact Hausdorff s p a c e con-

C,(T,C,C)~

h

Cb(T,_C,c)

,

and ( c ) e a c h

x ~ C ~ ( T , g , c )can b e extended c o n t i n u o u s l y t o a f u n c t i o n on Cb(T,C,c)

h

,

then

C,(T,C,C)~

must b e BT by Theorem 1.3-2.

We

now v e r i f y t h a t t h e s e t h r e e c o n d i t i o n s are i n d e e d m e t .

w e see t h a t

By ( 4 . 1 0 - 4 ) w i t h t h e space

M

of maximal i d e a l s of

e s t topology f o r h

5 ,

Cb(T,slc).

Moreover t h e

( f i r s t d i s c u s s e d a f t e r (4.10-4)) i s t h e weak-

Gelfand t o p o l o g y

Z:Cb(T,CIc)

may b e i d e n t i f i e d

C,(T,E,C)~

w i t h r e s p e c t t o which t h e maps

Cb(T,S,c)h

f+f(x)

a r e continuous f o r each

x&Cb(Trg).

S i n c e T i s a c o m p l e t e l y r e g u l a r Hausdorff s p a c e , i t s t o p o l o g y

i s t h e i n i t i a l t o p o l o g y d e t e r m i n e d by

on T ( ( 0 . 2 - 5 ) ) :

Cb(T,C)

a b a s i c neighborhood of a p o i n t to i n T i s t h e r e f o r e a set of t h e form where

V ( t o ; x l l . . . , x n I E ) = { t E T ]Ix. ( t ) - x . ( t 1

x l I . . . , x n ~ C b ( T , ~ ) and

E

1

)I

0


0.

A t y p i c a l b a s i c neighborhood of t h e e v a l u a t i o n map

t:ECb(T,&c)h

i n t h e r e l a t i v e Gelfand t o p o l o g y on T* would b e

I
0.

A

x i s a c o n t i n u o u s ex-

h h x ~ C ( C ~ ( T , s r c,g). )

it m u s t b e shown t h a t T* i s d e n s e i n

I f T* i s n o t d e n s e i n

T*




o r i g i n by ( 4 . 1 2 - 3 ) { x Xlsupl ~ < x,X ous

h

,

>I_
0. If the linear

transformation A maps X into the topological algebra Y and A is continuous when X and Y carry their homomorphism topologies then A is called homomorphically continuous. Suppose that X is a complex LMC

algebra.

Clearly if it

is strongly semisimple then Xh separates the points of X and a(X,Xh ) is Hausdorff. Conversely, if cs(X,Xh) is Hausdorff then for any non-zero x~ X there is some gc [Xhl where g = Caifi and fiE Xh , such that g(x)#O. Thus fi(x)#O for some i and the kernel of fif a closed maximal ideal, does not contain x.

Hence X is strongly semisimple. We summarize these

observations below. (4.13-1) HAUSDORFF HOMOMORPHISM TOPOLOGIES.

LMC algebra then

a(X,Xh)

If X is a complex

is Hausdorff iff X is strongly semi-

simple. The main result of this section is in three parts.

The

first two parts are concerned with continuity of a homomorphism when one or both spaces carry their homomorphism topologies. The third, which is established with the aid of the closed graph theorem is the result mentioned in the introductory remarks of this section on continuity of homomorphisms.

270

4 . COMMUTATIVE TOPOLOGICAL ALGEBRAS

Theorem 4.13-1.

C O N T I N U I T Y O F HOMOMORPHISM. L e t A b e a homomor-

phism t a k i n g t h e complex t o p o l o g i c a l a l g e b r a X i n t o t h e c o q l e x Then:

t o p o l o g i c a l a l g e b r a Y.

( a ) A i s homomorphically c o n t i n u o u s i f i t i s c o n t i n u o u s when X and Y c a r r y t h e i r o r i g i n a l t o p o l o g i e s : I f X i s a complete LMCH Q - a l g e b r a and Y i s a t o p o l o -

(b)

g i c a l a l g e b r a t h e n A i s c o n t i n u o u s when Y c a r r i e s i t s homomorphism topology and X i t s o r i g i n a l topology:

(c)

I f t h e complete LMCH Q-algebra X i s b a r r e l e d and t h e

LMCH a l g e b r a Y i s s t r o n g l y semisimple and f u l l y complete, t h e n A i s continuous.

(a)

Proof. X

h

.

F o r any

fll

C l e a r l y f o r each

...,f n & ) ,

V(O,fl,

E

...,fnE Yh , > 0,

from ( 4 . 1 2 - 3 )

...,f nA,&)

E

c

so A i s homomorphically c o n t i n u o u s .

that

(Xh)O

is a neighborhood of 0 i n X.

s i d e r any f i n i t e c o l l e c t i o n have t h a t

flA,.

h o A(6(X ) )cV(O,fl, u(Y,Y

(c)

h

fl,.

..,fnAE Xh u{O).

6 ( X h ) O I ( f i A ( x ) 15 6

carries

A(V(O,flA,

..,fnA

flA,.

S i n c e X i s a complete LMCH Q - a l g e b r a , it f o l l o w s

(b)

XE

note t h a t

for

.. , f , ~Yh;

...,f n , 6 ) .

t h e n by ( 4 . 1 0 - 5 )

Thus f o r any

i=l,.. .,n

Con-

6> 0

we

and

and w e see t h a t

Hence A is c o n t i n u o u s when Y

1.

By ( b ) A i s c o n t i n u o u s when Y carries

Since Y is s t r o n g l y s e m i s i m p l e ,

a(Y,Yh)

0

(Y,Y

h

)

.

is a Hausdorff topo-

l o g y , s o t h e g r a p h of t h e c o n t i n u o u s homomorphism A i s c l o s e d i n t h e p r o d u c t t o p o l o g y on t o p o l o g y and Y carries

XXY

h

u(Y,Y ).

when X carries i t s o r i g i n a l [Any c o n t i n u o u s map

f:S+T

where S i s a t o p o l o g i c a l s p a c e and T a Hausdorff s p a c e h a s a c l o s e d g r a p h i n SxT].

Now t h i s p r o d u c t t o p o l o g y i s c l e a r l y

4 . 1 3 C O N T I N U I T Y OF HOMOMORPHISMS

271

c o a r s e r t h a n t h e one induced by t h e o r i g i n a l t o p o l o g i e s o f X so t h e g r a p h of A remains c l o s e d i n the p r o d u c t o f t h e

and Y ,

o r i g i n a l topologies.

F i n a l l y , s i n c e X i s b a r r e l e d and Y i s

f u l l y c o m p l e t e , t h e c l o s e d graph theorem i m p l i e s t h a t A i s cont i n u o u s when X and Y c a r r y t h e i r o r i g i n a l topo1ogies.V A s an a p p l i c a t i o n o f

( c ) above, suppose t h a t X i s a s e m i -

s i m p l e f u l l y c o m p l e t e , b a r r e l e d , complex LMCH Q - a l g e b r a when X

carries e i t h e r of t h e topologies homomorphism

(XJ)+(X,a')

of t h e p r e v i o u s theorem and (4.13-2)

,

x+x = 0 we 2 may set x1 = (2/(4b-a2) (yl+ae/2). It follows that x1 =-e. {e,yl)

Furthermore this relation makes it impossible for

Ce,x,)

to be

a basis for X for in this event X is seen to be isomorphic to

5.

Let

y22=-e.

{e,xl,y21 Then

be linearly independent in X such that

x1+y2

and

xl-y2,

neither of which is a mul-

tiple of e, must have minimal polynomials of degree 2. there must be real numbers r(xl+Y2) + se

=

x1y2

and v such that

r,s,u

+ y2x1 - 2e and

Hence 2

(x1+y2)

( ~ ~ - =y u(xl-y2) ~ ) ~

=

+

-(x y +y x ) - 2e. By adding these equations and using 1 2 2 1 the linear independence of {e,xl,y21, we obtain r=u=O and

ve

=

s+v=4.

Thus

(1) x1y2

minimal polynomials of

+

y2x1

x1+y2

=

(s+2)e

and

=

xl-y2

-(v+2)e.

Since the

are irreducible, s

and v must be negative. It follows that 4< s < 0, SO that -s2-4s> 0. Let x2 = ( ( s + 2 ) / ( - s 2 -4s)1/2)x,+(2/(s2-4s)1’2)y~

so that

~e,x1,x2} is linearly independent, x2

x1x2 - -x2x1.

Finally, select

x3=x1x2.

2

=-e , and

To see that

{e,x1,x2,x3}

is linearly independent, suppose that x3=fe+gxl+ 2 hx2. Then x1x3 = x 1 x 2 - -x2 = fxl+gxl + hx1x2 = h(fe+gxl+hx2) so that f=g=h=O. Thus {e,x1,x2,x3) is linearly independent.

Furthermore

only remains to show that that

XEX and

assuming that

x12=x 2=x 2=x1x2x3

e,x1,x2,

2

3

and x3

span X.

=

-e.

~t

Suppose

x j% ge. Then there is no loss in generality in 2 x =-e so that equations analogous to (1) may be

EXERCISES 4

o b t a i n e d f o r t h e pa rs

(x,,x),

281

( x 2 , x ) , and ( x 3 , x ) , 1.e. t h e r e

are r e a l numbers

k m, and n such t h a t

and

Since

x x=xx =ne. 3 3

xx3 = (xx1)x2 = kx2 x x = kx -mx + x e 3 2 1 3 nx3). 4.9

Hence A-NORMED

LCHS.

-

x =x x - x x 3 1 2-- 2 1'

x ( x x ) = kx -mx 1 2 2 1

xx

x x=xx = k e , x X=XX = m e , 1 1 2 2

3

it follows t h a t

+

( x x ) x = kx2-mx + 1 2 1 2y = -(kxl+mx

which i m p l i e s t h a t

2

+

~ e I x 1 , x 2 , x 3 ~s p a n s X and t h e proof i s complete. ALGEBRAS.

L e t X b e a complex a l g e b r a which i s a

A seminorm p d e f i n e d on X w i l l b e c a l l e d an A-seminorm

i f f o r each

xcX and a l l

YE X

t h e r e e x i s t s a r e a l number

.

m > 0 such t h a t p ( x y ) 5 m p(y) I f t h e t o p o l o g y on X i s PIX PlX g e n e r a t e d by a s i n g l e A-seminorm, t h e n w e refer t o X as A-normI f t h e t o p o l o g y on X i s g e n e r a t e d by a f a m i l y of A - s e m i -

ed.

norms, t h e n w e r e f e r t o X as l o c a l l y A-convex. I f X i s a complete A-normed a l g e b r a , t h e n X i s a Banach

(a)

algebra.

x+Ax

Hint:

C o n s i d e r t h e l i n e a r s p a c e isomorphism

where

L(X,X)

H:X+L(X,X),

i s t h e Banach a l g e b r a o f bounded l i n e a r

t r a n s f o r m a t i o n s on X and

A y = xy

for all

X

Y E X.

Show t h a t

t h e mapping H h a s a c o n t i n u o u s i n v e r s e and t h a t H ( X ) i s c l o s e d in

L(X,X).

Then, assuming t h a t X i s complete, a p p l y t h e

c l o s e d g r a p h theorem. (b)

I f X i s a n A-normed d i v i s i o n a l g e b r a , t h e n X i s t o p o l o g i -

c a l l y i s o m o r p h i c t o t h e complex numbers. (c)

I f X i s an A-normed a l g e b r a and I a c l o s e d i d e a l i n X ,

t h e n t h e a l g e b r a X / I w i t h q u o t i e n t norm i s an A-normed a l g e b r a . A

m a x i m a l i d e a l i n a n A-normed a l g e b r a X i s t h e k e r n e l o f a

c o n t i n u o u s homomorphism of X o n t o (d)

Let

C[O,l]

g

i f f i t is c l o s e d i n X .

d e n o t e t h e a l g e b r a o f c o n t i n u o u s complex-

4 . COMMJTATIVE TOPOLOGICAL ALGEBRAS

282

valued f u n c t i o n s on the c l o s e d i n t e r v a l function

< 1. z ( t ) = t , 0 5 t i 1/2; 1-t, 1 / 2 5 t -

let

lbil =

sup z ( t ) x ( t ) t € 0111 A-normed b u t n o t normed. XE

[0,1].

Ct0,lI

Hint:

0, l / n < t.

.

Show t h a t

Consider t h e For any

x n ( t ) = 1-nt, 05

Consider t h e f u n c t i o n s Show t h a t t h e f u n c t i o n s

xn+O

is

C[O,l]

ts

l/n;

but the linear

2 f 0. Consider y , ( t ) = n t , 0 1 t i l / n 2 ; Xn 2 2 2 2 2-n t , l / n < t i 2/n ; 0 , 2/n < t and show t h a t

transformations

A

IFxn ( y n ) 11 /Ibnli> l - l / n . A-set

( e ) A s u b s e t U of X i s s a i d t o b e a n

if f o r e a c h x € X and some r e a l number

ax> 0 ,

xUcaxU.

Prove t h a t a l l of t h e f o l l o w i n g a r e A - s e t s : (a)

t h e c l o s u r e of a n A - s e t ;

(b)

t h e b a l a n c e d h u l l of an A - s e t ;

(c)

t h e convex h u l l o f an A - s e t ;

(d)

t h e b a l a n c e d convex h u l l of an A - s e t ;

(e)

t h e i n t e r s e c t i o n of two A - s e t s .

(f) Prove t h a t t h e gauge of an a b s o l u t e l y convex and a b s o r b i n g A-set

i s a n A-seminorm.

{xc X l p ( x ) s 11 (9)

I f p i s an A-seminorm show t h a t V

P

=

is an A - s e t .

Prove t h a t X i s l o c a l l y A-convex i f f there e x i s t s a b a s e

of neighborhoods of 0 c o n s i s t i n g of A - s e t s . (h)

(Michael, 1952) Prove t h a t a b a r r e l e d l o c a l l y A-convex

space i s an LMC a l g e b r a . Hint:

L e t U b e an a b s o l u t e l y convex c l o s e d A - s e t

which

is a neighborhood of 0 i n X and a b e a r e a l number such t h a t a> 0

and

ec

aU.

Consider

V = { x XlxUcU). ~

t h a t V i s a b a r r e l i n X , and t h a t (i) L e t

+

Co(13)

Show t h a t V c a U ,

2

V cV.

be t h e s t r i c t l y p o s i t i v e f u n c t i o n s i n

Cb(EIE)

EXERCISES 4

which vanish at infinity. the A-seminorm

px(y)

283

+

For each

xE Co(R)

let px denote

su {lx(t)y(t)[}. Show that the family t of seminorms {pX 1 cannot be replaced by a family of multiplica=

€E

tive seminorms generating the same topology. Hint:

Let x1 and x2 be two functions in

+ Co(R)

and sup-

pose that for some multiplicative seminorm p the set inclusions

V

PX2

c vP c vP

hold.

Let

x1 be a real number such that

x2(to)=b 2 xl(to) xl(to).

a

=

minC1, max x,(t)} t€S

O < b < a.

Then for some

and for some positive integer n,

Consider the function y defined by

and let b to€

E,

bn
0

p(xy)

=
0 ;

and ( y ) such t h a t n F i r s t w e d e a l w i t h t h e case

2

p ( x n y n ) > n p n ( x n ) p n ( y n ) => 0. pn(xn)pn(yn) = 0

where many

pn(xn)

=

f o r i n f i n i t e l y many n .

then

0,

and i n f a c t f o r any m,

x +O n

u l t i m a t e l y i s e q u a l t o 0.

Thus

x n / p ( x y ) = zn+O. n n

loss of g e n e r a l i t y t h e sequence (y,) that

yn+0.

Now

a contradiction. and

pn(yn)

and

w

n p(znwn) q1 ( x y )

zn+O

With no

and

yn+O.

For t h e case where w e can assume t h a t

are never zero, w e simply l e t

=>

ql=pl

Now l e t ( y ),

set

z

n

This is pn(xn)

= xn/npn(xn)

-to and z n -+O b u t n and f o r t h e f i r s t i such t h a t

Once a g a i n

=< Cpi ( x )pi

pm(xn)

c o u l d have been chosen s o

p(znyn) = 1 while

= yn/npn(yn).

1.

If infinitely

q2

=

w

Cpi.

C o n t i n u i n g i n t h i s way

w e g e n e r a t e t h e d e s i r e d seminorms. A F r e c h e t s p a c e which i s an a l g e b r a i n which m u l t i p l i c a -

t i o n is s e p a r a t e l y c o n t i n u o u s ( o r i n which t h e r e i s a f a m i l y of seminorms g e n e r a t i n g t h e t o p o l o g y which s a t i s f y the c o n d i t i o n s of ( a ) ) i s c a l l e d a F r e c h e t a l g e b r a . (b)

Show t h a t a complex F r e c h e t d i v i s i o n a l g e b r a i s t o p o l o g i -

c a l l y isomorphic t o Hint:

2.

Using Theorem 4 . 9 - 1 w e need o n l y p r o v e t h a t i n v e r -

s i o n i s continuous.

S i n c e t h e u n i t s are open t h e y a r e a G -set 6

Then Theorem 7 . 4 of Zelazko 1 9 6 5 may b e a p p l i e d . (c)

L e t X b e a complex F r e c h e t a l g e b r a and suppose t h a t t h e r e

293

EXERCISES 4

e x i s t s a c l o s e d maximal i d e a l i n X.

Show t h a t a s y s t e m of

seminorms ( p i ) can b e found s a t i s f y i n g t h e c o n d i t i o n s o f p i ( e ) = 1 f o r a l l i.

such t h a t Hint:

(qi)

Let

be a f a m i l y of seminorms g e n e r a t i n g t h e

t o p o l o g y o f X and s a t i s f y i n g t h e c o n d i t i o n o f closed i d e a l i n X then

m+Ae

+

i+l(m)

11.119

2. then

per

+ IAlqi(m') +

A€

with

y = m'

(a)

X = M

Define

{Aelhc

21.

wi(x) = qi(m)

If

I i l ]PI=
0 . X#O

(a)

An e l e m e n t o f a normed a l g e b r a X h a s no i n v e r s e i n any

s u p e r a l g e b r a o f X i f f it i s a t o p o l o g i c a l d i v i s o r o f 0 . Hint:

L e t t > 0 and

X(z;t)

( z transcendental over X) be

t h e a l g e b r a o f f o r m a l power series i n z w i t h c o e f f i c i e n t s from m

X s u c h t h a t f o r any such s e r i e s m

Let

Ilf ( z )

X(z;t)

11

= n&o llxn lit”

Ilf ( z ) + J

algebra with

11

Let

Y = X(z;t)/J

= i n f [If ( z ) + J ( z )

11.

W e show t h a t f o r s u i -

t a b l e t X i s i s o m e t r i c a l l y embedded i n Y . for all

(a 2 - to ’)

XEX

with

t o > 0.

tol/cx(I => IIxi/

Let

Let

x-(e-cz) n=O c xn zn = (x-xo) + (cxo-xl)z

II

(ti’llxo 11-

t

llxl

11)

+

+.

Iicxo-xl Ilt

.. =

IIcx1-x2 Ilt

+

IIx j j +

+

( c x1 -x 2 ) z 2

+..- =>

(tt,’) Ilf ( z )

11,

2

Ilx

g(z) =

and

XE X

0)

llg (2) II = IIx-xo

a.

be t h e quotient

j6J

-

n I/xnIlt
k)

Bckka r e

this i s equivalent t o requiring t h a t f o r A,

h o l d s i f f X i s hk-normal.

the closures Since BC Ghk

Summarizing t h e above d i s c u s -

s i o n w e s t a t e ( f o r t h e IMC a l g e b r a X): Theorem 5 . 4 - 2

M=

w(Mc&,2

a l l h k - c l o s e d s u b s e t s of

Ac.

a map w h i c h r e d u c e s t o (D : .& Ghk M c i f f X i s hk-normal.

L e t X s a t i s f y c o n d i t i o n hH and

Ghk d e n o t e

Then.& a n d w ( A c , G h k ) a r e homeomorphic v i a

-

w ( A c , '$',,),

M

+

dH={Ac

chkIMfA)

on

Going one s t e p f u r t h e r , i f w e c o n s i d e r o n l y r e g u l a r a l g e b r a s t h a t s a t i s f y c o n d i t i o n hH t h e n / - / may b e r e p l a c e d by t h e l a t t i c e

GG o f

a l l Gelfand

-

5.5

c l o s e d s u b s e t s of

4

(3.4-3).

Theorem 5 . 4 - 3

and shk= JG

Gelfand

Furthermore, i n t h i s case,w(

flc,

Thus:

WHEN I S

A

A STONE-CECH COMPACTIFICATION?

a l g e b r a X s a t i s f y c o n d i t i o n hH and

Mc.

c l o s e d s u b s e t s of

311

i f f X i s Gelfand normal f o r t h e n

normality is equivalent t o hk-normality.

JG)=sMc by

hfc

X-REPLETION OF

M

Then

L e t the regular


)

i s t h e i s o m e t r i c i s o m o r p h i c image o f X i n Y.

w h e r e X"

-1

I f @ i s a c o n t i n u o u s homomorphism and M =iP ({O]), a n o n z e r o 2 (bounded) p o i n t d e r i v a t i o n o f X e x i s t s i f f M #M ip (%+Mm). H i n t : L e t f b e a n o n t r i v i a l l i n e a r f u n c t i o n a l a n n i h i l a t i n g M2 a n d e . A s (b)

+

-

@l-

m

a n y xcX c a n b e w r i t t e n x = x ' + @ ( x ) e w i t h x'cM

t h e n f c a n b e shown t o b e t h e C' d e s i r e d p o i n t d e r i v a t i o n o f X a s s o c i a t e d w i t h $. (c)

I f X C Y and i f D i s a n o n z e r o bounded d e r i v a t i o n o f X i n t o Y b u t

D(X) i s n o t i n t h e r a d i c a l o f Y , t h e n t h e r e e x i s t s a n o n z e r o bounded p o i n t d e r i v a t i o n of X. By t h e c o n d i t i o n g i v e n a b o v e t h e r e e x i s t s a n o n t r i v i a l homomorphism

Hint:

whose r e s t r i c t i o n t o D(X) i s n o t z e r o . Assume t h a t X i s s e m i s i m p l e .

Define d

m

on X by d ( x ) = @ ( D ( x ) ) .

@

Then X h a s no n o n t r i v i a l ( c o n t i n u z ous d e r i v a t i o n s i n t o a s e m i s i m p l e c o m m u t a t i v e e x t e n s i o n Y o f X i f f M % (d)

-2-

(M =M)

-

t o r a l l maximal i d e a l s M C X .

(e)

I f T i s a compact H a u s d o r f f s p a c e , t h e n C(T,C,c) h a s no n o n t r i v i a l

d e r i v a t i o n s i n t o a n y s e m i s i m p l e commutative e x t e n s i o n Y.

5.7 1961) -

DERIVATIONS OF COMMUTATIVE REGULAR SEMISIMPLE BANACH ALGEBRAS ( C u r t i s I n t h i s e x e r c i s e i t i s shown t h a t i f t h e Banach a l g e b r a S i s r e g u l a r

324

5.

HULL-KERNEL TOPOLOGIES

and s e m i s i m p l e , t h e n t h e "boundedness" r e q u i r e m e n t o f (4.11-3) c a n b e dropped;

t h a t i s , a n y d e r i v a t i o n of X i n t o i t s e l f i s bounded and t h e r e f o r e

trivjal.

Moreover B.E.

Johnson (1969) h a s shown t h a t t h e " r e g u l a r i t y "

a s s u m p t i o n c a n be d r o p p e d . L e t X b e a complex commutative r c b u l a r s e m i s i m p l e Banach a l g e b r a w i t h identity.

A ' d e r i v a t i o n ' D from X i n t o B ( A

,C,c)

(bounded complex-valued

f u n c t i o n s 0n.M w i t h p o i n t w i s e o p e r a t i o n s and s u p norm) i s a l i n e a r t r a n s formation such t h a t Dxy = xDy 4- (Dx)y where

$:a

i s t h e G e l f a n d map d e t e r m i n e d by x .

(a)

If x i s a n i d e m p o t e n t i n X , t h e n Dx=O.

(b)

If

then l e t t i n g

Hint:

%

d e n o t e s t h e c h a r a c t e r i s t i c f u n c t i o n o f ( M I , M C N and

%=% w i t h

x=Q+Ae

% 0 s u c h t h a t

But 4

0.

By t h e

IDx(M) 15

kllxll f o r a l l xcX and a l l M€H.

To show t h a t H C F s u p p o s e Mo

-

-

0 there exists F

0 t h e n Dx n

such t h a t F

is f i n i t e

M i s i s o l a t e d , then Dx(M)=O f o r a l l xcX.

and M i s i s o l a t e d i n

isolated i n

CM

0 u n i f o r m l y on c l ( M n-Fn)

M

M

m

for a l l m

5

n then M i s

,

Once t h e s e s t a t e m e n t s have been e s t a b l i s h e d t h e n i t f o l l o w s t h a t Dx (M) 0 k f o r every whenever x -, 0 . S t a t e m e n t (1) t h e n i m p l i e s t h a t (Dx ) conk k ( ~ Kf i s compact i n M , v e r g e s t o o u n i f o r m l y on e v e r y compact s e t of M +

MFM

.

then K c M ,

sknce X is b a r r e l e d . )

f o r some n by (4.10-7)

To prove (1) f i x n € g and l e t Y be t h e seminormed a l g e b r a X s u p p l i e d w i t h seminorm P ( X > = pn(x>

L e t Fn={Mf

Mn

-

+

SUP

(Dx(M)

I

MFM f&(x)=Dx(M) i s n o t c o n t i n u o u s ] .

s p a c e and f o r e a c h xcX I s ( x )

I

-

Since X i s a Frechet

5

s u p IDx(M') I=kx, Dxk 0 u n i f o r m l y on M'dl i s n o t f i n i t e t h e n therc!? e x i s t s a sequence (M,) i n F and a

I f Fn Mn-F. sequence of m u t u a l l y d i s j o i n t neighborhoods V of Y, f o r e a c h k . S i n c e X k i s r e g u l a r t h e r e a r e sequences ( y ) and ( z ) such t h a t 9 ( ) = 1 , y z =y k k kMk k k k is a discontinuous l i n e a r functional and z z =O whenever k#p. S i n c e f k P t h e r e e x i s t s x €X, k€E,such t h a t k

Mk

If %

I=

AS X i s a F r e c h e t a l g e b r a , discontinuous.

Thus w e may

lDXk(%)

I > kPk(xk)

P,(Yk)

Pk(zk)'

r e s t r i c t e d t o any c l o s e d maximal i d e a l i s oose xk from

%,

L e t g =X y and \=z k k k k'

Then

EXERCISES 5

327

But t h i s c o n t r a d i c t s ( a ) . We n e x t o u t l i n e t h e p r o o f of ( 3 ) .

M

f o r m _> n .

that i f n

5

r

5

.

(e ) = e rs s r t h a t , w i t h Kn a s i n (4.10-7),

$(M')=O f o r a l l

2

n an idempotent

w h i l e $ ( M ' ) = O f o r a l l M ' # M w i t h M ' Q M ~ . Now show m A s , w i t h h r s a s i n Theorem 4.6-1, i t f o l l o w s t h a t h r s ( e s ) =

2m (M)=I

and t h e r e f o r e h

M'CM

Thus t h e r e e x i s t s a n i d e m p o t e n t e C X s u c h Hne=en f o r a l l n .

requirements of (3).

'2(M)=l while

L e t efX be a n i d e m p o t e n t s a t i s f y i n g t h e

Then by E x e r c i s e 5 . 7 ( b ) Dxe'=O f o r a l l xfX and

Dxe' (M)=O=x(M)De' (M)+e' (M)Dx(M)=Dx(M) L e t Cm(@

Clearly

w i t h M'#M.

We u s e (3) t o s k e t c h ( 2 ) .

(f)

and M i s i s o l a t e d i n

Then by p a r t ( d ) t h e r e e x i s t s f o r e a c h m

such t h a t

$

Suppose McY,

.

be t h e a l g e b r a o f a l l i n f i n i t e l y d i f f e r e n t i a b l e com-

p l e x - v a l u e d f u n c t i o n s on t h e r e a l l i n e i s t s a d e r i v a t i o n D of C

m

(E)

8.

Then f o r e a c h y p C m c ) t h e r e e x -

i n t o i t s e l f s u c h t h a t f o r a l l x€Cm(F.),

Dx(t)=

x ' ( t ) y ( t ) and c o n v e r s e l y .

Hint:

L e t C"(F).

for a l l n,k

2

c a r r y t h e t o p o l o g y d e f i n e d by t h e seminorms

0.

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

polynomial p , D p ( t ) = p ' ( t ) D ( t ) .

8s) and

f o r each

A s D is continuous, the r e s u l t follows.

This Page Intentionally Left Blank

sI X LB-Algebras

WE HAVE SEEN t h a t many o f the f e a t u r e s o f Banach a l g e b r a s a r e c a r r i e d o v e r t o t h e l a r g e r c l a s s o f IMCH Q - a l g e b r a s ,

e.g.,

o p e n n e s s o f the s e t o f u n i t s ,

c o n t i n u i t y of i n v e r s i o n , c o n t i n u i t y o f homomorphisms, c o m p a c t n e s s of t h e spectrum, etc. LB-algebra

-

The c e n t r a l n o t i o n o f t h i s a n d t h e n e x t two s e c t i o n s

-

the

p r o v i d e s a n o t h e r c l a s s o f s t r u c t u r e s c o n t a i n i n g t h e Banach

a l g e b r a s i n w h i c h some o f t h e i m p o r t a n t p r o p e r t i e s c a r r y o v e r , o n l y t h i s t i m e by way o f a more a l g e b r a i c a p p r o a c h .

W h i l e many of t h e IMCH Q - a l g e b r a s ,

namely the c o m p l e t e o n e s , a r e p r o j e c t i v e l i m i t s i n t h e TVS s e n s e o f t h e i r f a c t o r a l g e b r a s (Theorem 4 . 6 - 1 ) ,

a n LB-algebra i s a n a l g e b r a i c i n d u c t i v e

l i m i t o f a s y s t e m of Banach a l g e b r a s a n d c o n t i n u o u s u n i t a l i s o m o r p h i s m s .

The L B - a l g e b r a i t s e l f n e e d n o t b e s u p p l i e d w i t h a t o p o l o g y , however. 6.1

D e f i n i t i o n a n d Examples

I n t h i s s e c t i o n w e d e f i n e a n L B - a l g e b r a and

p r e s e n t some e x a m p l e s . BOUND STRUCTURES AND LB-ALGEBRAS*

Definition 6.1-1 algebra.

L e t X be a complex

A bound s t r u c t u r e f o r X i s a non-empty c o l l e c t i o n

dot

absolutely

m-convex s u b s e t s B o f X c o n t a l n i n g e s a t i s f y i n g t h e f o l l o w i n g s t a b i l i t y c o n dition :

€63

0 there e x i s t s a s e t B €@and a scalar h 2 3 s u c h t h a t B U B C hB 1 2 3' The p a i r ( X , 6 ) i s r e f e r r e d t o a s a bound a l g e b r a and i t i s s a i d t o be For each p a i r B

1'

B

complete p r o v i d e d e a c h of the s u b a l g e b r a s X(B)=[Qk \ L Y € ~ ,x€B} i s a Banach

a l g e b r a w i t h r e s p e c t t o t h e g a u g e pB o f B: p (x) B

X is an LB-algebra ture

63 i f

=

inf{a

>

0 b€aB] (x€B, B c Q .

(pseudo-Banach a l g e b r a ) w i t h r e s p e c t t o t h e bound s t r u c -

I f 6 i s under-

( X , 6 ) i s a c o m p l e t e bound a l g e b r a and X=UX(B).

s t o o d t h e n we simply s a y t h a t X i s an LB-algebra ("LB-algebra"

(pE$?do-Banach

i s u s e d i n l o o s e a n a l o g y w i t h "LF-space",

algebra).

e a c h LB-algebra b e -

i n g a n i n d u c t i v e l i m i t ( n o t n e c e s s a r i l y s t r i c t ) o f Banach a l g e b r a s . ) C e r t a i n l y e v e r y Banach a l g e b r a i s a n L B - a l g e b r a w i t h r e s p e c t t o t h e bound s t r u c t u r e

a consisting

s o l e l y of the closed u n i t b a l l .

d e r o f t h i s s e c t i o n w e p r e s e n t two e x a m p l e s o f L B - a l g e b r a s t';

F i r s t c o n s i d e r e d i n A l l a n , D a l e s , McClure 1971.

3 29

I n t h e remain-

which a r e n o t

330

6.

LB-ALGEBRAS

g e n e r a l l y Banach a l g e b r a s . Example 6 . 1 - 1

L e t X be a complex a l g e b r a equipped w i t h

p-BANACH ALGEBRAS

a p-norm, 0 5 p

5

1, i . e . a r e a l - v a l u e d f u n c t i o n

11 11

d e f i n e d on X such t h a t

( i ) ((xII 2 0 f o r each x€X and (\xJ\=Oi f f x=O, ( i i ) IlAII= 1 1 lpllxll f o r each ( i i i ) Ip+yII

5

ACS

and xcX,

IIxII+IIyII f o r each p a i r x , ycX.

I f , i n a d d i t i o n , llxyll 5 llxll llyll f o r each p a i r x , ycX and (lell=l, then

i s an a l g e b r a p-norm and X a p-normed a l g e b r a

*.

11 )I

I t i s a p-Banach a l g e b r a

i f X i s complete w i t h r e s p e c t t o t h e m e t r i c d(x,y)=llx-yll, x , y f X . A p-Banach a l g e b r a X i s a n LB-algebra w i t h r e s p e c t t o a c e r t a i n bound s t r u c t u r e @ w h i c h w e now d e s c r i b e .

I f xl,

...,x

a r e e l e m e n t s of X w i t h p-

norm l e s s t h a n one, and M(x.) d e n o t e s t h e c o l l e c t i o n o f a l l monomials i n x1

,...,x

i

(i.e.

i

e l e m e n t s of t h e form x l d I , x n

where il

,..., i n

a r e non-

n e g a t i v e i n t e g e r s and xo=e) t h e n 6 c o n s i s t s of a l l s e t s of t h e form

k

B(x.1 = cl(M(xi)bc). C l e a r l y each o f t h e s e t s B ( x . ) i s a b s o l u t e l y - c o n v e x and c o n t a i n s e .

Since

I(xy(\5 (IxI( l\y[\ f o r each p a i r x , y f X i t f o l l o w s j u s t a s i n t h e Banach a l g e bra c a s e t h a t m u l t i p l i c a t i o n i s ( j o i n t l y ) c o n t i n u o u s .

Using t h i s and t h e

f a c t t h a t t h e s e t s M(xi)bc a r e m u l t i p l i c a t i v e w e o b t a i n

Observing t h a t B ( x i ) i J B ( y . ) C B ( x i , y j ) w e conclude t h a t @ i s a bound s t r u c 3 To show t h a t @ i s complete i t i s f i r s t e s t a b l i s h e d t h a t each t u r e f o r X. l1 B=B(x.) i s bounded i n t h e m e t r i c s p a c e X . Indeed, i f b=CA(il, i )x 1n 1 .x lncM(xi)bc where t h e sum i s taken o v e r some f i n i t e c o l l e c t i o n of n-

. ..

t u p l e s ( i l , . . . , i n ) and ZlA(i,

...,

,..., i n ) I 5

1, then i

I f we choose 1 > r 2 0 such t h a t JIxkJI< r f o r e a c h k=l,.

.., n

lows from t h e above i n e q u a l i t y t h a t

*

For more on p-normed a l g e b r a s s e e Zelazko 1965, Chapter I.

then i t f o l -

6.1

s o e a c h M(xi)bc

331

BOUND STRUCTURES

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

i s bounded.

Next w e c l a i m t h a t pB i n d u c e s

s p a c e i s b o u n d e d , B=B(x.) i s a l s o bounded.

a s t r o n g e r t o p o l o g y on X(B) t h a n t h e r e l a t i v e t o p o l o g y f r o m X t h e r e b y i m S i n c e B i s bounded t h e r e e x i s t s a n c > 0 s u c h

p l y i n g t h a t pB i s a norm.

t h a t BcSC(o)nX(B)={xCX(B) l]lx)\


0 t h e r e i s an index N

> 0 such t h a t x -

It i s e a s y t o see t h a t CB i s c l o s e d i n X s o x

x’lim(xn-xm)€ cB. Hence xCX(B), p ( x -x) m B n c l u d e t h a t X(B) i s a Banach a l g e b r a .

5 c

-

whenever n _> N , and w e con-

The o n l y r e m a i n i n g t h i n g t o n o t e i s t h a t f o r e a c h xcX, x ~ X ( B ( h x ) ) ,

1~s h a s t h e p r o p e r t y t h a t llhxll < 1.

where

Thus X=UX(B) and X i s a n LBBC@

algebra . Example 6 . 1 - 2

L e t X b e a u n i f o r m a l g e b r a on t h e

X-HOLOMORPHIC FUNCTIONS

compact H a u s d o r f f s p a c e T ( D e f . 4 . 1 2 - 4 ) .

A f u n c t i o n y 0 there exists

The f u n c t i o n ycC(T,C) i s X-holomor-

p h i c on T i f i t i s X-holomorphic a t e a c h tCT.

The c o l l e c t i o n o f a l l X-

h o l o m o r p h i c f u n c t i o n s on T, d e n o t e d by H(T,X), c l e a r l y forms a complex a l gebra which c o n t a i n s X a s a s u b a l g e b r a .

It i s o u r c o n t e n t i o n t h a t H(T,X)

i s an LB-algebra. L e t d = ( u .)?- b e a f L n i t e open c o v e r of T a n d H ( T , Y , X ) b e t h e c o l J 3-1 l e c t i o n o f a l l ycC(T,L) s u c h t h a t y c a n be u n i f o r m l y a p p r o x i m a t e d on U. by J e l e m e n t s o f X f o r e a c h j=l, n. To see t h a t H ( T , U , X ) i s a u n i f o r m a l -

...,

g e b r a o n T i t s u f f i c e s t o show t h a t i t i s c l o s e d i n C ( T , C , c ) .

Then f o r e a c h c > 0 t h e r e i s a y < H ( T , U , X )

suppose t h a t z

<x

>+


i n d e x e d by s i s s p e c i f i e d t o b e x

and a l l t h e

i s e a s i l y s e e n t o be a n a l g e b r a i s o m o r p h i s m s o w e c a n

t r a n s f e r t h e norm o f X -k

I ( x )=O whenever t i s s o l a r g e t ts s

c xt-+x

Is:Xs--'

where t h e e n t r y o f

c < _

iff

t o Is(Xs),

i.e.

11

C xs

>+

Nlls=IFSIIS, whereby

One c a n m o t i v a t e t h e d e f i n i t i o n o f t h e i n d u c t i v e l i m i t i n t h e f o l l o w i n g

way.

The i d e a i s t o c o n s t r u c t a v e c t o r s p a c e Z c o n t a i n i n g t h e Z Is a s s u b t

is a

s p a c e s (more p r e c i s e l y i s o m o r p h i c images o f t h e Z ' s ) s u c h t h a t 2 t

whenever s 5 t a n d Z i s t h e i n c r e a s i n g l i m i t i n t h e s e t t t h e o r e t i c s e n s e o f t h e Z I s . The d i r e c t sum c 2 c e r t a i n l y c o n t a i n s i s o t tEM t m o r p h i c images o f t h e Z I s b u t t h e o t h e r d e s i r e d p r o p e r t i e s a r e l a c k i n g . subspace of Z

t

Thus w e a r e l e d t o c o n s i d e r a q u o t i e n t s p a c e o f t h e d i r e c t sum,

.?&Zt/N,

tc

and t h e f o l l o w i n g c a n d i d a t e s f o r e a c h index s f o r the r e q u i r e d isomorphism Is:Zs

z where < z

dexed by s .

- C Zt

- < z

Zt/N

> - < z

>+N

> is the tuple with zero e n t r i e s f o r It r e a d i l y f o l l o w s t h a t e a c h I

t#s and z

i n the entry in-

i s an isomorphism, Is(Zs)

s u b s p a c e of I (Z ) f o r s 5 t p r o v i d e d N i s d e f i n e d a s a b o v e (C I t s x s t t < x > EN f o r e a c h x cXs) and t h a t C Z t / N = Is(Zs).

u

s cM

is a

> -

334

6.

LB-ALGEBRAS

I (X ) becomes a Banach a l g e b r a . Suppose t h a t s 5 t ; s i n c e < I x > s s tS s > + N = < I x > + N f o r e a c h x FX With ts s s s t he a i d of t h i s e q u a l i t y i t follows t h a t Is(Xs) i s a subalgehra of I t ( X t )

.

<x

whenever s 5 t .

I( Ilt

F u r t h e r m o r e t h e c o n t i n u i t y of I

of I (X ) r e s t r i c t e d t o I s ( X s ) t

scalar

),

St

t

> 0 such t h a t

11 Itt

11 \Is whenever

g Ats

5 AtsJI

\Is

<e >+

i t follows t h a t B s C A

t € M such t h a t r , s

S i n c e X(Bs)=Is(Xs)

5

I( (Is:

s 1 t.

There e x i s t s a Let B

be t h e

We c l a i m t h a t & i s a complete

The e l e m e n t s of /ij a r e c e r t a i n l y a b s o l u t e l y m-convex

and e a c h c o n t a i n s t h e i d e n t i t y

(1 Ilt

i m p l i e s t h a t t h e norm

i s "weaker" t h a n

c l o s e d u n i t b a l l o f I (Xs) and @={Bs]scM. bound s t r u c t u r e f o r X .

tS

B

Moreover, i f s 5 t t h e n , s i n c e

N.

. t

Hence i f B

B

4w e

can choose

s' r t and o b t a i n t h e c o n c l u s i o n t h a t B U B C m a x ( k t r , X tS

)Bt. r s tS i s a Banach a l g e b r a , (X,@) i s a complete bound a l g e b r a .

I t remains t o show t h a t X=UX(B ) .

To t h i s end c o n s i d e r a n a r b i t r a r y

There a r e o n l y a f i n i t e number of i n d i c e s s s u c h t h a t n we s e e t h a t (x)tN=.C < x > -+ s 1=1 s. sn NCIt(Xt)=X(Bt) f o r any t 2 s l , We summarize t h e s e r e s u l t s i n : e l e m e n t (x )+N€X.

Denoting t h e s e i n d i c e s by s l , . . . , s

xs#O.

..., .

Theorem 6 . 2 - 1

AN LB-ALGEBRA IS AN INDUCTIVE LIMIT OF BANACH ALGEBRAS A

complex a l g e b r a X i s a n LB-algebra w i t h r e s p e c t t o some bound s t r u c t u r e i f f X i s t h e i n d u c t i v e l i m i t o f a n i n d u c t i v e s y s t e m of complex Banach a l g e b r a s

and c o n t i n u o u s u n i t a l isomorphisms. S i n c e any complex Banach a l g e b r a i s a Q - a l g e b r a e a c h n o n - t r i v i a l comp l e x - v a l u e d homomorphism o f a Banach a l g e b r a i s c o n t i n u o u s c ( 4 . 1 0 - 5 ) ] . (b.2-1)

HOMOMORPHISMS OF AN LB-ALGEBRA ARE "BOUNDED"

I f X i s a n LB-algebra

w i t h r e s p e c t t o t h e bound s t r u c t u r e

63

homomorphism h maps t h e e l e m e n t s o f

@ i n t o bounded s u b s e t s of

Proof

I f B&

t h e n e a c h n o n - t r i v i a l complex-valued

2.

t h e n , s i n c e e < B , h r e s t r i c t e d t o t h e Banach a l g e b r a X(B)

n o n - t r i v i a l homomorphism o f X ( B ) . B , a bounded s u b s e t o f X ( B ) ,

A s such i t is continuous.

is a

Thus i t t a k e s

i n t o a bounded s e t of complex numbers. V

I n view o f t h e f a c t t h a t t h e bound s t r u c t u r e s a s s o c i a t e d w i t h a n LBa l g e b r a X need n o t a r i s e from a t o p o l o g y , i t d o e s n ' t make s e n s e i n g e n e r a l t o a s k when a n o n - t r i v i a l homomorphism h i s c o n t i n u o u s .

Even i n t h e e v e n t

t h a t t h e e l e m e n t s o f a a r e bounded i n some c o m p a t i b l e t o p o l o g y on X , i t need n o t f o l l o w t h a t e a c h h be c o n t i n u o u s (see Example 6 . 3 - 2 ) . R e c a l l t h a t i f X i s a complex Banach a l g e b r a t h e n X h c a r r y i n g i t s h G e l f a n d t o p o l o g y , i . e . o(X ,X) i s a compact H a u s d o r f f s p a c e . This property i s c a r r i e d o v e r t o t h e LB-algebra

case, i . e .

i f X i s a n LB-algebra,

the col-

l e c t i o n o f a l l n o n - t r i v i a l complex-valued homomorphisms H(X) w i t h t h e

335

PROPERTIES OF LB-ALGEBRAS

6.2

O(H(X) ,X) topology ( t h e Gelfand topology) i s a non-empty compact Hausdorff

We w i l l prove t h i s by f i r s t showing t h a t H(X) i s a " t o p o l o g i c a l

space.

projective l i m i t "

(a n o t i o n which i s d e f i n e d below) o f non-empty compact

Hausdorff s p a c e s followed by a d e m o n s t r a t i o n o f t h e f a c t t h a t a t o p o l o g i c a l p r o j e c t i v e l i m i t of non-empty compact Hausdorff s p a c e s i s a non-empty comp a c t Hausdorff s p a c e . Theorem 6.2-2

H(X) I S A NON-EMPTY COMPACT HAUSDORFF SPACE

L e t H(X) be t h e

c o l l e c t i o n of a l l n o n - t r i v i a l complex-valued homomorphisms o f t h e LB-algebra

X w i t h bound s t r u c t u r e (a)

63 .

Then:

H(X) w i t h i t s Gelfand topology i s homeomorphic t o t h e t o p o l o g i c a l

p r o j e c t i v e l i m i t ( d e f i n e d i n t h e p r o o f ) of t h e non-empty compact Hausdorff w i t h B, B ' c a and B 5 B ' ; BB' H(X) w i t h i t s Gelfand topology i s a non-empty compact Hausdorff

s p a c e s X(B)h and c o n t i n u o u s r e s t r i c t i o n maps R (b) space.

_Proof

F i r s t we n o t e t h a t X(BIh i s n o t empty b e c a u s e X(B) i s a Banach

algebra.

Suppose now t h a t hCH(X) and B f . 6 ,

X.

t h e bound s t r u c t u r e a s s o c i a t e d w i t h

V(B)h

L e t hg=h

IX(B) and c o n s i d e r t h e e l e m e n t ( h ) BFI> of t h e p r o d u c t BF I t i s c l e a r t h a t i f B 5 B ' (where 0 ,

337

PROPERTIES OF LB-ALGEBRAS

h x o , c ) = ( h ~ X ( B o ) I J h ( x o ) - h B (x,)

BO

1 < €3,

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

in BO

Now i t i s c l e a r t h a t T ( V ( h , x o , c ) ) C < V(hBn,xo,C) >, from which

-

T i s b i c o n t i n u o u s s i n c e i t i s a 1-1 c o n t i n u o u s mapping

continuity follows.

o f a compact H a u s d o r f f s p a c e i n t o a H a u s d o r f f s p a c e . V

A s a n i m m e d i a t e c o n s e q u e n c e o f t h e f a c t t h a t H(X)#0

we o b t a i n a r e s u l t

f o r L B - a l g e b r a s s i m i l a r t o t h e Gelfand-Mazur t h e o r e m [Theorem 4.9-11.

(b.2-2)

I F THE LB-ALGEBRA X I S A FIELD, X"="C

I f X i s a n LB-algebra and

a f i e l d t h e n X i s a l g e b r a i c a l l y i s o m o r p h i c t o ,$. Proof

By Theorem b.2-2

( b ) t h e r e e x i s t s a n o n - t r i v i a l homomorphism h : X - s .

If w e assume t h a t h i s n o t 1-1 t h e n t h e r e e x i s t s a n o n - z e r o x i n t h e f i e l d X , s u c h t h a t h(x)=O.

T h u s , f o r e a c h yEX,

h(y) = h(ye) = h(yx

-1

-

x) = h(yx

Thus h i s 1-1 dnd X i s a l g e b r a -

w h i c h c o n t r a d i c t s t h e n o n - t r i v i a l i t y of h . i c a l l y isomorphic t o

-1 ) h(x) = 0

s. V

R e c a l l t h a t i n any Banach a l g e b r a t h e maximal i d e a l s a r e i n 1-1 c o r r e s p o n d e n c e w i t h t h e n o n - t r i v i a l complex h o m o m o r p h i s m . case t h a t the k e r n e l of a n o n - t r i v i a l i s a maximal i d e a l .

It i s always t h e

(complex) homomorphism o f a n a l g e b r a

On t h e o t h e r hand i f M i s a maximal i d e a l i n a n a l g e -

b r a X t h e n X/M i s c e r t a i n l y a f i e l d .

Hence i f w e c a n show t h a t X/M i s a n

L B - a l g e b r a whenever X i s a n L B - a l g e b r a and M i s a maximal i d e a l , i t w i l l f o l l o w by ( b . 2 - 2 )

t h a t M i s t h e k e r n e l o f t h e complex homomorphism h:X-+X/M.-

where x+M

-

x

+x+M+

A i s a n i s o m o r p h i s m between X / M and

c.

To do t h i s we p r o v e a more g e n e r a l r e s u l t : (b.2-3)

QUOTIENTS OF LB-ALGEBRAS

t h e bound s t r u c t u r e

&

L e t X be a n L B - a l g e b r a w i t h r e s p e c t t o

a n d I b e a n i d e a l of X w i t h t h e p r o p e r t y t h a t I ~ \ x ( B )

i s a c l o s e d i d e a l i n X(B)

f o r each B F 8 .

Then t h e a l g e b r a X / I i s a n LB-

a l g e b r a w i t h r e s p e c t t o t h e bound s t r u c t u r e

B/I=(B+I \Be&].

In particular

X / M i s a L B - a l g e b r a f o r a n y maximal i d e a l M of X . Proof

F i r s t w e n o t e t h a t by some r o u t i n e c o n s i d e r a t i o n s @/I

s t r u c t u r e on X / I .

i s a bound

I t i s a l s o c l e a r t h a t f o r e a c h B C O t h e s u b a l g e b r a of X / I

g e n e r a t e d by B + I i s j u s t X(B)+I=(x+I IxCX(B)) and X/I=Bu&(B)+I.

Thus i t

6.

338

LB-ALGEBRAS

remai,ns t o show t h a t e a c h a l g e b r a X(B)+I i s a Banach a l g e b r a w i t h r e s p e c t t o t h e gauge of B + I . 'B+I

To t h i s end s u p p o s e t h a t xcX(B) a n d c o n s i d e r

>0

( x + I ) = inf{A

Ix

= inf

-+

IE)\(B

+

>0

inf{h

I)]

+ zchB?

zcI n X ( B ) = inf

pB(x

zmX(B)

+

Z)

=

P,(x

+

I n x(B)).

Thus t h e mapping X(B)

+ I --+

X ( B ) / I n X(B)

+ I -+-

x

i s seen t o be an onto "isometric"

+ In X(B) isomorphism when X(B)+I c a r r i e s t h e semi-

a n d X ( B ) / I n X ( B ) t h e semi-norm pB i n d u c e d by p Since InX(B) norm p B+I B' i s c l o s e d i n X ( E ) , p i s a norm w h i c h , b e c a u s e X(B) i s c o m p l e t e i n t h e norm B pB, r e n d e r s X ( B ) / I n X ( B ) a Banach a l g e b r a . Hence e a c h a l g e b r a X(B)+I i s a Banach a l g e b r a and X / I i s a L B - a l g e b r a . Now i f M i s a maximal i d e a l i n X t h e n w e c l a i m t h a t e a c h i d e a l MnX(B) i s c l o s e d , i n t h e Banach a l g e b r a X ( B ) . i.e.

Suppose t h a t t h i s i s n ' t t h e c a s e ,

t h e r e e x i s t s a Bed) s u c h t h a t M n X ( B )

t h e r e e x i s t s a sequence (%)CMnX(B)

i s n o t c l o s e d i n X(B).

c o n v e r g e n t i n X(B)

Then

t o some X ~ M .

S i n c e M i s maximal i n X e l e m e n t s ycX and mcM e x i s t s u c h t h a t e=yx+m.

B'c@

s u c h t h a t X(B')XX(B)

t a k i n g X(B) y\+m

-+

Choose

A s t h e i n j e c t i o n mapping I B'B i n t o X(B') i s c o n t i n u o u s by Theorem 6 . 2 - 1 i t f o l l o w s t h a t

yx+m=e i n X ( B ' ) .

and y€X(B').

S i n c e a Banach a l g e b r a i s a Q - a l g e b r a

we may c h o o s e k s o l a r g e t h a t y\+m€M However y \+mtM

[(4.8-2)]

b e l o n g s t o t h e open s e t of u n i t s .

and M, b ei n g p ro p er, can c o n t a i n no i n v e r t i b l e el em ent s.

T h i s c o n t r a d i c t i o n i m p l i e s t h a t e a c h MnX(B) i s c l o s e d i n X ( B )

t h e r e b y con-

c l u d i n g the p r o 0 f . V A s a n immediate c o n s e q u e n c e of t h i s r e s u l t a n d t h e remarks p r e c e d i n g i t we h a v e

i6.2-5)

I N AN LB-ALGEBRA H(X)

"="hf

Each maximal i d e a l of a LB-algebra i s

t h e k e r n e l of some h c H ( X ) . I n l i g h t of t h e l a s t r e s u l t i t i s e v i d e n t t h a t whenever M i s a maximal i d e a l i n a n L B - a l g e b r a X e a c h e l e m e n t XCX c a n b e w r i t t e n u n i q u e l y i n the form x=)\e+m where h ~ and s mcM.

Thus M i s a s u b s p a c e of X of c o d i m e n s i o n

o n e c o n s i s t i n g of s i n g u l a r e l e m e n t s .

G l e a s o n 19b7 ( c f . Beckens t e i n , N a r i c i ,

6.2

339

PROPERTIES OF LB-ALGEBRAS

and Bachman 1971) p r o v e d t h e c o n v e r s e f o r Banach a l g e b r a s , i . e . any s u b s p a c e

of a Banach a l g e b r a c o n s i s t i n g s o l e l y o f s i n g u l a r e l e m e n t s a n d h a v i n g c o d i m e n s i o n one i s a maximal i d e a l .

We c o n c l u d e t h i s s e c t i o n w i t h a p r e s e n -

t a t i o n o f G l e a s o n ' s r e s u l t f o l l o w e d by a n e x t e n s i o n o f i t t o L B - a l g e b r a s . I N A N LB-ALGEBRA. M f l 0 =

(6.2-6)

0

AND COD(M)

= 1 IMPLIES pj&

If M is a

s u b s p a c e o f t h e LB-algebra X o f c o d i m e n s i o n o n e , c o n s i s t i n g of s i n g u l a r elements t h e n M i s a maximal i d e a l . F i r s t s u p p o s e t h a t X i s a compIex Banach a l g e b r a .

Proof

The s u b s p a c e M ,

b e i n g of c o d i m e n s i o n o n e , must b e e i t h e r d e n s e o r c l o s e d i n X. Q-algebra

(4.8-2),

Since X i s

Q, t h e s e t of u n i t s i n X , i s non-empty and o p e n ;

c o n t a i n i n g o n l y s i n g u l a r e l e m e n t s , must be c l o s e d .

so M,

I f we l e t h be t h a t

l i n e a r f u n c t i o n a l on X h a v i n g M a s i t s n u l l s p a c e and mapping e i n t o 1, t h e n s i n c e M i s closed, h i s continuous. homomorphism.

N o w i t s u f f i c e s t o show t h a t h i s a

Furthermore, the equation x+y) xy = (

2

-

x

2

- y

2

2 2

t o g e t h e r w i t h t h e l i n e a r i t y o f h r e d u c e s t h e p r o b l e m t o showing t h a t h ( x ) = (h(x))'

f o r each xcx.

To t h i s e n d c o n s i d e r t h e f u n c t i o n

S i n c e jlxnl/ 5 \lxlln i n a normed a l g e b r a i t f o l l o w s t h a t t h e d e f i n i n g s e r i e s

i s a b s o l u t e l y c o n v e r g e n t a n d , t h e r e f o r e , c o n v e r g e n t i n t h e Banach a l g e b r a X f o r each

LcG.

Thus, by t h e c o n t i n u i t y of h ,

and h ( e x p ( 1 x ) ) i s an e n t i r e f u n c t i o n w i t h no z e r o s .

Furthermore, s e t t i n g

M ( r ) = s u p I h ( e x p ( 1 x ) ) ) we o b t a i n I=r

IA

order (h(exp(hx))) = r

>+

l n l n M(r)* inr

-- m

The n o t i o n o f t h e o r d e r o f a n e n t i r e f u n c t i o n may be found i n M a r k u s h e v i c h

( 1 9 6 5 ) , V o l . 2 , p. 251.

6.

340

LB-ALGEBRAS

= 1.

the f a c t

Thus, by a weak v e r s i o n o f Hadamard's f a c t o r i z a t i o n theorem",

t h a t h ( e x p ( h x ) ) n e v e r assumes t h e v a l u e z e r o , a n d , h ( e x p ( h x ) ) = l , i t f o l l o w s that n n

m

h ( e x p ( h x ) ) = ecYh=

f o r some acs. f o r each n

n =

4 ~ n.

Now by t h e i d e n t i t y t h e o r e m f o r power s e r i e s

2 0;

?r*

, h(xn)=cyn

so

2 2 h(x ) = a = (h(x))' f o r e a c h xEX. Next s u p p o s e t h a t X i s a n L B - a l g e b r a w i t h r e s p e c t t o t h e bound s t r u c t u r e @ and M i s l i n e a r s u b s p a c e o f X a s i n t h e h y p o t h e s i s .

S i n c e t h e co-

d i m e n s i o n o f M i s o n e , w e may w r i t e e a c h xcX u n i q u e l y i n t h e form x=Ae+m where

AcL,

and mcM.

Now i f x€X(B), ( B E @ ) , t h e n , s i n c e e(B, mcX(B), and

i t f o l l o w s t h a t the c o d i m e n s i o n o f M n X ( B )

i s one i n X(B).

As t h e e l e m e n t s

o f M a r e a l l s i n g u l a r s o a r e t h e e l e m e n t s o f MnX(B) s i n g u l a r i n X(B). Hence by t h e r e s u l t e s t a b l i s h e d a b o v e w e c o n c l u d e t h a t MnX(B) i s a maximal i d e a l i n X(B).

It r e m a i n s t o show t h a t M i s a n i d e a l i n X .

l e t xcM and ycX and c h o o s e BE&

s u c h t h a t x , yCX(B).

To t h i s end

Then xcMnX(B) and

xyCMfl X(B)CM. V S e c t i o n 6.3-1.

The c o n c e p t of a n L B - a l g e b r a i s

Complete LMC LB-Algebras

primarily algebraic

-

i t n e e d n o t be a t o p o l o g i c a l a l g e b r a t o b e g i n w i t h ,

n o r d o w e h a v e t o add t o p o l o g i c a l s t r u c t u r e t o i t f o r i t s s a l i e n t p r o p e r t i e s ( s c e S e c . b.2;.

However, i t i s t h e c a s e t h a t w e c a n c o m p l e t e l y c h a r -

a c t e r i z e c o m p l e t e LMC L B - a l g e b r a s , i . e . 7k

t h o s e complete-LMCH-algebras

(Markushevich ( 1 9 b 5 ) , Vol 2 , p . 266) i f t h e e n t i r e f u n c t i o n f ( X )

o r d e r p n e v e r assumes t h e v a l u e

w c s then p

that of

i s a n i n t e g e r and f ( A ) i s o f t h e

form f(A)=w+eP(*) w h e r e p(A) i s a p o l y n o m i a l o f d e g r e e p . **r(Markushevich ( 1 9 6 5 ) , Vol. 1, p . 3 5 2 ) . I f t h e complex power s e r i e s a n a g r e e on a bounded i n f i n i t e s e t of complex numbers nEo a n ( z - z o > t h e n a =b f o r e a c h n 2 0. n n

6.3

a r e a l s o LB-algebras.

34 1

COMPLETE LMC LB-ALGEBRAS

A bound s t r u c t u r e o f a Banach a l g e b r a c o n s i s t s o f

t h e u n i t b a l l , a t o p o l o g i c a l l y bounded c l o s e d a b s o l u t e l y m-convex s e t cont a i n i n g the i d e n t i t y ;

a n a t u r a l bound s t r u c t u r e t o c o n s i d e r i n a n LMC a l -

g e b r a i s t h e c o l l e c t i o n @ o f a l l a b s o l u t e l y m-convex c l o s e d bollnded s e t s n w h i c h c o n t a i n t h e i d e n t i t y . W e s h a l l show i n t h e main t h e o r e m o f t h i s s e c t h a t a c o m p l e t e LMC a l g e b r a i s a n LB-algebra w i t h r e -

t i o n (Theorem 6.3-1)

s p e c t t o t h i s bound s t r u c t u r e whenever the s p a c e o f n o n - t r i v i a l cornplexv a l u e d homomorphism i s compact i n i t s G e l f a n d t o p o l o g y .

I t f o l l o w s from

t h i s and t h e f a c t t h a t t h e s p a c e o f n o n - t r i v i a l c o m p l e x - v a l u e d homeomorp h i s m s i s a l w a y s compact f o r a n L B - a l g e b r a (Theorem 6 . 2 - 2 )

t h a t i f a com-

p l e t e LMC a l g e b r a i s a n L B - a l g e b r a w i t h a bound s t r u c t u r e @ t h e n i t must b e an LB-algebra w i t h r e s p e c t t o

a.

It i s a l s o e s t a b l i s h e d t h a t t h e F r e c h e t

LB-algebras are p r e c i s e l y t h e F r e c h e t Q -al gebr as D e f i n i t i o n 6.3-1. i c a l algebra.

THE NATURAL BOUND STRUCTURE

(Theorem 6 . 3 - 2 ) . L e t X b e a complex t o p o l o g -

Then t h e a s s o c i a t e d n a t u r a l bound s t r u c t u r e , d e n o t e d by

Ban,

i s t h e c o l l e c t i o n of a l l s u b s e t s B C X such t h a t

(a)

B i s a b s o l u t e l y m-convex and e f B ,

(b)

B i s c l o s e d a n d bounded.

The c o l l e c t i o n Xo= gBnX(B) where X(B)={ay Icy&, bounded e l e m e n t s

of &

a n d i s a s u b a l g e b r a o f X.

yCB] i s r e f e r r e d t o a s t h e I f x f X w h i l e xdXo t h e n x

i s a n unbounded e l e m e n t .

P r i o r t o p r e s e n t i n g examples we c h a r a c t e r i z e t h e e l e m e n t s o f X (6.3-1) (i.e.

A CHARACTERIZATION OF Xo

.

The e l e m e n t o f xfX i s a bounded e l e m e n t

xfX ) i f f t h e r e e x i s t s a s c a l a r

s u c h t h a t {(Xx)"lnc>]

i s a bounded

set.

I f xfXo t h e n xfX(B) f o r some B€an.S i n c e X(B)={ay Iacs, y f B ] , x=ay

Proof

f o r some

afs a n d

yCB.

I f x=o t h e n ( ( A x ) " l n ~ g ] i s c e r t a i n l y a bounded s u b -1 s e t o f X. I f x # o t h e n y=Q xfB, and, s i n c e B i s m u l t i p l i c a t i v e , ynfB f o r -1 n a n y n . Thus {(a x ) In€gN]CB a n d i s t h e r e f o r e bounded. C o n J e r s e l y , s u p p o s e t h a t ( ( h ~ ) ~ l n € b J ]i s bounded. (Ax)" lnf,N]U

(el

i s m u l t i p l i c a t i v e and b o u n d e d .

I t i s c l e a r t h a t S=

S i n c e m u l t i p l i c a t i v i t y and

b o u n d e d n e s s a r e p r e s e r v e d when f o r m i n g t b e c l o s e d a b s o l u t e convex h u l l o f a s e t , B=cl(Sbc)ft2jn.

Thus x c X ( B ) C X o .

v

C e r t a i n l y a l l e l e m e n t s o f a normed a l g e b r a a r e bounded a s c a n b e s e e n d i r e c t l y from t h e d e f i n i t i o n of X (6.3-1).

o r from t h e c h a r a c t e r i z a t i o n g i v e n in

I n o u r n e x t example we c h a r a c t e r i z e t h e e l e m e n t s o f C(T,C,c) t h a t

a r e bounded.

A s m i g h t b e e x p e c t e d t h e y a r e j u s t t h e s e t of u n i f o r m l y bound-

e d c o m p l e x - v a l u e d f u n c t i o n s on T .

6.

342

Example 6 . 3 - 1 .

C(T,C,c)

LB-ALGEBRAS

"=" UNIFORMLY BOUNDED FUNCTIONS

L e t xFC(T,S,c) 2

we c l a i m t h a t s i s a bounded e l e m e n t i f f i t i s u n i f o r m l y bounded on T. deed i f x i s u n i f o r m l y bounded on

T by t h e number M 7 0, i . e . s u p Ix(T)

In-

I < M,

t h e n t h e f u n c t i o n (1/M)x and a l l o f i t s p o s i t i v e powers a r e u n i f o r m l y boundn Thus i t f o l l o w s t h a t e a c h f u n c t i o n ((I/Mx) , n = 1 , 2 , i s bounded

...,

ed by 1.

by 1 on e a c h compact s e t K.

S i n c e t h e seminorms p ( y ) = s u p l y ( K ) I,y

0 such t h a t

Hence p ( x ) , t h e r e s o l v e n t s e t , i s

and ( 4 . 8 - 6 ) , t h e r e s o l v e n t f u n c t i o n r O , ) = ( x - k e ) - '

is

F u r t h e r m o r e , a g a i n by t h e c o m p a c t n e s s o f ~ ( x ) ,t h e r e i s

(kc21 I h ~ > r ] c p ( x ) .

Thus, s e t t i n g

c=l/r,

w e can d e f i n e

t h e f u n c t i o n s a t A by

I

which i s c l e a i l y a n a l y t i c f o r 0 , 201.

so, 4 Sc ( 0 ) , open u n i t d i s c of r a d i u s

> 0 i n any normed space

f

U(X), 201 Q(X,X’), 4 supp ( s u p p o r t ) , 91, 92

302

JG,

am,300 Zhk’301 Jr,

255

*

Jr

T* = [ t ) t € T , t : C ( T , F ) N

VP.Jc1, Jwc

Y

+;,

x

-+

x(t))

190 248

T(X,X’), 4

201

U(O),

u T , 21 uxT, V -P

311

, 183

vP

= EXlP(X)

V(x),

5

13

neighborhood f i l t e r a t x i n any t o p o l o g i c a l space

wL, 143 w ( T , t ) , 143 (XI, 220 X(B), 329

xh,

223

X ’ , continuous d u a l of TVS X, 227

4, Gelfand map, 223 X

*, a l g e b r a i c

2, z e r o sets Z,

.-

integers

z(M), LO

z ( x > , 10

d u a l of v e c t o r s p a c e ,

x

INDEX

a b s o l u t e l y m-convex, 181 a b s o l u t e l y K-convex, 134 a d j o i n t , 227 A l l a n , 329 A lexandr ov, 84 a l g e b r a , 176 a l g e b r a homomorphism, 1 7 7 a l g e b r a w i t h i n v o l u t i o n , 259 symmetric, 259 almost open, 257 Alo, 170 u p - l a t t i c e , 148 a- l a t t i c e , 147 a n a l y t i c , 211 a n a l y t i c f u n c t i o n s on a d i s c , 195, 305 Arens, 199, 210, 296 A s c o l i theorems, 120 atom, 165 atomic, 166 Bachman, 51, 55 Bade, 317, 318, 320, 321 Bagley, 118, 120 B a i r e measure, 124 Banach a l g e b r a , 179 Banaschewski c o m p a c t i f i c a t i o n , 53 B a r r e l e d , 93 B a r r e l e d n e s s of C(T,F,c), 94 Beckenstein, 51, 55,-296, 339 p - l a t t i c e , 148 Boolean l a t t i c e , 137 bound a l g e b r a , 329 complete, 329 bounded i n a t o p o l o g i c a l r i n g , 49, 2 1 1 bounded element, 341 bounding s e t , 132 bound s t r u c t u r e , 329 b o r n o l o g i c a l , 99 b o r n o l o g i c i t y of C(T,E,c), 99 bornology, 110 Brooks, 1 7 0 , 172 Browder, 254 C-extension, 50 C - e x t e n s i o n , 50 b C-embedded, 2 1 C -embedded, 21 Ckandler, 50 c h a r a c t e r , 28, 52 clopen, 1 Comfort, 58 cornpactification, 8 compact-open topology, 3 , 186 compatible topology, 177 compatible u n i f o r m i t y , 5 complement, 137 365

366

INDEX

complemented l a t t i c e , 137 complete Boolean l a t t i c e , 166 completely r e g u l a r , 1 completely s e p a r a t e d , 12 completeness of C(T,E,c), 64 completion, 6 condition hH, 306 continuous i n v e r s e , 208 Correl, 49 C u r t i s , 241, 317, 318, 320, 321, 323 Dales, 329 d - d i s c r e t e , 44 6 - z - u l t r a f i l t e r , 42 d e r i v a t i o n , 237 De Wilde, 115 Dieudonne, 99 d i f f e r e n t i a b l e f u n c t i o n s , space o f , 187 discontinuous homomorphisms, 276, 345 dual of C (T,!), 77 dual of C?T,F,c), 88 E-closed, 171 E-compact, 54 E-compactification, 54, 171 E-completely r e g u l a r , 53 Engelking, 53, 54 Equivalent Wallman spaces, 161 ES-algebra, 293 f a c t o r a l g e b r a , 193 of C(T,;,c), 194 F-algebra, 290 f i l t e r , 140 f i n a l topology, 3, 180, 190 f i x e d i d e a l 1 7 , 51 formally r e a l , 218 f r e e i d e a l , 1 7 , 51 f r e e union, 1 1 7 , 252 F r o l i k , 58 f u l l a l g e b r a , 242 f u l l y complete 121, 258 f u n c t i o n a l l y continuous, 275 Gelfand, 175 Ge 1 fand-Kolmogorov theorem, 18 Gelfand map, 223 Gelfand-Mazur theorem, 212 Gelfand topology, 223 g e n e r i c p o i n t , 168 Gleason, 338 Glicksberg, 58 G-normal, 304 Goldhaber, 49 Gulick, 241

INDEX

hemicompac t , 62 Henriksen, 32, 4 9 , 59 HK, 300 hk, 300 hk-normal, 304 homomorphism, 28, 177, 222 homomorphism topology a ( X , X ), 269 HTVS, 4 h u l l , 300 h u l l - k e r n e l normal, 304 h u l l - k e r n e l topology, 300 i d e a l , 189 i n d u c t i v e l i m i t , 332 i n d u c t i v e system, 332 i n f norm, 222 i n f r a b a r r e l e d , 98 i n f r a b a r r e l e d n e s s of C ( T , F , c ) , 98 i n i t i a l topology, 2, 180,-188 i n i t i a l uniformity, 5 i r r e d u c i b l e c l o s e d s e t , 167 I s b e l l , 59 Jacobson complete, 168 Jacobson f i l t e r , 169 Johnson, 241 Kaplansky, 32, 4 9 , 175, 214 K - b a r r e l e d , 134 K - b o r n o l o g i c a l , 134 k e r n e l , 300 k - e x t e n s i o n topology, 70 Kowalsky, 4 9 , 320 K-pseudocompact, 134 k - s p a c e , 65 k r s p a c e , 65, 116 Kuczma, 296 kZ-space, 116 l a t t i c e , 32, 136 l a t t i c e f i l t e r , 140 l a t t i c e u l t r a f i l t e r , 140 LB-algebra, 329 LCHS, 4 LCS, 4 Lindelaf, 1 L i o u v i l l e ' s theorem, 211 LMC a l g e b r a , 184 LMCH a l g e b r a , 184 LMC topology, 184 i n i t i a l , 188 f i n a l , 190 l o c a l l y A-convex, 281 l o c a l l y compact, 1 l o c a l l y c o n s t a n t , 158 l o c a l l y m-convex a l g e b r a , 184

367

368

INDEX

l o c a l l y p-convex, 286 Il-uniform c o n t i n u i t y , 153 Mackey topology, 4 maximal f i l t e r subbase, 142 McClure, 329 m-convex, 181 measurable c a r d i n a l , 43 measurable f u n c t i o n s , space o f , 214 measure f i n i t e l y a d d i t i v e , 34 r e g u l a r , 35 signed, 34 0-1, 34 m e t r i z a b i l i t y of C(T,E,c), 62 M i l l e r , 241 Michael, 176, 242, 257, 282 m u l t i p l i c a t i v e convexity, 181 m u 1t i p l i c a t i v e seminorm, 182 Mrmka, 53, 54 Nachbin, 22, 99 Nanzetta, 57 N a r i c i , 51, 55, 296 n a t u r a l bound s t r u c t u r e , 341 Nielsen, 51 Noble, 119 nonarchimedean, 2 nonarchimedean INCH a l g e b r a , 296 nonarchimedean normed a l g e b r a , 296 normal a l g e b r a , 304 normal l a t t i c e , 145 normed a l g e b r a , 179 order-bounded l i n e a r f u n c t i o n a l , 128 p a r t i t i o n o f u n i t y , 82, 265 p-Banach a l g e b r a , 330 permanently s i n g u l a r , 296 Piacun, 172 Pierce, 48, 50, 53 Plank, 57 p-norm, 330 p-normed space, 285 point d e r i v a t i o n , 323 point-open topology, 3 p o l a r , 227 p o s i t i v e l i n e a r f u n c t i o n a l , 77 prime f i l t e r , 169 p r i n c i p a l i d e a l , 220 p r o j e c t i v e l i m i t , 198 t o p o l o g i c a l , 198 p r o j e c t i v e system, 198 t o p o l o g i c a l , 198 pseudo-Banach a l g e b r a , 329 pseudo-compact, 2 7 , 57

INDEX

r e l a t i v e l y , 28 P t a k , 258 Q - a l g e b r a , 204 q u o t i e n t t o p o l o g y , 189 r a d i c a l , 236, 289 r e a l m a x i m a l i d e a l , 29 r e f l e x i v i t y , 111 of C ( T , Z , c ) , 111 r e g u l a r , 201 r e g u l a r a l g e b r a , 241, 303, 318 replete, 21 example o f n o n - r e p l e t e s p a c e , 27 r e p l e t i o n u T o f T , 2 1 , 5 4 , 55 r e s o l v e n t map, 208 R o s e n f e l d , 241, 325 s a t u r a t e d , 183 Schmets, 115 s e m i s i m p l e , 236 s t r o n g l y , 236 s e p a r a b i l i t y o f C ( T , F , c ) , 107 separated by o p e n sets, 1 by a c o n t i n u o u s f u n c t i o n , 1 s e p a r a t i n g f a m i l y , 172 S h a p i r o , 170 S h e p h e r d s o n , 34 S h i r o t a , 3 2 , 99 S h i r o t a ' s theorem, 4 4 u -compact, 1 S i r k c l a i r , 322 S i n g e r , 241, 323 s i n g u l a r , 20 1 Sloyer, 51 s p e c t r a l l y c o m p l e t e , 168 s p e c t r a l norm, 287 s p e c t r a l r a d i u s , 201 spectrum, 201 s p h e r i c a l l y c o m p l e t e , 134 s q u a r e a l g e b r a , 250 s q u a r e - p r e s e r v i n g , 250 s t a r a l g e b r a , 259 s t a r homomorphism, 263 S t e i n e r , 172 S t o n e , 32 Stone-Cech c o m p a c t i f i c a t i o n , 9 , 48, 51, 58 S t o n e R e p r e s e n t a t i o n theorem, 165 S t o n e ' s theorem, 4 9 Su, 172 S u f f e l , 296 support o f a c o n t i n u o u s l i n e a r f u n c t i o n a l , 92, 132 of a s e t f u n c t i o n , 90, 132 supremum t o p o l o g y , 189

369

370

INDEX

topological algebra, 177 topological divisor of zero, 296 topological isomorphism, 177 topological projective limit, 335 TVS, 4 Ulam cardinal, 3 4 , 4 3 Ulam measure, 3 3 , 43 uniform algebra, 254 uniformizable, 5 unit, 201 u 1trabornologica1, 115 ultrafilter, 140 ultranormal, 53 ultraregular, 52 vague topology, 35 valuation, 2 valued field, 1 vanishing set, 132 Varadarajan, 33 very dense, 1 5 0 , 167 Wallman compactification, 1 4 3 , 1 7 1 , 308 Wallman space, 143 Warner, 5 1 , 55 weakened compact-open topology, 248 weakened topology, 4 weak-*normal, 304 weak topology, 4 Wermer, 241, 257, 323 Williamson, 214 Wolk, 49 X-holomorphic, 331 X-repletion, 3 1 1 Young, 118, 120 Zelazko, 285, 296 zero-dimensional, 1 zero-one (0-1) measure, 3 3 , 3 4 , 132 zero set, 10 z-filter, 13 z-ultrafilter, 13