Topological Algebras: Selected Topics

TO POLOGICAL ALGEBRAS SELECTED TOPICS

NORTH-HOLIAND MATHEMATICS STUDIES Notas de Matematica (109)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

124

TOPOLOGICAL ALGEBRAS SELECTED TOPICS Anastasios MALLIOS Mathematical Institute University of Ath ens Greece

1986

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

@ElsevierSciencePublishers B.V., 1986 Allrights reserved. No part of this publication may be reproduced, storedin a retrievalsystem, or transmitted, in any form or b y any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN:O 444 87966 8

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors forthe U.S.A.and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VAN DER BILT AVE N U E NEW YORK, N.Y. 10017 U.S.A.

Library of Congress Catalogingin-Publication Data

Ma l li o s, Anastasios. Topological a l g e b r a s . S e l e c t e d T o p i c s . (North-Holland mathematics s t u d i e s ; 124) (Notas d e m a t d t i c a ; 109) Bibliography: p . In c l u d e s index. 1. Topological al geb ras. I. T i t l e . 11. S e r i e s . 111. S e r i e s : Notas de matema'tica (Rio de Jan ei ro. B r a z i l ) ; 109. QAl.N86 n o . 109 [QA326] 510 s [ 5 1 2 ' . 5 5 ] ' 85-31544 ISBN0-444-87966-8

PRINTED IN THE NETHERLANDS

To the memory of

t h e l a t e Professor

DEMETRIOS A. KAPPOS and t o Professor Dr. D r . h.c. mult.

GOTTFRI ED KOTHE who b o t h i n s p i r e d and supported me i n my f i r s t s t e p s i n t h e profession

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" F u r vieZe Anwendungen i s t noch e i n anderes Prob-

lem wichtig: d i e ijbertragung der Theorie der nomi e r t e n AZgebren auf versehiedene Klassen topologischer ( n i c h t normierter) Algebren, wobei d i e von L . SCHWARTZ [.

.. ]

begriindete Theorie der Distribu-

tionen eine wesentliehe RolZe s p i e l e n wird". M . A . NEUMARK ( " N o r m i e r t e A l g e b r e n "

,

VEB

Deutscher V e r l a g d e r Wiss., B e r l i n , 1959; p. 7 ) . [From the Preface of the o r i g i n a l Russian t e x t of M.A. Naimark : Normed Rings ( e d i t i o n 1956, Moscow), as i t stands i n the above quoted German e d i t i o n ] .

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Preface

The book i s w r i t t e n from s c r a t c h c o n c e r n i n g t h e G e n e r a l Theory o f T o p o l o g i c a l A l g e b r a s , a t i t l e , t o o , w h i c h I was t e m p t e d t o a d o p t a t t h e o u t s e t . I t i s a d d r e s s e d t o a l l t h o s e who w i s h t o a p p l y t h e m e t h o d s and r e s u l t s of t h i s t h e o r y i n a v a r i e t y of d i s c i p l i n e s where one i s c o n f r o n t e d w i t h t o p o l o g i c a l a l g e b r a s e v e n i n some p a r t i c u l a r o r l e s s g e n e r a l form t h a n t h a t c o n s i d e r e d h e r e . A s i m i l a r s i t u a t i o n had app e a r e d a l r e a d y w i t h t h e v a r i o u s t y p e s o f c o n c r e t e o r a b s t r a c t Banach algebras. F u r t h e r m o r e , t h e s u b j e c t matter of t h e book m i g h t a l s o b e of i n -

terest t o t h o s e who w i s h , from an e n t i r e l y t h e o r e t i c a l p o i n t of v i e w , t o see how f a r one c a n g o beyond t h e c l a s s i c a l framework of Banach a l gebras t h e o r y w h i l e still r e t a i n i n g s u b s t a n t i a l r e s u l t s . Indeed, t h e l a t t e r i n q u i r y was, i n e f f e c t , t h e i n i t i a l m o t i v e f o r t h i s u n d e r t a k i n g . N e v e r t h e l e s s , t h e n e e d f o r s u c h a n e x t e n s i o n of t h e s t a n d a r d t h e o r y of normed a l g e b r a s h a s b e e n a p p a r e n t s i n c e t h e e a r l y d a y s of t h e t h e o r y of t o p o l o g i c a l a l g e b r a s , m o s t n o t a b l y t h e l o c a l l y c o n v e x o n e s . Moreover, it i s w o r t h n o t i c i n g t h a t t h e p r e v i o u s demand was due n o t only t o t h e o r e t i c a l reasons, but a l s o t o p o t e n t i a l concrete applicat i o n s o f t h e new d i s c i p l i n e . T h u s , f o r e x a m p l e , a program a l o n g t h e s e l i n e s o f t h o u g h t was a d v o c a t e d a l r e a d y i n t h e m i d - f i f t i e s b y t h e Sov i e t m a t h e m a t i c i a n M. A. Naimark. Among t h e p r e r e q u i s i t e s f o r r e a d i n g t h i s b o o k , some f a m i l i a r i t y w i t h t h e g e n e r a l t h e o r y of t o p o l o g i c a l v e c t o r s p a c e s ( l o c a l l y convex o r n o t ) i s a d v a n t a g e o u s , and i s a l s o presumed i n s e v e r a l p l a c e s . Howe v e r , f u l l r e f e r e n c e s are g i v e n f o r t h e r e a d e r ’ s convenience. S t a t e m e n t s ( w i t h o u t p r o o f ) a r e a l s o i n c l u d e d where more i n v o l v e d p a r t s of t h e l a t t e r t h e o r y a r e concerned. The m a t e r i a l i n hand h a s b e e n t e s t e d , i n i t s g r e a t e s t p a r t , i n a Seminar on T o p o l o g i c a l A l g e b r a s which t h e a u t h o r h e l d i n t h e Mathema-

t i c s Department of t h e U n i v e r s i t y of A t h e n s d u r i n g t h e academic y e a r s 1972-1975.

S o , a s i s most o f t e n t h e case ( ! ) ,

t h e whole p r o j e c t h a s

PREFACE

X

b e e n g r e a t l y i n f l u e n c e d by t h a t e x p e r i e n c e . The c h o i c e of t o p i c s l a r g e l y c o r r e s p o n d s t o t h e most b a s i c and now, s t a n d a r d a s p e c t s of t h e t h e o r y of t o p o l o g i c a l a l g e b r a s . More p a r t i c u l a r l y , localZy multiplicative-

ly-convex

( t o p o l o g i c a l ) aZgebras

ar'e s y s t e m a t i c a l l y employed.

(abbreviated t o

l o c a l l y m-convex algebras )

I n f a c t , t h i s l a t t e r c l a s s of t o p o l o g i -

c a l a l g e b r a s were among t h e f i r s t t o draw t h e a t t e n t i o n of mathemat i c i a n s and t h e y c o n t i n u e t o do so. Moreover, t h i s same c l a s s of a l g e b r a s h a v e r e c e n t l y a t t r a c t e d t h e a t t e n t i o n of a number of

mathema-

t i c a l p h y s i c i s t s . The i n i t i a t i o n of t h e s t u d y of t h e above c l a s s of a l g e b r a s i s d u e t o t h e American m a t h e m a t i c i a n s R . F . A r e n s a n d E . A. M i c h a e l , who i n t r o d u c e d t h e n o t i o n i n t h e e a r l y f i f t i e s , i n d e p e n d e n t l y of o n e a n o t h e r . With r e g a r d t o t h e p r e s e n t a t i o n , a n a t t e m p t h a s b e e n made t o k e e p b o t h i t , and t h e " s t a t e m e n t of theorems'' i n t h e i r " m o s t u s e f u l g e n e r a l i t y " . Thus, t o p o l o g i c a l a l g e b r a s which a r e n o t n e c e s s a r i l y l o c a l l y convex are a l s o s y s t e m a t i c a l l y a p p l i e d and t h i s i n a way t h a t , h o p e f u l l y , does n o t v i o l a t e t h e p r e v i o u s motto. I have a l s o t r i e d t o make t h e e x p o s i t i o n e n t i r e l y s e l f - c o n t a i n e d g i v i n g f u l l p r o o f s a n d , when n e c e s s a r y , p r e c i s e r e f e r e n c e s . An e x p l a n a t o r y and d e t a i l e d a p p r o a c h h a s b e e n p r e f e r r e d t o a s h o r t o n e , s o k e e p i n g t h e p a c e on a s elementary a l e v e l as p o s s i b l e . N a t u r a l l y , t h e r e f e r e n c e s c o n t a i n t h e work o f o t h e r mathema

-

t i c i a n s which h a s i n f l u e n c e d a n d / o r m o t i v a t e d t h e p r e s e n t w r i t i n g ; b o t h w i t h r e s p e c t t o t h e Banach a l g e b r a s t h e o r y and t h a t of non-normed t o p o l o g i c a l a l g e b r a s . I n p a r t i c u l a r , a n e n d e a v o u r was made f o r t h e b i b l i o g r a p h y t o c o m p r i s e t h e e x i s t i n g work r e l e v a n t t o t h e s u b j e c t

matter of t h i s book. ( B u t t h i s , t o o , w i t h i n a c e r t a i n c h o i c e ( ! ) 1 .

In

a d d i t i o n , it i n c l u d e s some c u r r e n t a p p l i c a t i o n s of t o p o l o g i c a l a l g e b r a s t o o t h e r a s p e c t s of M a t h e m a t i c a l A n a l y s i s and e v e n v i c e - v e r s a a p p l i c a t i o n s of t h e l a t t e r t o new d e v e l o p m e n t s i n t o p o l o g i c a l a l g e -

b r a s t h e o r y . B u t , a s it s t a n d s , it d o e s n o t i n c l u d e s u f f i c i e n t r e f e r e n c e s c o n c e r n i n g t h e e x i s t i n g c o n n e c t i o n s of t h e t h e o r y ( t h i s e s p e c i a l l y r e f e r s t o l o c a l l y convex o r y e t l o c a l l y m-convex a l g e b r a s ) w i t h r e c e n t a p p l i c a t i o n s t o quantum m e c h a n i c s ( r e l a t i v i s t i c o r n o t ) . T h i s

.

p a r t i c u l a r l y c o n c e r n s i n v o l u t i v e a l g e b r a s ( v i z topoZogica2 *-algebras)

.

The l a t t e r n o t i o n o n l y a p p e a r s i n t h e l a t e r s e c t i o n s of t h e book a n d it c e r t a i n l y d e s e r v e s s e p a r a t e t r e a t m e n t . A s was m e n t i o n e d a t t h e o u t s e t t h e book o r i g i n a t e d i n a Seminar

on t h e s u b j e c t , and i t i s a p l e a s u r e t o r e c o r d my i n d e b t e d n e s s b o t h

t o t h e a u d i e n c e and t h e a c t i v e p a r t i c i p a n t s i n t h a t S e m i n a r . I n d e e d ,

xi

PREFACE

i t was t h e i r e x t r e m e l y f a v o r a b l e r e a c t i o n and e n t h o u s i a s m t h a t i n i t i a l -

l y i n s p i r e d m e t o w r i t e t h e book w h i l s t t h e y c o n t i n u e d t o o f f e r e n couragement t o i t s c o m p l e t i o n . A l e a v e o f a b s e n c e from t h e U n i v e r s i t y of Athens a t t h a t t i m e a l s o e f f e c t i v e l y c o n t r i b u t e d t o t h e f u l f i l l m e n t of t h i s p r o j e c t . I c o n s i d e r it a v e r y p l e a s a n t d u t y t o e x p r e s s h e r e my i n d e b t e d -

n e s s t o P r o f e s s o r Leopoldo N a c h b i n , f i r s t l y , f o r a c c e p t i n g t h i s book

t o a p p e a r i n h i s p r e s t i g i o u s series "Notas d e Matemdtica" of N o r t h H o l l a n d , s e c o n d l y , a n d more s o , b e c a u s e t h r o u g h h i s v a l u a b l e r e m a r k s and s u g g e s t i o n s on a f o r m e r d r a f t of t h e m a n u s c r i p t , h e h a s g r e a t l y c o n t r i b u t e d t o b r i n g t h e book t o i t s p r e s e n t form. I a l s o r e c a l l my d e b t t o my v a r i o u s f r i e n d s and c o l l e a g u e s i n t h e

m a t h e m a t i c a l community w i t h whom I h a d , a t o n e t i m e or a n o t h e r , t h e o p p o r t u n i t y t o d i s c u s s c e r t a i n p a r t i c u l a r p o i n t s of t h e p r e s e n t ma-

t e r i a l a t t h e t i m e o f i t s i n c e p t i o n , a s w e l l a s f o r t h e i r s u p p o r t when t h e book t o o k i t s f i n a l form. F i n a l l y , I w i s h t o t h a n k M r s E l e n a Agathou-Kontzedaki,

who v e r y s k i l f u l l y t y p e d a f i r s t d r a f t of an e x t r e m e -

l y d i f f i c u l t m a n u s c r i p t , so making t h e s u b s e q u e n t t y p i n g of t h e f i n a l d r a f t ( c a m e r a - r e a d y c o p y ) much e a s i e r . November 1985 Athens

ANASTASIOS MALLIOS

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xiii

Contents

................................................

Preface PART I

.

GENERAL

CHAPTER I

1 2

. .

.

2.(2).The 2.(3).The

. 4.

2 . ( 4 ) . The Topologies

Continuity algebras

5

.

General Concepts

Preliminaries

. .

3

THEORY

. D e f i n i t i o n s ...............................

......................... a l g e b r a f ( E l ................................... a l g e b r a m(lR) ................................ a l g e b r a C G [ t ].................................. w Arens a l g e b r a L (LO. 2 1 ) ........................ d e f i n e d by s u b m u l t i p l i c a t i v e semi-norms . . . . . . . of t h e m u l t i p l i c a t i o n . Complete t o p o l o g i c a l ...............................................

Examples of t o p o l o g i c a l a l g e b r a s

2 ( 1 ) The

................................................... 5 . ( I ) . M i c h a e l ' s Theorem .................................

.

.................................

5.(2).A-convex a l g e b r a s C e r t a i n p a r t i c u l a r c l a s s e s of t o p o l o g i c a l a l g e b r a s

....... 6 .( 1 ) . L o c a l l y bounded a l g e b r a s .......................... 6 . ( 2 ) . & - a l g e b r a s ........................................ 6 . (3) . A d v e r t i b l y c o m p l e t e a l g e b r a s ...................... CHAPTER I 1

.

. Spectrum

10 10 11 12

13 21

31 31

37 39 39 43 44

(Local Theory)

.

.................. 2 . The r e s o l v e n t s e t ........................................ 3 . Topological algebras with continuous inversion . . . . . . . . . . . 4 . Waelbroeck a l g e b r a s ...................................... 5 . 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 . Gel'fand-Mazur Theorem .... 6 . Maximal i d e a l s ........................................... 7 . C h a r a c t e r s . Closed maximal i d e a l s ........................ 8 . Appendix: S c h u r ' s Lemma .................................. 1

1 9

T o p o l o g i c a l a l g e b r a s a d m i t t i n g l o c a l l y m-convex t o p o l o gies

6

ix

Spectrum of a n e l e m e n t

Spectral radius

47

50 51 54 61

63

67

77

xiv

CONTENTS

CHAPTER 111 . Projective Limit Algebras

.

1 . Initial topologies. Topological subalgebras Cartesian products ............................................... 2 . Projective systems of topological algebras . . . . . . . . . . . . . . . 3 . Representations of locally m-convex algebras as projective limits. Arens-Michael decomposition . . . . . . . . . . . . . . . 4 . Applications of the Arens-Michael decomposition . . . . . . . . . .

.

5 Advertibly complete locally m-convex algebras ............ 6 . Spectral properties of advertibly complete locally rn-convex algebras ......................................

79 82 85 91

94 99

CHAPTER IV . Inductive Limit Algebras

. Inductive systems of algebras. Algebraic preliminaries . . . . Final topologies. Inductive systems of topological algebras ................................................. 3 . Inductive limits of locally rn-convex algebras . . . . . . . . . . . . 4 . Examples of topological inductive limit algebras . . . . . . . . . 4.(1). The algebra K I X I ................................... 4 . (2). The algebra d)(X) as a topological subalgebra of C m i X l ................................................ 4. (3). The algebra O ( K l ...................................

1 2

109 113 120 127 127 129 134

CHAPTER V . Spectrum (Global Theory)

. Spectrum of a topological algebra ........................ . Spectrum of the completion of a topological algebra . . . . . . 3 . Spectrum of an inductive limit topological algebra . . . . . . . 4 . Envelopes of holomorphy .................................. 5 . The dual of the Arens-Michael decomposition . . . . . . . . . . . . . . 5.(I).Compactly generated topological spaces . . . . . . . . . . . . 6 . The dual of the Arens-Michael decomposition (contn'd.) . . . 7 . Spectrum of a projective limit topological algebra . Dense projective limit algebras ........................ 8 . Appendix: Generalized spectrum ........................... 8.(1). General theory ....................................

1 2

8.(2).Generalized spectrum of a topological projective limit algebra ........................................ 8.(3).Generalized spectrum of a topological inductive limit algebra ........................................

139 144 152 160 164

165 167 173 176 176 179 180

CONTENTS

CHAPTER

V I . The G e l ' f a n d Map

. C o n t i n u i t y of t h e G e l ' f a n d map ........................... 2 . B o u n d a r i e s ............................................... 3 . F u n c t i o n a l c a l c u l u s . Holomorphic f u n c t i o n s o f a s i n g l e e l e m e n t i n a t o p o l o g i c a l a l g e b r a ....................... 1

4

. Functional

calculus (contn'd.).

.

Appendix: G e n e r a l i z e d G e l ' f a n d map CHAPTER

1 2

.

....... .......................

S p e c t r u m of t h e a l g e b r a

. S p e c t r u m of 4 . S p e c t r u m of 5 . The l o c a l l y 3

the algebra the algebra

CfX) ............................

e ( X i ............................ O(X) . Stein algebras .............. m

t h e a l g e b r a L 1 (G)

.............................

m-convex a l g e b r a

c ( X ) (contn'd.)

. The

7.(2).Complexification

..................................

. Spectrally barrelled algebras (contn'd.) ................. . N a c h b i n - S h i r o t a a l g e b r a s ................................. 3 . F u n c t i o n a l r e p r e s e n t a t i o n s ............................... 4 . Topological algebras with a given dual . . . . . . . . . . . . . . . . . . . 5 . Uniform t o p o l o g i c a l a l g e b r a s ............................. 6 . A-convex a l g e b r a s ( c o n t n ' d . ) ............................. 7 . F i n i t e l y generated topological algebras . . . . . . . . . . . . . . . . . . 8 . F u n c t i o n a l c a l c u l u s ( c o n t n ' d . ) . The S i l o v - A r e n s - C a l d e r 6 n Waelbroeck t h e o r y ...................................... 1

. Miscellanea

.

215

224 228 231 240 246

251 251 251

V I I I . Some S p e c i a l Classes o f T o p o l o g i c a l Algebras

2

I0

212

Nach-

................................ 6 . The Nachbin Theorem ( s u f f i c i e n c y ) ........................ 7 . Appendix: V a r i a n t s of N a c h b i n ' s Theorem .................. 7 . ( 1 ) . D i f f e r e n t i a b i l i t y of c l a s s c".................... b i n Theorem ( n e c e s s i t y )

9

207

V I I . S p e c t r a o f C e r t a i n P a r t i c u l a r T o p o l o g i c a l Algebras

. S p e c t r u m of

CHAPTER

1aa

Holomorphic f u n c t i o n s

of f i n i t e many e l e m e n t s i n a t o p o l o g i c a l a l g e b r a 5

181

.............................................. 9.(1). fFQ-algebras ...................................... 9 . ( 2 ) . N u c l e a r a l g e b r a s .................................. 9 .(3) . ( 2 ) - a l g e b r a s ...................................... 9 . ( 4 ) . r n - i n f r a b a r r e l l e d a l g e b r a s ......................... 9 . (5) . Gel'fand-Mazur a l g e b r a s ........................... 9 . ( 6 ) . I n f r a - P t d k a l g e b r a s ...............................

I n f i n i t e d i m e n s i o n a l holomorphy topological algebras

.

253 262

265

270 274 280 283 294

301 301 302 304 306 308 308

S p e c t r a of p a r t i c u l a r

...................................

311

CONTENTS

mi ) . The . .

10 ..(I

a l g e b r a of c o n t i n u o u s p o l y n o m i a l s

P(E) . . . . . . . .

311

10 (2) T o p o l o g i c a l a l g e b r a s of holomorphic f u n c t i o n s on i n f i n i t e d i m e n s i o n a l s p a c e s

. .

..................

315

10 (3) Holomorphic f u n c t i o n s on i n f i n i t e dimen11

. .

s i o n a l spaces (contn'd ) The a l g e b r a H(Ui [ -to] . . . . . . . . . a l g e b r a s of e m - f u n c t i o n s ....................

. Convolution CHAPTER I X

. Maximal i d e a l s p a c e ( s t r u c t u r e s p a c e ) . hk-topology . . . . . . . . Regular. normal and s i l o v a l - g e b r a s ....................... 3 . H u l l s of i d e a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . H u l l s of i d e a l s : R e g u l a r , normal and s i l o v a l g e b r a s 1

(contn'd.)

. 6. 7.

S e t s of s p e c t r a l s y n t h e s i s

.

.

....... ............

Wiener-Tauber a l g e b r a s

Uniform a l g e b r a s ( c o n t n ' d . ) . Riemann a l g e b r a s

CHAPTER

.

............................................. . The L o c a l Theorem .......................

Further r e s u l t s

PART I I

3

.

329 332 335 338 347 348 352

TOPOLOGICAL TENSOR PRODUCTS

X . T o p o l o g i c a l Tensor P r o d u c t s o f T o p o l o g i c a l Algebras

.................................. 2 . T o p o l o g i c a l t e n s o r p r o d u c t s of l o c a l l y convex s p a c e s . . . . . 2.(1). The p r o j e c t i v e t e n s o r i a l t o p o l o g y IT . . . . . . . . . . . . . . . 2.(2).The i n d u c t i v e t e n s o r i a l t o p o l o g y i . . . . . . . . . . . . . . . . 2.(3). The b i p r o j e c t i v e t e n s o r i a l t o p o l o g y E . . . . . . . . . . . . . 2.(4). The E-product of L . Schwartz ....................... 1

325

. S t r u c t u r e Theory

2

5

321

Algebraic p r e l i m i n a r i e s

T o p o l o g i c a l t e n s o r p r o d u c t s of t o p o l o g i c a l a l g e b r a s Compatible t o p o l o g i e s

359 364 364

369 370 373

.

..................................

4 . Tensor p r o d u c t s of l o c a l l y bounded a l g e b r a s . . . . . . . . . . . . . . 5 . I n f i n i t e t o p o l o g i c a l t e n s o r product a l g e b r a s . . . . . . . . . . . . .

. T o p o l o g i c a l Tensor P r o d u c t Algebras . Examples a l g e b r a q X . Ej ...................................... a l g e b r a C"t.7, E ) .................................... a l g e b r a c m ( X . E ) ( c o n t n ' d . ) . N a c h b i n ' s Theorem

375 379 383

CHAPTER X I 1 2 3

. The

. The . The

........................................ c)(X. E ) ....................................... 1 L (G. El ( g e n e r a l i z e d g r o u p algebra) . . . . . . . . . .

(vectorization)

4 5

. The . The

algebra algebra

CHAPTER XI1

1

.

. Spectra

o f T o p o l o g i c a l Tensor P r o d u c t Algebras

Spectrum of a t e n s o r p r o d u c t of t o p o l o g i c a l a l g e b r a s

387 392 395 400 402

xvii

CCNTENTS

2

.

(numerical case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum of an infinite topological tensor product algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Generalized spectrum of a tensor product of topological algebras . Canonical decomposition ...................... 4 . Inductive limits and generalized spectra . . . . . . . . . . . . . . . . . 5 . Generalized spectra and "point evaluations" . . . . . . . . . . . . . .

407 414

3

419 424 429

CHAPTER XI11 . Properties of Permanence o f Topological Tensor Product A1 gebras

. Boundaries of topological tensor product algebras . . . . . . . . 2 . Continuity of the Gel'fand map . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Spectrally barrelled algebras ............................ 4 . Semi-simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

5 . Identity elements

........................................

. Regularity . silov algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Wiener-Tauber condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . Appendix: Generalized spectra (contn'd.). Canonical

6

decomposition

..........................................

433 438 439 441 441 450 453 456

CHAPTER XIV . Generalized Spectra in the Presence o f Approximate Identities . Representation Theory 1

.

2.

.

Topological algebras with approximate identities Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary measures of representations . . . . . . . . . . . . . . . . .

465 474

CHAPTER XV . Topological Algebras with Involution . Representation Theory (contn'd . ) 1 2

. .

. 4. 3

5.

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Certain particular (commutative) topological *-algebras and their representations .............................. SNAG Theorem (the classical case) ........................ Representations of generalized group algebras SNAG Theorem (extended form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract forms of "SNAG Theorem" type ....................

483 488

.

490 493

........

497

.............................................. List o f Symbols ........................................... INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

503

6 . Appendix: Enveloping locally m-convex C*-algebras

BIBLIOGRAPHY

527

531

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PART I GENERAL THEORY

This Page Intentionally Left Blank

1

General Concepts

CHAPTER I

1. Preliminaries. Definitions By an algebra

w e a l w a y s mean t h r o u g h o u t t h e s e q u e l a @-algebra,

t h a t i s a l i n e a r a s s o c i a t i v e algebra o v e r t h e f i e l d C o f complex numbers. So g i v e n an a l g e b r a E , w e r e c a l l t h a t a semi-norm on t h e u n d e r -

l y i n g (complex) v e c t o r s p a c e F i s a r e a l - v a l u e d f u n c t i o n p : E-IR

(the

f i e l d o f r e a l numbers) s u c h t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f ied: i) p

i s non-negative; i . e . , plxl 2 0 , f o r e v e r y Z E E .

ii) p i s subadditiue;i.e.,

p ( x + y ) S p ( x l + p l y l , f o r any x , y i n E .

i i i ) p i s positively-homogeneous ; i . e .

, plXxl

= IXlp(xl , f o r any

XE C

and x E E . Moreover, i f E i s an a l g e b r a , o n e may a l s o assume c o n c e r n i n g t h e above f u n c t i o n t h a t i v ) p i s submultiplicative ; i . e .

, plxyl

< p ( x i p l y l , f o r any e l e m e n t s

x,y in E. W e a b b r e v i a t e t h e above by s a y i n g t h a t p i s a

semi-norm

on E ; t h u s , a p a i r l E , p ) , a s b e f o r e ,

submultiplicative

i s s a i d t o be a semi-

normed algebra. NOW, s i n c e t h e f u n c t i o n p i s a semi-norm on t h e v e c t o r s p a c e 3 , t h e set u = { x E E : ~ ( z~) 1 1

(1.1)

is a b s o l k t e l y convex ( i . e . ,

b a l a n c e d and c o n v e x ) , a s w e l l a s an absorbing

s u b s e t of E . F u r t h e r m o r e , by t h e c o n d i t i o n i v ) , U i s a

multiplicative

s u b s e t o f t h e a l g e b r a E , i n t h e s e n s e t h a t U. U G U . ( W e a l s o s a y , b r i e f l y , t h a t U i s an m - s e t

i n E , o r even a n idempotent

s u b s e t of t h e a l -

.

gebra E ) On t h e o t h e r h a n d , f o r e v e r y a b s o r b i n g s u b s e t U o f t h e v e c t o r s p a c e E , one d e f i n e s t h e r e s p e c t i v e

Minkowski functional

gauge f u n c t i o n pu on E ( o r e l s e

o f U ) by t h e r e l a t i o n

I GENERAL CONCEPTS

2

p (x) =

, XEU

infX

X>O,

2 8

.

XU

Moreover, one wants to know the relation of a given submultiplicative semi-norm p on the algebra E with the respective gauge function of the "closed u n i t semi-ball" of E defined by p ; i.e. , of that subset U of E given by (1.1). NOW, the desired relation is, of course, the same asthat one between the respective functions on the underlying vector space 4 ; namely, the above two functions p and p U coincide on E. Thus, we have.

Lemma 1.1.

Let E be a vector space and U a balanced, absorbing subset o f E

whose gauge function i s p U . Then, one has (1.3)

{ x e E : p ( x ) < 1 } G U E { r € E : p U ( x )> I } . U

Proof. If x e E l with p U ( x )< I , there exist X > 0 and E 0, with X < p U ( x )+ E < 1 , and x e A U G U . Besides, for x e U , one has p U (XI 5 1 . I

Furthermore, one gets.

Lemma 1.2. Let E be a vector space. Besides, a s s m e t h a t p , q

: E

-

W

are two positively-homogeneous functions on E i n such a way t h a t the following condition i s s a t i s f i e d : pix) < I, x € E , implies q ( x )

(1.4)

1

.

Then, q ( z ) <= p ( x ) , f o r every x e F (we w r i t e the l a s t r e l a t i o n by q 2 p

).

Proof. If 0 I p(x) < q ( x ) , for some x e E l then for suitable a > 0, one would have 0 $ p ( x ) < a q ( x ) ; that is, p ( -1x ) < l and besides q ( 1c L x )> 1 1

which is a contradicti0n.R

As an application of the preceding lemma we first have the following useful result.

Proposition 1.1.

Let E be a vector space and U a balanced, absorbing subset of E whose gauge f u n c t i o n i s p U . Moreover, suppose t h a t p i s a positiuely-homogeneous f u n c t i o n on E and U (1) t h e respective "closed u n i t semi-ball" P (1.1)). Then, f o r every a>U, one has

(i)

a.lirUP ( 1 ) i f , and only i f , a . p ( x ) 4 p U (x), f o r every x e E

of p

(cf.

.

I n p a r t i c u l a r , i f p and q are (subirtultiplicative) semi-norms on a given algebra E , w i t h U ( 1 ) and U ( 1 ) as above, then, f o r every a>O, one has P 9 (ii) u.U ( 1 ) r U ( 1 ) i f , and only i f , a.p(xJ Iq ( x J , f o r every x e E 9 P Proof. If x e E l with p U ( x I
.

3

PRELIMINARIES. DEFINITIONS

1.

1 . 2 for the positively-homogeneous functions p U and a.p, with a > 0, which proves necessity of (i). Conversely, if xea.U, then p (xl 5 a , U

so that by hypothesis p ( x l 5 1, that is x e U ( l l , as well, which proves P (i); now (ii) is certainly straightforward. S

On the other hand, we also have the following.

Proposition 1.2. Let E be a vector space and p a positively-homogeneous f u n c t i o n on E. Moreover, l e t U be a balanced, absorbing subset of E whose gauge function i s p u ,

i n such a way t h a t the following r e l a t i o n holds trile :

{x& E

(1.5)

:

p(sl < I

)C_U

G { x e E : p(xl 5 1 1 .

Then, p u = p.

pu(xl 2 1 I so that (Lemma 1 . 2 ) one concludes pu(xl 6 p(xl, f o r every X E E . Similarly, if pu(xl
p (xl, for every x e E , that is p u = p . I

U

Now, an absolutely convex absorbing and multiplicative subset of an algebra E is called an algebraic barrel

(abbreviated to a - b a r r e l ) .

The respective notion for the underlying vector space is linear barrel (abbreviated to 2 - b a r r e l ) . We reserve the standard terminology concerning "barrels" and "m-barrels" for the topological counterparts of the previous notions (see Definition 1 . 2 ) . The following is an immediate consequence of ( 1 . 3 ) and ( 1 . 5 ) . That is, one gets.

Proposition 1.3. Let U be an a-barrel of an algebra E . Then, a p o s i t i v e l y homogeneous f u n c t i o n p : E + R is the gauge f u n c t i o n of U i f , and only i f , t h e following r e l a t i o n holds t r u e :

{ x € E : pix) < I I S U G { x € E : pix) 5 1 1 .

(1.6)

On the other hand, one obtains.

Proposition

1.4.

Let E be a given algebra. Then, a real-valued f u n c t i o n

p on E i s a submultiplicative semi-norm on E i f , and only i f , it i s the gauge func-

t i o n of an a-barre2 of E. Proof.

If p is the gauge function of an a-barrel of E l one proves

that p is a submultiplicative semi-norm on E l by a similar argument as in the "vector space case" (cf. , for instance, J . HORVkTh' [I : p.941) . Onthe other hand, if p is asubmultiplicative semi-norm on E , then it isthe qauge function of its "closed unit semi-ball" (i.e., of the set

4

I GENERAL CONCEPTS

( l . l ) ) , a s f o l l o w s by P r o p o s i t i o n 1 . 3 .

We f i n a l l y n o t e t h a t d i f f e r e n t a-barrels ( o r y e t I-barrels) may give the same gauge f u n c t i o n , a s t h i s becomes c l e a r by P r o p o s i t i o n 7 . 2 . Now, by r e f e r r i n g t o a t o p o l o g y on a given s e t , we s h a l l always asswne in the sequel t h a t t h i s i s H a u s d o r f f , unl e s s i t is indicated otherwise. Thus,we f i r s t have the following.

Definition 1.1. By a topoZogicaZ algebra w e s h a l l mean a n a l g e b r a E which 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 i n s u c h a way t h a t t h e r i n g m u l t i p l i c a t i o n i n E i s s e p a r a t e l y continuous ( i . e . , continuous i n each one

of t h e t w o v a r i a b l e s , t h e l a t t e r o p e r a t i o n b e i n g a map of F x E i n t c E ) , I f 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 E i s , i n p a r t i c u l a r , l o c a l l y convex, w e s h a l l s p e a k of a local%y convex ( t o p o l o g i c a l ) algebra.On t h e o t h e r h a n d , i f t h e m u l t i p l i c a t i o n of a g i v e n t o p o l o g i c a l a l g e b r a i s i n b o t h v a r i a b l e s ( i . e . , j o i n t l y ) c o n t i n u o u s , w e s p e a k of a t o p c t o g i c a l [ r e s p . , l o c a l l y convex) algebra w i t h a continuous m u l t i p l i c a t i o n .

The following i s a s t a n d a r d f a c t i n t h e t h e o r y of t o p o l o g i c a l v e c t o r s p a c e s ; t h e a n a l o g o u s r e s u l t i n o u r case i s a l s o u s e f u l .

Proposition 1.5. L e t E be a topoZogica1 algebra and p a isubmuZtiplicative1 semi-norm on E . Then, t h e following a s s e r t i o n s are equivalent: 11 p is continuous a t 0 ec”. 2) p i s uniformly continuous. 31 p is continuous. 4)

The s e t

(1.71

A

tzez

: pis)


i s an open subset of E ; hence, an open neighborhood o f zero. ( W e c a l l i t open u n i t semi-ball o f p j

.

Moreover, t h e c l o s w e of (2.7) is the set A = { z e B : p(x) 2 1 } (1.8) ( w e c a l l it closed u n i t semi-ball of p). FLna2,ly the i n t e r i o r o f 11.8) i s ( 1 . 7 ) . Proof. S i n c e p i s s u b a d d i t i v e , o n e h a s t h e r e l a t i o n lpiz) - p ( y ) ~ ( x - 3 )f o~r a n y

2,

* 4 ) : i n d e e d , by h y p o t h e s i s t h e s e t A

= p-’([-l,l))

of E

c o n t a i n i n g t h e z e r o e l e m e n t of E . Now,

f > O ,

one has

5

i s a n open s u b s e t

4)=>1),

s i n c e for every,

€ . A = { X € E : ~ ( ~ / < E } = ~ - ~ I I - E , where E / / , t h e set

open n e i g h b o r h o o d

I

y i n E , so t h a t 1 ) + 2 ) 5\31. On t h e o t h e r hand, 3)

&.A

i s an

of O e E , by h y p o t h e s i s . T h u s , t h e p r e c e d i n g f o u r

1.

PRELIMINARIES.

5

DEFINITIONS

a s s e r t i o n s a r e e q u i v a l e n t . F u r t h e r m o r e , one c e r t a i n l y h a s A C K , w h e r e A i s a c l o s e d s e t by 3); t h u s , t o v e r i f y t h e l a s t a s s e r t i o n i t s u f f i -

ces t o p r o v e t h a t , i f X E E , w i t h p(xl = 1 , t h e n x b e l o n g s t o t h e c l o s u re of A : I n d e e d , by t a k i n g a s e q u e n c e ( A n ) , w i t h 0
E

f o r every

E

0, there exists

A > 0,

< 1 , a n d x E A d , and hence p i x ) <= h < 1 , which f i n i s h e s

t h e p r o o f , s i n c e by 4 ) t h e s e t A i s a n open s u b s e t of

z.I

S e t s of t h e form ( 1 . 8 ) p l a y a f u n d a m e n t a l r o l e i n t h e s e q u e l , t h e r e f o r e w e s i n g l e them o u t t h r o u g h t h e f o l l o w i n g

Definition 1.2. L e t E be a t o p o l o g i c a l a l g e b r a . A s u b s e t U of E i s s a i d t o be a n m-barreZ, i f i t i s c l o s e d a b s o l u t e l y convex a b s o r b i n g and m u l t i p l i c a t i v e .

I n p a r t i c u l a r , i n a l l t h a t follows w e a r e concerned w i t h a c l a s s o f t o p o l o g i c a l a l g e b r a s h a v i n g t h e i m p o r t a n t p r o p e r t y t o p o s s e s s a lo-

c a l b a s i s ( f u n d a m e n t a l s y s t e m of n e i g h b o r h o o d s o f z e r o ) c o n s i s t i n g o f m-barrels.

To b e g i n w i t h , w e g i v e a s e e m i n g l y weaker d e f i n i t i o n ( c f .

Theorem 1.1)

.

Definition 1.3. By a locaZZy muZtipZicativeZy-convex algebra ( a b b r e v i a t e d t o l o c a l l y m-convex aZgebra), w e mean an a l g e b r a which i s , i n p a r t i c u -

l a r , a t o p o l o g i c a l v e c t o r s p a c e w i t h a l o c a l b a s i s c o n s i s t i n g of m -

convex ( i . e . , m u l t i p l i c a t i v e and convex) s e t s . D e s p i t e of t h e f a c t t h a t t h e p r e c e d i n g d e f i n i t i o n i s q u i t e g e n e r a l , m o t i v a t e d by i t s u s e f u l n e s s i n c o n c r e t e e x a m p l e s , it i s a c t u a l l y much s t r o n g e r t h a n it seems, a s it w i l l become c l e a r i n t h e s e q u e l . Thus, b e f o r e w e have a t o u r d i s p o s a l t h e n e c e s s a r y t e r m i n o l o g y t o see

i t s f u l l s t r e n g t h ( c f . Theorem 1 . 1 )

,

w e do h a v e t h e n e x t .

Proposition 1.6. L e t c" be a ZocalZy m-convex algebra. Then,

E is

a locally

convex topological algebra with a continuous m u l t i p l i c a t i o n . Proof. I f v e c t o r space E

a={ U } ,

i s a l o c a l b a s i s of t h e r e s p e c t i v e t o p o l o g i c a l

c o n s i s t i n g of m-convex s e t s , t h e n E i s , i n p a r t i c u -

l a r , a l o c a l l y convex s p a c e . On t h e o t h e r h a n d , t h e r e l a t i o n U . U r l l , f o r e v e r y UEUZ, i m p l i e s t h a t t h e m u l t i p l i c a t i o n i n E , when c o n s i d e r e d a s a n E-valued b i l i n e a r map on E x E ,

i s continuous a t t h e o r i g i n

of E x E , and h e n c e c o n t i n u o u s e v e r y w h e r e . I

I GENERAL CONCEPTS

6

In order to prove Theorem 1 . 1 below, the following preliminary material will be needed. Thus, we first have. Lemma 1.3.

Let U be a m u l t i p l i c a t i v e subset of an algebra E . Then, the ba-

lanced h u l l of U, i . e . , t h e s e t (1.9)

is also a m u l t i p l i c a t i v e subset of E , hence an m-balanced ( i . e . , m u l t i p l i c a t i v e and balanced) s e t E. i t

Proof.

It is obvious by the definition of an m-set in E and (1.9).1

Besides, the following holds a l s o true. Lemma 1.4. Let E be an algebra, and il a m u l t i p l i c a t i v e subset o f E . Then,

the convex h u l l of U, i . e . , the s e t

(1.10)

i s a l s o a m u l t i p l i c a t i v e subset of E , hence an m-convex s e t i n E . Proof. n

Indeed, if x , y are elements of < U > , then by ( 1 . 1 0 ) m

n

m

n

rn

(1.11)

with

i p j 2 0,

n

m

C

C Xipj = 1

,

i=I j = ]

and x i y j e U, for all indices i, j

in ( 1 1 1 ) ; hence, x y e < U > , which proves the assertion, the set ( 1 . 1 0 ) being certainly convex.^ Furthermore, one obtains. Lemma 1.5.

Let E be a topological algebra. Then, f o r any subsets A , B of E

one has the r e l a t i o n (1.12)

_ _

A.B E

A.B

(where "bar" means :opological cZosure). In p a r t i c u l a r , if A , B , C are subsets of E , then A.BzC,

implies A . B c C

so t h a t if U i s an m-set of E , i t s closure

i s a l s o an m-set of E ; i . e . , one has

1.

Proof. I t s u f f i c e s t o prove ( 1 . 1 2 ) ward. Thus, i f

7

PRELIMINARIES. D E F I N I T I O N S

X E A and y e 8, t h e r e

,

t h e rest being s t r a i g h t f o r -

lye)

e x i s t n e t s (xu) i n A and

in

w i t h x = lim xa and y = l h y 6 . Hence, by t h e s e p a r a t e c o n t i n u i t y of a 6 the multiplication i n one o b t a i n s

B,

(1.13)

where xa. y g €

6

m, which

therefore,

A.B;

lim(x,.y6

6

) E

z, so t h a t ,

by ( 1 . 1 3 ) , x.3

p r o v e s ( 1 . 1 2 ) , and t h i s f i n i s h e s t h e p r o o f . I

A u s e f u l c o n s e q u e n c e of t h e p r e v i o u s lemma i s

Corollary 1.1. L e t E be a topological algebra and F Iresp. I) a subalgebra iresp. an i d e a l ) of E. Then t h e closure F ( r e s p . 7 )i s a closed subalgebra Iresp. an i d e a l ) o f t h e algebra E . Proof. The a s s e r t i o n i s c l e a r by t h e d e f i n i t i o n s , s o w e o m i t t h e details.

N o w t h e f o l l o w i n g r e s u l t j u s t i f i e s t h e comment a f t e r D e f i n i t i o n 1 . 3 . T h a t i s , one h a s .

Theorem 1.1.

L s t E be an algebra. Then, t h e following a s s e r t i o n s are equi-

valent:

I) E i s a l o c a l l y m-convex algebra ( D e f i n i t i o n 1 . 3 ) . 2 ) E is a topological algebra u i t h a local b a s i s c o n s i s t i n g of m-barrels.

3 ) -?here e x i s t s a f i l t e r b a s i s on E c o n s i s t i n g o f a-barrels.

Proof. I f E i s a l o c a l l y m-convex a l g e b r a , t h e n by P r o p o s i t i o n 1 . 5 E is a

( l o c a l l y convex) t o p o l o g i c a l a l g e b r a , so t h a t l e t

%=

{ii}

be

a l o c a l b a s i s of t h e r e s p e c t i v e ( l o c a l l y convex) t o p o l o g i c a l v e c t o r s p a c e E ( i b i d . ) , c o n s i s t i n g of m-convex

s e t s ( D e f i n i t i o n 1 . 3 ) . Hence,

f o r e v e r y e l e m e n t U e a , t h e r e e x i s t s an a b s o l u t e l y convex and c l o s e d neighborhood of z e r o i n E as above, w i t h V C U . T h e r e f o r e , t h e r e e x i s t s W

e

a,w i t h

WGVEU, i n s u c h a way t h a t

(1.14)

EVGU

( c f . ( 1 . 9 ) snd ( 1 . 1 0 ) )

,

-<

( D e f i n i t i o n 1 . 2 ) o f E . Thus, t h e f a m i l y

i s an m - b a r r e l

i 1 , w i t h U e

e

,

where by t h e p r e c e d i n g t h r e e lemmas t h e s e t

0=

d e f i n e s a l o c a l b a s i s o f E l which hy ( 1 . 1 4 ) i s

equivalent t o t h e given b a s i s

%,

t h e members of w h i c h a r e m - b a r r e l s

of 5, ;,)'at i s 1 ) inpli2s 2 ) . N O W , i t i s c l e a r t h a t 2) im plie s 3 ) . F u r t h e r m o r e ,

a

I GENERAL CONCEPTS

if x i s a filter basis of E consisting of a-barrels, the family (1.15)

n'

= {XU:Ue?Z,

O < A < l l

constitutes a local basis of a uniquely defined locally convex vector space topology in E (see, for instance, J . HORV~TH[I; p. 87, Proposition Now, by (1.15), the elements of !%' are again a-barrels, hence 5)). in-convex sets, so that E becomes, in effect, a locally m-convex algebra (Definition 1.3) ; thus, 31 i m p l i e s 1 ) tog, and this terminates the proof. I The proof of the following useful corollary is a direct consequence of the previous argument.

Corollary 1.2. L e t E be an algebra. Then, E i s a l o c a l l y m-convex algebra i f , and only i f , t h e r e s p e c t i v e v e c t o r space E i s a l o c a l i y convex space w i t h a local

basis c o n s i s t i n g of m-sets.1

Scholium 1.1. It is in the form of the preceding Theorem 1.1 ;2) that the notion of a locally m-convex algebra is usually applied (in connection with Proposition 1.6); we constantly apply this in the sequel without any particular mention. On the other handlit also happens that the above form of Theorem 1.1 is taken for definition of a local-

ly m-convex algebra, in particular, when the topology of the algebra is defined in terms of a given family of (submultiplicative) seminorms, as we shall see in Section 3. However, Definition 1.3 or, in particular, Corollary 1.2 are, of course, more convenient tools to see whether a given algebra is locally m-convex (cf. Examples below). A s another consequence of Theorem 1.1;3), we remark that one can always consider on a given algebra E the " t r i v i a l Locally m-convex topology", which corresponds to the filter basis on E defined by the sin-

gleton { E l . On the other hand, one has the " j t r o n g e s t Locally m-convex t o pology" in E, which corresponds to the filter basis on E defined by the

family of all a-barrels of E (see Example 2.3 below). In this concern, we still note that not every m-barrel in a given t o l o g i c a l algebra, even a l o c a l l y m-convex one, is a neighborhood o f zero (cf.Example 2.2 below); hzwever, topological algebras having this property, or at least an appropriate weak form of it (cf.Definition V;1.3),play a significant role throughout the sequel. So we next consider formally this particular class of algebras.

Definition 1.4. By an m-barrelled algebra we shall mean a topological algebra, every

in-barrel of which is a neighborhood of zero.

2.

EXAMPLES OF TOPOLOGICAL ALGEBRAS

9

An i n - b a r r e l l e d t o p o l o g i c a l a l g e b r a which i s , i n p a r t i c u l a r , c a l l y convex ( r e s p . , l o c a l l y rn-convex)

,

lo-

w i l l be c a l l e d an rn - bcrrelled

l o c a l l y convex ( r e s p . l o c a l l y m-eonvex ) algebra. I n t h i s r e s p e c t , w e remark t h a t t h e l a s t t h r e e c l a s s e s of t o p o l o g i c a l a l g e b r a s are i n d e e d c o n w i t h an o b v i o u s " i n c l u s i o n r e l a t i o n " ; t h i s i s , i n e f f e c t a ge-

nected

one, a s it w i l l be shown by t h e Examples of t h e n e x t s e c t i o n .

nuine

However, b e f o r e w e t u r n i n t o c o n c r e t e examples of t h e k i n d of t o p o l o g i c a l a l g e b r a s d e f i n e d above, w e may c o n s i d e r c e r t a i n p a r t i c u l a r , neverthelessuseful,

i n s t a n c e s of D e f i n i t i o n 1 . 4 .

Thus, e v e r y t o p o l o g i c a l a l g e b r a f o r which t h e r e s p e c t i v e topol o g i c a l v e c t o r s p a c e i s b a r r e l l e d , we c a l l it a barrelled algebra, i s obviously

m-barrelled

( D e f i n i t i o n 1 . 4 ) ; namely, e v e r y m-barrel

is

a

Scrrel ( i . e . , a c l o s e d a b s o l u t e l y convex and a b s o r b i n g s e t ) . The p a r t i c u l a r c a s e of a barrelled

l o c a l l y convex ( r e s p . l o c a l l y m-convex) algebra

is

clear. I n p a r t i c u l a r , one h a s t h e above s i t u a t i o n by c o n s i d e r i n g

a

Baire topological algebrcr; t h a t i s , a t o p o l o g i c a l a l g e b r a f o r which t h e respective

t o p o l o g i c a l v e c t o r space i s a Baire space, s i n c e t h e l a t t e r

i s a barrelled space ( s e e , f o r i n s t a n c e , F.TREVE.5

[1: p . 3 4 6 1 ) .

On t h e o t h e r hand, by c o n s i d e r i n g a more p a r t i c u l a r , b u t import a n t , i n s t a n c e of t h e above w e come t o t h e n e x t

Definition 1.5. By a Fre'chet

( t o p o l o g i c a l ) algebra w e mean a t o p o l o -

g i c a l a l g e b r a f o r which t h e r e s p e c t i v e t o p o l o g i c a l v e c t o r s p a c e i s a FrGchet s p a c e ( i . e . , m e t r i z a b l e

and c o m p l e t e ) .

Every F r s c h e t s p a c e i s c e r t a i n l y a B a i r e s p a c e , hence a b a r r e l l e d o n e , so t h a t e v e r y F r E c h e t a l g e b r a i s a b a r r e l l e d a l g e b r a and, a f o r t i o r i , an m - b a r r e l l e d one. I n t h i s c o n c e r n , w e a l s o have t h e p a r t i c u l a r i n s t a n c e of a Fre'ehet l o c a l l y eonvex algebra o r , more p a r t i c u l a r l y , t h a t of a Fre'chet l o c a l l y m-convex algebra. The r e s p e c t i v e f o r m a l d e f i n i ' c i o n s a r e c l e a r , of c o u r s e , by t h e r e l e v a n t t e r m i n o l o g y a p p l i e d h i t h e r t o ( c f . D e f i n i t i o n s 1 . 3 and 1 . 5 ) . Moreover, t h e r e s u l t i n g c l a s s e s of t o p o l o g i cal

a l g e b r a s a r e n o t n e c e s s a r i l y t h e same, a s w e s h a l l see below ( c f .

Example 2 . 4 )

.

2. Examples o f topological algebras W e g i v e below some f i r s t examples of t o p o l o g i c a l a l g e b r a s which

s e r v e t o c l a r i f y t h e nocions a p p l i e d h i t h e r t o and, i n p a r t i c u l a r , t h e way one a p p l i e s a number of t h e p r e c e d i n g r e s u l t s ( f o r example, Theorem 1 . 1 and i t s C o r o l l a r y )

.

I

I0

GENERAL CONCEPTS

1 . Let E be a topological vector space and A ( E I

the vector space of

continuous l i n e a r maps of E i n t o i t s e l f (viz., continuous linear endomorphisms of E ) . This space equipped with the "topology of simple con-

vergence" in E l becomes a topological vector space, which we denote (see N . BOURBAKI [6: Chap. 1 ; p. 161). On the other hand, it is by & , ("E l easy to check that the "composition of linear maps" defines in the latter space a separately continuous ring multiplication, therefore Z S ( E ) i s a topological algebra. In particular, if E is a locally convex space, then Xs(EI i s a l o c a l l y convex algebra, since its topology can be considered as the "initial topology" on Z ( E ) defined on it by the "evaluation maps" B : d ( E ) + E l with x E E , where 4 1 u ) = u ( x I , for every u

€

&(E/.

2 . Let &(m) be the algebra of complexvalued bounded continuous funct i o n s on ( t h e r e a l l i n e ) IR (pointwise defined operations). The latter set endowed with the "topology of compact convergence" in IR becomes a topological algebra, denoted by a c ( R . J . Indeed, by considering the sets (2.1)

vK, E

=

E

x e & ~ ( l ~ )l x:( t ) l 5

E ,

for every t e K 1 ,

where K varies over the compact intervals oflR and 0 < E S I , one readily verifies that the family (2.1) constitutes a filter basis on the topologized by algebra &I (IR)consisting of a-barrels, so that &(IR) (2.1) becomes a locally m-convex algebra (Theorem 1.1). Moreover,the latter topology on &(IR) is, in effect, identical with the topology c of compact convergence in IR as follows by the same definitions and (2.2)

TIK,EI

{

x e & ( ~ ~ .: i I x ( t ) ( <

E,

for every

*EK}

c Y ~ , ~ ,

where K and E vary a s In ( 2 . 1 ) . Therefore, a c ( l R 1 i s a l o c a l l y m-convex algebra. On the other hand, by considering in ( 2 . 1 ) the K ' s varying over the intervals [ - n , n ] of IR, and E over I / n , with n EW (the natural numbers) , one actually obtains that fie(niRIi s a metrizable l o c a l l y m-convez algebra which is not m-barrelled, so that a fortiori i t i s not barrelled. The latter assertion is easily verified by remarkingthat the set (2.3)

tiO(?/ =

is an m-barrel i n

I x e 6 i 1 ~ i: Ixt.it)I

ac( RI

2 I

,

for every

, which i s not a neighborhood

ten1

of zero. Thus, the

to-

pology of uniform convergence" i n n , under which ds(IRI is a Banach algebra, is different from that of the compact convergence inIR, topologizing the latter algebra as before. Nonetheless, as we shall see c ( I R R 1 does retain a certain conlater on, the topological algebra a venient form of "barrelledness" (see Example VII1;l.l).

11

EWIPLES OF TCFOLOGICAL ALGEBRAS

2.

The l o c a l l y m-convex t o p o l o g y d e f i n e d on t h e a l g e b r a E of t h e n e x t example i s t h e s t r o n g e s t o n e , i n t h e s e n s e t h a t i t i s d e f i n e d by i n E (Scholium 1.1 ) . I n t h i s respect, wenote

t h e set of a l l a - b a r r e l s

t h a t every algebra topologized in the strongest l o c a l l y m-convex topology i s an m-convex a l g e b r a i s n o t ,

in-barrelled algebra. B u t , an m - b a r r e l l e d l o c a l l y

i n g e n e r a l , a b a r r e l l e d o n e , a s w e s h a l l p r e s e n t l y show.

3.

WARNER L2: p. 1 9 3 1 1 . L e t E be t h e aZgebra of polynomiaZs i n one in-

(S.

d e t e m i n c t e w i t h complex c o e f f i c i e n t s and without constant terms. W e assume Eurt h e r t h a t E i s e q u i p p e d w i t h t h e s t r o n g e s t l o c a l l y m-convex

topology,

s o t h a t a s we a l r e a d y n o t e d b e f o r e E is an m-barrelled l o c a l l y m-convex a l -

gebra. B u t , E i s not a barrelled algebra, t h a t i s , t h e r e s p e c t i v e l o c a l l y convexspace E i s n o t b a r r e l l e d : I n d e e d , consider t h e sets;

I antn

(2.4)

:nEJJ1,

o f r a t i o n a l numbers, w i t h a < l ; de-

where a v a r i e s o v e r t h e f i e l d Q

n o t e t h i s s u b s e t of Q by Q o / 2 1 , and l e t ;(a) b e t h e a b s o u t e l y convex ( i . e . , b a l a n c e d and c o n v e x ) h u l l of e a c h o n e o f t h e s e t s ( 2 . 4 ) . T h a t i s , w e have

~ ( a=) < [ f a n t n : n e m 1

(2.5)

>

. NGW

(see a l s o t h e r e l s . ( 1 - 9 ) and ( 1 . 1 0 ) )

it i s e a s y t o c h e c k t h a t t h e

family { V ( a ) : a e Q , ( l ! } c o n s i s t s of a - b a r r e l s ,

above

s o t h a t it d e f i -

n e s , i n f a c t , a l o c a l b a s i s o f t h e t o p o l o g i c a l a l g e b r a E. On t h e o t h e r hand, t h e s e t

1 tn

B = <({

(2.6)

2

n2

:

n e n }

) >

i s , of course, absorbing, therefore i t s closure for

every

E

> I , one h a s

-tn 6 B ; indeed, i f

0

a b a r r e l of 6'. Now, 1

a <-(E-I)"~,

2n2

then

2n

~n e a s i l y v e r i f i e s t h a t I T t + V ( a ) ) n B = P ) . Furthermore, f o r every 211 1 .x > 0 , o n e h a s l i m = O , s o t h a t t h e r e e x i s t s n eE';, s u c h that n 2

one

277 un

7

< 1

2

2n

,

that is,

an

r.

an

I

o r an =

Eiay:)

I

with E(a,ni

T h e r e f o r e , i f V t a l C B , € o r some a ~ Q ~ i lt h)e n , there wouldexist buch

that antn =

-9

tn 8 V i a ) C

1

2n

2

I

nEmJ,

t h u s a c o n t r a d i c t i o n t o khe abo-

zn

v e ; t h e r e f o r e , B c a n n o t be pological

a n e i g h b o r h o o d o f z e r o i.n E , s o t h a t t h e t o -

v e c t o r s p a c e E i s n o t b a r r e l l e d . Moreover, s i n c e t h e f a m i l y

12

I

GENERAL COi-CEPTS

{ V i a ) ) , with c x e Qo(l), is denumerable, one finally concludes that C i c an n,-baweZledm e t r i : . a b l ~ l o c a l l 3 n i - c m u e ~ a l d c b w ix;l,~ic/iI c nab b?;.,*rtj:led.

4. Most of the assertions stated below are proved in I?. A E N S [2J; they will also be explained occasionally in the following sections.So the space (2.7), considered below, can be treated, more naturally, as a “projective limit space” which is also an algebra suitably topologized (see Chapt.III;Sectionl). Its appearance at this place contributes, however, to the clarification of Definition 1 . 5 concerning the various classes of topological algebras under discussion. Thus, consider the set

~ ~ ( [ o , i =] i n ~ p ( [ u , ~ ,] i

(2.7)

p>1

of all comp2ex-valued fiinctions on t h e closed u n i t i n t e r v a l e [p,l] E IR which are p-s>-mable, f o r every p 2 I , w i t h r e s p e c t t o t h e Lebesgue measure on LO,]]. T i i i s set is certainly a vector space over the complexes, since each one of the L p ’ s

is: moreover, it is an algebra, due to the relation

1lf.s

(2.8)

with f , g e L w

, ar,d

2,

r

lip

5

II s l l q

. I/ g 1,

,

positive reals such that

3

7 = 1 +A ; here -I

one defines the respective “ L p - i ~ m . r i r fby the relation (2.9)

IIS

Ilp

= ( fqilfit,

1‘

dt )”p

for every f E L p ( [O, I] I , and p 2 1 (see also !V. the other hand, the sets (2.10)

U

P

(E)

= Ife.LW:

I/fJl<

,

BOURBAiiI [7 ; p. 121 )

.

On

E l ,

with p 2 1 and E > 0 , define a local basis of L w consisting of convex sets. NOW, since the measure space involved is bounded, we can restrict ourselves to a countable subset of p ’ s in ( 2 . 1 0 ) , getting thus a denumerable local basis of L w ; so the latter space becomesa netrizable

locally convex space and since it is also complete (cf. Chapt. 111; Section 1 ) , L’” it?a Tre‘chet locally convex algebra (with a continuous multiplication ; cf.Section 4) Furthermore, L w can not be topozogized as a ’LocalZy m-convex a l g e b m : in fact, an m-convex set U in L w is an open

.

neighbcrhood of zero if, and only if, (2.11)

U = L

w

(see also Proposition 1.5). In particular, L w can not be a normuble alg e b r a . On the other hand, L m

( [O, l ]

i (the essentially bounded measur-

able complex-valued functions on [ L 1 , 1 ] ) , being a Banach algebra inits own norm, is a dense subalgebra of L w , while L w is dense in each one uf i k

3. TOPOLOGIES BY SUBMULTIPLICATIVE SEMI-NORMS

spaces L p

,p

21

13

, a l l t h e r e s p e c t i v e canonical i n c l u s i o n maps being continuous.

3. Topologies defined by submultiplicative semi-norms Let E be an algebra and p a submultiplicative semi-norm on E . We recall that (cf.( 1 . 1 ) ) the u n i t seme-ball of E , corresponding to p, is the set U ( 1 ) = { x € E : p ( x )S E } .

(3.1)

P

On the other hand, one has (3.2)

U

P

(E)

= { x € E : p(xi S E } = E . U

P

(1)

,

for every real number E > 0 . In particular (cf. Section I ) , t h e s e t U Ill P i s an a-barrel of the given algebra E l so that the family { U ( -1 ) : n e n ) (3.3)

n'=

P n

yields a filter basis on the algebra E consisting of a-barrels. Therefore (Theorem 1 . I ) , E i s a ZocaZly m-eonvex algebra having %' as a local basis consisting of a-barrels. The latter assertion follows directly by the construction in the proof of Theorem 1.1 (see ( 1 . 1 5 ) and Proposition 1.4).

A pair ( E , p ) consisting of an algebra E and a submultiplicative semi-norm p on it is called a semi-normed algebra , while E is thus considered to be topologized as before. Thus one concludes by ( 3 . 3 ) that every semi-normed algebra ( E , p ) i s a metrizable l o c a l l y m-convex algebra. Moreover, a topological algebra E whose topology can be defined by a submultiplicative semi-norm is called a semi-normable algebra (cf. also Theorem 5.1). Now on the basis of the situation one has in the case of topological vector spaces, one concludes that a semi-normed (resp. seminormable) algebra E is Hausdorff if, and only if, the (submultiplicative) semi-norm p , which defines the topology of E , is a norm , i.e. ,

one has p ( a ) > 0, for every x € E l with x # O.The pair ( E , p ) is called then a normed (resp. normable ) algebra . (Following standard notation, we denote in that case p by 1 1 . 1 1 . In this concern, see also Section 5). A particular reason to consider semi-normed algebras is due, of course, to the fact that these algebras constitute, in effect, the "building blocks" in defining a locally m-convex algebra in terms of submultiplicative semi-norms as we shall see in the sequel (cf. Chapt. 111). But we still have to comment on some preliminary material. Thus, given the algebras E and F , a map u : E - + F is called an

I GENERAL CONCEPTS

14

aZgebra morphism, if it is a linear map between the respective vector

spaces and moreover (ring) "multiplication preserving" ,i..e., one has ulxy) = u1x)uiyl

(3.4)

,

for any elements x , y in E ; (just homomorphism

, or

yet morphism, are terms

also in use, if the category of the respective algebraic structures considered is clear).Moreover, if the algebras E and F have identity elements lE andlF , respectively, we always assume in the sequel that

.

, in the sense that u ( IE ) = lF Now the following result,often used in the seque1,is straightforward.

U?

algeb--a morphism i s i d e n t i t y preserving

Lemma 3.1.

Let u : E - F

be an algebra morphism and B an absolutely convex

absorbing and m u l t i p l i c a t i v e subset of t h e algebra F . Then, the s e t u-'(B)C E has t h e same p r 0 p e r t i e s ; i n o t h e r words, the inverse image, by an algebra rnorphkm, of an a-barrel i s again an a-barrel. I n particuZar, i f E and F are topological alge-

3ras and the given morphism u:E+F continuous, then t h e inverse image by u of an .a-barrsl of F is an rn-barrel of E too. I On the other hand, we also have the following.

Proposition 3.1.

Let E

be an algebra and (Ea, p I , a e I , a family o f se-

, aeI, be a family of algebra morphisms. on E , t h a t i s , t h e coarsest topology on it f o r

mi-normed algebras. Moreover, l e t ua:2-E Then, the

i n i t i a l topology

which the maps ua

Proof.

(3.5)

, n e I , are continuous, makes

E a l o c a l l y m-convex algebra.

Consider the sets V of the algebra E which are of the form v = n u - 1 (0 : € . ) I , a ci, z i=l

where the sets Ua 1 ~ ; )are varying over local bases of tne seai-ilormi with 1 S i 5 n , E i 2 1 ( i = l , . , n ) , and n elN (see ed algebras (Ea.,

z

psi),

..

also (3.3)).Now, it is clear (Lemma 3.1) that the sets (3.5) define a-barrels of E and, in fact, a filter basis of E. Therefore (TheoE is made into a locally m-convex algebra whose topology rem l.l), coincides with the initial topology defined on E by the maps ua, a e I , as this readily follows by ( 3 . 5 ) . I Remarks. -1) Since the sets Ua

semi- normecl a 1geb ra s l E a

( E ~ )

in (3.5) are m-barrels of the

i

, pol . I (cf. (3.3) and the relevant comments) , i z one concludes by (Lemma 3.1) that, if the algebra E is topologizedas

above, the s e t s 13.51 define a local b a s i s of E consisting o f m-barrels.

3.

TOPOLOGIES BY SUBMULTIPLICATIVE SEMI-NORMS

15

2 ) It is clear that t h e l o c a l l y m-convex algebra E of Proposition 3.1 i s h‘ausdorff i f , and o n l y i f , f o r every X E E , w i t h x # 0 , t h e r e e x i s t s a~ I such

t h a t p,iu,(xii

0.

The next corollary is a direct consequence of the above.

Corollary 3.1. L e t E be an algebra and

I a f a m i l y of s u b m l t i p l i c a -

be t h e r e s p e c t i v e f a a = f E , p c1 ), w i t h u E I , m i l y of semi-normed a l g e b r a s . Then, t h e i n i t i a l topology on E , w i t h r e s p e c t t o t h e t i v e semi-norms on E . Moreover, l e t E

i d e n t i t y maps i , :E-+(

E

,pa) , a e I, d e f i n e s

E a s a l o c a l l y m-convex algebra. I n

p a r t i c u l a r , t h e algebra E so t o p o l o g i z e d i s Hausdorff if, ond o n l y i f , f o r every x E E , w i t h x # 0 , t h e r e e x i s t s c1 E I such t h a t p a l x l # 0 (i.e., if , and only if , the respective locally convex space E is Hausdorff) . f i r t h e m o r e , one obt a i n s a l o c a l b a s i s of t h e topology of E , c o n s i s t i n g o f m-barrels, by t a k i n g s e t s

o f t h e form

w i t h neJU and 0 <

E.

5 1 (i= 1,.

.. , n ) .

I

,Thus, speaking within the previous context we say that E is a

r=

1 p,2 i a of s u b m Z t i p l i c a t i v e semi-norms. In this respect, we also say that r is a del o c a l l y m-convex algebra whose topology i s d e f i n e d by a f a m i l y

f i n i n g family

(of submultiplicative semi-norms) of the topology of E l

while the topology thus determined on E is said to be the l o c a l l y mconvex topology on E defined by the family r = ( p i a aci-’ Now suppose that E is a locally rn-convex algebra the topology of which is defined by a family r = ( p a l c l E I of submultiplicative seminorms as above. Thus, we shall say that r is a fundamental d e f i n i n g f a m i l y (or simply a fundamental f a m i l y ) of the topology of E , if the following condition is satisfied: For every f i n i t e s e t o f i n d i c e s HG I, therie e x i s t s an index aeI, and a r e a l number E > O , such t h a t (3.7)

p 1x1 2

E

. pz(xz’),

f o r e v e r y i 8 H , and f o r every element x

€

B.

The following useful result clarifies now the particular technical significance of the previous condition.

Lemma 3.2. L e t E b e a l o c a l l y rn-convex algebra whose topology i s d e f i n e d by r = ( p c iac I o f s u b m u l t i p l i c a t i v e semi-norms (Proposition 3 . 1 ) . Ti.en, ris a fundamental f a m i l y o f t h e topology of E i f , and only is, t h e s u b w q u e n t fa-

a family mily

I

16

n = c E.U

(3.8)

GENERAL CONCEPTS

(1) =

u

(E): a E I ,

0 <

2 1 },

E

pa pa determined by the respective unit semi-balls of the pa’s, a e I, defines a local basis of the (locally m-convex) tcpological algebra E. Proof.Taking

a n e i g h b o r h o o d o f z e r o i n E of t h e form ( 3 . 6 ) , one

obtains

with

E

= ’ (i=I 15isn ,...,n), so

,

Now, by ( 3 . 7 ) t h e r e e x i s t s E*> 0

infEi.

t h a t , s e t t i n g E = i n f ( ~ ~ & ~ , one l } , obtains

2~ ~ . p ,

i

n

E.U

and a E I , w i t h pa

pa

n

n u (11)

( i ) c E 1 (

i=l

G

Pai

n i=l

u

( E ~ J

pai

( c f . a l s o P r o p o s i t i o n l . l ) , t h e r e f o r e t h e n e c e s s i t y o f t h e s t a t e d cond i t i o n . C o n v e r s e l y , if t h e f a m i l y 3 . 8 ) i s a l o c a l b a s i s o f E and pa . e z n r (i=1 , n ) , then U ( 1 ) i s a n e i g h b o r h o o d of z e r o i n L7 ( C o r o l -

...,

i=l

lary 3.1),

psi

so t h a t t h e r e e x i s t s

E

n

Remarks.by assuming that

(1) E

2,

such t h a t

pa

.

( i = l , ., ,n) ,which f i n i s h e s t h e p r o o f .

h e n c e ( P r o p o s i t i o n 1 . 1 ) p 2 E. pa

a

u

i

1 ) Concerning t h e c o n d i t i o n ( 3 . 7 ) O<

E

,

it amounts to the same

5 1 ; t h i s h a s b e e n a p p l i e d a l r e a d y i n t h e p r o o f of

t h e above LemTa 3 . 2 . 2 ) Suppose t h a t E i s a l o c a l l y

a d e f i n i n g f a m i l y of i t s t o p o l o g y ;

we

m-convex

a l g e b r a and

r = {pi}iEI

t h e n we may aZways assume ( a n d t h i s

s h a l l do i n t h e s e q u e l ) that the given family

r

is a fundamentai: defining

famiZy. O t h e r w i s e , o n e c o n s i d e r s t h e f a m i l y r’ = {palaE F(li , where F I T ) d e n o t e s t h e set o f a l l f i n i t e s u b s e t s o f I s u c h t h a t one d e f i n e s f o r a,€ F I X ) : p (x) = supPi(x),

(3.9)

a for

iea

e v e r y x ~ E . N o w , on t h e b a s i s o f C o r o l l a r y 3 . 1 ( c f . ( 3 . 6 ) ) , it i s

easy

t o prove t h a t

r

d e f i n e s t h e same ( l o c a l l y m-convex)

a s t h e g i v e n family r’does

t o p o l o g y on

and, m o r e o v e r , i s a f u n d a m e n t a l f a m i l y .

3 ) I n p a r t i c u l a r , s u p p o s e t h a t a l o c a l l y m-convex t o p o l o g y on a

given tive

a l g e b r a E i s d e f i n e d by a d e n u m e r a b l e f a m i l y o f s u b m u l t i p l i c a semi-norms,

construction (cf

.

r = {pi } i e m; t h e n , (3.9) ) , one o b t a i n s an

say

by a p p l y i n g on

r

t h e previous

increasing sequence of submuZtipli-

3. TOPOLOGIES BY SUBMULTIPLICATIVE SEMI-NORMS

c a t i v e semi-norms,

say

r'

= { q n } n E w , where one sets qn(xL.) =

(3.10)

sup P i h l 15iSn

I

for every x & E , defining the same (locally rn-convex) topology on E . Therefore,i f E i s a l o c a l l y ni-convex algebua w i t h a denumerable d e f i n i n g f a m i l y of i t s t o p o l o g y , one may always suppose t h a t t h e g i v e n f a m i l y of (submultiplicative) semi-norms c o n s t i t u t e s an i n c r e a s i n g sequence ; moreover, the same family may always supposed to be a fundamental one (cf.the preceding remark 2 ) ) Thus, t h e r e s p e c t i v e f a m i l y of u n i t s e m i - b a l l s , defined by (3.10),

.

that is, { U q ( l L . ) l n E n , is a decreasing sequence of neighborhoods o f z e r o (Propon sition 1.1). Hence, t h e given topology of E is Hausdorff i f , and only i f , (3.11)

(cf. Corollary 3.1). Iii this respect, see also the next Proposition 3.3 The definition of a locally

r;-convex topology on a given alge-

bra E by means of a suitable family of submultiplicative semi-norms, as described above, characterizes in fact the class of topological algebras under consideration. That is, one gets, as we will presently see (cf. Theorem 3 . 7 ) , an alternative of Definition1.3 and of The oreml.l, as well as an inverse of the above Corollary 3.1. Thus, the " g e o m e t r i c " description of a locally m-convex algebra given in Section 1 , will be supplemented by an " a n a l y t i c " one. However, we need first the following easy, but useful, result.

Lemma 3 . 3 . Let E be a t o p o l o g i c a l aZgebra and U an m-barrel of E which i s

,,

neighborhood of z e r o . Moreover, l e t p u be t h e r e s p e c t i v e gauge f u n c t i o n of U. T h m , one has

(3.12)

u

=

I

X E

B : p U (x-i 6 1 1 ;

U c o i n c i d e s w i t h t h e r e s p e c t i v e c l o s e d u n i t semi-ball of its gauge f u n c t i o n . Hence, (Proposition 1.5), t h e i n t e r i o r of U i s t h e s e t

i.e.,

0

(3.13)

U = { x& E :pU(xl < 1 1.

Thus,there e x i s t s a one-to-one

and o n t o correspondence (i.e., a bijection)

between t h e s e t of m-barrels of t h e t o p o l o g i c a l algebra E,which are neighborhoods

of zero,aizd t h e s e t of continuous subnmltipZicatiue semi-norms on E , g i v e n

by

(3.12).

Proof. The first part of the assertion is a direct consequence of the hypothesis, Lemma 1.1, and Proposition 1.5. The rest is straightforward by the foregoing.m

I GENERAL CONCEPTS

18

Thus, we now have the following.

Proposition 3.2. L e t E be a l o c a l l y m-convex aZgebra (Definition 1 . 3 and/ or Theorem 1.1 ) Then , there e x i s t s a fundamental defining family (of submultiplicative semi-norms)of the topology o f E.

.

Proof. By hypothesis and Theorem 1 . 1 : 2 ) , there exists a localbasis, say { U 3 , of E consisting of m-barrels. Thus, the respective gauge functions of the members of xconstitute a family, say r =

z=

{ U IUE 3iT , of (continuous) submultiplicative semi-norms, the corresponding unit semi-balls of which are the given m-barrels (Lemma 3.3). Hence (Corollary 3 . 1 ; cf. ( 3 . 6 ) ) , the family rdefines the same locally m-convex topology on E as the family 2 . On the other hand, by (3.6) and Lemma 3 . 2 (see also ( 3 . 8 1 1 , one concludes that r is, in fact, a fundamental defining family of the topology of E l and this tcrminates the proof. I

Remark.- It-is clear by Lemma 3.3, in connection with Corollary and Lemma 3 . 2 , that t h e s e t of a l l continuous submultiplicative semin o m s on a given l o c a l l y rn-convex a2gebra c o n s t i t u t e s a fundamental fumilg of the topology of the algebra. In this concern, we note that the assertion of Proposition 3 . 2 is, in fact, a direct consequence of the laststatement; however, the significance of the above proposition (and of the proof given) lies, namely, in the possibility of choosing, suitably, an eventually smaller family (of submultiplicative semi-norms) than the last one above, if we have already a local basis of rn-barrels of E. (In this respect, see also Remark 2) after Lemma 3 . 2 ) . Of course, it would be equivalent to take as a local basis of the given locally m-convex algebra E the s e t of a l l m-barrels of E which are neighborhoods of the zero element (cf. Theorem 1 . 1 ; 3 ) , as well as Example 2 . 2 ) . 3.1,

As we mentioned it before, the following basic result combines now the preceding two equivalent definitions of a locally m-convex algebra; namely, the "geometric" and the "analytic"one. Thus, we have.

Theorem 3.1. Let E be an algebra. Then, the following a s s e r t i o n s are equivalent : 1 ) E i; ,z locally n-conitez a:,gcbrc ( l k f i n i t i o n I. 3 ) . 21 E i s a topological algebra w i t h a local b a s i s consisting o f m - b a r r e l s .

31 Them e x i s t s a fundamental defining f a m i l y o f submultiplicative semi-norms

on E I c f . ( 3 . 7 ) ) . Proof. Indeed, by Theorem 1 . 1 ,

1)

is equivalent to 2). Moreover,

3. TOPOLOGIES BY SUBMULTIPLICATIVE SEMI-NORMS

19

by Proposition 3.2, Corollary 3.1, and Lemma 3.2, 2) is equivalent to 3). 4 Thus, on the basis of the preceding result, by examining in the sequel whether a given altzebra, being a topological vector space, is, in particular, locally m-convex, one applies, of course, indifferently, prop.2) or prop.3) of the above Theorem 3.1, as the particular case may suggest. We conclude this section by considering, in particular, the case of a metrizabZe l o c a l l y m-convex aZgebra (i.e., a metrizable (locallyconvex) topological vector space underlies the given topological algebra). So by Proposition 1.6, this algebra is, in particular, a metrizable locally convex space, so that by Theorem 1.1 t h e given algebra a h i t s a denumerable local b a s i s of m-barrels. Thus, we obtained the first part of the

following proposition, while the rest is also concluded by what has been said above in connection with Remark 3) after Lemma 3.2 (see also the rels. (3.10) and (3.11)). Thus, one has the following.

Proposition 3.3. L e t E be a given algebra. Then, t h e following a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a rnetrizable ZocalZy m-convex algebra. 2 ) There e x i s t s a (decreasing) sequence of m-barrels d e f i n i n g a local b a s i s

of E.

31 There e x i s t s an increasing fundamental d e f i n i n g sequence of submultiplic a t i v e serni-norms

(of a locally m-convex topology) on E . 1

A similar statement to the above Proposition 3.3 is obviously

valid for a metrizabZe ZocaZly convex (topological) algebra. Indeed, the respective argument is certainly clear by the foregoing. Example 3.1.

Consider a (Hausdorff) topological space X, and let X

C i X ) be the algebra of complex-valued continuous functions on

(pointwise defined operations). Furthermore, for every compact subset X o f X, consider the following real-valued function on X; (3.14)

7 3

if) = suplfizcl

I

I

xeK where f E C(X1. It is easy to prove that ( 3 . 1 4 ) d e f i n e s a subrnultiplicat i v e semi-norm on t h e algebra C(X1; therefore, t h e f a m i l y r = { p K l , with K varying over the compact subsets of X, d e f i n e s on C i X ) a local.ly m-convex topoZogy (Corollary 3.1) NOW, a fundamental f a m i l y of " v i c i n i t i e s " (entourages), i.e., a basis

.

I

20

GENERAL C O N CE PT S

of the respective uniformity, for the topology of compact convergence in X ("compact-open topology" on the space f X l ) is given by

c

TIK,EI

(3.15)

= {(f,g):

(f(xi-g(x.J(~~,for every T e K l ,

where K is varying as above and E over the positive reals. Thus, one easily verifies that the topology defined on the algebra C t X i by the previous family r = {pK} coincides with the respective compact-open topology on it. We denote the topological space thus obtained by cc(X). Thus, ec(X) is a locally m-convex algebra. Further properties of this topological algebra, depending on the particular space X under discussion, will often be considered in the sequel. Example 3.2. Let E be a locally m-convex algebra and r' a subalgebra of E.Then, F is in the relative topology a locally m-convex algebra. The { U 3 , of the assertion follows by considering a local basis, say topology of E consisting of m-barrels (Theorem 1.1) and its "trace" on F;that is, the family n n F = {L!IIFIUen, whose members are m-barrels o f F i n the relative topology. On the other hand, if the topology of 8 is given by a (fundamental) family, say r = { p a l a e I , of submultiplicative semi-norms (The orem 3.1),then the restrictions to F of the members of r , that is the family

n=

(3.16) defines the relative locally m-convex topology on F induced on it by E . In this respect, it amounts to the same if one considers the relative topology on F as the initial topology defined on it by the maps

with

c1

E I ; here idiE Y

algebra f E , p , l

Pa'

denotes the identity map of the semi-normed

(cf.Proposition 3.1, as well as Chapt.III;Sectionl).

Example 3.3.Finally consider, in particular, an open subset R of

the (n-dimensional) complex numerical space Cn and let O ( 5 2 I be the algebra of all complex-valued holomorphic functions on R (pointwise defined operations; cf. R . C . G U N N I N G - H . R O S S I [ 1 ) ; we consider the latter space as a subalgebra of the locally rn-convex algebra ( R I(cf. Example 3.1). Thus, by the foregoing, O(s1 ) equipped with (the relative topology, that is) the compact-open topology becomes a locally m-eonvex algebra. A (fundamental) defining family of it is given by the respective relations to (3.14) (see also (3.16));thus, K is varying now over an in-

e

!+. COMPLETE

21

TOPOLOGICAL ALGEBRAS

creasing sequence of compact subsets of C n whose union is (En. Therefore ,

0 (R) is, in particular, a metrizable locally rn-convex algebra (Proposition 3 . 3 ) , the underlying (locally convex) topological vector space of which is, moreover, complete (Weierstrass Theorem). Thus, o:n), topologized as above, i s a Fre'chet l o c a l l y in-convex aZgebra (cf. Definition 1.5 and subsequent comment). Now, several properties of the algebra O i Q ) , which indeed represents an important class of topological algebras will also be considered in the sequel.

4. Continuity of the multiplication. Complete topological algebras We start with fixing up the relevant terminology. Thus, we first have the following.

Definition 4.1. A given topological algebra E is said to be

a

compZete topoZogicaZ algebra , if the underlying topological vector space of E is complete (i.e., every Cauchy f i l t e r in E converges)

.

Thus let E be a topological algebra. Then by looking at the underlying topological vector space E, one may consider the completion h

of the respective uniform space E, denoted by E , which as we know is naturally endowed with the structure of a vector space, having the corresponding "vector space operations" continuous, so that i i s a complet e topological. vector space (cf., for instance, F . TREVES [I ; p . 411 )

.

Moreover, the topological vector space E is (within a topological vector space isomorphism) a dense subspace of 2; thus, if {Ua} is a lo-

n=

h

cal basis of E , one obtains a local basis of E by taking the closures of the members of 72 in that is, by considering the family % =

{za)

(ibid., and/or J . HORVLTH [l; p. 1 3 4 1 ) . Now, if E is a coiixuiatiae topological. algebru (i.e., E is commutative, as an algebra) w i t h continuous m u l t i p l i c a t i o n , then the (ring) muZtiplica t i o n in E, considered as a bilinear map of E x E into E i s uniformly cont i n u o u s ; thus, by standard argument, it can uniquely be extended b3 continuit y t o , the completion of E, so that i s a topological algebra too (with continuous multiplication). But the preceding argument can not be applied if the algebra E is not commutative, even if the multiplication is continuous; the reason is simply due to the fact that the corresponding (left and right) uniform structures, defined by the (continuous) ring multiplication, are not the same, so that the multiplica-

tion is no more a uniformly continuous bilinear map as above (cf. N . BDURBAKI [4; Chap. 3 , p. 211 ) . Nonetheless, the continuity of the multiplication in a given topological algebra is still sufficient in or-

22

I GENERAL CONCEPTS

der the latter operation to be extended (preserving its continuity) to the completion E of E (ibid.;~. 5 0 , ThEorSme 1 ) . Therefore, A

i f E i s a topoZogica1 algebra w i t h continuous multiplica-

(4.1)

t i o n , then the completion of (the underlying topological vector space)E i s again a topological algebra w i t h continuous m d t i p l i c a t i o n .

Thus, by referring to the completion 2 of a topological algebra E l we shall always understand that of the underlying topological vector spa* ce E. This will also be the case even when E happens to be already a topological algebra (see, for instance, Theorem 4 . 2 ) . Now suppose, in particular, that E i s a l o c a l l y convex algebra w i t h continuous multipZication. S o its completion g , being a complete locally convex space, has by the previous argumenta continuous multiplication; hence, 2 is s t i l Z a ZocalZy convex algebra w i t h continuous m u l t i p l i c a t i o n . On the other hand, i f E is a l o c a l l y m-eonvex algebra, t h e n , b y the preceding and Proposition 1 . 6 , i t s completion E i s a l o c a l l y convex algebra (with { U } is a local basis of E concontinuous multiplication).Thus, if sisting of nl-barrels (Theorem l . l ) , let, on the basis of the above, R = { } be the local basis of , corresponding to ?7. Thus, since each member U of is an m-set of E l the same is true (Lemma 1 . 5 ) for each one of the sets (closure of U S E in? ) ; therefore, E i s , moreover, a local basis of (the locally convex space) 2 . Thuslone concludes (Corollary 1 . 1 ) the following much stronger result: A

n=

Lemma 4.1. Let E be a l o c a l l y m-convex algebra. Then, its completion is a = { U } is a local b a s i s o f (complete) l o c a l l y m-convex aZgebra. Moreover, i f of the E consisting of m-barrels, then the famiZy f i = { 5 } (closures in members of 7'Z) d e f i n e s a local basis of (the locally m-convex algebra) k consisting of m-barrels. Proof. The first part of the assertion has already been proved by the above comment. Now, since each U e Z i s an m-barrel of E and

is a neighborhood of zero in 2 , it is readily seen that each where U f z , is an m-barrel of 2,which is the assertion.1

e

2,

Remark 4.1.- Suppose that E is a locally m-convex algebra whose topology is given by a fundamental family r = { p a l a e I of submultiplicative semi-norms. Now, since each pa e T is a (real-valued) uniformly continuous function on E (Proposition 1 . 5 ) , it can uniquely be extended to a similar function iaon the completion 2 of E . Thus, by the

4.

COMPLETE T O P O L O G I C A L ALGEBRAS

23

principle of “extension of inequalities” (cf. N. BOURBAKI [4: Chap. IV; p. 1 8 1 ) , one easily verifies that each ;a , u E I , is a submultiplicative semi-norm on i. Pioreover, the family f = thus derived defines the topology of the locally m-convex algebra E . The proof is ~

the same as for locally convex spaces (see, for instance, J . HORVdTH [I: p. 1 3 4 1 ) . Now based, for instance, on (3.7) one readily proves that if the given family r is fundamental, then also ^r = { p a l U e I is a fundamental famizy of i. Thus it is important, if one has to consider complete topological algebras, to know whether the multiplization of the original algebra is continuous ( in both variables), since then, according tothe preceding, one may take “completions“. Now it seems that this is not simply a matter of convenience; indeed, as we shall see ‘%ompbeteness” of a given topological algebra already implies something more than j u s t separate c o n t i n u i t y o f the m u l t i p l i c a t i o n provided by the definition. In other words,t h e m u l t i p l i c a t i o n of a given topological algebra has t o be ”continuous enough” i n order t o be eztenduble (as a similar operation) t o the complet i o n of the given algebra. (In this respect, we note that the situation is

not quite clear except from special cases). Moreover, this may happen even for weaker forms of completeness, as it will become clear by the ensuing discussion. We start with fixing up the relevant terminology. Thus, we have. D e f i n i t i o n 4.2. Let E be a topological algebra. Then, we shall say

that E is a quasi-complete topological algebra, if every bounded and closed subset of E is complete (i.e., if the underlying topological vector space B is quasi-complete).Also, we say that E is a 0-complete (or else sequentialzy complete) topoZogieal aZgebra, if this is the case for the respective topological vector space E (i.e., if every Cauchy sequence in E converges (to some point of E ) ) . Now, since every Cauchy sequence in a topological vector space defines a bounded set and the closure of a bounded set is also bounded, one concludes that every quasi-complete topological. algebra is o-c;x;-Le:e. In particular, a metrizable topoZogica1 algebra i s conir,kte (hence FrEchet)

if, and only if, it i s a-complete. On the other hand, the multiplication in a Frschet algebra (Definition 1.5) is always continuous (in both variables). In fact, something more holds true on the basis of the following result from the general theory of topological vector spaces.

I

24

GENERAL CONCEPTS

Lemma 4 . 2 . Let E,F, G be topological vector spaces such t h a t E i s a Baire

space and F metrizable. Then, every separately continuous b i l i n e a r map u:ExF+G i s continuous.~~c:raover, the same holds t r u e i f E i s a barrelled rnetrizable (topological v e c t o r ) space and G any l o c a l l y convex space. Proof.

See H. SCHAEFER [l: p. 881 and F . TREVES [l: p. 4251. I

Thus, one has the following

Corollary 4.1. Every Baire metrizable topological algebra has (jointly) continuous r m l t i p i i c a t i o n . In p a r t i c u l a r , (since every complete metrizable topological space is a Baire space, one concludes that) evemj Frdcchet topological algebra (Definition 1 . 5 ) has a continuous m u l t i p l i c a t i o n . I On the other hand, by the second part of Lemma 4.2, one also obtains that every barrelled metrizable l o c a l l y convex algebra has a continuous mult i p l i c a t i o n (in both variables) NOW, i n a topological aZgebra F with continuous m u l t i p l i c a t i o n the product A. B of two bounded subsets A , B of E i s a l s o a bounded subset of E . Indeed, this follows directly by the (joint) continuity of the multiplication in E and the definition of a bounded set in the topological vector space E. In this respect, one gets a similar result for any given topological algebra, if one still assumes "some s o r t of completeness" for the particular algebra under consideration. In fact, this is a characteristic property in order the multiplication of the algebra to be (jointly) continuous, hence, the algebra itself to have a completion if, moreover, happens to be metrizable (see below Theorem 4.2 and its Corollary 4.7). In case of a locally convex algebra the latter is a necessary condition in order its o-completion (as a locally convex space) to be again a (locally convex) topological algebra (Theorem 4.1; see also Scholium 4.2). However, apart from these cases a similar characterization concerning, namely, the impact that a completeness hypothesis would have on the "amount of continuity" of the multiplication in a given topological algebra, seems to be lacking for more general topological algebras. On the other hand, and in conjunction with Theorem 4.2, it is m r t h noticing that there e x i s t complete (locally convex) topological algebras without (jointly)continuous m u l t i p l i c a t i o n . Now, to start with and for better understanding, we recall in some detail certain preliminary material from the general theory of (locally convex) topological vector spaces: Thus, suppose thatEis a locally convex space and let A be a

.

4.

25

COMPLETE T O P O L O G I C A L ALGEBRAS

bounded s u b s e t o f E . Then, t h e c l o s e d b a l a n c e d and convex h u l l of A , t h a t i s , t h e set B = <(A/>

(4.2)

,

i s a g a i n a bounded s u b s e t o f E . I t i s t h i s property of a bounded s e t i n a g i v e n l o c a l l y convex s p a c e w h i c h , i n f a c t , n e c e s s i t a t e s i n t h e f o l l o w i n g the Zocal convexity of a topological vector space, since t h e l a t t e r property i s not t r u e , i n general, f o r an a r b i t r a r y topological v e c t o r space. On t h e o t h e r h a n d , g i v e n a t o p o l o g i c a l v e c t o r s p a c e E l i f B i s a n a b s o l u t e l y c o n v e x and bounded s u b s e t of E , l e t (4.3) b e t h e v e c t o r s u b s p a c e of E g e n e r a t e d by B;now, s i n c e B i s a bounded s u b s e t o f E l EB i s a normed space whose norm i s d e f i n e d by t h e gauge f u n c t i o n of B , t h e l a s t set b e i n g a l s o t h e u n i t b a l l of E B (see, f o r i n s t a n c e , G. KSiTHE [l: p. 2521 o r J . HORVkTH [1: p. 2 0 7 1 ) . I n p a r t i c u l a r , one h a s t h e following.

Lemma 4 . 3 . Let

E be a topological vector space and B an absolutely convex

bounded closed and o-complete subset of E. Then, t h e space E i s o-complete, t h a t B i s a Banach space. Proof. W e

f i r s t remark t h a t t h e f a m i l y

n = {a.B:

(4.4)

a>C

c o n s t i t u t e s a l o c a l b a s i s o f t h e normed s p a c e E B , so t h a t s i n c e B i s a bounded s u b s e t of member o f

12;

E l e v e r y neighborhood U of z e r o i n E c o n t a i n s a

thus, t h e topology, say

f i n e r than t h e r e l a t i v e topology, say

z,i n

F B I g e n e r a t e d by

c, , i n d u c e d

?Z, i s

on E B b y t h e o r i g i -

fx I i s a Cauchy s e q u e n c e i n E E [ Z ] E . t h e r e e x i s t s n o € ilv s u c h t h a t x n - x , E aB , for

n a l t o p o l o g y of E. T h e r e f o r e , i f E,then for every a >0 , any n,rntn

. Thus,

s i n c e by h y p o t h e s i s x n + a B i s

a 0-complete s u b s e t

0

of E a n d ( x n l i s a l s o a Cauchy s e q u e n c e w i t h r e s p e c t t o t h e t o p o l o g y

zo
-x e

+ aB

o b t a i n s t h a t (xn I converges t o a p o i n t x e x

, so

that

no 2 0 3 , f o r any n 2 n

and a > 0 ; h e n c e , t h e s e q u e n c e ( x n ) c o n v e r g e s

a l s o t o x with respect t o

z,

so t h a t E B i s a Banach s p a c e . 1

Corollary 4.2. Let E be a topological vector space. Then, every barrel i n E absorbs every subset of E which i s balanced convex bounded closed and o-complete. Proof.

I f T i s a b a r r e l o f E and B i s a s u b s e t o f E a s s t a t e d ,

where EB i s t h e s p a c e d e f i n e d by ( 4 . 3 ) , t h e n (Lemma 4 . 3 )

, EB

i s a Ba-

I GENERAL CONCEPTS

26

nach s p a c e w i t h a (norm) t o p o l o g y f i n e r t h a n t h e r e l a t i v e t o p o l o g y i n duced o n i t by E ; t h u s , s i n c e T i s a b a r r e l , h e n c e a c l o s e d s u b s e t of E,

it follows t h a t T n E B i s a c l o s e d s u b s e t of E B , t h u s a barrel.Hen-

ce, s i n c e E H i s a b a r r e l l e d s p a c e , T n E B i s a n e i g h b o r h o o d o f z e r o i n E B so t h a t i t a b s o r b s B which i s also a bounded s u b s e t ,

u n i t b a l l , of EB ( c f . t h e f a m i l y ( 4 . 4 ) )

.1

in fact, the

Corollary 4.3. Let E , F be topological vector spaces where F i s , i n p a r t i -

a

cu%ar, l o c a l l y convex and l e t 5e a s e t consisting of balanced convex bounded closed and a-complete subsets of E. Moreover, l e t I : f E , F I be the space o f c o n t i -

nuous linear maps o f E i n t o F . Then, every simply bounded subset 2 o f I: ( E , F I i s a l s o bounded i n L G ( E , F l ( i . e . , t h e s p a c e I: ( E , F I endowed w i t h t h e 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 t h e members of ' I ) . p r o o f . I t s u f f i c e s t o p r o v e t h a t , f o r e v e r y n e i g h b o r h o o d V from

a l o c a l b a s i s of F, t h e s e t T = n u - ' f V ) a b s o r b s e v e r y B i n 6 ( c f . N. ueH BOLIRBAKI [7: Chap.3; p . 2 1 1 ) . Thus, s i n c e V i s by h y p o t h e s i s a convex

s u b s e t of

F , one c o n c l u d e s t h a t T

i s a b a r r e l of E , sc t h a t t h e as--

s e r t i o n f o l l o w s now by t h e above C o r o l l a r y 4 . 2 . 1 More p a r t i c u l a r l y , o n e h a s t h e f o l l o w i n g .

Corollary 4.4. Let E , F be %oca%Zyconvex spaces such t h a t E is a-complete. 1: ( E , F I i s a l s o bounded in L b ( E , F ) ( i . e . , t h e s p a c e Z f E , FI e q u i p p e d w i t h t h e t o p o l o g y ( n o t e d b ) of bounded conThen, every simply bounded subset of

vergence i n E ) .

Proof. I f l o g y b on

d e n o t e s t h e s e t of bounded s u b s e t s of E l t h e t o p o -

I::IE,FI

we replace

d e f i n e d on t h e l a s t s p a c e by

@I, i s n o t a l t e r e d i f

by t h e f a m i l y of a l l c l o s e d b a l a n c e d and convex ! l u l l s

o f t h e members of & , s o t h a t t h e a s s e r t i o n f o l l o w s by C o r o l l a r y 4 . 3 . 1

Thus, w e g e t now t h e f o l l o w i n g . Theorem 4.1. Let E be a o-complete l o c a l l y convex algebra. Then, i f A , B are any two bounded subsets of E , t h e i r product A.B i s a l s o a bounded subset of E. I n

p a r t i c u l a r , the conclusion holds t r u e f o r every quasi-complete l o c a l l y convex algebra.

Proof. F o r e u e r y x i n E l t h e r e s p e c t i v e " c a n o n i c a l r e p r e s e n t a t i on" 1 , : E + S s ( E /

of E i s g i v e n by t h e r e l a t i o n l x ( y l =xy, w i t h

.I.

Y E

E

,

t h a t i s , l x i s an ( a l g e b r a ) morphism of E i n t o t h e l o c a l l y convex a l g e bra

L s l E l ( c f . Example 2 . ? ) ,

b e i n g a l s o c o n t i n u o u s by t h e ( s e p a r a t e )

c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E. Hence, f o r e v e r y bounded sub-

4. COMPLETE TOPOLOGICAL ALGEBRAS

27

is simply bounded, thus (COset A of E , the set 2 = {lx: x E A } E rollary 4.4) bounded also in 1: (E,EI = l b ( E l . Therefore, for every bbounded subset B of E l the set AfB)={l X ( y ) = x y : x ~ A ,y~ B } = A . B E E is bounded. I The terminology thus introduced will presently be applied inthe ensuing discussion. Thus, we first have.

Lemma 4.4. Let E , F , G be topological vector spaces and u : E x F + G

a given

b i l i n e a r map. Then, the following a s s e r t i o n s are equivalent: 1 ) For any two bounded s e t s A G E and B C F , t h e s e t u ( A , B I ( : = u ( A x B ) ) C G i s

a l s o bounded. 2 ) For any

two nu21 sequences (x I i n E and (y,)

i n F , { ulx,, y n ) 1 n e m is a

bounded sequence in G. Proof. It is clear that 1 ) implies 2 ) , since every convergent sequence in a topological vector space is, certainly, bounded. On the other hand, suppose that A E E and B G F are bounded sets, and let ( z n ) be a sequence in u(A,B), that is, z = ulx , y , ) , with xneAand y n e B , n e n ; moreover, let ( 1 , ) be a null sequence of positive reals. Thus, 7

the sequence ( a I in IR, with a; = h n , n e W , is a null sequence too, so that by hypothesis for A and B, ( a n x n ) and ( a n y n ) are also null sequences in E and F , respactively. Therefore, by hypothesis,

u ( a n x n , a n y n I = 2a n u ( x n , y n ) , n e ~ , defines a bounded sequence in G , so that 3 a . u ( a n x n , a n y n l =an.u(xn, yn)=hn.u(xn,

yn), ncm,

is a null sequence in G I that is the set u ( A , B l C G is bounded. (Here, of course, one uses the fact that a s e t M i n a given topological vector space E i s bounded i f , and only i f , f o r every null sequence (x ) i n M G B and f o r every

null sequence (A,) of p o s i t i v e r e a l s , (A LC I i s a null sequence i n E ; see, for n n instance, G . K6THE [ l : p. 1 5 3 1 ) . Thus, 2) implies 1 ) , as well, and this finishes the proof.1

In particular, one has.

Corollary 4.5. Let E be a given algebra which i s , i n p a r t i c u l a r , a topologic-il vector space. Then, the next two a s s e r t i o n s are e q u i v a l e n t : 1 ) The product of two bounded subsets of E i s a l s o a bounded subset of E ;

(under the given circumstances for E l we also say, for short, that t h e m u l t i p l i c a t i o n i s bounded preserving or yet that E has a bounded preserving multiplication )

.

2 ) For any two null sequences (x ) and (y,)

i n E , their

(coordinate) pro-

28

I GENERAL CONCEPTS

duct (xn.yn) i s a bounded sequence in E. I

Now, suppose that E, F, G are topological vector spaces and let (resp. ) be a given family of bounded subsets of E (resp. F ) Thus a bilinear map u :E x F-+ G is called 6-hypocontinuous (resp. Z-hypocontinuous) if for every neighborhood W of 0 E G and A B G(resp. B B , there exists a neighborhood V of O e F (resp. U of 0 e E ) such that u ( A , V / c W (resp. u ( U , B I G W ) . In this respect, we further note that i f the family G (resp. z) i s a covering of E ( r e s p . F ) , it i s s t i l l equivalent t o say t h a t the bilinear map u : E x F+G is G(resp. Z)-hypocontinuous, if f o r every s e t A e G ! r e s p .

.

z

z)

B E

'ti,

the s e t

2

=

x

{kX:

5

e F ( r e s p . B = { u :y e Y

B}

A ) C L(F,G), with u ( y ) = u ( x , y J l for every y X G 1: (E,G), with u = u(x,y), for every x e E ) i s Y E

equicontinuous.

In particular, the bilinear map x is said to be t G , Z ) - h y i y p o - hypocontinuous. Moreover, if continuous , if it is both G and denotes the set of all bounded subsets of E (analogously for the is called hypocontinuous if it sets B F , fiG ) , a bilinear map u:E x F+G is ( aE,dFI - hypocontinuous. Thus, we come to the following result applied below.

z

aE

Lemma 4.5.

Let E , F, G be topological vector spaces and

6 (resp.

z a set )

of bounded subsets of E (resp. F I which y i e l d coverings of the respective spaces.

F i n a l l y , l e t u : ExF-G

ons : 1 ) The map u i s (

be a b i l i n e a r map. Thus, consider t h e next two propositi-

6,zI -

hypocontinuous.

z,

and B E t h e s e t u(A,BIG G i s bounded. 2 ) For any two s e t s A E Then, 1 ) implies 21. In p a r t i c u l a r , the two a s s e r t i o n s are equivalent i f the spaces E and F are metrizable. More p a r t i c u l a r l y , the conclusion holds t r u e f o r

= @E,

z=fiF.Thus,

i f E and F are metrizable, the map u i s hypocontinuous i f , and only i f , it i s bounded preserving; t h a t i s (applying the preceding notation), i f , and only i f , (4.5)

aEXfiF G

u -1

(aG).

Proof. By hypothesis for G and the previous comments, if 1) is true and A E , then A = { uX : x 8 A } G 1: ( F , G I is an equicontinuous subset of the last space, hence bounded in LZ(F,GI;tI1us, for every B EZ the s e t i i B ) = { u l c ( y )= u ( x , y ) : I C E A , y e B } = u ( A , B ) is a bounded subset of G, so 1) implies 2).(In this respect, we remark that we actually applied above the G -hypocontinuity of the map u I while a similar conclusion could be derived if u was only z- hypocontinuous;see also Remarks 4.2)

4.

29

COMPLETE TOPOLOGICAL ALGEBRAS

On t h e o t h e r h a n d , s u p p o s e t h a t E and F a r e m e t r i z a b l e and 2 ) i s val i d . Thus, i f 1 ) i s n o t t r u e , t h e r e would e x i s t a n e i g h b o r h o o d W o f 0

E G and a s e t A

E

GI

s u c h t h a t , f o r any V from a l o c a l b a s i s of F , w e

would h a v e u ( A , V ) n C W # @ , so t h a t t h e s e t u ( A , ( x n ) ) would h e n o t bounde d , which i s a c o n t r a d i c t i o n . Now, on t h e b a s i s of t h e m e t r i z a b i l i t y of E , a n a n a l o g o u s a r g u m e n t h o l d s t r u e c o n c e r n i n g t h e Z - h y p o c o n t i n u i t y of u, so 1 ) i s v a l i d , and t h i s f i n i s h e s t h e p r o o f . 1

Remarks 4.2.-

1 ) A s a c o n s e q u e n c e o f t h e p r o o f of Lemma 4.5,

o b t a i n s t h a t e i t h e r o n e of t h e c o n d i t i o n s : the map u : E x F +G i s

, implies proposition 2)

Z-hypocontinuous F are metrizable

one

6- , 0"

of t h a t lemma. Moreover, if E and

t o p o l o g i c a l v e c t o r s p a c e s , a l l the three a s s e r t i o n s are e-

quivalent. 2 ) Under t h e h y p o t h e s i s o f Lemma 4 . 5 ,

l i n e a r map u :E x F

----+

an I E , Z ) - h y p o c o n t i n u o u s

G i s , of course, separatezy continuous

maps u : F+G,

X E E l and u : E+G, X Y f i n e s u Iy) = u ( x ) = u I x , y ) , with X Y

(

i .e .

,

bi-

the partial

J E F , a r e c o n t i n u o u s , where one deX E

E and y e F ) .

I n p a r t i c u l a r , one g e t s f o r a m e t r i z a b l e t o p o l o g i c a l a l g e b r a t h e f o l l o w i n g r e s u l t which w e a p p l y below.

Corollary 4.6. Let E be a metrizable topoZogica2 algebra. Then, t h e next two a s s e r t i o n s are equivalent : 1 ) The m u l t i p l i c a t i o n i n E i s hypocontinuous ( i - e . ,

6 denotes

o u s , where

( a ,fi)- h y p o c o n t i n u -

t h e s e t of a l l bounded s u b s e t s of E ) . ( i . e . , one h a s

2 ) The m u l t i p l i c a t i o n i n E i s bounded preserving

63;c f .

a.fi C

Corollary 4.5). 1

W e are now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g r e s u l t s , con-

czrning

t h e p o s s i b i l i t y of e x t e n d i n g t h e m u l t i p l i c a t i o n i n a g i v e n t o algebra t o t h e completion of t h e algebra; t h i s w a s too our

pological ultimate

g o a l i n t h i s s e c t i o n . I n t h i s r e s p e c t , we need f i r s t t h e f o l -

lowing b a s i c l e m m a .

Lemma 4.6. Let Moreover,

E , F , G be topological vector spaces, w i t h G quasi-complete.

let E

and F be dense vector subspaces o f E and F, r e s p e c t i v e l y , and a family o f bounded subsets o f Eo (resp. F o ) , Whose closuves i n E ) c o n s t i t u t i n g a covering of E Iresp. Iresp. I') define a f a m i l y , say G I r e s p . G(resp.% I

z

z)-

which i s (@, hypocontinuous can be extended i n a unique way t o a (separately continuous) b i l i n e -

i l . Then, every separately continuous b i l i n e a r map u : E 0 x Fo+G

ar map A : E X F+G

which i s a l s o

I G ,Z ) - h y p o c o n t i n u o u s

Proof. C f . H . SZHAEFFrl [I: p. 901 o r N . BOURBAKT

"7:

.

Chap. 3 ; p . 4 1 1 . 1

T

30

Theorem 4.2. Let

GENERAL CONCEPTS

E be a topological algebra. Moreover, consider the next tuo

assertions: 1 ) The m u l t i p l i c a t i o n i n E i s hypocontinuous. 2 ) The 0-completion

E

of E i s a topological algebra.

Then, I) implies 2 ) . In p a r t i c u l a r , i f E i s a metrizable topologicaZ algebra, then the preceding two a s s e r t i o n s are equivaZent t o each o t h e r , as well as t o the f o l lowing one. 3 ) The muZtiplication i n E i s bounded preserving (hence, the equivalence of this statement with proposition 2 ) of Lemma 4 . 4 ) .

Proof. If the multiplication in E is hypocontinuous, then since every element of E is, by definition, the limit of a Cauchy, therefore, bounded, sequence in E l one concludes 2 ) by a direct application of Lemma 4.6. On the other hand, if E is metrizable, then E coincides with the usual completion (via Cauchy nets) of E, so that by the same argument E is a topological algebra which, moreover, has a continuous multiplication, as being Frechet (Corollary 4 . 1 ) . Thus, 3) follows by the conment after the said corollary. Finallylone further concludes, by Lemma 4 . 5 , that 3) implies 1 ) ; therefore, in that case allthe above three assertions are equivalent. h

Concerning the last part of Theorem 4.2, one may restate it in the following more constructive way.

Corollary 4 . 7 . Let E be a metrizabZe topological algebra. Then, i t s compleh

t i o n r' i s a topological algebra I t h e r c f o r e , F r & h e t ) i f , and only i f , t h e m u l t i p l i cation i n E i s bounded preserving. In p a r t i c u l a r , i f E i s a locally convex algebra, then the preceding c o n s t i t u t e s a converse t o Theorem 4 . 1 . Scholium

4.1. It is a consequence of Lemxa 4.6 and its application

in the proof of Theorem 4 . 2 , that one could also obtain an analogous result to the previous Corollary 4 . 7 , by considering a "bounded completion" of a given topological algebra, in the sense that every element of the "completion" should be adherent to some bounded subset of the initial algebra. Of course, this condition is satisfied if a-completions are considered, provided the multiplication to be "appropriately hypocontinuous" (see, for instance, proposition 1) in Theorem 4 . 2 , as well as that one in Lemma 4 . 5 ) . On the other hand, this would happen by assuming that a locaZ b a s i s of t h e given algebra admits a bounded member; but then one i s led to consider "locally bounded" topological algebraswhich, in fact, are metrizable as we shall see in the sequel (cf. Lemma 6 . 1 below) .

5.

31

LOCALLY m-CONVEX TOPOLOGIES

In this respect, we also remark that -if E' i s a I;arrelled l o c a l l y conv e z algebra, then its multiplication is hypocontinuous (cf. F.TAEKS [I: p. 4241 1, so that by Lemma 4.6, E admits a o-completion E , i.e., E is a o-comp l e t e ( b c m w l l d ) l o c a l l y convex algebra (see also Theorem 4.1 ) . Within this context, we finally remark that t h e completion of a g i ven topological algebra Z ~ m ~ l be c l ugicin a t . : p i ~ l ~ j ; c aalgebra, l Cj- t h e mzrltiplicat i o n i n E would s a t i s f y t h e condition t h n t t h e product o f two Cauchy f i l t e r s i n E

(i.e. , defines a basis of) a Cauchy p : l t e r one considers Cauchy sequences (metrizability, hence a given topological algebra), the above condition is the requirement the multiplication of the algebra to

s t i i l generates

.i'n E . Thus, if o-completion of strengthened by be bounded pre-

serving (cf. Theorem 4.2 and its concise form in Corollary 4.7: in this respect, see also N. BOURBAKI [5: Chap. VII; p. 231 ) .

5. Topological algebras admitting locally m-convex topologies Our aim in this section is to give sufficient conditions sothat a given locally convex (topological) algebra can be made, in effect, (within, namely, a topological algebraic isomorphism) into a locally m-convex algebra. In case of normed algebras (cf. Section 3 above), one has an analogous situation provided by a well-known result of I.M. Gel'fand which, for completeness' sake, is also stated below (cf.Theorem 5 . 1 ) . 5. ( I ) . Michael's

Theorem. We start with the following definition

motivated, in fact, by the said theorem above, being also in agreement with the usual terminology of topological algebras t-heory;that is,to name a topological

algebra after the corresponding properties

of the underlying topological vector space. The same context leads iinally to the formulation of the theorem in title of this subsection (see Theorem 5.2). Thus, we have.

Definition 5.1.

Given an algebra E , we say that E is a Banach alge-

b r a , if it is a topological algebra for which the underlying topologi-

cal vector space is a Banach space (i.e., a normed (vector) space which is also complete). Now it is a consequence of Theorem 5.1 that the previous terminology is, in fact, equivalent with the more standard usage of the term "Banach algebra". Thus, one is usually dispensed with the supposition that the algebra is a topological one, requiring instead something, which is seemingly stronger: namely, we suppose that the alge-

32

I

GENERAL CONCEPTS

bra E under consideration is endowed with a (vector space) norm, denoted by 1 1 . 11 , which is aslo an algebra norm (i.e., a submultiplicative norm; see also Section 11, in the sense that one has the relation (5.1)

IIXYII

2

ll~ll-ll9ll I

for any x , y in E . NOW, the situation is, in fact, equivalent with that of Definition 5.1, as we shall presently see. We first have the following result from the theory of normed algebras which we need below. It is actually given in a form reflecting the situation one has in case of more general topological algebras (by "adding an i d e n t i t y " element; cf. also Lemma 5.3) . Lemma 5.1.

Let E be a topological algebra for which t h e underlying topologi-

c a l vector space E i s a normed space.

(We call such a topological algebra, an A-normed algebra ; cf. also Definition 5.4) Then, there e x i s t s an A-normed algebra E 1 w i t h an i d e n t i t y element containing E as a subalgebra, w i t h i n a topological algebraic isomorphism, t h i s being, i n p a r t i c u l a r , a l i n e a r isometry (for the respective normed spaces E and E l ) .

.

Let E l be the vector space direct sum of the vector space E , underlying the given algebra E , and the 1-dimensional vector space C of the complexes; that is, we set Proof.

(5.2)

El= E 8 C .

The space E1 is further equipped with the structure of a (complex) algebra, where one defines the respective ring m u l t i p l i c a t i o n by (5.3)

( x , X ) . (y,pil = ( x y + X x + ~ y , x p ) '

for any ( x , X / and ( y , p ) in E l . (We write ( x , X I for a "generic element" of E l , where one may still write as a "formal sum" x+X = ( x , X / ) . NOW, the map (5.4)

E+El

:X

H

(x,O)

defines an i n j e c t i o n of E i n t o El which preserves the r e s p e c t i v e algebra structures

(i.e., it is an algebra (or aZgebraic) isomorphism of E into E l ) . Furthermore, it is readily seen that the relation (5.5)

II ( x ,

Ill= I1 x I1

+

1x1

with (x,?,) e E l , defines a norm on the vector space El making the multiplication in El separately continuous. (The last assertion is straightforward by the analogous property of the algebra E , in connection ) d e f i n e s El as an with (5.3) and (5.5)) . Therefore, t h e p a i r ( E l , 1 ) . A-normed algebra with ( 0 , l ) € E l as an i d e n t i t y element; moreover, E 2 con-

33

5. LOCALLY m-CONVEX TOPOLOGIES

t a i n s E as a s u b a l g e b r a , v i a t h e map ( 5 . 4 ) ,

which i s a l s o a t o p o l o g i -

c a l one ( i . e . , a homeomorphism o n t o i t s r a n g e ) . I n f a c t , i t i s , b y

( 5 . 5 ) , a n i s o m e t r y ( b e t w e e n t h e normed s p a c e s E and E l ) . I T h u s , w e come n e x t t o t h e f o l l o w i n g v e r s i o n of G e l ' f a n d ' s

re-

u l t m e n t i o n e d above (see a l s o Theorem 5 . 1 ) . T h a t i s , w e h a v e . Lemma 5.2. Let E be a Banach algebra ( D e f i n i t i o n 5 . 1 )

w i t h an i d e n t i t y

element e . Then, t h e r e e x i s t s an { i d e n t i t y preserving) topoZogical aZgebraic i s o morphism 1: E + C ( E l

(5.6)

where I: ( E l is t h e atgebra of bounded l i n e a r maps ( o p e r a t o r s ) of t h e r e s p e c t i ve Banach space E i n t o i t s e t f endowed w i t h the "operator norm topology".

Proof. The map ( 5 . 6 ) i s d e f i n e d by IL(x)

(5.7)

= xy

,

f o r any x , y i n E. NOW, it i s e a s y t o p r o v e t h a t , f o r e v e r y x € E , L i z )

i s a m u l t i p l i c a t i v e l i n e a r map of E i n t o t h e algebra LIE) of a l l l i n e a r (not necessarily continuous) l i n e a r o p e r a t o r s of E i n t o E l i n s u c h a way t h a t Z ( e ) = l e =i d E ( t h e i d e n t i t y l i n e a r o p e r a t o r on E , o r e l s e t h e i d e n t i t y e l e m e n t of t h e a l g e b r a L ( E ) ) Moreover, t h e c o n t i n u i t y o f the operator ( l i n e a r map) l X : E+ E l x e ( w h i c h w e a l s o d e n o t e by l x )

.

E , i s s t r a i g h t f o r w a r d by t h e s e p a r a t e c o n t i n u i t y o f t h e m u l t i p l i c a t i o n i n E l so t h a t ( 5 . 6 ) i s w e l l d e f i n e d by ( 5 . 7 ) . Now, i f 1X ( y ) = x y = G , f o r e v e r y y e E , one g e t s f o r y = e , 1 ( e l = x = G , so t h a t ( 5 . 6 ) i s an i n X j e c t i o n ; t h u s , ( 5 . 71 d e f i n e s an algebraic isomorphism of E onto 1 ( E l E I: ( E ) . On t h e o t h e r hand, Z(El i s a closed subspace of i: (E). I n d e e d , i f u n e E , n e B , a n d l ( u )+T E L (E), t h e n a f o r t i o r i lim lZ(anl ) ( x )= lima x = T ( x ) , f o r en n n n v e r y x E E l so t h a t T ( e ) = lim an . T h e r e f o r e , n

T ( x : 4 ) = lima ( x y ) = l i m ( u n x l y = ( l i m a , x ) . y = T ( x l . y

? I n f o r a n y x , y i n E , so t h a t by ( 5 . 7 )

T ( x ) = T ( e . x l = T ( e ) . x =Z T ( e ) ( x )

,

,

f o r e v e r y x e C ; t h u s , T = l T ( e l , so t h a t Z ( E l i s a weakly c l o s e d s u b s p a c e of

i(E)

and h e n c e a f o r t i o r i c l o s e d f o r t h e " o p e r a t o r norm t o p o l o g y "

in

I:(E)

( t h e t o p o l o g y of bounded c o n v e r g e n c e i n E )

a Banach subspace of map o f

(5.2)

, I-'

. Therefore, l(EI

is L (E). F i n a l l y , one e a s i l y v e r i f i e s t h a t t h e i n v e r s e

:l(E/+

E,

i s c o n t i n u o u s f o r t h e weak t o p o l o g y on t ( E l

C C i E ) h e n c e l a f o r t i o r i c o n t i n u o u s f o r t h e r e l a t i v e t o p o l o g y from I: ( E ) ; t h e r e f o r e (Open Mapping Theorem), Z i s a homeomorphism of E onto Z ( E l C I:(E).

34

I GENERAL CONCEPTS

W e a r e now i n a p o s i t i o n t o s t a t e t h e f o l l o w i n g ( t e c h n i c a l ! ) form

Gel'fand ( s e e a l s o C.E. RICKART [l ;p. 5 1 ) .

of t h e above t h e o r e m of I . M .

Theorem 5.1. Let ( E , /I . I\

be a given Banach algebra

( D e f i n i t i o n 5.1 ).

Then, t h e r e e x i s t s a topological algebraic isomorphism @ of E onto a Banach aZgebra

(El , (11 . (11 I having a subrnultipZicative norm. Moreover, i f t h e given algebra E has an i d e n t i t y element e , then ] ] ] $ / e l l l l= 1 . Proof. L e t E = E b C b e t h e Banach a l g e b r a o b t a i n e d by a p p l y i n g

-

Lemma 5 . 1 t o t h e g i v e n a l g e b r a E ( c f . ( 5 . 5 ) ) . Thus, by t h e f o r e g o i n g , t h e maps a: x

(x,O) :E-+

E and

I: E -+

C ( Eo

a r e topological algebraic

isomorphisms between t h e r e s p e c t i v e t o p o l o g i c a l a l g e b r a s : t h e r e f o r e , t h e map @ = l o o l : E-+Zfu.(E))

(5.8)

= El

l:(Eo)

i s , i n p a r t i c u l a r , a t o p o l o g i c a l a l g e b r a i c isomorphism of E o n t o E l ,

t h e l a t t e r b e i n g a Banach a l g e b r a (Lemma 5 . 2 ) ; m o r e o v e r , i t s norm s a -

, s i n c e t h i s i s c e r t a i n l y v a l i d f o r t h e " o p e r a t o r norm" (Eel ( d e n o t e d by 111 - 111). On t h e o t h e r hand, i f E h a s an i d e n t i t y

t i s f ies ( 5 . 1 ) in

1:

e l e m e n t e , t h e n by a p p l y i n g Lemma 5 . 2 o n e o b t a i n s 2 in

111 Z(e) / I (

= / I / i d E111 =

f C E ) . I n t h i s c a s e , o f c o u r s e , it s u f f i c e s t o u s e j u s t ( 5 . 6 )

,

s i n c e t h e above map $, g i v e n by ( 5 . 8 ) , i s now r e d u c e d t o t h e m a p Z , g i v e n by ( 5 . 7 ) . I The a b o v e d i s c u s s i o n l e a d s now t o t h e f o l l o w i n g s t a t e m e n t which

i s e s s e n t i a l l y Theorem 5 . l I g i v e n , however, i n a less t e c h n i c a l , t h u s m o r e i n t u i t i v e form. S o g i v e n a v e c t o r s p a c e E l two norms on E a r e s a i d t o b e equivaZent

,

i f t h e r e s p e c t i v e i d e n t i t y map of E l id,

, is a

t o p o l o g i c a l ( a l g e b r a i c ) isomorphism of E o n t o i t s e l f , where E i s eq u i p p e d w i t h t h e two g i v e n norms. Now t h e t e r m i s f u r t h e r e x t e n d e d t o t h e c a s e t h a t idE i s

r e p l a c e d by an a l g e b r a i c isomorphism, s a y $,

of

E o n t o @ ( E l G F , where F i s a n o t h e r normed s p a c e , t h e u n d e r l y i n g v e c t o r

s p a c e of which i s n o t n e c e s s a r i l y E ( c f . Theorem 5 . 1 ) . Thus, w e h a v e .

Corollary 5.1. L e t

IE,

11 - 1 1 )

be a Banach algebra

( D e f i n i t i o n 5.1). Then,

t h e r e e x i s t s ( w i t h i n a t o p o l o g i c a l a l g e b r a i c isomorphism, s a y @ ) an equiv a l e n t subnmltipZicative norm, say

111 111, *

E has an i d e n t i t y eZement e , t h e n one has

Scholium 5.1.-

on E such t h a t i f , moreover, t h e algebra

/I/ e //I

ii.e.,

jl/@iejl l j i = I . I

I t i s a d i r e c t c o n s e q u e n c e of t h e above t h a t D e f i -

n i t i o n 5.1 may b e r e p l a c e d , w i t h i n a t o p o l o g i c a l a l g e b r a i c isomorphism o f t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d , by t h e more s t a n d a r d t e r m i n o l o g y a c c o r d i n g t o which by a

Banach algebra, o n e means an algebra and a Ba-

5. LOCALLY

35

TOPOLOGIES

m-CONVEX

nach space whose norm s a t i s f i e s 15.11.

In this respect, by requiring the multiplication in the given algebra E to be (jointly) continuous as an E-valued bilinear map on E x E , one g e t s a f o r t i o r i t h e s i t u a t i o n described by Theorem 5.I; equivalently, this is expressed by the relation

I1 Z Y I1 5

(5.9)

II z II. I1 Ei II

0,.

I

for any x , y in E l where a is some positive constant. The last condition is also applied by defining a normed algebra; it is equivalent, of course, to the usual meaning of the term, as stated above (cf.Theorem 5.1 and its Corollary 5.1), if the underlying normed space is complete (i.e., a Banach space). Thus, a l l t h e t h r e e forms of t h e term Banach algebra, as appeared hitherto, are e s s e n t i a l l y equivalent in the sense of Corollary 5.1. Before we proceed further, we need the following lemma concerning the process of adding an i d e n t i t y element t o a given topoZogical aZgebra; a particular instance of it was given already by Lemma 5.1. Namely, we have. Lemma 5.3. Let E

be a topological algebra. Then, t h e v e c t o r space d i r e c t

SWn

E1 = E @ C

(5.10)

i s equipped w i t h t h e s t r u c t u r e of a topological algebra w i t h an i d e n t i t y element, i n such a way t h a t E i s a topologicaZ subalgebra o f El, w i t h i n a topological algeb r a i c isomorphism. In p a r t i c u l a r , E

1

i s a l o c a l l y convex ( r e s p . l o c a l l y m-convex)

algebra, i f t h i s i s t h e case f o r t h e algebra E . proof. By applying a similar argument to that in the proof of

Lemma 5.1, we see that (5.7) defines the ring multiplication inE. in such a way that (5.81 i s az algebra isomorphism of E onto E @

{O}S

El , the

identity ofEl being the element ( 0 , l l f El . Moreover, it is easy to prove that El equipped with the respective (vector space) direct sum topology (i.e., the Cartesian product topology on E x C ; cf. N . BOURBAKI [6: Chap. 1; p.51) is a topological algebra. Thus, by considering on E @ i O : the relative topology from E l , one proves that (5.8) i s a topological a l gebraic isomorphism of E i n t o El. On the other hand, if E is a locally convex (resp. locally rat-convex) algebra and i- = { p j is a fundamental family

of semi-norms (resp. submultiplicative semi-norms) defining the topology of El then the relation (5.11)

p 1 ( x c , h i= p ( d +

1x1 ,

5.

36

LOCALLY m-CONVEX

TOPOLOGIES

with (x,Xie E l , and p varying over T , yields a similar fanily of seminorms, to the particular case considered, definingthetopologyof E l . I Thus, we are now in the position to state the promised analogon for topological (non normed) algebras of Theorem 5.1; a particular form of it will also be considered below in terms of "A-convex algebras" (see Corollary 5.3). Namely, we have (see also E . A. MICHAEL [l :p.

18, Proposition 4.31).

Theorem 5 . 2 . Let E be an m-barrelled l o c a l l y convex algebra i n such a way t h a t the following condition i s s a t i s f i e d : There e x i s t s a local b a s i s (5.12)

z={ U 1 o f

E such t h a t

f o r any c e E and U E 72, there e x i s t s X > 3 (depending

on CI and L ) w i t h (5.12.1) a. UG A. U Then, E i s a l o c a l l y in-convex aZgebra.

E

Proof. Eased on the separate continuity of the multiplication in and by taking closures in (5.12.1), one obtains

0.8

I

-

(Z)cTTT7='E= )<.u,

for any a E E and U E z, varying as in (5.12). ( W e apply here the notation; 2 ( X I = 0.r t with a , x in E ; cf. (5.7)) . Thus, we may assume that the local basis 2 in (5.12) consists of closed sets. Now, consider the locally convex algebra El = E @ C obtained from E by adding anidentity element (see Lemma 5.3), and let A be the closed unit disc in C Uz 1 , such that = (i.e., A = { A e C :]XI 5 1 1 ) ; thus, the family

rl

(5.13)

Ul=TiXA

with U E E l defines a local basis of E l ,

, consisting of closed sets

(cf. also the proof of Lemma 5.3). On the other hand, consider the sets (5.14)

VlU1 = I x s E :

X.Ul

CU1

I,

with Ul given by (5.13) ( w e also set x = l x , O I E E I , with X E E ) . Thus, each one of the s e t s 15.141 , with U varying over i s an mbarrel of E . Indeed, we prove that these sets are absorbing subsets of 7 , while the rest of the assertion is easily obtained by (5.13) and

z,

(5.14): Thus, for any a e E and U e 2 , there exists v ( a , U ) > 0 , with a e pU. Moreover, by (5.12), there exists A(a,U/ > 0 , with a . U c U , so that one obtains, for every ( x , t ) U~1 = U x A , the relation

a. ( x , t i = (a,O/

. Ix,ti = lax + t a ,

0)

(cf. also (5.3)).Therefore, since It1 5 1 , one obtains

5. (2). A-CONVEX ALGEBRAS

ax + t a €XU + t U E f A + u ) U

i-e., a = (a,O)

8

37

,

( A + p ) V f U l . Thus, by hypothesis for the algebra E

,

the

sets (5.14) are neighborhoods of zero in E , so that, since ( 0 , l ) e U1 , one also obtains (5.15) V i U ) E u, for every U E ??. Therefore, t h e family

n, d e f i n e s a

n'= V t U , J ) , with

local b a s i s of E c o n s i s t i n g of m-barrels

,

U varying over and this finishes the

proof. I In particular, one gets the next.

Corollary 5.2. Every Fre'chet l o c a l l y convex algebra, which s a t i s f i e s ( 5 . 1 2 ) i s a (Frdcketl l o c a l l y m-convex algebra.

A stronger version of the previous condition ( 5 . 1 2 ) sently be considered in the next subsection. 5.(2).

will pre-

A-convex algebras. The class of (topological) algebras in ti-

tle of this subsection was introduced in the early 7 0 ' s by A.C. COCHThey present, as we shall see below, a RAN, R. KEOWN and C.R. WILLIAMS [l] convenient class of topological algebras satisfying the condition (5.

.

12) in a less tecnical (cf. (5.16)),hence, more managable form to apply in concrete examples (see Corollary 5.3 and Chapt. I V ; Section 4, (4.16)). We start with fixing up the relevant terminology. Thus we first have.

Definition 5.2. A semi-norm p on a given algebra E is said to be left absorbing if, for every element a e E , there exists X > 0 (depending on p and a ) such that

(5.16)

p ( a x ) S X.pix)

,

for every x 8 E . A r i g h t absorbing semi-norm F on E is defined analogously. Finally, we say that p is an absorbing semi-norm on E , if it is both left and right absorbing. Thus, we now have the following.

Definition 5.3. Given an algebra E, we shall say that E is an A-convex algebra ( A stands for "absorbing"), if there exists a family of absorbing semi-norms on E (Definition 5 . 2 ) defining it as a locally convex (topological vector) space.

38

I GENERAL CONCEPTS

The comment before Definition 5.2, as well as the terminology applied in the previous definition, are now well justified bythenext. Lemma 5.4. Every A-convex algebra i s a Zocally convex algebra whose topolo-

gy can be defined by a fundamental f a m i l y

nition 5 . 2 ) i n such a way t h a t , i f to

r, t h e n ,

r

z=

LI

of absorbing semi-norms ('3efi-

=

1 i s the local b a s i s of

E corresponding

t h e family ?z s a t i s f i e s t h e c o n d i t i o n IS. 1 2 ) .

Proof. Suppose that

r

= { q } is a family of absorbing semi-norms

defining E as a locally convex space (Definition 5.3). Then, it is a direct consequence of (5.16) that, for every a e E , the linear map l a : E+E , defining the left multiplication by n in E (i.e., I. (x)= a z , with x 6 E) is continuous;moreover, the same is certainly true, by hypothesis, for the right multiplication by a e E . Thus one concludes, in particular, that c v e r j A-convex algebra i s a l o c a l l y convex algebra. NOW, by considering the fundamental defining family of seminorms of the topology of B, say r = {p},which corresponds to the given family To , one concludes that every pe r i s a l s o an absorbing semi-norm on t h e algebra E . (In this respect, we recall that p = supqi , with c1 being a fi-

r,

iecl

a x ) <s,ipAia,iJq.(x)5 isupq.(x) = A;,:x), i i i " i with A = maxA(a,i), for any a , x in E ) . Therefore, if { : I ( E ) ) is the

nite subset of

; thus,

piax)=

SLiy~?(

x=

iecl

P

local basis of E l corresponding to r = { p ? , thenby the preceding, for any T E r and a & El there exists h ( a , p J E i > 0, in such a way that pinzi 5 X.piar)

(5.17)

,

f o r every x & E. Hence, by considering the unit semi-ball U ( 1 ) which P corresponds to per, one obtains by ( 5 . 1 7 )

a.U

(5.18)

for every

E

(EJ

EX.U

is)

,

P P > 0; therefore, t h e f a m i l y ?Z s a t i s f i e s t h e condition ( 5 . 1 2 ) . I

The next result is now straightforward by Theorem 5.2 ;"s5. 4.

and Lem-

C o r o l l a r y 5.3. Every m-barrelled A-convex algebra is, i n p a r t i c u l a r , a locall y m-convex algebra. I

In this respect, we finally note, foruse later on, that a particular instance of the above Definition 5.3 is also the next

Definition 5.4. Given an algebra E l we shall say that E 2-nomned

is

an

aZgebra, if the underlying vector space of E is a normed space

( E , ~ ~ ~in~ ~such ) , a way that the multiplication in E is separately con-

6.(1).

ti nilous

LOCALLY BOUNDED ALGEBRAS

39

.

Concerning the condition set forth by the above definition, this amounts, of course, to the requirement that, for every a e E , there e-

X(a) = X > 0 , such that

xists

llaxll 5 x . l l x l l

(5.19)

I

for every r c e E ; and the analogous condition too, forthe right multiplication by a e E . This justifies also the terminology applied above in connection with Definition 5.3. We have already encountered A-normed algebras in Lemma 5.1. Furthermore, we finally note that an equivalent formulation of the same notion runs as follows: An A-normed aZgebra i s a topoZogicaZ aZgebra f o r which

.

t h e underZying topozogical v e c t o r space is a normed space (cf Lemma 5 . 1 ) . On the other hand, the more general concept of an A-seminormed algebra ( E , p l is certainly clear. A further useful application of the concept of A-normed algebras in connection with general A-convex algebras and their relation

to locally m-convex ones will be considered in Chapt.VII1; Section 6. In this concern, it is clear, of course, that every locaZZy m-convex a2gebra i s an A-eonvex aZgebra.

6. Certain particular classes o f topological algebras The topological algebras considered in this section will be encountered several times in the sequel; we discuss here their most basic of their properties which we shall also apply below.

6.( 1 ) . Locally bounded algebras. The topological algebras tha.t we first consider yield, among other things, a non-trivial example of a class of togological algebras having the underlying topological vector spaces not locally convex, while sharing (at least in the commutative case, as we shall presently see) several basic properties of (commutative) Banach algebras. They were introduced in the early G O ' S , and systematically studied, by W. i E L A Z K 0 [ I

3.

We start with the relevant definition.

Definition 6.1. A topological algebra E is said to be locaZly bozmded

if there exists a neighborhood of the zero element, which is a bounded subset of E (i.e. , E has a bounded neighborhood o f the zero element). In other words, and in agreement with standard usage, a ZocaZly

I GENERAL CONCEPTS

40

bounded topological algebra i s , by definition, a topologicaZ algebra f o r which t h e underlying topological vector space i s localty bounded (i.e., the latter spa-

ce has a bounded neighborhood of zero). In this respect, we first have the following elementary, butbasic fact. Lemma 6.1.

Let E be a l o c a l l y bounded topological algebra. Then, E i s a me-

t r i z a b l e topological aZgebra. In p a r t i c u l a r , i f E i s , moreover, l o c a l l y

c-complete

(as a topological vector space), then it i s a Fre'chet topological algebra (Def inition 1 . 5 ) . Proof.

If U is a bounded neighborhood of 0 E E , then by reducing

it, if necessary, to a smaller neighborhood, we may suppose that U is also balanced. Thus, since U is a bounded subset of E , the family 1 {nU:n&ZV} yields a denumerable local basis of E, so that the topological vector space E is metrizable (cf. G. KUTHE [I : p. 162; S 15, 1 1 1 ) . Moreover, if the latter soace has a neighborhood of zero which is u-complete ( l o c a l l y o-complete topological vector space), then the last assertion is a direct consequence of the same definitions.1

The following extension of the notion of a norm is important in connection with the theory of l o c a l l y bounded (topological vector) spa ces and/or (topological) algebras.

Thus, given a (complex) vector space E , a real-valued function on E is said to be an u-norm, with 0 < u 5 1 if the following conditions are satisfied:

p

(6.1.1)

pix) t 0 , for every x

B El

while p i x ) = 0 implies x = 0 ( p is

strictly positive) (6.1.2) p / x + y l

p(x) + p ( y I ,

for any x , y in E ( p is s u b a d d i t i v e ) . a (6.1.3) p 0 . c ) = 1x1 p ( z ) , for any X € C and c c e E ( p is a-homogeneous). 5

Thus, a I-homogeneous norm p:E+lR+ is a usual norm on the given vector space E . On the other hand, one has the following characterization of a locally bounded space (cf. S. ROLEWICZ [I: p. 61 , Theorem 111. 2.11 and/or G. K6THE [l: p. 1 6 0 , ( 4 ) ] ) : Theorem 6.1. A topological vector space E i s l o c a l l y bounded i f , and only i f ,

there e x i s t s an u-norm, w i t h O < a <= 1

, defining the topology of

E.l

In this connection, one applies a similar method to the familiar one for normed spaces by defining a vector space topology on a given vector space through an a-norm (cf., for instance, G. K&HE [I: p . 1 5 9 f f . I ) . Within the same context, the following notion is also useful.

6. ( 1)

.

41

LOCALLY BOUNDED AJXEBRAS

Thus, if E is a locally bounded space and U a balanced andbounded neighborhoof of 0 in E l there exists X 2 2 such that U + UE XU. NOW, one defines the concavity module of 2, denoted by c t U l , by the relation c(u)= inf { A : U + U E A U ) .

(6.2)

On the other hand, the concavity module o f t h e space E is defined by c ( E l = inf { c ( U ) : U ! Z E ; open, balanced, bounded }

(6.3)

.

NOW, one concludes by (6.3) that c I E I 2 2 ; therefore, by setting

(hence, G < a C 1 1 , one obtains the following strengthening of the above theorem (cf. S.ROLEUCZ [I: p. 61, Theorem 111. 2.1'1 ) . Thus, we have. Theorem 6.2.

Let E be a l o c a l l y bounded space. Then, f o r every a , w i t h O < a

< aL (cf. (6.4)), t h e r e e x i s t s an a-norm d e f i n i n g t h e topology of E . 1

In connection with the above Theorem 6.2, still it is worth noting that f o r any two (or even finite many) l o c a l l y bounded spaces, one can a-norms d e f i n i n g t h e i r topologies in such a way t h a t the l a t t e r correspond t o

find

the same "order of homogenuity" (i.e., one has in (6.1.3)

a common a

,

with

O
NOW, if E is a locally bounded algebra and p an a-norm defining its topology (Theorem 6.1), it might happen that p be a s u b m u l t i p l i c a t i ve a-norm, in the sense that, besides the rels. (6.1.1) - (6.1.3), one has

also the relation p ( 5 y l 5 p(xl. p ( y l ,

(6.5)

for any x , y in E.The following result asserts that this is indeed the case, if E is complete. Namely, we have. Lemma 6.2,

L e t E be a complete ZocalZy bounded algebra. Then, i t s topoZogy can be defined by a s u b m u l t i p l i c a t i v e a-norm. Proof. This is based on the application to locally bounded spaces

of an argument similar to that in Theorem 6.1 for Banach algebras (cf. Definition 5.1):ThusI if E is a locally bounded space whose topology is defined by an a-norm (Theorem 6.1) , then the "operator norm" on the algebra L i E l (continuous linear endomorphisms of E ) defines a submultiplicative a-norm on it. Therefore, L ( E l is made into a locally bounded algebra which is also complete if E is. Thus, the argument applied in the proof Gf Theorem 5.1 can now be repeated. 1 In this respect, a pair (E,p) consisting of an algebra E and

a

42

I GENERAL CONCEPTS

submultiplicative a-norm on it, is said to define an a-normed aZgebra. Thus, in the proof of the previous lemma we used the fact that S ( E ) , endowed with the respective "operator norm", i s a complete a-normed algebra. (Here the proof is paterned after its analogon in case of normed vector spaces).On the i f E i s a l o c a l l y bounded space, then t h e algebra

other hand, a complete l o c a l l y bounded algebra being, in particular, a Fr&chet topological algebra (Lemma 6.1) , has a continuous m u l t i p l i c a t i o n ( C o rollary 4.1);this can also be derived by the previous Lemma 6.2 and (6.5) by applying a standard argument (namely, analogous to that one uses in the "normed case"). Thus, we have actually proved one half of the following. Lemma 6 . 3 . Let E be a l o c a l l y bounded algebra. Then, E has a continuous mul-

t i p l i c a t i o n i f , and only i f , E can be made i n t o an a-normed algebra (i.e., its topology can be defined by a submultiplicative a-norm).

Proof. W e have already proved the sufficiency of the condition,

as remarked above. On the other hand, if E has (jointly) continuous multiplication, then its completion E is a topological algebra (with continuous multiplication), which is also a complete locally bounded space (cf. G. K i T H E [I: p. 162, 10(7)]). Therefore (Lemma 6 . 2 )

,

there exists

a submultiplicative a-norm defining the topology of E , whose restriction to E yields now the desired a-norm.n The proof of the next theorem is now a direct consequence ofthe previous two lemmas, hence we may omit the details. Theorem 6 . 3 . Let E be a l o c a l l y bounded algebra. Then, t h e f o l l o m k g a s s e r t ions are equivalent : 1 ) The completion E o f E i s a (complete) l o c a l l y bounded algebra.

2) E i s ( w i t h i n a topological algebraic isomorphism) an a-normed algebra (with 0 a 5 1 1 . 3 ) E has a continuous m u l t i p l i c a t i o n . I

NOW, an important property of the class of Banach algebras is that t h e group o f t h e i n v e r t i b l e (i.e., regular) elements o f a Banach algebra w i t h an i d e n t i t y element i s open (see, for instance, C.E. RICKART [I: p. 12, Theorem (1-4.611) . In fact, t h i s i s s t i l l v a l i d f o r every complete l o c a l l y bounded algsbra w i t h an i d e n t i t y element: The proof is entirely analogous to that in the normed case, by using Lemma 6.2. Thus, in the terminology of the next subsection, every complete l o c a l l y bounded algebra (ui!.i? an i d e n t i t y e l e ment) is a &-algebra.

6.( 2 ) .

43

Q-ALGEBRAS

6 . ( 2 ) . Q - a l g e b r a s . I t i s e x a c t l y t h e b a s i c p r o p e r t y of Banach a l g e - b r a s m e n t i o n e d above, which i s e x t e n d e d t o more g e n e r a l t o p o l o g i c a l a l g e b r a s t h r o u g h t h e c l a s s of a l g e b r a s i n t i t l e of t h i s s u b s e c t i o n . The n o t i o n i s p r e s u m a b l y due t o I. KAPLANSKY [ I ]

who, i n f a c t , c o n s i d e r -

ed it f o r t o p o l o g i c a l r i n g s i n g e n e r a l . W e s t a r t with t h e relevant d e f i n i t i o n .

D e f i n i t i o n 6.2. A g i v e n t o p o l o g i c a l a l g e b r a

E w i t h an i d e n t i t y e-

l e m e n t i s s a i d t o b e a &-algebra, i f t h e s e t ( g r o u p , w i t h r e s p e c t t o m u l t i p l i c a t i o n ) of t h e i n v e r t i b l e e l e m e n t s of E i s an open s u b s e t o f E . The p r e v i o u s d e f i n i t i o n i s u s u a l l y a p p l i e d , i n a n e x t e n d e d f o r m , f o r t o p o l o g i c a l a l g e b r a s which d o n o t n e c e s s a r i l y h a v e i d e n t i t y e l e m e n t s , by employing " q u a s i - r e g u l a r i t y "

of t h e e l e m e n t s o f a g i v e n a l -

gebra. Thus, g i v e n an a l g e b r a

El

w e s h a l l s a y t h a t an e l e m e n t x i n E

i s quasi-reguZar ( o r e l s e quasi-invertibZe) (6.6)

X f

,

i f one h a s t h e r e l a t i o n

y-zy= y+z-yx= 5

I

for some e l e c i e n t y i n E . Such an eZement y E El i f t h e r e e x i s t s , i s necessa, o r e v e n advense of x ; c f . , f o r i n s t a n c e , F.F.

r i l y unique ( quasi-inverse

BONSALL - J . DUNCAN [I: p. 15 ff.] )

( r e s p . Z e f t ) quasi-reguZar

. In

t h i s r e s p e c t , t h e concept of a right

e l e m e n t of E i s c e r t a i n l y c l e a r by ( 6 . 6 ) .

Now, t h e b i n a r y o p e r a t i o n , which i s d e f i n e d on E by

xoy = X + y-zy

(6.7) with

x, y

i n E l i s u s u a l l y c a l l e d t h e c i r c l e operation

in E ; it converts the given algebra E i n t o a semi-group f o l l o w s by ( 6 . 6 )

, the

( ibid. )

( o r q-operation)

.

Moreover, a s

zero eZement of E i s t h e i d e n t i t y for ( 6 . 7 ) ; t h u s , i t i s

c l e a r t h a t the s e t of quasi-regular elements of E forms a group

,

w i t h res12ect

to t h e circle operation. So w e now h a v e t h e f o l l o w i n g e x t e n d e d form of t h e p r e v i o u s l y g i ven

Definition 6.2.

D e f i n i t i o n 6.3. W e s a y t h a t a t o p o l o g i c a l a l g e b r a E i s a &-algebra, if

t h e s e t of q u a s i - r e g u l a r e l e m e n t s of E i s open. Thus, t h e s e t of q u a s i - r e g u l a r e l e m e n t s o f a g i v e n Q-algebra E

i s by t h e same d e f i n i t i o n a n e i g h b o r h o o d of z e r o i n apparently

E, while t h i s

weaker c o n d i t i o n i s , i n f a c t , e q u i v a l e n t t o t h e l a s t d e f i -

n i t i o n , according t o t h e following.

Lemma 6.4.

Let E be a topological algebra and G q the s e t of quasi-regular

44

I GENERAL CONCEPTS

elements o f E . Then, t h e following a s s e r t i o n s are equivalent:

1,"G q i s an open subset o f E; i . e . , E i s a &-algebra (Definition 6.3). has a non-empty i n t e r i o r , say s s int ( ~ 9 ) .

2 ) ~q

3) G q

i s a neighborhood o f zero i n E.

Proof. It is clear that 1) implies 2 )

,

while since O B E is a quasi-

regular element, 1) implies 3 ) as well. Now, it is still obvious that 3 ) implies 2 ) , SO that it remains onlu t o prove t h a t 21 implies 1 ) : Thus, denoting by I, (resp. r3:) , with x € El the Left (resp. r i g h t ) "regular representations" of the semi-group ( E , o l , that is, the maps of E into itself defined by the relations 1 ( y l = s o y (resp. r x ( y ) = y o z ) ,

(6.8)

3:

for any x , y in E l one concludes by (6.7) that, for every 3: in E l the maps (6.81 are continuous.Now, if 3: is any element of G q with quasi-inverse y and a is any element of S , then one gets (3:) = a e S la o y (x)= r 9 oa

int (G") ;

therefore, one has

( s )n r - (si E s c ~q , soy YO a where the last non-trivial inclusion is readily verified by the same definitions, which proves the assertion, and this finishes the proof 3:E

I-'

of the lemma.1 A more complete form of the previous lemma will be considered in the sequel, after we have discussed the necessary preliminary material (cf.Chapt. II;Lemma1.5). However, an early use of it is made in the next subsection. 6 . ( 3 ) . A d v e r t i b l y c o m p l e t e a l g e b r a s . The topological algebras in title of this subsection were introduced by S. WARNER[3];in particular, he considered this notion in the context of the theory of locally mconvex algebras. Thus, he realized that "locally m-convex algebras possessing it advertible completeness possess also most of the fundamental properties of Banach algebras" (ibid.,~.1.See also Chapt. i I i ; Section 4 in the sequel). Thus, as we shall presently prove (see Theorem 6.4), &-algebras, for instance, possess already by their algebraic-topological structure a certain amount, so to say, of (topological, i.e., uniform) "completeness" (namely, "advertible completeness"; seethe next Definition 6.3). NOW, as we shall see (cf.Chapt. 111), this sort of completeness is wha.t one essentially needs, in several instances, rather than the

45

6 . ( 3 ) . ADVERTIBLY COMPLETE ALGEBRAS

more stringent hypothesis of the usual completeness for a given topological algebra (cf. Definition 4 . 1 , as well as the comment following Definition 6.3) We start with fixing up the terminology. Thus, we first have the following.

.

D e f i n i t i o n 6.4. A given topological algebra E is said to be aduer-

t i b l y complete, if a Cauchy filter F on E converges in E provided that

the following condition is satisfied: There exists an element x in E l in such a way that the respective filter bases lLCiFIand rx(FI both converge to the zero element of E.

(6.9)

(The maps l x , r are given by ( 6 . 8 ) , so that we simply write X and F o x , respectively, for the last filter bases)

xo

F

.

In other words, if an algebra E is advertibly complete, then a Cauchy filter F on E converges (to some elementx‘) in E (we write F-+z’),if there exists an element x in E , such that both of the f o l lowing conditions are satisfied: (6.10)

xoF+O

and

Fox-+O.

In this respect, it is readily seen that t h e element x’ (:= l i m F i s t h e quasi-inverse of x for which (6.10) is satisfied.

)

Furthermore, it is clear that every complete algebra i s a d v e r t i b l y complete. However, one still obtains the following.

Theorem 6.4. Let E be a &-algebra (Definition 6.2). Then, E is advertibZy comp1ei;e. Proof. Let F be a Cauchy filter on E , and suppose that x o F 0 and F o x - + O , for some element z e E . Therefore, if S denotes the set of quasi-regular elements of 6 , there exist a set A e F such that x o A S S and A o x C S . Thus, for any elements a,b in A and u , v in E , which satisfy the relations . f

( x o a ) o u = x o ( a o u ) =O = v o b o

LC)

= h o b ) ox

one concludes that a o u = v o b = z E E . Consequently, by hypothesis for the filter F , one finally gets that F = F o G = Foixozl= (Fox)oz+z; (we recall that the map r

2

in (6.8) is continuous, for every

LC

in E ) .

46

Thus, the filter

I GENERAL CONCEPTS

F converges to z E E, and this finishes the pro0f.l

Further properties of advertibly complete, as well as of Q-algebras, which are also u s e d in the sequel, will be discussed in subsequent sections.

47

Spectrum

(Local T h e o r y )

CHAPTER I 1

1. Spectrum of an element. Spectral radius Given any algebra with an identity element (no topology is considered), one sets the following.

Definition 1.1. Let E be an algebra with an identity element and x any element of E. Then, the spectmcm of x (denoted by SpE1xl) is the set of those complex numbers A , for which the element A-x in E does not have an inverse. Thus, one has, by definition, (1.1)

SpElxl = { X e C : A-x

is not invertible}.

Concerning the above notation, A-x stands, of course, for the element A.iE-x in t ' , where lE denotes the identity element of E . NOW, by ( 1 . 1 )

,

t h e zero element of @ belongs t o t h e spectrum of x i f , and

only i f , x i s a singular (i.e., non-invertible) element o f E. On the other hai,d, on the basis of the "circle operation" consi-

dered above (cf.Chapt. I; ( 6 . 7 ) ) ,

one extends the previous definition

to the case of an algebra which does not necessarily possess an identity element. Thus, we have the next.

Definition 1.2. Let E be an algebra and x any element of E . Then, the spectrum of x is the set of those non-zero complex numbers h (not. f o r which the element +z e E does not have a quasi-inverse in E , plus the zero element of @, unless x is regular (i.e., invertible).So

C,),

one has, by definition, (1.2)

SpE(xi = { AeC, : Ix x E E is not quasi-invertible]

{ o f @ , if x is singular}. In this respect, i f t h e algebra E does not have an i d e n t i t y element, we agree t h a t every xe E i s s i n g u l a r , so t h a t Oe SpE(x).

Thus, t h e above two d e f i n i t i o n s a r e , i n f a c t , equiualent i n ease t h e given algebra E does possess an i d e n t i t g element: This is easily seen, since a com-

48

11 SPECTRUM (LOCAL THEORY) 1

p l e x X EC, belongs t o SpE(x) i f , and only i f , - x

i s quasi-singular (i.e., does not have a quasi-inverse); now, this is readily proved by the relation

x

x - x = A ( l - - 1x )

(1.3)

A

(here the distinction between l e IR and 1 for l E € E is clear) and the fact that x o y = 0 i f , andonly i f ,( l - z ) ( l - y l = l ,

(1.4)

for any x , y in E. In connection with the previous Definition 1.2, we still note that it is also common to define the spectrum of an element x in an algebra E without identity, as the spectrum of x in the algebra E B C , obtained from E by adding an identity element (cf.Chapt.1;Section 5). Thus, if the algebra E does not have an identity element, the twosets SpE(xi and SpE8@ (x), with x e E , are naturally related (in fact, coincide). That is, we have.

Lemma 1 . I . Let E be an algebra (which does not necessarily have an identity element) and l e t E B C be t h e algebra obtained from E by "adding an i-

.

d e n t i t y " (cf Chapt. I; Section 5 I .

Then, one has

(1.5) f o r every element x i n E , where t h e two members 01 t h e l a s t r e l a t i o n are defined b y ( 1 . 2 ) and ( 1 . 1 ) , r e s p e c t i v e l y .

Proof. By the comments after Definition 1.2, one concludes that SpEeC (x) (which is given by (1.I) ) coincides with SpEec ( ix,O)) ( cf. (1.2)), for every element x in E l identified wi.th ( x , O l € E B C . Therefore, it suffices to prove that t h e l a s t s p e c s r m above coincides w i t h s p ( x l E

cJhich i s given by ( 1 . 2 ) . Thus, we first remark that a v e q element x s E i s a singular element o f t h e algebra E 8 C . ( Indeed, otherwise the relation (x,O). ( y , X I = ( x y +Ax,Ol =(O,l),

for some (y,X)€ E B C , would imply 0 = 1 i n C ,

.

which is acontradiction) Hence, 0 6 SpEBC(xl = SpEBC ( (x,O) I and similarly, 0 E SpE(xI (since we regard t h e algebra E a s lacking an i d e n t i t y e l e rnent; cf. also the comment after the en2 of this proof). Furthermore,

one verifies that an element x EE i s quasi-singular i n E i f , and only i f , (x,O) is quasi-singular i n E 8 C . (Otherwise, the relation ( x , O ) o ( y , A ) = (y,X)o(x,Ol =(O,O/, far some ( y , X I E E @ C , would yield x o y = y o x = 0,which is a contradiction to the hypothesis for x ; and conversely). NOW, the argument, as that after Definition 1.2, applied to the element with X#0, provides the desired conclusion. I

same 1 --€El

X

In connection with the preceding lemma, we note that if the gi-

SPECTRUM. SPECTRAL RADIUS

1.

49

ven a l g e b r a E has a n i d e n t i t y e l e m e n t , t h e n , s i n c e e v s r y e l e m e n t x € B

i s s i n g u l a r when c o n s i d e r e d a s a n e l e m e n t o f E B C (see a l s o t h e a b o v e , o n e c o n c l u d e s t h a t U ESpE 8 C ( x ) ; h o w e v e r , t h i s may n o t be t h e case f o r SpE(x), i f x h a p p e n s t o b e a r e g u l a r e l e m e n t o f E . I n t h a t ca-

proof)

se ( 1 . 5 ) i s n o l o n g e r t r u e , h e n c e , i f E has an i d e n t i t y element one g e t s , i n

general, t h e r e l a t i o n

=

s pE (2) 3pE@@ix)3

(1.6)

for every xEE. On t h e o t h e r h a n d , t h e f o l l o w i n g r e s u l t p r o v i d e s a c o n n e c t i o n b e t w e e n t h e s p e c t r a o f e l e m e n t s o f a g i v e n a l g e b r a a n d t h o s e of t h e s e e l e m e n t s i n a b i g g e r a l g e b r a . Namely, w e h a v e . P r o p o s i t i o n 1 . 1 . Let E , F be tuo algebras and b : E + F

an (algebra) mor-

ph.ism. Then, one has t h e r e l a t i o n

SPFl$iZi)E SPE(XCl,

(1.7)

f o r every element x i n E.

(The members o f t h e l a s t r e l a t i o n a r e d e f i n e d

.

b y t h e r e s p e c t i v e r e l a t i o n s t o ( 1 . 2 ) a b o v e ) In p a r t i c u l a r , i f E i s a sub-

algebra o f F, then one 7 b t a i n s (1

.a)

SPF(XI c SPEiX),

f o r every element x i n E. (The r e s p e c t i v e s p e c t r a a r e a l w a y s u n d e r s t o o d i n t h e sense of Definition 1 . 2 ) .

Proof. By Lemma 1 . 1

(see also i t s p r o o f ) , one h a s t h e r e l a t i o n

SPF(@(Xl)= SPF9@i!@(X/,0))

(1.9)

f o r every element

So a c o m p l e x number A # 0 b e l o n g s t o S p F ( @ ( d )

E.

J: f

i f , a n d o n l y i f , t h e e l e m e n t A-Qixi ( i . e . , A (O,l)-(@(x),O)) i n F BC

,

which i s t h e case o n l y i f A - 2 1

EBC:In fact,

i f -xox'=

t h e s i s f o r @,

one h a s

A

x @ix) 1

1 x'o-~:=O,

x

0

is singular ( i . e . , A(O,l)-(x,U)) i s s i n g u l a r i n

f o r some x ' ~ E , t h e n by h y p o 1

$ ( x ' ) = @ ( x ' l o -@(x)

A

=0 1

( i . e . , an algebra morphism "preserves t h e c i r c l e o p e r a t i o n " ) . Hence, -$(x) A

is

i n F o r , e q u i v a l e n t l y , A - $(x) i s

regular i n F @ C , thus a 1 c o n t r a d i c t i o n t o t h e h y p o t h e s i s f o r @(I) E F . T h e r e f o r e , - x is q u a s i -

quasi-regular

A

singular i n E ,

so t h a t A ~ S p ~ i M d .o r e o v e r , i f 0 6 S p F ( $ f x ) ) , t h e n $ ( x ) i s

s i n g u l a r i n F; h e n c e , by t h e f o r e g o i n g , x i s s i n g u l a r i n E as that is,

OeSpE(x), a n d

Now,

well,

t h i s finishes t h e proof. I

a n i m p o r t a n t n o t i o n which i s c o n n e c t e d w i t h t h e s p e c t r u m

of a n e l e m e n t i n a g i v e n a l g e b r a i s t h a t p r o v i d e d by t h e n e x t .

50

I1 SPECTRUM (LOCAL THEORY)

D e f i n i t i o n 1.3. Given a n a l g e b r a E and an e l e m e n t x e E l one d e f i n e s t h e spectral radius o f x

( d e n o t e d by r E ( x ) ) , by t h e r e l a t i o n

(1.10)

w e g e t , of course, t h e r e l a t i o n

Thus, by ( 1 . 1 0 ) and Lemma 1 . 1 ,

r (xl =

(1.11)

E

f o r every element x e E NOW,

l e m e n t of

P

Ef3cl:(x)

.

it i s c l e a r by ( 1 . 1 0 ) t h a t r (xJ may b e , i n g e n e r a l , a n eE Et ( : = IR, U {tm}), s o t h a t it i s i m p o r t a n t t o know t h a t i n a

g i v e n a l g e b r a E l r (zl i s , i n E

f a c t , a p o s i t i v e r e a l number. O f c o u r s e ,

t h i s i s t r u e i n e v e r y Banach a l g e b r a a n d , a s w e s h a l l s e e , t h i s i s a l -

so t h e case i n more g e n e r a l t o p o l o g i c a l a l g e b r a s ( Q - a l g e b r a s , f o r i n stance; c f . Corollary 4 . 5 ) . F i n a l l y , as we s h a l l see i n t h e e n s u i n g d i s c u s s i o n , t h e " a l g e

-

b r a i c " d e f i n i t i o n of s p e c t r a l r a d i u s , g i v e n by ( l . l O ) l h a s f o r s u i t a b l e t o p o l o g i c a l a l g e b r a s a l s o a " t o p o l o g i c a l " c o u n t e r p a r t (see, f o r i n s t a n c e , Theorem 7.2

and a l s o t h e n e x t c h a p t e r ) .

2 . The r e s o l v e n t s e t I n c o n n e c t i o n w i t h t h e n o t i o n of t h e s p e c t r u m of a n e l e m e n t

3:

i n a g i v e n a l g e b r a E , it i s s t i l l of i m p o r t a n c e t h e complement o f t h e s p e c t r u m i n 02, a n o t i o n which w e s h a l l a l s o a p p l y i n t h e s e q u e l . T h u s ,

w e f i r s t give the respective formal d e f i n i t i o n . T h u s , g i v e n a n a l g e b r a E w i t h an i d e n t i t y e l e m e n t , one d e f i n e s t h e resolvent s e t o f an e l e m e n t x e E

( d e n o t e d by p i x ) )

t o b e t h e com-

p l e m e n t i n C o f t h e s p e c t r u m of LC.T h a t i s , o n e h a s , by d e f i n i t i o n pix) =

(2.1)

c SPE(Xl .

I n o t h e r w o r d s , t h e s e t p i x ) c o n s i s t s o f t h o s e complex numbers A,

for

which t h e r e s p e c t i v e e l e m e n t A - x E E i s i n v e r t i b l e . T h e r e f o r e ,

one d e f i n e s t h e f o l l o w i n g E-vaZued map R(x;XI =

(2.2)

with

h

E pix);

(

A-xI-'

,

w e c a l l i t t h e resolvent f u n c t i o n

of t h e e l e m e n t x e 0". S o

i f G d e n o t e s t h e g r o u p of i n v e r t i b l e e l e m e n t s of E l one h a s

~ 1 x 1= R ( x ;

(2.3)

for

every element

ZE

E

'

)-I (Gl ,

.

The n e x t l e m m a ( f i r s t resolvent e q u a t i o n ) w i l l b e a p p l i e d below.

3.

Lemma 2.1. Let

ALGEBRAS WITH C O N T I N U O U S I N V E R S I O N

51

E be an algebra w i t h an i d e n t i t y element and x an element of

E. Then, t h e resolvent f u n c t i o n of x s a t i s f i e s t h e r e l a t i o n R(x;h)

(2.4)

-

R(x;p) =

- (A - p l

R ( x ; A l R ( ~ ; p l,

f o r every p a i r ( A , u i of elements of pix). Proof. At first a routine verification shows that R (x;X I .R(x; p) = R(x;p l .R(x; A/ ,

(2.5)

f o r any A , p in p i x ) . Therefore, one obtains R(x;Al-R(x;ul

- (h-21

= R l x ; h ) E l ' ~ ; p ) (( V - X )

)

= -IX-pIRlx;hlR(a;p),

which is the desired relation (2.4). I In this respect, we still note that,besides the relation (2.4) above,there is another formula referring to the resolvent function for different values of (the variable) x e E ; (we consider thus R ( x ; X I as a function of x too). This is valid f o r any algebra E with an i d e n t i t y element. Namely, it is easily proved first that (X-X)

(2.6)

( R l z ; A)

-

R(y;X)I

(A- y l =

3:

-y

,

for any x, y in E and A in both of the resolvent sets p(xl and ~ ( L J ) ; thus, multiplying the last relation on the left by R ( x ; X ) and on the right by R ( y ; X) , one obtains (2.7)

R(x;hi -R(y;X)

for every A

E

also E. HILLE

= R(z;X/(x- y)Rly;A),

p i x i n p(y1 and any 2 , in - R.S. PHILLIPS [I ; p. 126 ff.]).

(

second resolvent equation ; see

3. Topological algebras w i t h continuous inversion Our task in this and the following section is to examine towhat extent certain fundamental properties connected with the notions we have already defined above (cf. Section 1 ) and which are valid, of course, in every Banach algebra, are still true for appropriate topological algebras. Thus, we start with the following.

Definition 3.1. Given a topological algebra E with an identity element, we shall say that E has a continuous i n v e r s i o n , whenever the ( E valued) map z-x-'

is continuous on its domain of definition (i.e., the set of regular elements of E ) . On the other hand, a topological algebra E is said to have a continuous quasi-inversion , if the quasi-inversion in E defined by (Chapt.

52

11 SPECTRUM (LOCAL THEORY)

I;( 6 . 6 ) ) is a continuous (E-valued) map on the set of quasi-regular elements of E . In this respect, we first have the next.

Lemma 3.1. Let

E be a l o c a l l y rn-convex algebra with an i d e n t i t y element. T??en,

E has a continuous inversion. More generally, every l o c a l l y m-convex algebra has a

continuous quasi-inversion. Proof. This goes on verbatim as in the case of a normed algebra, by considering instead of a norm a family of submultiplicative seminorms defining the topology of E (cf. Chapt.1; Proposition 3.2):Thus.

see, for instance, C.E. RICKART [l :p. 13, Theorem 1.4.8, and p. 19, Theorem 1. 4.221. I The inversion in a given topological algebra with an identity element may be continuous without the algebra to be locally m-convex,

as this is shown by the following result. In fact, the same result may be considered as a rather concrete instance of a more general (nonetheless, more technical) situation (see the concluding Remark below). Thus, we have.

Theorem 3.1. Let

E be a metrizable topological algebra with an i d e n t i t y e l e -

ment, for which the group G of regular elements i s , i n the r e l a t i v e topology, a separable Baire space. Then, the inversion i n E i s a continuous (E-viluedl map on G,

so that G i s , i n p a r t i c u l a r , a topological group with respect t o the r e l a t i v e topology * Proof. By hypothesis the group G is a Baire space which is also

metrizable and separable, hence second countable; thus, the assertion is now a consequence of a known relevant (and more general) result (cf. T . HUSAIN [I : p. 3 6 , Theorem 8 and p. 37, Theorem 9 1 ) . I

As an application, one also gets the following.

Corollary 3.1. Let F be a Frgchet algebra (cf. Chapt.1;Definition 1 . 5 1 with an i d e n t i t y element for which the group G of regular elements i s a se,arable G -set.Then,

6

E has acontinuous i n v e r s i o n , so t h a t G i s , i n p a r t i c u l a r , a topologi-

caZ group i n t h e r e l a t i v e topology. Proof. The assertion follows by the previous theorem and the fact that G, being by hypothesis a G6-set in E , is "topologically complete"

(Alexandroff;cf. , for instance, J.G. HOCKING-G.S. YOUNG [ I : p. 85, Theorem 2-76]), hence a Baire space. I On the other hand, one also obtains.

3.

Corollary 3.2. Let

ALGEBRAS WITH CONTINUOUS INVERSION

53

E be a Baire metrizable Q-algebra w i t h an i d e n t i t y e l e -

ment such t h a t i t s group G o f regular elements i s a separable subspace. Then, G i s i n t h e r e l a t i v e topology a topological group, so t h a t E has, moreover, a continuous inversion. I n p a r t i c u l a r , a Frgchet separable 4-algebra w i t h an i d e n t i t y element i s a topological algebra w i t h a continuous i n v e r s i o n , having t h e group o f regular e l e ments a topological group i n t h e r e l a t i v e topology. Proof. By hypotesis G is an open subset of E

(cf. Chapt. I ; Definition 6 . 2 ) , hence a Baire space too, which is also separable, so that the first assertion follows by Theorem 3.1. On the other hand, if E is, moreover, a separable Q-algebra, then the group G of regular elements, being an open subset of E, is also a separable subspace; thus, one is led to the previous case, and this finishes the proof. I

Remark.- According to the result of T . HUSAIN [I] applied in the proof of Theorem 3 . 1 , a group G , endowed with a topology T making the (group-operation) "multiplication in G " separately continuous (i.e., a semi-topological group) becomes a topological group, if G i s a normal Baire second countable space in the topology T. This occurs, in particular, by the hypothesis of Theorem 3 . 1 and its corollaries above. Moreover, by a known result of D. MONTGOMERY [I :p. 881 , Theorem 21 , a semi-topological group whose underlying topological space is complete metrizable and separable, is a topological group. Now this condition is satisfied, of course, if one has

il

Frgehet algebra E w i t h an i d e n t i t y element

and group G o f i n v e r t i b l e elements a separable G - s e t .

6 ilore generally, the last conclusion is valid, if E is a metrizable (topological) algebra with an identity element and G locally complete and separable. In this respect, we still note that by T . HUSAIN [I: p. 3 6 , Theorem 71 (cf. also I".+. WU [I: p . 452, Theorem] ), one concludes that: Every topological algebra E w i t h an i d e n t i t y element and group G of regular elements a Baire second countable space, has a continuous i n v e r s i o n , G becoming t h u s a topological group i n t h e r e l a t i v e topology.

In particular, t h e l a s t conclusion i s v a t i d f o r a Baire Q-algebra E w i t h an i d e n t i t y element and G second countable o r , more p a r t i c u z a r l y , if E i s a Baire metrizable separable Q-aZgebra w i t h an i d e n t i t y element.

In conclusion, some extra conditions of the kind considered above seems to be necessary, as it concerns the group G of regular elements of a topological algebra E (not locally m-convex!) withan identity element, if one wants G to be a topological group, hence E to ha-

54

I1 SPECTRUM (LOCAL THEORY)

ve a continuous inversion. So this need appears even if E is a F r g c h e t locally convex algebra, as this is indicated by the "Arens algebra" Lw f [ O , l ] l (cf. Chapt. I; Subsection 2. ( 4 ) ) , which does not have a coztinuous inversion (cf. R. ARENS [4: p. 6301 ) . Now, topological algebras exhibiting the properties of the algebras appeared in the second part of Corollary 3 . 2 will play an important role in the sequel,therefore, they are singled out by the next section.

4. Waelbroeck algebras The algebras in title of this section seems to be the appropriate ones for an important branch of the whole theory of topological algebras, as it is the "Functional Calculus" (cf. Chapt.VI;Section 3 , as well as Chapt.VIII;Section 8). They have been applied in this context (in particular, as locally convex and/or locally m-convex ones) already in the early 5 0 ' s by L . WAELBROECK [ I ; 2 1 . On the other hand, the same algebras, at least the locally mconvex ones, seems to constitute that particular class of topological algebras in which one can find many of the fundamental results of the classical Banach algebras theory. This will become clear in several instances throughout the ensuing discussion. Thus, the same class of algebras plays a significant role too in other parts of the theory of topological algebras or of its applications, whose discussion, however, would be beyond our scope in this book. ( But see, for instance, A. MALLIOS [20-24; 2 7 1 ) . We start with giving the relevant formal definition. Namely, we have. Definition 4.1, By a Waelbroeck aZgebra , we mean a topological algebra E with an identity element, in such a way that the group G of the invertible elements is an open subset of E and the inversion a continuous (E-valued) map on G.

In other words, a Waelbroeck aZgebra is, by definition, a Q-aZgebrn w i t h n continuous inversion. Furthermore, by the foregoing (see the last Xemark above) , a separable Baire (hence, in particular, a separabZe Frgchet) &-algebra w i t h an i d e n t i t y element isa Waelbroeck algebra. Now, the following variant of Lemma I ; 6 . 4 is needed below, if one is going to apply Definition I ;6 . 2 . Thus , a topological aZgebra E with an i d e n t i t y element i s a 9-algebra (cf. Definition I;6.2) i f , and only i f , the

4. WAELBROECK ALGEBRAS

55

s e t of regular elements of E i s a neighborhood of t h e i d e n t i t y element of E:The assertion can easily be proved by suitably modifying the argument in the proof of Lemma I;6.4. We can now formulate another useful version of the above Defi-

nition 4 . 1 via the next.

Proposition 4.1. Let E be a topological aZgebra w i t h an i d e n t i t y element e . Then, t h e f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : 11 E i s a Waelbroeck algebra (Definition 4.1 ).

2) The group G of regular elements of E i s a neighborhood of e and t h e inver-

s i o n i n E i s continuous a t e . 31 The i d e n t i t y element e has a neighborhood c o n s i s t i n g of i n v e r t i b l e e l e -

rnen?s and the i n v e r s i o n i n E i s con-tinuous a t e . Proof. We first remark that 2 ) is equivalent to 3), while 1 ) implies 2) by Definition 4.1. Thus, based on the above comment and by assuming 2 ) , one concludes that E is a Q-algebra, so that it suffices to prove that the inversion in E is a continuous (E-valued) map on G ,

assuming its continuity at the point x = e . Indeed, if x e l ; , the map x b e + x - ' ( x - x O 1 , x E E , is continuous at x = x , so that :here exists O -7 a neighborhood U of zero in E , such that e +n: A i x - x 1 E G , €or every

x - x e U ; i.e., x e x O + U , where x + U is a neighborhood of xo in

E.

Thus,

due to the relation x = x0 i e + x0- ' i x - x O ) l

(4.1) valid for every

3:

e

E,

the map 3: + + i e + x - ' i x

(4.2)

,

-z0i)-',

defined for every x e x O +U , is by hypothesis continuous at refore, by (4.1), one obtains

3:

= :co

with r e x O +U , thus the desired continuity of the map x + - + x - ' , the point x = x and this finishes the proof. I

. The-

x e E , at

0'

On the basis of the previous proof, one concludes that cond.3) of Proposition 4 . 1 can be s t a t e d f o r every regular element 3: i n E , which is, in fact, an equivalent form of cond. 1 ) of the same proposition, i.e., of Definition 4.1. Now, as an application of the foregoing, we are in a positionto state the following. Theorem 4.1.

Let E be a Waelbroeck aZgebra and

3:

any element of E . Then, the

56

I1 SPECTRUM (LOCAL THEORY)

resolvent s e t

p1x) of the clement x i s an open subset of C and the respective resolvent f u n c t i o n R ( x ; J an E-valued holomorphic map on p ( ~ ) ,which also s a t i s f i e s the r e l a t i o n

-

lim RIx;AI = 0

(4.4)

A+

OJ

("bounded a t t h e point a t i n f i n i t y " o f C), so t h a t it i s , i n p a r t i c u l a r , a bounded (holomorphic) map on p l x l .

Proof. W e f i r s t remark t h a t R ( x ; X

i s continuous as a f u n c t i o n of A,

with X E p ( x I , b y ( 2 . 2 ) and t h e c o n t i n u i t y o f t h e i n v e r s i o n i n E. Hence, by h y p o t h e s i s and a p p l y i n g t h e n o t a t i o n of

( 2 . 3 ) , one c o n c l u d e s t h a t is an open s u b s e t o f C . On t h e o t h e r h a n d , t h e " f i r s t r e s o l v e n t e q u a t i o n " ( c f . Lemma 2 . 1 ) i m p l i e s t h a t

pix)

(4.5)

1

A-

IR1x;AI- R(x;Xoll = - R ( x ; X I R ( ~ : ; A , ) .

A,

T h e r e f o r e , by p a s s i n g t o t h e l i m i t i n t h e l a s t r e l a t i o n f o r A+h w i t h i n p ( x ) , and

0'

t a k i n g t h e c o n t i n u i t y of t h e i n v e r s i o n i n E i n t o ac-

c o u n t , one o b t a i n s

t h u s , RIx;AI i s an E-valued d i f f e r e n t i a b l e map on pixi

,

h e n c e by d e f i n i t i o n a

h o l o m o r p h i c map on p ( x ) . NOW, s i n c e , f o r e v e r y x e E , t h e map a c r e - a x :

i s c o n t i n u o u s , t h e r e e x i s t s a n e i g h b o r h o o d NE ( U l o f z e r o i n C , with e-ax e G , f o r every a e N E ( 0 ) (see a l s o P r o p o s i t i o n 4 . 1 ) Conse -

C+E

.

quently, the relation

makes s e n s e , f o r

1

(x(<E,

with a =-

1

X , i.e.,

l i m i t i n t h e l a s t r e l a t i o n f o r h + m i n @,

( X ( > E ; thus, by t a k i n g t h e

one g e t s by t h e c o n t i n u i t y

of t h e i n v e r s i o n i n E t h e d e s i r e d r e l a t i o n ( 4 . 4 ) . i o n i s now a c o n s e q u e n c e o f

(4.4)

S o t h e f i n a l assert-

and t h e c o n t i n u i t y of t h e map R ( x ; X ) ,

with X E p(xl. I

Scholium 4.1.-

I n t h e p r e v i o u s p r o o f an E-valued map d e f i n e d on an

open s u b s e t of C was t a k e n a s " h o l o m o r p h i c " , b e i n g d i f f e r e n t i a b l e on

i t s domain of d e f i n i t i o n ( a n d h e n c e c o n t i n u o u s too: t h i s w a s t h e case f o r t h e map R(x;XI b y ( 4 . 6 ) and ( 4 . 7 ) ) . NOW, t h i s r e m i n d s , o f c o u r s e , t h e c a s e E = C , namely, t h a t of complex-valued h o l o m o r p h i c maps, w h i l e t h i s i s s t i l l v a l i d i f one c o n s i d e r s t h e c o m p o s i t i o n s of a n E-valued h o l o m o r p h i c map a s above w i t h ( c o m p l e x - v a l u e d ) c o n t i n u o u s l i n e a r forms on t h e g i v e n t o p o l o g i c a l a l g e b r a E ,

i . e . , w i t h e l e m e n t s o f t h e topolo-

4. WAELBROECK

57

AJAGEBRAS

g i c a l dual E' of E . I n t h i s r e s p e c t , i t would b e t h u s o f i m p o r t a n c e t o h a v e i n E a "good s u p p l y " o f s u c h f o r m s , a s t h i s i s , f o r example, t h e

case i f E i s a l o c a l l y convex a l g e b r a (Hahn-Banach). B u t , a s w e s h a l l see i n s u b s e q u e n t s e c t i o n s , t h i s may s t i l l happ e n i n t o p o l o g i c a l a l g e b r a s which a r e n o t n e c e s s a r i l y l o c a l l y convex. Thus, what one r e a l l y n e e d s i n t h i s c o n t e x t i s

c(

topological algebra E

admitting a t o t a l s e t of continuous l i n e a r forms, i n t h e s e n s e t h a t , t h e r e e-

x i s t s a s u b s e t A of E ' , s u c h t h a t , f o r e v e r y e l e m e n t x E E , w i t h x # O , t h e r e e x i s t s a n e l e m e n t f e A , w i t h f(x)# 0 . T h u s , as a l r e a d y n o t e d , i f E i s , i n p a r t i c u l a r ,

a l o c a l l y con-

i s s u c h a t o t a l s e t f o r E (Hahn-Banach); i n f a c t , t h i s i s v a l i d f o r every subset A of E' whose linear hull i s "weakly dense" i n

vex a l g e b r a , t h e n E ' E', i.e.

,

i f A i s a total set

Finally,

i n E',

,

t h e "weak t o p o l o g i c a l d u a l " of E .

i n c o n n e c t i o n w i t h Theorem 4 . 1 ,

w e s t i l l remark t h a t

by t h e p r o o f o f t h e same t h e o r e m , t h e r e e x i s t s a n e i g h b o r h o o d of t h e "point a t i n f i n i t y " of C contained i n p ( x ) ,

so t h a t

p i x ) i s a neighbor-

hood of t h e p o i n t a t i n f i n i t y o f C . F u r t h e r m o r e , one c a n p r o v e t h a t R ( x ; X I i s holomorphic a t t h i s p o i n t t o o (see, f o r i n s t a n c e ,

M.A.

NAYMARK [I : p . 172,

111, and p. 66, 5 1 2 1 ) .

As a n a p p l i c a t i o n of t h e a b o v e d i s c u s s i o n , w e come now t o t h e f o l l o w i n g b a s i c r e s u l t which i s v a l i d , i n f a c t , f o r a n a p p r o p r i a t e

class of topological algebras ( e . g . ,

l o c a l l y convex o n e s , c f . C o r o l -

having a continuous i n v e r s i o n . That i s , w e g e t .

lary 4.1)

Theorem 4.2. Let

E be a topological algebra w i t h an i d e n t i t y element and con-

tinuous inversion admitting, moreover, a t o t a l s e t o f continuous l i n e a r forms ( c f . S c h o l i u m 4.1). Then, one has

SPE(XI # 0 ,

(4. 8)

f o r every element x i n E. Proof. S u p p o s e t h a t SpE(xLII = @ , f o r some x e E . T h u s ,

p(x)= a : ,

by ( 2 . 1 1 ,

so t h a t t h e r e s o l v e n t f u n c t i o n R ( x ; X I i s defined for every X E C

being, moreover, a holomorphic map ( c f . t h e Remark b e l o w ) . T h e r e f o r e , if A

cE'

i s a t o t a l s e t o f c o n t i n u o u s l i n e a r f o r m s on E l t h e n t h e

X++f(R(x;h))

,

map

w i t h f e A , i s a complex-valued h o l o m o r p h i c map on C (i.e.,

a n e n t i r e f u n c t i o n ) which i s a l s o bounded by ( 4 . 4 )

,

s i n c e f i s continu-

o u s . Hence ( L i o u v i l l e ' s T h e o r e m ) , t h e l a t t e r map i s c o n s t a n t , so t h a t by ( 4 . 4 ) (4.9)

one o b t a i n s

f(R1x;XII = f ( ( A - x ) - ' )

= 0 ,

58

with A

I1 SPECTRUM (LOCAL THEORY)

€@

,

and f o r e v e r y f E A . T h u s , by h y p o t h e s i s f o r A and ( 4 . 9 ) , one

o b t a i n s fA-x)-'=

0

,i.e. ,

a c o n t r a d i c t i o n t o o u r h y p o t h e s i s , and t h i s

terminates t h e proof. I

Remark.fact, a

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 w e have u s e d ,

in

strengthened form of Theorem 4 . 1 ; t h a t i s , concerning t h e r e s o l v e n t

f u n c t i o n R l x ; A ) , x € E , t h e analogous concZusion t o t h a t of Theorem 4.1 i s s t i l l val i d , f o r every element x , t h e r e s o l v e n t s e t of which i s an open subset of @. The a s s e r t i o n i s c l e a r by t h e a r g u m e n t i n t h e p r o o f of t h e same t h e o r e m , and i t w a s t h a t e x a c t l y , which h a s been a p p l i e d i n t h e p r o o f of t h e above Theorem 4 . 2 .

Of c o u r s e , it i s t r u e f o r e v e r y Waelbroeck a l g e b r a

(Theorem 4 . 1 ) Now, t h e n e x t f u n d a m e n t a l c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e above and Lemma 3.1.

Corollary 4.1. L e t

E be a l o c a l l y convex algebra w i t h an i d e n t i t y eZement

and continuous i n v e r s i o n . Then, t h e spectrum

SpEfx) of every element x i n E i s a

non-empty subset of C. In p a r t i c u z a r , f o r every l o c a l l y m-convex algebra E w i t h an i d e n t i t y element, every element x i n E has a non-empty s p e c t m SpEfxl.

Scholium 4.2.- I f a t o p o l o g i c a l a l g e b r a E d o e s n o t n e c e s s a r i l y hav e a n i d e n t i t y e l e m e n t , o n e c a n c o n s i d e r , o f c o u r s e , a k i n d of a gene-

raZized WaeZbroeck aZgebra,

i n t h e sense t h a t

E i s again a &-algebra

( c f . De-

f i n i t i o n I : 6 . 3 ) and t h e quasi-inversion i n E is continuous. I n t h i s r e s p e c t ,

by a p p l y i n g Lemma I ; 6 . 4 , tion 4.1

.

one o b t a i n s a n o b v i o u s a n a l o g u e o f P r o p o s i -

On t h e o t h e r hand , i f a generalized Waelbroeck aZgebra has already

an i d e n t i t y eZement, then i t i s a WaeZbroeck algebra ( D e f i n i t i o n 4.1); t h i s f o l -

l o w s by Lemma I ; 6 . 4

and P r o p o s i t i o n 4.1

(see a l s o ( 1 . 4 ) ) . Moreover,

t h e ( t o p o l o g i c a l ) adjunction of an i d e n t i t y ( c f . C h a p t . I ; 5. ( 1 ) ) t o a gene raZized Waelbroeck algebra makes it a WaeZbroeck algebra. Now by t h e f o r e g o i n g ( s e e a l s o D e f i n i t i o n 1 . 2 ) , one g e t s t h e f o l l o w i n g s t r e n g t h e n e d form of t h e above C o r o l l a r y 4 . 1 .

Corollary 4.2. Let E be a ZocalZy convex algebra w i t h a continuous quasi-inv e r s i o n . In p a r t i c u l a r , suppose t h a t E i s any locaZly m-convex algebra (cf.Lemm a 3.1 ). Then, i n e i t h e r c a s e , one has

0 # spE(x)G c

(4.10)

f o r every element x i n E . I I n t h i s r e s p e c t , one f u r t h e r o b t a i n s t h e f o l l o w i n g r e s u l t which

4. WAELBROECK

59

ALGEBRAS

p r o v i d e s supplementary i n f o r m a t i o n c o n c e r n i n g ( 1 . 8 ) . Thus, w e have.

Let E be a closed subazgebra o f a topoZogica1 algebra F , where F has an i d e n t i t y element and continuous i n v e r s i o n . Moreover, suppose t h a t , f o r eve-

Lemma 4.1.

r y x i n E , SpE(x) i s a closed subset o f C ( t a k e , f o r i n s t a n c e , a Waelbroeck a l g e b r a E). Then, concerning t h e boundaries o f t h e spectra of a given element x

i n E and F, r e s p e c t i v e l y , one has t h e reZation bd ( SpE(xl I 5 bd (SpF(xi )

(4.11)

.

Proof. By h y p o t h e s i s , one h a s t h e r e l a t i o n bd ISpElx) I = Sprixcl f o r every x E E. G C,

n el”,

n

,

T h e r e f o r e , i f A e b d ( S p E ( x ) ), t h e n A = l i m A n , w i t h A n € p ( x )

so t h a t one o b t a i n s

y = l h ( A - x)= A-x E E . n n NOW, i f y i s i n v e r t i b l e i n F , t h e n y - I = lirn(An-xl-l , by h y p o t h e s i s ; t h u s , y = A-x i s a l s o i n v e r t i b l e i n E , a c o n t r a d i c t i o n , s i n c e X€SpE(x). Therefore, y = A-x

i s n o t i n v e r t i b l e i n F , s o t h a t A E + ~ ( x ), which p r o by ( 1 . 8 ) and ( 2 . 1 ) . I

ves the desired relation (4.111,

Remark.-

I f t h e a l g e b r a s E a n d F i n t h e above lemma do n o t h a v e

n e c e s s a r i l y a n i d e n t i t y e l e m e n t ( n a m e l y , t h e same o n e ) , t h e n under t h e

r e s t o f t h e c o n d i t i o n s o f Lemma 4 . 1 , one o b t a i n s (4.12)

b d ( SpE(xJIC b d ( SpF(x)),

for every x i n E. ( C o n s i d e r t h e a l g e b r a s E O C and

FOC and

apply t h e pre-

v i o u s a r g u m e n t ) . Now t h e l a s t r e l a t i o n i s f u r t h e r j u s t i f i e d , s i n c e t h e a l g e b r a E may h a v e a n i d e n t i t y e l e m e n t which w i l l be n o t a n i d e n t i t y f o r F. The n e x t lemma w i l l p r e s e n t l y b e needed and s u p p l e m e n t s Lemma I; 6.4,

p r o v i d i n g a n o t h e r u s e f u l c h a r a c t e r i z a t i o n of & - a l g e b r a s .

Lemma 4.2.

Let E be a topologicaZ algebra and consider t h e foZZowing subset

of E (E) = {x E E : r,(x)

(4.13)

5 1),

where rE(x) is g i v e n by ( 1 . 1 0 ) . Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a Q-algebra. 2)

SIE) i s a neighborhood o f zero i n E.

Proof. I f E i s a & - a l g e b r a , t h e n (Lemma I ; 6.41,

Gq

is a neigh-

borhood of z e r o i n E , so t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U

60

I1 SPECTRUM (LOCAL THEORY)

of 0 6 E c o n t a i n e d i n G q .

We s h a l l show t h a t U

t h e n -1x is h G q , because

quasi-singular,

S ( E I : I n d e e d , i f x E E and

1x1 > 1 . B u t , t h a t i s a c o n t r a d i c t i o n , s i n c e -1x f U C

r E ( x l > I , t h e n by ( 1 - 1 0 ) t h e r e would e x i s t

A e S p E ( x l , with

x

1 and I, i s b a l a n c e d . Thus,

U C S ( E ) ,s o t h a t 1 ) i m -

l i e s 2 ) . On t h e o t h e r h a n d , i f 2 ) i s v a l i d , t h e n p o i n t of

s ( E l ; hence, t h e s e t

moreover i s c o n t a i n e d i n G q . E

1 --.S(EI

is quasi-singular,

2

1 .S(EI 2

is an i n t e r i o r

O& E

h a s a non-empty

i n t e r i o r t o o , and

1 T r u e , s i n c e o t h e r w i s e , i f a n e l e m e n t -x

2

then 2 ESpE(x)

( D e f i n i t i o n 1 .2), s o t h a t

by ( 1 . l o ) rE(xl 2 2 , t h a t i s , a c o n t r a d i c t i o n t o ( 4 . 1 3 ) : t h u s 2 ) i m p l i e s 1 ) a s w e l l (Lemma I ; 6 . 4 ) ,

and t h i s f i n i s h e s t h e p r o o f . I

The argument a p p l i e d i n t h e f i r s t p a r t of t h e p r e c e d i n g p r o o f i s s t i l l u s e d i n t h e p r o o f of t h e n e x t r e s u l t . Thus, w e h a v e .

Proposition 4.2. Let E be a c-aZgebra. Then, f o r every element x i n E , the spectrum of x is a ( p o s s i b l y empty) compact subset of @. On the other hand, f o r every l o c a l l y convex generalized Waelbroeck algebra ( S c h o l i u m 4 . 2 1 , hence, in part i c y l a r (see Lemma 3 . 1

),

f o r every l o c a l l y m-convex Q-algebra E , the spectrum of

every element x i n E is a non-empty compact subset of @ .

Proof. By h y p o t h e s i s , t h e s e t G q o f q u a s i - r e g u l a r e l e m e n t s o f i s a n e i g h b o r h o o d of z e r o i n E (Lemma I ; 6 . 4 ) , s o t h a t t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U o f 0 e E , w i t h U G G q . Moreover, i f x E E , t h e r e e x i s t s a > 0 , w i t h a x e U . T h u s , w e show t h a t r i x ) 2 .L

(4.14)

E

I n d e e d , i f r E ( x )>

1

, there

a

would e x i s t A € S p E ( x ) , w i t h I h / > 1

1

1 7 ,

so t h a t 1

s i n c e 1--1<1 and il i s b a l a n c e d , one c o n c l u d e s t h a t - -xa - - a x = -xx € Us xa G q ; t h a t i s , a c o n t r a d i c t i o n , and t h i s p r o v e s ( 4 . 1 4 ) , t h u s t h e f i r s t a s s e r t i o n of t h e s t a t e m e n t , t h e r e s t b e i n g s t r a i g h t f o r w a r d by t h e f o regoing (see C o r o l l a r y 4 . 2 ) . I The f o l l o w i n g e x h i b i t s a n o t h e r u s e f u l p r o p e r t y o f Waelbroeck a l g e b r a s , i n f a c t , of & - a l g e b r a s i n g e n e r a l , and i s a n immediate c o n s e quence o f

( 4 . 1 4 ) . Namely, w e h a v e .

Corollary 4.3. Let E be a Q-algebra. Moreover, consider the following subset of E (4.15)

B(E)={x&E: rE(x)<

+a).

Then,E=B(E); narneZy,the spectraZ radius of every element of E d e f i n e s a ( f i n i t e non-negative) r e a l number. F u r t h e r p r o p e r t i e s o f t h e above c l a s s of t o p o l o g i c a l a l g e b r a s

5. TOPOLOGICAL D I V I S I O N ALGEBRAS

61

w i l l repeatedly be encountered throughout the sequel.

5. Topological division algebras. Gel'fand-Mazur

Theorem

Our main o b j e c t i v e i n t h i s s e c t i o n i s t o d i s c u s s a " c o n t i n u o u s analogon" of a classical r e s u l t of F.G.Frobenius

referring to f i n i t e

d i m e n s i o n a l ( a s s o c i a t i v e ) d i v i s i o n a l g e b r a s . Namely, t o p r o v e , w i t h i n t h e g e n e r a l c o n t e x t o f t o p o l o g i c a l a l g e b r a s t h e o r y , t h e Gel'fand-Maazur

Theorem ( s e e Theorem 5 . 2 ) ,

i n i t i a l l y g i v e n f o r normed a l g e b r a s .

I n t h i s r e s p e c t , by a d i v i s i o n aZgebra w e mean, o f c o u r s e , a n a l g e b r a E w i t h a n i d e n t i t y e l e m e n t which i s a l s o a d i v i s i o n r i n g ( e v e r y n o n - z e r o e l e m e n t of E i s r e g u l a r ) . W e s i m p l y s t a t e t h e F r o b e n i u s t h e o r e m ( w i t h o u t p r o o f ) f o r con-

v e n i e n c e of r e f e r e n c e ; ( s e e , f o r i n s t a n c e , 14. KOCHENDORFFER [I : p. 353, The-

orem 1 0 . 5 . 1 1 ) .

Theorem 5.1. Let E be a f i n i t e dimensional r e a l ( l i n e a r a s s o c i a t i v e ) d i v i s i o n algebra. Then t h e only p o s s i b l e dimensions f o r E are 1 , 2 , and 4 . I n p a r t i c u l a r , t h e only f i n i t e dimensional d i v i s i o n aZgebra over t h e complexes i s the complex number f i e Zd i t s e I f . 1

Schol ium.-

The h y p o t h e s i s of a s s o c i a t i v i t y of t h e a l g e b r a s i n

-

v o l v e d i n t h e above t h e o r e m i s q u i t e e s s e n t i a l . T h u s , it i s a r a t h e r r e c e n t r e s u l t t h e s e t t l e m e n t of t h e q u e s t i o n of e x i s t e n c e o f d i v i s i o n a l g e b r a s which a r e n o t n e c e s s a r i l y a s s o c i a t i v e . i n g methods of A l g e b r a i c Topology it w a s p r o v e d t h a t

mensional r e a l

S o by a p p l y -

t h e only f i n i t e d i -

( n o t n e c e s s a r i l y a s s o c i a t i v e ) d i a i s i o n aZgebras are those o f

aimension 1 , 2 , 4 , and 8 ( c f . , f o r i n s t a n c e , R. B O T T - J . lary I]).

(real)

MILNOR [I: p. 87, Corol-

On t h e o t h e r h a n d , i t seems t h a t t h e r e d o e s n o t e x i s t a s y e t

a n y " s i m p l e a l g e b r a i c " ( ! ) p r o o f o f t h e l a s t r e s u l t , namely one t o a v o i d t h e m a c h i n e r y of A l g e b r a i c Topology i n v o l v e d i n t h e p r o o f s e x i s t -

ed so f a r . Now, our fundamental r e s u l t i n t h i s s e c t i o n , i . e . , t h e Gel'fand-

Maziir Theorem ( t h e n e x t Theorem 5 . 2 a n d / o r i t s C o r o l l a r y 5 . 1 ) w i l l be c o n c l u d e d a s a d i r e c t a p p l i c a t i o n of t h e above Theorem 4 . 2 rollary 4.1).

and i t s C o -

Thus, w e h a v e .

Theorem 5.2. Every topological d i v i s i o n algebra E having a contintnous i n v e r s i o n and a totaZ s e t of continuous l i n e a r forms i s , w i t h i n a topological-algebraic isomorphism, t h e f i e l d o f complex numbers.

Proof. I t

s u f f i c e s t o p r o v e t h a t e v e r y e l e m e n t x i n E i s of t h e

11 SPECTRUM (LOCAL THEORY)

62

form

x =Ae,

(5.1)

f o r some X E C (where e d e n o t e s t h e i d e n t i t y e l e m e n t o f El. O t h e r w i s e , s u p p o s e t h a t 3: - X e # 0 , f o r e v e r y h e @ ; t h e n , s i n c e E i s by h y p o t h e s i s a d i v i s i o n a l g e b r a , o n e c o n c l u d e s t h a t S p E ( x l = Q , which i s a c o n t r a d i c t i o n by h y p o t h e s i s a n d Theorem 4 . 2 .

Thus, t h e r e l a t i o n ( 5 . 1 ) h o l d s

t r u e f o r e v e r y x i n E and s u i t a b l e X e C , i n o t h e r words, t h e r e l a t i o n (5.2)

X-Xe:C-+E,

d e f i n e s an onto map between t h e algebras C and E. Moreover, i t i s r e a d i l y s e e n ( E i s supposed t o be n o n - t r i v i a l )

t h a t (5.2) i s a continuous ( a l -

g e b r a ) isomorphism of t h e ( t o p o l o g i c a l ) a l g e b r a s C a n d E . Now, t h e i n v e r s e map o f

( 5 . 2 ) is continuous t o o . Indeed,

f o r every a > 0 , s i n c e

a e # O i n E , t h e r e e x i s t s a b a l a n c e d n e i g h b o r h o o d U of z e r o i n E , w i t h a e e U ( t h e t o p o l o g i c a l a l g e b r a E i s H a u s d o r f f , by h y p o t h e s i s ) . T h e r e f o r e , f o r e v e r y x = h e e U , one o b t a i n s have

01

01

a

5 1 , h e n c e -x = - * Xe = a e

x

t h e i n v e r s e map o f

x

I X ( < a, s i n c e o t h e r w i s e o n e would

e U , t h a t i s a c o n t r a d i c t i o n . Thus ,

( 5 . 2 ) i s c o n t i n u o u s a t O e E , h e n c e c o n t i n u o u s , and

t h i s t e r m i n a t e s t h e proof o f t h e theorem. I

Schol ium.of

(5.1); then,

The p r e c e d i n g p r o o f i s e s s e n t i a l l y t h e v e r i f i c a t i o n ( 5 . 2 ) i s , i n f a c t , a n a l g e b r a isomorphism o f C o n t o E

( w e assume t h a t e f 0 ) . T h u s , E i s a I - d i m e n s i o n a l

(Hausdorff topolo-

g i c a l ) v e c t o r s p a c e o v e r C , so t h a t it i s i n f a c t i s o m o q h i c t o C , as a t o -

pological v e c t o r space; i n d e e d , t h e l a s t p a r t of t h e above p r o o f w a s ess e n t i a l l y t h e v e r i f i c a t i o n o f t h i s f a c t . I n t h i s r e s p e c t , see a l s o , f o r i n s t a n c e , J . HORVhh! [l:p. 107, P r o p o s i t i o n 6 1 . NOW,

t h e f o l l o w i n g c o n c l u s i o n i s a d i r e c t c o n s e q u e n c e of t h e a-

bove Theorem 5.2 and C o r o l l a r y 4 . 1 .

Corollary 5.1. Every l o c a l l y convex d i v i s i o n algebra having a continuous i n -

.

version i s (algebra) isomorphic (hence, t o p o l o g i c a l l y isomorphic; c f t h e p r e c e d i n g Scholium) t o t h e f i e l d of complex numbers. In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e for every Locally m-convex d i v i s i o n algebra. I C o n c e r n i n g t h e c o n c l u s i o n o f t h e Gel’fand-Mazur

Theorem g i v e n by

t h e above Theorem 5 . 2 , w e remark t h a t t h i s r e m a i n s t r u e u n d e r t h e weak-

er h y p o t h e s i s t h e s p a c e E ’ ( t h e t o p o l o g i c a l d u a l o f E ) t o be n o n - t r i v i a l . I n f a c t , i n t h e p r e s e n c e of t h e rest c o n d i t i o n s f o r E , w e a c t u a l l y g e t a n e q u i v a l e n c e e x p r e s s e d by t h e f o l l o w i n g lemma. Moreover,

6. MAXIMAL IDEALS

63

it i s i n t h e l a t t e r form t h a t w e s h a l l e n c o u n t e r Theorem 5 . 2 i n s u b -

s e q u e n t s e c t i o n s . Thus, w e have. LemM 5.1.

L e t E b e a topological d i v i s i o n algebra. Then, t h e following as-

s e r t i o n s are equivalent 1 ) The topological dual E’ o f E i s n o n - t r i v i a l .

21 E admits a t o t a l s e t of continuous %inear forms ( c f . Scholium 4 . 1 ).

Proof. I t i s c l e a r t h a t 2 ) i m p l i e s 1 ) ( f o r e v e r y t o p o l o g i c a l v e c t o r s p a c e E ) . O n t h e o t h e r h a n d , it i s e a s i l y s e e n t h a t 1 ) i m p l i e s 2 ) b y t h e s e p a r a t e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E and t h e f a c t t h a t E i s a division algebra. I T h u s , a n e q u i v a l e n t f o r m u l a t i o n o f Theorem 5 . 2 i s t o s a y t h a t :

The only (complex) topologica2 d i v i s i o n algebra w i t h a continu

-

ous i n v e r s i o n and n o n - t r i v i a l topological dua% i s t h e complex nwnber f i e l d i t s e l f .

(5.3)

O r yet,

assuming t h a t by a

d i v i s i o n topological algebra w e s h a l l mean a

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 w i t h c o n t i n u o u s i n v e r s i o n , one c o n c l u d e s by t h e p r e v i o u s d i s c u s s i o n t h a t : A (complex) d i v i s i o n topological aZgebra is isomorphic t o C ( h e n -

c e , t o p o l o g i c a l l y i s o m o r p h i c ) i f , and only if, it possesses

(5.4)

a non-trivial

topological dual.

T h u s , some e x t r a c o n d i t i o n s a r e n e c e s s a r y , i n d e e d , t h a t a g i v e n (complex) topological d i v i s i o n algebra t o be t h e f i e l d C . In f a c t , t h e r e e x i s t d i v i s i o n t o p o l o g i c a l a l g e b r a s , i n t h e above s e n s e , t h a t a r e d i f f e r e n t from t h e complex number f i e l d i t s e l f . T h i s h a p p e n s , € o r i n s t a n ce, i n t h e (complex) a l g e b r a C t t l o f a l l r a t i o n a l f u n c t i o n s of one complex v a r i a b l e ; t h u s , t h e l a t t e r a l g e b r a s h o u l d a d m i t no n o n - t r i v i a l c o n t i n u o u s l i n e a r f o r m s ( c f . J.H.

WILLIAMSON [I : p. 7311 a n d / o r L . WAELBROECI:

[4:p. 134 f f . ] ) . 6. Maximal i d e a l s Given a n a l g e b r a E , a s u b s e t I of E i s s a i d t o be a l c f t ( r e s p . right) i d e a l of E l i f I i s a v e c t o r s u b s p a c e of E s u c h t h a t a x ( r e s p . x a ) b e l o n g s t o I , f o r a n y a e E and x el. Thus, I i s , i n p a r t i c u l a r , a suba l g e b r a of E s a t i s f y i n g a l s o t h e l a s t c o n d i t i o n . A s u b s e t of E which i s b o t h a l e f t and r i g h t i d e a l i s s a i d t o b e

a

2-sided i d e a l

of E .

I n a l l t h a t f o l l o w s by an i d e a l of an algebra E, We s h a l l always mean a

64

11 SPECTRUM (LOCAL THEORY)

2-sided i d e a l unless i t i s stated otherwise. Moreover, w e s h a l l a l w a y s unders t a n d , a s p a r t of t h e d e f i n i t i o n , t h a t a g i v e n i d e a l i n E w i l l b e a

proper non-trivial i d e a l , i.e.,

an i d e a l d i f f e r e n t from b o t h t h e t r i v i a l

i d e a l {Ol ( o r e l s e zero-ideal) The set of

and t h e whole a l g e b r a E .

( 2 - s i d e d ) i d e a l s o f E c a n be " p a r t i a l l y o r d e r e d " by

i n c l u s i o n , sc) t h a t by a mmimal ( 2 - s i d e d ) i d e a l w e s h a l l mean a n y maximal e l e m e n t of t h e l a t t e r s e t , i . e . ,

an element n o t g e n u i n l y c o n t a i n e d i n

any o t h e r element o f t h e s e t . Thus, on t h e b a s i s o f Lemma 6.1.

Zorn's Lemma

one now o b t a i n s .

Let E be an algebra w i t h an i d e n t i t y element. Then, every i d e a l

of E i s contained i n a maximal i d e a l .

Proof.

The set o f i d e a l s of E o r d e r e d by i n c l u s i o n i s a p a r t i a l -

l y o r d e r e d s e t , i n s u c h a way t h a t , f o r e v e r y chain (Le., t o t a l l y order-

ed s e t ) of i d e a l s o f E , t h e i r u n i o n i s e a s i l y s e e n t o b e a g a i n an i d e -

a l of E ; t h u s , t h e a s s e r t i o n f o l l o w s now by a n a p p l i c a t i o n of Z o r n ' s Lemma (see, f o r i n s t a n c e , N . BOURBAKI [l :Chap. 3 ; p. 3 5 , C o r o l l a i r e 11) I

.

On t h e o t h e r h a n d , t h e f o l l o w i n g p r o v i d e s a u s e f u l c r i t e r i o n t h a t a g i v e n i d e a l i n a commutative a l g e b r a t o b e maximal. Lemma 6.2.

Let E be a c o m t a t i v e algebra w i t h an i d e n t i t y element. Then,

the folloljing a s s e r t i o n s are equivalent: 1 ) M i s a maximal i d e a l of E. 2) The quotient algebra E/M i s a (conunutative) d i v i s i o n algebra (hence, a

f i e l d over 0 .

Proof. Suppose t h a t M i s a maximal i d e a l of E and l e t a € E l w i t h a $ M I s o t h a t a + M = [ a ] # 0 i n E / M . Moreover, i f I i s t h e ( 2 - s i d e d ) id e a l of E g e n e r a t e d by M U { a ) , one h a s M G I , p r o p e r l y , so t h a t I =E , hence e = m t a x , w i t h m E M and X E E . T h u s , by t a k i n g t h e l a s t r e l a t i o n "modulo M"

one o b t a i n s

[el = [ax! = [a]-[zJ ,

s o t h a t [ a ] i s an i n v e r t i b l e e l e m e n t i n E / M , t h a t i s 1) implies 2). On t h e i s a d i v i s i o n a l g e b r a and I i s a n i d e a l of E which

o t h e r h a n d , i f E/M

p r o p e r l y c o n t a i n s M I t h e n t h e r e e x i s t s a n element a E i , w i t h a#M.Thus, (6.1)

i s t r u e , f o r some x e E l so t h a t e - a x E M G I ; h e n c e , e e l s i n c e a E I ,

t h a t i s I = E which p r o v e s t h a t 2) implies 1 ) a s w e l l ,

and t h i s f i n i s h e s

the proof. NOW, given a c o m t a t i v e algebra E w i t h an i d e n t i t y element, an element x in E i s i n v e r t i b l e i f , and o n l y i f , it does not belong t o any i d e a l of E : The con-

6.

65

MAXIMAL IDEALS

f o r a n y a € E , t h e principal i d e a l (c)= { a x : x e B } i n E g e n e r a t e d by a c o n t a i n s a , f o r x = e l so t h a t

d i t i o n i s o b v i o u s l y n e c e s s a r y . Moreover,

by h y p o t h e s i s a must be i n v e r t i b l e , and t h i s p r o v e s t h e a s s e r t i o n . Thus, a s a c o n s e q u e n c e of t h e l a s t s t a t e m e n t and Lemma 5 . 1 ,

we

conclude t h a t :

An element x of a given c o m t a t i v e algebra E w i t h an i d e n t i t y element i s i n v e r t i b l e i f , and only i f , it i s not contained i n

(6.2)

any maximal i d e a l o f E . The l a s t p r o p o s i t i o n i s o f t e n a p p l i e d i n t h e s e q u e l . The p r e c e d i n g c a n b e e x t e n d e d t o a l g e b r a s which do Rot n e c e s s a r i l y have i d e n t i t y elements: Thus, by a regular i d e a l

o f a g i v e n a l g e b r a E w e s h a l l mean a

( 2 - s i d e d ) i d e a l I o f E i n s u c h a way t h a t t h e q u o t i e n t a l g e b r a E / I has an i d e n t i t y element. T h e r e f o r e , i f

[ a ] = a + I i s s u c h i n E/I, we s h a l l

u s u a l l y c a l l a E E an i d e n t i t y of E modulo I ; hence , one g e t s by d e f i n i t i o n

[.I

(6.3)

=

rxi.1.r

=

,

o r , equivalently,

ax-xeI

(6.41

and x a - x e I ,

f o r e ve ry element x i n E . N o w it i s c l e a r t h a t

i f the given algebra E has already an i d e n t i t y e -

lement, t h e n t h i s i s a n i d e n t i t y o f E modulo any ( 2 - s i d e d ) i d e a l o f E , h e n c e i n t h a t c a s e every i d e a l i n E i s regular. Thus, t h e d e s i r e d e x t e n s i o n of t h e p r e c e d i n g d i s c u s s i o n i s now subsummed i n t o t h e f o l l o w i n g . Lemma 6 . 3 . Let E be an algebra. Then, every regular i d e a l of E i s contained

i n a maximal regular i d e a l . In p a r t i c u l a r , a regular i d e a l M o f a c o m t a t i v e algebra E i s maximal i f , and only i f , E/M i s a l c o m t a t i v e l d i v i s i o n algebra (hence, i n p a r t i c u l a r , a f i e l d over C). Proof. Lemma 6 . 1 ,

The p r o o f o f t h e f i r s t a s s e r t i o n g o e s on s i m i l a r l y a s i n s i n c e t h e s e t of r e g u l a r i d e a l s o f E o r d e r e d by i n c l u s i o n

i s s t i l l a n " i n d u c t i v e o r d e r e d s e t " ( i . e . , e v e r y c h a i n i n t h e s e t ad-

m i t s an u p p e r bound; c f ; N . BOURBAKI [l: Chap. 3 ; p. 34, § 4]),

so t h a t t h e

a s s e r t i o n f o l l o w s a g a i n by Z o r n ' s Lemma. On t h e o t h e r h a n d , t h e

se-

cond p a r t of t h e lemma f o l l o w s by a p p l y i n g a s i m i l a r argument a s i n t h e proof o f Lemma 6.2,

t a k i n g t h u s a s e i n ( 6 . 1 ) a n i d e n t i t y of

E

modulo M . I F u r t h e r m o r e , i n a n a l o g y w i t h t h e comment f o l l o w i n g Lemma 6 . 2 ,

66

I1 SPECTRUM (LOCAL THEORY)

one g e t s t h e f o l l o w i n g r e s u l t . An “ a n a l y t i c ” form of it ( i n p a r t i c u -

l a r , of t h e n e x t C o r o l l a r y 6 . 1 ) , and of t h o s e r e m a r k s t o o , w i l l be c o n s i d e r e d i n s u b s e q u e n t s e c t i o n s (see C h a p t . 111).

Lemma 6.4. Let E be a commutative algebra. Then an element x of E i s quasisingular i f , and only i f , t h e following subset of E , (6.5)

I = { ~ - x Y :y e E 1

i s an i d e a l . Moreover, I i s , in f a c t , a regular i d e a l o f E not containing x, t h e l a t t e r being an i d e n t i t y of E modulo I.

Proof. I f I = E , t h e n by

X‘E

E is quasi-singular,

(6.5) x o y = O ,

t h e n I #E . T r u e , b e c a u s e

,if

f o r some y f E which i s a c o n t r a d i c t i o n .

T h u s , i f I f E , one e a s i l y p r o v e s t h a t I i s , i n f a c t , a r e g u l a r i d e a l of E w h i l e t h e g i v e n e l e m e n t x i s a n i d e n t i t y of E modulo I . M o r e o v e r , x & I , because, i f x e I

I , f o r e-

t h e n by ( 6 . 5 ) o n e g e t s y = ( y - x y l + x y

v e r y y e E , so I = E which i s a g a i n a c o n t r a d i c t i o n t o o u r h y p o t h e s i s . On t h e o t h e r h a n d , i f x e E i s q u a s i - r e g u l a r ,

t h e n it i s e a s i l y proved

b y ( 6 . 5 ) t h a t I = E , which i s s t i l l a c o n t r a d i c t i o n i f w e assume t h a t I i s an i d e a l , and t h i s f i n i s h e s t h e p r o o f . I

Now, it i s c l e a r b y t h e p r e c e d i n g p r o o f t h a t Lemma 6 . 4

is still

v a l i d € o r a non-commutative a l g e b r a E , b y t a k i n g i n s t e a d t h e e l e m e n t

a e E t o be r i g h t ( r e s p . l e f t ) q u a s i - s i n g u l a r ;

then, I is a r i g h t o r

l e f t i d e a l i n E , r e s p e c t i v e l y . F o r s i m p l i c i t y ’ s s a k e , however, w e a s sumed F commutative. W e c o n c l u d e w i t h t h e f o l l o w i n g .

Corollary 6.1. Let E be a c o m t a t i v e algebra.Then, an element

3:

in E

is

quasi-regular i f , and only i f , f o r every niazimal i ~ p t ? iud e~a l M o f E , there e x i s t s an element y e E i n such a way t h a t

x + y-xy = x

(6.6)

0

Proof. I f x e E i s q u a s i - r e g u l a r ,

y

E M.

t h e n x o y = O E M , f o r some y € E ,

and f o r e v e r y (maximal r e g u l a r ) i d e a l M of E . C o n v e r s e l y , i f x e E i s quasi-singular,

t h e n t h e i d e a l I of E g i v e n by ( 6 . 4 )

i s r e g u l a r , so

that(Lemma 6 . 3 ) , it i s c o n t a i n e d i n a maximal r e g u l a r i d e a l , s a y M , o f E . T h e r e f o r e , t h e r e e x i s t s by h y p o t h e s i s y E E , w i t h x+ y - x y e M I so t h a t s i n c e by ( 6 . 4 ) y - x y e I ~ M , f o r e v e r y ~ E E one , concludes t h a t x M ; thus, for

every y e E ,

one h a s y = ( y

-

€

x y l + x y e M , t h a t i s M = E . Name-

l y , a c o n t r a d i c t i o n and t h i s f i n i s h e s t h e p r o o f .

I

W e c o n s i d e r now, and t h i s i s of c o u r s e o u r c o n c e r n , maximal ide-

a l s i n t o p o l o g i c a l a l g e b r a s (see a l s o t h e n e x t s e c t i o n ) . Thus,we know

7.

61

CHARACTERS

f o r i n s t a n c e t h a t e v e r y maximal i d e a l i n any Banach a l g e b r a i s c l o s e d . On t h e o t h e r h a n d , a s w e p r e s e n t l y see, t h i s f a c t i s s t i l l v a l i d i n t h e more g e n e r a l c l a s s of 4-(topologica1)algebras. T h a t i s , w e h a v e .

Theorem 6.1. Let E be a &-algebra. Then, every mazima2 regu2ar i d e a l of E i s closed. In particu2ar, every maximal i d e a l o f a Q-a2gebra with an i d e n t i t y element i s closed. C o n s i d e r a maximal r e g u l a r i d e a l M of E and suppose t h a t

Proof. -

M =E.

Thus, i f a i s a n i d e n t i t y of E modulo M and Gq t h e s e t of q u a s i -

r e g u l a r e l e m e n t s of E , t h e s e t a + G q i s a n ( o p e n ) neighborhood of a i n E (Lemma I ; 6.4) ; h e n c e , one o b t a i n s ia+Gq)

that is, a

-m

n

M

i0

,

f o r some e l e m e n t m E M . T h e r e f o r e , i f y e E

E Gq,

quasi-inverse of a

is the

one g e t s

-in,

la-rnioy= (a-m)+y- ia-m)y=O that is, a =rn+(ay- y l hypothesis f o r M .

-

M , that

-

my,

t h u s a E M , which i s a c o n t r a d i c t i o n t o t h e

Thus, M f E , so t h a t ( C o r o l l a r y I ; 1.1)

we obtain M =

i s t h e a s s e r t i o n , w h i l e t h e r e s t of t h e s t a t e m e n t i s c e r t a i n -

l y clear. I On t h e o t h e r hand, t h e s i t u a t i o n d e s c r i b e d by t h e above theorem

may be d i f f e r e n t f o r an a r b i t r a r y t o p o l o g i c a l a l g e b r a ( e v e n a l o c a l l y ri-convex

one) which i s n o t a & - a l g e b r a ; namely, t h e r e may e x i s t maxi-

m a l i d e a l s which a r e dense s u b s e t s of t h e g i v e n t o p o l o g i c a l a l g e b r a . Whether t h i s may n o t happen in case of a commutative Frgchet l o c a l l y in-eonvex

algebra seems t o be s t i l l an open q u e s t i o n (Michae2’s conjeetttre; see T . HUSAIN[2]). In t h i s respect, c f . a l s o t h e next section, i n p a r t i c u l a r ,

Theorem 1 . 1 .

7. Characters. Closed maximal ideals The s i t u a t i o n d e s c r i b e d b y Lemma 6 . 2 of t h e p r e v i o u s s e c t i o n i n c o n n e c t i o n w i t h t h e Gel’fand-Mazur Theorem ( c f . Theorem 5 . 2 o r i t s l e s s t e c h n i c a l form g i v e n by C o r o l l a r y 5 . 1 ) ,

leads u s t o our next o b j e c t i -

v e ; namely, t h e r e l a t i o n between maximal i d e a l s of a g i v e n t o p o l o g i c a l a l g e b r a E and (complex) a l g e b r a morphisms o f E i n t o t h e complexes, i n p a r t i c u l a r , c o n t i n u o u s o n e s . Thus, we s t a r t w i t h t h e f o l l o w i n g .

Definition 7.1. mean

a non-zero

L e t E be an a l g e b r a . Then, by a

character of E , we

(complex) morphism of E i n t o t h e ( a l g e b r a o f ) comple-

x e s C . The s e t of c h a r a c t e r s of E i s c a l l e d t h e algebraic speetiwv

of E

11 SPECTRUM (LOCAL THEORY)

68

and denoted by M ( E I . ( I t i s o n l y i n Chapt. V where w e c o n s i d e r

M(EI, i n

f a c t , a subspace of it a s an a p p r o p r i a t e t o p o l o g i c a l s p a c e ) . F u r t h e r more, t h e s e t M(E)+ = M(E)U {

(7.1)

u I,

i s c a l l e d t h e extended algebraic spectruv of t h e a l g e b r a E .

In t h i s respect, w e note t h a t

M(EJ may be t h e empty

s e t , even

i f L" i s a Banach a l g e b r a . However, t h e s i t u a t i o n i s q u i t e d i f f e r e n t , of c o u r s e , f o r a commutative Banach a l g e b r a ( i n t h e p r e s e n c e of a n i -

d e n t i t y e l e m e n t ) ; a s w e s h a l l see ( c f . Chapt. V ) t h i s i s s t i l l v a l i d f o r much more g e n e r a l (commutative!) t o p o l o g i c a l a l g e b r a s . Now, i f f i s a c h a r a c t e r of a g i v e n a l g e b r a E one h a s by d e f i n i t i o n the r e l a t i o n flxyl = f(xi.f(y),

(7.2)

f c r any x , y i n E l f b e i n g a l s o a (complex) l i n e a r form on t h e r e s p e c t i v e v e c t o r s p a c e E ; moreover, t h e r e i s an e l e m e n t x i n E , w i t h f i x ) # 0 . In p a r t i c u l a r ,

i f t h e a l g e b r a E has an i d e n t i t y element, say e , s a y -

i n g t h a t f i s " n o n - i d e n t i c a l l y z e r o " i s , of c o u r s e , e q u i v a l e n t w i t h f ( e l = l i n @,

thus f i s then " i d e n t i t y preserving".

On t h e o t h e r hand, one c o n c l u d e s t h a t every f e M ( E ) i s e s s e n t i a l l y a map of E onto C . I n d e e d , i t s u f f i c e s t o c o n s i d e r , f o r e v e r y X E C,

x

element - x

E E , where

f(x) =

c1

the

# 0 i n C by h y p o t h e s i s f o r f . (So one ob-

t a i n s , i n f a c t , t h a t e v e r y non-zero

(complex) l i n e a r form o n E i s an

o n t o map). Thus, our f i r s t statement r u n s a s follows.

Lemma 7.1.

Let E be an algebra and f a character of E . Then, the kernet of 5

the set

i.e.,

k e r ( f ) = C x e E : f ( x l = 01,

(7.3) ?:B

a

2-sided regular ninceimat i d e a t of E.

Proof. By ( 7 . 3 ) and t h e h y p o t h e s i s f o r f one c o n c l u d e s t h a t ker If) i s a 2 - s i d e d i d e a l of E . Moreover,

S E

if

f ( a ) = X # O i n C,

f o r some a

1

E l t h e e l e m e n t ~a E E i s an i d e n t i t y of E modulo ker if), so t h a t t h e l a t t e r i s a r e g u l a r i d e a l of E. On t h e o t h e r hand, s i n c e f i s a (comE

p l e x ) l i n e a r form on E , i t s k e r n e l

E of ximal be

codimension 1 (i. e . ,

( 7 . 3 ) w i l l be a v e c t o r subspace of

dim (Elker(f)l= I ) , h e n c e , e q u i v a l e n t l y , a ma-

v e c t o r subspace of E ; s o s i n c e ker (f) i s an i d e a l of E l it w i l l

t h e n a maximal i d e a l of E l and t h i s p r o v e s t h e a s s e r t i 0 n . I

7.

Remark.-

69

CHARACTERS

C o n c e r n i n g t h e l a s t p a r t of t h e above p r o o f , w e c o u l d

a p p l y , o f c o u r s e , a more e l e m e n t a r y ( h o w e v e r , n o t s i m p l e r ! ) argument t o p r o v e t h a t t h e regular i d e a l of E defined by (7.31 i s maximal.

Thus, sup-

p o s e t h a t I i s a n i d e a l o f E which p r o p e r l y c o n t a i n s k e r ( f I ; t h e r e f o -

re, t h e r e e x i s t s x e I , w i t h f ( x l $ 0.

T1a e E

of

t h e preceding proof

s i n c e f l u l = l , one o b t a i n s s i n c e f ( x ) # 0, E

x-f(x)u

E

k e r ( f l C I . Thus, f l x l u e I so t h a t ,

one c o n c l u d e s t h a t u e I , t h a t i s , a c o n t r a d i c t i o n t o

the hypothesis f o r I .

lement u

Hence, by u s i n g t h e e l e m e n t u =

( i . e . , an i d e n t i t y of E modulo k e r i f l ) ,

( I n t h i s r e s p e c t , w e u s e d t h e f a c t t h a t i f a n e-

E: i s an i d e n t i t y of E modulo an i d e a l I of E , then u

E

I , unless I = E .

Moreover, u i s a l s o an i d e n t i t y o f E modulo any other idea2 J of E containing I ) . T h u s , d e n o t i n g by M l E l t.he s e t of 2 - s i d e d r e g u l a r maximal ideals o f an a l g e b r a E , one g e t s by Lemma 7 . 1 t h e f o l l o w i n g map

9:

(7.4) Now,

M(EI-MIEI:

f w @(f/:= ker(fI.

t h e n e x t s t a t e m e n t s a y s t h a t t h e map 9 i c , i n f m t , one-to-one.

Namely, w e h a v e . Lemma 7.2.

Let E be an a2gehra and f , g any

TWO

characters of E. Then, f = g

if, and o n l y i f . k e r ( f l = kerigl. I n p a r t i c u z a r , t h e map @ defined by ( 7 . 4 ) i s an

injection.

Proof. Of c o u r s e , it s u f f i c e s t o p r o v e o n l y t h e " i f " p a r t of t h e a s s e r t i o n . Thus, i f a e E then f ( a ) =g(al - 1

,

i s an i d e n t i t y of E modulo ker(fi = k e r , ' g I ,

b e c a u s e by h y p o t h e s i s f, g a r e n o n - i d e n t i c a l l y z e r o .

Hence, one h a s

x - f ( x ) aF k e r i f ) = kerig) ,

(7.5) f o r every x e E

,

s o t h a t gix) = f f x ) g ( a l , i.e. ,f (x)= g ( x ) , f o r e v e r y x

E

E

,

and t h i s p r o v e s t h e a s s e r t i o n . I W e c o n s i d e r n e x t continuous characters of a g i v e n t o p o l o g i c a l a l -

g e b r a , i n p a r t i c u l a r , a s it c o n c e r n s t h e i r r e l a t i o n t o t h e ( c l o s e d reg u l a r ) maximal i d e a l s of t h e a l g e b r a ; t h e i r i n t e r r e l a t i o n i s e s t a b l i s h e d by a r e s t r i c t i o n o f t h e map ( 7 , 4 ) ( s e e Theorem 7 . 1 b e l o w ) . Thus, w e f i r s t have t h e f o l l o w i n g . Definition 7.2.

L e t E b e a g i v e n t o p o l o g i c a l a l g e b r a . Then by t h e

( o r s i m p l y spectrum) o f E l w e s h a l l mean t h e s e t of c o n t i n u o u s c h a r a c t e r s o f E , d e n o t e d by m l E i . (Thus an e l e m e n t f e mfE) topo2ogicaZ s p e c t m

w i l l b e a non-zero c o n t i n u o u s ( c o m p l e x ) morphism of E i n t o t h e ( a l g e b r a C of t h e ) complexes).

I1 SPECTRUM (LOCAL THEORY)

70

I t i s c l e a r t h a t a s i m i l a r comment t o t h a t f o l l o w i n g D e f i n i t i o n

7 . 1 i s h e r e i n o r d e r . We a l s o d e f e r f o r C h a p t e r V t o c o n s i d e r m ( E )

a s a s u i t a b l e t o p o l o g i c a l s p a c e , r e t a i n i n g however t h e t e r m i n o l o g y of Moreover, s i m i l a r l y t o ( 7 . 1 ) , w e s t i l l con-

t h e above D e f i n i t i o n 7 . 2 . sider the set

V"Y(El+ =

(7.6)

m(E.,u( 0 3 , ( o r s i m p l y extended spec -

which w e c a l l t h e extended topoZogica1 spectrum

tmun) of t h e t o p o l o g i c a l a l g e b r a E . Thus, w e c o n c l u d e t h a t ; t h e r e s t r i c t i o n o f t h e map 1 7 . 4 ) t o m ( E ) C M ( E ) d e f i n e s a oneto-one correspondence between continuous characters of E ( i . e . ,

(7.7)

e l e m e n t s of 97UE)) and closed regular maximal

( 2 - s i d e d ) i-

deals o f E . W e d e n o t e t h e l a s t s e t i n ( 7 . 7 ) by h4iE). Thus, it i s c l e a r t h a t , f o r e v e r y f e m ( E ) , one h a s kerifi e M ( E ) G M ( E ) , a s f o l l o w s from Lemma 7.1

and

t h e c o n t i n u i t y of f. Now,

it i s o u r main o b j e c t i v e i n t h e e n s u i n g d i s c u s s i o n t o en-

s u r e , under s u i t a b l e r e s t r i c t i o n s

on t h e p a r t i c u l a r t o p o l o g i c a l a l g e -

b r a E c o n s i d e r e d , t h a t t h e above c o r r e s p o n d e n c e ( 7 . 7 ) i s , i n f a c t , a b i j e c t i o n (see Theorem 7 . 1 and i t s C o r o l l a r y 7 . 1 .

On t h e o t h e r h a n d ,

t h e n e x t c h a p t e r p r o v i d e s a more g e n e r a l framework f o r t h e t o p o l o g i c a l algebras considered h e r e ) . W e n e e d f i r s t some more p r e l i m i n a r y m a t e r i a l . Th.us, s u p p o s e t h a t E i s a t o p o l o g i c a l a l g e b r a and I

a ( 2 - s i d e d ) closed i d e a l o f E . S o t h e

q u o t i e n t a l g e b r a E/I, b e i n g a H a u s d o r f f t o p o l o g i c a l v e c t o r s p a c e , i s

also a t o p o l o g i c a l a l g e b r a ; i . e . , t h e m u l t i p l i c a t i o n i n E / I i s separately continuous. I n d e e d , s i n c e t h e t o p o l o g y i n E / I

is a " f i n a l ( v e c t o r space)

t o p o l o g y " w i t h r e s p e c t t o t h e c a n o n i c a l ( q u o t i e n t ) map @ : E - + E / I , one h a s t o p r o v e t h e c o n t i n u i t y o f t h e map l a o @ : E+

E/I

,

f o r every a E E . Therefore, s i n c e

l.(@(x)) = &-@(xc)= @(a).@(xc) = @(ax) = @ ( l a ( x ) ) , with

x e E , one o b t a i n s la o @ = @ o La : E-+

(7.8) where

@ o l a , with a e E ,

E/I ,

i s c e r t a i n l y c o n t i n u o u s , and t h i s p r o v e s t h e

a s s e r t i o n . ( I n t h i s r e s p e c t , see a l s o C h a p t . I V ; S e c t i o n 2 ) . I n p a r t i c u l a r , i f E i s a l o c a l l y convex a l g e b r a and I a s b e f o r e ,

7.

CHARACTERS

71

t h e n one p r o v e s t h a t E/I i s a l o c a l l y c o n v e x a l g e b r a t o o , by a p p l y i n g a s i m i l a r argument to t h e above c o n c e r n i n g t h e ( q u o t i e n t ) l o c a l l y convex s p a c e E I I . F u r t h e r m o r e ,

i f E i s a l o c a l l y m-convex algebvaa, t h e n

this i s

s t i l l t r u e f o r a quotient algebra E / I : T h i s i s e a s i l y s e e n , s i n c e (7.9) with k = [x] e E / I ,

y i e l d s a s u b m u l t i p l i c a t i v e semi-norm on E / I , f o r any

s u c h p on E ( c f . a l s o Theorem 1;3.1). T h u s , w e now h a v e t h e f o l l o w i n g .

Lemma 7.3. Let E be a topological algebra with a continuous quasi-inversion and I a closed (2-sided) ideal of E. Then, the quotient algebra E / I i s a topologic a l algebra w i t h a continuous quasi-inversion t o o .

Proof. By t h e p r e c e d i n g E/I i s a t o p o l o g i c a l a l g e b r a . Moreover, i f x++x'denotes

t h e quasi-inversion

i n E , t h e n t h e c o n t i n u i t y of t h e

r e s p e c t i v e map i n E / I i s e q u i v a l e n t w i t h t h a t o f t h e map x -

[x']:E+E/I.

T h i s i s e a s i l y s e e n b y h y p o t h e s i s f o r E and t h e c o n t i n u i t y of t h e quot i e n t map @ : B + E / I ,

s i n c e @ i s a n a l g e b r a morphism p r e s e r v i n g

the

circle operation. I We a r e now i n t h e p o s i t i o n t o s t a t e t h e f i n a l b a s i c r e s u l t o f t h i s s e c t i o n . Namely, w e h a v e .

Theorem 7.1. Let E be a commutative l o c a l l y convex algebra w i t h a continuous quasi-inversion and M a closed regular maximal i d e a l of E . Then, the quotient algebra E/M i s the algebra C o f the complexes, w i t h i n a topological algebra isoniorphism. I n p a r t i c u l a r , the conclusion holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra E .

Proof. By Lemma 7 . 3 t h e q u o t i e n t a l g e b r a E/M i s a l o c a l l y convex algebra with a continuous quasi-inversion,

r e s p . , a l o c a l l y in-convex

a l g e b r a ( c f . a l s o ( 7 . 9 ) ) . Moreover, by h y p o t h e s i s f o r M a n d L e m m a 6 . 3 ,

E/M i s a d i v i s i o n a l g e b r a , s o t h a t B/M i s , i n f a c t , a (commutative) l o c a l l y convex d i v i s i o n algebra w i t h a continuous inversion

(the l a s t assertion is

e a s i l y d e r i v e d by Lemma 7 . 3 and ( 1 . 4 ) 1 , r e s p . , a ZocaZZy m-emvex d i v i s i o n

algebra.

S o t h e a s s e r t i o n f o l l o w s now f r o m an a p p l i c a t i o n of t h e G e l ' -

fand-Mazur Theorem ( s e e C o r o l l a r y 5 . 1 ) . I

Corollary 7.1. Let

E be a commutative l o c a l l y convex algebra w i t h a continu-

ous quasi-inversion, i n p a r t i c u l a r ( c f . Lemma 3 . 1 ) , any c o m t a t i v e l o c a l l y mconvex algebra. Then, every closed regular maximal i d e a l M of E gives r i s e t o a continuous character f of E ; namely, t h a t one defined by t h e following commutative d i -

72

I1 SPECTRUM (LOCAL THEORY)

agramm :

(7.10)

iiere j denotes t h e topological algebra isomorphism given by t h e above Theorem 7 . 1 . 1

F i n a l l y , one g e t s t h e f o l l o w i n g u s e f u l r e s u l t .

I t is mostly i n

t h i s form t h a t w e a r e g o i n g t o a p p l y Theorem 7 . 1 i n t h e s e q u e l . T h a t i s , we have.

Corollary 7.2. Let E be a c o m u t a t i v e localZy convex aZgebra w i t h a continuous quasi-inversion.

Then, t h e r e e x i s t s a one-to-one

and onto correspondence be -

tween t h e s e t o f continuous characters o f E ( i . e . , t h e spectrwn m(E)o f t h e algebra El and t h e s e t M(EI of cZosed regular maximal ideaZs o f E, given by t h e r e l a t i o n ( 7 . 4 1 . I n p a r t i c u l a r , t h i s i s t m e f o r every l o c a l l y m-convex algebra, so t h a t i n t h i s case one has t h e r e l a t i o n m(E)= i/d(El ,

(7.11)

within a bijection. Proof.

I t i s c l e a r ( c f . Lemma 7 . 2 )

i n d e e d one-to-one. f

&

t h a t t h e map i n q u e s t i o n i s

F u r t h e r m o r e , i f MeM:i’)

m(Eiw i t h k e r ( f ) = M , a s follows f r o m t h i s finishes the proof.1

t h e n by ( 7 . 1 0 ) o n e g e t s t h e r e l a t i o n f = jo@ i n ( 7 .

l o ) , and

On t h e o t h e r hahd, a u s e f u l c o n s e q u e n c e o f t h e p r e c e d i n g , i n c o n j u n c t i o n w i t h Theorem 6 . 1 ,

is t h e following r e s u l t extending t h e

s i t u a t i o n one h a s i n ( c o m m u t a t i v e ) Banach a l g e b r a s .

Corollary 7.3. L e t E be a &-algebra. Then, every character of

E is continu-

ous, i.e., one has

M(E) = m ( E l C E’,

(7.12)

where E’ denotes t h e topoZogicaZ dual o f E. In p a r t i c u l a r , if E i s a c o m t a t i v e Zocally convex Q-aZgebra w i t h a c o n t i nuous quasi-inversion, o r y e t any commutative Locally m-convex Q-algebra ( h e n c e , a

l o c a l l y convex i r e s p . l o c a l l y m-convex.) generazized Waelbroeck aZgebra ( S c h o l i u m then one o b t a i n s

4.2)),

M(EJ = MIE!

(7.13)

where

‘1

I

”

Pi-ocf.

‘m(El= M ( E 1 ,

stands f o r a b i j e c t i o n . I f f i s a c h a r a c t e r of E , t h e n i t s k e r n e l , b e i n g a r e g u l a r

7.

CHARACTERS

maximal ( 2 - s i d e d ) i d e a l of E: (Lemma 7 . 1 ) c l o s e d s u b s p a c e of E .

In particular,

73

,

i s a l s o by Theorem 6 . 1 a

ker(fI

i s a c l o s e d ( v e c t o r ) sub-

s p a c e of E of c o d i m e n s i o n 1 , so t h a t f i s a c o n t i n u o u s l i n e a r form on E

( c f . N . HORVhH [I: p. 107, P r o p o s i t i o n 71 )

,

h e n c e , i n p a r t i c u l a r , a con-

t i n u o u s c h a r a c t e r of E which p r o v e s ( 7 . 1 2 ) . Now, t h e r e s t o f t h e a s s e r t i o n i s a g a i n a c o n s e q u e n c e of Theorem 6 . 1 ,

Corollary 7 . 2 ,

and of

( 7 . 1 2 1 , and t h i s t e r m i n a t e s t h e p r o o f . I Scholium 7.1.

The h y p o t h e s i s i n t h e f i r s t p a r t of t h e p r e v i o u s co-

r o l l a r y t h a t E i s a 4 - a l g e b r a was made o n l y i n c o n n e c t i o n w i t h Theorem

6.1.

Thus, more g e n e r a l l y , t h e f o l l o w i n g i s t r u e : For every t o p o l o g i c a l

algebra E w i t h t h e p r o p e r t y t h a t every r e g u l a r maximal ( 2 - s i d e d ) i d e a l i n E is c l o s e d , one concludes t h a t every character o f E i s continuous. I n o t h e r words, i.he r e l a t i o n M ( E ) = M I E ) i m p l i e s MIEI = m ( E I . F o r c o n v e n i e n c e w e have u s e d i n s t e a d t h e s t r o n g e r , however l e s s t e c h n i c a l , h y p o t h e s i s t h a t t h e a l g e b r a E i s a &-algebra; t h i s w i l l be a l s o t h e case i n several instances below, w h i l e t h e more g e n e r a l s i t u a t i o n d i s c u s s e d h e r e w i l l be a l w a y s c l e a r from t h e c o n t e x t .

W e c o n c l u d e t h e s e c t i o n and t h i s c h a p t e r t o o w i t h c e r t a i n comments and some ( p a r t i a l ) r e s u l t s r e f e r r i n g t o Q - a l g e b r a s ; more p r e c i s e l y , t h i s h a s t o do w i t h a c e r t a i n p a r t i c u l a r n o t i o n , t h e Gel’fand

transform,

which a l s o w e a r e g o i n g t o a p p l y s e v e r a l t i m e s i n s u b s e q u e n t

c h a p t e r s of t h i s book. Thus, w e f i r s t have t h e n e x t .

Definition 7.3. L e t E b e a n a l g e b r a a n d M I E ) i t s a l g e b r a i c s p e c t r u m (i.e., t h e s e t of c h a r a c t e r s of E ; c f . D e f i n i t i o n 7 . 1 ) .

l e t x b e a n e l e m e n t of E. Then, t h e

Moreover,

Gel’fand transform o f x , d e n o t e d by

3 , i s d e f i n e d t o b e t h e map

2 : M(EI * C : f ct 2 i f I : = f 1x1 .

(7.14)

Thus, i f

F ( M I E 1 , CI d e n o t e s t h e s e t o f a l l complex-valued maps on

M I E ) , t h e r e s u l t i n g map

(7.15)

:E

+

F ( M ( E I , C I : x -%(xi

( g i v e n by ( 7 . 1 4 ) ) i s c a l l e d t h e

:=

2

Gel’fand map of t h e g i v e n a l g e b r a E.

I f E i s a topological algebra, the preceding terminology i s , i n particular,

r e s e r v e d f o r t h e r e s t r i c t i o n of

(7.15) to

t h e ( t o p o l o g i c a l ) s p e c t r u m of E ( s e e D e f i n i t i o n 7 . 2 ,

7 7 U E ) G M(E)

,

and C h a p t . V ) .

On t h e o t h e r h a n d , i f w e c o n s i d e r F ( M ( E I , C 1 a s a (complex) a l g e -

11 SPECTRUM ( L O C ~THEORY)

74

bra, with pointwise defined operations, we readily see by the preceding relations that t h e Gel'fand map o f an algebra E i s an algebra rnorphism of E into FfM(EI, C).

Thus, the relation

I~~S)CFIM(EI,CI

E* =

(7.16)

defines E A as a subalgebra of F I M f E I , C I , called the Gel'fand transform algebra of E .

NOW, if we are given a topological algebra E l the respective Gel'fand transform algebra E* consists, in effect, of (complex-valued) continuous functions on mIEI, the latter set being suitably topologized; as we shall see (cf. Chapt. V), one at least natural topology, we usually consider on T I T l E I , is the weakest one making the elements of E ^ continuous functions ( G e l ' f a n d t o p o l o g y ; but cf. also Chapt. IX) . Conclusively, we just apply (7.15) to derive some useful relations concerning the spectrum SpE (x) of an element x in a given algebra E. But first we need the next lemma.(A form of a converse is also valid but deferred for Chapt. 111; see there Theorem 5.1). So we have.

Lemma 7.4. Let E be an aZgebra and x a quasi-regular element o f E. Then, one has t h e reZation f(xl # 1 ,

(7.17) f o r every character f of E.

Proof. If x o y = 0, for some y e E l

(7.18)

one gets

f ( x o y I = f ( x +y - x y I = f f x I + f f y I - f f x I f ( y I = O ,

for every f e M f E I . Thus, if f ( x l = l , for some f e M f E J , one gets by (7.18) I=O,i.e., a contradiction, and this proves the assertion. 1 Thus, a useful consequence of the previous lemma is now the following result, a "continuous analogon" of which will be given lateron (see Chapt. 111; Theorem 6.2, or even Corollary 7.5 below). That is, we have.

Corollary 7.4. L e t form

2

E b e an aZgebra and

R(O/ t h e range o f t h e Gel'fand trans-

o f an element x in E (cf. (7.15)). Then, one has

(7.19)

R(2)L S p E ( x c ) .

I n p a r t i c u l a r , one has t h e r e l a t i o n

(7.20)

~ ( M ( E I += I spE(x)

u {OI,

f o r every x i n E. Furthermore, one o b t a i n s

(7.21)

r E fxl =

sup J j Y x I J ,

S EM I E I

7. CHARACTERS

75

f o r every x i n E. Proof. If fixl=O, for some f EM~EI, then x can not be a regular element of E, so that (Definition l .2) 0 e SpElxl. On the other hand, if ?if) = ffxl= X f ; O in C , where f E MlEl, then one has f ( 1T z l = 1 ; hence (Lem1 should be a quasi-singular element of E, so A f SpE(xl and ma 7.4) , h x this proves (7.19). NOW if A E SpElxl , with A # 0, then (cf. Definition 1

1.2) - x

x

must be quasi-singular, so that (Lemma 7.4) there exists f 1

e MfEI, with f ( - xx ) = l , thus f l x l = ~ i f i = X ;hence, X E R i ; ) , finally obtains the relation (7.22)

spE(x)

so that one

u I O I = R( 2 i u {OI = ;IMIEI +i ,

which together with (7.19) proves (7.20). Finally (7.21) is now immediate by (1.10) and (7.20). I

Now the following proposition is still another application of the above Lemma 7.4 and will repeatedly be used in subsequent chapters.

Proposition 7.1. Let E be a Q-algebra. Then, i t s ( t o p o l o g i c a l ) spectmm mfEl i s an equicontinuous subset of E' ( t h e topologicaZ dual of E l and hence r e l a t i v e l y compact i n E,' (the weak topological dual of E ; see Chapt. V )

.

If U is a balanced neighborhood of O E E consisting of quasi-regular elements of E (Lemma I ; 6.4), then one has t h e r e l a t i o n Proof.

UE. f?7YfEllo,

(7.23)

where the second member of the last relation denotes the polar s e t of mrElr E'in

E: Indeed, if -3 E U and Iffx)I 2 1 , f o r some f €?;rZ(E), then 1 1 1 ffx) = A # 0 and 1-ix < 1. Thus -AX E U and f ( T z l = 1 , a contradiction ::i.nce --31 x is quasi-regular (Lemma 7.4). Therefore, one has ( f f d 51

(7.24)

,

for every f f 772(E/ and x E U which proves (7.23). Hence, (rrZ(E)IO is a neighborhood of 0 e E , therefore 772: f E l C ( m(El! O 0 an equicontinuous subset of E'. I Finally, we do have the next "continuous analogon" of Corollary 7.4, a stronger version of which will be derived in Chapter 111, just when the necessary preliminary material will be available. Thus, we have. Theorem 7 . 2 . Let E be a c o m t a t i v e localZy convex &-algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n ( i . e . , a commutative l o c a l l y convex WaeZbroeck

.

algebra; cf Definition 4.1). Then, for every element x i n E, one g e t s t h e r e l a -

11

76

SPECTRUM (LOCAL THEORY)

t i o n (cf. also the notation of Corollary 7.4)

R ( G I = 3 ( r n t E / ) = SPE(Xi.

(7.25)

In p a r t i c u l a r , one g e t s (7.26) Hence, the a s s e r t i o n holds t r u e f o r every commutative l o c a l l y m-convex 4-algebra with an i d e n t i t y element ( i . e . , a commutative l o c a l l y m-convex Waelbroeck algebra). I n t h i s r e s p e c t , if t h e given algebra E does not have an i d e n t i t y , one g e t s

S(7^n(EIf) = SpE(xl,

(7.27)

for every element x in E, while the r e l a t i o n ( 7 . 2 6 ) i s s t i l l t r u e . Boof.

By hypothesis and (7.12) one has M(EI = ??Z ( E l ,

so that by (7.19) R(;)rSpE(x), for every X E E. Moreover, for every h e SpElx), with h # 0, one concludes, based on the proof of (7.20), that h e R ( G ) (see also the comment after Definition 1.2). Thus, suppose that 0 e SpE(x:); then (Definition l.l), x can not be aregular element of E , so that x E M , for some maximal ideal M in E (cf.(6.2)). Hence, by (7.4) and (7.13) one obtains f(x1 =G(f) = O , for some f € MIE) = m(El, and this proves (7.25). Now (7.26) is immediate by (7.13) and (7.21) which actually finishes the proof, the rest of the assertion being now straightf0rward.m Now, given an algebra E and an element x of E, we denote by Z ( s ) the zero-set of the function 2 , the Gel'fand transform of x ; that is, one has by definition (7.28)

Z(?)=

{ f E M I E ) : G ( f ) = flx)=O}.

The following is now a direct consequence of the argument plied in the preceding proof. Namely, we have.

ap-

Corollary 7 . 5 . Let E be a c o m t a t i v e l o c a l l y convex Waelbroeck algebra (cf.

.

the above Theorem 7.2) Then, an element x i n E i s regular ( i . e . , i n v e r t i b l e ) if, and only i f , one has

Z ( 2 ) = G1;

(7.29) t h a t i s , equivalentzy, the r e l a t i o n

(7.30)

.;'.tf,

=f(x) f 0,

for every (continuous) character f of E. 1 In this respect, we finally note that a stronger version of the last result is still true, in case one has locally m-convex algebras. In fact, it suffices then to consider just advertibly complete alge-

3.

77

SCHUR'S LEMMA

b r a s i n s t e a d o f & - a l g e b r a s ( c f . T h e o r e m I ; 6.4. S e e a l s o C h a p t . 1 I I ; C o rollary 5.2). 8. Appendix: Schur's Lemma W e p r e s e n t i n t h i s a p p e n d i x a form of t h e c l a s s i c a l lemma

in

t i t l e which o f t e n a p p e a r s m a i n l y i n q u e s t i o n s c o n c e r n i n g t h e r e p r e s e n t a t i o n t h e o r y o f t o p o l o g i c a l a l g e b r a s . Though w e a r e n o t i n p a r t i c u l a r c o n c e r n e d w i t h t h a t p a r t of t h e t h e o r y i n t h i s book ( s e e , h o w e v e r , t h e f i n a l C h a p t e r X V ) , w e do d i s c u s s t h e r e s u l t i n q u e s t i o n m a i n l y as an a p p l i c a t i o n i n o u r c a s e o f t h e Gel'fand-Mazur

Theorem ( S e c -

tion 5). W e s t a r t w i t h a p r e l i m i n a r y comment on t h e r e l e v a n t t e r m i n o l o g y .

T h u s , g i v e n a (complex) a l g e b r a E and an ( a l g e b r a ) moryhism

of E i n t o t h e a l g e b r a o f l i n e a r endomorphisms ( o p e r a t o r s ) o f a (comp l e x ) v e c t o r s p a c e X , one d e f i n e s t h e map 41 a s a representation of t h e algebra

E

in

X

( t h e representation space). A l t e r n a t i v e l y , t h e s p a c e X

becomes a n E-mo&h*e v i a t h e r e l a t i o n

f o r a n y a 6 E and x e X .

( T h a t i s , o n e t a k e s a s a d e f i n i t i o n o f t h e re-

levant notion t h e r e l a t i o n (8.2) i n e i t h e r d i r e c t i o n , r e s p e c t i v e l y ) . Thus w e s p e a k o f an irreducible representation X I

of

(the algebra) E i n

i f t h e E-module X d e f i n e d by ( 8 . 2 ) i s i r r e d u c i b l e . T h a t i s , t h e r e

do n o t e x i s t o t h e r sub-E-modules

(03rX and t h e s p a c e X i t s e l f .

o f X e x c e p t t h e t r i v i a l o n e s , namely, (Thus i n t h i s case $ d o e s n o t decompose

( "reduce") t o any o t h e r " s i m p l e r "

,

non-trivial

,

(sub-) r e p r e s e n t a t i o n s

of E ) . Now a n i m p o r t a n t " s t r u c t u r a l i n g r e d i e n t " of t h e " g e o m e t r y " of t h e r e p r e s e n t a t i o n $ ( h e n c e , of t h e g i v e n a l g e b r a t o o ) i s t h o s e e l e ments T

€

L(X) which "commute w i t h t h e e l e m e n t s of E " ; i n f a c t t h e y f u l -

f il t h e r e l a t i o n

f o r every a e E . W e

d e n o t e by c($) t h e s e t of t h o s e T f L i X ) which s a t i s -

f y ( 8 . 3 ) and c a l l them intertwining operators o f t h e r e p r e s e n t a t i o n $ . Thus o n e d i s t i n g u i s h e s now t h a t p a r t of LIX) which b e h a v e s r e l a t i v e t o t h e "domain o f c o e f f i c i e n t s " E ( X b e i n g c o n s i d e r e d a s a n Emodule), e x a c t l y a s i f

E was a " n u m e r i c a l domain". Now, t h a n k s t o I .

Schur, t h i s d e s i r e i s , i n f a c t , v e r y c l o s e t o t h e r e a l i t y concerning

11 SPECTRUM (LOCAL THEORY)

78

at least "simple" (read irreducible) representations. Thus, his classical lemma says that: For every i r r e d u c i b l e representation 6 , as in (8.1), c ( @I

(8.4)

i s a d i v i s i o n subalgebra of L I X I .

(Here the identity operator on X , idX= 1 , is the identity element of ~ ( 4 ) . For a proof of (8.1) see, f o r instance, F.F. BONSAL- J . DUNKAN [l:p. 121, Proposition 6 1 ) . Now in case of a c o m t a t i v e topological algebra, the previous picture is further amended by appealing to Gel'fand-Mazur. In this respect, by looking at continuous representations one considers on L ( X ) any topology Gmaking the latter space a topological algebra (Definition I; 1.1. See also Example I;2.(1)). Thus we now have. Lemma 8.1. Let E be a c o m t a t i v e topo2ogical algebra w i t h an i d e n t i t y e l e -

ment and continuous i n v e r s i o n . Then, every continuous representation of E , say

@:

f o r which E/ker(@) (in t h e q u o t i e n t topology) has a non-trivial topological dual, i s one-dimensiona2.

E -Le(X),

Proof, By hypothesis ker(6) is a closed ideal of E, so E/ker($), endowed with the respective quotient topology, is a topological algera of the same type as E (see also Lemma 7.3). Moreover, one has

(8.5)

E/ker(O) = @(E) E L ( X ) ,

within an (algebra) isomorphism, while $ ( E l is a (complex)division algebra (Schur's Lemma; cf. (8.3)). Thus E/ker(@l is a division topological algebra satisfying, by hypothesis, (5.4), hence (topologically) isomorphic to C, and this proves the assertion. In addition, (8.5) becomes, in effect, a topological (algebraic) isomorphism.l

As an illustration of the above, we still give the following less technical result. Namely, we have.

Corollary 8.1. L e t E be a coirnnutatiue Zocally convex algebra w i t h an i d e n t i t y element and continuous i n v e r s i o n . Then, t h e only continuous i r r e d u c i b l e r e p r e s e n t a t i o n s of E are i t s continuous characters ( i . e . , t h e elements of ??Z(Ei, t h e ( t o p o l o g i c a l ) spectrum of E; see Definition 7.2).

In p a r t i c u l a r , t h e a s s e r t i o n holds t r u e f o r every c o m t a t i v e l o c a l l y m-convex algebra w i t h an i d e n t i t y element ( see also Lemma 3.1 ) . I

79

Projective

Limit

Algebras

CHAPTER 111

1 , I n i t i a l topologies. Topological subalgebras. Cartesian products Suppose w e a r e g i v e n a n a l g e b r a E and a f a m i l y { ( E a , f a ) ) a c I p a i r s ( E , f a ) where, f o r e a c h i n d e x a e I, E, i s a topological algebra a Ea an algebra morphisrn.

of

and

fa: E -

Thus, one can c o n s i d e r t h e i n i t i a Z topology on E d e f i n e d on it by t h e maps f a , a~ I ( c f . , f o r i n s t a n c e , N . BUURBAKI [4: Chap. 1 ; p. 1 2 , Propos i t i o n 4]),

namely, t h e weakest t o p o l o g y on E f o r which t h e s e maps a r e

c o n t i n u o u s . Of c o u r s e , t h i s t o p o l o g y i s c o m p a t i b l e w i t h t h e v e c t o r s p a c e s t r u c t u r e of t h e g i v e n a l g e b r a E , t h u s a vector space topology on E . I n d e e d , a l o c a l b a s i s of E c o n s i s t s of sets of t h e form (1.1)

where t h e sets U,

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

i

.

a B I( c f . N. BUURBAKI [ 6 :Chap. 1 ; p. 1 2 , P r o p o s i t i o n 71 ) Moreover, t h e same v e c t o r s p a c e t o p o l o g y on E i s a l s o compatible w i t h t h e a l g e b r a s t r u c t u r e of E , t h a t i s E i s t h u s made i n t o a t o p o l o g i c a l a l g e b r a , on t h e b a s i s of t h e f o l l o w i n g .

Proposition 1.1. Let E be a given algebra, IEa:acl

a fa-iil:] of topological

algebras and, f o r each index a I , an algebra morphism f : E - E a a' Then, the i n i t i a l topology defined on E by the maps f a , a e I , makes E a topological algebra. I n p a r t i c u l a r , the algebra E thus topologized has a continuous m u l t i p l i c a t i o n , if the given topological algebras Ea

, a e I , have.

Proof. By t h e p r e v i o u s comment t h e t o p o l o g y under c o n s i d e r a t i o n

is a v e c t o r s p a c e t o p o l o g y on E , a l o c a l b a s i s of which i s g i v e n by t h e s e t s of t h e form ( 1 . 1 ) .

F u r t h e r m o r e , f o r e v e r y x € E l the map ( l i n e a r

endomorphism of E ) (1.2)

lx:E-E:y-lx7^y):=

XY

i s continuous. I n f a c t , it s u f f i c e s t o prove ( s e e , f o r example, ,V. BOURBAKI [4: Chap. 1; p. 1 2 , P r o p o s i t i o n 4 1 ) t h a t , f o r e v e r y a~ I , t h e composite map 79

80

f a o 1,

I11 PROJECTIVE LIMIT ALGEBRAS

,a s

i n ( 1 . 3 ) , is continuous w h i c h , however , i s s t r a i g h t f o r w a r d by

t h e h y p o t h e s i s f o r t h e family (Ea,f'll,aeI.

On t h e o t h e r h a n d , t h e l a s t

p a r t of t h e a s s e r t i o n i s now an immediate c o n s e q u e n c e of t h e f o l l o w i n g r e s u l t ( s e e a l s o t h e n e x t Remark).

-

Lemma. L e t E be an algebra, F a topological algebra w i t h a continuous multip l i c a t i o n , and h : E

F an algebra morphism. Then, t h e i n v e r s e image topoZogy on

E defined by h is a v e c t o r space topology on E making i t a topological a l g e b r a w i t h

a continuous muZtiplication. Proof. I n f a c t , t h e t o p o l o g y i n q u e s t i o n makes E i n t o a t o p o l o

-

g i c a l v e c t o r s p a c e (see, f o r i n s t a n c e , N . BOURBAXI [ 6 : Chap. 1 ; p . 13, Lem-

me] )

.

Besides, i f

h-ltW) i s a n e i g h b o r h o o d of z e r o i n E , w h i l e

d

is

s u c h i n F ( c f . ( l . l ) ) , t h e n by h y p o t h e s i s t h e r e e x i s t n e i g h b o r h o o d s o f 0 e F, say

U ar,d V , w i t h U . VE W , i n s u c h a way t h a t

and t h i s p r o v e s t h e a s s e r t i o n .

Remark.- C o n c e r n i n g t h e u s e o f t h e above Lemma i n t h e l a s t p a r t of t h e p r o o f of P r o p o s i t i o n 1 . 1 , w e remark ( a s e a s i l y f o l l o w s from t h e s a m e d e f i n i t i o n s ) t h a t t h e i n i t i a l topology on E defined b y t h e maps f a , a f I , coincides w i t h t h e " i n v e r s e image topology" defined on E b y t h e map

h e r e t h e r a n g e of ( 1 . 4 ) i s t a k e n t o b e t h e r e s p e c t i v e C a r t e s i a n p r o d u c t s p a c e endowed w i t h t h e u s u a l ( c a r t e s i a n ) p r o d u c t t o p o l o g y d e f i n e d on it b t h e t o p o l o g i c a l s p a c e s E, , ae I ( c f . a l s o N . BOUREAKI Chap. 1 ; p.12, ~ r o p o s i t i o 71 n )

G:

.

NOW, if t h e given topological algebras E , a el, are l o c a l l y convex,

one

c o n c l u d e s by ( 1 . 1 ) t h a t t h e i n i t i a l topology on E d e f i n e d by t h e maps f a , a e I , makes it i n t o a l o c a l l y convex algebra too. F u r t h e r m o r e , a s a g a i n f o l lows b y t h e p r e v i o u s

Lemma, t h e l o c a l l y convex algebra E t h u s defined has a

a e I, have. I n t h i s r e s p e c t , t h e same Lemma i m p l i e s t h a t any subalgebra of a given topological algebra w i t h a continuous m u l t i p l i c a t i o n is i n t h e r e l a t i v e topology (i.e., t h e " i n v e r s e image t o p o l o g y " d e f i n e d on it by t h e r e s p e c t i continuous m u l t i p l i c a t i o n in ease t h e algebras E

a'

1.

INITIAL TOPOLOGIES

81

ve injection (inclusion) map) a topologicaZ algebra of t h e same type as t h e given one. (If the topological algebra considered has a separately con-

tinuous multiplication, a similar argument is applied using an analogous diagram to ( 1 . 3 ) ) Similarly, one concludes that any subalgebra of a l o c a l l y convex aZgebra i s i n t h e r e l a t i v e topoZogy a topological algebra of t h e same t y p e a s t h e o r i g i n a l one.

.

On the other hand, one still obtains that: The i n i t i a l topology defined on a given algebra E by t h e maps fa, c1 e I, as above, i n case t h e algebras,,?i u E I, arz l o c a l l y m-convex ones, makes E i n t o a l o c a l l y m-con-

(1.5)

v e x algebra too.

This is again straightforward by ( 1 . 1 ) in conjunction with Theorem I; 1.1,2) and Lemma I; 3 . 1 . In fact, one proves that the inverse image toF with F bepology, corresponding to a given algebra morphism h : Eing a locally m-convex algebra, still defines E as a locally m-convex algebra. In particular, one concludes again that any subalgebra of a l o c a l l y m-convex algebra i s i n t h e reZative topology a topological algebra of t h e same type. N o w suppose we are given a topological algebra E endowed with the initial topology defined on it by a family {(E,, f, ) I , c1 e I, as above, where t h e topology of each one of t h e algebra.s E, , a € 1, i s Fluusdorff. 30 t h e topology of E i s Hausdorff i f , and o n l y i f , for every element x # G in E , t h e r e e x i s t s an index a E I suck t h a t f a (xi # 0 i n Ea. Indeed, if E is HausdorfL' and x # O in E , there exists a neighborhood U of G E E with x & E, so that by ( 1 . 1 ) x 6 l! f a - . ' ( U a m ) t U ; thus, x $ fa-1 (U i , for some aiE I,hence f (x)* a. -v -_ 2 2 i ai -1 U. Conversely, if falix:l # G for some index a E I, then x e' f, (U,) while the latter set belongs to a local basis of E (cf. ( l . l ) ) , which yields the assertion. (An indirect, but less elementary, proof could also be given by using ( 1 . 4 ) and realizing E as a subspace of the Cartesian product space n E, Cf. N. BOURBAKI [ 6 :Chap. 1 ; p. 13, Corollaire 21). a €I In particular, suppose we are given a family ( E ) a , € I of topological algebras and let us consider the respective Cartesian product s e t

.

(1.6)

E=

n E a e r a'

defined by the sets E a E I. Thus, by "coordinatewise" defined operations t h e a' s e t ( 1 . 6 ) becomes an algebra, so that the respective (canonical) "projec tion maps" (1.7)

n :E-E,,

UEI,

are used to define t h e i n i t i a l topology on E which y i e l d s , by d e f i n i t i o n , t h e

a2

I11 PROJECTIVE L I M I T ALGEBRAS

"cartesian pro&& a €I .

topology"

on E , d e f i n e d by t h e t o p o l o g i c a l s p a c e s

Eu

,

On t h e o t h e r hand, by d e f i n i t i o n of t h e o p e r a t i o n s i n t h e (cart e s i a n p r o d u c t ) a l g e b r a E l one r e a d i l y sees t h a t t h e projection maps TI a' a e I , are (algebra) mo2rphisms of t h e r e s p e c t i v e a l g e b r a s ; t h u s , one conc l u d e s ( P r o p o s i t i o n 1 . 1 ) t h a t E endowed with t h e cartesian product topology becomes a topological algebra which, i n p a r t i c u l a r , has a continuous m u l t i p l i c a t i o n i f t h i s i s the case f o r each one of t h e given topological algebras E

a' a e I . I n t h i s r e s p e c t , o n e g e t s a l o c a l b a s i s of t h e t o p o l o g i c a l alge-

b r a E t h u s d e f i n e d by c o n s i d e r i n g , a c c o r d i n g t o ( l . l ) , f i n i t e i n t e r s e c t i o n s of t h e form

where t h e s e t s Ua.

run o v e r l o c a l b a s e s of t h e t o p o l o g i c a l a l g e b r a s

?,

E a , a e I. E q u i v a l e n t l y , one c o n s i d e r s s e t s of t h e form

w i t h Ua = E a

,

e x c e p t of f i n i t e many i n d i c e s a f I f o r which t h e r e s p e c -

t i v e s e t s Ua a r e v a r y i n g a s i n ( 1 . 8 )

.

S i m i l a r l y , a s f o l l o w s from t h e p r e c e d i n g d i s c u s s i o n ,

the c a r t e -

( r e s p . , with continuous ml-

sian product ( a l g e b r a ) of l o c a l l y convex algebras

t i p l i e a t i o n ) i s a topological algebra of the same type as t h e f a c t o r algebras.

I n p a r t i c u l a r , t h i s i s v a l i d f o r l o c a l l y m-convex a l g e b r a s , a s f o l l o w s from t h e n e x t .

Lemma 1 . 1 . Let ( E a i a

I be a family of l o c a l l y m-convex algebras. Then, the

cartGsfczn prs&ct glgebra E = a ! I E a

, endowed with the cartesian product topology,

i s a l o c a l l y m-convex a l g e b r a . I n p a r t i c u l a r , t h i s i s v a l i d f o r any subalgebra of E. I

On t h e o t h e r hand, a cartesian product topological algebra i s Hausdorff i f , and only i f , t h i s i s the case f o r each one of the f a c t o r algebras; h e r e one

a p p l i e s a s i m i l a r argument t o t h a t f o r a c a r t e s i a n p r o d u c t s p a c e . F i n a l l y , a cartesian product topological algebra i s complete i f , and only i f , t h i s holds true f o r each one of t h e f a c t o r algebras. O f c o u r s e , t h i s i s a

consequence o f t h e r e s p e c t i v e more g e n e r a l s i t u a t i o n one h a s i n c a s e o f a c o m p l e t e c a r t e s i a n p r o d u c t s p a c e ( s e e , f o r i n s t a n c e , N . EOURBAKI [4: Chap. 2 ; p . 17, P r o p o s i t i o n 101).

2. Projective systems of topological algebras Suppose w e a r e g i v e n a f a m i l y I Ea i

u6

I

of a l g e b r a s , where t h e

2. P R O J E C T I V E SYSTEMS OF T O P O L O G I C A L ALGEBRAS

83

i n d e x s e t I i s now a directed s e t ; t h a t i s , a p a r t i a l l y o r d e r e d set ( s ) such t h a t f o r any i n d i c e s a , ,

i n I t h e r e e x i s t s an i n d e x y e I w i t h ci


-

such t h a t

fa,

(2.1)

:E,

( a l g e b r a ) morphisms

Ea '

f o r a n y a,B i n I , w i t h a < R ; m o r e o v e r , f o r e v e r y a € I , l e t

f,,=

(2.2)

,

idE

a

namely, t h e " i d e n t i t y map (morphism) " of t h e a l g e b r a Ecl

.

Thus, a p r o j e c t i v e ( o r inverse ) system of algebras i s , by d e f i n i t i o n ,

a family {IE,,

f,, ) } a s above w h i c h , i n a d d i t i o n t o t h e above t w o re-

lations, s a t i s f i e s a l s o the condition

=

(2.3) f o r any a,R,y

B 'y

i n I, with aCRCy.

Now c o n s i d e r t h e C a r t e s i a n p r o d u c t a l g e b r a F = f l . E cl€I

t h e following s u b s e t of F : E = {x=(x ) € F : z

(2.4)

c1

a

as w e l l as

i n I).

= f (x ) , i f aB R

Thus by d e f i n i t i o n o f t h e a l g e b r a o p e r a t i o n s i n F and t h e hypoE is a subalgebra of F ; it may happen t o be

t h e s i s f o r the maps f a , ,

the

t r i v i a l one. I n t h i s respect, i f t h e algebras Ea,acI,have i d e n t i t y elements, say 1

0'

c1 E

, w i t h a < D , are

& I , and t h e a l g e b r a morphisms f

i d e n t i t y p r e s e r v i n g ( t h a t w e do assume)

,

t h e i d e n t i t y 1 = (3,)

i n F be-

l o n g s a l s o t o t h e s u b a l g e b r a E o f F d e f i n e d by ( 2 . 4 ) . The a l g e b r a E t h u s d e f i n e d i s c a l l e d t h e

p r o j e c t i v e ( o r inverse )

l i m i t algebra of t h e g i v e n p r o j e c t i v e s y s t e m (E,,faB) E=lim(Ea, f

(2.5)

aB

c

and i s d e n o t e d by

)

o r s i m p l y by E = l > E a . On t h e o t h e r h a n d , g i v e n a p r o j e c t i v e s y s t e m of a l g e b r a s , o n e

of (algebra) morphisrns by t h e r e l a t i o n

d e f i n e s a family

fa = n,

(2.6)

f o r every a E I ,

IE:

E

-

Ea '

namely, by c o n s i d e r i n g t h e r e s t r i c t i o n t o E o f t h e ca-

n o n i c a l p r o j e c t i o n map

IT

a o f F o n t o Ea

, a € I ;t h u s

era1 g i v e an unto map. F u r t h e r m o r e , b y ( 2 . 4 ) and ( 2 . 6 )

f a = fa@O f B

(2.7)

I

f o r a n y a,R i n I w i t h a 5 R . Thus, w e come n e x t t o t h e f o l l o w i n g .

( 2 . 6 ) does nut in gen-

,

one o b t a i n s

84

111 PROJECTIVE LIMIT ALGEBRAS

Definition 2.1.

A p r o j e c t i v e ( o r i n v e r s e ) system of topological algebras i s

a p r o j e c t i v e s y s t e m of a l g e b r a s { I E a , f I } a s a b o v e , where t h e a l g e aa bras E a , a E I , a r e t o p o l o g i c a l o n e s a n d t h e maps f a @ , w i t h LY 2 3! i n I , c o n t i n u o u s ( a l g e b r a ) morphisms. The r e s p e c t i v e p r o j e c t i v e l i m i t a l g e b r a E ( c f . ( 2 . 5 ) ) i s f u r t h e r endowed w i t h t h e i n i t i a l t o p o l o g y d e f i n e d on it by t h e maps f a , a E I ( c f . ( 2 . 6 ) ) . The t o p o l o g i c a l a l g e b r a t h u s d e f i n e d ( P r o p o s i t i o n 1 . 1) is called the

p r o j e c t i v e ( o r i n v e r s e ) l i m i t topological algebra.

Thus by t h e f o r e g o i n g and Lemma 1 . 1 w e c o n c l u d e t h a t : A p r o j e c t i v e l i m i t ( a l g e b r a ) of l o c a l l y m-convex algebras

is a topological algebra of t h e same t y p e . Of c o u r s e , t h i s i s c e r t a i n l y v a l i d i f t h e a l g e b r a s E a , a ~ I , a r e norme d o r , i n p a r t i c u l a r , Banach a l g e b r a s as t h i s w i l l be t h e c a s e i n t h e n e x t s e c t i o n . The i m p o r t a n t t h i n g h e r e i s t h a t t h e c o n v e r s e i s , i n f a c t , t r u e , so t h a t l o c a l l y m-convex algebras appear t o be only o f t h i s type of

algebras

( o r a t l e a s t a p p r o p r i a t e s u b a l g e b r a s of t h i s t y p e , i f t h e y a r e

n o t c o m p l e t e ) . T h i s i s , i n e f f e c t , a f u n d a m e n t a l r e s u l t i n t h e whole t h e o r y of l o c a l l y m-convex

a l g e b r a s (see S e c t i o n 3 ) .

W e f i n a l l y remark t h a t a given p r o j e c t i v e l i m i t topological algebra E =

l i m ( E a , fa,) ( b e i n g a

t o p o l o g i c a l subalgebra of F =

t

i f t h i s i s the case f o r each one of t h e algebras Ea , C L E I .

n

E

aer a

)

i s Hausdorff

W e c o n c l u d e t h i s s e c t i o n w i t h t h e f o l l o w i n g lemma which i s u s e d

below. Namely, w e h a v e . Lemma 2.1.

Every p r o j e c t i v e l i m i t topological algebra E = P ( E a , f a B I i s

a CloSQd subalgebra of the cartesian product topological algebra F = fl E

aer a

.

In

p a r t i c u l a r , E is complete Iresp., quasi-complete), i f t h i s i s the case f o r each one of the given topologicaL algebras E Proof. (2.4),

a’

a

t?

I.

The f i r s t p a r t of t h e a s s e r t i o n i s a d i r e c t c o n s e q u e n c e of

t h e c o n t i n u i t y of t h e maps f a B

,

with

c1 S

B i n I, and t h e d e f i n i -

t i o n o f t h e t o p o l o g y of E l a s t h e r e l a t i v e o n e i n d u c e d on it by F. ( O f c o u r s e , t h e p r e v i o u s argument i s v a l i d , i n e f f e c t , f o r e v e r y p r o j e c t i v e l i m i t e t o p o l o g i c a l s p a c e E c o n c e r n i n g i t s r e l a t i o n w i t h t h e resp e c t i v e c a r t e s i a n p r o d u c t s p a c e F ) . NOW, t h e l a s t c o n c l u s i o n t o g e t h e r w i t h t h e h y p o t h e s i s f o r t h e a l g e b r a s E,, i n t h e s e c o n d pa:?:

yields the desired assertion

o f t h e s t a t e m e n t , s i n c e t h e n E w i l l b e a c l o s e d sub-

space o f - t h e complete ( r e s p . , quasi-complete) BOURBAKI[7: Chap. 3 ; p . 9

s p a c e F (see also

and p . 10, P r o p o s i t i o n 7 1 ) . I

81.

3 . ARENS -MICHAEL DECOMPOSITION

3. Representation o f locally m-convex algebras as projective limits. Arens -Michael decomposition Our main o b j e c t i v e i n t h e f o l l o w i n g d i s c u s s i o n i s t o g i v e a rep r e s e n t a t i o n o f any c o m p l e t e l o c a l l y m-convex a l g e b r a E a s a p r o j e c t i v e l i m i t o f Banach a l g e b r a s ( c f

position of E )

.

.

D e f i n i t i o n 2 . 1 ; Arens-Michael decom-

A s a l r e a d y mentioned above t h i s r e p r e s e n t a t i o n c o n s t i -

t u t e s , i n f a c t , a f u n d a m e n t a l c o n c l u s i o n of t h e whole t h e o r y of l o c a l l y rn-convex a l g e b r a s a n d was c o n c e i v e d a l r e a d y i n t h e e a r l y d a y s o f t h e t h e o r y (see R . ARENS [ 5 ; 6 ] and E. A . MICHAEL [ l ] ) . A s w e s h a l l s e e , t h r o u g h t h i s d e v i c e b a s i c q u e s t i o n s a b o u t a g i v e n l o c a l l y rn-convex a l g e b r a c a n b e reduced t o analogous ones f o r t h e r e s p e c t i v e factor-Banach a l g e b r a s where t h e y may b e known. I n t h i s r e s p e c t , w e a r e g o i n g t o a p p l y some of t h e c o n s e q u e n c e s o f t h e above r e s u l t i n t h e n e x t s e c t i o n and f u r t h e r on i n C h a p t e r VI, a f t e r w e h a v e d i s c u s s e d t h e n e c e s s a r y p r e l i m i nary material. Thus t o b e g i n w i t h , s u p p o s e t h a t we are given a l o c a l l y m-emvex a-ge-

2ra

,?

z= (Ualaer

together w i t h a local b a s i s

o j E c o n s i s t i n g of rn-barrels ( s e e

Theorem 1 ; 3 . 1 ) . Now b y c o n s i d e r i n g t h e r e s p e c t i v e g a u g e f u n c t i o n s of t h e aembers of

x , one

g e t s a family

r=

(p

c1

)

a€ I

of

( c o n t i n u o u s ) sub-

m u l t i p l i c a t i v e semi-norms d e f i n i n g t h e t o p o l o g y of E ( i b i d . )

.

On t h e o t h e r h a n d , f o r e v e r y i n d e x a e l , t h e p a i r (E,p,,)i s a semi-normed a l g e b r a , so t h a t o n e c a n d e f i n e t h e r e s p e c t i v e ( q u o t i e n t )

normed algebra E,

;

i.e., one h a s

where

of

,

Ea = E / p i ' ( O )

(3.1)

p l ' ( 0 ) = ker i p I = { z e E : p (x)= 0)

a

=

N

a is a closed (2-sided) i d e a l

( t h e t o p o l o g i c a l a l g e b r a ) E . So t a k i n g c o m p l e t i o n s i n ( 3 . 1 ) one g e t s ,

f o r every a € I ,

t h e c o r r e s p o n d i n g Banach algebra

F u r t h e r m o r e , c o n s i d e r t h e r e s p e c t i v e ( c a n o n i c a l ) q u o t i e n t map

(3.3)

p a : E-Ea:=

E/Na, a & I ,

which i s a n ( o n t o ) a l g e b r a morphism, and l e t a l s o

b e t h e map d e -

f i n e d by

(3.4)

t h i s is again

a n a l g e b r a morphism of t h e g i v e n a l g e b r a E i n t o t h e Ba-

86

111 PROJECTIVE LIMIT ALGEBRAS

h

nach algebra E, , € I. Now one defines a partial order in the index set I by the following relation: I f a , B are elements of I , we s e t

a < B,

(3.5)

i f UoG U,; ?

equivalently, i f (X) 2

‘a f o r every X E E.

p (xi,

B

-

(See also Proposition I; 1.1, (11)). S o by hypothesis for one con cludes that I endowed with t h e above p a r t i a l order becomes a directed s e t . Thus, f o r any a , B i n I , with a < $ , one obtains by (3.5), p 2 p a , (pointwise on E ) , so that N B kerlp Ic ker(pal = N a . Therefore, t h e r e l a B tion faB : x + N B = f (x+W,):= x + N a = (3.6) B, with x E E Zefines an onto algebra morphism

[XI,

[XI,

(3.7)

Go : EB-

Ea

I

such that one gets by (3.3) the relation (3.8)

Pa =

faB0

Pa

for any a , @ in I, with a 5 6. So the maps fa,, with a 5 a , are continuous as well. Moreover, one readily verifies that (2.2) and (2.3) are satisfied, so that one concludes that: The family

(3.9)

(Ea

, fa,

j

I a I

e I,

d e f i n e s a p r o j e c t i v e system of normed algebras,

corresponding to the given local basis 72 of E. Furthermore, by conof the maps f (cf. (3.711, sidering the continuous extensions r ‘a6 aB one g e t s a p r o j e c t i v e system of Banach algebras

t

(3.10)

.f;a,i

I, a e r ,

while the respective relation to (3.8) is true as well. Now as follows from the discussion in the previous section, one thus obtains the l o c a l l y m-convex algebra (3.11)

lim Ea t

as well as t h e complete locallyr m-eonvex algebra (3.12)

limi

c

a

a7

3 . ARENS -MICHAEL DECOMPOSITION

( s e e a l s o Lemma 2 . 1 ) .

Thus o u r main c o n c e r n i n t h e f o l l o w i n g l i n e s i s

now t o f i n d t h e c o n n e c t i o n s between t h e p r e c e d i n g two a l g e b r a s and t h e g i v e n a l g e b r a E ( s e e Theorem 3 . 1 ) . F i r s t w e need t h e f o l l o w i n g two lemmas. (Both o f t h e m a r e v a l i d ,

i n d e e d , f o r any p r o j e c t i v e l i m i t s p a c e s ; c f . , f o r i n s t a n c e , N . BOURBAKI [4: Chap. 1 ; p . 29, P r o p o s i t i o n 9 and C o r o l l a i r e ] . However, t h e i r c i t a t i o n

h e r e l i n t h e s p e c i a l case c o n s i d e r e d , i s made j u s t f o r c o n v e n i e n c e o f r e f e r e n c e ) . Thus, w e have. Lemma 3.1. Let E = l i m ( E c

a

, faB ) be a p r o j e c t i v e l i m i t topological algebra

( s e e D e f i n i t i o n 2 . 1 ) . Then a local b a s i s of E i s given by t h e family

I f a-1

(3.13)

where

?ra

ma):U a E 7 - z a ,

a

E

I),

i s a local b a s i s of t h e algebra E a , w i t h a e I .

Proof. By ( 1 . 8 ) and ( 2 . 6 ) , one o b t a i n s a l o c a l b a s i s of E by cons i d e r i n g s e t s o f t h e form (3.14)

n,

where t h e s e t s Ua run over l o c a l b a s e s of t h e a l g e b r a s E a , s a y , i w i t h a e l . On t h e o t h e r h a n d , s i n c e I i s a d i r e c t e d s e t , t h e r e e x i s t s a n i n d e x a e I s u c h t h a t ai5 a (2.7) f a

i

= faia

,

w i t h i = I,.

,, ,n

,

so t h a t one g e t s by

o f a : h e n c e , one h a s

'(U ) i s a n e i g h b o r h o o d of z e r o i n E a . where t h e s e t Va = i = I f -aia "i

t h e r e e x i s t s U a e n a , w i t h U" C-V u b a s i s of

,

Thus,

which p r o v e s t h e a s s e r t i o n on =he

( 3 . 1 4 ) and ( 3 . 1 5 ) . 1

Now b a s e d on t h e p r e v i o u s lemma w e a l s o g e t t h e f o l l o w i n g u s e f u l r e s u l t which w e n e x t a p p l y i n Theorem 3 . 1 .

Lemma 3.2. Let E = l i m ( E a , f I be a p r o j e c t i v e l i m i t algebra as i n t h e prec aO vious Lema 3 . 1 , and B any vector subspace of E . Then, one has

where the maps f a' a € I, are given by ( 2 . 6 ) . In p a r t i c u l a r , if B i s closed one g e t s (3.17)

-

B = lim f (B) = l i m f (BI . c a t a

Proof. W e f i r s t remark t h a t t h e f a m i l y

88

I11 PROJECTIVE L I M I T ALGEBRAS

d e f i n e s a p r o j e c t i v e system of t o p o l o g i c a l

(vector) spaces, a s t h i s w i t h a 5 B. O n

f o l l o w s from ( 2 . 7 ) a n d t h e c o n t i n u i t y o f t h e maps f aB t h e o t h e r hand, s i n c e

f o r any a ' @ , one o b t a i n s t h a t t h e f a m i l y (3.19)

d e f i n e s a l s o a p r o j e c t i v e s y s t e m of t o p o l o g i c a l

(vector) spaces; thus

t h e second e q u a l i t y i n ( 3 . 1 6 ) i s now a d i r e c t c o n s e q u e p c e of t h e same d e f i n i t i o n s . T h e r e f o r e , s i n c e t h e one d i r e c t i o n i n t h e f i r s t r e l a t i o n o f ( 3 . 1 6 ) i s , of c o u r s e , o b v i o u s , it s u f f i c e s t o p r o v e t h a t e v e r y ele-1 ment x i n fI f a (fa(<))i s a n a d h e r e n t p o i n t o f B . Thus i f f ( X I = x E U a a fa(B), f o r e v e r y U E I , t h e n f o r e v e r y 'I basic open neighborhood" U of xa i n

a

E

a

an e l e m e n t o f a f u n d a m e n t a l s y s t e m o f open n e i g h b o r h o o d s of

(i.e.,

the point i n question) where f - ' ( U , I U

,

one h a s

U, n f a ( B ) # 0, so t h a t fil(UuI n B $ 0 , ( c f . Lemma 3 . 1 ) ,

i s a b a s i c open n e i g h b o r h o o d of x i n E

which p r o v e s t h e a s s e r t i o n . On t h e o t h e r h a n d , one h a s

so t h a t by t h e p r e c e d i n g ,

and ( 3 . 1 9 ) one g e t s

(3.18),

-

= ( b y (3.16)) % = B ,

B G 1 3 J 2 ( B j = l&nfa(31

( i f B i s c l o s e d ) which p r o v e s ( 3 . 1 7 ) , a n d t h i s t e r m i n a t e s t h e p r o o f o f t h e lemma. I

s o w e a r r i v e d by now a t t h e main r e s u l t s of t h i s s e c t i o n . T h u s , w e f i r s t have t h e f o l l o w i n g b a s i c f a c t . Theorem 3.1. Let E be a (Hausdorff) l o c a l l y m-convex algebra and

n=(

~

A

a local basis of B consisting of m-barrels. Moreover, denote by E the completion o f E and l e t E

a (resp.,

a I , a e I, be the normed ( r e s p . , Banach) algebras correspond-

ing t o t h e given local b a s i s

defined by ( 3 . 1 ) ( r e s p . , ( 3 . 2 1 1 . Then, one has the

relations (3.20)

E

5

limEa t

5 lci m ? u = 2 ,

within topological algebraic isomorphisms. I n p a r t i c u l a r , f o r every complete locall y m-eonvex algebra E , w i t h

a s above, one g e t s the r e l a t i o n s E = limE

h

= limE c u t u ' w i t h i n topological aZgebraic isomorphisms.

(3.21)

;

i

~

89

3. AIiENS-MICHAEL DECOMPOSITION

Proof. By c o n s i d e r i n g t h e l o c a l l y m-convex limi

t a map

(cf.

,

( 3 . 1 1 ) and ( 3 . 1 2 )

(3.22)

algebras

limE

c

a

and

one g e t s t h e f o l l o w i n g

@(x):= (pa(x))

1 2 E a : x-

@ :E-

,

respectively)

I

where w e s e t pa ( ~ 1 = x + k e r l p ~ l = ~ x ~ ~ , ~ ~ I ,

(3.23)

( c f . ( 3 . 3 ) ) , t h a t i s , @ ( x ) , x e E ,i s a n e l e m e n t of

,

as t h i s foll o w s f r o m ( 3 . 8 ) , so t h a t t h e map ( 3 . 2 2 ) i s w e l l d e f i n e d . Now it i s limEa t

s t r a i g h t f o r w a r d t h a t ( 3 . 2 2 ) d e f i n e s a morphism

of t h e respective algebras. S o

w e n e x t p r o v e t h a t ( 3 . 2 2 ) d e f i n e s , i n f a c t , a topological

([XI,,

( a l g e b r a ) isomor-

[XI,

phism ( i n t o ) : I n d e e d , i f @fx1= = O r then = 0 , with a E I , t h a t i s p ( x i = 0 , f o r e v e r y a e l ; t h u s since E i s Hausdorff o n e g e t s x = 0 , so a t h a t the map @ i s 1-1. F u r t h e r m o r e , o n e o b t a i n s (3.24)

fa 0 @ = P,

I

aE II

t h a t i s , t h e c o m m u t a t i v i t y of t h e f o l l o w i n g d i a g r a m

(3.25)

-

which i m p l i e s t h a t @ i s continuous. On t h e o t h e r h a n d , t h e inverse map

@-I: @ ( E l 5 limE

(3.26)

i s also continuous. I n d e e d , i f U a s i s of E , t h e s e t

ex

+-a

E

i s any member of t h e g i v e n l o c a l ba-

(3.27) w i t h V, =

1

Ua)

a n d V6 = ER

,

f o r every

B E I, with 6 # a ,

n e i g h b o r h o o d of z e r o i n @ ( E ) , i n s u c h a way t h a t V C @(Ua).

defines a Thus , @ i s a l s o

an open map, so t h a t , f i n a l l y , w e h a v e p r o v e d t h a t ( 3 . 2 2 ) d e f i n e s a topolog i c a l algebraic isomorphism of t h e given algebra E i n t o l>Ea. Now,

za, a~ I, ( b e i n g t o p o l o g i -

t h e c a n o n i c a l i n j e c t i o n s j,: E -

c a l a l g e b r a i c i s o m o r p h i s m s ) commute w i t h t h e maps fa@ a n d t h e i r e x t e n s i o n s faR : g6;a , w i t h a 5 6, so t h a t o n e o b t a i n s an ( a l g e b r a ) isomor-

phism (3.28)

(see a l s o N . BOURBAKI [I

j= limj :l h E +-a +-a

-

h

Ea

: Chap. 3; p. 78, C o r o l l a i r e 1 a n d p.

80, C o r o l l a i r e ] ) ;

t h i s i s a l s o a topological one, a s it i s e a s i l y s e e n by d e f i n i t i o n of t h e

90

I11 PROJECTIVE LIMIT ALGEBRAS

t o p o l o g i e s of t h e a l g e b r a s i n v o l v e d i n ( 3 . 2 8 ) . On t h e o t h e r h a n d , by ( 3 . 2 4 ) and ( 3 . 3 ) , one h a s

f o r e v e r y ~ E Iso; b y Lemma 3.2 a n d t h e p r e c e d i n g c o n c l u s i o n f o r

@,

one o b t a i n s

Thus, s i n c e t h e l a s t s p a c e i s c o m p l e t e ( c f . Lemma 2 . 1 ) , o n e f i n a l l y concludes t h a t

w i t h i n t o p o l o g i c a l a l g e b r a i c isomorphisms ( a s i n d i c a t e d i n ( 3 . 3 0 ) ) ; now, t h i s t o g e t h e r w i t h ( 3 . 2 8 ) p r o v e b o t h of t h e d e s i r e d r e l a t i o n s ( 3 . 2 0 ) and ( 3 . 2 1 ) , a n d t h i s f i n i s h e s t h e p r o o f of t h e t h e o r e m . I A d i r e c t c o n s e q u e n c e o f t h e p r e c e d i n g i s now t h e f o l l o w i n g .

Corollary 3.1. Every (Hausdorffl l o c a l l y m-convex algebra i s , w i t h i n a topological algebraic isomorphism, a subalgebra of a cartesian product of Banach algebras. These provide, i n f a c t , a p r o j e c t i v e system of topological (Banach) algebras, such t h a t the given algebra i s a l s o a dense subalgebra of the respective p r o j e c t i v e l i m i t ( t o p o l o g i c a l ) algebra. I I n p a r t i c u l a r , i f E has a countable l o c a l b a s i s % = ( U n I n e m be., E i s a m e t r i z a b l e l o c a l l y m-convex a l g e b r a ) , o n e g e t s t h e f o l l o w i n g c o r o l l a r y of Theorem 3 . 1 . I n t h i s r e s p e c t , it i s a l s o c l e a r t h a t one may assume t h e l o c a l b a s i s % t o c o n s i s t o f a d e c r e a s i n g s e q u e n c e of s u b s e t s of E .

Thus, w e h a v e .

Corollary 3.2. Every Frgchet l o c a l l y m-convex algebra i s ( w i t h i n a topologic a l algebraic isomorphism) the p r o j e c t i v e l i m i t of a "decreasing sequence" of Banach algebras. 1

Remark.-

The s e q u e n c e

En

,n

E

N, o f Banach a l g e b r a s c o n s i d e r e d i n

t h e p r e v i o u s c o r o l l a r y i s s u c h t h a t , f o r e v e r y m l n i n IN, o n e h a s a ( c o n t i n u o u s ) algebra rnorphism

c,i

:

En-

gm

; t h i s is n o t , i n genera1,l-1,

t h e r e f o r e , t h e q u o t a t i o n marks a p p e a r e d i n t h e above s t a t e m e n t . I n conclusion, we s i n g l e o u t f o r later use t h e s i t u a t i o n

de-

s c r i b e d by t h e p r e v i o u s Theorem 3 . 1 t h r o u g h t h e n e x t

Definition 3.1. Suppose w e a r e g i v e n a c o m p l e t e l o c a l l y m-convex I of E c o n s i s t i n g o f m - b a r r e l s IUula a l g e b r a E and a l o c a l b a s i s

z=

91

4. APPLICATIONS (Theorem I ; 3 . 1 ). Now w e c a l l an Arens - Michael decomposition of E , c o r

-

responding t o t h e p a r t i c u l a r l o c a l b a s i s ?Z under c o n s i d e r a t i o n , an e x p r e s s i o n of t h e form E = lhi

(3.32)

a , v a l i d by t h e above Theorem 3 . 1 w i t h i n a t o p o l o g i c a l a l g e b r a i c isomort

phism of t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d . F u r t h e r m o r e , by a b u s i n g t e r m i n o l o g y , w e s t i l l r e f e r t o an Arens -Michael decomposition of any g i v e n l o c a l l y m-convex algebra E l n o t n e c e s s a r -

i l y c o m p l e t e , by c o n s i d e r i n g any r e l a t i o n of t h e form ( 3 . 2 0 ) c o r r e sponding t o a g i v e n l o c a l b a s i s

z=( U a l a E I

of E .

A p p l i c a t i o n s of Theorem 3 . 1 w i l l p r e s e n t l y be g i v e n i n t h e n e x t s e c t i o n , a s w e l l a s i n subsequent r e l e v a n t d i s c u s s i o n s ( s e e , f o r i n s t a n c e , Chapter V I ) .

4. Applications o f the Arens -Michael decomposition On t h e b a s i s of Theorem 3 . 1 of t h e p r e v i o u s s e c t i o n one suc

-

ceeds i n obtaining s e v e r a l r e s u l t s t r u e already i n t h e c l a s s i c a l theo r y of Banach a l g e b r a s f o r t h e more g e n e r a l c l a s s of t o p o l o g i c a l a l g e b r a s r e p r e s e n t e d a s p r o j e c t i v e l i m i t s of Banach a l g e b r a s ( i . e . , l o c a l l y m-convex a l g e b r a s ) ; s o by t h i s p r o c e d u r e t h e d e s i r e d p r o p e r t i e s of t h e a l g e b r a s under d i s c u s s i o n a r e r e d u s e d , i n f a c t , t o a n a l o g o u s q u e s t i o n s f o r Banach a l g e b r a s , w h e r e they may b e known o r even have a s i m p l e r answer. A s a f i r s t basic a p p l i c a t i o n of t h i s t e c h n i q u e , w e cons i d e r Theorem 4 . 1 below. W e repeatedly

of Theorem 3.1.

a p p l y i n t h e s e q u e l t h e n o t a t i o n and terminology

Moreover, by r e f e r r i n g t o a l o c a l b a s i s

2

of a g i v e n

l o c a l l y m-convex a l g e b r a E l w e s h a l l always u n d e r s t a n d t h a t s i s t s of rn-barrels

con-

(Theorem 1 : 3 . 1 ) . Thus w e have.

Theorem 4.1. Let

E be a complete l o c a l l y m-convex algebra and E =

Arens-Michael decomposition of E corresponding t o a local b a s i s

z=

fUa

lht t a

Icl I

an

of E

( D e f i n i t i o n 3.1 ) . Then, one concludes the following: 1 ) An element x i n E i s quasi-regular i f , and only i f , [x]a is quasi-regular

in

ka , f o r

every a e I .

2) The algebra E has an i d e n t i t y element i f , and only i f , t h i s i s true f o r each one o f the algebras

Za, a E I .

3 ) An element x i n E i s i n v e r t i b l e i f , and o n l y i f , [x]a i s i n v e r t i b l e i n f o r every a E I . ( I n p a r t i c u l a r , E has by 21 an i d e n t i t y element).

Proof. A s follows from t h e same d e f i n i t i o n s and t h e d i s c u s s i o n

a’

I11 PROJECTIVE LIMIT ALGEBRAS

92

In t h e p r e v i o u s s e c t i o n , t h e c o n d i t i o n s s t a t e d

i n a l l t h e preceding three

p r o p o s i t i o n s are necessary. Thus, w e p r o v e t h a t t h e y are suffieient t o o : S o 1 ) suppose t h a t x =

,

h

e l e m e n t o f Ea

fa

(yo,) e

([XI,)

,

e E = ljm,?, t

w i t h [x],

a quasi-regular

f o r e v e r y a € I . T h e r e f o r e , t h e r e e x i s t s an e l e m e n t y =

such t h a t

cx], o y a =

(4.1)

= O , f o r every a e ~ .

$,O[CG],

F u r t h e r m o r e , f o r any a < i3 i n I , one h a s t h e r e l a t i o n

j&'ya)

[.]a

=

cfi ["]@)"&@(Yo' (

=

.$a'

96) =

'

w h i l e a similar r e l a t i o n i s a l s o v a l i d f o r l e f t "multi-

(cf. (4.1)),

7a0 ( yB )_ . Thus,

p l i c a t i o n " by so t h a t y = ( y

a

one o b t a i n s

t

Ga ( y R l = y ,

,for

any a 5 D i n I ,

one c o n c l u d e s by ( 4 . 1 ) t h e rela-

E; t h e r e f o r e ,

E limZa=

J,

t i o n x o y = y o x = 0 , which p r o v e s 1 ) . A

2 ) Now s u p p o s e t h a t e = (e,) E II E,

ga ,

of p,(x)

€

Ea

,

f o r e v e r y a € I. T h e r e f o r e , :ince

,

w i t h e,

an i d e n t i t y element

by ( 3 . 2 9 ) E , = p , ( E / ,

i f xa =

t h e n one g e t s by h y p o t h e s i s and ( 3 . 8 ) t h e r e l a t i o n s

x

7

(e

a a D B

I

( c G ) .( ~ e I = c1 a R B

= p

=f

a4 ( p a (x)) =

Ta613 (p ix)I.~,(egl=~aB(pR(").eg/ p (2) = 5 ci

a

i'or a n y a < B i n I , and s i m i l a r l y f o r l e f t m u l t i p l i c a t i o n by G a l e ) ; s o

B

t h e l a t t e r e l e m e n t i s an i d e n t i t y of t h e a l g e b r a E,, hence a l s o of i, - E a . Thus, by h y p o t h e s i s , one c o n c l u d e s t h a t ( e ) = e a , f o r any a < B aa i n I, s o t h a t e = ( e i e l i m i = E. So o n e r e a d i l y sees by " c o o r d i n a t e w i s e "

e

01

+ a

v e r i f i c a t i o n t h a t e i s a n i d e n t i t y e l e m e n t of E . 3) I f x = ( [ z ] p E = l . T

a ' w i t h [cG], f o r every a €I, then there e x i s t s y =(yal e

"[7;

(4.2)

where by 2 ) ( e a ) = e I

,

Y, = Y,.

c.1,

a r e g u l a r e l e m e n t of

n %,

a

= e,

i,,

such t h a t

I

i s a n i d e n t i t y e l e m e n t of E . Now,

f o r any a < B i n

one o b t a i n s

[CGlC1..faB(YB) that is, f

aa

(y

)

B

pothesis that f

=faa'rXla)-fcla(Y4)

i s a n i n v e r s e of

= e, =

faa(Y&[4,

;

[x], i n E a , so t h a t one g e t s by hyh

( ya I = y,,

f o r a n y a < 6 i n I. Hence, y = (y,)

so t h a t o n e c o n c l u d e s by ( 4 . 2 ) t h a t y i s an i n v e r s e of

CG

€

lim Ea = 3 c

i n E , and

t h i s f i n i s h e s t h e proof of t h e theorem. I Remark.-

W e h a v e u s e d t h e h y p o t h e s i s of c o m p l e t e n e s s o f t h e a l -

g e b r a E i n t h e above Theorem 4 . 1

o n l y i n what c o n c e r n s t h e r e l a t i o n

( 3 . 3 2 ) of D e f i n i t i o n 3 . 1 . So it i s e s s e n t i a l l y needed o n l y i n t h e p r o o f of t h e " i f " p a r t o f t h e a s s e r t i o n , w h i l e t h e " o n l y i f " p a r t of it re-

93

4. APPLICATIONS

mains a l w a y s t r u e f o r any l o c a l l y m-convex a l g e b r a . However, f o r s i m plicity's

s a k e a c o m p l e t e a l g e b r a h a s b e e n u s e d i n s t e a d . ( I n t h i s re-

s p e c t , see a l s o t h e n e x t s e c t i o n ) . The f o l l o w i n g i s now an i m m e d i a t e c o n s e q u e n c e o f Theorem 4 . 1 i n conjunction with Definition I; 6.2. C o r o l l a r y 4.1.

Let E be a complete ZocaZEy m-convex aZgebra. Then, appZying

t h e n o t a t i o n of Theorem 4 . 1 , one has

spEix) =

(4.31

u

spg

acI

,

([.TI,,

@"

for every element x i n E. Moreover, t h e spectra2 radius of an element cc i n E

<s

given by

.

r (x) = ~ u p r g ( [ c c ] ~ = ) suplim(pcl(xn E a n+m 0.

(4.4)

is Proof. I f h # O , t h e n by D e f i n i t i o n I ; 6 . 2 , A e S p (x) i f + 3 : 1 E l q u a s i - s i n g u l a r ; s o by Theorem 4 . 1 t h e e l e m e n t [Tx], = T[x], , for some a E l must b e q u a s i - s i n g u l a r . Hence, X E Sp;([x],, , and c o n v e r s e l y a by t h e same t h e o r e m . F u r t h e r m o r e , g u l a r e l e m e n t of

f o r some e l e m e n t [x], Theorem 4 . 1 , 3 )

i f O E SpE(x), t h e n x must b e a s i n -

E: ( D e f i n i t i o n I[; 1 . 2 ) ,

in

2a'a t h e r e f o r e ,

so t h a t t h i s should be t r u e OE

Spf ( [ x ] ~and ) c o n v e r s e l y by c1

and t h i s p r o v e s ( 4 . 3 ) .

O n t h e o t h e r h a n d , b y ~ I I : ~ l . l O a) n) d ( 4 . 3 ) , one r e a d i l y c o n c l u d e s

t h e f i r s t of t h e r e l a t i o n s ( 4 . 4 ) . Moreover, t h e s e c o n d i s a c o n s e q u e n -

ce of t h e r e s p e c t i v e f o r m u l a g i v i n g t h e s p e c t r a l r e d i u s of an e l e m e n t 3:

a i n t h e Banach a l g e b r a ( ga '

11 * l l a , J ,

~ E I and , t h e f a c t t h a t by ( 3 . 1 )

a n d ( 3 . 2 3 1 , one g e t s

11

(4.5) f o r any x

E

E

Remark.-

,

(Ia

=

(I ~

~ ( x )= ~11 IP ~ l (~" L ~ I I = I ~~

~

(

3r :

n~ e )N

r

and a e I , and t h i s f i n i s h e s t h e p r o o f . I C o n c e r n i n g t h e a b o v e r e l a t i o n ( 4 . 5 ) , w e n o t e t h a t by

c o n s i d e r i n g a n Arens-MichaeZ

decomposition o f a g i v e n a l g e b r a E o n e g e t s ( s e e a l s o t h e same r e l a t i o n above f o r n = I )

F u r t h e r m o r e , w e c o u l d g i v e s t i l l some a p p l i c a t i o n s of

(3.32), a

t y p i c a l and b a s i c a s w e l l example of s u c h a n a p p l i c a t i o n b e i n g , of c o u r s e , t h e p r e v i o u s Theorem 4 . 1 .

However, w e p r o c e e d t o t h e n e x t sec-

I11 PROJECTIVE LIMIT ALGEBRAS

94

t i o n , where w e c o n s i d e r a more g e n e r a l s i t u a t i o n , namely,

complete

(or yet

advertibly

a-complete; c f . Scholium 6 . 2 below) l o c a l l y m-convex al-

g e b r a s ( D e f i n i t i o n I;6.3). Thus t h e l a t t e r t h e y do s u f f i c e f o r t h e app l i c a t i o n s u n d e r d i s c u s s i o n i n s t e a d of c o n s i d e r i n g c o m p l e t e ( l o c a l l y m-convex) a l g e b r a s . A h i n t t o t h i s t y p e of r e s u l t s w e have had a l r e a d y i n Chapt.11; S e c t i o n 7 i n c o n n e c t i o n w i t h Theorem 11; 7 . 2 (see a l s o Cor o l l a r y 11; 7 . 2 ) ; a n e x t e n d e d form of t h i s t h e o r e m , a s w e l l a s of t h e above Theorem 4 . 1 ,

w i l l b e g i v e n below ( f o r l o c a l l y m-convex a l g e b r a s ) .

5. Advertibly complete l o c a l l y m-convex algebras W e g i v e i n t h i s s e c t i o n a f i n e r a n a l y s i s of t h e s i t u a t i o n d e -

s c r i b e d by t h e p r e v i o u s Theorem 4 . 1 .

So it h a s been n o t e d a l r e a d y t h a t

t h e h y p o t h e s i s of b e i n g t h e a l g e b r a s i n v o l v e d i n t h a t t h e o r e m coinplete i s n o t n e c e s s a r y i n t h e " o n l y i f " p a r t o f t h e a s s e r t i o n (see t h e R e -

mark f o l l o w i n g t h a t t h e o r e m ) . Now it i s o u r a i m i n t h e n e x t l i n e s t o g i v e a c h a r a c t e r i z a t i o n of t h e c o n d i t i o n s t h a t a r e , i n d e e d , n e c e s s a r y i n t h e " i f " p a r t of t h e same t h e o r e m i n s t e a d o f c o n s i d e r i n g c o m p l e t e a l g e b r a s . So i t i s p r o v e d t h a t a d v e r t i b l y complete algebras s u f f i c e ; i n ad-

d i t i o n , t h i s a l s o characterizes the respective s i t u a t i o n

(see , f o r i n s t a n c e ,

C o r o l l a r y 5.1 a s w e l l a s S c h o l i u m 5 . 1 ) . S e v e r a l a p p l i c a t i o n s of t h i s f a c t w i l l b e g i v e n i n t h i s and t h e n e x t s e c t i o n , i n c l u d i n g a n e x t e n -

sion of t h e p r e v i o u s f o r m u l a (4.4). Thus, w e f i r s t h a v e t h e f o l l o w i n g . h

Lemma 5.1. Let E be a l o c a l l y m-convex algebra, lc i m E a an Arens-Michael decomposition of E corresponding t o a local basis (Ua l a €I of E ( c f . D e f i n i t i o n 3.1 1, and x an element of E . Then, the following a s s e r t i o n s are equiva-

r=

lent: 11 The given element x i n E i s "coordinatewise quasi-regular" i n the sense

that, f o r every ~ E It h,e element

p (21 i s quasi-regular i n

a* there e x i s t s an element z = l z 1~ n i.: i n such a way t h a t a aeIa' p,(x) o z a = z c 01 pa Ixl = 0, (5.1)

2%

GP,

equivalently,

f o r every a E I . I n p a r t i c u l a r , one obtains (by I S . 111 t h a t z e lzEa.

2) There e x i s t s a Cauchy f i l t e r

on E such t h a t

(5.2)

and, moreover, (5.3)

f o r every a~ I . I n p a r t i c u l a r ,

limpafF1 = za E E a ,

7

converges t o the element z = l z a 1 i n fl ga a €I

.

5. ADVERTIBLY COMPLETE ALGEBRAS

95

Proof. Suppose t h a t ( 5 . 1 ) i s t r u e : t h e n , o n e h a s pJx) o & ( z B I = 7

aB

-

so t h a t

163(z)

7aB ( z BI = za ,

j&(ZB0

( p ( x ) l o f ( z ,=f( p ( G aB B aB G

PG(X))

=

0

p,(xl

X j O Z

R

I

= 0

f o r a n y a 5 B i n I , h e n c e z = (z

.

limi NOW, i f + a d e n o t e s t h e f i l t e r o f n e i g h b o r h o o d s of z i n t h e l a t t e r a l g e b r a ,

a

)€

s i n c e E = @ ( E ) i s d e n s e i n it ( C o r o l l a r y 3 . 1 ) , w e c o n c l u d e t h a t @-’(&(z) n @ i E l l i s a Cauchy f i l t e r b a s i s on E , so l e t ‘it b e t h e r e s p e c t i v e Cauc h y f i l t e r on E g e n e r a t e d b y t h i s . W e show n e x t t h a t ( 5 . 3 ) i s s a t i s f i e d , which e n t a i l s t h a t t h e f i l t e r

on E t h u s c o n s t r u c t e d conver-

g e s t o t h e e l e m e n t z = ( z ) €12; C n E a . I n d e e d , i f W i s a n e i g h b o r h o o d a a a t h e s e t V= n V , w i t h V = W a n d V = of za i n , f o r every B#a i n BEI 6 a B I , d e f i n e s a n e i g h b o r h o o d of z = l z i n fl E^ So $J-’(V n ( l i m k a i n @ i E ) I be-

zB

c1

l o n g s t o ‘F, h e n c e o n e o b t a i n s

pa(c~-’(vn(izE”a)n @(EI) =

c1a

.

t

( by (3.24) and

(2.6))

which p r o v e s t h e a s s e r t i o n . T h e r e f o r e ,

f o r e v e r y a e I , which p r o v e s ( 5 . 2 ) suppose t h a t 2 ) i s v a l i d ( i . e . t h e map p

,

,

t h a t i s 1 i implies 2). C o n v e r s e l y

,

( 5 . 2 ) and ( 5 . 3 ) a r e t r u e ) ; t h e n , s i n c e

a i s c o n t i n u o u s , one g e t s 0 = p a( l i m ( x o n ) = l i m p a ( x o F ) = pa(x)olinpa(Tl

= pa(xl

0 za

= pa(lim

(70x)) = za

f o r e v e r y a E I , t h a t i s w e g e t (5.11,

0

pJx)

from which a l s o f o l l o w s , a s i n

t h e f i r s t p a r t o f t h e p r o o f , t h a t z = ( z a ) f lim; c

and t h i s t e r m i n a t e s

a’

t h e p r o o f of t h e lemma. I

Remark.- C o n c e r n i n g t h e e l e m e n t 5.1,

lc hfa

z

3

(z l e

n

h

E

a aeI a

i n t h e a b o v e Lemma

i t was c o n c l u d e d i n b o t h of t h e s t a t e m e n t s of t h i s lemma t h a t

Z E

s u c h t h a t t h e f i l t e r ’;Gin 2 ) c o n v e r g e s t o z w i t h r e s p e c t t o t h e

same a l g e b r a limz c a

.

F u r t h e r m o r e , one o b t a i n s by ( 5 . 2 ) lim(xo;rl= x o l i m 7 = x o z = 0

a n d s i m i l a r l y z o x = 0 , t h a t i s z is the quasi-inverse

iiz

=

l.Fa

of the

I11 PROJECTIVE LIMIT ALGEBRAS

96

given element x

E E.

Thus, Lemma 5.1 provides already a strengthening of

Theorem 4.1,l) informing us that if an element X E E is "coordinatewise quasi-regular'' (see the terminology applied in the above lemma), then it does have a quasi-inverse, say z , but belonging to the completion of the algebra E l that is z e i = l i m i In fact, we get a better infort a' mation (thatis, location of the element z in question) through the next Corollary 5.1 ; namely, we r e a l i z e t h a t z belongs t o E , i f t h e given a l gebra E i s advertibly complete, i.e., "complete with respect to taking adverses" in it (cf. the relevant terminology in Section 11; 6).

Corollary 5 . 1 . Let

E be a l o c a l l y m-convex algebra. Then, t h e following pru-

p o s i t i o n s are equivalent: 1 ) E i s a d v e r t i b l y complete (Definition I; 6.3).

(and only if) it i s coordinate(cf. the terminology of the previous Lemma 5.1 ) .

2) An element x i n E i s quasi-regular,if

wise quasi-regular

Proof. If an element x E E , with $ ! X I = ( ~ , ( x l ) e l i m ~ ,(cf. the notat tion of Section 3 ; (3.22)) is quasi-regular, then (Lemma 5.1),there exists a Cauchy filter 7 on E satisfying (5.2). Hence (cf. Definition I; 6.3 and subsequent comments there), i f E i s a d v e r t i b l y complete 7 converges to an element y in E l so that as easily follows from (5.2) one gets x o y = y o x = 0 ; thus y is the quasi-inverse of x in E , which proves t h a t 1 ) implies 2 ) . Now suppose that P i s a Cauchy filter on E which satisfies (5.2) for some element x in E . On the other hand, by hypothesis and Theorem 3.1, 7 converges to an element z = ( z 01I in lc b E a 2 E l so that by (5.2) one obtains 2 o x = x o z = 0 in E , hence also the relation (5.1). That is, the given element x € E i s coordinatewise quasi-regular, therefore if 2) i s v a l i d then x is quasi-regular in E; namely, there exists y e E 5 E , with ; c o y = y o x = 0 , thus by the relation just proved before one has z = ~ E E , and this finishes the proof.(In this respect, we finally remark that the condition in 2 ) is always necessary; see also the Remark following Theorem 4.1). I h

r

_

h

The next lemma is needed below (cf.Theorem 5.1). It exhibits a stronger condition for the quasi-regularity of a given element in a locally m-convex algebra than the two equivalent propositions of COrollary 5.1. Thus, we have.

Lemma 5 . 2 . Let E be a l o c a l l y m-convex algebra. Moreover, consider the f o l lowing condition : An element x i n E i s quasi-regular i f

Lemma 11; 7.4) one has the r e l a t i o n

(and only if; cf.

97

5 . ADVERTIBLY COMPLETE ALGEBRAS f l x l z 1, f o r every continuous character f of E .

(5.4)

Then, ( 5 . 4 ) i m p l i e s e i t h e r one o f t h e two equivalent propositions i n Corollary 5 . 1 .

7 b e a Cauchy f i l t e r on

Proof. L e t

E such t h a t ,

f o r some e l e m e n t

x i n E l t h e f o l l o w i n g r e l a t i o n t o be t r u e :

lim(xoF,J=lim(%oxl = o .

(5.5)

Thus, a s s u m i n g t h a t ( 5 . 4 ) i s v a l i d w e s h a l l show t h a t x i s a q u a s i r e g u l a r e l e m e n t o f E: F o r , o t h e r w i s e , t h e r e would e x i s t , by ( 5 . 4 ) , an e l e m e n t f e m l E ) , w i t h f ( x ) = l . NOW, by t h e c o n t i n u i t y o f f , t h e set

U= I y

1 f (yl I < 1 )

EE :

%,

thesis for

i s an open

there exists B

n e i g h b o r h o o d of 0 i n E ; hence I by hypo8

?,

w i t h xoB = { x o y: y E B )C U, h e n c e

\ f i x o y l l < I , f o r e v e r y y e B . But o n e s t i l l o b t a i n s f ( x 0 y ) = f ( x + y - x y ) = f(x) +f(yi-f(xif(yl = f ( x i = 1 , f o r a n y y e B , t h u s a c o n t r a d i c t i o n t o t h e l a s t c o n c l u s i o n . Hence, x i s i n d e e d a q u a s i - r e g u l a r e l e m e n t o f E ; t h e r e f o r e , t h e r e e x i s t s an e l e ment z e E , w i t h x

02

=

?:

o x = 0,

so t h a t one h a s

= p,lzl

p,(xlo p J z l

(5.6)

0

p,(xi = 0 ,

f o r e v e r y a e I . F u r t h e r m o r e , by h y p o t h e s i s f o r

F,

p c l ( F ) i s a Cauchy

h

filter in E

a'

while t h e l a t t e r is

a complete a l g e b r a , hence

l i m p p - i = Y,E

(5.7)

Ecl

f o r e v e r y ,€I. T h e r e f o r e , by ( 5 . 5 ) a n d ( 5 . 7 1 , l i m pa(

X O F )

= l i m ~ p , ( x i o p a ( m = pcl(x)o(limpc1(?,) = (limp,(

t h a t i s , p,(xl one h a s y

one o b t a i n s

F,)0 pa(x)

o y , = y, o p,(x) = 0 ,

=0

f o r every

c1 E

I. C o n s e q u e n t l y , by ( 5 . 6 )

c =i pa ( z ) , w i t h a z I ; t h u s w e f i n a l l y g e t y = ( y a ) = ( p c1( z ) l =

;

E

E,

so t h a t by ( 5 . 7 ) t h e f i l t e r 7 c o n v e r g e s t o a n e l e m e n t y e E , t h a t i s t h e algebra E i s a d v e r t i b l y complete. Thus, w e h a v e p r o v e d t h a t t h e c o n d i t i o n ( 5 . 4 ) i m p l i e s p r o p o s i t i o n 1 ) i n C o r o l l a r y 5 . 1 , and t h i s f i n i s h e s t h e p r o o f o f t h e lemma. I N o w by c o n s i d e r i n g commutative a l g e b r a s t h e p i c t u r e p r o v i d e d by

t h e p r e v i o u s Lemma 5 . 2 i s f u r t h e r s u p p l e m e n t e d t h r o u g h t h e n e x t t h e o -

r e m . The same r e s u l t may b e c o n s i d e r e d a " c o n t i n u o u s a n a l o g o n " o f Lemma 11; 7 . 4

a n d y i e l d s a l s o a c h a r a c t e r i z a t i o n of commutative a d v e r t i b -

l y c o m p l e t e l o c a l l y m-convex a l g e b r a s . Thus w e h a v e .

Theorem 5.1.

Let E be a commutative l o c a l l y m-eonvex aZgebra. Then the f o l -

98

111 PROJECTIVE LIMIT ALGEBRAS

lowing a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a d v e r t i b l y complete. 2 ) An element

x in E i s quasi-regular if (and only i f ) one has t h e r e l a t i o n f(x/ # 1 ,

(5.8)

f o r every continuous character f of E , i . e . , f o r every element f

E

m(E).

We first remark that by Lemma 5.2 c o n d i t i o n (5.8) (which, anyhow, is necessary: see Lemma 11; 7 . 4 ) always implies I / , i.e., independently of the commutativity of the algebra E ; thus the only thing we have to prove is that under t h e supplementary hypothesis t h e algebra E t o be c o m t a t i v e , 1 ) i m p l i e s 21. Indeed, let x be an element of E which satisfies ( 5 . 6 ) and suppose, otherwise, that x is quasi-singular. Then (Lemma 5.2 in conjunction with Corollary 5.1), there would exist an index cic I such that p,lxl to be a quasi-singular element of the c o m t a t i v e Banach algebra ?a S o there would exist a character of 2, , i.e. , a continuous morphism h : -C, with h ( p (xll = 1 (see, for instance, 0. a01 Chapt. 11, Lemma 7.4 and Corollary 7.3): thus, one gets an elementf= h,o p, € , with f ( x l = I , hence a contradiction to the hypothesis for x , and this finishes the proof of the theorem.1 Proof.

.

m(E/

In particular, for algebras having an identity element one gets the following

Corol1 a r y 5 . 2 . L e t E be a c o m t a t i v e a d v e r t i b l y complete local l y m-eonvex algebra w i t h an i d e n t i t y element. Then, an element x e E i s i n v e r t i b l e i f (and only i f ) one has (5.9)

SIX)

# 0,

f o r every f e m(E). Proof. Since c o n d i t i o n 15.91 is necessary (even for not necessarily commutative algebras: cf. Lemma 11: 7.4 and 11: (1.4) or yet 11;(7.19)), it remains to prove the "if" part of the assertion. Thus, if (5.9) is valid for some x e E , and e denotes the identity element of E , then for every f E ~ R ~ Eone ) gets the relation

(5.10)

fle-x)=fle)-flxcl

= l-flxl # I .

Hence, by ( 5 . 8 ) , e - x is quasi-regular, therefore, x a regular element of E(cf. 11:(1.4)), and this finishes the proof. In connection with the above result, we still remark that i n t h e presence of an idenCity element,in a given c o m t a t i v e l o c a l l y m-convex algebra E , t h e above two conditions ( 5 . 8 ) and (5.91 are indeed e q u i v a l e n t : The fact that

6. SPECTRAL PROPERTIES

99

f o l l o w s a l r e a d y from t h e l a s t p r o o f . Moreover,

(5.8) implies (5.9)

if

( 5 . 9 ) i s v a l i d , t h e n a p p l y i n g a s i m i l a r argument t o t h a t i n ( 5 . 1 0 1 ,

o n e now g e t s

f 1 e - 2 ) = f 1 e l - f1xl = 1 - f ( x l

(5.11)

+ 0,

f o r e v e r y f & m 1 E ) ; h e n c e , by h y p o t h e s i s , e - x i s a r e g u l a r element of E l t h e r e f o r e , x q u a s i - r e g u l a r .

Thus ( 5 . 8 ) i s f u l f i l l e d .

So one c o n c l u d e s t h a t

under t h e supplementary assumption t h a t E is a

commutative a l g e b r a , e i t h e r one of t h e c o n d i t i o n s 15.81

(5.12)

or 15.9) may be taken a s an equivalent statswent t o t h e hyp o t h e s i s t h e algebra E t o be a d v e r t i b l y complete.

.-

Schol ium 5.1 The above d i s c u s s i o n p r o v i d e s a l r e a d y an extension of L?ropositions 1 ) and 3 ) of Theorem 4 . 1 of t h e p r e v i o u s s e c t i o n t o t h e case of a d v e r t i b l y complete l o c a l l y m-convex algebras. Thus , p r o p o s i t i o n 1 ) of t h a t t h e o r e m i s c o n t a i n e d i n C o r o l l a r y 5 . 1 , which a l s o i n d i c a t e s t h a t t h e p r o p e r t y i n q u e s t i o n c o n s t i t u t e s , i n e f f e c t , a c h a r a c t e r i z a t i o n of " a d v e r t i b l e c o m p l e t e n e s s " f o r a g i v e n l o c a l l y m-convex a l g e b r a . F u r t h e r m o r e , p r o p o s i t i o n 3 ) of t h e same t h e o r e m c a n b e d e r i v e d , w i t h i n t h e c o n t e x t of t h i s s e c t i o n , by r e m a r k i n g i n a n a l o g y w i t h C o r o l l a r y 5.2 t h a t i f t h e algebra considered 710s an i d e n t i t y element, then p r o p o s i t i o n s 1 )

and 3 ) o f t h e same t h e o r e m are equivalent for any l o c a l l y m-convex algebra.

6. Spectral properties o f advertibly complete locally m-convex a1 gebras W e e x t e n d i n t h i s s e c t i o n t h e p r e c e d i n g r e l a t i o n s ( 4 . 3 ) and ( 4 . 4 )

t o a d v e r t i b l y c o m p l e t e l o c a l l y in-convex a l g e b r a s . I n p a r t i c u l a r , i f t h e a l g e b r a s i n v o l v e d a r e commutative, w e a l s o o b t a i n f u r t h e r c h a r a c t e r i z a t i o n s of t h e s a m e c l a s s of a l g e b r a s v i a Lemma 5 . 2 a n d Theorem 5.1.

F u r t h e r m o r e , c e r t a i n s p a c i a l f e a t u r e s o f l o c a l l y m-convex Q-al-

g e b r a s are d e r i v e d on t h e b a s i s of b e i n g a d v e r t i b l y c o m p l e t e ( c f . Theorem I ; 6 . 4 ) . Thus w e s t a r t w i t h t h e f o l l o w i n g e x t e n s i o n of C o r o l l a r y 4 . 1 .

Theorem 6.1. Let E be an a d v e r t i b l y complete l o c a l l y m-convex algebra. Then, by considering an Arens-Michael decomposition of E ( c f . D e f i n i t i o n 3 . l ) , t h e s p e c t r a l radius of an element x i n E ( c f . 11; ( 1 . 1 0 ) ) i s given by (6.1)

r (x) = s u p r i I E

a c 1

[.el c1 I

= sup lim ( p C 1 ( x nI ,lh c1 n-tm

.

I n p a r t i c u l a r , if E has an i d e n t i t y element, then t h e spectrum of an element x i n E

100

I11 PROJECTIVE LIMIT ALGEBRAS

f u l f i l s the relation (6.2)

hoof. 1 11; 1 - 2 1 , - x

some

M

If A # 0 , then

AeSpE(x) i f , and o n l y if ( c f . D e f i n i t i o n

is quasi-singular;

A € I ,s u c h t h a t

tTxIM 1

S ~ - ( [ X ] ~ and ) , conversely,

hence ( C o r o l l a r y 5 . 1 1 ,

there exists

= [z], t o b e q u a s i - s i n g u l a r , t h u s X E s o t h a t by a p p l y i n g ( I I ; ( l . l O ) ) one g e t s

( 6 . 1 ) . On t h e o t h e r h a n d , i f E h a s a n i d e n t i t y e l e m e n t , t h e above a r gument, c o n c e r n i n g a n e l e m e n t A # 0 i n SpE(x), w i t h x e E , s t i l l a p p l i e s ; moreover, i f

U E SpElxl t h e n x i s s i n g u l a r , h e n c e t h e same i s t r u e , f o r

some [ z ] , , w i t h M e I ( c f . C o r o l l a r y 5.1 and Scholium 5.1 ) i s v a l i d and t h i s c o m p l e t e s t h e p r o o f . 1

.

Thus,

(6.2)

Now a n i m m e d i a t e c o n s e q u e n c e o f ( 6 . 1 ) i s t h e f o l l o w i n g r e s u l t r e f l e c t i n g t h e f a m i l i a r s i t u a t i o n o n e h a s i n case o f Banach a l g e b r a s .

Corollary 6.1. L e t E be an a d v e r t i b l y complete l o c a l l y m-convex algebra. Then, concerning t h e s p e c t r a l radius r (x) of an element x i n E , one has t h e r e l a t i o n s :

E

1 ) r E ( A x ) =\Al*rE(x),w i t h

A€@.

2) rE (xy) = r E ( y . L ' ) , f o r any elements x, y i n E . 3 ) r E ( x n )= (r ( z ) j n ,f o r every E

n e IN.

In p a r t i c u l a r , f o r every p a i r ( x , y ) of commuting elements of E one o b t a i n s : 4) r (x + y ) 2 rE(x) + r f y ) . E

E

5) rE ( z y ) 5 rEix).rE(g).

Proof.

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

r ( x j = suprA ( [ x

(6.3)

E

0.

which c o r r e s p o n d s , by ( 6 . 1 )

,

EM

t o a n A r ns-Michael

d e c o m p o s i t i o n of E

a n d t h e r e s p e c t i v e r e l a t i o n s one h a s f o r any normed a l g e b r a ( c f . , f o r i n s t a n c e , C.E. RICKART [I: p. 10, Theorem 1.4.11). 1 I n t h i s r e s p e c t , it i s c l e a r , of c o u r s e , t h a t

( 6 . 3 ) ( o r y e t 11;

( 1 . 1 0 ) ) may g i v e i n g e n e r a l r E ( x / = + m ( c f . , f o r i n s t a n c e , E . A . MICHAEL 11:

p. 58, Examples 13.3; b, c])

.

The f o l l o w i n g i s now a n i m m e d i a t e c o n s e q u e n c e o f C o r o l l a r y 6 . 1 . Namely, w e h a v e .

Corollary 6.2. For every commutative a d v e r t i b l y complete l o c a l l y m-convex a l gebra E , t h e r e s p e c t i v e s p e c t r a l radius rE d e f i n e s a subrriultiplicative semi-norm on

he s e t B ( E ) = { I I : : e E : r,(xi

6.4)

<+-I,

h i c h t h u s becomes a subaZgebra o f E. 1

On the other hand, we still conclude by the preceding discussion .he next useful result. P r o p o s i t i o n 6.1. L e t E be an a d v e r t i b l y complete l o c a l l y m-convex ulgebra.

'hen, t h e s e t N = { x e E : rE(x)< 1 )

6.5) s

contained i n t h e group G'

o f quasi-regular elements o f E.

Proof. By considering an Arens-Michael decomposition of E (cf.

)efinition 3 . 1 ) ,

one concludes by hypothesis and (6.3) the relation r" i [ x I a i < I

16.6)

,

EU

:gr every ole I. Thus (cf. C.E. RICKART [I: p. 18, Lemma 1.4.18]), [x], is 1 quasi-regular element of the Banach algebra for every index aEI. a' ;o the assertion is now a consequence of Corollary 5.1.1

As an application of the previous result (or yet of Corollary 5.1), we further yet the following

Corollary 6 . 3 . L e t E be an a d v e r t i b l y complete l o c a l l y m-convex algebra. Furr = { p }, CL e l , be a (fundamental) f a m i l y of s u b m u l t i p l i c a t i v e seminorms d e f i n i n g t h e topology o f E (cf.Theorem I;3.1 ). Then, t h e following condithermore, l e t

tion i s satisfied: For any element x i n E , t h e r e l a t i o n

pa(")

(6.7.1) f o r every a

E

I,

I, i m p l i e s t h a t

II:

i s a quasi-regular

element o f E.

Proof. By (4.6) and the hypothesis one has

for every a e I (see also C . E . RICKART[l:p.IO,Theorem

1.4.11). Thus,

[XI,

is a quasi-regular element of Ea (ibid., p. 18,Corollary 1.4.19), so that again the assertion follows from Corollary 5 . 1 (or yet, already from the previous Proposition 6 1, since then rE (II:) < I ; cf. (6.3) and (6.8)) I

.

In this respect, we further note that recently B . H. DAYTON [I]

I

studying I:-Theory of (commutative) Banach algebras, has considered

I11 P R O J E C T I V E L I M I T ALGEBRAS

102

normed a l g e b r a s ( c o m m u t a t i v e w i t h i d e n t i t y e l e m e n t s ) , s a y ( E , which f u l f i l t h e a n a l o g o u s c o n d i t i o n t o ( 6 . 7 )

(1 11 , *

above; namely, s u c h t h a t ,

f o r any X E E , t h e r e l a t i o n

(6.9) implies t h a t r is a quasi-regular cf.

(11; ( 1 . 4 ) ) . Such a l g e b r a s

(hence, 1-:c i s regular;

e l e m e n t of E

a r e named s p e c i a l algebras ( i b i d . ) .

N o w a p p l y i n g t h e p r e c e d i n g t e r m i n o l o g y , one c a n p u t i n s t e a d t h e

r e l e v a n t t h e o r y w i t h i n t h e framework of m-convex

( a d v e r t i b l y complete) l o c a l l y

a l g e b r a s : Thus i t i s p r o v e d , f o r i n s t a n c e , t h a t every l o c a l l y

m-convex algebra w i t h an i d e n t i t y element, whose compZetion i s a &-algebra ( h e n c e , a d v e r t i b l y c o m p l e t e ; c f . Theorem I ; 6 . 4 ) , has a " l o c a l i z a t i o n " which

sat-

i s f i e s t h e c o n d i t i o n ( 6 . 7 ) . C o n s e q u e n t l y , every normed algebra with an i d e n t i t y element has a " l o c a l i z a t i o n " which i s a s p e c i a l algebra ( B . H . Dayton ; i n t h i s r e s p e c t , see a l s o A . MALLiOS [28]). F i n a l l y , w e s t i l l

remark t h a t f o r

any normed a l g e b r a E l t h e p r o p e r t y of b e i n g E a s p e c i a l a l g e b r a i s e q u i v a l e n t w i t h E t o b e a & - a l g e b r a , o r y e t a d v e r t i b l y complete ( c f . S. EIARNER [3: p. 8 , Theorem 71 )

.

The f o l l o w i n g d i s c u s s i o n i s now a n o t h e r r e s u l t of t h e p r e v i o u s t e r m i n o l o g y . We p u t it i n t o t h e form of t h e n e x t

Scholium 6.1.-

Thus, s u p p o s e w e t a k e t h e above r e l a t i o n ( 6 . 1 ) f o r

d e f i n i t i o n o f r E f x ) , x e E . Then, one

proves t h a t :

For any l o c a l l y in-convex algebra E , and f o r every element x e E, w i t h r E ( x ) (6.10)

+m

, t h e r e e x i s t s A € SpElx)

such t h a t

rE ( x ) 6

(6.10.1)

1x1.

The p r o o f i s p a t t e r n e d a f t e r t h a t v a l i d f o r normed a l g e b r a s ( s e e , f o r i n s t a n c e , C. E . RICKART [l: p. 28, Theorem 1.6.31). b a s i s of P r o p o s i t i o n 6 . 1 ,

On t h e o t h e r hand, on t h e

w e conclude t h a t : For every a d v e r t i b l y complete l o c a l l y in-convex a'lge-

(6.11)

bra E , and f o r any xE E , t h e r e l a t i o n (6.11.1)

x e SpEfxi

implies

( C f . a l s o C o r o l l a r y 6 . 1 , l ) ) . Thus a c o m b i n a t i o n o f

1x1 5

rE(x).

( 6 . 1 0 ) and ( 6 . 1 1 )

y i e l d s now t h e f o l l o w i n g :

For every a d v e r t i b l y complete Zocal'ly in-convex alge(6.12)

bra E and f o r any r e E , w i t h rE(xcl < + a the relation

, one has

6.

103

SPECTRAL PROPERTIES

f o r some A,

8

SpE(z).

The p r e v i o u s r e l a t i o n , a u s e f u l f a c t i n c l a s s i c a l Banach a l g e b r a s a l g e b r a s t h e o r y (see, f o r example, C. E . RICKART [l: p. 30, Theorem 1.6.4.]), i s h e r e r a t h e r c i r c u m v e n t e d by t h e same d e f i n i t i o n w e p r e v i o u s l y as-

sumed, namely, t h e r e l a t i o n 11; ( 1 . l o ) . Moreover, by D e f i n i t i o n 1 I ; l .2, SpEfx), X E E , i s a non-empty

s u b s e t o f t h e complexes ( s e e a l s o Chapt.11;

Theorem 4 . 2 a n d t h e r e l e v a n t d i s c u s s i o n t h e r e ) . NOW,

t h e f o l l o w i n g s u p p l e m e n t s Lemma 11; 4 . 2 .

Thus, w e h a v e .

Proposition 6 . 2 . Let B be a l o c a l l y In-convex algebra. Then, t h e following a s s e r t i o n s are equivalent: 11 The s e t S l E l = { x E E : r (xi =< 1 E

(6.13)

i s a neighborhood of zero i n E . 21 E i s a 4-algebra. 31 The spectral radius rE ( c f

. 11; ( 1 . l o ) )

i s continuous a t 0 E E .

In p a r t i c u l a r , .
+ m ,

a n d C o r o l l a r y 11; 4 . 3 ) ; t h e r e f o r e , borhood of O e E (Lemma 11; 4 . 2 ) , t i n u i t y of rE a t 0

€

E

,

t h a t is

for

every

XE

E ; c f . Theorem I ; 6 . 4

s i n c e rE(O1= 0 and

SlEl is a neigh-

one g e t s by C o r o l l a r y 6 . 1 , 1 ) t h e con2) implies 3 ) . F u r t h e r m o r e , i f 3 ) i s t r u e

t h e n by t h e r e l a t i o n r E ( 0 ) = 0 and o u r h y p o t h e s i s , t h e r e e x i s t s a n e i g h borhood V of z e r o i n E , w i t h rF(x71=< 1

,for

every x

€

V , so t h a t V C S I E ) ,

t h a t i s E i s a & - a l g e b r a (Lemma 11; 4 . 2 ) ; t h u s 31 implies 2) a s w e l l . On t h e o t h e r h a n d , i f t h e a l g e b r a E i s c o m m u t a t i v e , t h e n 3 ) i m -

p l i e s 4 1 . I n d e e d , b y h y p o t h e s i s and t h e f a c t t h a t 3 ) = + 2 ) , rE i s by Coroll a r y 6 . 2 a s u b m u l t i p l i c a t i v e semi-norm on E , s o t h a t t h e a s s e r t i o n

follows t h e n from P r o p o s i t i o n I ; 1 . 5 . NOW, t h e o t h e r i m p l i c a t i o n i s obv i o u s , and t h i s t e r m i n a t e s t h e p r o o f . I A s a n o t h e r a p p l i c a t i o n of t h e p r e c e d i n g d i s c u s s i o n , w e c o n s i d e r

n e x t some f u r t h e r c h a r a c t e r i z a t i o n s of a d v e r t i b l y c o m p l e t e a s w e l l a s

of & - a l g e b r a s , i n p a r t i c u l a r , when t h e a l g e b r a s i n v o l v e d a r e commuta-

I11 PROJECTIVE LIMIT ALGEBRAS

104

tive. Thus, we first have.

Theorem 6.2. L e t E be a l o c a l l y m-convex algebra. Moreover, consider t h e following two a s s e r t i o n s : 1 ) Suppose t h a t t h e following r e l a t i o n i s tr ue .

(6.14)

SpE(xi U C O ) = 2 IMIE)'),

with x e E (cf. the notation applied in Corollary 11; 7.4 ) .

21 E i s a d v e r t i b l y complete. Then, l / i m p l i e s 2 / . In p a r t i c u l a r , if t h e algebra E is commutative, then t h e preceding two a s s e r t i o n s are equivalent.

Indeed, 16.141 implies ( 5 . 4 ) . Thus, if flx) # 1 , for every f e by (6.14) 1 e'SpE(xi, that is x is a quasi-regular element of E , which proves the assertion (see also Lemma 11; 7.4). Thus, 1 ) i m p l i e s Proof.

m(E),then

2) byLemma 5.2. Furthermore, if the algebra Z is commutative and advertibly complete, then b y Theorem 5.1, h € SpE(xl i f , and o n l y i f , there I exists an element fem(El, with f(x;cc =j1 , that is f i x ) = 2Cfj = X ; so A

belongs to the second member of (6.14). Thus, in case E is commutative, then 2) implies 1 ) as well, and this finishes the proof. I Now on the basis of the preceding proof, we still conclude the following useful result.

Corollary 6.4. L e t E be a c o m t a t i v e l o c a l l y rn-convex algebra w i t h an identi t y element. Then, E i s a d v e r t i b l y complete i f , and only i f , one has t h e r e l a t i o n (6.15)

G(m(E)l

= SpE(x) #

0

( s e e also Corollary II; 4.2), for every element

ic E

E. I

Furthermore, one gets the following corollary of Theorem 6.2, which also provides a strengthening ofTheorem II;7.2, as well as a supplement of the previous Corollary 6.2. Thus, we have.

Corollary 6.5. L e t E be a commutative a d v e r t i b l y complete l o c a l l y m-convex a l g e i r a (hence, in particular, a topological algebra for which (6.14)

is valid!. Then, one hcs t h e r e l a t i o n (6.16)

for every element x € E . Moreover one obtains (6.17)

S I E ) = (m(El)'

;

t h u s t h e s e t S 1 E ) (cf. (6.13) i s an m-barrel of t h e l o c a l l y m-convex algebra B l E )

6. SPECTRAL

(cf

. (6 .4 ) 1 Proof.

105

PHOPERTiES

when t h e l a t t e r is endowed w i t h t h e r e l a t i v e topoZogy from E . The d e s i r e d r e l a t i o n ( 6 . 1 6 )

t i o n of t h e s p e c t r a l r a d i u s and ( 6 . 1 4 ) ,

i s s t r a i g h t f o r w a r d by d e f i n i while (6.17) again follows

e a s i l y from ( 6 . 1 6 ) a n d t h e same d e f i n i t i o n s . Thus one now c o n c l u d e s t h a t S(El is

a c l o s e d a b s o l u t e l y c o n v e x and i d e m p o t e n t s u b s e t of E ,

b e i n g b y ( 6 . 1 7 ) t h e p o l a r s e t i n E of m ( E I C E ’ ; m o r e o v e r , i t i s by d e f i n i t i o n a subset of

B(El.

Thus, f o r e v e r y element x

E

B ( E ) , with rE(x)

0 ( t h e c a s e r ( 2 ) = 0 i s o b v i o u s ) , one g e t s by C o r o l l a r y 6 . 1 ; 1 ) E 1 t h a t y x E S ( B ) , s o t h a t t h e l a t t e r s e t i s an a b s o r b i n g s u b s e t of t h e

=Y

( l o c a l l y m-convex) a n m-barrel

a l g e b r a B i E l , hence

b y what h a s b e e n p r o v e d b e f o r e

o f t h e same a 1 g e b r a . D

I n p a r t i c u l a r , one g e t s t h e f o l l o w i n g r e s u l t , a more i n c l u s i v e v e r s i o n of which i s p r o v i d e d t h r o u g h t h e n e x t t h e o r e m (see a l s o C h a p t . V,

Scholium 2 . 2 ) .

Corollary 6 . 6 . L e t E be a commutative a d v e r t i b l y complete l o c a l l y m-convex a2gebra. Then, t h e following p r o p o s i t i o n s are e q u i v a l e n t : 1 ) E is a &-algebra.

2) m ( E ) ( t h e t o p o l o g i c a l s p e c t r u m o f E ) is an equicntinu&&s s u b s e t

of E’.

Proof- BY P r o p o s i t i o n 11; 7 . 1 , 1 ) i m p l i e s 2 ) f o r e v e r y & - ( t o p o l o g i c a l ) a l g e b r a . The c o n v e r s e a s s e r t i o n f o l l o w s from ( 6 . 1 7 ) and Lemma 11; 4 . 2 . I

The f o l l o w i n g i s a n a p p l i c a t i o n o f t h e above C o r o l l . a r y 6 . 4 and w i l l f u r t h e r b e s u p p l e m e n t d by Theorem 1 . 3 of C h a p t e r V I

(see a l s o t h e

r e l e v a n t d i s c u s s i o n t h e r e ) . Thus, w e h a v e .

Lemma 6.1. L e t

,O

b e a dornnctntiv; a d v c r t i b l y complete m-barrelled

(Def i n i -

t i o n I ; 1 . 4 ) l o c a l l y m-convex algebra. Then, t h e f o l l o w i n g tuo a s s e r t i o n s are equiv-

alent: 1 ) E is a Q-algebra.

2) E = B i E ) ( c f . ( 6 . 4 ) ) .

Proof. I n d e e d , 1 ) i m p l i e s 2 ) f o r e v e r y t o p o l o g i c a l a l g e b r a ( c f . C o r o l l a r y 11; 4 . 3 ) . C o n v e r s e l y ,

i f 2 ) i s v a l i d , t h e n by C o r o l l a r y 6 . 4

S ( E ) i s an m-barrel o f E , so t h a t by h y p o t h e s i s

Thus E i s a Q - a l g e b r a by Lemma I I ; 4 . 2 , a n d

a n e i g h b o r h o o d of 0 e E

.

t h i s f i n i s h e s t h e pro0f.D

W e c o n c l u d e t h i s s e c t i o n by s u p p l y i n g a f u r t h e r c h a r a c t e r i z a t i o n

o f commutative & - a l g e b r a s , which w e h a v e p r o m i s e d b e f o r e , s u p p l e m e n t i n g t h u s t h e above C o r o l l a r y 6 . 5 .

Thus, w e h a v e .

I11 PROJECTIVE LIMIT ALGEBRAS

106

Theorem 6.3. Let E be a l o c a l l y m-convex algebra whose ( t o p o l o g i c a l ) spect m i s ???(El. Moreover,consider t h e following statements: 1 ) WiE) i s an equicontinuous subset of E’ and c o n d i t i o n ( 5 . 4 ) holds t r u e . 2 1 E i s a Q-algebra (DefinitionI; 6 . 3 ) . 31 E i s adverti.bly complete having an equicontinuous spectrum

m(E).

4)

m(E) i s

equicontinuous and every regular maximal i d e a l of E i s c l o s e d .

Then, one has t h e following i m p l i c a t i o n s : 1 ) 3 21 => 3) and 4 1 . I n particul a r , i f t h e algebra E i s c o m t a t i v e , then a l l t h e preceding four statements a r e equivalent. 1 Proof. If 1 ) is valid, then U=7lm(E1I0 is by hypothesis a 1 neighborhood of zero in E , such that if x e U , then If(xl I S 7, that is

f(x) # I , for every f e m ( E l . Thus by ( 5 . 4 ) x must be a quasi-regular element of E , so UC G q (group of quasi-regular elements of E 1 ; hence, 1 ) implies 2) according to Lemma I ; 6 . 4 . On the other hand, by Theorem I; 6 . 4 and Proposition 11: 7 . 1 , 21 i m p l i e s 31 while by Theorem 11; 6 . 1 2) i m p l i e s 4) as well. NOW, if E i s c o m t a t i v e , then by Theorem 5 . 1 , 3 ) i m p l i e s I). Furthermore, in that case too, 4) i m p l i e s 11. Indeed, suppose that fix1 # I, for every f em(E), while x is a quasi-singular element of E ; thus, x will belong to a regular maximal ideal, say M I of E being also an identity of E modulo M (cf. Chapt. 1I;Lemmas 6 . 3 and 6 . 4 ) . Therefore, since by hypothesis x is closed it will give rise to a continuous character, say f , of E (Corollary 11; 7 . 2 ) such that f(xl= I , namely, a contradiction to the hypothesis for x ; so x must be a quasiregular element of E, but this is the assertion, and this finishes the proof of the theorem. I

Remark.- Concerning the first statement of the above theorem, we note that t h e topolog i c a l adjunction of an i d e n t i t y element to the algebra E (cf. Lemma I; 5 . 3 ) r e s p e c t s proposition 1 ) , so that the algebra Ef( := E O C ) is d Q-aijebra too. The assertion follows from Lenma 5.2 and Corollary 5 . 2 in conjunction with 11; (1.4): besides, concerning the equicontinuity of ??L?iEtl see also Chapt. VIII; Lemma 1.1. Scholium 6.2.- It is a result of the previous discussion that, by considering a locally m-convex algebra E and an element x in E , i f x i s quasi-regular i n E, then it i s a l s o coordinatewise quasi-regular; this, of course, in the sense that, by taking an Arens-Michael decomposition of of E ; E , say l>?a (corresponding to a given local basis { U,la cf. Definition 3.1 ) , [ x ] , is a quasi-regular element of the respective

n=

Banach algebra

2, , for

every a e I. Consequently, one obtains

6. SPECTRAL

f o r e v e r y x in E ,

PROPERTIES

107

so t h a t o n e g e t s

(6.19)

for e v e r y x e E

.

On t h e o t h e r h a n d , i f the l o c a l l y m-convex algebra E i s s e q u e n t i a l l y com-

p l e t e (or e l s e 0 - c o m p l e t e ) , o n e can p r o v e t h a t ( 6 . 1 9 ) i s , i n e f f e c t ,

on

e q u a l i t y (i.e., ( 6 . 3 ) a g a i n h o l d s t r u e ) , so t h a t one then concludes the above Corollaries 6 . 1 and 6.2 f o r any a-complete l o c a l l y m-convex algebra ( c f . E . KILLAM [l : p. 584, Theorem 1.51 )

.

This Page Intentionally Left Blank

109

Inductive

Limit

Algebras

CHAPTER I V

1 . Inductive systems o f algebras. Algebraic preliminaries Suppose w e a r e g i v e n a (non-empty) s e t of i n d i c e s , s a y I, direct-

ed ( u p w a r d s ) ; w e d e n o t e t h e p a r t i a l o r d e r i n I b y a S B ( e q u i v a l e n t l y , w e also w r i t e a t B).So

,with a , R

in I

f o r any a , B i n I t h e r e e x i s t s ,

by d e f i n i t i o n , an i n d e x y El, w i t h a 5 y and

5y

.

F o r c o n v e n i e n c e o f r e f e r e n c e , w e r e c a l l t h e d e f i n i t i o n of an " i n d u c t i v e s y s t e m ( r e s p . , l i m i t ) of s e t s " . T h u s , l e t (E,ia

I be a

f a m i l y of s e t s ( i n d e x e d by a d i r e c t e d s e t I as a b o v e ) and assume t h a t , f o r e v e r y p a i r of i n d i c e s (a,R,' i n I, w i t h a < 6,

w e are g i v e n a map

i n s u c h a way t h a t t h e f o l l o w i n g two c o n d i t i o n s a r e s a t i s f i e d :

f o r e v e r y i n d e x a E I , and (1.3)

f o r a n y ct,B,y

i n I, with a S B ' y .

Thus u n d e r t h e p r e c e d i n g c i r c u m s t a n c e s , w e s h a l l s a y t h a t w e a r e g i v e n a n inductive system of s e t s

w i t h r e s p e c t t o an i n d e x s e t I a s a b o v e . Now c o n s i d e r t h e s e t (1.5)

Eo =

>

Ea

uei ( d i s j o i n t union, i . e . ,

sum, o f

t h e g i v e n sets Ea,a e l ) , a s w e l l a s t h e

f o l l o w i n g ( e q u i v a l e n c e ) r e l a t i o n i n E, :

If x , y a r e e l e m e n t s o f E,, w e s a y t h a t t h e y ( w e w r i t e x-.y ) , i f t h e r e e x i s t s

a r e equivalent

an i n d e x y E I, w i t h y 2 a and y 2

a,

where x E Z

c1

IV INDUCTIVE LIMIT ALGEBRAS

110

a n d Y E EB ( c f . (1.5)) s u c h t h a t one h a s

f 12) = f l y i . Ya YB

(1.6.1) N o w it i s r e a d i l y s e e n by

( 1 - 2 ) a n d ( 1 . 3 ) t h a t t h e above c o n d i t i o n

, so

( 1 . 6 ) d e f i n e s an equivalence r e l a t i o n i n t h e s e t E,

t h a t one c a n c o n s i d e r

t h e quotient set

a

T h u s , l e t i a : Ea(1.5)

.

E = l ZEa)/-

(1.7)

,

d e n o t e t h e canonical i n j e c t i o n map d e f i n e d by

E,

w i t h a € I ( h e r e one u s u a l l y i d e n t i f i e s a n element x

€

F, w i t h t h e

u n i q u e l y d e f i n e d e l e m e n t .-,ri€Ea, f o r some a E I , w i t h z = i ix

a

i n s t a n c e , t h e p r e v i o u s c o n d . ( 1 . 6 ) ) . M o r e o v e r , i f .rr:E,-

a ) ; see, f o r Eo/- i s t h e

r e s p e c t i v e q u o t i e n t map d e f i n e d by ( 1 . 7 ) , o n e c a n c o n s i d e r t h e canoni-

c a l map fa =

(1.8)

~i

o ia: Ea-

E

f o r e v e r y a e I , namely t h e r e s t r i c t i o n t o E

CY

, of t h e q u o t i e n t map

n

;

t h u s one a c t u a l l y o b t a i n s t h e r e l a t i o n E =

(1.9)

u

c,€I

fa(Ec,).

The s e t E' g i v e n by ( 1 . 9 ) i s c a l l e d t h e i n d u c t i v e ( o r e l s e d i r e c t )

l i m i t of t h e

g i v e n i n d u c t i v e system (Ea, f

Ba

)

(with r e s p e c t t o t h e in-

d e x s e t I ) a n d i s d e n o t e d by E = limIE

(1.10)

-+

o r s i m p l y by E = l i m E +CY

Now,

a ¶ faa'

.

from ( 1 . 2 ) and ( 1 . 6 ) , o n e h a s

(1.11)

fB0fBa

= fa

f o r a n y a,, i n I , w i t h a <

4; t h e r e f o r e ,

(1.12)

f, (Ea) c f ,( E B l ,

one g e t s

whenever a IB i n I . T h u s , ( 1 . 9 ) may be considered as t h e union of an increas-

.

of s e t s ( s e e a l s o S c h o l i u m 1 . 1 b e l o w ) ing family (fa(Ea)i a F u r t h e r m o r e , w e n o t e t h a t f o r a n y x , y i n Eo, w i t h x e E a a n d y the relation (1.13)

€

E

a'

f a & = f, l y i

i m p l i e s , by ( 1 . 8 ) , t h e r e l a t i o n ~ ( x ) = n l y )i n E ; n a m e l y , t h e g i v e n e l e ments x, y i n Eo f o r which ( 1 . 1 3 ) i s v a l i d are indeed equivalent. T h e r e f o r e ( c f .

111

1. INDUCTIVE SYSTEMS OF ALGEBRAS

,

(1.6) )

t h e r e e x i s t s an i n d e x y E I, w i t h y 2 a. a n d y 2

f

(1.14)

Ya

6, such t h a t

(xi = fYBiY).

I n p a r t i c u l a r , w e n e x t c o n s i d e r t h e c a s e t h a t t h e s e t s Ea, e I, w i t h a 5 6, algebra morphisms. as above, are algebras and t h e r e s p e c t i v e maps f Ra' Thus w e s p e a k t h e n of an i n d u c t i v e system a f algebras ( E a , fBa), so t h a t t h e

r e s p e c t i v e i n d u c t i v e l i m i t ( s e t ) E ( g i v e n by ( 1 . 1 0 ) ) can uniqueZy be endowed w i t h t h e s t r u c t u r e of an algebra. I n t h i s r e s p e c t , t h e f o l l o w i n g r e s u l t w i l l b e needed ( c f , N . BOUBBAKI [l: Chap. 3 ; p. 90, Lemme I ] )

Lemma 1 . 1 . L e t ( E a , f l i m i t s e t is E = l s E a

Ra

i

. So

w e have.

be an i n d u c t i v e system of s e t s whose i n d u c t i v e

. Moreover,

l e t (xi), ci6

of E . Tken, t h e r e e x i s t s an index a

€

be a f i n i t e sequence of eZements

I , a s w e l l a s a f i n i t ; e sequence ( y i , c l i I s i C n

o f elements of Ea i n such a way t h a t one has t h e r e l a t i o n

(1.15)

2.

uith 15i

= fa (y i , c l ) '

n.

) , w i t h xu. € Eai , 1 5 i C n , S O i "i t h a t by h y p o t h e s i s f o r I t h e r e e x i s t s a n i n d e x cxe I , w i t h a a < , 1 5 i

Proof.

By ( 1 . 9 ) one g e t s xi = fa (x

5 n . Thus, i f

w i t h 1 <,i5 n , t h e n o n e g e t s by (

with 1 5 i i n , t h a t i s t h e a s s e r t i o n . I Next w e d e f i n e a l g e b r a i c o p e r a t i o n s i n t h e i n d u c t i v e l i m i t s e t E = limE

+ a

of a g i v e n i n d u c t i v e s y s t e m of a l g e b r a s (Ea, f B c l j , by assum-

i n g t h a t any two e l e m e n t s x , ( 1.16)

J

i n E a r e of t h e form

x = f i x ) and a c 1

f o r some i n d e x cx € 1( c f . Lemma

.1 ) .

y = f l~i

a

. ) I

T h u s , one d e f i n e s i n E t h e f o l l o w i z g

operations : h.x

(1 .17)

f o r any x, y i n E ( c f . ( 1 . 1 6 ) ) and every h e C . I n f a c t , t h e preceding r e l a t i o n s (1.171 are independent

are valid.

of t h e choise of a € I , f o r which t h e r e l a t i o n s ( 1 . 1 6 )

IV INDUCTIVE LIMIT ALGEBRAS

112

F o r , suppose o t h e r w i s e t h a t w e have, b e s i d e s ( 1 . 1 6 ) ,

the rela-

tions

x = fc1i3:I = f Cx I c1

where (1.13 with

3:

R

and

B

y = f (2,) = fBl yB i

,

,y a r e two g i v e n e l e m e n t s i n E ( s e e a l s o Lemma 1 . 1 )

.

Thus, by

and t h e comment f o l l o w i n g i t , t h e r e would e x i s t a n i n d e x y e I , and Y > B, s u c h t h a t one now c o n c l u d e s by ( 1 . 1 4 )

T h e r e f o r e , c o n c e r n i n g f o r example t h e s e c o n d of t h e r e l a t i o n s ( 1 . 1 7 ) , one o b t a i n s , on t h e b a s i s o f

( 1 . 1 1 ) a n d t h e h y p o t h e s i s f o r t h e maps

fBcl

fcl(xcl+3,) = f

is

Y C f Ya a

- sy‘fyB ixgi + f y o l y B ) )

+

y 0,‘

’

= f i f izaI +fy,Cyclil Y Ya

= fY if YB 12 R + y a l I = f B (3:B + 3R I

which p r o v e s t h e a s s e r t i o n c o n c e r n i n g t h e “ a d d i t i o n ” i n E . Of c o u r s e , a s i m i l a r argument c a n b e a p p l i e d f o r t h e r e s t two o p e r a t i o n s i n ( 1 . 1 7 ) . Accordingly,

( 1 . 1 7 1 d e f i n e s uniquely the s t m c t u r e of an ulgebru in the

indz,.ctivc 1im:;t E = l-f i m Z a o f u g i v e x inductive system of algebras ( Ecl, fB,i, i n such a way t h a t the respective canonical maps f :E,E , c1 € I , become algebra wi-phisms. The l a s t

a s s e r t i o n i s a g a i n a d i r e c t c o n s e q u e n c e o f t h e same

relations ( 1 . 1 7 ) , defining the algebra structure i n E . Thus, on t h e b a s i s o f t h e p r e v i o u s d i s c u s s i o n , w e now s e t t h e following.

Definition 1 . 1 . Given a n i n d u c t i v e s y s t e m of a l g e b r a s

(E,, fa,), w e c a l l inductive l i m i t algebra o f t h e s y s t e m , t h e r e s p e c t i v e i n d u c t i v e l i m i t s e t E = l i m E ( c f . ( 1 . 1 0 ) ) endowed w i t h t h e a l g e b r a s t r u c t u r e d e f i n e d i n --+a it ( u n i q u e l y ) by t h e r e l a t i o n s ( 1 . 1 7 ) .

Scholium 1.1.-

A s a c o n s e q u e n c e of

(1.I21

,

t h e remarks f o l l o w i n g

i t , and t h e l a s t c o n c l u s i o n a b o v e , e v e r y i n d u c t i v e l i m i t a l g e b r a E = limE

+ a

may b e c o n s i d e r e d a u n i o n of a n i n c r e a s i n g f a m i l y of s u b a l g e b r a s

i n s u c h a way t h a t o n e g e t s t h e r e l a t i o n E = 12Ea =

(1.18)

u

CieI

f (E I ,

On t h e o t h e r h a n d , the union of an increasing family of s e t s ( r e s p . , aZ-

gebrasi may be considered an inductive l i m i t s e t ( r e s p . , clgebral jection (resp.,

, within

,ith s t o o d i f one c o n s i d e r s t h e c a n o n i c a l i n j e c t i o n s i B aw E

a bi-

a l g e b r a isomorphism). I n f a c t , t h i s i s r e a d i l y under-

5 E B , “ c o n n e c t i n g maps“ ( c f . ( 1 . 1 )

) defining the

a s B, i . e . ,

r e s p e c t i v e induc-

113

2. INDUCTIVE LIMIT TOPOLOGICAL ALGEBRAS

tive limit set. (In this respect, see also N.BOURBAKI Remarque 1 1 )

[1: Chap. 3; p. 92,

.

We come now to examine in the next section inductive systems of algebras, where now the algebras involved are topological ones.

2 . Final topologies. Inductive systems o f topological algebras We first recall briefly some rudimentary facts from the general theory of inductive limit topological vector spaces. Thus suppose we are given a vector space E and a family (Ec1 ) c1 I of topological vector spaces together with a family ( f a i a e I of linear maps, i.e., (2.1)

with

fa: Ea-Z c1

,

€I.

N o w , one can consider the f i n a l v e c t o r space topology on E defined

by the naps f aeI,that is, the strongest vector space topology on E a' for which the maps f a , a e I , are continuous. In this concern, we note that the trivial topology on E is one of the vector space topologies thus considered, so the supremum of the above set of topologies is,in fact, the previously defined (final) topology on E. Furthermore, it is easily seen that the preceding topology on E can be characterized as well by the fact that: A l i n e a r map g of E i n t o a topological v e c t o r space F i s continuous i f , and only i f , each one o f t h e mops g o fa : Zcl -t r' , c1 e I, i s continuous, In particular, if we are given a family of locally convex spaces and linear maps as in ( 2 . 1 ) , then we can consider, of course, (EaIael the finest locally convex (vector space) topology on E for which the linear maps fa, a e I, are continuous, namely, the f i n a l l o c a l l y convex topoZogy on E defined by the maps fa,c1 f I . In this respect, one g e t s a Zoc a l b a s i s o f E by considering t h e s e t of a l l balanced convex and absorbing s u b s e t s o f E , say V , i n such a way t h a t , f o r every a e I, f-'iVl

.

i s a neighborhood o f zero Thus, it is easy to see that the sets in question constitute a c1

in E filter basis in E invariant under all the homothesies, with ratio A > O ; hence they actually define a local basis of a locally convex (vector space) topology on E making the maps fa, a e I , continuous. Moreover, the same topology is the finest among those sharing the last property, which is the assertion. On the other hand, suppose we have the relation

that is, assume that E coincides with the l i n e a r h u l l of the (vector)

114

IV INDUCTIVE LIMIT ALGEBRAS

s u b s p a c e s f o l ( E a 12 E , a a i. I n p a r t i c u l a r , t h i s happens a c c o r d i n g t o (1.9) ifE=lhE

+ a

i s t h e i n d u c t i v e l i m i t of a g i v e n i n d u c t i v e s y s t e m o f v e c t o r

spaces ( r e s p . , a l g e b r a s :

see D e f i n i t i o n 1 . 1 ) .

Thus i f E , g i v e n by ( 2 . 2 ) ,

i s equipped with t h e f i n a l l o c a l l y

convex ( v e c t o r s p a c e ) t o p o l o g y d e f i n e d on it by t h e l i n e a r maps ( 2 . 1 ) , t h e n a l o c a l basis o f E consists of s e t s of t h e f o r m

v

(2.3)

=

r ( U f tu ) I a a a

where t h e s e t s Ua a r e r a n g i n g o v e r l o c a l b a s e s of t h e g i v e n t o p o l o g i -

c a l v e c t o r s p a c e s ( o r y e t l o c a l l y convex s p a c e s ) E

a'

u ~ I , w h i l er de-

n o t e s t h e a b s o l u t e l y convex h u l l o f a s e t i n E. I n d e e d , f o r e v e r y a a I , one o b t a i n s

ua E ri

(2.4)

i.e., fi'(V)

u fi5f(U

!3e1

))) =

f;h I

i s a n e i g h b o r h o o d of z e r o i n E a , where V i s by

( 2 . 3 ) and

( 2 . 2 ) a n a b s o l u t e l y convex and a b s o r b i n g s u b s e t of E . C o n v e r s e l y , by c o n s i d e r i n g a s e t V of t h i s t y p e b e l o n g i n g , by what h a s b e e n

said

a b o v e , t o a l o c a l b a s i s of E l one o b t a i n s t h e r e l a t i o n (2.5)

r ( u fa (f;* (v)i I

t h i s together with (2.4)

proves t h e a s s e r t i o n .

a

G

v;

The n e x t lemma c o n c e r n s t h e c o n t i n u i t y of a g i v e n b i l i n e a r map between i n d u c t i v e l i m i t s p a c e s and w i l l b e a p p l i e d below. I n t h i s res p e c t , t h e i n d u c t i v e l i m i t s p a c e s i n v o l v e d a r e assumed t o s a t i s f y ( 2 . 2 ) t h e i r ( f i n a l l o c a l l y c o n v e x ) t o p o l o g i e s b e i n g d e f i n e d by l i n e a r maps a s i n ( 2 . 1 ) above. Thus, w e h a v e .

Lemma 2.1. L e t E = l i m E and F = l i m F be l o c a l l y convex 7;nduuc?i v e Z . J m i u + a -6 spaces, G a localZy convex space and I$ : E x F - + G a b i l i n e a r map. Then @ is continuous if, and o n l y i f , f o r every p a i r of i n d i c e s ( u ,B ), t h e map (2.6)

$ 0

(f,XgB)

: E

a

x

F

B

+

G : ( x ,y)

-@(f,(x),

ga(y)i

i s continuous. Proof.

The n e c e s s i t y o f t h e c o n d i t i o n i s c e r t a i n l y c l e a r s i n c e ,

37 h y p o t h e s i s , e a c h one of t h e maps f : E - E f

a

ir,

a

a e I , and g B : F B - - t F ,

BeK,

c o n t i n u o u s . On t h e o t h e r hand, i f W i s any convex ( a n d b a l a n c e d )

of z e r o i n G , t h e r e e x i s t by h y p o t h e s i s n e i g h b o r h o o d s of i n E a , V a I r e s p e c t i v e l y , such t h a t

ceighborhood

zero Ua, V B (2.7)

f o r every ( a , B ) € I x K .

@(f(U&

q V p G

w,

But t h e s e t s U = r ( $ l J f a ( U a ) ) a n d V = r ( U g , : ( V B ) )

13

115

2. I N D U C T I V E LIMIT TOPOLOGICAL ALGEBRAS

d e f i n e d f o r e v e r y ( a , R ) e I X K by ( 2 . 7 ) a r e by ( 2 . 3 ) n e i g h b o r h o o d s o f i n s u c h a way t h a t o n e h a s

z e r o i n E and F, r e s p e c t i v e l y ,

4tu, V l G W , and t h i s f i n i s h e s t h e p r o o f . I W e come n e x t t o c o n s i d e r f i n a l v e c t o r s p a c e t o p o l o g i e s ( l o c a l l y

convex o r n o t ) d e r i v e d from t o p o l o g i c a l a l g e b r a s a s a b o v e , i n I s a r t i c u -

l a r , i n c a s e t h e a l g e b r a s i n v o l v e d form i n d u c t i v e s y s t e m s , i n t h e s e n s e of t h e following

Definition 2.1. Given a n i n d u c t i v e s y s t e m o f a l g e b r a s

(E

ci

’ f3a ) ’

w i t h r e s p e c t t o a n i n d e x s e t I, w e s a y t h a t t h i s i s a n i n d u c t i v e system

of topological algebras i f f o r any a , B

,

f o r e v e r y a e I, E

i n I , with a 5

8, f,,

: Y0,-Ea

Q

i s a t o p o l o g i c a l a l g e b r a and i s a c o n t i n u o u s a l g e b r a mor-

phism. Thus t h e f i n a l v e c t o r s p a c e t o p o l o g y , d e f i n e d by t h e above d a t a , on t h e i n d u c t i v e l i m i t a l g e b r a E c o n v e r t s it n a t u r a l l y i n t o a t o p o l o g i c a l a l g e b r a , w i t h or without c o n t i n u o u s m u l t i p l i c a t i o n , may b e f o r t h e g i v e n t o p o l o g i c a l a l g e b r a s E o f t h e n e x t lemma (see a l s o Scholium 2 . 1

a’

a s t h e case

a e I . T h i s i s t h e theme

below).

Lemma 2.2. L e t (Ea, faa I be an i n d u c t i v e system of topoZogica2 algebras and E = l i m E t h e r e s p e c t i v e i n d u c t i v e l i m i t algebra ( D e f i n i t i o n 1 . 1 ) .

T;!?:;,t h e f i n i l l * a v e c t o r space topology on E , defined b y t h e canonical maps fa, a E Z , makes E a topol o g i c a l a2gebra. I n p a r t i c u l a r , t h e f i n a l l o c a / l y convex v e c t o r space topology on I:, defined

by t h e maps f

,’ aEI, makes

E a l o c a l l y convex algebra which, moreover, has a con-

tinuous m l t i p l i c a t i o n i f t h i s is t h e case for each one of t h e topological algebras Ea

, ci e I. Proof. The f i n a l v e c t o r s p a c e t o p o l o g y on t h e i n d u c t i v e l i m i t a l -

g e b r a E = 1 2 E a makes i t , by d e f i n i t i o n , i n t o a t o p o l o g i c a l v e c t o r s p a c e ;

so w e h a v e t o p r o v e , u n d e r t h e above d a t a , t h e s e p a r a t e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E , t h a t i s , t h e c o n t i n u i t y of t h e maps

Zx ( r e s p . , r 3 : ) : f o r every x s a y t h e map

E Z .

E-E:

y-l,(y)=xy

(resp., rxiy) = y x ) ,

I t s u f f i c e s , of c o u r s e , t o c o n s i d e r o n l y one of them,

Zx, w i t h x E E : now, by t h e p r e v i o u s d i s c u s s i o n , it amounts

t o t h e p r o o f of t h e c o n t i n u i t y of e a c h one of t h e maps l x o fa : E a + E ,

withciEI.Thus,if

z = f ( x I ( s e e ( 1 . 1 8 ) ) , then f o r e v e r y y E I , with B G

IV INDUCTIVE L I M I T ALGEBRAS

116

y 5 a and y <

a,

one o b t a i n s by ( 1 . 1 1 )

i.f ( y l l I R ya f o r e v e r y y E E a . Hence, i f lay.=y i n Ea, t h e n by ( 2 . 8 ) and t h e c o n t i z-zand f as w e l l a s t h e h y p o t h e s i s f o r t h e n u i t y o f t h e maps f a I fya ' YB ' a l g e b r a s Eal a E I, o n e g e t s x.fa(yi=- f

(2.8)

R

(3:

B

i . f a ( y l = f if

Y YR

(3:

fa(yi)i = x . f a ( y l r

l*f(x.

z

and t h i s p r o v e s t h e a s s e r t i o n . F u r t h e r m o r e , t h e p r e v i o u s argument i s a l s o v a l i d i n p r o v i n g t h a t E i s a l o c a l l y convex algebra (see D e f i n i t i o n I; l . l ) , i n case one considers on it the f i n a l l o c a l l y convex topology d e f i n e d by a €I. t h e c a n o n i c a l maps f a : E -+El a NOW, s u p p o s e t h a t t h e g i v e n t o p o l o g i c a l a l g e b r a s E a , a € I ,

have

c o n t i n u o u s m u l t i p l i c a t i o n and E i s endowed w i t h t h e f i n a l l o c a l l y convex t o p o l o g y a s b e f o r e . Then, E i s a l o c a l l y convex algebra w i t h continuous

a € I )d e n o t e s t h e m u l t i p l i c a t i o n multiplication. Indeed, i f u ( r e s p . , u a' i n E ( r e s p . , Ea, a c I ) , t h e n a c c o r d i n g t o Lemma 2 . 1 t h e a s s e r t i o n i s r e d u c e d t o t h e p r o o f o f t h e c o n t i n u i t y of the map

v

= u o (fax fR): E,

X

Ea+

(2.9)

E : (x,y)

+-+ i

(fa(x),fa ( y i j

= fa(xl. fa ( y ) ,

f o r every pair ( a , 13)

of i n d f c e s i n I . Thus, i f y E I, w i t h y 2 a and y 2 6

,

t h e n one g e t s t h e f o l l o w i n g commutative d i a g r a m

t h a t i s , e q u i v a l e n t l y , one c o n c l u d e s t h e r e l a t i o n

with y 2

Y

and y ? (3 i n I . T h u s , by h y p o t h e s i s f o r t h e a l g e b r a s E a , a E I,

t h e preceding r e l a t i o n (2.10)

d e f i n e s t h e map ( 2 . 9 ) a s c o m p o s i t i o n of

c o n t i n u o u s maps, and t h i s p r o v e s t h e d e s i r e d c o n d i t i o n ; so t h e p r o o f o f t h e lemma i s f u l l y e s t a b l i s h e d . 1

Scholium 2.1.-

117

INDUCTIVE L I M I T TOPOLOGICAL ALGEBRAS

2.

I n c o n n e c t i o n w i t h t h e l a s t p a r t o f Lemma 2 . 2 ,

remark t h a t an e s s e n t i a l t o o l f o r i t s p r o o f was Lemma 2 . 1

we

formulated

for l o c a l l y convex i n d u c t i v e l i m i t s p a c e s ; t h i s i n t u r n i s b a s e d o n t h e use of

( 2 . 3 ) which g i v e s t h e form of a l o c a l b a s i s of t h e l o c a l l y

convex i n d u c t i v e l i m i t s p a c e u n d e r c o n s i d e r a t i o n . But f o r more g e n e r a l i n d u c t i v e l i m i t v e c t o r s p a c e t o p o l o g i e s t h i s i s a drawback w h i c h , howe v e r , could be e l i m i n a t e d i n c a s e t h e r e s p e c t i v e ( d i r e c t e d ) index s e t

I had a denumerable c o f i n a l subset J ( i . e . , a d e n u m e r a b l e J G I s u c h t h a t , f o r e v e r y a~ I , t h e r e e x i s t s a € J , w i t h E 2 a ) . Namely, i n t h e s p e c i a l c a s e c o n s i d e r e d , s u p p o s e w e a r e g i v e n an i n d u c t i v e s y s t e m o f t o p o l o g i c a l v e c t o r s p a c e s ( E a , f o a ) and l e t E = limE

4

a

b e t h e c o r r e s p o n d i n g i n d u c t i v e l i m i t ( v e c t o r ) s p a c e . Moreover ,

c o n s i d e r t h e f i n a l vector space topology d e f i n e d on E by t h e c a n o n i c a l maps f a : Ea-

E , a € I ( s e e S e c t i o n 1 ) . NOW,

one o b t a i n s a local b a s i s o f

t h e l a t t e r by c o n s i d e r i n g s e t s of t h e form (2.11) where F(Il d e n o t e s t h e s e t o f f i n i t e s u b s e t s of I , w h i l e t h e s e t s U a i-ange o v e r l o c a l b a s e s 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 s E

a'

CIE

I. Thus,

it i s clear t h a t again, i n the p a r t i c u l a r case under consideration, an analogue

of t h e previous Lema 2 . 2 could be obtained.

( I n t h i s r e s p e c t and t o t h e

ex-

t e n t t h a t t h i s c o n c e r n s t o p o l o g i c a l v e c t o r s p a c e s , see S.O. IYAHEN [l : p . 287, P r o p o s i t i o n 2.21,

and G . KOTHE [l: p. 221, (2)] ) .

Thus, w i t h t h e e x c e p t i o n o f t h e l a s t c a s e c o n s i d e r e d a b o v e , it

ma:;

happen t h a t the f i n a l l o c a l l y convex topology t o be diJ'feiwnt from the 5 i n a l

vector space topology on an i n d u c t i v e l i m i t v e c t o r s p a c e , t h e l a t t e r bei n g d e f i n e d by a g i v e n i n d u c t i v e s y s t e m of l o c a l l y convex s p a c e s ( c f . S . O . IYAHEN [l: p. 2881). F u r t h e r m o r e ,

-

it i s worth n o t i c i n g t h a t the f i n a l

topology on E = 1 2 E a d e f i n e d by t h e r e s p e c t i v e c a n o n i c a l maps f :E E, c i a a E I , may not be compatible with t h e vector space structure of E ( c f . A . (XOTKENDIECK [2: pp. 98,993 and

Remark 2.1.-

a l s o N . BOURBAKI [6: Chap. 2 ; p. 135, E x e r . 14 f f ] ) .

I n t h e l a s t p a r t of t h e p r e v i o u s Scholium 2.1 t h e f o l -

lowing argument h a s been a p p l i e d : I f w e a r e g i v e n an i n d u c t i v e system of t o p o l o g i c a l a l g e b r a s , a s a b o v e , and one r e s t r i c t s himself t o cr c o f i n a l

subset J of the respective index s e t I, then one o b t a i n s , w i t h i n a t o p o l o g i c a l a l e b r a i c isomorphism, t h e same inductive l i m i t topological algebra

as w i t h

t h e o r i g i n a l i n d e x s e t I. Of c o u r s e , t h i s i s t r u e f o r t h e c o r r e s p o n d i n g i n d u c t i v e l i m i t s e t s ( s e e , f o r i n s t a n c e , N . BGUI?BAKI [l:Chap. 3 ; p . 95, P r o p o s i t i o n 8 1 ) ; now, it i s e a s i l y v e r i f i e d t h a t t h e r e s p e c t i v e s e t -

theoretic identification (canonical bijection) preserves the algebraic

118

IV INDUCTIVE LIMIT ALGEBRAS

s t r u c t u r e s u n d e r c o n s i d e r a t i o n . I t i s a l s o a homeomorphism by d e f i n i t i o n o f t h e r e s p e c t i v e f i n a l t o p o l o g i e s and t h e c o f i n a l i t y of J E I . W e come n e x t t o t h e f o l l o w i n g p r o p o s i t i o n which w i l l b e a p p l i e d

i n s u b s e q u e n t s e c t i o n s . Namely w e h a v e .

Proposition 2.1. Let (Ea,f g i c a l algebras

Ba

I be an i n d u c t i v e system of m-barrelled topolo-

( D e f i n i t i o n I; 1.4), and E = l i m E

t h e corresponding i n d u c t i v e

+ a

l i m i t algebra endowed w i t h t h e f i n a l l o c a l l y convex ( v e c t o r space) topology defined

on it by t h e r e s p e c t i v e canonical maps f : E -+E,

a

a

a € I . Then, E i s an m-barrelled

l o c a l l y eonvex algebra. Proof. I f U i s a n m - b a r r e l of E t h e n by h y p o t h e s i s f o r t h e maps fa, fa-’(U) i s an m - b a r r e l o f t h e a l g e b r a E,, s o t h a t by h y p o t h e s i s it is a n e i g h b o r h o o d of z e r o i n E,, f o r e v e r y a ~ I . H e n c e , by d e f i n i t i o n of t h e t o p o l o g y i n E ( s e e a l s o t h e comment a t t h e b e g i n n i n g o f t h i s sect i o n ) U i s a n e i g h b o r h o o d of z e r o i n E a s w e l l , which i s t h e a s s e r tion. I W e c l o s e t h i s s e c t i o n w i t h t h e f o l l o w i n g comment c o n c e r n i n g t h e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n a g i v e n ( t o p o l o g i c a l ) a l g e b r a E t o p o l o g i z e d i n t h e f i n a l v e c t o r s p a c e t o p o l o g y d e f i n e d on i t by a fami l y ( E a , fa) o f t o p o l o g i c a l a l g e b r a s i n a n a l o g y w i t h t h e c a s e c o n s i d e r e d a t t h e b e g i n n i n g of t h i s s e c t i o n . Thus, s u p p o s e w e a r e g i v e n a f a m i l y

( Ea , fa) o f t o p o l o g i c a l a l -

g e b r a s t o g e t h e r w i t h a l g e b r a morphisms f a : E a + E

i n t o a given algebra

E , where a v a r i e s o v e r an i n d e x s e t I. I n f a c t , w e may assume t h a t I

i s directed upwards

(S

I and, moreover, t h a t f o r any a ,

a

t h e r e e x i s t s a c o n t i n u o u s a l g e b r a morphism f B a : Ea-E

i n I , with a 2 B ,

a

such t h a t t h e

r e s p e c t i v e r e l a t i o n s t o ( 1 . 2 ) and ( 1 . 3 ) t o be s a t i s f i e d , a s w e l l a s t h a t one c o r r e s p o n d i n g t o ( 1 . 1 1 )

f o r t h e g i v e n maps f a , a € I . F u r t h e r -

more, w e may r e s t r i c t o u r s e l v e s t o a c o f i n a l s u b s e t J o f I

( c f . also

t h e above R em a r k 2 . 1 ) . F o r , o t h e r w i s e , i f F(I) d e n o t e s t h e s e t of f i n i t e s u b s e t s o f I and H E F ( I ) , t h e n t h e farr.ily (2.12)

E

=

fl

aeH

Ea,

t o g e t h e r w i t h t h e f a m i l y of maps f H : EH -+E

,

g i v e n by

(2.13) w i t h H E F(I), s a t i s f y

the

p r e c e d i n g c o n d i t i o n s (see a l s o S e c t i o n 1 ) .

Now, i t i s e a s y t o v e r i f y t h a t t h e f i n a l v e c t o r s p a c e t o p o l o g y ( r e s p . , l o c a l l y convex o n e ) on E making t h e maps f a , a E I , c o n t i n u o u s

119

2. I N D U C T I V E L I M I T T O P O L O G I C A L ALGEBRAS

c o i n c i d e s w i t h t h a t one making t h e maps f H , H e F(II c o n t i n u c u s . ( I n t h i s r e s p e c t , c f . a l s o A . GROTHENDIECK [4: p. 136 f f ] ) . F u r t h e r m o r e , w e n o t e t h a t t h e t o p o l o g y d e f i n e d on k', as a b o v e ,

a s t h a t one o b t a i n e d by r e p l a c i n g e a c h one of t h e t o p o l o g i c a l a l g e b r a s F a by t h e q u o t i e n t a l g e b r a E m /

by t h e maps f a , a e I , ker(f

cc

i s t h e same

( c f . a l s o t h e l a s t Ref. above; p. 1 3 7 ) .

)

In t h i s respect, we fur-

t h e r remark t h a t the r e s u l t i n g quotient topological algebras are of the same type as the given ones. T h i s h a s been c o n s i d e r e d a l r e a d y i n Chapt. I1 ;

Section 5 f o r topological algebras i n general

( l o c a l l y convex o r n o t

and a l s o f o r l o c a l l y m-convex o n e s ) . Now i n t h e p r e s e n c e Gf a ' k o n t i n u o u s m u l t i p l i c a t i o n " t h e a s s e r t i o n f o l l o w s from t h e n e x t Lemma 2.3. Let E be a topological algebra with a continuous m u l t i p l i c a t i o n and l e t I be a closed (2-sided) i d e a l of E . Then, the quotient algebra E I I i s ( i n t h e quotien! uector space t o p o l i g y ) a topological algebra with continuoi,.s m k l t i p l i cation.

w

i s a neighborhood of O E E I I , t h e n n(W),where W i s a neighborhood of 0 e E from a l o c a l b a s i s , s a y 7 2 , of E and TI: F - E I I Proof.

If

t h e c a n o n i c a l q u o t i e n t map ( c f . a l s o A . GROTHENDIECK [4: p. 15, Theorem 11). Thus t h e r e e x i s t by h y p o t h e s i s e l e m e n t s U , 1' i n J . HORVLTH [l: p. 356, P r o p o s i t i o n 11 )

,

w i t h U.VCW ( s e e a l s o

s o t h a t one o b t a i n s

T r t U - V ) = ~rr(UI*TrriV)=.rriwi= G ,

which p r o v e s t h e a s s e r t i o n . I In p a r t i c u l a r , a s i m i l a r r e s u l t holds c e r t a i n l y t r u e i n case t h e t o p o l o g i c a l a l g e b r a E i s l o c a l l y convex w i t h c o n t i n u o u s m u l t i p l i c a t i o n . Now one c o n s i d e r s , of c o u r s e , on

E I I the respective quotient

l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y . Thus, a r g u i n g w i t h i n t h e p r e v i o u s c o n t e x t , w e may suppose t h a t t h e given family ( E a , f a )

c o n s i s t s of t o p o l o g i c a l a l g e b r a s which a r e ,

i n e f f e c t , s u b a l g e b r a s of E ( w i t h t h e i r own t o p o l o g i e s ) and t h e maps f a , a E I , t h e a s s o c i a t e d c a n o n i c a l i n j e c t i o n s . N o w , if E i s spanned by t h e union of ImIf,),

a. e I ( t h i s i s u s u a l l y t h e c a s e i n a p p l i c a t i o n s ) , one I i s assumed t o be d i r e c t e d as above)

f i n a l l y g e t s ( a g a i n t h e index set E =

(2.14)

u

a e I

Ea

= 12Ear

where t h e l a s t r e l a t i o n i s v a l i d w i t h i n a ( t o p o l o g i c a l a l g e b r a ) i s o morphism

( c f . a l s o N. BOURBAKI [ I : Chap. 3 ; p. 92, Remarque 11 )

w e a l s o have a simia family (Fa, fa) of l o c a l l y eon-

A s a consequence of t h e p r e v i o u s d i s c u s s i o n ,

l a r r e s u l t w i t h Lemma 2 . 2

i f we consider

.

120

I V INDUCTIVE L I M I T ALGEBRAS

vex algebras, a s above ( r e s p . , with continuous m u l t i p l i c a t i o n ) , w h i l e t h e g i v e n a l g e b r a E i s t h e u n i o n of Im(&),

ae I .

So t h e a l g e b r a E , e q u i p p e d w i t h

t h e f i n a l l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y d e f i n e d by t h e maps

fa ,a I , i s then a l o c a l l y conoex algebra ( r e s p . , with continuous multiplication). F i n a l l y w e n o t e t h a t an analogous s i t u a t i o n t o t h a t d e s c r i b e d by t h e above S c h o l i u m 2 . 1 cal-algebraic

is s t i l l appeared, concerning t h e topologi-

s t r u c t u r e of an i n d u c t i v e l i m i t a l g e b r a d e f i n e d by an

i n d u c t i v e s y s t e m o f t o p o l o g i c a l a l g e b r a s , when t h e l a t t e r a r e , i n p a r t i c u l a r , l o c a l l y m-convex o n e s , as w e s h a l l see i n t h e n e x t s e c t i o n .

3. Inductive limits of locally m-convex algebras Suppose w e a r e g i v e n a n i n d u c t i v e s y s t e m of a l g e b r a s ( E where e v e r y E

a’ fBa’ 1 a’ a e I, i s a l o c a l l y m-convex algebra; m o r e o v e r , l e t E = l+i maE

be t h e r e s p e c t i v e i n d u c t i v e l i m i t a l g e b r a ( D e f i n i t i o n 1 . 1 )

-E,aeI,

and fa : Ea

t h e c o r r e s p o n d i n g c a n o n i c a l maps. Thus, one may c o n s i d e r

on E t h e f i n a l v e c t o r s p a c e t o p o l o g y a s w e l l a s t h e f i n a l l o c a l l y convex t o p o l o g y ( w i t h r e s p e c t t o t h e l i n e a r maps fa and t h e l o c a l l y convex s p a c e s E ~ € 1 . )making E a t o p o l o g i c a l a l g e b r a o r a l o c a l l y cona’ vex a l g e b r a , r e s p e c t i v e l y , w i t h c o n t i n u o u s m u l t i p l i c a t i o n ( c f . Lemma 2.2).

On t h e o t h e r h a n d , t h e r e e x i s t s a l o c a l l y m-convex t o p o l o g y on E f o r which t h e maps f a , a

e l , a r e continuous; t h a t i s , t h e t r i v i a l

one on t h e a l g e b r a E , so t h a t o n e may s p e a k a b o u t t h e f i n e s t amongst t h e s e t o p o l o g i e s on E . T h u s , w e s e t t h e f o l l o w i n g .

Definition 3.1. family

Suppose w e a r e g i v e n a n a l g e b r a E t o g e t h e r w i t h

( E a , f a ) , a e I , where f o r e v e r y i n d e x a e I , E

v e x a l g e b r a and f : Ea-E

a

c1

3

i s a l o c a l l y rn-con-

a n a l g e b r a morphism. Now t h e f i n e s t l o c a l -

l y m-convex t o p o l o g y on E making t h e maps f a , a e I , c o n t i n u o u s i s c a l l e d t h e f i n a l l o c a l l y m-convex topology

on E d e f i n e d by t h e g i v e n f a m i l y

(Ea, f a ) , a e I . F u r t h e r m o r e , w e s p e a k of a l o c a l l y m-convex inductive l i m i t t o pology on E , i f E i s t h e i n d u c t i v e l i m i t a l g e b r a of a g i v e n i n d u c t i v e s y s t e m of l o c a l l y m-convex a l g e b r a s ( E a , f,,), w h i l e t h e a l g e b r a E i s e q u i p p e d w i t h t h e f i n a l l o c a l l y m-convex t o p o l o g y d e f i n e d on it by t h e r e s p e c t i v e c a n o n i c a l maps f : Ea-

E,

aeI .

I n t h i s r e s p e c t , w e f i r s t h a v e t h e f o l l o w i n g “ l o c a l l y rn-convex a n a l o g o n “ of a s i m i l a r r e s u l t v a l i d f o r f i n a l l o c a l l y convex t o p o l o g i e s ( s e e t h e comment a t t h e b e g i n n i n g of t h i s s e c t i o n ) .

Proposition 3.1. Let E be an algebra and (Ea , fa),a e I , a family of l o c a l l y

3. INDUCTIVE LIMIT LOCALLY a-CONVEX ALGEBRAS

m-convex algebras together w i t h algebra morphisms f, : .T, -E,

121

C(E I .

Then an a-bar-

r e l , say V , of E i s a neighborhood of zero i n E f o r t h e f i n a l l o c a l l y m-convex t o pology defined on E by t h e preceding family of maps i f , and only i f , f o r every a I , t h e s e t f i l ( V I i s a neighborhood of zero i n E,.

Proof. The c o n d i t i o n i s , of c o u r s e , n e c e s s a r y b y t h e same d e f i n i t i o n o f t h e f i n a l l o c a l l y m-convex t o p o l o g y on E ( D e f i n i t i o n 3 . 1 ) . On t h e o t h e r h a n d , c o n s i d e r t h e f o l l o w i n g f a m i l y of s u b s e t s o f E : V S E : V i s an a - b a r r e l o f E l w i t h f i l ( V ) a

NOW,

it i s c l e a r t h a t 72 i s a f i l t e r b a s i s on E c o n s i s t i n g of a - b a r -

r e l s , so t h a t it d e f i n e s a l o c a l b a s i s f o r a l o c a l l y m-convex t o p o l o g y on E (Theorem I ; 1 . 1 ) f o r which t h e maps f a , u e I , a r e c o n t i n u o u s . So t h i s t o p o l o g y i s weaker t h a n t h e f i n a l l o c a l l y m-convex t o p o l o g y o n E d e f i n e d by t h e same maps. T h u s , w e h a v e a l s o e s t a b l i s h e d t h e s u f f i c i e n c y o f t h e s t a t e d c o n d i t i o n and w i t h it t h e proof of t h e p r o p o s i tion. I

Remark.-

, given by

I n d e e d , the family

( 3 . 1 ) , yieLds a local b a s i s of

the f i n a l l o c a l l y m-convex topology defined on E by t h e family ( E a , fu), a &I. Thus, i f T i s any l o c a l l y m-convex t o p o l o g y on E f o r which t h e maps ffl, aE I , a r e c o n t i n u o u s a n d U an m-barrel o f E from a l o c a l b a s i s o f T ( c f . a l s o then

T h e o r e m I ; 3.1), Eal

fil(UI i s by h y p o t h e s i s a n e i g h b o r h o o d of z e r o i n

f o r e v e r y u E I , so t h a t by ( 3 . 1 ) U E E , which p r o v e s t h e a s s e r t i o n .

Corollary 3.1. Let E and ( E a , fa I , w i t h

CL

E I,

be as i n the above Proposition

3 . 1 . Moreover, l e t F be a given l o c a l l y m-convex algebra and g : E+F

an algebra

morphism. Then g i s continuous, with respect t o t h e f i n a l l o c a l l y m-convex topology on E defined by t h e given family (ED, f a ) , maps g o f a : E,

GE

I , i f , and only i f , each one of the

F, a E I , is continuous.

+

Proof. The c o n d i t i o n i s c e r t a i n l y n e c e s s a r y . On t h e o t h e r h a n d , i f i t i s t r u e , and V i s a n m - b a r r e l g y of F (Theorem I ; 3.1)l

then g

-1

from a l o c a l b a s i s of t h e t o p o l o -

(V) is an a-barrel

i n E.

So by hypo-

t h e s i s t h e set f a-2 ( g-1 ( V I ) = ( g o f , I - l ( v )

i s a n e i g h b o r h o o d of z e r o i n E

a'

f o r e v e r y a E I , so

neighborhood of z e r o i n E ( P r o p o s i t i o n 3 . 1 ) l

that

g-l(V)

is a

which p r o v e s t h e c o n t i n u -

i t y of t h e map g , a n d t h e p r o o f i s f i n i s h e d . I

Remark.-

Suppose w e a r e g i v e n a n a l g e b r a E a n d a f a m i l y (E,, f a ) ,

122

IV INDUCTIVE LIMIT ALGEBRAS

a e l , of l o c a l l y m-convex a l g e b r a s and a l g e b r a morphisms fa : E,-E , a.e I. F u r t h e r m o r e , c o n s i d e r a l o c a l l y m-convex topology on t h e algebra E such that the

following condition is s a t i s f i e d :

For any l o c a l l y m-convex algebra F and f o r any aZgebra morphism g : E-F, t h e map g is continurns i f , and onZy if, each one of t h e maps gof,:E,-F, ~ E I is , continuous.

(3.2) NOW it

i s a c o n s e q u e n c e of t h e p r e v i o u s C o r o l l a r y 3 . 1 t h a t t h e

c1 f i n a l l o c a l l y m-convex t o p o l o g y d e f i n e d on c" by t h e f a m i l y ( E , , f , ) , e I ( w e d e n o t e it by T ~ ,) s a t i s f i e s t h e above c o n d i t i o n ( 3 . 2 ) On t h e

.

i s t h e only l o c a l l y m-convex topology on E s a t i s f y i n g ( 3 . 2 ) . I n any l o c a l l y m-convex topology, say T , on E for which (3.2) i s v a l i d makes t h e maps f,: EaE [ T ] , M. E I , continuous. F o r , by

o t h e r hand,

T~

d e e d , w e f i r s t remark t h a t

c o n s i d e r i n g t h e i d e n t i t y map i: E = fa : Ea-

E

map i: E[T]-

[TI,c1 B I , a r e

[T]

one concludes t h a t t h e maps i o fa

-E[T]

continuous, i . e . ,

T

5

T~

; moreover, t h e i d e n t i t y

E [ T ~ ] i s c o n t i n u o u s , s i n c e by h y p o t h e s i s and ( 3 . 2 ) t h e

maps i o f : E -+ E [ T ~ ]a , E I, a r e c o n t i n u o u s . Hence, one o b t a i n s T~ 5 T a s , a w e l l , and t h i s p r o v e s t h e a s s e r t i o n . Thus, a s a c o n s e q u e n c e of t h e p r e v i o u s d i s c u s s i o n , we may assume, of course, as d e f i n i t i o n of t h e topology To

t h e above c o n d i t i o n (3.21. A s an a p p l i c a t i o n o f t h e p r e c e d i n g w e now o b t a i n t h e f o l l o w i n g

" l o c a l l y m-convex

a n a l o g o n " of P r o p o s i t i o n 2 . 1 . Namely, w e h a v e .

Proposition 3.2. Let E be an algebra endowed w i t h t h e f i n a l l o c a l l y m-convex topology defined on it by a f a m i l y ( E

a' f , ) , a c

I , of ZocaZly m-convex algebras and

corresponding algebra morphisms (see D e f i n i t i o n 3.1 1 . Moreover, suppose t h a t each one of t h e algebras Fa,"€ I , i s m-barrelled.

Then, E is an m-barrelled ( l o c a l l y m-

convex) algebra too.

Proof. I f V i s any m-barrel of t h e g i v e n l o c a l l y m-convex a l g e b r a E

,

t h e n fil(V) i s a n m - b a r r e l o f t h e a l g e b r a E 0.'

a neighborhood of z e r o i n E,,for of zero i n E ( P r o p o s i t i o n 3.1),

h e n c e by h y p o t h e s i s

e v e r y a e I . Thus V i s a n e i g h b o r h o o d b e i n g a l s o , by h y p o t h e s i s , a n a r b i -

t r a r y m - b a r r e l of E , and t h i s p r o v e s t h e a s s e r t i o n . 1 Now a n o t h e r u s e f u l p r o p e r t y which i s r e t a i n e d ( a t l e a s t i n t h e commutative case) by c o n s i d e r i n g f i n a l l o c a l l y m-convex

topologies is

t h a t of b e i n g t h e a l g e b r a s i n v o l v e d & - a l g e b r a s . More p r e c i s e l y , one g e t s t h e f o l l o w i n g r e s u l t which w e s h a l l a l s o u s e i n t h e s e q u e l .

(In

t h i s r e g a r d , a more g e n e r a l s i t u a t i o n c o u l d a l s o b e c o n s i d e r e d , i n c o n n e c t i o n w i t h C o r o l l a r y 111;6.5, a s t h i s i s i n d i c a t e d i n t h e s e q u e l , a f t e r t h e n e c e s s a r y m a t e r i a l i s worked o u t ; see C h a p t . V ; S e c t i o n s 2 , 3 ) .

3.

123

INDUCTIVE LIMIT LOCALLY m-CONVEX ALGEBRA2

Lemma 3 . 1 . Let E be a commutative algebra and ( E a l a e I a family of l o c a l l y m-convex algebras, each one of them being a subalgebra of E such t h a t E =

LJ E a .

a€I h r t h e r m o r e , suppose t h a t E i s equipped w i t h t h e f i n a l l o c a l l y m-convex topology E, defined on it by t h e algebras Ea and the respective i n c l u s i o n maps i :Ea~ E I i n, sueh a way t h a t the algebra E t o b e , moreover, advertibly complete. Thus, i f t h e algebras Ea , a e I, are i n p a r t i c u l a r &-algebras, then t h e same hol.ds t r u e f o r the aZgebra E endowed with t h e previous topology. Proof. By Lemma 11; 4 . 2 , it s u f f i c e s t o p r o v e t h a t S ( E ) = { x € E :

i s a n e i g h b o r h o o d o f z e r o i n E . Now by h y p o t h e s i s , f o r e v e r y E x e E, o n e g e t s x e E a' f o r some a € I , so t h a t r (xi < I )

r,(x) 5 rEojxl

(3.3)

+

m

,

a s t h i s f o l l o w s from II;l.B, t h e h y p o t h e s i s f o r t h e a l g e b r a s E a r " € I , and C o r o l l a r y 11;4 . 3 . T h e r e f o r e , S = B(E) so t h a t S ( E / i s an m - b a r r e l

of

E ( c f . C o r o l l a r y 111; 6 . 4 ) . Thus t h e p r o o f i s now r e d u c e d t o t h e v e r i f i c a t i o n t h a t S ( E l n E a i s a n e i g h b o r h o o d of z e r o i n E a r f o r e v e r y c1 E I ; b u t t h i s i s , i n f a c t , s t r a i g h t f o r w a r d b y h y p o t h e s i s and t h e r e l a t i o n (3.4)

w i t h a e I ( c f . 11; (1.8), t h e p r e v i o u s r e l . ( 3 . 3 ) , as w e l l as Lemma 1 1 ; 4 . 2 ) , and t h i s f i n i s h e s t h e p r o o f . I

Schol ium 3.1

.-

C o n c e r n i n g t h e above D e f i n i t i o n 3 . 1

,

w e remark t h a t

s i n c e t h e f i n a l l o c a l l y m-convex t o p o l o g y on t h e a l g e b r a E i s c e r t a i n l y a l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y on it f o r which t h e maps

,a e I ,

fa :Ea-E

a r e c o n t i n u o u s ( t h e same d e f i n i t i o n )

, the

f i r s t topology

i s weaker t h a n t h e c o r r e s p o n d i n g f i n a l l o c a l l y convex t o p o l o g y on E def i n e d b y t h e same f a m i l y ( E a , f a ) , a e I (see a l s o t h e comment a t t h e b e g i n n i n g of t h e p r e v i o u s s e c t i o n ) . NOW, it may happen t h a t the preceding two topologies on E t o be d i f f e r e n t :

I n t h i s r e s p e c t , c f . S . WARNER [2: p. 195ff.l f o r r e l e v a n t e x a m p l e s , a s

w e l l a s f o r c o n d i t i o n s g u a r a n t e i n g t h e c o i n c i d e n c e of t h e above two t o p o l o g i e s . I n f a c t , w e are g o i n g t o r e p r o d u c e t h e s e c o n d i t i o n s p a r t -

,a s

well a s C h a p t . VII; S e c t i o n 9 ( 1 ) i n t h e s e q u e l ) . On t h e o t h e r h a n d , t h e res p e c t i v e s i t u a t i o n i n c a s e t h e i n d i v i d u a l a l g e b r a s Ea, a e I ( D e f i n i t i o n 3.1) a r e , i n p a r t i c u l a r , normed algebras seems t o b e s t i l l u n s e t l e d ( c f . l y i n t h e e n s u i n g d i s c u s s i o n ( s e e a l s o E. A . MICHAEL [l: p. 64 ff.]

A . ilROSI0

[I]

)

.

Thus t h e f o l l o w i n g p r o p o s i t i o n a n d i t s c o r o l l a r y p r o v i d e a n i n s t a n c e o f t h e s i t u a t i o n d e s c r i b e d i n t h e p r e c e d i n g Scholium 3 . 1 .

They

124

IV

INDUCTIVE LIMIT ALGEBRAS

w i l l a l s o b e a p p l i e d i n s u b s e q u e n t s e c t i o n s of t h i s book ( s e e C h a p t e r

VII )

. P r o p o s i t i o n 3 . 3 . Let E be a l o c a l l y m-convex algebra and (Enln e~~

an i n -

creasing sequence of i d e a l s i n E such that

U E . n€N n

E =

(3.5)

Moreover, suppose t h a t eack one o f t h e i d e a l s En, n E It?, i s equipped w i t h t h e relat i v e ( l o c a l l y m-convex) topology from E. Then, thg f i n a l l o c a l l y m-convex topology as well as t h e f i n a l l o c a l l y convex ( v e c t o r space) topology defined on E by t h e sequence IF I (and t h e r e s p e c t i v e i n c l u s i o n maps E S E , n € N ) are t h e same. n n e N

Proof.

L e t T~ and T~

d e n o t e t h e l o c a l l y convex a n d l o c a l l y rn-con-

vex t o p o l o g y , r e s p e c t i v e l y , d e f i n e d on E by t h e c a n o n i c a l i n j e c t i o n s ( i n c l u s i o n s ) , s a y i :E + E l n e IN, of t h e l o c a l l y m-convex a l g e b r a s En n n i n t o t h e a l g e b r a E ( t h e l a t t e r been understood w i t h o u t topology).Now s i n c e b y t h e same d e f i n i t i o n s T~ 5 Tl, one has t o prove only t h a t T~ C T ~ : Thus s u p p o s e t h a t U i s an a b s o l u t e l y convex n e i g h b o r h o o d o f z e r o of t h e l o c a l l y convex s p a c e E I T l ] . Then, f o r e v e r y m € I N , U E E m i s a n e i g h borhood of z e r o i n Em

r

( c f . a l s o t h e comment a t t h e b e g i n n i n g of Sec-

i s a f a m i l y of s u b m u l t i p l i c a t i v e semia a c l norms d e f i n i n g t h e t o p o l o g y of E (Theorem I ; 3.1), t h e n by h y p o t h e s i s

t i o n 2 ) ; hence, i f

= (p )

and C o r o l l a r y 1 : 3 . 1 , and a p o s i t i v e r e a l

there E

e x i s t s a f i n i t e set of i n d i c e s , s a y I m C I ,

,with 0 <

m Um = { x e E : p (2)< m a

(3.6) where t h e

E

rn

E ~ ,f

5 I , such t h a t

o r every a

E

I m ]G U n Em

,

s e t Urn i s a n e i g h b o r h o o d of z e r o i n E m , m e N . F u r t h e r m o r e ,

consider t h e sets

w i t h n E N , and l e t

w = u wn.

(3.8)

ne N

Now, i t i s c l e a r by h y p o t h e s i s t h a t W n i s a n e i g h b o r h o o d o f z e r o i n

E n , n e 7 N , so t h a t W i s a n a b s o r b i n g s u b s e t of E ; m o r e o v e r , one h a s W n C Un

, for

e v e r y n E N , so t h a t W G U. F u r t h e r m o r e , W i s an m-set o f E , i. e .

,

f o r a n y x , y i n W , one h a s x . y e W . I n d e e d , i f x € Wk and y e W l , w i t h k 5 1 ( c f . (3.8)), s i n c e E k , EE ‘ .

. On

f o r e v e r y k e N , i s an i d e a l of E l o n e g e t s r a y

t h e o t h e r hand, s i n c e

min E 5 1 , Ism
one o b t a i n s

k

ul

U Irn C I m ( i f k 6 1 ) as w e l l as rn = I m =I

3. INDUCTIVE LIMIT LOCALLY m-CONVEX ALGEBRAS

p ( x y ) 5 p a ( x ) p ( y ) 5 p a( x ) 5

k

for every a e

u

E ( c f . Chapt.

I ; Lemmas 1 . 3 and 1 . 4 )

min

125

E

16msk

1 , I s o t h a t xa f W G W , which i s t h e a s s e r t i o n . Thus, k rn = I t h e a b s o l u t e l y convex h u l l of W i n E , s a y V = < ( W ) > I i s a n a - b a r r e l of

such t h a t

WnCWnEnCV/En

I

t h a t i s , V n E n i s a neighborhood o f z e r o i n E n ] (Proposition 3.1),

, so

that since

V=

< i W ) > r U , one c o n c l u d e s t h a t U i s also a n e i q h -

borhood o f z e r o i n E f o r t h e t o p o l o g y T ~ t;h u s , minates t h e proof.

f o r e v e r y n e J N . Sence

V i s a neighborhood of z e r o i n E f o r t h e topology T ~ and ]

t h i s ter-

a

Corollary 3.2. Let E be an algebra and (En i n EJN

an i n c r ~ e c z s i ~ sequence g of

i d e a l s of E such t h a t

Moreover, suppose t h a t each one of t h e i d e a l s En i s a l o c a l l y m-convex algebra i n such a way t h a t t h e topology of E i s t h e r e l a t i v e one induced on it from Enil Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) The f i n a l l o c a l l y convex topology d e f i n e d on E by t h e l o c a l l y convex

n’ nE 7 N , Ian2 t h e corresponding i n c l u s i o n maps), a s we21 as t h e r e s p e c t i v e f i n a l l o c a l l y m-convex topology on it a r e t h e same. spaces E

2) The l o c a l l y m-convex topology o f En c o i n c i d e s w i t h t h e r e l a t i v e one i n duced on En by E , when t h e l a t t e r aZgebra i s equipped w i t h t h e f i n a l l o c a l l y m-conv e x topoZogy a s above. Proof.

I t i s c l e a r by P r o p o s i t i o n 3 . 3 t h a t 2 ) i m p l i e s 1 ) . On t h e

o t h e r h a n d , 1 ) i m p l i e s 2 ) by Dieudonng-Schwartz’s Theorem (see, f o r i n s t a n c e ] N . BOURBAKI [ 6 : Chap. 2 ; p . 7 9 , P r o p o s i t i o n 91 )

.I

I n a n a l o g y w i t h t h e s i t u a t i o n one h a s i n c a s e o f l o c a l l y convex s p a c e s , a ( l o c a l l y m-convex)

a l g e b r a E s a t i s f y i n g t h e c o n d i t i o n s of

t h e p r e c e d i n g C o r o l l a r y 3 . 2 i s s a i d t o be a s t r i c t i n d u c t i v e L i m i t ( l o c a l -

l y m-convex) algebra. W e a r e g o i n g t o e n c o u n t e r s u c h a l g e b r a s i n t h e n e x t s e c t i o n , a s w e l l as i n Chapter V I I below. W e c o n c l u d e t h i s s e c t i o n w i t h t h e f o l l o w i n g p r o p o s i t i o n which

s t i l l p r o v i d e s a n o t h e r i n s t a n c e o f t h e s i t u a t i o n d e s c r i b e d by t h e p r e -

v i o u s Scholium 3 . 1 . ( I t i s a l s o i m p l i c i t l y i n v o l v e d i n

9 4.(1) below).

Proposition 3.4. L e t E be u normed aZgebra and ( Ea ) o l € I a f a m i l y o f i d e a l s o f E such t h a t

126

IV

INDUCTIVE LIMIT ALGEBRAS

(3.10)

u E a .

E =

ClEI

Moreover, suppose t h a t each one of t h e algebras E 5 E , ~ E I i,s equipped w i t h t h e r e l a t i v e inormed a l g e b r a ) topology induced on it by E. Then, t h e r e s p e c t i v e f i n a l

a' a e I , and t h e correspond-

l o c a l l y m-convex topology on E d e f i n e d by t h e f a m i l y F

ing i n c l u s i o n maps c o i n c i d e s w i t h t h e f i n a l l o c a l l y conuex ( v e c t o r space) topology d e f i n e d on E b y t h e same f a m i l y .

I f U i s an a b s o l u t e l y convex n e i g h b o r h o o d of O f E f o r t h e

Proof.

f i n a l l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y d e f i n e d on E by t h e fami l Y ( E a i a e I 1 t h e n b y t h e same d e f i n i t i o n of t h i s t o p o l o g y U n E a i s a n e i g h b o r h o o d of z e r o i n Ea , f o r e v e r y a e l , so t h a t t h e r e e x i s t s a n e i g h b o r h o o d of O E E a

with O< ca

1

,

, say

Wa

,

such t h a t

and a E I ( s e e a l s o C o r o l l a r y I ; 2 . 1 )

. Now

if W =

avI

Wa,

t h e n b y (3.10) and (3.11) W S U , b e i n g a l s o an a b s o r b i n g s u b s e t of E. Moreover, W i s an m-set of E : T h a t i s , if x , y a r e e l e m e n t s of E l w i t h

X E

Wa and y e Wa t h e n s i n c e E a is a n i d e a l of B, one h a s t h a t x y E Ea , so t h a t one g e t s

s i n c e by y e E D and (3.11) o n e h a s

/ l y ( (2 ~ ~ 2 T 1 h.e r e f o r e , z y e W sW,i.e.,

a

t h e a s s e r t i o n . So by c o n s i d e r i n g t h e a b s o l u t e l y convex h u l l of W,say I/=

<(W)

, one

gets

VGU

,

t h e s e t V b e i n g a l s o an a - b a r r e l

of t h e a l -

g e b r a E ( c f . C h a p t . I ; L e m m a s l . 3 and l . 4) s u c h t h a t one h a s W a S W n Ea G V n E a

w i t h a e I. Thus, V

n F a i s a n e i g h b o r h o o d of z e r o i n B a' f o r e v e r y a E

I;

h e n c e ( P r o p o s i t i o n 3.1), V i s a n e i g h b o r h o o d o f z e r o i n E f o r t h e l o c a l l y rn-convex

i n d u c t i v e l i m i t t o p o l o g y on t h i s a l g e b r a , so t h a t t h e

same h o l d s t r u e f o r U , a n d t h i s f i n i s h e s t h e p r o o f . I

Remark.family

(EajaeI

The above p r o p o s i t i o n i s s t i l l v a l i d i n c a s e t h e g i v e n consists

i d e a l s ) while t h e

o n l y of s u b a l g e b r a s o f

i n d e x s e t I i s now

E (not necessarily

of

t o t a l l y ordered ( t a k e , f o r e x a m p l e ,

a s e q u e n c e ( E n I n E N o f s u b a l g e b r a s of E ) . The h y p o t h e s i s t h a t E i s a normed a l g e b r a c a n n o t b e weakened ( w i t h a r e s p e c t i v e weakening o f t h a t f o r t h e t o p o l o g i c a l a l g e b r a s E a , a e l ; see a l s o t h e comment a t t h e e n d of t h e p r e c e d i n g Scholium 3.1 )

.

I n t h i s r e g a r d , c f . S. WARNER [2: p.

195, P r o p o s i t i o n 2 and p. 194, Example 61

.

4.

EXAMPLES. THE ALGEBRA

K(Xj

127

4 . Examples o f topological inductive l i m i t algebras W e c o n s i d e r i n t h i s s e c t i o n c e r t a i n c o n c r e t e e x a m p l e s of t o p o l o g i c a l a l g e b r a s of t h e t y p e e n c o u n t e r e d i n t h e f o r e g o i n g w h i c h , m o r e o v e r , a r e most f r e q u e n t l y o c c u r r e d i n a p p l i c a t i o n s . Thus w e s t a r t w i t h t h e f o l l o w i n g example of an i n d u c t i v e l i m i t l o c a l l y m-convex a l g e b r a of t h e t y p e c o n s i d e r e d i n t h e a b o v e P r o p o s i t i o n 3 . 4 , which i s a l s o of a f u n d a m e n t a l i m p o r t a n c e i n I n t e g r a t i o n Theory (Radon m e a s u r e s i n l o c a l l y compact s p a c e s ) .

The algebra KtXl.- L e t X b e a t o p o l o g i c a l s p a c e ( H a u s d o r f f K ( X l t h e algebra (with pointwise defined o p e r a t i o n s ) of a l l complex-valued c o n t i n u o u s f u n c t i o n s on X w i t h com4.(1).

i s assumed t h r o u g h o u t ) and

pact supports. In t h i s r e s p e c t , we r e c a l l t h a t t h e

support

of a g i v e n map f de-

f i n e d on X and t a k i n g v a l u e s i n a ( t o p o l o g i c a l ) v e c t o r s p a c e E i s , by d e f i n i t i o n , t h e set Supp(fi = c x e X: f(xi i0 1

(4.1)

.

T h a t i s , t h e ( t o p o l o g i c a l ) c l o s u r e of t h e s e t o f t h o s e p o i n t s of X a t which f d o e s n o t v a n i s h , o r y e t t h e s m a l l e s t c l o s e d s u b s e t of X i n t h e complement of which t h e map f v a n i s h e s . Now, f o r e v e r y compact s u b s e t K of X , one c o n s i d e r s t h e s u b a l gebra in K;

K (XI of t h o s e f u n c t i o n s f r o m KlX) whose s u p p o r t i s c o n t a i n e d K i . e . , o n e h a s by d e f i n i t i o n

KKfXI = { f e KlXl: Supp if1 E K 1 .

(4.2)

So s i n c e S u p p ( f g i c S u p p ( f l n S u p p i g ) , f o r any f , g

i n K t X i , (4.2) d e f i n e s ,

i n e f f e c t , f o r e v e r y compact s u b s e t K of X , an i d e a % of t h e algebra KlX). F u r t h e r -

more, s i n c e ( 4 . 2 ) i s " i n c l u s i o n p r e s e r v i n g " w i t h r e s p e c t t o t h e comp a c t s u b s e t s of X ,

t h e l a t t e r s e t of s u b s e t s of X being a l s o " s t a b l e

f o r f i n i t e u n i o n s " , one g e t s an upwards d i r e c t e d f a m i l y of i d e a l s

I K K ( X I 1 of

the algebra KlXl such that (4.3)

where K v a r i e s o v e r t h e compact s u b s e t s of X ( c f . a l s o ( 2 . 1 4 ) a b o v e ) . O n t h e o t h e r hand,

a n a t u r a l " a l g e b r a norm" ( c f . C h a p t . 1 ; ( 5 . 1 ) )

c a n b e d e f i n e d on K K ( X i by t h e

relation

(4.4)

with f

E

K K I X ) . Thus, s i n c e t h e l a t t e r s p a c e i s a c l o s e d s u b s e t of

BtXi,

t h e s p a c e ( a l g e b r a ) of all complex-valued bounded a n d c o n t i n u o u s f u n c -

IV INDUCTIVE LIMIT ALGEBRAS

128

t i o n s on X i n t h e uniform "sup-norm''

t o p o l o g y , one c o n c l u d e s t h a t

KK (Xi i s , i n p a r t i c u l a r , a Banach algebra ( i d e a l i n K ( X i i w i t h respect t o t h norm defined on it by ( 4 . 4 ) . F u r t h e r m o r e , t h e a l g e b r a K I X I , b e i n g a s u b a l g e b r a

of

W X ) , c a r r i e s t h e r e l a t i v e "sup-norm t o p o l o g y " t h u s becoming

normed a l g e b r a ; so e a c h one of i t s s u b a l g e b r a s ( i d e a l s ) K,(X)

a

may be

endowed w i t h t h e c o r r e s p o n d i n g r e l a t i v e t o p o l o g y , i n d e e d , induced on

it by ( 4 . 4 ) . Thus, w e have r e p r o d u c e d so f a r t h e s i t u a t i o n e x h i b i t e d by t h e

so t h a t one c o n c l u d e s t h a t

above P r o p o s i t i o n 3 . 4 ,

t h e algebra K l X ) , given by 14.31, i s a ( c o m t a t i v e i l o c a l l y m-convex algebra, whenever one considers on it the f i n a l l o c a l l y convex o r , e q u i v a l e n t l y , l o c a l l y m-convex topology defined on it by the family

(4.5)

{

KK(XI 1

o f Banach algebras ( i d e a l s i n K ( X ) i given by ( 4 . 2 ) .

F u r t h e r m o r e , a s a l o c a l l y c o n v e x i n d u c t i v e l i m i t of Banach s p a c e s l a n d hence b a r r e l l e d o n e s ,

K ( X i i s , i n p a r t i c u l a r , a barrelled

( l o c a l l y m-con-

vex) algebra ( s e e a l s o P r o p o s i t i o n 3 . 2 ) . Now an i m p o r t a n t s p e c i a l i n s t a n c e of t h e f o r e g o i n g c o n s t i t u t e s t h e case t h a t t h e g i v e n t o p o l o g i c a l s p a c e X

a d m i t s a fundamental se-

quence of compact s u b s e t s , s a y ( K E i n e m : t h a t i s , w e assume t h a t e v e r y compact K E X

i s c o n t a i n e d i n K n , f o r some n E N ("k-covering sequence

I'

of X ;

s e e a l s o Chapt.V; S e c t i o n 5 ) . Thus, s i n c e w e may always suppose t h a t t h e sequence ( X n ) i s i n c r e a s i n g , one o b t a i n s by t h e p r e c e d i n g d i s c u s -

( X i , n E N , of the algebra K l X i f o r which Kn the conditions of Corollary 3.2 are s a t i s f i e d . T h e r e f o r e , one s t i l l concludes

s i o n an increasing sequence of i d e a l s K

( 4 . 5 ) by a p p l y i n g a l s o C o r o l l a r y 3 . 2 .

S i m i l a r l y , i n t h i s c a s e t h e con-

d i t i o n s of K o t h e ' s Theorem f o r t h e c o r r e s p o n d i n g i n d u c t i v e l i m i t l o c a l l y convex s p a c e K(X) VkTH [l : p. 164, C o r o l l a r y

are a l s o f u l f i l l e d (see, for i n s t a n c e , J . HOR-

])

;

h e n c e , the algebra

-

KCil = l i m K (Xi , Kn t o p o l o g i z e d a s above, i s a ( c o m t a t i v e ) complete l o c a l l y m-convex algebra (bei n g a l s o a barrelled

one, a s a b o v e ) . Moreover, s i n c e each one of t h e

a l g e b r a s K ( X i , n E N , i s a Banach a l g e b r a , so a Q - a l g b r a , one a l s o con-

%

eludes (Lemma 3.1 ) t h a t K t X i , given by (4.5), i s a &-algebra. I n t h i s respect, w e s t i l l note t h a t i f t h e algebra

KlXl i s g i v e n

by ( 4 . 6 ) , t h e n it i s a p a r t i c u l a r i n s t a n c e of a c l a s s of t o p o l o g i c a l a l g e b r a s c a l l e d LFQ-aZgebras

which w i l l be c o n s i d e r e d l a t e r on ( s e e

4.(2). EXAMPLES. THE ALGEBRA

.

a(X)

129

.

Chapt V I I ; S e c t i o n 9 . ( 1 ) ) The p r e v i o u s s i t u a t i o n i s a p p e a r e d , f o r example, i f w e a r e g i v e n

a l o c a l l y compact space which i s a l s o o-compact ( i . e .

,

a denumerable u n i o n of

compact s u b s e t s ) which i s o f t e n t h e case i n I n t e g r a t i o n Theory. I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t the topological dual of the l o c a l l y con-

vex

( i n d u c t i v e l i m i t ) space

i s , by d e f i n i t i o n , t h e s p a c e o f

( c o m p l e x ) R a d o n measures on a g i v e n

( c f . N . BOURBAKI [8: Chap.3;p.47, D e f i n i t i o n 2 1 ) .

l o c a l l y compact s p a c e X

So t h e t o p o l o g y d e f i n e d by

( 4 . 7 ) i s t h e n the "measure topology" on t h e

K ( X l which is, by t h e above d i s c u s s i o n , a l o c a l l y m-convex topology on t h e respective algebra K ( X I . space

a(X)a s

The a l g e b r a

4.(2).

a t o p o l o g i c a l subalgebra o f

c"(X).We

con-

s i d e r n e x t c e r t a i n t o p o l o g i c a l a l g e b r a s which are m a i n l y a p p e a r e d i n t h e Theory of D i s t r i b u t i o n s

( a s w e l l as i n i t s v a r i o u s a p p l i c a t i o n s ,

i n c l u d i n g Mathematical P h y s i c s ) . A s t a n d a r d r e f e r e n c e i s , of c o u r s e , t h e c l a s s i c a c o u n t of L . SCHWARTZ [I];

i n t h i s r e g a r d , w e a l s o l e a n on

c e r t a i n a s p e c t s , a s t h e y h a v e b e e n p r e s e n t e d i n F . TREVES [2]. Thus, w e c o n s i d e r complex-valued f u n c t i o n s d e f i n e d on a g i v e n d i f f e r e n t i a l m a n i f o l d X which i s f u r t h e r assumed t o b e f i n i t e - d i m e n s i o n a l a n d of " c l a s s

c""( i n t h e

sense t h a t t h e corresponding "tran-

e d e n o t e by s i t i o n f u n c t i o n s " are s u c h o n e s ) . W

c"fX)t h e

sot of a l l e m -

on X ; t h a t i s , t h o s e f u n c t i o n s f : X C for plex-valued C " - & n c t i o n s which t h e r e e x i s t s , f o r e v e r y p o i n t x E X I a l o c a l chart ( U , $ l of X a t x i n s u c h a way t h a t t h e ( c o m p l e x - v a l u e d ) f u n c t i o n

i s of c l a s s

c" c o n s i d e r e d

a s a f u n c t i o n on t h e open s e t @ ( U ) G I R n .

So w i t h o p e r a t i o n s d e f i n e d p o i n t w i s e

C " ( X l i s a (complex) c o m t a -

t i v e algebra. On t h e o t h e r h a n d , f o r e v e r y l o c a l c h a r t ( U , @ I of t h e manif o l d X I c o n s i d e r t h e map

+*: c " i u )

(4.9) where t h e s p a c e

,while NOW,

,

C"(G c a n b e d e f i n e d a s b e f o r e f o r X = U

s e t i s e s s e n t i a 2 l y I i.e., Wn

- c ~ ~ Q ~ u ~ I : ~ - + * I ~ If o: b=d 1

; the latter

w i t h i n a d i f f e o m o r p h i s m , t h e open s e t $ ( U l

t h e n u m e r i c a l s p a c e a p p e a r e d i s t h e ( e u c l i d e a n ) "model

of of X.

it i s e a s y t o see t h a t $* d e f i n e s a b i j e c t i o n o f t h e correspond-

ing spaces which a l s o preserves the r e s p e c t i v e algebraic s t r u c t u r e s of the spaces involved. T h e r e f o r e , w e c a n t r a n s f e r , v i a $*, t h e t o p o l o g i c a l - a l g e b r a i c

130

IV INDUCTIVE LIMIT ALGEBRAS

t o e"(UI;

s t r u c t u r e of t h e s p a c e e " ( $ ( U l )

the c o m e -

i n t h i s regard,

sponding structure t h u s imposed on c"iUl i s independent of the p a r t i c u l a r local chart considered o f t h e m a n i f o l d x. T h a t i s , i n t h e common domain ( i n t e r s e c t i o n ) of two l o c a l c h a r t s , s a y ill,$) and s p e c t i v e s t r u c t u r e s d e f i n e d on

( V , $ ) , of X

t h e re-

by c " ( U l and e m ( V l a r e

e"(UnV)

the

same: T h i s i s e a s i l y s e e n by ( 4 . 9 ) and t h e r e l a t i o n

@*= l $ 0 $ - I ) *

(4.10)

0

@*,

-

i f w e a l s o t a k e i n t o account t h a t t h e r e s p e c t i v e transition function (4.11)

$ 0

i s a local diffeornorphisrn of

@-I: $(UnvI

Giun V )

IRn.

T h u s o u r n e x t o b j e c t i v e i s t o show t h a t

e m i U I is a FrLchet l o c a l l y m-convex algebra.

(4.12)

So, € o r e v e r y compact s u b s e t K o f U , and e v e r y m

N ( t h e set of

non-

.

n e g a t i v e i n t e g e r s ) , i f p = ( p l , . . , pnl E N n ( a g i v e n multi-index ) w i t h Ip 1 - p I + . . .+ p n ( t h e length of t h e m u l t i - i n d e x p ) , one d e f i n e s (4.13)

f o r every f

E

C " t U l ; h e r e one s e t s n2fi'xI = iOpfI(xl = ( D p f I x

,

with

( i = l ,..., n ) d e n o t e t h e ( g l o b a l ) c o o r d i n a t e f u n c t i o n s of IRn i (canonical p r o j e c t i o n s ) c o r r e s p o n d i n g t o t h e g i v e n l o c a l c h a r t (U, @ I of X

where u

a t t h e p o i n t x. Moreover, w e d e n o t e by

xi = u .

(4.15)

o

t h e r e s p e c t i v e local coordinate functions NOW,

since U

,

i$

ii=I

,..., nI

(or else

,

local c o o r d i n a t e s ) on U. (cf.,

b e i n g homeomorphic v i a $ t o $(U), is u--onpact

f o r i n s t a n c e , J . DUGUI'JDJI [1:p. 241,Theorem 7 . 2 ] ) ,

one g e t s by ( 4 . 1 3 ) a

s e q u e n c e of semi-norms (4.16)

making c " ( U l i n t o a m e t r i z a b l e l o c a l l y convex s p a c e : t h e l a t t . e r i s also c o m p l e t e , s o t h a t is a Frgchet l o c a l l y convex space. ( I n f a c t , one i s l e d t o t h e l a s t c o n c l u s i o n c o n c e r n i n g t h e s p a c e c " I $ l U ) l ; see

e?U)

L . SCfiWARTZ (I:

p. 8 8 1 . Hence, by t h e p r e v i o u s d i s c u s s i o n , t h e same h o l d s

t r u e a p p l y i n g t h e isomorphism ( 4 . 9 ) )

.

4.(2). EXAMPLES.

THE ALGEBRA

131

d)(X)

Furthermore, applying Leibniz's r u l e ( t o t h e d i f f e r e n t i a l opera t o r d e f i n e d b y ( 4 . 1 4 ) ) one o b t a i n s

f o r any f , g i n

eco(U)

, and

w h i l e a i s a p o s i t i v e constant

(rn, K) a s i n ( 4 . 1 3 )

any p a i r of i n d i c e s

( d e p e n d i n g on m a n d n = d i m X ) . Thus

,

the

m l t i p l i c a t i o n i n e o o ( X ) i s continuous, so t h a t t h e l a t t e r s p a c e becomes a F r e c h e t l o c a l l y convex a l g e b r a . Now i t i s r e a d i l y s e e n t h r o u g h ( 4 . 1 7 ) ' s i s an absorbing semi-norm on e r n i U ) ( c f . D e f i m,K s o t h a t one f i n a l l y c o n c l u d e s ( C o r o l l a r y I ; 5 . 3 ) t h a t

t h a t e a c h one of nition I;5.2),

N

C"(U)is, i n f a c t , a Frdcket l o c a l l y m-convex algebra

which was t h e a s s e r -

tion. In t h i s concern, we s t i l l note t h a t (4.13) j u s t i r i e s a l s o t h e t e r m i n o l o g y o f t e n a p p l i e d t o t h e t o p o l o g y d e f i n e d by t h e f a m i l y of semi-norms

( 4 . 1 6 ) : t h u s w e u s u a l l y speak of t h e

topology of uniform conver-

gence on compacta of t h e functions i n e r n t U ) and all of t h e i r d e r i v a t i v e s . t e r m Schwartz topology on t h e

space e m ( U )

The

i s a l s o a p p i i e d . So, Eormu-

l a t e d o t h e r w i s e , w e w e r e l e d by t h e p r e v i o u s d i s c u s s i o n t o t h e c o n c l u s i o n t h a t the space (algebra) c " ( U )

logy is a

endowed with t h e respective Schwartz topo-

(complex c o m m u t a t i v e ) Frdchet l o c a l l y m-convex algebra ( w i t h a n

identity element). On t h e o t h e r h a n d , s u p p o s e now t h a t t h e g i v e n m a n i f o l d X i s sebe a cond countable ( t h i s a l s o i s u s u a l l y t h e case) , and l e t ( U n I n E m b a s i s of i t s t o p o l o g y ; (we may a l w a y s assume t h a t t h i s c o n s i s t s of l o -

c a l c h a r t s b e l o n g i n g t o t h e maximal a t l a s d e f i n i n g t h e d i f f e r e n t i a l structure of X )

.

Thus, c o n s i d e r n e x t

t h e i n i t i a l topology defined on c " i X ) by

-

the canonical r e s t r i c t i o n maps ( a l g e b r a morphisms) j n : C"(X)

(4.18)

e"(un I ,

nem.

So ( s e e , f o r i n s t a n c e , Lemma 111; l . l ) , w e a r e l e d t o t h e f o l l o w -

ing conclusion: The s p a c e

c " i X ) beeoms a Frdchet l o c a l l y m-convex algebra , where i t s

t o p o l o g y i s g i v e n by t h e r e l a t i o n

cmixi=

(4.19)

lim

cw, i

nTm The l a s t r e l a t i o n h o l d s t r u e w i t h i n a l-opological -algebraic isomorphism, X being a

second countable ( f i n i t e dimensional) e"-manifold.

I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t t h e correspondence (4.20)

u-

emiui,

132

IV INDUCTIVE LIMIT ALGEBRAS

w i t h U r a n g i n g o v e r t h e open s u b s e t s o f X ( i n p a r t i c u l a r , o v e r t h e l o c a l c h a r t s of t h e maximal a t l a s o f

algebras; i n d e e d , t h i s i s a sheaf

x),

d e f i n e s a presheaf of topologica2

which f a c t , i n t u r n , y i e l d s ( 4 . 1 7 ) t o o .

T h e r e f o r e , one g e t s , i n p a r t i c u l a r , by ( 4 . 2 0 ) a topological algebra sheaf on X . ( I n t h i s r e s p e c t , c f . a l s o R . GODZMENT [l] a n d / o r F . HIRZEBRUCH [l. p. 24, Example 3 1 ) . However, w e a r e n o t g o i n g t o t o u c h upon t h i s f a c t any further.

Instead we r e f e r t o the pertinent l i t e r a t u r e f o r applications

of t h i s n o t i o n i n t o p o l o g i c a l a l g e b r a s t h e o r y : t h u s see, f o r example, A . KYRIAZIS [3] and A . MALLIOS [lo, 16, 1 7 , 2 7 1 . Now, t h e strong topological dual of t h e s p a c e ( a l g e b r a )

"fX), the

l a t t e r b e i n g t o p o l o g i z e d b y ( 4 . 1 9 1 , i s i d e n t i f i e d w i t h t h e s p a c e of on t h e m a n i f o l d X ( u s u a l l y d e n o t e d by

d i s t r i b u t i o n s w i t h compact support

&'(X)

; t h u s o n e h a s , by d e f i n i t i o n ,

(cf.,

&(Xi = em(X)

where w e a l s o s e t , f o l l o w i n g s t a n d a r d n o t a t i o n ,

f o r i n s t a n c e , L . SCHWARTZ [l: p.98, Th6orSme XXV] a n d / o r F . TREVES [2: p. 117, r e l . (1.811. See a l s o t h e e n s u i n g d i s c u s s i o n c o n c e r n i n g t h e s p a c e

d)fXi). Thus, w e c o n s i d e r n e x t t h e s p a c e d , ( X ) , e r n ( X ) = &(XI,

namely, t h e s u b s p a c e of

c o n s i s t i n g of t h o s e ( c o m p l e - v a l u e d )

ernfunctions

on

x

having compact supports ( c f . a l s o t h e p r e v i o u s S u b s e c t i o n 4. ( I ) ) . W e a l s o s p e a k o f t h e s p a c e of t e s t f u n c t i o n s ( o r S e h a r t z space ) on X . Now g i v e n a compact s u b s e t K of X I w e d e n o t e by a,(X:

t h e sub-

s p a c e o f 0 r Z ) f X l t h e e l e m e n t s of which h a v e s u p p o r t s c o n t a i n e d i n K . So

(see a l s o S u b s e c t i o n 4 . ( 1 ) ) it i s e a s y t o see t h a t d ) ( X ) b r a ( i n f a c t , a n i d e a l ) of

i s a subalge-

c m f X ) and a,fX/a n i d e a l i n a t X l

,

for

a n y compact K C X , s u c h t h a t (4.22) ( t h e s e t - t h e o r e t i c u n i o n i n t h e s e c o n d m e m b e r of

( 4 . 2 2 ) may a l w a y s b e

t a k e n w i t h r e s p e c t t o t h e upwards d i r e c t e d f a m i l y of t h e compacta of X I t h e c o r r e s p o n d i n g d i r e c t l i m i t b e i n g v a l i d w i t h i n a b i j e c t i o n ) . So

i f t h e m a n i f o l d X ( b e i n g l o c a l l y compact) i s a l s o a g e t s a n increasing sequence of i d e a l s d)(Xl;

hence,

one o b t a i n s

Kn

(Xi),

~

(I-compact s p a c e , one

whose union is the space

(4.23) w i t h i n a b i j e c t i o n ( a l g e b r a i c isomorphism) of t h e s p a c e s ( a l g e b r a s ) involved.

4.( 2 ) . EXAMPLES.

afXi

THE ALGEBRA

133

Now, f o r e v e r y compact s u b s e t K of X I one c o n s i d e r s on a K ( X ) t h e t o p o l o g y d e f i n e d on i t by t h e sequence of semi-norms

{ "l'm,K ~

g i v e n by ( 4 . 1 3 ) ;

thus,

(4.24) becomes a m e t r i z a b l e l o c a l l y convex s p a c e which i s a l s o complete ( c f . L . SCHWARTZ [I : p. 64

3).

Hence, by ( 4 . 1 7 ) and C o r o l l a r y I ; 5 . 3

,

a K ( X , J be-

comes, i n e f f e c t , a (complex commutative) Fre'chet l o c a l l y m-convex algebra. Moreover, by c o n s i d e r i n g t h e sequence of t h e s u b s p a c e s d (X) , Kn nE IN, appeared i n ( 4 . 2 3 ) , one has t h e s i t u a t i o n occured i n K6the's Theorem ( c f . , f o r example, J . HOHVhH [I: p. 164, C o r o l l a r y ] ) , so t h a t w e g e t by ( 4 . 2 3 ) a complete l o c a l l y convex s p a c e , hence, i n p a r t i c u l a r , an

LF-space.

On t h e o t h e r hand, c o n s i d e r i n g a D X ) a l o c a l l y m-convex sub-

e m ( X l , t h e l a t t e r a l g e b r a b e i n g t o p o l o g i z e d a s above, one

a l g e b r a of

e a s i l y v e r i f i e s t h a t t h e c o n d i t i o n s of P r o p o s i t i o n 3.3 a r e s a t i s f i e d ,

so t h a t one c o n c l u d e s t h a t

3 IX) becomes, i n f a c t , an 1F - l o c a l l y m-convex aZgebra, whose l o c a l l y m-convex topology c o i n c i d e s w i t h

(4.25)

t h a t of t h e L F - s p a c e a ( X l , g i v e n by ( 4 . 2 3 ) . W e

S c h a r t z topology on

c a l l t h i s topology t h e

a(X).

by a p p l y i n g s t a n d a r d a r g u m e n t a t i o n ( s e e , f o r i n s t a n c e , L .

NOW,

SCHWARTZ [I: p. 8 9 1 ) ,

one

f u r t h e r concludes t h a t

w h i l e t h e given topology on

one indilced on a(Xi b y

d)(X)

i s dense i n

a(X)defined

C"IXl.

e m ( X l,

by ( 4 . 2 3 ) i s f i n e r than t h e r e l a t i v e I n t h i s c o n c e r n , w e s h a l l a l s o see ( c f

.

Scholium 4 . 1 below) t h a t n e i t h e r e m ( X l nor 3 i X l are &-algebras, i f one considers

3 ( X i a t o p o l o g i c a l s u b a l g e b r a of C"(Xl, w h i l e t h e l a t t e r

a l g e b r a i s , however, by t h e p r e v i o u s d i s c u s s i o n , a F r e c h e t l o c a l l y mconvex a l g e b r a . O n t h e o t h e r hand, t h e topological dual

S ' I X l

vex) s p a c e a D X ) i s d e f i n e d t o be t h e s p a c e of

f o l d X; i.e.

,

one c o n s i d e r s , a c c o r d i n g t o ( 4 . 2 3 )

forms on d ) ( X l

whose

,

t h o s e (complex) l i n e a r

r e s t r i c t i o n s t o t h e subspaces

continuous.

of thc ( l o c a l l y con-

d i s t r i b u t i o n s on t h e mani-

3 (Xi, nEN, Kn

are

Now, w i t h i n t h e same c o n t e x t , w e f i n a l l y n o t e t h a t i n c a s e t h e g i v e n manifold X i s , i n p a r t i c u l a r a d i s t r i b u t i o n s on X I r e s t r i c t e d t o

L i e group t h e n , t h e

a ' ( X ) ,

convolution of

makes t h e l a t t e r s p a c e i n t o

a (complex c o r i m i t d t i x ; i n c a s e X i s a b e l i a n ) conplete b a r r e l l e d l o c a l l y eon-

vex algebra w i t h continuous m u l t i p l i c a t i o n , which a l s o a d m i t t s a s an i d e n t i t y

element t h e Dirac measure on X

( c f . , f o r i n s t a n c e , L . SCHWARTZ [I:

p. 89, a s

134

IV

INDUCTIVE LIMIT ALGEBRAS

w e l l a s p . 157, Th6orSme I V , and p . 158, Th6orSme V I I ] )

.

T o p o l o g i c a l a l g e b r a s of t h e k i n d c o n s i d e r e d i n t h e p r e v i o u s d i s c u s s i o n , t h e e l e m e n t s of which a r e a l s o p a r t i c u l a r t y p e s of d i s t r i b u t i o n s on a g i v e n m a n i f o l d a r e of u s e , e s p e c i a l l y , i n c o n n e c t i o n w i t h ( t o p o l o g i c a l ) t e n s o r p r o d u c t s p a c e s a n d / o r a l g e b r a s ( nuclear algebras; s e e a l s o P a r t I1 of t h i s b o o k ) . Moreover, t h e same c l a s s of t o p o l o g i c a l a l g e b r a s h a s q u i t e a b r o a d a r e a of a p p l i c a t i o n s i n c l u d i n g , f o r i n s t a n c e , Mathematical P h y s i c s a s t h i s c o n c e r n s t h e m a t h e m a t i c a l found a t i o n of Quantum Mechanics. A s w e noted already t h e algebra c ” i X ) ,

Scholium 4.1.with i t s subalgebra

together

d ) t X i i n t h e r e l a t i v e t o p o l o g y , p r o v i d e examples

of l o c a l l y m-convex a l g e b r a s of t h e p a r t i c u l a r t y p e c o n s i d e r e d above, which are not &-algebras. Indeed, i f e m ( X )

was a Q - a l g e b r a , t h e n ( P r o p o s i t i o n 11; 7.1 ) i t s

spectrum would be a n e q u i c o n t i n u o u s s u b s e t of t h e r e s p e c t i v e weak t o p o l o g i c a l d u a l s p a c e ; t h u s i f t h e same a l g e b r a i s , i n p a r t i c u l a r , a F r d c h e t ( l o c a l l y m-convex)

algebra ( c f . ( 4 . 1 9 ) )

,

a ( X )

since

dense s u b a l g e b r a of e ” ( X i ( c f . L . SCHWARTZ [l: p. 89]),

is a

t h e l a s t conclusion

would t h e n be t r u e f o r t h e spectrum of d ( X ) t o o ( c f . Chapt. V; Scholium 2.2)

.

But t h e a l g e b r a 3 ( X i is c o m t a t i v e and advertibly complete ( c f . Def i-

nition I;6.4.

See a l s o

s. WARNER

[3: p. 811, s o t h a t d ( X )

w i l l be t h e n a

Q - a l g e b r a a s w e l l ( C o r o l l a r y 111; 6 . 5 ) , which i s a c o n t r a d i c t i o n

s.

~/MIVER [3] )

.

However, t h e s i t u a t i o n can be amended by c o n s i d e r i n g a (second countable)

C “-manifold; t h e n ,

e

(see

compact

i s , indeed , a comtative Fr6-

chet l o c a l l y m-convex Q-algebra w i t h an identity element. T h i s i s

a d i r e c t con-

sequence of t h e p r e v i o u s d i s c u s s i o n t o g e t h e r w i t h Lemma 2 . 1

of Chapter

V I I and Lemma 1 . 3 of Chapter V I .

4 . ( 3 ) . The algebra O(X).-

We c o n s i d e r i n t h i s f i n a l Example s p a c e s

of holomorphic f u n c t i o n s on a g i v e n (fi n i t e - d i m e n s i o n a l ) complex (anal y t i c ) m a n i f o l d a s a l g e b r a s s u i t a b l y t o p o l o g i z e d s o t h a t t o be induct i v e l i m i t t o p o l o g i c a l a l g e b r a s i n t h e s e n s e of t h e p r e v i o u s S e c t i o n 3 . Thus, suppose t h a t X i s an n-dimensional

complex ( a n a l y t i c ) mani-

fold and l e t 0:X) be t h e a l g e b r a ( w i t h p o i n t w i s e d e f i n e d o p e r a t i o n s ) of a l l complex-valued holomorphic functions on X. Namely , a complex-valued f u n c t i o n f on X b e l o n g s t o O ( X ) i f t h e r e e x i s t s a l o c a l c h a r t ( U , I$) of X such t h a t t h e map (4.26)

f0qI-I:

Q(0i-C

4.(3).

i s a holomorphic f u n c t i o n

EXAMPLES.

135

THE ALGEBRA c)(k')

i n t h e u s u a l s e n s e ( t h a t i s , a s a complex-

v a l u e d f u n c t i o n of s e v e r a l complex v a r i a b l e s d e f i n e d on t h e open s e t i$(U)LCn )

. Now

c o n s i d e r i n g O ( X l a s u b a l g e b r a of CiX)( t h e a l g e b r a of com-

p l e x - 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 X ) , w e f u r t h e r assume O ( X ) endowed w i t h t h e compact-open topology

i n d u c e d on i t by e ( X ) . Thus a d e f i n i n g

f a m i l y o f s u b m u l t i p l i c a t i v e semi-norms o f t h i s t o p o l o g y i s g i v e n by

f o r every f E O ( X ) ,

w i t h K v a r y i n g o v e r t h e compact s u b s e t s of X ( c f .

a l s o Examples I ; 3 . 1 and 3 . 2 ) :

becomes a (complex c o m m u t a t i v e )

so ()(A')

l o c a l l y m-convex algebra ( w i t h a n i d e n t i t y e l e m e n t ) . Moreover, a s t h i s assumed t o b e seeon2 countable, s o t h a t i t a d m i t s denumerable k-covering, s a y (Kn ) n ( c f . S u b s e c t i o n 4 . ( 1 ) o r y e t Chapt.

u s u a l l y happens, X i s a

~

( 4 . 2 7 ) a denwnerable fundamental fam-

V; S e c t i o n 5 ) . T h e r e f o r e , one g e t s by

iZy of

(submultiplicative)

semi-norms

(4.28)

d e f i n i n g t h e t o p o l o g y of O ( X l , w h i l e t h e l a t t e r ( l o c a l l y c o n v e x ) s p a c e i s a l s o c o m p l e t e ( W c i e r s t r a s s Theorem) ; h e n c e ,

m u t a t i v e ) Frgehet Zocally m-convex algebra NOW,

OiXl i s a (complex com-

( w i t h an i d e n t i t y element)

i f w e a r e g i v e n a compact subset K

of X one d e f i n e s (l(K), i.e.,

( c o m p l e x - v a l u e d ) hoZomo2rphic functions on K

t h e algebra of L e t QiKi

.

a s follows:

b e a f u n d a m e n t a l s y s t e m of open n e i g h b o r h o o d s o f K i n

X i n s u c h a way t h a t , f o r e v e r y U E n ( K ) , l e t O l U l b e t h e a l g e b r a of h o l o m o r p h i c f u n c t i o n s on t h e open s e t U C X , where t h e l a t t e r s e t i s c o n s i d e r e d a s an

open submanifold o f X

( h e n c e , t h e p r e v i o u s argumenta-

t i o n c a n b e a p p l i e d ) . T h u s , w e g e t a n inductive system of algebras { O i U i , pUv

(4.29)

1

,

U

-

E

163(K),

w i t h t h e c a n o n i c a l r e s t r i c t i o n maps ( a l g e b r a morphisms) (4.30)

puv :

f o r any U , V

OiV)

Otui,

i n 0 1 K i w i t h U E V ; so one

s e t s , by d e f i n i t i o n ,

(4.31)

T h e r e f o r e , f o r any Ue OiKl and f

E

OlU), pU(f)

B

O l K ) ( c f . a l s o (1.8))

c o n s i s t s of t h o s e h o l o m o r p h i c f u n c t i o n s , e a c h one of which i s d e f i n e d on some open n e i g h b o r h o o d o f K a n d c o i n c i d e s w i t h f i n some n e i g h b o r hood of K from 163iK) ( g e r m of a h o l o m o r p h i c f u n c t i o n f

equivalence class )

. Thus,

on K , o r y e t i t s

w e may c o n s i d e r on O ( K ) t h e f i n a l l o c a l l y convex

136

IV INDUCTIVE LIMIT ALGEBRAS

( v e c t o r s p a c e ) topology, d e f i n e d b y t h e canonical maps pu : o ( U l -

(4.32)

O(Kl,

U

E

m(k'),

a s i n d i c a t e d a b o v e , and t h e r e s p e c t i v e F r e c h e t s p a c e s ( i n f a c t , F r e c h e t l o c a l l y m-convex a l g e b r a s ) O l U l

,

with

U e n ( K l . On t h e o t h e r h a n d ,

t h i s topology on O(K! i s independent of t h e p a r i c u l a r open b a s i s of neighborhoods o f K ( n a m e l y , two s u c h s y s t e m s d e t e r m i n e a t h i r d one w i t h r e s p e c t t o

which e a c h one of t h e g i v e n two d e f i n e s a

cofinal

system, concerning

t h e analogous r e l a t i o n t o ( 4 . 2 0 ) . Now by assuming f u r t h e r t h a t t h e g i v e n m a n i f o l d X i s second count-

a b l e , one g e t s a denumerable open b a s i s of neighborhoods of K , s a y ( U n ln

.

Moreover, t h i s may b e c h o s e n so t h a t t o y i e l d a d e c r e a s i n g s e q u e n c e

3vn+I

( o f r e l a t i v e l y compact s e t s ) , w i t h U n , n E F;?, whose i n t e r s e c t i o n i s K . Thus by ( 4 . 3 1 ) one o b t a i n s a strict i n d u c t i v e l i m i t of Frgchet ( l o c a l l y convex) spaces, s o t h a t ( K o t h e ' s Theorem) (4.33)

complete ( H a u s d o r f f l o c a l l y c o n v e x ) s p a c e , i . e . ,

is, in effect, a

an

LF-space. On t h e o t h e r h a n d , o n e may c o n s i d e r , f o r e a c h U n , n E I N , g e b r a a ( z n l of

(complex-valued) c o n t i n u o u s f u n c t i o n s on

h o l o m o r p h i c on U n'

s o t h a t one o b t a i n s

c#(cnI C OlU, ),

(4.34) Now t h e "sup-norm''

the alwhich a r e

w i t h n E IN.

t o p o l o g y on a ( y n l makes it i n t o a Banach algebra, so

t h a t t h e l a s t r e l a t i o n a b o v e becomes a c o n t i n u o u s i n j e c t i o n ( a l g e b r a morphism) f o r t h e r e s p e c t i v e t o p o l o g i e s . Thus, by t a k i n g a l s o t h e rel a t i o n Un+I C U n , n e IN, i n t o a c c o u n t , o n e h a s t h e f o l l o w i n g s e q u e n c e of continuous i n j e c t i o n s

So one f i n a l l y c o n c l u d e s t h a t

where t h e s e c o n d of t h e l a s t r e l a t i o n s i s v a l i d w i t h i n a topological ( l i n -

earl isomorphism of t h e r e s p e c t i v e l o c a l l y convex spaces. F u r t h e r m o r e , one g e t s a l o c a l b a s i s of t h e l o c a l l y convex s p a c e (4.37) b y c o n s i d e r i n g s e t s of t h e f o l l o w i n g form (see a l s o ( 2 . 3 ) )

4.( 3 ) . EXAMPLES. THE ALGEBRA O(K)

137

(4.38) where B nE7N.

i s t h e u n i t s p h e r e of t h e Banach a l g e b r a a f u n l and O < E ~ < I ,

(By an a b u s e of n o t a t i o n we i d e n t i f y i n ( 4 . 3 8 1 , v i a t h e r e s p e c -

t i v e c a n o n i c a l map, Bn w i t h i t s image i n (4.37)). NOW,

it i s c l e a r t h a t t h e

sets ( 4 . 3 8 ) a r e a c t u a l l y m-sets of t h e a l g e b r a O I K ) , i . e .

,

one h a s

v.vsv.

(4.39)

Therefore, t h e algebra

O(Kl,which i s by t h e f o r e g o i n g a l o c a l l y con-

vex s s a c e , a d m i t s a l s o a l o c a l b a s i s c o n s i s t i n g of m-sets; orem I ; 1 . 1 )

,

O(KI

hence (The-

is a c t u a l l y a l o c a l l y m-convex algebra. I n d e e d , O ( 1 0 is a

(complex commutative) LF-locally m-convex algebra ( w i t h a n i d e n t i t y e l e

-

m e n t ) . The t o p o l o g y on O ( K i d e f i n e d by ( 4 . 3 7 ) ( s e e a l s o ( 4 . 3 6 ) ) i s c a l l e d a l s o t h e van Hove topology on

O(KI; t h e same t o p o l o g y c o i n c i d e s w i t h

t h e f i n a l v e c t o r s p a c e t o p o l o g y d e f i n e d on O(K) by t h e Banach a l g e b r a s

a ( E n ) ,n e I N , a s

above ( c f . L . VAN HOVE [l: p. 3411, a s w e l l a s L . WAELBPOOECK

[2: p. 1 5 6 1 ) .

On t h e o t h e r h a n d , O ( K ) b e i n g

an LF-algebra

i s a l s o a E'arrelled

( l o c a l l y rn-convex) a l g e b r a . Moreover, one c o n c l u d e s by ( 4 . 3 6 ) and Lem-

m a 3 . 1 t h a t O ( K i is s t i l l a Q-aZgebra; so w e have a n o t h e r i n s t a n c e of an LFQ- ( l o c a l l y rn-convex) algebra. ( I n t h i s r e s p e c t , s e e a l s o t h e above Subsection 4 . ( 1 ) ,

as w e l l a s Chapt.VI1; Section 9 f o r an a b s t r a c t set-

t i n g , c o n c e r n i n g t h e l a t t e r t y p e of t o p o l o g i c a l a l g e b r a s ) . Within t h e same framework, a s above, w e f i n a l l y n o t e t h a t one c a n c o n s i d e r , i n g e n e r a l , and i n a s i m i l a r way w i t h ( 4 . 3 1 ) ,

t h e topolo-

g i c a l algebra of holomorphic functions on any subset S of X , g i v e n by t h e r e l a t ion (4.40)

OfS) :=

13OiU)

u2; ( c f . a l s o Theorem 3 . 2 of C h a p t e r V I ) .

This Page Intentionally Left Blank

139

Spectrum (Global Theory)

CHAPTER V

1 . Spectrum

of a t o p o l o g i c a l a l g e b r a

Given a t o p o l o g i c a l a l g e b r a E , w e r e c a l l t h a t by a

continuous

character o f E we mean a non-zero c o n t i n u o u s (complex) morphism of E i n t o t h e a l g e b r a C of t h e complexes. The s e t of a l l c o n t i n u o u s char a c t e r s of E i s c a l l e d t h e

topological s p e c t r o of

E and i s d e n o t e d by

m ( E ) ( D e f i n i t i o n 11; 7 . 2 ) .

Thus i t i s o u r o b j e c t i v e i n t h e e n s u i n g d i s c u s s i o n t o endow t h e p r e v i o u s s e t w i t h an a p p r o p r i a t e t o p o l o g y and examine f u r t h e r some of t h e p r o p e r t i e s of t h e t o p o l o g i c a l s p a c e t h u s o b t a i n e d . T h i s e n r i c h e d

s e t i s o f t e n t a k e n t o be a l s o t h e spectrum of t h e t o p o l o g i c a l a l g e b r a under c o n s i d e r a t i o n ,

unless it i s s p e c i f i e d otherwise: t h u s , w e s t i l l

r e t a i n t h e t e r m i n o l o g y of D e f i n i t i o n 11; 7.2.More

p r e c i s e l y , w e adopt

h e r e a f t e r t h e t e r m i n o l o g y s e t f o r t h by t h e n e x t

D e f i n i t i o n 1 . 1 . L e t E be a t o p o l o g i c a l a l g e b r a . W e c a l l topoZogieaZ

spectrum, o r simply spectrum, of E t h e s e t m ( E l of a l l c o n t i n u o u s c h a -

r a c t e r s of E e q u i p p e d w i t h t h e r e l a t i v e t o p o l o g y induced on it by t h e weak t o p o l o g i c a l d u a l

E‘

of E ; t h a t i s , one h a s , by d e f i n i t i o n ,

(1.1)

m ( E l C E,‘

w i t h t h e r e s p e c t i v e r e l a t i v e t o p o l o g y on m(E).The same t o p o l o g i c a l space, say (??T(E), s ) , i s c a l l e d t h e logical algebra by ( 1 . 1 )

GeZ’fand space

E , w h i l e t h e weak t o p o l o g y

of t h e g i v e n t o p o -

oiE’, E l r e l a t i v i z e d on it

t h e GeZ’fand topology on m(E)(we s t i l l d e n o t e i t by

In t h i s respect,

i n s t e a d of t h e t o p o l o g i c a l s p a c e

s

T

) *

I m ( E l , s)

,

where w e have s e t s = o(E’, E l 1 , i . e . , t h e weak t o p o l o g y s i E ’ , El r e 1772 ( E ) l a t i v i z e d t o ??Z(E), we s h a l l simply w r i t e mIEl (which i s , of c o u r s e , a common p r a c t i c e ) by r e f e r r i n g t o t h e s p e c t r u m ( G e l ‘ f a n d s p a c e ) of a given topological algebra. Thus, a s a f i r s t consequence of t h e above D e f i n i t i o n 1 . 1

(cf.

v

140

SPECTRUM(GLOBAL THEORY)

( 1 . 1 ) ) one c o n c l u d e s t h a t

t h e spectrum (Gel’fand space) m ( E l of a g i v e n t o p o l o g i c a l algebra E i s a Hcusdorff compZetely r e g u l a r t o p o l o g i c a l space. However, it i s c l e a r , of c o u r s e , t h a t i n t h e most g e n e r a l case

7 1 2 i E I might b e t h e empty s e t , a s t h i s c a n b e t r u e e v e n f o r a commutat i v e Banach a l g e b r a w i t h o u t a n i d e n t i t y e l e m e n t ( c f . , f o r i n s t a n c e ,

N. BOURBAKI [ I l : Chap. 1 ; p . 9 5 , E x e r c . 4 ) ] ) . But w e a d o p t h e r e t h e q u i t e g e n e r a l framework w e a p p l y i n t h e s e q u e l t o h a v e t h u s t h e “most u s e f u l g e n e r a l i t y ” ( ! ) ( I n t h i s c o n c e r n , see a l s o , f o r example, I.E . SEGAL

-

R . A . KUIVZE [l: p. 2091).

On t h e o t h e r h a n d , w e h a v e d e f i n e d i n C h a p t . 11; S e c t i o n 7 t h e

Gel’fand map of a g i v e n t o p o l o g i c a l a l g e b r a E , t o be t h e map

Moreover, t h e v a l u e of

at x e E

,

-

3 (xi = 2 : m(E/

t h a t i s t h e map

C , g i v e n by ( 1 . 3 ) i s , by d e f i n i t i o n , t h e

Gel’fand transform of t h e e l e -

ment x i n E ( s e e D e f i n i t i o n 11; 7 . 3 ) . Now, s i n c e m l E l

c a r r i e s t h e r e l a t i v e t o p o l o g y from t h e weak

( t o p o l o g i c a l ) d u a l s p a c e E,’ of E one r e a d i l y v e r i f i e s , by ( 1 . 3 ) ,

that

t h e range o f t h e Gel’fand map i s contained i n t h e (1.4)

space ( i n f a c t , a l g e b r a ) of complex-valued continuous f u n c t i o n s on t h e spectrum of E .

So one g e t s e s s e n t i a l l y t h e map (1.5) with

C l = c1172(E)),

>:E+C(?7(E),

[g(x)](f) = s(f) = f f x ) , f o r

a n y x e E and f e m(E); i n o t h e r words,

f o r eoeru elome?i: x i n E , t h e r e s p e c t i v e Gel’fand transform

2

o f x d e f i n e s a com-

plex-valued continuous f u n c t i o n on m i h i . F u r t h e r m o r e , it i s a l s o a d i r e c t c o n s e q u e n c e o f

( 1 . 3 ) t h a t the

Gel’fand map d e f i n e s an ( a l g e b r a ) morphism of E i n t o the aZgebra t h a t one d e f i n e s t h e image of

5

c (m(E)),so

a s t h e Gel’fand transform algebra o f t h e

g i v e n ( t o p o l o g i c a l ) a l g e b r a E , d e n o t e d by E ^ ; t h u s , one g e t s by ( 1 . 3 ) (1.6)

E^

f

Im(gIG C ( ? T ( E l ) .

I n p a r t i c u l a r , ihu aZgebra morphism d e f i n e d by

9 preserves

t h e i d e n t i t y eZe

-

men&, i n c a s e t h e g i v e n a l g e b r a E h a s o n e . T h i s f o l l o w s d i r e c t l y by t h e h y p o t h e s i s f o r t h e e l e m e n t s o f M ( E ) (see a l s o C h a p t . 11; S e c t i o n 7 ) . A s a m a t t e r of f a c t , c o n c e r n i n g t h e above r e l a t i o n

(1.6),

one

1.

TOPOLOGICAL SPECTRUM

141

actually gets something much more, according to the following

Lemma 1.1. Let E be a topological algebra. Then t h e Gel’fand topology on 7 7 7 1 E ) (Definition 1.1) i s the weukest topology on it f o r which the maps &, w i t h x e E , are continuous ( i . e . , t h e i n i t i a l topology defined on m(E)by the maps 2 , xEE).

Proof. The weak topology on E is, by definition, the initial topology defined on it by the elements of E l where these elements are

considered as complex-valued functions on E l via the canonical d u a l i t y

for any x €

:= x‘lx) = l l x ’ )

2, x‘>

(1.7) E

E and x‘eE’.

So the Gel’fand transform

E is the restriction to ???‘(Z)

(1.7),

, 2 of an element x

E’ of the map 2 : E’-+

,defined by which proves the assertion on the basis of the same Definition @

1.1.1 On the other hand, by considering with every element x of a given topological algebra E the associated evaluation map i, as a complex-valued continuous function on the weak dual E’, of E , by (1.7), one concludes that the extended spectmm of E (cf. Chapt. 11; (7.6)) defined by the relation m(El’

(1.8)

i s a (weakly) close? subset of F’

= m(E) U

{ O ] C E’

. Indeed, this

follows immediately from

the relation

since each one of the sets appeared in the second member of (1.9) is a weakly closed subset of E’, being the “kernel“ in Es’ of the continuous (complex-valued) function x^y - 2.;; that is, one has (1.lo)

{ f e ~ ’ : f(syi=fix)f(yl)=

ker(x?-2i)r~;,

for any elements x,y in E . In this respect, one actually concludes that (1.9) is related to the closure of m(El in E,‘ ; that is, one obtains (1.11)

m ( E ) C m ( E l U { 0 } = m(E)+ C E,’

Indeed, we remark that for any net ( f 6 I

f e m(U.)cE;,

in mlEl, s o that one gets

.

one has f = lhf6, for some 6

nuity of the multiplication in @ ! f ( x ) f ( y ) ,

v

142

SPECTRUM (GLOBAL THEORY)

f o r any e l e m e n t s x,y i n E , so f e m I E l f .

Remark 1.1.I n case the topoZogical aZgebra E has an i d e n t i t y eZement, t h e p r e v i o u s r e l a t i o n ( 7 . 1 2 ) a l w a y s y i e i d s a n e l e m e n t of ???(El, so t h a t one c o n c l u d e s t h a t t h e spectrwn of E is a closed subset of EL. I n t h i s r e s p e c t , see a l s o R.M. BROOKS [ 3 ] . A p a r t i c u l a r l y u s e f u l n o t i o n connected w i t h t h e spectrum of a

t o p o l o g i c a l a l g e b r a , which a l s o w i l l o f t e n b e a p p l i e d i n t h e s e q u e l ,

i s t h a t p r o v i d e d by t h e n e x t

Definition 1.2. W e s h a l l s a y t h a t t h e s p e c t r u m t o p o l o g i c a l a l g e b r a E i s l o c a l l y equicontinuous

m(ElC E ' .

of a g i v e n

i f f o r every element f

i n M ( E ) t h e r e e x i s t s a n e i g h b o r h o o d U of f i n e q u i c o n t i n u o u s s u b s e t of

??Z(E)

m ( E ) which

i s an

(Thus, b r i e f l y , w e r e q u i r e every

e l e m e n t of t h e s p e c t r u m t o have a n e q u i c o n t i n u o u s n e i g h b o r h o o d ) . I t i s c l e a r t h a t every t o p o l o g i c a l a l g e b r a with an equicontinu-

o u s s p e c t r u m , f o r e x a m p l e , e v e r y & - a l g e b r a ( P r o p o s i t i o n 11; 7 . 1 1 , i s a

topoZogicaZ aZgebra w i t h a ZocaZZy equicontinuous s p e c t m . More g e n e r a l t o p o logical algebras pcssessing t h i s property w i l l be encountered i n t h e ensuing d i s c u s s i o n . Thus, a b a s i c f a c t from t h e g e n e r a l t h e o r y of t o p o l o g i c a l v e c t o r s p a c e s , which i s n a t u r a l l y i n v o l v e d i n t h i s framework i s , of c o u r s e , t h e Theorem o f A l a o g l u - B o u r b a k i a c c o r d i n g t o which ( s e e , f o r i n s t a n c e , J . HORVLTH [l: p. 201, Theorem I] ) : Given a topoZogicaZ vector space E every equi-

continuous subset o f p' i s weakZy relativeZy compact ( i . e . , r e l a t i v e l y compact i n t h e weak t o p o l o g y aiE', E ) of E ' ) .

An a p p l i c a t i o n of t h i s t h e o r e m i n

c o n j u n c t i o n w i t h t h e above D e f i n i t i o n 1 . 2 e x h i b i t s , p a r t l y , t h e s i t u a t i o n one h a s i n case of Banach a l g e b r a s , a s t h i s i s e x p r e s s e d by t h e n e x t Theorem 1 . 1 .

F u r t h e r m o r e , o n e g e t s a more s a t i s f a c t o r y p i c t u r e

(see a l s o P r o p o s i t i o n 1 . 1 )

by c o n s i d e r i n g , f o r example, t h e p a r t i c u -

l a r c l a s s of t o p o l o g i c a l a l g e b r a s s e t f o r t h by t h e f o l l o w i n g d e f i n i t i o n ; t h e same c l a s s of a l g e b r a s w i l l a l s o p l a y a p a r t i c u l a r

rale

in

t h e s e q u e l . Thus w e now h a v e .

Definition 1.3.

A t o p o l o g i c a l a l g e b r a E i s s a i d t o b e spectraZZy

b a r r e l l e d , whenever e v e r y bounded s u b s e t of i t s s p e c t r u m ( c o n s i d e r e d a s a s u b s e t of E ' ) CYF '

_

is

is equicontinuous.

( I n o t h e r words, a s u b s e t B of

e q u i c o n t i n u o u s i f , and o n l y i f

,

i t i s ( w e a k l y ) bounded)

m(E)

.

S e v e r a l p r o p e r t i e s of t h e p r e c e d i n g c l a s s of t o p o l o g i c a l a l g e b r a s w i l l b e c o n s i d e r e d t h r o u g h o u t t h e s e q u e l : however, w e p r e s e n t l y

143

TOPOLOGICAL SPECTRUM

1.

g i v e t h e f o l l o w i n g p r o p o s i t i o n , a f i r s t c o n n e c t i o n o f t h e c l a s s of t o p o l o g i c a l a l g e b r a s under c o n s i d e r a t i o n w i t h t h o s e a l r e a d y f a m i l i a r from t h e p r e c e d i n g d i s c u s s i o n . Thus, w e h a v e .

P r o p o s i t i o n 1.1. Every m-barrelled topological algebra i s s p e c t r a l l y b a r r e l -

l e d . I n p a r t i c u l a r , every b a r r e l l e d , hence every Frbcchet, topological algebra i s spectrally barrelled. Proof. I f B i s a bounded s u b s e t o f then i t s p o l a r set B'CE

mIE/ ( w i t h r e s p e c t t o E,' )

,

i s an a b s o r b i n g a b s o l u t e l y convex c l o s e d and

i d e m p o t e n t s u b s e t of E ( s i n c e B C ??'Z(E))

, that

i s , an m-barrel

of

E

( D e f i n i t i o n I ; 1 . 2 ) , s o by h y p o t h e s i s a n e i g h b o r h o o d o f z e r o i n E.Thus t h e ( b i p o l a r ) s e t Boo

s e t of

E',

and a f o r t i o r i B G B o O

i s an e q u i c o n t i n u o u s s u b -

which p r o v e s t h e f i r s t p a r t o f t h e a s s e r t i o n t h e r e s t be-

i n g o b v i o u s l y v a l i d by t h e same d e f i n i t i 0 n s . I I n f a c t , i n d u c t i v e l i m i t s of F r g c h e t a l g e b r a s (see Chapter I V )

a r e s t i l l s p e c t r a l l y b a r r e l l e d a l g e b r a s , a s w e s h a l l see l a t e r o n ; t h e l a s t c l a s s i s , m o r e o v e r , g e n u i n l y l a r g e r t h a n a l l t h e o t h e r o n e s cons i d e r e d i n t h e above P r o p o s i t i o n 1 . 1

( c f . C h a p t . V I I 1 ; Scheme ( 2 . 1 ) ) .

The f o l l o w i n g h a s a l r e a d y b e e n p r o m i s e d a b o v e .

(However, see

a l s o the next chapter, Corollary 1.3, f o r a s t i l l stronger v e r s i o n ) . Thus, w e have.

Theorem 1.1. Let E be a topological algebra w i t h spectrum m(El. Moreover,

consider t h e following a s s e r t i o n s : 1 ) m ( E l i s l o c a l l y equicontinuous. 2 ) ??Z(E/

i s l o c a l l y compact.

Then, 1 ) i m p l i e s 2 ) . I n p a r t i c u l a r , if E i s a s p e c t r a l l y b a r r e l l e d a l g e b r z , then t h e preceding two a s s e r t i o n s are equivalent. Proof. L e t

f e m ( E l and U a n e q u i c o n t i n u o u s n e i g h b o r h o o d of f i n

T ? Z ( E ) , s o t h a t U i s a r e l a t i v e l y compact s u b s e t of E,'

- __ k i ) . That i s , U S n T ( E l C E i ( 1 -13)

(Alaoglu-Bourba-

i s weakly compact i n E', t h u s (cf. ( 1 . 1 1 ) ) -

un m ( ~ = U) n C { o }

i s a l o c a l l y compact s u b s p a c e of

m(E).T h e r e f o r e , t h e r e e x i s t s a com-

p a c t n e i g h b o r h o o d , s a y K , of f w i t h f e K C - u fl VY(E), t h a t i s 1 ) i m p l i e s 2). On t h e o t h e r h a n d , i f t h e a l g e b r a E i s s p e c t r a l l y b a r r e l l e d , t h e n i t

i s r e a d i l y s e e n t h a t 21 i m p l i e s 1 ) a s w e l l , by t h e same d e f i n i t i o n s , and t h i s f i n i s h e s t h e proof. I Now l o c a l e q u i c o n t i n u i t y of t h e s p e c t r u m of a g i v e n t o p o l o g i c a l a l g e b r a p r o v i d e s , a s w e s h a l l see t h r o u g h o u t t h e s e q u e l , s e v e r a l u s e -

v SPECTRUM (GLOBAL THEORY)

144

f u l p r o p e r t i e s of t h e a l g e b r a i n q u e s t i o n , a f a c t which i n c a s e of Banach a l g e b r a s i s a u t o m a t i c a l l y v e r i f i e d , on t h e b a s i s of t h e p r e v i o u s t h e o r e m , s i n c e t h e l a t t e r a l g e b r a s h a v e , of c o u r s e , t h e i r s p e c t r a loc a l l y compact s p a c e s ( c f . a l s o P r o p o s i t i o n 1 . 1 ) . Thus, it i s , i n f a c t , v i a t h e l o c a l c o m p a c t n e s s t h a t l o c a l e q u i c o n t i n u i t y of t h e s p e c t r u m i s u s u a l l y a p p e a r e d i n s e v e r a l c o n c r e t e c a s e s , where t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d a r e , i n e f f e c t , s p e c t r a l l y barrelled.

Similarly, another instance t h a t i s often occurred i s t h e s t r o n g e r a s s u m p t i o n of b e i n g t h e s p e c t r u m e q u i c o n t i n u o u s , ex-

p r e s s e d v i a i t s c o m p a c t n e s s , i n t h e p r e s e n c e of s u p p l e m e n t a r y hypot h e s e s f o r t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d : it h a p p e n s

for

in-

s t a n c e , i n case of F r 6 c h e t o r e v e n b a r r e l l e d a l g e b r a s . Thus a s a n app l i c a t i o n of t h e p r e c e d i n g t h i s m i g h t b e r e p l a c e d by t h e weaker hypot h e s i s the algebras involved t o be only s p e c t r a l l y b a r r e l l e d , a s t h i s w i l l become c l e a r i n t h e s e q u e l ( s e e C h a p t . V I : S e c t i o n 1 ) . A p a r t i c u l a r a p p l i c a t i o n of t h e p r o p e r t y of b e i n g t h e spectrum

of a g i v e n t o p o l o g i c a l a l g e b r a l o c a l l y e q u i c o n t i n u o u s i s p r e s e n t l y made i n t h e n e x t s e c t i o n .

2 . Spectrum o f t h e completion o f a topological algebra Suppose w e a r e g i v e n a t o p o l o g i c a l a l g e b r a E , t h e c o m p l e t i o n of

2

i s a g a i n a t o p o l o g i c a l a l g e b r a . We know t h a t t h i s a s s u m p t i o n a l r e a d y e n t a i l s c e r t a i n c o n d i t i o n s on t h e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n E (see, f o r i n s t a n c e , Corollary 1 ; 4 . 7 ) . T h u s i n t h e m o s t which

g e n e r a l case, i f f i s a c o n t i n u o u s c h a r a c t e r of E , i t s c o n t i n u o u s extension t o

?

which i s u n i q u e l y d e t e r m i n e d by

m u l t i p l i c a t i v e l i n e a r form on

f , may n o t be a l w a y s a

i.

However, i f t h e topological algebra E has a continuous m u l t i p l i c a t i o n one

g e t s , of c o u r s e , t h e r e l a t i o n

miE) =

(2.1)

m (%),

within a b i j e c t i o n , b y i d e n t i f y i n g a n e l e m e n t of ?%(El extension t o

2.

with i t s (unique)

Thus d e n o t i n g by

j : E'-i'

(2.2)

t h e c a n o n i c a l b i j e c t i o n which i d e n t i f i e s a n e l e m e n t of

E'

with

its

h

( c o n t i n u o u s ) e x t e n s i o n t o E l s u p p o s e t h a t t h e r a n g e of t h e r e s t r i c t i o n l e tius i s c o n t a i n e d i n ?? T h ai t? is, ( )assume . the rela-

o f j t o ??2iE)G E'

tion

145

2. SPECTRA OF COMPLETED ALGEBRAS

of c o u r s e , it i s e q u i v a l e n t t o assume t h a t the map

j:

(2.4)

m(E)

---L

mi.?)

/ .

is a b i j e c t i o n

(supposing s t i l l t h a t E i s a topological a l g e b r a ) .

T h u s , o u r main o b j e c t i v e i n t h e r e s t of t h i s s e c t i o n i s t o examine t o what e x t e n t t h e canonical b i j e c t i o n (2.4) (whenever t h e r e

ists !)

ex-

is, in f a c t , a homeomorphism of t h e r e s p e c t i v e G e l ' f a n d s p a c e s

( s p e c t r a ) of t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d . However, a s w e s h a l l

s e e , t h i s i s n o t a l w a y s t r u e e v e n f o r l o c a l l y m-convex a l g e b r a s : on t h e o t h e r h a n d , by c o n s i d e r i n g t h e i n v e r s e map of

( 2 . 4 ) , we s t i l l g e t

a continuous b i j e c t i o n , namely, t h e map (2.5)

i

:

miff)- ZT(E)

( s e e t h e n e x t lemma). Thus t h e f o l l o w i n g i s a d i r e c t c o n s e q u e n c e of t h e preceding.

Lemma 2.1.

2 is

Let E b e a topological algebra f o r which the respective completion

s t i l l a topological algebra. Moreover, suppose t h a t t h e canonical bi,jection

(2.4) holds t r u e , arid consider its inverse map

i :

(2.6)

m(2)+??Z(Ej.

Then, i d e f i n e s a continuous b i j e c t i o n of the r e s p e c t i v e topological spaces; i . e . , one has t h e r e l a t i o n

= m(E),

(2.7)

v a l i d w i t h i n a con'?nuous Proof. Assuming

b i j e c t i o n given by t h e map i . ( 2 . 4 ) a n d on t h e b a s i s of Lemma 1 . I ,

it i s i m -

A

m e d i a t e t h a t i i s c o n t i n u o u s , s i n c e E E E , and t h i s p r o v e s t h e assertion. I

Remark.- Suppose t h a t ( 2 . 7 ) i s v a l i d and cons i d e r , f o r i n s t a n c e , ;he c o n t e x t of 11; ( 7 . 2 5 ) . Then, f o r e v e r y x e E S D , o n e g e t s t h e r e l a t i o n (2.8)

Sp-(x) =

E

2(??'L(?))

= G(TC(E)).

T h i s i s immediate from t h e same d e f i n i t i o n of the bijection (2.7). Now on t h e b a s i s of t h e p r e v i o u s lemma, one o b t a i n s t h a t

i n case

the spectrum m(E) o f t h e c o m p l e t i o n of E i s a compact (Hausdorffl space, then the r e l a t i o n ( 2 . 7 ) is v a l i d w i t h i n a homeomorphism. An i m p o r t a n t example of t h e l a s t s i t u a t i o n i s s t i l l a p p e a r e d i f t h e s p e c t r u m of t h e o r i g i n a l a l g e b r a i s l o c a l l y e q u i c o n t i n u o u s , s i n c e t h e n (see Theorem 2 . 1 ) t h e relation (2.7)

y i e l d s a homeomorphism. A s a m a t t e r of f a c t , s o m e t h i n g

v

146

SPECTRUM (GLOBAL THEORY)

more i s v a l i d , a c c o r d i n g t o t h e n e x t

Lemma 2 . 2 . Let E be a topological algebra whose completion E i s s t i l l a to-

pological algebra, and i n such a way t h a t one has f o r the respectiue spectra the relation

= miE),

"-?C;)

(2.9)

m (El i s

within a (continuous) b i j e c t i o n . Then, o n l y i f , t h i s is the case f o r

l o c a l l y equicontinuous i f , and

In p a r t i c u l a r , t h e conclusion holds t r u e f o r

??Y(;).

every l o c a l l y m-convex algebra.

Proof. I f

m(E) i s

l o c a l l y e q u i c o n t i n u o u s and U i s a n e q u i c o n t i n u -

o u s n e i g h b o r h o o d of f =if?) i n m(E),w i t h -1

n u i t y of t h e map i (Lemma 2 . I ) , i

f E m ( i ) t,h e n

(U)= j(Ul

SG

??Z(k)

by t h e c o n t i -

i s a neighborhood

=j(f)= 7 in

??Z(i), which i s a l s o e q u i c o n t i n u o u s ( c f . N. BOURBAKI [5: Chap. 10; p. 15, P r o p o s i t i o n 4 1 ) On t h e o t h e r h a n d , i f ??Z(i) i s of

i-'(f)

.

l o c a l l y e q u i c o n t i n u o u s and f € ? ? U E ) , e q u i c o n t i n u o u s neighborhood a n e q u i c o n t i n u o u s s u b s e t of

t h e r e e x i s t s , by h y p o t h e s i s , a n

f = jlf) i n mii),so t h a t i i ; ) = U i s m(EI , w i t h i i f ) = f E: U ( c f . a l s o t h e p r e v i -

G

of

o u s R e f . ) T h u s , t h e two t o p o l o g i e s i n d u c e d on U by E and

2

coincide

(see Lemma VI; 4 . 2 i n t h e s e q u e l ) ; t h a t i s , U i s a n e q u i c o n t i n u o u s neighborhood of f i n m(E),a n d t h i s f i n i s h e s t h e p r o o f . I W e come n e x t t o o u r a s s e r t i o n t h a t l o c a l e q u i c o n t i n u i t y of m1EI i m p l i e s a l r e a d y ( 2 . 7 ) w i t h i n a homeomorphism: Thus b y t a k i n g t h e map

j g i v e n by ( 2 . 4 ) , it i s c l e a r ( c f . a l s o Lemma 1 . 1 ) t h a t t h i s i s cont i n u o u s i f ( a n d o n l y i f ) , f o r e v e r y e l e m e n t z e E , t h e map 2 ( G e l ' f a n d t r a n s f o r m of z ) i s a c o n t i n u o u s f u n c t i o n on m(2)when t h e l a t t e r s p a c e c a r r i e s t h e weak t o p o l o g y d e f i n e d on it by t h e maps ; , w i t h

so d e n o t i n g b y

TE

xeE.

t h e l a s t topology, t h e preceding c o n d i t i o n i s equi-

valent t o the relation 12.10)

f o r every z e

2.

On t h e o t h e r h a n d , s i n c e the Gel'fand map of E l a s g i v e n by

, is

always continuous r e l a t i v e t o t h e topology of simple convergence on i t s range, one g e t s (1.3)

(2.11)

p i ) c S),

t h e c l o s u r e being taken with r e s p e c t t o t h e l a s t topology. (2.10)

i s f u l f i l l e d whenever t h e s e c o n d member of

( c o m p l e x - 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 ???@I, i s endowed w i t h t h e p r e v i o u s t o p o l o g y

However, one o b t a i n s , i n f a c t ,

Therefore,

(2.11) still y i e l d s

w h i l e t h e l a t t e r space

T"

E ' (2.10)

i f , f o r example, f o r every

2. SPECTRA OF COMPLETED ALGEBRAS

141

h

z € E, t h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d :

Z^ is the uniform limit on

(2.12)

m(ij of

a net (x6 is € I i n

m(Ej ( i . e . , rcse E , with &€I!.

I n f a c t , it s u f f i c e s t h e a b o v e c o n d i t i o n ( 2 . 1 2 ) t o b e t r u e o n l y

localty on

m(g) and

e v e n more j u s t f o r a s e t o f g e n e r a t o r s of t h e t o -

p o l o g i c a l a l g e b r a E u n d e r c o n s i d e r a t i o n . Thus one i s n a t u r a l l y l e d t o Lemma 2 . 3 below. But w e n e e d f i r s t some more t e r m i n o l o g y , which w i l l a l s o b e u s e f u l i n some o t h e r c o n t e x t i n t h e s e q u e l . So w e f i r s t h a v e .

Definition 2 . 1 . Given a t o p o l o g i c a l a l g e b r a E , a s e t of tcpcZogZca2 generators

o f 6 i s a s u b s e t , s a y S , of E s u c h t h a t E

t h e s m a l l e s t c l o s e d s u b a l g e b r a of

coincides with

it c o n t a i n i n g t h e s e t S. That i s ,

e v e r y e l e m e n t o f E is t h e l i m i t o f a n e t i n E c o n s i s t i n g of f i n i t e l i n -

e a r c o m b i n a t i o n s ( w i t h complex c o e f f i c i e n t s ) of p r o d u c t s of f i n i t e many e l e m e n t s o f S. T h u s , e v e r y

xCE= 1 X . a . z

z'

z B E i s o f t h e form z = l i m x g , w i t h

6

ai i s a f i n i t e p r o d u c t of e l e m e n t s of S .

such t h a t each

Now, s i n c e by Lemma I ; 1 . 5 , the closure of a subalgebra of E is a closed

subalgebra of it, it i s c l e a r t h a t S c Z i s a s e t o f t o p o l o g i c a l g e n e r a t o r s o f E i f , and o n l y i f , t h e a l g e b r a E c o i n c i d e s w i t h t h e c l o s u r e of t h e t s u b a l g e b r a o f i t , t h e e l e m e n t s of which a r e o f t h e form 1 A . a .

2 2'

with a .

a f i n i t e p r o d u c t of e l e m e n t s of S . Furthermore, i f t h e given t o p o l o g i c a l a l g e b r a E has an i d e n t i t y e l e m e n t and t h e s e t S i s a "commuting f a m i l y o f e l e m e n t s of E : " ,

say

i x c l ) a e I , t h e n c o n c e r n i n g t h e above d e f i n i t i o n , i s e q u i v a l e n t t o s a y t h a t t h e algebra of polynomials with complex c o e f f i c i e n t s defined by t h e famiZy S =(x

c1

Ia e 1 ( w e

[ix a )n e I ] ) i s d e n s e i n E ; i . e . , one h a s E = c[(s,laE J .

d e n o t e it by

(2.13)

(I:

I n g e n e r a l , o n e c o n s i d e r s t h e s u b a l g e b r a of Q: [ i x a ) a

,] ,

c o n s i s t i n g of

polynomials without constant term ( w e d e n o t e t h i s a l g e b r a by C 0 S o t h e p r e v i o u s r e l a t i o n ( 2 . 1 3 ) i s now e x p r e s s e d by (2.14)

i5

= c ,[ ( X a J a

I]

[

i].

I

which a c t u a l l y c o r r e s p o n d s t o t h e above D e f i n i t i o n 2 . 1 . I n p a r t i c u l a r , i f w e have t h e r e l a t i o n s

E = c [ ( x a ~ a E (I r]e s p . , B =

(2.75)

i n s t e a d of

c,[(xa

iu

E I ] ) ,

( 2 . 1 3 ) and ( 2 . 1 4 ) , r e s p e c t i v e l y , t h e n w e s p e a k o f a s e t of

generators ( o r y e t

of t h e a l g e b r a E'.

algebraic generators) , s a y S = (xu)

with x a e E , a e I ,

So i n t h i s c a s e , t h e a l g e b r a E c o i n c i d e s w i t h t h e

148

V

SPECTRUM (GLOBAL THEORY)

smallest subalgebra of it containing the given set (family) S . The notation of ( 2 . 1 3 ) is applied in the sequel essentially in case of a commutative (locally m-convex) algebra E (see, for example, C E considered Chapt. VIII; Section 7) , so that the family S = above is certainly " c o m t i n g " (its elements are pairwise commuting) Nevertheless, for convenience of the notation applied we presently use the same terinology, as above, in the general case; butthis could also be viewed in the sense of the above Definition 2 . 1 . (In this re-

.

spect, cf. , for instance, N . BOURBAKI [3: Chap. 4; p. 18, Proposition 1 1 ) . NOW, consider a topological algebra E, whose spectrum is (El, and l e t f be a (continuous) character o f E ; moreover, consider any element of t h e algebra Q: [ (2,)" ,] , say P({x 1 , 1 6 i 6 n l (i.e., a polynomial with 'i complex coefficients and without constant term, in a finite number of elements ("indeterminates")x in S ) Thus, one g e t s t h e r e l a t i o n "i (2.16) f ( P f { z c i . } ) ) =P ( { f i z a 1 1 ) = P ( { ; . ( f ) I , 1 5 i ~ n ;) 7, i "7,

.

that is, concerning the respective Gel'fand transforms of the elements of C, [ ( x . , ) ~ I]~, one obtains

Thus, we are now in the position to state the following basic result, which we have promised before. Lemma 2.3. Let E be a topological algebra whose spectrum i s m r E l , and S =

( x , ) , ~I a s e t of topological generators of E. Moreover, suppose t h a t t h e following condition i s s a t i s f i e d :

(2.18)

For every element xE E and every n e t (z6)from C 0 [ ( z a ) " E I] converging t o x (cf. ( 2 . 1 4 ) 1, t h e n e t f;&) converges t o 2 l o c a l l y u n i f o r m l y i n m(E).(That is, for every f in mfE), there exists a neighborhood, say U , of f

in m(Ej such that t h e n e t (x6) converges t o x uniformly on

U).

Then, t h e topology o f nTfE) c o i n c i d e s w i t h t h e weak ( i . e . ,

i n i t ! : a l ) topology de-

f i n e d on it by t h e maps &", w i t h x a e S . Proof. Since S = ( x a l c i B I C E l it is clear that the initial (weak) c i e I, is weaker topology, say T~ , on m(E) defined by the maps than the given topology on m(E)(cf. also Lemma 1 . 1 ) . T h u s , the assertion is reduced to the proof that every map 2 : mfE)-Q:, w i t h I C E E , i s a l s o continuous w i t h r e s p e c t t o t h e topology T~ on m(EI:So by hypothesis and ( 2 . 1 4 ) one gets, for every Z E E, the relation

149

SPECTRA OF COMPLETED ALGEBRAS

2.

x = limx

(2.19)

6 where (x6 ) i s a n e t i n t h e a l g e b r a C, [ ( x c1I map

3

I]. Thus c o n c e r n i n g t h e x i s g i v e n by ( 2 . 1 9 ) , one c o n c l u d e s by ( 2 . 1 8 ) m ( E l , t h e r e e x i s t s a neighborhood u of f i n

: m(E)-@, where

t h a t , f o r every f E

m(E),

s u c h t h a t t h e n e t (xg I

,corresponding

t o (2.19), converges t o

uni-

3:

f o r m l y on U C m ( E ) . NOW, s i n c e by h y p o t h e s i s f o r t h e n e t (x6) and ( 2 . 1 7 ) e a c h one of t h e maps

i s c o n t i n u o u s f o r t h e t o p o l o g y Ts

one

I

s t i l l o b t a i n s t h e c o n t i n u i t y of 3 f o r t h e s a m e t o p o l o g y , and t h e p r o o f

is finished. I

Schol ium.-

T a k i n g t h e comment c o n c e r n i n g ( 2 . 1 1 ) i n t o a c c o u n t , one

g e t s , c o n c e r n i n g t h e p r e c e d i n g Lemma 2 . 3 ,

that

(2.20)

-

so t h a t , by ( 2 . 1 8 ) , t h e c l o s u r e i n t h e l a s t r e l a t i o n i s s t i l l con tained i n a

c ( ~ ( E ) [ T ~ .] )Moreover,

by c o n s i d e r i n g

C [(xu:UEI] E E a s

topological subalgebra o f E , i t i s c l e a r t h a t i f 12.181 holds t r u e and E

has a continuous m u l t i p l i c a t i o n , one g e t s t h e r e l a t i o n (2.21)

w i t h i n a homeomorphism

o f t h e r e s p e c t i v e G e l ’ f a n d s p a c e s of t h e t o p o l o -

g i c a l a l g e b r a s u n d e r c o n s i d e r a t i o n . Now, a s w e s h a l l p r e s e n t l y see, t h e above s i t u a t i o n i s p r o v i d e d , i n f a c t , i n case t h e t o p o l o g i c a l a l g e b r a E h a s a l o c a l l y e q u i c o n t i n u o u s s p e c t r u m ( c f . Lemma 2 . 4 )

.

The n e x t r e s u l t e x h i b i t s a n i m p o r t a n t s p e c i a l i n s t a n c e of t h e p r e c e d i n g c o n d i t i o n ( 2 . 1 8 ) , g i v i n g a t t h e same t i m e t h e f o r m u n d e r which t h i s c o n d i t i o n i s u s u a l l y a p p e a r e d .

Lemma 2.4. Let

E be a topological algebra, whose spectrum ??Z(E)

equicontinuous ( c o n s i d e r e d a s a s u b s e t of E i )

. Moreover,

i s locally

suppose t h a t

S =

( x a J u E I i s a s e t of topological generators of E . Then, t h e c o n d i t i o n ( 2 . 1 8 ) is s a t i s f i e d , so t h a t one has

mici

(2.22)

= 7 7 2 ( C 0 [(Xa).

,

w i t h i n a homeomorphism of t h e Gel’fand spaces involved ( c f . Scholium 2 . 1

)

.

Proof. By h y p o t h e s i s one g e t s x=limx

(2.23)

6 f o r every x

Now,

&

E

,with

(x6

6

b e i n g a n e t i n t h e a l g e b r a C, [(x:,ia a

,I.

f o r e v e r y f e r ) 2 ( E ) , t h e r e e x i s t s by h y p o t h e s i s a n e q u i c o n t i n u o u s

V

150

SPECTRUM (GLOBAL THEORY)

n e i g h b o r h o o d , s a y U , o f f i n m(E)s u c h t h a t i t s c l o s u r e

i n E,'

is

s t i l l a n e q u i c o n t i n u o u s s u b s e t o f t h e l a t t e r s p a c e ( c f . N . BOURBAKI 15: -

Chap. 10; p. 15, P r o p o s i t i o n 6 1 ) ; h e n c e , U i s i n p a r t i c u l a r a ( w e a k l y ) compact s u b s e t of E i ( A l a o g l u - B o u r b a k i ) . So t h e same h o l d s t r u e f o r t h e s e t V = u n ?rC(EI which i s t h e r e f o r e a ( w e a k l y ) compact and e q u i c o n t i n u o u s n e i g h b o r h o o d of f i n ???'(El.

C o n s e q u e n t l y , s i n c e by ( 2 . 2 3 ) $ =

l i m 4 6 , w i t h r e s p e c t t o t h e t o p o l o g y o f s i m p l e c o n v e r g e n c e (weak t o p o 6

and Lemma 1 . 1 ) , one c o n c l u d e s t h a t 2 = lim; 6 6 i s s t i l l t r u e on V f o r t h e t o p o l o g y of compact c o n v e r g e n c e , h e n c e f o r

l o g y ) i n m(EI ( c f . ( 2 . 1 1 )

t h a t o f t h e u n i f o r m c o n v e r g e n c e as w e l l ( i b i d . ; p.29,ThSorSme l ) . B u t t h i s i s , i n d e e d , t h e d e s i r e d cond. ( 2 . 1 8 ) . O n t h e o t h e r hand,

i s now a d i r e c t c o n s e q u e n c e of t h e above Scholium 2 . 1 ,

(2.22)

and t h i s t e r -

minates the proof.4 A s an a p p l i c a t i o n o f t h e p r e v i o u s d i s c u s s i o n w e a r e now i n t h e

p o s i t i o n t o s t a t e t h e following u s e f u l r e s u l t .

Theorem 2 . I. Let E be a topological algebra whose completion

2 ic

still a

topological algebra and, moreover, concerning t h e i r spectra, suppose t h a t we hava the r e l a t i o n I ? Z ( E l = 172 (E), w i t h i n a (continuous) b i j e c t i o n of the topological spaces involved

( s e e Lemma 2 . 1 )

. Fincrlly,

suppose t h a t m(El i s Locally equi-

continuous. Then, one g e t s the r e l a t i o n m(E)=

(2.24)

???(z),

within a homeomorphism of the respective topological spnces.

Proof. S i n c e , b y h y p o t h e s i s 2.2;

MIE)

m(E)of

same i s t r u e f o r t h e s p e c t r u m

i s l o c a l l y equicontinuous , t h e t h e c o m p l e t i o n of E

( c f . Lemma

i n f a c t , t h e p r o o f of t h e " o n l y i f " p a r t of t h e a s s e r t i o n ) . More-

o v e r , it i s c l e a r t h a t E may b e c o n s i d e r e d a s a s e t of t o p o l o g i c a l h

g e n e r a t o r s of i t s c o m p l e t i o n E , s o t h a t t h e a s s e r t i o n i s now a d i r e c t consequence of t h e above Lemma 2 . 4 . 1 The f o l l o w i n g r e s u l t c l a r i f i e s t h e p r e v i o u s s i t u a t i o n i n case of a n i m p o r t a n t c l a s s of t o p o l o g i c a l a l g e b r a s .

Corollary 2.1. Let E be a l o c a l l y m-convex algebra and suppose t h a t i t s completion

k

i s a ( l o c a l l y m-convex) Q-algebra. Then, one has t h e r e l a t i o n T / z ( E ) = mlg),

(2.25)

w i t h i n a homeomorphism of the topological spaces involved. Proof. vex

By h y p o t h e s i s and P r o p o s i t i o n 11; 7 . 1 , t h e l o c a l l y m-con-

&-algebra

2

h a s an e q u i c o n t i n u o u s s p e c t r u m m(klC

2..

Therefore,

2.

m ( E ) G E'

SPECTRA OF COMPLETED ALGEBRAS

151

i s a l s o e q u i c o n t i n u o u s ( c f . N . BOURBAKI [5: Chap. 10; p. '15, P r o -

p o s i t i o n 41,

a s w e l l a s t h e n e x t Scholium 2 . 2 ) ;

hence, a f o r t i o r i , lo-

c a l l y e q u i c o n t i n u o u s , s o t h a t t h e a s s e r t i o n f o l l o w s now d i r e c t l y from t h e above Theorem 2 . 1 lation (2.2))

( s e e a l s o t h e comment c o n c e r n i n g t h e above re-

.I

Scholium 2.2.-

Regarding t h e p r e c e d i n g C o r o l l a r y 2 . 1

,

w e s t i l l re-

mark t h e f o l l o w i n g , more g e n e r a l , f a c t : I f E i s a topological algebra t h e

completion of which

2

is again a topological algebra, while E i s , moreover, a Q-al-

gebra, then n^rlE) i s an equicontinuous subset o f E ' . T h i s i s a c t u a l l y p r o v i d e d by t h e same argument a s t h a t i n t h e p r e c e d i n g p r o o f . F u r t h e r m o r e , i n case of a c o m t a t i v e l o c a l l y m-convex algebra, t h e c o n v e r s e of t h e l a s t s t a t e m e n t i s s t i l l t r u e . Namely, i n t h a t c a s e one g e t s t h e f o l l o w i n g more c o m p l e t e p i c t u r e :

Lemma. Let E be a commutative l o c a l l y m-convex alge-

2

i s a Q-algebra i f , and only i f , t h e spectrum mIE) of t h e i n i t i a l algebra i s equicontinuous.

bra. Then, i t s completion

(2.26)

The " i f " p a r t of t h e l a s t a s s e r t i o n i s c o n t a i n e d i n C o r o l l a r y I I I i 6 . 5

i n c o n j u n c t i o n w i t h t h e f a c t ( a l r e a d y remarked, a t l e a s t , f o r l o c a l l y m-convex

case f o r

a l g e b r a s ) t h a t miEl i s equicontinuous i f , and only i f , t h i s i s t h e

m(ii ( s e e

Lemma 2 . 2 ) ; y e t t h e same argument y i e l d s t h e r e s t of

t h e above Lemma, i n c o n n e c t i o n w i t h P r o p o s i t i o n 11; 7.1. h

I n t h i s r e s p e c t , w e a l s o n o t e t h a t t h e completion E o f a given t o -

pological algebra E ( e v e n of a l o c a l l y m-convex o n e ) may be a Q-algebra ( f o r i n s t a n c e , a Banach a l g e b r a ) without t h i s t o be t r u e f o r t h e i n i t i a l algebra t o o

( i . e . , t h e g i v e n normed a l g e b r a , i n t h e example c o n s i d e r e d . See, f o r i n s t a n c e , S . W A R N E R [3 : p. 8, Theorem 71)

.

T h e r e f o r e , by c o n s i d e r i n g com-

m u t a t i v e a l g e b r a s , one o b t a i n s , e q u i v a l e n t l y ( s e e a l s o C o r o l l a r y 111;

6.5)

,

t h a t a ( c o m t a t i v e ) 7,ocaZly m-convex algebra E may have an equicontinuous

spectrum without t h e given algebra E t o b e a d v e r t i b l y complete. However, i t i s a c o n s e q u e n c e of t h e p r e v i o u s d i s c u s s i o n and t h e s a m e C o r o l l a r y 111; 6 . 5 t h a t : A c o m t a t i v e a d v e r t i b l y complete l o c a l l y m-convex a l -

gebra E i s a Q-algebra i f , and only i f , t h i s i s t h e

(2.27)

case f o r i t s completion Now,

t h e s i t u a t i o n d e s c r i b e d by ( 2 . 2 7 )

s i d e r s non-commutative

E.

i s n o t y e t c l e a r , i f one con-

algebras.

We c o n c l u d e by r e m a r k i n g , a s w e a l r e a d y done it a t t h e b e g i n n i n g

152

V

SPECTRUM (GLOBAL THEORY)

o f t h i s s e c t i o n , t h a t w e a r e g o i n g t o see i n s u b s e q u e n t s e c t i o n s of t h i s book e x a m p l e s of

( l o c a l l y m-convex)

topological algebras

which t h e c o n t e n t of t h e p r e v i o u s Theorem 2 . 1 ,

for

h e n c e of t h e whole of

t h e p r e c e d i n g d i s c u s s i o n , w i l l b e p u t i n a more c o n c r e t e form.

3. Spectrum o f an inductive limit topological algebra W e f i r s t comment on some p r e l i m i n a r y m a t e r i a l which w e n e e d be-

1 o w ; i t i s r e f e r r e d t o t h e " c o n t r a v a r i a n c e " o f t h e c o r r e s p o n d e n c e between s p a c e s of c o n t i n u o u s l i n e a r maps o f i n d u c t i v e l i m i t t o p o l o g i c a l v e c t o r s p a c e s t o a c e r t a i n f i x e d t o p o l o g i c a l v e c t o r s p a c e and t h e res p e c t i v e p r o j e c t i v e l i m i t s p a c e s p r o v i d e d by s i m i l a r s p a c e s of c o n t i nuous l i n e a r maps of t h e f a c t o r s of t h e i n d u c t i v e l i m i t s y s t e m s

un-

d e r c o n s i d e r a t i o n ( c f . Lemmas 3 . 1 and 3.2, a s w e l l as 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 a1gebras;Theorem 3 . 1 ) . Thus, s u p p o s e w e a r e g i v e n a n i n d u c t i v e s y s t e m of t o p o l o g i c a l vector spaces f E a , f l , with r e s p e c t t o a d i r e c t e d index set I

ecr

(cf.

Chapt. I V ; S e c t i o n s 1 , 2 f o r t h e a n a l o g o u s c a s e of t o p o l o g i c a l a l g e b r a s ) , and l e t

-

E = l i m ( E c1 4 a ) b e t h e c o r r e s p o n d i n g inductive l i m i t topological vector space; i.e. , t h e ( i n d u c t i v e ) l i m i t v e c t o r s p a c e E endowed w i t h t h e f i n a l v e c t o r s p a c e t o (3.1)

p o l o g y making t h e c a n o n i c a l maps (3.2)

fa: E a - E ,

aEI,

continuous. Furthermore, c o n s i d e r a f i x e d t o p o l o g i c a l v e c t o r space P and l e t

s s ( E a , F) , c1 e I ,

(3.3)

b e t h e r e s p e c t i v e s p a c e s of c o n t i n u o u s l i n e a r maps o f e a c h one of t h e s p a c e s Ea

, with

c1

eI , into

F equipped w i t h t h e topology

( p o i n t w i s e ) convergence i n E

t

(3.4)

. Thus,

r s i E a , Fl

"

s " of simple

one o b t a i n s a f a m i l y

1 ,a E I ,

of t o p o l o g i c a l v e c t o r s p a c e s ( c f . N . BOURBAXI [7: Chap. 3 ; p. 1 7 , C o r o l l a i r e ] ) i n such a

c o r r e s p o n d i n g transpose maps (3.5)

t fea = f , , : S s ( E e , F)

-

o f t h e g i v e n c o n t i n u o u s l i n e a r maps (3.6)

-

manner t h a t , by c o n s i d e r i n g f o r a n y a < B i n I , t h e

fscl: Ea,-

EB '

S (E

s

a' Fl

3.

SPECTRA OF INDUCTIVE LIMIT ALGEBRAE

153

o n e g e t s a f a m i l y of c o n t i n u o u s l i n e a r maps ( c f . a l s o H . H . SCHAEFER p. 130; (2.4)])

[I:

such t h a t t h e family

(3.7)

(LS(Ea,

t

F) ,

f S a L a €1,

y i e l d s a p r o j e c t i v e s y s t e m ; i . e . , c o n c e r n i n g t h e maps (3.5), one g e t s t h e a n a l o g o u s r e l a t i o n s w i t h (2.2) and (2.3) i n C h a p t . 111; S e c t i o n 2 . So l e t

a ’ F)

H = l h 1: ( E

(3.8)

s

c

b e t h e c o r r e s p o n d i n g p r o j e c t i v e l i m i t topoZogica1 vector space ( see C h a p t . 1II;Section 2

f o r t h e c a s e of t o p o l o g i c a l a l g e b r a s ) .

T h u s , w e now h a v e t h e f o l l o w i n g .

Lemma 3.1. Let ( E ~ f ,o a ) be an i n d u c t i v e system of topological vector spaces and E = 1 2 E a t h e respective inductive l i m i t topological vector space. I4oreover, t given a topological vector space F , l e t ( LS(Ea, FI, faa i be the corresponding p r o j e c t i v e system of topological vector spaces defined by ( 3 . 7 ) . F i n a l l y , l e t S

( E , F l be t h e (topological v e c t o r ) space of continuous l i n e a r maps of E i n t o F

endowed w i t h t h e topology of simple convergence i n E. Then, one g e t s t h e r e l a t i o n

-

S s ( E , F) = 1: ( l i m E

(3.9)

s

a’

F) = lt h Ls(Ea , F ) ,

w i t h i n a topological vector space isomorphism of the spaces involved. Proof. W e f i r s t remark t h a t , f o r e v e r y

(3.10)

ha :

1: ( E , F )

--+

c1 E I ,

o n e h a s a l i n e a r map

1: ( E a J F) : u w h c l l u l := u o fa

,

where t h e map fa i s g i v e n b y (3.2) , so t h a t b y (3.5) ( c E . a l s o IV;(l.ll)) one g e t s

(3.11)

t f s a ( h s ) = hS 0 $a = ha

-

I

f o r a n y a 5 B i n I ; h e n c e , o n e o b t a i n s a u n i q u e l y d e f i n e d map h = l i m h : I: ( E , F)

(3.12)

t

a

1 2 1: (Ea,FI

,

i n s u c h a manner t h a t

(3.13)

ha = p a o h ,

f o r e v e r y a € I . H e r e pa : H-

1: ( E

a’

F), w i t h a E I , d e n o t e s t h e c o r r e

-

s p o n d i n g c a n o n i c a l l i n e a r map ( c f . 111; (2.6));thus, t h e map ( 3 . 1 2 ) i s g i v e n by t h e r e l a t i o n

h= ( h a j a I

,

s u c h t h a t ( c f . (3.10)) h f u j = ( h a ( u ) l a

(3.14) =(uofaIaeI

w i t h u e L ( E , F). The

ear map, w h i l e t h e

I

I

l a s t r e l a t i o n i m p l i e s , of c o u r s e , t h a t h i s a l i n -

same r e l a t i o n , i n c o n n e c t i o n w i t h IV; (1.9) , y i e l d s

154

V

SPECTRUM (GLOBAL THEORY)

a l s o t h a t h i s one-to-one; b e s i d e s , h i s an onto map. T h u s , g i v e n a n e l e ment ( u , I u E I

i n H I one d e f i n e s a n e l e m e n t u e

ua=uofa

(3.15)

w i t h a e I , w h i l e t h e same r e l a t i o n

-

I: ( E , F I by t h e r e l a t i o n

I

(3.15) y i e l d s a l s o t h e continuity

o f t h e map u , a s e a s i l y f o l l o w s from t h e same d e f i n i t i o n o f t h e t o p o logy i n E = l i m E

.

Thus

the map h d e f i n e s a vector space isomorphism of t h e

s p a c e s i n v o l v e d i n ( 3 . 9 ) . On t h e o t h e r h a n d , by d e f i n i t i o n o f t h e t o -

the c o n t i n u i t y of h , f o r t h i s i s r e d u c e d , i n

pology i n H , one c o n c l u d e s f a c t , by ( 3 . 1 3 )

,

t o t h e c o n t i n u i t y o f t h e maps k

a'

a ~ I , w h i c hhowever

a r e c e r t a i n l y c o n t i n u o u s by ( 3 . 1 0 ) ( s e e a l s o I V ; ( 1 . 9 ) ) . F i n a l l y , w e p r o v e t h a t the inverse map o f h i s a l s o continuous: T h i s map i s g i v e n , i n e f f e c t , b y ( 3 . 1 5 ) ; SG s u p p o s e t h a t ( u a I6a E I = (6 u ~ ) ~ ~ ~ 6 e K , i s a n e t i n H c o n v e r g i n g t o z e r o ; e q u i v a l e n t l y , t h i s means t h a t , f o r e v e r y a e I, t h e n e t ( u : ) 6 E K c o n v e r g e s t o z e r o i n I: s ( E , , P i , s o t h a t by ( 3 . 1 5 ) a n d I V ; ( l . 9 )

one h a s

(3.16) f o r e v e r y x E E . Thus one c o n c l u d e s t h a t u

-

6 6

0

in

f s ( E , F ) which i s

t h e a s s e r t i o n , and t h e p r o o f o f t h e l e m m a i s f u l l y e s t a b l i s h e d . NOW,

t e m (E,,fa,I

in particular,

suppose t h a t w e are g i v e n an i n d u c t i v e sys-

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

morphisms fBa: Eol-+LB

,for

IV, Definition 2.1).

M o r e o v e r , c o n s i d e r a f i x e d t o p o l o g i c a l a l g e b r a F.

a n y ,I

in I

( a d i r e c t e d i n d e x set; c f . Chapt.

Thus s i m i l a r l y t o ( 3 . 5 ) it i s e a s y t o see t h a t one g e t s a fami l y o f continuous maps (3.17)

t

f B a = f,B

: Homs(EB, FI

-

ffoms(Ea,

FI,

where, f o r e v e r y i n d e x a E I , (3.18) denotes the

Homs(E,,

FI C L s ( E a , Fl

space o f continuous algebra morphisms o f Ea i n t o F endowed w i t h , t h a t is t h e r e l a t i v e

the topology of simple ( p o i n t w i s e ) convergence i n Ea o n e i n d u c e d on HomiE,, PI

by

t h e space

.Cs(Ea,

F I , a s above.

(In this

r e s p e c t , w e d e f e r a more d e t a i l e d d i s c u s s i o n o n t h e s p a c e s ffom ( E , F ) , " g e n e r a l i z e d s p e c t r a " , f o r S e c t i o n I below)

.

Thus , ( 3 . 1 7 ) c o n s t i t u t e s

i n d e e d t h e r e s t r i c t i o n of t h e c o r r e s p o n d i n g r e l a t i o n t o ( 3 . 5 ) above t o (3.18). Furthermore, w e s t i l l note t h a t (3.17) e n t a i l s t h a t

3. SPECTRA OF INDUCTIVE LIMIT ALGEBRAS

d e f i n e s a p r o j e c t i v e system o f topological spaces

155

( s u b s p a c e s of t h e members

of t h e r e s p e c t i v e system ( 3 . 7 ) ) , s o t h a t one may c o n s i d e r t h e c o r r e sponding p r o j e c t i v e l i m i t ( t o p o l o g i c a l ) space H = limHom ( E

(3.20)

4.-

s

a’ F )

( c f . a l s o N . BOURBAKI [4: Chap. 1 ; p. 28, and 1 : Chap. 3; p. 80, rel. ( 8 ) ] ) . Now, t h e f o l l o w i n g i s a d i r e c t consequence of Lemma 3 . 1 and t h e preceding discussion.

Corollary 3.1.

Let ( E a , f

a u I be an i n d u c t i v e system o f topological algebras

( w i t h r e s p e c t t o a d i r e c t e d index s e t I ) and F a f i x e d

topological alge-

brn. Moreover, i f E = l h E

a i s t h e r e s p e c t i v e i n d u c t i v e l i m i t topological algebra and Horns ( E a , F / denote t h e spaces of continuous algebra rnorphisms o f t h e topological algebras under consideration, endowed w i t h h

(Lemma I V ; 2 . 2 ) l e t

Horns ( E , FI

t h e corresponding t o p o l o g i e s of simple convergence. Then, one g e t s ( r e s t r i c t i o n

of ( 3 . 9 ) ) Hom ( E , F) = Hom ( l i m E FI = limHom (E F1 s + a’ c s a’

(3.21)

,

w i t h i n a homeomo2-.phism o f t h e r e s p e c t i v e topological spaces. I n p a r t i c u l a r , by taking F = C ( t h e complex numbers), and considering t k e ertendzd spectra of t h e topological algebras involved

( c f . ( 1 . 8 ) ), one g e t s

(by ( 3 . 2 1 ) ) t h e r e l a t i o n m(EI’=

(3.22)

m(1lp:Ea I i

= @I m ( E a ji

w i t h i n a homeomorphism of t h e r e s p e c t i v e topological spaces. I W e come n e x t t o d i s c u s s t h e a n a l o g o u s r e l a t i o n t o ( 3 . 2 2 ) by con-

s i d e r i n g s p e c t r o of t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d , e x c l u d i n g thus z e r o homomorphisms. T h e r e f o r e , one i s n a t u r a l y l e d t o s e e k c o n d i t i o n s guaranteing t h a t t h e corresponding r e l a t i o n t o ( 3 . 1 7 ) preserves t h e r e s p e c t i v e s p e c t r a ; t h a t i s , one should have

t

(3.23)

foa 5

Go : m ( E o I

--*

m (Ea 1 ,

f o r any a < 3! i n I , w i t h i n t h e c o n t e x t of t h e above C o r o l l a r y 3 . 1 , t a k ing F = C

.

In t h i s respect,

it i s e a s i l y seen t h a t : 1 . ; . 2 3 ) holds t r u e

if, and only if, one has t h e r e l a -

tion (3.24)

(3.24.1)

foa(EaInCkex(v) # 0 ,

f o r every ( c o n t i n u o u s c h a r a c t e r ) v e m ( E j and R any a < R i n I .

V SPECTRUM (GLOBAL THEORY)

156

under t h e above cond. ( 3 . 2 4 . 1 ) t h e family

Thus,

(m(Ea), tfsnl

(3.25)

a p r o j e c t i v e zystem of tcprlogical spaces, so t h a t one g e t s f o r t h e r e s p e c t i v e p r o j e c t i v e l i m i t t o p o l o g i c a l space t h e r e l a t i o n cmstitUtr?s

S = l>m(E,I

(3.26)

= (limm(E,)+) n t

n

.

m(Ea)

U€I

( I n t h i s c o n c e r n , see a l s o N . BOURBAKI [I: Chap.3; p. 80, rel. NOW,

(a)]).

c o n s i d e r t h e canonical i n j e c t i o n

j : 1 2 m(E, I

(3.27) d e f i n e d by r e s t r i c t i n g

( 3 . 1 5 ) ( w i t h F = C ) t o t h e space S = l*m(E,I

(3.28)

m(E)

+

C 12xs(Ea, @I.

Thus, o n e r e a d i l y v e r i f i e s t h a t ( 3 . 2 7 ) d e f i n e s a b i j e c t i o n i f , and only i f , the s o l lowing r e l a t i o n holds t r u e

(3.29)

fa(Ea ) n

(3.29.1 )

c ker (uI

# 0,

f o r every index u e I and every ( c o n t i n u o u s c h a -

racter) u e m l E I . Therefore, under the previous c o n d f t i o n s ( 3 . 2 4 . 1 ) and (3.23. I / , one g e t s

m(E)= r n ( l * E a )

(3.30)

= l $

m(EaI ,

within a b i j e c t i o n . On t h e o t h e r h a n d , t h e a r g u m e n t a p p l i e d t o p r o v e t h a t ( 3 . 2 1 ) h o l d s t r u e w i t h i n a homeomorphism i s c e r t a i n l y v a l i d h e r e a s

w e l l , w i t h F = @ , whence o n e i s l e d t o a s i m i l a r c o n c l u s i o n f o r (3.30). Thus, t h e p r e v i o u s d i s c u s s i o n p r o v i d e s ,

i n e f f e c t , t h e proof of t h e

following.

Theorem 3.1. Suppose we are given an inductive system of topological a l g e bras i E a , f B a I , w i t h respect t o a directed index s e t I , and l e t E = l % E a be the corresponding inductive l i m i t topological algebra i n such a manner t h a t the above conds. ( 3 . 2 4 . 1 )

and ( 3 . 2 9 . 1 ) are s a t i s f i e d . Then, one g e t s the r e l a t i o n

(3.31)

m ( E I = E ( l 2 Ea ) = 1 2 m(E,I

,

within a homeomorphism of the respective topological spaces.

In p a r t i c u l a r , t h e l a s t conclusion i s v a l i d , i f the topological algebras E a' u e I , have i d e n t i t y elements. ( T h u s t h e same i s t r u e € o r t h e a l g e b r a E = l i m E , w h i l e b o t h of t h e a b o v e c o n d s . (3.24.1) a n d (3.29.1) a r e t h e n t r i -+ a vially satisfied).

3. SPECTRA OF INDUCTIVE LIMIT ALGEBRAS

157

Next w e s u p p o s e t h a t t h e i n d i v i d u a l a l g e b r a s of a g i v e n i n d u c t i v e s y s t e m o f t o p o l o g i c a l a l g e b r a s h a v e e q u i c o n t i n u o u s o r even l o c a l l y e q u i c o n t i n u o u s s p e c t r a and examine t h e same p r o p e r t y f o r t h e s p e c t r u m of t h e c o r r e s p o n d i n g i n d u c t i v e l i m i t a l g e b r a . However, w e s h a l l need f i r s t t h e f o l l o w i n g .

Lemma 3.2. L e t E b e a t o p o l o g i c a l v e c t o r space, i n d u c t i v e l i m i t of a g i v e n i n d u c t i v e system of t o p o l o g i c a l v e c t o r spaces (Ea,f B a ) , endowed w i t h t h e respect i v e f i n a l v e c t o r space topology d e f i n e d on it b y t h e canonical maps fU:Ea-E l h E , ,a€ I ( c f . C h a p t . I V ; S e c t i o n 1 )

=

.

Moreover, l e t F be a l o c a l l y convex + space.Then a s e t M C S (E,FI of continuous l i n e a r maps of E i n t o F i s eyuicontinu-

ous i f , and only i f , t h e s e t Mofa = { u o f a : U E M }

(3.32)

i s an equicontinuous s u b s e t of

S (Ea, F I , f o r e v e r y u e I .

Proof. Suppose t h a t M i s e q u i c o n t i n u o u s ; so f o r e v e r y n e i g h b o r h o o d N of

zero i n F t h e set

(3.33) i s a n e i g h b o r h o o d of z e r o i n E ; t h u s , b y t h e c o n t i n u i t y of t h e maps

fa , a e I , t h e s e t

E, , t h a t i s t h e s e t MofaE L ( E a , F ) i s e q u i c o n t i n u o u s , f o r e v e r y aeI. On t h e o t h e r h a n d , f o r e v e r y a b s o l u t e i s a n e i g h b o r h o o d of z e r o i n

l y convex n e i g h b o r h o o d of z e r o i n F , s a y N , t h e s e t (3.33) i s a n abs o l u t e l y convex s u b s e t of E l so t h a t by h y p o t h e s i s t h e s e t f;l(M-'(N))

= ( b y (3.34)) (Mofa)-'(NI

i s a n e i g h b o r h o o d of z e r o i n E,

,

f o r e v e r y a e I, whence a n a b s o r b i n g

s e t . T h e r e f o r e , M-'(N) i s a b s o r b i n g f o r t h e s e t f a ( E a ) C E, f o r e v e r y a as well; E I. C o n s e q u e n t l y , it i s a n a b s o r b i n g s u b s e t o f E = (J f a ( E a ) h e n c e , b y t h e f o r e g o i n g , M-'(NI i s a n e i g h b o r h o o d ofol z e r o of t h e f i n a l l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y d e f i n e d on E by t h e maps f u , a E

I, thus a f o r t i o r i f o r t h e stronger f i n a l vector space topology d e f i n e d on E by t h e same maps, and t h i s t e r m i n a t e s t h e p r o o f . I T h u s , w e now h a v e t h e f o l l o w i n g . P r o p o s i t i o n 3.1. L e t ( E f I b e an i n d u c t i v e system of t o p o l o g i c a l algebras a' l3a and E = l i m E t h e corresponding i n d u c t i v e l i m i t t o p o l o g i c a l a l g e b r a , such t h a t + a ( 3 . 2 9 . 1 ) i s s a t i s f i e d . Moreover, suppose t h a t each one of t h e algebras Ea, a e I,

v

158

SPECTRUM (GLOBAL THEORY)

has a l o c a l l y equicontinuous s p e c t m i n such a manner t h a t f o r a l l , except of f i nite-many i n d i c e s ci €1, ? Z ( E a I i s an equicontinuous subset of E‘ Then, the algec1

bra E has

.

l o c a l l y equicontinuous spectrum m(E1.

In p a r t i c u l a r , i f each one of t h e algebras Ea,a € I , has an equicontinuous

spectrum, then the same i s true f o r the algebra E = l i m E

--+a

.

Proof. By c o n s i d e r i n g t h e c a n o n i c a l map ( c o n t i n u o u s a l g e b r a mor-

phism) (3.35) with

E = limEci

f a :Ea-

ci

€ I , w e f i r s t remark t h a t

(3.36)

*

( 3 . 2 9 . 1 ) is equivalent t o the r e l a t i o n

m(E)C E,‘

tfci :

,

m(E,) s (E,);

.-+

,

f o r every a € I ; o r y e t to t h e c o n d i t i o n t h a t t h e range of t h e r e s t r i c t i o n of the continuous transpose map

t

f a : E’, f o r every a € I , so t h a t one h a s

(3.37)

tf

---+

iu)=uof =

0.

a

ti

( E a :)

t o m(E1 i s contained in m(EaI,

a’ with c i e ~ ,

€or every element u e m ( E 1 . Therefore, i f

m(E .), w i t h Z S i S n , i s a n 012

enumeration of t h e given finite-many l o c a l l y equi cont i nuous s p e c t r a ,

l e t V Em(Ea 1 be t h e e q u i c o n t i n u o u s n e i g h b o r h o o d s o f u = t f a . ( u ) i n ‘i i “i z l < i S n , c o r r e s p o n d i n g b y h y p o t h e s i s a n d ( 3 . 3 7 ) to a g i v e n e l e ment u & m ( E iT . h u s , b y t h e c o n t i n u i t y of t h e maps ( 3 . 3 6 ) , t h e s e t (3.38)

is a neighborhood of u i n m(Ei which, moreover, i s an equicontinuous subset of the same space. The l a s t a s s e r t i o n i s a c o n s e q u e n c e o f t h e f o l l o w i n g f a c t : A s e t V r T / L ( E ) = 771 (1%

E a l i s equicontinuous i f ,

and only if, f o r every index

(3.39) (3.39.1)

ci

€ 1 , the s e t

t f c i ( v ) = v o f a = E u o f c i : u Ev l

is an equicontinuous subset of E i .

(Cf. a l s o ( 3 . 3 6 ) ) . NOW, ( 3 . 3 9 ) i s a d i r e c t c o n s e q u e n c e o f Lemma 3 . 2 f o r F = C . Hence, one g e t s b y ( 3 . 3 8 ) t h e r e l a t i o n

Thus, w i t h t h e e x c e p t i o n o f t h o s e f i n i t e - m a n y i n d i c e s ail as a b o v e , f o r a l l t h e r e s t a ’ s i n I o n e g e t s , by h y p o t h e s i s a n d ( 3 . 4 0 ) a n e q u i c o n t i n u o u s s u b s e t of E&

,respectively.

,

But f o r t h e same c i i ’ s ,

I < i < n , a s w e l l , one o b t a i n s by ( 3 . 3 8 )

T h a t i s , by h y p o t h e s i s , a n e q u i c o n t i n u o u s s u b s e t of mIE, ) S E L ; , w i t h i

159

3. SPECTRA OF INDUCTIVE LIMIT ALGEBRAS

I S i S n . S o by ( 3 . 3 9 ) t h e s e t V , g i v e n by ( 3 . 3 8 ) , is an equicontinuous neighborhood of

1*

i n ??7(E),

t h a t i s M ( E ) i s l o c a l l y equicontinuous, as i t w a s t o

be proved. Finally,

i f ? Z ( E a ) i s an

e q u i c o n t i n u o u s s u b s e t of E' , f o r e v e r y

a

i n d e x a e I , o n e c o n c l u d e s by a p p l y i n g ( 3 . 3 6 ) 'fa(??L(l?)) = m(E)o f a = ? 7 Z l E a )

(3.41) t h a t is,

C Ei,

Z T ( E ) is an equicontinuous subset of E' ( a p p l y Lemma 3 . 2 , f o r F =

C ), a n d t h i s t e r m i n a t e s t h e p r o o f . I

Scholium 3.1.-

P r o j e c t i v e l i m i t s p a c e s may happen t o b e r e d u c e d

t o t h e empty s e t ( e v e n i f e a c h o n e of t h e f a c t o r s p a c e s i s non-empty a n d t h e r e s p e c t i v e " t r a n s i t i o n maps" faa , w i t h a 5 R i n I , i s s u r j e c t i v e ) . T h u s , it would b e , of c o u r s e , of a p a r t i c u l a r s i g n i f i c a n c e t o know t h a t t h e p r e c e d i n g f o r m u l a s ( 3 . 2 2 ) and ( 3 . 3 1 ) h a v e a r e a l meaning. On t h e o t h e r h a n d , the p r o j e c t i v e limit of a given pjrojective system of

non-void compact ( H a u s d o r f f t o p o l o g i c a l ) spaces i s a topological space of t h e same kind ( c f . , f o r i n s t a n c e , N . BOURBAKI [4: Chap. 1 ; p. 64, P r o p o s i t i o n 81). Thus c o n c e r n i n g t h e a b o v e Theorem 3.1

( i n connection with Proposition

3 . 1 ) , one o b t a i n s an im portant s p e c i a l i n s t a n c e : Namely, s u p p o s e w e h a v e a n i n d u c t i v e s y s t e m o f t o p o l o g i c a l a l g e b r a s with i d e n t i t y elements i E a ,

Ga)

such t h a t t h e corresponding spec-

t r a m(E,), c1 E I , are e q u i c o n t i n u o u s . Thus (Theorem 3.1 ), t h e s p e c t r u m m(E)o f t h e r e s p e c t i v e i n d u c t i v e l i m i t t o p o l o g i c a l a l g e b r a E = 1 2 E a i s g i v e n by t h e r e l a t i o n (3.42) w i t h i n a homeomorphism of t h e t o p o l o g i c a l s p a c e s c o n s i d e r e d . F u r t h e r -

more, by h y p o t h e s i s f o r t h e a l g e b r a s E,, cc € I, m(Ea)i s a compact subs e t of E& , f o r e v e r y a e I. ( T h i s i s a c o n s e q u e n c e o f t h e Alaoglu-Bourb a k i Theorem i n c o n j u n c t i o n w i t h R m a r k 1 . 1 ) T h u s , i n case m(Ea)i s n o n - t r i v i a l , f o r every a e I , the same holds true for ??Z(E), where t h e l a t t e r space i s g i v e n by ( 3 . 4 2 ) . Now, one h a s a n i m p o r t a n t n o n - t r i v i a l a p p l i -

.

c a t i o n o f t h e p r e c e d i n g by c o n s i d e r i n g t h e f o l l o w i n g d a t a : Suppose w e are g i v e n

(3.43)

an inductive system of com-

mutative Banach algebras w i t h i d e n t i t y elements (Ea' fBa ) and l e t E = l h E , be the respective inductive limit al+

-

gebra endowed w i t h t h e finaZ vector space topology makE , c( E I , continuous (cf. a c t Lemma IV;2 . 2 ) . Tken, E is a ( c o m m u t a t i v e ) topologicaZ

ing the canonical maps f : E

160

SPECTRUM (GLOBAL THEORY)

V

algebra ( w i t h a n i d e n t i t y e l e m e n t ) , whose spectrum m(E)i s a non-empty compact ( H a u s d o r f f ) space , i n s u c h a way t h a t one h a s m ( E l = 1 2 ??Y(E,), w i t h i n a homeomorphism. I n t h i s r e s p e c t , w e h a v e a p p l i e d , of c o u r s e , i n t h e p r e v i o u s argument t h e f a c t t h a t e a c h one o f t h e s p a c e s m ( E , l , a ~ I , i s by h y p o t h e s i s a (non-empty) compact ( H a u s d o r f f ) s p a c e ; t h i s i s a l s o c o n c l u d e d by t h e p r e c e d i n g and P r o p o s i t i o n 11; 7 . 1 (see also Theorem V I ; 1 . 3

i n t h e se-

q u e l a n d i t s C o r o l l a r y 1 . 5 ) . A s i m i l a r case w i t h t h e p r e c e d i n g one w i l l s t i l l be considered i n t h e next section, i n connection with t h e example i n S u b s e c t i o n I V ; 4 . ( 3 ) . Applications of t h e preceding discussion w i l l a l s o be given i n s u b s e q u e n t c h a p t e r s . However, w e s t i l l c o n s i d e r one more a p p l i c a t i o n of t h e f o r e g o i n g i n t h e n e x t s e c t i o n .

4. Envelopes o f holomorphy Suppose w e are g i v e n a complex ( a n a l y t i c ) m a n i f o l d X and l e t K be a compact s u b s e t of X. Then, one d e f i n e s t h e algebra ( o f germs) of

holomorpphic functions on K by (4.1) ( f o r t h e t e r m i n o l o g y a p p l i e d i n t h e l a s t r e l a t i o n c f . C h a p t . IV; Subs e c t i o n 4 . ( 3 ) ) . Thus, one g e t s by ( 4 . 1 ) a topological algebra w h i c h , i n p a r t i c u l a r , f o r X 2nd c o u n t a b l e , i s an LFQ- l o c a l l y m-convex algebra ( i b i d . ) . S o w e come t o t h e f o l l o w i n g

D e f i n i t i o n 4.1.

L e t X b e a complex

m a n i f o l d a n d K a compact sub-

s e t of X . Then, t h e envelope of holomorphy of K i s t h e s p e c t r u m of t h e , g i v e n by ( 4 . 1 ) , i . e . , one s e t s

topological algebra O(K)

h u l l o ( X ) ( K l :=

(4.2) A l s o we say t h a t ( 4 . 2 )

Thus, b y ( 3 . 3 1 )

,

(OlK)).

d e f i n e s t h e c)(X)-huzz of K .

w e now h a v e .

(4.3) which i s v a l i d w i t h i n a homeomorphism o f t h e t o p o l o g i c a l s p a c e s i n v o l v e d . The same r e l a t i o n r e p r e s e n t s a l s o a f a m i l i a r s i t u a t i o n , f o r s u i t a b l e K , a s w e s h a l l p r e s e n t l y see.

4. ENVELOPES OF HOLOMORPHY

161

So, i n p a r t i c u l a r , s u p p o s e t h a t K i s a compact S t e i n s e t

i n X. (The

t e r m i n o l o g y a n S s e t ( c f . H . ROSSI [ l ] ) o r y e t a holornorphic s e t ( c f . P . 0 . 6 WELLS, Jr. [I]) h a s a l s o b e e n a p p l i e d i n t h i s c o n t e x t ) . Namely, w e assume t h a t K i s t h e i n t e r s e c t i o n o f a f u n d a m e n t a l s y s t e m of open S t e i n ( s u b m a n i f o l d s ) n e i g h b o r h o o d s of K i n X. I n t h i s r e s p e c t , s i n c e a f i n i t e i n t e r s e c t i o n o f open S t e i n sets ( s u b m a n i f o l d s ) i n X i s s t i l l an o p e n S t e i n s e t , i n case t h e g i v e n m a n i f o l d X i s s e c o n d c o u n t a b l e , w e may a l w a y s c o n s i d e r w i t h a compact S t e i n s e t K i n X a fundamental decreas (Un I n

ing sequence of open S t e i n neighborhoods of K , s a y

,in

~~

s u c h a manner

t h a t o n e g e t s , by d e f i n i t i o n , K =

(4.4)

u

n = ~n

.

=limrJ t n '

h e r e t h e s e c o n d of t h e p r e c e d i n g r e l a t i o n s i s v a l i d w i t h i n a b i j e c t i o n ( a s t h i s is r e a d i l y seen, with (Un)nEN

being a decreasing

se-

q u e n c e of s e t s ) . NOW, g i v e n a complex S t e i n m a n i f o l d

( i n f a c t , a n y complex anczly-

t i c S t e i n s p a c e ) X one h a s t h e r e l a t i o n

rn(O(X1) =

(4.5)

x

I

w i t h i n a homeomorphism ( c f . c h a p t . V I I ; S e c t i o n 3 )

.

Thus, o n e o b t a i n s

by t h e f o r e g o i n g

T h i s i s a l s o e x p r e s s e d by s a y i n g t h a t : A compact S t e i n s e t ( K of a p l e x m a n i f o l d ( s p a c e ) X ) i s holomorphically convex ; i . e . w i t h i t s own e n v e l o p e of holomorphy ( c f .

,

com-

it c o i n c i d e s

( 4 . 2 ) ) . The c o n v e r s e i s n o t ,

i n g e n e r a l , t r u e ( s e e J . E . BJORK [ l ] ) . But t h e two n o t i o n s d o , i n f a c t , c o i n c i d e i f K i s a compact subset of a S t e i n manifold X (see, f o r i n s t a n c e ,

H. ROSSI [ I : p. 476, Lemma 2 . 9 1 ) . On t h e o t h e r h a n d , b y c o n s i d e r i n g t h e Banach a l g e b r a

e ( K 1 of ( c o m p l e x - 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 o n K ( i n t h e "sup-norm'' t o p o l o gy)

,

where K i s a compact subset of a complex manifold X , t h e r e l a t i o n A ~ K := )

(4.7)

~ l r (El

e( K )

y i e l d s a Banach a l g e b r a ; t h u s , w e a c t u a l l y c o n s i d e r b y ( 4 . 7 ) t h e " u n i form c l o s u r e ' ' i n

CIK) of

t h e algebra

K o f t h e " a l g e b r a o f germs" g i v e n by

O(K1 ( i n f a c t , t h e r e s t r i c t i o n t o (4.1) )

.

F u r t h e r m o r e , t h e foZlowing

r e lat i o n (4.8)

m(A(K)) =

m(O(K))

holds also t r u e , w i t h i n a homeomorphism of t h e t o p o l o g i c a l s p a c e s u n d e r con-

s i d e r a t i o n ( c f . R. HARVEY-R.O. WELLS, J r . [I: p. 512, P r o p o s i t i o n 2 . 4 1 and

162

V

a l s o H . PORT.4

- L. RECHT

SPECTRUM (GLOBAL THEORY)

[I : p. 518, P r o p o s i t i o n 3.61 )

.

I n t h i s r e s p e c t , one a l s o d e f i n e s t h e holomorphically convex h u l l o f a compact K i n a g i v e n complex ( a n a l y t i c ) s p a c e X by t h e r e l a t i o n

(4.9)

f o r e v e r y f e Otx)}

.

(The u s e o f t h e same n o t a t i o n a s w i t h t h e " O t X I - h u l l " o f K i n ( 4 . 2 ) i s n e x t j u s t i f i e d by t h e i m p o r t a n t s p e c i a l case where X i s a S t e i n

m a n i f o l d , a s w e s h a l l p r e s e n t l y s e e ) . T h u s , c o n c e r n i n g ( 4 . 9 ) , i n case of a S t e i n space X , one g e t s

m(A(WI =

(4.10)

i ,

w i t h i n a homeomorphism ( c f . R . C . GUNNING - H. ROSSI [I: p. 213, C o r o l l a r y 7 1 ) . So, ( 4 . 8 ) , t h e two notions o f hoZomorphic convexity, g i v e n by (4.9) ( o r , e q u i v a l e n t l y , ( 4 . 1 0 ) ) and ( 4 . 2 ) , coincide i n ease of a S t e i n manifoZd

on t h e b a s i s of

X . ( I n t h i s c o n c e r n , we can s a y t h a t ( 4 . 2 )

indicates, i n effect, the

" s h e a f t h e o r e t i c " a s p e c t o f t h e n o t i o n i n q u e s t i o n which i s , i n d e e d , more s u i t a b l e i n some o t h e r more g e n e r a l c o n t e x t : s e e , f o r i n s t a n c e , A . MALLIOS [16, 1 7 1 o r y e t A . KYRIAZIS [ 3 ] ) .

I n p a r t i c u l a r , if K is a compact subset o f

one g e t s the r e l a t i o n

n T ( A ( K I ) = m(O(K/) = m ( P ( K ) ) = I;. ,

(4.11)

A

within a homeomorphism o f t h e t o p o l o g i c a l s p a c e s i n v o l v e d : h e r e K d e n o t e s t h e polynomially convex h u l l o f K

where

a: [ z I ,. . . , z ] i s

C n d e f i n e d by

t h e algebra of polynomials ( i n t h e v a r i a b l e s z l , .

z , w i t h complex c o e f f i c i e n t s ) . T h e r e f o r e ,

.. ,

one o b t a i n s

(4.13)

( i . e . , t h e s e t (Banach a l g e b r a ) o f " u n i f o r m l i m i t s " on K of polynomiso t h a t w e a c t u a l l y g e t a l s from @ [ z l , ..., z

I),

'FZ(A(KII =

(4.14)

m(P(KI) =

El,

w i t h i n a homeomorphism ( c f . , f o r i n s t a n c e , H.ROSSIL1: p. 473, Lemma 2.41

as w e l l a s R. C . GUNNING - H. ROSSI [1: p. 58, P r o p o s i t i o n 81 )

.

On t h e o t h e r h a n d , o n e d e f i n e s t h e envelope of hoZomorphy o f a com-

p l e x manifold

( a c t u a l l y o f a n y ( r e d u c e d ) complex a n a l y t i c space) X by t h e

relation (4.15) i.e.

,

E (XI = rniO(XI I , a s the spectrum o f t h e ( F r g c h e t ) l o c a l l y !.-convex

algebra

OIX) ( w i t h

4. ENVELOPES OF

163

HOLOMORPHY

X s e c o n d c o u n t a b l e ) . Thus, i n c a s e X i s , f o r i n s t a n c e , a Riemann d o main ( o r e l s e manifold spread ) over g e t s by ( 4 . 1 5 ) a Riemann domain over

C n ( y e t o v e r any S t e i n m a n i f o l d ) one

C n ( o r over t h e S t e i n manifold under

c o n s i d e r a t i o n ; see H . ROSSI[2: p. 15, C o r o l l a r y 3 . 1 2 , Theorem 4 . 6 3 )

as w e l l a s p. 16,

.

I n p a r t i c u l a r , i f U i s an open s u b s e t ( s u b m a n i f o l d ) o f a S t e i n manifold X , the relation

c = rn(O(U/I

(4.16)

y i e l d s a S t e i n manifold. I n t h a t c a s e one o b t a i n s , i n f a c t , t h e f o l l o w i n g commutative d i a g r a m

u-u

P

-

\I

(4.17)

X

where p i s a b i h o l o m o r p h i c map o n t o an open s u b s e t o f

l y biholomorphic

and n a l o c a l -

map, i n s u c h a way t h a t one h a s

OIU) = O i U ) ,

(4.18)

within a

( t o p o l o g i c a l a l g e b r a i c ) isomorphism of t h e r e s p e c t i v e Fre'chet ( l o -

c a l l y m-convex) algebras ( i b i d . I n f a c t , t h e same r e s u l t c a n s u i t a b l y b e d e s c r i b e d w i t h i n t h e c o n t e x t of

( r e d u c e d ) complex a n a l y t i c s p a c e s ;

see G . TOMASSIN1 [I: p. 62, P r o p o s i z i o n e 21 )

.

I n t h i s c o n c e r n , i f K i s n compact subset o f a S t e i n manifold X

(UniilfN

and

i s a n open b a s i s of n e i g h b o r h o o d s of K i n X I t h e n by c o n s i d e r -

d e f i n e d by ( 4 . 1 6 ) , one may a l i n g t h e r e s p e c t i v e sequence ('n In e N ways s u p p o s e t h a t t h e r e e x i s t s , i n e f f e c t , a decreasing sequence of S t e i n

manifolds containing K. T h e r e f o r e , c o n c e r n i n g t h e e n v e l o p e of holomorphy of K , one o b t a i n s , on t h e b a s i s of t h e p r e v i o u s d i s c u s s i o n , t h e r e l a t i o n

??Y(O(K)I (4.19)

= limm(L)(Un ) ) = 1 2 m(O(gn /I t

-

= limo =

w i t h i n a homeomorphism

,

n u,,,

n=l concerning t h e respective " e q u a l i t i e s " involved.

C o n s e q u e n t l y , b y t a k i n g t h e c o u n t e r e x a m p l e of J . E . BJORK [I: p. 511 i n t o a c c o u n t , one e s s e n t i a l l y g e t s by t h e l a s t r e l a t i o n a k i n d o f " b e s t a p p r o x i m a t i o n " t o t h e s i t u a t i o n which o n e h a s i n c a s e of a S t e i n com-

p a c t by c o n s i d e r i n g S t e i n manifold

i n s t e a d a compact holomorphically convex subset of a

( c f . a l s o ( 4 . 6 ) above).

On t h e o t h e r h a n d , l e t u s c o n s i d e r t h e e n v e l o p e s of holomorphy

V

164

SPECTRUM (GLOBAL THEORY)

of a compact s e t K i n a S t e i n manifoZd X , c o r r e s p o n d i n g t o t h e s e t s Un

,n

e N,

h

as above: t h a t i s , c o n s i d e r t h e sets K , , n f X ,

kn

(4.20)

g i v e n by

= h u l l o ( 4 ) ( a l : = I x E ~ n : J f i x ) ) 'IlflI,V

f€OIGn)}.

Thus, one o b t a i n s a p r o j e c t i v e system of compact s e t s

(in, T ~ , , ~w ) ith

(4.21)

n h ,

i n such a manner t h a t Lha folZowing reZation

i

(4.22)

= rn(O(.K)i = I..& r,

is t r u e , w i t h i n a homeomorphism, w h i l e

i s d e f i n e d by ( 4 . 2 ) and ( 4 . 3 )

( c f . R . HARVEY-R.O. WELLS, J r . [ l : p. 513, Lemma 3 . 1 1 ) .

NOW,

t h e same r e l a -

t i o n ( 4 . 2 2 ) ,w i t h K a compact holomorphicaZZy convex subset of a S t e i n manifold X , h a s been a p p l i e d f o r e x t e n d i n g t o s u c h K t h e c l a s s i c a l Theorems A

and B of H . CARTAN which a r e v a l i d , of c o u r s e , f o r S t e i n m a n i f o l d s ( i b i d . ; p. 513, Theorem 3 . 3 ) . S i m i l a r c o n s i d e r a t i o n s t o t h e above a r e a l s o found i n Chapt.VI1; S e c t i o n 3 and C h a p t . V I I 1 ; S e c t i o n 8 . On t h e o t h e r h a n d , f u r t h e r a p p l i c a t i o n s of t h e p r e c e d i n g t e r m i n o l o g y c o u l d a l s o b e g i v e n by c o n s i d e r i n g "vector-valued"

holomorphic maps i n c o n n e c t i o n w i t h " t o p o l o g i c a l ( t h e a l g e b r a o(X, Z)).

t e n s o r product a l g e b r a s " ( c f . Chapt. X I ; S e c t i o n 4 I n t h i s c o n c e r n , see a l s o A . MALLIOS [16]).

5. The dual o f the Arens-Michael decomposition Suppose w e a r e g i v e n a t o p o l o g i c a l a l g e b r a E , t o g e t h e r w i t h a local basis

rt: = (u,ia

I o f E . I n t h i s r e s p e c t , one may a l s o assume, (downwards) by set-

by h y p o t h e s i s f o r ?Y, t h a t the index s e t I i s directed t i n g a < R i f U R 'Ua

( c f . a l s o C h a p t . 111; ( 3 . 5 ) ) .

F u r t h e r c o n s i d e r t h e t o p o l o g i c a l d u a l E'of

E and t h e c o r r e s p o n d -

i n g f a m i l y of t h e p o l a r s o f t h e s e t s U a , a e I , i n E ' ; t h e n , one g e t s an

upwards directed famiZy

(U,. )a E I

(5.1)

of equicontinuous, h e n c e nition

waakZy bounded subsets of E'

which are by t h e i r defi-

absoZuteZy convex and weakZy cZosed i n E ' ( a s b e i n g t h e p o l a r s of

n e i g h b o r h o o d s of zero i n E ) . F i n a l l y , c o n s i d e r t h e s p e c t r u m m l E ) E E'

of t h e g i v e n t o p o l o g i -

c a l a l g e b r a E l a s w e l l a s t h e f a m i l y of s u b s e t s of m ( E ) c o r r e s p o n d ing t o (5.1) (5.2)

,

given by i??"Z(E)nU:

lae I

.

5.

165

DUAL ARENS-MICHAEL DECOMPOSITION

T h u s , o u r n e x t o b j e c t i v e i s t o examine t h e ( t o p o l o g i c a l ) decomposition

of

m(E)i n terms of t h e family ( 5 . 2 1 .

T h i s mihgt b e viewed, i n g e n e r a l , a s to a

t h e " d u a l o f t h e Arens-Michael d e c o m p o s i t i o n " , c o r r e s p o n d i n g g i v e n ( c o m p l e t e ) l o c a l l y m-convex

a l g e b r a ( c f . Theorem 111; 3.1 a n d / o r

D e f i n i t i o n 1 1 1 ; 3 . 1 ) . S o , s i n c e t h e f a m i l y ( 5 . 2 ) may b e assumed t o be

an upwards directed family of subsets of

T ? Z ( E l , one g e t s t h e r e l a t i o n

which i s v a l i d w i t h i n a ( s e t - t h e o r e t i c ) b i j e c t i o n . ( I n t h i s r e g a r d , see N . BOURBAKI [l: Chap. 3 ; p. 92, Remarque 13 )

??Z(El b e l o n g s t o some :U Lemma 5.1.

.

Therefore, since every f e , w i t h ,€I, w e first conclude t h e following.

Let P be a topological algebra w i t h spectrum

n Z i E l and (U, la I

a local b a s i s of E. Then, one g e t s (5.4) (where t h e s e c o n d e q u a l i t y i s v a l i d w i t h i n a b i j e c t i o n ) . I Now by c o n s i d e r i n g on e a c h

miE)n U t G m l E l , a.€ I, t h e r e l a t i v e

t o p o l o g y from ? X { . T l , one g e t s by

( 5 . 2 ) and ( 5 . 3 ) a n i n d u c t i v e system of

topological spaces s u c h t h a t t h e canonical b i j e c t i o n (5.5) d e f i n e d by ( 5 . 4 ) , i s , i n f a c t , c o n t i w o u s . However, j i s n o t , i n general, a ho-

meomorphism (see M . BONNARC, [l: p. 2656, Remarque] )

. So

w e seek next condi

-

t i o n s u n d e r which t h i s i s t r u e , on t h e c o n t r a r y , w h i l e w e e s s e n t i a l l y o b t a i n a c h a r a c t e r i z a t i o n of t h e r e l e v a n t s i t u a t i o n f o r a p a r t i c u l a r c l a s s of t o p o l o g i c a l a l g e b r a s ( n a m e l y , s p e c t r a l l y b a r r e l l e d t o p o l o g i -

cal algebras with an i d e n t i t y element). T h u s , " c o m p a c t l y generateGI' t o p o l o g i c a l s p a c e s seems t o b e t h e c r u c i a l f e a t u r e i n t h i s c o n t e x t . So w e d i g r e s s s l i g h t l y f r o m o u r theme i n t h e f o l l o w i n g s u b s e c t i o n by commenting on t h e r e l e v a n t t e r m i n o l o g y , c o n c e r n i n g t h i s c l a s s of t o p o l o g i c a l s p a c e s , t o t h e e x t e n t t h a t t h i s

i s needed i n t h e s e q u e l ( S e c t i o n 6 ) .

5 . ( 1 ) . Compactly g e n e r a t e d t o p o l o g i c a l spaces.-

We s t a r t w i t h t h e f o l -

lowing D e f i n i t i o n 5.1. Given a t o p o l o g i c a l s p a c e X , a f a m i l y s u b s e t s of X i s s a i d t o b e a k-covering family

o f X i f t h e f o l l o w i n g two

conditions a r e s a t i s f i e d : (5.6)

i l The s e t s S

a'

a.

E I,

)a E I of

are compact subsets of X .

166

V

SPECTRUM (GLOBAL THEORY)

i-i) For every compact K C X , t h e r e e x i s t s some a E I , w i t h K CSa.

I n o t h e r words, a k - c o v e r i n g

f a m i l y of X may b e c o n s i d e r e d a s

a c o f i n a l subfamily o f t h e f a m i l y K of a l l compact s u b s e t s of X ( t h e l a t t e r f a m i l y b e i n g d i r e c t e d , upwards, "by i n c l u s i o n " ) . S e e , f o r i n

-

s t a n c e , Lemma 5 . 2 below. I t i s a l s o c l e a r by ii) of t h e above d e f i n i -

I i s a c o v e r i n g o f X.

t i o n t h a t the family

Now, by c o n s i d e r i n g t h e f a m i l y K of a l l compact s u b s e t s of a

, one d e f i n e s t h e k - topology ( o r e l s e t h e compactly generated topology) on X , w e d e n o t e i t by k X , v i a t h e relation g i v e n t o p o l o g i c a l s p a c e I ' X , T j = X [T]

. KTK

(X, k X ) := lrnll,

(5.7) That i s ,

k x i s t h e f i n a l topology on X making t h e c a n o n i c a l i n j e c t i o n s

iK:K-X,

w i t h K E K , c o n t i n u o u s , where e a c h compact K L X

t h e r e l a t i v e t o p o l o g y from ( X , T / .

i s taken i n

S o it amounts t o t h e same, s a y i n g

t h a t k X i s the f i n a l topology on X determined by t h e compact s u b s e t s ; e q u i v a l e n t l y , one c o n c l u d e s by (5.1) t h a t a s u b s e t B of X i s kX-closed open) i f of

1

,

(and only i f ) t h e set B G K

(resp. i s a ~ - 2 l o s e d ( r e s p . open) s u b s e t

f o r every K E K . F u r t h e r m o r e , i t i s c l e a r by (5.7) t h a t T L k X , whence t h e k - t o p o -

logy on X i s Hausdorff i n case

T

is.

Thus, a p p l y i n g t h e p r e c e d i n g t e r m i n o l o g y , v e come now t o t h e next D e f i n i t i o n 5.2. A g i v e n t o p o l o g i c a l s p a c e ( X , r l

k-space i f

T

i s s a i d t o be a

= k X . T h a t i s , whenever i t s t o p o l o g y i s defined

( o r else

g e n e r a t e d ) by t h e compact s e t s of t h e s p a c e , so t h a t one a l s o s p e a k s i n t h a t case of a compactly generated topological space. The n e x t s t a t e m e n t a s s u r e s t h a t a compactly generated topology i s determined, i n f a c t , by any given k-covering f a m i l y of t h e space. T h a t i s , w e have. Lemma 5.2. Let X be a Hausdorff topological space and l S , l a E I

a k-cover-

ing f a m i l y of X . Then t h e kX-topology i n X c o i n c i d e s w i t h t h e f i n a l topology on i t determined by t h e f a m i l y is, ), I , where each S,, a e I , is endowed w i t h t h e r e l a t i v e topology from X ( i. e .

,

t h e kX-topology

i s t h e "weak t o p o l o g y " of

d e t e r m i n e d by t h e g i v e n f a m i l y ) . Equivalently, one g e t s t h e r e l a t i o n (5.8)

X[kX] = l i m K = 12sa, K T K

CAEI

& t h i n a homeomorphism ( r e l a t i v e t o t h e i d e n t i t y map of X )

X

5. ( 1 )

.

Proof. By h y p o t h e s i s f o r t h e f a m i l y

kX 5

167

COMPACTLY GENERATED S P A C E S

s = f S a i a e I , it

T, ( t h e "weak t o p o l o g y " d e t e r m i n e d by S )

.

i s clear t h a t

T h e r e f o r e , i f -ro d e n o t e s

t h e g i v e n t o p o l o g y of X I one h a s To 5

(5.9) On t h e o t h e r hand,

kx

5 T

S'

i f B i s -rS-closed and K a

co compact

t h e n t h e r e e x i s t s by h y p o t h e s i s a n i n d e x a € I , w i t h K E S a , (5.10)

s u b s e t of X,

so t h a t

BnKCBnSaZSCY.

Thus, s i n c e B n K

is a

and hence kX-closed,

closed that is

T~

s u b s e t o f Sa it i s a l s o

compact

< k X , and t h i s p r o v e s t h e a s s e r t i o n . 1

The f o l l o w i n g i s a s t r a i g h t f o r w a r d consequence of t h e p r e v i o u s lemma; it w i l l p r e s e n t l y b e a p p l i e d below. Namely, w e have.

Corollary 5.1. L e t

fX,T)

= X [ T ] b e a Hausdorff t o p o l o g ' c a l space and

s

=

( S a i a E I a k-covering f a m i l y of X . Then, one has t h e r e l a t i o n

x

(5.11)

= 13Sa I*& I

[IT]

if, m d o n l y i f , X is a k-space, i.e., T = kX ( c f . D e f i n i t i o n 5 . 2 ) . I

6. The dual o f the Arens-Michael decomposition (contn'd.) NOW,

suppose w e a r e g i v e n a t o p o l o g i c a l a l g e b r a E t o g e t h e r w i t h

a l o c a l b a s i s (Ua

i a e I of

E ( t h e latter is

always u n d e r s t o o d i n t h e

s e q u e l as d i r e c t e d , downwards, by i n c l u s i o n ) . Thus, by c o n s i d e r i n g t h e extended spectrum ??T(E)+

of F ( c f .

( 1 . 8 ) and ( 1 . 1 1 ) ) i n c o n j u n c t i o n

w i t h ( 5 . 1 ) , one o b t a i n s a f a m i l y { ~ i E i + f l U1 , ~ w i t h a e I ,

(6.1)

of c l o s e d e q u i c o n t i n u o u s s u b s e t s of E'

and hence (Alaoglu-Bourbaki)

(weakly) compact s u b s e t s of t h e same s p a c e ; so t k e ~a r e a l s o , b y ( 1 . l l ) ,

weakly compact s u b s e t s of ??Y(EI+. I n t h i s c o n c e r n , w e s t i l l n o t e t h a t i n c a s e of a t o p o l o g i c a l a l g e b r a E w i t h an i d e n t i t y element one

I, ?WE')

(6.2)

g e t s , of c o u r s e , a n a n a l o g o u s f a m i l y

n:u I ,

CY e

r,

of compact s u b s e t s o f " C t E J . F u r t h e r m o r e , s i m i l a r l y a s w i t h ( 5 . 4 ) one g e t s (6.3)

m i ~ i +U=( m i c i + n y l , O i =1 2 i m i ~ ) + n u , O i ,

where t h e second e q u a l i t y holds t r u e w i t h i n a b i j e c t i o n ( t h e f i r s t one bei n g an e a s y consequence of

( 5 . 4 ) and ( 1 . I 1 ) ) .

N o w by r e s t r i c t i n g o u r a t t e n t i o n t o s p e c t r a l l y b a r r e l l e d a l g e -

168

V

SPECTRUM (GLOBAL THEORY)

r a s , w e a c t u a l l y o b t a i n b y 1 6 . 1 ) a k - c o v e r i n g of t h e e x t e n d e d s p e c

-

t r u m o f t h e p a r t i c u l a r a l g e b r a u n d e r c o n s i d e r a t i o n . But w e s h a l l need f i r s t t h e f o l l o w i n g t e c h n i c a l f a c t which, i n e f f e c t , supplements t h e above D e f i n i t i o n 1 . 3 . :

Remark 6.1.-

A given topological algebra E i s s p e c t r a l l y barrelled i f , and

only i f , it i s such w i t h respect t o i t s extended spectrum

mi€)+( i . e . ,

every

bounded s u b s e t o f m ( E l + i s e q u i c o n t i n u o u s ) : I t i s c l e a r t h a t t h e s t a t e d c o n d i t i o n i s s u f f i c i e n t i n o r d e r t h a t t h e a l g e b r a F t o b e spect r a l l y b a r r e l l e d ( c f . D e f i n i t i o n 1 . 3 ) . On t h e o t h e r h a n d , i f t h e a l g e b r a E i s s p e c t r a l l y b a r r e l l e d ( n a m e l y , w i t h r e s p e c t t o ??Z(EI) and B i s

a bounded s u b s e t of m ( E ) + , t h e n B n ? ? Z ( E ) E B

is

a bounded s u b s e t of

7 7 Z ( E i , h e n c e by h y p o t h e s i s e q u i c o n t i n u o u s . T h u s , t h e s e t BG(B

n miEliuC o 1 E B n m ( z ) u l a 1 E E,'

i s equicontinuous a s w e l l ( c f . N. BOURBAKI [IS: Chap. 10; p. 28, P r o p o s i t i o n 6]), that is, the assertion. Thus, w e now h a v e t h e f o l l o w i n g .

Lemma 6.1. Let E be a s p e c t r a l l y barrelZed algebra ( D e f i n i t i o n 1 . 3 ) and U i CY )a€I a ZocaZ b a s i s of E. Then, t h e family (6.4)

{

m(E)+n U:},

t'itk a E I ,

c o n s t i t u t e s a k-covering family of P L ( E ) + . Proof. W e h a v e a l r e a d y remarked t h a t ( 6 . 4 ) y i e l d s a f a m i l y o f compact s u b s e t s of VZ(Eif. On t h e o t h e r hand, i f K i s a compact s u b s e t of " Z ( E ) + , t h e n i t i s a f o r t i o r i bounded, h e n c e by h y p o t h e s i s and R e mark 6 . 1 an e q u i c o n t i n u o u s s u b s e t of m ( E l + G E ' . T h e r e f o r e , i t s p o l a r

set

i s a n e i g h b o r h o o d o f z e r o i n E , so t h a t t h e r e e x i s t s by hypoc( € I ,w i t h U a E K o ; thus, KGKooGU: , t h a t i s , K E n T ( E ) + n

KO

t h e s i s an index U,"

,for

some C ~ IE , which i s

the assertion. I

The f o l l o w i n g p r o v i d e s a n example of t h e s i t u a t i o n one g e t s by c o n s i d e r i n g s p e c t r a l l y b a r r e l l e d a l g e b r a s , i n g e n e r a l , a s it concerns t h e ( k - c o v e r i n g ) f a m i l y ( 6 . 4 ) and t h e f a c t of b e i n g t h i s a g e n e r a t i n g f a m i l y of t h e t o p o l o g y of mlEl'.

Thus, w e n e x t o b t a i n an a n a l o -

gous ( i n f a c t , more g e n e r a l ) r e s u l t f o r ? ? Z ( E ) ,

by c o n s i d e r i n g a l g e -

bras with i d e n t i t y elements.

Lemma 6.2. Let E be a s p e c t r a l l y barrelled algebra, whose spectrum i s m(El, and l e t (U, i a I be a Local b a s i s of E . Furthermore, consider t h e following assertions:

6. DUAL ARENS-MICHAEL DECOMPOSITION (CONTN'D. ) 1) The extended spectrwn of E , i.e., ??Z(E,+,

169

i s a k-space.

2) One has t h e iqelation

?7Z(El+ = 1 3 (??Z(E)'n

(6.5)

Ut) ,

w i t h i n a homeomorphism. 3 ) The r e l a t i o n

-

~ ~ ( ~ ) = i i r n (u:)~ ~ i ~ i n

(6.6)

holds t r u e , w i t h i n a homeomorphism. Then, one g e t s t h e following i m p l i c a t i o n s : 1 ) <=> 21

(6.7)

Proof. By Lemma 6 . 1 ,

* 3).

( 6 . 4 ) y i e l d s a k-covering

so t h e h y p o t h e s i s t h a t ??Z(Ei+

i s a k-space

f a m i l y o f PT(E)';

i s a 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 i n o r d e r ( 6 . 5 ) t o b e v a l i d , w i t h i n a homeomorphism ( c f . C o r o l l a r y 5 . 1 ) . Thus, t h e c o n d i t i o n s 1) and 2) a r e , indeed, e q u i v a l e n t . On t h e o t h e r h a n d , one o b t a i n s by ( 6 . 3 ) m:Ei+nCto}=

VUE)=

i u i m i ~ i n+u ; i i n c { o } c1

(6.8) =

v i nT ( E l n u,"i = %1

I nr ( E i n u,"I ,

where t h e l a s t r e l a t i o n i s v a l i d w i t h i n a homeomorphism ( w e a p p l y h e r e t h e argument f o r t h e o p e n n e s s of a s e t r e l a t i v e t o a f i n a l t o p o l o g y ) . So one g e t s ( 6 . 6 ) , and t h i s a c t u a l l y a c c o m p l i s h e s t h e p r o o f of

Schol i u m 6.1

.-

Suppose t h a t E i s a s p e c t r a l l y b a r r e l l e d a l g e b r a

having an equicontinuous spectrum m(E! ( e. g . Thus, t h e e x t e n d e d s p e c t r u m m ( E ) ' s u b s e t o f E,' by Lemma 6 . 2 ,

(6.7).1

,

a n LFQ-algebra; see Chapt .VIII).

b e i n g a c l o s e d and e q u i c o n t i n u o u s

( c f . ( 1 . 8 ) ) i s a l s o c o m p a c t , whence a k - s p a c e ;

therefore,

one concludes t h e relation V - Z ~ E ! = ~ ~ I V L ( En )u,Oi,

(6.9)

which i s v a l i d w i t h i n a homeomoirrphism. I n t h i s

r e s p e c t , w e f u r t h e r remark

t h a t t h e f a m i l y { m ( E ) n U o } , a e I, which c o r r e s p o n d s t o a g i v e n l o c a l bac1

s i s ?Z= i

rn(E),

~ o f E may ~ n o t )b e , i n ~g e n e r a l~, a k - c~ o v e r i n g f a m i l y of

s o t h a t one c a n n o t a l w a y s a p p l y d i r e c t l y C o r o l l a r y 5 . 1 .

e v e r , t h e s p e c t r u m of E i s by h y p o t h e s i s a k - s p a c e

How-

( c f . Theorem 1 . 1 ) .

The p r e v i o u s s i t u a t i o n i s improved by c o n s i d e r i n g a l g e b r a s w i t h i d e n t i t y e l e m e n t s , s i n c e t h e n t h e r e s p e c t i v e s p e c t r a a r e c l o s e d sub-

s e t s of t h e (weak) d u a l s p a c e s of t h e a l g e b r a s c o n s i d e r e d ( c f . Remark 1 . 1 ) . That i s , w e have. Theorem 6 . 1 . Let E be a t o p o l o g i c u 2 aZgebra w i t h an i d e n t i t y eZement and con-

v

170

SPECTRUM(GLOBAL THEORY)

m(E/be

tinuous Gel'fand map. Furthermore, l e t

t h e spectrum and

n=( U a ) O 1

I a

local b a s i s of E. Then, t h e f a m i l y

(?&Z(El

(6.10)

nu:),

a EI,

d e f i n e s a k-covering f a m i l y of m(Ei, i n such a way t h a t t h e r e l a t i o n

??Z(E) =

(6.11)

i%r

???YE) n

u," i

holds t r u e w i t h i n a homeomorphism i f , and only i f , m(E)i s a k-space.

Proof. S i n c e t h e sets U o , O I E I ,a r e c l o s e d e q u i c o n t i n u o u s s u b s e t s of E'

s'

set

t h e y a r e ( w e a k l y ) compact t o o ( A l a o g l u - B o u r b a k i ) ; so s i n c e t h e

m(E)i s

by h y p o t h e s i s a c l o s e d s u b s e t of Ei

,

t h e family (6.10)

c o n s i s t s of compact s u b s e t s of m l E l . On t h e o t h e r h a n d , i f KG???(E .l

i s compact, t h e n it i s by h y p o t h e s i s e q u i c o n t i n u o u s , so t h a t

KO

n e i g h b o r h o o d of z e r o i n E . T h e r e f o r e , t h e r e e x i s t s u e I , w i t h

U- C K o ,

is a

'

s o t h a t .KGKoo-t Ua ; t h a t i s , KC X l E l n U: which p r o v e s t h a t ( 6 . 1 0 ) def i n e s a k - c o v e r i n g f a m i l y of m(El. Now, t h e l a s t p a r t of t h e a s s e r t i o n f o l l o w s d i r e c t l y from C o r o l l a r y 5 . 1 , and t h i s f i n i s h e s t h e p r o e f . 1 The f o l l o w i n g i s a n immediate c o n s e q u e n c e o f t h e argument a p p l i e d 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 . Namely, w e h a v e .

C o r o l l a r y 6.1. Let E be a metrizable topological algebra w i t h an i d e n t i t y miE: :s a hems-compact s,Jace ( i . e . , miEl has a denumerable k-covering). I element and continuous Gel'fand map. Then, i t s spectrum

Remark 6.2.-

Within t h e p r e v i o u s c o n t e x t , w e f u r t h e r n o t e t h a t i n

c a s e E i s a l o c a l l y convex algebra w i t h continuous Gel'fand map, t h e n t h e p r e c e d i n g argument c a n b e r e v e r s e d i n i h e f o l l o w i n g s e n s e :

Suppose we are g i v e n a k-covering f a m i l y ( K a i a e I of ??Z(E).

Then, t h e r e e x i s t s a local b a s i s 7?=

I of

E i n such a way t h a t ~ ? X ( E i n U ~ aI e, I ,

(6.12)

i s a k-covering f a m i l y of m(El w i t h

K ~ C ? ? Z ( nE U); , f o r every

01 E

I.

I n t h i s r e s p e c t , it s u f f i c e s t o t a k e Ua

=_

Ki,

l a r Theorem: c f . J . HORVLTH [l: p. 192, Theorem 11

01

€ I , and a p p l y t h e Bipo-

.

N O W , s u p p o s e i n p a r t i c u l a r t h a t w e a r e g i v e n a l o c a l l y m-convex a l -

gebra E , t o g e t h e r w i t h a l o c a l b a s i s ??= ( U O 1 ) A e I of E ( c o n s i s t i n g of mb a r r e l s ) . Further consider the corresponding t o

Arens-Michael de-

6.

DECOMPOSITION (COIITN’D. )

DUAL CLHGNS-MICHAEL

171

c o m p o s i t i o n o f E (see D e f i n i t i o n 111; 3 . 1 ) ; t h u s , one g e t s a f a m i l y 0

(6.13)

Ea ( r e s p .

, Ea 1 , a e l ,

o f normed ( r e s p . , Banach) a l g e b r a s , s o t h a t l e t

(6.14)

V Z ( E ~ )( r e s p .

, mifa)), a

E

I,

b e t h e c o r r e s p o n d i n g f a m i l y of t h e s p e c t r a of t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d . T h u s , f o r e v e r y i n d e x a € I ,one h a s t h e canonica2 map

j,

(6.15)

: ???(E/

n :U

+

= fa

m(Ea): f -ja(fI

defined by t h e re2ation fafx+Na)=fci(pci(xCi! = [jci(fI](pa(x)):= f ( x ) ,

(6.16)

f o r every x e E . ( W e apply here constantly

t h e r e s p e c t i v e n o t a t i o n of

C h a p t . 111; S e c t i o n 3 ) . N O W , t h e l a s t r e l a t i o n y i e l d s , i n d e e d , a w e l l d e f i n e d map, s i n c e f o r any f € T??(El n U:,

if p

d e n o t e s t h e gauge

c1

func-

t i o n of U c i e 2 , one o b t a i n s

(6.17)

N

Namely, f o r e v e r y x

E

ci

I

k e r i pCY i s k e r ( f ) .

a n d h e n c e If(-x/j
1

= 2, so

1

E Ua l f ( x i j < ~ ,f o r e v e r y E>O, whence f(xci=O. So

E , w i t h p ( x ) = O , one h a s

1

i.e.,

pa(--)

--3:

one g e t s , by d e f i n i t i o n , fci0Pa =

(6.18)

f

F u r t h e r m o r e , s i n c e pa i s a “ q u o t i e n t map“, t h e l a s t r e l a t i o n c h a r a c -

t e r i z e s a l s o t h e c o n t i n u i t y of f fa E m(Ea). Conversely,

( f o r any g i v e n c o n t i n u o u s

f o r any f a F “ Z ( E a i ,

f ) , hence

o n e d e f i n e s by ( 6 . 1 8 ) a n e l e m e n t

f e m ( E ) such t h a t one h a s , i n p a r t i c u l a r , t h a t f

E

??Z(E)n U:.

This f o l -

lows f r o m

Iffxi

(6.19)

. C o n s e q u e n t l y , t h e map ja which is i n j e c t i v e , by t h e s a m e a d e f i n i t i o n s , i t i s a l s o an onto map. Now it i s r e a d i l y s e e n by ( 6 . 1 6 ) t h a t t h e map j i s continuous. Fura t h e r m o r e , by c o n s i d e r i n g t h e t r a n s p o s e map of t h e c a n o n i c a l quot i e n t map pa: E +E = E / p l 1 ( O i , r e s t r i c t e d t o m f E a ) C ( E i i s , one g e t s f o r every

2

EU

ci

(6.20)

w i t h fa

tpa(fcl) E

=fa 0 Pa

=

fr

m i E c i ) . T h u s , on t h e b a s i s of ( 6 . 1 9 ) , one e s s e n t i a l l y o b t a i n s

-

by t h e p r e c e d i n g r e l a t i o n ( 6 . 2 0 ) t h e fo22owing continuous map (6.21)

:

m i ~ ~ a ,~

( E n Iui

,

172

with

V

c1 E

I. Therefore

SPECTRUM (GLOBAL THEORY)

,

(6.22)

i, o

t

-

Pa

-

idm(E,)

I

f o r e v e r y a E I ; t h u s , the map j,, g i v e n b y ( 6 . 1 6 )

, is

.

a homeomorphism The

p r e v i o u s d i s c u s s i o n p r o v i d e s , i n e f f e c t , t h e p r o o f of t h e f i r s t p a r t of t h e n e x t

Lemma 6.3. Let E be a l o c a l l y m-convex algebra and 71:= IU,iae I a local I Iresp., i,, c1 E I ) be t h e corresponding fami l y o f normed ( r e s p . , Banachl algebras provided by t h e given local b a s i s 72. Then,

basis of E. Furthermore, l e t (E,lae one obtains

m(El n U:

(6.23)

= m(E,i

=

m(ia),

f o r every ,€I, within a homeomorphism of the r e s p e c t i v e topological spaces. Moreover, concerning the s p e c t m of E, one g e t s t h e r e l a t i o n

~T(E)=~S(VZ(E)~U,O)=

(6.24)

1 2 rni~,)=i$ 1 ? ~ ( E ^ ~ i ,

within a continuous b i j e c t i o n of the fiiocr-id rrierbe-rs o?ztcJ M ( E l ( s e e

( 5.5) )

.

Proof. The p r e v i o u s d i s c u s s i o n p r o v i d e s a l r e a d y t h e p r o o f o f t h e f i r s t r e l a t i o n i n ( 6 . 2 3 ) . Furthermore, s i n c e t h e spectrum

m(?,)of

t h e Banach a l g e b r a ~ a : = E / p ~ l ( 0()c f . C h a p t . 111; ( 3 . 2 ) ) i s l o c a l l y comp a c t , it i s a l s o l o c a l l y e q u i c o n t i n u o u s (Theorem 1 . 1 ) ; so t h e s a m e i s t r u e f o r ??Z(E,l

(Lemma 2 . 2 )

(6.25)

,

h e n c e one obtains =

m(E,l

within a homeomorphism (Theorem 2 . 1 )

,

m(ic1),

f o r e v e r y c1 E I, a n d t h i s f i n i s h e s t h e

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

(6.24)

is a direct

c o n s e q u e n c e o f Lemma 5 . 1 a n d t h e comment f o l l o w i n g i t , a n d t h i s t e r m i n a t e s t h e p r o o f of o u r lemma. I

Remark 6.3.vex a l g e b r a E , (6.26)

If

2

d e n o t e s t h e c o m p l e t i o n of a g i v e n l o c a l l y m-con-

then applying t h e n o t a t i o n of

m&)=

( s e e a l s o Lemma 2 . 1 ) ,

( 6 . 2 4 ) one g e t s

m(Ec1 )

??TIE) = l h n m(Ecl) = 1 3 -t

within a continuous b i j e c t i o n ( f o r t h e f i r s t

t h r e e s p a c e s ; c f . a l s o ( 6 . 2 3 ) ) . On t h e o t h e r h a n d , i f t h e a l g e b r a

E

h a s an i d e n t i t y element and c o n t i w e u s rJeZ’fand map w h i l e i t s s p e c t r u m m(E)

i s , moreover, a k-space, t h e n t h e p r e c e d i n g r e l a t i o n s a r e a c t u a l l y t r u e w i t h i n homeomorphisms ( c f . Theorem 6 . 1

i n connection with Corollary

5 . 1 ) . I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t t h e h y p o t h e s i s of b e i n g ??Y(El a k - s p a c e i s , of c o u r s e , weaker t h a n l o c a l e q u i c o n t i n u i t y ( a n d h e n c e l o c a l c o m p a c t n e s s ) o f t h e same s p a c e , g i v e n t h a t a ( H a u s d o r f f )

7. SPECTRA OF PROJECTIVE L I M I T ALGEBRAS

l o c a l l y compact s p a c e i s c e r t a i n l y a k - s p a c e

Scholium 6.2.-

NOW,

173

( s e e a l s o Theorem 2 . 1 ) .

a r g u i n g w i t h i n t h e p r e v i o u s framework, l e t u s

c o n s i d e r i n s t e a d t h e extended s p e c t r a of t h e t o p o l o g i c a l a l g e b r a s inv o l v e d . Then,

one obtains t h e r e l a t i o n nT(El+nUE =

(6.27) IJLi ?:in

m(E,)'=m(ia)',

c1 e

a homearnorphisn;, f o r e v e r y

I. T h i s i s immediately d e r i v e d u s i n g

a n o b v i o u s m o d i f i c a t i o n of t h e a r g u m e n t a p p l i e d i n t h e p r o o f of (6.23) a n d ( 6 . 2 5 ) ; t h e same r e l a t i o n s c o u l d a c t u a l l y be d e r i v e d from ( 6 . 3 7 )

a s well, i n c a s e t h e l a t t e r would h a v e b e e n p r o v e d i n t h e way i n d i c a t e d . Thus, c o n c e r n i n g t h e e x t e n d e d s p e c t r u m o f E l one g e t s i n con

-

j u n c t i o n w i t h ( 6 . 3 ) the relation

(nZ( E l ' n Ua I = % 1

m(E)' = 1 %

(6.28)

??T(Ea)+ = 1%

m(ia),

v a l i d w i t h i n a b i j e c t i o n , having a continuous inverse p r o v i d e d by t h e c o r r e sponding e x t e n s i o n of

(5.5)

-

.

I n p a r t i c u l a r , f o r s p e c t r a l l y b a r r e l l e d l o c a l l y m-convex a l g e b r a s having equicontinuous s p e c t r a , o r a s extended s p e c t r a k-spaces, one o b t a i n s t h e r e s p e c t i v e r e l a t i o n s ( 6 . 2 4 ) a n d ( 6 . 2 8 ) , v a l i d n o w w i t h i n ( c a n o n i c a l ) homeomorphisms ( c f . Lemma 6 . 2 i n c o n n e c t i o n w i t h S c h o l iuin 6 . 1 and Remark 6 . 3 )

.

7. Spectrum o f a p r o j e c t i v e l i m i t topological algebra. Dense p r o j e c t i v e l i m i t algebras W e e x h i b i t i n t h i s s e c t i o n a more g e n e r a l and a b s t r a c t f r a m e -

work of t h e p a r t i c u l a r , a l t h o u g h p r o b a b l y most i m p o r t a n t ( ! ) ,

c a s e de-

s c r i b e d by t h e r e l a t i o n ( 6 . 2 4 ) o f t h e p r e v i o u s Lemma 6 . 3 . T h u s , s u p p o s e w e a r e g i v e n a p r o j e c t i v e system (7.1)

'Ea

3

fa$

o f topological algebras, i n d e x e d by a d i r e c t e d s e t I ( c f . C h a p t . 111; Sec-

tion 2),

and l e t E = 12(Ea, f

(7.2)

aB

be t h e r e s p e c t i v e projective l i m i t

)

( t o p o l o g i c a l ) algebra.

Furthermore, suppose t h a t f o r any a , @ i n

I , w i t h a < 4 the re-

s p e c t i v e (connecting) map (7.3)

fa4 : Eo-

has a dense image, i.e., E sume t h a t f o r a n y a

,3

a

= Im(f

).

a4

Ea

(Applying a n o t h e r t e r m i n o l o g y , w e a s -

i n I , w i t h a c b , t h e a l g e b r a s Ea

,E

B

form a

174

V

SPECTRUM (GLOBAL THEORY)

&nge p a i r ; see, f o r i n s t a n c e , H . GRAUERT-R. REWERT [l: p. 131 , S t e i n s c h e r S a t z ] ) . Thus, w e s p e a k of a dense p r o j e c t i v e system ( o f t o p o l o g i c a l a l g e b r a s ) and E i s t h e n c a l l e d a dense p r o j e c t i v e l i m i t algebra. NOW, b y c o n s i d e r i n g t h e t r a n s p o s e of C

,

(EL)S

-

( 7 . 3 ) r e s t r i c t e d t o ??Z(E,)

one g e t s by t h e h y p o t h e s i s f o r t h e l a t t e r map

-

- $a

t

(7.4)

*

m(Ea)

n ( E BI

,

f o r a n y a 5 B i n I. Thus, one o b t a i n s a n i n d u c t i v e system of topologicaZ

spaces

( r n ( E a I , fBa 1

(7.5)

so t h a t l e t S = 1 3 m(EaI

(7.6)

b e t h e c o r r e s p o n d i n g i n d u c t i v e l i m i t s e t endowed w i t h t h e r e s p e c t i v e f i n a l

topology ( i n d u c t i v e l i m i t space o f t h e s y s t e m ( 7 . 5 ) ; see a l s o Chapt. I V ; Section 1 ) . On t h e o t h e r h a n d ,

s u p p o s e t h a t each one of t h e canonicaZ maps

a' withael,

pa:E=limEa-E

(7.7)

c

has a dense image, i . e .

,

,

Ea = I m ( p a )

f o r e v e r y a € I. The l a s t h y p o t h e s i s

t o g e t h e r w i t h t h e a n a l o g o u s one f o r t h e maps ( 7 . 3 ) c h a r a c t e r i z e s a g i v e n p r o j e c t i v e s y s t e m ( 7 . ) a s a s t r i c t Z y dense p r o j e c t i v e system and t h e a l g e b r a E a s t h e r e s p e c t i v e s t r i c t l y dense p r o j e c t i v e l i m i t algebra. Now assuming t h e l a s t c o n d i t i o n f o r t h e map ( 7 . 7 1 , i t s t r a n s pose r e s t r i c t e d t o

m(E,) y e l d s a continsous map

(7.8) f o r every a e I . (7.1)

,

Thus i n case o f a s t r i c t l y d e n s e p r o j e c t i v e s y s t e m

one g e t s t h e f o l l o w i n g commutative d i a g r a m

(7.9)

t h a t i s , one has t h e r e l a t i o n (7.10)

h =h of

a

B

Ba'

for any a < B i n I. T h e r e f o r e ( c f . N . BOURBAKI [l: Chap. 3 ; p. 95, Remarque 2)] )

,

one o b t a i n s a ( c a n o n i c a l ) continuous map

7. SPECTRA OF PROJECTIVE LIMIT ALGEBRAS

h = limk

(7.11)

+ a

: S = lim T??iEaI

+

--+

175

m(Ei

i n s u c h a manner t h a t

-

k o @ C= I iz

(7.12)

for e v e r y a e I , where

: 772(Ea)

a'

S denotes t h e r e s p e c t i v e canoni-

c a l map ( c f . I V ; ( 1 . 8 ) ) . T h u s , o n e c o n c l u d e s t h a t

= f u a o p )(x) = u a i p ( z i ) = u , ( x a )

a

a

,

f o r a n y e l e m e n t s x = f z a l e E, and u E S . N o w , t h e map ( 7 . 1 1 ) i s one-to-one : T h a t i s , f o r a n y u , v i n S, w i t h h ( u ) = h ( v ) , o n e g e t s by [ h i u ) ](x) = L L ~ ~ x=, )[ h ( ~ ) ] f o r every w i t h ua

,v

3:

=

L,

a n d v = @ ( u 1, a a (Lemma I V ; 1 . 1 ) . S o t h e l a s t r e l a t i o n y i e l d s t h a t

m(E,)

u = @ fu,)

so t h a t e x t e n d i n g b y c o n t i n u i t y and b y t h e h y p o t h e s i s

uC =L ~j a on p,(El, for

= va(xa)I

ix a ) e E l where o n e may assume t h a t

in

(7.13)

one g e t s u = v

a

,

a on Ea = p a f E /

t h a t is, u = v .

F u r t h e r m o r e , f o r topoZogica2 algebras w i t h continuous m u l t i p l i c a t i o n t h e p r e c e d i n g map ( 7 . 1 1 ) i s an onto map : T h a t i s , f o r e v e r y e l e m e n t X E 9?Z(El GE', there exists u

a

e E'a s u c h t h a t

x=

(7.14)

t p (u i = u 0 p a a a a

( c f . , f o r e x a m p l e , H . H . SCHAEFER [l: p. 139, 4 . 4 1 ) . T h e r e f o r e , u i s c e r t a i n a ly m u l t i p l i c a t i v e on p ( E l G E t h a t i s u C I ~ ~ ( p , f E l )t ;h u s , it can u n i q u e a a' l y be extended by c o n t i n u i t y t o a n e l e m e n t It e ??Z(E,) , i n s u c h a way t h a t C

CI

=-i

a'

Hence o n e o b t a i n s , b y ( 7 . 1 4 )

x=

(7.15)

,

t ~ a ( E a ) = ha(Ea) = ( b y (7.12)) h ( $ a ( u a l l = k f u ) ,

where u = @ (ii 1 E S . a a T h e p r e v i o u s argument p r o v i d e s a l r e a d y t h e proof of t h e n e x t .

Lemma 7.1.

Let ( E a , f

aR

)

be a s t r i c t l y dense p r o j e c t i v e system of topologi-

c a l algebras aizd E = l z E a t h e r e s p e c t i v e s t r i c t l y dense p r o j e c t i v e l i m i t algebra. T h e n , by considering t h e spectra of t h e topological algebras i n v o l v e d , one o b t a i n s

an i n d u c t i v e system of topological spaces one g e t s a continuous i n j e c t i o n (7.16)

(

??'YfEa), f B a = t f a B l in such a way t h a t

- -

h :S = lim ??Z(Eal

mfEl.

I n p a r t i c u l a r , t h e previous mzp becomes a ( c o n t i n u o u s ) b i j e c t i o n in case o f topol o g i c a l algebras w i t h continuous m u l t i p l i c a t i o n . I The p r e c e d i n g i n c l u d e s n a t u r a l l y t h e case of a l o c a l l y m-convex

a l g e b r a E and t h e r e s p e c t i v e Arens-Michael decomposition

o f it (The-

v

176

SPECTRUM (GLOBAL THEORY) h

orem 111; 3 . 1 ) d e f i n e d v i a t h e Banach a l g e b r a s E , a given local b a s i s

z=(U,icle

I

which c o r r e s p o n d t o

of E . Thus, b y d e f i n i t i o n , each one of

the canonical maps

has a dense image. Moreover, f o r a n y a,B i n 1 w i t h a

<

a,

the respective (con-

n e c t i n g ) map

has a dense image. Thus, one h a s by t h e same d e f i n i t i o n s ( c f . a l s o 111; 13.7) I (7.17) f o r any a( B i n I . T h e r e f o r e , o n e c o n c l u d e s t h a t

t h e Arens - Michael decomposition ( c f . D e f i n i t i o n (7.18)

111; 3.1 ) provides a s t r i c t l y dense p r o j e c t i v e sys-

tem of Banach algebras, a s s o c i a t e d w i t h any l o c a l l y m-convex a l g e b r a whose c o m p l e t i o n i s t h e r e s p e c t i v e ( p r o j e c t i v e ) l i m i t a l g e b r a o f t h e s y s t e m (see Theorem 111; 3 . 1 ) . Thus, on t h e b a s i s o f Lemma 7 . 1

( c f . a l s o Lemma 2 . 1 and C o r r o -

l a r y 2 . 1 ) one g e t s , i n p a r t i c u l a r , the re2atioi:s

- m(;,)

-.

(7.19)

1 3 m(Ea) lim hole0

=

m(i)= m(E).

The l a s t two r e l a t i o n s are vaZid w i t h i n continuous b i j e c t i o n s , for every l o c a l l y m-convex algebra E , w i t h r e s p e c t t o a n y Arens-Michael d e c o m p o s i t i o n of it. In t h i s respect,

(6.24)

s h o u l d b e compared t o t h e a n a l o g o u s re-

l a t i o n ( 7 . 1 6 ) of t h e p r e v i o u s lemma, b o t h of these r e l a t i o n s providing con-

tinuous b i j e c t i o n s . On t h e

o t h e r h a n d , t h e p o s s i b i l i t y of b e i n g ( 7 . 1 6 ) a

homeomorphism c o u l d b e examined i n a s i m i l a r way a s i n Lemma 6 . 2 . T h u s one c o u l d t h i n k , f o r e x a m p l e , of a denumerable p r o j e c t i v e l i m i t a l g e b r a of F r 6 c h e t a l g e b r a s , f o r t h e s p e c t r u m of which a s u i t a b l e k - c o v e r i n g f a m i l y c o u l d be d e f i n e d .

8. Appendix: G e n e r a l i z e d spectrum For l a t t e r u s e concerning, i n p a r t i c u l a r , t o p o l o g i c a l t e n s o r p r o d u c t s o f t o p o l o g i c a l a l g e b r a s (see P a r t I1 of t h i s b o o k ) , w e exami n e i n t h i s a p p e n d i x t h e c a s e t h a t t h e c h a r a c t e r s of t h e t o p o l o g i c a l a l q e b r a s i n v o l v e d t a k e t h e i r v a l - J e s from a n o t h e r f i x e d t o p o l o g i c a l a l g e b r a which t h u s r e p l a c e s t h e complex numbers f i e l d 6. 8.(1).

General theory.-

W e start with the following

8.

177

GENERALIZED SPECTRUM

Definition 8.1. L e t E l F b e t o p o l o g i c a l a l g e b r a s . A n o n - i d e n t i c a l l y z e r o c o n t i n u o u s a l g e b r a morphism of E i n t o F i s c a l l e d a generalized ( v i z . , F - v a l u e d ) character of E ( w i t h r e s p e c t t o t h e g i v e n t o p o l o g i c a l algebra F)

.

The s e t Horn ( E , F )

of a l l s u c h c h a r a c t e r s , e q u i p p e d w i t h

t h e t o p o l o g y of s i m p l e c o n v e r g e n c e i n E , i s d e f i n e d t o b e t h e general-

i z e d spectrum o f E ( r e l a t i v e t o F ) , and w e d e n o t e it by Hums(E, F l . Thus, one h a s by d e f i n i t i o n t h e r e l a t i o n Horn IE, FI

(8.1)

s

L n IE, FI ,

t h e s e c o n d member d e n o t i n g t h e s p a c e of c o n t i n u o u s l i n e a r maps of

E

i n t o F w i t h t h e r e s p e c t i v e topology of simple convergence i n Z . I n t h i s c o n c e r n , i t i s c e r t a i n l y c l e a r t h a t t h e f i r s t m e m b e r of

( 8 . 1 ) may

iiappen t o b e , i n g e n e r a l , t h e empty s e t . On t h e o t h e r h a n d , t h e n o t i o n of t h e extended generalized spectrum of E i s i m m e d i a t e l y u n d e r s t o o d i n a n a l o g y w i t h t h e p r e v i o u s r e l a t i o n ( 8 . 1 ) , t h a t i s , o n e h a s , by d e f i n i t i o n , (8.2)

Uoms(E, FI'

:= Horns(E, FIU { O )

Ss(E, FI.

T h u s , by t h e f o r e g o i n g , and an o b v i o u s e x t e n s i o n of

(1.9) to the

case t h a t t h e complex numbers a r e r e p l a c e d by t h e p r e c e d i n g t o p o l o g i cal algebra F (cf. a l s o (1.10) )

,

one c o n c l u d e s t h a t

t h e extended generat-


Ss(E, FI.

Thus,

i n c a s e t h e g e n e r a l i z e d spectrum of E i s n o t a l r e a d y a c l o s e d s u b s e t

of

1: ( E , F) one g e t s the r e l a t i o n

(8.3) Now, i f t h e topotogica2 algebras E and F have i d e n t i t y elements ( h e r e u p o n

w e a l w a y s assume t h e a l g e b r a morphisms t o b e " i d e n t i t y p r e s e r v i n g " t s ( E , F). ( T h e r e f o r e , ( 0 ) i s an i s o l a t e d p o i n t of U o m s ( E , F i t ; i n f a c t , w e h a v e {O} = Homs(E, F ) + n C U o r n s ( E , FI).

maps) Uorns(E, FI i s a c l o s e d subset of

N e x t , w e c o n s i d e r t h e r e l a t i o n between t h e s p a c e s HomsIE, F I ,

denotes t h e corqtetion

where

Homs(E, F ) a n d

of t h e g i v e n t o p o l o g i c a l a l g e -

b r a E . I n t h i s r e s p e c t , w e want t o a v o i d t e r m i n o l o g i c a l c o m p l i c a t i o n s i n case the multiplication i n ( i f , of c o u r s e , Cf.

2

2

is, eventually, separately continuous

t h e r e does e x i s t a t a l l , as a t o p o l o g i c a l a l g e b r a !

C h a p t . I ; S e c t i o n 4 ) . However, i n t h a t c a s e t o o , t h e r e l e v a n t r a m i -

f i c a t i o n s c o u l d b e i n d i c a t e d by t h e a n a l o g o u s c o n s i d e r a t i o n s i n Sect i o n 2 . T h u s , w e r e s t r i c t o u r a t t e n t i o n t o t h e c a s e t h a t the topoZogi-

c a l algebras involved have continuous m u l t i p l i c a t i o n . (Complete t o p o l o g i c a l a l g e b r a s t h e r e d o e x i s t , however, a n d a r e c e r t a i n l y of i m p o r t a n c e ! ) . So s u p p o s e t h a t E and F a r e topologica2 algebras w i t h continuous mu&

v

178

SPECTRIM (GLOBAL THEORY)

tiplieation, t h e algebra F being ing t h e completion

;o f

Hums

(8.4)

m o r e o v e r , complete. Then, b y c o n s i d e r -

E , one g e t s t h e r e l a t i o n

(f,F)

= Homs(E, F) ,

w i t h i n a b i j e c t i o n , d e f i n e d b y t h e n a t u r a l " e x t e n s i o n by c o n t i n u i t y " of t h e continuous ( g e n e r a l i z e d ) c h a r a c t e r s involved (see a l s o ( 2 . 1 )

,

as

w e l l a s N . BOURBAKI [4: Chap. 3 ; p. 521). F u r t h e r m o r e , on t h e b a s i s of t h e

(see a l s o Lemma V 1 : 5 . 1 ) , t h e b i j e c t i o n , g i v e n by (8.41, i s a continuous map.

a n a l o g o u s argument u s e d i n Lemma 1 . 1 cludes t h a t

o n e con-

Nod, i n a n a l o g y w i t h t h e s i t u a t i o n d e s c r i b e d i n S e c t i o n 2 , w e f u r t h e r c o n s i d e r t h e p o s s i b i l i t y of b e i n g t h e p r e c e d i n g c o n t i n u o u s b i j e c t i o n a homeomorphism between t h e r e s p e c t i v e t o p o l o g i c a l s p a c e s . So t h e f o l l o w i n g n o t i o n ( w h i c h n a t u r a l l y e x t e n d s t h a t of D e f i n i t i o n 1 . 2 ) w i l l b e of i m p o r t a n c e .

D e f i n i t i o n 8.2. Given t h e t o p o l o g i c a l a l g e b r a s E and F , w e s a y t h a t t h e g e n e r a l i z e d s p e c t r u m of E , HumsiE, F) , i s l o c a l l y equicontinuous if e v e r y e l e m e n t U E Hum (E, F) h a s a n e i g h b o r h o o d i n Hum (E, F) which i s an e q u i c o n t i n u o u s s u b s e t of Horn ( E , F) C S (B,F). The s i g n i f i c a n c e of t h e p r e c e d i n g c o n c e p t , i n c o n n e c t i o n w i t h o u r main t h e o r e m b e l o w , l i e s i n e f f e c t i n t h e f o l l o w i n g

Lemma 8.1. Let E and F be topological v e c t o r spaces. Moreover, l e t L ( E , F ) be t h e space of continuous l i n e a r maps o f E i n t o F and M C I^ ( T , F) an zquicsi.tiouous subset. Then, t h e following two t o p o l o g i e s on M are i d e n t i c a l .

1 ) The topology of simple convergence in a t o t a l subset o f E

( a s u b s e t of

E whose c l o s e d l i n e a r h u l l e q u a l s E ) . :)

The topology of simple convergence i n E.

Proof. C f . N . BOURBAKI [7: Chap. 3 ; p . 2 3 , P r o p o s i t i o n 51

.I

Thus, w e now have t h e f o l l o w i n g .

Theorem 8.1. Let E and F be topological algebras having continuous m u l t i p l i c a t i o n , w i t h t h e algebra F being complete. Moreovet', suppose t h t t h e generalized spectrum o f E w i t h respect t o F , Honis(E,F) is l o c a l l y equicontinuous. Then, one gets the relation (8.5)

H u m ( E , F) = How

(E,

F),

w i t h i n a homeomorphism of the r e s p e c t i v e topological spaces.

-

Proof. C o n s i d e r t h e c a n o n i c a l b i j e c t i o n (8.6)

j : tlonis (E,FI

Hums

(E,

F) ,

8.

g i v e n by ( 8 . 4 ) ;

GENERALIZED SPECTRUM

179

it h a s a l r e a d y b e e n remarked t h a t i t s i n v e r s e i s a

c o n t i n u o u s map. NOW, w e s h a l l show t h a t j i s continuous t o o : I n d e e d , i f u

i s a n e l e m e n t of Homs(E, FI t h e n t h e r e e x i s t s , by h y p o t h e s i s , an open

e q u i c o n t i n u o u s n e i g h b o r h o o d 'v' o f u i n Homs(E, FI ( D e f i n i t i o n 8 . 2 ) ,

so

t h a t (Lemma 8 . 1 ) t h e two t o p o l o g i e s j-nduced on V from t h e s p a c e s

Hum 1 3 , P I and ffom (2,F) a r e , i n f a c t , i d e n t i c a l . Thus, t h e r e s t r i c t i o n of j t o t h e open s e t VEHorn I E , Fl i s c o n t i n u o u s , and t h i s p r o v e s t h e assertion. I

Remark 8.1.- I t i s a c o n s e q u e n c e of t h e p r e c e d i n g p r o o f ( c f . a l s o t h a t of Lemma 2 . 2 ) t h a t , w i t h i n t h e same c o n t e x t a s t h a t of t h e above Theorem 8 . 1 , one c o n c l u d e s t h a t : The generalized spectrum Hom,,(E, F) i s l o c a l l y equicontinuous if, and only if, t h i s i s t h e case f o r Honis(E, FI. In t h i s regard, we further note t h a t

t h e argument appZied i n t h e

proof of Theorem 8 . 1 could a l s o be used f o r t h e anaZogous r e s u l t of Theorem 2 . l . W e f o l l o w e d , however, a d i f f e r e n t a p p r o a c h b a s e d on Lemma 2 . 3 ,

since it

exhibited, a t l e a s t , a special interest i n i t s e l f .

Generalized spectrum o f a topological projective limit algebra.- W e c o n s i d e r n e x t t h e case of a p r o j e c t i v e l i m i t t o p o l o g i c a l a l g e b r a E = 8.(2).

1 S E

a

and i t s g e n e r a l i z e d s p e c t r u m , v i z . ,

t h e extension within t h e

p r e s e n t framework of o u r p r e v i o u s c o n s i d e r a t i o n s c o n c e r n i n g t h e a l g e bra (7.2). I n p a r t i c u l a r , suppose t h a t w e a r e g i v e n

a dense p r o j e c t i v e l i m i t

topological algebra (8.7)

E = l ~ ( U c u , fcla

( c f . t h e p r e v i o u s S e c t i o n 7 ) and a f i x e d t o p o l o g i c a l a l g e b r a F. Then, b y a p p l y i n g s i m i l a r t h o u g h t s t o t h o s e of S e c t i o n 7 , one o b t a i n s f o r

t h e corresponding g e n e r a l i z e d s p e c t r a (w t h r e s p e c t t o F ) of t h e a l g e b r a s u n d e r c o n s i d e r a t i o n a n i n d u c t i v e system of topological spaces (8.8)

(

3

F)9

ts, 6 - foa i.

So l e t

H = l$Homs

(8.9)

(Ea, F )

b e t h e C o r r e s p o n d i n g i n d u c t i v e l i m i t ( t o p o l o g i c a l ) space. S i m i l a r l y , assumi n g f u r t h e r t h a t E i s a s t r i c t l y dense p r o j e c t i v e l i m i t algebra one g e t s a continuous map (8.10)

4 :H = lim Hum (Ea, F) + s

-

Horn (E, F ) ,

(cf. (7.7)),

180

V SPECTRUM (GLOBAL THEORY)

which i s one-to-one; m o r e o v e r , f o r topological algebras w i t h continuous m u l t i p l i cation (8.101 i s an onto map ( h e n c e , a b i j e c t i o n ) . I n t h i s r e s p e c t , t h e ext e n s i o n t o t h e p r e s e n t c a s e of t h e a n a l o g o u s argument a p p l i e d i n Sect i o n 7 i s , of c o u r s e , s t r a i g h t f o r w a r d . Thus, w e o b t a i n

Lemma 8.2. Let E = l*Ea

be a s t r i c t l y dense p r o j e c t i v e l i m i t algebra and

F a given topological algebra, the m u l t i p l i c a t i o n i n a l l the topological algebras

involved being assumed continuous. Then, concerning t h e generalized spectra ( w i t h respect t o F) of the algebras E and Ea (8.11)

, a € I (cf.

(8.7 1 , one has the r e l a t i o n

Horn (E, F) = l ~ H o n r s ( E a , F),

within a b i j e c t i o n , having a continuous inverse ( c f . ( 8 . 1 0 ) ) . I A possible strengthening of

( 8 . 1 1 ) t o a homeomorphism, by s u i t -

a b l y r e s t r i c t i n g t h e t o p o l o g i c a l a l g e b r a s involved, could b e considere d i n a n a l o g y w i t h t h e c o n c l u d i n g r e m a r k s of t h e p r e v i o u s S e c t i o n 7 .

8 . ( 3 ) . Generalized spectrum o f a topological inductive limit algebra.- n'sxt w e c o n s i d e r a " d u a l i z a t i o n " of t h e p r e v i o u s s e c t i o n , as it i s t h e gene r a l i z e d spectrum o f a t o p o l o g i c a l i n d u c t i v e l i m i t a l g e b r a E = 1 2 E a ( c f . Chapt. I V ; S e c t i o n 2 )

.

I t c o n c e r n s , i n f a c t , a " v e c t o r i z a t i o n " of o u r p r e v i o u s

s i d e r a t i o n s i n S e c t i o n 3 , while C o r o l l a r y 3.1 t h e extended generalized spectrum of E

con-

is actually referred t o

i n t h e s e n s e of t h i s s e c t i o n ( c f .

( 3 . 2 1 ) ) . O f c o u r s e , it i s c l e a r t h a t t h e same r e l a t i o n ( 3 . 2 1 ) r e d u c e s t o generalized spectra (Definition 8.1),

i n case the topological a l -

g e b r a s c o n s i d e r e d have i d e n t i t y e l e m e n t s . On t h e o t h e r h a n d , t h e g e n e r a l case of " g e n e r a l i z e d s p e c t r a " ( n o i d e n t i t y e l e m e n t s a r e n e c e s s a r i l y p r e s e n t ) c a n s i m i l a r l y b e exami n e d , a s t h a t of t h e u s u a l ( i . e . , " n u m e r i c a l " )

s p e c t r a , by an a p p r o -

p r i a t e f o r m u l a t i o n , w i t h i n t h e p r e s e n t c o n t e x t , of t h e r e l s . ( 3 . 2 4 . 1 ) and ( 3 . 2 9 . 1 ) . Thus, a similar r e l a t i o n us ( 3 . 3 1 ) can be derived. F u r t h e r m o r e , one c a n a l s o c o n c l u d e f o r t h e e x t e n d e d c a s e cons i d e r e d h e r e t h e c o r r e s p o n d i n g r e s u l t t o P r o p o s i t i o n 3.1.

181

The G e l 'fand M a p

CHAPTER V I

1. Continuity of the Gel'fand map We have already defined the notion of the Gel'fand map of a given topological algebra E (Chapt. I; Section 8 and Chapt.V; (1.3), ( 1 . 5 ) ) , as well as the GeZ'fand transform of an element x e E , that is, the value

2 , obtaining thus a continuous

at x of the Gel'fand map, denoted by

(complex-valued) map on the spectrum of E (ibid.). Now, our main objective in the following lines is to examine t h e c o n t i n u i t y o f t h e Gel'fand map (Chapt.V; (1. 5 ) ) when i t s range, i.e., the algebra C I m ( E I J , c a r r i e s t h e topology o f compact convergence (cf.Chapt. V; (1.4)j.

Namely, the continuity of the map

5 : E-+

(1.1)

cc(m(E))

for a given topological algebra E l where

[ $? (x)]( p i = a f i

(1.2)

:= f

,

(5)

for any x € E and f e m(E).Furthermore, we also consider certain basic consequences on F;' which this property entails, in connection with the terminology applied hitherto. Further applications of the same notion will also be exhibited in subsequent sections (see, for instance,Chapter VIII). Thus, we first have the following general result which is of importance for the sequel. In this concern, the terminology applied in Chapt.V;(3.21) seems to be the appropriate one for this type of results, and it has been also systematically applied in the last appendix in the previous chapter.

Lemma 1.1. (1.3)

Let E , F be topological algebras and

Hams ( E , F l

t h e space of aZl non--7sro continuous (algebra) morphisms o f E i n t o F , endowed w i t h t h e topoZogy o f simple convergence i n E ("generalized spectrum" of E with respect to F ; cf. Definition V;8.1 ) . Furthermore, consider t h e map

VI ?'HE GEL'FAND MAP

182

g F : E -C(Hams(E,

F), F ) : x

(1.4) u

-[$FIXI](U1

= G(ui:=

-

@x):

UIXi.

Then, i t s range i s contained i n t h e s e t ( a l g e b r a ) o f a l l F-valued continuous maps on t h e space (1.31. Finally, l e t

-

be a f a m i l y o f s u b s e t s o f H u m f E , F I .

sF

:E

(1.5)

Then, t h e map

CG((Hams(E, F), F),

d e f i n e d by ( 1 . 4 ) , i s continuous ( i t s r a n g e b e i n g e q u i p p e d w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e members o f G I

only i f , t h e f a m i l y

"

- topology

"

) i f , and

c o n s i s t s of equicontinuous s u b s e t s o f H a m ( E , F ) .

Proof. I t i s a n immediate c o n s e q u e n c e of t h e r e s p e c t i v e d e f i n i -

t i o n s t h a t t h e r a n g e of

S F, defined

b y ( 1 . 4 ) , i s t h e s p a c e of F-va-

l u e d c o n t i n u o u s maps on t h e s p a c e ( 1 . 3 ) . On t h e o t h e r h a n d , one g e t s by d e f i n i t i o n of t h e G - t o p o l o g y

where

Cu(A,FI

d e n o t e s t h e r e s t r i c t i o n of t h e r a n g e of

FF

ffam ( E , F ) endowed w i t h t h e 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

to

A

=

i n A; fur-

( I . 6 ) i n d i c a t e s t h e i n i t i a l topology on (cf. t h e f i r s t member d e f i n e d on i t by t h e s p a c e s C u ( A , F ) , w i t h A e

t h e r m o r e , t h e s e c o n d m e m b e r of

sF

i s continua l s o N . BOURBAKI [5: Chap. 1 0 ; p. 1 0 , D e f i n i t i o n 2 1 ) . Thus, o u s i f , and o n l y i f , t h e map i A o S F of t h e f o l l o w i n g commutative d i a gram i s c o n t i n u o u s , f o r e v e r y

AEG.That is, we

have

(1.7)

NOW, the l a s t c o n d i t i o n i s e q u i v u l e n t w i t h t h a t o f being A G H a m ( E , F l an equicontinuous s e t , f o r every A

€

( c f . t h e l a s t Ref.

a b o v e ; p . 23, C o r o l l a i r e 1) ,

and t h i s c o m p l e t e s t h e p r o o f . 1 NOW, t h e n e x t r e s u l t i s a d i r e c t c o n s e q u e n c e o f t h e f o r e g o i n g

( f o r F = @ , and

6=

t h e f a m i l y o f compact s u b s e t s o f m ( E I ) and t h e re-

s p e c t i v e d e f i n i t i o n s c o n c e r n i n g e q u i c o n t i n u o u s sets. T h a t i s , one h a s .

Theorem 1.1. Let E be a t o p o l o g i c a l algebra whose s p e c t m i s m ( E 1 . Moreover, l e t

1.

CONTITKIITY OF GEL’FAND MAP

183

(1.8)

be t h e r e s p e c t i v e Gel’fand map of E ( d e f i n e d b y ( 1 . 2 ) ) whose range i s endowed w i t h t h e topology of compact convergence i n mfE).Then t h e foLlowing a s s e r t i o n s are e q u i v a l e n t : 1 ) The map

5 i s continuous.

2) Every compact subset of ?nisi i s equicontinuous. 3 ) Every r e l a t i v e l y compact subset of m(E)i s equicontinuous. I

On t h e o t h e r h a n d , s i n c e e v e r y compact s u b s e t of t h e s p e c t r u m i s a f o r t i o r i bounded, one o b t a i n s .

Corollary 1.1. Let E be a s p e c t r a l l y b a r r e l l e d algebra ( c f . D e f i n i t i o n V ; 1 . 3 ) . Then, E has a continuous Gel’fand map. 8 The n e x t c o r o l l a r y i s , i n f a c t , a s t r e n g t h e n i n g of t h e above Theorem 1 . 1 , i n c a s e o n e c o n s i d e r s t o p o l o g i c a l a l g e b r a s w i t h G e l ‘ f a n d maps c o n t i n u o i s .

Corollary 1.2. L e t

E be a topological algebra, having t h e r e s p e c t i v e Gel’-

fand map continuous. Then, t h e spectrum of E is l o c a % l y equicontinuous i f , and o n l y i f , i t is l o c a l l y compact. I n p a r t i c u l a r

( c f . c o r o l l a r y 1 . 1 ), t h e a s s e r t i o n holds

t r u e f o r s p e c t r a l l y b a r r e l l e d algebras.

Proof. The c o n d i t i o n i s , of c o u r s e , n e c e s s a r y f o r e v e r y t o p o l o g i c a l a l g e b r a ( c f . Theorem V; 1 . 1 ) . On t h e o t h e r h a n d , i f mfE) i s l o c a l l y compact, t h e n i t i s a l s o l o c a l l y e q u i c o n t i n u o u s a s f o l l o w s from t h e h y p o t h e s i s and Theorem 1.1.1 A s a m a t t e r of

f a c t , l o c a l e q u i c o n t i n u i t y of t h e s p e c t r u m of a

g i v e n t o p o l o g i c a l a l g e b r a i m p l i e s a l r e a d y t h e c o n t i n u i t y o f t h e res p e c t i v e G e l ‘ f a n d map of t h e a l g e b r a . But w e f i r s t n e e d t h e f o l l o w i n g . Lemma 1.2.

Let A G f f v m s f E , F ) be a l o c a l l y equicontinuous subset o f t h e space

( 1 . 3 ) ( c f . a l s o D e f i n i t i o n V ; 6 . 2 ) . Then, t h e map

(1.9)

i s continuous when A c a r r i e s t h e r e l a t i v e topology from ( 1 . 3 1 . Proof. I f

(x,

, f,

continuous neighborhood (1.lo) i.e.,

)e E

x A,

t h e r e e x i s t s by h y p o t h e s i s a n open equi-

s a y U , of f ,

6 :E x U - F

i n AEHvms(E, FI s u c h t h a t t h e map

: fx, j)+-+

$(x, f ) = f ( x ) ,

t h e r e s t r i c t i o n o f 41 ( c f . ( 1 . 9 ) ) on E X U I i s continuous ( s e e N . BUUR-

184

THE GEL’FAND MAP

VI

BMI [5:Chap. 10; p. 25, C o r o l l a i r e ] ) which a c t u a l l y p r o v e s t h e a s s e r t i 0 n . d A s a n immediate c o n s e q u e n c e o f t h e p r e c e d i n g lemma one o b t a i n s .

Theorem 1.2. Let E be a topological algebra whose s p e c t m i s ??T(E) i s loc a l l y equicontinuous. Then, t h e respective Gel’fand map of E is continuous. Proof. I t i s c l e a r t h a t t h e a s s e r t i o n f o l l o w s from t h e p r e v i o u s

Lemma 1 . 2 , b y t a k i n g F = @ and A = m(E);thus,

t h e G e l ’ f a n d map of E be-

comes, i n f a c t , t h e p a r t i a l map c o r r e s p o n d i n g , f o r any f i x e d r e E l (1.9).

to

B u t , b y t h e same lemma, t h e l a t t e r map i s c o n t i n u o u s . 1 The f o l l o w i n g p r o v i d e s , a t l e a s t , a c o n v e r s e t o t h e s i t u a t i o n

d e s c r i b e d by Lemma 1 . 2 and Theorem 1 . 2 GUNDJI [l : p. 261 , Theorem 3 . I ] )

. Thus , w e

(see a l s o , f o r example, J . D U have.

Corollary 1.3. Let E be a topological algebra whose spectrum i s ??Z(E).Then, the following two statements are equivalent: 1) m ( E ) i s l o c a l l y equicontinuous. 2) m ( E /

i s l o c a l l y compact and the Gel’fand map of E i s continuous.

Proof. The a s s e r t i o n 1 ) = > 2 ) f o l l o w s a l r e a d y from Theorem V ; 1 . 1 and t h e above Theorem 1 . 2 .

Furthermore, under t h e h y p o t h e s i s t h a t t h e

G e l ’ f a n d map o f E i s c o n t i n u o u s , t h e p r o p e r t y t h a t

m(E) t o b e l o c a l -

l y compact i s a l r e a d y e q u i v a l e n t w i t h t h a t of b e i n g i t l o c a l l y e q u i c o n t i n u o u s ( C o r o l l a r y 1 . 2 ) , and t h i s t e r m i n a t e s t h e p r o o f . On t h e o t h e r h a n d , t h e f o l l o w i n g i s a d i r e c t c o n s e q u e n c e o f t h e

same d e f i n i t i o n s . Namely, w e h a v e .

Corollary 1.4. Every topological algebra w i t h an equicontinuous spectrum, hence any &-algebra ( P r o p o s i t i o n 11;I . 11, is s p e c t r a l l y barrelled having thus

. .

the respective Gel’fand map continuous ( C o r o l l a r y 1 1 ) I n p a r t i c u l a r , t h e c l a s s of

l o c a l l y m-convex algebras w i t h Gel’fand

maps continuous e x h i b i t s s e v e r a l b a s i c p r o p e r t i e s a l r e a d y f a m i l i a r from t h e c l a s s i c a l t h e o r y of Banach a l g e b r a s , o r y e t f r c m t h a t of t h e more g e n e r a l class of

FrEchet Q- ( l o c a l l y m-convex) algebras (see E . A . MICHAEL

[ l ] ) . The l a t t e r c a s e i s a c t u a l l y c l a r i f i e d by t h e p r e v i o u s c o r o l l a r y i n c o n j u n c t i o n w i t h t h e e n s u i n g d i s c u s s i o n . W i t h i n t h e same c o n t e x t ,

w e s h a l l also need t h e f o l l o w i n g terminology.

Definition 1 . 1 .

So w e s e t

A g i v e n t o p o l o g i c a l a l g e b r a E i s s a i d t o b e a bound-

ed algebra, i f t h e r a n g e of t h e r e s p e c t i v e G e l ’ f a n d map o f E ( c f . ( 1 . I ) ) i s c o n t a i n e d i n t h e set ( a l g e b r a ) of a l l bounded ( c o n t i n u o u s , complex-

v a l u e d ) f u n c t i o n s on 7??(E).Namely, t h e map

1.

CONTINUITY OF GEL'FAND MAP

g ( x 1 = x : m(E/

(1.11)

+

185

C

d e f i n e d by ( 1 . 2 ) i s a ( c o n t i n u o u s ) bounded f u n c t i o n on ?7Y(E) , f o r e v e r y

x

E

E

.

E q u i v a l e n t l y t h e Gel'fand transform algebra E A =

3 (El G c ( ? ? L ( E ) )

( c f . Chapt.V; ( 1 . 6 ) ) of E c o n s i s t s o f complex-valued continuous bounded func-

t i o n s on m(E); i . e . , one h a s t h e r e l a t i o n (1.12)

E* = ~ ( E ) G E ( v Y ( E ) )

(where t h e l a s t set of t h e p r e v i o u s r e l a t i o n d e n o t e s t h e a l g e b r a o f bounded f u n c t i o n s , a s a b o v e ) . Remark 1.1.-

I t i s known t h a t , f o r any g i v e n l o c a l l y convex s p a c e

E , one h a s t h e r e l a t i o n

IE'I'=E,

w i t h i n a v e c t o r s p a c e isomorphism(cf.,

f o r example, N . BOURBAKI [7: Chap. 4 ; p . 50, P r o p o s i t i o n 1 1 ) . Thus, it i s a n immediate c o n s e q u e n c e o f t h e p r e v i o u s D e f i n i t i o n 1 . 1 t h a t a given

l o c a l l y convex algebra E i s a bounded algebra i f , and only i f , i t s spectrum m ( E l is a bounded subset of E,' . C o n s e q u e n t l y , on t h e b a s i s o f t h e n e x t Theorem 1 . 3 , o n e o b t a i n s f o r s p e c t r a l l y b a r r e l l e d l o c a l l y in-convex

algebras a

s t r e n g t h e n e d form o f Lemma I I I i 6 . 1 . On t h e o t h e r h a n d , i f E i s a c o m t a t i v e a d v e r t i b l y complete l o c a l l y

in-convex algebra, t h e n t h e s p e c t r a l r a d i u s of a n e l e m e n t

X E

E

i s g i v e n by

the relation (1.13)

( c f . Chapt.111; ( 6 . 1 6 ) ) . T h e r e f o r e I i f E = B ( E ) , i . e . , element x

€

E ( c f . 111; ( 6 . 4 ) )

,

t h e given algebra

r E ( x ) < + m ,f o r every

E i s a bounded algebra i n

t h e s e n s e o f t h e above D e f i n i t i o n 1 . 1 . F u r t h e r m o r e ,

(1.12) certainly

i m p l i e s t h a t E = 81E) s o t h a t , i n t h e c a s e u n d e r c o n s i d e r a t i o n , the two

notions c o i n c i d e , a u s e f u l f a c t i n c o n n e c t i o n w i t h Lemma 111; 6 . 1 . Thus, w e come n e x t t o t h e f o l l o w i n g . Theorem 1.3. Let E be a given topological algebra whose s p e c t m i s m(E).

Moreover, consider t h e next propositions: *

1 ) The completion E of E i s a &-algebra.

2) m(E) i s equicontinuous. 3) m(E) i s a r e l a t i v e l y compact subset of 41 n^r(El+ i s a compact subset of

E,'

E,'.

.

51 E i s a bounded algebra. Then, one has the following i m p l i c a t i o n s : (1.14)

11=>21=> 3 ) = > 4 1 = > 5 1 .

In p a r t i c u l a r , i f t h e Gel'fand map of E i s continuous, then 4 ) implies that

186

VI THE GEL'FAND MAP

m(E)i s l o c a l l y equicontinuous. On t h e o t h e r hand, i f E i s a c o m t a t i v e l o c a l l y

m-convex algebra, then 2) => 1 ) a s w e l l , so t h a t i n case E i s , i n p a r t i c u l a r , a commutative s p e c t r a l l y b a r r e l l e d l o c a l l y m-convex algebra, then a l l t h e above f i r s t f o u r a s s e r t i o n s are equivalent. F i n a l l y , i f E i s a c o m u t a t i v e a d v e r t i b l y complete ( D e f i n i t i o n I ; 6 . 4 ) s p e c t r a l l y b a r r e l l e d l o c a l l y m-eonvex algebra, then a l l t h e above f i v e a s s e r t i o n s are equivalent. Proof.

s e t of

If

2

i s a & - a l g e b r a , t h e n m(El i s a n e q u i c o n t i n u o u s sub, s o t h a t t h e same i s t r u e f o r ??Z(E) ( c f . h e n c e 1 ) = = > 2 ) .Moi-eover, f o r a commutative l o c a l l y

i'(Proposition 1;7.1)

Scholium V ; 2 . 2 ) ,

m-convex a l g e b r a E , 2 ) = > 1 ) a s w e l l by t h e same S c o l i u m V ; 2 . 2 ( ~ e m m a ) . NOW, t h e i m p l i c a t i o n s 2 ) = > 3 ) = > 4 ) a r e a c o n s e q u e n c e o f t h e A l a o g l u -

B o u r b a k i Theorem a p p l i e d t o

m(E) and

V ; ( 1 . 1 1 ) . F u r t h e r m o r e , one cer-

t a i n l y h a s t h a t 4) =>5) s i n c e , f o r e v e r y

3: E

E , the

map $ : m(E)'.

C,

w i t h & O ) = 0 , i s by h y p o t h e s i s bounded, whence a f o r t i o r i i t s r e s t r i c t i o n t o m(El; i.e., t h e u s u a l G e l ' f a n d t r a n s f o r m

2:

??Z(E)-

C of

X ,

which p r o v e s t h e f i r s t p a r t o f t h e t h e o r e m . Now, i f t h e G e l ' f a n d map o f E i s c o n t i n u o u s , t h e n 4 ) i m p l i e s t h a t m(El i s l o c a l l y compact ( c f . V ; ( 1 . 1 1 ) )

, so

t h a t it i s a l s o l o c a l -

l y e q u i c o n t i n u o u s ( C o r o l l a r y 1 . 3 ) . Moreover, w e h a v e a l r e a d y remarked t h a t , i n c a s e of a commutative l o c a l l y m-convex a l g e b r a E , 2 ) = > 1 ) ; so i f E i s a l s o s p e c t r a l l y b a r r e l l e d ( w i t h r e s p e c t t o m(E);c f . D e f i n i t i o n V; 1 . 3 ) , t h e n it i s a l s o s u c h r e l a t i v e t o m(E)+(Remark V; 6 . 1 ) . T h e r e f o r e , 4 ) i m p l i e s t h a t m(EICi s e q u i c o n t i n u o u s t i o r i ??Y(El

, i.e. , 4)=>2),

and hence a f o r -

so t h a t i n t h e c a s e u n d e r c o n s i d e r a t i o n t h e

f i r s t four propositions are a l l equivalent. F i n a l l y , i f , i n a d d i t i o n t o t h e l a s t hypothesis above, t h e a l g e b r a E i s a l s o a d v e r t i b l y c o m p l e t e , t h e n by a s s u m i n g 5 ) o n e c o n c l u d e s t h a t ? ? T I E ) i s e q u i c o n t i n u o u s ( c f . t h e above Remark 1 . 1 ) . T h u s , t h e

s a m e h o l d s t r u e f o r m i k ) ( c f . S c h o l i u m V ; 2 . 2 ) , a n d h e n c e E i s a &-algebra ( C o r o l l a r y 111; 6 . 5 , t h e g i v e n a l g e b r a E b e i n g a l s o a & - a l g e b r a , b y h y p o t h e s i s and t h e same c o r o l l a r y ) . T h a t i s 5 ) = > 1 ) a s w e l l , and t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . I The f o l l o w i n g p r o v i d e s a p a r t i c u l a r u s u f u l i n s t a n c e of t h e above r e s u l t , i n c a s e t h e a l g e b r a s i n v o l v e d h a v e i d e n t i t y e l e m e n t s . Namely, one h a s

Corollary 1.5. Let E be a topological algebra w i t h an i d e n t i t y element and spectrum ??Z(E). Moreover, consider t h e following statements: 1 ) The completion

2 of

E i s a &-algebra.

1.

CONTINUITY OF GEL'FAND MAP

21 ??Y ( E l i s equicontinuous. 31 T?Y(E) i s a compact subset

of E,'

.

41 E i s a bounded algebra.

Then, one has the following i m p l i c a t i o n s : ( 1 -15)

1)=>2)

*Z)

=>4).

I n p a r t i c u l a r , i f E i s a commutative a d v e r t i b l y complete l o c a l l y m-convex algebra w i t h Gel'fand map continuous, then t h e above f i r s t t h r e e a s s e r t i o n s are equivalent ( w h i l e t h e g i v e n a l g e b r a E i s a Q - a l g e b r a t o o ) . F i n a l l y , i f E i s a commutative a d v e r t i b l y complete s p e c t r a l l y barrelled l o c a l l y m-convex algebra, then a l l the preceding f o u r statements are equivalent ( i n t h e l a s t c a s e t o o , E h a s a l s o a c o n t i n u o u s G e l ' f a n d map; s e e

Corollary I . l ) . I

The f o l l o w i n g lemma i s needed below and h a s , i n f a c t , i . m p l i c i t l y been u s e d b e f o r e . Namely, w e have. Lemna 1.3. Let E be a topological aZgebra w i t h an i d e n t i t y element and spect r u m m ( E 1 . Moreover, consider t h e following a s s e r t i o n s : 1 ) E i s a Q-algebra.

2) m i E l i s equicontinuous.

31 m(E) is a compact subset of E,'

.

Then, one has the following i m p l i c a t i o n s : 1 ) => 2 ) => 3 ) .

(1.16)

I n p a r t i c u l a r , i f E i s a c o n m t a t i v e a d v e r t i b l y complete l o c a l l y m-eonvex algebra w i t h Gel'fand map continuous, then 31=> 1 ) as w e l l , so that a l l the precedi n g three a s s e r t i o n s are equivalent. fioof.

1 ) => 2 ) by P r o p o s i t i o n I ; 1 . 1

weakly c l o s e d s u b s e t of E,' Bourbaki) t h a t

, so

that since ? Z ( E ) i s a

( c f . RemarkV;I.l),

m(E) i s (weakly) compact, i . e . ,

it f o l l o w s ( A l a o g l u 2)=>3).

-

Now by assum-

i n g 3 ) one c o n c l u d e s , when t h e p a r t i c u l a r c a s e of t h e l a s t p a r t of t h e s t a t e m e n t i s c o n s i d e r e d , t h a t m ( E ) i s e q u i c o n t i n u o u s (Theorem 1 . 1 ) Thus, w i t h i n t h i s c o n t e x t ,

3)*1)

.

a s w e l l , by C o r o l l a r y 111; 6 . 5 , and

t h i s terminates the proof. I F u r t h e r a p p l i c a t i o n s of t o p o l o g i c a l a l g e b r a s h a v i n g c o n t i n u o u s Gel'fand maps o r , i n p a r t i c u l a r , o f s p e c t r a l l y b a r r e l l e d a l g e b r a s w i l l be

considered i n Chapter V I I I .

However, w e s t i l l g i v e one more d i r e c t

a p p l i c a t i o n of t h e above d i s c u s s i o n , c o n c e r n i n g i n p a r t i c u l a r Waelbroeckalgebras

( c f . D e f i n i t i o n 1 ; 4 . 1 ) . Thus, w e h a v e .

Theorem 1.4. Let E be a topologieal aZgebra w i t h an i d e n t i t y element and spectrum

m ( E ) . Furthermore, consider t h e folZowing statements:

VI THE GEL'FAND MAP

188

1 ) E i s a Waelbroeck algebra. 2 ) E i s a Q-algebra w i t h a continuous inversion.

m(E)i s a compact subset of E i .

31 E has a continuous Gel'fand map and

Then, one g e t s the following iniplications:

I)=> 2) - 3 ) .

(1.17)

In p a r t i c u l a r , i f E is a c o m u t a t i v e advertibly complete l o c a l l y m-convex algebra, then a l l the preceding three a s s e r t i o n s are equivalent. Proof. I t i s c l e a r t h a t 1)=>2) , by t h e same d e f i n i t i o n s ( c f . D e f i n i t i o n 11; 4 . 1 and t h e comment a f t e r i t ) , w h i l e 3 ) is t r u e f o r any Q-alg e b r a w i t h a n i d e n t i t y e l e m e n t f c f . C o r o l l a r y 1 . 4 and Theorem 1 . 1 ) . F i n a l l y , u n d e r t h e c o n d i t i o n s of t h e l a s t p a r t o f t h e t h e o r e m 3) 4 2 ) as w e l l , by t h e above Lemma 1.3 i n c o n n e c t i o n w i t h Lemma 1 1 ; 3.1 and t h i s terminates the proof. I W e c o n c l u d e t h i s s e c t i o n w i t h t h e f o l l o w i n g r e s u l t which w i l l

a l s o b e of u s e i n t h e s e q u e l ( c f . , f o r i n s t a n c e , C h a p t . XII; S e c t i o n 2)

.

Thus, w e h a v e .

4aI be an inductive system of topological aZgebras and the corresponding inductive l i m i t topoZogica2 algebra (Lemma IV; CI 2.2) such t h a t each one of the i n d i v i d u a l algebras E a , a € I , t o have t h e respective Lemma 1.4. Let I E a , f

E = 1 S E

Gel'fand map continuous. Then, t h e same holds t r u e f o r the algebra E i t s e l f . Proof. I t s u f f i c e s t o show (Theorem 1 . 1 )

t h a t any compact K G m ( E )

i s a l s o e q u i c o n t i n u o u s . T h u s , by Lemma V;3.2, t h i s i s r e d u c e d , i n f a c t ,

t o the verification that 'fCI(K) = Kof,={uof,

:uEK}

i s a n e q u i c o n t i n u o u s s u b s e t o f ? ? Z ( E C I ) + s E ~ ,f o r e v e r y u e I ( w e a p p l y h e r e t h e n o t a t i o n o f V;(3.32)). But t h i s i s a c t u a l l y t r u e by t h e cont i n u i t y of t h e maps t f , , C I C I , and t h e h y p o t h e s i s f o r t h e s e t K and t h e CI € 1 ( t h e same Theorem 1.1 ; see a l s o Remark V ; 6.1). I algebras E

a'

The a n a l o g o u s r e s u l t w i t h t h e p r e c e d i n g lemma i n c a s e of l o c a l l y convex a l g e b r a s ( c f . t h e same Lemma V ; 2.2) a n d / o r l o c a l l y m-convex ones ( D e f i n i t i o n IV;3.1) i s c e r t a i n l y clear.

2. Boundaries We d i s c u s s i n t h i s s e c t i o n , and w i t h i n t h e g e n e r a l framework of t h i s t r e a t i s e , t h e a n a l o g o n of a n i m p o r t a n t n o t i o n i n t h e t h e o r y of Banach a l g e b r a s , namely, t h a t of t h e

s i l o # boundary. However, t h e p a r t

o f t h e g e n e r a l t h e o r y of t o p o l o g i c a l a l g e b r a s r e f e r r e d t o t h i s n o t i o n ,

2.

189

BOUNDARY

i n a n a l o g y w i t h t h e c l a s s i c a l Banach a l g e b r a s t h e o r y c o u n t e r p a r t , i s s t i l l v e r y l i m i t e d . S i m i l a r r e m a r k s are a l s o i n f o r c e , c o n c e r n i n g t h e

a n a l o g o u s c o n c e p t w i t h i n t h e p r e s e n t framework

,

t o t h e Choquet boundary

which, however, w e a r e n o t g o i n g t o t o u c h upon i n t h e s e q u e l . I n s t e a d

w e r e f e r , f o r i n s t a n c e , t o t h e r e l e v a n t work o f W. DIETRICH, Jr. [3],where b o t h of t h e p r e c e d i n g n o t i o n s a r e t r e a t e d i n case of s u i t a b l e " f u n c t i o n a l g e b r a s " on c o m p l e t e l y r e g u l a r s p a c e s . W e s t a r t w i t h t h e f o l l o w i n g d e f i n i t i o n which i s a t t h e b a s i s of

t h e ensuing discussion.

Definition 2.1. ?7L(E).

L e t E b e a t o p o l o g i c a l a l g e b r a whose s p e c t r u m i s

A c l o s e d s u b s e t B of

a maximizing s e t ) f o r

E,

PZ(E) i s s a i d t o b e a boundary s e t ( o r e l s e 3: e E , t h e r e e x i s t s a n e l e m e n t f e

i f for e v e r y

B S V Z ( E / I such t h a t (2.1)

I n t h i s r e s p e c t , it i s c l e a r t h a t ( 2 . 1 )

implies t h e r e l a t i o n

(2.2)

f o r e v e r y e l e m e n t x e E , whenever a c l o s e d s e t

BC??Y(E)

i s a boundary

s e t f o r E. Of c o u r s e , t h e above two conditions are equivalent i n cuss the given

algebra E has a compact spectrum ( e . g .

, when

E i s a & - ( l o c a l l y m-convex) a l -

g e b r a w i t h a n i d e n t i t y e l e m e n t ) , a n i n s t a n c e which w i l l a l s o b e exami n e d below. NOW, m o t i v a t e d by t h e p r e v i o u s r e l a t i o n

(2.1)

and i n o r d e r t o

a v o i d a more c o m p l i c a t e d t e c h n i c a l l a n g u a g e , we shaZZ exclusiueZy consider

i n the sequez bounded uZgebras ( c f . D e f i n i t i o n 1 . I )

.

Thus , s t a t e d o t h e r w i s e ,

a b o u n d a r y set f o r a g i v e n (bounded) t o p o l o g i c a l a l g e b r a E is a c l o s e d subset, say B ,

of i t s s p e c t r u m m(E) s u c h t h a t , f o r e v e r y e l e m e n t X E E ,

t h e r e s p e c t i v e G e l ' f a n d t r a n s f o r m ( b e i n g by h y p o t h e s i s a bounded, and c o n t i n u o u s ( s e e V ; (1.4)), f u n c t i o n on m ( E l ) t o a t t a i n i t s maximum on B ( h e n c e , t h e t e r m i n o l o g y t h a t B i s a "maximizing s e t " ) . N o w t h e s m a l l e s t s u c h s e t i n m(E1 i s d e f i n e d t o b e t h e boundary of t h e g i v e n a l g e b r a E . That i s , w e set t h e following.

Definition 2.2. L e t E be a g i v e n bounded t o p o l o g i c a l a l g e b r a ( D e f i n i t i o n 1 . 1 ) whose s p e c t r u m i s m ( E I . T h e n t h e i n t e r s e c t i o n of a l l ( c l o s e d ) boundary s e t s f o r E (whenever t h i s s e t i s n o n - e m p t y ; c f . C o r o l l a r y 2 . 1 b e l o w ) , i . e . , t h e l e a s t b o u n d a r y set f o r E (see t h e same c o r o l l a r y ) , i s c a l l e d the boundary ( o r e l s e the SiZov boundary) of the aZge-

bra E , a n d i s d e n o t e d by

VI THE GEL'FAND MAP

190

I n t h i s c o n c e r n , one r e m a r k s t h a t i n c a s e

m(E) i s a

compact

s p a c e ( h e n c e , E i s a f o r t i o r i a bounded a l g e b r a ) , t h e n m(E) i s i t s e l f a boundary set f o r E ;

so i n t h a t case aE i s , i n f a c t , t h e nonempty i n t e r s e c t i o n o f a l l t h e boundary s e t s f o r E ( c f . Theorem 2 . 1 and Corollary 2 . 1 ) .

The f o l l o w i n g s e t w i l l f r e q u e n t l y b e c o n s i d e r e d i n

t h e s e q u e l ; namely, w e s e t , f o r e v e r y x € E ,

M(&) := { f €

(2.4) NOW,

m ( E l : /;if/ 1

=

Supl;(h/ h e nZ:(E)

I3.

it i s c l e a r t h a t , by t h e c o n t i n u i t y o f t h e map

t h e set ( 2 . 4 )

i s a c l o s e d s u b s e t of

2,

f o r every X E E ,

??'ZiEl; h e n c e , it w i l l a l s o b e com-

p a c t i n c a s e t h e a l g e b r a E h a s a compact s p e c t r u m . On t h e o t h e r h a n d , t h e n e x t " l o c a l d e s c r i p t i o n " of t h e boundary of a g i v e n t o p o l o g i c a l a l g e b r a w i l l p r e s e n t l y be a p p l i e d below.

Lemma 2 . 1 . Let E be a bounded topological algebra whose spectrum i s m(E), and l e t aE be i t s boundary ( D e f i n i t i o n 2 . 2 ) . Then, t h e following two a s s e r t i o n s are equivalent: 1 ) f eaECnrcE!.

m(E),there

2) For every open neighborhood V o f f i n E such t h a t (2.5)

Mi';)

Proof.

Suppose t h a t f e a E

e x i s t s an element

3:

C V.

and c o n d . 2 )

i s n o t t r u e ; t h a t i s , as-

sume t h a t t h e r e e x i s t s an open n e i g h b o r h o o d V of f i n m(E)s u c h t h a t

gin Cv = 0

(2.6)

f o r every

X E

,

E . So t h e c l o s e d s e t C V E M I E I

(Definition 2.1)

,

i s a boundary s e t f o r E f€aEGCV,

h e n c e by h y p o t h e s i s ( c f . D e f i n i t i o n 2 . 2 )

t h a t i s , a c o n t r a d i c t i o n , so 1 ) = > 2 ) .

F u r t h e r m o r e , by assuming cond. 2 )

f o r an e l e m e n t f EmiE), s u p p o s e a l s o t h a t f E aE.Thus,

there exists a

( c l o s e d ) boundary s e t B f o r E l w i t h fee.'; t h a t i s , one g e t s an open neighborhood o f f i n

m(E)w i t h t h e p r o p e r t y t h a t

~ ( $ 1n B # 0

(2.7)

,

f o r e v e r y e l e m e n t x E E , which i s a c o n t r a d i c t i o n t o ( 2 . 5 ) ,

so t h a t 2 )

= > 1 ) a s w e l l and t h i s c o m p l e t e s t h e p r o o f o f t h e 1emma.I T h u s , w e come n e x t t o t h e p r o o f of t h e e x i s t e n c e of t h e s i l o v b o u n d a r y f o r a n a p p r o p r i a t e c l a s s of t o p o l o g i c a l a l g e b r a s ( c f . C o r o l l a r y 2.1 b e l o w ) , i n analogy w i t h t h e c l a s s i c a l s i t u a t i o n one h a s i n

case of

( c o m m u t a t i v e ) Banach a l g e b r a s . So w e f i r s t h a v e t h e f o l l o w i n g

2. BOUNDARY

191

r e s u l t which a l s o p r o v i d e s s u p p l e m e n t a r y i n f o r m a t i o n a b o u t t h e l o c a l d e s c r i p t i o n of 3E t o t h a t g i v e n by t h e above Lemma 2 . 1 .

Theorem 2.1. Let E be a topoZogical algebra w i t h an i d e n t i t y elerne-.;t and compact spectrum mlE). Furthemore, denoting by F t h e s e t of a l l ( c l o s e d ) bound-

ary s e t s f o r E , consider t h e s e t

s = ~BC??UE),

(2.8)

BE F

and suppose t h a t f, e C S . Then, t h e r e e x i s t s an open neighborhood V of f, i n m(E) such t h a t B n C V e F,

(2.9)

for every B e F.

Proof. If f, 6 S , t h e n by ( 2 . 8 ) t h e r e e x i s t s B , e F , w i t h f, T h e r e f o r e , f o r every element f e B ,

:(So)

(2.10)

, there

= 0 and $(f)

E

.

B,

e x i s t s an element x e E , w i t h 1

( b y h y p o t h e s i s t h e a l g e b r a E A CC (miEI)" c o n t a i n s t h e c o n s t a n t s " and " s e p a r a t e t h e p o i n t s " i n m(E)). Thus , t h e o p e n s e t s U ( f ; x) :={ h E m(E): I z ( h ) 1 > I

>- ,

w i t h f e B , and x e E s a t i s f y i n g ( 2 . 1 0 ) , c o n s t i t u t e a c o v e r i n g of t h e compact s e t B , G m ( E ) ; so one g e t s a f i n i t e s u b c o v e r i n g , namely, f i n i t e many e l e m e n t s x i E E , 1 5 i In

, such

t h a t the set

. . ,n 1 V n Bo = @ . Next

V = { h E m(EI : I cci(h) 1 < 1, w i t h i = 1 , .

(2.11)

i s an open neighborhood of f, i n ??Z (El , w i t h t h e s e t V g i v e n by ( 2 . 1 1 )

w e prove t h a t

f u l f i l l s f u r t h e r t h e rest o f t h e c o n d i t i o n s

r e q u i r e d hy t h e s t a t e m e n t , i . e . ,

the relation

( 2 . 9 ) , f o r every B e

So suppose , o t h e r w i s e , t h a t t h e r e e x i s t s B E F , w i t h B n

exists Z E E

cV $ F.

F.

Then, t h e r e

such t h a t

(2.12)

( e v e n t u a l l y , a f t e r a s u i t a b l e m u l t i p l i c a t i o n of e v e r y h E B fi

cV.

2

by a c o n s t a n t ) , f o r

T h e r e f ore , by s e t t i n g c1

where t h e e l e m e n t s xi

E

= m a x { 12.1

..., c I ,

;i=l,

miE) E ( i= 1 , . . . , n ) a r e t h o s e i n ( 2 . 1 1 ) , one g e t s f o r

s u i t a b l e m e IN t h e r e l a t i o n IG(h)l". ci < 1

, for

every h E B

ncV

; hence,

one

h a s l.?7..2i(h)l < I (i=Z,..., n ) , f o r e v e r y h e B n C V . On t h e o t h e r h a n d , one c o n c l u d e s , by ( 2 . 1 1 ) and ( 2 . 1 2 )

,

the relation

12m2i(h)1

< 1 , f o r every

h e I/, w i t h 1 Z i S n , so t h a t one g e t s , i n f a c t , t h e r e l a t i o n l ; m 2 i ( h ) l < 1 ,

192

VI THE GEL'FANE MAP

f o r e v e r y h € B , and h e n c e

limGi

IB

< 2

( c f . a l s o ( 2 . 2 ) and t h e comment

f o l l o w i n g i t ) . C o n s e q u e n t l y , s i n c e B e F , one o b t a i n s

1 'm'i \ M ( E )

(2.13)

,.

- ^rn - Iz z i l B < I

However, s i n c e B,E F , t h e r e e x i s t s g e B ,

(i=I,

...,n ) .

such t h a t , b y ( 2 . 1 2 )

= Ii?(gl I = 2 , and h e n c e by ( 2 . 1 3 )

I = I 3m2i(g)I

13i(g) I = IIm(&iigl with 2 5 i S n

,

2

I;m;.

7,

Im(El

, I' I m ( E )

< I ,

t h a t i s , g c B 0 n V : b u t t h i s i s a c o n t r a d i c t i o n t o t h e re-

l a t i o n . R o n V = 0 o b t a i n e d b e f o r e (see ( 2 . 1 1 ) ) , a n d t h i s c o m p l e t e s t h e proof o f t h e t h e o r e m . I The f o l l o w i n g r e s u l t p r o v i d e s t h e " e x i s t e n c e " ( a n d " u n i q u e n e s s " as w e l l ,

see a l s o t h e n e x t Remark 2 . 1 )

of a E f o r a n y a l g e b r a E satis-

f y i n g t h e c o n d i t i o n s of t h e a b o v e Theorem 2 . 1 .

That i s , w e have.

Corollary 2.1. Let E be a topoZogicaZ aZgebra w i t h an i d e n t i t y eZement and compact spectrum m(Et. Then, t h e s e t ( 2 . 8 ) i s a hou~zdarys e t f o r E , so t h a t i t is, i n f a c t , t h e l e a s t boundary s e t f o r E ; t h u s , one has (2.14)

( w e a p p l y t h e n o t a t i o n of

(2.8)).

Proof. I n f a c t , w e p r o v e t h a t , f o r e v e r y x e E ,

one g e t s t h e re

-

lation

~ ( 2 n) s

(2.15)

( I n t h i s r e s p e c t , w e remark t h a t

f 0. the same r e l a t i o n ensures t h a t S # 0

, and

a l s o M(2) # 0 , f o r e v e r y x e E , a l t h o u g h t h i s i s a l r e a d y t r u e , by ( 2 . 4 ) and t h e h y p o t h e s i s f o r t h e a l g e b r a E ) : Thus s u p p o s e , o t h e r w i s e , t h a t t h e r e e x i s t s some x e E , w i t h M ( i ) C

c S : then,

by a p p l y i n g t h e argument

o f t h e p r e c e d i n g p r o o f , o n e o b t a i n s a n open c o v e r i n g of t h e compact

s e t M(?) E n ( E . 1 c o n s i s t i n g of s e t s of t h e form ( 2 . 1 1 ) a f i n i t e subfamily, say V

,...,

V n , s u c h t h a t by ( 2 . 9 )

. Hence,

one g e t s

and t h e f a c t t h a t

? ? Z ( F I e F , one c o n c l u d e s t h a t t h e sets CV,

,c ( V ,

V2)

,..., C l V , ...

V I

B

are a l l boundary s e t s f o r E . S o , s i n c e MI21 n B = 0

, one

has

t h a t i s , a c o n t r a d i c t i o n t o t h e f a c t t h a t B e F , and t h i s t e r m i n a t e s t h e proof. I

193

2. BOUNDARY

Remark 2.1 (2.4)

.- By

t h e above C o r o l l a r y 2 . 1

,

t h e set 5 defined

i s a boundary s e t f o r t h e g i v e n a l g e b r a E l b e i n g a l s o

c1

by

non-em-

the smallest ( w i t h r e s p e c t t o set-

p t y subset of ? Z ( E i ; m o r e o v e r , it i s

i n c l u s i o n ) boundary s e t of B. Thus, i n t h e c a s e u n d e r c o n s i d e r a t i o n , o n e a c t u a l l y c o n c l u d e s b y t h e same r e s u l t the ( n o n - t r i v i a l ! ) unique existence

of a E . I n t h i s r e s p e c t , w e a l s o n o t e t h a t F i n Theorem 2 . 1

i s a n induc-

t i v e s e t , i.e., it s a t i s f i e s t h e c o n d i t i o n s o f Z o r n ' s Lemma, so t h a t it r Thus, on t h e b a s i s of t h e p r e v i o u s

p o s s e s s e s a minimal element, s a y

.

Theorem 2 . 1 a n d i n c o n j u n c t i o n w i t h C o r o l l a r y 2 . 1 , t h a t T C S which a c t u a l l y i m p l i e s t h a t

r

o n e c a n e a s i l y see

= S , so the unique existence of

t h e &lov boundary of E i n t h e case u n d e r c o n s i d e r a t i o n .

( For

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

(commutative)

an account

Banach a l g e b r a s ( w i t h i d e n t i t y e l e m e n t s ) , see, f o r e x a m p l e , M.A.NA7MARK [1: p. 211, Theorem 13

. Cf.

also the relevant presentations i n K.

HOFFMAN[l:p.59, Theorem 7 . 1 1 a n d F . F . BONSAL-J. DUNCAN[l:p.l12ff.]).

Now,

t h e following provides a non-trivial

t i o n d e s c r i b e d by t h e p r e v i o u s Theorem 2.1 c e r n i n g t h e class of t h e t o p o l o g i c a l

i n s t a n c e of t h e s i t u a -

( a n d i t s c o r o l l a r y ) , con-

(non-normed) a l g e b r a s i n v o l v e d .

I n t h i s r e g a r d , see a l s o C h a p t . I V ; S u b s e c t i o n 4 . ( 3 ) .

Example 2.1.-

Suppose w e h a v e a n i n d u c t i v e s y s t e m (Ea, fRa) of com-

mutative Banach algebras w i t h i d e n t i t y elements, and l e t (2.16)

E = lz(Ea, f

Ra

be t h e respective

I

inductive l i m i t algebra ( c f . C h a p t . I V ; S e c t i o n s 1 , 2 )

Thus, t h e final vector space topology

.

i n E makes i t i n t o a ( c o m m u t a t i v e )

topologica2 algebra w i t h an i d e n t i t y element ( i b i d . ; Lemma 2 . 2 ) , whose spectrum (2.17)

is a (non-empty) compact (Hausdorffl space ( c f . Theorem V ; 3.1

i n connection

w i t h S c h o l i u m V; 3 . 1 ) . W e a r e t h u s w i t h i n t h e p r e v i o u s framework, so t h a t one a c t u a l l y c o n c l u d e s f o r t h e g i v e n a l g e b r a S ( C o r o l 1 a r y 2 . l ) t h e r e l ati o n (2.18) Q, # a E c m ( E ) . ( A n example o f t h e p r e v i o u s a l g e b r a ( 2 . 1 6 ) i s p r o v i d e d by t h e a l g e b r a O ( S ) i n Chapt.IV;

4. (3): cf. (4.36)).

W e c l o s e t h e p r e s e n t s e c t i o n w i t h a number o f a p p l i c a t i o n s of

t h e p r e c e d i n g t e c h n i q u e , a l r e a d y f a m i l i a r f r o m Banach a l g e b r a s t h e o r y . The a r g u m e n t a t i o n u s e d i s a l s o s i m i l a r t o t h e one a p p l i e d i n t h a t c a s e

194

THE GEL'FAND MAP

VI

t o o , b u t i t i s h e r e p r e s e n t e d m o s t l y a s a n example o f t h e a p p r o p r i a t e m o d i f i c a t i o n s n e e d e d w i t h i n t h e p r e s e n t framework. Thus, c o n c e r n i n g t h e n e x t lemma, see, f o r i n s t a n c e , W. k L A Z K O [2: p. 62, Theorem 15.51 a n d /

or C . E . RICKAIiT[I:p. 143, Theorem 3.3.191,

p . 114,

F . F . B O N S A L - J . DUNCAN[I:

P r o p o s i t i o n 7 1 ) . So w e f i r s t h a v e t h e f o l l o w i n g . Lemma 2 . 2 . Let E be a c o m t a t i v e a d v e r t i b l y complete locaLly in-eonvex alge-

bra with an i d e n t i t y element and compact spectrum m ( E ) ( c o n s i d e r e . g . a com& - a l g e b r a w i t h an i d e n t i t y e l e m e n t ) and Let

m u t a t i v e l o c a l l y m-convex

x € E . Then, one g e t s the r e l a t i o n bd( SpE(x)) C ;(dE).

(2.19)

(The f i r s t member o f

( 2 . 1 9 ) d e n o t e s t h e t o p o l o g i c a l boundary of t h e

s e t SpE(xl i n C ) .

proof. W e f i r s t remark t h a t a F i s c o m p a c t , b e i n g a c l o s e d s u b s e t o f t h e compact s p a c e m ( E ) ( C o r o l l a r y 2 . 1 ) . S o i f Aoe . G ( a E ) , t h e n t h e r e e x i s t s 6 > 0 such t h a t

(2.20) Now, i n c o n t r a d i s t i n c t i o n t o ( 2 . 1 9 ) , s u p p o s e a l s o t h a t

A, e bd (SpE(x))= SpE(x) fl

(2.21)

c SpE(x)

(see a l s o I I I ; ( 6 . 1 5 ) ) ; t h u s , e v e r y n e i g h b o r h o o d of A,

c o n t a i n s elements

o f cSpEix), so t h a t t h e r e e x i s t s a n e l e m e n t A€SpE(x), w i t h IA,-hl

< 6

.

T h e r e f o r e , one g e t s by ( 2 . 2 0 )

/2(f,-AI21G(f)-A0I-

(2.22)

IAo-A1>26-6=6

f o r e v e r y f e a E . Now, by h y p o t h e s i s f o r A , i b l e i n E , so t h a t i f y=(z--X)-', rollary I I I ; 6 . 5 (2.23)

the

rE(y) =

f

t h e e l e m e n t x-X

relation

sup Iijif) 1 = sup l i j ~ I ~=/ i n f 12(fi - A 11-l < 6-' Z"(E) f e a E f eaE

=;if,)

Sp ix) = ; ( r r C ( E ) ) , so t h a t A, E ( s e e a l s o 111; ( 6 . 1 6 ) )

~

.

&

On t h e o t h e r h a n d , by ( 6 . 2 1 ) and 111; ( 6 . 1 5 ) ,

rE(y)> ~

is invert-

t h e n o n e c o n c l u d e s by ( 2 . 2 2 ) and C o -

~

~

,

~

one o b t a i n s t h a t

A, e

f o r some f 0 & m ( E l . T h e r e f o r e , o n e has

~

= l (~x c i~f , ~~- ~ )-- ' ~ =~I A , - )A J - '-

~

>c-' i ~,

,

~

t h a t i s , a c o n t r a d i c t i o n t o ( 2 . 2 3 ) , and t h i s p r o v e s t h e a s s e r t i 0 n . I

W e examine n e x t t h e b e h a v i o u r of t h e s i l o v boundary o f a g i v e n t o p o l o g i c a l a l g e b r a E u n d e r t h e a c t i o n on E o f a n a l g e b r a morphism. T h u s , s u p p o s e w e a r e g i v e n a c o n t i n u o u s a l g e b r a morphism

~

=

195

BOUNDARY

2.

@:E-+F,

(2.24)

where E , F a r e t w o g i v e n t o p o l o g i c a l

‘@

(2.25)

: m(F)

a l g e b r a s , and l e t

-m(E)

b e t h e r e s p e c t i v e t r a n s p o s e map of @ d e f i n e d by t h e r e l a t i o n (2.26)

t@(g)= g

0

@

,

f o r e v e r y g E m(F).S o t h e l a t t e r is a continuous map between t h e r e s p e c t i v e

topological spaces ( c f . a l s o C h a p t . V ; S e c t i o n 7 )

.

I n t h i s r e s p e c t , w e now g e t t h e f o l l o w i n g .

Theorem 2.2. Let E , F be commutative a d v e r t i b l y complete l o c a l l y m-convex algebras w i t h i d e n t i t y elements and compact spectra. Moreover, l e t @ : E+

F a con-

tinuous (algebra) rnorphism and ’ $ : m(F)+ m(EI t h e corresponding transpose map. F i n a l l y , suppose t h a t t h e following r e l a t i o n holds t r u e

r,fzl = r F ( @ ( x ) ) ,

(2.27)

for every x e E ( i . e relation

., t h e

map

i#~

.

p r e s e r v e s t h e s p e c t r a l r a d i i ) Then, one has t h e

(2.28)

aE G t @ i a F ) .

I n p a r t i c u l a r , i f t h e algebra E has t h e r e s p e c t i v e Gel’fand map continuous,

then

a E i s characterized by t h e property o f being t h e l a r g e s t closed subset of

? ? Z ( E ) s a t i s f y i n g ( 2 . 2 8 1 , f o r any given t r i a d (E,@, F ) , a s above.

Proof. F o r e v e r y x e E , o n e o b t a i n s

.

sup l ; ( h ) \ = ( b y ( 2 . 2 6 ) )

(2.29)

g

h e t@taE)

sup I $ ( x l ( g ) I = rF(@(x)l = ( b y (2.27)) r E ( x ) e aF

i s a compact s u b s e t

T h e r e f o r e , s i n c e by h y p o t h e s i s a n d ( 2 . 2 5 ) t@(aFI

of

7 X ( E ) , i t i s , i n f a c t , a s f o l l o w s from ( 2 . 2 9 )

,

a boundary s e t f o r E

and t h i s p r o v e s ( 2 . 2 8 ) . NOW,

i n o r d e r t o p r o v e t h e l a s t p a r t of t h e t h e o r e m , s u p p o s e

t h a t B i s a ( c l o s e d ) subset of

‘I%(E)

w i t h t h e property t h a t

B G ~ @ ( ~ F ) ,

f o r any g i v e n t r i a d (E,(I,F )

,as

i n t h e s t a t e m e n t . A c c o r d i n g l y , one may a p -

p l y ( 2 . 2 9 ) f o r t h e t r i a d ( E , g , F ) , where F = c u l a E l ( t h e Banach a l g e b r a

of complex-valued c o n t i n u o u s f u n c t i o n s on t h e compact s p a c e aE endowed w i t h t h e “sup-norm‘‘ ( u n i f o r m ) t o p o l o g y ( i n aE)) and t h e map g i s g i v e n by t h e r e l a t i o n

g : E d Cu(aE): x-

g ( x ) :=

f I3E

;

s o a c o n t i n u o u s map, by h y p o t h e s i s f o r E . M o r e o v e r , one h a s

196

VI

THE GEL'FAND MAP

(2.30) ( t h e l a s t r e l a t i o n i s t r u e w i t h i n a homeomorphism d e f i n e d by t h e res p e c t i v e " e v a l u a t i o n ( D i r a c ) map"; see t h e n e x t c h a p t e r , S e c t i o n 1 ) ; Thus, o n e o b t a i n s (2.31)

8~

= aE,

w i t h i n t h e same homeomorphism, a s above ( i b i d . ) . T h e r e f o r e , t h e t r a n s p o s e map of g

,

i.e.

, tg : ~ ( F =I a E -+m(~i

i s a c t u a l l y t h e r e s p e c t i v e ( c a n o n i c a l ) i n j e c t i o n of a E i n t o ??YIEl, so

t h a t one c o n c l u d e s , by ( 2 . 3 1 1 ,

tgiaFl = tglaEl = a E . T h u s , o n e f i n a l l y g e t s by t h e h y p o t h e s i s f o r B , i . e . , (2.29)

the relation

I

B

r t g ( a E i = ar

which i s e s s e n t i a l l y t h e a s s e r t i o n i n t h e l a s t p a r t of t h e t h e o r e m , and t h i s completes t h e p r o o f . I An a p p l i c a t i o n of t h e p r e v i o u s t h e o r e m p r o v i d e s u s w i t h a cond i t i o n e n s u r i n g t h e extension of

(continuous) characters from a g i v e n t o -

p o l o g i c a l a l g e b r a t o a b i g g e r o n e ; w e t h u s h a v e a "Hahn-Banach t y p e " r e s u l t i n t o p o l o g i c a l a l g e b r a s t h e o r y , i n analogy w i t h t h e f a m i l i a r s i t u a t i o n one h a s i n c a s e o f Banach a l g e b r a s ( s e e , f o r example, C . E . RICKART [ l : p. 1471 a n d / o r I. GEL'FAND-D. R A I K O V - G . SHILOV [I: p.79,Theorem

I]).

Thus, w e have.

Corollary 2 . 2 . Let F be a commutative a d v e r t i b l y complete l o c a l l y m-convex algebra w i t h an i d e n t i t y element and cornpct spectmm nZIF). Moreover, l e t E be a subalgebra of F endowed w i t h t h e reZative ( l o c a l l y m-convex) topology such t h a t t o be a l s o advertibly compZete, have a compact spectrwn m ( E l , and the same i d e n t i t y a s F . Then, every element of ? l Z ( E ) which belongs t o t h e &Zov boundary of E can be extended t o an element of m ( F ) ; i . e . , for every f € a E E m(E),there e x i s t s an element ge m(FI such t h a t (2.32)

'IEEF

= f.

Proof. By h y p o t h e s i s and Theorem III;6.1, one concludes t h e r e l a t i o n (2.33)

f o r every X E E. T h e r e f o r e , by a p p l y i n g t h e p r e v i o u s Theorem 2 . 2 t o t h e t r i a d I S , i , F I , w i t h i : E z F t h e c a n o n i c a l i n c l u s i o n map, one g e t s from

197

BOUNDARY

2.

(2.28) t h e r e l a t i o n (2.34)

aEsti(aF).

Thus, f o r e v e r y f E aE, t h e r e e x i s t s g

€

aF G m ( F ) , w i t h

f = t i ( g ) = ( b y (2.26)) g o i

(2.35)

=g lEGF

t h a t i s , one h a s (2.32) and t h i s completes t h e proof. I The f o l l o w i n g s h o u l d be compared b o t h w i t h Lemma 11; 4 . 1 , w h o s e

a l s o c o n s t i t u t e s a p a r t i a l improvement, and t h e analogous r e s u l t from Banach a l g e b r a s t h e o r y ( c f .

,

f o r i n s t a n c e , C. E . RICKART [l: p. 147, The0

-

r e m 3 . 3 . 2 7 1 ) . Namely, w e h a v e . Theorem 2.3. Let E , F be commutative a d v e r t i b l y complete l o c a l l y m-convex

algebras w i t h cornpact spectra VYiE), m(F),r e s p e c t i v e l y , and such t h a t E t o be a subalgebra of F having with it a common i d e n t i t y element and the canonical i n c l u sion map i:E

5F

continuous. Furthermore, suppose t h a t we have

r,(cci = r F (x),

(2.36)

f o r every X E E . Then, t h e following r e t a t i o n s hold t m e : Sp ( x l F

(2.37)

C

SpE(x) , and

bd SpEIx)C bd SpF(x),

(2.38)

f o r every x € E.

Proof. W e a l r e a d y know t h a t ( 2 . 3 7 ) i s t r u e , i n g e n e r a l , w h e n e v e r E i s a s u b a l g e b r a o f F ( c f . P r o p o s i t i o n 1 I ; l . l ) . NOW, b y c o n s i d e r i n g t h e c a n o n i c a l i n j e c t i o n i : E S F o n e g e t s (Theorem 2 . 2 ) aE G t i ( a F )

(2.39)

,

a n d by Lemma 2 . 2 (2.40)

b d SpE(cc) C ; ( a E i .

T h e r e f o r e , i f AebdSpEix), t h e r e e x i s t s b y ( 2 . 4 0 ) a n e l e m e n t f eaE E

? ? Y i E ) , w i t h X = $ i f ) , s o t h a t t h e r e e x i s t s , by ( 2 . 3 9 ) , a n e l e m e n t g € a E , with f = ti(g) = g o i = g one obtains

IE

A =

( a l s o by h y p o t h e s i s f o r t h e map i : E - + F ) .

2(f! = ; ( t i ( g l l

=?is IE )

= gix)

,

w h e r e gix) = G(g) E SpF(x) = (;

? Z ( F ) ) ( c f . C o r o l l a r y 111; 6 . 4 ) , t h a t i s

(2.41)

bdSpEfxl C SpF(x).

Thus, one f i n a l l y g e t s

bdSpE(xl = SpE(rl nCSpE(x) E ( b y (2.41) a n d (2.37))

Hence

198

VI THE GEL’FAND MAP

spF(x) n

C spE(x) =

t h a t is, t h e desired re1.(2.38)

,

bd SpF(x)

,

and t h i s c o m p l e t e s t h e p r o o f . I

The t o p o l o g i c a l a l g e b r a s c o n s i d e r e d s o f a r i n t h i s s e c t i o n f u l f i l l e d t h e c o n d i t i o n s r e q u i r e d b y Theorem 2 . 1

(see a l s o Lemma 2 . 2 ) .

and i t s C o r o l l a r y 2 . 1

On t h e o t h e r h a n d , one c o u l d g e t , o f c o u r s e , a

s i m p l e r form of t h e p r e c e d i n g r e s u l t s by c o n s i d e r i n g , f o r example, ( l o c a l l y rn-convex)

&-algebras. T h u s , w e a l r e a d y know t h a t & - a l g e b r a s

( i n t h e p r e s e n c e of a n i d e n t i t y e l e m e n t ) have compact s p e c t r a (Lemma 1 . 3 ) , a r e a d v e r t i b l y c o m p l e t e (Theorem I ; 6.41,

t h e i r respective Gel’-

f a n d maps a r e a l w a y s c o n t i n u o u s ( C o r o l l a r y 1 . 4 ) , and b e s i d e s e v e r y c h a r a c t e r ( o f a & - a l g e b r a ) i s c o n t i n u o u s ( C o r o l l a r y 11; 7 . 3 ) . T h u s , a s

a c l a r i f i c a t i o n of t h e above comment, w e s i m p l y s t a t e t h e f o l l o w i n g c o n s e q u e n c e of t h e above Theorem 2 . 3 i t s p r o o f b e i n g q u i t e c l e a r by t h e previous discussion. I n t h i s r e s p e c t , w e a l s o n o t e t h a t f o r any t r i a d (E, 4, F ) , w i t h

$:F-+F

a g i v e n a l g e b r a morphism between & - a l g e b r a s , one g e t s f o r t h e

r e s p e c t i v e t r a n s p o s e map o f 4 t h e r e l a t i o n (2.42)

ti$

: M r F ) = m(F)-

M(El’

=

m(E)’

( c f . C o r o l l a r y 11; 7 . 3 ) . So w e now h a v e .

Corollary 2.3. Let E , F be c o m t a t i v e l o c a l l y m-convex &-algebras such t h a t a comnion i d e n t i t y element, and moreover suppose t h a t r E ( x ) = rF (xi, f o r every x e E . Then, t h e previous r e l a t i o n s ( 2 . 3 7 ) and ( 2 . 3 8 ) are v a l i d , f o r every element x e E . a

E t o be a subalgebra of F, having w i t h it

3. Functional calculus. Holomorphic functions o f a single element in a topological algebra W e d i s c u s s i n t h i s s e c t i o n t h e form of a n i m p o r t a n t b r a n c h of

Banach a l g e b r a s t h e o r y , namely, t h a t o f t h e “Symbolic C a l c u l u s “ , w i t h i n a f a i r l y g e n e r a l c l a s s of t o p o l o g i c a l a l g e b r a s amongst t h o s e cons i d e r e d h i t h e r t o . W e a r e o n l y c o n c e r n e d , however, w i t h t h e most f u n d a m e n t a l f e a t u r e s of t h e t h e o r y which a r e a l s o u s e d i n s u b s e q u e n t c h a p t e r s of t h i s d i s c u s s i o n . T h u s , w e p r e s e n t l y c o n s i d e r a p u r e l y “ a l g e b r a i c ” a s p e c t of t h e theory, t h a t i s, t h e so-called

“

Polynomial s p e c t r a l mapping theorem

“

which

a l s o i l l u m i n a t e s t h e k i n d of p r o b l e m s o n e i s c o n f r o n t e d w i t h . T h u s , w e f i r s t have.

Lemma 3.1. Let E be an algebra ( n o t o p o l o g y i s c o n s i d e r e d on E ) w i t h an i d e i ’ t l t y element and x an element o f E . Furthermore, l e t @ [ tdenote ]

t h e alge-

3.

199

FUNCTIONAL CALCULUS

bra o f polynomials i n one indeterminate w i t h complex c o e f f i c i e n t s . Then, one has the relation PfspEIxclI = SPE(P(21I ,

(3.1)

f o r every polynomiaZ P e c [ t ].

Scholium ( o n t h e t e r m i n o l o g y a p p l i e d ) IT?

c l e a r t h a t , f o r a n y x e E and P l t j = 2 a t" n=On

one o b t a i n s

.E

It is

C[ b ]

,

T h u s , a s u s u a l l y s a i d , " t h e polynomials i n C [ t ] opera t e on t h e elements of t h e algebra E ",i n t h e s e n s e t h a t FIz) d e f i n e d by ( 3 . 2 ) i s a n e l e m e n t o f E , f o r e v e r y P ( t ) e C [ t ] . The f i n a l t a r g e t o f t h e t h e o r y i s t h u s t o broaden up t h e class of f u n c t i o n s which may " o p e r a t e " on t h e e l e m e n t s o f E , f o r s u i t a b l e t o p o l o g i c a l a l g e b r a s E and s u i t a b l y chosen ( o p e r a t i o n a l ) f u n c t i o n s ; t h e l a t t e r are, a s w e s h a l l see, a p p r o p r i a t e l y d e f i n e d E-valued h o l o m o r p h i c maps. Moreover, t h i s e x t e n d e d " o p e r a t i o n a l c a l c u l u s " s h o u l d r e t a i n , o f c o u r s e , cert a i n basic properties already present, for ins t a n c e , i n t h e above r e l . ( 3 . 1 ) . (See a l z o t h e s t a t e m e n t o f t h e r e s p e c t i v e Lemma 3.2 below and o f c o u r s e Theorem 3 . 1 ) . Fi*oof of Lemma 3.1.

W e f i r s t prove t h a t

P I X ) e SPE(P(SiI,

(3.3)

f o r e v e r y A E Sp,Ix), w h i l e PIX) i s g i v e n by ( 3 . 2 ) : T h u s , by c o n s i d e r i n g Y

t h e polynomial

Q ( t ,= J P l t i -P(AI

(3.4)

or.e h a s . & ( A / = 0 . So i f

,...,

\,'

E C[t]

a r e t h e r e s t o f t h e m-many r o o t s o f

& i t ) ,o n e g e t s Qitl =

#.

m

It-hiit-hl)...It-hI?:-Zi

and hence a l s o t h e r e l a t i o n

&(XI=

(3.5)

~1

rn

(x-h)13:-hl)

T h u s , s i n c e by h y p o t h e s i s f o r A ,

... ( x - h m-1 I

t h e e l e m e n t x-h

. i s singular i n E, the

same i s v a l i d , by ( 3 . 5 ) , f o r t h e e l e m e n t QIx) = P l c l - P ( h /

€

E

( c f . ( 3 . 4 ) ) , due t o t h e f a c t t h a t t h e f a c t o r s 2 - h

a n d x-Xi ( i = l ..., , r7-2)

a r e commuting e l e m e n t s of E , which p r o v e s ( 3 . 3 ) . On t h e o t h e r hand,

lynomial

s u p p o s e t h a t h e S p E I P 1 d l and c o n s i d e r t h e po-

VI THE GEL'FAND MAP

200

&,It) = P:t)-X= Clm(t-All(t-XzI...It-T.ml i n i t s c a n o n i c a l f a c t o r d e c o m p o s i t i o n . Thus, o n e o b t a i n s Q , ( x ) = P/x)- X =

(3.6)

s o t h a t , by h y p o t h e s i s f o r A ,

~ ~ ( 3 X1l : - (3:

-Az).

..(x-Xm)

I

( 3 . 6 ) y i e l d s a s i n g u l a r e l e m e n t of E .

T h e r e f o r e , one a t l e a s t o f t h e f a c t o r s x-Xi

,with

l < i S m , in

(3.6) i s

a s i n g u l a r e l e m e n t of E ; s o if t h i s i s t h e case € o r t h e e l e m e n t , s a y 3;-A;

, t h e n A3.

e SpE(x). Moreover, one g e t s by ( 3 . 6 ) PIX .) = A 3

,

that is,

AeP(SpE(x)) and t h i s c o m p l e t e s t h e p r o o f o f t h e l e m m a . I

Scholium 3.1 .- I f t h e a l g e b r a E i n t h e above Lemma 3 . 1 d o e s n o t h a v e a n i d e n t i t y e l e m e n t , t h e n one pi~oves ?he r e l a t i o n ( 3 . 1 ) f o r t h e algebra

C o [ t ] CC [ t ]

,i.e. ,

t h e a l g e b r a of p o l y n o m i a l s i n C [ t ] w i t h P i ( ; ) = 0. ( I n

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

i s what w e a l s o assume i n t h e s e q u e l ) . F u r t h e r m o r e , t h e l a s t case i s a l s o r e d u c e d t o t h e p r e c e d i n g Lemma 3.1 by c o n s i d e r i n g t h e a l g e b r a E C

=F@C

for which t h e same lemma h o l d s t r u e ; s o t h e d e s i r e d a s s e r t i o n

f o r t h e algebra C o [ t ]

f o l l o w s now from t h e d e f i n i t i o n of t h i s a l g e b r a ,

t h e r e l . ( 3 . 1 ) b e i n g a p p l i e d t o E + , and Lemma 11; 1 . I

.

Thus s u p p o s e t h a t E i s a commutative a d v e r t i b l y complete Zocally rn-con-

v e x algebra w i t h a n i d e n t i t y e l e m e n t and s p e c t r u m m(E). Then, a p p l y i n g 1 1 1 ; ( 6 . 1 5 ) , o n e g e t s by ( 3 . 1 ) t h e r e l a t i o n P ( 2 I f l l = P I X ) If),

(3.7)

f o r e v e r y f e m(E). T h u s , e q u i v a l e n t l y ,

( 3 . 1 ) i s e x p r e s s e d , i n t h e case

u n d e r c o n s i d e r a t i o n , by t h e r e l a t i o n (3.8)

P o ; = PIX),

f o r a n y x E E and P e C [ t ], where P I X ) e E

i s g i v e n by ( 3 . 2 )

. Thus, one

w a n t s t o r e p l a c e P i n ( 3 . 8 ) t h r o u g h c e r t a i n s u i t a b l e more g e n e r a l f m c t i o n s ( t h a n t h e polynomials i n C [ t ] )

.

I n d e e d , t h e s e a r e (for s u i t a b i s 3 )

t h e E-valued h o l o m o r p h i c f u n c t i o n s d e f i n e d on open n e i g h b o r h o o d s of Sp ( x i s C , w i t h x & E ( c f . Theorem 3 . 1 b e l o w ) . T h e r e f o r e , one i s led t o E c o n s i d e r "holomorphic f u n c t i o n s of a s i n g l e element i n a given topoZogica2 a2ge-

b r a " , a s i t u a t i o n which becomes much more c o m p l i c a t e d i f o n e w a n t s t o c o n s i d e r a n a l o g o u s f u n c t i o n s , a s b e f o r e , b u t now o f

algebra elements

'I

( c f . Theorem 3 . 3 )

several topological

.

Now, t h e g e n e r a l case t r e a t e d h e r e i n i s a p p r o p r i a t e l y r e d u c e d t o t h a t of Banach a l g e b r a s t h e o r y s o , f o r c o n v e n i e n c e of t h e e x p o s i t i o n , w e q u o t e t h e c o r r e s p o n d i n g r e s u l t from t h a t t h e o r y ( w i t h o u t p r o o f ) . Thus, w e f i r s t h a v e .

3. FUNCTIONAL CALCULUS

201

Lemma 3.2. L e t E be a Banach algebra w i t h an i d e n t i t y element and x an eZe-

ment o f E . Furthermore, l e t h be a holomorphic f u n c t i o n defined on an open neighborhood o f

Sp,(xl C C

. Then,

t h e r e e x i s t s an element y e E such t h a t h(SPEiXl) = SPE(Xi,

(3.9)

where we s e t h ( x ) = y ( S p e c t r a l mapping Theorem). In p a r t i c u l a r , i f E i s commutative, one has

6= ho2,

(3.10)

namely, y ^ ( f i = i i ( ; ( f ) ) , f o r every f e VZ(E). Proof. C f . F. F. BONSAL - J . DUIZJCAN [I: p. 33, Theorem 41

.I

T h e r e i s a n e x t e n s i o n of t h e p r e c e d i n g r e s u l t t o t h e c a s e t h e a l g e b r a E does n o t n e c e s s a r i l y have an i d e n t i t y element ( w e a l s o cons i d e r below i t s

"

l o c a l l y m-convex a n a l o g o n " ) .

A s a m a t t e r of f a c t ,

one i s a c t u a l l y l e d t o examine w h e t h e r t h e q u o t i e n t a l g e b r a E , I R ( E ) does have an i d e n t i t y element; h e r e R(EI d e n o t e s t h e

" r a d i c a l " of C , so w e

f i r s t have t o e x p l a i n t h e r e s p e c t i v e t e r m i n o l o g y . Thus, w e have t h e following.

Definition 3.1. Suppose we a r e g i v e n a commutative l o c a l l y m-convex a l gebra E whose s p e c t r u m i s m(E).Then t h e r a d i c a l of E l d e n o t e d by R ( E ) , i s d e f i n e d by t h e r e l a t i o n

i.e., a s t h e i n t e r s e c t i o n of a l l c l o s e d r e g u l a r maximal i d e a l s o f E ( c f . a l s o Corollary 11:7.2). Thus , t h e q u o t i e n t a l g e b r a Z/F?(E)

equipped with t h e r e s p e c t i v e

q u o t i e n t t o p o l o g y ( c f . , f o r i n s t a n c e , Chapt.IV; S e c t i o n 3 )

i s a (com-

m u t a t i v e ) l o c a l l y m-convex a l g e b r a : s o w e a r e n e x t c o n s i d e r t h e case t h a t t h e l a t t e r a l g e b r a d o e s n o t h a v e a n i d e n t i t y e l e m e n t , namely, equivalently, the case that

t h e r a d i c a l o f E i s not a regular i d e a l i n E.

I n p a r t i c u l a r , i n c a s e E i s a commutative Banach a l g e b r a , t h e n one h a s t h e following. Lemma 3.3. Let E be a commutative 2anach algebra and x an element f.

E . More-

o v e r , l e t h be a holomorphic f u n d i o n defined on an open neighborhood o f SpE(x) , I a ; , if the r a d i c a l o f E i s not a r e g u l a r i d e a l o f E, t o have hiOl = O . Then,

.:~rn;'

t h e r e e x i s t s an element y e E such t h a t (3.12)

h(SpEiz)) = SpE(h(xll,

w i t h hix1 = y; i . e . , e q u i v a l e n t l y , one has

202

VI THE GEL'FAND MAP

hof=G.

(3.13)

.

Proof. C f . G . Fi. MACKEY [l: p. 53, Theorem -161 I

Scholium 3.2.-

It i s a s t a n d a r d d e v i c e t h e element hfxi = y e E

of

t h e p r e c e d i n g s t a t e m e n t t o b e g i v e n a s t h e v a l u e of a "Riemann t y p e i n t e g r a l " o f a n a p p r o p r i a t e l y d e f i n e d E-valued

f u n c t i o n ( a n analogous

a r g u m e n t i s a l s o v a l i d f o r t h e p r e v i o u s Lemma 3 . 2 ) . I n t h i s r e g a r d , cf.,

f o r i n s t a n c e , t h e l a s t R e f . a b o v e a n d a l s o F . P . BONSAL- J . DUNCAN

[l: p. 27 f f . ,

55 6 ,7] ) .

S i n c e w e are g o i n g t o u s e i n t h e s e q u e l t h e c o n c r e t e form t h a t has the element y

E under c o n s i d e r a t i o n , we recall, i n t h i s r e s p e c t ,

( i b i d . ) t h a t one h a s t h e r e l a t i o n (3.14) 1

where f - x i '

d e n o t e s the

x

quasi-inverse

1

of - x

x

a n d h i s a (complex-

v a l u e d ) h o l o m o r p h i c map d e f i n e d on a n o p e n n e i g h b o r h o o d S2 o f

Sp

E (2) i n a , w i t h h 1 0 ) = 0 i n t h e case c o n s i d e r e d by Lemma 3 . 3 ; f i n a l l y , r i s a s u i t a b l y d e f i n e d " c l o s e d r e g u l a r c u r v e " c o n t a i n e d i n R n cSpEfxl ( o r r w i l l be a f i n i t e sum o f s u c h c u r v e s , i n case Sp ( x i i s n o t c o n n e c t e d : E y e t it would be e q u i v a l e n t to s a y , i n g e n e r a l , t h a t r i s a 1 - d i m e n s i o n a l e m - s u b m a n i f o l d o f C. I n t h i s r e g a r d , see a l s o N. BOURBAKI [ I I : p . 39, Lemma 71 )

.

NOW, o n e p r o v e s t h a t t h e element y

independent of t h e c h o i s e of

r

E d e f i n e d by ( 3 . 1 4 1 i s , i n e f f e c t ,

and h a s t h e p r e c e d i n g p r o p e r t i e s , a f a c t

which w e s h a l l p r e s e n t l y n e e d . F u r t h e r m o r e , a p p l y i n g t h e a b o v e d e f i n i t i o n o f y , o n e s t i l l p r o v e s t h a t for every continaous l i n e a r form @ on E,

one g e t s t h e r e l a t i o n

( c f . G. W. MACKEY [ I : p. 52, Theorem]

,

a n d / o r F . F . BONSAL- d. DUNCAN 11: p. 271 ) .

A s i m i l a r r e l a t i o n t o ( 3 . 1 5 ) f o r s u i t a b l e a l g e b r a morphisms w i l l

be considered

i n t h e s e q u e l ( c f . (3.19) i n t h e proof of t h e n e x t t h e -

orem). T h u s , w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g r e s u l t ,

a " l o c a l l y rri-convex a n a l o g o n " o f t h e p r e c e d i n g . Theorem 3.1. L e t E be a c o m u t a t i v e complete loca2ly m-convex algebra whose

; ~ e e t m i ni s ? ? Z f E i , and l e t cc be an element of E . Furthermore, l e t h be a holomorT h i c f u n c t i o n defined on an open neighborhood of SpEfz) such t h a t , i n case R I E i is

3.

203

FUNCTIONAL CALCULUS

not a r e g u l a r i d e a l of E , t o have h ( 0 ) =O. Then, t h e r e e x i s t s an element 3

€

E such

that

(3.16)

= SpE(h(xii,

h(SpE(xII

w i t h hixi = y e E ( S p e c t r a l mapping Theorem). That i s , e q u i v a l e n t l y , 7 7 0 2 = hlx) =

(3.17)

or y e t li($(fI) = h ( x l ( f ) , fobr e v e r y f Proof.

Let E=limE

E

ij,

%(El.

be an Arens-Michael decomposition of E cori U a ) a E I of it (cf. Theorem 111; 3.1). responding to a local basis Moreover, if x is an element of E l one has (Corollary 111; 4.1)

t

a

z=

SpElxi =

u Sp.2 (xaI a€I

with - z = ( x a J e l ~andx,=[.z] ~ a' ael(cf. 111;(3.23)). Therefore, any map h satisfying the hypothesis is a holomorphic map defined on an open neighborhood s2 of Spn(xa), for every a E I ; thus, if i a / R ( 2 a ) does

zclc

not have an identity element, that is, if R(2aI is not a r e g u l a r i d e a l in 2 ( L E I ) ,t h e n t h e same holds t r u e for t h e i d e a l R(E) in E (see the next Remark). So by hypothesis for 1 1 , one has h l O l = O .

Remark.- Concerning the last statement above about R f E ) , we note that: If z_is an " i d a n t C t y of E modtilc R(E)", t h e n z i s an i d e n t i t y of E, modulo U [ Z ), f o r every a €I . In-

deed,aone has the relation

TZ(E,)= m ( E I g U , O , a € I , within 2 homeomorphism (Lemma V;6.3); so, for every fa e m f E , I , one gets the relation (cf. also V; (6.16)) (3.18) fa(zaza-xa)= f a l [ z x - x ] c l i = f ( z x - d = 9 I for every xeE,due to the hypothesis for z € E and (3. 11),and this proves the assertion. Thus, concerning the Banach algebra 2 r , ~ e I , o n has e the situation described by Lemma 3.3, so that one concludes the existence of an element yc, e 2 ," E I , satisfying the corresponding relation to (3.12) Hence we actually get an element y = iy, ) E fl Ra such that, in particular, y %€I E E : That is, for any a, Bin I , such that B>a,onegets by (3.14)

.

VI THE GEL'FAND MAP

204

(Concerning the last relations, we have applied the fact that an algebra morphism, like fclB , "preserves advertible elements", as well as 11;(1.7) in connection with Scholium 3.2 concerning the independence of ( 3 . 1 4 ) from the curve r ; finally, we have also applied an exten,with & E ) . sion of ( 3 . 1 5 ) to continuous algebra morphisms such as aB Finally, by ( 3 . 1 3 ) applied for a e I, and V; ( 6 . 1 6 ) , one obtains y^(fi= f ( y J = f a ( y a ) = i a l f c l ! = h ( 2 a ( f a ) ) = h(fa(xclJ)

,

= hif(xli = h(2ifl) = Iho4)If)

for every f e ? Z ( E l , that is, ;= h o ; the theorem. I

,

and this completes the proof of

Remark.- A direct proof of the above Theorem 3 . 1 without use of the Arens-Michael decomposition could be given as well, by an appropriate adaptation within the present context of the respective argument applied in the proof of Lemma 3 . 3 (ibid.).

A straightforward application of the foregoing is now the next.

Corollary 3.1. Let E be a commutative complete l o c a l l y m-convex algebra having a compact spectrum m i E ) . Moreover, suppose t h a t t h e r e e x i s t s an element x€E, w i t h 3 never vanishing on ?YY(E). Then t h e algebra E/RIE) has an i d e n t i t y element. I n p a r t i c u l a r , i f t h e Gel'fand map of E i s 1-1, then t h e algebra E has an i d e n t i t y element.

Proof. By hypothesis the set

RiGI = 2 ( ??Z(ElI

is a compact subset of

C, with OeRI~I.Therefore,there exists an open neighborhood, say R , of Rig) and a holomorphic function h , with h = 1

I

and h ( O l = 0. Thus (Theorem 3 . 1 ) , one gets an element y E E such that ; i f ) = h l 2 ( f l i = l r for every f e ? ? Z ( E I , that is ; = 1 . Hence, f i y z - 2 ) = y^(f)z^(fl- 2 l f ) = 0 ,

for any z e E and f €m(E) , so that the element y € E is an identity of E modulo RIE) (cf. also ( 3 . 1 1 ) ) . Now, the last assertion is clear in case the Gel'fand map of E is 1 - 1 (i.e., R ( E ) = 0; see also ( 3 . 1 1 ) ) ! and this terminates the pro0f.I We close this section by commenting a bit more on the map (3.20)

h-hlxl,

with x e E

,

which is defined by the preceding theorem. The relevant discussion is already standard within the familiar framework of Banach algebras theory. Thus, being within the context of Theorem 3 . 1 , let X E E ; then,

3.

205

F U N C T I O N A L CALCULUS

w e d e n o t e by (3.21)

t h e a l g e b r a of

(complex-valued)

holomorphic f u n c t i o n s on

considered a s a t o p o l o g i c a l algebra a c c o r d i n g t o C h a p t e v e r y open neighborhood R o f

. IV:

4. ( 3 )

SpE(xl i n C ( w e d e n o t e by

SpE(x) C C

. Thus ,

,

for

;63(SpE(x1) a

f u n d a m e n t a l s y s t e m o f s u c h n e i g h b o r h o o d s ) , o n e o b t a i n s a map $I:

( 3- 2 2 ) where h(xl

: HoL(R)-E:

h*$E(hl:=

i s g i v e n b y Theorem 3 . 1 .

h(xl,

( I n t h i s r e s p e c t , w e remark t h a t

one can d i s p e n s e w ith t h e c o n s t r a i n t

on

the functions h considered

a b o v e ( i . e . , t h e y do n o t h a v e a p o l e a t O E C ) when t h e a l g e b r a E h a s a n i d e n t i t y e l e m e n t , a s t h i s w i l l b e t h e case i n t h e s e q u e l ( s e e a l s o G . W. MACKEY [l: p. 55, l a s t p a r t o f t h e p r o o f

o f Theorem 1 6 1 . F u r t h e r m o r e ,

t h e r e l s . ( 3 . 1 4 ) and ( 3 . 1 5 ) t a k e now s u i t a b l y m o d i f i e d f o r m s : i b i d . ) . On t h e o t h e r h a n d ,

f o r any

R, , R, i n ~ ( S p E ( x l ) ,w i t h R , s R , ,

by ( 3 . 1 4 ) and t h e independence of of

r,

hixi E E

one g e t s , from t h e p a r t i c u l a r c h o i s e

t h e f o l l o w i n g commutative d i a g r a m

(3.23)

t h a t is, the relation (3.24)

Qx =; ; @ R2

PQ

Q

1 2

-

T h u s , o n e a c t u a l l y o b t a i n s a map (3.25)

QX : HaL(SpE(xl) -E

i n s u c h a way t h a t t h e f o l l o w i n g complement t o Theorem 3.1 t o be t r u e : Theorem 3.2. L e t E be a complete Zocally m-eonvex algebra w i t h an i d e n t i t y

element and x an element of E. Then, t h e map (3.26)

I$x: ffoL(SpE(x)) =

1 2Hve(R) -E

,

R E y3(SPE(X)1 d e f i n e d by ( 3 . 2 5 / , is a continuous algebra rnorl-hism. In p a r t i c u Z a r , one has t h z r e lations (3.27)

bX"rii = z ,

THE GEL'FAND MAP

VI

206

uhere 5 = i d c

denotes t h e i d e n t i t y map on C, and

= lE ,

(3.28)

with I c , lE denoting t h e i d e n t i t y elements of t h e algebras C and E, r e s p e c t i v e l y . A p p l y i n g t h e t e r m i n o l o g y which w e have u s e d i n t h e p r o o f

Proof.

o f Theorem 3 . 1 , w e g e t t h e f o l l o w i n g c o m m u t a t i v e d i a g r a m

RISPE(ccl

.A

pa

( c f . IV;(4.32)) pQ

( c f . 111;(3.4))

,4;

(3.29)

p E I z I ) ( s e e IV; , i s reduced

concerning t h e Banach a l g e b r a E ,

( c o r r e s p o n d i n g t o any g i v e n Arens-Michael

decomposi-

t i o n o f E ) , where t h e a l g e b r a HaLIR) i s t o p o l o g i z e d i n t h e compact-open topology ( c f .

f o r i n s t a n c e , F . F. BONSAL - J . DUNKAN [I : p.33, Theorem 4 , i i i ) ] 1.

F u r t h e r m o r e , on t h e b a s i s of t h e p r e c e d i n g diagram, one g e t s f o r any f u n c t i o n s g , h i n

HuClSp,lz))

P,i@X(g.hli = Pal$xip,(gh)))

= :@ f o r every

01

,, i gi , :@

,( h ) = Pa ($z( 3I I

= i5,($g(gh))=@:,,(g.k) *

P, ( 0" ( h) I = b, ($X (gl

a

i

@2 71)

)

€ 1( t h e map ( 3 . 3 0 ) b e i n g a n a l g e b r a morphism; i b i d .

, Theo-

rem 4 , i i ) ) . Thus , o n e o b t a i n s Q X ( g * h= ) @Xig)-$X(h/,

(3.31) t h a t w a s t o be p r o v e d .

S i m i l a r l y , o n e now g e t s Pa(@X(SI)

=5,(@X(PR(sIiI

=P,(@;(sI)

=

= x,=

P,lz),

f o r e v e r y a e I , and h e n c e (9X(5) = n:

( c f . a l s o F. F . BONSAL

- J . DUNKAN

[I: p. 31, Lemma I ] c o n c e r n i n g t h e r e l . ( 3 . 2 7 )

4. FUNCTIONAL

h

i n c a s e of a Banach a l g e b r a ,

207

CALCULUS (CONTN'D. )

such a s E , , a e I ) .

F i n a l l y , by a s i m i l a r

argument, one p r o v e s ( 3 . 2 8 ) r e d u c i n g it t o t h e Banach a l g e b r a ( i b i d . ; p. 32, Lemma 2 )

,

Ear

cleI

and t h i s f u l l y e s t a b l i s h e s t h e proof of t h e

theorem.

4. F u n c t i o n a l c a l c u l u s ( c o n t n ' d . ) .

Holomorphic f u n c t i o n s o f f i n i t e

many elements i n a t o p o l o g i c a l a l g e b r a W e c o n s i d e r i n t h i s s e c t i o n complex-valued

holomorphic f u n c t i o n s

d e f i n e d on t h e j o i n t spectrum ( c f . ( 4 . 4 ) below) of f i n i t e many e l e m e n t s

of a g i v e n c o m t a t i v e complete l o c a l l y m-convex algebra E w i t h an i d e n t i t y e l e m e n t , i n what c o n c e r n s t h e p o s s i b i l i t y of g e t t i n g a n a n a l o g o u s r e s u l t t o t h e above r e l . ( 3 . 1 7 ) . T h i s i s i n o u r c a s e t h e ( a b s t r a c t form of t h e ) " s p e c t r a l mapp i n g theorem" f o r a n a l y t i c f u n c t i o n s of several elements of t h e p a r t i c u l a r t o p o l o g i c a l a l g e b r a s c o n s i d e r e d . The c o r r e s p o n d i n g r e s u l t s w i t h i n t h e framework of Banach a l g e b r a s t h e o r y a r e s t a n d a r d a l r e a d y and c o n s t i t u t e , i n e f f e c t , t h e f u n d a m e n t a l s of t h e Silov-Arens-Calder6n-Waelbroeck Theory ( c f . , f o r i n s t a n c e , E . L . STOUT [I] a n d / o r N . BOURBAKI [ l l : Chap. I ] ) . I n t h i s r e s p e c t , more g e n e r a l , t o a c e r t a i n e x t e n t , t o p o l o g i c a l a l g e b r a s t h a n Banach a l g e b r a s have a l s o been c o n s i d e r e d ; see R . ARENS [ 7 ]

and I;. WAELBROECK 112, 51 ( c f . a l s o Scholium 4 . 1 b e l o w ) . On t h e

o t h e r hand, w e w i l l m o s t l y b a s e d , r e g a r d i n g t h i s e x p o s i t i o n , on t h e r e l e v a n t c o n s i d e r a t i o n s i n M . BONNARD [2] ; a p p l i c a t i o n s of t h i s d i s c u s s i o n w i l l a l s o be g i v e n l a t e r ( c f . C h a p t . V I I 1 ; S e c t i o n 7 ) . Thus, i n a l l t h a t f o l l o w s E w i l l d e n o t e a commutative complete lo-

c a l l y rn-convex algebra w i t h an i d e n t i t y element and spectrum

m(E).B e s i d e s ,

w e w i l l mostly suppose, f o r convenience, t h a t t h e t o p o l o g i c a l algebras involved have compact s p e c t r a (see Theorem 4 . 1 below) ; of c o u r s e , t h i s

i s t h e case i n Banach a l g e b r a s and w i l l a l s o i n t h e a p p l i c a t i o n s which

a r e g i v e n i n t h e s e q u e l ( C h a p t . V I I 1 ; S e c t i o n 8 ) . However, see, i n t h i s r e s p e c t , Scholium 4 . 2 below. Now, f o r e v e r y f i n i t e sequence of e l e m e n t s of E , s a y

x = i sI " ' . ,

(4.1)

xn I

one d e f i n e s t h e f o l l o w i n g ( c a n o n i c a l ) map (4.2)

@ x : ???(E/-

C n :f

@,if)=j;ifi:=I2,if),..., ? n ( f I I .

++

T h a t i s , one h a s , i n f a c t , t h e r e l a t i o n

VI THE GEL'FANE MAP

208

..

..

= (xl,. , x n ) e E n = E x . x E n-times w i t h i t s e l f . Now, t h e s e t

with

X

,

the

n-fold C a r t e s i a n product of E

(4.4)

t h a t i s , t h e image o f t h e map ( 4 . 2 ) , i s c a l l e d t h e t h e e l e m e n t s x~,...,x n

, with

joint spectrm

of

r e s p e c t t o t h e given a l g e b r a E . (See a l s o

Chapt. V I I I ; ( 7 . 2 8 ) ) . Applying t h e p r e v i o u s c o n t e x t , t h e a b o v e n o t i o n i s r e d u c e d , o f c o u r s e , by 111;(6.15) and f o r n = I , t o t h a t of t h e c s u a l s p e c t r u m , SpE(x), of a n e l e m e n t xceE. NOW,

s i n c e e a c h one of t h e f u n c t i o n s

gi

..., n )

(i=l,

is continu-

, one g e t s by ( 4 . 3 ) t h a t QX is a Cn-vaZued continuous map on m(E);t h e r e f o r e , in case m ( E l i s compact, t h e j o i n t spectrum

o u s on m(E)( c f . V; ( 1 . 4 ) ) of t h e element x = ( xI,. .

. ,xn ) e E

is a compact subset of Cn. F u r t h e r p r o p e r t i e s

of t h e map @ x w i l l b e d i s c u s s e d i n s u b s e q u e n t s e c t i o n s . On t h e o t h e r h a n d , w e s t i l l n e e d i n t h e s e q u e l some more t e r m i n o l o g y from t h e t h e o r y of S e v e r a l Complex V a r i a b l e s which w e come now t o e x p l a i n : Thus, g i v e n a t o p o l o g i c a l a l g e b r a E w i t h s p e c t r u m m f E l , a t r i p l e t for E is a triad

( x , v,

(4.5)

where x = i sI,.

. . ,x n I e E n l

P i

I

and

(4.6)

71:C n.V

i s a manifold spread ( o r Riemann domain) o v e r C n ; i . e . , a t r i a d ( V , TI , C n ) , TI a l o c a l homeomorphism ( i n s u c h a manner t h a t V t o b e c a n o n i c a l l y endowed w i t h t h e s t r u c t u r e of a complex a n a l y t i c m a n i f o l d modeled on C n , t h e map TI d e f i n i n g "glo-

where V i s a ( H a u s d o r f f ) t o p o l o g i c a l s p a c e a n d

-

bal-local coordinates" i n V ; c f . R . C. G U N N I N G - H . ROSSI [ l : p. 4 3 , ff.]) p : ?rr(EI

(4.7)

V

i s a continuous map making t h e f o l l o w i n g d i a g r a m commutative

(4.8)

t h a t i s , one o b t a i n s (4.9)

where

QX

i s g i v e n by ( 4 . 2 ) .

. Finally,

209

4. FUNCTIONAL CALCULUS ( CONTN 'D. )

T h u s , t h e a n a l o g o n of Lemma 3 . 2 , w i t h i n t h e p r e c e d i n g c o n t e x t , r e a d s now a s f o l l o w s .

Lemma 4.1.

and

L e t E be a c o m m t a t i v e Banach algebra w i t h an i d e n t i t y element

V , p ) a given t r i p l e t for E . Then, t h e r e e x i s t s a continuous algebra mor-

(X,

phism $ E ( x , V , p ) : HaklImip)) - E ,

(4.10)

i n such a way that $ E ( ~ V, , p ) f h ) = h o p

(4.11)

,

f o r e.,;ery h € Hot(Im(P)I. Moreover, the corresponding r e l a t i o n s t o ( 3 . 2 7 ) and 1 3 . 2 8 ) hold a l s o t r u e .

Proof. Cf. M. BONNARD [3: p. 4 0 7 , ThSorSme I ] ,

a s w e l l a s N . BOURBAKI

111: Chap. 1 ; p. 32, ThdorSme 1 1 . I

Scholium 4.1.-

R e g a r d i n g t h e t e r m i n o l o g y a p p l i e d i n t h e above Lem-

ma 4.1,

w e n o t e t h a t , by h y p o t h e s i s f o r t h e a l g e b r a E , I m ( p ) = p ( m ( E ) ) GI/ i s a compact s u b s e t of t h e complex m a n i f o l d V , so t h a t one d e f i n e s

the topological algebra

HaC(p(VZlE)) :=

(4.12)

12Ha.LtUi U2Im(pi

a c c o r d i n g t o t h e c o n s i d e r a t i o n s i n Chapt. I V ; ( 4 . 3 ) . T h e r e f o r e , t h e f u n c t i o n 11 i n ( 4 . 1 1 ) s t a n d s , o f c o u r s e , f o r a " r e p r e s e n a t i v e " of a

a h o l o m o r p h i c f u n c t i o n " d e f i n e d on a n open n e i g h b o r h o o d of

"germ o f Im(p) i r i

V , and i t i s t h i s germ t h a t i s a c t u a l l y u n d e r s t o o d

f i r s t member o f

in

( 4 . 1 1 ) . However, f o r c o n v e n i e n c e , w e r e t a i n e d

the the

s a m e n o t a t i o n . On t h e o t h e r h a n d , t h e same r e l a t i o n i s e s s e n t i a l l y i n d e p e n d e n t from t h e i n d i v i d u a l r e p r e s e n t a t i v e of t h e p a r t i c u l a r germ u n d e r c o n s i d e r a t i o n ( c f . M. EONNARD [3: p. 4081). NOW,

t h e a n a l o g o n of

( 3 . 2 7 ) a l l u d e d t o i n t h e s t a t e m e n t of Lem-

m a 4 . 1 means, of c o u r s e , t h a t i f z= denotes t h e

( Z ] ,

..., 2

)

caizonicaZ chart of C n , where zi ( i =I , .

. ., n )

are t h e r e s p e c -

t i v e c o o r d i n a t e f u n c t i o n s of C n , one g e t s by ( 4 . 1 1 ) $ ( ~ , V , p i f z ~ l = z i o p = z , I: S , i s n ,

(4.13) w i t h x = (x,, .

E

.. ,x n j .

I n t h i s r e s p e c t , z i i n t h e f i r s t member of ( 4 . 1 3 ) s t a n d s , i n e f f e c t , f o r t h e germ o f t h e r e s p e c t i v e c o o r d i n a t e f u n c t i o n zi

d e f i n e d by ( 4 . 1 2 ) on a n open n e i g h b o r h o o d of

Im(p). Furthermore, a

s i m i l a r argument, w i t h i n t h e p r e v i o u s c o n t e x t , concerning (3.28) i s

VI THE GEL'FAND MAP

210

c e r t a i n l y clear.

W e come now t o o u r main o b j e c t i v e i n t h i s s e c t i o n , namely, t h e e x t e n s i o n o f Theorem 3 . 1 t o t h e case o n e c o n s i d e r s " a n a l y t i c f u n c t i o n s o f s e v e r a l a l g e b r a e l e m e n t s " . T h a t i s , one g e t s t h e f o l l o w i n g .

Theorem 4.1. L e t E be a commutative complete l o c a l l y m-convex algebra w i t h ( X , V , 0 I be a given t r i p l e t f o r E. Then, t h e r e e x i s t s a continuous algebra morphism

an i d e n t i t y element and compact spectrum m(E).Moreover, l e t

~ ~ ( v x, e, ) : H o u ~ ( v z i ~ -E i))

(4.14)

i n such a m y t h a t

= hot?,

(4.15)

f o r every h e H d ( I m ( 8 I I

.

That i s , one has hf0ifi).

(4.16)

f o r every f e 772 ( E l ( s p e c t r a l mapping theorem)

I n p a r t i c u l a r , one o b t a i n s t h e re-

s p e c t i v e r e l a t i o n t o ( 4 . 1 3 1 , a s w e l l a s t h a t e x p r e s s i n g t h e p r e s e r v a t i o n of t h e i d e n t i t y elements o f t h e algebras i n v o l v e d i n 14.14).

Proof. C o n s i d e r a n Arens-Michael d e c o m p o s i t i o n of E , c o r r e s p o n d i n g t o a g i v e n l o c a l b a s i s ???= ( U c l l n e I (Theorem 111; 3 . 1 ) ; m o r e o v e r ,

let

(X,

of E , t h a t i s , l e t E = lim?, t

Y , 8 / be t h e g i v e n t r i p l e t f o r E .

Thus, one g e t s a t r i p l e t

ha' v, 0 , )

(4.17)

f o r t h e Banach algebra

2a '

(4.18)

f o r every i n d e x a E I , where one d e f i n e s

xa = ( k J c 1 & i S n

( c f . a l s o ( 4 . 1 ) a n d 1 1 1 ; ( 3 . 2 3 ) ) ; m o r e o v e r , t h e c o n t i n u o u s map t?

is

g i v e n by t h e r e l a t i o n

ea =

a e ~ , a' where p a : ??Z(k,) m(E,i--+??Z(E)a c c o r d i n g t o V i ( ( 6 . 2 1 ) and ( 6 . 2 2 ) ) . honieo Moreover, b a s e d on Lemma 4 . 1 , w e o b t a i n by V i ( 5 . 4 ) t h e f o l l o w i n g com(4.19)

eotp

m u t a t i v e diagram (non-broken a r r o w s )

(4.20)

pa

c1

4. FUNCTIONAL CALCULUS (CONTN’D.) f o r any a, 6 i n I, w i t h a < (4.21)

&+(X,

a.

21 1

Thus, one f i n a l l y g e t s a map

V , B ) : Hd(B(m(E/)) -E

making t h e ( w h o l e of t h e ) p r e c e d i n g d i a g r a m ( 4 . 2 0 ) c o m m u t a t i v e , w h i l e t h e s a m e map i s , i n f a c t , t h e d e s i r e d o n e i n ( 4 . 1 4 ) . I n t h i s r e s p e c t , one f u r t h e r p r o v e s t h a t t h e p r e v i o u s argument i s e s s e n t i a l l y i n d e p e n d e n t of t h e p a r t i c u l a r l o c a l b a s i s 72: c o n s i d e r e d a b o v e . F i n a l l y , t h e

r e l s . ( 4 . 1 5 ) and ( 4 . 1 6 ) , a s w e l l a s t h e rest ones claimed by t h e t h e orem a r e p r o v i d e d , on t h e b a s i s of Lemma 4 . 1 , r e l a t i o n s v a l i d i n t h e Banach a i g e b r a s

by t h e c o r r e s p o n d i n g

a E I , of t h e r e s p e c t i v e

Arens-Michael d e c o m p o s i t i o n of E. I n t h i s c o n c e r n , c f . M. EONNARD [3: p. 4 1 2 , ThGorSme 31 a s w e l l a s N . EUURBAKI [12: Chap. 1 ; p. 32, f f . ] .

Scholium 4.2.-

The h y p o t h e s i s made i n t h e p r e v i o u s d i s c u s s i o n t h a t

i s a compact s p a c e was o n l y f o r c o n v e n i e n c e , a s it c o n c e r n s t h e

m ( E )

d e f i n i t i o n of t h e t o p o l o g i c a l a l g e b r a ( 4 . 1 2 )

. (However,

c f . M . BUNNARD

[3]). B e s i d e s t h i s w i l l a l s o b e t h e c a s e i n t h e a p p l i c a t i o n s which w e c o n s i d e r i n t h e s e q u e l ( s e e C h a p t e r V I I I ) . On t h e o t h e r h a n d , assumi n g f u r t h e r t h e c o n t i n u i t y of t h e G e l ’ f a n d map of t h e a l g e b r a E Theorem 4 . 1 ,

one o b t a i n s s o m e o t h e r f a m i l i a r and s t a n d a r d , as w e l l ,

f a c t s of t h e r e s p e c t i v e t h e o r y of Banach a l g e b r a s ( e . g . ,

of

in

”Arens-CuZderdn Lcma’’ ) ; t h e s e

t h e analogon

we s h a l l a l s o have t h e o p p o r t u n i t y t o

a p p l y i n t h e s e q u e l ( c f . C h a p t . V I I 1 ; Theorem 8 . 2 ) .

Remark 4.1.-

Assuming t h e c o n d i t i o n s of Theorem 4 . 1

f o r t h e topo-

l o g i c a l a l g e b r a E c o n s i d e r e d , s u p p o s e f u r t h e r t h a t t h e a l g e b r a E is

f i n i t e l y generated and has t h e r e s p e c t i v e GeZ’fand map c o n t i n u o u s . Thus ( c f . D e finition Vi2.1

t e m of

and t h e comment f o l l o w i n g i t ) , i f

( x],..., x n ) 1 s a s y s -

( t o p o l o g i c a l ) g e n e r a t o r s of E , one h a s t h e r e l a t i o n

(4.22)

E = C[irc,,

. . . , 2, ,]

( s e e a l s o C h a p t . V I I 1 ; S e c t i o n 7 ) . F u r t h e r m o r e , t h e above s u p p l e m e n t a r y hypothesis f o r E i n conjunction with Corollary 7.1

of C h a p t e r V I I I

i m p l i e s now t h a t ??Z(E), t h e spectrum o f E , is (homeomorphic t o ) a compact po-

lynorniaLLy convex s u b s e t of

C n ( t h e “ j o i n t s p e c t r u m ” of t h e p r e v i o u s s y s -

t e m of g e n e r a t o r s ; c f . V I I I i ( 7 . 5 ) ) . Now a p p l y i n g t h e c o r r e s p o n d i n g c a n o n i c a l map V I I I ; ( 7 . 3 ) ( c f . a l s o

-

( 4 . 3 ) a b o v e ) , one c o n c l u d e s t h a t , for every element x e E, t h e map

2 0 @-I: @ ( i T ( E ) )=

(4.23)

belongs

LO

S

C

the uniform c2osure ( i n C i s ) ) of t h e algebra PlS) o f polynomials on S :

I n d e e d , by h y p o t h e s i s f o r E and ( 4 . 2 2 ) , one o b t a i n s

VI THE GEL'FAND MAP

212

The " c l o s u r e " i n t h e l a s t r e l a t i o n i s t a k e n , of c o u r s e , i n q S l and t h i s p r o v e s o u r a s s e r t i o n c o n c e r n i n g t h e map ( 4 . 2 3 ) . I n t h i s r e s p e c t ,

see a l s o L . H2RM4NDER [l: p. 66, Theorem 3.1.151;

t h u s , the p r e c e d i n g have

a s p e c i a l b e a r i n g on t h e r e s u l t j u s t q u o t e d . F u r t h e r a p p l i c a t i o n s of t h e above d i s c u s s i o n , i n c o n j u n c t i o n with our considerations i n Chapt.VII1; Section 7 , a r e a l s o given i n A . MALLIOS [26].

5. Appendix: G e n e r a l i z e d G e l ' f a n d map W e f i x h e r e , f o r m a l l y , t h e t e r m i n o l o g y which w e h a v e a l r e a d y ap-

p l i e d i n Lemma 1 . 1 .

Thus, g i v e n a t o p o l o g i c a l a l g e b r a E , one d e f i n e s

t h e generalized Gel'fand transform

, with

o f a n e l e m e n t xe E

r e s p e c t t o an-

o t h e r ( kept f i x e d throughozct t h i s a p p e n d i x ) t o p o l o g i c a l a l g e b r a F , by t h e relation

P I L L 1 := u l x l

(5.1) f o r e v e r y u e Hams(E, F l

.

,

(Cf. a l s o D e f i n i t i o n V; 8.1

o f t h i s c h a p t e r ) . So one a c t u a l l y o b t a i n s

Homs(F, F )

,

,a s

w e l l a s Lemma 1 . 1

a continuous F-valued map on

a s e a s i l y f o l l o w s from t h e s a m e d e f i n i t i o n s (see a l s o Lem-

ma 1 . 1 ) . Thus, one c o n s i d e r s t h e c o r r e s p o n d i n g generaZized Gel'fand map of E , d e n o t e d by

gF; i.e.,

gF : E

(5.2) with

one h a s , by d e f i n i t i o n , t h e map

$Fix) = 2 ,

--+

C (Hams (E,F), F) ,

f o r e v e r y x € E , where 2 i s g i v e n b y ( 5 . 1 ) .

On t h e o t h e r h a n d , t h e image of t h e p r e v i o u s map ( 5 . 2 ) i s , by

generalized Gel'fand transform algebra of E , i n f a c t , a sub-

definition, the

a l g e b r a of t h e r a n g e of

2F, t h e

l a t t e r map b e i n g of c o u r s e an a l g e b r a

morphism f o r t h e a l g e b r a s i n v o l v e d i n ( 5 . 2 ) . Now, one h a s a s i m i l a r i n t e r p r e t a t i o n o f t h e t o p o l o g y i n t h e space Hoin(E,F)

t o t h a t p r o v i d e d by Lemma V; 1 . 1

i n c a s e of t h e u s u a l

s p e c t r u m of a g i v e n t o p o l o g i c a l a l g e b r a E . Indeed, t h e topology of simple convergence i n

C. ( E , F 1

is essen-

t i a l l y t h e i n i t i a l t o p o l o g y d e f i n e d on t h i s s p a c e by t h e maps (5.3) and f o r every

2 : u - 2 1 ~ ) := u(x1 , w i t h u J: E

E

€

I: ( E , F)

,

( c f . a l s o N . BOURBAKI [5: Chap. 10; p. 1 3 1 ) . The r e s t r i c -

5.

GENERALIZED GEL’FAND MAP

t i o n s of t h e s e maps on HomiL’, F i

213

a r e , of c o u r s e , t h e maps d e f i n e d by

( 5 . 1 ) . Thus, one g e t s t h e f o l l o w i n g . Lemma 5.1.

Let E , F be topological algebras and Hum (E,F) t h e r e s p e c t i v e gen-

e r a l i z e d spectrum of E ( w i t h r e s p e c t t o F ; c f . D e f i n i t i o n V ; 8 . 1 1 . Then, t h e t o p o l o g y of t h e Latter space coincides w i t h t h e i n i t i a l topology defined on i t by t h e

generalized Gel’fand transforms of t h e elements o f E . I Remark 5.1.-

I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t t h e s a m e Lem-

m a 1 . 1 above p r o v i d e s a l r e a d y a c r i t e r i o n f o r t h e c o n t i n u i t y of t h e generali z e d Gel’fand map of a g i v e n t o p o l o g i c a l a l g e b r a izing, i n particular,

t h e family

6

E ; namely, by s p e c i a l -

( i b i d . ) t o t h a t of a l l t h e compact

s u b s e t s of t h e r e s p e c t i v e g e n e r a l i z e d s p e c t r u m (see a l s o , f o r i n s t a n c e , Theorem 1 . 1 )

.

This Page Intentionally Left Blank

21 5

Spectra o f Certain P a r t i c u l a r Topological Algebras

CHAPTER

VII

W e c o n s i d e r i n t h i s c h a p t e r t h e s p e c t r a of c e r t a i n p a r t i c u l a r

t o p o l o g i c a l a l g e b r a s , which a r e i m p o r t a n t i n t h e a p p l i c a t i o n s . Thus, a l l t h e a l g e b r a s examined a r e , i n e f f e c t , a l g e b r a s of complex-valued f u n c t i o n s h a v i n g e x t r a supplementary p r o p e r t i e s , w h i l e t h e i r s p e c t r a a r e i n most c a s e s c a n o n i c a l l y i d e n t i f i e d w i t h t h e p a r t i c u l a r domain of d e f i n i t i o n of t h e f u n c t i o n s i n v o l v e d ( c f . , however, t h e a l g e b r a 1 L (G) i n t h e s e q u e l ) .

1 . Spectrum o f t h e algebra

c(X) i s a compact space. I n t h i s re-

W e consider f i r s t t h e case t h a t X

s p e c t , w e r e c a l l t h a t a l l t o p o l o g i c a l s p a c e s c o n s i d e r e d a r e assumed t o be Hausdorff

u n l e s s it i s i n d i c a t e d o t h e r w i s e .

Now, t h e a l g e b r a

C i X ) , f o r any t o p o l o g i c a l s p a c e X , h a s been

c o n s i d e r e d a s a t o p o l o g i c a l a l g e b r a a l r e a d y i n Example I ; 3 . 1 . So i n t h e p a r t i c u l a r case c o n s i d e r e d h e r e i n , t h e "compact-open t o p o l o g y " i n

CIXl c o i n c i d e s w i t h t h e "sup-norm

(i.e.

,

u n i f o r m ) t o p o l o g y " on

X ,

g i v e n by t h e r e l a t i o n

/IfII

(1.1)

:=

SuPlf(Z)I T.

f o r every

f

f

c ( X 1 . The a l g e b r a

e

x

CiX)

I

t h u s t o p o l o g i z e d becomes

a

(complex) commutative Banach algebra w i t h an i d e n t i t y element; we d e n o t e t h i s a l g e b r a by

Cutxi, a s

well.

Now a s a f i r s t s t e p towards t h e c o n c r e t e d e s c r i p t i o n of

mtCzl(X)),

one r e a l i z e s t h a t X i s c a n o n i c a l l y imbedded i n t h e l a t t e r s p a c e , a f a c t t h a t i s a c t u a l l y v a l i d f o r e v e r y c o m p l e t e l y r e g u l a r s p a c e X. So one h a s t h e f o l l o w i n g g e n e r a l r e s u l t .

Lemma 1 . 1 . Let X be a completely regular space and c c ( X ) t h e l o c a l l y mconvex algebra of complex-valued continuous f u n c t i o n s on X, endowed w i t h the compact-open topology ( c f

.

Example I ; 3 . 1 )

.

Thee, by considering t h e weak topolo-

g i c a l dual ic c ( X i I ' o f t h e previous l o c a l l y convex space, t h e following map

216

(1.2)

SPECTRA OF PARTICULAR ALGEBRAS

VII

6 :x

-

(ectx)I;

-

:

6 ( x ) = 6,

:f

-

&,if)

:= f ( x )

d e f i n e s a homeomorphism of X i n t o t h e range of

6 ; i n p a r t i c u l a r , the l a t t e r space i s e s s e n t i a l l y contained i n t h e spectrum of q X l . That i s , 6x = 61x) y i e l d s ( b y ( 1.2) )

a (continuous) character o f q X i , f o r every x E X . Proof.

I t i s c l e a r by

6x :

(1.3)

-

( 1 . 2 ) t h a t , f o r e v e r y x E X , t h e map

c(X)--+

c :f

6x(f) = fix)

d e f i n e s a l i n e a r form on C l X i , which i s c o n t i n u o u s f o r t h e t o p o l o g y o f s i m p l e c o n v e r g e n c e i n X I h e n c e a f o r t i o r i f o r t h e s t r o n g e r compactopen t o p o l o g y ; so t h e r a n g e of 6 i s , i n d e e d , g i v e n b y ( 1 . 2 ) . Furthermore,

t h e map (1.2) i s o n e - t 2 - o n e :

T h i s i s , of c o u r s e , a

c o n s e q u e n c e o f t h e same d e f i n i t i o n of a c o m p l e t e l y r e g u l a r s p a c e , a c c o r d i n g t o which " 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 on X s e p a r a t e p o i n t s and c l o s e d s u b s e t s o f X " ( c f . , f o r i n s t a n c e , J . R . MUNKRES [l: p. 236, D e f i n i t i o n ] ) . I t i s s t i l l a n e a s y c o n s e q u e n c e o f t h e d e f i n i t i o n o f t h e t o p o l o g y i n ( c c l X ) l L t h a t 6 i s a continuous map a s w e l l . On t h e o t h e r h a n d , i f i n g t o an e l e m e n t 6,eIm(6/

xi-x

i n X:Otherwise,

(6,

i

i i s a n e t i n I m ( 6 I C - i e c l X l ) ; converg-

i n t h e r e l a t i v e t o p o l o g y , w e s h a l l show t h a t

t h e r e would e x i s t an open n e i g h b o r h o o d U o f x

i n X s u c h t h a t t h e n e t ( z i J i e I t o b e e v e n t u a l l y i n C U (i.e., f o r e v e r y i E I , t h e r e would e x i s t j S i , w i t h z E C U ) . B e s i d e s by h y p o t h e s i s f o r X j ( U r y s o n ' s L e m m a ) , t h e r e would e x i s t a n e l e m e n t f E C t X ! , w i t h f i x ) = 1 and f ( y ) = 0, for e v e r y y e U. Hence, b y h y p o t h e s i s for (6, ), one g e t s i l=flx)=6,(fl=l~6 (f)=l+f(xi), so t h a t \ f ( x i ) \ > l ,f o r e v e r y i ? i o, .b xi 7, f o r some i , E I ; so a c o n t r a d i c t i o n t o t h e f a c t t h a t (xi) i s e v e n t u a l l y i n cii and t h e d e f i n i t i o n of f. T h e r e f o r e , t h e inverse map of 6 i s continuous ( o n t h e r a n g e of 6). I n t h i s r e s p e c t , w e s t i l l n o t e t h a t t h e c o n t i n u i t y of 6 - l i n ( 1 . 2 ) i s a c o n s e q u e n c e o f t h e f a c t t h a t t h e topology of X i s e x a c t l y t h e "weak topology" of i t s continuous f u n c t i o n s : C f . "Embedding L e m a Ir ; J . L . KELLBY [ l : p. 1161. (We h a v e an a n a l o g o u s s i t u a t i o n i n case of a S t e i n s p a c e , where now t h e h o l o m o r p h i c f u n c t i o n s p l a y t h e r61e t h a t do h e r e t h e c o n t i n u o u s f u n c t i o n s ; see S e c t i o n 3 b e l o w ) . F i n a l l y , it i s c l e a r by ( 1 . 3 ) t h a t 6 x , x E X , d e f i n e s a complex a l g e b r a morphism o f C i X ) w h i c h , a s n o t e d b e f o r e , i s a l s o c o n t i n u o u s f o r t h e t o p o l o g i c a l a l g e b r a cc(XI; c e r t a i n l y i t i s non-zero, since t h e algebra

c ( X /

by ( 1 . 3 ) ,

c o n t a i n s t h e c o n s t a n t f u n c t i o n s . Hence, we

c o n c l u d e t h a t 6,~722( CclXl) , f o r e v e r y x E E , a n d t h i s c o m p l e t e s t h e p r o o f o f t h e lemma. I T h e map ( 1 . 2 )

i s a l s o c a l l e d t h e Dirac ( o r e l s e e v a l u a t i o n ) map on

217

X; i t s v a l u e a t a p o i n t x e X i s t h u s t h e Dirac ( o r p o i n t (Radon))measur'e a t x ( s e e , f o r i n s t a n c e , N . BOURBAKI [8: Chap. 3 ; p . 481 ) . Thus, a s a consequence of t h e p r e c e d i n g lemma, one o b t a i n s t h e relation

I~(U= UX)=

(1.4)

rnr c c ( x l ) ,

w i t h i n a homeomorphism ( i n t o ) ; i t i s a c t u a l l y an onto homeomorphism s h a l l see, f i r s t f o r X compact ( C o r o l l a r y 1 . 2 )

, as

we

and t h e n f o r e v e r y com-

p l e t e l y r e g u l a r s p a c e , i n g e n e r a l (Theorem 1 . 1 ) . T h a t i s , one r e a l i z e s t h a t t h e p o i n t s of X are t h e only characters of t h e Banach a l g e b r a q

X

)

, with

X compact , o r y e t

cc(X) i n the

rn-convex a l g e b r a

t h e only continuous ones

of t h e l o c a l l y

g e n e r a l c a s e of a c o m p l e t e l y r e g u l a r

s p a c e X. W e f i r s t comment a l i t t l e b i t more on t h e t e r m i n o l o g y a p p l i e d

i n t h e s e q u e l . Thus g i v e n a t o p o l o g i c a l s p a c e X and t h e r e s p e c t i v e algebra

c ( X ) a s above, w e s e t f o r any A

I ~ c=f

(1.5)

E

cX =

c ( X ) : f

!A

oI

;

y e t by a p p l y i n g t h e n o t a t i o n of I I ; ( 7 . 2 8 ) one h a s =

(1.6)

tf e

m ) z:( f ) =

A1

,

where one d e f i n e s

Zif) = I x e x : f (x) = 0 1

(1.7) t h a t is, the zero-set

I

of f e CiX).

Now, it i s c l e a r t h a t IA i s a (%sided) i d e a l of t h e (commutative)

algebra c ( X ) , f o r every A c a l l y rn-convex a l g e t r a

c X . F u r t h e r m o r e , I A i s a closed subset of t h e loc c i X ) ( s e e Example I ; 3.1 ) ; t h i s i s o b v i o u s l y

t r u e , by ( 1 . 5 ) , f o r t h e t o p o l o g y s of s i m p l e convergence i n X and s o a f o r t i o r i f o r t h e s t r o n g e r t o p o l o g y c of compact convergence i n X . Thus, IA i s a c l o s e d

( 2 - s i d e d ) of

cciX).

Moreover, i f X i s a complete2y regular space

and A a non-empty closed

IA i s a n o n - t r i v i a l closed proper i d e a l of q X 1 . B e of c o u r s e , t h a t I@ = CIX) and I = {O} C C t X ) . However, X

proper s u b s e t of X , t h e n s i d e s , one h a s ,

t h e i m p o r t a n t t h i n g h e r e i s c e r t a i n l y t h e f a c t t h a t , i n case of a comp l e t e l y r e g u l a r s p a c e , t h e c o n v e r s e of t h e l a s t s t a t e m e n t i s a c t u a l l y t r u e ( c f . Lemma 1 . 5 ) . W e f i r s t prove i t f o r compact s p a c e s ( C o r o l l a r y 1.1).

Thus, w e s t a r t w i t h t h e f o l l o w i n g a u x i l i a r y lemmas. Lemma 1.2.

Let X be a compact space and I an idea2 of t h e algebra e(X). Be-

s i d e s , assume t h a t I "separates p o i n t s of X " ( i . e . , we assume that, f o r every p o i n t x E X , t h e r e e x i s t s an element f E I such t h a t f (xi # 0). Then, I =

c(Xl .

218

VII

SPECTRA OF

PARTICULAR ALGEBRAS

Proof. By t h e c o n t i n u i t y o f t h e f u n c t i o n f E I and t h e h y p o t h e s i s

f o r t h e e l e m e n t x f X , one g e t s a n open n e i g h b o r h o o d U, of x s u c h t h a t f n e v e r v a n i s h e s on L i . Hence, t h e r e e x i s t s b y h y p o t h e s i s f i n i t e many s u c h f u n c t i o n s , s a y fl,... f n , Of I , c o r r e s p o n d i n g t o t h e f i n i t e o p e n c o v e r i n g of X d e f i n e d by t h e open c o v e r i n g U-, x e X . So t h e f u n c t i o n (1.a)

which c e r t a i n l y b e l o n g s t o I , h a s t h e p r o p e r t y t h a t gtx)

(1.9)

and h e n c e i f

h=-

1

0, f o r e v e r y x E X ,

e e ( X i , one g e t s h . g = l e I , i.e., I = C t X ) . I

9

Remark 1.1.A s f o l l o w s from t h e p r e c e d i n g p r o o f , u n d e r t h e h y p o t h e s i s of t h e above Lemma 1 . 2 , t h e r e ( X i , g i v e n by ( 1 . 8 ) , which e x i s t s a function g i n I E never v a n i s h e s on X . ( S o t h e r e e x i s t s t h e i n v e r s e f u n c t i o n of g and 1 = g . L E I). I t i s t h i s a u x i l i a r y re9 s u l t , d e r i v e d from t h e p r e c e d i n g p r o o f , which w i l l c o n s t a n t l y b e a p p l i e d i n t h e s e q u e l . I n t h i s re s p e c t , i t i s o f c o u r s e e q u i v a l e n t w i t h Lemma 1 . 2 t o say t h a t :

e

I i s a proper i d e a l of C t X ) i f , and o n l y if, t h e r e e x i s t s a p o i n t x f X , w i t h f ( x i = 0 , f o r every f E I.

Y e t a p p l y i n g t h e n o t a t i o n of ( 1 . 1 5 ) below, t h e previous statement is equivalent with t h e r e l a t i o n

I GI, f o r some x e X . ( I n t h i s c o n c e r n , see a l s o t h e n e x t Remark 1 . 2 )

.

Lemma 1.3. Let X be a compact space and I an i d e a l of t h e algebra e t X ) . Bes i d e s , consider the s e t (1.10)

and an element @ of C (x), w i t h t h e p r o p e r t y t h a t t h e r e e x i s t s an open neighborhood U of A in X on which @ v a n i s h e s ; t h a t i s , assume t h a t

(1.11)

A G U G Z(@).

Then, @ E I .

Proof. By ( 1 . 1 1 ) and t h e d e f i n i t i o n of A

,

one v e r i f i e s € o r t h e

compact s p a c e K a C l i c X and t h e " r e s t r i c t i o n o f t h e i d e a l I" t o K

(de-

IIK) t h a t t h e c o n d i t i o n s of t h e p r e v i o u s Lemma 1 . 2 a r e s a t i s f i e d . T h e r e f o r e , t h e r e e x i s t s a f u n c t i o n g e C ( K ) ( = I ) which n e v e r IK v a n i s h e s on K (see a l s o Remark 1.1 ) Thus a p p l y i n g l ' i e t z e ' s Extension n o t e d by

.

Theorem ( c f . , f o r i n s t a n c e , J . DUGUNDJI [1: p. 1 4 9 , Theorem 5.11) one ob1 t a i n s a n e l e m e n t h e C t X l e x t e n d i n g - e C I K ) s u c h t h a t one h a s ( s e e g

1.

219

SPECTRUM OF c ( X )

also ( 1 . 1 1 ) ) (1.12)

(we actually consider here the extension of g to the hole of X ) , and this finishes the proof. I Lemma 1.4. Let X be a compact space and I an i d e a l of t h e algebra

CtX)

.

Then, one g e t s t h e r e l a t i o n

(1.13)

" I A ,

where A C X is given by ( 2 . 1 0 ) and

7 denotes

t h e closure of I in the Banach alge-

bra e i x ) . Proof. Let @ $,

8 IA

, with

@

# 0 , and

0. Then, by hypothesis for

E

the sets

(1.14)

M = { z 8 X :

I@(zII5 E } and N = { r e X : ( @ ( rE)] ( >

define two non-empty closed subsets of X , with M n N = 0. Hence, since X is, in particular a normal space, there exists ( Uryson's L e m ) an element g E ClX),with 0 < g 6 1 , and in such a manner that M E Z ( g l and

(1.15)

g = l

on N .

Thus, by definition of M and (1.15), one obtains set u = C r e X : (qdrllc 1

$ g = 0 on the open

-$-

hence, in particular, A C U E Ziggl.

Therefore (Lemma 1.3) , $9 €I, so that for any and (1.15))

/ 1 ~ - s ~= /ll@(1-9)ll /

E

> 0 , one gets (cf. (1.14)

< E r

i.e., $E?, and hence I A C I . Moreover, I E I A (cf. (1.6) and (l.lO)), S O that since I A is a closed ideal of the preceding yields already the proof of the asserti0n.I

e(X),

Thus, the previous discussion provides already the following basic result. That is, we have

Corollary 1 . 1 . Let X be a compact space and C (XI t h e respective Banach a l gebra of comptex-vatued continuous functions on X i n t h e uniform (sup-norm) topology in X. Furthermore, l e t F t X ) be the s e t of a l l non-empty closed and proper subs e t s of X, and

J ( C t x ) ) t h a t of non-trivial ctosed and proper i d e a l s of t h e Ba-

nach algebra C i X l . Then, the map

(1.16)

8: F(X)-

J i C ( X ) ) : A -eiAl:=

'A '

220

VII

SPECTRA OF PARTICULAR ALGEBRAS

&ere IA i s given by ( 1 . 5 ) , y i e l d s a b i j e c t i o n between t h e r e s p e c t i v e s e t s . t h e range o f 8 i s J I

Proof. W e h a v e a l r e a d y n o t i c e d t h a t

f o r a completely regular space X )

. NOW,

,B

if A

CIX))(even

a r e any two members of F ( X )

with A # F , then t h e r e e x i s t s ( X is a f o r t i o r i a completely regular

C I X l i n s u c h a way t h a t o n e h a s , f o r i n s t a n c e , A G Z I f ) and f ( x ) = l , f o r some X E B ~ C A T . h u s , i n any c a s e , o n e o b t a i n s I A # IB, t h a t i s t h e map 8 i s o n e - t o - o n e . F u r t h e r m o r e , i f I e l J ( C ( X ) ) and I , i s t h e c o r r e s p o n d i n g i d e a l of C i X ) d e f i n e d by ( 1 . 5 ) and ( l . l O ) , s p a c e ) a n e l e m e n t f~

H

one h a s

Lemma I . 4 ) e(A)=I =I=i, A

(1.17)

and t h i s f i n i s h e s t h e p r o o f . 1

I n p a r t i c u l a r , w e now g e t t h e f o l l o w i n g .

Theorem 1.1. Suppose we have t h e c o n t e x t o f t h e preceding Corollary 1.1. Then, t h e r e e x i s t s a o n e - t o - o n e and onto CGrrespondenCe between t h e s e t o f a l L maximal i d e a l s of t h e Banach algebra C I X ) and t h e p o i n t s o f X ( d e r i v e d from t h e res p e c t i v e r e s t r i c t i o n of t h e above map ( 1 . 1 6 ) ) .

I I f I i s a maximal i d e a l of C I X l , t h e n (Lemma 1 . 4 ) , iA= i s g i v e n b y ( 1 . 1 0 ) ( e v e r y maximal i d e a l of a & - a l g e b r a w i t h a n i d e n t i t y e l e m e n t and h e n c e of t h e Banach a l g e b r a C l X ) , i s c l o s e d ; Proof.

where A E X

c f . Theorem I I ; 6 . 1 ) . N o w I I = I=~I ~ . (~s o ) since

e

f o r e v e r y x e A , one g e t s

Furthermore, f o r any x

€

I A = II x } I h e n c e

one g e t s A = { z > a s w e l l ) .

i s 1-1,

the set

X

I =I ={fEetxi:frxi=oi x tx}

(1.18)

d e f i n e s a maximal i d e a l of C i X l : I n d e e d s u p p o s e , o t h e r w i s e , t h a t I i s a maximal i d e a l of C i X ) w i t h I GI; t h e n

by t h e p r e c e d i n g o n e h a s

5

i G I = I X

f o r some p o i n t

Y

Y E X . Now, t h e l a s t r e l a t i o n e n t a i l s t h a t I GI x Ix,yl

Ix

'

s o t h a t one h a s 8 1 1 d ) = I, = 'i'hat

is (Corollary 1.1)

, x=y,

Yl =

e(b, ~ 1 ) .

a n d hence I = I which i s t h e a s s e r t i o n , X

and t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . 1 The f o l l o w i n g i s now a d i r e c t a p p l i c a t i o n o f Lemma 1.1 and t h e p r e v i o u s Theorem 1 . 1 , i n c o n j u n c t i o n w i t h C o r o l l a r y 11; 7 . 3 . T h u s I w e have.

221

Corollary 1 . 2 . L e t X be a compact space and C(X/ t h e r e s p e c t i v e Banach a l gebra, as above. Then, concerning t h e s p e c t r m of C(X/,one has t h e r e l a t i o n

m(e(x)) = X

(1.19)

,

w i t h i n a homeomorphism o f t h e r e s p e c t i v e topological spaces ( g i v e n by

(

.I

1.2) )

.-

Schol i u m 1 . I S i n c e e v e r y maximal i d e a l of a Banach a l g e b r a ( a n d y e t , more g e n e r a l l y , of a Q-alg e b r a ) w i t h a n i d e n t i t y e l e m e n t i s c l o s e d (Theorem 11; 6 . 1 ) , i t i s a c o n s e q u e n c e of t h e p r e v i o u s r e l a t i o n ( 1 . 1 9 ) t h a t t h e topology of a compact space X i s compZeteZy determined by t h e algebra ( a c t u a l l y r i n g ) structure of i t s r e s p e c t i v e “function algebra” C I X l ; t h i s e s s e n t i a l l y amounts t o t h e c l a s s i c a l Banach-Stone Pheorern, a c c o r d i n g t o which two compact spaces are homeomorp h i c if, and o n l y i f , t h e i r r e s p e c t i v e f u n c t i o n algebras are isomorphic ( a s r i n g s ) . Now, w i t h i n t h e p r e c e d i n g framework, t h e l a s t r e s u l t t u r n s o u t t o be t h e b e s t p o s s i b l e ; namely, t h e l o c a l l y m-convex algebra C,(Xl, w i t h X 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 a &-algebra i f , and only i f , X i s a compact space ( c f . , f o r i n s t a n c e , W . DIETRICH, J r . [3: p. 58, Theorem 2.1.5, i)])

.

T h u s , w e come now t o t h e p r o m i s e d e x t e n s i o n of the p r e v i o u s C o r o l l a r y 1 . 2 t o t h e case one h a s a n a r b i t r a r y compZeteZy regular space

x; b u t

w e f i r s t g e t a s i m i l a r e x t e n s i o n of C o r o l l a r y 1 . 1 . T h a t i s , w e h a v e .

Lemma 1.5. Let X be a completely regular space and C (Xl t h e l o c a l l y m-conv e x algebra of complex-valued continuous f u n c t i o n s on X i n t h e compact-open topology. Then, there e x i s t s a one-to-one and onto correspondence between t h e s e t o f a l l non-empty closed and proper s u b s e t s o f X and t h a t o f n o n - t r i v i a l closed and proper i d e a l s o f c c I X 1 , g i v e n by t h e r e s p e c t i v e map t o (1.161. Proof. W e h a v e a l r e a d y remarked i n t h e p r o o f of C o r o l l a r y 1 . 1

t h a t t h e map 0 , g i v e n by ( 1 . 1 6 ) , d e f i n e s a n i n j e c t i o n f o r e v e r y comp l e t e l y r e g u l a r s p a c e X ; so it r e m a i n s a c t u a l l y t o p r o v e t h a t ‘8 i s an

onto map: Thus, a p p l y i n g t h e n o t a t i o n of C o r o l l a r y I . 1 , w e

must p r o v e t h a t

i s of t h e form IA , where t h e s e t A -C X i s g i v e n by ( 1 . l o ) . Moreover, s i n c e w e a l w a y s h a v e t h a t I C IA, we are a c t u a l l y led t o prove t h e every l e J l e c ( X I I relation

I r?=1 A

(1.20)

( t h a t i s , t h e r e s p e c t i v e r e l a t i o n t o ( 1 . 1 3 ) ) . Thus, by d e f i n i t i o n o f t h e topology i n

cc(Xl

,

i f I$e IA one h a s t o p r o v e t h a t , f o r any

and K a compact s u b s e t of X I t h e f o l l o w i n g r e l a t i o n i s v a l i d (1.21)

p K ( b- h / <

E

,

E

>0

222

VII

f o r some h e r , where

SPECTRA OF PARTICULAR ALGEBRAS

i s d e f i n e d by I ; ( 3 . 1 3 ) :

p,

t o t h e compact set

Thus, r e s t r i c t i n g t h e g i v e n i d e a l I e J ( C c ( X ) I K : c X ( c o n s i d e r e d by ( 1 . 2 1 ) ) , i . e . ,

transpose

t a k i n g t h e image of I u n d e r

j, = tiK o f t h e c a n o n i c a l i n j e c t i o n

the

i, : KS X, one g e t s t h e

set

fl,

j,(~) = { j K ( f ): f E I 3 = c

(1.22)

CIKI.

which, i n f a c t , i s an i d e a l o f the algebra

S

:f e

r1 g e C ( K l , there

Indeed, i f

c ( X l e x t e n d i n g g (“every compact K C X is c ( X i embedded”; c f . L . GILLMAN-M. JERISON [l: p. 43, (c)] , o r y e t S. WARNER [5: p.

e x i s t s a function

E

2661); so one o b t a i n s

K if) = jK (S)-j K (f) = j,(g.f) E jK(I) , f o r e v e r y f 8 I ( t h e “ r e s t r i c t i o n map“ j, i s , of c o u r s e , an a l g e b r a g.j

morphism), and t h i s p r o v e s t h e above a s s e r t i o n . NOW,

C(K),

a p p l y i n g Lemma 1.4 t o t h e Banach s l g e b r a

(cf. (1.13))

-

,

jK(I)= IA,

(1.23)

one g e t s

where one d e f i n e s

(1.24) F u r t h e r m o r e , one h a s

n z(j,(j-)~

=

f €1

n

(zifi

n

KI = (

f E I

n

z(f)) n

I(

= A nK

,

f E I

so t h a t one c o n c l u d e s by ( 1 . 2 4 ) t h a t ___

j (I) = I A n K K

(1.25) Therefore, s i n c e

4 € I A‘ I A n K

.

t h e r e e x i s t s by ( 1 . 2 5 )

,

f o r any g i v e n E > O

( a s i n ( l . 2 1 ) ) l an e l e m e n t g e j K ( I ) , t h a t i s , g - j K I h ) , with h E I , i n such a way t h a t

Ilm-4,

= PK(Q-g) = p , ( @ - h h

So t h e d e s i r e d r e l a t i o n ( 1 . 2 1 )

:

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

t h e proof of t h e 1emma.I Remark 1.2.We c a n a c t u a l l y e x t e n d t h e map (1.16) t o a l l c l o s e d s u b s e t s of ( t h e c o m p l e t e l y r e g u l a r s p a c e ) X and c l o s e d i d e a l s of by s e t t i n g

c,(XI,

(1.26)

I@=

CIx)

and

I~ =

to}.

Thus, one o b t a i n s , w i t h i n t h e c o n t e x t of t h e precedi n g Lemma 1 . 5 , a one-to-one c o r r e s p o n d e n c e of t h e s e t of a l l c l o s e d s u b s e t s of X o n t o t h e s e t of c l o s e d i d e a l s of C J X I ( a s o - c a l l e d “GaZois correspondence” 1 .

223

Now, i n a n a l o g y w i t h Theorem 1 . 1 ,

one g e t s , i n p a r t i c u l a r , t h e

following.

C o r o l l a r y 1.3. Assume t h a t ~3 have t h e c o n t e x t of t h e preceding Lemma 1 . 5 . and onto correspondence between t h e s e t Gf c l o s e d

Then, t h e r e e x i s t s a one-to-one

maximal i d e a l s of ( t h e l o c a l l y m-convex a l g e b r a ) q X l and t h e p o i n t s o f X , g i v e n by t h e r e l a t i o n

f o r every x E X .

Proof.

I t i s a consequence of t h e p r e v i o u s Lemma 1 . 5 t h a t t h e

map (1.27) i s one-to-one,

where

I, i s a c l o s e d maximal i d e a l of

ec(Xi.

For by (1.27) one h a s t h e r e l a t i o n

I3: = kerf&,)

(1.28)

,

where S , E ~ ( ~ ~ ( X (, c) f . (1.4)) i s g i v e n by (1.3)(.,-c-e also Lemma 11:7.2). O n t h e o t h e r hand,

if

I

€

J ( c c ( X i ) i s , i n p a r t i c u l a r , a ( c l o s e d ) maxi-

m a l i d e a l of e c f X i , t h e n by Lemma 1.5 one h a s I =IA , f o r some ( u n i q u e l y defined) A

€

F I X ) . Thus, f o r e v e r y

A , one h a s by h y p o t h e s i s f o r I

L€

the relation 1=1 = I A

x'

and t h i s p r o v e s t h e a s s e r t i 0 n . I Thus, w e now g e t t h e f o l l o w i n g fundamental r e s u l t , a n a p p l i c a t i o n of t h e p r e v i o u s C o r o l l a r y 1.3, i n c o n j u n c t i o n w i t h Lemma 1 . 1 and C o r o l l a r y II;7.2 r e f e r r e d t o t h e l o c a l l y m-convex a l g e b r a

c (Xl. Name-

l y , we h a v e . Theorem 1.2. L e t X be a completely r e g u l a r space and

c,(X)t h e

l o c a l l y m-

convex algebra of complex-valued continuous f u n c t i o n s on X i n t h e compact-open t o pology. Then, t h e spectrum of

(1.29)

eJX)i s g i v e n by

the relation

m(ccixi)= X ,

w i t h i n a homeomorphism of t h e r e s p e c t i v e spaces ( d e f i n e d by t h e map ( 1 . 2 ) )

.

A s a m a t t e r of f a c t , one c o n c l u d e s i n p a r t i c u l a r t h a t : The map 6 ( c f . (1.2)) i s a homeomorphism i f , and only if, t h e t o p o l o g i c a l space X i s completely r e g u l a r .

(The "holomorphic analogon" of t h e l a s t s t a t e m e n t is g i v e n by Theorem 2.1 b e l o w ) . Now, by c o n s i d e r i n g t h e G e l ' f a n d map of t h e a l g e b r a has

(1.30)

?(xi =

xff)

= Axif) = f(x)

,

Cc(Xi,one

224

VII SPECTRA OF PARTICULAR ALGEBRAS

f o r any f E q ous c h a r a c t e r by (1.29)

.

) a n d x e ~ ( c c ( X i l w; e h a v e i d e n t i f i e d h e r e a c o n t i n u -

X

x

of

CJX) w i t h

the c o r r e s p o n d i n g p o i n t x e X

defined

Thus, t h e r e s p e c t i v e Gel'fand map of t h e algebra C J X i i s t h e

-

.

i d e n t i t y map, and t h e r e f o r e continuous. Namely, w e h a v e (see a l s o V I ; ( 1 1 ) )

g : cctx)

(1.31)

c p.

ecim(c c i X ) i i

So i f X i s , i n p a r t i c u l a r , a locaZZy compact space

then consider-

i n g X I v i a ( 1 . 2 9 ) , as t h e s p e c t r u m of t h e a l g e b r a C c I X I , o n e o b t a i n s by t h e p r e v i o u s c o n c l u s i o n , c o n c e r n i n g ( 1 . 3 1 ) , t h a t

X i s l o c a l l y equi-

continuous ( c f . C o r o l l a r y V I ; 1 . 3 . See a l s o t h e n e x t c h a p t e r , Example 1.1). I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t it may happen t h a t e v e r y c h a r a c t e r o f a n a l g e b r a o f t h e form q

X

I

t o b e g i v e n by ( t h e r e -

s p e c t i v e " e v a l u a t i o n map" a t ) some ( u n i q u e l y d e f i n e d ) p o i n t of X

,

w i t h o u t X t o be n e c e s s a r i l y a compact s p a c e . So t h i s i s , f o r i n s t a n c e , t h e case i f X i s a c o m p l e t e l y r e g u l a r Lindellif space ( c f . E . A . MICHAEL [l: p. 54, P r o p o s i t i o n 12.51

,

as w e l l as S e c t i o n 3 i n t h e s e q u e l ) .

cm(X)

2. Spectrum o f t h e algebra

w e consider next t h e al-

A s t h e t i t l e of t h i s s e c t i o n i n d i c a t e s ,

gebra of

( c o m p l e x - v a l u e d ) e m - f u n c t i o n s on a g i v e n ( f i n i t e d i m e n s i o n a l ) e " - m a n i f o l d X. Thus, w e h a v e s e e n a l r e a d y i n C h a p t e r IV;4. ( 2 ) t h a t b y assuming X t o b e s e c o n d c o u n t a b l e ( c f . IV; ( 4 . 1 9 ) ) c"iXi m u t a t i v e ) Frgchet l o c a l l y m-convex algebra NOW,

i s a (com-

(with an i d e n t i t y element)

.

t h e canonical i n j e c t i o n

i : C"(x)--Cctxi

(2.1)

i s a continuous map

by t h e same d e f i n i t i o n of t h e t o p o l o g i e s o f t h e t o -

p o l o g i c a l a l g e b r a s i n v o l v e d ; namely t h e S c h w a r t z t o p o l o g y i n c " l U l

,

w i t h U open i n X I i s by i t s d e f i n i t i o n s t r o n g e r t h a n t h e compact-open t o p o l o g y i n C t U i (see I V ; ( 4 . 1 3 ) 1 . So o n e g e t s by ( 1 . 2 ) a

continuous map

6 :X

(2.2)

-

(canonical)

(c"cX),;

which i s g i v e n by t h e a n a l o g o u s r e l a t i o n t o ( 1 . 3 ) ; i . e . ,

by " e v a l u a t -

i n g " a t any g i v e n p o i n t x E X ( i n f a c t , b y a n o b v i o u s a b u s e o f n o t a t i o n , w e h a v e i d e n t i f i e d t h e a b o v e map 6 w i t h t h e map ti o 6 1 . F u r t h e r m o r e , i t i s a l s o c l e a r t h a t t h e r a n g e o f 6 i s c o n t a i n e d i n t h e s p e c t r u m of C m ( X / , i.e.

(2.3)

,

one h a s

rm (6)

=_

6(;:) G r n 1 e m t x ) )

.

On t h e o t h e r h a n d , s i n c e X i s l o c a l l y compact ( a s b e i n g " l o c a l -

2.

SPECTRUM OF

c"(X)

225

ly e u c l i d e a n " ) , i f it i s , m o r e o v e r , s e c o n d c o u n t a b l e t h e n X i s a l s o a paracompact s p a c e ; t h i s w i l l b e q u i t e f u n d a m e n t a l f o r t h e s e q u e l ( c f . ,

f o r i n s t a n c e , S. STERNBERG[I: p. 55,Lemma 4 . 1 1 f o r a p r o o f of t h e l a s t a s s e r t i o n , as w e l l a s f o r t h e p e r t i n e n t d e f i n i t i o n s of t h e terms u c e d ) . T h u s , w i t h i n t h e p r e c e d i n g f rarnework, o n e c o n c l u d e s t h a t e m ( X ) separates t h e p o i n t s of X : I n

f a c t , t h i s is a

Cw-analogon of Uryson's Lem"em-particion of

ma", which i n t u r n i s d e r i v e d from t h e e x i s t e n c e of a unity" i n e v e r y paracompact

for every p o i n t x

C " - d i f f e r e n t i a l manifold ( i b i d . )

X , and every neighborhood

e m ( X ) , w i t h O S f 6 1 , f(x) = I , and f = O l

Lemma I ] ,

.

Thus,

U of 2, t h e r e e x i s t s a f u n c t i o n f

E

( c f . S . KOBAYASHI-K. NOMIZU[l:p.272,

cu

a n d / o r Y . MATSUSSHIMA [1:p. 69, Lemma I ] f o r a n o t h e r v e r s i o n

more a k i n t o t h e p r e v i o u s a n a l o g o n ) . T h e r e f o r e , t h e above map 6 i s a continuous i n j e c t i o n ; a s a m a t t e r of f a c t , i t i s e s s e n t i a l l y a homeomorphism o n t o i t s r a n g e , w i t h r e s p e c t to t h e r e l a t i v e t o p o l o g y . T h a t i s , one h a s t h e f o l l o w i n g Lemma 2.1.

emanalogon

of Lemma 1 . 1 .

L e t X be an n-dimensional

e m - d i f f e r e n t i a Z manifold whose under-

Lying topoLogica1 space X i s (Hausdorff connected a n d ) second countable (and hence paracompact). Moreover, l e t e m ( X ) be t h e Fre'chet locally m-convex algebra of complex-vaZued

C--functions

on X i n t h e r e s p e c t i v e C"-topoZogy

( C h a p t e r N . 4 . ( 2 )1.

Then, t h e corresponding Dirac ( i . e . , e v a l u a t i o n ) map

( g i v e n by ( 1 . 3 ) ) d e f i n e s a homeomorphism between t h e r e s p e c t i v e t G p h g i c a i s, *:;es i n j v c h a m y ;hut its rai?gc fo b e contained i n t h e spectrum of e m ( X / . Proof. I t s u f f i c e s t o p r o v e , a c c o r d i n g t o t h e p r e v i o u s d i s c u s s i o n ,

t h a t t h e i n v e r s e map of 6 i s c o n t i n u o u s when 31x1 c a r r i e s t h e r e l a t i v e t o p o l o g y from ( c " ( X ) I i ; e q u i v a l e n t l y , w e h a v e t o p r o v e t h a t , f o r any point x e X

and a n e i g h b o r h o o d V o f

Ii

n X , t h e r e e x i s t s a neighbor-

h o o d , s a y h i , of x i n t h e r a n g e of 6 , a s a b o v e , which i s c o n t a i n e d i n V . T h a t i s , W must b e d e t e r m i n e d b y ;he ( i n i t i a l ) topology d e f i n e d on X by

t h e e m - f u n c t i o n s . Now, t h i s i s a s s u r e d by t h e f o l l o w i n g : Lemma 1. (Whitney's Imbedding Theorem). Euery C m - p a r a compact manifold i s diffeomorphic t o a c l o s e d submunifoZd of t h e 12n+li-dimensionul a f i n e space B 2 ' ' + I . ( C f . , f o r i n s t a n c e , S. STERNBERG [ I : p. 6 3 , Theorem 4.4:).

Thus, i f V i s a n y n e i g h b o r h o o d of a p o i n t x e X , t h e r e e x i s t s a a l o c a l c h a r t , s a y ( U , $ ) , of t h e m a n i f o l d X a t x , w i t h U C V . Hence, i f )': : X - - t h ( X ) = Y C IR2'+I i s t h e d i f f e o m o r p h i s m ( embedding) of X , p r o v i d e d by t h e p r e v i o u s L e m m a 1

,

t h e n t h e r e e x i s t s a l o c a l c h a r t (H, x) o f

VII SPECTRA OF PARTICULAR ALGEBRAS

226

h ( x ) (which w e may t a k e , of c o u r s e , t o b e t h e o r i g i n of

a t the point BZn+I)

,

such t h a t one h a s

B n h l X i C h(Ui C k i V ) . T ~ u s ,t h e s e t

W = h-I(Bnhtxi) = C x e x

(2.5)

: j u i o 7 z I < & ; i =I,.

. . ,2n+1 I

i s t h e d e s i r e d neighborhood of x , a s above; h e r e t h e l o c a l c h a r t l B , x ) ia

I R ~ ~c a+n ~b e t h u s c h o s e n ( b y s u i t a b l y r e s t r i c t i n g

E

> O ) so a s t o

b e an a p p r o p r i a t e open b a l l a t h l z i = O d e f i n e d by t h e r e s p e c t i v e c o o r d i n a t e f u n c t i o n s (ui

of

)

Thus, t h e c m - f u n c t i o n s kio h

€

ewlXi,

I < i < 2 n + l , w i l l b e t h e n t h o s e d e f i n i n g W i n ( 2 . 5 ) , and t h i s c o m p l e t e s t h e proof. I The p r e v i o u s lemma f o r m u l a t e d i n a less t e c h n i c a l manner s a y s , t h e r e f o r e , t h a t t h e topology o f a

em( p a r a c o m p a c t ) manifold

i s determined

by ( i t i s , namely, t h e i n i t i a l topoZogy o f ) its e m - f u n c t i a n s , v i a t h e r e s p e c t i v e D i r a c map 6 . W e h a v e a l r e a d y e n c o u n t e r e d t h e a n a l o g o u s f a c t i n

case of t h e a l g e b r a

q

X

l (Lemma 7 - 1 ) ; y e t t h i s w i l l be also t h e case,

as w e s h a l l see i n t h e n e x t s e c t i o n , f o r a n i m p o r t a n t c l a s s of c o m plex a n a l y t i c manifolds ( i n f a c t , s p a c e s ) , concerning the respective t o p o l o g i c a l a l g e b r a s of h o l o m o r p h i c f u n c t i o n s .

Now suppose, i n p a r t i c u l a r , t h a t X i s a t i a l ) manifold

t h a t i n t h e p r o o f of Lemma 1 . 2 ,

niz's m l e

compact

Cm-(differen-

of d i m e n s i o n n. Thus, a p p l y i n g a n a n a l o g o u s argument t o o n e g e t s ( a s a n a p p l i c a t i o n of

Leib-

on a d i f f e r e n t i a l o p e r a t o r a p p l i e d on t h e p r o d u c t of two

Cm-functions; cf.

IV; ( 4 . 1 4 ) ) t h a t ,

an i d e a l I G C " i X ) is proper i f , and

o n l y i f , it has a t l e a s t one z e r o i n X ; namely, t h e r e i s a t l e a s t one p o i n t o f X a t which a l l f u n c t i o n s of I v a n i s h . A c c o r d i n g l y , one o b t a i n s t h a t

every mazimal i d e a l o f C " l X i i s o f t h e form I,,

f o r some ( u n i q u e l y d e f i n e d )

p o i n t x e X ; hence, it is c l o s e d . Of c o u r s e , t h i s e n t a i l s e v e r y character of C " t X l

in particular that

is continuous ( s e e a l s o C o r o l l a r y I I ; 7 . 3 i n con-

n e c t i o n w i t h Scholium I I ; 7 . 1 ) . I n t h i s c o n c e r n , w e f u r t h e r r e m a r k t h a t one c o u l d a l s o o b t a i n

s i m i l a r r e s u l t s t o t h e p r e v i o u s Lemmas 1 . 3 a n d 1 . 4 , h e n c e t o C o r o l l a r y 1.1 a s w e l l .

So one h a s , i n d e e d , t h e f o l l o w i n g .

Lemma 2 . L e t X be a paracompact C " - m a n i f o l d and A a c l o s e d s u b s e t o f X . Then, every e w - f u n c t i o n on A can be extended t o a C w - f u n c t i o n on X. ( S e e , f o r i n s t a n c e , S. KOBAYASHI-K. NOMIZU [I: p. 273, Theorem 2 1 ) . Thus, t h e p r e v i o u s d i s c u s s i o n p r o v i d e s a l r e a d y t h e p r o o f of t h e

2.

227

SPECTRUM OF e - ( X )

following.

Theorem 2.1. Let X be an n-dimensional campact

c"(X)

e m - m a n i f o l d and

t h e r e s p e c t i v e Fre'chet l o c a l l y m-convex algebra a s i n Lemma 2 . 1 . Then, concerning t h e s p e c t m o f t h t s algebra, one has t h e r e l a t i o n

m( c v ) )= X ,

(2.6)

w i t h i n a homeomorphism o f t h e r e s p e c t i v e topological spaces ( g i v e n by ( 2 . 4 ) )

.

Furthermore, t h e l a s t r e l a t i o n y i e l d s , i n e f f e c t , t h e s e t o f a l l characters

em(xl.

of t h e algebra

On t h e o t h e r h a n d , t h e p r e c e d i n g p r o v i d e s , i n f a c t ,

a character-

i z a t i o n o f X i n terms o f e m i X j . T h a t i s , o n e h a s t h e f o l l o w i n g lemma, a "

C"-analogon"

of Banach-Stone Theorem ( c f

.

Scholium 1 . 1 )

.

Lemma 2 . 2 . Suppose t h a t t h e Cm-manifoZds X , Y s a t i s f y t h e c o n d i t i o n s o f

t h e previous Theorem 2 . 1 . Then, one has t h e r e l a t i o n

ew

(2.7)

= e"(yj

,

w i t h i n a t o p o l o g i c a l algebra isomorphism, i f , and only if, t h e r e l a t i o n

x =Y

(2.8)

holds t r u e , w i t h i n a diffeomorphism. Proof.

I t i s c l e a r , of c o u r s e , t h a t w e h a v e o n l y t o p r o v e t h e

" o n l y i f " p a r t o f t h e a s s e r t i o n . So i f j d e n o t e s t h e isomorphism i n (2.7)'

t h e n o n e g e t s by ( 2 . 6 )

respective manifolds X,Y

a homeomorphism, s a y i : X - Y ,

of

the

i n s u c h a way t h a t o n e h a s

[j-'(g)](x) = g(i(x!) = ( g o i ) ( x i ,

(2.9) f o r any x e X

a n d g E e m f Y ) . Thus one c o n c l u d e s , i n d e e d , by ( 2 . 9 ) t h e

re l a t i o n (2.10)

g oiE

for every g

E

em(Y)

which

C"(X),

c e r t a i n l y i m p l i e s t h a t i i s a e"-map

of X

c n t o Y , a n d t h e same i s s i m i l a r l y p r o v e d f o r t h e i n v e r s e map of i ; i . e . , the assertion. I I n t h i s r e s p e c t , w e s t i l l n o t e t h a t , i n view of t h e l a s t s t a t e m e n t o f Theorem 2 . 1 , i t s u f f i c e s , i n e f f e c t , t h e isomorphism i n ( 2 . 7 ) t o be only an algebraic one i n order (2.8) t o hold t r u e .

Scholium 2.1.-

The argument a p p l i e d i n t h e p r e c e d i n g c a n b e ex-

t e n d e d , i n f a c t , t o t h e g e n e r a l c a s e of n o t n e c e s s a r i l y compact manif o l d s , by means, however, o f a more i n v o l v e d t e c h n i q u e from t h e p a r t

228

VII SPECTRA OF PARTICULAR ALGEBRAS

of D i f f e r e n t i a l A n a l y s i s ( o r T o p o l o g y ) . Thus, one c a n o b t a i n t h e analogous r e s u l t t o t h e above Theorem 2 . 1

( u s i n g an a p p r o p r i a t e c h a r a c -

t e r i z a t i o n of " l o c a l sets'' of i d e a l s i n e m ( X ) : " S p e c t r a l Theorem" ; H . WHITNEY [l]. Cf. a l s o B . MALGRAIANGE [l: p. 2 5 , C o r o l l a r y 1 . 7 1 a n d / o r J . C . TOUGERON [l: p. 89, Th6orGme 1 - 3 1 1 .

F u r t h e r m o r e , s i m i l a r r e s u l t s t o Theorem 2 . 1 a r e a l s o a v a i l a b l e (even f o r n o t n e c e s s a r i l y compact m a n i f o l d s ) by c o n s i d e r i n g o t h e r " d i f f e r e n t i a l s t r u c t u r e s " of " h i g h e r o r d e r " on a g i v e n d i f f e r e n t i a l manifold S t h a n algebra

Xo(X)

E. PURSELL-M.E. §XI

emlXl; this

is, f o r instance, the case f o r the L i e

e c a - v e c t o r f i e l d s on

of a l l

X w i t h compact support. Cf. L .

SHANKS [l] a n d / o r A . KORIYAM.4 e t aZ.[l];

y e t H . OMORI [l:p. 123,

*

3. Spectrum o f the algebra O ( X ) .

Stein algebras

W e c o n s i d e r i n t h e s e q u e l t h e spectrum of t h e t o p o l o g i c a l a l g e -

bra

0lXl = T(X, 0,)

(3.1)

t h a t i s , of t h e Fmkhet

,

ZocalZy m-eonvex algebra

of

(complex-valued) holo-

morphic f u n c t i o n s on a complex a n a l y t i c space (X, Ox) w i t h

struetiire

sheaf 0, (we simply w r i t e X , h o w e v e r ) ; t h i s w i l l b e , i n p a r t i c u l a r , a S t e i n s p a c e . I n t h i s r e g a r d , w e r e f e r t o R . C . G U N N I N G - H . ROSSI [l] f o r t h e d e t a i l s of t h e t e r m i n o l o g y a p p l i e d . ( S e e a l s o C . ANDREIAN CAZACULl] and/ o r L . KAUP - B. KAUP [I]

.

w e mean a complex a n a l y t i c s p a c e X i n such a way t h a t X i s , i n p a r t i c u l a r , second c o u n t a b l e holomorS p e c i f i c a l l y , by a S t e i n space

p h i c a l l y s e p a r a b l e r e g u l a r and convex. I n t h i s r e s p e c t , one means by hoZomorphicaZZy separabZe algebra (3.I )

that the

"separates the p o i n t s of X". F u r t h e r m o r e , t h e same a l g e b r a

p r o v i d e s "ZocaZ-gZobaZ coordinates" f o r X ( holomorphicaZZy r e g u l a r ) , and f i n a l -

i s a g a i n a compact s e t Z c X . ( F o r t h e t e r m i n o l o g y a p p l i e d h e r e c f . also Chapter V;(4.9), as w e l l as Chapter I V ; 4 . ( 3 ) ) . For s i m p l i c i t y w e s h a l l c o n s i d e r i n t h e s e q u e l o n l y reduced comp l e x spaces which amounts, w i t h i n t h e p r e s e n t c o n t e x t , t o t h e assumpl y the

holomorphically convex h u l l o f a compact K S X

t i o n t h a t the respectioe GeI'fand map of the algebra ( 3 . 2 ) i s i n j e c t i v e . T h i s p e r m i t s , among o t h e r t h i n g s , t o c o n s i d e r t h e same a l g e b r a a s a s u b a l g e b r a of

c c ( X l and t h e n w i t h t h e c o r r e s p o n d i n g r e l a t i v e t o p o l o g y , a s

we s h a l l p r e s e n t l y see i n t h e s e q u e l .

Now, a t o p o l o g i c a l a l g e b r a E i s s a i d t o be a e v e r one h a s t h e r e l a t i o n

S t e i n aZgebra, when-

2.

229

SPECTRUM OF c ) ( X ) . STEIN ALGEBRAS

E =

(3.2)

r(x, o x ) ,

within a topological algebraic isomorphism, where ( X , O , )

is a Stein

space. Thus, our main conclusion will be the fact that t h e spectrum of a given S t e i n aZgebra E is homeomorphic t o t h e S t e i n space X of ( 3 . 2 ) (Theorem 3.1).

Indeed, much more is essentially valid (ibid.; see also Scholium 3.1 below). In this respect, we first note that the usual evaluation map f -4Jf)

(3.3)

for any xe X and f e O t X l , defines 6,

:=

f(.C.‘,

,x e X ,

as a complex algebra mor-

phism of O ( X ) which is also continuous,according to the inclusion (3.4)

and Lemma 1.1.

OIX)

c

ep

Thus, t h s map 6:X-

(3.5)

defined by (3.3)(with 6 i x ) =

)

1OtXil’ i s one-to-one;

namely, X being, by hypo-

thesis, a Stein space, it is O(XI-separabZe, that is, for any x , y in X, with z # y , there exists an element f e O ( X ) such that f l x ) # f l y ) . Besides, t h e map 1 3 . 5 ) i s continuous with respect to the weak topological dual of OtXl , i.e. , the space This follows certainly from Lemma 1 . 1 and the continuous injection (3.4)(thereforeI the continuity of the

(O(X))i .

respective transpose map and of its composition with 6, the resulting map being denoted still by 6). NOW, one verifies that 6 i s essentiaZZy a homeomorphism onto its image in (O(X))i This is based on the O(X)-reguZarity of X , which

.

amounts to the fact that t h e (original)topology of X i s determined by t h e s e t o f i t s hoZomorphic f u n e t i o n s : Indeed, this is a consequence of the following Embedding L e m a (Remmert-Bishop-Narasimhan)

.

Lemma 1. Let X be a (reduced) S t e i n space of dimens i o n n. Then, t h e r e e x i s t s an i n j e c t i v e proper immersion (and hence a homeomorphism) of X onto a complex analyt i c subvariety of ~ 2 ’ 2 ~. 1 ( See, for instance, R . C. GUNNING- H . ROSSI 224,Theorem 101

.

[I:p.

Therefore, by a similar argument to that used in the proof of the analogous statement in Lemma 2.1, one now gets the assertion (see also 0. FORSTER [ 1: p. 3121). So it remains only to prove that 6 t X ) E 1 O t X i i ~ is indeed t h e whole of t h e spectrum o f O i X l ; this is based, in fact, on some of the deepest

230

VII SPECTRA OF PARTICULAR ALGEBRAS

r e s u l t s o f Complex A n a l y s i s ( C a r t a n ' s Theorems A and B , Cartan-Oka Theory of c o h e r e n t a n a l y t i c s h e a v e s ; see R. C'. G U N N I N G - H . ROSSI [ I ] ) .

More s p e c i f i c -

a l l y , one h a s t h e f o l l o w i n g . Lemma 2. ( H . C a r t a n ) . Let I be an i d e a l of t h e algebra O(X). P x n , m e gets (3.6) I = rix, J ( V ( I ) ) )

( t h e c l o s u r e i s t a k e n i n O ( X 1 ) ; here J ( V ( I ) ) denotes the coherent a n a l y t i c s4eaf o f i d e a l s of t h e anaZytic vari e t y V ( I 1 of t h e given i d e a l I .

(Cf. H. CARTAN [I] a n d / o r 0. FORSTER [ I : p. 312, S a t z S e e a l s o h'. FIHITNEY [2: p. 280, S e c t i o n 9 1 o r R . C . GUlVNINC- H. ROSSI [ I : p. 138, Theorem 21 ) I]

.

.

T h u s , a s a c o n s e q u e n c e of t h e p r e v i o u s Lemma 2 , o n e r e a l i z e s t h a t e u e q proper closed idea2 o f t h e algebra O ( X ) has a t l e a s t one zero ( p l a c e : N u l l s t e l l e ) i n X . Hence, f o r e v e r y c l o s e d maximal i d e a l Iof c ) ( X ) , one c o n c l u d e s t h a t I C I z , f o r some X E X , so t h a t by h y p o t h e s i s f o r I, one has the relation

I = I = ker(6,)

(3.7)

3:

,

and t h i s , i n connection w i t h C o r o l l a r y I I i 7 . 2 ,

p r o v e s t h a t t h e image

of 6 i n ( 3 . 5 ) d e s c r i b e s , i n f a c t , t h e s p e c t r u m o f O ( X ) . So w e h a v e a c t u a l l y p r o v e d by t h e p r e v i o u s d i s c u s s i o n t h e f o l lowing. Lemma 3.1.

Let X be a S t e i n space. Then, t h e spectrum of t h e algebra O ( X ) i s

given by

i r r r i O ( X ) ) = mirrx, ox ) ) = x ,

(3.8)

w i t h i n a homeomorphism ( d e f i n e d b y t h e map ( 3 . 5 ) ) Furthermore,

(x, 0X

)

t h e above r e l a t i o n

.

I

(3.8) c h a r a c t e r i z e s , i n f a c t ,

a s a S t e i n space, i n v i e w o f t h e f o l l o w i n g r e s u l t (Igusa-Remert-

Iwahashi Theorem )

.

Theorem 3.1. Let (X, 0,)

be a complex a n a l y t i c space. Then, X i s a S t e i n

space if, and o n l y i f , t h e canonical map (3.9)

6 :x

-mtr(x,

ox 1 ) ,

given by ( 3 . 5 1 , i s a homeomorphism ( o n t o ) . Proof. The n e c e s s i t y of t h e s t a t e d c o n d i t i o n i s d e r i v e d a l r e a d y f r o m t h e p r e v i o u s Lemma 3 . 1 . F o r t h e " i f " p a r t of t h e a s s e r t i o n c o n s u l t , f o r i n s t a n c e , 0 . FORSPER [4: p. 139, S a t z 7 1 . I

4, SPECTRUM OF L1 ( G) Schol ium 3.1

.-

23 1

The p r e c e d i n g Theorem 3.1 y i e l d s a c h a r a c t e r i z a

-

t i o n o f S t e i n s p a c e s i n t e r m s of t h e r e s p e c t i v e S t e i n a l g e b r a s . A s a

matter of f a c t , t h e two n o t i o n s are c a t e g o r i c a l l y ( a n t i ) e q u i v a l e n t

( c f . 0.

FORSTER [3: p. 378, S a t z 11 o r C. BANICA- 0. STANASILA [l: p. 46, Theorem 4.111).

I n t h i s c o n c e r n , w e a c t u a l l y have t h a t t h e a l g e b r a i c equivalence o f two

S t e i n algebras i r r p l i ? ~ in , e f f e c t , t h e i r t o p o l o g i c a l one a s w e l l , h e n c e t h e homeomorphism o f t k L e r e s p e c t i v e S t e i n s p a c e s by ( 3 . 8 ) ; t h e r e f o r e , t h e i r e q u i v a l e n c e by t h e f o r e g o i n g ( c f . a l s o 0 . FORSTER [2: p. 161, C o r o l l a r y 11). On t h e o t h e r h a n d , i n t h e p a r t i c u l a r c a s e t h a t mann domain o v e r

enrwhich

tX,p) is a Rie-

i s a l s o a S t e i n m a n i f o l d ( i . e . , a domain of

holomorphy; c f . C h a p t e r V ; S e c t i o n 4 )

,

one c o n c l u d e s t h a t e v e r y c h a r a c t e r

o f t h e corresponding S t e i n algebra 0lXl i s continuous ( s e e R . C . GUNNING - H . IiOSSI [ l : p. 283, Theorem 4 1 ) . T h i s amounts t o t h e same t h i n g a s t h a t

every

maximal ideal o f L)(Xl is c l o s e d a n d h e n c e f i n i t e l y g e n e r a t e d ( c f . C. FOtiSTER [2: p. 159, Theorem 21,

a s w e l l a s E . A . MICHAEL [l: p. 5 4 , P r o p o s i t i o n 12.51).

F i n a l l y , w e a l s o h a v e t h a t a g i v e n S t e i n space ( X , 0,) l c c a l r i n g ( a l g e b r a ) 0,

, with

i s reduced ( t h e

EX, d o e s n o t c o n t a i n n i l p o t e n t e l e

m e n t s ) i f , and o n l y i f , t h e r e s p e c t i v e GeZ'fand map o f O t X ) , i . e . ,

-

t h e map

(3.10)

i s one-to-one

( c f . 0 . FORSTER

[ 1:

p. 3101). I n t h i s case w e a l s o s a y t h a t

t h e (commutative) S t e i n a l g e b r a

O ( X ) i s semi-simple

( c f . a l s o i.n t h e se-

q u e l Chapt. V I I I ; D e f i n i t i o n 3 . 2 ) . 1 4. Spectrum o f t h e a l g e b r a L (G) The a l g e b r a i n t i t l e of t h i s s e c t i o n i s o f c o u r s e a Banach a l g e b r a , t h e c l a s s i c a l a l r e a d y "group algebra" ( a b e l i a n ) group G .

of a g i v e n l o c a l l y compact

But t h e main r e a s o n o f i n c l u d i n g it h e r e i s r a t h e r

f o r purpose o f l a t e r a p p l i c a t i o n s , s p e c i f i c a l l y , i n c o n n e c t i o n w i t h t o p o l o g i c a l t e n s o r p r o d u c t s . So w e i n c l u d e i n t h e p r e s e n t s e c t i o n t h e r e l e v a n t d i s c u s s i o n i n o r d e r t o have t h e r e s p e c t i v e e x p o s i t i o n l a t e r more " s e l f - c o n t a i n e d " . Thus, w e a r e c o n s i d e r i n g i n t h e e n s u i n g d i s c u s s i o n a l o c a l l y compact ( t o p o l o g i c a l a b e l i a n ) g r o u p G , t o g e t h e r w i t h t h e a s s o c i a t e d

Haar measure on i t ; w e d e n o t e t h e l a t t e r b y dx a n d c o n s i d e r it a s a complex-valued Radon measure on t h e s p a c e ( a l g e b r a ) K ( G i of complex-valued c o n t i n u o u s f u n c t i o n s on G w i t h compact s u p p o r t . The r e s p e c t i v e v e c t o r space

L ' I G ) of complex-valued

dx ( i . e .

,

s u ma b l e f u n c t i o n s on G , w i t h r e s p e c t t o

t h e Hausdorff completion o f K ( G )

,

with respect t o t h e s e m i -

normed t o p o l o g y d e f i n e d on it by t h e n e x t r e l a t i o n ( 4 . 1 ) ) i s made i n t o

232

VII

SPECTRA OF PARTICULAR ALGEBRAS

a Banach s p a c e whose n o r n i s g i v e n by

iv,(fl=(I”fI/, = ( I f l d s = J l P ( d l d 3 : = u ( 1 f l I

(4.1) 1

f o r e v e r y f E L ( G I . ( W e d e n o t e by u = d z

as an e l e m e n t of ( K ( G I ) ‘ ,

i.e.,

I

t h e Haar measure on G c o n s i d e r e d

o f t h e t o p o l o g i c a l d u a l of

K(GI

where

t h e l a t t e r s p a c e i s t o p o l o g i z e d a s i n C h a p t . I V ; 4. ( 1 ) ; cf. N ;( 4 . 6 ) ) . Now, d e n o t i n g t h e g r o u p o p e r a t i o n i n G a d d i t i v e l y , one d e f i n e s 1

t h e convolution

o p e r a t i o n ( m u l t i p l i c a t i o n ) i n L (GI b y t h e r e l a t i o n

(f* g ) ( X I =

(4.2)

with

J f (3: - ylg(yi d y

e G , and f o r a n y f , g e L ‘ I G ) , w h i l e t h e l a s t r e l a t i o n i s a s s u r e d

3:

1

by an a p p l i c a t i o n of ELbini’s Theorem on L ( G I ( c f . , f o r example, L . H . LOOMIS [I:

P. 122, C o r o l l a r y ] ) . I n t h i s r e s p e c t , s i n c e t h e Haar measure

i n is by d e f i n i t i o n l e f t

( a n d s i n c e G i s a b e l i a n , a l s o r i g h t ) transla-

tion invariant,

one a c t u a l l y g e t s by ( 4 . 2 ) t h e r e l a t i o n

(4.3)

(f * g I i ~ I = ~ f ( ~ - y I g ( y I- d( fyl y ) g i x - y I d y ,

w i t h z f G , and f o r a n y f , g

1

in LiG) ( w e

refer, for instance, t o L.H.

LOOMIS [l: Chapt. V I ] f o r t h e r e l e v a n t t e r m i n o l o g y a p p l i e d h e r e ) . Thus,

(4.2)

p r o v i d e s a n ( a l g e b r a ) m u l t i p l i c a t i o n i n L ’ i G l (whit\

i s a l s o commutative i n c a s e t h e g r o u p G i s a b e l i a n , and o n l y t h e n of 1

,

so t h a t L (GI becomes a Banach algebra. F u r t h e r m o r e , i t a l w a y s h a s a (bounded) approximate i d e n t i t y , w h i l e it h a s an i d e n t i t y e l e m e n t i f (and o n l y i f ) t h e group G carries t h e d i s c r e t e topology ( i b i d . ) . 1 NOW, w e a r e f u r t h e r i n t e r e s t e d i n i d e n t i f y i n g t h e s p e c t m of L ( G l course)

when G i s commutative. T h a t i s , t h e ( G e l ‘ f a n d ) s p a c e ?l‘i!(L1(GI) (Definition o r what amounts t o t h e s a m e ( C o r o l l a r y 11; 7 . 3 ) t h e s p a c e of

V;l.l),

1

( c l o s e d ) r e g u l a r maximal i d e a l s ( “maximal i d e a l space” ) of L ( G I (endowed w i t h t h e r e s p e c t i v e G e l ’ f a n d t o p o l o g y ) . A s w e s h a l l see, t h i s i s ( w i t h i n a homeomorphism) c a n o n i c a l l y i d e n t i f i e d w i t h t h e G (Theorem 4 . 1 )

.

character group of

I n t h i s r e s p e c t , g i v e n a t o p o l o g i c a l g r o u p G , one means by a

character of G I a complex-valued c o n t i n u o u s f u n c t i o n on G , s a y a : G + C , o f modulus 1 ( i . e . , l a ( s ) I = I , f o r e v e r y 3 : E G ) which i s a l s o a morphism of G i n t o t h e ( m u l t i p l i c a t i v e , “ u n i t a r y “ ) group

u = I x € c : ( x I = z l,

(4-4)

t h u s a continuous morphisrn

t e r s of G by

2.

o f G i n t o U. W e d e n o t e t h e s e t of a l l c h a r a c -

S o , t h i s is,

by d e f i n i t i o n , a s u b s p a c e of

C c ( G , U)

where t h e l a t t e r s p a c e c a r r i e s t h e compact-open t o p o l o g y , as i n d i c a t e d . Thus,

2

e q u i p p e d w i t h t h e r e l a t i v e t o p o l o g y becomes a n ( a b e l i a n ) t o p o -

l o g i c a l g r o u p ( p o i n t w i s e d e f i n e d o p e r a t i o n s ) , which i s a l s o l o c a l l y

4. SPECTRUM OF

233

L'(G)

c o m p a c t , whenever G i s ) . W e c a l l i t t h e c h a r a c t e r group

c; c f .

n o t e d by

(4.5)

iz e

1

= f o I , :G +G

f2

so t h a t

ment

L.H. LOOMIS [ I : C h a p t . VII]).

f o r a n y g i v e n f e 5 (G) and x e G ,

NOW,

of G ( s t i l l d e -

:y

I (y) = f (2,(y) I

o I,

-(f

one d e f i n e s t h e map

f (x+y )

:=

w e s t i l l t a k e , by t h e t r a n s l a t i o n i n v a r i a n c e of d z , an e l e 1 L (G)

.

Moreover

,

t h c mu;;

( 4 .6)

5:

1

-&

: G d L (G)

( w i t h r e s p e c t t o t h e L1-norm ( 4 . 1 ) .

i s continuous

I b i d . : p. 118, Theorem

30C).

On t h e o t h e r h a n d , we a l s o o b t a i n t h e r e l a t i o n

(4.7)

fz*g =f*gx

f o r every

LCE

G , and any

Thus, by 1 4 . 2 )

f, g i n L 1 ( G 1, which w e s h a l l p r e s e n t l y u s e below.

and ( 4 . 5 ) , one g e t s f o r e v e r y a e C

(fa

* q ) (x) = I fa (x - y l g ( y ) d y = / f (a+.,- - y ) g i y i d y

= ( v i a t h e t r a n s f o r m a t i o n y-a = /f(.-yig,iy)dy with x EG,

+ y ) Jfiz-ylgia +yidy

= (f*gn

)(XI ,

which p r o v e s ( 4 . 7 ) .

Thus, w e come n e x t t o t h e f o l l o w i n g

Lemma 4.1. Let L1(G1 be the group algebra of a l o c a l l y compact a b e l i a n group G and m ( L 1 ( G I J

i t s spectrum. Moreover, l e t

8

be t h e c h a r a c t e r group of G . Then,

the relation

1

where f i s any element of L ( G I , w i t h

a@):=

(4.9)

@(f) # 0

,

p r o v i d e s a we22 d e f i n e d map between t h e r e s p e c t i v e spaces. Proof. W e f i r s t remark t h a t 14.81 1

of f e L (GI s a t i s f y i n g 1 4 . 9 1 , @ emIL1(G)).

i s , i n d e e d , independent of t h e c h o i s e

and t h e e x i s t e n c e o f which i s a s s u r e d , s i n c e 1

T h u s , f o r a n y o t h e r e l e m e n t g e L (G) w i t h

g e t s , b y ( 4 . 7 ) and t h e h y p o t h e s i s f o r @,

@ i f z l @ ( g= l @(f)@(gziI f o r e v e r y x € G ; t h a t i s , one h a s

g ( @ 1= @ ( g 1 # 0 , one

234

VII

SPECTRA OF PARTICULAR ALGEBRAS

which i s t h e a s s e r t i o n , so t h a t t h e map a S : G + C

is w e l l defined.

F u r t h e r m o r e , we a l s o have t h e r e l a t i o n

a (x* y ) = a (xi.a@(y),

(4.10)

rp

for any x , y i n G : 1

G and f e L ( G ) ,

rp

(f = f z + y , f o r any Z , Y in setting g=f

I n d e e d , s i n c e by ( 4 . 5 )

one o b t a i n s by ( 4 . 7 1 , fx

Y‘

* fy = f*(fy)z= f *f,+, = f * f z + y-

T h e r e f o r e , one g e t s , f o r e v e r y @ e m f L ’ ( C 1 ) ,

@(f)Ufz+ y ) = @(fz)@(fy) * (rp(f)J2

Thus, d i v i d i n g t h e l a s t r e l a t i o n by @ (fz iy

-

,

one h a s

-.@ (fx)

@(&,)

@if)

(P(f)

@if)

that is, the desired relation ( 4 1 0 ) .

-.

la (x)1 > 1 , f o r e v e r y z e G , one g e t s by (4.10)

NOW, assuming t h a t

@

la (nx)l= / a(x)(”

rp

@

--+

m ,

with n

m

But t h i s i s a c o n t r a d i c t i o n , s i n c e by ( 4 . 8 ) one h a s la

(xi1 =

6

1 ~

I o(f) I

IUf,)l ~ W f z l l =1k * l l f I I I = M

f o r every x e G ( w i t h k = ( l r p ( f I ( ) - ’ ;

m( L 1 ( G I ) ,

as w e l l a s t h e r e l .

t i o n i n v a r i a n c e of dx). Thus,

we t a k e here i n t o account t h a t

(4.1) la

(P

1

rp

E

i n connection with t h e t r a n s l a i s a bounded f u n c t i o n . T h e r e f o r e ,

one o b t a i n s

la (z)l
Hence, la fz) I = I , f o r every z~G . Q, F i n a l l y , one e a s i l y c o n c l u d e s by ( 4 . 8 ) and t h e c o n t i n u i t y of t h e map ( 4 . 6 ) t h a t t h e map a : G+

@

U G C i s continuous a s w e l l , a n d t h i s com-

p l e t e s t h e p r o o f of t h e lemma. I W e come now t o p r o v e t h a t 1

(4.8) provides, i n e f f e c t , a b i j e c t i o n

b e t w e e n t h e s p e c t r u m of L ( G ) and t h e c h a r a c t e r g r o u p of G .

However, w e

n e e d f i r s t some more p r e l i m i n a r y m a t e r i a l . Thus, s i n c e e v e r y e l e m e n t

1

@ e??Z( L (G)) i s , i n p a r t i c u l a r , a con-

t i n u o u s l i n e a r form on L ‘ I G ) , one h a s t h e r e l a t i o n (4.11)

MIL’IGI

= ZWL’(G)

I c: ( L ~ ( G Ij

*

2 LYG) .

4.

SPECTRUM OF L ' ( G )

235

The last relation in ( 4 . 1 1 ) yields, within an i s o m e t r i c isomorphism 0 , the 1 topological dual of (the Banach space) L f G / (in the respective "strong topology") as the Banach space L c ' ( G ) of all e s s e n t i a l l y bounded measurable on G . In parti.cular, this is expressed, through the Radon-Nikodini Theorem, by the relation

functions

(4.12)

@ I f ) = Jflzin(ccl d z

,

1

with f e L 1G1 , for a uniquely defined u e L " ( G I , which corresponds, via o, 1 to a given @ € ( L ( G l i ' . (In this regard, cf. L . H . LOOMIS [I: 7 5 C , D ] , and/ or M. A . NA?MARK 11: p. 140, 5 1 6 , Theorem 3 1 ) In this respect, we still remark that ~ ~ L c 4 ( G an ) , element of ?

.

being, by definition, a bounded and continuous (therefore, measurable) h

function on G. Furthermore, t h e i n v e r s e map o f a in 14.111 r e s t r i c t e d t o G 1 1 has range i n t o MIL ( G l l c ( L f G ) ) ' : thus, ( 4 . 1 2 ) yields a (continuous) character Q, of L 1(G), for every u e 6 : Thus, for any elements f , g in L 1i G ) , one gets by ( 4 . 1 2 ) (through a repeated application of Fubini's Theorem, and the translation invariance of d z ) Q , ( f * g i = J ( f * g ) f z i m d z = (by (4.2)) J(Jf(z-y)giyidy =

JJ : i r , ' p ( g ) a i z

ZZdx =

JJ f i z - y i g i y i Z Z d z d d y ___

+yiJzdg = ~

~

;i.t.Jy ~

i

~

= J I J f ( x l a l c ! ? z i g ( y i m d y = I J f ( x l ~ d x ) l J g f y l a ( y / d y / = @1fi@(g)

which is the assertion. On the other hand, the converse of the last assertion concerning ( 4 . 1 2 ) is also true. First we shall need, however, some more comment on the respective terminology. So if i e 6 i is an approximate i d e n t 1 ity of L (G) one has, by definition, the relation l i m ( f * e 6 i = l i m i e * f i =f

(4.13)

6

6

6

1

,

1

for e ery f E L ( G l . Thus, f o r every @Em(L fG)),

one gets

@if)=@llimif*e6!)= limr$lf*e61 6 6

(4.14

= l i m @ i f i @ ( e 6=1 @ffi.lim@1e6

6

Hence

6

.

since @ # 0 , one concludes t h e r e l a t i o n limQ,ie6 i = 1 6

(4.15)

,

1

f o r every @ E m ( L(GI). (In this respect, we remark that the last conclu-

sion concerning ( 4 . 1 5 ) is certainly valid for any topological algebra whatsoever, withintof course, the appropriate context).

~

236

VII

NOW,

SPECTRA OF PARTICULAR ALGEBRAS

a p p l y i n g ( 4 . 7 ) w i t h g = e 6 one g e t s , f o r e v e r y

1

$€mfL(G)),

$(fxl$(e6 I = $ifi$((e6lz I

so t h a t f o r

@(f)

# 0

one o b t a i n s

Hence, by ( 4 . 8 ) , w e have $ f ( e 6 1 x I = $ ( e ) a (x)

(4. 6)

6

$

f o r e v e r y x e G . I n p a r t i c u l a r , o n e h a s by ( 4 . 1 5 ) l i m $ f i e 6 1 z I = a (z),

(4. 7)

$

6

f o r every x E G . Thus, w e come f i n a l l y t o o u r p r e v i o u s a s s e r t i o n t h a t , namely, t h e i n v e r s e map of 0-1 I

(4.18)

:

2

-

1

M(L'(GI ) = 17il ( L ( c i

,

which i s , i n d e e d , t h e a s s e r t i o n . ( I n t h e p r e v i o u s argument w e h a v e a l s o a p p l y t h e f a c t t h a t , f o r e v e r y a e ; , one h a s t h e r e l a t i o n d - x / = a(x) , w i t h X E G I a s f o l l o w s from t h e same d e f i n i t i o n s ) , Thus, w e c a n summarize t h e p r e c e d i n g i n t o t h e form o f t h e n e x t . Lemma 4.2.

map (4.20)

Suppose t h a t t h e c o n d i t i o n s of L e m a 4 . 1 are s a t i s f i e d . Then, t h e 0 :$

-

a$: ~T(L'(GII

d e f i n e d by (4.81, y i e l d s a one-to-one

-E,

and o n t o correspondence between t h e r e s p e c t i v e

4.

231

SPECTRUM OF L?G) 1

of the canonicaZ

spaces. I n f a c t , it i s t h e r e s t r i c t i o n t o the s p e c t r m of L (G) 1

( i s o m e t r i c ) isomorphism (L (GII '

Lm(GI.

Proof. I f e ( @ ) = a = 9(J,) = a

t h e n by ( 4 . 1 9 )

@ one g e t s @ =

i),

J,

with

@, J,

that is,

, g e t s by ( 4 . 1 2 ) hand, f o r any a E ~ ~ L m f G Ione

1

1

i n g it a n e l e m e n t 9 € M(L (G)) € f L (GI)', 9(6) = a i s related t o @ v i a (4.19) 9

+; theref ore,

.

elements

of

0 i s one-to-one.

,

mfL1(GI)

On t h e o t h e r

and t h e comment f o l l o w -

i n s u c h a way t h a t t h e r e s p e c t i v e Hence, one h a s o-'faI = o-"(a I =

@

= 0( @I = o ( @ ), t h a t i s 9 i s an onto map a s w e l l . The a =a+ l a s t a s s e r t i o n of t h e s t a t e m e n t i s c l e a r a l r e a d y by t h e p r e v i o u s d i s c u s s i o n , and t h i s t e r m i n a t e s t h e proof o f t h e 1emma.I W e come n e x t t o prove t h a t t h e p r e c e d i n g map 9 i s , i n f a c t , a

homeomorphism. Thus, w e h a v e . Theorem 4.1.

Let t h e conditions of Lemma 4 . 2 be s a t i s f i e d . Then, the map

e

(4.21)

: ~ ( L ' I G ) -Z, ) 1

defined by ( 4 . 8 1 , i s a homeomorphism of t h e spectmun of L ( G I

onto the character

group of G. Proof. According t o Lemma 4 . 2 ,

f i r s t that

6 - l i s continuous. Namely,

9 i s a b i j e c t i o n . Thus, w e prove w e must prove t h a t f o r any n e i g h -

borhood v ( @ O ; f , E= ) { cp

(4.22)

E

I f^rw -?(ao) I < E I

~ ( L ~ I G I ) :

1

from a fundamental system of such a t t h e p o i n t

$,=9-l(a0) e ?YZ(L I G ) ) ,

t h e r e e x i s t s a neighborhood

U(ao;K,61={af2:

(4.23)

of a. E (4.24)

ECcfGI

Ic~(T)--c~~(x)~<~

V T ~ K I

such t h a t

O-*(U(aO ; K, 6 1 ) E V ( @ o ; f ,

E

I.

Thus, one has t o prove t h a t (4.25)

l@(f)-@o(Jc)l

'€3

f o r every @ = €I-l(aI, w i t h a € U(ao;K , 6 1 . T h e r e f o r e , i n view of

obtains (4.26)

F u r t h e r m o r e , one h a s by d e f i n i t i o n t h e r e l a t i o n (4.27)

K(GI = L?GI

( 4 .19), one

238

VII SPECTRA OF PARTICULAR ALGEBRAS

with respect t o t h e since

L1 -norm

1 L (GI d e f i n e d by

(topology) of

(4.1 )

. So

1

f € L f G ) , o n e c o n c l u d e s t h a t f o r a n y E > O t h e r e e x i s t s an e l e m e n t such t h a t

g€KfG)

N1(f-g)

=!I

5.

f(s)-gfz)\dx<

Thus, f o r e v e r y compact K C G , o n e h a s t h e r e l a t i o n (see a l s o N . BOUR-

BAKI [8: Chap. 4 ; p. 185, C o r o l l a i r e ]

)

jIf(x)-g(z)ldx= JKIf(zi-g(xiIdr+j

CK

Consequently, t a k i n g

/f(xl-g(xiIdz < + .

K = Supp ( : I ) , w i t h g a s b e f o r e , s i n c e g = O ( C K

One

obtains

Thus, from ( 4 . 2 6 ) a n d t a k i n g K a s b e f o r e a n d i n ( 4 . 2 3 ) 6 =

E

211fl11

one h a s f o r e v e r y a e

c'(clo;

K, 6) ( c f . a l s o

t i o n 1)

1 @(f)- 4Jo

If) I S

I If

I

t h e above R e f . ; p . 205, P r o p o s i -

- -

-

(sl I */adz)~1 (xi I d x

K

$0

- -

If(x)I.la ( x i - a lxlldx 4J 40

( w e a l s o r e c a l l t h a t la ixll = 1 , w i t h z e G , f o r e v e r y 4 J € ?7?(L'(G)l). T h a t 4J i s , w e do h a v e ( 4 . 2 5 ) , which w a s t o b e p r o v e d . The p r e c e d i n g p a r t of t h e p r o o f e s t a b l i s h e s , i n f a c t , one h a l f o f t h e a s s e r t i o n . The rest w i l l b e d e r i v e d from t h e f o l l o w i n g two l e m -

mas.

Lemma 1 . L e t G be a ZocaZly compact a b e l i a n group, L ' f G ) 1

t h e r e s p e c t i v e group algebra and ?Z(L ( G I ) Then, t h e f u n c t i o n (1)

h : (x,4 )

-

i t s spectrum.

h(x, $11:=a f x ) ,

4J

d e f i n e d b y 1 4 . 8 1 , is a (complex-valued) continuous f u n c t i o n

Iroof. F o r e v e r y f e LI(G)

1

,

( w e assumed t h a t $ E mfL(G)) )

one o b t a i n s

,

which p r o v e s ( c f

.

4. SPECTRUM OF

d(G)

239

-

a l s o ( 4 . 6 ) ) the continuity of the function

(z,@ ) j p :) G x VY(L’(G)I

(2)

1

f o r every f e L ( G ) . Now, f o r e v e r y

( x ~ , @t ~ h e) r e e x i s t s

# 0 . Thus, o n e g e t s

an e l e m e n t f , e L’fG), w i t h t h e c o n t i n u i t y of

( 1 ) a t a given point

t i o n of a

$

FJ$)+t(@o)#

fxo ,@o) b y where w e

t($),

d i v i d i n g t h e map a p p e a r e d i n ( 2 ) b y a l s o have t h a t

c ,

0 (see a l s o t h e d e f i n i -

i n Lemma 4 . 1 ) . 1

Furthermore, w e a l s o need t h e f o l l o w i n g t o p o l o g i c a l f a c t .

Lemma 2 . Suppose we upG g i v e n t h e t o p o l o g i c a l spaces X, a s w e l l as a continuous f u n c t i o n

Y,Z

h :X x Y d Z . Then, f o r any compact s u b s e t K of X , and any open subset A o f 2, t h e s e t V=

y e Y :h f z , y )

E

A

V

z e K}

i s an open s u b s e t o f Y. Proof. C f . L . H . LOOMIS [ I : p. 12, Lemma

5 ~ 1 1.

End o f t h e proof o f Theorem 4.1. W e p r o v e now t h a t t h e map 0 - I , which i s g i v e n by ( 4 . 2 1 ) , i s an o p m map: Thus, g i v e n aoeG a n d any open n e i g h h

borhood

K,

U(clo;

t h a t E-l~i!!’ao;K,

ao= N$,i = a

)

.

E)

E))

from a l o c a l b a s i s a t a .

,

a s i n ( 4 . 2 3 ) , one p r o v e s

1

is an open s e t i n 5 W f L fG)) c o n t a i n i n g $o= 8 - l ( a 0 ) ( s o

Namely, w e h a v e

$0

a-I(uia,;

NOW,

K,

E

)I =

c + e V Z ( L1(GI) : ei4) =

t h i s i s a n open s e t i n

2

1 m ( L f G ) ) containing

u(a0;K, EJI

@o

,

a s t h i s follows

from t h e c o n t i n u i t y o f t h e map ( c f . t h e above Lemma 1 )

h :(z,ql)-hfz,f$,):=

ja f x ) - a

@

(.El $0

a n d Lemma 2 , which i s j u s t t h e a s s e r t i o n . T h i s c o m p l e t e s t h e proof of Theorem 4 . 1 . I A d i r e c t a p p l i c a t i o n of t h e i d e n t i f i c a t i o n provided by t h e p r e -

c e d i n g Theorem 4 . 1

i s t h a t t h e c l a s s i c a l Fcurier-Gel’fand

transform of t h e

1

g r o u p a l g e b r a L f X n l a c t s on t h e c h a r a c t e r g r o u p of IRn ( c o r r e s p o n d i n g

t o t h e c a n o n i c a l i n n e r product of I R n )

,

n a m e l y , t h e g r o u p IRn i t s e l f ( w i t h -

i n a n isomorphism of t o p o l o g i c a l g r o u p s ) by t h e r e l a t i o n

240

V I I SPECTRA OF PARTICULAR ALGEBRAS

(4.28) 1

with f e L ( E n ) . (See, for instance, E.M. S T E I N - G . WEISS [I:pp. 2 , 31). Thus, one also gets, straightforwardly, the classical fact (ibid.) that the Fatrier transform of t h e convolution of two functions i s the (pointwise) product of t h e i r Fourier transforms (the Gel'fand map is an (algebra) morphism). Now consider the given group G , as above, and its character group G ; thus the latter, being a locally compact (abelian) group,has its own character group, denoted by 2 (second character group of G ) Now this group is, within a homeomorphism, the initial group G itse1f;in other words, l o c a l l y compact abelian group i s (within a homeomorphism) i t s own second character group ( Pontrjagin Duality Theorem; cf . , for example , L . H. LOOMIS [I: p.151, 37D]). h

.

h

C"(X) (contn'd.). The Nachbin Theorem ( n e c e s s i t y )

5. The l o c a l l y m-convex algebra

We consider in this and the following section the Cm-analogon" of the classical Stone-Waierstrass Theorem (see, for instance, L . NACHBIN [4: p. 48, Corollary 21) : it is due to L . NACHBIN [l] . We need first, however, still more terminology from the theory of Differential Manifolds than the one already applied in Section IV;4.(2) and Section 2 above, which thus we are going next to supply. Thus suppose we are given a d i f f e r e n t i a l manifold X which we assume, thereinafter, to be connected second countable and Hausdorff, of course. Furthermore, we suppose henceforth that

cw

(5.11

C ;

(XI

(see also Section IV;4.(2)), that is, we consider exclusively r e a l valued ( em-) functions on X . (The complex-case can be treated in a similar way to the analogous one used for Stone-Weierstrass Theorem; namely, " s e l f - a d j o i n t " subalgebras of ( X I should be considered instead. Cf. Section 7.2 below) Thus, for every x e X , denote by T ( X , X I the tangent space of the manifold X at x ; this is (isomorphic to) the n-dimensional numerical space IRn , where n =dimX S o , for every element V E T ( X , x ) , tangent vector of X at x , one defines a (real) linear form on C " t X I , say l,.,, by the relation

ccm

.

.

(5.2)

5.

the notation applied i n (5.2) , ( A i )

CmIX).C o n c e r n i n g

f o r every f E

24 1

N A C H B I N THEOREM (NECESSITY)

w i t h l S i < n , d e n o t e s t h e s e t of

,

( l o c a l ) C o o r d i n a t e s of veT'IX,x:I a t t h e

p o i n t x € X ( c f . a l s o ( 5 . 7 ) b e l o w ) , which c o r r e s p o n d s t o t h e l o c a l c h a r t

iu, @) = (U;X I ,. . . , xni

(5.3)

of t h e m a n i f o l d X a t x , where (5.4)

xi=uio@, I S i S n ,

1SiSn

a n d ui : W n+iR,

,

are the canonical coordinates (projections

o n t o t h e r e s p e c t i v e a x e s ) of

so one h a s , by d e f i n i t i o n ,

IRn;

(5.5) with i = I ,

...,

Now,

eqression"

n , f o r e v e r y f &?Xi.

t h e expression of an element v E T ( X , x ) i n local coordina*es ( " l o c a l of v )

, which

c o r r e s p o n d s t o any g i v e n c h a r t of the manifold

X a t x , l i k e ( 5 . 3 ) , i s g i v e n by t h e r e l a t i o n n

(5.6)

1)

= t A i ( Za J x i=l 7,

.

Furthermore, one o b t a i n s A . = v(xiI = idxilz

(5.7)

(vl ,

( 5 . 4 ) ) . T h u s , t h e ( r e a l ) numbers X i , l S i 6 n ,

with 16i<,n ( c f .

i n (5.7)

a

Local coordinates of v , w i t h r e s p e c t t o t h e canonical b a s i s {C-) j ax-. s I S i S n , of T I X , x ) d e f i n e d , f o r i n s t a n c e , b y ( 5 . 5 ) and c o r r e s p o n d i n g " t o

are the

. ,snI

t h e g i v e n c h a r t (U;xI ,. .

of X a t z.

In p a r t i c u l a r , t h e r e l a t i o n ( 5 . 2 ) defines T(X, X I ,

as a (real-valued)

emlX) (viz.

,a

"Leibniz m a p " )

. Thus,

2

V

E

by c o n s i d e r i n g t h e l a t t e r a l g e b r a

equipped with t h e c a n o n i c a l em-topology ( c f readily verifies that

Zv , f o r e v e r y v

derivation ( a t t h e p o i n t x ) of t h e a l g e b r a

.

Section IV; 4 . ( 2 ) )

,

one

i s , i n f a c t , a continuous l i n e a r f o m on c m ( X ) ( c f .

( 5 . 2 ) and I V ; ( 4 . 1 4 ) ) . W e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g lemma which

a c t u a l l y c o n s t i t u t e s t h e " o n l y i f " p a r t of o u r main a s s e r t i o n b e l o w (Theorem 6 . 1 ) . Lemma 5.1.

Thus, w e h a v e .

Let X be u ( n o n - t r i v i a l ) finite-dimensional

1 5 n =dimX < m ) and

em-munifoZd ( v i z . ,

~ " t x lt h e algebra o f (real-valued) C m - f u n c t i o n s on X en-

dowed w i t h t h e canonical

c -topology m

( c f . S e c t i o n IV; 4 . ( 2 ) ) . Moreover, l e t A

be a dense subalgebra o f c m i X I ; i . e . , we assume t h a t

Then, t h e following t h r e e c o n d i t i o n s are s a t i s f i e d :

242

VII SPECTIiA OF PARTICULAR ALGEBRAS

1 ) The algebra A i s %on-vanishing

on X"; namely, not a l l f u n c t k n s from A

vanish a t a l l p o i n t s o f X . ( E q u i v a l e n t l y , f o r every p o i n t ment f € A ,

X , t h e r e e x i s t s an e l e -

with f l x l # 0 ) .

2 ) The algebra A '!separates t h e p o i n t s o f X"; i . e . ,

x #y,

X E

there e x i s t s an element f

E A,

f o r any x , y i n X , w i t h

with f i x ) # f ( y l .

3 ) For any p o i n t X E X and tangent v e c t o r v E T l X , x l , w i t h v # 0, t h e r e i s f E A , such t h a t

( d f J X ( v l = v ( f I # 0.

(5.9)

Thus, t h e ( p o i n t ) d i f f e r e n t i a l s o f t h e f u n c t i o n s i n A , a t any p o i n t X E X , separate t h e F o i i t i s of t h e r e s p e c t i v e tangent space T ( X , x ) .

Proof. By c o n s i d e r i n g t h e t o p o l o g i c a l d u a l s o f t h e ( l o c a l l y convex) spaces appeared i n ( 5 . 8 ) , (5.10)

A'

o n e h a s , of c o u r s e , t h e r e l a t i o n

= (~OO(XI)',

w i t h i n a l i n e a r s p a c e isomorphism ( s e e , f o r i n s t a n c e , G . KOTHE [l: p. 158, ( I l ) ] ) . On t h e o t h e r h a n d , i n view o f Lemma 2 . 1 ,

one g e t s

(5.11) where t h e map

6:

is the Xs c"(X))'

e v a l u a t i o n map ( "Dirac map") ; t h a t

i s , one o b t a i n s

f o r any x e X

and

f 6 e m ( X ) . So t h e f i r s t r e a t i o n i n

(5.11) is valid

w i t h i n a homeomorphism ( Whitney's Imbedding Lemma; c f . t h e p r o o f o f t h e s a n e lemma above (Lemma l ) ) , w h i l e t h e p o i n t s of X c o r r e s p o n d t h r o u g h ( 5 . 1 2 ) t o non-zero c o n t i n u o u s l i n e a r f o r m s on

C"(X)( t h e

l a t t e r alge-

bra contains the constants). Thus, t h e f i r s t two of t h e s t a t e d c o n d i t i o n s f o r A a r e now a d i r e c t c o n s e q u e n c e of

( 5 . 1 0 ) and ( 5 . 1 1 ) . Furthermore,

f o r every v f 0

i n T ( X , x ) , one g e t s from ( 5 . 2 ) c o n c e r n i n g t h e r e s p e c t i v e d e r i v a t i o n , lv#O

( t h e map v ~ - + l ~ , v ~ T ( X , xd )e ,f i n e d by ( 5 . 2 ) , i s a b i j e c t i o n ) . So

s t i l l from ( 5 . 1 0 ) a n d t h e f a c t t h a t lv€ lC"lX)I'

( s e e t h e comment be-

f o r e Lemma 5 . 1 ) , o n e c o n c l u d e s t h a t t h e r e e x i s t s f € A , w i t h %,if) = v ( f )

= tdf),(v)

# 0 ; i.e., one o b t a i n s c o n d . 3 ) a s w e l l , a n d t h i s f i n i s h e s t h e

proof. The f o l l o w i n g i s a n e s s e n t i a l t e c h n i c a l i s s u e i n t h e p r o o f o f t h e c o n v e r s e of t h e p r e v i o u s Lemma 5 . 1 , which w e p r o v e i n t h e s e q u e l . Lemma 5 . 2 . Consider a Cm-rnanifot"oZd X and t h e r e s p e c t i v e ( r e a l ) algeb1.a Cm(X)

5. NACHBIN THEOREM

24 3

(NECESSITY)

a s in t h e above L e m a 5 . 1 . Moreover, l e t A be a subalgebra of

C m ( X ) satisfying

cond. 3 ) of t h e same lemma. FinaZZy, l e t T ( X , x ) be t h e tangent space of X a t a p o i n t z~X . Then, t h e r e e x i s t s a b a s i s

c v l , . . . , 'n'

(5.13)

of T(X, x i , t o g e t h e r w i t h a b a s i s of (T(X, xi)* = T*IX,2) ( a l g e b r a i c d u a l of TiX, x), o r y e t t h e cotangent space o f t h e m a n i f o l d X a t z),c o n s i s t i n g of d i f f e r e n t i a l s a t x of f u n c t i o n s from t h e algebra A S C " ( X 1 ;

i . e . , of t h e form

{(dfi)x)lsi gn

(5.14)

w i t h f. E A, 1 5 i 5 n , in such a manner t h a t one has

(df.) z xI v7,. )

(5.15)

=1, 1 S i L n ,

a d (df.)(v.l=O, l S i < j S n .

(5.16)

Froof.

I f vl # 0 i n

zx 3 T(X, x ) , t h e r e e x i s t s b y h y p o t h e s i s ( c o n d . 3 )

o f Lemma 5 . 1 ) a n e l e m e n t g l f A s u c h t h a t

IdglIx(v,) = u l ( g l ) # 0

(5.17) Now,

-

i f d i m X = n > 2 , o n e g e t s from ( 5 . 1 7 ) t h a t dimiker idgl

(5.18)

Ix I

(see, f o r i n s t a n c e , J . HORVLTH [I: p.411);

21

hence, t h e r e e x i s t s v I E T / X , x )

i n s u c h a way t h a t

C # u2e k e r

(5.19) Therefore,

(dgl)x )

.

s t i l l by h y p o t h e s i s , o n e f i n d s a n e l e m e n t g 2 e A , w i t h (dg,

(5.20)

Now,

(

).-

iv,) = u 2 ( g g I # 0 .

i f dimX23, then ( i b i d . ) dimfker(dgiiZ) 2 2 , i = l , 2

(5.21)

,

so t h a t o n e h a s (5.22)

dim(ker(dgl),n

Thus , t h e r e e x i s t s v3 E TIX, (5.23)

2)

ker(dg2Sc) t 1 .

such t h a t

0 # u3 E ker(dgI),

n ker(dg2ix ,

w h i l e o n e o b t a i n s , by h y p o t h e s i s ,

(dg3!,1v3) = v3(g3) # O r

(5.24)

f o r some g3 E A . So r e p e a t i n g t h i s a r g u m e n t one f i n a l l y o b t a i n s a ( f i n i t e ) s e q u e n c e { v2,. quence

.., v n }

i n T(X, x ) , t o g e t h e r w i t h a c o r r e s p o n d i n g se-

{g,, . . . , g n } i n A ; now by s e t t i n g

244

VII

SPECTRA OF PARTICULAR ALGEBRAS

(5.25)

fi

with a . = v . ( f . ) # 0 2

2

a sequence

2

{f,,

,1 S i S n

..., f n }

:=I gieA,

(cf. (5.17)

,

lziln, (5.20), e t c . )

,

one f i n a l l y o b t a i n s

i n A , which b y t h e p r e c e d i n g s a t i s f i e s t h e re-

lation ( d f .) ( v . ) = v , . ( f . ) = 6 . . , zx 3 d 2 23

(5.26)

w i t h l < i < j S n . T h a t i s , w e h a v e ( 5 . 1 5 ) and ( 5 . 1 6 ) . F u r t h e r m o r e , it i s r e a d i l y s e e n from ( 5 . 2 6 ) t h a t t h e two f a m i l i e s { v i } and

{(dfiIx 1

,

1 S i S n , a r e l i n e a r l y i n d e p e n d e n t i n T(X, x ) a n d i t s a l g e b r a i c d u a l , re-

s p e c t i v e l y , a n d t h i s c o m p l e t e s t h e p r o o f of t h e 1emma.i NOW, t h e f o l l o w i n g s t a n d a r d f a c t s from t h e r u d i m e n t s of

Rieman-

nian D i f f e r e n t i a l Geometry w i l l n e x t b e n e e d e d . T h u s , o u r m a n i f o l d X bei n g a l w a y s l o c a l l y compact a n d , by h y p o t h e s i s , 2nd c o u n t a b l e , it i s pa-

racompact ( c f . , f o r i n s t a n c e , J . Dugundji [l: p. 174, Theorem 6 . 5 1 ) . So :he manifold X admits a Cm-Riemannian metric o r , e q u i v a l e n t l y , it i s , i n f a c t , a Cm-Riemannian manifold (see S . KOEAYASHI -K. NOMIZU

[ 1 : p.

6 0 1 ) . Thus , a s a

c o n s e q u e n c e of t h e r e s p e c t i v e t h e o r y f o r t h e "exponential function" i n X I one g e t s t h e f o l l o w i n g l e m m a . ( C f . ,

f o r e x a m p l e , t h e l a s t R e f . : p. 148,

P r o p o s i t i o n 8 . 3 ; o r y e t F . BRICKELL- R.S. CLARK [I: p. I 8 O f l ) . Lemma. (Normal coordinates). Let X be a Cm-Riemannian

manifold and x a point of X . Then, f o r every b a s i s { v l , ...,

vn} of T ( X , x ) , there e x i s t s a local chart (17, $ ) = (U; x l , . .

.

, 3cn) (see ( 5 . 3 ) ) of the manifold X a t x such t h a t t h e basis ( v . ) coincides w i t h the Ncanonical basis" (-1 a oxi x ' 1 <= i S n , of T(X,x ) which corresponds t o the chart (U,@). i-kt i s , one has the r e l a t i o n (5.27)

vi =

a

(-)axi

x

w i t h 1 5 i S n. The f o l l o w i n g r e s u l t i s now a c o n s e q u e n c e o f t h e p r e v i o u s Lem-

m a and Lemma 5 . 2 . Namely, w e h a v e .

Corollary 5.1. Let the conditions of L e m 5.2 be s a t i s f i e d . Then, f o r every point x

X , there e x i s t s a local chart of the manifold X a t x , whose "coordinate

functions" are (appropriate r e s t r i c t i o n s of f u n c t i o n s ) from t h e given algebra A C

C"(X).Y e t

which, i n f a c t , amounts t o t h e same t h i n g , the algebra A provides "glo-

bal-local coordinates" a t every p o i n t of the manifold X . Woof. If ( V i ) l 5 i 6 n i s t h e b a s i s o f Ti'X, x ) , p r o v i d e d from Lemm a 5 . 2 , t h e n i n v i e w o f t h e p r e v i o u s Lemma t h e r e e x i s t s a l o c a l c h a r t

5.

245

NACHBIN THEOREM (NECESSITY)

vie T(X,x), I S i S n , as t h e c o r r e s p o n d i n g c a n o n i c a l b a s i s . Moreover, t h e f u n c t i o n s f . E A , 1 < i=C n , p r o (U, $1 a t x h a v i n g t h e g i v e n v e c t o r s

v i d e d by t h e same Lemma 5 . 2 ,

satisfy the relation

((=Is a

(djy,

(5.28)

)

= 6i

,

3 w i t h I < i l n ( c f . ( 5 . 2 6 ) and ( 5 . 2 7 ) ) . Now, by c o n s i d e r i n g t h e v e c t o r -

valued function -f

f := (fi I I < i S n : x-

(5.29)

a s a b o v e , o n e o b t a i n s , c o n c e r n i n g t h e r e s p e c t i v e Jacobian

w i t h fi",

matrix o f

lRn,

p

a t x, the relation

(5.30) w i t h I < i , j S n . So o n e g e t s (5.31)

from ( 5 . 2 8 ) ( s e e a l s o ( 5 . 2 ) ) clij

with 1 < iS j < n

.

,

Hence, d e t J z ( f I # O F so t h a t t h e map

id$),

(5.32)

= €iij

+

: T(X,x ) - T ( I R n ,

?(XI)

d e f i n e s a n isomorphism. T h e r e f o r e , b y t h e

3

IRn

Inverse Function Theorem ( f o r

-

m a n i f o l d s ( !) ; see , f o r i n s t a n c e , F. BRICKELL R. S. CLARK [l : p. 63, Prop+ c s i t i o n 4.4.11 ) , t h e map f i s a "local diffeomorphism": Namely, t h e r e ex-

i s t s a n open n e i g h b o r h o o d V of x , i n s u c h a way t h a t t h e r e s t r i c t i o n -+ t o V i s a d i f f e o m o r p h i s m o f V o n t o ( t h e open s e t ) f ( V l c IR'I T h e r e -

of

f'

fore, the pair

+

( V , f J , ) = (L';

(5.33)

(f.1z V ) l 5 i S n 1

y i e l d s a l o c a l c h a r t of t h e manifold X a t t h e p o i n t x , s a t i s f y i n g t h e r e q u i r e d c o n d i t i o n s ( c f . a l s o ( 5 . 3 ) ) ; moreover, t h e f u n c t i o n s

fiIv

(5.34)

, IsiSn,

correspond, of c o u r s e , t o t h e l a s t s t a t e m e n t of t h e c o r o l l a r y ( c f . a l s o t h e r e l . (5.35) i n t h e s e q u e l ) ,and t h e proof i s f i n i s h e d . I I n v i e w o f t h e p r e v i o u s lemma, o n e o b t a i n s a e m - a t l a s m a n i f o l d X which i s , of c o u r s e , c o m p a t i b l e w i t h t h e g i v e n

t u r e of X .

I t b e l o n g s , namely, t o t h e

maximal e m - a t l a s

of t h e

em-struc-

o f X . Thus, w e

are l e d t o t h e following. D e f i n i t i o n 5.1. Keeping t h e n o t a t i o n of C o r o l l a r y 5.1

,

the

em-

a t l a s o f t h e m a n i f o l d X which i s p r o v i d e d , a c c o r d i n g t o t h e p r e v i o u s comment, from t h i s c o r o l l a r y , i s c a l l e d t h e Nachbin a t l a s o f X associated

with +,he algebra A C C " ( X ) . Any c h a r t o f t h i s a t l a s i s a l s o c a l l e d a Nachof X , a t t h e p o i n t I C E X u n d e r c o n s i d e r a t i o n , corresponding t o the

b i n chart

246

VII

SPECTRA OF PARTICULAR

ALGEBRAS

algebra A . NOW, g i v e n a n y c h a r t (U, $I) = (U; x

,..., x

)

( c f . ( 5 . 3 ) ) o f t h e mani-

f o l d X a t x E X , i t s coordinate functions xi, 1 < i < = nmay , be considered as appro-

priate restrictions

( e v e n t u a l l y , however, t o an open s u b s e t o f V ) of func-

t i o n s from the algebra e m ( X ) ( c f . , f o r i n s t a n c e , S. HELGASON [ I : p . 6 , Lemma

1.2, a n d t h e comment a t t h e end o f p . 71 ) terminology ( c f . C o r o l l a r y 5.1)

,

. Thus,

applying t h e previous

one c o n c l u d e s t h a t , for every em-rizani-

C r n t X )provides global-lo-

f o l d X ( n o t n e c e s s a r i l y p a r a c o m p a c t ) , t h e algebra cal coordinates a t every point x e X .

The p r e c e d i n g c h a r a c t e r i z e s , i n f a c t , c o n d . 3 ) of Lemma 5 . 1 . So one h a s t h e following f a c t :

Cond. 3 ) of Lema 5 . 1 i s equivalent with t h e assumption t h a t the algebra A C e r n t X ) provides global-ZocaZ coor-

(5.35)

d i n a t e s , for every p o i n t x e X. Namely ( s e e a l s o C o r o l l a r y 5 . 1 ) , f o r e v e r y p o i n t x E X , t h e r e exi s t a n open n e i g h b o r h o o d U o f x a n d f u n c t i o n s C e A , w i t h 1 S i S n = dim X, such t h a t t h e p a i r (5.36)

cu; (f:I u )l
d e f i n e s a l o c a l c h a r t of t h e m a n i f o l d X a t x

( t h e c h a r t (5.36) belongs

t o t h e maximal e m - a t l a s o f X). The r e s p e c t i v e by t h e f o l l o w i n g map ($: u-IRn:

(5.37)

x

-

$ ( x i := I

= ( f 1 ( x ) ,..., f

chart-map o f U i s g i v e n

(filU)(X)ll'

.<

-t=n

n ( X H .

Thus, o n e h a l f of t h e a s s e r t i o n i n ( 5 . 3 5 ) i s d e r i v e d a l r e a d y from C o r o l l a r y 5 . 1 .

F u r t h e r m o r e , s u p p o s e now t h a t t h i s c o n d i t i o n

is

t r u e , and l e t v e T ( X , x ) , w i t h v f 0 ; so f o r any c h a r t , l i k e (5.31, a t x , one a t l e a s t of t h e r e s p e c t i v e c o o r d i n a t e s of

1)

( c f . ( 5 . 7 ) ) i s non-

z e r o . Hence, c o n s i d e r i n g a t x t h e p r e v i o u s c h a r t ( 5 . 3 6 ) , o n e g e t s ( c f . a l s o (5.37))

v(xi)=v(filu) =v(fil =idfilx(vl# 0 ,

(5.38)

f o r o n e i n d e x i , a t l e a s t , a s above; i . e . ,

f o r some f .

E

A , which p r o v e s

the assertion.

6 . The Nachbin Theorem (sufficiency) W e prove f u r t h e r i n t h i s s e c t i o n t h a t t h e t h r e e c o n d i t i o n s f o r

a n a l g e b r a A S e " ( X i , s e t f o r t h b y Lemma 5 . 1 , a r e , i n d e e d , s u f f i c i e n t

6. NACHBIN THEOREM (SUFFICIENCY)

247

i n o r d e r t h a t ( 5 . 8 ) t o hold t r u e . W e f i r s t h a v e t h e f o l l o w i n g r e s u l t , which a l s o s u p p l e m e n t s t h e

p r e v i o u s Lemma 5 . 2 .

Lemma 6.1

. Let

X be an n-dimensional (paracompact) e m - m a n i f o I d and K a e m ( X ) , satisfying the

compact s u b s e t of X . Moreover, l e t A be a subalgebra of

-

t h r e e c o n d i t i o n s of Lermna 5 . 1 . Then, t h e r e e x i s t s a diffeomorphism

0: V

(6.1)

@(V)C B N ,

for an appropriate p o s i t i v e i n t e g e r N , where V i s an open neighborhood of K and @(V) c a r r i e s t h e submanifold s t r u c t u r e induced on it from B N under t h e n a t u r a l imbedding 0 ( V j E R N . In p a r t i c u l a r , t h e map 0 i s made of f u n c t i o n s from t h e g i v e n algebra A

C

c"(X!

( c f . a l s o t h e n e x t C o r o l l a r y 6.1 )

.

S i n c e t h e m a n i f o l d X i s a l o c a l l y compact s p a c e , t h e r e ex-

Prcof.

i s t s an open n e i g h b o r h o o d V of t h e g i v e n compact K E X which i s r e l a t i v e l y compact, i . e . ,

one h a s

(6.2)

K C V Cv,

w i t h V C X a compact s e t ( s e e , f o r i n s t a n c e , N . EOUREAKI [4: Chap. I ; p . 65, P r o p o s i t i o n

lo] ) . NOW, f o r e v e r y x

€ 7 ,t h e r e

e x i s t s by h y p o t h e s i s

, so

a f u n c t i o n f e A , w i t h f ( x l # O ( c f . c o n d . 1 ) of Lemma 5 . 1 ) f o r e v e r y y e N J I l w i t h Nx

b e i n g a n open n e i g h b o r h o o d o f

a r e f i n i t e many s u c h n e i g h b o r h o o d s T d x . l l S i S k , i n g of t h e compact

v,

Thus, t h e r e

p r o v i d i n g a n open c o v e r -

7

i n s u c h a way t h a t t h e r e s p e c t i v e f u n c t i o n s f .

d e f i n e a map

-

(6.3)

F = ( f i ) I S{$

which n e v e r v a n i s h e s on

7. T h a t

z: : v-

k

IR

,

i s , one h a s

F ( z l := ( f i (xt.))15i6k # OelR

(6.4)

2.

t h a t f(y)#O,

k

,

f o r every x E V . On t h e o t h e r h a n d , by C o r o l l a r y 5 . 1 , c o v e r i n g of

7,

t h e r e e x i s t s a f i n i t e open

c o n s i s t i n g o f Nachbin c h a r t s , s a y

I ( u ~ ;( $ ) l S j 5 n ) ~ , ~ ~ i z l ,

(6.5)

so t h a t , b y t a k i n g ( 6 . 4 )

and ( 6 . 5 ) i n t o a c c o u n t , w e f u r t h e r s e t

i

$

(6.6)

with 1 6 i S l

atd

= 'k+(i-I)i1+j '

ISj6n.

Furthermore, f o r any x, y i n T l w i t h x f y , t h e r e 2 ) o f Lemma 5 . 1 a f u n c t i o n f ~ A , w i t h f ( x l # f ( y ) .

a n open n e i g h b o r h o o d W o f

(2,y

i

s u c h t h a t f(x7 # f ( y ' i ,

T h u s , t h e compact s e t ( c f . a l s o ( 6 . 5 ) )

e x i s t s by c o n d .

T h e r e f o r e , one f i n d s f o r any (x', y') e W.

248

VII SPECTRA OF PARTICULAR ALGEBRAS

n=~x~-(ru,xu1lv...uIUzxVz)~

(6.7)

d o e s n o t c o n t a i n t h e d i a g o n a l of 7 x 7 . T h u s , o n e c o n c l u d e s by t h e p r e c e d i n g t h a t t h e r e e x i s t f i n i t e many f u n c t i o n s , s a y h,,.. ., hm i n A , i n s u c h a way t h a t one h a s h . ( x f # h.(yl, Z < i < m

(6.8) f o r every

(2,y l

E R

. Thus,

,

setting

h . = fk+ln+i

(6.9)

, 1ai5m ,

we f u r t h e r consider t h e function =lR N

Q = ( f c i ) l < a < N: V-Q(V)

(6.10)

I

w i t h N = k + l n +m e m . NOW, a p p l y i n g ( 6 . 8 )

f o r e l e m e n t s of 0 , o r a n a p p r o p r i a t e Nach-

b i n c h a r t from ( 6 . 5 ) , o t h e r w i s e ( i . e . , f o r n o n - d i a g o n a l e l e m e n t s of - V x V b e l o n g i n g , however, t o t h e s e t ( U 1 x U 2 1 U . U (Ul x U,)) , o n e r e a d i l y

..

r e a l i z e s t h a t t h e map Q i s one-to-one. B e s i d e s , it i s a c L n - m a p definition ( c f . (6.6) inverse map CP-'

, (6.9)

by t h e same

and ( 6 . 1 0 ) ) . F u r t h e r m o r e , c o n s i d e r i n g t h e

o f Q , l e t x = @ - ' ( @ ( x ) ) GV ; t h u s t a k i n g a g a i n a Nachbin

c h a r t a t x , o n e c o n c l u d e s t h a t Q i s indeed a diffeomorphism. T h a t i s , i f

x e V S v b e l o n g s t o a Nachbin c h a r t , s a y ( U i ; f l '

,...,ft I , from

( 6 . 5 ) one

e s s e n t i a l l y c o n s i d e r s t h e diffeomorphism CP

(6.11)

1 ui

@(U.) c R n S R N

; ' iU

w i t h r e s p e c t t o t h e induced C " - s t r u c t u r e t h e " c a n o n i c a l imbedding" that

,

since f

ci

€

A Cc"(X)

r e s t r i c t i o n o f t h e map

lRnS I R N .

,1Sa 2 N ,

,

on lRn from R N I d e f i n e d by

( I n t h i s concern , w e f u r t h e r n o t e

one a c t u a l l y c o n s i d e r s b y ( 6 . 1 0 ) t h e

(fa):X+IRN

to

V;

a s i m i l a r remark h o l d s t r u e

f o r t h e map ( 6 . 1 1 ) 1 . I A s a c o n s e q u e n c e o f t h e p r e v i o u s Lemma 6 . 1 ,

i n g f a c t which, a s w e s h a l l see ( c f . Theorem 6 . 2 1 ,

w e have t h e f o l l o w characterizes, in

e f f e c t , the r e l a t i o n (5.8) concerning t h e given algebra A E e m f X ) .

So

one has

e

"-manifoZd and A Coroll ary 6.1. Let X be an n-dimensional (paracompact) a subalgebra of e " t X ) s a t i s f g i n g the three conditions of Lema 5 . 1 . Then, t h e foZlowing f a c t holds t r u e :

For every compact K G X , there e x i s t s a canonical imbedding (6.12.1 )

@:v-@(v)GlR

N- I a ) E n N

249

6. NACHBIN THEOREM (SUFFICIENCY)

of an open neighborhood V of K i n t o a ( f i n i t e dimensional) R N , whose coordinate functions

eucZidean space

0.= u . 0 O : V -W,

(6.12.2)

z

(see also ( 5 . 4 )

z

,

ISiSlV,

a r e r e s t r i c t i o n s t o V of ( r e a l -

v a l u e d ) C m - f u n c t i o n s on X which) belong t o the algebra A.

The p r e v i o u s a s s e r t i o n f o l l o w s , o f c o u r s e , d i r e c t l y from t h e p r o o f o f Lemma 6 . 1

c a l imbedding" NOW,

( c f . ( 6 . 1 0 ) ) and t h e same d e f i n i t i o n of a "canoni-

( s e e , f o r i n s t a n c e , F. BRICKELL- R.S. CLARK [ l : p. 811)

-

t a k i n g t h e d e f i n i t i o n o f t h e e m - t o p o l o g y i n t h e algebra c w ( X )

.

i n t o a c c o u n t ( s e e C h a p t e r Iv; S e c t i o n 4. ( 2 ) : t h e r e l s ( 4 . 1 4 ) a n d (4.191 ) , it i s on t h e image o f a compact K C X t h r o u g h t h e p r e v i o u s imbedding (6.12.1)

on which o n e a c t u a l l y c o n s i d e r s t h e a p p r o x i m a t i o n s e t f o r t h

by t h e r e l a t i o n ( 5 . 8 ) . So w e have now t h e f o l l o w i n g .

Theorem 6.1. (L. Nachbin). Consider a finite-dimensional 2nd countable Hausd o r f f e m - m a n i f o l d X , and l e t e " ( X ) be the algebra o f real-valued C m - f u n c t i o n s

ew(X). Then,

on X endowed w i t h t h e e m - t o p o l o g y . Moreover, l e t A be a subalgebra oJ the following two a s s e r t i o n s are equivalent: 1) A i s a dense subalgebra of i . e . , one has the r e l a t i o n (6.13) A = em(X).

e"(X),

2 ) A i s " s u f f i c i e n t l y separating"

ordinates a t every point of

on X providing, moreover, global-locaZ co-

x. Note.-

S a y i n g t h a t t h e a l g e b r a A i s suffici.enton X , w e mean t h a t it i s non-vanishi n g and s e p a r a t i n g ; c f . Lemma 5.1 f o r t h e relev a n t terminology.

ly separating

Proof of Theorem 6 . 1 , I f

( 6 . 1 3 ) i s v a l i d , t h e n it h a s a l r e a d y b e e n

p r o v e d i n t h e p r e v i o u s s e c t i o n (Lemma 5 . 1 ) t h a t A i s s u f f i c i e n t l y s e p a r a t i n g , y i e l d i n g a l s o global-local coordinates a t every p o i n t x e X ( c f . C o r o l l a r y 5.1 a n d a l s o ( 5 . 3 5 ) ) . Thus, 1 ) * 2 ) . On t h e o t h e r h a n d , from what h a s b e e n s a i d above, t o p r o v e (6.13) it s u f f i c e s t o t a k e any compact K C X on

which one c o n s i d e r s , i n t u r n ,

t h e a p p r o x i m a t i o n c l a i m e d by ( 6 . 1 3 ) . Thus,

by assuming 2), o n e g e t s from

C o r o l l a r y 6.1 t h a t , c o n c e r n i n g t h e g i v e n compact K C X ,

Condition ( 6 . 1 2 )

i s satisfied.

-

T h e r e f o r e , by c o n s i d e r i n g t h e i n v e r s e map o f (6.14)

,-I

:0 ( V l

V L X

(6.12.1), i.e.,

250

VII

SPECTRA OF PARTICULAR ALGEBRAS

one o b t a i n s t h a t f o @-I E C " ( @ ( V ) ) , f o r e v e r y t i o n t o t h e compact s e t

f E c"(X). Thus by r e s t r i c -

@ ( K I G @ ( V ) G i R None , f i n d s a function $ee"tiRN)

( c f . S e c t i o n 2 ; Lemma 2 ) s u c h t h a t (6.15) T h e r e f o r e , one h a s (6.16)

s o t h a t one g e t s f r o m ( 6 . 1 2 . 2 ) f(x) = @(O(xil = $4@l(x),...,@N(x)),

(6.17)

f o r e v e r y xEK.Furthermore, due t o (6.12.1),

one c o n c l u d e s t h a t O / Q . ( V )

and h e n c e 0 d @(K) S @ ( V ) .

(6.18)

Thus, b y c o n s i d e r i n g C O ( K ) a s a n open n e i g h b o r h o o d of 0 E X N , one may f i n a l l y assume t h a t

$to) = 0

(6.19)

( s e e a l s o , f o r example, S. KOBAYASHI-K. NOMIZU [ l : p. 272, Lemma 2 1 1 . NOW, a n a p p l i c a t i o n o f t h e c l a s s i c a l Weierstrass Theorem y i e l d s , i n view of ( 6 . 1 7 ) , t h a t t h e g i v e n f u n c t i o n f e c"(Xl i s approximated i n t h e Cm-topology of

ern(X) through

f u n c t i o n s from t h e g i v e n a l g e b r a A .

Namely, by p o l y n o m i a l s ( w i t h o u t c o n s t a n t terms; c f . (6.19) ) i n t h e v a r i ables

Q1 ix),.

(6.20)

.. , b N ( X ) ,

w i t h x e K , h e n c e i n t h e f u n c t i o n s Qi, 1 5 i S N . Thus, t h e f u n c t i o n s c"(X)

are a p p r o x i m a t e d i n t h e

C"-topo:ogy

in

by e l e m e n t s from t h e

g i v e n a l g e b r a A ( c f . ( 6 . 1 2 ) ; t h e a l g e b r a A d o e s n o t n e c e s s a r i l y cont a i n t h e i d e n t i t y e l e m e n t of r e l a t i o n (6.13), i.e.,

e m ( X ) ) . S o w e f i n a l l y have t h e d e s i r e d

2 ) = > 1 ) as w e l l ,

and t h i s c o m p l e t e s t h e p r o o f

of t h e theorem. 4 The s i t u a t i o n set f o r t h by t h e p r e c e d i n g p r o o f i s f u r t h e r c l a r i f i e d b y t h e n e x t t h e o r e m . However, w e do s e t f i r s t t h e f o l l o w i n g .

Definition 6.1. L e t X b e a f i n i t e - d i m e n s i o n a l C " - m a n i f o l d a s u b a l g e b r a of

C"(X).Now,

and A

w e s h a l l say t h a t t h e algebra A s a t i s f i e s

Condition ( N l , i n c a s e t h e t h r e e c o n d i t i o n s i n Lemma 5 . 1 h o l d t r u e f o r A (Namely, A i s t h u s s u f f i c i e n t l y s e p a r a t i n g on X (Theorem 6 . 1 ) and a l s o y i e l d s g l o b a l - l o c a l c o o r d i n a t e s a t every p o i n t of X ( c f . ( 5 . 3 5 ) ) .

.

The f o l l o w i n g i l l u m i n a t e s t h e meaning o f

Cond. (6.12). So w e h a v e .

7.

251

V A R I A N T S OF N A C H B I N ’ S THEOREM

Theorem 6.2. Let X be an n-dimensional (paracompact) e m - m a n i f o l d and

C”(X).Then,

subalgebra o f

A a

t h e following three a s s e r t i o n s are e q u i v a l e n t :

1 ) A is a dense subalgebra o f

Cml.XI, where t h e l a t t e r algebra c a r r i e s t h e

.

C”-topotogy

21 A s a t i s f i e s Condition ( N I ( c f . D e f i n i t i o n 6.1 )

.

5) A s a t i s f i e s Condition ( 6 . 1 2 ) .

Proof. The f a c t t h a t 1 ) - 2 ) i s Nachbin’s Theorem ( i - e . , Theorem 2)+3) a s f o l l o w s from C o r o l l a r y 6 . 1 . On t h e o t h e r h a n d ,

6 . 1 ) . NOW,

b y a s s u m i n g 3 ) and a p p l y i n g t h e argument u s e d i n t h e s e c o n d h a l f o f t h e p r o o f of Theorem 6 . 1

,

one o b t a i n s t h a t

=

Cm(X).

T h a t i s , 3 ) => 1 )

a s w e l l , and t h i s f i n i s h e s t h e p r o o f . 1

7. Appendix: Variants o f Nachbin’s Theorem W e c o n s i d e r , i n b r i e f , i n t h i s a p p e n d i x some v a r i a n t s o f Nach-

b i n ’ s Theorem r e f e r r i n g t o t h e a l g e b r a s e c t i o n 7.1 below)

,

and t h e a l g e b r a

CgtX),

cl(X)

w i t h 15rn<m ( c f . Sub-

o f complex-valued

em-

f u n c t i o n s on a C m - m a n i f o l d X (see S u b s e c t i o n 7 . 2 ) .

7.1. Differentiability of class t h e previous Sections 5, 6 e d a l r e a d y i n L . NACHBIN [ I ] .

t h a t t h e space X i n

is a (finite-dimensional) d i f f e r e n t i a l

d i f f e r e n t i a b i l i t y class

m a n i f o l d of

em.The case

ern ,w i t h

1 2 rn < 03,

h a s been c o n s i d e r -

Thus, a s f a r a s o n e c o n s i d e r s real-valued

functions, t h e argument a p p l i e d i n t h e p r e v i o u s s e c t i o n s i s s i m i l a r l y

C g l X ) of r e a l - v a l u e d d i f f e r e n t i a b l e f u n c on a ( p a r a c o m p a c t ) C”-rnanifoId X. So one g e t s t i o n s of “cZass t h e a n a l o g o n o f Theorem 6 . 1 i n t h i s c a s e too. Of c o u r s e , t h e a l g e b r a

used h e r e f o r t h e a l g e b r a

em

i s endowed, i n t u r n , w i t h t h e r e s p e c t i v e c a n o n i c a l

Cm-topology,i.e.,

t h e topology of uniform convergence on cornpacta of t h e ( d i f f e r e n t i a b l e ) f u n c t i o n s on X and of t h e i r d e r i v a t i v e s u p t o order rn (see, f o r i n s t a n c e , F. TREVES [ I :

p. 85 f f ] )

.

The t o p o l o g i c a l a l g e b r a

( r e a l ) Frgehet localZy m-eonuex algebra a l s o Corollary I;7.2 compact X )

.

CrnfX) thus

defined is s t i l l a

( w i t h an i d e n t i t y e l e m e n t ; i b i d . See

o r y e t C o r o l l a r y I ; 7 . 3 of t h i s book i n c a s e o f

I n t h i s r e s p e c t , w e s t i l l n o t e t h a t t h e r e s p e c t i v e m a t e r i a l of C h a p t e r IV, S e c t i o n 4 . ( 2 ) u s e d above h o l d s t r u e , of c o u r s e , f o r t h e ‘I

( r e a l ) Crn-caseIt as w e l l .

7.2. Comp1exification.-

W e consider next the case t h a t the differ-

252

VII

SPECTRA OF PARTICULAR ALGEBRAS

e n t i a b l e f u n c t i o n s i n v o l v e d i n t h e p r e c e d i n g d i s c u s s i o n a r e complexv a l u e d . That i s , w e s e t

c"(x) =

(7.2)

The c a s e of t h e a l g e b r a

e,"(X)

cCrncx).

is then treated similarly a s i n the

previous s e c t i o n 7 . 1 . Thus , c o n c e r n i n g Condition ( N l ( D e f i n i t i o n 6.1 ) f o r every f E e , " ( X )

, we

remark t h a t ,

,

t h e c o r r e s p o n d i n g d i f f e r e n t i a l of f a t some p o i n t i s now, i n g e n e r a l , a complex-valued map on T(X, x); i.e., one h a s

x EX,

( d f )x : F(X, X I

(7.3)

-

Q:

t h i s b e i n g , o f c o u r s e , IR-linear ( s a t i s f y i n g , moreover, t h e L e i b n i z c o n d i t i o n ; c f . ( 5 . 2 ) ) . So one may c o n s i d e r ( 7 . 3 ) a s a Leibniz map on t h e compZexified tangent space o f X a t x of the aZgebra 1 7 . 2 ) ) . T h a t i s , w e have

( o r y e t a co.nplex derivation a t X E X ,

,

( d f l , : T (X,x) = T I X , x) 8 C -C C IR i n s i c h a manner t h a t (7.4)

(dfl,

(7.5)

f o r a n y v e T ( X , x I and X E C . NOW, s i n c e C % I R + i l R imaginary part of

(7.4)

,

(V

8 A) =

, with

X. t d f l , ( v )

I

i = R , by c o n s i d e r i n g t h e r e a l and

namely, R e ( ( d f & ) and l r n ( ( d f ) x ) , r e s p e c t i v e l y ,

one g e t s t h e a n a l o g o u s r e s u l t to Lemma 5 . 2 , h e n c e , f i n a l l y , a Nachbin c h a r t a t x e x c o r r e s p o n d i n g t o R e ( i d f 1 , ) . [ I n t h i s r e s p e c t , see a l s o N . BOURBAKI [2:

Chap. 2 ; p . 1 4 , C o r o l l a i r e I] )

i n t h e proof of Theorem 6 . 1

. So

t h e argument a p p l i e d

i s s t i l l v a l i d , assuming moreover t h a t t h e

a l g e b r a A G C Z t X ) i s seZf-adjoint; t h a t i s , w e assume t h a t P E A , f o r e v e r y f e A

(7d e n o t i n g

t h e complex-conjugate of t h e (complex-valued) f u n c -

t i o n f e A ) . Thuslone obtains t h e following. Theorem 7.1. Let X be an n-dimensional paracompact Cm-manifoZd and c " ( X l the l o c a l l y m-convex algebra o f complex-valued e " - f u n c t i o n s

on X i n the C " - t o -

.

p o l o g y ( c f . S e c t i o n I V ; 4 . ( 2 ) ) Moreover, l e t A be a s e l f - a d j o i n t subalgebra of Then, one has the r e l a t i o n (7.6) A = em(X)

c"(X).

i f , and o n l y i f , the corresponding t h r e e conditions of Lema 5.1 are s a t i s f i e d . I

253

Some S p e c i a l C l a s s e s o f Topological Algebras

CHAPTER VIII

W e c o n t i n u e i n t h i s c h a p t e r t h e s t u d y of c e r t a i n p a r t i c u l a r c l a s s e si of t o p o l o g i c a l a l g e b r a s , some of which have a l r e a d y been c o n s i d e r e d i n t h e f o r e g o i n g . The same c l a s s e s of a l g e b r a s a r e a l s o p a r i c u l a r 1 Y significant for the applications.

1. Spectrally barrelled algebras (contn'd.) The t o p o l o g i c a l a l g e b r a s i n t i t l e have been d e f i n e d a l r e a d y (Def i n i t i o n V;1.3)

and u s e d i n v a r i o u s i n s t a n c e s i n t h e f o r e g o i n g d i s c u s -

s i o n (see C h a p t e r V). I n t h e f o l l o w i n g w e s t i l l g i v e some f u r t h e r b a s i c p r o p e r t i e s of t h i s c l a s s of a l g e b r a s which a r e u s e f u l i n l a t e r applications. Thus, w e c o n s i d e r n e x t s e v e r a l " p r o p e r t i e s of permanence" which t h e s e a l g e b r a s p o s s e s s and f i r s t t h a t o f " a d j o i n i n g an i d e n t i t y e l e ment".

However, w e s h a l l t u r n , f o r a w h i l e , t o a r e l e v a n t s u b j e c t con-

c e r n i n g " t o p o l o g i c a l d i r e c t sums" of s p e c t r a l l y b a r r e l l e d a l g e b r a s . So suppose w e are g i v e n a f i n i t e s e q u e n c e IEi)l<,isn

of t o p o l o g i -

c a l a l g e b r a s , and l e t

be the Ei

topological &%re& sim of t h e r e s p e c t i v e t o p o l o g i c a l v e c t o r s p a c e s

( I S i S n ) which,of

course, coincides, within a topological vector

s p a c e isomorphism, w i t h t h e c o r r e s p o n d i n g C a r t e s i a n p r o d u c t t o p o l o g i n c a l v e c t o r s p a c e I7 Z . (see Chapt. I; S e c t i o n 1 and Chapt. 11; S e c t i o n s 4 , 5. Cf. a l s o

i=l Z, J . EOEVATH [I: p. 1 2 1 1 ) . Thus by c o n s i d e r i n g t h e c o o r d i n a t e -

w i s e d e f i n e d m u l t i p l i c a t i o n i n E one o b t a i n s from ( 1 . 1 ) a topological aZgebra

( c f . P r o p o s i t i o n 111; 1 . 1 ) which h a s , i n p a r t i c u l a r ,

continuous

m u l t i p l i c a t i o n o r i s l o c a l l y convex o r l o c a l l y m-convex a s t h e case may be of t h e i n d i v i d u a l a l g e b r a s Bi , 1 6 i < n ( i b i d . ; see a l s o Lemma 111; 1.1).

The t o p o l o g i c a l a l g e b r a B t h u s d e f i n e d by ( 1 . 1 ) i s s t i l l c a l l e d t h e topologica2 d i r e c t sum

( c f . a l s o 111; ( 1 . 5 ) ) of t h e g i v e n ( f i n i t e ) fam-

VIII

254

SPECIAL TOPOLOGICAL ALGEBRAS

i l y of t o p o l o g i c a l a l g e b r a s ( E i j l Z i s n . So w e f i r s t h a v e t h e f o l l o w i n g .

Lemma 1 . 1 . The s p e c t r m of a t o p o l o g i cal d i r e c t sum of a f i n i t e sequence of

topologicaZ algebras is, w i t k i n a homeomorphism, the topologic al sun of the spectra of t h e f a c t o r algebras. That is ( a p p l y i n g t h e a b o v e n o t a t i o n ) , cne has

m(E)=m(.g E.) ~ = zl

(1.2)

= :nL'iEi),

w it h i n a homeomorphism of t h e Y es p ect i ve topological spaces. Proof. One may p r o c e e d by i n d u c t i o n , h e n c e it s u f f i c e s t o p r o v e ( 1 . 2 ) f o r n = 2 . So f o r e v e r y h e ?1Z(Ei@E2), c o n s i d e r t h e f o l l o w i n g d i a -

gram

(1.3)

w i t h i = l , 2, a n d t h e o b v i o u s n o t a t i o n c o n c e r n i n g t h e ( c a n o n i c a l ) proj e c t i o n s and i n j e c t i o n s ( t h e s e b e i n g , i n f a c t , continuous aZgebra mor-

phisms). Thus , t h e r e l a t i o n p . = j,.opr.

(1.4)

7,

I.

(

z

i= 1 , 2 ) ,

y i e l d s t h e r e s p e c t i v e "components" o f h , t h a t i s , o n e h a s

h.= hopi= h o l j . o p r . ) ,

(1.5)

2

2

i=1,2.

T h e r e f o r e , one o b t a i n s

h i e? ' ? Z ( E l @ E 2 ) + ( i = 1 , 2 ) , w i t h h = h1 + h 2 .

(1.6)

Thus, o n e g e t s from t h e l a s t r e l a t i o n t h a t o n e o f t h e h.'s h e n c e i f , for example,

hq#O

l e t a e E l @ E 2 such t h a t

hl l a ) # 0

(1.7)

and

h2(al = 0

( W e may t a k e , f o r i n s t a n c e , a = (x, 0) 6 El @ E2

,

. f o r some z6 El

t h a t hl aj = hl iz, 0) #O, w h i l e from ( 1 . 5 ) one h a s h2(al = 0 ) . NOW, element

z e E @E2 1

, one

i s non-zero;

,

such

f o r every

o b t a i n s from ( 1 . 6 )

h ( a z l = h ( a l . h l z l = ( h l l a l + h 2 1 a l ) - ( h l ( z )+ h 2 ( z ) ) (1.8)

= hl laihl

(2) t

hl la)h2 l z i + h2 ( a l h , ( z ) + h2 (ajh2( z l = hllalhllz) +h,la)h21zl= h , l a z ) .

T h e r e f o r e , by ( 1 . 7 ) ,

one h a s h 2 1 z l = 0 , t h a t i s h = 0 . C o n s e q u e n t l y , every 2

1.

S P E C T R A L L Y BARRELLED ALGEBRAS ( CONTN'D. )

255

element h e ?YK(E, @ E 2 1 i s of t h e form h =); .o p r .

(1.9)

1.

2-

(for some i = I , 21,

withxi

6 nT(Ei/ (i=l,2;cf.( 1 . 3 ) ) . Furthermore, it is clear that ( 1 . 9 ) nonical) bijection

provides, in effect, a (ca-

(1.10)

Besides, t h e topology of m i E l @ E 2 1 i s c l e a r l y , by the same definitions (cf. Lemma V; 1 . 1 ) , t h a t of the d i s j o i n t union m l E I V m i E , ) , and this completes 1 the proof. 1 Remark 1 . 1 .-Concerning the preceding proof , we note that the rels. ( 1 . 7 ) and ( 1 . 8 ) which led to the form of h e TnlEZBE21 given by ( 1 . 9 ) , do not actually depend on t h e coordinatewise definition of the multiplication in ( 1 . 1 ) Besides, regarding the topological conclusion at the end of the same proof, see also ( 1 . 1 1 ) in the sequel.

.

NOW, the next is an immediate application of the foregoing.

Proposition 1.1. The topological d i r e c t swn o f a given f i n i t e sequence ( E . ) , ISiSn, of topological algebras is a s p e c t r a l l y b a r r e l l e d algebra if, and onl;;' if, t h i s is t h e case f o r eaciz one of t h e f a e t o r algebras E i , 1 5 i S n . Suppose that the algebras E . ( l < i < n ) are spectrally barrelled and let B be a bounded subset of m i E l c E t ; so each one of the sets B T1 ??Z(EiI 2 Bi is a bounded subset of F?'Z(E .if ISiSn, (where the latter space is identified with its image in the topological sum given by ( 1 . 2 ) ) . Hence they are, by hypothesis, equicontinuous; moreover, the Proof.

i.lBi

polar set :B is thus a neighborhood of Oe E i , and hence the set is a neighborhood of zero in J 1 E i 2 .8 E . = E . Furthermore, one has by z = l 2( 1 . 9 ) .i 3: E B O G E , that is Bo is a neighborhood of 0 B E , and hence B E z = l 2m(E)is equicontinuous, so E is spectrally barrelled. n NOW, if E = %.@= I Ei is spectrally barrelled and B is a bounded subset of m ( E i I z 7 7 Z ( E l , then by ( 1 . 2 ) the latter is also a bounded subset of ? ? Z l E ) , hence by hypothesis equicontinuous. Thus, its polar set B O G E is a neighborhood of O E E. Therefore, since B0n Ei is a neighborY

Ei

hood of 06 Ei , one concludes that BS(BOn Eil0 C is an equicontinuous subset of m(E.), that is Ei is a barrelled algebra ( 1 6 i S n ) , and the proof is complete.1 Our next result (Proposition 1 . 2 ) ,

referred to at the beginning

256

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

o f t h i s s e c t i o n , i s c o n c e r n e d w i t h a t o p o l o g i c a l ( d i r e c t ) sum of t h e form c o n s i d e r e d i n ( l . l ) , where now t h e m u l t i p l i c a t i o n i s d e f i n e d n o t ( S e e , however, t h e p r e v i o u s Remark 1 . 1 ) .

coordinatewise a s before.

Thus, g i v e n a t o p o l o g i c a l a l g e b r a E , t h e a l g e b r a o b t a i n e d by t h e ( t o p o l o g i c a 1 ) a d j u n c t i o n o f a n i d e n t i t y e l e m e n t i s , by d e f i n i t i o n , 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 ( d i r e c t ) sum E+=E$c, t h e m u l t i p l i c a t i o n i n E+ b e i n g d e f i n e d by I ; ( 5 . 3 ) ( c f . a l s o LemmaI;5.3). NOW, c o n c e r n i n g

the spectrum of E: w e f i r s t h a v e t h e f o l l o w i n g re-

l a t i o n , r e f e r r i n g t o t h e r e s p e c t i v e weak t o p o l o g i c a l d u a l s p a c e s : (EfBCg

(1.11)

E'eC

E,'x

S

E,

w i t h i n a n isomorphism o f t h e l o c a l l y convex s p a c e s i n v o l v e d (see, f o r SCHAEFER [ I : p. 138, C o r o l l a r y 1 1 ) . Thus, one obtains

instance, H.H.

??Z(E+) = m ( E ) + = ??Y(E) U { O ]

(1.12)

w i t h i n a homeomorphism: T h a t i s , g i v e n any

fE

,

??Z(El, one d e f i n e s

?€

m(E'l

by t h e r e l a t i o n f(x,

(1.13)

X) = f ( x l + x ,

f o r e v e r y ix, XJE E+. Thus, o n e a c t u a l l y o b t a i n s

p= f @ i d c , f 6

(1.14)

m(El ,

t h e n o t a t i o n a p p l i e d b e i n g o b v i o u s ( s e e a l s o ( l . l O ) , ( l . l l ) , and R e -

it i s c l e a r t h a t o n e h a s , i n t h e n o t a t i o n o f t h e l a s t r e l a t i o n , 5 = id, as it c o n c e r n s ( 1 . 1 2 ) . F i n a l l y , it i s o b v i o u s from ( 1 .11) t h a t t h e c o r r e s p o n d e n c e (1.15) f (f, id,) , f E m ( E l ,

mark 1 . 1 ) . Moreover,

-

g i v e n by ( 1 . 1 3 ) ( o r ( 1 . 1 4 ) ) d e f i n e s ( 1 . 1 2 ) w i t h i n a homeomorphjsm. I n t h i s r e s p e c t , i t i s a l s o c l e a r from ( 1 . 1 2 ) , a n d i n c o n j u n c t i o n w i t h V; ( 1 . 8 )

,

t h a t one has

mi~+i n c 0; = V Z ( E ~,

(1.16)

& t h i n a homeomorphism. The f o l l o w i n g i s a d i r e c t c o n s e q u e n c e o f comment made i n Remark V; 6 . 1 .

(1.12) and t h e r e l e v a n t

Thus, w e h a v e .

Proposition 1 . 2 . A topological algebra E i s s p e c t r a l l y barrelled i f , and only if, t h e algebra E + = E f3 Q:

obtained from E by "adjoining an i d e n t i t y element" (see Lemma I ; 5 . 3 ) is spec-

1.

SPECTRALLY BARRELLED ALGEBRAS (CONTN 'D. )

257

t r a l l y barrelled. On t h e o t h e r h a n d , t h e a n a l o g o n o f P r o p o s i t i o n 1 . 1 f o r a n a r b i t r a r y ( n o t n e c e s s a r i l y f i n i t e ) i n d e x s e t I i s now g i v e n by t h e f o l lowing.

Proposition 1.3. L e t E be the inductive l i m i t algebra of a given i n d u c t i v e fscl), a E I , of s p e c t r a l l y barrelled algebras endowed w i t h t h e respeca' t i v e f i n a l vector space topology ( c f . Lemma IV; 2.2). Then,

system ( E

(1.17)

E = ld im(E,,

fsa)

i s a spectralZy barreZled algebra. Proof.

I n view o f P r o p o s i t i o n 1.2, it s u f f i c e s t o p r o v e t h e asThus, one g e t s (see V ; (3.22) a n d a l s o V;

sertion f o r t h e algebra

E'.

(1.12)) (1.18)

??Y(E+) = lzm(E'),

w i t h i n a homeomorphism. T h e r e f o r e ,

i f B i s a bounded s u b s e t o f

m(E')

t h e sets t f , (B) = B o f , := { h of, : h E B

(1.19)

1

a r e bounded s u b s e t s o f ?Tr(E+),ue I, h e n c e by h y p o t h e s i s and P r o p o s i t i o n a 1.2 e q u i c o n t i n u o u s . So B i s an e q u i c o n t i n u o u s s u b s e t o f miE+)a s w e l l i c f . V;(3.39)), and t h i s p r o v e s t h e a s s e r t i o n . 1

Remark 1.2

.-

I n p a r t i c u l a r , t h e analogous co n clu s io n t o Proposi-

t i o n 1 . 3 i s c e r t a i n l y v a l i d , i f o n e c o n s i d e r s l o c a l l y convex o r l o c a l l y m-convex

(spectrally barrelled) algebras.

Moreover, c o n c e r n i n g t h e c o n t e x t o f t h e p r e v i o u s P r o p o s i t i o n 1.3, it would b e e q u i v a l e n t t o c o n s i d e r i n s t e a d , t a k i n g Scholium I V ; 2.1 i n t o a c c o u n t , a s e e m i n g l y more g e n e r a l s i t u a t i o n by t a k i n g a n a l g e b r a E e q u i p p e d w i t h t h e f i n a l v e c t o r s p a c e t o p o l o g y d e f i n e d on i t by a

given family

(E,,

f , ) , u e I, of b a r r e l l e d a l g e b r a s ; w e h a v e a l s o t o as-

sume t h a t t h e r e s p e c t i v e r e l a t i o n t o IV;(2.2) h o l d s t r u e , w h i l e t h e

, :E,+E, maps f

c1

e I , a r e of c o u r s e a l g e b r a morphisms.

The p r o p e r t y of b e i n g a t o p o l o g i c a l a l g e b r a s p e c t r a l l y b a r r e l l e d i s a l s o p r e s e r v e d by " p a s s i n g t o i t s c o m p l e t i o n " , when o f c o u r s e t h e

l a t t e r h a s a meaning ( c f . C h a p t . 1 ; S e c t i o n 4 ) . Thus, w e h a v e .

Lemma 1.2. Suppose t h a t E i s a s p e c t r a l l y barrelled algebra whose completion is

2

such t h a t one has

m&)= m(E)( w i t h i n

V;2.1). Then, i t s completion

2'

a c o n t i n u o u s b i j e c t i o n ; Lemma i s a l s o a s p e c t r a l l y barrelled algebra.

2 58

VIII

Proof.

SPECIAL TOPOLOGICAL ALGEBRAS

I f B i s a ( w e a k l y ) bounded s u b s e t of

r i it i s a bounded s u b s e t of

??Z ' (El

(Sm(2)) , and

so B i s an e q u i c o n t i n u o u s s u b s e t o f o u s s u b s e t s of V; 2 . 1

m(E) and

m(i1 a r e

m($),t h e n

a fortio-

hence equicontinuous;

a s w e l l ("the equicontinut h e same". C f . t h e p r o o f o f Lemma

a n d / o r N . BOURBAXI [ S : Chap. 10; p. 27, P r o p o s i t i o n 4 1 )

. That

is,

k

is

a s p e c t r a l l y b a r r e l l e d a l g e b r a t o o , which i s t h e a s s e r t i o n . 1 NOW, a p p l y i n g f o r example t h e t e r m i n o l o g y of Theorem V;2.1,

we

a c t u a l l y c o n c l u d e t h a t E i s a s p e c t r a l l y b a r r e l l e d algebra i f , and only if, h

t h i s i s t h e case f o r i t s completion E. On t h e o t h e r h a n d , w e s t i l l remark t h a t any topological algebra E

w i t h an equicontinuous spectrum ?7Y(El is s p e c t r a l l y b a r r e l l e d . I n p a r t i c u l a r , i n case o f l o c a l l y m-convex a l g e b r a s w e h a v e a l r e a d y c o n s i d e r e d i n C h a p t . V I ; S e c t i o n l s e v e r a l p r o p e r t i e s of a t o p o l o g i c a l a l g e b r a d e r i v e d from t h e h y p o t h e s i s o f b e i n g t h e a l g e b r a s p e c t r a l l y b a r r e l l e d ( c f . , f o r i n s t a n c e , Theorem V I ; l . 3 ) . So i t i s a cons e q u e n c e of Lemma V I ; 1 .3 t h a t a c o m t a t i v e (complete) l o c a l l y m-convex alge-

bra with an i d e n t i t y eZement and Gel'fand map continuous ( h e n c e a f o r t i o r i any s p e c t r a l l y b a r r e l l e d a l g e b r a o f t h e t y p e c o n s i d e r e d ) has a compact

spectrum i f , and only if, m(E1 is equicontinuous. F u r t h e r m o r e , t h i s is equivale-t u i t h E being a Q-algebra ( t h e l a s t c o n d i t i o n i m p l i e s , of c o u r s e , t h a t E is advertibly complete).

Now, t h e l a s t p r o p e r t y of t h e a l g e b r a E y i e l d s , i n t u r n , t h a t M ( E ) = m(E)( C o r o l l a r y I I ; 7 . 3 ) ; t h u s , t h e given algebra Z i s e s s e n t i a l l y ex-

pressed

( w i t h i n , namely, a n a l g e b r a isomorphism) as an i n d u c t i v e l i m i t of

( a s u i t a b l e i n d u c t i v e s y s t e m o f ) Banach algebras, i . e . , E i s a pseudo-Banach algebra i n t h e s e n s e of G . R . ALLAN e t a l . [l : p. 58, P r o p o s i t i o n and p. 59,

.

~roposition] Thus, w e may now summarize t h e f o r e g o i n g i n t o t h e form of t h e following. Lemma 1.3. Let E be a commutative complete l o c a l l y m-convex algebra w i t h an i d e n t i t y element and Gel' fund map continuous. Then, t h e following a s s e r t i o n s are

equivalent: 1) E i s a Q-algebra. 2)

??'Y(El is a compact space.

3 ) E is a pseudo-Banach algebra. In p a r t i c u l a r , if t h e algebra E is s p e c t r a l l y b a r r e l l e d , then a l l t h e pre-

Leeding t h r e e e7iiLluLe71 stotemenls are s t i l l such w i t h t h e following one: 4 ) E is a bounded algebra (i.e., every p o i n t of E has a bounded spectrum).

Proof. The e q u i v a l e n c e of t h e f i r s t t h r e e a s s e r t i o n s f o l l o w s from

1

.

SPECTRALLY BARRELLED ALGEBRAS ( CONTN 'D. )

259

t h e p r e c e d i n g d i s c u s s i o n . Moreover, w e a l w a y s h a v e , o f c o u r s e , t h a t

1)=>4)(cf. Lemma VI;l.3). N o w s u p p o s e , i n p a r t i c u l a r , t h a t E i s s p e c t r a l l y b a r r e l l e d . Then, i f i t i s a l s o a bounded a l g e b r a ( D e f i n i t i o n VI;l.l), one g e t s by t h e f o l l o w i n g r e l a t i o n

R l P ) = G(???(E)) = SpE(x),

(1.20)

w i t h X E E ( c f . C o r o l l a r y 111;6.4) t h a t m ( E I i s ( w e a k l y ) bounded, a n d h e n c e by h y p o t h e s i s e q u i c o n t i n u o u s . So 4 ) + 1 )

as w e l l ( c f . C o r o l l a r y

111;6.6), and t h i s c o m p l e t e s t h e p r o o f . 4 A s a m a t t e r o f f a c t , t h e a b o v e Lemma 1.3 i s a l r e a d y subsummed

i n t o t h e more g e n e r a l framework o f Theorem VI;1.3. Its p a r t i c u l a r q u o t a t i o n h e r e i s d u e , however, t o i t s c o n n e c t i o n w i t h p r o p . 3) o f t h e s t a t e m e n t ; t h e l a t t e r s h o u l d a l s o b e compared w i t h t h e s i t u a t i o n d e s c r i b e d by S c h o l i u m V;3.1, a s w e l l a s w i t h a s i m i l a r one i n S e c t i o n VIII;9. ( 3 ) i n t h e s e q u e l ( ~ Z k ) - a Z g e b r a s ) .

On t h e o t h e r h a n d , u n d e r t h e a s s u m p t i o n t h a t a g i v e n t o p o l o g i -

c a l algebra i s s p e c t r a l l y barrelledlone g e t s t h e following "multiplic a t i v e ( i . e . , a l g e b r a i c ) a n a l o g o n " o f t h e well-known s i t u a t i o n one h a s i n t h e weak t o p o l o g i c a l d u a l o f a b a r r e l l e d ( l o c a l l y convex t o p o l o g i c a l v e c t o r ) s p a c e . S e e , f o r i n s t a n c e , J . HORVA'TH

[I:p.

212, Coral

-

l a r y ] ) . That i s , w e have. Theorem 1.1. Let E be a s p e c t r a l l y b a m e l l e d algebra whose spectrum i s ??I(E/.

Then, t h e following c o l l e c t i o n s o f s u b s e t s of m(E) are i d e n t i c a l : 1 ) The equicontinuous s e t s ,

2) t h e (weakly) r e l a t i v e l y compact s e t s , 31 t h e (weakly) bounded s e i s .

Proof.

By t h e A l a o g l u - B o u r b a k i Theorem ( c f . J . HORVLTH

[ 1:

p. 201 ,

Theorem 11) , 1 ) = > 2 ), a n d it i s a l s o c l e a r t h a t 2)=>3) a s w e l l ,

for

any topological algebra not necessarily s p e c t r a l l y b a r r e l l e d .

(Here

II=> II

means, o f c o u r s e , " i n c l u s i o n "

b e t w e e n t h e r e s p e c t i v e c l a s s e s of

s e t s ) . F u r t h e r m o r e , b y h y p o t h e s i s f o r E , 3)=> 1 ) too ( D e f i n i t i o n V;1.3), so t h a t t h e t h r e e c l a s s e s o f s e t s a r e i n d e e d t h e s a m e . 4 So d e n o t i n g now by Ai ( i = l ,2 , 3 ) t h e a b o v e c l a s s e s of s e t s a p

-

w e c o n c l u d e t h a t a spectralZy barrel2ed algebra i s characterized by t h e r e l a t i o n

p e a r e d i n Theorem 1 . 1 ,

A = A = A

(1.21)

1

2

3'

Y e t t h e d e f i n i t i o n of a s p e c t r a l l y b a r r e l l e d a l g e b r a a m o u n t s , i n e f -

fect, t o the relation

Al = A , .

260

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

On t h e o t h e r hand, t h e r e l a t i o n

A

(1.22)

1

= A

2

characterizes t h e topological algebras f o r which the r e s p e c t i v e Gel'fand map i s continuous. T h i s i s a consequence o f Theorem VI: 1.1. I n t h i s r e s p e c t , w e remark t h a t t h e r e l a t i o n

(1.23) may hold t r u e , i n general; t h i s i s e a s i l y u n d e r s t o o d by c o n s i d e r i n g a l o which i s n o t a b a r r e l l e d c a l l y m-convex a l g e b r a of t h e form ectX) ( l o c a l l y convex) s p a c e (see t h e n e x t s e c t i o n ) G e l ' f a n d map of

i s continuous, ee(X)

.

Thus, by VII; (1.31) t h e

while t h e algebra i t s e l f i s not

s p e c t r a l l y b a r r e l l e d by h y p o t h e s i s and Theorem 1.1

( c f . a l s o t h e next

s e c t i o n : Nachbin-Shirota Theorem). F i n a l l y , the r e l a t i o n

(1.24)

A1

9

A, = A3

may c e r t a i n l y hold t r u e , i n general, a s w e s h a l l see i n t h e n e x t s e c t i o n ( Nachbin-Shirota algebras)

.

The above s i t u a t i o n i s f u r t h e r c l a r i f i e d by t h e f o l l o w i n g example which p r o v i d e s a s p e c t r a l l y barrelled algebra which i s not m-barrelled, i n a n t i c i p a t i o n of a more complete p i c t u r e s u p p l i e d by t h e n e x t section.

Example 1.1.1.(1) w i t h X = I R .

L e t cc(lR b) e t h e l o c a l l y m-convex a l g e b r a of VII;

N o w , s i n c e t h e sequence of compact i n t e r v a l s [ - n , n ] ,

w i t h n e IN, p r o v i d e s a

denumerable k-covering of IR ( s e e D e f i n i t i o n V:5.1, IR i s a hemicompact s p a c e ) , t h e a l g e b r a q I R ) i s a Fre'chet l o c a l l y m-convex algebra whose spectrum i s homeomorphic t o IR ( c f . Theorem VI1;l .2). On t h e o t h e r hand, t h e algebra BfW) of complex-valued c o n t i n u o u s and bounded f u n c t i o n s o n R , c o n s i d e r e d a s a t o p o l o g i c a l s u b a l g e b r a of e,tIR),i s a metrizable l o c a l l y m-convex algebra with an i d e n t i t y element

t h a t one h a s

BO

(1.25)

=

such

e p

( W e i e r s t r a s s - S t o n e Theorem, p l u s Uryson's Lemma: see, f o r i n s t a n c e , L. NACHBIN [4: p. 48, C o r o l l a r y 21 ) ; hence, one g e t s B ( I R ) 2 So,

ccIIR).

since

IR i s l o c a l l y compact and hence l o c a l l y e q u i c o n t i n u o u s , c o n s i d e r Cc(IRI ( c f . Theorem VI1:l. 1 ) , one has

ed a s t h e spectrum of

n z ( B c t r n ) ) = IR,

(1.26)

w i t h i n a homeomorphism ( c f . Lemma V; 2.2 and Theorem V; 2.1 ) spective Gel'fand map of

Be(=)

.

Hence , t h e re-

i s continuous ( b e i n g t h e r e s t r i c t i o n t o B ( I R J

1.

SPECTRALLY BARRELLED ALGEBRAS (CONTN'D.)

261

of the map VII; ( 1 . 3 1 ) ) .

NOW, it is clear that every (weakly) bounded subset of W i s (weakly) r e l a t i v e l y compact (Bolzano), where t h e weak topology of IR (the latter space being considered as the spectrum of B c t W ) ) i s , i n f a c t , t h e usual topolog y of R

according to the relation

(1.27)

1~

=m

e ( IR))

G

I

epq

(Theorem VI1;l . 2 ) . Thus, the algebra BcfJR) satisfies the relation A2 =A3,in the terminology of Theorem 1.1. Hence one concludes from the same theorem and ( 7 . 2 1 ) that Bc(IR) i s a s p e c t r a l l y barrelled algebra. On the other hand, t h e s e t

I

V = { ZE B ( B ) : z(tlI5 1

(1.28)

v teW

1

i s c e r t a i n l y an m-barrel of

B (IR) which i s not a neighborhood of zero. Therefore, B,( IR) i s not an m-barrelled algebra, which proves the assertion. (In this respect, see also the next two sections).

We conclude this section with the following "geometric" (i.e., structural) characterization of spectrally barrelled algebras in analogy with the familiar situation one has, for instance, in the case of m-barrelled algebras (cf. Definition 1;1.4). The idea is due to J . ARAHOVITIS [3: Theorem 1 . 3 ] . Thus , we have. Theorem 1.3.

Let E be a topological algebra whose spectrwn i s 9?Y(El. Then,

t h e following two a s s e r t i o n s are e q u i v a l e n t : 1) E i s s p e c t r a l l y b a r r e l l e d .

2) Every m-barrel of E which contains t h e polar of a bounded subset of n T ( E ) i s a neighborhood of 0 i n E. Proof. Suppose that E is spectrally barrelled, and let T be an m-barrel of E such that B O G T I where B G m ( E ) is (weakly) bounded. Then by hypothesis, B is an equicontinuous subset of m(E),and hence its polar B o a neighborhood of zero in E . So the same holds for T, i.e., 1)-2).

A s a matter of fact, every subset of E s a t i s f y i n g cond. 2 ) i s a neighborhood of zero i n E , under the hypo-

thesis that the algebra E is spectrally barrelled. On the other hand, if 2 ) is valid and B E ? ? Z ( E ) i s (weakly)bounde d , then i t s polar s e t Bo is an m-barrel of E (cf., for example, the proof of Proposition V; 1.1 ) Therefore, by hypothesis , Bo is a neighborhood of Oe E l so that B G B o o is equicontinuous; thus, 2)=> 1 ) as well, and the proof is complete. g

.

We remark that cond. 2 ) is, in fact, equivalent

262

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

with the seemingly more stringent hypothesis that: Every m-barrel of E which i s t h e polar of a bounded subset of m(E) i s a neighborhood o f zero i n E . NOW, this combined with the remark made in the preceding proof entails that 1) is essentially equivalent with the (seemingly) much more stringent hypothesis that: Every subset of E which contains ( o r i s ) t h e polar of a bounded subset of miEl i s a neighborhood o f 0 e E.

2 . Nachbin - S h i r o t a algebras The class of topological algebras in title of this section is, in fact,characterized by the rel. (1.24) of the preceding section. However, we first need some preliminary material. Thus, suppose we are given a (Hausdorff) completely regular

cc(X)

be the corresponding locally m-convex algebra space X , and let of Chapt.VII;Section 1. So by considering the (evaluation) map 6: x

(2.1)

-

(ecixii: v

defined by VII;(l.2), one gets a homeomorphism of X onto the spectrum of f X ) (Theorem V I I :1 .2 ) . On the other hand, the classical Nachbin-Shirota Theorem proves

cc

cc(X)

i s b a r r e l l e d if, a d only i f , every bounded that t h e l o c a l l y convez space subset o f X i s r e l a t i v e l y compact, where the latter space is identified v i a t h e map ( 2 . 1 ) with the spectrum of the algebra C,IX);that is, when the , topology of X is (identified with) the relative one from I

ec(X));

the weak topological dual of q X ) . Therefore, according to the classical terminology, q X ) is a barrelled (locally convex) space if, and only if, every closed and "relatively precompact" ( T. SHIROTA [I] ) subset of X is compact. In this respect, a set B G X is said to be r e l a t i v e -

Zy p r e c o q a c t , if every real-valued continuous function on X is bounded. Thus, for every ff q X ) , one gets that I f ( B ) I G l R is a bounded subset, so that the set (2.2)

is a weakly bounded subset of X, if one applies the homeomorphism defined by (2.1). Solin view of the previous terminology, we call a Hausdorff completely regular space X a Nachbin - Shirota space, if every bounded subset of X is relatively compact, the topology of X being defined by (2.1). Equivalently (Nachbin-Shirota Theorem), if the locally convex space

ccIX)is barrelled. Accordingly, we set the following.

2. NACHBIN-SHIROTA ALGEBRAS

263

D e f i n i t i o n 2.1. A g i v e n t o p o l o g i c a l a l g e b r a E i s s a i d t o b e a Nachb i n - Shirota algebra, i f

i t s spectrum ?i7(E) i s a Nachbin-Shirota s p a c e .

A s a n i m m e d i a t e c o n s e q u e n c e of t h e p r e v i o u s d i s c u s s i o n , w e h a v e

now t h e f o l l o w i n g " t h e o r e m - ( d e f i n i t i o n ) ' I .

Theorem 2.1.

L e t E be a g i v e n topologica2 algebra. Then, t h e f o l l o w i n g as-

s e r t i o n s are e q u i v a l e n t : 1 ) E i s a Nachbin-Shirota algebra.

2) Every bounded s u b s e t o f miE) i s r e l a t i v e l y compact ( i . e . , one h a s the relation A,=A 3) C c ( 7 & ( E I )

Thus,

3

i n t h e t e r m i n o l o g y o f S e c t i o n 1).

is a b a r r e l l e d ( l o c a l l y m-convex) algebra. I

( 1 . 2 1 ) c a n now b e e x p r e s s e d by t h e f o l l o w i n g .

C o r o l l a r y 2.1.

A t o p o l o g i c a l algebra i s s p e c t r a l l y b a r r e l l e d i f , and only

i f , i t i s a Nachbin-Shirota algebra having, moreover, a continuous GeZ'fand map. I On t h e o t h e r h a n d , not every Nachbin-Shirota algebra i s s p e c t r a l l y bar-

r e l l e d . I n o t h e r words, t h e r e l . (1.24) may h o l d , i n g e n e r a l , a s t h i s i s s e e n from t h e f o l l o w i n g .

Example 2.1

.-

L e t E = Ps ( [ O , I]) b e t h e a l g e b r a of C-valued

polyno-

m i a l f u n c t i o n s ( p o l y n o m i a l s ) on t h e compact i n t e r v a l [0, I ] G R endowed w i t h t h e t o p o l o g y o f s i m p l e c o n v e r g e n c e i n [O, l ] . T h u s , i n g t h e a l g e b r a E a s a s u b a l g e b r a of CrgY'] p o l o g y , one o b t a i n s t h a t E i s made i n t o a

by c o n s i d e r -

i n t h e r e l a t i v e product to-

( c o m m u t a t i v e ) l o c a l l y m-convex

algebra ( w i t h a n i d e n t i t y e l e m e n t ; see a l s o Lemma 1 I I ; l . l ) i n such a way t h a t one has rn'PS([O,

(2.3)

21)) =

[o, 13 ,

w i t h i n a homeomorphism. T h a t i s , t h e a l g e b r a P([O, r e l a t i v e t o p o l o g y from t h e Banach a l g e b r a

11) e q u i p p e d w i t h t h e

cis([O,

I]) i s a d e n s e sub-

a l g e b r a of t h e l a t t e r ( W e i e r s t r a s s Theorem; see, f o r i n s t a n c e , L . NACHB I N [4: p. 49, Theorem 1 1 )

.

Hence, i t s s p e c t r u m i s homeomorphic t o [ O , 13

s o t h a t one g e t s

(2.4)

(PILO,

1111

=mceui[o, 1 1 ) ) = p, 1 1 ,

w i t h i n homeomorphisms of t h e s p a c e s u n d e r c o n s i d e r a t i o n ( c f . Theorem

V;2.1, i n c o n j u n c t i o n w i t h Lemma V;2.2 and C o r o l l a r y V I I ; 1 . 2 ) .

Now,

(2.4) i m p l i e s t h e d e s i r e d r e 1 . ( 2 . 3 ) , s i n c e a n y c o n t i n u o u s c h a r a c t e r of

Ps([O,

11) is a f o r t i o r i

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

P I [ O , I ] / E C ~ ( [ O ,I ] ) . T h e r e f o r e , i n

view o f

(2.4) , it i s g i v e n by a n

,

VIII SPECIAL TOPOLOGICAL ALGEBRAS

264

e l e m e n t of [ O , l ] , morphism) of

t h e l a t t e r s p a c e b e i n g a s u b s p a c e ( w i t h i n a homeo-

m(Ps[O,I]j).

So e v e r y bounded s u b s e t of t h e l a s t s p a c e i s , of c o u r s e , b y

( 2 . 3 ) , r e l a t i v e l y c o m p a c t , so t h a t (Theorem 2 . 1 ) $ ( [ O , I ] i

Shirota algebra. On t h e o t h e r hand,

i s a Nachbin-

t h e r e s p e c t i v e Gel’fand map

( c f . a l s o V I I ; ( 1 . 3 1 ) ) cannot be continuous, t h e t o p o l o g y of p o i n t w i s e conv e r g e n c e i n [0,2] f o r t h e a l g e b r a P([O,l]I

b e i n g a c t u a l l y s t r i c t l y smal-

l e r t h a n t h a t of compact ( u n i f o r m ) c o n v e r g e n c e i n [ O , I ] t h a t t h e algebra P , ( [ O , l ] )

;t

his entails

i s not s p e c t r a l l y b a r r e l l e d ( c f . C o r o l l a r y 2 . 1 ) .

More g e n e r a l l y , w e s t i l l remark t h a t any topological algebra w i t h a

compact spectrum a n d d i s c o n t i n u o u s G e l ‘ f a n d map, l i k e t h e a l g e b r a PJ [0,1]) of t h e above Example 2 . 1

,

i s a Nachbin-Shirota algebra (Theorem 2 . 1 )

, which

cannot be s p e c t r a l l y b a r r e l l e d (Corollary 2 . 1 ) . F u r t h e r m o r e , a Nachbin Shirota algebra which does n o t necessarily have a

compact spectrum

b u t d o e s h a v e a d i s c o n t i n u o u s G e l ’ f a n d map i s c e r t a i n -

l y p o s s i b l e ; t a k e , f o r i n s t a n c e , t h e a l g e b r a Ps(7R) ( p o l y n o m i a l f u n c t i o n s ) p o l y n o m i a l s on convergence i n

W.

of a l l C-valued

IR w i t h t h e t o p o l o g y of s i m p l e

So a p p l y i n g a s i m i l a r argument t o t h a t of t h e p r e -

v i o u s example, a n a n a l o g o u s r e l a t i o n t o ( 2 . 4 ) (see a l s o L . NACHBIN 4 9 , Theorem l ] ) , a n d t a k i n g Theorem V ; 2 . 1

[a:

p.

i n t o a c c o u n t , one c o n c l u d e s

t h a t t h e spectrum of Ps(XZl 7:s homeomorphic t o IR. Hence ( D e f i n i t i o n 2 . 1 ), Ps(IR/

i s a Nachbin-Shirota algebra h a v i n g , moreover , a d i s c o n t i n u o u s Gel’fand map. The p r e v i o u s d i s c u s s i o n l e a d s now t o t h e f o l l o w i n g scheme.

Fr6cchet ( t o p o l o g i c a l a l g e b r a s (t.a.1; I.ocall y convex o r l o c a l l y m-convex)

infra-Pta7c ( c f . 5 9 . ( 6 ) )

d

s p e c t r a l l y barre Zled

J

t.a. w i t h Gel’fand map continuous (Scheme 2 . 1 )

265

3. FUNCTIONAL REPRESENTATIONS

The p r e v i o u s scheme e x h i b i t s t h e i n t e r r e l a t i o n s among t h e v a r i o u s c l a s s e s of t o p o l o g i c a l a l g e b r a s c o n s i d e r e d h i t h e r t o (Ptdk a l g e b r a s w i l l b e d e f i n e d i n t h e n e x t s e c t i o n ) , which a r e r e l a t e d w i t h t h e c o n c e p t of " b a r r e l l e d n e s s "

( s p a t i a l , a l g e b r a i c o r s p e c t r a l ) . An a r r o w

means " g e n u i n l y l a r g e r c l a s s " a c c o r d i n g t o t h e e x a m p l e s c o n s i d e r e d i n t h e f o r e g o i n g . B e s i d e s , t h e l a s t two c l a s s e s o f a l g e b r a s

t o e a c h o t h e r ( c f . t h e p r e v i o u s Example 2 . 1 ),

are different

t h e i r " i n t e r s e c t i o n " be-

i n g t h e class of s p e c t r a l l y b a r r e l l e d a l g e b r a s ( C o r o l l a r y 2 . 1 ) . W e c o n c l u d e w i t h t h e f o l l o w i n g comment r e l a t e d w i t h t h e above

Example 2 . 1 ,

and which a l s o c l a r i f i e s t h e c o n s i d e r a t i o n s i n C h a p t . V;

Section 2 .

Scholium 2.1.-

L e t E = P,([D,l]I

t h e p r e v i o u s Example 2 . 1

,

and l e t

b e t h e l o c a l l y rn-convex

E^

a l g e b r a of

be i t s completion; (i.e., equival-

e n t l y , one c o n s i d e r s i t s c l o s u r e i n t h e complete s p a c e C[oyl]).Thus, it h a s b e e n p r o v e d b e f o r e ( c f . morphic t o

[O,l].

( 2 . 3 ) ) t h a t t h e s p e c t r u m of E i s homeo-

Hence, one o b t a i n s

(2.6) where t h e f i r s t e q u a l i t y i s t r u e w i t h i n a c o n t i n u o u s b i j e c t i o n (cf. Lemma V ; 2 . 1 ) .

NOW, t h e s e q u e n c e of p o l y n o m i a l s

p (x)= x ( 1 - x ) ( 1 . + x i...+ z n ) , n E N ,

(2.7)

with x E [ 0 , 1 ] , converges t o a function f (2.8)

,

€E^

d e f i n e d by t h e r e l a t i o n

with

O<x
with

x=l. h

F u r t h e r m o r e , u s i n g t h e p r e c e d i n g f u n c t i o n f € E , it i s c l e a r t h a t

m(.??),

VY(E) i s a n i s o l a t e d p o i n t o f so t h a t t h e c o n t i n u o u s b i 1 e [0,1] j e c t i o n ( 2 . 6 ) cannot be a homeomorphism of t h e r e s p e c t i v e t o p o l o g i c a l s p a c e s . C o n c e r n i n g t h e example d i s c u s s e d h e r e , see a l s o B . GUENNEBAUD [I : p.901).

A p p l y i n g t o p o l o g i c a l t e n s o r p r o d u c t t e c h n i q u e s (see P a r t I1

o f t h i s b o o k ) , W. DIETRICH, Jr. [I: p. 21G] h a s a l s o g i v e n a n example i n t h e

same s p i r i t a s t h e l a s t o n e l r e f e r r i n g , namely, t o t h e p o s s i b i l i t y o f b e i n g a c o n t i n u o u s b i j e c t i o n , l i k e ( 2 . 6 ) , i n g e n e r a l , n o t a homeomorphism.

( I n t h i s r e s p e c t , see a l s o C h a p t . V ; ( 2 . 5 ) and A . MALLIOS[l:p. 1731).

3. Functional representations W e consider i n t h e following topological algebras admitting a

f u n c t i o n a l r e p r e s e n t a t i o n ; t h u s t h e s e t o p o l o g i c a l a l g e b r a s may b e i d e n t i f i e d , w i t h i n a t o p o l o g i c a l a l g e b r a i c isomorphism, w i t h t h e l o c a l l y

VIII

266

SFECIAL TOPOL9GICAL ALGEBRAS

m-convex a l g e b r a s o f complex-valued c o n t i n u o u s f u n c t i o n s on t h e i r s p e c t r a , t h e l a t t e r a l g e b r a s b e i n g t o p o l o g i z e d i n t h e compact-open t o p o l o g y . So w e f i r s t h a v e t h e f o l l o w i n g . D e f i n i t i o n 3.1.

L e t E b e a t o p o l o g i c a l a l g e b r a whose s p e c t r u m i s

m(EI. W e s h a l l s a y t h a t

E i s a full algebra,whenever t h e r e s p e c t i v e

G e l ' f a n d map

i s a b i j e c t i o n , h e n c e an algebraic onto isomorphism of t h e c o r r e s p o n d i n g algebras. Thus, t h e G e l ' f a n d t r a n s f o r m a l g e b r a F A = $ ( E l

of a f u l l a l g e -

C c ( ? 7 ( E I ) , where t h e c o r r e s p o n d e n c e e s t a b l i s h e d by i s a l s o 1 - 1 . Now, t h e l a s t c o n d i t i o n e n t a i l s , of c o u r s e , t h a t t h e a l g e b r a E is actually commutative, f o r it i s

b r a E i s t h e whole o f t h e " f u n c t i o n a l g e b r a "

F

i d e n t i f i e d , v i a t h e a l g e b r a morphism (3.1) , w i t h t h e commutative f u n c t i o n a l g e b r a E A , w h i l e t h e l a t t e r i s a l s o , by h y p o t h e s i s , t h e whole

of

??Z(E).

So the algebra E has necesazrily an identity element t o o .

On t h e o t h e r h a n d , s u p p o s e t h a t w e a r e g i v e n a commutative l o c a l l y rn-convex a l g e b r a E ; t h e n , t h e h y p o t h e s i s t h a t to-one

the map

$ is

one-

implies t h a t

n kdf) c

(3.2)

fEM(EI

n ker(f) fE M ( E )

n k e r ( f I = 101

=

f

€

m(E)

( c f . 11; (7.4) i n c o n n e c t i o n w i t h 11;(7.11)). T h u s , t h e commutative a l g e b r a E i s semi-simple,

i n the usual

a l g e b r a i c s e n s e ; namely, o n e t h e n

o b t a i n s by ( 3 . 2 ) RE) =

(3.3)

( s e e Lemma II;7.3)

,

n MIEI

=

COI

where R ( B J d e n o t e s t h e Jacobson radical of t h e (com-

mutative) r i n g E . Thus, b y an o b v i o u s " a b u s e of l a n g u a g e " and e x t e n s i o n , as w e l l , of t h e p r e v i o u s s i t u a t i o n , one s e t s now t h e f o l l o w i n g . D e f i n i t i o n 3.2. A g i v e n t o p o l o g i c a l a l g e b r a E i s s a i d t o b e semi-

simple, i f t h e r e s p e c t i v e G e l ' f a n d map o f E i s one-to-one. words, d e n o t i n g by

m(EI t h e s p e c t r u m o f E ,

In o t h e r

o n e h a s , by d e f i n i t i o n ,

the relation (3.4)

I n v i e w o f t h e p r e v i o u s r e l a t i o n (3.4), one m i g h t c a l l a s e m i -

3.

FUNCTIONAL REPRESENTATIONS

267

s i m p l e a l g e b r a , i n t h e p r e v i o u s s e n s e , a l s o s p e c t r a l l y semi-simple y e t f u n c t i o n a l l y o r strongly semi-simple).

(or

F o r b r e v i t y ' s s a k e , however, w e

r e t a i n i n t h e s e q u e l t h e s i m p l e r t e r m i n o l o g y of t h e p r e v i o u s D e f i n i t i o n 3 . 2 , a n y o t h e r case b e i n g e x p l i c i t l y s t a t e d d i f f e r e n t l y . I n t h i s r e s p e c t , w e a l s o have t h e f o l l o w i n g .

Lemma 3.1.

Let E be a topological algebra whose spectrum i s m(E). Further-

more, assume t h a t ( t h e u n d e r l y i n g t o p o l o g i c a l v e c t o r s p a c e ) E i s in a "separated d m Z i t y " w i t h r e s p e c t t o i t s topological dual E' ( t a k e , f o r i n s t a n c e , a ( H a u s d o r f f ) l o c a l l y convex s p a c e E; c f . J . HORVdTH [ I : p. 183, D e f i n i t i o n I ] ) . Then, t h e given algebra E i s semi-simple i f , and only if, i t s spectrum is a t o t a l subset o f Ei ; i.e., i f , and only i f , one has t h e r e l a t i o n

[mil?)] = EL

(3.5)

.

Proof. W e f i r s t remark t h a t

v

VZ(E)':=I~EE :f(z) = O

(3.6) Therefore, i f

f E ~ ( E ) =I

n kerif) f e m(E) m(EiL =

( 3 . 4 ) i s t r u e , t h e n one o b t a i n s

.

{Oj,

so t h a t

( s e e , f o r i n s t a n c e , J . HORVdTH [I: p. 193, P r o p o s i t i o n 3 1 ) one g e t s ['LTiE)] =

(3.7)

m(E)" =

,

{OIL = E,'

t h a t i s , t h e d e s i r e d r e l . ( 3 . 5 ) . O n t h e o t h e r hand, i f

(3.5) i s v a l i d ,

t h e n ( i b i d . ) one h a s [VZiE)]

(3.8)

= ?TZ(E)"

= E,'

,

and h e n c e ( i b i d . ) , "Z(E)'

(3.9) T h u s , i n view of Now,

= ??Y(E)L''

= (Ei)'

= {O}

.

( 3 . 6 ) , o n e h a s ( 3 . 4 ) , and t h i s c o m p l e t e s t h e pro0f.M

our next objective i s t o e x h i b i t conditions ensuring t h a t

a f u l l a l g e b r a , which by i t s d e f i n i t i o n i s a s e m i - s i m p l e

(commutative)

t o p o l o g i c a l a l g e b r a w i t h G e l ' f a n d map o n t o , i s , i n f a c t , t o p o l o y i c a 1 . l y i d e n t i f i e d , v i a ( 3 . 1 ) , w i t h t h e r a n g e o f t h e l a t t e r map. Accordingl y , t h e analogous s i t u a t i o n t o t h a t connected with t h e c 1 a s s i c a l " o p e n mapping t h e o r e m " ( i n l o c a l l y convex s p a c e s ) i s h e r e n a t u r a l l y i n v o l v e d , h e n c e t h e a p p e a r a n c e of t h e p a r t i c u l a r class o f t o p o l o g i c a l a l g e b r a s i n t r o d u c e d by t h e f o l l o w i n g d e f i n i t i o n ( s e e a l s o Scheme 2.1 i n t h e p r e c e d i n g ) . So w e h a v e .

Definition 3.3. s a i d t o be a

A g i v e n l o c a l l y convex ( t o p o l o g i c a l ) a l g e b r a E i s Pt5k algebra ( o r e l s e fuZZy complete) i f t h e u n d e r l y i n g l o -

c a l l y c o n v e x s p a c e E i s a PtSk s p a c e ( c f -

,

f o r example, J . HORVdTH [I:

268

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

p. 299, D e f i n i t i o n 2 1 ) . A c r u c i a l p r o p e r t y which t h e d e f i n i t i o n of a P t 5 k s p a c e

F im-

p l i e s i s , of c o u r s e , t h e f a c t t h a t e v e r y c o n t i n u o u s l i n e a r map +:E-+F, where P i s any b a r r e l l e d l o c a l l y convex s p a c e , i s , i n e f f e c t , a n open map ( i b i d . ; p. 299, P r o p o s i t i o n 2 ) . I n p a r t i c u l a r , i f

@I

i s a continuous l i n -

ear onto isomorphism, then i t i s , indeed, a topological vector space isomorphism of Lp

( P t 5 k ) onto F ( b a r r e l l e d ) . Furthermore,

e-Jery P t d k space, and hence every Ptdk algebra, i s complete.

Y e t , every Frgchet ( l o c a l l y convex) space is a Ptcik space ( i b i d . ;p. 299 ,Prop-

osition 3 ) . On t h e o t h e r hand, i f t h e l o c a l l y convex space

c c ( X l , w i t h X com-

p l e t e l y r e g u l a r , f s a Pta'k space, then X i s , i n p a r t i c u l a r , a k-space(Ptdk's

Theorem;see, f o r i n s t a n c e , S. WARNER [5 : p. 2671). Thus, g i v e n a t o p o l o g i C , ( ? ? ( E ) ) i s a Ptdk ( l o c a l l y m-convex) algebra, then t h e spectrum i'7Z.E) of the given algebra i s a k-space.

c a l algebra E , i f

So w e come n e x t t o t h e f o l l o w i n g r e s u l t ( s e e a l s o Theorem 9.1 i n the sequel).

Theorem 3.1. Let E be a f u l l Ptcik s p e c t r a l l y barrelled ( l o c a l l y convex) algebra. Then, the respective Gel'fand map (3.10)

defines a topological algebraic isomorphism of E onto

e c ( I r r ( E l ) . In p a r t i c u l a r ,

the spectrum V Y ( E l of the given algebra ( h e n c e a l s o of t h e range of

: see

Theorem V I I : 1 . 2 ) is a k-space. Proof. By h y p o t h e s i s and C o r o l l a r y 2 . 1

,5

is a c o n t i n u o u s o n t o

map whose range ~ c ( 7 T Z ( E I li s a b a r r e l l e d ( l o c a l l y convex) s p a c e (Theorem 2 . 1 ) . Thus, on t h e b a s i s of t h e p r e v i o u s d i s c u s s i o n ,

gis

anopen

map t o o , so t h a t one a c t u a l l y h a s (3.11)

within a topological algebraic isomorphism given by

F . NOW, t h e

r e s t p a r t of

t h e a s s e r t i o n f o l l o w s d i r e c t l y from t h e l a s t c o n c l u s i o n and t h e p r e c e d i n g comments, and t h i s t e r m i n a t e s t h e p r o o f . S t a t e d d i f f e r e n t l y , t h e p r e v i o u s r e s u l t s a y s t h a t any f u l l P t d k s p e c t r a l l y barrelled ( l o c a l l y convex) algebra "admits a functional representa-

t i o n " , i n t h e s e n s e t h a t ( 3 . 9 ) h o l d s t r u e . I n p a r t i c u l a r , such an a l g e b r a i s t h u s a c o m t a t i v e ( c o m p l e t e ) Pta'k barrelled

(and hence, m-barrel-

l e d ) l o c a l l y m-convex algebra w i t h an i d e n t i t y element ( i t s spectrum b e i n g , moreover, a k - s p a c e ) . More p a r t i c u l a r l y , one h a s t h e n e x t .

269

3. FUNCTIONAL REPRESENTATIONS

Corollary 3.1. Let E be a f u l l Frdchet l o c a l l y convex algebra. Then, t h e res p e c t i v e Gel'fand map d e f i n e s a topological algebraic isomorphism

cc(m(E)f. I n particular, Proof.

of

r'

onto

t h e spectrum o f E i s a hemicornpact k-space.

I n v i e w o f t h e p r e v i o u s d i s c u s s i o n and Theorem 3 . 1 , t h e

o n l y t h i n g o n e s t i l l h a s t o p r o v e i s t h a t m ( E l is a hemicompact space. (That i s , a t o p o l o g i c a l s p a c e a d m i t t i n g a denumerable k-covering; c f .

.

i s a Frdchet ( l o c a l l y c o n v e x ) space i f , and only i f , m ( E l i s a hemicompact k-space ( c f . S . WARNER [5: p. 267, Theorem 21 ) which, i n f a c t , p r o v e s t h e a s s e r t i o n .

D e f i n i t i o n Vi5.1)

Scholium 3.1.-

But,

ec(71iT(E))

By t h e p r e c e d i n g and t h e argument a p p l i e d i n Chapt.

V i S e c t i o n s 5 , 6 ( i n f a c t l a form o f U r y s o n ' s Lemma i s s t i l l needed: see,

for i n s t a n c e , t h e p r o o f o f Lemma V I 1 ; l .5), one c o n c l u d e s , f o r any topologicaZ algebra E s a t i s f y i n g t h e c o n d i t i o n s of Theorem 3 . 1 , t h e f o l l o w i n g (3.12)

icl= Cu(m(Ec1)), a e L . T h u s , u s i n g a n Arens-Michael d e c o m p o s i t i o n of t h e g i v e n a l g e b r a E , ( i a l a E I , c o r r e s p o n d i n g t o a k - c o v e r i n g of

with

??Z(E), s a y ( K a l a e I a k-space,

I w h i l e n ( E J i s by t h e p r e c e d i n g a n d t h e h y p o t h e s i s one a l s o g e t s t h e following

m(E)= l z K a =

(3.13)

w i t h i n a homeomorphism (Theorem V ; 6 . 1 ) . b r a E i s Fre'chet ( s e e C o r o l l a r y 3 . 1 )

lsm(Ea,l

In p a r t i c u l a r , i f t h e given alge-

,

t h e l a s t two r e l a t i o n s s p e c i a l -

i z e , o f c o u r s e , t o analogous ones w i t h denumerable decompqsitions. The f o l l o w i n g t e r m i n o l o g y w i l l b e u s e f u l i n t h e s e q u e l . T h u s ,

we f i r s t set. D e f i n i t i o n 3 . 4 . A t o p o l o g i c a l a l g e b r a E i s s a i d t o b e a Michael a l -

gebra, i f E h a s t h e i n i t i a l ( l o c a l l y m-convex)

t o p o l o g y d e f i n e d on it

b y t h e r e s p e c t i v e G e l ' f a n d map

9 :E

(3.14)

+

Ce(7nlE/);

h e r e one c o n s i d e r s i n t h e r a n g e of

5 the

( l o c a l l y m-convex,cf.The-

orem 3 . 2 b e l o w ) t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e ( c l o s e d ) e q u i c o n t i n u o u s s u b s e t s o f ??Y(E) ( Michael topology i n e f ? Y T ( E ) ) , d e n o t e d by e ). Indeed,

t h e previously defined Michael topology i n C ( ? ? ( E l ) makes it

i n t o a l o c a l l y m-convex algebra, a c c o r d i n g t o t h e f o l l o w i n g . Theorem 3 . 2 . Let E be a t o p o l o g i c a l algebra whose s p e c t m i s ??Z(E).

Then,

270

VIII

S P E C I A L T O P O L O G I CA L ALGEBRAS

t h e topology of uniform convergence on t h e equicontinuous (hence, e q u i v a l e n t l y , closed and equicontinuous) s u b s e t s o f 97Z (El d e f i n e s

c(VYlE))

as a (Hcusdorff)

l o c a l l y m-convex algebra ( d e n o t e d b y e e ( ? ? ( E ) ) ) . I n p a r t i c u l a r , i f t h e given algebra E has a continuous Gel'fand map, then t h e Michael topology i n C ( m ( E ) ) ( D e f i n i t i o n 3 . 4 ) c o i n c i d e s w i t h t h e compact-open topozogy i n i t ; hence, one has Cc(m(El) =

(3.15)

Ce(m(El)

,

i n t h e sense o f topological algebras. Proof.

I f A i s a n e q u i c o n t i n u o u s s u b s e t o f ??Z(E), l e t

(3.16) w i t h $ e C l T W E ) ) ; now s i n c e by h y p o t h e s i s A i s ( w e a k l y ) r e l a t i v e l y compact, t h e l a s t r e l a t i o n d e f i n e s a

s u b m u l t i p l i c a t i v e semi-norm

of t h e

M i c h a e l t o p o l o g y . E q u i v a l e n t l y , one may c o n s i d e r t h e c l o s e d and e q u i continuous s u b s e t s of

m ( E l , t h e r e s p e c t i v e t o p o l o g y of u n i f o r m con-

v e r g e n c e b e i n g a H a u s d o r f f one ( t h e s i n g l e t o n s of ??Z(E)

a r e equicon-

t i n u o u s s u b s e t s ) . T h i s p r o v e s t h e f i r s t p a r t of t h e a s s e r t i o n , t h e

rest b e i n g a d i r e c t c o n s e q u e n c e of Theorem V1;l.l. I A p p l y i n g t h e p r e c e d i n g t e r m i n o l o g y , w e may now r e s t a t e Theorem 3.1 and i t s c o r o l l a r y a s f o l l o w s :

Any f u l l Ptdk s p e c t r a l l y b a r r e l l e d ( h e n c e , i n p a r t i c u l a r , any f u l l Fr:'chet ) l o c a l l y convex algebra i s a M i -

chael algebra. More g e n e r a l l y (Theorem 3 . 2 ) , every t o -

(3.17)

pological algebra admitting a f u n c t i o n a l r e p r e s e n t a t i - x , i s a Michael algebra. Thus, i n p a r t i c u l a r , any algebra o f o f t h e fom

c c ( X l , w i t h X a completely r e g u l a r space.

On t h e o t h e r h a n d , a s p e c t r a l l y b a r r e l l e d Michael algebra i s not neces-

s a r i l y m-barrelled

;

t a k e , f o r i n s t a n c e , t h e a l g e b r a 8,tlRl

of Example 1 . 1 .

Michael a l g e b r a s w i l l a l s o b e c o n s i d e r e d i n t h e n e x t s e c t i o n .

4. Topological algebras w i t h a given dual Our o b j e c t i v e i n t h i s s e c t i o n r e m i n d s , o f c o u r s e , t h e f a m i l i a r s i t u a t i o n which one h a s i n t h e t h e o r y o f l o c a l l y c o n v e x spaces. ( T h a t i s , " l o c a l l y convex v e c t o r s p a c e t o p o l o g i e s c o m p a t i b l e w i t h a g i v e n

d u a l i t y " ; s e e , f o r i n s t a n c e , J . H O W h [I: p. 198, D e f i n i t i o n I ] ) . Thus , g i v e n a t o p o l o g i c a l a l g e b r a

E I T o ] whose t o p o l o g i c a l d u a l w e l o o k f o r c o n d i t i o n s on t h e g i v e n a l g e b r a E , which are r e l a t e d w i t h i t s a l g e b r a i c - t o p o l o g i c a l s t r u c t u r e , and e n s u r e t h a t

s p a c e i s CE[T

I)',

t h e p r e v i o u s d u a l s p a c e r e m a i n s t h e same u n d e r some o t h e r t o p o l o g y ,

4. ALGEBRAS WITH GIVEN DUAL

271

s a y T , making E i n t o a t o p o l o g i c a l a l g e b r a ( o f

the

same

type

as

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

E [ T ~ ] ) ;i . e . ,

(4.1)

(a set-theoretic e q u a l i t y ) . Thus, w e f i r s t s e t t h e f o l l o w i n g .

Definition 4.1. A g i v e n ( H a u s d o r f f ) l o c a l l y m-convex a l g e b r a E

Warner algebra, when i t s t o p o l o g y i s t h e s t r o n g e s t l o -

i s s a i d t o be a c a l l y rn-convex

t o p o l o g y on E amongst t h o s e h a v i n g a s d u a l s p a c e t h a t

of t h e g i v e n a l g e b r a E ; i . e . , i f E[T] i s

Now,

[T]

t h e space

(E[T])’.

a Mackey ( l o c a l l y c o n v e x ) s p a c e , w e know t h a t

i t s ( l o c a l l y convex v e c t o r s p a c e ) t o p o l o g y i s t h e s t r o n g e s t one comp a t i b l e w i t h t h e n a t u r a l d u a l i t y < E , E’> osition gebra

41 ) .

- we

( c f . J . HORVA?H

[ I : p. 2 0 6 , Prop-

So i f t h e same s p a c e i s , m o r e o v e r , a l o c a l l y rn-convex

al-

c a l l t h e n E a Mackey algebra - t h e n , it i s c l e a r , b y t h e s a m e

d e f i n i t i o n s , t h a t E i s a Warner algebra. The c o n v e r s e i s n o t t r u e , however;

t h a t i s , not every Warner alge-

.

bra i s a Mackey algebra ( c f . A . C. COCHRAN [2 : p. 1 1 7 1 ) O n t h e o t h e r h a n d , i f E[ro] i s a ( H a u s d o r f f ) l o c a l l y m-convex a l g e b r a w i t h t o p o l o g i c a l d u a l E’,

t h e n t h e r e d o e s e z i s t a f i n e s t (Haus-

dorff) l o c a l l y m-convex topology, s a y x , ( w e also w r i t e x ( E , E ’ ) ) , i n such a manner t h a t t h e r e s p e c t i v e r e l a t i o n t o ( 4 . 1 ) h o l d s t r u e ; so f o r t h i s t o p o l o g y E [ x ] is a Warner algebra: Thus, s u p p o s e t h a t T i s t h e s e t o f a l l (Haus-

d o r f f ) l o c a l l y rn-convex t h e t o p o l o g y on E

,

t o p o l o g i e s on E , a s above ( h e n c e , T~ € T ) : t h e n , say

x,

E [ X ] = limE[rl]

,

“ g e n e r a t e d by T ”

,

i.e.

,

a m

i s t h e d e s i r e d one. Indeed,

it i s c e r t a i n l y weaker t h a n t h e Mackey

on E l s i n c e t h i s i s t h e case f o r e a c h one o f t h e t o e T ( c f . a l s o 5’. WAil7NCK [2: p. 2021). F u r t h e r m o r e , a l o c a l b a s i s i s obtaiiied by taking t h e polars i n E of all a b s o l u t e l y o f t h e topology x ( E , E ‘ ) convez and weakly compact s u b s e t s of E’,whose p o l a r s are idempotent s u b s e t s of E (Mackey-Arens Theorem; s e e , f o r i n s t a n c e , u’. HORVLTH [ l : p. 2 0 5 , Theorem 1 3 ,

topology ?(E,E‘) pologies

c1

i n connection with Corollary 1 ; 1 . 2 ) . I n p a r t i c u l a r , w e now h a v e t h e f o l l o w i n g . Theorem 4.1.

L e t E[T,]

be a given semi-simple

( D e f i n i t i o n 3 . 2 ) topologi-

c a l algebra, having a l s o a continuous GeZ’fand map. Furthermore, consider t h e fol-

lowing two a s s e r t i o n s : 1) E i s a Warner algebra ( D e f i n i t i o n 4 . 1 )

with respect t o the (locally

VIII SPECIAL TOTOLOGICAL ALGEBRAS

272

-

m-convex) topology, say T, defined on i t by t h e Gel'fand map

5 :E

e , ( ~ ( E ) ) .

I'= { p } of s u b m u l t i p l i c a t i v e semi-norms d e f i n i n g

21 There e x i s t s a f a m i l y

t h e topology of E , i n such a way t h a t one has (4.3)

p(x

2

2 = (pix)) ,

)

for any Z E E and p f f . Then, one g e t s t h e following i m p l i c a t i o n s :

I)=> 2) =>

(4.4)

=

T

.

T

I n p a r t i c u l a r , under e i t h e r one of 11 o r 2 1 , one obtains =

E#

(4.5)

(E[T])'

Proof. By assuming l ) , s i n c e plicit that

ous, o n e g e t s

one h a s

T

~

(EITo]l'

E[T]

= E'.

i s a Warner a l g e b r a , it i s i m -

H a u s d o r f f l o c a l l y m-convex a l g e b r a , s o t h a t

is a

EITo]

(Definition 4.1)

=

T

, iT. e . ,~

Accordingly, s i n c e

~ T<. T

=

s is

a l s o continu-

Thus, by h y p o t h e s i s and Theorem 3 . 2 ,

T .

E is, i n p a r t i c u l a r , a Michael algebra s u c h t h a t o n e h a s

E

(4.6)

$ ( E I E C c ( m ( E ) )= C e ( ? ? U E ) ) ,

w i t h i n a t o p o l o g i c a l a l g e b r a i c isomorphism of E o n t o i t s G e l ' f a n d t r a n s form a l g e b r a E"

=

d e f i n e d by

IEI

g.

t o p o l o g y i n t h e r a n g e o f 3 i s g i v e n by

Now, t h e compact-open

t h e f o l l o w i n g f a m i l y of semi-norms

(4.7) w i t h K any compact s u b s e t of m(E).Hence, o n e h a s

p

(4.8)

f o r any

c (m(E))and

Banach a l g e b r a

CJKI);

p

2

I

= (p

K

I

K a s above ( c f . a l s o V I I ; ( 1 .1) c o n c e r n i n g t h e thus, 1)+

2).

F u r t h e r m o r e , c o n d . 2 ) i m p l i e s t h a t E[T,]

i s a Michael a l g e b r a ( c f .

Theorem 5 . 1 of t h e n e x t s e c t i o n ) . Thus, it f o l l o w s from t h e h y p o t h e -

sis f o r E

that

(4.9)

T

r i v e d from l ) , i m p l i e s ( 4 . 5 ) ,

and t h i s f i n i s h e s t h e p r o 0 f . l

= T

= T ,

o e t h a t i s t h e a s s e r t i o n . F i n a l l y , t h e l a s t r e l a t i o n , which w a s a l s o de-

S o , i n p a r t i c u l a r , w e now h a v e t h e n e x t .

Corollary 4.1. L e t

EITo]

be a semi-simple topoLogicaZ algebra w i t h spectrum

4. ALGEBRAS WITH GIVEN DUAL

273

?YZ(Ei having, moreover, a continuous Gel'fand map. Besides, consider t h e following two a s s e r t i o n s : 1 ) There e x i s t s an algebra-norm,

say

11 . 11,

on E compatib2e w i t h t h e i n i t i a l

topology T~ o f E ( i . e . , E i s normable) such t h a t one has

11. . 2 / (

(4.10)

= ( ( I z I l I2

,

for, eyer2 x € E . 2) The algebra E i s bounded

( D e f i n i t i o n V ; 1 . 1 ) such t h a t one a l s o has

the relation

E# =

(4.11)

E',

(cf. (4.5)).

Then, l ) + 2 ) .

I n p a r t i c u l a r , i f t h e given algebra E i s , moreover, a Nach-

bin-Shirota algebra (hence, by h y p o t h e s i s , s p e c t r a l l y b a r r e l l e d ( C o r o l l a r y 2.1)), then t h e preceding two a s s e r t i o n s are e q u i v a l e n t . Proof. Assuming 1 ) , w e g e t from t h e p r e v i o u s Theorem 4 . 1 , 2 ) t h a t T

= ? , a n d hence

t h e d e s i r e d r e l . ( 4 . 1 1 ) a s w e l l . B e s i d e s , E i s , by

h y p o t h e s i s , a normable a l g e b r a , i t s c o m p l e t i o n t h e r e b y a B a n a c h a b l e a l g e b r a and h e n c e a & - a l g e b r a ; s o E i s , i n p a r t i c u l a r , a bounded a l g e b r a t o o ( c f . Theorem VI;1 . 3 , 1 ) * 5 )

). That i s ,

one g e t s t h a t 1 ) * 2 ) .

On t h e o t h e r h a n d , i f c o n d . 2 ) i s s a t i s f i e d a n d E i s , m o r e o v e r ,

a N a c h b i n - S h i r o t a a l g e b r a , t h e n one c o n c l u d e s (Theorem 2 . 1 , 2 ) ) t h a t 7TZ(El+ i s a ( w e a k l y ) compact s u b s e t of Es'; h e n c e , (4.12)

i s a normed a l g e b r a . T h e r e f o r e , t h e same i s t r u e f o r t h e g i v e n a l g e b r a E endowed w i t h t h e t o p o l o g y T , d e f i n e d i n i t by

i s by h y p o t h e s i s 1 - 1 ) ; so E [ T ]

( t h e l a t t e r map

i s a bornological algebra ( t h a t i s , t h e

u n d e r l y i n g l o c a l l y c o n v e x s p a c e E [ T ] i s b o r n o l o g i c a l ; c f ., f o r instance, J . HORVLTH [l: p. 222, P r o p o s i t i o n 3 ] ) , and h e n c e Mackey. Thus, E i s a for-

t i o r i a Warner algebra, so t h a t a s f o l l o w s from ( 4 . 1 1 ) ~~5 T ( c f . D e f i n i tion 4 . 7 ) ,

and h e n c e b y h y p o t h e s i s f o r

one a c t u a l l y h a s

T

~

T= . N O W

t h e f i n a l a s s e r t i o n f o l l o w s from t h e p r e v i o u s Theorem 4 . 1 ( 1 ) = > 2 ) ) t o g e t h e r w i t h t h e f a c t t h a t t h e topology

T ~ ( = T )is

n o r m a b l e , and t h e

p r o o f i s comp1ete.B Now, an immediate c o n s e q u e n c e and c l a r i f i c a t i o n a s w e l l of t h e previous Corollary 4.1

i s t h e following.

COrol l a r y 4.2. Every ( c o m m u t a t i v e ) semi-simple complete 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 i c a l ) algebra w i t h an i d e n t i t y element which s a t i s f i e s 14.111

and i s , moreover, bounded and s p e c t r a l l y b a r r e l l e d , i t i s , i n f a c t , a commutative

VIII SPECIAL TOPOLOGICAL ALGEBRAS

274

Banachable semi-simple ”uniform” algebra (it s a t i s f i e s , namely, ( 4 . 1 0 ) ) w i t h an i d e n t i t y element, and conversely

(see a l s o t h e n e x t s e c t i o n f o r t h e t e r m i -

nology a p p l i e d ) . 1 I t h a s a l r e a d y b e e n remarked t h a t a b o r n o l o g i c a l a l g e b r a i s , i n

p a r t i c u l a r , a Warner a l g e b r a . I n d e e d , more g e n e r a l l y , t h e Hausdorff lo-

c a l l y rn-convex i n d u c t i v e l i m i t algebra of a given i n d u c t i v e system o f (Hausdorff) l o c a l l y m-convex bornoZogical algebras ( c f . C h a p t . IV; S e c t i o n 3 ) is a Warner algebra. T h i s i s a c o n s e q u e n c e o f S. WARNER [2: p. 203, Theorem 4 , and p. 204

,

P r o p o s i t i o n 61 )

. Moreover,

t h e c a r t e s i a n product of any f a m i l y of War-

ner aZgebras is again a Warner algebra

( i b i d . ; p. 2 1 0 , P r o p o s i t i o n 9 )

.

F u r t h e r i n f o r m a t i o n c o n c e r n i n g t h e above m a t e r i a l i s c o n t a i n e d i n t h e work o f S . Warner q u o t e d a b o v e , a s w e l l a s i n A . C . COCHRAN [ 2 ] and A . K. CHILAAM [2]

.

5. Uniform topological algebras T o p o l o g i c a l a l g e b r a s , i n p a r t i c u l a r , l o c a l l y m-convex o n e s , f o r which t h e d e f i n i n g f a m i l y of semi-norms s a t i s f i e s a r e l a t i o n l i k e ( 4 . 3 ) h a v e b e e n c o n s i d e r e d i n t h e p r e c e d i n g d i s c u s s i o n t h r o u g h Theorem 4 . 1 and i t s c o r o l l a r y . On t h e o t h e r h a n d , t h e c l a s s i c a l c a s e o f Banach a l g e b r a s c o r r e s p o n d s t o a n i m p o r t a n t c l a s s of a l g e b r a s , t h e uniform (Ban a c h ) a l g e b r a s , f o r which

the respective

literature is certainly vast

and i s s t i l l i n c r e a s i n g c o n s t i t u t i n g , i n f a c t , a p a r t of t h e g e n e r a l t h e o r y of

( c o m m u t a t i v e Banach) f u n c t i o n algebras. C f . , f o r i n s t a n c e , T .

i’. GAMELIN [ l ] ,

E. L . STOUT [ l ] ,

I. SUCIU [ I ] .

However, w e s h a l l b e c o n c e r n e d w i t h o n l y t h e most b a s i c f e a t u r e s of t h e c o r r e s p o n d i n g n o t i o n f o r t o p o l o g i c a l a l g e b r a s ( n o t n e c e s s a r i l y n o r m e d ) , and i n p a r t i c u l a r t o t h e e x t e n t t h a t t h i s i s c o n n e c t e d w i t h t h e p r e v i o u s Theorem 4 . 1 and i t s c o r o l l a r y . T h u s , w e f i r s t s e t t h e following.

Definition 5.1. L e t E b e a

( H a u s d o r f f ) l o c a l l y m-convex

(topolo-

g i c a l ) a l g e b r a such t h a t t h e r e e x i s t s a fundamental f a m i l y of m u l t i p l i c a t i v e ) semi-norms, ( c f . Proposition I; 3.2)

say

r

={ p ) ,

, with p(x 2 I = (pix))2 ,

(5.1)

f o r every p e

r.

(sub-

d e f i n i n g t h e topology of E

X€E,

Then, w e s a y t h a t E i s a

uniform a l g e b r a .

The n e x t lemma p o i n t s o u t some c o n s e q u e n c e s i m p l i c i t i n t h e p r e vious d e f i n i t i o n .

5.

275

UNIFORM ALGEBRAS

Lemma 5 . 1 . Let E be a uniform algebra D e f i n i t i o n 5 . 1 ) . Then E i s , i n f a c t , a commutative semi-simple

( l o c a l l y m-convex) algebra.

Proof. By c o n s i d e r i n g a n Arens-Michael d e c o m p o s i t i o n o f E , s a y (fa)&

one o b t a i n s

I

(5.2)

.,

l Z E a

ES

r

w i t h i n a t o p o l o g i c a l a l g e b r a i c isomorphism ( i n t o ; c f . I I I : ( 3 . 2 0 ) ) . N o w

I/ - I I a

t h e norm

11

pa(;c)

lla

on e a c h one o f t h e a l g e b r a s ~ € 1( g, iven by [ ~ ] Z € E : see a l s o 111; ( 4 . 6 ) ) s a t i s f i e s t h e r e l a t i o n

ia,

:= pa(cc)

/I r.1:

(5.3)

lla

=

II ,I.[

IiZ

~ l l ~ =

r

f o r e v e r y z €E (combine ( 5 . 1 ) w i t h 111; ( 4 . 5 ) ) . T h e r e f o r e , e a c h one of

2a' a € I , i s a commutative a l g e b r a ( Theorem Hirschf e l d - i e l a z k o - L e Page: s e e , f o r i n s t a n c e , F . F . BONSAL-J. DUNKAN [ l : p. 17, C o t h e Banach a l g e b r a s r o l l a r y 81.

I t f o l l o w s from ( 5 . 2 ) t h a t E i s a commutative algebra t o o .

F u r t h e r m o r e , ( 5 . 3 ) i m p l i e s a l r e a d y t h a t e a c h o n e of t h e a l g e b r a s Ea,

CY E

I, i s a s e m i - s i m p l e

( c o m m u t a t i v e Banach) a l g e b r a ( i b i d . ; p. 8 3 ,

C o r o l l a r y 7 , a n d C. E . RICKART [ I : p. 1 1 , Lemma (1.4.2)]).

simple aZgebra

(Definition 3.2)

osition 7.3, b)

],

,

Thus, E i s a semi-

a c c o r d i n g t o E . A . MICHAEL [ I : p.29,

Prop-

and t h i s f i n i s h e s t h e p r o o f . 4

A c o n s e q u e n c e o f t h e p r e v i o u s Lemma 5 . 1 a n d Theorem 4 . 1

i s now

t h e next.

Lemma 5.2. Let E b e a uniform ( l o c a l l y m-convex) algebra having t h e re-

$

s p e c t i v e Gel'fand map considers

qiia

i i ,

5 , as

continuous. Then, t h e r e l a t i v e topology on E , when one a subalgebra of e c ( ? 7 ( E ) ) , i s i d e n t i f i e d w i t h t h e i n -

i t i a l topology of E . Thus i f t h e algebra E i s , moreover, compZete, t h e n i t i s ( w i t h i n t h e p r e v i o u s t o p o l o g i c a l a l g e b r a i c i s o m o r p h i s m ) a c l o s e d subalgebra o f

Cc(m(E) I. 4 A c o n v e r s e s t a t e m e n t t o t h e p r e c e d i n g lemma i s a l s o v a l i d i n

case of

zi

Frdcchet uniform algebra having an i d e n t i t y element. Thus, one h a s

t h e following.

Corollary 5.1. A g i v e n algebra E w i t h an i d e n t i t y eZement i s a Frdehet u n i form algebra i f , and o n l y i f , i t i s a c l o s e d subaZgebra ( w i t h r e s p e c t t o

the

r e l a t i v e t o p o l o g y ) of some cciXi " c o n t a i n i n g t h e c o n s t a n t s " , where X is a hemi-

compact ( c o m p l e t e l y r e g u l a r Hausdorff) space. W o o f . The s t a t e d c o n d i t i o n i s n e c e s s a r y , s i n c e t h e n E i s a c l o s e d s u b a l g e b r a of CcJ???iE)) hypothesis f o r E

a

(Lemma 5 . 2 ) , w h i l e i t s s p e c t r u m m(E) i s by

hemicompact ( 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

2 76

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

Now t h e rest of t h e a s s e r t i o n i s s t r a i g i i t f o r -

(cf.Corollary V ; 6 . 1 ) .

ward from t h e same d e f i n i t i o n s and t h e r e l a t i o n (4.8). On t h e o t h e r h a n d , a s t r o n g e r r e s u l t , i n t h e same s p i r i t t h a t of C o r o l l a r y 5.1,

-

5 :E [ T ~ ]

c o n t e x t of Lemma 5 . 2 t h e map

morphism. Thus (see a l s o Theorem VI;1 . l )

C c f 7 7 ( E ) ) i s a c t u a l l y a homeo-

,

t h e i n v e r s e map of

tinuous when i t s domain c a r r i e s t h e r e l a t i v e Michael topology T t h e range of

as

i s s t i l l p o s s i b l e by n o t i n g t h a t w i t h i n t h e

-

$

i s aZso con-

induced on i t from

( c f . D e f i n i t i o n 4 . 2 ) ; t h a t i s , t h e map

5-l : E A

(5.4)

L

$? ( E ) [T,]

E[ T , ]

i s continuous. I n f a c t , t h i s i s t h e c r u x i m p l i c i t i n C o r o l l a r y 5 . 1 ,

as

w e s h a l l see i n t h e n e x t r e s u l t , which a c t u a l l y h a s b e e n a p p l i e d a l Thus, w e have.

r e a d y i n Theorem 4 . 1 .

Theorem 5.1.

L e t E be an algebra w i t h an i d e n t i t y element. Then, t h e follow-

ing a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a uniform algebra ( D e f i n i t i o n 5 . 1 ) . 2 ) E i s a topological algebra such t h a t

(5.5)

w i t h i n a topclogical algebraic isomorphism ( i n t o ) , where ( X c l ) , e l

i s a given fam-

i l y of compact ( H a u s d o r f f ) spaces. 3 ) E i s a topological subalgebra of some

c c ( X 1 , w i t h X a l o c a l l y compact

(Hausdorf f ) space. That i s , one has (5.6)

ec(x),

E~

w i t h i n a topological algebraic isomorphism ( i n t o ) . 41 E i s a topological algebra such t h a t (5.6) h o l d s , where

X is a completely

regular ( H a u s d o r f f ) space. 5) E i s a topological algebra w i t h spectrum m(E1 such t h a t t h e map

i n ( 5 . 4 ) i s continuous. 61 E i s a topological algebra whose spectrum i s V Z ( E ) , i n such a manner

that the

( G e l ' f a n d ) map

5 :E [ T ~ ]

(5.7)

-c

Ce(??(EI)

( e d e n o t e s t h e r e s p e c t i v e Michael t o p o l o g y i n t h e r a n g e of a topological

5 )d e f i n e s

( a l g e b r a i c ) isomorphism onto i t s image (where t h e l a t t e r

e q u i p p e d w i t h t h e r e l a t i v e M i c h a e l t o p o l o g y T ~c f; . ( 5 . 4 ) ) (Definition 3 . 4 ) , E[T

Proof. 1 )

*2) :

3

is

. Equivalently,

i s a Michael algebrcz.

By h y p o t h e s i s and Lemma 5 . 1

,E

i s a commutative

s e m i - s i m p l e l o c a l l y m-convex a l g e b r a ; so a p p l y i n g ( 5 . 2 ) o n e g e t s

5.

277

UNIFORM ALGEBRAS,

(5.8)

lt imza,

Ec+

ia ,a~ I ,

w i t h i n a t o p o l o g i c a l a l g e b r a i c i s o m o r p h i s m , where

i s a commuta-

t i v e Banach algebra w i t h an i d e n t i t y element, which a l s o is uniform, i n view o f ( 5 . 3 ) . C o n s e q u e n t l y ( s e e a l s o t h e p r o o f o f Lemma 5.1 and Theorem 4 . I ) ,

one h a s a n i s o m e t r i c isomorphism o f

ka

into

Cu(??(qp

m :si

d e f i n e d by t h e r e s p e c t i v e G e l ' f a n d map ( c f . a l s o v;(6.23)).

Cu(m(Ea)) So it f o l -

lows from ( 5 . 8 ) t h a t

E Z i2iusit cU(mEa)) E n eU(m(2, 1)

(5.9)

;

cl€I

t h a t i s , one g e t s ( 5 . 5 ) , w i t h X

m ( i a ?homeo m ( E a ? , a € I .

T h i s i s a c o n s e q u e n c e of

2)*3):

-c

ES iim

(5.10)

=

a

( 5 . 5 ) and t h e r e l a t i o n

_eu(Xcl) cc(12xa)

(X I C n a a€,

q x )

which i s t r u e w i t h i n t o p o l o g i c a l ( a l g e b r a i c ) isomorphisms , a s i n d i

-

cated.

3)* 4) : T h i s i s o b v i o u s . 4)*5):

By h y p o t h e s i s E i s a u n i f o r m a l g e b r a which i s , w i t h i n a

t o p o l o g i c a l a l g e b r a i c isomorphism, a t o p o l o g i c a l s u b a l g e b r a o f q X ) . Therefore,

-

( 5 . 4 ) i s well-defined;

j:

(5.11)

E[T,]

thus, i f

Ce(xl

d e n o t e s t h e a b o v e i s o m o r p h i s m , where X i s a c o m p l e t e l y r e g u l a r s p a c e , t h e n by c o n s i d e r i n g t h e c o r r e s p o n d i n g t r a n s p o s e map ( r e s t r i c t e d t o t h e s p e c t r a of t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d ) , one obtains a continuous map (5.12)

i = tj :TZ(C~(X))

( c f . a l s o v I I ; ( 1 . 2 9 ) ) ; t h i s map

3

x---+~(E)

i s not necessarily injective (unless

is "point separating" f o r X 1 . NOW, f o r a n y compact K C X and E > O ,

j(El c ( X l

U ( ' K ; & ) = { $ € j ( E ) : l @ f z ) 6~ \

(5.13)

d e f i n e a l o c a l b a s i s of j ( E ) r e l a t i v e t o p o l o g y from E [ T ~ ] as w e l l .

t h e sets V z€K)

E l when t h e l a t t e r

Cc(X)and

i s endowed w i t h t h e

h e n c e , v i a ( 5 . 1 1 ) , from t h e s p a c e

T h e r e f o r e , o n e o b t a i n s from ( 5 . 1 2 ) t h a t

(5.14) so t h a t one h a s (5.15)

U(K;E)

= E.UIK;I/ =

E.(iiK/)O

i i ~ E) (iiK))O0 =

,

IUIK;IIIO.

Thus, a s f o l l o w s from ( 5 . 1 3 ) , t h e compact s e t i ( K ? G m(E) i s a c t u a l l y an

278

VIII SPECIAL TOPOLOGICAL ALGEBRAS

equicontinuous subset o f m(E). On t h e o t h e r h a n d , t h e s e t s V ( H ; E ) = {Z€$?(E) : I Z ( f ) I S E V f € H )

(5.16)

I

where H i s a n y ( c l o s e d ) e q u i c o n t i n u o u s s u b s e t o f

??Z(El, and & > O r form a l o c a l b a s i s o f S ( E ) = E ^ , w i t h r e s p e c t t o t h e r e l a t i v e Michael t o -

p o l o g y i n d u c e d on it from (5.17)

V(B; EI =

Ce(9?'(E));thus,

E . V ( H ; I ) = E . ~ Z ~ H O )

o n e c o n c l u d e s from ( 5 . 1 6 ) =$(~.iiOi.

So from ( 5 . 1 4 ) and ( 5 . 1 7 ) o n e now o b t a i n s U(X;E) =

E.

(iiK))O))

(i(KlI0=$-'(S(s.

=F-I(E.$((~(K).I'IJ

= ~ - ' ( V ( ~ ( K ) ; E) )

,

t h a t i s , t h e c o n t i n u i t y o f t h e map ( 5 . 4 ) , which i s t h e a s s e r t i o n . 5 ) *6) ?,
: By h y p o t h e s i s ~~5

T~

, so

t h a t it s u f f i c e s t o prove t h a t

Thus, i f K is a n y ( c l o s e d ) e q u i c o n t i n u o u s s u b s e t o f ???3E), one

g e t s from ( 5 . 1 6 ) and ( 5 . 1 7 ) t h e f o l l o w i n g V(;(;E)

n

S(E,'={geS(E):

I$(f)I

SEV f € K )

= E . ( V ( K ; ~ I ~ ~ I E / I = E . ~ ( K ~ ) = ~ ( E . K ~ ) . T h e r e f o r e , one f i n a l y o b t a i n s (5.19)

$-'(ViK;E

/ ) n$(E)

= E.K3,

t h e s e c o n d member b e i n g , by h y p o t h e s i s f o r K , a n e i g h b o r h o o d o f

Oe

E [ T ~ ] ,a n d t h i s p r o v e s t h e a s s e r t i o n ( s e e a l s o t h e n e x t Scholium 5 . 1 ) .

6 ) + l ) : When K r u n s o v e r t h e ( c l o s e d ) e q u i c o n t i n u o u s ( a n d h e n c e c o m p a c t ) s u b s e t s of m(E), o n e g e t s from ( 4 . 7 ) a f a m i l y o f p l i c a t i v e ) semi-norms

f o r t h e Michael t o p o l o g y i n

(submulti-

e(m(E)), t h e

latter

a l g e b r a b e i n g by ( 4 . 8 ) a u n i f o r m o n e . Hence, t h e same i s t r u e f o r i t s s u b a l g e b r a S ( E ) , w i t h r e s p e c t t o t h e r e l a t i v e t o p o l o g y , and a l s o f o r

1,

t h e g i v e n a l g e b r a E[T = T by o u r h y p o t h e s i s . But t h i s i s t h e assero e t i o n , and t h e t h e o r e m i s p r 0 v e d . I

Scholium 5.1.(5.20)

-

I t i s a c o n s e q u e n c e of

3 :E[ro]

( 5 . 1 8 ) t h a t t h e map

G(m(EII

i s continuous; t h a t is, t h e Michael topology

T~

on E ( D e f i n i t i o n 4 . 3 ) i s a l m y s

weaker than t h e i n i t i a l topology T ~ Y. e t , what amounts of c o u r s e t o t h e same t h i n g , g i v e n a n y t o p o l o g i c a l a l g e b r a E [ T ~ ] ,t h e r e s p e c t i v e Gel'fand map

is always continuous w i t h r e s p e c t t o t h e Michael topology in t h e range o f

$

8 . (An-

o t h e r way t o r e a l i z e p a r t of t h e s i t u a t i o n d e s c r i b e d by Theorem V I ; 1 . 1 ) . T h u s , t a k i n g a n e i g h b o r h o o d of Oa$(El,

i n t h e r e l a t i v e topology

5.

from

ce(m(E)I,of

279

UNIFORM ALGEBRAS

the form (5.18) one obtains

(5.21) (extending (5.19) for $? not necessarily one-to-one); here

KO

is a

neighborhood of zero in E [ T , ] by hypothesis for K E ' M ( E ) . Furthermore, as a consequence of Theorem V;6.1 and the previous Theorem 5.1 (prop. 3 ) : cf. (5.9), (5.10)), one concludes the following: Suppose t h a t E i s a uniforni algebra having an i d e n t i t y element and c o n t i n u ous Gel'fand map (take, for example, E to be spectrally barrelled). Then, i t s spectrum m(E)i s a k-space ithus (6.11) holds) i f , and o n l y i f , i t i s loc a l l y compact.

NOW, the topological algebraic isomorphism of a given uniform algebra with a (topological) subalgebra of some c c i X ) , with X a completely regular space, (established by Theorem 5.1, cf. (5.9),through an Arens-Michael decomposition) is the only one we have, concerning a similar " i d e n t i f i c a t i o n " of the given algebra through the Gel'fand map (cf., however, the rel. (5.7));this also clarifies the situation in the above Lemma 5.2. On the other hand, by the same lemma and its Corollary 5.1 , we further conclude that: I f t h e g i v e n uniform algebra E [ T , ] has a continuous Gel'fand map (hence,T~ = T< ), t h e n t h e preceding TGeorem 5.1 spec i a l i z e s t o t h e following equivalent statement: 6'1 The Gel'fand map

(5.22) d e f i n e s a t o p o l o g i c a l a l g e b r a i c isomorphism o n t o i t s image.

-

Finally, for convenience, we still give the last equivalence of the previous Theorem 5.1 (i.e., 6) 7 ) ) in the (more transparent) form of the following corollary (semi-simplicity is needed only in the one half of it (cf. Lemma 5.1 and the above Scholium 5.1)). Thus,we have.

Corollary 5.2. A semi-simple (locally m-convex) topologicaz algebra E i s a uniform algebra i f , and o n l y i f , i t i s a Michael algebra. I

As a final comment on the preceding discussion, we still note that the theory of uniform t o p o l o g i c a l a l g e b r a s , not necessarily normed ones, lacks, indeed, a systematic development. However, its usefulness is apparent in conjunction with important topological "function algebras", which are uniform in the sense of (5.2), but whose individual elements are "unbounded functions". Thus, this is the case, for

280

VIII S P E C I A L TOPOLOGICAL ALGEBRAS

instance, in Complex Analysis (one or several complex variables), and more generally, in algebras of “continuous functions on non (necessarily) compact spaces”. (In this respect, see, for example, F . T . BIRTEL [ l ] , W. DIETRIC, Jr. [3: Chapt. 3 1 ) . Particular classes of (Frechet) uniform algebras will also be considered in the next chapter (cf. Section IX; 4 ) .

6. A-convex algebras (contn’d.) The class of topological algebras in title has been defined already in Chapter I (Definition 5.3). Yet we are going to consider in this section its connection with the class of locally m-convex algebras further on, after we set fixed the necessary terminology.(Thus, see also Lemma Ii5.4 and Corollary 1;5.3). In this concern, the appearance of A-convex algebras in Chapter I was actually due to their natural relation with an abstract form of Gel’fand’s Theorem (Theorem 1 ; 5 . 7 ) , concerning the usual definition of a Banach algebra (cf. Theorem I;5.2 and the relevant comment). On the other hand, as we shall presently see, the same concept gives rise to a different, however, equivalent analysis of the notion of a locally m-convex algebra. Thus, we start with the following.

Definition 6.1. A r-complete algebra is a topological vector space E and an algebra in such a way that the following conditions are satisfied: 1 ) There exists a family r = { p ) of (continuous) semi-norms defining the topology of E , such that,for every p E r , the set p-’(O) is a (closed) 2-sided ideal of the algebra E . 2) The quotient algebra := E / p - l ( O ) P has the property that i t s completion 2 , as a Banach space, has a sepai*P a t e l y continuous m u l t i p l i c a t i o n (hence, in fact, a jointly continuous one; E

(6.1)

cf. Lemma I;4.2). NOW, it is clear that every l o c a l l y m-convex algebra E is a r-corrplete algebra: Thus, it suffices to consider the elements of an Arens-Michael decomposition of E , say ( g a l c r e l (cf. Chapt. 1II;Section 3). As a matter of fact, the two notions are, in effect, equivalent, as we shall presently see. Thus, we have.

Theorem 6.1. Let E be a given algebra. Then, t h e following two a s s e r t i o n s

6. A-CONVEX ALGEBRAS ( CONTN 'D. )

28 1

are equivalent: 1) E i s a r-complete algebra.

2) E i s a topological vector space whose topology can be defined (i.e. , e q u i v a l e n t l y , within a topological vector space isomorphism) by a f a m i l y

r

= { p l of submultiplicative semi-norms.

(We briefly say that E is topologiz-

able as a l o c a l l y m-convex a l g e b r a ) .

Note.- The last statement offers an example indicating that one might define a l o c a l l y m-convex algebra as a topological algebra whose topology can be defined (i.e., within a topological vector space isomorphism) by a family of submultiplicative semi-norms.(In this respect, see also N. BOURBAKI [ll : p. 9 , definition of a Banach space] , or yet E. VASSILIOU [l : p. 227, Remarks] )

.

Proof of Theorem 6.1. 1);s 2) : If r = { p l is a family of semi-norms on E satisfying the conditions of Definition 6.1, then the Banach space E , p e r , which is in fact by hypothesis a topological algebra, P may be retopologized so as to become a Banach algebra (Gel'fand Theorem; cf. Theorem I; 5.1) , say ' E , p e r . Consequently, since E is esP sentially (Definition 6.1) a locally convex space, it is (within a topological vector space isomorphism) a (dense) subspace of the cor(this eventually under a responding projective limit space 1 . k P' suitable rechoice of the given family r . Cf. , for instance, G. K8Z'HE [l: p. 231, 8 9 , (l)] 1. So one actually gets A

-

Now the last (projective limit) algebra in (6.2) is in fact locally m-convex (see Chapt. 111; Section 2, as well as Lemma 111; l.l), and this proves the assertion. 2)*1): This is clear, of course, according to the above Definition 6.1 and the comment following it, and this completes the proof of the theorem. I In particular , one concludes that: Every r-complete algebra can be (re)topologized (within a topological vector space isomorphism) so as t o become an A-convex algebra. More specifically, every l o c a l l y m-convex algebra i s an A-convex algebra (cf. Definition I; 5 . 3 and subsequent comments there). On the other hand, the property of the normed spaces (6.1) in the definition of a r-complete algebra, the fact namely that the corresponding completions of these spaces have separately continuous multiplication, seems to be the crucial feature which an A-convex algebra fails to have , under an analogous "Arens-Michael decomposition", in

282

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

i n c o n t r a d i s t i n c t i o n t o what h a p p e n s i n t h e c a s e o f a l o c a l l y m-convex a l g e b r a . T h i s , i n d e e d , i s p o i n t e d o u t b y t h e n e x t . Theorem 6 . 2 . Let E be an algebra. Then,the following two statements are

equivalent: 1 ) E i s an A-convex algebra ( D e f i n i t i o n I ; 5 . 3 ) 2 ) There e x i s t s a f a m i l y

r = { p } of

.

semi-norms on E making it i n t o a local-

l y convex algebra, and i n such a way t h a t , f o r every p e r , t h e s e t p - I ( o ) = ker ( p ) =

(6.3)

txeE

:p ( x ) =

o1

i s a (closed 2-sided) i d e a l o f t h e ( t o p o l o g i c a l ) algebra E . Proof. 1 ) * 2 )

: T h i s f o l l o w s d i r e c t l y from ( 6 . 3 ) and t h e f a c t t h a t

t h e t o p o l o g y of a n A-convex a l g e b r a ( D e f i n i t i o n I ; 5 . 3 ) i s g i v e n by a f a m i l y of absorbing semi-norms, s a y S =

I?);

namely, one h a s

f o r e v e r y T E E , and f o r a n y a~ E l w i t h k , X i n g on a e E and p e S), and f o r e v e r y p e

2)*1):

r

p o s i t i v e c o n s t a n t s (depend-

(see a l s o D e f i n i t i o n I ; 5 . 2 ) .

We remark t h a t ( 6 . 3 ) i s e q u i v a l e n t w i t h t h e f a c t t h a t p(x)= O , implies

(6.5)

piax) = O , with x e E

,

and f o r e v e r y a e E ( w i t h p e T ) . E q u i v a l e n t l y , one o b t a i n s t h a t , f o r e v e r y a e E l t h e map l :E-E

(6.6)

:x-la(x)

:= a x

,

which b y h y p o t h e s i s f o r t h e a l g e b r a E i s c o n t i n u o u s , i s i n f a c t " c o m p a t i b l e w i t h t h e e q u i v a l e n c e r e l a t i o n on E d e f i n e d by ( 6 . 3 ) " . T h u s ,

, say

t h e "quotient map" (6.7)

i s a l s o continuous ( c f . N . BOURBAKI [4: Chap. 1 ;p. 21 , C o r o l l a i r e ] ; a s i m i l a r >a 1 . S o t h e normed space E which by ( 6 . 3 ) P t i s a n a l g e b r a , h a s a s e p a r a t e l y c o n t i n u o u s m u l t i p l i c a t i o n , so t h a t it i s an A-normed algebra ( c f . D e f i n i t i o n I ; 5 . 4 ) . F u r t h e r m o r e , s i n c e E i s

argument h o l d s f o r t h e map

by h y p o t h e s i s a l o c a l l y convex s p a c e , o n e g e t s

w i t h i n a t o p o l o g i c a l vector s p a c e isomorphism ( c f . G . K6l'HE [l: p. 231, 09, (I)]), with E

P

,p e r ,

being

A-normed a l g e b r a s . N O W , it i s e a s i l y

p r o v e d t h a t : The c a r t e s i a n product o f any given f a m i l y of A-eonvex algebras ( a n d h e n c e a l s o of A-normed o n e s ; c f . ( 6 . a ) ) , as well as any subalgebra of i t , i s

7.

an A-convex algebra

283

FINITELY GENERATED ALGEBRAS

which p r o v e s t h e d e s i r e d a s s e r t i o n , and t h i s com-

p l e t e s t h e p r o o f of t h e t h e o r e m . I I n t h i s r e s p e c t , w e f i n a l l y n o t e t h a t Fre'chet l o c a l l y convex a l g e b r a s may f a i l , i n g e n e r a l , t o b e localLy m-convex algebras

(4) ) , and hence (Theorem 6 . 1

)

( c f . Example I;2.

l7-complete algebras t o o . On t h e o t h e r h a n d ,

complete A-convex algebras may not, i n g e n e r a l , be l o c a l l y m-convex algebras ( c f . ,

f o r i n s t a n c e , A . C . COCHRAN [ l : p. 7 4 , Example ( 2 . 6 ) ] ) ; t h i s c l a r i f i e s f u r t h e r t h e s i g n i f i c a n c e of D e f i n i t i o n 6 . 1

i n connection with t h e pre-

c e d i n g Theorem 6 . 2 , p r o p . 2 ) . F u r t h e r information about t h e preceding c l a s s of a l g e b r a s i s c o n t a i n e d i n t h e work o f A . C . COCHRAN e t a l . [l], A.C. COCHRAN and A.K. CHILANA [2].

7 . F i n i t e l y generated topological algebras The c l a s s of t o p o l o g i c a l a l g e b r a s i n t i t l e a n d , more g e n e r a l l y , i n d u c t i v e l i m i t s of s u c h , seems t o c o n s t i t u t e , u n d e r f u r t h e r s u i t a b l e r e s t r i c t i o n s , t h e a p p r o p r i a t e framework w i t h i n which one c a n o b t a i n a n a l o g o u s r e s u l t s t o t h o s e v a l i d h i t h e r t o i n Banach a l g e b r a s , a s t h i s c o n c e r n s t h e S p e c t r a l Theory of " s e v e r a l v a r i a b l e s " ( c f . , f o r example, t h e n e x t s e c t i o n ) o r a p p l i c a t i o n s of A l g e b r a i c Topology and t h e l i k e i n t h e " s t r u c t u r e t h e o r y " o f Banach a l g e b r a s a s t h e y a r e p a r t i c u l a r l y e n c o u n t e r e d i n r e c e n t y e a r s . The p r e v i o u s comment c o n s t i t u t e s , i n e f f e c t , t h e " L e i t f a d e n " f o r t h e r e s t of o u r d i s c u s s i o n i n t h i s and t h e next section. Thus, w e s h a l l b e i n t e r e s t e d n e x t i n d e r i v i n g , u n d e r s u i t a b l e r e s t r i c t i o n s on a g i v e n f i n i t e l y g e n e r a t e d t o p o l o g i c a l a l g e b r a E l a c e r t a i n " g e o m e t r i c " p r o p e r t y o f i t s s p e c t r u m V t Y E l , as w e l l a s t o i n d i c a t e ( s e e , i n p a r i c u l a r , t h e n e x t s e c t i o n ) some of i t s a p p l i c a t i o n s . But f o r c o n v e n i e n c e , w e s h a l l h a v e f i r s t t o r e c a l l ( c f . C h a p t . V ; S e c t i o n 2 ) a n d / o r e s t a b l i s h some more t e r m i n o l o g y n e c e s s a r y t o t h e sequel. So t o s t a r t w i t h w e r e c a l l ( c f . D e f i n i t i o n V ; 2 . 1 )

that E is said

t o b e a f i n i t e l y generated topologicaL a l g e b r a , i f t h e r e e x i s t s a f i n i t e s e t , s a y A G E , i n s u c h a manner t h a t E c o i n c i d e s w i t h t h e smallest c l o s e d s u b a l g e b r a of i t c o n t a i n i n g A . T h u s , s u p p o s e t h a t E i s a g i v e n f i n i t e l y , s a y , n-generated g i c a l algebra whose s p e c t r u m i s

(7.1) b e a s y s t e m of

topolo-

miE/, a n d l e t

A = (xl , . . . , x t o p o l o g i c a l generators

1 = x € En of E ; i . e .

,

w e assume as g i v e n a

284

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

a finite sequence of elements of E such that,by applying the notation of Chapt. V; Section 2,one has E = Co [(x,,

(7.2)

. .. , xn )]

(cf. V; (2.14). The given algebra E does not necessarily have an identity element). NOW, the following canonical map is defined @ : !VZ(E)

(7.3)

-

Cn:

f -@(f)

:= ( f ( x l ) , . . ., f / x n ) ) = r

q f ) ,... , P n l f ) l

.

Furthermore, one defines the j o i n t spectrwn of the element x = (2,

,... ,x n l E E n l

as in (7.11, by the relation

(7.4)

That is, one has, by definition (7.5)

SpE(lx

.

x n ) ) := Im(@l = @(?VIE)) C C n

NOW, it follows directly from the definition that the Cn-vaZued map @, given by (7.3) , is continuous. Furthermore, by applying (7.21, if x = li$nPg(xl,..., xnl is any element of X , one gets for every f e m(EI f(x) = limp (@(f)), X E E . 6 6 So one gets by (7.3) that the same map Cp y i e l d s i n f a c t a b i j e c t i o n onto i t s image in (7.5). (The previous argument specialize, of course, to that for the map VI;(4.2). Cf. also Complement 7.(1) below). (7.6)

Now suppose, in particular, that E is a given n-generated (commutative) complete l o c a l l y m-convex algebra with an identity element and spectrum m(El. Furthermore, let be a local basis of E providing an Arens-Michael decomposition of E’, say !E,las1 (cf.111; (3-10)). S o one now defines

n=

h

(7.7) :=

@a(m(8a)) = Qa(m(Eall

in such a way that one has (7.8)

QJfJ := ‘ f a ( [ x i l a ) ) l s is n

.

with fa e ???(gal 2 mrt.,) (cf. also V ; (6.23) and/or V; (2- 7 )) Thus, applying the notation of V;(6.16), one now gets the following commutative diagram

7.

285

F I N I T E L Y GENERATED ALGEBRAS

,-1

So w e a c t u a l l y h a v e

(7.10)

f o r e v e r y i n d e x a e I , where now t h e e l e m e n t x = (xl,..

., x n ) E E n

satisfies

the r e l a t i o n (7.11)

(see a l s o V; ( 2 . 1 3 ) ) . A

On t h e o t h e r h a n d , w e s t i l l n o t e t h a t each one of t h e algebi.as

Ea '

aeI, i s an n-generated Banach algebra, so t h a t i f ( 7 . 1 1 ) h o l d s , then one g e t s (7.12)

-

for every a e I ( t h e a n a l o g o u s deed, if p :E

r e l a t i o n t o (7.2) is c e r t a i n l y c l e a r ) . In-

i'/ker(pal = E a , a f l , d e n o t e s t h e c o r r e s p o n d i n g (cano-

a c a l l q u o t i e n t map ( c f . 111; ( 3 . 3 ) ) , t h e l a t t e r b e i n g i n f a c t a c o n t i n u ous a l g e b r a morphism, o n e h a s from ( 7 . 1 1 ) P,(E)

= paIC[(xl,.

..,zn,]) c

pa(C[(xl,. ..,xnI],

(7.13) A

-

@[(pa(zl),...,pa(znll] c Ea= pa ( E l , f o r e v e r y aeI, which o f c o u r s e y i e l d s t h e a s s e r t i o n . T h e n e x t r e s u l t g i v e s a c r i t e r i o n i n o r d e r t h a t t h e map ( 7 . 3 ) t o

be a homeomorphism o n t o i t s image i n C n . A s a m a t t e r o f f a c t , one i m -

p l i c i t l y e n c o u n t e r s a n a n a l o g o u s s i t u a t i o n when d e a l i n g w i t h Banach ( o r y e t F r e c h e t l o c a l l y m-convex) Lemma 7.1.

a l g e b r a s . Thus, w e have.

Lei: E be an n-generated l o c a l l y m-convex algebra w i t h an i d e n t i t y

element and spectrum ??T(E),

and l e t

z= iUala e I

be a local b a s i s of E providing

G

corresponding Arens-Michael decomposition o f E . Furthermore, consider t h e folZowing two a s s e r t i o n s : 1 ) The c o l l e c t i o n of s e t s

(7.14)

{

$a(nr(2all 1,

c1 E

I,

d e f i n e d by ( 7 . 8 ) , c o n s t i t u t e s a k-covering famiZy f o r I m ( $ ) C C n ( c f . ( 7 . 5 ) and a l s o D e f i n i t i o n V; 5 . 1 ) .

286

VIII

S P E C I A L T O P O L O G I CA L ALGEBRAS

2) The (canonical) miip @ defined by (7.3) i s a homeomorphism ( o n t o i t s image i n

en).

Then, 1 ) = > 2 ) .

I n p a r t i c u l a r , i f t h e given algebra E has t h e r e s p e c t i v e Gel'-

f u n d map continuous, then t h e preceding two a s s e r t i o n s are in f a c t e q u i v a l e n t .

Note.-A commutative f i n i t e l y g e n e r a t e d l o c a l l y m-convex a l g e b r a w i t h an i d e n t i t y e l e m e n t s a t i s f y i n g , m o r e o v e r , c o n d . 1 ) o f t h e p r e v i o u s lemma i s s a i d t o b e a ( k l - a l g e b r a (see a l s o D e f i n i t i o n 9 . 3 i n the sequel).

-

Proof of Lema 7.1. I t h a s b e e n n o t e d b e f o r e t h a t

4 : m(E)

(7.15)

t h e map

@(m(E)) G Cn

d e f i n e d by ( 7 . 3 ) provides a continuous b i j e c t i o n ( o n t o i t s image; i n f a c t , t h i s i s t r u e f o r a n y t o p o l o g i c a l a l g e b r a ) . NOW, @ ( ? 7 Z ( E ) ) b e i n g a s u b s p a c e of

en

i s 1 s t c o u n t a b l e , a n d h e n c e a k - s p a c e ( s e e e . g . J . DUGUNDSO @ i s a homeomorphism i f (and o n l y i f ) i t i s a proper map

JI [I: p. 248, 9 . 3 1 ) .

(i.e.,

$-'(K)Cm(E/

b y assuming I ) ,

i s compact, f o r e v e r y compact K G @ ( l n r ( E / I ) : Thus,

i f K E @ ( ? T X l E ) I i s compact one g e t s

K

(7.16)

.L

@a(m(ka)) ,

f o r some i n d e x a e l , so t h a t ( b y ( 7 . 7 ) and ( 7 . 1 0 ) , see a l s o V ; ( 6 . 2 3 ) ) one h a s

@-I ( K ) C ??Z ( E a )

(7.17)

m(kc!)

t h e l a t t e r s p a c e b e i n g ( w i t h i n a homeomorphism) a compact s u b s e t of

m ( E l ( c f . C h a p t . V; ( 6 . 2 ) and ( 6 . 2 3 ) ) . T h e r e f o r e , @-'(KI pact t h a t

i s a l s o com-

was t o b e p r o v e d , which f i n a l l y y i e l d s t h a t 1 ) + 2 ) .

On t h e o t h e r h a n d , i f t h e G e l ' f a n d map o f E i s c o n t i n u o u s , t h e

sets

m(ea), a e I, c o n s t i t u t e

a k - c o v e r i n g f a m i l y of

m(E)( c f . C h a p t . V ;

Theorem 6 . 1 a n d Lemma 6 . 3 ) . Thus, i f K i s a compact s u b s e t of @ ( ? Y Z ( E ) ) and c o n d . 2 ) h o l d s , @ - ' ( K ) G n Z ( E l i s a l s o compact, so t h a t i t i s cont a i n e d by t h e p r e c e d i n g i n some

m(ia).So

o n e g e t s ( 7 . 1 6 ) , which means t h a t 2 ) * 1 )

a p p l y i n g ( 7 . 7 ) and ( 7 . 9 )

a s w e l l , and t h i s c o m p l e t e s

t h e proof. I W e come n e x t t o t h e f o l l o w i n g u s e f u l r e s u l t o f t e n a p p l i e d i n

t h e sequel ( i n f a c t , i t s Corollary 7 . 1 ) .

Theorem 7.1.

Thus, w e h a v e .

Let E be a commutative complete

n-generated l o c a l l y m-conuelc

algebra w i t h an i d e n t i t y element and s p e c t m Z'Z(E). Moreover, assume t h a t t h e following condition h o l d s : (7.18)

il; ihe c c n t e x t of 17.111 t h e map @ defined by ( 7 . 3 1

7 . FINITELY GENERATED ALGEBRAS

28 7

gtcZds a homeomorphism onto i t s image, say S =

( T h i s w i l l be, f o r i n s t a n c e , t h e c a s e i f

Im ( $ ) - c C n .

7?T(E) i s c o m p a c t ) . F i n a l l y , sup-

pose t h a t t h e given algebra F has the r e s p e c t i v e Gel'fand map continuous. Then, (7.19)

S = $(m(EllG C n

,.

i s a polynomially convex s e t ( i . e .

,for

e v e r y compact K C S , i t s p o l y n o m i a l -

l y convex h u l l K i s c o n t a i n e d i n S ( c f . V ; ( 4 . 1 2 ) ) .

Proof. By ( 7 . 1 8 ) and t h e c o n t i n u i t y o f t h e G e l ' f a n d map, one conc l u d e s (Lemma 7 . 1 ) t h a t t h e s e t s (7.20) c o r r e s p o n d i n g b y ( 7 . 7 ) t o a g i v e n Arens-Michael d e c o m p o s i t i o n of E , c o n s t i t u t e a k-covering

f a m i l y of S . T h e r e f o r e , i f K i s any compact

s u b s e t of S , one o b t a i n s

f o r some i n d e x ~ E I . NOW, i n view o f

( 7 . 7 ) and ( 7 . 1 2 ) , t h e s e t

C C n i s t h e j o i n t s p e c t r u m o f t h e ( t o p o l o g i c a l ) g e n e r a t o r s I p,(xiil, l < i S n , o f t h e Banach a l g e b r a so t h a t i t i s a c t u a l l y , b y h y p o t h e s i s a'

f o r t h e a l g e b r a E , a compact polynomially convex subset of ample , L . H8RMANDER [ l : p . 66 , Theorem 3.1 . I s ] ) t h e p o l y n o m i a l l y convex h u l l

K^

.

C n ( c f . , f o r ex-

Hence , by c o n s i d e r i n g

of K , o n e g e t s by ( 7 . 2 1 )

(7.22)

which e x a c t l y w a s t o b e p r o v e d , a n d t h i s f i n i s h e s t h e p r o o f o f t h e theorem. I

I n a l e s s t e c h n i c a l l a n g u a g e one c o n c l u d e s w i t h t h e f o l l o w i n g r e s u l t ( c f . a l s o t h e comment a f t e r ( 7 . 1 8 ) ) .

Corollary 7.1. Let

E be a commutative complete n-generated

l o c a l l y m-convex

algebra w i t h an i d e n t i t y element and compact spectrwi ??Z(E). Moreover, assume t h a t t h e r e s p e c t i v e Gel'fand map of E i s continuous. Then, m(El i s homeomorphic t o a compact polynomially convex subset of C n . I Note.W e remark t h a t t h e c o n c l u s i o n of t h e p r e c e d i n g C o r o l l a r y 7 . 1 i s more g e n e r a l l y v a l i d f o r an a l g e b r a E , which i s a commutative complete n-generated l o c a l l y m-convex algebra w i t h an i d e n t i t y element and c o n F x t spectrum m ( E ) a d m i t t i n g , moreover , a k-covering f a m i l y of mi€) ( c f . c o n d . ( 7 . 1 4 ) ) ; i . e . , a complete (ki-algebra w i t h compact spectrum. ( I n t h i s r e s p e c t , c f . a l s o t h e Note a f t e r Lemma 7 . 1 , a s w e l l a s t h e p r o o f of t h e p r e v i o u s Theorem 7.1 )

.

288

BIII

S P E C I A L TOPOLOGICAL ALGEBRAS

Regarding t h e r e q u i r e m e n t s set f o r t h by t h e h y p o t h e s i s of t h e p r e c e d i n g C o r o l l a r y 7.1, w e s t i l l remark t h a t t h e s e imply a l r e a d y t h a t the given algebra E i s , i n f a c t , s p e c t r a l l y barrelled ( c f . Chapt. VI; Theorem 1.1

and C o r o l l a r y 1.4).

On t h e o t h e r hand, i t i s well-known t h a t a compact subset o f

Cn

( c a n o n i c a l ) homeomorphism; c f . ( 7 . 3 ) ) t h e spectrum of an n-

i s (within a

generated (commutative) Banach algebra ( w i t h an i d e n t i t y e l e m e n t ) i f , and only i f , it i s poZynomiaZZy convex ( c f . , f o r i n s t a n c e , K . HOFFMAN [I: p. 35, ( t h e l a s t ) Theorem and t h e comment f o l l o w i n g i t ] ) . Now, t h e a n a l o g o u s s i t u a t i o n , i n c o n n e c t i o n w i t h t h e p r e c e d i n g C o r o l l a r y 7.1, f o r f i n i t e -

l y g e n e r a t e d t o p o l o g i c a l a l g e b r a s , i n g e n e r a l , i s n o t c l e a r . However, w e do have an e x t e n s i o n of t h e above r e s u l t f o r Banach a l g e b r a s i n c a s e of f i n i t e l y g e n e r a t e d F r e c h e t ( l o c a l l y m-convex) a l g e b r a s , due t o R . M . BROOKS [4: p. 149, Theorem 2.21. Thus, w e have.

Theorem 7 . 2 . Let S be any given subset of C n . Then, t h e following two ass e r t i o n s are equivalent: 1 ) There e x i s t s a commutative n-generated Frdchet l o c a l l y m-convex algebra E w i t h an i d e n t i t y element and spectrum m(E),i n such a way t h a t i f x = ( x l ,

...,x

1

is a system of generators f o r E ( i . e . , (7.11 ) i s v a l i d ) , then t h e respective map Q defined by ( 7 . 3 ) yieZds a homeomorphism of m(EI onto S. 2 ) The s e t

Proof. 1 )

S i s a hemicompact polynomiaZly convex subset of

3

Cn.

2) : W e f i r s t remark t h a t m(E) i s a hemicompact s p a c e

( C o r o l l a r y V;6.1), hence t h e s e t S a s w e l l , s i n c e 4 i s by h y p o t h e s i s

a homeomorphism, so t h a t cond. (7.18) of Theorem 7.1 i s s a t i s f i e d . So t h e s e t S i s , by t h e same theorem, p o l y n o m i a l l y convex s i n c e t h e res p e c t i v e G e l ' f a n d map o f E i s c o n t i n u o u s , E b e i n g a F r e c h e t a l g e b r a (cf

. C o r o l l a r y VI; 1 .1 Now assuming 2 )

and P r o p o s i t i o n V ; 1 . 1 ) : i . e . , t h e a s s e r t i o n . t h e a l g e b r a A = ec(S) which i s t h u s , by

, consider

h y p o t h e s i s f o r S (hemicompact) and t h e f a c t t h a t S i s a l s o a k-space, a commutative Frgchet ZocaZly m-convex algebra w i t h an i d e n t i t y element and spectrum ? Z ( A ) =

s, w i t h i n a homeomorphism (Theorem VII;1.2). Furthermore,

if

i s a k-covering

(KmImem

( a s c e n d i n g ) sequence f o r S , one g e t s

(7.23) ( c f . , f o r i n s t a n c e , Chapt.V;Lemma 5 . 2 a n d / o r C o r o l l a r y 5 . 1 ) . Theref o r e , s i n c e S i s b y h y p o t h e s i s p o l y n o m i a l l y convex, one may assume t h a t each one of the s e t s K m , m e I N , i s polynomially convex. NOW, c o n s i d e r t h e s u b a l g e b r a , s a y E l of A g e n e r a t e d by t h e "co-

, n ) of o r d i n a t e f u n c t i o n s " zi ( i = l ...,

Cn r e s t r i c t e d t o S ( i . e . , z . ( X ) =

7.

Xi, w i t h X

= (A,,

.. ., A n )

e Cn)

E =

(7.24)

28 9

F I N I T E L Y GENERATED ALGEBRAS

;thus

, by

c [ ( z l , .. . , Z n ) l

a p p l y i n g (7.11 )

c

,

one o b t a i n s

c p=A .

S o E i s an n-generated Frgchet ( t o p o l o g i c a l ) subalgebra of A ( w i t h t h e same

i d e n t i t y element a s A ) . C o n s e q u e n t l y , by c o n s i d e r i n g t h e Arens-Yichael d e c o m p o s i t i o n o f

, s a y (a:7!)mEN , as ( g m l m E m , one o b t a i n s

A c o r r e s p o n d i n g t o ( 7 . 2 3 ) ( c f . a l s o Remark V ; 6 . 2 )

w e l l a s t h a t one f o r t h e a l g e b r a E , s a y

i z m = ;K m

= S = SpA((zl,.

.., z n ) )

,

t h e map 4 b e i n g a c o n t i n u o u s b i j e c t i o n ; t h e f o l l o w i n g f a c t h a s a l s o been used i n ( 7 . 2 5 ) : (7.26)

m('m)

VYIA^,/, h o i e o 'rn - K m home0

mem.

Accordingly, t h e s e t s K m , m € l N I c o n s t i t u t e a k-covering

sequence

€or @ ( ? ? Z t E / ) , E b e i n g a F r s c h e t a l g e b r a ; so Lemma 7.1 c a n now be u s e d y i e l d i n g t h a t @ i s i n f a c t a homeomorphism. 'Thus, 2)=>1)

a s w e l l , and

t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . W e c l o s e t h i s s e c t i o n by e x h i b i t i n g a f u r t h e r a p p l i c a t i o n of t h e

previous d i s c u s s i o n t o t h e e x t e n t t h a t a given ( f i n i t e l y g en er ated ) l o c a l l y in-convex a l g e b r a E c a n b e r e a l i z e d as a n a l g e b r a o f holomorp h i c f u n c t i o n s on a NpoZynomially convex" region

(open s u b s e t ) o f

en.

The

m o t i v e t o t h i s argument was t h e r e l e v a n t s t u d y of F . R . HEAL-M.P. WINDHAM [ l ]

. Thus, w e f i r s t g i v e a n o t h e r c r i t e r i o n i n o r d e r t h a t t h e map (7.3)

t o be a homeomorphism, which we s h a l l p r e s e n t l y a p p l y i n t h e s e q u e l . So c o n s i d e r t h e f o l l o w i n g d a t a :

Suppose we have t h e c o n t e x t of Lemma 7.1 w.L+/. r k g i x n algebra B h w i n g , mareaver, a continuous Gel'fand map. Now, is a s y s t m of generators of E , suppose t h a t if (xi)] t h e following c o n d i t i o n is s a t i s f i e d : ~

(7.27)

For any E > O ,

t h e set

KE = 1 f e m ( E / : I 2. (f)I

=I i ( z . I 5 E , 16 i 5 n 1

is a compact subset of m(E). Then, t h e map @ ( c f . ( 7 . 3 ) is a homeomorphism. Thus, under t h e assumption of t h e c o n t i n u i t y of t h e r e s p e c t i v e Gel'fand

290

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

map of t h e algebra E , t h e previous c o n d i t i o n i s i n f a c t equivalent t o t h e o t h e r two

(equivalent t h e n ) conditions of Lemma 7 . 1 . Now, c o n c e r n i n g o u r a s s e r t i o n i n ( 7 . 2 7 ) , s u p p o s e t h a t K i s any compact ( a n d h e n c e bounded) s u b s e t of @ ( m ( E ) ) L C nSo . there exists

E > 0

@-I ( K )

(7.28)

such t h a t

.

E K t G m(E)

Thus, s i n c e i$ i s c o n t i n u o u s and K coinpact, o n e g e t s t h a t

is

@-'(K)

s t i l l a compact s u b s e t o f K E ( c o n d . ( 7 . 2 7 ) ) ; t h e r e f o r e I by h y p o t h e s i s

f o r E ( G e l ' f a n d map c o n t i n u o u s ) and Theorem V i 6 . 1 , ??Z&,

@-'(K)S

,

one o b t a i n s t h a t

f o r some i n d e x a E I. Thus, KG@(??Z(.?a)l=

means t h a t c o n d . ( 7 . 1 4 )

which

is valid, that i n turn entails the assertion,

i n view of t h e h y p o t h e s i s f o r E and t h e same Lemma 7 . 1 . NOW, l e t

E

be

an n-generated

l o c a l l y m-convex

algebra with

a n i d e n t i t y e l e m e n t and s p e c t r u m n Z ( E ) , and l e t (xi)'
b e a set of

g e n e r a t o r s of E. Moreover, l e t

O(@(m(E))) = lLm$(U) U ?$(??l(E))

(7.29)

be t h e a l g e b r a o f h o l o m o r p h i c f u n c t i o n s on @ ( (4.40)),

(E))

c Cn

( c f . Chapt. I V ;

t h e ( c a n o n i c a l ) map @ b e i n g g i v e n a g a i n by ( 7 . 3 ) . On t h e o t h e r

h a n d , one h a s (7.30)

c f . ( 7 . 9 ) and C h a p t . V ; Lemma 5.1 and Lemma 6 . 3 . T h e r e f o r e ,

one c a n

c o n s i d e r on t h e a l g e b r a ( 7 . 2 9 ) t h e l o c a l l y m-convex t o p o l o g y d e f i n e d on i t b y t h e s u b m u l t i p l i c a t i v e semi-norms

t h a t i s , t h e "sup-norms" o n t h e compact s e t s @a(m(ia!)CCn, a€I. The f o l l o w i n g y i e l d s a g a i n a c r i t e r i o n i n o r d e r t h a t t h e map ( 7 . 3 ) t o be a homeomorphism. T h u s , w e h a v e .

Lemma 7 . 2 . Let E be an n-generated l o c a l l y m-convez algebra w i t h an i d e n t i t y element and continuous Gel'fand map. Then, t h e canonical map @, given by ( 7 . 3 1 , i s a homeomorphism i f , and only i f , t h e ( l o c a l l y m-convex) topology of t h e algebra

O($I??YiE)l), defined on it by t h e family ( 7 . 3 1 1 , coincides w i t h t h e compact-open topology induced on i t by t h e r e l a t i o n

cc(@(m(E).

(7.32)

o(@(nZ(El))~ ?roof.If

))

t h e map @ i s a homeomorphism, t h e n b y h y p o t h e s i s a n d

Lemma 7 . 1 t h e compact-open t o p o l o g y i n

C ( @ ( m ( E c) o) )i n c i d e s

with t h e

t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e members of t h e f a m i l y ( 7 . 1 4 ) .

291

7. FIWITZLY GENERATED ALGEBRAS

But this is, by our hypothesis, the topology considered on the algebra (7.29), which thus proves the "if" part of the assertion. On the other hand, if (7.32) holds within a homeomorphism, then by suitably adapting into the present context the argument applied by E . R . HEAL-M.P. WINDHAM [1: p. 462, Theorem 3.61, one concludes that (7.27) is valid, so that @ is a homeomorphism as this follows from what has been proved by that condition. Thus, we are now in the position to state the following result for "uniform (locally m-convex) algebras" (cf. Definition 5 . 1 ) .

Theorem 7 . 3 . Let E be an n-generated complete uniform ( l o c a l l y m-convex) algebra w i t h an i d e n t i t y element having, moreover, a continuous Gel'fand map. Then, t h e following two a s s e r t i o n s are Equivalent: 1 ) E i s isomorphic, a s a topological algebra, t o t h e algebra o f holomorphic

f u n c t i o n s on a polynomially convex open subset of 21 The spectrum o f E ,

??Z(El,

cn.

i s homeomorphic t o a 2n-dimensional topologi-

c a l manifold (without boundary). Proof. If U is an open polynomially convex subset of C n , then it is, in particular, a S t e i n submanifoid of C n (cf. , for instance, Chapt. VI1;Section 3 and V;(4.14)). Thus, by assuming 1 1 , we obtain

(7.33)

m(E)=

m(O(U)) = UCCn

I

within homeomorphisms (see Lemma VII; 3.1).So 1)*2). On the other hand, suppose that m(E) is (within a homeomorphism) a 2n-dimensional topological manifold (without boundary). Thus, let (Uf, xf i be a (topological) chart of m(E) at any given f € W E ) , where U is a relatively compact subset of ??Z(E);then the restriction s map 4 (cf. (7.3)) is a homeomorphism, and hence @ as well. So @iU

f

1is gf an open subset of

IUf

C" (see also J . DUGUNDJI [I: p. 358, Theorem

3.1]),that is @ is a local homeomorphism and hence a homeomorphism, since @ is 1-1. Therefore, if (7.34)

U=@(7Z(E))S Cnl

then U i s an open s ~ t t n a t~f C n which i s a7s.s po7,ynomiaZly convex as this follows from the hypothesis for the algebra E and Theorem 7 . 1 . Furthermore, by Lemma 7.2 the topology of the algebra (7.35)

O ( $ ( V ( E I ) ~ OW

c

ep

is that of the compact convergence in U. But this is also the topology of E , by means of its identification with $ ( E ) E c'^ (see Lemma 5.2). Thus, one actually obtains

292

VIII

S P E C I A L T O P O L O G I C A L ALGEBRAS

E = E" =

(7.36)

o($fm(E)))o(Ul , E

w i t h i n isomorphisms of t o p o l o g i c a l a l g e b r a s , i . e . ,

2)+1)

a s wel1,and

t h i s c o m p l e t e s t h e proof o f t h e theorem. I The f o l l o w i n g i s now a d i r e c t a p p l i c a t i o n of t h e p r e v i o u s t h e orem. That i s , w e h a v e .

Corollary 7.2. L e t E be an n-generated complete uniform ( l o c a l l y m-convex) algebra w i t h an i d e n t i t y element and continuous Gel'fand m a p , i t s s p e c t m being, moreover, homeomorphic t o a dn-dimensional topological manifold (without boundary). Then, E i s ( w i t h i n a topological algebraic isomorphism) a Fre'ehet l o c a l l y m-convex algebra ( s i n c e i t i s , modulo t h e l a s t isomorphism, t h e a l g e b r a of holomorphic f u n c t i o n s on a p o l y n o m i a l l y convex r e g i o n of C n ) . I For f u r t h e r a p p l i c a t i o n s w i t h i n t h e same c o n t e x t , a s above, w e r e f e r t o A. .~&L,IOS [ 2 6 ] .

7.(1).

Complement: Infinite many topological generators.-

W e consider i n

t h i s appendix of t h e p r e v i o u s d i s c u s s i o n an " i n f i n i t e - d i m e n s i o n a l ' ' v e r s i o n of t h e f o r e g o i n g (mainly Lemma 7 . 1 and Theorem 7 . 1 ) ,

extend-

ing t h u s , within t h e present context, analogous r e s u l t s i n t h e theory of

(commutative) Banach a l g e b r a s .

(See, f o r i n s t a n c e , E . L . STOUT [ I : p.

2 5 , Theorem 5.81 a n d / o r C . r 7 . RICKART [l : p. 151 , Theorem ( 3.3.4 ) ] )

. As

an

a p p l i c a t i o n of t h i s p o i n t of view, one would o b t a i n an a b s t r a c t t h e o r y o f holomorphic f u n c t i o n s of an " i n f i n i t e number of v a r i a b l e s " w i t h i n t h e framework of g e n e r a l f u n c t i o n a l g e b r a s t h e o r y . C f . , ample, C . E .

f o r ex-

RICKART [ 2 ] .

The f o l l o w i n g c o n s i d e r a t i o n s s p e c i a l i z e , of c o u r s e , t o t h e p r e c e d i n g m a t e r i a l . However, w e p r e f e r r e d above t o g i v e a s e p a r a t e t r e a t -

m e n t f o r a b e t t e r c l a r i f i c a t i o n o f t h e argument a p p l i e d . Thus, suppose w e a r e g i v e n a topological algebra E whose spectrum i s ??Z(E). Furthermore, l e t ( x i l i e I be a s e t of t o p o l o g i c a l g e n e r a t o r s of E l s o t h a t one h a s E =

(7.37)

ii

( c f . D e f i n i t i o n V; 2 . 1 ) . NOW, e x t e n d i n g t h e t e r m i n o l o g y a p p l i e d i n VI;(4.4) as w e l l a s

i n ( 7 . 4 ) above, one d e f i n e s t h e r e s p e c t i v e canonical map

which, s i m i l a r l y t o t h e f o r e g o i n g , i s proved t o be a continuous b i j e c -

t i o n o n t o i t s image i n

&.

Besides, t h e l a s t set i s again defined t o

7 . ( 1)

.

29 3

INFINITE MANY GENERATORS

j o i n t s p e c t m of t h e g i v e n s e t of g e n e r a t o r s o f E ; i . e .

be the

, one

h a s , by d e f i n i t i o n , SpE((xilieII

(7.39)

:=

I

=

Im(@)

@(m(E)}CC

.

I n t h i s r e s p e c t , it i s c e r t a i n l y c l e a r t h a t whenever

compact sFace, t h e preceding canonical map

@

??Z(E)

is a

is in f a c t a homeomorphism.

F u r t h e r m o r e , i f t h e given topological algebra B has a l o c a l l y equicontinu-

ous spectrum ( D e f i n i t i o n V;1.2), t h e n (Lemma V;2.4) t h e same map $ p r o v i d e s a hcmeomorphism of ? Z ( E ) onto i t s image i n C

I

.

I n p a r t i c u l a r , if E is a commutative complete l o c a l l y m-convex algbera

w i t h an i d e n t i t y element and l o c a l l y equicontinuous spectrum, f o r which an analogous r e l a t i o n t o ( 7 . 3 7 ) h o l d s , then t h e r e s p e c t i v e map @ is a homeomorphism; b e s i d e s , t h e s e t s ( 7 . 2 4 ) c o n s t i t u t e a k-covekng f a m i l y for $(7?2(El)

(cf. also Corollary

VI; 1.3). A s a matter of f a c t , assuming t h a t t h e map @ i s a homeomorphism,

one p r o v e s t h a t

$(m(E))G @I i s a polynornially convex s e t . However, w e s t i l l n e e d a b i t more t e r m i n o l o g y .

Thus, by e x t e n d i n g t h e f a m i l i a r t e r m i n o l o g y one h a s i n c a s e of t h e complex n u m e r i c a l s p a c e C n

( c f . , f o r e x a m p l e , T.W. GAMELIN [I:

f f . ; i n p a r t i c u l a r , p. 67, Theorem 1.2]),

K G Cr t h e polynomially convex h u l l

2

p. 66

one d e f i n e s f o r a n y compact

of K b y t h e r e l a t i o n

K := ~ ( P I K I ) ;

(7.40)

P(K) d e n o t e s t h e uniform closure on K of t h e algebra of polynomials in I ( w i t h zi E C , i e I ) . T h a t i s , a commutative Banach a l g e b r a w i t h a n (zi )i

here

i d e n t i t y e l e m e n t such t h a t

On t h e o t h e r h a n d , a s u b s e t S C C I i s s a i d t o b e polynomially eonvex,whenever one h a s t h e r e l a t i o n

(7.42)

S=

.

1Ji 5 3 K , compact

T h a t i s , whenever t h e s e t S , t o g e t h e r w i t h a n y compact s u b s e t K of i t , contni.ns

al.so

its

p o l y n o m i a l l y convex h u l l

?.

Thus, w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g r e s u l t , summarizing a s w e l l p a r t of t h e p r e v i o u s d i s c u s s i o n .

Let E be a tcpoZogical algebra whose spectrum is P Z i E ) , and l e t of topological generators for E . Then, t h e r e s p e c t i v e canonical be a s e t ixi)ie I Theorem 7.4.

” ’ap

(7.43)

$ : 1?Z(E)-+

@I : f

W

@(fl:=(f(x.))

i E I

294

BIII

S P E C I A L TOPOLOGICAL ALGEBRAS

d e f i n e s a continuous b i j e c t i o n o f ??Z(El

onto Im($)= SpE((x 1 . C C'. i % e l )J n particular,suppose t h a t E i s a commutative l o c a l l y m-convex algebra w i t h

an i d e n t i t y element having, moreover, t h e r e s p e c t i v e Gel'fand map continuous. Also assume t h a t t h e map $ i s a homeomorphism ( o n t o i t s image; t a k e , f o r i n s t a n c e , a s p e c t r a l l y b a r r e l l e d a l g e b r a w i t h %(El

c o m p a c t . C f . a l s o Remark 7.1

below). Then, t h e s e t Im($) :=$(??UE))

(7.44)

i s a polynomially convex subset of @' Proof.

= SpE((x.) %

.

ier

I

The f i r s t p a r t o f t h e a s s e r t i o n f o l l o w s a l r e a d y from t h e

p r e c e d i n g d i s c u s s i o n . On t h e o t h e r h a n d , b y c o n s i d e r i n g a n Arens

,

M i c h a e l d e c o m p o s i t i o n of 6 , s a y (E",jaeL aeL,

-

each one o f t h e a l g e b r a s

2a'

i s a c o m m u t a t i v e Banach a l g e b r a w i t h a n i d e n t i t y e l e m e n t f o r

which ample,

is a family of t o p o l o g i c a l g e n e r a t o r s ( c f .

(pa(xili

,

f o r ex-

( 7 . 1 2 ) ) . Thus, the s e t

(7.45)

is a compact poZynomiaZZy convex subset o f

@I ( c f . E. L . STOUT [I: p. 25, Theorem 5 . 8 1 ) . F u r t h e r m o r e , f o r a n y g i v e n compact s e t K Z $ ( ? ? l ( E l J S C I , one g e t s

by h y p o t h e s i s f o r E

K

(7.46)

f o r some i n d e x a

f

L.

c g,crni2,

)

Therefore, one o b t a i n s

(7.47 ? ( b y (7.45)) $ a ( V 2 ( g a ) ) G @ ( T Z ( E I l C C I , which p r o v e s t h e a s s e r t i o n , a n d t h i s c o m p l e t e s t h e p r o o f o f t h e t h e -

orem. I

.-

Remark 7.1 I n connection with t h e preceding Theorem 7 . 3 , w e f u r t h e r n o t e t h e f o l l o w i n g f a c t : Under t h e assumption t h a t a given algebra E s a t i s f i e s t h e c o n d i t i o n s of Theorem 7 . 3 and t h e index s e t I is countable, t h e map $, g i v e n b y ( 7 . 4 3 ) , is a homemorphism i f , and only i f , t h e r e e x i s t s a local b a s i s of E, say i U a ) a e i , , providing a k-covering f a m i l y f o r @ ( m ( Ei)n) t h e sense o f ( 7 . 1 4 ) : The a s s e r t i o n i s a s t r a i g h t f o r w a r d c o n s e q u e n c e o f t h e a r g u m e n t a p p l i e d i n t h e p r o o f o f Lemma 7.1.

8. Functional calculus (contn'd.). The s i l o v - Arens Wael broeck theory

- Calderdn -

W e c o n s i d e r i n t h i s s e c t i o n a s t r e n g t h e n i n g o f a number o f basic

8. FUNCTIONAL CALCULUS ( CONTN 'D . )

295

c o n c e p t s a p p l i e d i n t h e f o r e g o i n g , by u s i n g t h e t e r m i n o l o g y and res u l t s of C h a p t . V 1 ; S e c t i o n 3 . Among t h e a p p l i c a t i o n s which one c a n h a v e w i t h i n t h e c l a s s of t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d ( i n f a c t , i n d u c t i v e l i m i t s of such) a r e r e s u l t s connected with t h e " i n v e r t i b i l i t y of matrices" w i t h e n t r i e s i n s u c h t o p o l o g i c a l a l g e b r a s . These a r e v a l i d s o f a r i n commutative Banach a l g e b r a s . However, w e s h a l l r e f e r f o r t h i s k i n d o f a p p l i c a t i o n s t o t h e p e r t i n e n t r e c e n t l i t e r a t u r e : See V . Y A . LIN [l]

, N . SIEONY- J .

WERMER [l]

a n d / o r A . IVALLIOS [20].

T h u s , l e t E b e a g i v e n t o p o l o g i c a l a l g e b r a whose s p e c t r u m i s

m(E)a n d

X a ( f i n i t e d i m e n s i o n a l ) complex ( a n a l y t i c ) m a n i f o l d . NOW, a

c o n t i n u o u s map (8.1)

:

m(Ei

-

x

is s a i d t o b e a s p e c t r a 2 map, i f f o r e v e r y h o l o m o r p h i c

(complex-valued)

f u n c t i o n h d e f i n e d o n an open n e i g h b o r h o o d , s a y U , of Im($) z $ ( m ( E ) ) CX,one has the relation $,(hi := h

(8.2)

o@

,

=

f o r some e l e m e n t X E E . Thus, a p p l y i n g a more c o n c i s e n o t a t i o n ,

if

(8.3) denotes the

a!gorbm of hoZomorphic functions

on I m

($1 C X ( c f . C h a p t .

IV;

4 . ( 3 ) , a s w e l l a s Scholium V1;4.1), t h e n t h e p r e c e d i n g c o n d . ( 8 . 2 ) amounts t o t h e r e l a t i o n $,(O(Imi@i)js z A 5 $(E)

(8.4)

(see a l s o V; ( 1 . 6 ) ) . O n t h e o t h e r h a n d , t h e map $ i n ( 8 . 1 ) i s s a i d t o b e a weakly spec-

t r a % map, i f , b y s t i l l a p p l y i n g t h e r e s p e c t i v e n o t a t i o n of

(8.21

,

one

has the relation

$,(o(X)l

(8.5) I t i s c l e a r by

C EA.

( 8 . 3 ) t h a t every s p e c t r a l map is weakly s p e c t r a l . On

t h e o t h e r hand, a l t h o u g h t h e converse does n o t i n g e n e r a l h o l d , i t is v a l i d , h o w e v e r , a s w e s h a l l p r e s e n t l y see (Theorem 8 . 1 1 , i n c a s e X i s a Stein manifold. In p a r t i c u l a r , t h e case X = C n represents the s i t u a t i o n p r o v i d e d a l r e a d y by t h e " & l c v - Arens

- Calderdn - WaeZbrceck

( c f . , f o r i n s t a n c e , I. COLOJOARd [ l : C h a p t . 11;

-

Namely, s u p p o s e t h a t t h e continuous map (8.6)

$ : mlE)

Cn

5 51).

Theory

I'

296

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

is weakly s p e c t r a l . Thus, by c o n s i d e r i n g t h e c o o r d i n a t e f u n c t i o n s

z

i

of

C n , s i n c e o f c o u r s e z . E O I C ” ) , w i t h 1 5 i < n , t h e r e e x i s t s by ( 8 . 5 ) a f i n i t e sequence o f elements of E , say

x

(8.7)

a

(xI,...,xn) e F”

i n s u c h a way t h a t z . 0 ~ =

(8.8)

2i,

with

i = ,..., ~ n.

C o n s e q u e n t l y , one f u r t h e r o b t a i n s

I m ( @ ) = @(m(E)I = {(2i(f))lzi ~n e C n : f € 7 Y I m (8.9) 5

(cf. also VI;(4.4)

SpE(lxl,...’xni)

a n d / o r t h e above r e l . ( 7 . 3 9 ) ) .

Thus, by c o n s i d e r i n g t h e L r i p l e t

( x , C n , 0 ) a s above ( c f . a l s o V I ;

( 4 . 5 ) ) , one c o n c l u d e s t h e r e l . ( 8 . 4 ) f o r any member o f t h e c a t e g o r y -

domain o f d e f i n i t i o n of t h e S i l o v - Arens - CaZderdn - Waelbroeck f u n c t o r , t h a t

i s f o r e v e r y commutative c o m p l e t e l o c a l l y m-convex a l g e b r a w i t h an i d e n t i t y e l e m e n t : C f . Theorem V 1 ; 4 . 1 : ( 4 . 1 5 ) , a s w e l l as Scholium V I ; 4.2

i n c o n n e c t i o n w i t h M . EONNARD [3: p. 412, ThEorSme

31.

F o r c o n v e n i e n c e of l a t t e r r e f e r e n c e , w e summarize t h e l a s t a r gument i n t o t h e n e x t . Lemma 8.1. Let E be a commutative complete l o c a l l y m-convex algebra w i t h an

i d e n t i t y element and spectrum ??Z(E).

Moreover, l e t @ :m ( E i + C n

be a given con-

tinuous map. Then, @is a s p e c t r a l map if, and o n l y if, it is weakZy s p e c t r a l . I A s a c o n s e q u e n c e of t h e p r e c e d i n g , a weakly s p e c t r a l map $:?WE)+- C n i s defined by a f i n i t e sequence of elements of E s u c h t h a t ( 8 . 9 ) t o b e t r u e .

I n t h i s r e s p e c t , t h e s e q u e n c e ( 8 . 7 ) i s uniquely determined c o u r s e t h e g i v e n t o p o l o g i c a l a l g e b r a E i s i.emi-simple

by @ i f of

( D e f i n i t i o n 3.2),

a s it i s r e a d i l y s e e n b y ( 8 . 9 ) . W e come n e x t t o t h e p r o m i s e d e x t e n s i o n of t h e p r e v i o u s Lemma

8.1

t o t h e c a s e of a n a r b i t r a r y S t e i n m a n i f o l d X. Thus, w e h a v e . Theorem 8.1. Let E be a commutative complete localZy m-convex algebra w i t h

an i d e n t i t y eZernent and spectrum m(E).Moreover, suppose t h a t X is a S t e i n manifold, aiid l e t (8.10)

@ :m(E)+X

be a given continuous map. Then, Q i s a s p e c t r a l map if, and only if, it is weakZy

spectra I .

8.

Proof.

F U N C T I O N A L CALCULUS ( C O N T N ' D . )

291

I t s u f f i c e s t o p r o v e o n l y t h e " i f " p a r t of t h e a s s e r t i o n :

Thus, by a p p l y i n g R e m e r t ' s Embedding Lemma ( c f . C h a p t . V I I ; S e c t i o n 3 , Lemma 1 1 , w e may s u p p o s e t h a t t h e given S t e i n manifold X i s ( c a n o n i c a l l y i s o m o r p h i c t o ) a cZosed (complex a n a l y t i c ) submanifoZd of some compZe3: numerical space C n . NOW, w i t h i n t h i s c o n t e x t , i f ( 8 . 1 0 ) i s a weakly s p e c -

t r a l map r e l a t i v e t o X , it i s a f o r t i o r i

,

by ( 8 . 5 ) , weakly s p e c -

t r a l r e l a t i v e t o Cn, so t h a t (Lemma 8 . 1 ) a spectra2 map r e l a t i v e t o

en.

F u r t h e r m o r e , s u p p o s e t h a t ti i s a n open n e i g h b o r h o o d o f i$(??Z(El) i n X and l e t h E O f t i )

. Thus

we t h e n a p p l y Docquier - Grauert Theorem ( a c c o r d -

i n g t o which f o r every closed submanifold X o f Cn, t h e r e ~cr:i.tn an open neighborhood V of X i n C n and a holomorphic w t r a c i f h n p : V - X ; .:f.

:.

G:iN?.'I:fG-h'.

53SdT

[I:

p. 257, Theorem 8 1 )

.

c f . , for i n s t a n c e ,

T h e r e f o r e , one g e t s a h o l o -

morphic e x t e n s i o n of h o n t o an open n e i g h b o r h o o d , s a y W , of $ ( m ( E l ) . i n C n such t h a t U C W.

Thus, by t h e p r e c e d i n g , t h e l a t t e r map i s r e a l -

i z e d by a r e l a t i o n l i k e ( 8 . 2 ) ; t h a t i s , $ i s a s p e c t r a l map r e l a t i v e t o X a s w e l l , and t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . I W e c o n s i d e r n e x t , w i t h i n t h e p r e s e n t framework,

the extension,

a l r e a d y p r o m i s e d i n t h e f o r e g o i n g , of t h e c l a s s i c a l " A r e n s - C a l d e r h Lemma" ( s e e , f o r example, T. W. GAMELIN [I: p. 85, Lemma 5.21 ) w e s t i l l need some more comments.

. However,

T h u s , s u p p o s e w e a r e g i v e n a c o n t i n u o u s map (8.11)

@ :

m(E)---tX

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

values

i n a t o p o l o g i c a l s p a c e X I a n d a f i n i t e s e q u e n c e o f e l e m e n t s of E l s a y

..., xn .

x

s (XI,

x

E E n i s , by d e f i n i t i o n , t h e

NOW,

(8.12)

an E-modification

$': ??Z(E)--+

of @ c o r r e s p o n d i n g t o t h e g i v e n s e q u e n c e

-

( c o n t i n u o u s ) map

X x Cn : f

@'if)

:=

($if)

;~

~ i ( f ) ~ l s)i .s

That i s , one h a s i$'=

(8.13)

$x(;

1

X...X$

Now, a p p l y i n g t h e n o t a t i o n of that X is a

complex manifold and

b'

) .

(8.111,

suppose, i n p a r t i c u l a r ,

an open neighborhood of i $ f V Z ( E / ) i n X .

-

T h u s , w e s h a l l s a y t h a t a n E-modification i s subordinated t o U , i f one h a s

n(@'(ZniE/l)

(8.14)

where

T T :X x

Cn-

X denotes t h e 7

t h e f i r s t f a c t o r and

$I'

of @,

a s i n (8.12),

the relation E

U

,

respective (canonical) projection onto

@'(fl(E)I t h e h o l o m o r p h i c a l l y c o n v e x h u l l of 1n1'N

VIII

Si'3ClAL

T O P O L O G I C A L ALGEBRAS

in X x C n (cf.also Vi(4.2)). Thus, we first have the following. Lenuna 8.2. Let E be a commutative complete l o c a l l y m-convez algebra w i t h an i d e n t i t y element and spectrum

m(F).Moreover,

let X,Y

be S t e i n manifolds and @,

Q s p e c t r a l maps o f 73;!(E) i n t o X , Y , r e s p e c t i v e l y . Then, t h e (Cartesian product) vap

-

@XqJ :mlE)

(9.15)

X X Y

i s a s p e c t r a l map t o o . In p a r t i c u l a r , every E-modification o f a s p e c t r a l map of '??ZlE)

i n t o a S t e i n manifold i s a l s o a s p e c t r a l map.

Proof. The Cartesian product X x Y is again a Stein manifold (cf., for instance, C. ANDREIAN CAZACU [ I : p. 248, Beisp. 3 1 ; as we shall see in

a later chapter of this book (topological tensor products), a "functional analysis-type'' of proof of this fact can also be given). Thus, it suffices to prove first , in view of Theorem 8.1 , that t h e map ( 8 . 1 5 1 i s weakly s p e c t r a z : N o w , on the basis of the previous discussion, one may

consider that X x 1 7 is "regularly embedded" (Remmert) as a closed submanifold of the complex (numerical) space, say C n x C m ; so applying Cartan's Theorem B any holomorphic function on X x Y can be extended to a holomorphic function on C n x C m (cf- , for example , R . C. G U N N I N G - H. ROSSI [I: p. 245, Theorem 181). Consequently, since by hypothesis each one of the maps @ , $ is given by a finite sequence of elements of E l considered as a C n l resp. C m , -valued map (cf. (8.8) as well as the comment following Lemma 8.I ) , the same holds for the C n x @"-valued map 4 x w . Thus (Theorem VI;4.1) the latter is a weakly spectral map relative to C n x C m ; equivalently, it is by the foregoing a weakly spectral map relative to XxY,which is the assertion. On the other hand, suppose that

4'

(8.16)

: m(E/--+ Xx

en

is a given E-modification of a spectral map 4 of ??Z(E/ into a Stein manifold X . Thus, by embedding X (Remmert) in the complex space, say, C m and considering the map (8.17)

2

I

.., 2 n ) : m ( E ) - + C n :

f M % ( f ) : = ( 2 i ( f ) )I l

i < l l

'

which corresponds to the given modification of @ (cf. [8.12)), one concludes by the previous argument (see also Theorem VI;4.1) that (8.17) yields a weakly spectral map. Therefore, by the first part of the statement, the same holds for the map ( 8 . 1 6 ) which,infact, yields the assertion (Theorem 8 . 1 ) , and this completes the proof of the lemma. I

8.

299

F U N C T I O N A L CALCULUS (CONTN 'D. )

W e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g b a s i c r e s u l t ;

it p r o v i d e s , w i t h i n t h e p r e s e n t framework, a n e x t e n s i o n o f t h e ArensC a l d e r 6 n Lemma, i n d e e d , of a n a l r e a d y g e n e r a l i z e d form of it ( f o r t o p o l o g i c a l a l g e b r a s ) . T h u s , o n e g e t s f o r t h e c l a s s of t o p o l o g i c a l a l g e b r a s c o n s i d e r e d i n t h e s e q u e l a n e x t e n d e d v e r s i o n o f t h e s a i d res u l t , a s t h i s i s a p p l i e d i n Banach a l g e b r a s t h e o r y ( c f . 14. BONNARD [2: p. 29, Lemma 5 1 ) . T h i s c l a s s i c a l a n a l o g u e o f t h e same r e s u l t w i l l a l s o b e a p p l i e d i n t h e n e x t p r o o f . T h u s , w e now h a v e t h e f o l l o w i n g . Theorem 8.2. Let E be a commutative complete l o c a l l y m-convex algebra w i t h

an i d e n t i t y element and compact spectrum m(E)having, moreover, t h e r e s p e c t i v e

Gel'fand map continuous. Furthermore, suppose t h a t X is a given S t e i n manifold,and let (8.18)

@:

??ZiEi-+X

bc o wec'tly s p e c t r v i r q . F i n a l l y , suppose t h a t U is an open neighborhood of $(77l'f'LJ

in X . Then, t h e r e e x i s t s an E-modification of @ subordinated t o U. C o n c e r n i n g t h e h y p o t h e s i s of t h e p r e v i o u s t h e orem, w e s t i l l n o t e t h a t i t would be e q u i v a l e n t t o assume t h a t E i s a commutative ccnplete l o c a l l y m-eonvex Q-aigebra w i t h an i d e n t i t y element ( D e f i n i t i o n I ;6 . 2 ) : The a s s e r t i o n f o l l o w s from Theorem V : l . l , C o r o l l a r y 1 1 1 ; 6 . 6 , P r o p o s i t i o n I 1 ; T . l and C o r o l l a r y V I ; 1 . 4 . I n t h i s r e s p e c t , w e f u r t h e r remark t h a t t h e t o p o l o g i c a l a l g e b r a s under c o n s i d e r a t i o n are, i n p a r t i c u l a r , Arens - Calderdn algebras i n t h e s e n s e of M . BONNARD [2: p. 29, P r o p o s i t i o n 81 ( c f . a l s o S e c t i o n 9.(3) below)

.

Proof o f Theorem 8 . 2 . Suppose t h a t X i s embedded (Remmert's Theorem) a s a c l o s e d s u b m a n i f o l d of t h e complex s p a c e C n s u c h t h a t (8.19)

$ : ?7i?(E)-XcjCn

i s determined by t h e e l e m e n t s , s a y , x ~ E E w , ith i = Z

,..., n

( c f . ( 8 . 8 ) ) , so

t h a t one h a s t h e r e l a t i o n

@ ( f i= (Zi( f ) l l < i 5

(8.20)

I

f€rniEi

.

T h e r e f o r e , by t h e p r e v i o u s embedding and t h e h y p o t h e s i s f o r U , o n e h a s

g t m i ~ i u= ) ~x n v c v s c n

(8.21) where

v

i s a n open s e t i n

cn.

T h u s , a c c o r d i n g t o t h e e x t e n d e d form of Arens-CalderBn Lemma m e n t i o n e d a b o v e , one g e t s a f i n i t e s e q u e n c e of e l e m e n t s o f E , s a y ,

iyl,.

. .,yF 1

t h e map

s u c h t h a t t h e c o r r e s p o n d i n g E-modif i c a t i o n of

4,

namely,

VIII SPECIAL TOPOLOGICAL ALGEBRAS

300

@':

(8.22)

m(E)+Xx

C"

5

Cnx C m

g i v e n by t h e r e l a t i o n

@'(f)= ( 5 p , . . . , q f ) ;

(8.23) with f E

m(E),

s a t i s f i e s the condition T T ' ( @ ' ( ~ ( E ) )C ) V :

(8.24) here

G1(f) )...)$ * ( f I I ,

: Cn x

TT'

Cm-

C n d e n o t e s t h e r e s p e c t i v e c a n o n i c a l p r o j e c t i o n ap-

p l i e d t o t h e p o l y n o m i a l l y convex h u l l of t h e compact s e t Cn+m ( c f . a l s o

34. BONNA??D 12: p. 29, Lemme 51 )

@'(m(E)) in

.

On t h e o t h e r h a n d , t h e l a t t e r h u l l c o i n c i d e s w i t h t h e holomorn+ m p h i c a l l y convex h u l l o f t h e same s e t i n C , i . e . , one h a s ( c f . a l s o (8.22) 1 (8.25)

@'(??Z(E)) =

@'(m(?))= X x

Cm

5

Cn

Cm

(see C h a p t . V ; ( 4 . 1 0 ) , ( 4 . l l ) a n d ( 4 . 1 4 ) ) . Hence, by a p p l y i n g t h e can o n i c a l p r o j e c t i o n I T : X X C " - + X , a s w e l l a s ( 8 . 2 1 ) and ( 8 . 2 4 ) , one obtains

c xnv

(8.26)

T T ( ~ ' ( ~ ( E ) ) )

= u

which i s j u s t t h e a s s e r t i o n , by ( 8 . 1 4 ) , a n a t h i s c o m p l e t e s t h e p r o o f of t h e theorem. I W e have a l r e a d y r e f e r r e d a t t h e b e g i n n i n g o f t h i s s e c t i o n t o

p o s s i b l e a p p l i c a t i o n s of t h e p r e s e n t m a t e r i a l c o n c e r n i n g "complement a t i o n of matrices t o i n v e r t i b l e o n e s " , t h e e n t r i e s of t h e m a t r i c e s i n v o l v e d b e i n g t a k e n from t o p o l o g i c a l a l g e b r a s of t h e t y p e c o n s i d e r e d above. On t h e o t h e r hand, t h i s k i n d o f a r g u m e n t i s b a s e d , i n e f f e c t , on a v e r s i o n of t h e s o - c a l l e d "Oka's p r i n c i p l e "

as t h i s i s a p p l i e d t o

t o p o l o g i c a l a l g e b r a s . I n t h i s r e s p e c t , " s p e c t r a l maps are ( i n a secse)

equivalent t o continuous maps"

when t h e y are c o n s i d e r e d

as

"sections"

of s u i t a b l e " h o l o m o r p h i c v e c t o r b u n d l e s " o v e r ( o p e n ) n e i g h b o r h o o d s of t h e s p e c t r a of t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d : m o r e o v e r , t h e l a t t e r s p a c e s are a p p r o p r i a t e l y "embedded" i n complex n u m e r i c a l s p a c e s (act u a l l y S t e i n m a n i f o l d s ) . Thus, t h e c o n t r i b u t i o n of t h e p r e v i o u s Theorem 8 . 2 t o s u c h k i n d of argument i s h a r d l y n e c e s s a r y t o b e f u r t h e r pointed out. However, f o r s u c h a p p l i c a t i o n s a s w e l l a s f u r t h e r o n e s b a s e d on t h e p r e c e d i n g p r i n c i p l e a n d r e l a t e d t o t h e " s t r u c t u r e t h e o r y " of t o p o l o g i c a l a l g e b r a s i n a n a l o g y w i t h t h e s i m i l a r s i t u a t i o n one h a s i n Banach a l g e b r a s t h e o r y , w e r e f e r t o t h e r e c e n t l i t e r a t u r e : Thus, f o r Banach a l g e b r a s , see e . 9. 0 . FORSTER [5]

, V.

Ya. LIN [l]

,

M. E. NOVOODVORSKTI [l]

9. ( 1 ) . L FQ-ALGEBRAS

301

a n d / o r A . MALLIOS [20] f o r t o p o l o g i c a l a l g e b r a s of t h e t y p e c o n s i d e r e d above.

( C f . a l s o t h e p e r t i n e n t comment i n t h e n e x t S u b s e c t i o n 9.(3)).

9. Mi scel 1anea W e g a t h e r i n t h i s s e c t i o n some complementary comments on c e r t a i n t y p e s of t o p o l o g i c a l a l g e b r a s which a r e c l o s e l y r e l a t e d t o t h o s e a l r e a d y c o n s i d e r e d i n t h e f o r e g o i n g . W e a l s o comment on t h e t e r m i n o l ogy used i n t h e c a s e of c e r t a i n o t h e r t o p o l o g i c a l a l g e b r a s which w e s h a l l have t h e o p p o r t u n i t y t o c o n s i d e r i n s u b s e q u e n t c h a p t e r s of t h i s discussion. Thus, w e s t a r t w i t h t h e f o l l o w i n g c l a s s of t o p o l o g i c a l a l g e b r a s , a p a r t i c u l a r i n s t a n c e of which h a s been c o n s i d e r e d a l r e a d y i n Chapt. IV; Section 4 ( c f . Subsections

9.(1).

L F Q - algebras.-

4. ( 1 ) and 4 . (3)).

The t o p o l o g i c a l a l g e b r a s i n q u e s t i o n a r e

c o i n e d by t h e f o l l o w i n g .

Definition 9.1.

An

a l g e b r a E i s s a i d t o be an L F Q - a l g e b r a , when-

e v e r t h e r e e x i s t s an i n c r e a s i n g sequence ( E n i n e m

of F r d c h e t

locally

rn-convex a l g e b r a s ( s u b a l g e b r a s of E ) i n such a manner t h a t t h e f o l l o w ing conditions a r e s a t i s f i e d :

1) E = i:

U t;; 2 4 IN

( w i t h i n a b i j e c t i o n ) l Z E n (see also Scholium I V ;

1.1).

2) E i s a ( 2 - s i d e d ) i d e a l o f t h e a l g e b r a En+l , f o r e v e r y n e N . 3) The t o p o l o g y of En c o i n c i d e s w i t h t h e r e l a t i v e one induced on it from E f o r every n f N . n+l ’ 4 ) The f i n a l l o c a l l y rn-convex

-

( a l g e b r a ) t o p o l o g y on B , d e f i n e d

on it by t h e c a n o n i c a l i n c l u s i o n maps in : E E , n e IN ( c f . D e f i n i t i o n IV;3.1), i n d u c e s on e a c h one of t h e a l g e b r a s E n , n e I N , i t s i n i t i a l t o POlogY5 ) The ( l o c a l l y rn-convex) cond. 4 ) i s a

a l g e b r a E t o p o l o g i z e d by t h e preceding

&-algebra.

I n c o n n e c t i o n w i t h t h e p r e c e d i n g d e f i n i t i o n , one now remarks t h e following f a c t s :

1 ) I n t h e p r e s e n c e of cond. 1 ) and t h e h y p o t h e s i s t h a t t h e given s e q u e n c e E , n E N , i s i n c r e a s i n g , cond. 2 ) i s e s s e n t i a l l y e q u i v a l e n t t o t h e seemingly more s t r i n g e n t r e q u i r e m e n t t h a t each one of t h e algebras E n ,

n e m , i s a ( 2 - s i d e d ) i d e a l of t h e algebra E . Thus, a c c o r d i n g t o C o r o l l a r y IV;3.2, t h e t o p c l o g y on E c o n s i d e r e d by cond. 4 ) c o i n c i d e s w i t h t h e f i n a l l o c a l l y convex ( v e c t o r s p a c e )

302

V'I T

CFF C I.4L T O P O L O G I C A L ALGEBRAS

t o p o l o g y i n d u c e d o n it by t h e c a n o n i c a l maps i : E - E , n 8 I N . Theren n f o r e , by h y p o t h e s i s f o r t h e a l g e b r a s E n , n e N , t h e g i v e n a l g e b r a E

is an CF-space

thus topologized

( i . e . , a s t r i c t i n d u c t i v e l i m i t of a n

i n c r e a s i n g sequence of F r g c h e t ( l o c a l l y convex) s p a c e s s a t i s f y i n g t h e a n a l o g o u s c o n d i t i o n s t o 1 ) and 3 ) o f t h e p r e v i o u s d e f i n i t i o n ) . T h u s ,

it i s a c o m p l e t e l o c a l l y convex s p a c e (Kothe's Theorem ) ; h e n c e ,

E

is

f i n a l l y a complete l o c a l l y m-convex a l g e b r a , which i s a l s o by d e f i n i t i o n (cond. 5 ) ) a

&-algebra.

2 ) I t may happen t h a t e a c h o n e o f t h e a l g e b r a s E n e N , i s t o n' p o l o g i z e d a s a l o c a l l y m-convex s u b a l g e b r a o f E , w h i l e t h e a l g e b r a c"

i s e q u i p p e d w i t h a l o c a l l y m-convex from t h e o u t - s e t ,

( o r y e t normed) t o p o l o g y g i v e n

d i f f e r e n t i n g e n e r a l from t h e ' l i n d u c t i v e l i m i t " o n e ;

t h u s , w e c a n t h e n a p p l y P r o p o s i t i o n IV;3.3

( o r y e t P r o p o s i t i o n IVi3.4)

a s it c o n c e r n s t h e p r o p e r t y of b e i n g t h e a l g e b r a E an SF-space.

(See,

f o r i n s t a n c e , t h e CFQ-algebra c o n s i d e r e d i n C h a p t . I V ; 4 . ( 3 ) ) .

3) F i n a l l y , w e n o t e t h a t cond. 5 ) c a n b e provided

(as t h i s i s the

c a s e i n I V i ( 4 . 3 ) ) b y a p p l y i n g Lemma I V ; 3 . 1 . T h u s , i n t h e c o n t e x t of Definition 9 . 1 ,

c o n d . 5 ) i s s a t i s f i e d i n c a s e one h a s an LFQ-algebra

E f o r which t h e i n d i v i d u a l a l g e b r a s a l g e b r a s ( s e e y e t C h a p t . I V ; 4 . ( 3 ))

E n , n e I N , a r e i n p a r t i c u l a r Banach

.

I n t h i s r e s p e c t , w e s t i l l n o t e t h a t a more g e n e r a l s i t u a t i o n , t h a n t h e o n e p r o v i d e d by t h e same Lemma I V ; 3 . 1 ,

c o u l d be

f o l l o w s : Take, f o r i n s t a n c e , a ( c o m m u t a t i v e ) a l g e b r a E

o b t a i n e d as

(with an ident-

i t y e l e m e n t ) , which i s a l s o a c o m p l e t e ( o r y e t a d v e r t i b l y c o m p l e t e ) l o c a l l y m-convex i n d u c t i v e l i m i t ( D e f i n i t i o n I V ; 3.1) o f a g i v e n fami l y (E,iaEI

o f commutative l o c a l l y m-convex & - a l g e b r a s w i t h i d e n t i t y

e l e m e n t s . Cf. C o r o l l a r y I I I ; 6 . 6 P r o p o s i t i o n V;3.1.MoreoverI

9 . ( 2 ) . Nuclear algebras.-

i n c o n j u n c t i o n w i t h Theorem Vi3.1 and

see Lemma 1 . 3 o f t h i s c h a p t e r . Given t h e s i g n i f i c a n c e o f t h e t h e o r y of

( l o c a l l y convex) n u c l e a r s p a c e s , t h e c l a s s o f t o p o l o g i c a l a l g e b r a s i n t i t l e i s , of c o u r s e , a n i m p o r t a n t one a p p e a r e d i n s e v e r a l c o n c r e t e c o n t e x t s , i n c l u d i n g t h e m a t h e m a t i c a l f o u n d a t i o n of Quantum Mechanics

(see e.g.

K. MAURIN [ I : p. 291 ff.]

).

The l a t t e r i s m a i n l y d u e t o t h e con-

n e c t i o n of t h e s e a l g e b r a s w i t h l o c a l l y convex s p a c e s ( i n f a c t , a l g e b r a s ) of

( S c h w a r t z ) d i s t r i b u t i o n s a n d / o r s p a c e s ( a l g e b r a s ) o f holomor-

phic functions. Now, t o f i x o u r t e r m i n o l o g y , w e f i r s t s e t t h e f o l l o w i n g .

Definition 9.2. A g i v e n l o c a l l y convex a l g e b r a E ( D e f i n i t i o n 1 ; l . l ) i s s a i d t o be a

nuclear algebra, i f t h e u n d e r l y i n g l o c a l l y convex ( v e c -

9

- ( 2) .

303

NUCLEAR ALGEBRAS

t o r ) s p a c e E i s n u c l e a r . ( F o r t h e d e f i n i t i o n of t h e l a t t e r c l a s s of s p a c e s c f . , f o r i n s t a n c e , A . PIETSCH [l: p. 6 9 1 a n d / o r K . MAURIN [ I : p. 67,

S

91 ) . I n t h i s c o n c e r n , w e s h a l l have t h e o p p o r t u n i t y t o c o n s i d e r l a t e r

on ( l o c a l l y c o n v e x ) n u c l e a r a l g e b r a s i n t h e framework of t o p o l o g i c a l t e n s o r p r o d u c t s , i n terms o f w h i c h , of c o u r s e , n u c l e a r s p a c e s were f i r s t i n t r o d u c e d by A . GROTIiENDIECK [3].

(For another equivalent version

o f t h e same n o t i o n , b a s e d however on t h e t h e o r y of " H i l b e r t - S c h m i d t o p e r a t o r s " see a l s o , a p a r t from t h e a b o v e c i t e d a u t h o r s , I . M . GEL'FAND

-A'. Ya. VILENKIN [I])

.

On t h e o t h e r h a n d , i n t h e s a m e manner t h a t one c o n s i d e r s a (comp l e t e ) l o c a l l y convex s p a c e a s a p r o j e c t i v e l i m i t of Banach s p a c e s ,

w e a l s o g e t , r e s p e c t i v e l y , t h a t every complete nuclear space

( t h e comple-

t i o n o f a n u c l e a r s p a c e i s a g a i n a n u c l e a r s p a c e ; c f . e . g . F. TREVES

[I :p.514, (50.2)]) i s ( w i t h i n a t o p o l o g i c a l v e c t o r s p a c e isomorphism) t h e p r o j e c t i v e l i m i t of H i l b e r t spaces. E q u i v a l e n t l y , t h e t o p o l o g y of t h e s p a c e c a n b e d e f i n e d by a f a m i l y o f semi-norms, e a c h one o f which i s d e r i v e d from a p o s i t i v e s e m i - d e f i n i t e h e r m i t i a n form ( s e e , f o r i n s t a n c e , H . H . SCHAEFER [I: p. 102, C o r o l l a r y 2 , and p. 103, C o r o l l a r y 31 ) Now,

.

s e v e r a l of t h e i m p o r t a n t e x a m p l e s o f n u c l e a r a l g e b r a s a r e ,

i n e f f e c t , l o c a l l y m-convex a l g e b r a s , so t h a t g i v e n a complete nuclear

locally m-convex algebra ( h e n c e , by t h e p r e c e d i n g , t h e c o m p l e t i o n of a nuc l e a r l o c a l l y m-convex a l g e b r a ) one a s k s w h e t h e r s u c h a n a l g e b r a c a n a l w a y s b e e x p r e s s e d as a p r o j e c t i v e E m i t of HiZbert algebras; by t h e

last

t e r m o n e means, of c o u r s e , a p r o j e c t i v e l i m i t of Banach a l g e b r a s , t h e r e s p e c t i v e norms of which are d e r i v e d from " i n n e r p r o d u c t s " . Y e t , view of G e l ' f a n d ' s

in

Theorem ( c f . Theorem I ; 5 . 1 ) , one c o n s i d e r s , i n a

s e e m i n g l y m o r e g e n e r a l way, a p r o j e c t i v e l i m i t of Banach s p a c e s w i t h separately continuous multiplication I ; 5 . 1 ) whose

( a l g e b r a s ; see a l s o D e f i n i t i o n

t o p o l o g i e s may b e d e f i n e d by H i l b e r t s p a c e norms. A s a

m a t t e r o f f a c t , w e i g n o r e w h e t h e r t h i s i s a l w a y s t h e c a s e ; however, it d o e s n o t seem t o b e t r u e , i n g e n e r a l . orem 6 . 1 )

( I n t h i s r e s p e c t , c f . a l s o The-

.

F u r t h e r m o r e , a n i m p o r t a n t p r o p e r t y which s h a r e t h e n u c l e a r a l g e b r a s ( t h e l a t t e r b e i n g , by d e f i n i t i o n , n u c l e a r s p a c e s ) , and which w i l l a l s o b e u s e f u l l a t e r on , i s t h e s o - c a l l e d

property

Banach-Grothendieck approximation

( f o r l o c a l l y convex s p a c e s ) : T h u s , a l o c a l l y convex s p a c e

E

i s s a i d t o s a t i s f y t h e ( Banach-Grothendieck ) approximation p r o p e r t y , whenever

by c o n s i d e r i n g t h e s p a c e L ( E l ( c o n t i n u o u s l i n e a r endomorphisms of E c endowed w i t h t h e t o p o l o g y c of p r e c o m p a c t c o n v e r g e n c e i n E ) , t h e i d e r i t -

304

VIII

SPECIAL TOPOLOGICAL ALGEBRAS

i t y o p e r a t o r l E e l : ( E l i s a p p r o x i m a t e d i n L ( E l by c o n t i n u o u s l i n e a r o p e r a t o r s of f i n i t e r a n k

( i - e . , by e l e m e n t s u e L ( E l , w i t h I m ( u 1 a f i -

n i t e - d i m e n s i o n a l s u b s p a c e of E ; c f . A . GROTHENDIECK [3:

Chap. I ; p. 1651).

I n t h i s r e s p e c t , c f . a l s o Chapt. X ; D e f i n i t i o n 2 . 4 i n t h e s e q u e l . T o p o l o g i c a l a l g e b r a s w i t h t h e above p r o p e r t y , i n p a r t i c u l a r ,

$ilov algebras w i l l a l s o b e c o n s i d e r e d i n t h e e n s u i n g d i s c u s s i o n ( s e e Chapt. X I ; S e c t i o n 4 ) . F i n a l l y , w e n o t e t h a t c o n c r e t e e x a m p l e s o f n u c l e a r ( l o c a l l y mconvex) a l g e b r a s have a l r e a d y been g i v e n i n Chapt. I V ; S u b s e c t i o n s 4.(2)

and 4 . ( 3 ) ; t h e s e a l g e b r a s were, i n p a r t i c u l a r , Fre'chet-Schwartz m-convex) algebras (see a l s o Chapt 9 . ( 3 ) . ( 1 k)-algebras.-

. X;

S e c t i o n 4)

(locally

.

The c r u c i a l f e a t u r e o f t h e t o p o l o g i c a l a l g e -

b r a s i n t i t l e i s t h e f a c t t h a t t h e y a r e " s t r u c t u r a l l y approximated" by a l g e b r a s o f t h e t y p e a p p e a r e d i n C o r o l l a r y 1 . 1 .

This property i s ,

i n d e e d , f u n d a m e n t a l f o r t h e s o r t of a p p l i c a t i o n s w e a r e c o n c e r n e d w i t h , and which a l s o w e r e a l l u d e d t o a t t h e b e g i n n i n g of t h e p r e v i o u s S e c t i o n 8 . F u r t h e r m o r e , a number of o t h e r u s e s i s r e l a t e d t o r e c e n t a p p l i c a t i o n s of A l g e b r a i c Topology t o t h e " s t r u c t u r e t h e o r y " of Banach a l g e b r a s , t h e s e b e i n g t h u s e x t e n d e d t o t h e more g e n e r a l c l a s s of t o p o l o g i c a l a l g e b r a s u n d e r d i s c u s s i o n ; see 0. FORSTER [5] [l]

, A . MALLIOS

[20]

,M . E .

NOVODVORSKII

.

W e f i x up t h e r e l e v a n t t e r m i n o l o g y . T h u s , w e f i r s t h a v e t h e f o l -

lowing.

Definition 9.3. An a l g e b r a E i s s a i d t o b e a s t r i c t l y ikl-algebra, i f it s a t i s f i e s t h e c o n d i t i o n s o f C o r o l l a r y 1 . 1 : Thus, E i s , by d e f i n i -

t i o n , a commutative c o m p l e t e f i n i t e l y g e n e r a t e d l o c a l l y m-convex a l g e b r a w i t h an i d e n t i t y e l e m e n t and compact s p e c t r u m h a v i n g , moreover, t h e r e s p e c t i v e G e l ' f a n d map c o n t i n u o u s .

( S e e a l s o t h e Note a f t e r Lem-

ma 7.1). NOW,

a commutative c o m p l e t e l o c a l l y m-convex a l g e b r a E w i t h t h e

r e s p e c t i v e G e l ' f a n d map c o n t i n u o u s , i s s a i d t o b e a n

Ilk)-algebra

if

t h e following t w o c o n d i t i o n s are s a t i s f i e d : 1 ) The n e x t r e l a t i o n i s t r u e E = limE

(9.1)

4

( E , i,,

a'

I i s a g i v e n i n d u c t i v e f a m i l y of s t r i c t l y ( k i - a l g e b r a s which a r e s u b a l g e b r a s of E ( c f . a l s o D e f i n i t i o n I V ; 1 . 1 )

where

.

2 ) The f o l l o w i n g r e l a t i o n i s v a l i d , c o n c e r n i n g t h e s p e c t r a o f

t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d i n t h e above c o n d . 1 ) ; i . e . , w e

9. (3). ( lk)-ALGEBRAS

305

have (9.2 w i t h n a homeomorphism of t h e t o p o l o g i c a l s p a c e s i n v o l v e d . Thus, a s a c o n s e q u e n c e of t h e p r e v i o u s d e f i n i t i o n , one c o n c l u d e s t h a t : An ilk)-algebra is a commutative complete locally m-convez algebra with an

identity element and compact spectrum having, moreover, t h e respective Gel'fand map continuous. T h e r e f o r e , it is, i n p a r t i c u l a r , a Q-aZgebra and hence an ArensCalderdn algebra (see a l s o t h e comment f o l l o w i n g Theorem 8 . 2 )

.

it i s c l e a r t h a t any commutative Banach a l g e b r a

In t h i s respect,

w i t h a n i d e n t i t y e l e m e n t i s an

(consider its f i n i t e l y

ilk)-algebra

g e n e r a t e d s u b a l g e b r a s , i n c o n n e c t i o n w i t h ( 9 . 1 ) and ( 9 . 2 ) ) . On t h e o t h e r h a n d , t h e s t r u c t u r a l e p r o p e r t i e s d e r i v e d from t h e p r e c e d i n g t w o r e l a t i o n s makes a n ( l k ) - a l g e b r a t h e s u i t a b l e s u b s t i t u t e of a Banach a l g e b r a f o r t h e a p p l i c a t i o n s a l l u d e d t o i n t h e f o r e g o i n g . I n t h i s res p e c t , a b a s i c argument f o r t h i s t y p e o f a p p l i c a t i o n s i s p r o v i d e d by t h e f o l l o w i n g c o n s i d e r a t i o n s ( c f . a l s o A. MALLIOS [20] ) : T h u s , s u p p o s e w e a r e g i v e n an ilk)-algebra E whose s p e c t r u m i s m(E), a n d l e t

( E a J C L E I be t h e respective

bras defining E (Definition 9.3).

f a m i l y of s t r i c t l y ( k i - a l g e Moreover, assume t h a t

@:mlE)-%

(9.3)

i s a weakly spectral map with X being a Skein manifold. Thus, w e s h a l l s a y t h a t t h e map $ is subordinated to an index whenever t h e r e s p e c t i v e s t r i c t l y ( k l - a l g e b r a by t h e e l e m e n t s

xl,...,

s

of E,with X

l?

a

i s n-generated,

IUE I

,

say,

( R e m m e r t ' s embedding; see

Cn

"R S e c t i o n 8 ) , s u c h t h a t one h a s , m o r e o v e r , t h e r e l a t i o n @ = @ otia

liR0l$)

(9.4)

Here $ =ti : m(E)-m(E,)

a

a

c l u s i o n ) map ia :Ea-E

=

l$ao$a

.

d e n o t e s t h e t r a n s p o s e of t h e c a n o n i c a l ( i n d e f i n e d by ( 9 . 1 )

,

and

( s e e ( 9 . 5 ) below) t h e

map g i v e n by t h e r e s p e c t i v e r e l a t i o n t o ( 7 . 3 ) ( f o r t h e s e t o f g e n e r a t o r s { s,,..., xn} of Ea). T h a t i s , w e assume t h e f o l l o w i n g commutative diagram

(9.5)

a

m (EJ G.

=ti

a

.

= ( z . o i E io$ ,'Zi

0

iR

Cn

VIII S P E C I A L T O P O L O G I C A L ALGEBRAS

306

with 1 6 i < n , while one has, by hypothesis, the relation

Ea = @ [ ( x l , .. . ,xn ,]

(9.6)

(cf. also V;(2.13)). The significance of the above lies, among other things, in the fact that a given "continuous l i f t i n g " of $J to an appropriate holomorphic vector bundle over an open neighborhood of Im(@) in X i s , i n f a c t , equiv a l e n t t o a s i m i l a r " s p e c t r a l Z i f t i n g " of $J ("Oka's principle"; in this respect, cf. A. MALLIOS [20: p. 466, Theorem 4.11 , as well as V . Y a . L I N [ I : p. 123, Theorem 21). In turn, this entails the aforementioned applications to "complementing matrices" (of topological algebra elements) "to invertible ones" (see also the relevant comment at the end of Section 8 ) . As an indication of the type of applications one obtains, we quote (without proof) the relevant result from A. MALLIOS [20: p. 470, Theorem 5.11 . For the proof and more details on the terminology applied we refer instead to that discussion. S o we have.

Theorem 9.1. L e t E be an Ilk)-algebra (Definition 9 - 3 ) w i t h spectrum a = ( a . . ) bc a given m x n matrix w i t h e n t r i e s i n E , i n such ZJ a way t h a t t h e r e e x i s t s an i n d e x acI such t h a t t h e r e s p e c t i v e s t r i c t l y (k)-algebra ??Z(E). Moreover, l e t

E

c1

i s m.n - g e n e r a t e d . Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1) The given matrix a i s complemented i n E. 2) The Gel'fand t r a n s f o m matrix

2

of

a i s complemented i n t h e algebra

e'(m(E)).I In particular, i f t h e spectiw" m(E) of the given algebra E in the above Theorem 9.1 has t h e homotopy type of a p o i n t ( " c o n t r a c t i b l e space" ) , then one proves that a matrix a, as above, is complemented in E if, and only if, it is of "maximum rank" (ibid.; p. 472, Corollary 5.1). The situation reminds, of course, the familiar one, that one has in case of the classical "function algebras" (see, for instance, 1. WAkEWSKI

tll

).

9 . ( 4 ) . m-infrabarrelled algebras.-

We consider next a wider class of topological algebras than that of m-barrelled ones (see Definition I; 1.4) in analogy with the situation one has in the theory of locally convex spaces. Now, we know already (Proposition Vil.1) that every m-barrelled algebra is, in particular, spectrally barrelled, so that the respective Gel'fand map is continuous (Corollary V1;l.l). The topological algebras under discussion maintain this property if they are complete

9. (4)

( i n f a c t , o-complete,

.

m-INFRABARRELLED

hence a

307

ALGEBRAS

f o r t i o r i

quasi-complete ; c f . Defi-

nition I i 4 . 2 ) . T o f i x t h e terminology a p p l i e d below, w e say t h a t a s u b s e t B of a topological algebra E is

m-set

m-borniuorous if B a b s o r b s e v e r y bounded

of E ( b r i e f l y m-bounded

s u b s e t of E ; i . e . , f o r e v e r y A C - E which

i s bounded and i d e m p o t e n t t h e r e e x i s t s X > O w i t h A r X B ) . Thus, w e s e t t h e following.

Definition 9.4. m-infrabarrelled

A given

l o c a l l y convex a l g e b r a E i s s a i d t o b e

i f e v e r y m-bornivorous m - b a r r e l

( c f . Definition I; 1 - 2 )

i n E i s a n e i g h b o r h o o d of z e r o . The f o l l o w i n g f a c t w i l l b e u s e f u l below. T h a t i s , w e h a v e Lemma 9.1.

Let E b e a g i v e n t o p o l o g i c a l algebra whose spectruni is V Z i Z l .

Then, by considering t h e strong t o p o l o g i c a l dual E' of E , a g i v e n s e t KC- mfE) i s b bounded i n EZ; ( i. e . , "strongly bounded" ) if, and o n l y i f , KC 3' where B i s a bornivorous ( a b s o r b s t h e bounded s e t s ) m-barrel i n E. Proof. If K G m ( E ) i s s t r o n g l y bounded, it i s a f o r t i o r i weakl y bounded h e n c e i t s p o l a r s e t K ' G E

a n a b s o r b i n g a b s o l u t e l y convex

and i d e m p o t e n t s u b s e t of E ( t h u s , a n m-set, m - b a r r e l of E .

s i n c e KEm(E)).So K i s a n

Moreover, it i s a b o r n i v o r o u s s u b s e t o f E ; t h a t i s , f o r

a n y bounded s e t B G E t h e r e e x i s t s , b y h y p o t h e s i s f o r K , X > O such t h a t KGXB',

h e n c e EGB"

C XK"

which i s t h e a s s e r t i o n . Thus, t h e r e l a t i o n

K G ( K o i o p r o v e s t h e n e c e s s i t y of t h e s t a t e m e n t . O n t h e o t h e r hand,

i f KCB'

,

t h e lemma, t h e n K i s a f o r t i o r i

w i t h B C E as i n t h e s t a t e m e n t Of contained

i n t h e p o l a r of a born-

i v o z o u s b a r r e l o f E ; so K i s a s t r o n g l y bounded s u b s e t of m(E) ( c f . J . HORVLTH

[7:

p . 2 1 0 , P r o p o s i t i o n 8]).So

t h e stated condition i s s u f f i

-

c i e n t t o o , and t h i s t e r m i n a t e s t h e p r o o f . I W e a r e now i n t h e p o s i t i o n t o s t a t e t h e a f o r e m e n t i o n e d p r o p e r t y

o f t h e t o p o l o g i c a l a l g e b r a s u n d e r c o n s i d e r a t i o n . Thus, w e h a v e .

Proposition 9.1. L e t

F be a a-complete

m-infrabarreZled ( l o c a l l y convex) a l -

gebra. Then, E i s a s p e c t r a l l y b a r m I Z e d algebra so t h a t , i n p a r t i c u l a r , E has t h e r e s p e c t i v e Gel' Sand map continuous.

Proof, I f K G n Z f E ) i s a weakly bounded s u b s e t , i t i s a l s o s t r o n g l y bounded by h y p o t h e s i s € o r B and C o r o l l a r y I ; 4 , 4 . 9.1),

o n e h a s KGB'

a f or t io r i

T h e r e f o r e (Lemma

where B i s a b o r n i v o r o u s rn-barrel

i n E and h e n c e

an m-bornivorous m - b a r r e l i n E ; t h u s by h y p o t h e s i s K i s

a n e i g h b o r h o o d o f z e r o i n E. So t h e r e l a t i o n K c q ' a l r e a d y

implies t h a t

308

KC

VIII S P E C I A L T O P O L O G I C A L ALGEBRAS

m(EI is an equicontinuous subset, so that E is a spectrally bar-

relled algebra (Theorem VI; l.l).The rest of the assertion is now a consequence of Corollary VI; 1.1, and the proof is complete. I 9.(5).

Gel 'fand-Mazur algebras.-

The topological algebras in title generalize the situation one encounters in Theorem 1I;T.l to the extent that any commutative locally convex algebra with continuous quasi-inversion (Definition 11;3.1), has the important property that the quotient algebra E / M , where M is any closed regular maximal ideal in E, is essentially (i.e., within a topological algebraic isomorphism) the field C of complex numbers. One identifies this property as the GeZ'fand-Mazur c o n d i t i o n for a given topological algebra. Thus, we are led to the following.

D e f i n i t i o n 9.5. A given topological algebra E is said to be a Gel'fand-Mazur aZgebra, if E satisfies the Gel'fand-Mazur condition, as

above. That is, for every (2-sided proper) closed maximal regular ideal M G E , one has the relation (9.7)

E/M

C

,

within a topological algebraic isomorphism. According to the same Theorem 11;7.1

,

every commutative l o c a l l y m-

convex algebra i s a Gel'fand-Mazur algebra. Yet , more generally , every topo-

logical algebra of the type considered by that theorem. On the other hand, every commutative complete locally bounded algebra (cf. Definition Ii6.1) with an identity element is a Gel'fand -Mazur algebra (cf. W. iELAZK0 [2: p . 1 1 , Theorem 3.8, and/or p. 12, Prop-

.

osition 4.11 ) NOW, concerning the previous definition it would be, of course, equivalent to say that every (2-sided) closed regular maximal i d e a l of E i s t h e kernel of a non-zero continuous complex homomorphism ( i.e., continuous character) of E. In this respect, cf. a l s o e.g. J. A . LINDBERG J r . 11: natural aZgebras] and/or I . M. GEL'FAND- D . A . KAZ:'iDAii e t aZ. [l]

.

9.(6). Infra-Ptdk algebras.- A locally convex (topological) algebra (Definition 1;l-1) is said to be an infra-Ptci? algebra, if the underlying locally convex vector space is infra-Ptbk (or else Br-compZete ; cf. H . H. SCHAEFER [1: p. 1627 1. On the other hand, an equivalent definition for an infra-Ptdk space E is that every i n j e c t i v e continuous l i n e a r map u : E+F

, where

F is

any locally convex (Hausdorff) space, which also is nearly open ( i .e.,

9. (6).

309

I N F R A - P T ~ ALGEBRAS

t h e c l o s u r e of t h e image u n d e r u o f a n y n e i g h b o r h o o d of O E E i s a n e i g h b o r h o o d o f 0 i n u ( E I ) i s u s t r i c t morphism (i.e. , r e l a t i v e Z y open: It maps c n y open s e t o f E o n t o an open s e t of

u(E) i n t h e r e l a t i v e t o p o l o g y

from P . E q u i v a l e n t l y , u n d e r t h e a b o v e h y p o t h e s i s f o r u , one h a s t h e r e l a t i o n E = I m ( u i , w i t h i n a n isomorphism of l o c a l l y convex s p a c e s ) . Cf. H.H. SCHAEFER [l: p. 163, Theorem 8 . 3 1 a n d / o r J . HORVA’TH [ l : p. 106, Theorem 1

a n d p . 310, Ex. 17.51. I n p a r t i c u l a r , i f u:E+F c a l l y convex s p a c e s , w i t h

is a

s u r j e c t i v e l i n e a r map between l o -

F being a barrezled space, t h e n u is nearly open

(6.HORVATH [I: p. 296, P r o p o s i t i o n I] )

. Consequently,

one g e t s t h e f o l l o w -

i n g p a r t i a l s t r e n g t h e n i n g of Theorem 3 . 1 .

Theorem 9.2. Let E be a f u l l infra-Ptbk s p e c t r a l l y b a r r e l l e d algebra. Then, one has t h e r e l a t i o n

with:? an isomorphism of topological algebras defined by t h e r e s p e c t i v e Gel’fand map of E . In p a r t i c u l a r , i f E i s a P t b k aZgebra ( c f . D e f i n i t i o n 3 . 3 ) , t h e n i t s spectrum ? Z ( E ) i s a k-space. Proof. The G e l ‘ f a n d map $?: E -

C ImiE)) i s by h y p o t h e s i s o n e - t o

-one s u r j e c t i v e a n d c o n t i n u o u s ( c f . D e f i n i t i o n 3.1 a n d C o r o l l a r y 2 . 1 ) . F u r t h e r m o r e , by t h e p r e v i o u s d i s c u s s i o n and Theorem 2 . 1

$ is

actual-

l y a n o p e n map, which p r o v e s ( 9 . 8 ) . Now, t h e rest of t h e a s s e r t i o n i s p a r t o f Theorem 3 . 1 . I

.-

Schol i u m 9.1 In connection with t h e foregoing w e s t i l l remark t h a t an infra-Ptbk algebra i s complete ( c f . H.H. SCHAEFER [l: p. 162, (8.1)]). T h i s p r o p e r t y i s t r u e , of c o u r s e , i n t h e s m a l l e r c l a s s of P t d k a l g e b r a s . However, a n i m p o r t a n t p r o p e r t y s h a r e d b y

t h e l a t t e r c l a s s o f a l g e b r a s i s t h e f a c t t h a t Pt6k

algebras r e s p e c t separated q u o t i e n t s . Thus , t h e q u o t i e n t o f a P t d k a l g e b r a modulo a c l o s e d ( 2 - s i d e d ) i d e a l i s a g a i n a P t d k a l g e b r a and h e n c e c o m p l e t e ( c f . e.g. H.H. SCHAEFER [l: p. 163, C o r o l l a r y 31

,

i n connection

w i t h C h a p t . 11; S e c t i o n 7 , a n d C h a p t . 1V;Lemma 2 . 3 w i t h t h e comment f o l l o w i n g i t ) . NOW,

t h i s property

w i l l b e e s p e c i a l l y u s e f u l i n t h e n e x t c h a p t e r (see e.9.

S e c t i o n 2 ) . On t h e o t h e r h a n d , t h e c o r r e s p o n d -

i n g s i t u a t i o n f o r t h e case o f i n f r a - P t 6 k a l g e b r a s

310

VIII

S P E C I A L TOPOLOGICAL ALGEBRAS

seems t o b e s t i l l u n s e t t l e d : c f . i n s t e a d G. KOTHE [2: p. 29, (211 f o r t h e case of i n f r a - P t d k s p a c e s . C f . a l s o t h e following discussion concerning "algebrai c a l l y i n f ra-Pt8k a l g e b r a s " . Thus, a f u r t h e r e x t e n s i o n o f t h e p r e c e d i n g Theorem 9 . 2 i s s t i l l o b t a i n e d by

considering Ptdk or infra-Pta3c algebras i n t h e "algebraic sense", a s

t h e l a t t e r h a v e r e c e n t l y b e e n a p p l i e d b y D. ROSA [ I ] . Namely, 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 i c a l ) a l g e b r a E w i t h continuous multiplication

b r a i c a l l y Ptdk

( c f . D e f i n i t i o n I ; 1 . 1 ) i s s a i d t o b e alge-

( r e s p . , infra-Ptdk ) , i f e v e r y c o n t i n u o u s ( r e s p .

,

one-to-

o n e ) n e a r l y o p e n a l g e b r a morphism of E i n t o a n y l o c a l l y convex a l g e ( w i t h c o n t i n u o u s m u l t i p l i c a t i o n ) i s a s t r i c t morphism.(The t e r m

bra F

B-complete, r e s p .

, Bp-complete,

algebra

h a s a l s o been a p p l i e d ; i b i d . )

.

I n t h i s r e s p e c t , every a l g e b r a i c a l l y Ptdk a l g e b r a i s , i n p a r t i c u l a r , a l g e b r a i c a l l y infra-Ptdk,

while t h e class of Ptak a l g e b r a s

(with continuous m u l t i p l i c a t i o n ) , i n t h e sense of D e f i n i t i o n 3 . 3 ,

is

g e n u i n l y c o n t a i n e d i n t h a t o f a l g e b r a i c a l l y P t d k o n e s . But algebraicall y Ptdk algebras need n o t , i n general, be complete ( i b i d . : p. 202, C o r o l l a r y

2.5,

a s w e l l as t h e comment f o l l o w i n g C o r o l l a r y 2 . 6 ) . T h u s , a l t h o u g h

t h e p r e c e d i n g c l a s s of a l g e b r a s " d o e s form q u o t i e n t s " ( i b i d . )

,

the

p r e v i o u s f a c t i s a d i s a d v a n t a g e when l o o k i n g a t a p p l i c a t i o n s of t h e t y p e c o n s i d e r e d , f o r example, i n S e c t i o n 2 of t h e n e x t c h a p t e r . On t h e o t h e r h a n d , a l g e b r a i c a l l y PtSk a l g e b r a s " a d m i t t i n g f u n c t i o n a l r e p r e s e n t a t i o n s " ( S e c t i o n 3 ) are c o m p l e t e : More p r e c i s e l y ,

algebra

ihe

C c ( X l , w i t h X c o m p l e t e l y r e g u l a r , i s a l g e b r a i c a l l y Ptdk i f , and only

.

i f , X i s a k-space (see D. ROSA [I : p. 204, Theorem 3.21 ) Thus , w e now h a v e t h e f o l l o w i n g " a l g e b r a i c " e x t e n s i o n of Theorem 9.2 : Theorem 9.3. L e t E be a f u l l a l g e b r a i c a l l y infra-Ptdk s p e c t r a l l y b a r r e l l e d algebra. Then, one g e t s t h e r e l a t i o n

w i t h i n an isomorphism of topological algebras d e f i n e d b y t h e r e s p e c t i v e GeI'fand map of E. I n p a r t i c u l a r , i f t h e given aZgebra E i s a l g e b r a i c a l l y Ptdk, t h e n i t s spec-

trum

(El i s a k-space.

Proof. By h y p o t h e s i s t h e G e l ' f a n d map o f E i s a c o n t i n u o u s oneto-one

a l g e b r a morphism o f E o n t o t h e b a r r e l l e d a l g e b r a

qWEl)(The-

orem 2.1 a n d C o r o l l a r y 2 . ? ) : h e n c e , by t h e comment b e f o r e Theorem 9 . 2 ,

5 is

a l s o n e a r l y open and h e n c e a n open map, by h y p o t h e s i s f o r

E,

lO.(l).

31 1

ALGEBRA OF CONTINUOUS POLYNOMIALS

which p r o v e s ( 9 . 9 ) . Now t h e r e s t f o l l o w s d i r e c t l y from t h e r e s u l t nent i o n e d j u s t b e f o r e t h e s t a t e m e n t of t h e theorem. I Thus a n a l g e b r a s a t i s f y i n g t h e c o n d i t i o n s of Theorem 9 . 3 i s , i n

e c t X ) " w i t h X c o m p l e t e l y r e g u l a r , t h a t i s , a commutative a l g e b r a i c a l l y infra-Ptdk barrelzed l o c a l l y in-convex aZyebra ( w i t h an f a c t , "of t h e type

i d e n t i t y e l e m e n t ) b e i n g a l s o complete i n c a s e E i s . I n p a r t i c u l a r , it

i s an a l g e b r a i c a l l y Ptdk a l g e b r a when E i s , i t s s p e c t r u m b e i n g t h e n a k-space

( c f . J . C . KELLEY [l: p. 231, Theorem 1 2 1 ) .

A more p r e c i s e c h a r a c t e r i z a t i o n ( e x c e p t , o f c o u r s e , of t h e c a s e "

ccIX)"

) of

complete a l g e b r a i c a l l y P t d k Iresp., infra-Pta'k) algebras which form

"complete q u o t i e n t s " s h o u l d c e r t a i n l y b e of i n t e r e s t (see e . g .

Chapt .IX;

Section 2 ) .

10. I n f i n i t e dimensional holomorphy. S p e c t r a o f p a r t i c u l a r topological algebras W e consider i n t h i s section certain particular topological a l -

gebras, i n f a c t ,

( c o m m u t a t i v e ) l o c a l l y m-convex o n e s ( w i t h i d e n t i t y

e l e m e n t s ) which are i n v o l v e d i n t h e t h e o r y of

(complex-valued) h o l o -

morphic f u n c t i o n s o n "domains" c o n t a i n e d i n i n f i n i t e d i m e n s i o n a l ( l o c a l l y convex) s p a c e s , i .e.

. Furthermore,

morphy"

,

i n t h e so-called

" i n f i n i t e dimensional holo-

we i d e n t i f y t h e i r spectra ( c f . C h a p t . V )

.

F i r s t w e comment b r i e f l y on t h e t e r m i n o l o g y a p p l i e d , b u t w e a l s o r e f e r t o t h e p e r t i n e n t l i t e r a t u r e f o r more d e t a i l s . T h u s , see e . g . NACHBIN

[71,

P. NOVERRAZ [I] and

J . -F. COLOMBEAU [l]

L.

f o r b a s i c p r o p e r t i e s of

h o l o m o r p h i c f u n c t i o n s and p o l y n o m i a l s on i n f i n i t e d i m e n s i o n a l s p a c e s . Thus, w e s t a r t w i t h t h e a l g e b r a

PIE)

of " c o n t i n u o u s p o l y n o m i a l s " on

a g i v e n l o c a l l y convex space E .

l O . ( l ) . The a l g e b r a o f c o n t i n u o u s p o l y n o m i a l s

T(E).-Suppose

g i v e n a ( c o m p l e x ) l o c a l l y convex s p a c e E . A complex-valued on E ,

say, P : E - + C

,

we a r e function

i s s a i d t o b e a continuous homogeneous polynomial of

order n , o r y e t a continuous n-homogeneous polynomial, n a continuous n-livzar map

E

N, if

there exists

(10.1) such t h a t P ( x ) = uxn = u i x , .

(10.2)

f o r e v e r y x E E (see e . g . L . W L ' H B I N [ 7 : p. 6 ,

s e t mz'

z . 4 .

. . , xci , 9 31). I n t h i s r e s p e c t , w e a l s o

312

VIII SPECIAL TOPOLOGICAL ALGEBRAS

W e d e n o t e by q ( n E )

t h e ( c o m p l e x ) v e c t o r s p a c e of a l l c o n t i n u o u s

n-homogeneous p o l y n o m i a l s on E . Thus d e n o t i n g by 6 n , n € N , t h e r e s p e c t i v e " d i a g o n a l map" 6n : E-En:x-6

(10.3)

(2)

..., x) ,

: = (x,

one h a s by ( 1 0 . 2 ) t h e r e l a t i o n P=uo6

(10.4) f o r every P E V ( n E ) , with u tinuous

E

n '

L ( n E ) ( v e c t o r s p a c e of complex-valued con-

n - l i n e a r maps ( f o r m s ) on E ; c f . ( l O . l ) ) . F u r t h e r m o r e , w e s e t

(10.5)

q(E)

=

@ P(nE)

nSO

continuous poZynomiaZs on E ,

t h e (complex) v e c t o r s p a c e ( d i r e c t sum) of where w e a l s o s e t (10.6)

q(OEl

N

c.

On t h e o t h e r h a n d , by c o n s i d e r i n g t h e p o i n t w i s e ( r i n g ) m u l t i p l i c a t i o n i n ( 1 0 . 5 ) t h e l a t t e r s e t i s made i n t o a (complex c o m m u t a t i v e ) a l g e b r a (with an i d e n t i t y element; c f .

inEl.

(10.7)

(mE) G

( 1 0 . 6 ) ) such t h a t one h a s

9 (n+mE)

I

f o r a n y n , m i n IN. W i t h i n t h e p r e v i o u s c o n t e x t w e d e n o t e by (10.8) t h e subalgebra of

V ( E ) g e n e r a t e d by E' ( t h e t o p o l o g i c a l d u a l of E ) =

T ( ' E ) $ ~ ( E ) ( c f . a l s o (10.4) f o r n = l ) .

On t h e o t h e r h a n d , i t i s c l e a r from ( 1 0 . 4 ) and ( 1 0 . 5 ) t h a t P(ElG Cc(El

(10.9)

( a s complex a l g e b r a s ) , w h i l e o u r f i r s t g o a l i s t o p r o v e t h a t ( 1 0 . 8 )

i s a d e n s e s u b a l g e b r a of ( 1 0 . 5 ) w i t h r e s p e c t t o t h e ( l o c a l l y m-convex) ( c f . C h a p t . 1 ; E x a m p l e 3 . 1 ) , when t h e l o c a l l y convex s p a c e E p o s s e s s e s t h e " a p p r o x i m a t i o n p r o p e r t y " . Thus, w e f i r s t s e t t h e

algebra e J E l following.

Definition 10.1. Suppose t h a t E i s a l o c a l l y convex s p a c e and l e t L ( E l b e t h e s p a c e of c o n t i n u o u s l i n e a r endomorphisms of E endowed w i t h

t h e t o p o l o g y o f compact c o n v e r g e n c e i n E . Moreover, l e t s u b s p a c e o f L ( E l c o n s i s t i n g of

.C

f (El

be t h e

( c o n t i n u o u s ) l i n e a r endomorphisms of E

o f f i n i t e r a n k . Then, w e s a y t h a t E h a s t h e approximation property one h a s (10.10)

if

31 3

ALGEBRA OF CONTINUOUS POLYNOMIALS

lO.(l),

T h a t i s , w e assume t h a t f o r e v e r y compact s u b s e t K of E and e v e r y neighborhood V o f 0 e E , t h e r e e x i s t s an e l e m e n t u e f ( E l

f ( E ) such

f

t h a t ~ ( x I - x E V ,f o r every X E K .

I t i s known t h a t every nuclear ( l o c a l l y convex t o p o l o g i c a l v e c t o r )

space has the approximation property ( s e e e . g . F . TREVES [ I : p. 520, P r o p o s i t i o n 5 0 . 3 ) ) . On t h e o t h e r hand, i t i s a r a t h e r r e c e n t r e s u l t ( P . E n f l o , 1973) t h a t not every Banach space s a t i s f i e s ( 1 0 . 1 0 ) . Moreover, it i s clear

i s j u s t the "Banach-Grothendieck apspace and J . HORVA?H [i:p. 146, scholium f o l l o w -

t h a t t h e above c o n d i t i o n ( 1 0 . 1 0 )

p r o x i m a t i o n p r o p e r t y " ( c f . 9 . ( 2 ) ) f o r E a t l e a s t a quasi-compZete ( s e e Chapt. I ; D e f i n i t i o n 4 . 2

,

i n g P r o p o s i t i o n 71 ) , Thus, w e come n e x t t o t h e f o l l o w i n g a u x i l i a r y r e s u l t i n s t a n c e , J . MUJICA;[4:p.569,Lemma

Lemma 10.1. Let

(see, f o r

3 . 4 1 a n d / o r [5: p.171,Lemma 2 7 . 2 1 ) .

E be a l o c a l l y convex space w i t h t h e approximation property

(cf. Definition 10.1)

. Moreover,

l e t U be an

F i n a l l y , l e t K beacompact subset of U and

open subset of E and f

8

f

t h a t u I K I G U and

1 f(xI - f ( u ( x ) l I <

(10.11)

f o r every x

f

E,

K.

Proof. S i n c e f any E > O

C(l1).

Then, there e x i s t s u e f (El such

E>O.

IK

i s u n i f o r m l y c o n t i n u o u s , it f o l l o w s t h a t f o r

t h e r e e x i s t s a neighborhood V of O E E such t h a t K + V = U

I f ( x l - f ( y ) I< E

,for

and

any x e K and y e x + V . Hence, by h y p o t h e s i s f o r

E

,

t h e r e e x i s t s u E I: ( E l , w i t h u f x l - x e V , f o r e v e r y x e K ; t h a t i s , u(K) C

f

K + V G U . Moreover, one h a s

If(xl - f ( u f x ) ) I

<E,

f o r e v e r y X E K , from what h a s

been proved b e f o r e . I A s a consequence of t h e p r e v i o u s lemma, one g e t s t h e f o l l o w i n g .

Corollary 10.1. Let E be a localZy convex space w i t h t h e approximation prope r t y . Then, one has the r e l a t i o n (10.12)

f

fEI = q f E l E C c ( E )

(cf. (10.8)).

Cc(E) ,KCE,

Proof. L e t P e ? ( E ) S 10.1)

,

compact, and E > O .

t h e r e e x i s t s u a f f E ) such t h a t I P f x ) - P f u i x l l

f

Thus, it s u f f i c e s t o prove t h a t P o u a

9f ( E ) .

I <E,

Then (Lemma f o r every x e K .

Indeed, s i n c e

14 e

f (El, it

f

f o l l o w s t h a t PI In!f u l E q ( u ( E ) ) . T h e r e f o r e , t h e r e e x i s t s a f i n i t e sequence

fgi)16isn

i n (u(EII'

such t h a t

314

VIII SPECIAL TOPOLOGICAL ALGEBRAS

P(yl=

(10.13)

1 (@i(yl)"i

I

i=I f o r e v e r y y e u f E ) . Thus ,

E

1

P(u(x)l = I @ i ( u ( x l )Imi= l ( @ o u l ( x l l m i, i =I i=l

(10.14)

€or e v e r y

X E

E , w h i l e @i o u E E ' ,

l S i 6 n

, which

b y ( 1 0 . 1 4 ) p r o v e s t h e as-

sertion. 1 NOW,

i f t h e g i v e n l o c a l l y convex s p a c e E i s qAasi-compZete, t h e n

the topoZogy of compact ( o r , e q u i v a l e n t l y , precompact) convergence i n E cons i d e r e d as a l o c a l l y convex t o p o l o q y on ( t h e t o p o l o g i c a l d u a l s p a c e ) E'

i s coarser than the Mackey topology on i t . T h e r e f o r e (Mackey -&ens

The -

orern), one h a s t h e r e l a t i o n (10.15)

(EL

(see G. K b h E [I:

I' = E

p. 264, ( I ) ] ) .

Thus, w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g t h e o r e m

(see a l s o J . ISIDRO [I: p. 409, P r o p o s i t i o n 31 ) Theorem 10.1.

.

Let E be a quasi-compZete locaZZy convex space possessing the

Banach-Grothendieck approximation property. Moreover, l e t LS)CE) be t h e (ZocaZZy mconvex) aZgebra of continuous polynomials on E endowed w i t h the topoZogy of compact convergence i n E ( c f . (10.9)). Then, i t s spectrum i s givsn by the r e l a t i o n

m($(EII= E ,

(10.16)

within a homeomorphism.

Proof. L e t h E ? ? Z ( V ( E ) ) ; t h e n i t s r e s t r i c t i o n t o E ' = q ( ' E l 5 Q ( E ) i s a c o n t i n u o u s l i n e a r form on E' w i t h r e s p e c t t o t h e t o p o l o g y r c ( E ) ( t o p o l o g y of compact c o n v e r g e n c e i n E ; c f . t h e comment b e f o r e (10.15)). T h e r e f o r e ( c f . ( 1 0 . 1 5 ) ) , t h e r e e x i s t s a u n i q u e e l e m e n t x € E such t h a t h

h i @ )=

(10.17)

@fXhI

I

f o r e v e r y @ E E ' . O n t h e o t h e r h a n d , by h y p o t h e s i s f o r h , one h a s f o r every

P E

P/E) k h(Pl = h f .I

%=I

= ( b y (10.17))

Thus, t h e r e s t r i c t i o n of P t o

f( E )

@T?i)

=

5z $i(xh

z

Jni

h($i I n {

= P(zh

I

.

i s given by evaluation a t x h e E. T h e r e f o r e ,

by C o r o l l a r y 10.1 a n d t h e f a c t t h a t t h e t o p o l o g y c i n

e l E ) is

finer

t h a n s , t h e same h o l d s t r u e f o r a n y P e q ( E ) . Now, t h e rest of t h e ass e r t i o n f o l l l o w s from ( 1 0 . 9 ) and Theorem V I I ; 1 . 2 . 1

10. ( 2 )

.

315

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

1 0 . ( 2 ) . Topological algebras o f holomorphic f u n c t i o n s on i n f i n i t e diniensiona1 spaces.-

W e c o n s i d e r i n t h i s and t h e n e x t s u b s e c t i o n l o c a l l y m-con-

v e x ( t o p o l o g i c a l ) a l g e b r a s of complex-valued h o l o m o r p h i c f u n c t i o n s o r

of "germs" of s u c h on " i n f i n i t e d i m e n s i o n a l d o m a i n s " , and t h e n i d e n t i f y t h e i r s p e c t r a . The e n s u i n g d i s c u s s i o n i s m a i n l y b a s e d on t h e work of J . WJICA ( c f . t h e Bibliography). W e s t a r t w i t h t h e n e c e s s a r y d e f i n i t i o n s and background m a t e r i a l . So w e h a v e .

D e f i n i t i o n 10.2. L e t E b e a g i v e n ( c o m p l e x ) l o c a l l y convex s p a c e

a n d U E E , o p e n . A hoZomorphic f u n c t i o n on U i s a complex-valued map f : U-C

s u c h t h a t , f o r e v e r y z e U , 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 of

z

and an element (10.18) s u c h t h a t one h a s (10.19) u n i f o r m l y on V . The s e q u e n c e ( P n ) i n ( 1 0 . 1 8 )

i s u n i q u e l y d e f i n e d and g i v e n by

(10.20)

( c f . P. NOVERRAZ [ l : p. 2 5 , Remarque a f t e r ThEorSme 1 . 2 . 8 ] ) . N o w , a f u n c t i o n f : U+C

i s s a i d t o b e G - holomorphic ("GI' f o r GC-

t e a m ) if i t s r e s t r i c t i o n t o e a c h complex l i n e i s h o l o m o r p h i c ; i . e . , f o r any (I,y l e U x ( E - { O ) ) t h e map A -

f(x+hy)

is holomorphic f o r e v e r y

X E C , w i t h x+XyeU.

i s holomorphic ( D e f i n i t i o n 10.2) if, and only if, it is G-holomorphic and continuous. ( S e e P. NOVERRAZ [l: p. 25, Th6orSme T h u s , a function f : U -

C

1.2.81). W e d e n o t e by H ( U i

t h e (complex c o m m u t a t i v e ) a l g e b r a ( w i t h a n

i d e n t i t y e l e m e n t ) o f a l l h o l o m o r p h i c f u n c t i o n s onU.We f i r s t s e t o n t h e

l a t t e r a l g e b r a t h e s o - c a l l e d "compact p o r t e d t o p o l o g y " ; it w a s f i r s t c o n s i d e r e d by L . NACHBIN [7] ( i n case of Banach s p a c e s ) . Thus, w e h a v e : D e f i n i t i o n 10.3. A semi-norm p on

f f l U l i s s a i d t o b e ported by a com-

pact subset K of U , i f f o r a n y open s e t V w i t h K S V S U , t h e r e e x i s t s c ( V ) >D

such t h a t

(10.21)

316

VIII SPECIAL TOPOLOGICAL ALGEBRAS

for every fEH(Ul.We denote by T~ the locally convex (vector space) topology on H t U l defined by the totality of such semi-norms and call it the compact ported topology on

HtUl.

As a matter of fact, we will show that H t U l

is a locally rnconvex algebra, for suitable spaces E , and identify its spectrum (Theorem 10.5). However, we still need some additional terminology. Thus, [T,]

we further set the following. D e f i n i t i o n 10.4. Let K be a compact subset of a given locally con-

vex space E . Then we set (10.22) as an algebra, with respect to the inductive system (10.23) with U varying over the open neighborhoods of K (cf. also Chapt. IV; Subsection 4. (3): (4.31)). We call an element of (10.22) a germ of a

.

holomorphic f u n c t i o n on K (or yet hotornorphic germ on K ) Furthermore, we consider on H(K) the i n d u c t i v e l i m i t l o c a l l y convex topotogy defined on it by the canonical maps

with U varying as above. In this respect, we note that the concept of a hozomorphic f u n c t i o n i s of a "Zocal character" (see, for instance, L . hrlCHBIN [7: p. 17, Remark 21 ) ;

hence (10.22) is well-defined. Furthermore, the final (locally convex) topology on H t K I defined by the maps (10.24) does not change if we assume that each connected component of U E ' K meets K. Thus, due to the uniqueness of holomorphic continuation, the canonical maps (10.24)are, in effect, one-to-one so that one has (10.25) (See also J . MUJKA [2: p. 71). Now if U S E is open, we denote by H " ( U I

the algebra of (complex-

valued) bounded holornorphic functions on U , endowed with the "sup-norm topology". In this respect, it is to be noticed that f E H t U l if, and only if, it is G - hoZomorphic and ZocaZly bounded on U (cf.P. NOVERRAZ [I: p. 25, The( U l i s a Banach algebra (see J . orem 1.2.81 ) Thus, it is proved that

.

MUJICA [2: p. 7, Proposition 2.21 1 .

In particular, for every compact K C -

10. ( 2 )

.

ALGEBRAS OF HOLOMGRPHIC FUNCTIONS

3'11

U, one concludes that ( 10.26)

As a matter of fact, the preceding relation is, indeed, an isonyorphism of l o c a l l y convex spaces

(see J . MUJIKA [2: p. 8, Proposition 2.31) : Thus, it is clear that the (canonical) inclusion maps H " i U l 5 H ( U ) [ T ~ ] are continuous (cf. ( 1 0 . 2 0 ) ) ,

so that one gets the continuity of the

identity map (10.27)

(cf. also Chapt. IV; Section 2 ) . On the other hand, if p is any continuous semi-norm on lim H " ( U ) , one proves that p is yet a continuous U 2 K

semi-norm on H(U), ported by K . Finally, H l K l topologized by ( 1 0 . 2 6 ) i s Hausdorff (ibid.;p. 9, Proposition 2.5). NOW, the following lemmas will presently be applied below, complementing the information supplied by our discussion in Chapt. IV; Section 3. Thus, we first have. Lemma 10.2. L e t ICEa,

11

*

f

Ba

3

be an i n d u c t i v e system of normed algebras

w i t h r e s p e c t t o a l i n e a r l y ordered i n d e x s e t I such t h a t , f o r any a
Ba

traction; i.e.,

one has

/I SBa(Z) I I B

(10.28)

f o r e v e r y x I?E

in I , t h e

E B i s a con-

:E,+

5

I/ x 1,

,(cf. Chapt. IV; Definition

2.1).

Then, t h e corresponding i n -

d u c t i v e l i m i t aZgebra (10.29)

E a U S ( E l = l i m E

, a a

-a+ a

(cf. IV; (1.9)) endowed w i t h t h e r e s p e c t i v e f i n a l ZocaZly convex ( v e c t o r space) t o pology i s a localZy m-convex algebra. (That i s , t h e former topology c o i n c i d e s w i t h t h e f i n a l l o c a l l y m-convex topology on E d e f i n e d on it by t h e normed a l g b r a s E a' ~ E I ibi.d., ; Section 3 ) . Proof. Let ( E ~ be) a ~family ~ ~ of positive reals with any a € I . Moreover, let (10.30)

U,

2

S o ( E a ) = { x E E : l/ZII <

be the open ball in E, centered at O s E , sider the absolutely convex hull (10.31)

IE 1

S l , for

EC1>

with radius

E

. Finally, con-

318

VIII

in E (cf. IV; ( 2 . 3 ) )

,

S P E C I A L TOPOLOGICAL ALGEBRAS

describing a local basis of E for the final topo-

logy defined on E by the normed spaces Ea,aeI,when the family ica) varies as above (ibid.) NOW, we shall prove that U i s an idempotent s e t in E ; i.e., U 2 I U.U G U : So let x = z X . x and y = 7 i.l.y (finite sums) be elements of U 2 2 cr. i J 3 Bj with X i , p

j

in C such that

3

2

2 1 . Thus, one obtains

3

r - y = ,'l.A.y.x .y 1 , 2 ~ 3 ai @j

(10.32)

with h i p j

2 \Xi( , S l p . 1

in C and

2 . I X . i . l . 1 2 1 .Therefore, it suffices to prove that 3

i,j

(10.33)

E

Xa..Ya

z

u,

j

for any i, j as taken in ( 1 0 . 3 2 ) . Indeed, by hypothesis for I, there exists an index

I, with a i , @ . < a , for any i ,j , as above. Thus, we 3

have

so that x a i

. yGi

,

e UE

which proves the assertion. 1

Bj

Furthermore, we shall also need the following. Lemma 1 0 . 3 . Let E be an algebra and ( E a ) a E I

an (upwards) directed family

of l o c a l l y eonvex &-algebras, ssLbalgebras o f E whose union is E. Then, the a lgebra (10.34)

E= U a Ea zi limE + a

endoded c L t 7 : the f i n a l l o c a l l y convex ( v e c t o r space) topology (cf. Lemma Iv; 2 . 2 )

is a l o c a l l y convex &-algebra t o o .

Proof. By hypothesis there exists an absolutely convex neighborhood of O E E , , say, Va consisting of quasi-regular elements of Ea, for every c1 E I (cf. Lemma I; 6.4 and J . HORVAh [l:p. 8 7 , Proposition 4 1 ) . So due to the hypothesis for the family (Ea), the set (10.35)

v = uva a

is an absolutely convex neighborhood of zero in E (cf. I V ; ( 2 . 3 ) ) consisting of quasi-regular elements of E. This proves the asserti0n.I A S a consequence of the preceding, one now gets the following corollary. See also J . MUJICA; [5: p. 163, Proposicidn 25.2 , p.166, Pro-

10. ( 2 )

.

posicidn 25.51 and/or

31 9

ALGEBRAS OF HOLOMORPHIC FUNCTIONS

[4: p. 567, Proposition 2.61.

Thus, we have.

/I l l a ) ,

a e l , be an (upwards) d i r e c t e d famiZy of Zocally convex &-algebras, w i t h r e s p e c t t o a l i n e a r l y ordered i n d e x s e t I , which Corollary 1 0 . 2 . Let f E a ,

a r e subaZgebras of a g i v e n algebra E and aZso exhaust E. Moreover, suppose t h a t , f o r any a
(see also Chapt. IV; Section 1 .l)endowed w i t h t h e f i n a l Zocally convex f u e c t o r space) topology d e f i n e d on i t by t h e n o m e d algebras E convex

a' a e l , i s a l o c a l l y m-

Q-a?gehra. I

Now, suppose that E is a metrizable locally convex space and let K Z E be compact.So we may consider a decreasing fundamental sequence of neighborhoods of K, say, (Un in e~ , so that one has

Thus, the next result is now a direct consequence of the previous Corollary 10.2.

Theorem 10.2. L e t E be a m e t r i z a b l e ZocaZly convex space and X a compact s u b s e t of E . Then, H(K)

(cf. (10.22)) i s a (commutative Hausdorff) l o c a l l y m-convex

&-algebra ( w i t h an i d e n t i t y e l e m e n t ) . I

Next we identify the spectrum of the algebra HfK), as above, for suitable locally convex spaces E and compact K E E . Thus, we first have the following.

Definition 10.5. Let E be a locally convex space and K a compact subset of E. Then, the polynorniaZly convex huZ1 of K is given by

K^= C x e E :

/ p ( ~ - i I ~ I ( p ( l K V P e ~ ( E , , }

where I \ P I I K = sup I P f x ) 1 . (It is clear that KE? 1 . The xeK is said to be poZynomiaZly convex if one has K = i .

(compact) set K

In this respect, one has the following "Runge t y p e " result.

Theorem 10.3. L e t E be LI rwtrizable ZocaZly convex space having t h e approximation property (cf. Definition 10.1) and K a compact polynomially convex subset of E. Then, every f E H I K I can be approximated

nomials on E.

uniformly on K b y continuous poZy-

320

VIII

Proof.

S P E C I A L TOPOLOGICAL ALGEBRAS

C f . M. SCHOTTENLOHER [2: p. 386, C o r o l l a r y 2 1 .

I

T h u s , w e come n e x t t o t h e f o l l o w i n g r e s u l t c o n c e r n i n g t h e s p e c [4: p. 571,Theorem 4-31).

( S e e a l s o J.MUJICA

trum of t h e a l g e b r a H(K).

Theorem 10.4. L e t E be a Fre'chet l o c a l l y convex space w i t h t h e (Banach-Grothendieck) approximation property ( c f . D e f i n i t i o n 10.1 and t h e comment a f t e r i t ) and K a compact po%ynomia%lyconvex subset of E ( D e f i n i t i o n 10.5). Then, t h e spectrum of ( t h e l o c a l l y m-convex &-algebra; c f . Theorem 10.2) H ( K ) i s given by

77Z(H(K)I = K ,

(10.39)

w i t h i n a (continuous) b i j e c t i o n (given by t h e "evaluation map"). Proof.

Since H(K)

i s a ( l o c a l l y m-convex)

Q - a l g e b r a , one h a s

( c f . C o r o l l a r y 11; 7.3)

M ( H ( K ) ) = ???(HiK)).

(10.40)

T h u s , l e t h b e a ( c o n t i n u o u s ) c h a r a c t e r o f t h e a l g e b r a H ( K l = lim H " ( U l USK E 7YC(H"(U)l , f o r e v e r y open ( c f . ( 1 0 . 2 6 ) ) , so t h a t o n e g e t s h

I Hrn(Ul

n e i g h b o r h o o d U of K ; i . e . , w e

have

IMf) I 5

(10.41)

with c ( U ) > O f f o r every f E H"(U)

C(U) (

11.

norm" i n t h e Banach a l g e b r a H " ( U ) ) . e x t r a c t i n g t h e n-th

I

d e n o t e s , of c o u r s e , t h e "supThus , a p p l y i n g ( 1 0 . 4 1 ) t o f E H"(U),

r o o t , and l e t t i n g n-+m

lh(f)/ 5

(10.42) f o r every f f H " ( U / .

/ I fIIu

IIfllu

,

one h a s

f

T h e r e f o r e , one g e t s

Ih(f)l

(10.43)

IIf IIK

I

f o r every f E H ( K ) . ( W e note t h a t t h e last r e l a t i o n is t r u e f o r every compact K E E ) . Thus, t h e r e s t r i c t i o n o f h t o

. (10.15) ) , h

t i n u o u s i n E 6 , t h e r e f o r e (Mackey-Arens; c f by a u n i q u e p o i n t a e E ; i . e . ,

IK

C

-

H l K ) i s con-

"is realized"

one h a s

h ( @ l= .@(a),

(10.44) f o r e v e r y QEE'.

Now,

g e n e r a t e d by E ' , s o

( E l i s , by d e f i n i t i o n , t h e s u b a l g e b r a o f H ( K I

f

t h a t one g e t s t h e r e l a t i o n h(PI = P(al

(10.45)

,

f o r e v e r y P E y ( E l . Hence ( C o r o l l a r y 1 0 . 1 )

f

(10.46)

E.5 V(E)

h(Pl = P ( a )

,

one f i n a l l y has

,

f o r e v e r y P € p ( E J . T h a t i s , w e have ( c f .

(10.43))

10.(3).

(10.47)

THE ALGEBRA

( H a ) ( = Ih(PII 5 IIPllK

I

for every P E Q ~ E ) , so that (Definition 10.5) a cludes from Theorem 1 0 . 3 that

€if?=

K . Finally, one con-

,

hlf) = f(al

( 1 0 -48)

32 1

ff(U)[Tw]

for every f E H ( K I , which in fact proves the assertion. (Concerningthe last statement in the theorem, see also the proof of Theorem IV; 1.2).1 10.(3). a l g e b r a H(U)[T

Holomorphic f u n c t i o n s on i n f i n i t e dimensional spaces ( c o n t n ' d . ) .

The

1.w

We continue in this subsection our previous study on topological algebras occurring in "infinite dimensional holomorphy". Thus, suppose as above that U is an open subset of a given (Hausdorff) locally convex space E (over the complexes), and also let H ~ U I [ T ~be ] the algebra of complex-valued holomorphic functions on U endowed with the "compact ported topology" T~ (cf. Definition 1 0 . 3 ) . Now, for every compact K G U , we denote by (10.49)

j, : HtU) + H l K )

the canonical injection (cf. ( 1 0 . 2 5 ) ) (10.50)

;

moreover, let

H K (U) = Im(jKI C H ( K l

.

On the other hand, for every open V S E , with K C V C U , let (10.51)

H F ( U l = H K ( U I n Hm(Vl

,

while we also set ( 10.52)

Finally, we define (10.53)

as well as (10.54)

Furthermore, we endow ( 1 0 . 5 1 ) and ( 1 0 . 5 2 ) with the relative "normed (algebra) topologies" from Hm ( V ) , while one further considers the respective final locally convex (vector space) topologies on the algebras ( 1 0 . 5 3 ) and ( 1 0 . 5 4 ) . S o we first have the following. Lemma 10.4. Let E be a metrizable l o c a l l y convex space, K C E compact, and I.: G

E opsn , w i t h K E U . Then, one has t h e r e l a t i o n s

(10.55)

VIII SPECIAL TOPOLOGICAL ALGEBRAS

322

within isomorphisms of l o c a l l y convex algebras. Proof.

Cf. J . MUJICA [5: p. 134, Lemma 19.41. I

In this respect, one still obtains the relation 1Lm H f K l , KSU within an algebra isomorphism (ibid.; p. 133, Lemma 19.3 ) . NOW, on the basis of Corollary 10.2 one concludes from (10.46) K that H (U) is, i n f a c t , a l o c a l l y m-convex algebra. S o it is a consequence of Lemma 10.4 (see also Chapt. 1II;Section 2) that

H(u) =

(10.56)

3

H ~ U [)T w is a Locally m-convex algebra, for every open subset U of a metrizable locally convex space Z .

(10.57)

Thus we come next to our final target, namely, the identification of the spectrum of the latter algebra. Indeed, this is done for appropriate open sets U C E , as before. However, we first set the following. D e f i n i t i o n 10.6. Let E be a locally convex space and U an open sub-

set of E . Then, we say that U is polynomially convex if for every com(Definition 10.5) is containpact K CU its polynomially convex hull ed in U. The nition of ROSSI [l: p. stands, a

preceding definition reminds, of course, the classical defia polynomially convex domain of C n (see e.g. R . C. G U N N I N G - H . 38, Definition 31 1; however, the above concept is , as it consequence of a more general notion, in case E is a Fr6-

chet locally convex space with the approximation property. Namely, one defines an open set U E E as polynomially convex, if for every compact K CU the set is compact (or yet precompact in U). Cf. J.MUJICA; [4: p. 568, Definition 3.11 and also [5: p. 174, and p. 177, Corolario 27-91. The form of this notion that we adopted by the previous definition is

just what one really applies in the next result. The proof of the latter is actually similar to that of Theorem 10.4, so is omitted (see also J . MUJICA [5: p. 184, Teorema 29.21). Theorem 10.5. Let E be a Fre'chet l o c a l l y convex space w i t h the (Banach-Gro-

thendieck) approximation property and U a polynomially convex open subset of E (cf.

Definition 10.6). Then, the spectrum of ( t h e l o c a l l y m-convex algebra; see is given by the r e l a t i o n (10.57)I H(U)

IT,]

(10.58)

(H(Ui [T~]) = U ,

z i t h i n a (continuous1 b i j e c t i o n (defined by t h e "evaluation map"). I

THE ALGEBRA

10.(3).

323

H(U)[To]

As a consequence of the previous theorem one gets, for instance,

the following analogue of a classical result in "finite-dimensional holomorphy" (cf. H . CARTAN [ 21). That is, we have.

Corollary 10.3. L e t UGE be as i n t h e previous Theorem 20.5, and F a family o f f u n c t i o n s i n H t U ) without common zeros i n U. Moreover, l e t J be t h e i d e a l o f HtU) generated by F. Then, one has

7

(1 0.59)

.

= H(UI[T~]

Proof. Suppose that 7 is a proper ideal in H(Ul [T,] . Hence, there exists a maximal closed ideal M in the same algebra with j G M (cf. Lemma 10.5 below). Therefore, one has M = ker(hi , where h E * M I H I U I [ ? , ] ) (see Chapt. 11; Corollary 7.2).That is (Theorem 10.5), there exists a unique point a e U such that h ( f I = f ( a ) = O f for every f E F, which is a contra-

diction. I In the proof of the previous corollary the following result has been applied, supplementing our considerations in Chapt. 11; Section 7 (see also A . GUICHARDET [ 2 : p. 163, Proposition 1.41). Lemma 10.5.

L e t E be a c o m u t a t i v e l o c a l l y m-convex algebra and I a closed

regular i d e a l of E . Then, t h e r e e x i s t s a c l o s e d regular maximal i d e a l M of E containing I . Proof. By hypothesis there exists an element x e E with x 6 I = 7. Therefore, there is a neighborhood V of O E E with (x+VI nI = @. On the

other hand, consider an Arens-Michael decomposition of E , say, E

5

1Lm fa (cf. Chapt. 111; (3.20)); furthermore, the same decomposition provides a "strictly dense projective system" (cf. Chapt.V; (7.18)), so that each one of the canonical maps pol : E-+ icl(cf.Chapt. IIIi(3.25)) has a dense image. That is, one obtains palE) = EU ,U E I . Therefore, the ideal p U ( I 1 of the (normed) algebra p c l I E I E f yields a closed ideal of the Banach algebra = k , in such a way that one has withc1 in the algebra p a ( E l -

A

pcl(ll

p

p

n PJX+V)

= PJI) + ' x a +p , ( W

= p,II

n(x +Vi)= @

;

thus, ci is a proper closed regular ideal of the Banach algebra ,?a , and, therefore, is contained in a closed regular maximal ideal, say,

M of the Banach algebra g a . But then Ma= kerlhcl) , with h a € U e.g. Chapt. 1I;Corollary 7.2), so that one gets I E ker(h o

(10.60)

with hU o pa

p

c 1 U

€

m(E)

M(EI (ibid.)

, which

IIM

m(ga)(cf.

I

proves the assertion. I

324

VIII SPECIAL TOPOLOGICAL ALGEBRAS

Schol i u m 10.1. - Concerning Theorem 10.3 (Oka - V e i l Theorem) , there is still another version of it: Namely, one considers instead a quasicomplete l o c a l l y convex space E w i t h t h e approximation property (cf. J . MUJICA [I: Theorem 3.21). O f course, the two versions in question specialize to the case one considers a Frbchet l o c a l l y convex space w i t h the (Banach-Grothendieck) approximation property. However, applying the afore-mentioned type of Theorem 10.3, one c t i z l g e t s a strengthened form o f the previous Theorems 1 0 .4 and 10.5, f o r m e r y quasi-complete local Zy convex space having the approximation property ( ibid. : Theorems 5.4, 5.5 ) .

Finally, by considering the algebra H ( K ) (cf. (10.26)), one gets a similar result to the preceding Corollary 10.3. Indeed, since H I K ) is a &-algebra (Theorem 10.2), one actually has the following more pleasant form of the same result (see also J . MUJICA [5: p. 182, Corolario 28.31).

Corollary 10.4. Let E be a Fre'chet l o c a l l y convex space w i t h t h e (BanachGrothendieck) approximation property and K a compact polynomially convex subset of E . Moreover, consider a family F of (germs of) holomorphic functions on K without

conunon zeros i n K, and l e t J be the i d e a l i n H ~ K I generated by F. Then, one has the relation

(10.61)

J = HIKI.

I n p a r t i c u l a r , there e x i s t s functions fi,.

. .,f n

.. .,gn

i n F and gl,

i n H ( K I , such

t h a t one has n 2 f.g. = I , i=I 7, z

( 10.62)

on some open neighborhood

U of K.

Proof. Suppose that J # H ( K I . Then, there exists a maximal ideal M of H ( K I , with J G M . But, by hypothesis for E , and Chapt. 1I;Corol-

laries 7.2 and 7.3, one has the relation (10.63)

M = ker(hl

,

for some h e m l H [ K I ) . That is (Theorem 10.4), there exists a (unique) point a € K with

(10.64)

hif) = fial

,

for every f e H ( K 1 ; hence, from JGM=ker(hi and (10.641, one gets f ( a l = 0, for every f e F , which is a contradiction. Thus, (10.61) is valid. Now, the rest of the assertion follows easily from (10.26). I We conclude by still remarking that analogous results to The-

11.

325

CONVOLUTION ALGEBRAS

orem 1 0 . 5 , pertaining to other kinds of (natural) topologies on the algebra HIU), are still valid. We refer instead to the pertinent literature. See, for instance, J . MUJICA [5] and also J.-M. ISIDRO [ I ] . 11. Convolution algebras o f C m - f u n c t i o n s

We end the present chapter by further commenting on a particular class of (locally m-convex) topological algebras which have recently been considered in connection with potential theory on Lie groups and representation theory of such in spaces of (Schwartz) distributions. See, for instance, P. E . T . JQRGENSEN [ 2 ] . The latter work has also been our motive for what follows. Thus, suppose we have a L i e group G (em-functions are considered; cf. e.g. F . W. FlARNER [ I ] ) , arid let d ) ( G ) he the (csmplex) vector space of (complex-valued)C m - f u n c t i o n s on G w i t h compact support. See Chapt. IV; Subsection 4 . ( 2 ) for the space (algebra) a(Xl,where X is a givendifferential manifold. Althouqh we retain the same symbol for the latter (vector)space, we do change, however, the ring structure in a>(G) considering as ring multiplication the "convolution" of functions in (G): Thus, suppose that d x denotes a l e f t i n v a r i a n t Haar measure in G(see, for instance, Chapt. VII; Section 4 ) . So since a ( G l G L 1 ( G l (ibid.), one defines a D I G )by restricting to the latter space the (ring) multiplication in a "convolution operation" in L1(Gl ; i .e . , one has

for any f , g in B i G ) (see also VII; (4.3).We denote here multiplicatively the group operation in G) . Thus, one actually proves thatd)(G) is, indeed, a subalgebra of ( t h e convolution algebra) Li(G1 (cf. e.g. L . SCHWARTZ [I: p. 151, or p. 22, proof of ThGorSme I]). NOW, one considers on d ) ( G ) the final locally convex (vector space) topology defined on it by the relation (11.2)

a ( G ) = U aK(G) c 1 2

K

ax(G)

where S K ( G ) denotes the subspace (actually "convolution ideal") of those functions in a ( G l whose support is contained in K E G , the latter set being varied over the compact subsets of G. In this respect, aK(G) is a Frgchet locally convex space in the (canonical)Cm-topology (see L . SCHWARTZ [I: p. 641) which makes the convolution algebra a,(G) into a Frgchet locally convex algebra (with a continuous multiplication; cf. Chapt. 1;Corollary 4 . 1 ) . As a matter

VIII SPECIAL TOPOLOGICAL ALGEBRAS

326

of fact, a D , ( G ) i s a Fre'chet A-convex algebra (cf. Definition I ; 5.31, hence (cf.Corollary I; 5.3 ) . Therefore, it is

a Fr4chet l o c a l l y m-convex algebra

a direct consequence of the previous Lemma 10.3 that t h e convolution algebra a ( G ) ( S i l ( G ) ) topologized

(11.3)

1 1 1 . 2 ) becomes a l o c a l l y m-convex algebra.

It is now our next task to identify t h e s p e c t m o f t h e l a t t e r a l gebra.

Now, since the topological dual (a(G)I' of a ( G )

is, by defini-

tion, the space o f (Schwartz) d i s t r i b u t i o n s on G I it is reasonable to call an element of the spectrum of a ( C i , i.e., of the space (11.4)

m ( a ( G ) ) E (d)(G/)i

.

. ,

(cf Chapt. V; Definition 1 1 )

a multipZicative distribution

on G .

Thus, it is proved that t h e only m u l t i p l i c a t i v e d i s t r i b u t i o n s on G are e x a c t l y t h e (continuous) characters o f G (see P . E . T . J O R G E N S E N [2: p. 28, Thecrem 1 1 ) . Namely, we have the next.

Theorem 11.1. L e t G be a given L i e group and a ) ( G ) t h e l o c a l l y m-convex con-

v o l u t i o n algebra o f

c -functions m

on G w i t h compact support, defined by ( 1 1 . 3 ) . i s given by t h e r e l a t i o n

Then, t h e spectrum of a > ( G l

??'Z(a(G))

(11.5) ;.,;tp:. ' ~

=

8,

a homeomorphism (cf. the rels. (11.6)and (11.9) below), where

i 2

2 is

ti:e

character group of G (see Chapt. VLI; (4.4)).

Proof. Let a c 2 , that is, a continuous (group) morphism of G into the (multiplicative) group 117 = S i of complex numbers A , with I A / = 1. 1 Thus, a is, in f a c t , a C m - f u n c t i o n between the Lie groups G and S . G L ( l , @ ) (cf., for instance, F . W . W A R N E R [I: p. 109, Theorem 3.39, and p . 86, 53.10

(b)]).

Thus,

the relation h a ( f ) = : =

(11.6)

I,

f(x)a(3:)d.cI

with f e S ( G ) d e f i n e s h a s a m u l t i p l i c a t i v e d i s t r i b u t i o n on G. We shall show ) of the form (11.6), for a uniquely defined that every h ~ m ( a ( G Iis ct e G: First uniqueness of t h e correspondence h

a

(11.7)

. That

- ha:

m t a ( G ) ) ,

one has ha(f)=kfi(.f), for every f e a(G). Then, a = 6 on G , modulo a set of measure zero (cf. L . SCHWARTZ [I: p. 251), hence, due to their continuity, one obtains a = f i given by (11.6)

on G .

is, suppose that for a,13 in

11.

NOW, let

every

[I (f)](x) Y

327

CONVOLUTION ALGEBRAS

:= f ( y - ' z ) ,

with x , y

in G and f e B(G).

Thus, for

h E m(da(Gl), one has

for any f,g in a(G).The last integral in (11.8) is well-defined since the map y W h ( 1 (9)) is a (complex-valued) continuous map on Y fact, C"). Thus, by setting

,

a ( y l := h ( f ) - ' h ( Z 9 ( f ) )

(11.9)

G ( in

for any f E B D G ) , with h ( f ) # O , one gets a continuous character of G . Indeed, one has a(xyi.h(f.1 = h l l (f)) = a l l if)) Y XY = a(i! 1 if)) = a ( x ) h ( l (f)) = a(xl-a(ylh(fl, Y

X Y

therefore, from the hypothesis for f E 2 ( G ) , it follows that a(xy) = a ( x ) . a(y), for any : , g in G.Moreover, by (11.8) and (11.9), one proves that (11.6) is satisfied; finally, from the uniqueness of (11.7), it follows that (11.9) is actually independent of the particular element f

a(;),for which

h ( f l # 0. On the other hand, we prove that ( 1 1 . 7 ) is a homeomorphism: Thus, writing a.dx for the "measure o f d e n s i t y a" wit.h respect to the Haar E

measure d x chosen on G , and with a

E

h,(f)=
(11.10)

2,

one has from (11.6) a.dx>

I

for every feB(G) (see N . BOURBAKI [8:Chap. 3 ; p. 511). NOW, one easily proves that :7w correspondence (1l.lli

,5 : Cc(G1+ W G ) [s] = (K(G));

u

,

where 6 If)= f - p , with f e Cc(G),is continuous. Here for any given Radon P measure i-1 on G , the rzmqe of 6 is the space of Radon measures on G in LJ the topology of simple convergence in K(GI ( "vague topology" : ibid., p. 59, s 9 ) Now, this entails the continuity of (11.7). Furthermore, if a net ( h

'z'

converges to ?z

in ? ? Z ( a ( G l ) , there

h c l ! f ) # LJ ; so one may suppose that the

exists an element f E b , ( G ) , with This now easily same net has " e v e n t u a l l y " the property that h ( f ) # O . 'i implies, by (1 1.9) , the c o n t i n u i t y of t h e i n v e r s e map o f ( 1 1 . 7 1 , and this terminates the proof of the theorem. I In particular, suppose that ;' = l? acter cleg is given by the relation

.

Then, every (continuous)char-

VIII SPECIAL TOPOLOGICAL ALGEBRAS

328

(11.12)

a(t) =eXt s expiXt) I t E JR I

f o r a uniquely defined Xed:. On the other hand, if f E a ( I F . ! ,

j(<) = I

(11 .13)

let

f ( t i expl-ittidt

1R

denote the ("normalized") Fourier transform of f, with C E C . Thus, applying Theorem 1 1 . 1 , every h E Z(d(IR)) is of the form h

(11.6)l

for some (uniquely defined) a e I R ; i.e., we have h(f)

= IIFi f(tla(t)dt

= (by ( 1 1 .1 2 ) )

= (for < = i h f C ) I f(tlexp(-iCt)dt IR

IIR f(t)exp(Xtldt

= (by ( 1 1 .1 3 ) )

;(<).

So the foregoing proves the following (see also P. E . T. JORGENSEN 12: p. 29, Corollary I

I) .

Corollary 11.1. Let the notation of the previous Theorem 11.1 be valid. Ther,, one has the relation 312(8(1Rll= c ,

(11.14)

within a homeomorphism, given by "point evalrtation" of the Fourier transform. That is, we have h ( f l = s^(51

(11.15)

f o r every f E

Z(lP.), where h E

and conversely (cf. ( 1 1 . 1 2 ) )

??Y(d,fIR))

.I

and

,

< eC

are uniquely determined by h ,

Further examples of (locally m-convex) convolution algebras of (complex-valued) ew-functions on IR and determination of their spectra are still given in the work of P.E.T. Jargensen cited above. These are further applied to study the groups of (algebra) automorphisms of the respective topological algebras.

329

S t r u c t u r e Theory

CHAPTER I X

We examine in this chapter another aspect of the fundamental notion of the (global) spectrum (Chapt.V) of a given topological algebra, namely, that one corresponding to what might be called " s t m e t u r e theory of t o p o l o g i c a l algebras ;this is in analogy with the familiar situation one has in the classical theory of normed (actually Banach) algebras. The ingredients of the theory are, of course, the ideals of the particular topological algebra under consideration, mostly, the maximal (regular closed) ones. Thus, the afore-mentioned variance of the notion of the spectrum is now the "maximal ideal space" of the algebra endowed with the so-called "hk-topology"(cf. Section 1 below). Yet some other particular kinds of ideals will also be considered which play, as we shall see, a particularly preponderant r61e in the sequel. 1 . Maximal ideal space ( s t r u c t u r e space). hk-topology

The context within which we shall be working in the sequel, is the set M(E) of all (2-sided) maximal regular closed ideals of a given topological algebra E endowed with the "hk-topology" (see Definition below) . Thus, given a c o m t a t i v e l o c a l l y m-convex algebra E whose spectrum ("Gel'fand space") is we already know that the correspondence

1.2

-

m(E),

(1.1)

j :f

i s always one-to-one

(cf. Corollary 11; 7.2). In fact, this is valid f o r

kercf) :??UE)-

McE)

any t o p o l o g i c a l algebra E (cf. Lemma 11; 7.2 and the comment following it). However, j is, in particular an "onto" map under the assumption

of the "Gel'fand-Mazur condition" for the particular topological algebra under consideration (see Chapt. VIII; Subsection 9.(5) in connection with Corollary 11; 7.1). Thus , by considering the i n j e c t i o n (7.2)

m ( E l $ MIE)

,

as above, we look further at the spectrum of E as being a subspace of h l ( E ) , when the latter space is equipped with the afore-mentioned hk-

topology. So one essentially defines the latter in terms of the ele-

IX STRUCTURE THEORY

330

ments of ?'YYiE) rather than of M ( E I , the transition however to the ambient space being standard concerning the terminology applied. We start with fixing the relevant terminology, so we first have the following. D e f i n i t i o n 1.1. Let E be a topological algebra whose spectrum is m ( E ) and let A be any subset of E . The set

hlA) = { f

(1.3)

€

,

m(E): Z l f ) 1 A }

with Z(f)=ker(f), is called the h u l l of A in m'(E). On the other hand, if B is any subset of m ( E ) , the set k(B) = { ; c € E : Z ( P ) 1 B } ,

(1.4)

with Z(3) = { f e m(E): s(f)= f (x) = 0 } is called the kernel of B in E . Thus, a set B C m(EI is said to be a h u l l , if B = hlk(6)) ,

(1.5)

while a set A C E is said to be a k e r n e l , if one has A = k(h(A))

(1.6)

.

Applying the previous notation, we already have.

Lemma 1 . I . Let E be a topologicaZ algebra whose s p e c t m i s m(Ei. Then, t h e correspondence

,

B-hlklBII

(1.7)

w i t h BC ???+(El, d e f i n e s a "Kuratowski closure-operator" on t h e s u b s e t s of ??Tit.). Hence, t h e hulls of i.e., t h e s e t s

m(E),

B = h(k(B))

(1.8)

may be taken a s t h e closed s e t s of a topology on m ( E / the l a t t e r being, moreover, a T -topology. 1

Proof. We first have to verify the following:

B G h ( k ( B I I , f o r any

(1.9)

For any two subsets A , B of

m ( E l , the relation

A C B implies

(1. l o )

BEm(E).

hlk(A)I C h(k(B)).

Moreover, one has (1.11)

h l k ( [ h l k ( B ) l l l l = h ( k ( B ) I , for any B E

Finally, for any two s u b s e t s A , B of miE), one has (1.12)

h ( k (A[!B l ) = h ( k ( A / ) U 7) ( k ( B I )

.

m(E).

331

MAXIMAL lDEAL SPACE. hk-TOPOLOGY

1.

Now, the rels. ( 1 . 9 ) and the fact that:

and ( 1 . 1 0 )

are immediate from ( 1 . 3 ) ,

(1.4)

For any two subsets A], A 2 of E , t h e r e l a t i o n

A]

(1.13)

.G A2

implies

Similarly, if B I

C

h(AI)

.

,B2 are subsets of ??Z(EI, then

BI L Be i m p l i e s

(1.14)

h(A,I

k(B21 L k ( B I )

.

(The preceding relations are straightforward from ( 1 . 3 ) On the other hand, ( 1 . 1 1 ) follows immediately from k ( h f k ( B ) I ) = k f B I , f o r any

( 1 .15)

and ( 1 . 4 ) )

.

B G ??Z(EI.

Thus, from ( 1 . 9 ) and ( 1 . 1 4 ) one gets k f h f k ( B ) ) ) Ck(B1. Furthermore, since by ( 1 . 1 3 ) f E k ( k f B ) ) implies k f B J C Z ( f ) , one gets k(BI E k(h(k(B))l as well, that is the assertion. Finally, ( 1 . 1 2 )

is

a

consequence of the fact that ( 1 . 7 )

yields

a (univalent) map from the set of all subsets of M f E l into itself. NOW, the last assertion means, of course, that every singleton of %(El

is a h u l l : That isf equivalently, one has

hlk(fl)

(1.16)

=f ,

for every f e mlEl; for if g E i Z ( k ( f I ) then from ( 1 . 3 ) one gets kerIg) which implies by Lemma 11; 7 . 2 that g = f , which is the assertion and this completes the proof of the lemma. I

1 ker ff)

The foregoing enables us to set the following.

Definition 1.2. Given a topological algebra E with spectrum ?7ZfE), we call h u l l - k e r n e l topology (or for short hk-topology ) on the (TI-)

m(E),

topology defined on it by Lemma 1.1; namely, that topology having as closed sets the hulls in m(El (Definition 1 . I ) ( The terminology StoneJacobson, or yet Jacobson, topology is also applied). The set mfEl endowed with the hull-kernel topology is called the s t r u c t u r e space of the given topological algebra E and denoted by

.

.

m(E) [ T ~ ~ ]( The term maxima2 i d e a l space of E i n t h e h u l l - k e r n e l topology is also applied, on the basis of the (canonical) injection ( 1 . 2 ) ) .

Note (on the terminology). - Since M(EI 2 a i m ( E i (the set of prime i d e a l s of E ) , where E is a commutative (topological) algebra with an identity element, the hk-topology on M(El, and hence on ??Z(E), is the relativized one of the standard Zar i s k y topo2ogy on Prim ( E ) . Thus, the last term is also applied when referring to the toFslogy given by the previous Definition 1 . 2 .

332

IX STRUCTURE THEORY

The next result exhibits the relation between the two basic towhen E is a given topological algebra. That is, thee pologies on relation between the Gel'fand (Definition V; 1 . 1 ) and the hull-kernel

m(E)

topology in m(El.

Lemma 1.2. Let E be a topological algebra whose spectrum is m(El. Then, t h e hull-kernel topology on ??i?(El

is weaker than t h e Gel'fand topology on it.

Proof. It suffices to prove that any closed set of m(El in the hk-topology, i.e., any h u l l i n m(El (Lemma 1 . I ) is a weakly closed s e t in c E,' : Thus, if B is any hull in m(El and f E ?? G m ( E l C E,' , then f =limf8, with f6 E B , so that one gets, from ( 1 . 5 ) ,

m(E)

2 ( f l = f(1imf I =limf(f*l = 0 , 6 6 6 since f e C(?i'i?(EI), with x E E (cf. Chapt. V; ( 1 . 5 ) )

for every x e k(B), f e h(k(B)l= B (Definition 1. I ) ,

. So

and this proves the assertion. 1

The preceding t w o topologies in the spectrum of a given topological

algebra do n o t , in general, coincide (even in the particular case of a Banach algebra): A counter example is provided, for instance, through the (commutative unital) Banach algebra A(Dl of all complex-valued continuous functions on the "unit disc" D € Q: which are holomorphic in the interior of the disc, in the "sup-norm'' topology ( " d i s c algebra" ; cf. , for instance, C.E . RICKART [I: p. 303, ff.]).

2. Regular, normal and s i l o v algebras We consider next conditions on a given topological algebra E guaranteing the coincidence of the preceding two topologies. Thus, we start with fixing up the following basic notion.

Definition 2.1. Let E be a topological algebra with spectrum ??Y(E). We shall say that E is a regular algebra, whenever for any (weakly)closed subset A of 1?Z(El and f E c.4 , there exists an element x e E such that

G(f)=l

(2.1)

algebra

and ? = 0

1A

.

That is, E is a regular algebra, whenever t h e Gel'fand transform E* C e ( ? ? U E l ) (cf Chapt. V; ( 1 .6)) of E is a regular f a m i l y of func-

.

t i o n s on m(El. The significance of the preceding terminology is now revealed

by the next.

Theorem 2.1. L e t E be a topological algebra whose spectrum i s

m(E/.Then,

2.

REGULAR, NORMAL,

333

SILOV ALGEBRAS

t h e f o l l o w i n g two a s s e r t i o n s are e q u i v a l e n t : 1 ) E i s a r e g u l a r algebra 2 ) The h u l l - k e r n e l

(Definition 2 . 1 ) .

topology c o i n c i d e s w i t h t h e Gel'fand topology on m(E).

proof. 1 ) + 2 ) : Since B C h ( k ( B ) ) , for any B E mfE) (Lemma 1 . 1 ) , it suffices to prove that for every closed subset 8 of ~ ! E ) [ T ~(see ]

also Definition V; l.I),one has the relation

CB E Ch(klBI)

(2.2

.

Thus if f E C B , then from the hypothesis there exists an element x € E such that (2.3) 3(fl = 1 and G(gJ = 0 , g e B ,

so that one gets from ( 1 . 3 ) that x k ( B I . Therefore, one concludes from the first of ( 2 . 3 ) that f 6 hikfE)), that is ( 2 . 2 ) is valid. 2 ) s l ) : Suppose that E 5. mlE) [ T ~ ] is a closed subset and let f E C B . Then, by hypothesis, B = h l k f B ) ) (Lemma 1 . 1 ) ; so since f t ! B , one obtains from ( 1 . 3 ) that there exists x € k(BI, with x $ Z i f ) . Therefore, we have 2 = OIB and GCf) = f(x) = a # 0 . Thus, the element z c =c L1 z € E fulfills ( 2 . 1 ) plete. I

,

and the proof is

com-

Now in analogy with the situation one has in Banach algebras theory, we obtain another equivalent version of the two equivalent assertions of Theorem 2 . 1 for any commutative complete l o c a l l y in-convex algebra E having it6 eztended spectrwn 7YC(Elt compact and t h e r e s p e c t i v e Gel'fand map continuous (equivalently,for any c o m u t a t i v e complete l o c a l l y in-convex Qa l g e b r a ) . This is given in the form of the following statement: For every element f & m(EI, t h e r e e x i s t s a neighborhood, say,

u of

f i n 712 (E)[ T ~ ~ ]i n such a way t h a t k i u ) (see

( 1 . 4 ) ) i s a r e g u l a r ( 2 - s i d e d ) i d e a l i n E.

We shall omit the details of this argument; in this respect, cf. however Lemma V I I I ; 1 . 3

and the comment preceding it. See also G.R. and F . F . B O N S A L - J . DUNCAN [I: p. 115, 8

ALLAN e t a Z . [ I : p. 6 7 , Corollary 4.31

231).

Furthermore, we remark that within the context of the previous Theorem 2 1 , i.e., f o r a g i v e n r e g u l a r t o p o l o g i c a l algebra (or its equivalent form given by (2.4), in the particular case considered), t h e h u l l - k e r n e l topology on ??UE) m(E)[T

F

]

5

is Hausdorff. (Here, of course, one still assumes that

Z ( E )C

E,' is a Hausdorff (completely regular) space).

334

2.1,

IX STRUCTURE THEORY

Regarding the terminology applied in the preceding Definition it is to be noticed that this is reminiscent, of course, of the

fact that t h e l o c a l l y m-convex algebra q X ) , where X is a completely regular (Hausdorff) space (cf. Example I ; 3.1), being t h e Gel'fund transform algebra of i t s e l f (cf. VII; (1.31)) fulfilZs ( 2 . 1 ) . Therefore, c c ( X l i s a "regular algebra" in the sence of the above Definition 2 . 1 . Yet the term "compZeteZy r e g u l a r algebra" is also of use in the literature. Furthermore, by considering the same algebra, as before, if is, in particular, a normal Hausdorff space, one is led to the following.

Definition 2.2. Given a topological algebra E whose spectrum is ? ? Z i E l , we shall say that E is a normal a l g e b r a , if for any two disjoint

(weakly) closed subsets Bz

, B2

of

m(E)there

exists an elerrent

3:

eE

such that and

(2.5)

; = O

I Bt' .

Equivalently, if t h e Gel'fund transform aZgebra E " C e ( ? ? Z ( E ) ) of E

is a

normal f a m i l y of (complex-valued continuous) f u n c t i o n s on ? ? Z ( E l , or yet it

"separates closed subsets" of m ( E ) . We shall not have thus much the opportunity to apply the preceding class of topological algebras. Nevertheless, we notice the already standard result that every commutative Frgehet l o c a l l y m-convex algebra (with an identity element) is normal. In this respect, cf. M . ROSENFELD [ l : p. 1601 and/or R . M . BROOKS [3:p. 2661; see also Corollary 4.5 below. On the other hand, the particular class of topological algebras (not necessarily Frgchet), which is singled out by the following definition, has several applications some of which will be considered in the next section. Thus, we first have.

Definition 2.3. By a SiZou algebra we shall mean a commutative complete regular semi-simple (cf. Definition VIII;3.2) locally m-convex algebra. Particular classes of silov algebras will be considered in the remaining part of this chapter, as well as in subsequent chapters.Yet a different (although equivalent) terminology has also been applied, concerning the preceding class of algebras, within the framework of Banach algebras theory (cf., for instance, A . MIRKIL [I: p. 18 ff.]); its extension, however, within the present context will also be considered in the ensuing discussion.

3.

335

HULLS OF IDEALS

Furthermore, we observe that it does not affect the generality when assuming that a silov algebra has an identity element (we shall not do it, however). Indeed, one actually has the following. Lemma 2.1. A t o p o l o g i c a l algebra E is a S i l o v algebra (cf. Definition

2.3) if, and onZy if, t h i s i s t h e case f o r t h e algebra E C = E B C (cf. I; (5.10)). Proof.

The assertion follows immediately from the same definitions and Lemma VIII; 8.1. I Finally, it is still to be noticed that the (non normed) locally m-convex algebra C J X l considered in Chapt.VII;Section 1 constitutes an outstanding particular example forcing us to consider S i l o v algebras which a r e n o t n e c e s s a r i l y normed ones.

3. Hulls o f ideals We apply in the sequel the technique of the preceding two sections by considering interrelations of (closed 2-sided) ideals in a given topological algebra with their hulls (Definition 1 . 1 ) in the respective structure space of the algebra (Definition 1.2). In this respect, the class of Silov algebras (Definition 2.3) play a preponderant r81e. Thus, applying the terminology of Section 1, we first have the following. Lemma 3.1.

L e t E be a t o p o l o g i c a l algebra whose spectrwn is m(E),and l e t

A C E. Then, t h e h u l l of A , h ( A ) (Definition 1 .I), i s a (weakly) cLoscc! subset

o f ???(E).

(3.1)

That is, one has

-

h(A) = h(A) . Proof. The assertion is an immediate consequence of the relation

(3.2)

h(Ai =

n Z(?l

xe A

E m(Ej,

which follows from Definition 1.1 and the fact that each one of the (complex-valued) functions 2 , x e E , is continuous (cf. Chapt.V;(1.5)). E Now, it is clear from ( 1 . 4 )

that t h e kerneZ of any set B G

el i s ,

e q u i v a l e n t l g , expressed by t h e r e l a t i o n (3.3)

k i ~ =i n kerif) =

f

n zif) f € B

.

Therefore, kin) i s a cZosed ( 2 - s i d e d ) i d e a l o f E. Besides, as follows from Lemma 3.1, any hull. i n m(EI i s a c l o s e d s e t . The following indicates that these are, in fact, t h e only c l o s e d s u b s e t s of E (resp. ? ? Z ( E j ) which are given

336

IX

STRUCTURE THEORY

by ( 1 . 3 ) ( r e s p . ( 2 . 4 ) ) . More precisely, one has the following.

Lemma 3.2. Let E be a t o p o l o g i c a l algebra w i t h spectrwn m(El. Then, t h e f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : 1) B = h(A/, with A G E

(resp. I = k ( B ) , w i t h B

C

m(E)).

2 ) B = h ( k ( B ) ) (resp. I = k ( h ( I i l ) .

Proof. It suffices only to prove that 1 j=> 2) : Thus, if B = h l A l s C E , we prove that

m(E), with A

h ( k ( B I ) G 6.

(3.4)

Now, by hypothesis, one has (see also (1.3) and (3.3)) A & k ( h ( A l l = k ( B ) , so that (cf. (1.13)) h ( k ( 6 ) ) h~( A ) = B , i.e., (2.4). A similar argument can be applied for the kernels (see, for instance, (1.15)), and this finishes the proof. I

Corollary 3.1. L e t E be a t o p o l o g i c a l algebra whose spe ctrwn is ??'Z(E). Moreover, l e t h(?R(E)) ( r e s p . k(E))

b e t h e s e t o f a l l h u l l s i n m(E) ( r e s p . k e r n e l s

i n E l . Then, between t h e s e two s e t s t h e r e e x i s t s a ( c a n o n i c a l ) b i j e c t i o n g i v e n by the re la tio n h(A)-

(3.5)

k(h(A)I, uith

AGE,

("Galois correspondence"), t h e l a s t map being a l s o " i n c l u s i o n i n v e r t i n g " . A i r t h e r -

more, i f t h e g i v e n algebra E i s , i n p a r t i c u l a r , a Gel'fand-Mazur algebra (cf. Defi-

nition VIII; 9.5), t h e n t h e s i n g l e t o n s of Z'ZiE) correspond, by ( 3 . 5 1 , t o c l o s e d maximal regular ( 2 - s i d e d ) i d e a l s o f E.

Proof. From Lemma 3.2 one gets h(k(h(Al)) = h(A),

(3.6)

for every A L E , which actually proves that (3.5) is one-to-one. Furthermore, (1.15) entails, in fact, that (3.5) is an onto map too.Now the same map is inclusion inverting, as this follows already from (1.14) Finally, since the hk-topology in m(E)is a T,-topology (cf.

.

.

Lemma 1 .l), t h e s i n g l e t o n s o f m(E) are a c t u a l l y h u l l s (see (1.16)) Thus, the last assertion follows already from the same Definition VIII; 9.5 in conjunction with the argument applied in Corollary 11; 7.1, and this completes the proof. I The previous Corollary 3.1 provides a particularly useful descriptior. of certain (closed) ideals of a given topological algebra: namely, those which are, in fact, characterized by Lemma 3.2. On the other hand, the situation is far, of course, of being realizable in the most general case (this being true even for Banach algebras). But

3. HULLS OF IDEALS

337

we do have it for certain important topological algebras, as they are algebras of the form C c ( X l , where X is a compact, or even a completely regular Hausdorff space. Thus, as a consequence of the reasoning in Section VII; 1, the preceding terminology already yields the following.

Theorem 3.1.

Let X be a Hausdorff completely regular space and cc(XI the

r e s p e c t i v e l o c a l l y m-convex algebra of (complex-valued) continuous f u n c t i o n s on X

.

i n t h e compact-open topology (ibid.) Then, every closed i d e a l o f

ec(X) is a

k e r n e l , while every closed subset o f X i s a h u l l . Furthermore, t h i s e s t a b l i s h e s a one-to-one

and onto correspondence between k e r n e l s an% h u l l s , i n such a way t h a t

t h e p o i n t s of X correspond t o t h e closed maximal ( r e g u l a r ) i d e a l s o f

C.(Xl.

Proof. The assertion is, in fact, a reformulation of Lemma VII; 1.5 in connection with Remark VII; 1.2 and the fact that ?%?(cc(Xl)= X , true within a homeomorphism (Theorem VII; 1.2) : Thus, if Ie J I c c ( X ) ) (i.e., I is a non-trivial proper closed ideal of (X)), by considering the set (3.7) B = n ker(f) I n Z(f), feI feI in fact, a closed subset of X = m ( C c ( X ) l , one concludes from ( 1 . 4 ) and Chapt. VII; ( 1 . 6 ) , ( 1 . 3 1 ) that

c

(3.8)

k l a ) =I f e Cc(XI: Z(fl 3 B } = I B '

Besides, one gets from (1.3) and ( 3 . 7 ) B = h(Il.

(3.9) Therefore, from VII; (1.20) (3.10)

one further concludes that

I = F = I B = k(B) = k(h(I)),

that is, I i s a kernel (cf. also V I I ; ( 1 . 2 6 ) ) . On the other hand, by Theorem 2 . 1 , .?very closed subset of X i s a h u l l course, a regular algebra (cf. the comment after the same theorem above). In particular, the latter is a commutative local-

Cc(X)being, of

ly m-convex algebra with an identity element, therefore, a f o r t ior i a Gel'fand-Mazur algebra (cf. Theorem II;7.1);nowI on the basis of Corollary 3 . 1 , this proves the last part of the statement, andthe proof of the theorem is complete. I

Remark 3.1. - Applying the notation of Theorem 3.1, if B is any closed subset of X and I B : = { f € e nu( X ) : Z ( f 1 2 B ) , then one obtains (see a l s o ( 3 . 3 )

and VII; ( 1 . 2 )

IX

338

STRUCTURE THEORY

(3.11 Thus , by the preceding one has (3.12 B = h ( k ( B ) )= k i I B )

.

NOW, although ( 5 . 1 2 1 t u r n s out t o be valid f o r any regular topological algebra E l the description of a closed ideal I in E through a relation like ( 3 . 1 0 ) is proved to be, in general, an extremely complicated problem going back essentially to the very structure of the topological algebra under consideration. In this respect, the following notion has certainly an intuitive appeal by pointing out those topological algebras with no (non-trivial) closed ideals at all. More precisely, one has the following.

Definition 3.1. A given topological algebra E is said to be simple (or more precisely, t o p o l o g i c a l l y simple) , whenever there are no non-trivial closed ideals in E (i.e., besides the zero ideal ( 0 ) and the algebra E itself). The preceding class of algebras is frequently appeared in various applications of the theory of topological algebras (Banach or more general). Thus it happens, for instance, in case of "topological algebras of kernels" (cf. J. GIL DF; LAMADRID [3] and/or A. KYRIAZIS [l]) as well as in certain applications in Differential Geometry; "abstract cohomology theory" , see I.M. GEL'FAND-D.A. KAZHDAN-D.B. FUKS 111. However, we are not going to be dealt with this class of algebras, at least at the present sts.ge of this discussion. At the other extreme one is interested in the extent to which a given (non-trivial proper) closed ideal of a topological algebra E admits the "geometric realization" of being the intersection ("kernel") of the set ("hull") of closed maximal regular ideals which contain it. Within the classical terminology (Banach algebras theory) the problem is known as that of s p e c t r a l synthesis, concerning the topological algebra E under consideration. A s a matter of fact, one is dealt with the relevant problem of whether a closed subset of % W E ) is the hull of more than one closed ideals in E. This will be the subject matter of Section 6 in the sequel. However, still as a preliminary material thereof, we presently consider in the next section further applications of the machinery developed sofar. 4. Hulls of ideals: Regular, normal and s i l o v algebras (contn'd)

As a first application of the preceding material one can evalu-

4. HULL:: OF

339

IDEALS (CONTN'D. )

a t e t h e spectrwn of a t o p o l o g i c a l q u o t i e n t algebra by means

of t h a t of t h e

g i v e n a l g e b r a . Namely, one h a s t h e f o l l o w i n g b a s i c r e s u l t .

Theorem 4.2. L e t E be a t o p o l o g i c a l algebra whose spectrwn i s m(E)and I a c l o s e d ( 2 - s i d e d ) i d e a l o f E . Then, t h e spectrwn o f t h e q u o t i e n t t o p o l o g i c a l a l gebra E/I ( c f . C h a p t . 11; ( 7 . 8 ) ) i s g i v e n by t h e r e l a t i o n ??'Z (E/I) = h ( I ) C m(E),

(4.1)

w i t h i n a homeomorphism; q u o t i e n t map

ii:E

-+E/I,

(4 - 2 )

he l a t t s r , is d e f i n e d by t h e tuJanspose of t h e canonical

i.e., by t h e map t p = 71 : m(E/I) -?&?(El

Proof. I t i s c l e a r t h a t t h e map (4.3)

pi$)=$ o n ,

i s one-to-one.

~ e s i d e s ,f o r any

$ E

g i v e n by

b e ??Z(E/I),

? ? Z ( E / I I , one g e t s

I = kerinrl E k e r ( $ o n )

(4.4)

?

.

~7

k e r ( p i @ ) )I Z i p ( $ ) )

,

so t h a t one h a s , by ( 1 . 3 ) , I m ( p i = p(m(E/I)iC h ( I ) .

(4.5) Furthermore,

if

feh(I)E

lation

m(E),s i n c e

p(J)

(4.6)

y i e l d s an element

7e

by ( 1 . 3 ) Z ( f ) 2 I , t h e re-

= f 0 n := f

? ? Z ( E / I ) , s o t h a t p i s a c t u a l l y an onto map. T h e r e -

f o r e , one o b t a i n s P ( ~ ( E / I ) )= !:(I).

(4.7)

On t h e o t h e r h a n d , by t h e f o r e g o i n g and t h e r e l a t i o n /---

n i x ) = ; ~ ~ x, e E ,

(4.8)

o n e c o n c l u d e s t h a t t h e GeZ'fand t o p o l o g i e s on m ( E / I I and h ( I ) a r e e s s e n t i a l l y

g i v e n by t h e same f a m i l y of maps, when t h e t w o s p a c e s a r e i d e n t i f i e d v i a ( 4 . 7 ) ; so the topologies i n question coincide

Thus,

( c f . a l s o Lemma V; 1 . 1 ) .

( 4 . 7 ) p r o v i d e s i n f a c t a homeomorphism, and t h i s c o m p l e t e s t h e

proof. I A s a b y p r o d u c t of t h e p r e v i o u s t h e o r e m ,

t h a t "quotients preserve regularity". T h a t

one f i r s t c o n c l u d e s

i s , w e have.

Corollary 4.1. L e t E be a r e g u l a r t o p o l o g i c a l algebra whose spectrim is ??'l(S) and I a c l o s e d ( 2 - s i d e d ) i d e a l o f E . Then, t h e q u o t i e n t t o p o l o g i c a l algebra E / I is a l s o r e g u l a r .

Proof. I f B C

mrr/I) i s

any c l o s e d s u b s e t , s i n c e /?(I)i s a c l o s e d

340

IX STRUCTURE THEORY

subset of m(EI (Lemma 3.1) , one gets by (4.1) that the set h ( I ) C ???XE/

BG??Z(E/I)

is (homeomorphic to) a closed subset of MlEl. Now, let @ € mIE/II, with , so that by (4.3) p ( @ ) # p ( B ) C ??Z(E). Thus, by hypothesis for (cf. Definition 2 . 1 ) , there exists xeE,with ; ( , ( @ I ) = 1 and 2 = I ; ( p ( B l ; i.e., @#E

by (4.3) and (4.8), one has

i i P ( @ i i ( p ( @ i l i x l =@ ( T ( X ) l = Til:cI ($1 = 1 as well as

/--

TI(X)

=0

IB

,

which proves the assertion. I

Furthermore, one also gets.

Corollary 4.2. Suppose t h a t t7ie c o n d i t i o n s o f t h e preceding Theorem 4 . 1 a r e s a t i s f i e d . Then, the q u o t i e n t t o p o l o g i c a l algebra E/I i s semi-simple i f , and o n l y i f , t h e given i d e a l

Proof.

I i s a k e r n e l (Definition 1.1) .

Suppose that E / I

is semi-simple (Definition VIII; 3.2), and

.

let x E k(h(I)). Then (Definition 7 1 ) , Z ( 2 ) 3 h ( I ) , i.e., ;(f) = f ( x l = 0, for any f e h ( I ) . Therefore, by applying (4.8), one gets

n?i?(@)

(4.9)

= s i p ( + ) ) = 0 , a E v~(E/I),

f i

i.e., ~ ( x=) 0 . Hence, by hypothesis, n i x ) = 0, so that x € I . Thus, k ( h ( I l l E I , so that since I G k ( h ( I ) ) , one concludes that I is a kernel, i.e., I = k(h(Il). fi

NOW, if the last condition is true, and nlx) = 0, then by (4.8) ; ( p i @ ) ) = O r for every @ e m(E/Ii. Therefore, one gets from (4.7) that ;(f) = O r €or every f E h ( I l , that is Z(;lZ h ( I ) , so that z.fk(h(III = I;so nix) =0, that is, E/I is semi-simple, and this terminates the proof. I The following result will be needed next. That is, we have.

Corollary 4.3. L e t E be a t o p o l o g i c a l algebra w i t h spectrwn 7?Z(E) having b e s i d e s t h e r e s p e c t i v e Gel’fand map continuous, and l e t I be a c l o s e d (2-sided) ideal i n E. Then, t h e q u o t i e n t t o p o l o g i c a l algebra E/I has t h e corresponding Gel’fand map continuous. In p a r t i c u l a r , if E i s a

cwr-nutative l o c a l l g m-convex algeA

bra and B i h l I l i s a compact s u b s e t o f ?Z?(E), tlie algebra E/I i s a (commutative complete) l o c a l l y n-conuex Q-aZgebra. Proof. The first part of the statement is provided by Theorem 4.1 and the commutative diagram, given next, where one considers in E/I the quotient topology. NOW, by the same theorem and the hypothesis for B = h ( I ) , one concludes from the preceding and Theorem VI; 1.1 A

that m(E/I) is equicontinuous. Therefore (cf. also Theorem V; 2.1) E/I is a commutative complete locally rn-convex algebra with an equicon-

4. HULLS OF IDEALS (CONTN’D. )

341

(4.10)

tinuous spectrum, hence (Corollary 111;6.6) a &-algebra, which is the assertion and the proof is complete. I On the other hand, we now have the following auxiliary result. Theorem 4.2.

Let E be a commutative Ptcik (DefinitionVIII; 3.3 ) ZocaZZy

tn-eonvex aZgebra with spectrum m(E)and B a compact huZZ in nZ(E).

suppose that there exists an element x E E whose Gel’fand transform on B. Then, there exists y EE such that (4.11)

9 = 2 ) ,

i.e., x?(f!

Furthermore,

2 never vanishes

,

=f(xy) = I , f o r every f E B C 1?Z(E!.

Proof. I f I=k i B ) , then by hypothesis for B , one gets I = kihll)),

the latter set being by its definition a closed (2-sided) ideal in E. So by hypothesis and Corollary 4 . 2 , the quotient algebra E/I is a commutative complete (the hypothesis of E being Ftdk is applied just here; see e.g. J . HORVAkH [l : p. 300, Proposition 51 ) semi-simple locally m-convex algebra, in such a way that ??.?(E/I!

(4.12)

E

hlI) = B ,

where the first relation holds true within a homeomorphism (Theorem 4 . 1 ) . Thus, the algebra E/I has a compact spectrum. On the other hand, the element n

i E C (m(E/II I

never vanishes on VY(E/I), as this follows from the hypothesis for x eE and the rels. ( 4 . 8 ) , ( 4 . 1 2 ) . Therefore (Corollary VI; 3.1 ) , the algebra

E/I has an identity element, so that by the foregoing and Corollary III;5.2 the element nix) is invertible in the same algebra. Hence, there exists an element n l y ) eE/I, with TI(x).TI(~)= 7ilxyl = I , so that (4.13)

71

Therefore, one has from ( 4 . 8 ) 5jj(f) =

-

G ( P ~ $=)n)( x y ) ( $ I =

for any fehfl) = B E ???(El

(by (4.13) 1

,

which is, in fact, the desired relation (4.11),

342

IX

STRUCTURE THEORY

and this completes the proof.1 Furthermore, we still have the following. Theorem 4.3. Let E be a c o m u t a t i v e Pta'k l o c a l l y m-convex algebra w i t h spectrwn m(E) having b e s i d e s t h e r e s p e c t i v e GeZ'fand map continuous, and l e t I be any

idea2 (not necessarily closed) in E. Furthermore, l e t B be a compact h u l l i n m(E/ such t h a t (4.14)

B

n h ( I l = @,

and x an element of E w i t h 2 never vanishing on B. Then, t h e r e e x i s t s an element y e I such t h a t

ij =

(4.15)

;IB.

Proof. By considering the canonical quotient map (4.16)

n: E

-

EIklB)

the element n(xl never vanishes on (4.17)

(E/k(Bl)

=P h ( k ( B / ) = B .

(The last relation is true, by ( 4 . 1 ) , within a homeomorphism; see also ( 4 . 3 ) ) . Now this follows from the hypothesis for x e E and the respective relation to ( 4 . 8 )

,

concerning the quotient algebra E / k ( B I .

Furthermore, the latter is a commutative complete ( E being Ptdk; cf. Scholium VIII; 9.1 ) semi-simple (Corollary 4 . 2 ) locally m-convex algebra having, moreover, by ( 4 . 1 7 ) , a compact spectrum: so i t is a Q-algebra (cf. Theorem V I ; 1 . 1

in conjunction with Corollary 111; 6 . 6 and the previous Corollary 4 . 3 ) . Therefore (Corollary VI; 3.1 ) , t h e algebra E/k(BI has an i d e n t i t y e l e -

ment, so that (Corollary 111; 5.2) the element n ( x ) is, by the forego-

ing, invertible in the same algebra, i.e., one gets (4.18)

n(rl.n(zl = n ( x z J = 1

for some n l z l e E / k ( B I . Now, if T ( I ) ,being an ideal in E / k ( B I , is proper then it will be contained in a maximal ideal, say, M of E/k(B) (see Lemma 1 1 ; 6 . 1 ) , while M will also be a closed ideal, the algebra E / k f B ) being by the preceding discussion a &-algebra with an identity element (cf. Theorem I I ; 6 . 1 ) . Consequently, if f E ? ? z f E / k ( B ) I ~ (by ( 4 . 1 7 ) ) B with kerlf) = M (cf.Corollary 11; 7 . 2 ) , one obtains n(IIG

M = kerifl

E

Z(f)

i.e., f ~ h ( n ( l J J (by = ( 4 . 1 7 ) ) h ( 1 ) ; so a contradiction, by ( 4 . 1 4 ) e B . Therefore, one concludes t h a t

f

,

since

4.

343

HULLS OF IDEALS (CONTN'D.)

n i l ) = E/k(B)

(4.19)

so that the element n f z ) in element y =xu0 E I satisfies (4.20)

.$if) = x$(f)

(4.18) may be taken with z e I . Thus, the fi

= (by (4.8))~

( z z )p l @ ) l

= (by (4.181)1

denotes, of course, the transpose map of (4.16)): thus, ; = I / * which in fact is (4.15), and this completes the proof of the theorem. I (p

We come next to another application of the foregoing by considering conditions on a given locally m-convex algebra which guarantee the existence of an identity element. Thus, we have. Theorem 4.4. L e t E be a given Ptbk-SiZov algebrn (take e.g. a Frgchetsilov algebra), having a compact spectrzm m(E).Then, t h e algebra E has an i d e n t i t y element. Proof.

By considering the set

B= { B G ~ ~ E 3 ) :z e E , with 2 = 1 l B 1

(4.21)

one concludes that f o r any p a i r is,, B2) of c l o s e d s e t s in B , t h e ( c l o s e d ) s e t BI U B,u is an element of 8 as well: Namely, by hypothesis (see also Remark 3.1 and the comment following it) the sets Bl , Be are compact hulls in R ( E ) , so that one has (4.22)

B = hiII)

,

B

and hence (4.23)

I . = k(Bil

,

with i = I , 2.

1

2

= hi12),

Therefore, the respective quotient algebras E / I i ( i = 1 , 2 ) are commutative complete (by hypothesis for E ) locally m-convex algebras which are also semi-simple (Corollary 4 . 2 ) and have compact spectra (cf. Theorem 4.1), that is, (E/I.) h i I i i = Bi ( i = l ,2) (4.24)

.

Consequently, by (4.21) and Corollary VI; 3.1, the algebras E/ii ( i = 1 , 2) have identity elements, in such a way that if e . (i=I , 2 ) are the elements of E , such that ei + I i are the identities in E / I i ( i = l , 2 ) , one obtains that F . Z - z E 1. ( i = 1 , 21, (4.25) for every element (4.26)

X E

E . Thus, one further obtains that

( e1 + e 2 - e ,1e 2 )x-x = - f e 1 ( e 2x - z i - le 2x-x)) e i1 n I 2 '

for every x € E , so that the element (4.27)

(C

1

+ e -e e )+I1nIII? 2 1 2

344

IX STRUCTURE THEORY

is an identity of the algebra E/(Il

n I,).

Therefore (cf. also ( 1 . 3 ) )

,

since (4.28)

Bl U Be = h(1,) U h ( I a ) = h ( I l UI,)

,

one finally concludes that (4.29)

e l + e2 - e I e z = ~ ~ B ~ u B ~ ~

that is the closed set B I U B z C ??,?(El

is again an element of 8 , which

is exactly the assertion. Now, for every element f e m(E), there exists x e E with $(f) f 0, hence by hypothesis for m(E) and the continuity of $ there exists a compact neighborhood, say, K f' of f in m(El, in such a way that 2 never vanishes on K the latter being in fact a compact hull in m(E); f' so (Theorem 4 . 2 ) one concludes that

K

(4.30)

f

E

m(E).

B , with f e

Therefore, by the compactness of m(E),there exist finite many K fDS covering m(E),hence by ( 4 . 3 0 ) and the foregoing m(E) itself is an element of B ; thus, there exists an element z E E , with z^ = I on m(E) , which of course yields the assertion by thesemi-simplicity of E . B As an application of the previous Theorem 4 . 4

one further ob-

tains the foolowing result, in fact, a weak form of "normality" for a given topological algebra E; see also Corollary 4 . 5 below.

Corollary 4.4. L e t E be a given commutative Pta'k regular l o c a l l y m-convex algebra w i t h spectrum ??Z(E) having, moreover, t h e r e s p e c t i v e Gel'fand map continu-

ous. Furthermore, assume t h a t B i s a compact and F a closed subset of m(E) u i t h B n F = 0 . Then, t h e r e e x i s t s an element X E E such t h a t 2 = 1 l B , and

(4.31)

2=0

IF

.

Proof. By ( 3 . 3 ) k(B) is a closed (2-sided) ideal of E l so that E/k(B) i s a commutative Pta'k (cf. Scholium VIII; 9.1) regular (Corollary 4.1) semi-simple (Corollary 4.2) l o c a l l y m-convex algebra ( t h a t i s , a Ptdk-Silov a l g e b m ) having, moreouer, a compact spectrum m(E/k(B))

i z hik(B))

= B

(cf. Theorem 4.1 in connection with Lemma 1 . 1 ) . Therefore (Theorem E/k(B) has an identity element; i.e., there exists an element y e E , with n ( y ) = l on B (where IT:E--*E/~(B)), so that (cf. ( 4 . 8 ) ) y^ never 4.4),

vanishes on B . On the other hand, by considering the (closed) ideal I = k ( F ) in El one gets by hypothesis

4.

345

HULLS OF IDEALS (CONTN'D)

B n h ( I l = B fl h ( k ( F l ) = B n F =

0.

Hence (Theorem 4.3) , there exists an element x € I = k ( F I , thus G = O I F , such that ? = Z I B . So one gets, in fact, the desired element x € E satisfying (4.31), and this completes the proof. I In particular, in any topological algebra E of the type considered in the previous corollary, which besides has a compact spectrum ? ? T ( E ) , the functions of the Gel'fand transform algebra E A "separate closed sets" in ? K ( E ) . Namely, one has.

Corollary 4.5. Let E be a c o m u t a t i v e Ptdk regular ZocaZZy m-convex algebra having a compact spectrum m(E)and t h e r e s p e c t i v e Gel'fand map continuous. Then, E is a normal algebra (Definition 2.2); i . e . , f o r any two d i s j o i n t closed subsets B I , B2 of

m(E),t h e r e

e x i s t s an element x € E such t h a t

Applying another terminology, the preceding conclusion admits the equivalent version (see also Corollary 111; 6.6 ) that every c o m t a t i v e regular Pta'k-Waelbroeck (ZocalZy m-convex) algebra i s nomaZ. On the other hand, we still set the following. D e f i n i t i o n 4.1. Let E be a regular algebra whose spectrum is

NOW, for any B E

m(E),we

m(E/.

denote by J ( B ) the following subset of E:

2 has compact support, Suppl.?) and, moreover, there exists an open neighborhood U

X E J I B ) , if

(4.32)

of B in ? n ( E ) such that Supp(2) G CU ; i.e., one has (4.32.1)

BcUG CSuppl?).

In particular, we denote by J ( m ) the set of those elements x in transforms 2 have compact supports. Thus, applying the preceding terminology, as well as the notation in IV;(4.1) and

E whose Gel'fand

V; ( 1 . 5 )

, one has , by definition,

(4.33)

J ( @ )= J ( m ) = $ - ' ( K ( ? ? ? ( E ) ) I

.

In this respect, we first remark that J ( B ) is a 2-sided i d e a l i n E , f o r every B L f f Z I B ) : Namely, one has, for any x e J ( B I and y e E , supp(x2) = supp(y5)

G

supp(2)

n ~upp(G)E ~uppi.i?;i ECLI

(cf. also IV;(4.1)), so that the elements xcy and y x belong to J ( B I . Thus, one now obtains the following. Lemma 4.1.

Let E be a Ptdk-Silov algebra w i t h spectrwn m ( E l having, more-

IX STRUCTURE THEORY

346

o v e r , t h e r e s p e c t i v e Gel’fand map continuous, and l e t I be any idea7 of E. Then, one has t h e r e l a t i o n J ( h ( I ) )C I .

(4.34) Froof.

If x € J ( h ( I ) ) , then by (4.32.1) !::I)c_

(4.351

u E c supp ( 3 ),

where U is an open set and Supp ( 2 ) a compact set in ??Z(E). Hence, Supp ( 2 ) n hil) = 0

(4.36)

,

the set Supp(2) being, in particular, a compact hull in m(E) (cf.Remark 3.1 and the comment after it). Thus (Corollary 4.4), there is an element y € E such that y^ never vanishes on Supp($),and 4.3) there exists z E I with

Therefore, since 2=0 on cSupp(2), one concludes that by the semi-simplicity of E (Definition 2.3) one gets this completes the pr0of.M

hence (Theorem

f = G I so that x = z z EI,and

Thus, from the preceding discussion, one concludes t h a t J ( h ( I ) ) E I C k ( h ( I ) ),

(4.37)

f o r every i d e a l I i n a g i v e n Ftd;c-Silov algebra, having t h e r e s p e c t i v e Gel’fand map continuous.

On the other hand, under the further assumption that the topological algebras involved have l o c a l l y compact s p e c t r a , one obtains the result that follows. In this respect, when the topological algebras under consideration have continuous Gel‘fand maps (this is often the case in this section, and in the next one too), the above assumption is equivalent, of course, with that of being the considered spectra locally equicontinuous (cf. Corollary VI; 1.3) This is, indeed, a most convenient property in some other context too (topological ten-

.

sor products, for instance, as we shall see in subsequent chapters). Thus, we now have. Lemma 4.2. L e t E be a r e g u l a r algebra (Definition 2.1 ) having a l o c a z l y compact s p e c t m

m(E), and

Let B be any c l o s e d

s u b s e t of ???(E).

the relation

(4.38)

h ( J ( B ) ) = h ( J ( s l ) = B.

Proof. We first remark that one has, from (4.32.1), (4.39)

B

E.

hlJ(3))

Then, one has

5.

341

'THE LOCAL THEOREM

(this is true, in fact, for any regular algebra E and B E m ( E ) ) , so let us suppose that ff!B.Then from the hypothesis for Z ( E ) l there exists an open relatively compact neighborhood, say, U of f in ??Z(E) such that ? n B = 0. Furthermore, there exists an element z € E with 2(f) = I (cf. Definition 2 . 1 ) , and hence a f o r t i o r i ? = O and 2 = 0 ICT ' CU Thus, z e J ( B ) so that since ;(f) # 0, one finally so that Supp(2) G

I

c.

obtains that f , d h ( J ( B ) ) , which together with ( 4 . 3 9 ) B = k(J(B))

(4.40)

proves that

.

On the other hand, one has hlA) = h ( z )

(4.41)

,

f o r any s u b s e t A of a g i v e n t o p o l o g i c a l algebra E (cf. ( 1 . 3 )

one already concludes from ( 4 . 4 0 )

and ( 1 . 9 ) )

the desired rel. ( 4 . 3 8 ) ,

. Thus,

and this

finishes the proof. I An application of the foregoing (see also Theorem VI; 1.2) yields now the following.

Corollary 4.6. Let E be a PtOk-&lov

algebra w i t h spectrum ??Y(E)

and con-

t i n u o u s C e l ' f a n d map, and suppose t h a t B i s a c l o s e d s u b s e t of M ( E ) . Then, t h e s e t J ( E ) (cf. Definition 4 . 1 ) ,

resp.

JIB),

is t h e s m a l l e s t i d e a l , r e s p . c l o s e d

i d e a l , i n E suck t h a t ( see also (4.41)) B = h ( J ( B ) )= klJ(B))

(4.42)

.

Proof. If I is any ideal (resp., closed ideal) in E, with B =h(I),

then from ( 4 . 3 4 )

one gets J(h(I)) = J(B) E J

(4.43

(resp

I

JmC 7 = I

)

,

which proves the assertion. I

5. Further r e s u l t s . The Local Theorem The machinery developed so far enables one to get, within the same context, analogous results to those already valid in case of a commutative regular semi-simple Banach algebra (i.e., in a so-called Banach-&Zov

algebra )

.

Thus concepts like "elements t h a t l o c a l l y belong t o an i d e a z " and the like can be considered, of course, providing an analogue, for example,

of D i t k i n ' s Theorem in the case considered herein. However, we are not going to look at this, but we refer instead to the classical framework of Banach algebras theory for motivation leading to the aforementioned considerations (see e.g. G.W.MACKEY [ l : p. 110 ff.] and/or

IX STRUCTURE THEORY

348

R. LARSEN [I: p. 22 ff.]). Applications of the preceding material in the "structure theory" of topological algebras (not normed ones) are further possible, in connection with methods of Commutative Algebra and Differential Geometry: see, for instance, A. MALLIOS [21: 2 5 1 as well as A . V . FERREIRAG. TOMASSIN1 [I , and E . BALLICO - A . V . FERRE'IRA [l] . In this respect, the following result, the so-called "Local Theo r e m " , has a particular significance: it can be derived from an application of the preceding argument (cf. Corollary 4.5) to the standard methods of commutative Banach algebras theory: See, for instance, I. GEL'FAND-D. R A I K O V - G . SHISOV [I: p. 201, Theorem 1 1 , as well as A . MALLIOS [21] and A . GUICHARDET [2: p. 611 . Thus, we have.

1

Theorem 5.1. Let E be a Ptu'k-Silov algebra w i t h an i d e n t i t y element and equim(E). ( E q u i v a l e n t l y (cf. Theorem 4 . 4 ) , E i s a Psn'k-SiZov algebra having 772IE) compact and t h e r e s p e c t i v e Gel'fand map continuous (see Theorem V I ; 1.1 ) ) . Then, every continuous complex-valued f u n c t i o n h on ??Z(E), which "1ocaZly belongs t o E ^ " (i.e., for every f E m:E), there exists a neighborhod U of f in m(EJ and an element X E E , with h i u = : l l u ) is r e a l i z e d by an element of E (namely, t h e r e e x i s t s an element y EE, w i t h h=y^ on m(E1 1 . I continuous spectrum

We close this section with the following ( separation theorem; cf. (5.2))

-

Theorem 5.2. L e t E be a Ptdk-&Zov

algebra w i t h spectrum

m(E).Then,

the

following condition i s s a t i s f i e d : For any element fem(El and a c l o s e d s u b s e t B of V Y ( E ) , (5.2)

w i t h f B , t h e r e e x i s t s a neighborhood U of f i n U l l B = @ , and an eZement x E E such t h a t

mtE), w i t h

The proof of the previous theorem is based on results from the theory of "continuous l o c a l i z a t i o n " of topological algebras (or yet "Commutative Topological Algebra"); this is, of course, akin to the preceding considerations, being also an important area of applications of the same material. We are not going to consider it any further in this discussion. However, see e.g. A. MALLIOS [21]

.

6. Sets of spectral synthesis. Wiener-Tauber algebras Continuing the argument at the end of Section 3 , we start now

6. WIENER

-

349

TAURRR ALGEBRAS

with the following. D e f i n i t i o n 6.1. Suppose that we have a topological algebra E with spectrum m(E).Now, a closed set B C m(E) is said to be a s e t of spect r a l s y n t h e s i s , whenever I=k(BI (cf. (1.4)) is the only closed ideal in E having B as hull, i.e., such that

B = h ( k ( B ) ) = h(I).

(6.1)

On the other hand, we say that a given topological algebra E is an algebra of spectra2 s y n t h e s i s , if every closed subset of 7?Z(E) is a set of spectral synthesis in the sense of (6.1). Thus, within the context of Theorem 3.1, we conclude that every cZosed subset o f X i s a s e t of s p e c t r a l s y n t h e s i s : Namely, if B C X is closed, then by ( 3 . 1 2 ) B is a hull, in such a way that if IE J ( c c ( X ) l (cf.VI1;

(1.16)), with B=h(l),

one gets

k(BI = k i h ( I ) i = (Theorem 3.1) I ,

(6.2)

that is the assertion (cf-also ( 3 . 1 1 ) ) . Therefore, in connection with Remark VII; 1.2, one concludes with the following. Lemma 6.1.

m-convex algebra synthesis. I

Let X be a completely regular Hausdorff space. Then, t h e l o c a l l y

c (X) (cf.Chapt. VII;Section 1 )

i s an algebra of s p e c t r a l

On the other hand, t h e r e a l s o e x i s t important topological algebras which are not of s p e c t r a l s y n t h e s i s : This is, for instance, the case for the

group algebra L 1 ( G ) of any locally compact (but non compact) abelian topological group G ( MalZiavin’s Theorem; see e.g. R. LARSEN [l : p. 199, Theorem 8.3.41 and/or Y. KATZ’ZNELSON [1:p. 231 ff.] ) . In the sequel we give, in terms of “Wiener-Tauber algebras” (cf. Definition 6.2), a criterion in order the empty set in m(E)to be a set of spectral synthesis (Theorem 6.1). Furthermore, similar criteria are also given for any closed subset of??Z(El and, in particular, for a singleton; see Theorems 6.2 and 6.3, respectively. Thus, we first have the following. D e f i n i t i o n 6.2. A given topological algebra E is said to be a Wiener-Tauber algebra, if one has the relation (see also Definition 4.1) (6.3)

JF) = E.

The last relation is also called the Wiener-Tauber c o n d i t i o n f o r E. That is, we assume that the elements of E whose Gel’fand transforms have

350

IX STRUCTURE THEORY

compact supports are dense in E (see (4.33)). N o w , on the basis of Definition 6.1 and (4.41), one concludes that the previous rel. 16.31 means, e q u i v a l e n t l y , t h a t t h e empty s e t i n ??Z(E) i s a s e t of s p e c t r a l s y n t h e s i s . (Thus, an equivalent d e f i n i t i o n f o r a Wiener-

Tauber a l g e b r a ) . Furthermore, we still have the following criterion (Wiener-Tauber Theorem) :

Theorem 6.1. Let E be a Ptdk-&lov algebra, having a l o c a l l y equicontinuous spectrum ???(El.

m e n , t h e following two a s s e r t i o n s

are equivalent:

1 ) E i s a Wiener-Tauber algebra (Definition 6.2).

2 1 Every proper closed i d e a l i n E has a non-empty h u l l ; i . e . ,

equivalently,

i t i s contained i n a closed regular maximal i d e a l o f E .

Proof. Suppose that J T I = E , and let I be a proper closed ideal in E with h ( I l = # . Then (Corollary 4.6; see also Theorem VI; 1.2), J X = T G I , thus a contradiction to the hypothesis for I, so that one finally gets that h f I l # # , i.e., 1) + 2 ) . On the other hand, by assuming 2), if 1-J the comment following Definition 4.1 )

# E, then (see also

J(B)

would be a proper closed ideal in E l so that by hypothesis would have a non-empty hull. Thus, h ( J ( / I ) f #, which is a contradiction according to (4.38)(see also C o rollary vI; 1.3), that is 2)=> 1 ) as well, and the proof is complete (cf. also Corollary 11; 7.2). 1 Since in the spectrum of a spectrally barrelled algebra bounded

and relatively compact sets are the same (Theorem VIII; 1.1 ), while the same is true concerning locally compact and locally equicontinuous sets (Theorem V; 1.1) , one has in particular the following.

Corollary 6.1. Suppose that E i s a Ptdk-SiZov s p e c t r a l l y b a r r e l l e d algebra w i t h a l o c a l l y compact spectrum 1 7 2 ( E l . Then, t h e f o l l o w i n g c o n d i t i o n s are equivaZent: 1 ) The s e t of those elements of E whose GeZ’fand transforms have bounded

supports i s dense i n E. 2 ) Every proper closed i d e a l i n E i s contained

i n a closed regular maximal

i d e a l o f E. I

As we promised already we give next two further criteria in order a singleton and an arbitrary closed set in the spectrum of a given (suitable) topological algebra to be sets of spectral synthesis (see Definition 6 . 1 ) . The previous discussion provides, as we remarked before, a simi-

6.

WIENER-TAUBER

ALGEBRAS

351

lar criterion for the empty set Theorem 6.1 and its corollary); hence in this case one thus obtains a one-to-one correspondence between closed s e t s i n m(E) and closed i d e a l s in E (a particular instance of the situation described by the rel. (3.10). In this connection, we still remark that the terminology pointed out by the next definition is also applied (cf. e.g. C . E . R I C W T [I: p. 92, Definition (2.7.6)] ) . D e f i n i t i o n 6.3. Let E be a topological algebra with spectrum ??Z(E). A closed ideal I in E is called a primary i d e a l , if its hull h ( I ) G m(E) (Definition 1.1) consists exactly of one element. (In case of a Gel'fand-Mazur algebra, it is equivalent to say (see (1.3)) that I is contained in exactly one closed regular maximal (2-sided) ideal of E ) . On the other hand, we shall say that a given topological algebra E is a primary algebra, if m(E) consists of just one point. (The term local algebra is also applied).

In this respect, it should be noticed that local (topological) algebras play an important r61e in the general theory of topological algebras as it concerns the respective structure theory. We are not going, however, to consider this class of algebras at all, at least, at this stage of this discussion. In this concern, see e.g. A . MIRKIL [I] for the case of local Banach algebras and/or G. TOMASSINI [I; 2; 3 1 for more general topological algebras alludded to before. Now, we first have the following. Theorem 6.2.

Let E be a Pta'k-&lov

algebra w i t h a locally eqtiicontinuous

s p e c t m m(E), and l e t f be an element of

m(E).Then, t h e following two asser-

t i o n s are equivalent: 1 ) The one-point

s e t {f}'

m(E) i s a s e t of s p e c t r a l s y n t h e s i s .

2) One has t h e r e l a t i o n

kerff) = J ( S ) . (Applying the notation of Definition 4.1, one sets here J(f) = J ( { f } ) ) .

Proof. Assuming that I = ker(f) P k ( f ) is the only closed ideal in E l with h ( k ( f l l = { f } = f , one concludes (6.4) by Lemma 4.2, so that 1 ) +2).On the other hand, if (6.4) is valid and I is a closed ideal in E with h ( I ) = f (thus a primary ideal in E satisfying the last relation), then by Lemma 4.1 (cf. also Theorem V I ; 1.2) one gets

J(fl G

I G

(by (1.3)) kerif)= J c f ) .

Thus, I = ker (f), that is 2)*I)

as well and the proof is complete. I

352

IX STRUCTURE THEORY

Thus, stated otherwise, the preceding result provides a criterion in order an element f enT(EI to fulfil the condition: There does not e x i s t any primary ideal I i n E w i t h I E kerlf) (except of course of the maximal ideal ker(fl itself). Now as a consequence of the next Theorem 6.3 and Corollary 4 . 4 , one concludes that: If every one-point s e t of ??Z(E) i s a s e t of spectral synt h e s i s , t h e same i s v a l i d f o r every f i n i t e subset of 7 7 2 ( E ) . (In this respect, cf., for instance, G.W. MACKEY [ I : p. 109, Corollary]). We come finally to the following promised result (being, in effect, an extension of the preceding two Theorems 6.1 and 6 . 2 ) .

Theorem 6.3. Let E be a Pta%-.%lov algebra w i t h a l o c a l l y equicontinuous spectrum m(E),and l e t B be any closed subset

of m(E).Then, the following two

propositions are equivalent: 1 ) The s e t B G m(E/ i s a s e t of spectral s y n t h e s i s . 2) m e following r e l a t i o n i s v a l i d , i . e . ,

-

k(B/ = J ( B I

(6.5)

.

Proof. Assuming 1) we conclude (see Definition 6 . 1 )

that k ( B I is the only closed ideal in E , with B = h(k(BII; thus ( 6 . 5 ) is now a consequence of Lemma 4 . 2 (cf. also Corollary VI; 1.3).On the other hand, if ( 6 . 5 ) is true and I is a closed ideal in E with h l I ) = B , then one concludes from Lemma 4 . 1 and (1.4) that

-

J ( B ) C I E k(B)

= (by ( 6 . 5 ) )

J(B) ,

Therefore, I = k ( B I that is B is thus a set of spectral synthesis, and this completes the proof. I

7. Uniform algebras (contn’d.).

Riemann algebras

We consider in this final section of the present chapter a certain particular class of topological algebras, which are uniform (cf. Definition VIII; 5.1) and consist of holomorphic functions. Thus, the ensuing discussion may be considered as another application of the argument used in Chapt.VII1;Section 5 . On the other hand, it was indeed the particular “structural property” referred to the (maximal) closed ideals of the preceding class of algebras (cf. Theorem 1.1 below) that exhorted us to deal with it in the present chapter. So the algebras under consideration consist of (complex-valued) holomorphic functions defined on 1-dimensional complex (analytic)manifolds. We first recall some relevant terminology which lies at the basis of their definition (see also the relevant discussion in Chapt.

7.

VII: Section 3 )

R I E M A " ALGEBRAS

353

.

D e f i n i t i o n 7.1. By a Riemann surface we mean a complex analytic manifold X of (complex)dimension 1 (or,equivalently,of real dimension 2 . S o X will be, by definition, a second countable and connected Hausdorff topological space which is a 2-dimensional (topological) manifold endowed with the pertinent compZex s t m c t u r e ; cf. , for example, 0.FORSTER

[6: p. 3, Definition 1.41 ) . NOW, a given topological algebra E is said to be a Riemann algeb r a , if it is isomorphic (as a topological algebra) with the algebra

O t X ) of a Riemann surface IX, 0) (see also Chapt. VII; (3.1)). On the other hand, we are exclusively concerned in the sequel with non-compact Riemann surfaces, hence one may regard it as part of the previous definition. NOW, since a (non-compact) Riemann surface (X, O x ) i s a S t e i n manifold (see R . C . GUNNING - H . ROSSI [I : p. 270, Theorem 101 ) , one concludes that a given Riemann algebra E is, i n p a r t i c u l a r , a S t e i n algebra (cf. Chapt. VII; (3.2)) hence by its definition a commutative FrEchet lovally m-convex algebra with an identity element. Furthermore, by Lemma VII; 3.1, i t s spectmvn m l E ) i s homeomorphic t o the given Riemann surface X , thus a (second countable) l o c a l l y compact and connected (Hausdorff) space. Moreover, since any complex manifold is, in particular, a reduced complex analytic space (cf., for instance, C. ANDREIAN CAZACU [1: p. 328 ff.] or yet L.lL4UP-B. KAUP [l: p. 107]), one concludes that every Riemann algebra is semi-sinple (see also Scholium VII; 3.1, (3.10)). Thus (Definition 7.1), for every Riemann algebra E one gets

E

(7.lJ

0(X) 5

ec(X)

(cf. also VII; (3.10)), within a topological algebraic isomorphism of the topological algebras involved. S o t h e given Riemann algebra E i s , i n f a c t , a uniform algebra (a consequence of the previous discussion and Theorem VIII; 5.1). Thus, we can summarize the preceding into the form of the following. Lemma 7.1.

Let E be a given Riemann algebra. Then, E i s a c o m u t a t i v e Frgehet

uniform ( l o c a l l y m-convex) algebra w i t h an i d e n t i t y element whose spectrwn

(E) i s

a l o c a l l y compact and connected space (different from the one-point space).I

On the other hand, a Riemann algebra Ebeing by the preceding a Stein algebra, has further the property that: Any closed i d e a l (hence,in particular, any closed maximal i d e a l ) i n E i s a principal i d e a l . (In this re-

IX STRUCTURE THEORY

354

spect, we recall that an ideal I in E is said to be a p r i n c i p a l i d e a l

,

whenever there exists an element x in E , with (x)= E= = I ) . The last statement is a consequence of a more general result (valid, namely, for any I-dimensional complex space) due to 0 . FORSTER [3:p. 395, Satz 5.21. As a matter of fact, the last property of a Riemann algebra,in conjunction with those exhibited in the preceding Lemma 7.1, characterizes the class of Riemann algebras among FrEchet uniform algebras. This rather recent result is due in the form given below to B . KRAIM [I: p. 394, Theorem]. In this concern, see also 1. RICHARDS [ I ] for an expository account on the subject that actually started with a question of S. Kakutani at the early fifties, who asked for a "characteri z a t i o n o f Riemann algebras v i a i n t r i n s i c properties".

Thus, more precisely, one has the following.

Theorem 7.1. Let E be a Frkchet u n i f o m algebra w i t h an i d e n t i t y element and

m(E) a l o c a l l y compact and connected space (we assure that a singleton). Then, t h e following a s s e r t i o n s are e q u i v a l e n t : spectrwn

m(E) is not

1 ) E i s a Riemann algebra. 2 ) Every closed maximal i d e a l i n E i s p r i n c i p a l .

3 ) Every closed i d e a l i n E i s p r i n c i p a l .

Proof. It has already been noted in the previous discussion that 1 ) implies 3) , hence 2) as well, so that -it s u f f i c e s t o prove t h a t 21=>1).

Thus, one proves that m ( E ) i s endowed w i t h t h e s t r u c t u r e of a Riemann surface in such a manner that E is isomorphic, as a topological algebra, to 0I

nz (EII :

NOW, as a consequence of a previous result of R . CARPENTER [ I : p. 562, Corollary 2.71, prop. 2) entails, under the hypothesis of the present theorem, that m(E)has t h e s t r u c t u r e o f a I-dimensional complex (analytic) manifold such that one has (see also Lemma VIII: 5.1 )

E

(7.2) where $ i E ) (7.3)

S i E l E O(??Y(E))

i s a closed subalgebra o f o C % ( E ) ) . E"

=

,

Indeed, one has

g(E)= O(%?(E))

proving that t h e Gel'fand t r a n s f o m algebra of E i s (via (7.2)) a dense subalgebra of O(m(E)).Thus, concerning the algebra $ ( E l , following facts:

one concludes the

1 ) ?&?(El is S ( E 1 - s e p a r a b l e ; i.e., the algebra $ I E )

=

E"

separates

the points of ?n(E). This is easy by the same definitions. 2) m(E) is $(E)-convex; i.e., for every compact K L m ( E ) , the set

7.

?

(7.4)

355

R I E M A " ALGEBRAS

= {f&???(EJ: l?(fl[ 5

I/ 2

(IK

V ZE

E1

(i.e., the $(E)-convex hull of K ) is also compact: Indeed, if KC W(E) is any compact subset, let (7.5)

Ex= E

m

be the respective term of the corresponding Arens-Michael decomposition of E , when the latter is, by hypothesis, identified with $(El; SO one sets by definition (7.6) Therefore, one concludes the relation (cf. also Theorem Vi5.1, as well as Scholium VIII; 3.1) (7.71

within a homeomorphism (where K denotes a "fundamental family" of compact subsets of ?7Z(El). NOV, since the relation which defines (7.4) is essentially that which characterizes the continuous multiplicative linear forms (characters) of the commutative Banach algebra (with an identity element) E K , one finally gets ??Z(EK) =

(7.8)

thus,

C

;

2

is a compact subset of m(E), which is the assertion. 3 ) m(EI is s(E/-regular; i.e., the algebra E n provides (via (7.2)) "local-global coordinates" for m ( E l . This is essentially true according to I?. CARPELTER [ I : p. 562,Corollary 2.71 ) .

Now, the preceding three properties of $ ( E l G ( l ( m ( E ) ) , in conjunction with the fact that ??ZIEl is a non-compact 1-dimensional complex manifold, hence a Stein manifold, guarantee already through the classical Oh-veil-Cartan fieorern (see, for instance, H . BEHNKE-P. THULLEN [I: p. 145, Satz lo]) that $ ( E l is a dense subalgebra of O W Z C E l ) . Thus from the hypothesis for the algebra E , one actually obtains the desired rel. (7.3), and this completes the proof of the theorem. I Concerning the previous Theorem 7.1, it is clear that cond. 1) implies already the properties for E asserted by the above Lemma 7.1. Thus, the conditions for E set forth by the same theorem refer, ineffect, to the rest two equivalent assertions of the statement.

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PART I I

TOPOLOGICAL TENSOR PRODUCTS

This Page Intentionally Left Blank

359

Topological Tensor Products o f Topological Algebras

CHAPTER X

We apply in this chapter the machinery developed hitherto to the case of suitably topologized tensor products of topological algebras, and in fact of locally convex ones. The latter notion is mainly motivated by the classical work of A. GROTHENDIECK [3] concerning locally convex topological vector spaces (see also L . SCHWARE [2]). Thus, we start with the necessary algebraic background material.

1. Algebraic preliminaries We begin with recalling the algebraic definition of the tensor product of two (complex) vector spaces (and hence of any finite number of them. "Infinite tensor products", suitably topologized, will also be considered in the sequel). Furthermore,we give the (algebraic) decinition of the tensor product of two (and hence of any finite number) of (complex) algebras. Thus, if E , F are vector spaces over the field Q: of complex numbers, we denote by Co(ExF) = C (EX F)

(1. I )

the free (complex) vector space defined by the set E x F , equivalently, the set of all complex-valued functions on Ex F having finite supports (they vanish except of a finite subset of ExF);the same is vector space isomorphic to the (vector space) direct sum of the second member of ( 1 . 1 ) . Now, denoting by

6: ExF-

(1.2)

Co(ExF)

the canonical injection map, the set Im6G Co(ExF) defines a basis of (1.1) ; moreover, let H be the (vector) subspace of Co(Ex F ) spanned by all elements of the form G(x;c+px',

9 ) - Uix, y ) -pG(x', y l

(1.3) d(x, Xycpy'l

with

-

Xd(x, y l -uG(x, 9')

,

(2,x ' ) e

E x E , (y, y') E F X F , and (X,p) E C x C . Thus, by considering the quotient space eo(ErF)/h', if

p

denotes

360

X TOPOLOGICAL TENSOR PRODUCTS

the respective quotient map, one defines a canonical map $=po6:ExF+G=C,(EXF)/H

(1.4)

which, in fact, is a b i l i n e a r map (linear in each variable) between the corresponding vector spaces. This follows immediately from the fact that the respective "linearization" of 4 through C o I E x F ) , being by definition the quotient map p , has obviously the property that ker(p)2 H . Now the last relation is, by the same definition of H , a necessary and sufficient condition in order that a map from E x F into a vector space G to be bilinear. Thus, given the vector spaces E , P one defines a pair (G,$), as above, characterized (within a vector space isomorphism for G), by the following "universa2 property" : For any b i l i n e a r map $ of E X F i n t o a vector space M , there e x i s t s a unique linear map ? : G+M which " f a c t o r i z e s " through I$, i . e . , one has T = ? c @ .

The vector space G thus defined is called t h e tensor product of the given pair ( E , F ) and is denoted by E 8 F . Thus, for any vector space M , one gets a canonicaZ (vector space) isomorphism (1.5)

B ( E , F; M ) = L(EBF, M I

of the vector space B ( E , F ; M ) of bilinear maps of E x F into M onto the vector space L I E @ F , M) of linear maps of E 8 F into M. Moreover, G is spanned by Im@, so that the general element of E Q F is of the form (1.6)

with x. E E and y . E F ( i = l , .. .,n ) . Besides, for any Ix,y) E E x F , one defines the respective decomposable tensor by x 8 y $(x, y ) , so that one actually gets (1.7) E @ F = [Im$] (linear hull of Im@ in E B F ) . In the sequel for any A C E and B C F , we set (1.8)

A 8 B = { x @ y e E B F : x € A ,y E B } .

Before we give the definition of the tensor product of two algebras, we shall briefly discuss some basic properties of the tensor product of two given vector spaces, which we already defined, and which will be useful in the sequel. Thus, one has the following basic lemmas. Lemma 1 . 1 . Given tuo vector spaces E , F and a decomposable tensor x a y E

1.

ALGEBRAIC PRELIMINARIES

E 8 F , one has z@y # 0 if, and o n l y if, x # 0 and y

36 1

# 0.1

One half of the above lemma is evident, since one has from the bilinearity of @ the relation xa0 = 0ay = 0 ,

for any

XE

E , y e F . On the other hand, we still have

Lemma 1.2. For any element t # 0 in E Q F , one g e t s n i=l z

t = t x.ay.

(1.9)

where t h e fami1;os (x .) C E and f y i i

(1.9)

C

z’

F are l i n e a r l y independent. I

The positive integer n for which a decomposition of t e E Q F like holds true is called the rank of t , and is characterized as the

least nelN for which such a decomposition is valid, concerning the element t # 0 in E B F .

Lemma 1.3. Suppose t h a t (xi)l5i5n is a l i n e a r l y independent family in E , and let

t

(1.lo)

Then, one g e t s y

1

=

... = yn = 0

n

t cC.ay.=o. i = ] 7, z

in F . I

As a consequence of the preceding lemmas one obtains.

Theorem 1.1. An element t E B Q F is d i f f e r e n t from zero if, and only if, one has

n

t = 1 x.ay i=l z

i

’

where (x.) and (y .I are Linearly independent f a m i l i e s i n E and F , r e s p e c t i v e l y . 1

For proofs of the previous lemmas see, for instance, L.CHAMEADAL-J.L. OVAERT [I] , where we also refer for further details on the preceding material. (See a l s o G. KGTHE [I: p. 76 ff.1 ) .

We come next to the definition of the tensor product of two algebras E and F , supplying thus the respective tensor product vector space E Q F , defined above, with a natural multiplication. That is we have the following lemma-definition.

Lemma 1.4. L e t E and F be two given algebras, and E 8 F t h e corresponding t e n s o r product v e c t o r space. Then, f o r any p a i r ( s , t l o f elements in E Q F , w i t h n rn (1.11) s = t a . a b . and t = t x.ayj, j=1 3 i = I 7,

X TOPOLOGICAL TENSOR PRODUCTS

362

the r e l a t i o n

s.t =

(1.12)

I a . x . @ bi y j

'J

i,j

defines a "multipZication" on E 8 F making i t into a (complex linear a s s o c i a t i v e ) algebra. Proof.

For any ( a . ) E E and

(b,l

-C F

( i= I , .

6

.. ,n )

and every

(x, y ) a

E x F , the relation C $ ~ ( X y,

(1 .13)

n l = .I aa. x a b.y z.

defines a bilinear map Q S : E x F + E @ F , so that by the "universal property" of E @ F there exists a unique linear map $ : E 8 F + E @ F such m that qS (x @ yl = C$s ( x , y I . Hence , for every .1= . t x . @ y . a E @ F , one obtains 3=1J

3

from ( 1 - 1 3) in

n

(1.14)

in such In a similar way one gets the linear map $ t : E 8 F + E 8 F iT 5 ( a @ b )= $ t ( a , b ) = .I ax.@ by for every ( a , b l e E x F , so that

a way that

t

3=1

3

J-'

in connection with (1.14), one fi-nally obtains m n (1 . 1 5 ) $ , ( t )= . I I a . x . a biyj = $ t ( ~ l. j=i

i=I z3

Therefore, one may define 5.t =

(1.16)

6 ( t l = 6,lSl ,

for every pair (s, t l of elements in E 8 F , given by ( 1 .I 1). Thus, (1.12) is "well-defined", independently of the particular representations of s , t in E a 9 F according to (1.11). Now, the bilinearity of the multiplication map defined by (1.16) follows immediately from the linearity of the respective maps 6, and $t .Moreover, its associativity is easily checked through (1.16) arguing on the decomposable tensors in E@F,taking also the associativity

of the given algebras E and F into account, and this terminates the proof. I NOW, analogously with the case of two vector spaces, one further defines the tensor product of any finite family ( E i l i s i s s of (complex) vector spaces as a solution to the "universal problem" concerning the "transcription of multilinear maps (n-linear maps) to linear ones". Thus, one defines uniquely (in fact, within a vector space isomorI?

phism) a vector space 8 Ei ( t e n s o r product of the given (finite) family i=1 (E.)) together with a canonical n-linear map

'

n

I$: fl Ei-

(1 .17)

8 Ei i=1

i=I

i n s u c h a way t h a t ,

n 8

2.

i=I z 11

w i t h (xi) E

.n

n T

Ei+

:

F

,t h e r e

exists

li

a u n i q u e l i n e a r map ?:

(1. l a )

,

f o r e v e r y n - l i n e a r map

so t h a t c o n c e r n i n g a

363

ALGEBRAIC PRELIMINARIES

1.

f a c t o r i z i n g t h r o u g h I$; i . e . , n decomposabZe tensor i n .;=I 8 E. z we a l s o w r i t e 8 Ei+F i=l

= x o m 2 @ . ..ox 1

?=TO$,

I$(xl,. .., z n ) ,

B. .

z=l 2 I f t h e v e c t o r s p a c e s Ei

( i = l ..., , n ) a r e a l g e b r a s , one c o n s i d e r s n on t h e r e s p e c t i v e t e n s o r p r o d u c t v e c t o r s p a c e &Ei an a l g e b r a s t r u c -

t u r e by d e f i n i n g a m u l t i p l i c a t i o n i n t h e l a t t e r s p a c e i n a s i m i l a r way a s i n t h e c a s e of two a l g e b r a s (Lemma 1 . 4 ) .

Thus, f o r decompos-

a b l e t e n s o r s one h a s , i n p a r t i c u l a r , t h e r e l a t i o n n ( O a . ) . ( S x . ) = . @ a.x. (1.19) ,;

n

f o r any ( a i ) , ( z i I

in

n

Ei

i=l

zz

,i z

7,

.

F u r t h e r m o r e , the formation of ( f i n i t e ) tensor products i s associative i n t h e s e n s e of t h e f o l l o w i n g .

Lemma 1.5. Let I be any f i n i t e s e t and I=KUL a partstior? oJ it. Furthermore, l e t (EilieK , ( E j I j e gebras). Then, one obtains

be two families of (complex) vector spaces ( r e s p . , al-

(1.20)

w i t h i n a vector space (resp., algebra) isomorphjsm h, given by t h e relation (1.21)

f o r every element (zctJct I e

n

E,.

ct € I

I

W e r e f e r t h e r e a d e r t o L . CHAMBADAL - J . L. OVAERT [I : p. 87 f f .] f o r a

p r o o f of t h e p r e c e d i n g leinma, a s w e l l a s f o r f u r t h e r d e t a i l s c o n c e r n i n g f i n i t e t e n s o r p r o d u c t s of v e c t o r s p a c e s . W e close t h e p r e s e n t s e c t i o n w i t h f u r t h e r r e c a l l i n g s o m e b a s i c

f a c t s a b o u t inductive systems of tensor product algebras

which, i n p a r t i c u -

l a r , w i l l b e c o n s i d e r e d i n t h e s e q u e l (endowed w i t h s u i t a b l e t o p o l o gies). T h u s , g i v e n t h e i n d u c t i v e s y s t e m s of a l g e b r a s ( E U ,

(FA,SPA; h E K ) (see C h a p t . IV;S e c t i o n 1 ) ing (tensor product) family

, one

f,,;u € I ) and

c o n s i d e r s t h e correspond-

364

X TOPOLOGICAL TENSOR PRODUCTS

(1.22)

(Ea@FA),

(a,A/ E

IxK,

where, of course, the index set I x K is directed (upwards) by the "Cartesian product order" (cf. e.g. N. BOURBAhT [l: p. 231 ), together with the corresponding family of "connecting algebra morphisms" (1.23)

f a C l @ g u AEa@Fx+ :

E QF

B P I in IxK.NOW, it is a direct consequence of the

for any ( a , A)<(B,u) very definitions that (1.24)

(Ea@ FA,faa@guAI, (a,A) E I x K ,

i s an i n d u c t i v e system of algebras (see Chapt. IV; Seciion 1 ) , (1.25)

H

6

lim(Ec@F 4

TO

that let

a)

be the respective i n d u c t i v e l i m i t algebra (cf. Definition IV; 1.1 1. This will further be endowed with the corresponding final topology making it a topological algebra, in case the tensor product algebras in ( 1 . 2 2 ) are suitably topologized (cf. Lemma IV; 2.2).

2. Topological tensor products of locally convex spaces We discuss in this section briefly three important ways of topologizing the tensor product EBF of two locally convex spaces E and F ( 2 la Grothendieck [ 3 ] ) . These will be also at the basis of our subsequent considerations concerning the respective topologization of the tensor product of two given (locally convex) topological algebras. True we shall use a more general notion characterizing the "compatibility" of a locally convex (vector space) topology on E @ F (also due to Grothendieck; ibid.). The analogous notion for topological tensor product algebras will suffice for the most of our applications. Finally, we consider the "&-(tensor) product" of L. SCHWARTZ [ Z ] in connection with the so-called "Banach-Grothendieck approximation condition" (cf. Definition 2.4 below). This tensorial topology will also be applied in the sequel within the same context of topological tensor product algebras. 2 . ( 1 ) . The projective tensorial topology IT.- We start with the follow-

ing definition (cf. also A . GROTHENDIECK [3: Chap. I; p. 88, no 3 1 ) .

Definition 2.1. Let E l F be locally convex spaces and E BF the respective tensor product of E,F. A locally convex vector space topology, say, T on E Q F is said to be a compatible topology (with the tensor product vector space structure of E@F,we denote the locally convex

2.(1).

space thus defined by E 8 F and its completion by T lowing two conditions are satisfied: (2.1)

365

PROJECTIVE TENSOR PRODUCTS

E 6 F ), T

if the fol-

The canonical b i z i n e a r map @ : E XF + E @ F

T

i s separately (i.e., in each variable) continuous.

(2.2) One has t h e r e l a t i o n E ' 0 F'

5 (E

0F)' T

concerning the r e s p e c t i v e topoZogical dual spaces.

Concerning the above cond. (2.2),we remark that one mainly requires that for every pair (x', y ' l e E ' x F' the linear form x ' m y ' e ( E 8 F ) " is continuous when E 8 F is endowed with T , therefore, one essentially gets, by definition, a "canonical embedding" (linear injection) of the vector spaces indicated in (2.2). In this respect, we are going to apply in the sequel the following stronger compatibility condition than is (2.2); i.e. , (2.3) For any equicontinuous subsets ACE' and BC- F

',

the set A 0 B

(

{ x ' e y ' : x ' € A , y ' e B } ) i s an equicontinuous subset of ( E 8 F ) ' . T

It is certainly clear that (2.3) implies (2.2); furthermore,we are going to apply Definition 2.1 in the sequel under the form of conds. (2.1) and (2.3) mainly. Thus, as a first fundamental example of the previous Definition 2.1 (in fact, of its stronger form, provided by (2.3)), we consider next the projective tensorial topology on E 8 F . That is, we have.

1 be locaZly convex spaces B BEK together w i t h corresponding f a m i l i e s of (continuous) semi-norms defining t h e i r t o pologies. Then, t h e r e l a t i o n n r ( 2 ) = inf 2 pa(xi!q ( y .) (2.4) a,B i=l 6 % n where "inf i s taken over a l l expressions of t h e element z i n the f o m z = .I x.my i=i t i E F B F, provides a family ( rcl, I B) I of semi-norms on E @ F such tkaiat t h e Lemma 2.1. Let

E, (pa ,Ja

and

{ F, (q I

respectiue 'uocally conz.'1;x vector .?pace topolog9 thus defined i s a compatible topology on E B F (conds. (2.1), (2.3)). I n p a r t i c u l a r , f o r every decomposable tensor x e y E E 8 F , one has t h e relation r (2.5) ( X O Y ) = p ( x ) q (yl , a,6 a B f o r every p a i r ( a , @ ) of i n d i c e s from I x K .

366

X TOPOLOGICAL TENSOR PRODUCTS

FinalZy, the semi-norm on E Q F defined by ( 2 . 4 ) i s t h e gauge of r ( U a 8 V B ) , the balanced and convex (”absoZuteZy convex”) huZl of t h e s e t U a Q V

i n E @ F , where B l U a l a E I , resp., ( V a l B f K ,are ZocaZ bases i n E and F , r e s p e c t i v e l y , defined by the given f a m i l i e s of semi-norms, a s above.

The locally convex (vector space) topology defined on E @ F by the preceding lemma is denoted by

TI

and called the p r o j e c t i v e tensor

product topoZogy (or yet p r o j e c t i v e t e n s o r i a l topology) on E Q F I while the

corresponding locally convex space is denoted by E 8 F and its completion by E @ F . (The notation E & F is also used seve:a1 times). 71

Proof of Lema 2.1. We first on E Q F , for any pair ( p , 4 ) of n a % is, for any z = t xioyi and z ‘ = i=l

note that (2.4) defines a semi-norm semi-norms on E , F , respectively. That m t x.’oy: in E Q F , one gets from (2.4) j=] 3 J

for every E > O and suitable expressions of z and z‘ corresponding to the above sums. Hence, if z + z ’ = tains applying again (2.4)

Ia oBX = ( ~ x i o y i ) + ( ~ x ’ a y : )one , obA x

2

j

3

3

(2.7)

for any

E>

0; thus, one gets

Furthermore, it is clear that

for any X E C and z E E Q F ,which proves the assertion about (2.4). Thus, the same relation defines a family

of semi-norms on E @ F making it a locally convex space. Moreover, f o r every decomposable tensor x o y E E @ F one has ( 2 . 5 ) : Indeed, it is clear from (2.4) that r ( z @ y ) Z p a ( x ) q ( y ) , for any decomposable tensor x a y e E 8 F a,B B and any pair of indices ( a , @ ) eI x K. Besides (Hahn-Banach;cf. A.P. ROBERTSON - W.J. ROBERTSON [ I : p. 29, Corollary 21 ) there exist x’f E’ and y ’ e F’ with x ’ ( x J = p a ( x l and I x ’ ( u ) ( S p a l ( u ) , u e E , resp., y ’ ( y l = q B ( y ) and ly’(u)l
PROJECTIVE TENSOR PRODUCTS

2.(1).

) I r ' o y 7 ( s o y l I ~ ~ l e ' ( x i l Ily'(yill
,

Thus, f r o m ( 2 . 4 )

.

B ( 2y ..)

one concludes t h a t I(z'o y'j(xmy)I S r

a,B

therefore,

367

(xoy)

,

one h a s from t h e p r e c e d i n g

= pa(x)-qB(y) s r

I(r'oy3(soyj(=lx'ix)(.ly'(y)l

a,B

(xoy)

which t o g e t h e r w i t h t h e f i r s t o f t h e r e l a t i o n s p r o v e d a b o v e y i e l d s (2.5). NOW, c o n c e r n i n g t h e c a n o n i c a l b i l i n e a r map

(2.11)

=x@y

@ : E x F - - t E B F : (x,y)-@(x,y)

o n e h a s from ( 2 . 5 ) t h e r e l a t i o n

r

(2.12)

a,

f@(x,y ) ) = r ( x o y ) = pu(xlq f y l a,B R

a p p l y i n g a s t a n d a r d a r g u m e n t f o r t h e c o n t i n u i t y of a b i l i n e a r map ( c f . e.g. N . BOURBAKI [6:Chap. 2 ; p. 43, P r o p o s i t i o n 4 1 ) .

Thus, t h e map ( 2 . 1 1 ) is

continuous ( i n b o t h v a r i a b l e s ) , h e n c e a f o r t i o r i c o n d . ( 2 . 1 )

of D e f i -

nition 2.1 is satisfied. NOW,

b e f o r e w e proceed t o t h e v e r i f i c a t i o n of

( 2 . 3 ) , w e come

n e x t t o t h e f i n a l p a r t of t h e a s s e r t i o n : Namely, f o r any p a i r ( p a , q ) of semi-norms

on E , F,

respectively,

are

i f U,, V

a

t h e corresponding " u n i t

B (see Chapt. I ; ( l . l ) ) , w e p r o v e t h e r e l a t i o n

pseudo-spheres"

p r i u a 8 v B l = ru,B

(2.13)

*

Here t h e f i r s t member d e n o t e s t h e r e s p e c t i v e g a u g e f u n c t i o n d e f i n e d by t h e set r ( U a @ V B ) as i n t h e

( c f . a l s o C h a p t . I ; ( 1 . 2 ) ) : SO

statement

t h e l a s t s e t b e i n g , by i t s d e f i n i t i o n , a b s o l u t e l y c o n v e x , o n e e a s i l y p r o v e s t h a t t h e same i s a l s o a n a b s o r b i n g s u b s e t o f

E 6 F , hence t h e

(2.13) i s well-defined ( i b i d . ) . n O n t h e o t h e r h a n d , f o r any z = . Z x . @ y . € E B Fand p a , q a

f i r s t member o f

-L

a

given s e m i -

norms o n E, F , r e s p e c t i v e l y , a s a b o v e , o n e h a s

s u c h t h a t a 7,.

. .,n )

(i=l,.

xi

I

___ P p i l E Ua and b .

and

A

=_

n 2 Ai > 0 i=I

,

I

Yi ClB

'Yi j

e Va

.

Hence,

one o b t a i n s

t h a t i s z e A r ( U a @ V ) , s o t h a t from 1 ; ( 1 . 3 ) o n e h a s

B

if

Xi= pa(xi)q;(yi) ,

X

368

hence, p

(2) rtuaava)

5 r

a,4

TOPOLOGICAL TENSOR FBODUCTS

(2).

Furthermore, one easily verifies on the

basis of 1;(1.3) and (2.5), that r a - g l z l < A , for every A > O , X T ( U B V I , so that by I; (1.2) one obtains r a , % ( z l 5 p fZI a

rtuclev6

%

with z E which t e r -

minates t h e proof o f ( 2 . 1 3 1 .

A

G

Now, if ASE'and BEF' are given equicontinuous sets, one has U o and B E V i , for some fa,@)E I x K , so that o n e gets A8B C U Z B V ;

C (TIu,BVB))o;

the last relation is true, since one has ) ( x ' o y ' ) ( ~ 2X.. (2x . o2.y . ) )5) I ~Xi~.~x'(xiJ~.~y'(~.J~ 5 1 \Xi\ 6 2 , n for any x ' @ y ' € A B B and z =.I X . ( x i o y i ) € r ( U a B V I . Therefore, since from 2=1

a

'7

(2.13) r(UaC3V ) is a neighborhood of zero in E B F (cf., for instance, a 71 Proposition I; 1.3 ) , one concludes from ( 2 . 1 4 ) that A B B is an equicontinuous subset of (EOF)' and t h i s proves (2.3). We have proved by now 71 that the projective tensorial topology 1~ is a compatible topology on E B F (in the stronger sense), so that the proof of the lemma is fully established. I The following is now a direct consequence of the previous proof. Thus, we have. Corollary 2.1. Let E and F be l o c a l l y eonvex spaces w i t h (U,), ( V I , B E K , local bases of E and F, r e s p e c t i v e l y . Then, t h e family

aE1,

and

B

(2.15)

{T([email protected])}(a,B)E I X K

d e f i n e s a local b a s i s of the l o c a l l y eonvex space E8F, where r ( U a @ V 1 denotes the D C E8F. I

balanced and convex kul2 of the s e t U a @ V

8-

NOW, from the universal property of E 8 F (see Section I), one concludes that every continuous bilinear map u of E x F into a locally convex space G gives rise to a uniquely defined continuous linear map :EBF--+G; indeed, one has IT

(2.16)

v ( i i ( x o y l ) = v(u(x, y l ) 5

x. p ( x l q ( y l

= X.r(xoyi

.

Here v is any continuous semi-norm on G I and r the continuous seminorm on E 8 F defined by (2.4) with p , q continuous semi-norms on E l F, respectively. (We also say that r s p @ q is the (projective)tensor prod-

I N D U C T I V E TENSOR PRODUCTS

2.(2).

a t of t h e semi-norms

369

p , 9).

Thus, t h e r e e x i s t s a vector space isomorphism of B(E,F; G),

of c o n t i n u o u s b i l i n e a r maps of E onto L(E8F, G) TI

,t h e

t h e space

F i n t o a l o c a l l y convex s p a c e

G,

s p a c e of c o n t i n u o u s l i n e a r maps between E 0 F and

G.

I n p a r t i c u l a r , f o r G = C, t h e t o p o l o g i c a l d u a l o f E Q F c o i n c i d e s w i t h t h e s p a c e of c o n t i n u o u s b i l i n e a r forms on E

x

F : i.e.

, one

has the r e L f f t i o n

(EQFl‘ = BIE, F),

(2.17)

TI

within a vector space isomorphism. A s a c o n s e q u e n c e one v e r i f i e s t h a t the p r o j e c t i v e tensor product to-

pology IT on E Q F i s the f i n e s t l o c a l l y convex ( v e c t o r s p a c e ) topology on EQF, TI n making t h e canonical b i l i n e a r map 6 ( c f . ( 2 . 1 1 ) ) continuous: I n d e e d , i f T i s s u c h a t o p o l o g y on E Q F , t h e n b y t h e commutative d i a g r a m

(2.18) E@F IT

one c o n c l u d e s , from t h e p r e c e d i n g , t h a t t h e i d e n t i t y map, a s i n d i c a t ed i s c o n t i n u o u s , hence

T < IT.

T h u s , a s s t i l l f o l l o w s from t h e p r e c e d i n g , the p r o j e c t i v e tensor

product topology

TI

on E @ F i s uniquely defined (by t h e f a m i l y ( 2 . 1 0 ) o r , e q u i -

v a l e n t l y , (2.15) )

, when t h e l o c a l l y convex spaces

E, F are given.

On t h e o t h e r hand, w e s t i l l h a v e t h e f o l l o w i n g u s e f u l r e s u l t .

Lemma 2.2.

The Locally convex space E Q F (Lemma 2 . 1 ) i s Hausdorff i f , and

n

only i f , t h i s i s t h e case f o r each one o f t h e spaces E, F.

Proof. I t i s a d i r e c t c o n s e q u e n c e o f ( 2 . 5 ) t h a t t h e s a i d c o n d i t i o n i s n e c e s s a r y . On t h e o t h e r hand, i f t h e s p a c e s E and F a r e Hausd o r f f t h e n one h a s E = ( E i ) ’ and F = ( F i

I‘

( w i t h i n ( v e c t o r s p a c e ) isomor-

phisms: see J . HORVLTH [I: p. 187, P r o p o s i t i o n 2 1 ) . T h e r e f o r e , o n e g e t s E Q F r (EL )’8

(2.19)

E@F’ g i v e s r i s e t o a n o n - z e r o c o n t i n u o u s

Accordingly, any z f 0

in

l i n e a r form on E ’ 8 Fi

,

(2.20)

z’ay‘e

SIT

f F i ) ’ 5 ( c o n d . ( 2 . 2 ) ) ( E i fFs’,”.

h e n c e t h e r e e x i s t s an e l e m e n t E’8 F’

5 (cond. (2.2))

( E @F)’ IT

s u c h t h a t (x‘ay71z) # 0 , i.e., E 8 F i s a l s o H a u s d o r f f . I

n

2.(2).

The inductive tensorial topology i .- The t e n s o r i a l t o p o l o g y w e

370

X TOPOLOGICAL TENSOR PRODUCT;

c o n s i d e r n e x t i s thw s o - c a l l e d " i n d u c t i v e t e n s o r p r o d u c t t o p o l o g y " ( c f . A . GROTHENDIECK [3: Chap. I ; p. 7 3 ,

8

3 1 ) . The r e l e v a n t d e f i n i t i o n , i n

analogy w i t h t h e p r e v i o u s " c a t e g o r i c a l d e f i n i t i o n " of t h e topology

TI,

i s b a s e d on t h e s e p a r a t e c o n t i n u i t y o f t h e c a n o n i c a l b i l i n e a r map @

( c f . ( 2 . 1 8 ) ) . Thus w e h a v e .

Definition 2.2. Given t h e l o c a l l y convex s p a c e s E , F , w e c a l l ind u c t i v e t e n s o r product topology on E @ F ( w e d e n o t e it by i), t h e f i n e s t l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y on E @ F f o r which t h e c a n o n i c a l b i l i n e a r map I$ : E

X

F-+E@ F i s s e p a r a t e l y c o n t i n u o u s . ( W e

r e s p e c t i v e l o c a l l y convex s p a c e by E ? F t h e n o t a t i o n Eh5F

denote t h e

and i t s c o m p l e t i o n b y

i s a l s o a p p l i e d for" t h e l a t t e r s p a c e ) .

EGF; 2

From t h e p r e c e d i n g d e f i n i t i o n a n d t h e r e m a r k s on ( 2 . 1 8 ) , i t i s clear that

r < i ,

(2.21)

so t h a t ( 2 . 3 ) ( a n d h e n c e a f o r t i o r i ( 2 . 2 ) ) i s o b v i o u s l y f u l f i l l e d . T h e r e f o r e , i i s a compatible t e n s o r i a l topology on E @ F a s w e l l , and by i t s d e f i n i t i o n , t h e f i n e s t one. I n t h i s c o n c e r n , one o b t a i n s from Lemma I ; 4 . 2

t h a t : If t h e Zo-

ca%%yconvex spaces E and F are such t h a t E i s b a r r e l l e d and metrizable and F metr i z a b l e , hence, i n p a r t i c u l a r , if E and F are both Fre'chet spaces, then i = n. Furthermore, concerning D e f i n i t i o n 2 . 2 ,

( l o c a l l y convex)

one r e a d i l y c o n c l u d e s

t h a t it i s ec!uil;alent t o consider t h e topology i as t h e f i n a l l o c a l l y convex ( v e c t o r s p a c e ) topology on E 8 F defined on i t by t h e ( p a r t i a l ) l i n e a r maps

(pro-

v i d e d by @ ) :

@Y

(2.22)

: E-+E

8 F : x -I$

@x:F-E@F:y-Q) f o r any y € F

and

3: €

E, r e s p e c t i v e l y

Y X

(xl:= afx, y ) (y):=@(x,y)

(see a l s o Chapt. IV;S e c t i o n 2 ) .

On t h e o t h e r h a n d , t h e r e l a t i o n c o r r e s p o n d i n g t o ( 2 . 1 7 )

takes

now t h e form (2.23)

(E? Fl ' = &E,

Fl

2

,

w i t h i n a v e c t o r s p a c e isomorphism, where t h e s e c o n d member of t h e l a s t r e l a t i o n d e n o t e s t h e s p a c e of s e p a r a t e l y c o n t i n u o u s b i l i n e a r f o r m s on E X

F.

2 . ( 3 ) . The biprojective tensorial topology

E.

- W e come now t o t h e d e f i -

n i t i o n of t h e " b i p r o j e c t i v e t e n s o r p r o d u c t t o p o l o g y " on E @ F . H o w e v e r ,

2. ( 3 ) .

BIPROJECTIVE TENSOR PRODUCTS

371

we do need first some more terminology: Thus, suppose we are given the locally convex spaces E , F and let E‘,F’be their weak topological dual spaces, respectively. Then, by s

s

i

considering the space S I E ~ F ,

) of separately continuous bilinear forms on E,‘ x Fi , one gets a canonical l i n e a r i n j e c t i o n

(2.24)

j : E@F+&fE’,

,

Fil

which is given hy the relation (2.25)

fX@Y)(X’,

y’) : = x ’ f x ) y ‘ f y l

,

for every pair (x‘, y ’ ) e E,‘ x F: and any decomposable tensor x @ y e E 8 F ; now (2.25) is extended by linearity to all of E8F,while the separate continuity of (2.24), as a bilinear form on E‘x F’, is easily derived s s as well, from the same definitions. Besides, the same map is certainly 1-1 when the given spaces E and F are Hausdorff (apply Hahn-Banach; this, in fact, will be the case in the sequel) Nonetheless, the general case is still valid as this is derived from the following useful result. Namely, we have. Lemma 2 . 3 . Given the local-ly convex spaces E and F ( n o t necessarily Haus-

d o r f f l , one has

(2.26)

E 8 F = B(Ei, Fs’)

,

within a vector space isomorphism, the second member of the l a s t r e l a t i o n denoting t h e space of continuous b i l i n e a r maps on E’xFL.

Proof. See F. TREVES [I : p. 432, Proposition 42.41

.I

Thus we obtain, of course, (2.27)

E 8 F n B(Ei, F i i E S f E , ‘ , $‘I

that is, another (more precise!) estimation of the range of (2.24). On the other hand, one further considers on &(Ei,FL) the socalled topology of biequicontinuous convergence which we denote by e ; i.e. , the (locally convex vector space) topology of uniform convergence on the sets A x B C E’ x F ; with A , B varying over the equicontinuous subsets of E’,F’, respectively. We denote the locally convex space, thus defined, by S e i E i , F i i . Thus, we come to the following. D e f i n i t i o n 2.3. Given the locally convex spaces E and F, we call

b i p r o j e c t i v e tensor product topology on E 8 F , the relative topology induced on the latter space by the relation

(2.28)

E8F

5 %(Ei, F i )

.

X TOPOLOGICAL TENSOR PRODUCTS

372

The l o c a l l y convex s p a c e t h u s d e r i v e d i s d e n o t e d b y EBF and i t s comE

p l e t i o n by E 6 F . ( B u t t h e n o t a t i o n E 6 F E

.

space)

is a l s o applied f o r the latter

I t i s r a t h e r i n s t r u c t i v e t o r e c a l l , b r i e f l y , a t t h i s p l a c e some

b a s i c f a c t s , concerning t h e biequicontinuous convergence topology e on &(E;,

F; 1.

Thus, o n e g e t s a l o c a l b a s i s f o r e l by c o n s i d e r i n g t h e s e t s lu(x',y')lS~,with ( x ' , y ' ) e A x B ) ,

(2.29)

T(A,B;E) =iue&(E;,Fs'):

where A , B

r a n g e o v e r t h e e q u i c o n t i n u o u s s u b s e t s o f E ; F',

l y , a n d E o v e r t h e p o s i t i v e r e a l s . Thus, a s a e i s a l o c a l l y convex ( v e c t o r s p a c e ) t o p o l o g y t h e r d e t a i l s , see, f o r i n s t a n c e , A . GROTHENDIECK Moreover, it i s a d i r e c t c o n s e q u e n c e of

c o n s e q u e n c e of on &(Ei,

F').

(2.29),

(For fur-

[4: p. 122 f f . ] ) . the preceding Definition

2.3 and (2.29) t h a t t h e canonical b i l i n e a r map @ :E x F hence ( 2 . 1 ) i s s a t i s f i e d ; b e s i d e s , o n e h a s

respective-

+

E Q F i s continuous, E

Now t h e same, a t l e a s t ,

E -=T.

c a n b e d e r i v e d from t h e f o l l o w i n g argument: F o r a n y e q u i c o n t i n u o u s s u b s e t s A , B of E ' , F ' , d e f i n e s v i a (2.29) a semi-norm p cording t o t h e r e l a t i o n

f o r any element z

A,B

on E 8 F

r e s p e c t i v e l y , one

f o r t h e topology

el

ac-

n 2 x i @ y i e E B F (see a l s o (2.25)). i=l

Thus, w i t h A , B r a n g i n g o v e r t h e e q u i c o n t i n u o u s s u b s e t s of E',F; respectively, t h e family

(2.31) "A,B

p r o v i d e s a " d e f i n i n g f a m i l y o f semi-norms''

sorial topology

E

AQB

(2.32)

f o r t h e biprojective ten-

on E @ F ; m o r e o v e r , o n e h a s

and t h i s p r o v e s (2.3)

c

. In

( { z ~ E B F : ~( 2 ) 5 A, B

11)'

E (E~F)', E

p a r t i c u l a r , s i n c e A c U" and B C V",

for suit-

a b l e n e i g h b o r h o o d s of z e r o U S E and V'EF, a p p l y i n g t h e n o t a t i o n of

(2.15) one o b t a i n s r ( U @ V) C ( U 0 @ V o ) "

(2.33) therefore,

E <TI ( C o r o l l a r y

E (ABB)"

,

2 . 1 ) . T h e p r e c e d i n g argument i m p l i e s a l s o

t h a t i t would be s u f f i c i e n t t o t a k e i n (2.30) and (2.31) U", V" i n s t e a d of A , B,

where U , V v a r y o v e r l o c a l b a s e s of E , F , r e s p e c t i v e l y .

2 . ( 4).

THE E-PRODUCT

373

Thus, t h e topology E i s a compatible t e n s o r i a l topology on EBF as well (Definition 2.1), in such a way that, concerning the three compatible tensorial topologies on E6F considered hitherb, one obtains the relation (2.34)

E < T < ~ .

In fact, on the basis of Definition 2.2 and cond.(2.3) of Definition 2.1 in connection with (2.321, one g e t s t h e r e l a t i o n (2.35)

E
< i,

f o r er.ery l o c a l l y convex compatible topology

T

on E@F. In this respect, we

further note that t h e r e l a t i o n (2.36)

E < T

i s , i n f a c t , equivalent w i t h cond. ( 2 . 3 ) of Definition 2.1 (see also (2.32)).

2. ( 4 ) . The € - p r o d u c t o f L. Schwartz. The Banach-Grothendieck a p p r o x i m a t i o n p r o p e r t y . - We finally come to consider another locally convex (vec-

tor space) topology on E@F, associated with a given pair (E,F) of locally convex spaces, and which is due to L . SCHWARTZ [2]. This topology coincides, in some important paricular instances, with the above defined biprojective tensorial topology E on [email protected] the other hand, it is rather under this particular form that the €-product is actually applied in some important cases (cf. e.g. A. MALLIOS [lo; 171 and/or A . KYRIAZIS [2] ) . Thus, given the locally convex (Hausdorff) spaces E,F, let E , ' be the topological dual of E equipped with the (locally convex) topology of uniform convergence on the absolutely convex and compact subsets (compact d i s c s ) of E . Moreover, let .CE(E,', F) be the space of continuous linear maps of E' into F endowed with the (locally convex) topology of uniform convergence on the equicontinuous subsets of E'. Now, the E-product of the spaces E and F (denoted by E E F) is defined by the relation (2.37)

E E F 11:E f EC' , F/.

In fact, the above relation is valid within an isomorphism of (locally convex) topological vector spaces; i.e.,one defines, indeed, the first member of (2.37) as a space of bilinear forms on EixFi satisfying suitable continuity conditions, and topologized appropriately (in a similar way to the preceding space & ( E i , Fill. However, for convenience, we took for definition the previous relation (2.37).(See also L . SCHFIARTZ [4: p. 18 ff., and p. 34, Corollaire 21).

3 74

X TOPOLOGICAL TENSOR PRODUCTS

On t h e o t h e r h a n d , t h e r e i s an i m p o r t a n t p a r t i c u l a r case where t h e € - p r o d u c t d e f i n e d above c o i n c i d e s w i t h t h e b i p r o j e c t i v e t o p o l o g i c a l t e n s o r p r o d u c t o f D e f i n i t i o n 2 . 3 . But f i r s t it i s u s e f u l t o ( r e ) t u r n (see a l s o Chapt. 1 X ; D e f i n i t i o n 1 0 . 1 ) t o t h e f o l l o w i n g . D e f i n i t i o n 2.4. i s f ies t h e

W e s a y t h a t a g i v e n l o c a l l y convex s p a c e E s a t -

condition of approximation

mation property i s a l s o i n f o r c e ) (2.38)

( t h e t e r m Banach-Grothendieck approxi-

, if t h e -

idE€E'8E

5

following r e l a t i o n holds t r u e Lc(E/.

T h a t i s , w e s u p p o s e t h a t t h e i d e n t i t y l i n e a r endomorphism of E b e l o n g s t o t h e c l o s u r e o f E'C3 E , where t h e l a t t e r s p a c e i s i d e n t i f i e d w i t h t h a t o f c o n t i n u o u s l i n e a r endomorphisms of E o f f i n i t e r a n k , t h e a l g e b r a .C(E) b e i n g endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e compact d i s c s o f E . I n t h i s r e s p e c t , one has t h e following. LeMna 2.4.

Let E , F be complete l o c a l l y convex spaces such t h a t one of them

s a t i s f i e s the condition of approximation. Then, one has the r e l a t i o n E&F = E E F ,

(2.39)

E

within an isomorphism of l o c a l l y convex spaces.

Proof. c f . I;.SCHWARTZ [4: p. 4 7 , C o r o l l a i r e I ] . Thus, u n d e r t h e a s s u m p t i o n s o f t h e p r e c e d i n g Lemma 2 . 4 , one g e t s from ( 2 . 3 7 ) and ( 2 . 3 9 ) t h e f o l l o w i n g . E ~ F = E E F = L( E , F I .

(2.40)

E

E

C

F u r t h e r m o r e , f o r any complete l o c a l l y convex spaces E , F o n e o b t a i n s (2.41)

E$F

s %(E;,

F ; ) = s ~ ( E ; , F) ,

w i t h i n ( l o c a l l y c o n v e x ) v e c t o r s p a c e i s o m o r p h i s m s ; h e r e E;

stands f o r

t h e t o p o l o g i c a l d u a l of E endowed w i t h t h e Mackey topology

i.e.,

T,

the

( l o c a l l y c o n v e x ) 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 t h e a b s o l u t e l y convex and weakly compact s u b s e t s o f E . ( F o r a p r o o f o f

( 2 . 4 1 ) , see F .

TREVES [1: pp. 429, 430; P r o p o s i t i o n s 42.2 a n d 4 2 . 3 1 ) .

Now,

it a s t a n d a r d f a c t t h a t every nuclear ( l o c a l l y c o n v e x ) space

s a t i s f i e s the condition of approximation ( c f . , f o r i n s t a n c e , F. TREVES [ l : p. 520, P r o p o s i t i o n 5 0 . 3 1 ) . T h e r e f o r e , for any two complete l o c a l l y convex spaces E and F , one of which i s nuclear, one o b t a i n s A

(2.42)

EBF EE TI

E

~ EF E E F

r E ( EC' ,

F I c_.c€(E;, F )

,

3.

375

T O P O L O G I C A L TENSOR PRODUCT ALGEBRAS

within (locally convex) topological vector space isomorphisms, as indicated. In this concern, the first of (2.42) is a classical relation that is more generally valid, for any locaZZy convex spaces E and F one of which is nuclear. That is, one has (2.43)

E ~ =F E ~ ,F E

TI

within an isomorphism (of locally convex spaces). For convenience, this is often taken, as well, for definition of the nuclearity of one of the (locally convex) spaces involved in (2.43). (For the proof of the last relation cf. A. GROTHENDIECK [3: Chap. 11; P. 34, ThEorSme 6 1 , and/or F. TREVES [l: p. 511 , Theorem 50.11). On the other hand, it is under this form of nuclearity that the relations in (2.42) will be applied in the sequel. Moreover, the situation described by (2.41), in connection with (2.43),will be considered below, as well.

3. Topological tensor products o f topological algebras, Compatible topologies The topological algebras considered in this section will be at least locally convex ones having separately continuous multiplication, according to the general definition (cf. Definition I; l.l),or yet a (jointly) continuous one (ibid.); in particular, we also consider locally m-convex algebras. We start with fixing first a basic notion for all that follows, provided by the following.

Definition 3.1. Let E,F be two locally convex algebras (Definition I; 1.1) and E 8 F the respective tensor product algebra of E,F (Lemma 1.4). A locally convex vector space topology T on E 8 F is said to be a compatible topoZogy with the tensor product algebra stmcture of E 8 F (or yet a compatible topology on E 8 F , if no ambiguity occurs), if the following conditions are satisfied: 1 ) The locally convex space E 8 F is a locally convex algebra. T 2) The topology T is a compatible topology with the tensor prod-

uct vector space structure of E 8 F

(Definition 2.1).

We denote the resulting locally convex algebra still by E 8 F ; T

its completion (when the latter exists, as a topological algebra!, cf. Chapt. 1 ; Section 4 ) is denoted by E g F . T

The analogous notion of a locally m -convex tensorial topology on E 8 F ,

376

X TOPOLOGICAL TENSOR PRODUCTS

where now E , F are locally rn-convex algebras is clear, after an obvious modification of c o n d . 1 ) of the previous Definition 3 . 1 (see also Definition 3 . 2 below). NOW, we are going to consider certain particular instances of the preceding Definition 3 . 1 in connection with the tensorial topologies dealt with in the previous section. S o we first have.

Lemma 3.1. Let E , F be l o c a l l y convex algebras and E 8 F the respective pro71

j e c t i v e tensor product l o c a l l y convex space of E , F (Lemma 2 . 1 1 . Then, E 8 F i s a TI locally convex algebra i n such a way t h a t 71 i s a compatible topology w i t h t h e tensor product algebra s t m t u r e of E 8 F (Definition 3 . 1 ) . In p a r t i c u l a r , i f the given l o c a l l y convex algebras E , F have continuous m u l t i p l i c a t i o n (Definition I;1 . l ) , then the same i s t r u e f o r t h e l o c a l l y convex algebra E 8 F. TI

Proof. The separate continuity of the multiplication in E , F is equivalent with the continuity of the linear operators la:E-+ E : x - l a ( x l := a x , with a € E l and l b : F - F : y - l b ( y l : = by, with b e F (as well as that of the analogous operators for the "multiplication on the right"). So one gets a corresponding continuous linear operator on E 8 F via the 71 relation z l 81b : E 8 F - E 8 F : x @ y Hl a o b ( x a y ) 'aab a (3.1)

: = ( a a b J ( x a y l =(by ( 1 . 9 ) ) a x e b y I

for every decomposable tensor x o y e E 8 F , hence 1a e b is extended by linearity to E 8 F . Moreover, l a a b= la@$,: E B F + E @ F is a continuous linear endomorphism of the locally convex space E 8 F being, by defi71

nition, the tensor product of the continuous l i n e a r operators 1a , l b I as above, with ( a , b ) e E x F (see A. GROTHENDIECK [3: Chap. I; p- 37, no 2 1 ) . On the other hand, the relation (3.2)

implies the separate continuity of the multiplication (Ifonthe left") n in E 8 F , for every element t = *=I -2 a2. e b2. e E 8 F . (An analogous argument is valid of course when "multiplying on the right" in E 8 F ) . Thus, E 8 F is a locally convex algebra, the topology TI being also a compatible tensorial topology on E 8 F in the sense of Definition 2 . 1 (Lemma 2 . 1 ) ; hence, all the conditions of the preceding Definition 3 . 1 are satisfied. On the other hand, if the locally convex algebras E , F have continuous multiplication and ( p ) , ( q ) are families of (continuous) semi-

3.

TOPOLOGICAL TENSOR PRODUCT ALGEBRAS

311

norms defining the topologies of E and F , respectively, then for any tensor product semi-norm r p o q on E 8 F (see (2.4)) , one gets the following, according to the hypothesis for E , F ; r ( ( a o b ) . ( x o y l ) = r ( a x @ b y l= p ( a x l . q ( b y )

(3.3) S Xpl (alp2 (xlql (b)q2 ( y l

=

ATl

( a o b ) r 2 (x oy)

for suitable semi-norms pi e (pl and qi E ( 4 1 , with ri= pioqi ( < = l3, ) and X > 0 , and for any decomposable tensors a o b , x o y in E 8 F . Therefore, applying an argument analogous to ( 2 . 7 ) and from ( 1 . 1 2 ) , one gets (3.3) , for any elements s , t in E @ F , hence the continuity of the multiplication in E B F , and this completes the proof. I TI

In case of locally m-convex algebras the projective tensorial topology TI provides still a stronger result. So we first have the following. Lemma 3.2. Let E , F be two given algebras and p , q submultiplicative seminorms on E , F , r e s p e c t i v e l y . Then, "the tensor product semi-norm" r = p o q defined on (the vector space) E Q F

(3.4)

by the r e l a t i o n n r l t ) = inf 1 p l x i ) q ( y i ) , i=l

n t = 1 x . o y . E E 8 F , yields 2 2 a submultiplicative semi-norm on the tensor product algebra E 8 F . Furthermore, if U = { x € E : p ( x ) S l }and V = { y € F : q ( y l < l }are the respect i v e u n i t "semi-balZs" defined by p and q, then these s e t s are a-barrels of E , F , r e s p e c t i v e l y (cf. Chapt. ; Section 1 ), in such a way t h a t r is t h e gauge f u n c t i o n of t h e a-barrel I'(U8V). where trinfrr i s taken over a l l expressions of t h e form

Proof. It has been proved already (cf. Lemma 2.1) that (3.4) defines a (vector space) semi-norm on E 8 F . S o concerning the first part of the assertion, it suffices to prove only the submultiplicativity of r = p @ q ; i.e., the relation

(3.5)

r(s.tl 5 r(s).r(tl

,

for any elements s , t in the (tensor product) algebra E 8 F , as in ( l . l l ) , where the element s - t is then defined by (1.12) : Thus, for every E > O , one concludes from (2.4) that

which certainly proves (3.5). NOW, by hypothesis for p , q , the sets U E E

and V G F are a-bar-

X TOPOLOGICAL TENSOR PRODUCTS

378

rels (absolutely convex absorbing m-sets); hence, U @ V is a balanced m-set as well, SO its (absolutely) convex hull r ( U 8 V ) an a-barrel in the algebra E 8 F (cf.also Chapt. 1;Lemmas 1.3, 1 . 4 ) . On the other hand, it has been proved already (Lemma 2 . 1 ) that the semi-norm r given by (3.4) is the gauge function of r ( U @ V l , and this terminates the proof. I Thus, we now have the following.

Proposition 3.1. Let

E , F be l o c a l l y m-convex algebras and E 8 F the corre-

sponding tensor product algebra of E , F. Moreover,

let (pal,

I , lqalg

be fam-

i l i e s of submultiplicative semi-norms defining t h e topologies of E , F , r e s p e c t i v e l y

(Theorem I; 3.1

) . Then,

(Iu.,CIE I x K ) given by ( 3 . 4 1 , i s a family of submultiplicative w i t h p a. @ q p= semi-norms f o r the p r o j e c t i v e t e n s o r i a l topoZogy T on E B F . Thus, E 8 F i s madeinto TI

a ZocalZy m-convex algebra (we denote its completion by E g F ) , and c a l l y m-convex) compatible topology on E 8 F (Definition 3.1).

~i

a 120-

Furthermore, one obtains a local basis of E 8 F c o n s i s t i n g of m-barrels, by taking t h e family of convex h u l l s ( r ( U @ V I ) , w i t h U, V running over local bases of E, F,

respective&j, c o n s i s t i n g of m-barrels. FinaZly, t h e topology on E @ F thus

defined i s t h e strongest Zocally m-con-

vex topology on t h e tensor product algebra E 8 F , which makes t h e canonical b i l i n e a r map @ : E x F - + E @ F

continuous (hence, i t s uniqueness).

Proof. Taking Lemma 2 . 1 and the previous Lemma 3 . 2 into account, one concludes that the projective tensorial topology TI (being also uniquely defined, cf. the comment on (2.18)) is actually a locally mconvex topology on the algebra E @ F ( s e e also Theorem I; 3.1). Thus, a f o r t i o r i cond. 1 ) of Definition 3 . 1 is satisfied, hence n is a compatible topology on E B F in the sense of the same definition (cf. Lemma 2 . 1 ) . Now, the second part of the assertion is still a direct consequence of the proof of the preceding Lemma 3.2 in conjunction with Corollary 2 . 1 . Finally, if T is a locally m-convex topology on E 8 F making the map @ continuous, then T TI since then T is a f o r t i o r i a locally convex topology on E 8 F with the same proprty (cf. the comment following (2.17)). Thus, taking a l s o the uniqueness of the topology TI into account, this proves the last assertion of the proposition, and the proof is finished. I

In this respect, we still note that: The l o c a l l y m-convex topology on E 8 F defined by (3.61

4.

379

TENSOR PRODUCT LOCALLY BOUNDED ALGEBRAS

i s Hausdorff i f , and only i f , t h i s i s the case f o r t?ze to-

(3.7)

pology

(See Lemma 2.2)

of each one of the algebras E and F.

.

4. Tensor products o f l o c a l l y bounded algebras We consider in this section another instance of topologizing a tensor product algebra E 8 F in a "compatible way" within a context analogous to that of Definition 3.1; however, this one here is still more general, the topological algebras involved being now not necessarily locally convex. (The theoretical significance, at least, of these considerations is further discussed in Scholium 4.1 below). Thus, suppose we are given two l o c a l l y bounded algebras E and F (cf. Chap. I; Definition 6.1) w i t h continuous m u l t i p l i c a t i o n . Now this is equivalent with the assumption that each one of the algebras E , F is an anormed algebra, with 0 < a I I (Theorem I;6 . 3 ) , so that one may further assume that both of E , F are of the same "order of homogenuity" cy. (cf. Theorem I;5.2 and the comment following it). As a matter of fact, this is more generally valid for locally bounded spaces (ibid.), a fact which we come first to apply in the following lemma, a result analogous to our previous Lemma 2.1.

Lemna 4 . 1 . Let E , F be a-norrned spaces (we assume the same a ) w i t h p , q t h e corresponding a-norms on E , F , r e s p e c t i v e l y , and l e t E 8 F be the tensor product vector space of E,F.Furthermore,

suppose t h a t one of t h e spaces E , F i s i n a "sep-

arated d u a l i t y " w i t h i t s topological dual (cf. J . HORVATH [I : p. 183, Definition I] 1 . m e n , t h e r e l a t i o n n r i t ) = inf 1 p ( x i ) q ( y i ) , (4.1) i= 1 n where "inf'I i s taken over a l l expressions of t h e form t = -2 xi o y . e E B F , d e f i n e s

%=I

on E 8 F an a-norm (cf. Chap.I;(6.1.1, 2,311.

Furthermore, i f U, V are t h e corresponding u n i t semi-balls defined by p and q, then one g e t s t h e r e l a t i o n

(4.2)

r ( t ) = I g ( t ) 1'

, teE

where g i s the gauge f u n c t i o n of t h e balanced

~ F ,

a-convex h u l l

r c y . ( U @ V ) of the s e t

U @ V C E @ F; t h a t i s one has, by d e f i n i t i o n ,

(4.3)

g ( t ) :=inf{h>~:tehr~(u~vI},

with t e E 8 F .

In this concern, we recall that a subset A of

X TOPOLOGICAL TENSOR PRODUCTS

380

of a (complex) v e c t o r s p a c e E i s s a i d t o b e a-conv e x , 0 a 5 I , whenever A i s s u c h f o r t h e r e s p e c t i v e r e a l v e c t o r s p a c e E ; Le., if f o r any x , y i n Alone h a s t h a t Xx+pye A , w i t h X,p 2 0 and Aapl+pa = 1 . On t h e o t h e r hand, t h e balanced and a-convex h u l l ( o r y e t absolutely a-convex h u l l ) o f a s e t B C E , den o t e d by r a ( B ) , i s by d e f i n i t i o n t h e s e t

( c f . , f o r i n s t a n c e , G . K8THe [ I : p. 160 f f . ] ) .

Proof of Lema 4.1.

The argument a p p l i e d t o p r o v e t h a t ( 2 . 4 ) d e -

f i n e d a semi-norm c a n b e a p p l i e d h e r e , as w e l l , w h i l e I ; ( 6 . 1 . 1 )

is

proved a n a l o g o u s l y t o t h e normed c a s e ( s e e , f o r i n s t a n c e , R . SCHATTEN [ l : p. 37, Lemma 2.131)

,

taking a l s o t h e hypothesis f o r the algebras E ,

F i n t o account.

On t h e o t h e r hand, i n o r d e r t o p r o v e ( 4 . 2 ) , we remark t h a t z e and o n l y i f , z = > X . ( x . a p y . ) , where p ( x . I Z l a n d

ATa(UQVIrwith A > O , i f ,

z z 2 is equivalent t o z = Ea,@bi ,with wherz a . = X . x . and b . = h . y . ( i = 1 , ...,n ) . Thus, o n e o b t a i n s

q f y . ) < l, while 1 I X . I a $ X a . 2p(ailq(bi) S A,'

2

Now, t h i s

2 2

7,

r(zI= inf{

2 2

A ~ : x > o ,Z

E X ~ ~ ( ~ @ V ) )

(4.4)

= ( i n f { X > 0 : zeXra(UQV13)" = ( b y (3.9)) I g ( z I ) " which c o m p l e t e s t h e p r o o f o f t h e lemma. I The a-norm on E 8 F d e f i n e d b y ( 4 . I ) i s c a l l e d t h e

a-norm

tensor product

.

of p , q and i s d e n o t e d by r E p @ q NOW, t h e f o l l o w i n g r e s u l t o f f e r s t h e a n a l o g o u s s i t u a t i o n t o Lem-

m a 2 . 1 and i t s c o n s e q u e n c e s , which one h a s w i t h i n t h e p r e s e n t c o n t e x t . Thus, w e have. Lemma 4.2.

Let E, F be l o c a l l y bounded spaces, considered a s a-nomned spaces

w i t h the same order of homogenuity a (Theorem I; 6 . 2 ) . Then, there e x i s t s on E Q F a uniquely defined l o c a l l y bounded vector space topology T making E Q F [ T ] s E Q F T i n t o an a-normed space. Moreover, f o r every l o c a l l y bounded space G whose topology can be defined by an a-norm, t h e canonical isomorphism of t h e v e c t o r space B(E,F;GI o f b i l i n e a r maps of E x F i n t o G onto t h e vector space L ( E Q F , GI of l i n e a r maps of E Q F i n t o G ( c f . (1.5)) corresponds t o each o t h e r t h e equicontinuous subsets of

t h e s e two spaces, i n such a manner t h a t one has (4.5)

B(E,F; G ) = f ( E Y F , G I ,

within a vector space isomorphism. Furthermore, T i s the f i n e s t l o c a l l y bounded v e c t o r space topology on E B F making it an a-normed space f o r which t h e canonical b i l i n e a r map $ : E x F + E Q F

is

4.

38 1

TENSOR PRODUCT LOCALLY BOUNDED ALGEBRAS

continuous. Proof. Applying Theorem I; 6.2 we may assume that the given locally bounded spaces E , F are a-normed spaces whose topologies are defined by a-norms, say, p and q , respectively, corresponding to the same ("order of homoqenuity") a , with 0 < ci < 1 Thus, the tensor product a-nom r = p a q defined by the previous Lemma 4.1 yields on E 8 F a locally bounded (vector space) topology, say,^ (Theorem I; 6.1),in such a way that E 8 F satisfies the required conditions. T Indeed, if H G B ( E , F ; G ) is an equicontinuous subset, then for every W from a local basis of G I the set

.

n u-'

( 4 -6) u

(WI

eFI

is a neighborhood of zero in E x F , therefore one gets u ( A x B I C W, with u € H , where A , B belong to local bases of E , F , respectively. Thus, ap-

plying (1.5) together with Lemma 4.1, we conclude that (4.7)

for some X >O; i.e., H" (the image of H by (1.5)) is an equicontinuous S. subset of L ( E @ F , G). Conversely, for every equicontinuous set T S l E B F , G ) and every W as above, there exists a neighborhood of O E E % F T of the form 'I a ( U 8 VI (Lemma 4 . 1 ) such that one h a s (4.8)

~ ( U X V )= i i ( u 8 v 1 E r a ( i i ( u a v ) ) = i i ( r a ( u s v i )

cw

for every ii E H " , and hence u E H. S o H E B ( E , F ; GI is an equicontinuous set as well, which proves the first part of the assertion. NOW, the identity map of E B F into itself is continuous, hence T from what has been said above the canonicaZ biZinear map $ : E X F - + E 8 F is T continuous. On the other hand, for any locally bounded (vector space) topology 'T on E 8 F defined by an a-norm and making $I : E x F d E B F continu4 ous, one concludes from (4.5) that the identity map of E @ F into E @ F T T1 is continuous, hence T < T ~which proves the last assertion, and the proof is complete. I NOW, according to M. LANDSBERG [ 1 : p. 107, Satz I ] , any normed space is an a-normed ( a-convex) space, f o r every 0 < a < 1 , so that specializing (4.5) to the case G = C, one obtains an analogous relation to ( 2 . 1 7 ) ; i.e., one has (4.9)

B ( E , FI = ( E 8 F ) ' ,

within a vector space isomorphism, where the (locally bounded vector space) topology T on E 8 F is defined as in the previous Lemma 4.2. Moreover,

X TOPOLOGICAL TENSOR PRODUCTS

382

one concludes from ( 4 . 9 ) that, for any two locally bounded topologies T. (i= I , 2 ) on E 8 F defined via an application of Lemma 4 . 2 , the respect i v e topologica2 dual spaces ( E B F ) ' and ( E B F ) ' are t h e same (within a vector 1'

*2

space isomorphism), t h e equicontinuous subsets of t h e s e t u o spaces being i n t o a b i j e c t i v e correspondence.

The following result was, in fact, our main motivation to the preceding argument (see also Scholium 4 . 1 that follows). S o we have. Theorem 4.1.

plication

and

Let E , F be l o c a l l y bounded algebras having a continuous multi-

E Q F t h e respective tensor

product algebra. Then, f o r any a-norms

p , q defining t h e topologies of E , F , respectiveZy, t h e tensor product a-norm r =

p @ q , given by ( 4 . 1 ) , makes E B F

1TI

(cf. the proof below) i n t o a l o c a l l y bounded

algebra with a continuous m u l t i p l i c a t i o n (we denote it by E B F ) , i n such a manT

ner t h a t i t s completion E 6 F can be made i n t o a complete 6-nomed algebra, w i t h 'I

0. R.1.

Proof.

If p , q are a-norms that may define the topologies of E l the corresponding tensor product anorm r p @ q supplies E 8 F with a (uniquely defined, see Lemma 4 . 2 ) locally bounded (vector space) topology, say, T , in such a way that E B F is a (locally bounded) topological algebra with a continuous T multiplication. Indeed, the last assertion can be derived froman analogous argument to the "locally convex case" (cf. Lemma 3 . 1 ) taking also the rel. (4.1) into account. Thus (Lemma I; 6 . 3 ) , the resulting topology T on E @ F can be given through a submultiplicative R-norm ( O < B < l , cf. 1;(6.5)), in such a way that its completion E G F becomes a T complete 0-normed algebra, and this terminates the proof. 1 F , respectively (cf. Lemma 4 . 2 ) ,

Schol ium 4.1.

- The preceding discussion provides a (non-trivial)

example of a tensor product algebra E 8 F of a given pair f E , F l of suitable topological algebras (locally bounded ones with continuous multiplication), which is further supplied with a compatible algebra topology (Theorem 4 . 1 ) in the following generalized (i.e., "not necessarily locally convex") sense. Indeed, this will be applied, systematically, in several instances below (see Chapt.XI1;Section 1 ) . Thus, we set. D e f i n i t i o n 4.1. Let E , F be two given topological algebras and E 8 F the respective tensor product algebra. A topology T on E 8 F is said to be compatible (with the tensor product algebra structure of E B F ) , if the following conditions are satisfied: I E 8 F , TI E B F is a topological algebra. T The canonical bilinear map @ : E x F + E B F is separately con-

1 ) The pair 2)

5.

383

I N F I N I T E TENSOR PRODUCT ALGEBRAS

tinuous. 3 ) The f o l l o w i n g r e l a t i o n h o l d s t r u e , w i t h i n a l i n e a r i n j e c t i o n ;

i.e.

,

(4.10)

E'8F'

$ (E8F)'. T

E v e n t u a l l y , w e r e q u i r e a s t r o n g e r v e r s i o n of t h e p r e v i o u s r e l a tion: t h a t is, we ask f o r t h e following condition: 3 a ) F o r a n y e q u i c o n t i n u o u s s e t s A C E ' and B C F ' , t h e s e t A 8 B i s s t i l l an e q u i c o n t i n u o u s s u b s e t o f NOW,

(E8F)'.

c o n c e r n i n g t h e l o c a l l y bounded a l g e b r a s c o n s i d e r e d i n t h e

p r e v i o u s d i s c u s s i o n i n c o n n e c t i o n w i t h t h e above D e f i n i t i o n 4 . 1 ,

the

f o l l o w i n g remark i s i n o r d e r : A s w e a l r e a d y n o t i c e d a t t h e b e g i n n i n g of t h i s scholium, cond. 1 ) of t h e p r e v i o u s d e f i n i t i o n i s f u l f i l l e d , w i t h i n t h e c o n t e x t of t h e a b o v e Theorem 4 . 1 . 2)

I

4.1

I n t h e same c o n t e x t , c o n d .

a s w e l l a s c o n d . 3 a ) , h e n c e a f o r t i o r i c o n d . 31, o f D e f i n i t i o n

a r e a l r e a d y d i r e c t c o n s e q u e n c e s o f Lemma 4 . 2 i n c o n n e c t i o n w i t h B ( E , F); t h e l a t t e r i s v a l i d w i t h i n a l i n -

( 4 . 9 ) and t h e r e l a t i o n E ' @ F ' $

e a r i n j e c t i o n , which r e s p e c t s " t e n s o r i n g w i t h e q u i c o n t i n u o u s s e t s " .

5. I n f i n i t e topological tensor product algebras W e consider i n t h i s section a suitable topologization

of t h e

( a l g e b r a i c ) " i n f i n i t e t e n s o r product" of a given family of t o p o l o g i c a l a l g e b r a s . F o r c o n v e n i e n c e , w e f i r s t commnent, however, on t h e necessary algebraic preliminaries. Thus, suppose t h a t (Eiii

I i s a given f a m i l y of

(complex l i n e a r

a s s o c i a t i v e ) algebras ( w i t h r e s p e c t t o a n a r b i t r a r y ( n o t n e c e s s a r i l y f i n i t e ) i n d e x s e t I ) each one having an i d e n t i t y element e i ' i e I . F u r t h e r m o r e , l e t F(I1 b e t h e s e t of f i n i t e s u b s e t s o f I , s u c h t h a t , f o r every (5.1)

c1 E

F(II, let Ea

=

8 E< i e a

is, of c o u r s e , uniquely d e f i n e d , i n d e p e n d e n t l y o f a n y e v e n t u a l " e n u m e r a t i o n " of t h e s e t ci a s a f i n i t e s u b s e t of I ( " a s s o c i a t i v i t y of t h e t e n s o r product";

b e t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a ( c f . (1.19)).The l a t t e r

see Lemma 1 . 3 )

.

Thus, f o r e v e r y p a i r la, R !

w G 6, one g e t s a ( c a n o n i c a l ) algebra morphism (5.2)

Ea-

E0 '

d e f i n e d by t h e r e l a t i o n ( s e e a l s o ( 1 . 1 8 ) )

o f f i n i t e s u b s e t s of I w i t h

384

X

TOPOLOGICAL TENSOR PRODUCTS

where y = x f o r every j = i e a E B , and y = e i f j e R n C a , f o r every j i' j j decomposable t e n s o r @ x i e E - ( w e t h e n e x t e n d ( 5 . 3 ) by l i n e a r i t y ) .

iea

a'

F u r t h e r m o r e , c o n s i d e r i n g F(I) a s a d i r e c t e d i n d e x set r e l a t i v e t o t h e " i n c l u s i o n r e l a t i o n " , o n e g e t s from t h e uniqueness o f

(5.2) (cf.

( 1 . 1 4 ) ) t h e f o l l o w i n g r e l a t i o n , f o r a n y a r B l y i n F(I) w i t h a G B C y ,

Moreover, it i s s t i l l clear from ( 5 . 3 ) t h a t o n e h a s

fa, = idE

(5.5)

I

a

f o r e v e r y a E F(I). Thus, the f a m i l y

ma, ;ba)

(5.6)

c o n s t i t u t e s an inductive system o f algebras ( s e e C h a p t . I V ; S e c t i o n l ) , w i t h r e s p e c t t o t h e ( u p w a r d s ) d i r e c t e d i n d e x s e t F(I). A c c o r d i n g l y , w e may

s e t now t h e f o l l o w i n g .

D e f i n i t i o n 5.1. L e t ( E i l i e I

be a g i v e n f a m i l y o f a l g e b r a s w i t h

and l e t ( E a , f I b e t h e r e s p e c t i v e i n d u c Ba t i v e s y s t e m o f a l g e b r a s d e f i n e d by ( 5 . 6 ) . Then t h e c o r r e s p o n d i n g i n i d e n t i t y elements e i , i e I ,

d u c t i v e l i m i t a l g e b r a E = l i m E a (see D e f i n i t i o n IV; 1.1 )

i s called t h e

i n f i n i t e tensor product aZgebra o f t h e g i v e n f a m i l y of a l g e b r a s and i s s t i l l d e n o t e d ( b y a n a b u s e o f n o t a t i o n ! ) by nition

,

€3

Ei.

ieI

Thus o n e s e t s , by d e f i -

I n t h i s r e s p e c t , it i s c l e a r t h a t , i n c a s e one h as a f i n i t e index s e t I, t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a

8 Ei i E I

( c f . (1.16)) c o -

i n c i d e s ( w i t h i n a n a l g e b r a isomorphism) w i t h t h e a l g e b r a E d e f i n e d by (5.7). On t h e o t h e r h a n d , f o r e v e r y i E I and a 8 F f I ) , w i t h i E a , one a l s o c o n s i d e r s . t h e c a n o n i c a l ( a l g e b r a ) isomorphism

which i s g i v e n by t h e r e l a t i o n

such t h a t x . = x

= e f o r every j # i ( j e a ) , w i t h z e E . j j' We come n e x t t o c o n s i d e r a t o p o l o g i z a t i o n o f t h e a l g e b r a E deand

3:

5.

38 5

INFINITE TENSOR PRODUCT ALGEBRAS

f i n e d b y ( 5 . 7 ) , i n case t h e g i v e n f a m i l y (Eil

, as

above, c o n s i s t s of

topological algebras: Thus, l e t ( E i I i e 1

l o c a l l y convex algebras w i t h i d e n t t h e respective ten-

be a family of

i t y e l e m e n t s and suppose t h a t , f o r e v e r y a E F ( I ) ,

sor p r o d u c t a l g e b r a Ea = 8 Ei i s t o p o l o g i z e d w i t h t h e p r o j e c t i v e t e n iea s o r i a l t o p o l o g y TT; so Ecl becomes a l o c a l l y convex algebra ( c f . Lemma 3.1 ) . I n t h i s c o n c e r n , w e f u r t h e r n o t e t h a t t h i s topologization of E a , a E. F(Il, i s uniquely defined ( w i t h i n a l o c a l l y convex v e c t o r s p a c e isomorphism, r e s p e c t i n g t h e a l g e b r a s t r u c t u r e as w e l l , c o n c e r n i n g any " e n u m e r a t i o n " o f a e F(I); " a s s o c i a t i v i t y of t h e t o p o l o g y T " : C f . A . GROTHENDIECK [l:Chap. I ; p. 50 ff.] , a s w e l l a s Lemma 3 . 1 )

.

Consequently, one o b t a i n s a n i n d u c t i v e system o f l o c a l l y convex a l g e b r a s , a s i n ( 5 . 6 ) (see a l s o D e f i n i t i o n IV;2.1), where, f o r any aGB i n F ( I ) , the respective algebra morphisms f are continuous. I n d e e d , it s u f f i c e s , Ba

o f c o u r s e , t o c o n s i d e r o n l y t h e decomposable t e n s o r s of t h e a l g e b r a s

Ea ,a e F ( I I . Thus, f o r e v e r y x

(5.10)

a

= 8xieEa iea

o n e g e t s from ( 5 . 3 ) t h a t f B a ( . 8x i ) = ( 8 x i ) @ (

(5.11)

iea

ZEc1

such t h a t e form

8 ei

a iea

,with

c1

j

8 ej e BRCa

1= x

c1

8 t

e F ( I ) . T h e r e f o r e , t h e map

,nCa 5 . 2 ) i s of t h e

- x 8e a ~1 anca c o n c e r n i n g t h e decomposable t e n s o r s x,EEcl. So it i s c e r t a i n l y c o n t i n u o u s d u e t o t h e c o m p a t i b i l i t y of t h e t e n s o r i a l t o p o l o g y T ( c f . c o n d .

x

(5.12)

(2.1)

i n D e f i n i t i o n 2 . 1 and Lemma 3 . 1 ) . F u r t h e r m o r e , t h e above ( a l g e b r a ) morphisms fBa a r e , i n d e e d , ho-

meomorphisms when r e s t r i c t e d t o decomposable t e n s o r s . ( I n t h i s r e s p e c t , t h e r e w a s a n o b v i o u s a b u s e of n o t a t i o n i n ( 5 . 1 1 ) a p p l y i n g , i n f a c t , a s s o c i a t i v i t y of t h e r e s p e c t i v e t e n s o r p r o d u c t s ) . So w e may s e t now t h e f o l l o w i n g .

D e f i n i t i o n 5.2.

L e t (EilieI

b e a g i v e n f a m i l y of l o c a l l y c o n v e x a l -

g e b r a s w i t h i d e n t i t y e l e m e n t s a n d (Ea, f B a ) , a e F ( I ) , t h e r e s p e c t i v e i n d u c t i v e s y s t e m of l o c a l l y convex a l g e b r a s ( a n d c o n t i n u o u s a l g e b r a morp h i s m s ) , a s a b o v e . Then, w e c a l l i n f i n i t e ( p r o j e c t i v e ) topological tensor

product ( o r s i m p l y i n f i n i t e tensor product l o c a l l y convex ) a l g e b r a , t h e c o r r e s p o n d i n g l o c a l l y c o n v e x i n d u c t i v e l i m i t a l g e b r a E = l i m E a ( Lemma I V ; 2.2).Thus

( b y a n a b u s e of n o t a t i o n ) , w e s t i l l s e t , b y d e f i n i t i o n ,

386

X TOPOLOGICAL TENSOR PRODUCTS

(5.13) The p r e c e d i n g a l g e b r a ( 5 . 1 3 ) i s a ZocaZZy convex aZgebra w i t h continuif t h e a l g e b r a s Ei

ous mZtipZication,

,ieI,

a r e s u c h ( c f . Lemma 3 . 1 i n

c o n j u n c t i o n w i t h Lemma I V ; 2 . 2 ) . On t h e o t h e r h a n d , i f t h e a l g e b r a s E i , i

e I , a r e 2ocaZZ.y m-convex ~ ~ F ( (I t oI p o l o g i z e d

o n e s , t h e n t h e same i s t r u e f o r t h e a l g e b r a s E with

7;

U '

c f . P r o p o s i t i o n 3 . 1 ) . T h u s , one c a n f u r t h e r c o n s i d e r on t h e

a l g e b r a ( 5 . 1 3 ) t h e c o r r e s p o n d i n g f i n a l l o c a l l y m-convex t o p o l o g y ( c f . D e f i n i t i o n I V ; 3 . 1 ) , so t h a t E becomes ( b y d e f i n i t i o n ) a ZocaZZy m-convex a Zgebra

. I n b o t h t h e p r e c e d i n g two c a s e s w e d e n o t e t h e c o r r e s p o n d i n g com-

p l e t e l o c a l l y convex ( r e s p . , h

E

(5.14)

l o c a l l y m-convex) h

a l g e b r a by

A

B E . ( : = l+ imEu).

iei

I n t h i s r e s p e c t , one actm22y gets A

(5.15)

E:

=

h

C3

ieI

e.

A,-

zi,

Ei := I* i m E a = limEa = ( D e f i n i t i o n 5 . 2 ) 8 iei

w i t h i n an isomorphism of the topoZogicaZ algebras under

consideration: The a s s e r -

t i o n i s e a s i l y d e r i v e d from t h e same d e f i n i t i o n s o f t h e r e l a t i o n s i n v o l v e d i n ( 5 . 1 5 ) and t h e d e f i n i t i o n o f t h e t o p o l o g y of t h e a l g e b r a ( 5 . 1 3 ) . See a l s o C h a p t . 1 V ; S e c t i o n 2 , a s w e l l a s P r o p o s i t i o n I V ; 3 . 1 , c o n c e r n i n g t h e t w o p a r t i c u l a r cases of t o p o l o g i c a l a l g e b r a s which a r e considered i n (5.14)

.

The above w i l l b e a p p l i e d i n s u b s e q u e n t s e c t i o n s of t h i s book, i n p a r t i c u l a r , i n what c o n c e r n s s p e c t r a of t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s (see C h a p t . X I I ; S e c t i o n 2 ) .

387

Topological Tensor Product A1 g e b r a s . Examples

CHAPTER X I

We consider in this chapter some special instances of topological tensor product algebras, which are of a particular significance for the applications within several contexts. Thus, the algebras involved will be , in particular, algebra-vaZued f u n c t i o n algebras which thus may be considered, in an appropriate way, as topological tensor product algebras in the sense of the previous chapter.

1. The algebra

cc(X, E)

The algebra in title is that of continuous functions defined on a Hausdorff topological space X with values from a given (complex) topological algebra E and pointwise defined operations, endowed with the topology of compact convergence in X (denoted, as usually, by c ) . In particular, E will be a locally convex algebra, eventually, with continuoas multiplication. Thus, if r = ( P I is a fundamental family of semi-norms defining the (locally convex) topology of E and K the family of compact subsets of X , one gets a fundamental defining family of semi-norms for the topology c of the algebra C I X , E l by the relation

with f E CCX, E l , and for any p E r and K E K . Now, as follows from ( 1 . 1 ) and an application of a familiar argument, the l o c a l l y convex space c c ( X , IE) i s Hausdorff i f , and only if, t h i s i s t h e case f o r B. On the other hand, i f t h e m u l t i p l i c a t i o n i n E i s separately, o r j o i n t l y , continuous then the same is t r u e f o r the algebra CctX,E l . Indeed, taking (1.1) into account, this is a direct consequence of the continuity of the appropriate linear endomorphisms, or the bilinear map, respectively, defining the multiplication in the latter algebra. Furthermore, it is straightforward too, from (1 . I ) , that if the given algebra E i s l o c a l l y m-convex, then t h e same holds t r u e f o r t h e topological algebra c , C X , X ) : just consider submultiplicative semi-norms p in (1.1) (cf. Theorem I; 3 . 1 . See also Chapt. 1;Example 3.1).

XI EXAMPLES

W e come now t o t h e main p o i n t o f t h e p r e s e n t example f o r which,

however, w e n e e d f i r s t t h e f o l l o w i n g b a s i c r e s u l t . Lemma 1.1. Let X be a Hausdorff completely regtdar k-space ( c f . C h a p t . V ; S e c t i o n 5 ) and E a complete l o c a l l y convex space. Furthermore, l e t q X , E ) be

t h e l o c a l l y convex space of E-valued continuous maps on X ( w i t h p o i n t w i s e def i n e d o p e r a t i o n s ) endowed w i t h the topology of compact convergence i n X. Then,

one has t h e r e l a t i o n (1.2) within an isomorphism of l o c a l l y convex spaces ( c f . D e f i n i t i o n X ; 2.3)

.

Proof. W e f i r s t remark t h a t one g e t s " c a n o n i c a l l y " a b i l i n e a r map (1.3)

:

a x ) x E+

e t x , E)

: If,

8)-

z ) : =f z ,

%(f,

where (fzl(x1 = f ( x l z € E , x e X , so t h a t by t h e " u n i v e r s a l p r o p e r t y o f t e n s o r p r o d u c t s " one o b t a i n s a c a n o n i c a l l i n e a r map ii : C(X)@E+etX,E) g i v e n by

(1.4)

u(

.? fi@ 2i 1 = ! f i 2i

-L=l

I

1 fi @Zi e C(XlC4E. NOW, t h e l i n e a r map ii i s 1-1 : Namely, f o r e v e r y i 148 e e ( X ) @ E, w i t h z #0, o n e g e t s a c o r r e s p o n d i n g e x p r e s s i o n f o r

f o r any z

I

zi

z with l i n e a r l y independent

(4)a n d (zi)

( c f . Lemma X;1.2), so t h a t t h e r e l a t i o n imply G ( x ) = O r

f o r e v e r y x e x and i = 1 ,

in

C(X) and E , r e s p e c t i v e l y

1 f . f ~=) 0~ , w . i t h x e X , would Z

..., n ;

-

L

but t h i s

is a contradic-

t i o n t o z # 0 , which p r o v e s t h e a s s e r t i o n . C o n s e q u e n t l y , o n e g e t s , v i a ii, t h e f o l l o w i n g " f u n c t i o n a l r e a l ization", valid within a linear injection; i.e., C(X)@iE 5

(1.5)

CIX, E l .

ii Thus, Lhe two ( l o c a l l y c o n v e x ) topoZogies E and c i n d u c e d on I m ( i i l r C ( X ) @ E by & e ( ( C ( ~ ) E;) ) i , a n d C (x,E ) , r e s p e c t i v e l y , are i d e n t i c a l . I n d e e d , C

from ( 1 . 1 ) and ( X ; (2.33)), o n e c o n c l u d e s f i r s t t h e r e l a t i o n

concerning t h e respective "unit-pseudospheres", would imply o f c o u r s e t h a t c < longs to t h e f i r s t member of

E

on I m ( i i ) .

a s i n d i c a t e d , which

In fact, i f z =

;.

f.@zi

-L=l %

be-

(1.6), t h e n f o r e v e r y x e K o n e g e t s t h a t

f i ( x ) @ Z i € E ; t h e r e f o r e (Hahn-Banach) , t h e r e e x i s t s X ' E (S ( 1 ) ) O (= ix'€ + I 6 p(u', , w i t h $ e E l , see a l s o Lemma I ; 1.2) , s u c h t hPa t , from t h e E': Ix'(c) h y p o t h e s i s for z and s i n c e K C ( S ( l ) ) O ( c f . a l s o Lemma VII;l.l), one h a s

pK

I.

=

THE ALGEBRA

cp,E )

?2 & xrf.)x‘c2i!=(s c4X‘I)(1fiOZiti) = 2 X

)((sxLsx’)(z)I 5 1 .

i

Thus, from ( 1 . 1 ) , we still obtain that inu‘

(2)

22

PK

(1.6).

389

,

as well, which proves

On the other hand, consider the equicontinuous sets A G ( e c ( X ) l ‘ and B E E ’, given as follows; (1.7) A = {ue lV(f)1 ~ ~ ~ . f~e ~ ( f =) Is, 1 I

(ep),:

ectx))

PK for some (1.8)

E

1

> O

1

and K E X compact, as well a s

B = {x*eE,:lx’(Z)

1 5 E2. p m , a’ e E 1

= (S (F) 1 10 p

2

,

for some E ~ > Oand a continuous semi-norm p on B . Thus, one proves the relation

or, equivalently, from ( 1 . 7 )

and ( 1 . 8 ) ,

the relation

(1.10)

wkich, in turn, implies of course that 1 (-) : then one has f.@Zi e S i=l 2 ‘p,K €1‘2

E G C :

Indeed, suppose that

t e

So on the basis of : X ; ( 2 . 3 0 ) ) one now obtains the desired rel. ( 1 . 9 ) , and this proves our assertion. Thus, we proved so far t h a t ~ = c on ImlGI Y ClXl B E ; accordingly, 6 d e f i n e s a topoLogical Linear isomorphism for t h e r e s p e c t i v e locally convex spaces. Furthermore, it is a direct consequence of the preceding relations ( 1 . 6 ) and ( 1 . 1 0 ) that

390

XI EXAMPLES

(1.11) concerning the respective unit-pseudospheres, namely, a "geometric realization" of our previous conclusion. Now, by hypothesis for X , one has that X =limK (see Chapt. V;

KFK

(5.7)); therefore, one obtains (1.12)

valid within allnatural homeomorphism" of the topological spaces involved (cf. also N . BOURBAKI [5: Chap. 10; p. 2, Definition 2 1 ) . Thus, since each one of the spaces C u ( K , B ) , with K E K , is complete (ibid.;p. 9, Corollaire I), t h e space i n t h e first member of (1.12) is complete as well (see N . BOUREAKI [ 4 : Chap. 2; p. 17, Corollaire]). We come next to the proof that c ( X ) @ Bi s a dense subset of C _ ( X , B): E ) deT h u s , let f e C ( X , S ) and S (E) a neighborhood of zero in NP,K fined by a semi-norm N given by (1.1). Now, by the continuity of P,K f , one finds an open neighborhood U for every element x e X such that 1

cc(X,

(1.13) there exist finite many such neighborhoods Ui ( i = l , .. ., n ) supplying an open covering of the compact set K G X , so let x.E Ui , l
So

more, there exists an associated "finite partion of unity, subordinated to (Uil15i5nn , i.e., a family ( h . ) c C ( X ) such that O < h i S l , Supp(hi) E

U. ( i = l ,..., n ) , and t h . = l . Therefore, one gets z z

I thi("C)p(f(x)-f(xi))

S(by (1.13))z h i ( X ) E = E ,

2

2

for every x E X . Thus, one finally gets from (1.1) the relation n If- 1 h.flX.)J S E, N (1.14) P,K i=I 2 ,7with

h . f ( x i l = G ( t h i 8 f ( x i ) ) e I m ( Z ) , which proves the assertion.

z z

z in view of the preceding "identifications", one has already the relation So,

c,iX)GE

(1.15)

E

= C l X ) @ l E=

ccixB) ,

and this completes the proof of the lemma. I T h u s , specializing to topological algebras, we now obtain the

following.

391

Theorem 1.1. Let X be a (Hausdorff) completely regular k-space and E a com-

p l e t e l o c a l l y convex algebra. Then,

C'tx,

E ) (Lemma 1.1) i s a complete l o c a l l y con-

v e x algebra, i n such a way t h a t one has t h e r e l a t i o n (1.16)

w i t h i n an isomorphism o f ( l o c a l l y convex) topological algebras. I n p a r t i c u l a r , t h i s i s t h e case i f E

i s a l o c a l l y convex algebra w i t h con-

tinuous m u l t i p l i c a t i o n , or y e t a Zoeally m-convex one, such that (1.16) y i e l d s then an isomorphism w i t h i n t h e r e s p e c t i v e category o f topological algebras.

Proof. I n view o f t h e p r e c e d i n g Lemma 1 . 1 , t h e map (1.17)

g i v e n by ( 1 . 4 )

p r o v i d e s a t o p o l o g i c a l l i n e a r isomorphism o n t o i t s

r a n g e . Moreover, i t c e r t a i n l y p r e s e r v e s t h e r i n g s t r u c t u r e s of t h e a l gebras

C(X)@Z( c f .

a subalgebra of

X ; ( 1 . 9 ) ) and Imlii), when t h e s e c o n d one i s viewed as

CtX, E ) .

On t h e o t h e r hand, t h i s t o p o l o g i c a l ( v e c t o r

s p a c e ) isomorphism i s ( u n i q u e l y ) e x t e n d e d ( s t i l l d e n o t e d b y ii) t o t h e r e s p e c t i v e complete s p a c e s

e c f X ) $ E and

Imlii) = ( b y (1.15))e.(X, Ej,

where t h e l a t t e r s p a c e i s , i n p a r t i c u l a r , a complete l o c a l l y convex algebra ( s e e a l s o t h e comment a t t h e b e g i n n i n g of t h i s s e c t i o n ) ; m o r e o v e r , t h e

same i s i s o m o r p h i c , a s a l o c a l l y convex s p a c e , t o q " l X ) b , t h i s l a s t E

s p a c e b e i n g t h u s c o n s i d e r e d , v i a t h e e x t e n d e d map 5 , t o b e s t i l l an algebra. Finally,

i f t h e given algebra E has a continuous multiplication

o r i s , i n p a r t i c u l a r , a l o c a l l y m-convex o n e , t h e t o p o l o g i c a l a l g e b r a isomorphism ii, a s a b o v e , g i v e n by ( 1 . 1 7 ) , c a n b e e x t e n d e d a s s u c h t o t h e r e s p e c t i v e completions ( c f . Chapt. I; S e c t i o n 4 , i n p a r t i c u l a r , Lemma 4.l);so w e s t i l l g e t ( 1 . 1 6 )

i n t h e c a s e u n d e r c o n s i d e r a t i o n , and

t h i s terminates the proof. I The f o l l o w i n g lemma i s now n e e d e d i n o r d e r t o g i v e a u s e f u l apl i c a t i o n of t h e p r e c e d i n g t h e o r e m .

Lemma 1.2. L e t X , Y be completely regular ic-spaces such t h a t t h e c a r t e s i a n produet space X x Y t o be a ( c o m p l e t e l y r e g u l a r ) k-space a s w e l l ( X ,Y m i g h t b e , f o r i n s t a n c e , l o c a l l y compact s p a c e s ) . Then, one g e t s

w i t h i n a topological algebraic isomorphism

of t h e corresponding topological alge-

bras. Proof.

I n d e e d , t h e map ( 1 . 1 8 ) i s g i v e n by

XI EXAMPLES

392

(1.19)

f

I

+$,

with 1 ? ( x ) I ( y ) = f (x,y l

,

for every f E c ( X , Y ) , where the correspondence ( 1 . 1 9 ) is clearly an algebra isomorphism with domain and image those indicated by (1.18):See e.g. J. DUGUNDJI [ I : p. 261, Corollary 3.2, and p. 265, Theorem 5.31. I So

we now have the following.

Corollary 1.1. L e t the hypothesis of t h e preceding Lemma

2.2 be s a t i s f i e d .

Then, one has (1.20)

ec(xxY) =

qx,

e p = q x )2 ~

Y

I

,

within a topological algebraic isomorphism of t h e ( Z o c a l l y m-convex) topological algebras involved. Proof. The first part of ( 1 . 2 0 ) is already concluded from ( 1 . 1 8 ) , while the second relation in ( 1 . 2 0 ) is a direct consequence of Theorem 1 . 1 and this completes the proof. I

An interesting application of the previous relation ( 1 . 2 0 ) will be given in the next chapter (see Scholium XII: 1 . 1 as well as Lemma XII; 1.4).

2. The algebra

c"(X,E)

We have already examined in Chapt. IV; Subsection 4 . ( 2 ) the algeof complex-valued "Cm-functions" on a given differential bra Crn(X) ((?-)manifold X, its topology as a topological algebra being defined by (IV; ( 4 . 1 9 ) ) . On the other hand, replacing the scalar field C, range of the functions involved, by an arbitrary complete l o c a l l y convex space B one applies (for convenience!) the following definition: f E

ern(X, E) i f ,

and o n l y i f (by definition), a ' o f

e CrnfXI, f o r every a ' f l E ' (the topological dual of El.

The preceding condition(-definition) is, indeed, a theorem, for one can define directly the "partial derivatives" of an element f E C"(X,B) extending, in an obvious manner, the corresponding "limiting process" of the scalar case: so one is then led to ( 2 . 1 ) ("Grothendieck's Lema". See e.g. L . SCHWARTZ [3: p. 146, Lemme 111 1 . NOW, in complete analogy with the scalar case, one defines for every open subset U of a given (o-compactem-) manifold X, t h e l o c a l l y convex space c : ( U ,

El

of E-valued Crn-functions on U Z X endowed with

2.

THE ALGEBRA c ( X , E)

393

t h e "topology of compact convergence of the functions and a l l o f

t i v e s " . Thus i f

r=(pl

their

deriva-

i s a fundamental f a m i l y of semi-norms d e f i n i n g

t h e t o p o l o g y of B , one g e t s a n a n a l o g o u s f a m i l y f o r t h e p r e v i o u s 1.0c a l l y convex t o p o l o g y by t h e r e l a t i o n

f o r e v e r y f e e " ( U , E ) , where p e r , w h i l e t h e m , a, K's v a r y a s i n ( I V ; (4.12) 1 .

Thus one o b t a i n s a complete l o c a l l y convex space which is, i n particu-

l a r , a Frgehet space i n case E i s ( i b i d . , and F . TR.WES [ l : p. 446, P r o p o s i t i o n 44.11). I n p a r t i c u l a r ,

c:(U,

E ) i s a complete l o c a l l y convex algebra,

i n c a s e E i s such which, moreover, has a separately or j o i n t l y continuous

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

E . This i s eas-

i l y c o n c l u d e d , i n a s i m i l a r way a s f o r t h e a l g e b r a c(X,E )

i n the pre-

v i o u s s e c t i o n , u s i n g now ( 2 . 2 ) . I n p a r t i c u l a r , one h a s t h e f o l l o w i n g .

Lemma 2.1.

U an open subset o f X and E

Let X be a a-compact C"-manifold,

a Frdcchet l o c a l l y m-convex algebra. Then, the Frdcket l o c a l l y convex a % g d m C"fX,E) topologized v i a ( 2 . 2 ) may be given a family of s u b m l t i p l i c a t i v e semi-norms d e f i n i n g i t s topology, i n such a way t h a t c : ( U , El becomes a Fre'chet l o c a l l y m-convex algebra. Proof. I t i s c l e a r from t h e p r e c e d i n g t h a t e;(tv,El, t o p o l o g i z e d by t h e f a m i l y o f semi-norms g i v e n by ( 2 . 2 )

I

i s a Frgchet ZocalZy eonzvx

algebra. Now a p p l y i n g a s i m i l a r r e a s o n i n g t o t h a t i n t h e s c a l a r c a s e , one g e t s from t h e " l o c a l m-convexity''

of t h e a l g e b r a B a n a n a l o g o u s

r e l a t i o n t o ( I V ; ( 4 . 1 7 ) , s o t h a t t h e c o n d i t i o n s of C o r o l l a r y I ; 5.3 a r e s a t i s f i e d , and t h i s p r o v e s t h e a s s e r t i o n . 1 W e come n e x t t o examir.e t h e p o s s i b i l i t y of "approximating" t h e

e l e m e n t s of t h e p r e v i o u s a l g e b r a , a p p r o p r i a t e l y , by e l e m e n t s of t h e t e n s o r p r o d u c t a l g e b r a C"tUl@E. Thus w e f i r s t have.

Lemma 2.2.

Let X be a C - - m a n i f o I d ,

U an open subset of X and B a corn-

pLete ZocaZly convex algebra. Then, one has (2.3)

w i t h i n an isomorphism of l o c a l l y convex spaces. Proof. I n

f a c t , one p r o v e s t h a t e * ( U l @ E i s

( l i n e a r l y isomor-

p h i c t o ) a d e n s e s u b s e t of c z ( U , BI and t h e n , s i n c e 6 i s c o m p l e t e , the re1.(2.3),

r e a l i z i n g t h a t t h e two t o p o l o g i e s

i c a l o n ~ w ( U ) B BFor . the d e t a i l s

E

and c a r e i d e n t -

see F. TREVES [I: p. 448, P r o p o s i t i o n

394

XI EXAMPLES

44.2, and p. 44, Theorem 44.11 . I

Now, suppose, in particular, that E is a locally convex algebra. Then applying a similar argument to that used in the proof of Theorem 1.1, we conclude with the following result, remarking besides that i s a nuclear l o c a l l y eonvex space (see A . GROT€iENDIFCK [3: Chap. 11; p. 54 ff.]) such that (X; (2.42)) to be true. Thus we get.

ep

Theorem 2.1.

Let X be a Cm-manifold, U an open subset of X and E a com-

p l e t e l o c a l l y convex algebra. Then, one g e t s

(2.4)

ep$E= e p i g ~= ey,3) ,

within an isomorphism of l o c a l l y convex algebras, t h i s being a l s o t r u e i n ease B

has a ( j o i n t l y ) continuous m u l t i p l i c a t i o n . I n paricular, i f X i s +compact and E a Fre'chet l o c a l l y m-convex algebra, one obtains

(2.5)

C"ctU)$E =

eyJ, E ) ,

within an isomorphism of (Frdchet) l o c a l l y m-convex algebras, where T i s any one of the (compatible tensorial) topologies considered i n Section X ; 2 . I

Now, let f U c i ) a e I be an open covering of X. (We may assume that this consists of local charts of X belonging to the maximal Cw-atlas defining the differential structure of X; cf. Subsection IV;4.(2)). So one gets the relation

where X is a paracompact em-manifotd and E a (complete) locally convex space, resp., algebra. So from (2.4) and (2.6) one now obtains

within an isomorphism of locally convex spaces (cf. also A. GROTHENDIECK [3: Chap. I; p. 46, Proposition 6 1 ) . Of course, t i i r precoding isomorphism i s "topological algebra preserving", as it was also the case with the previous Theorem 2.1. Thus, in the context of the last part of that theorem, one concludes that (2.8)

em tx, S ) = cm IX)& E T

,

w i t h i n an isomorphism of FrLchet l o c a l l y m-convex algebras; here again

T

is an$

5.

NACHBIN'S THSOREM (VECTORIZATION)

395

one of t h e ( c o m p a t i b l e ) t e n s o r i a l t o p o l o g i e s d e f i n e d i n S e c t i o n X ; 2 .

We c o n c l u d e w i t h s t i l l c o n s i d e r i n g t h e a l g e b r a aIX) o f t e s t funce m - f u n c t i o n s on X ( s e e C h a p t . I V ; S u b s e c t i o n 4 . ( 2 ) ) . Thus, one g e t s

t i o n s on X , a s a b o v e , i . e . , t h a t o n e of complex-valued w i t h compact support

m

(2.9) ( c f . L . SCHWARTZ [l: p. 891 ) .

Now,

= emcx,

combining t h i s w i t h ( 2 . 8 )

,

one r e a d i l y

obtains t h a t

a ( X ) @ E= m x , EI

(2.10)

= e m ( X l 633 = TT

e."tx, x?) ,

where t h e r e s u l t i n g r e l a t i o n s r e s p e c t , i n e f f e c t , t h e algebra s t m c t u r e s of t h e s p a c e s i n v o l v e d . Here one a p p l i e s , o f c o u r s e , t h e f a c t t h a t .aIX)@E 5 C ( X , E ) ,

(2.11)

w i t h i n a n a l g e b r a isomorphism (see a l s o ( 1 . 5 ) ) NOW,

t h e preceding rels.

(2.9)

and ( 2 . 1 0 ) c a n b e a p p l i e d , f o r

example, i n t h e i d e n t i f i c a t i o n of t h e s p e c t r a of t h e ( l o c a l l y m-convex) a l g e b r a s a ( X l and/or

P(X,E). ( S e e

Chapts. VI a n d / o r

XII, a s w e l l

a s Chapt. V I I ; S e c t i o n 2 ) .

3. The algebra

cm(X, h)(contn'd.):

Nachbin's Theorem ( v e c t o r i z a t i o n )

W e p r e s e n t i n t h e f o l l o w i n g l i n e s a " v e c t o r i z a t i o n " of Nachbin's

Theorem a l r e a d y d i s c u s s e d , 5-7).

i n t h e "scalar c a s e " , i n C h a p t . V I I ( S e c t s .

Moreover, a n o t h e r " i n f i n i t e d i m e n s i o n a l " form

ium 3 . 1 below) h a s b e e n g i v e n a l r e a d y b y L . Nachbin

of

it (see S c h o l -

himself

(see, f o r

i n s t a n c e [ 6 ] 1, as w e l l a s by o t h e r a u t h o r s ( c f . , f o r example, J . B . PROLLA [l

]

and J . G . LLAVONA [l] )

.

Thus, w e f i r s t have t h e f o l l o w i n g r e s u l t b e i n g , i n f a c t , oneh a l f of o u r f i n a l t h e o r e m

( n e c e s s i t y of Nachbin's c o n d i t i o n ; c f . Lemma V I I ;

5.1).

Lemma 3.1.

L e t X b e a (paracompact 2nd countable Hausdorffl e m - m a n i f o l d ,

E

a I n o n - t r i v i a l complex) complete l o c a l l y convex algebra w i t h continuous m u l t i p l i c a t i o n , and A a subalgebra of c"IX,lE) such t h a t -

A =

(3.1)

where t h e algebra

e"(X, E) ,

m X , IE) i s t o p o l o g i z e d by (2.6). Then, t h e f o l l o w i n g c o n d i t i o n s

hold t m e (see a l s o Lemma V I I ; 5.1

f o r t h e terminology a p p l i e d ) :

1 ) The algebra A i s non-vanishing on X . 2) The algebra A s e p a r a t e s t h e p o i n t s o f X . (The p r e c e d i n g two c o n d i -

t i o n s c h a r a c t e r i z e , b y d e f i n i t i o n , A a s s u f f i c i e n t l y s e p a r a t i n g o n ,Y; c f . Theorem V I I ; 6 . 1 ) .

396

XI EXAMPLES

3) The (point) differentials of the functions in A, at any point x : X , separate the points of the tangent space T ( X , x ) of the rnanifotd X at x .

B e f o r e embarking on t h e p r o o f of t h e p r e v i o u s l e m m a , it i s r a t h e r i n s t r u c t i v e t o make some commer.ts on t h e t e r m i n o l o g y u s e d ( c o n d . 3) o f Lemma 3 . 1 ) , when c o n s i d e r i n g " v e c t o r - v a l u e d "

f u n c t i o n s on t h e

g i v e n m a n i f o l d X. Thus b y c o n s i d e r i n g t h e tangent space T(X, x ) o f X a t t h e p o i n t xeX

...,xnl ,

and a local chart, s a y , ( U , $ ) ~ ( U ; x l y

at x

( c f . Chapt. VII; ( 5 . 3 ) ) ,

one h a s , f o r e v e r y v f TtX, x ) , t h e e x p r e s s i o n

( i b i d . ; (5.6)). So o n e may d e f i n e f o r e v e r y f u n c t i o n f E c ( X , E) t h e ( p o i n t ) differential of f at

xeX

by t h e r e l a t i o n

(3.3) where n = d i m X , w h i l e one s t i l l s e t s , b y d e f i n i t i o n , (3.4)

see a l s o V11;(5.5)). C o n c e r n i n g t h e p a r t i a l d e r i v a t i v e of t h e f u n c t i o n fo@-' :

@(Ul+E which i s i n d i c a t e d i n t h e s e c o n d member o f ( 3 . 4 ) , one

applies the standard definition

( c f . e.g.

L . SCHWMTZ [3: p. 93, ( l . l ) ] ) .

Thus, a p p l y i n g t h e n o t a t i o n ( c f . (3.3)) 1 J f l := ( d f l x ( v l ,

(3.5)

w i t h v e T(X, x l , f e e m ( X , El and x e X , a n d t h e f a c t t h a t (3.6)

= Ifoa-1) @ a,

(faalo

f o r any f u n c t i o n f o a

E

c"CX)@E, one g e t s from ( 3 . 3 ) t h e r e l a t i o n

(3.71 the (point) where v e T I X , x) i s g i v e n by ( 3 . 2 ) . Thus, d e n o t i n g by Z; VII; S e c t i o n 5) which i s g i v e n by C h a p t . d e r i v a t i o n of e * ( X l a t x ( c f . (VII;( 5 . 2 ) ) , o n e o b t a i n s from ( 3 . 7 ) t h e r e l a t i o n Z = Z' @ i d E

(3.8)

'

Therefore (cf

.

j

'

j

.

A . GROTHENDIECK [3 : Chap. I ; p. 37, § 21 ) , o n e h a s

(3.9) Furthermore, s i n c e f o r every v E T(X, x ) with

2)

f 0 , one h a s Z' f 0 V

5.

NACHBIN'S THEOREM (VECTORIZATION)

397

(cf. the proof of Lemma VII; 5.1), one gets from (3.8) and (3.9) the relation

T(X,x) 5

(3.10)

C(

cw(x, B ) , E ) ,

w i t h i n a l i n e a r isomorphism. Thus, we come now to the Proof of L e m 3.1. The desired properties 1 ) and 2 ) for the alge-

bra A are straightforward from (3.1) and the relation (3.11)

valid within an algebra isomorphism (see (1.5)). In this respect, we still take into account that, by hypothesis,E#(O), as well as the fact that C"(X) is "sufficiently separating" on X (cf. Lemma V11:2.1). On the other hand, one has by (3.1) (3.121

so that the desired cond. 3) for A is now a direct consequence of (3.10), and the proof is complete. I We suppose next that the algebra E l range of the cm-functions involved is a (complex) complete locally convex (topological) algebra with continuous multiplication having, moreover, one at least non-trivial closed s i n g l y generated subalgebra. The last condition on B is equivalent, of course, with the assumption that t h e r e e x i s t s an element x e E such t h a t x2 # O.For example, this is true i f E has an identity element or yet in case of a commutative algebraE with non-trivial multiplication. On the other hand, we shall further consider a subalgebra A of the algebra e m ( X , El, as above, in such a way that the following condition is satisfied: For any x ' e E ' f

(topological dual of E ) , f e A and a

E , one has

(3.13)

(x'of)@aeA,

where one defines (3.14)

( ( x / o f ) @ a ) ( x := ) x'(frx)f.a

,

for every x e X. We call (3.13) the ProlZa-Machado c o n d i t i o n , or simply (PM)-condition, for the algebra A . It is cl-ear that any subalgebra A of e m ( X , 33) with e " ( X l C 3 E E A satisfies the (PMI-condition, in view of (3.14). Furthermore, the significance of (3.13) for the sequel is essentially revealed through the following.

398

XI EXAMPLES

Lemma 3.2. Suppose t h a t thtl hypothesis of Lenuna 3.1 i s s a t i s f i e d concerning X and E , except of cond. ( 3 . 1 ) . Moreover, assume t h a t E has a n o n - t r i v i a l closed s i n g l y generated subalgebra. F i n a l l y , l e t A b e a subalgebra of w X , EI s a t i s f y Then, t h e s e t

ing t h e fPM1-condition.

M={x’of:x’eE’,

(3.15)

i s a subalgebra o f

w X 1 , i n such a way t h a t one has

M@E E A .

(3.16)

Proof. S i n c e A E

ew(X, E ) , it

L . SCHWARTZ [3: p. 146, Lemma 111 MC

feA)

.

i s clear

( G r o t h e n d i e c k ’ s Lemma; see

Cf. a l s o ( 2 . 1 ) i n t h e preceding) t h a t

ewfX) Now . t h e h y p o t h e s i s s e t f o r t h f o r t h e a l g e b r a s B and A i m -

p l i e s t h a t A i s , i n e f f e c t , a polynomial algebra, i n t h e s e n s e o f J . B . PROLLA and S . MACHADO [l:

p. 250, D e f i n i t i o n 2 . 3 1 ; t h i s i s d e r i v e d from t h e i r

Lemma 2 . 2 ( i b i d . ; p . 2 4 9 ) , a f a c t w h i c h , i n t u r n , e n t a i l s t h e a s s e r t i o n

of o u r lemma, a c c o r d i n g t o t h e same r e s u l t . B The n e x t lemma e x p l a i n s t h e r61e p l a y e d by t h e same (PMI-condition

a s above i n o u r “ v e c t o r i z a t i o n ” o f N a c h b i n ’ s t h e o r e m . Thus w e h a v e . Lemma 3.3. Suppose t h a t t h e c o n d i t i o n s of t h e previous L e m 3.2 are

i s J i e d , i n such a way t h a t t h e algebra A G cw(X, B) f u l f i l s , d i t i o n s 11 -31 o f Lema 3.1. Then, t h e algebra M C =

sat-

moreover, t h e con-

ew(X)eiCX)( c f . (3.15)) sat-

i s f i e s t h e Condition IN) (see D e f i n i t i o n V I 1 ; G . l ) . Proof.

The f a c t t h a t ir!

i s s u f f i c i e n t l y separating on X ( i . e .

,

conds.

o f Lemma V I I ; 5.1 a r e s a t i s f i e d ) f o l l o w s d i r e c t l y from t h e re-

11, 2)

s p e c t i v e p r o p e r t i e s o f t h e a l g e b r a A ( c f . t h e p r e v i o u s Lemma 3 . 1 ) t h e hypothesis f o r

and

6 ,t h e l a t t e r b e i n g , i n p a r t i c u l a r , a H a u s d o r f f

l o c a l l y convex s p a c e so t h a t one may a p p l y Hahn-Banach

( c f . , f o r in-

s t a n c e , J . HORVkH [I: p. 108, P r o p o s i t i o n 2 , and p . 1 8 5 1 ) . On t h e o t h e r h a n d , f o r e v e r y u e T ( X , x) w i t h v # O , t h e r e e x i s t s a n E - v a l u e d f u n c t i o n f e A such t h a t

lu(fl

(3.17)

Hence, a p p l y i n g Hahn-Banach, element

X’E

(3.18)

=

( d f Ixlv) # 0 .

a s above ( c f . a l s o ( 3 . 9 ) ) , one g e t s a n

E’, with

x‘l(dflx(vl) # 0 .

So o u r a s s e r t i o n f o r t h e a l g e b r a M S e m ( X ) c o n c e r n i n g c o n d . 3 ) of Lem-

ma V I I ; 5.1

f o l l o w s now d i r e c t l y from ( 3 . 1 8 ) a n d from t h e o b v i o u s re-

l a t i o n (see a l s o ( 3 . 3 ) and ( 3 . 4 ) ) (3.19)

x ’ ( l d f k ( v l l = d(x‘o f l x ( v ) . I

5.

RliCHBiii ' S THEOREM ( V E C T O R I Z A T I 3 ? J )

399

NOW, t h e f o l l o w i n g i s a d i r e c t c o n s e q u e n c e of N a c h b i n ' s Theorem

( " s c a l a r c a s e " ; see Theorem V I I ; 7.1) and t h e p r e v i o u s lemma.

Corollary 3.1. Let t h e hypothesis of t h e preceding Lema 3.3 be s a t i s f i e d and, moreover, assume that t h e algebra M i s a s e l f - a d j o i n t subalgebra of c m ( X ) ( c f . a l s o (3.21) b e l o w ) . Then, one has

z = e-ix, .

(3.20)

I

Now c o n c e r n i n g o u r assumption i n t h e p r e v i o u s c o r o l l a r y t h a t

t h e a l g e b r a M w a s a s e l f - a d j o i n t s u b a l g e b r a of

e"(X)

Cg(X), w e

remark

t h a t t h i s i s s a t i s f i e d i f the following condition ( s u p p l e m e n t a r y t o (3.13))

holds t r u e : For any (x', f , a I E E ' x A

x

E , one has the r e l a t i o n

G'ofloa E A .

(3.21) I n t h i s r e s p e c t , one d e f i n e s

,

( ( 2 ' 0 f l c a a ) ( x ) := x ' ( f ( x l l . a

(3.22) with x E X I the

(PM)'-condition NOW,

"bar" d e n o t i n g c o m p l e x - c o n j u g a t i o n .

W e c a l l (3.21) t h e

f o r A.

g i v e n an a l g e b r a

A C C"fX,

E ) , a s above, w e s h a l l s a y

t h a t t h i s s a t i s f i e s t h e f u l l (PM)-condition, i f b o t h

the

conditions

( 3 . 1 3 ) a n d ( 3 . 2 1 ) are v a l i d i n A . On t h e o t h e r h a n d , a p p l y i n g t h e t e r m i n o l o g y w e j u s t i n t r o d u c e d ,

i f the algebra A s a t i s f i e s the (PM)'-condition, then f o r every x'of EM, one g e t s ;'of€

M

,

(3.23)

i n E ' , l e t a e E w i t h x'(a) = 1.Hence

a s w e l l : Thus, i f

x'f

0

w i t h x'f

0,

one g e t s

f o r every x'of

EM,

,'of

= ~ ' ~ ( ( E ' oaf ia )

E

M,

which p r o v e s t h e a s s e r t i o n . T h u s , w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g " v e c t o r i z a t i o n " o f N a c h b i n ' s Theorem.

Theorem 3.1. Let X be a paracompact 2nd countable Hausdorff e m - m a n i f o l d , B a (complex) complete l o c a l l y convex algebra w i t h continuous m u l t i p l i c a t i o n hav-

i n g a n o n - t r i v i a l closed singly generated subaZgebra, and A a subalgebra of e"(X,E)

(cf. S e c t i o n 2 ) s a t i s f y i n g t h e f u l l (PMI- condition. Thsn, t h e s e r t i o n s are equivalent : 1)

T=

following two as-

e"cx, El.

2 ) The algebra A s a t i s f i e s the t h r e e conditions

of Lemma 3 .1 .

Proof. I t i s a c o n s e q u e n c e o f Lemma 3.1 t h a t 1 ) + 2 ) ( e v e n u n d e r

400

XI EXAMPLES

the milder hypothesis On the other hand, if ceding discussion and c " ( X l . Therefore, one

of that lemma, concerning the algebras 2: and A ) . 2) is valid, then one concludes from the preCorollary 3 . 1 , that M is a dense subalgebra of has

(3.24)

that is, M8E is a dense subalgebra of e"(X EI , so that the same is true for the algebra A ; thus, 1) is valid since M 8 X C A (cf. ( 3 . 1 6 ) ) , and this completes the proof. I

Scholium 3.1.- As we have already mentioned there are also other ("infinite dimensional") versions of the previous Theorem 3.1. Thus, L. Nachbin [6] gave a treatment of his initial result [ I ] (extending also previous work on this subject) in terms of Cm-functions ( r n = l , . ..,m) defined on an open subset of a given (real) locallyconvex space E and taking values from another such space F. Here the notion of polynomial algebra is applied (see e.g. J . B . PROLLA [ I ] ) . In this respect, there is also another point that merits a particular attention: Namely, the interference of the Banach-Grothendieck approximation property (cf. Definition X; 2 . 4 ) of the spaces, "source" of the (vector-valued) functions considered (see also Chapt. VII1;Section 1 0 for a similar situation in "infinite dimensional holomorphy"). Thus, Condition ( N I (see Definition ViI;6.1)seems to be intimately related to the afore-mentioned property; so while Condition ( N ) i s stronger (cf. L. NACHBIN [6: p. 201]), it has been conjectured as well (ibid.) that these two a r e , i n e f f e c t , equivale n t . However, for further details we refer to the pertinet literature cited above. 4. The algebra U ( X , E) In the sequel E will denote a complete locally convex space or, in particular, algebra and X a complex (analytic)manifold which is assumed to be second countable so that the space (resp., algebra) O ( X ) of complexvalued holomorphic functions on X carries its natural Frdchet l o c a l l y convex space (resp., l o c a l l y m-convex algebra) structure inherited from c c ( X ) (see Chapt. IV; Subsection 4 . ( 3 ) ) . Thus, we denote by O ( X , El the space of iE-valued holomorphic functions on X , in the sense of the following condition(-definition): f e O ( X , E ) i f , and only i f (by definition), a'o f e (4.1) O ( X ) , f o r every a'eB'(topologica1 dual of B ) .

4.

401

THE ALGEBRA o ( X , IE)

The previous cond. (4.1), like (2.1), is in fact a theorem (cf. [I: p. 37, Theorsme 1 1 ) ; indeed, the notion of an E-valued holomorphic map on X can be defined directly via a suitable extension to such functions of the classical "Cauchy's integral formula" (ibid.).

A . GROTEETVDIECK

Furthermore, one considers on O ( X , I E )

the relative topology in-

duced on it by the canonical inclusion

otx,

(4.2)

El E

q x ,

EI

(see (1.1)) ; hence considering further the relation (see also (1.5)) O ( X ) ~ Ec e p @ Es

(4.3)

ep,E )

U

one concludes, from (1.4) and (1.17), that (4.4) Therefore (by still applying an obvious abuse of notation for i i ) , and since OlXlSE

(4.51

s

O i X , El

G

CJX,

Ej ,

U

one obtains (see also the proof of Lemma 1 .1 ) that t h e r e l a t i v e topology ( X , E ) and t h e b i p r o j e c t i v e t e n s o r i a l topology E on i t induced on O ( X ) @ E by (cf. Definition X; 2.3) are t h e same. Moreover, since O ( X l (with its ca-

c

i s a nuclear space (cf. A . nonical Frechet space topology from cc(X)) GROTHENDWCK 13: Chap. 11; p. 56, Corollaire] ), one finally gets O(X)bIE

(4.6)

5 O ( X , IE) s Cctx,S ) ,

within an isomorphism of the locally convex spaces involved; yet one has T = E = T I = ~ , whenever X is second countable and E a Frechet locally convex space. On the other hand, if E is a complete l o c a l l y convex space, then the space O ( X , E ) C C (X, 6)i s a l s o complete ("generalized Monte1 Theorem; see A. GROTHENDIECK [I: p. 441). Thus, from (4.5) and the preceding remarks (see also X; (2.42)) one now obtains

orxiib E

(4.7)

= O(X)@B

= (by (1.2))

c

E

Cc(Xl&E E

Furthermore, one has (4.8)

O(X, E)

s

S e l E;

, OlX))

=

le((a(x));,

E ) = O(Xl$E

I

within (locally convex space) isomorphisms as indicated; the first relation in (4.8) is given by (4.9)

[f(a'i](x)= a'(f(x))

402

X EXAMPLES

with x B X, and for any f E O ( X , El and a’eE‘. Indeed,the assertion follows immediately from (4.1) and the fact that the relation a ’ ( f ( x ) ) = 0 for any x:eX and a ’ E E ’ , implies f = O . Therefore, from (4.7) and (4.8) , one finaZZy g e t s (4.10)

O ( X , El = O ( X ) C 3 S = O ( X ) $ E ,

w i t h i n an isomorphism of Zocally convex spaces.

In particular, in case of a topological algebra E , using an analogous technique to that of Sections 1, 2,we conclude now with the following.

Lemma 4.1. Suppose t h a t t h e loca2Zy convex space E i n ( 4 . 1 0 ) is, i n part i c u l a r , a Zocally convex algebra. Then, t h e isomorphism e s t a b l i s h e d b y ( 4 . 1 0 ) preserves t h e r e s p e c t i v e topologicaZ aZgebra s t r u c t u r e s , whiZe t h i s i s aZso t r u e i n case IE has a continuous m u l t i p l i c a t i o n . I n p a r t i c u l a r , ( 4 . 1 0 ) y i e l d s an isomorphism o f Locally m-convex algebras, i f E is a complete ZocaZZy m-eonvex algebra, such t h a t t h e topological algebras i n -

volved a r e , i n p a r t i c u l a r , Fre’chet ones, i n case X i s second countable and B a Fre‘chet ZocaZZy convex ( r e s p . , ZocaZly m-convex) algebra. I

NOW, concerning the last proposition in the preceding Lemma 4.1, since O ( X l is a nuclear algebra (see also ( 4 . 6 ) ) one may consider, of course, in the second member of (4.10) any one of t h e compatible t e n s o r i a l topoZogies which have been examined already in Section X; 2.

5. The algebra L1 (G, IE) (generalized group algebra) Concerning the algebra in title of this section, we have considered already in Chapt. VII; Section 4 the classical “group algebra“ 1 1 L (GI and its spectrum m(L(G))=

2

(ibid.; (4.21)), where G was an abel-

ian ZocaZZy compact

(topological) group. Thus our concern in this section will be to examine the previous algebra in case the functions involved take values from a given topological algebra E, the algebra thus resulted being then considered as a suitable topological tensor product algebra. Thus, suppose that E is a given (commutative complete) ZocaZly m-conv e x algebra and let K I G , E ) be the algebra (with point-wise defined op-

erations) of E-valued continuous functions on G with compact support (cf. also Chapt. IV; Subsection 4.(1)). Now if T = ( p ) is a fundamental family of (submultiplicative) semi-norms defining the topology of E (cf. Theorem 1;3.1), the relation

5.

GENERALIZED G R O W ALGEBRA

403

N (fl = ~ p I f I x l l d ; c , F with f E K(G, E l and p e T I defines a semi-norm on (the vector space) K(G, B ) ; here dx denotes the Haar measure on G. Thus, one has (5.1)

p o f E KfG) ,

(5.2)

for every p e r , with f e K (G, El , for concerning the respective supports of the functions involved, one has of course the relation Suppfpof)

c Supp(fl. 1 E

Now, let A (G) be the space of all E-valued fiinctions on G for which (5.1) yields a (finite non-negative) real number; thus one sets, by definition, 1

LE(G)

(5.3)

1

= KIG, El G AB(G),

and calls it the space of 3 - v a l u e d (absolutely)swmnable functions on G, while one considers on A 1 ( G I the locally convex topology defined on B it by the semi-norms ( 5 . 1 ) , with p E. ' l In this respect, it is well-known that for a locally convex space E the space (5.3) is n e i t h e r Hausdorff nor complete, even with B complete (cf. A. GROTHENDIECK [3: Chap. I; p. 591 ; it is complete if B is a Banach or even a Frgchet (locally convex) space (ibid.)). Thus, we denote by (5.4)

L'(G,

zi

the Hausdorff completion of the locally convex space SA(Gl given by (5.3). (See,for instance, N. BOURBAKI [4: Chap. 11; p. 21, ThEorSme 3, and p. 24 ff.] ) .

A particular difficulty at this point is the fact that an arbi1 trary function f E L (G, El may not be measurable; however, this is not the case if E is a metrizable, hence a f o r t i o r i a Frgchet space (cf. L . SCHWARTZ "3:Chap. 11; p. 4 7 1 and/or A. GROTHENDIECK [3:Chap. I; p. 64, ftn. 1 6 1 ) . On the other hand, one needs the measurability of the 1 elements of L ( C , E ) when applying e.g. Fubini's Theorem, a crucial

fact in defining the "convolution multiplication" in L1(G/ (see Chapt.

IV; (4.2)). To bypass this phenomenon in case of an arbitrary locally m-convex algebra E l we first consider the tensor product algebra K(G)8B on which one defines the afore-mentioned multiplication, and then extend it to the completion (under TI) of the latter algebra; this is in fact the (locally m-convex) algebra (5.4). Thus, we first have the following. Lemma 5.1.

L e t G be a g i v e n locally compact abelian ( a d d i t i v e ) group and E

404

XI EXAMPLES

a commutative complete l o c a l l y m-convex algebra, w i t h r = ( p ) being a fundamental f a m i l y of s u b m L t i p 1 i c a t i v e semi-norms defining i t s topology. Furthermore, l e t UG,E ) be the space of IE-valued continuous f u n c t i o n s on G w i t h compact support. Then, the r e l a t i o n

(5.5) wtth f,

(f

*g

)(xl =

If(.

x eG,

-y)g(y)dy, 1

i n K I G , E l , d e f i n e s an element of AE(Gl i n such a way t h a t one has Np(f* g) < N ( f ) - N (g),

(5.6) f o r every p e r , where N

P

P i s given by (5.2).

P

Proof. Since the "translations" define homeomorphisms of G I we

first remark that the map y c-tf(X-ylg(y),

(5.7)

y EG,

provides an element of K ( G , E l , for every Z E G , and for any f, g in K(G,E). Therefore, for every x E G , ( 5 . 5 ) yields a well-defined element of E (cf. N . BOURBAKI [8: Chap. 111; p. 80, Corollaire 2 1 ) . On the other hand, in view of (5.1) and (5.5), one still gets, for every p e r , N p ( f * g l = j p ( l f ( x - y l g ( y l d y l d x S (ibid.;p. 79, Proposition 6)

5 I f l p l f I x - y ) g ( y l l d y ) d x 5 (sukmlt/ity of p )

( f p ( f ( x- y ) I . p ( g ( y l l d y ) d x = (Fubini)

(by (5.1)) N ( f l - N ( g )

(fp(f(x))~).(lp(g(y))dy)

P

by hypothesis for the functions f , g ,

P

<+

co

and this completes the proof. a

Thus, (5.1) yields, when p ranges over T I a family of submultiplicative semi-norms on KIG, El making t h e l a t t e r space i n t o a ZocaZZy m-eonvex aZgebra. In particular, one gets the following useful

Lemma 5 . 2 . Suppose t h a t t h e conditions of t h e previous Lema

5 . 1 are s a t -

i s f i e d . Then, concerning the tensor product algebra K ( G ) @ E , one g e t s the r e l a t i o n

(5.8)

K(G)@B

S K(G, El,

within an isomorphism o f algebras, when one considers on K(G, El the convolutionm u l t i p l i c a t i o n defined by ( 5 . 5 1 . I n p a r t i c u l a r , K ( G ) @ E becomes a l o c a l l y m-convex algebra when endowed w i t h the r e l a t i v e topology fr o m K ( G , E l , w?tile the l a t t e r algebra i s topologized through t h e famiZy (.5.1), w i t h p e r . Proof. We first observe that the restriction of the corresponding map ii, given by (1.4), to the algebra K(Gl@lE C c ( G ) @ Ehas a range contained in K ( G , El C C(G, E ) ; hence, one gets an isomorphism of the

5.

405

GENERALIZED GROUP ALGEBRA

v e c t o r s p a c e s i n v o l v e d i n ( 5 . 8 ) , which w e s t i l l d e n o t e by c . F u r t h e r more, f o r any decomposable t e n s o r s f @ a , g e b i n K ( G ) B E , one g e t s

ii(fea)*ii(gab)= i i ( ( f * g l e a b ) , t h a t i s (see a l s o ( 1 . 4 ) ) , t h e r e l a t i o n (5.91

f a

*

gb = ( f k g i a b .

T h i s i s f u r t h e r e x t e n d e d by l i n e a r i t y t o t h e whole o f I m ( C ) E. K ( G , B I . Therefore, K ( G ) @ E

E

I m ( 6 ) i s a c t u a l l y a subalgebra of K ( G , B ) w i t h respect t o

t h e convolution(-multiplication) defined by ( 5 . 5 ) . N o w , t o prove (5.9) w e f i r s t o b s e r v e t h a t o n e g e t s from ( 5 . 5 ) (5.10) T h e r e f o r e , f o r e v e r y a‘€ E‘,

one o b t a i n s ( c f . N . BOURBAKI [8:Chap.

111; p .

741 1 a‘(JJ@(x -y)g(y)abdy =

1a ’ ( f ( x

- y ) g ( y ) a b )dy

= jf(x-y)g(yla’(ab)dy = a’(abl/f(x-ylg(y)dy = a ’ ( ( j f ( x -y)g(yldylab )

,

so t h a t one g e t s jf(x-y)g(ylabdy = Thus, i n view o f

( j f ( -~y i g ( y ) d y ) a b .

( 5 . 5 ) and ( 5 . 1 0 ) , one f i n a l l y o b t a i n s ( 5 . 9 ) : h e n c e ,

s i n c e f * g e K ( G ) (see N . B O U R B N [9: Chap. 7; p. 166, P r o p o s i t i o n l l ] ) , one g e t s ( 5 . 8 ) . O n t h e o t h e r hand, t h e l a s t a s s e r t i o n of t h e lemma i s now a d i r e c t c o n s e q u e n c e o f t h e f o r e g o i n g and (5.6), and t h i s t e r m i n a t e s t h e proof. I Thus, w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g .

Theorem 5.1.

Let G be a l o c a l l y compact abelian group

and E a c o m t a t i v e

1

complete l o c a l l y m-convex algebra. Then, L ( G , E ) (see (5.4)) i s an aZgebra of the same type w i t h ( r i n g ) m u l t i p l i c a t i o n defined by a (continuous) extension of t h e conv o l u t i o n given by ( 5 . 5 ) .

Proof. W e f i r s t remark t h a t , f o r e v e r y e l e m e n t f e K ( G , E ) , f o r any E > O a n d a c o n t i n u o u s semi-norm p onlE, t h e r e e x i s t s a “ c o n t i n u o u s p a r t i t i o n of u n i t y ” , s a y , (‘QiiljiSn , s u c h t h a t t h e s u p p o r t s o f t h e Q i J s are c o n t a i n e d i n Supp(f) a n d , moreover, one h a s n plf(x:) - I Q i ( X ) f ( X i l ) s E , (5.11) i=l

f o r every

X E

G , w i t h xi E

sup^(@^),

l l i l n ( s e e N . BOURBAKI [ 8 : Chap. 3 ; p. 44,

XI EXAMPLES

406

Lemme 2 1 ) . Thus, with an obvious abuse of notation concerning Im(61g K ( G 1 8 E (cf. also (5.8)) , one gets the relation K ( G ) 8 E = K(G, E l .

(5.12)

with respect to the uniform topology in G. Therefore, a f o r t i o r i , 1 the same is true for the relative topology of K ( G , El from A E ( G 1 (see ( 5 . 3 ) as well as N . BOURBAKI [ 8 : Chap. 4; p. 127, Proposition 4 1 ) . SO from ( 5 . 3 ) and ( 5 . 4 ) (an abuse of notation is still applied to the latter) one finally gets L+G, E ) = K ( G I B E .

(5.13) 1

Consequently, L ( G , E1 becomes a (commutative complete) l o c a l l y mthe completion of the locally m-convex algebra K ( G 1 8 . E (Lemma 5. I ) , and this completes the proof. I

convex algebra , being by ( 5 . 1 3 )

On the other hand, one gets with A . GROTHENDIECK [3: Chap. I; p. 59, ThBorSme 21 the relation (5.14)

L ~ GE , ) = L' ( G I

4E ,

within an isomorphism of locally convex spaces. NOW, this follows from ( 5 . 1 3 ) and the relation (5.15)

K ( G ) 8~ = L'(G)

$ IE

by also remarking that t h e topologies 71 and t h a t one defined by 15.31 coincide i n f a c t on K ( G ) @ E (ibid.). Therefore, under the conditions of the previous Theorem 5.1 , the preceding isomorphism ( 5 . 1 4 1 becomes, indeed, an isomorphism of t h e respective l o c a l l y m-convex algebras (see also Proposition X; 3 . 1 ) -

407

S p e c t r a of Topological T e n s o r P r o d u c t A1 g e b r a s

CHAPTER X I 1

1. Spectrum o f a tensor product o f topological algebras (numerical case) W e examine i n t h i s s e c t i o n a b a s i c r e l a t i o n c o n n e c t i n g t h e spec-

trum ( C h a p t e r V ) of a g i v e n t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a w i t h t h e s p e c t r a of t h e t o p o l o g i c a l a l g e b r a s ,

f a c t o r s of t h e t e n s o r prod-

u c t considered. Now, a l t h o u g h t h e c o m p a t i b l e t e n s o r i a l t o p o l o g i e s a l r e a d y cons i d e r e d i n C h a p t . X ; S e c t i o n 2 a r e , by d e f i n i t i o n , l o c a l l y convex o n e s , t o be more c o n s i s t e n t w i t h t h e c o n t e n t of C h a p t e r V we s h a l l n o t a s sume a p r i o r i t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d t o be n e c e s s a r i l y l o c a l l y convex o n e s , u n l e s s t h i s i s e x p l i c i t l y s t a t e d . Thus, more gene r a l l y , concerning a given "compatible t e n s o r i a l topology", w e argue w i t h i n t h e c o n t e x t of Scholium X ; 4.1

and t h e c o r r e s p o n d i n g D e f i n i t i o n

X; 4.1.

I n t h i s r e s p e c t , w e f i r s t have t h e f o l l o w i n g b a s i c r e s u l t .

Lemma 1.1.

Let E, F be topological algebras and E 8 F t h e respective tensor

product algebra endowed w i t h a compatible ( t e n s o r i a l ) topology

T

( D e f i n i t i o n X;

4.1). Then, concerning t h e spectra of t h e topological algebras under considerat i o n , one has t h e r e l a t i o n

m(EBT F ) = m(E)x m(F) ,

(1.I)

w i t h i n a b i j e c t i o n of t h e r e s p e c t i v e (Gel'fand topological) spuces. Proof,

-

The b i j e c t i o n a s s e r t e d i n ( 1 . l ) i s d e f i n e d by t h e c o r r e -

spondence h

(7.2)

( f , gl

such t h a t h = f a g . Here one d e f i n e s (1.3) where a

f(x)= h(a@bb)

h ( a x a b ) , x e E , and g ( y ) =

I h ( a a bl

b i s any decomposable t e n s o r i n E 8 F , w i t h

h(a @ b y ) , y

E

F,

h ( a a b ) # 0. The l a s t

408

XI1 SPECTRA

condition on h E m ( E 8 F l is satisfied of course for some a @ b € E B F , T since by hypothesis h f 0 . Moreover, the same condition on h y i e l d s the relation h(axobl = hfxaobl,

(1.4)

f o r e v e q X E E . Indeed, this follows at once from 2

h ( a x o b l h ( a c b l = hlaxa o b l = h ( a e b l h ( x a e b l .

So after this remark, we further note that the rels. (1.3) are in the sense that they are, in fact, independent of the particular element a o b e E 8 F for which one has h ( a e b l # 0. Thus, suppose that, for some a7 e b E E ~ F we , also have h ( a l e b l ) # 0;then one has 7 from ( 1 . 4 ) well-defined,

h(axob)h(alebll = h(alebllh(xaab) = h(alxob71h(aebl,

so that one obtains (1.5)

h ( a o bl

h ( a x o bl =

h ( a p b,)

h(alxebl)

,

for every x B E , which proves the assertion for f , while a similar argument is valid for g too. On the other hand, the same rels.(1.3) provide, of course, linear forms on E , F, such that one has ( 1 -6)

h =feg.

Now one proves the last relation for the decomposable tensors in E 8 F and then extend it by linearity. So, for every x o y e E 8 F , one has (1.7)

h l x m y ) = f ( x l g ( y ),

where the last relation is a direct consequence of ( 1 . 3 )

and the next

h ( a ; c e b ) h ( a e b y l = (by (1.4)) h ( a o b y l h ( x a @ b b ) = h(axaobybl = h ( a o b l h ( x o y l h ( a o b l .

Thus, one concludes, in particular, that (1.8)

h(aapb) = f ( a l g ( b l # 0

,

hence both of f , g are non-zero. Moreover, they are indeed m u l t i p l i c a t i v e (linear forms). This is proved as follows: h ( a x e b l h ( a x l e bl = (by (1.4)) h ( a x o b b l h ( x l a o b l = h(aml e b ) h ( a o b l ,

for any x , x 7 in E . (An analogous argument holds true for g , in connection with the respective relation to ( 1 . 4 ) ) . Furthermore, f , g are continuous: this follows from ( 1 . 3 ) in view

409

NUMERICAL SPECTRUM

1.

of the separate continuity of the map $I : E x F - + E 8 F (cf. Definition T X ; 4.1 ) and the continuity of h e m ( E 8 F l . Thus, we have proved so far T that ( f , g ) e m ( E ) x m(F!. We finally prove that t h e correspondence h

(1.9)

H(f, g

) :m ( E 8 F ) -m(E) T

x m(F),

given by ( 1 . 3 ) , and in such a way that ( 1 . 5 ) to hold true, i s i n f a c t a b i j e c t i o n : Thus, (1.91 i s 1 - 1 ; that is, the relation (f, g ) = (fl,g l ) in m ’ ( E ) x m ( F ) implies obviously by ( 1 . 5 ) that h = f e g = f l o gl = kl , i.e., the assertion. Furthermore, ( 1 . 9 ) i s an onto map. Namely, for any (f, g ) in m ( E / x m ( F ) E E i X ‘F, , the relation h = f @ g defines an element of ( E 8 F ) ’ T (cf. Definition X ; 4 . 1 , 3 ) ) which obviously is non-zero and multiplicative too, as follows (for decomposable tensors in E 8 F ) from the relation h f xoy)h(xl m y l ) = f(x)g(y)f(xllg(yl I = f(x)f(xl)g(y)gIylI

( 1 .lo) = f f x x l ) g ( y y1 I = h ( xxl o y y , ) = k f ( x o y ) ( x l o y l ) )

-

(It is worth noting here that the commutativity of the algebra C is a crucial point in the previous argument; see however Lemma 3 . 1 below). On the other hand, the element f o g = h e m ( E @ F I thus defined, yields T via ( 1 . 3 ) the initial pair ( f , g ) , as follows straightforwardly, which proves the assertion, and the proof of the lemma is complete. I Indeed, one gets the following topological supplement to the previous result. That is, we have.

Theorem 1.1. Under t h e conditions of t h e previous Lemma 1.1, we a c t u a l l y have the r e l a t i o n (1.11)

m(E?F/

= m(E) X m ( F ) ,

w i t h i n a homeomorphism of t h e r e s p e c t i v e Gel’fand topological spaces which i s given by t h e m p ( 1 . 9 ) .

Proof.

It is easily seen that the inverse map of ( 1 . 9 ) ,

i.e., the

correspondence

(1.12)

( f , 9)-

fog

s

k : ??Z(E/ x m ( F ) - m ( E @ F )

T

i s continuous: Thus, for every net (f, g)i

in m ( E ) X m ( F ) converging to an and gi-g in m(E) and m ( F ) I reelement (f, g ) , one gets that fi-f z spectively, so that €or every x o y B E @ F one obtains (1.13)

= f (x)g(y) h(X@ y )

I

which proves the assertion, by an obvious extension of the last rela-

410

XI1 SPECTRA

tion to an arbitrary element

z

n 1 x . a y . e EOF).

j=1 3

3

On the other hand, t h e map given by the previous r e l a t i o n ( 1 . 9 ) i s continuous. Equivalently, this is true for the partial maps h - f and h u g . We prove it, for instance, for the map h - f : m(EfFI-+?X(E) given by the first relation in (1.3). Thus, suppose we have h.?h

(1.14)

in m(E@FI, T

2 2

so that, since h ( a a b 1 f 0 , for some a @ b s E O F , one gets from (1.14) hi(acob)

(1.15)

+ 0,

for every index i > i o Therefore, . for every xE E and for every neighborhood U of f ( x : l = h l a x a b b ) , there exists an index i such that h ( a apt) U

’

(1.16)

for every index Z > i,, iu : i.e., f.(x)* fix), for every 3: e E . So f.- f z z 2 2 in m(E) which is the assertion, and this completes the proof of the theorem. I NOW, complete topological tensor product algebras of the form n E O F are encountered in abundance in the applications (see, for inT stance, the previous Chapter XI), so that our next objective will be to get the preceding formula (1.11) for this type of topological algebras too. However, we shall need first the following useful result. Lemma 1.2.

Let E , F be topological algebras and E8F t h e respective tensor T

product algebra of E , F endowed with a compatible t e n s o r i a l topology T (Definition X; 4.1: 1 ) -3a) ) . Then, concerning t h e spectra of the topological algebras under consideration, the following two a s s e r t i o n s are equivalent: 1 ) The spectra ? n ( E ) , M ( F ) are l o c a l l y equicontinuous

2 1 The s p e c t m m(E8F) T

(Definition Vi1.2).

i s l o c a l l y equicontinuous.

Proof. If 1 ) is valid and h is any element of the spectrum of E 8 F , then (Lemma 1 .I ) h = f a g such that (f,g) e m(El X m(F/. Thus , there T exist by hypothesis equicontinuous neighborhoods, say, U of f in ???is) and V of g in m(F1 such that the set U x V ,being a neighborhood of (f, g ) in ??Z(El x m(F1 , has an image under the inverse map of (1.9) , that

is the set U 8 V , which is a neighborhood of h = f a g in m ( E 8TF I (Theorem 1.1). Furthermore, U B Y is an equicontinuous subset of m(EOF! (cf. T Definition X; 4.1, 3.3) ) Thus, the spectrum of E 8 F is locally equiconT tinuous, that is 1)* 2) Conversely, suppose that m(E$F) is locally equicontinuous, and

.

.

1.

41 1

NUMERICAL SPECTRUM

let ( f , g) E m(E)x m ( F ) , hence (Lemma 1 . 1 ) h = f o g e 7 n f E 8 F ) . Therefore, T there exists by hypothesis an equicontinuous neighborhood W of h in m ( E : F l , such that (Theorem 1 . 1 ) there are neighborhoods U of f in m ( E l and V of g in ??.?(FI, with (1.17)

h

E

fogeU8Vc

w,

where U @ V is thus an equicontinuous neighborhood of h r f og in V Z i E Q F I . T Hence, there exists a neighborhood N of 0 E E 8 F such that U 8 V C N o . So T (Definition X; 4.1, 2 ) ) there exists a neighborhood A of 0 E E such that, for every fixed, however arbitrary, y E F , one gets the relation

uov s

(1.18)

NO

c (AO~)~.

Thus, one obtains I ( f @ g ~ f x e y ! 1= If(rcll* IgCy,] s I ,

(1.19)

for any f @ g s U @ Vand x@yeA@y. Thus one finally gets, for any g E V , because of ( 1 . 1 9 ) , the relation If(xlI 5

X , with X > O ,

Hence, U is an equicontinufor any X E A and f E U , so that U E (-A)'.1 A ous neighborhood of f in m ( E ) . NOW, an analogous argument can be applied to g , for V, so that we have actually proved that 2 ) * 1 ) as well, and this terminates the proof. I

.-

Remark 1 .I The argument applied to prove Lemma 1 . 2 can be used as well in the particular case where the spectra of the given topological algebras are equicontinuous; so one concludes that: ? ? Z ( E ~ Fi )s equicontinuous if, and only i f , t h i s i s the case T f o r each one of t h e spectra m(E)and m(Fl. We are now in the position to state the following basic fact.

Theorem 1.2.

Let E , F be topoZogical algebras w i t h locaZly equicontinuous

spectra and l e t E @ F be t h e associated tensor product algebra equipped w i t h a comp a t i b l e topology

such t h a t E i i F i s a topoZogical algebra w i t h M ( E @ F ) = m ( E b F ) T

T

v a l i d w i t h i n a b i j e c t i o n (cf. Chapt. V; ( 2 . 5 ) ) . Then, one g e t s the r e l a t i o n (1.20)

PZ(E$FI

= ZT(E) x ~ ( F I ,

w i t h i n a homeomolrphism of t h e Gel'fand topological spaces under consideration.

Proof. By hypothesis and the previous Lemma 1 . 2 , m ( E 8 F ) is loT cally equicontinuous, so that one obtains (see Theorem V i 2 . 1 ) (1.21)

~ ( E G F =) X W E T F ) ,

412

XI1

SPECTRA

within a homeomorphism of the respective Gel'fand spaces. So the assertion is now a direct consequence of (1.21) and the previous Theorem 1.1, and this terminates the proof. I

As a clarification of the preceding argument, we may also state the following.

Corollary 1.1. Let E , F be locally m-eonvex algebras whose completions

8, 2,

respectively, are &-algebras. Then, f o r every compatible topology T on the tensor product algebra E 8 F (Definition X ; 3.1), one gets

(1.22)

m I E 6? F ) =

EWE)

x

PUF),

within a homeomorphism of the topological spaces involved. Proof. By hypothesis and Proposition 11; 7.1, m(8),??'L(.$) are equicontinuous subsets of the corresponding dual spaces, so that the same is valid for the spectra m(E)," Y ( F ) (cf. Chapt. V; Scholium 2.2). Hence, the assertion is now immediate from the previous Theorem 1.2. I

On the other hand, as a further illustration of the preceding, we have the next.

Corollary 1.2. Let E , F be ( c o m t a t i u e ) Banach algebras and

T

a (ZoeaLly

convex algebra) topology on the respective tensor product algebra E B F , such that

(1.23)

E
(cf. Section X; 2) mking E & F i n t o a Banach algebra. Then, one gets T

(1.24)

P ~ ( E $ F I= W E )

x

WF),

within a homeomorphism of the ("maximal ideal") spaces

involved.

Proof. From (1.23) and the relation I T
Lemma 1.3. Let E, F be commutative locally m-convex algebras and E T F the

1.

413

NUMERICAL SPECTRUM

corresponding (commutative) l o c a l l y m-convex t e z s o r product algebra of E , F w i t h respect t o a compatible topology T (Definition X ; 3.1). a s s e r t i o n s are equivalent 1 ) The completions

2,F^

Then, t h e following two

of E , F, r e s p e c t i v e l y , are Q-algebras.

21 The completed algebra E 6 F i s a Q-algebra. T

Proof. Suppose that 1 ) is true. Then, applying the argument of the proof of Corollary 1 . 1 , we conclude that the spectra m ( E ) G E’, m(F)GF’ are equicontinuous subsets, so that by hypothesis for T (cf. Definition X; 2.1, cond. (2.3)) the set (1.25)

m(E8F) = ??‘ZIE) 63 T

m(F)C

(EYF)’

is an equicontinuous subset of the respective topological dual space. Here we still remark that the first of the previous relations follows from the (canonical) bijection ( 1 . 9 ) (Lemma 1 . 1 ) . Thus, proposition 2) of the statement is again a consequence of Scholium V;2.2:Lemma. Conversely, from an application of the afore-mentioned lemma, one still concludes, assuming 2 ) , that ??Z(E@Fl is equicontinuous, so T that the same is true concerning the spectra m(El,??Z(F) (Remark 1 . 1 ) Now this implies, in turn, that E l ? , are Q-algebras, i.e., assertion l)(cf. the same Lemma, as above), and this completes the proof. I

.

Scholium 1.1.The hypothesis of being the spectra of the topological algebras involved inthe previous Theorem 1 .2 l o c a l l y equicontinuous , was actually due to our intention of having ( 1 . 2 1 ) truewithin a homeomorphism; this was supplied in turn by our hypothesis, in view of Theorem V; 2.1. However, the same hypothesis is not likely to be necessary in order ( 1 . 2 0 ) to hold true; this is pointed out, for instance, by the next proposition. On the other hand, the question of having minimal conditions for the topological algebras involved i n order (1.20) t o be t r u e seems to be, so far, not entirely clear.

We close this section with the following consequence of the preceding discussion, referred to also in the previous Scholium 1 . 1 . So we have. Lemma 1.4. Suppose t h a t t h e conditions of Lema XI; 1.2 are s a t i s f i e d . Then, concerning t h e spectra of t h e topological algebras involved i n XI;(1.201, one has

414

XI1

(1.26)

SPECTRA

rn(ep$p)) = x ~ Y =~

rnccc(~)),

( C ~ ( X xI I

within a homeomorphism of the r e s p e c t i v e topoZogicaZ spaces.

Proof. Indeed, the assertion is straightforward from Corollary XI; 1 . I and Theorem VII; 1.2. I

2. Spectrum of an infinite topological tensor product algebra We consider in this section the spectrum of an infinite topological tensor product algebra (cf. Chapt. X;Section 5) in connection with the corresponding spectra of the individual algebras, "factors" of the tensor product considered (see Theorem 2.1 and its Corollary 2.1 below). Furthermore, we examine a particular application to Q-algebras (Corollary 2.2). However, we shall need first the following easy lemma. be a famiZy of topoZogica2 spaces and Zet F(II denote the s e t of f i n i t e subsets of I . Moreover, Zet X 6 ll X . w i t h a e F f I ) , and f o r a iea" any a E fi i n F ( I I , l e t fsa :X -+ x be the corresponding (canonicaZ) projection R a map. Then, t h e famiZy

Lemma 2.1. Let

( x a , f,,),

(2.1)

a E F(I),

c o n s t i t u t e s a p r o j e c t i v e system of topotogica2 spaces in such a m y that one has

n xi

(2.2)

ieI

= lLm(.ll x i ) , u :€a

within a homeomorphism of t h e r e s p e c t i v e topological spaces. Proof. We first remark that, for any a,fi in F(II with a E fi

,

the

map (canonical projection) (2.3)

f R a :X B 7

xa : (xi)iefi-(xi)ie

a

is continuous, as easily follows from an application of a standard argument, concerning continuity of maps with values in product spaces (see e.g. N . B O U R B W [4: Chap. 1; p. 25, Proposition 1 1 ) . Thus the corresponding relations to 111; (2.2) and (2.3) are certainly satisfied, so that the family (2.1 ) defines a p r o j e c t i v e system of topological spaces (ibid.;p.28). S o if Y = l L m X a is the respective projective limit space and X = . n Xi the Cartesian product space of the given family of topo"€I logical spaces, one has a continuous map ua :X+Xa :(xi li c----+ (xilie, ; in fact, the corresponding (canonical) projection map: (2.4)

ua :x+xa

:

Hxa

(Xilie

c1

(Xi,aJa

I

for every a e F ( I ) ; (here we eventually write for distinction xi = x .

2, a

€

2.

SPECTRA OF INFINITE TENSOR PRODUCTS

415

Xi’ with i e a ) . So one has

faa o u g --

(2.5) for any a ED in F(I).

-

I

Hence, there exists a uniquely defined continuous map

x

u : 4

(2.6)

Ua

Y : (2.)

z is1

such that

(“a)aeFIr)

fao u = u

(2.7)

a‘ for every a ~ F f I l .(Here fa stands for the respective “canonical map“ defined by the analogous relation to 111; (2.6); cf. also N. BOURBAKI [l: Chap. 3; p. 77, Proposition I]). N O W , it is clear from (2.6) that the map u i s 1 - 1 (see also ibid.). On the other hand, the correspondence (xa)a E FlI)

(2.8) with

3:

a

= ix.z).z e u ’ a e F(I),

any a S D in F(I),

-

(xi)i

I

is a well-defined map of Y into X. Indeed, for

one gets from (2.5)

for every i e a G D (see also (2.4)). Thus, for every i E I, there exists a uniquely defined a E F(I), with x.= xi,a , which proves our assertion for the map (2.8). Therefore, (2.6) d e f i n e s in f a c t a b i j e c t i o n of X onto Y (having as inverse the map (2.8)). We noticed already that the map (2.5) is continuous: similarly,via an analogous argument about continuity of maps into product spaces, one proves the continuity of (2.8) too, and this completes the proof of the lemma. I We are now in a position to prove the following. Theorem 2.1. Let lEili

be a family of l o c a l l y convex algebras with i d e n t -

i t y elements, and l e t E = - 8 E . be the corresponding i n f i n i t e tensor product local,LEI

Zy convex algebra (cf. Definition X ; 5.2). Then, concerning the spectra of t h e topoZogical algebras involved, one has

within a homeomorphism of the r e s p e c t i v e topological spaces. Proof.

In view of (X; (5.13)) and Theorem V; 3.1, one has

NOW, by definition of the locally convex algebra E a one has (2.12)

m(Ea I = m( e E i ) = ~ ( 8 71I E i : i € a } j = m ( Ts (I E2 . : l < i <1n) . iea

416

XI1

SPECTRA

As f o l l o w s from t h e p r e c e d i n g , t h e l a s t r e l a t i o n i s v a l i d w i t h i n a

homeomorphism, which c o r r e s p o n d s t o an "enumeration" of t h e f i n i t e s e t a G I . Thus, v i a an easy extension of Theorem I . 1 ( t o f i n i t e many f a c t o r s ) i n case

'I =TI

, one

obtains

t h e l a s t of t h e above r e l a t i o n s i s v a l i d i n view of t h e " a s s o c i a t i v i t y of C a r t e s i a n p r o d u c t s p a c e s " . Thus, ( 2 . 1 1 ) now y i e l d s

w i t h i n homeomorphisms of t h e c o r r e s p o n d i n g t o p o l o g i c a l s p a c e s , and t h i s completes t h e proof. I Now, i f w e c o n s i d e r t o p o l o g i c a l a l g e b r a s w i t h l o c a l l y equicont i n u o u s s p e c t r a , w e o b t a i n of c o u r s e an a n a l o g o u s r e l a t i o n t o ( 2 . 1 0 )

X; (5.15). However, w e g i v e

c o n c e r n i n g t h e a l g e b r a s X ; (5.14) a n d / o r

n e x t a s l i g h t l y d i f f e r e n t t y p e of a r e s u l t , which h a s an independent i n t e r e s t , assuming t h a t t h e i n d i v i d u a l t o p o l o g i c a l a l g e b r a s have, i n p a r t i c u l a r , c o n t i n u o u s G e l ' f a n d maps ( c f . C h a p t . V I ; S e c t i o n 1 ) . Thus,

w e have.

Theorem 2 . 2 . Let (EilieI

be a given family of l o c a l l y convex algebras a l l

having i d e n t i t y elements and continuous Gel'fand

maps, and l e t E = - 8 Ei be the

zeI

corresponding i n f i n i t e tensor product l o c a l l y convex algebra. Then, t h e following ~ I J Oassertions

are equivalent:

I ) The s p e e t m of E i s l o c a l l y equicontinuous. 2) The algebras Ei,

i € I , have l o e a l l y equicontinuous spectra such t h a t , i n

p a r t i c u l a r , a l l except of f i n i t e many of them have equicontinuous spectra.

I n p a r t i c u l a r , t h e algebra

E = - 8 Ei

has an equicontinuous spectrum i f ,

Z € I

and only i f , t h i s i s the case for each one of t h e individual algebras E i , i €I.

Proof. I f 1 ) i s v a l i d , t h e n from ( 2 . 1 0 )

and Theorem V : l . l ,

one

concludes t h a t

i s a l o c a l l y compact space ( i n t h e r e s p e c t i v e G e l ' f a n d t o p o l o g y ) . So each one of the p a r t i c u l a r spaces m(E.), i E I , i s l o c a l l y compact , and hence Z

i n p a r t i c u l a r a l l t h e spaces i n question, except of f i n i t e many of them, are

compact ( " l o c a l compactness of a C a r t e s i a n p r o d u c t s p a c e " ; see e . g . N. BOURBAKI [4: Chap. 1 ; p. 66, P r o p o s i t i o n 141 )

. Thus, t h e

a s s e r t i o n 1)

f o l l o w s now from t h e h y p o t h e s i s f o r t h e a l g e b r a s E i , i € I , VI;1.3,

and Theorem V 1 ; l . l .

3

2)

Corollary

2.

SPECTRA OF I N F I N I T E TENSOR PRODUCTS

417

NOW, if 2) is in force one concludes that each one of t h e locally convex algebras Ea=

.C3'Ei

,

Z€Cl

with a e F ( I ) , has a l o c a l l y equicontinuous s p e c t m

(see also (2.13)). Indeed, the assertion is an easy extension (to finite many factors) of Lemma 1.2, with ther there fore, concerning the inductive system fEa, f, ) which defines the algebra E (Definition X; La 5.2), one has from the hypothesis (cf. also Remark 1.1) that the conditions of Proposition V ; 3.1 are fulfilled, hence m(E) is locally equicontinuous. So 2) implies 1) as well, and the first part of the statement is thus established. Now, the rest of our assertion is easily concluded, applying an analogous argument to the previous one, on the basis of the corresponding part of the same Proposition V;3.1 in connection with Remark 1.1, and this completes the proof of the theorem. I Applying Lemma V;2.2, in conjunction with Theorem V;2.1, we now have,as an immediate consequence of the foregoing,the next.

Corollary 2.1. Suppose t h a t the conditions of t h e preceding Theorem 2.2 are s a t i s f i e d and l e t t h e l o c a l l y convex algebras Ei, i e I , have, moreover, continuous muZtiplication. Finally assume that e i t h e r one of t h e two ( e q u i v a l e n t ) a s s e r t i o n s o f t h e same theorem hoZds t r u e . Then, one g e t s (within homeomorphisms of the respec-

t i v e topological spaces)

(2.14)

Scholiurn 2.1.- The previous Theorem 2.2 has been stated, in fact, with the above Corollary 2.1 in mind. So the hypothesis "Gel'fand maps continuous" for the algebras Eil i E I , has been applied only to prove that l)+ 2). Thus, 1 2 . 1 4 ) is s t i l l i n f o r c e under t h e weaker hypothesis t h a t o n l y cond. 21 o f Theorem 2.2 i s s a t i s f i e d , without asszuning t h e c o n t i n u i t y of t h e

r e s p e c t i v e Gel'fand maps of t h e algebras Ei, i s I . Nevertheless, we do assume

"continuous multiplication" in the algebras under consideration (see also Chapt. I; Section 4 ) . Furthermore, one has another (stronger) version of the two equivalent assertions of the same Theorem 2.2, without namely invoking continuity of the Gel'fand maps considered therein. Thus, we do have the following result:

418

XI1 SPECTRA

The spectrum of t h e algebra E = 8 E . i s a l o c a l l y iaI compact ( r e s p . , compact) space i f , and only i f , t h i s i s the case for a l l t h e spectra of t h e individual algebras

(2.15)

E i E I , i n such a manner t h a t a l l except of f i n i t e i' many of them ( r e s p . , without exception) are compact spaces.

(See the proof of the first part of the previous Theorem 2.2). In this respect, we further remark that: The c o n t i n u i t y of t h e Gel'fand maps of the algebras Ei,

i €1,e n t a i l s already the same property for the Gel'(2.16)

This is a direct consequence of the definition of the infinite tensor product locally convex algebra (see Chapt. X; Section 5), in view of Theorem 2.1 of the next chapter and Lemma VI;1.4.(See also Scholium XIII;2.1 below in case that complete algebras are considered). We close this section with a further application of the preceding discussion referred to (locally m-convex) &-algebras. Thus, one has the following.

Corollary 2.2. Let (EilieI

be a family of c o m t a t i v e l o c a l l y m-convex a l -

gebras w i t h i d e n t i t y elements and l e t

E = . @ Ei

be t h e corresponding i n f i n i t e ten-

Z E I

sor product l o c a l l y m-convex algebra (cf.Definition X ; 5.2 and the comment following it). Moreover, consider t h e following two a s s e r t i o n s 1 ) The completion 2i of each one o f t h e algebras Ei, i E I , i s a &-algebra. 2 ) The completion 2 of the algebra E , a s above, i s a &-algebra. m e n , 1 ) implies 2 ) . I n p a r t i c u l a r , t h e preceding two a s s e r t i o n s are equiv a l e n t , i n case we f u r t h e r assume t h a t each one of t h e individual algebras Ei, i e I , has the respective Gel'fand map continuous (hence, the same is true for the algebra E; cf. (2.16)).

Proof. If 1 ) is valid, then

m(gi) is

an equicontinuous subset of

for every i a I (Proposition 11; 7.1), so that the same is true for the spectra m ( E i ) , i e I (cf. Scholium V;2.2:Lemma). Thus, each one of the algebras Eu = @ Ei , with u e F I I ) , has an equicontinuous spectrum ieu too, ??L(Ea) (cf. Remark 1.1) . Hence, m ' ( E ) = m(lAmEu) = 1 - m M ' ( E u ) is a l s o equicontinuous (Proposition V;3.1), so that from the same Lemma of Scholium V;2.2, one concludes that is a &-algebra, so 1) 3 2). On the other hand, if 2 is a &-algebra, then m(E) is an equi-

3.

GENERALIZED SPECTRA

419

continuous subset of E'(Lemma of Scholium V;2.2), which is thus (weakly) compact too, since the algebra E has from its definition an identity element (cf. Lemma VI;l.3). Therefore (Theorem 2.I),all the spectra nZ(Ei) of the algebras E i r i & I , are compact spaces too (the"loca1ly rn-convex case" considered here is treated analogously to that in Theorem 2.1 ) . Thus, under t h e f u r t h e r assumption t h a t the algebras Ei, i E I, have the r e s p e c t i v e Gel'fand maps continuous, one gets (Theorem VI; 1.1) that the spectra m(EiI,ieI,of these algebras are also equicontinuous. So (in view of the same Lemma, as above) the respective completions Ei, i e l , of all the given algebras are Q-algebras; thus, in the particular case considered, 2) implies 1) as well, and this completes the proof. I h

The last corollary should be compared with Lemma 1.3 whose thus might be considered as an "infinite tensor product" analogon.

3. Generalized spectrum o f a tensor product o f topological algebras. Canonical decomposition Generalized spectra of topological algebras (i-e., spaces of continuous algebra morphisms) have been considered already in the appendix to Chapter V (ibid.; Section 8).We apply here this notion in the particular case of topological tensor product algebras, indicating also some relevant applications. So by considering a topology T on the tensor product algebra E 8 F associated with two given topological algebras E , F, it is reasonable to consider as a first condition of "compatibility" of T the following:

(3.1)

The p a i r ( E 8 F , T I E 8 F i s a topoZog7~caZalgebra, beT longing to the same category of topological algebras as E and F (concerning, in particular, the underlying topological vector space structure and the continuityof the respective (ring) multiplication).

(See also Definition Xi4.1). Now by an obvious extension of the analogous terminology applied in the preceding, we simply refer to a topology T satisfying (3.1) as a compatible topology on E 8 F , where E , F are given topological algebras. This will be further specialized, imposing eventually supplementary conditions on the topology T , which will depend, of course, on the particular applications considered.

420

XI1

SPECTRA

So the second natural condition that one imposes on a compatible topology ‘I, as above, is that: The canonical b i l i n e a r map

t o be separately continuous.

Furthermore, we also frequently require of lowing condition:

T

to satisfy the fol-

F o r every topological algebra G , and for any pair

(f,g ) e f f o m ( E , G / (3.3)

x

f f a m f F , G)

(cf. Definition V; 8 . 1 for the notation used), one has

f @ g S S ( E @ F , G). T

On the other hand, the following stronger version of the previous cond. (3.3) will be needed in the sequel. This happens, in particular, when one has to consider completed topological tensor product algebras. So we require that: F o r any equicontinuous subsets A C f f a m l E , G) and B C

(3.4)

ffomfF, G),

t h e s e t A 8 B t o be an equicontinuous subset

of S ( E 8 F , G I . T

So arguing within the category of locally convex (topological) algebras, t h e p r o j e c t i v e t e n s o r i a l topology TI i s a compatible topology on a tens o r product algebra E 8 F s a t i s f y i n g cond. ( 3 . 2 ) and ( 3 . 4 ) (and hence, a f ort i o r i , cond. (3.3). Cf. also Chapt. X;Lemmas 2.1 and 3 . 1 , in conjunction with Lemma 4.1). The remark is still in force within the category of locally convex algebras with a continuous multiplication, or yet inthat of locally rn-convex ones (ibid.; see also Proposition X;3.1). On the other hand, our previous discussion in Chapt. X;Section 4 (see also the relevant Scholium X;4.1) shows that the preceding context is still possible within also the category of non-necessarily locally convex topological algebras. This will be constantly our concern in the sequel, unless it is specified otherwise. So we first have the following basic result which specializes to Lemma 1.1 in case of topological algebras with identity elements and G = C . Thus, one gets. Lemma 3.1. Let E , F , G, be topological algebras w i t h i d e n t i t y elements and

421

3. GENERALIZED SPECTRA

l e t G have continuous m u l t i p l i c a t i o n . Moreover, l e t

T

be a compatible topology on

t h e tensor product algebra E 8 F s a t i s f y i n g ( 3 . 2 ) and ( 3 . 3 ) . Then, there e x i s t s a homeomorphism

i :H a m s f E B F , G)-+

(3.5)

f f u m s ( E , GI

x

Hams(F, G /

(cf. Definition V;8.1 for the notation applied) whose range are those p a i r s ff, g ) f o r which one has f ( x ) g f y )= g ( y l f ( x ) ,

(3.6)

f o r every

(2, y

) E E *F.

In p a r t i c u l a r , i f t h e algebra G i s conmtutative, then the preceding ( i n t o ) homeomorphism becomes an onto m a p .

For simplicity we considered in ( 3 . 5 ) the respective generalized spectra (relative to G) of the topological algebras involved; nevertheless, the corresponding trivial case (of zero homomorphisms) could be included as well (see also the next proof). Proof of L e w 3.1. If h

E

H o m f E B F , G)

then the relations

T

f ( x ) = h ( x @ l ) , with x € E ,

(3.7)

S l y ) = h ( 1 Q Y ) , with y e F ,

define f , g as continuous (algebra) morphisms of E, F i n t o G , r e s p e c t i v e l y . (By an obvious abuse of notation, we denoted here by 1 the identity element in both the given algebras E and F ) . Now the continuity of the maps in (3.7) follows from (3.2) and the continuity of h . Moreover, concerning f l for example, one gets from the first of (3.7) ( 3.8)

f I s ) .f ( x I = h ( x @1 l h f x Q 1 ) = h ( f x @1 1 . 1 1

(2, Q

1 I) = h f x x l Q 1 ) = f ( x x l ) ,

for any elements x , x I inE; an analogous argument is true for g , and this proves the assertion. On the other hand, for every decomposable tensor x ~ € yE B F one obtains (3.9) h(xmyJ = h f ( x Q 1 / . ( l Q y ) l = h l x e l 1 . h ( la P y ) = f ( x ) g f y l =( f Q g i ( x Q y ) , so that by linearity (of the map h ) one has

(3.10) n

for every element

z

E

2.0 7,

Ply,by (3.11)

y , E E B F . We denote the last relation, sim7,

h = fmg.

Consequently, f o r every h E H o m ( E @ E , G) one g e t s ( 3 . 1 1 ) , where the p a i r ff, g ) E Ham(E, G) x Horn(F, G) i s given by ( 3 . 7 ) . Moreover I on the basis of (3.10) and

422

(3.11)

XI1

SPECTRA

, one proves that ( 3 . 5 ) i s a one-to-one correspondence. Furthermore, one g e t s from (3.7) f ( x ) g ( y ) = h ( x o l ) * h ( lm y ) = h ( f x o I ) ( l e y l l = hlxoyyl

(3.12) = h ( ( 1 a y Y ) ( J : o I ) )= h l l o y l * h ( x o I ) = g ( y ) f ( x ) ,

for every (x, y l

€

E x F, so (3.6) is proved. Now, this is also a s u f f i c i e n t

condition i n order an element i n the range of ( 3 . 5 ) t o y i e l d , v i a ( 3 . 1 1 ) , an e l e ment i n tluni(E@F, G I . Indeed, for every pair If, gl in the range of ( 3 . 5 ) T which also satisfies (3.6), one concludes from (3.3) that the map h e

f o g given by

(3.10)

yields an element in S(E8F, GI. NOW, this is by

(3.6) a continuous (algebra) morphism of E 8 F into G; i.e. , one has T

h ( ( x o y l . ( 2 , @ y p = h ( x x , o y y l ) = f ( x x l ) g ( y y l )= f ( x ) f ( x 1 ) g ( y ) g ( y 1 l

(3.13) = (by (3.6)) f ( x ) g ( y ) f f X l ) g ( Y I I= h ( ~ o y ) * hm yf l ~) ~

for any decomposable tensors x m y , x l o y1 in E8F, which proves the assertion (cf. Lemma X;1.4) On the other hand, every element h = f o g

.

in the range of ( 3 . 5 ) , thus defined, yields in view of (3.7) the initial pair ( f , g ) , which proves that (3.5) is a map onto that subset of its range which is characterized by (3.6). Thus, we have proved that: A given p a i r ( f , g ) i n the range of ( 3 . 5 1 y i e l d s an element h f o g i n i t s domain of d e f i n i t i o n i f , and only if, ( 3 . 6 ) is s a t i s f i e d . Now the last condition is always satisfied if the algebra G is commutative; so t h i s a l s o proves t h e l a s t p a r t of the statement, once we have proved the bicontinuity of the map (3.5), to which we now turn. NOW, the continuity of (3.5) is reduced, in fact, to that of the respective partial maps, in analogy with the map ( 1 . 9 ) ; thus, the latter maps are obviously continuous, in view of (3.7). On the other hand, the continuity of the inverse map of (3.5), whose domain ofdefinition is characterized, in view of the preceding, by (3.61, is now concluded from an argument similar to that applied to the map (1.12); here we also resort to the continuity of the (ring) multiplication of the algebra G (see (1.13), and this terminates the proof of the lemma. I We come next to our main objective in this section, namely to extend (3.5) to the case of completed topological tensor product algebras. This generalizes, within the present framework, the analogous situation yielded by Theorem 1.2. But first we need an auxiliary result, analogous to the above Lemma 1.2. So we have. Lemma 3.2. Let E , F , G be topological algebras s a t i s f y i n g the conditions of

3.

t h e previous Lemu 3.1 and

T

423

GENERALIZED SPECTRA

a compatible topology on t h e respective tensor product

aLgebra E @ F s a t i s f y i a g ( 3 . 2 ) and ( 3 . 4 1 .

Furthermore, consider t h e following asser-

tions : 1) The Prneralized spectra Hams(E, G) and Homs(F, G) are both Locally equi-

continuous (cf. Definition V;8.2). 2 ) The generalized spectrum H v m s I E @ F , G) T

i s l o c a l l y equicontinuous.

Then, 1 ) implies 2 ) . On t h e other hand, f o r every pair ( f , g ) i n the image

of ( 3 . 5 ) ( i . e . , which s a t i s f i e s ( 3 . 6 ) ) , t h e r e e x i s t equicontinuous neighborhoods U of f i n Hums(E, G ) and V of g i n Honis(F, G) such t h a t U @ V is an equicontinuous neighborhood of h E f a g i n H o m s ( E 8 F , G). T

I n particuZar, i f t h e algebra G is c o m t a t i v e then the preceding two assert i o n s are equivalent. proof. 1 ) *2)

: On the basis of the previous Lemma 3 . 1

,

this is an

easy adaptation to the present case of the corresponding part of the proof of Lemma 1 . 2 , so we omit the details. On the other hand, consider an element (f,g ) E Homs(E, G) x Homs(F, G) satisfying (3.6), hence (Lemma 3 . 1 ) h = f a g e H o m s ( E @ F , G). Therefore, T assuming 2), there exists an equicontinuous neighborhood W of h in Homs(E8F, G) such that there is a neighborhood U x V of I f , g ) in the 'I range of ( 3 . 5 ) with ( U x V 1 n I r n ( i ) C i ( W ) , so that U @ V C W (cf. ( 3 . 1 1 ) ) . So for every neighborhood N of O e G , there exists a neighborhood T of O E E8F T

and hence by ( 3 . 2 ) a neighborhood A of O e E such that one has

(3.14)

Thus, by ( 3 . 9 ) and keeping fixed the element g e V

A

(3.15)

, as

above, one has

= n f - l ( ~, ) f

so that U is an equicontinuous neighborhood of f in Homs(E, G).

An analogous argument is certainly in force for V , hence U B V is an equicontinuous neighborhood of k = f a g in H a m s ( E 8 F , G) , which proves the second part of the assertion. Finally, if the algebra G is commutative, then in view of Lemma 3 . 1 , ( 3 . 5 ) is an onto map, so that the previous reasoning entails already that 2 ) implies l ) as well, and this finishes the proof of the

lemma. I Thus, we are now in the position to state the following basic result. Theorem 3.1.

Let E, F , G be locaLly convex algebras having i d e n t i t y eLements

and continuous multipLication (Definition I;l . l ) , while t h e aZgebra G i s com-

424

XI1 SPECTRA

p l e t e . Moreover, suppose t h a t t h e generalized spectra Hom(E, G) C S s ( E , GI

and

HomlF, GI C I: IF, GI are locaZly equicontinuous (Definition V; 8.2) and l e t T be

a compatible topology on t h e tensor product algebra E 8 F (cf. (3.1))s a t i s f y i n g ( 3 . 2 ) and ( 3 . 4 ) .

(3.16)

-

m e n , one g e t s a homeomorphism

5 : H o m s ( E T6 F ,

GI

Homs(E, G ) x Homs(F, G )

of t h e f i r s t space i n t o the second as indicated, which, i n p a r t i c u l a r , i s an onto map i n case G i s a c o m t a t i v e algebra.

Schol ium. - In order to avoid complicated statements, by considering weakened assumptions in the above Theorem 3.1, that might be suggested however by the preceding discussion (see e.g. Chapt.V; Section 8), we have restricted ourselves to the previous framework, considering thus l o c a l l y convex a l gebras w i t h continuous m u l t i p l i c a t i o n . Such algebras occur often in the applications (e.g. locally m-convex algebras). Thus, E 6 F is by hypothesis a locally convex algebra with 'continuous multiplication, being the completion of E Q F . Proof of Theorem 3.1. By hypothesis for the algebras E , F , G one has (3.17)

HOM

S

( E ~ F G, I = H O M ~ ~ E T GF I ,, T

u i t h i n a (canonical)b i j e c t i o n (see also the comment on V; (8.4)). On the

other hand, by hypothesis and the preceding Lemma 3.2, H u m s ( E 8 F , G) is T locally equicontinuous, hence (3.17) i s v a l i d within a homeomorphism of the respective topological spaces (cf. Theorem V;8.1). Thus, our assertion concerning (3.16) and the particular case where G is commutative, is now a direct consequence of the previous Lemma 3.1, and this completes the proof. I We are going to consider a further extension of the previous homeomorphism (3.16) to the case where the topological algebras involved have instead "approximate identities" (see Chapt. XIV; Section l), together with applications to "generalized group algebras" (Chapt. XV; Section 4 ; cf. also Chapt. XI; Section 5 ) .

4. Inductive limits and generalized spectra We deal in this section with generalized spectra of inductive limit topological algebras (cf. Chapt. IV; Section 2), where the terms of the respective inductive systems are topological tensor product algebras. In this respect, it is worth noticing that in spite of the well-known subtleties which are involved with the "commutativity" of tensor products and inductive limits, if suitable topologies have to

4. INDUCTIVE LIMITS AND GENERALIZED SPECTRA

425

be considered (see A . GROTHENDIECK [3: Chap. I; p. 47, and pp. 76,77]), the situation is nicely simplified when (generalized) spectra are involved. This is primarily due to the “dualization” of the “lim-functors” in case the “Hani“-functor is involved, as we shall see below (cf. e.g. Lemma 4 . 1 ) . An indication of a possible application of this material is offered by our considerations in Section 5 below in connection with the work of H. PORTA-J.T. SCHWMTZ [l] . NOW, our results, as they are Lemma 3 . 1 and Theorem 3 . 1 , of the previous section describe, in effect, the framework within which we are going to argue in the sequel. Thus our first analogue to Lemma 3 . 1 is the following crucial result.

Lemma 4.1. Let (Ea)aeI, ( F A ) h e be i n d u c t i v e systems of topological algebras (Definition IV; 2 . 1 ) w i t h i d e n t i t y elements, and l e t E = l i m E, ,

(4.1)

F = lim FA

be t h e respective topological inductive l i m i t algebras (Lemma IV; 2 . 2 ) suppose t h a t , f o r every p a i r of i n d i c e s (,,A) pology

T,,~

also l e t

T

. Moreover,

e I x K , we are given a compatible t o -

on t h e tensor product aZgebra Ea8 FA s a t i s f y i n g (3.2) and ( 3 . 3 1 , and be a s i m i l a r topology on E8F. F i n a l l y , l e t G be a topological algebra

w i t h an i d e n t i t y element and continuous m u l t i p l i c a t i o n . Then, one g e t s t h e following homeomorphisms ( i n t o ) between t h e respective topologicaL spaces; i . e . , p : lfums(

(4.2)

1.m (E OF I , G) %$A

a s well a s v : ttoms(EfF, G)

(4.3)

-

-

ffams(E, G)

X

ffffmsiF, GI,

Hffms(E, G)xHoms(F, G) ,

i n such a way t h a t one has (4.4)

Im

= Im (v),

w i t h i n a (canonical) homeomoqhism, which i s characterized by an analogous r e l a tion t o (3.6).

By V; Subsection 7.(3) and V; ( 3 . 2 1 ) , homeomorphism) Proof.

one gets (within a

5 (bicontCnuous i n j e c t i o n , by (3.5); see a l s o V; (3.11)) (4.5) lim(Hams(Ea, G) =

Hums (limE 3 a’

X

H f f m s ( F A , G)) = ( limHams(E,,

G) x

C)) x

(limffams(FA, G))

GI = Homs(E, GI x Homs(F, G) Hams(limF 4 A’

.

426

XI1

SPECTRA

(The last equalities are valid, of course, within homeomorphisms). So one obtains, in effect, a homeomorphic injection as indicated by (4.2), the image of which is identified by a relation analogous to ( 3 . 6 ) . Furthermore, one gets from Lemma 3 . 1 a homeomorphism as asserted by ( 4 . 3 ) whose image is again characterized by a relation like (3.6). NOW, this together with our previous analogous conclusion for ( 4 . 2 ) implies already ( 4 . 4 ) , and the proof is complete. I In particular, one gets the following, as a consequence of (4.4).

Corollary 4.1. Let a l l t h e conditions of the previous Lema 4 . 1 be s a t i s f i e d . Then, one g e t s t h e r e l a t i o n

where p denotes the homeomorphism of t h e respective topological spaces, given by p s (v-1

(4.71

IIm(v) lo’

*

Thus, the map (4.8) V O P defines a homeomorphism of the f i r s t space i n ( 4 . 6 ) i n t o the l a s t one t h e r e .

is well-defined, i n view of ( 4 . 4 1 . So

Proof. The relation ( 4 . 7 )

( 4 . 7 ) yields a homeomorphism of the respective topological spaces (cf.

Lemma 4 . 1 ) , and this proves the assertion. I More particularly, if the given algebra G is commutative, the following more satisfactory picture is then a direct consequence of the foregoing and Lemma 3 . 1 . That is, we have.

Corollary 4.2. Let the conditions of Lemma 4.1 be s a t i s f i e d and, moreover, suppose t h a t the given algebra G i s c o m u t a t i v e . Then, one has Hams(lim(E

@Fl,), GI = Hams(ETF, G) a,

a T

(4.9)

= (by ( 4 . 1 ) ) Homs((l&mEa)

@

( 1 3 F A ) , GI = Hams(E, G )

x

Homs(F, G) ,

within homeomorphisms, i n p a r t i c u l a r , of the topological spaces i n t h e f i r s t and the l a s t of the pretiious r e l a t i o n s . I

Now Proposition V;3.1 suggests the way of extending the previous results to the case one considers completed topological tensor product algebras. Nevertheless, we have again to resort to the local equicontinuity of the (generalized) spectra considered. Thus, we first have the following.

4.

INDUCTIVE LIMITS AND GENERALIZED SPECTRA

Theorem 4.1. Let

,

(Eajae I

427

be i n d u c t i v e systems o f l o c a l l y convex

algebras where t h e i n d i v i d u a l topological algebras have i d e n t i t y elements and continuous m u l t i p l i c a t i o n s , and l e t E = 1Lm E

(4.10)

FA

F= & lm

a’

b e t h e r e s p e c t i v e l o c a l l y convex i n d u c t i v e l i m i t algebras ( b e i n g of t h e same t y p e a s t h e g i v e n o n e s ; Lemma 4 . 2 ) . Furthermore, assume t h a t f o r every p a i r on E B F s a t i s f y o f i n d i c e s ( a , A ) E I x K , we are given a compatible topology T a,h ci x i n g ( 3 . 2 ) and ( 3 . 4 ) and, moreover, l e t G be a complete l o c a l l y convex algebra w i t h an i d e n t i t y element and c o n t i n m u s m u l t i p l i c a t i o n . F i n a l l y , suppose t h a t , f o r every p a i r o f i n d i c e s (a,x )

€

IxK,

t h e respec-

t i v e generalized spectra

+toms ( E a , G)

(4.11)

and

Horns (FA,GI

are l o c a l l y equicontinuous. Then, one g e t s t h e following (canonical) homeomorphism

( i n t o ) between t h e

r e s p e c t i v e topological spaces; i . e . , : Hum (lim(Ea6

(4.12)

s -

Ei 1,

C) -+

+ton7

(E, G I

x

U o m s ( F , G)

.

T

L,A

Proof. The f i r s t of t h e f o l l o w i n g r e l a t i o n s f o l l o w s from an a p p l i c a t i o n a g a i n o f V; (3.21) , w h i l e t h e s u b s e q u e n t o n e s a r e d e r i v e d from t h e p r e c e d i n g d i s c u s s i o n , a s i n d i c a t e d ; t h a t i s , w e h a v e

(4.13)

5

( b i c o n t i n u o u s i n j e c t i o n ; c f . Theorem 3 . 1 )

This proves ( 4 . 1 2 )

.I

Now a p p l y i n g , i n p a r t i c u l a r , t h e argument used i n P r o p o s i t i o n

3 . 1 , we c a n g e t f u r t h e r t h e p r e c e d i n g f o r m u l a ( 4 . 1 2 ) f o r c o m p l e t e d ( t o p o l o g i c a l ) i n d u c t i v e l i m i t a l g e b r a s . So w e h a v e .

Corollary 4.3. L e t t h e c o n d i t i o n s of t h e preceding Theorem 4 . 1 be s a t i s f i e d . I n p a r t i c u l a r , suppose moreover t h a t a l l t h e generalized spectra involved a r e equicontinuous, except ( a t most) o f f i n i t e many o f them which are s t i l l l o c a l l y equicontinuous. Then, one g e t s t h e f o l l o w i n g homeo:?orphism ( i n t o ) o f t h e topological spaces under consideration; i . e . ,

4 28

XI1 SPECTRA

such that (4.15)

Proof. T o s t a r t w i t h w e l e a n upon, f i r s t l y , o u r h y p o t h e s i s con-

,

cerning t h e generalized spectra appeared i n ( 4 . 1 1 )

and s e c o n d l y , on

a n extension of Propositzhn V ; 3 . 1 t o t h e present ease (which a l s o c o u l d be der i v e d from a d i r e c t a d a p t a t i o n , w i t h i n t h e p r e s e n t c o n t e x t , of t h e SO one c o n c l u d e s t h a t the following two generalized spectra are locally equicontinuous, i . e . ,

proof o f t h a t r e s u l t ) :

(4.16)

Hum (E,

(4.17) NOW,

G ) = Hums(l&mEa,

GI = l i m Homs(Ea, G ) , and

Hum (F, G) = l i m Horns (FA,GI c

.

from t h e s a m e c o n d i t i o n o n t h e s p e c t r a i n ( 4 . 1 1 ) and t h e r e s p e c -

t i v e r e l a t i o n t o V : ( 3 . 2 1 ) ( s e e a l s o t h e p r o o f o f Theorem 3 . 1 , t i c u l a r , V;(3.17)),

i n par-

one c o n c l u d e s t h e f o l l o w i n g ;

*

G) = Horn ( l i m Ea , G l s + = ( b y ( 4 . 1 6 ) ) l i m f f o m s ( E a , G I = 1>mHoms(~,, G)

Hum (E, G ) = Hums(,?,

(4.18)

4-

.

= Ham ( l i m i a , G ) = H o m ( 1 i m E GI s + s 4 a' A s i m i l a r r e s u l t i s v a l i d f o r t h e a l g e b r a F , i . e . , w e have (4.19)

Hom ( F , GI = Hums(?,

A

G I = Horns( & lm

4,.

F A , G I = Hum I l i m P A , G I

s +

.

A l l t h e p r e c e d i n g r e l a t i o n s a r e t r u e w i t h i n homeomorphisms.

On t h e o t h e r h a n d , by h y p o t h e s i s and Lemma 3 . 2 , e a c h one o f t h e ( l o c a l l y c o n v e x ) a l g e b r a s Ea :FA

w i t h (a,A) e I X

K

,

has a l o c a l l y equi-

a,

c o n t i n u o u s g e n e r a l i z e d s p e c t r u m , w i t h r e s p e c t t o G: m o r e o v e r , a l l exc e p t o f f i n i t e many of them h a v e e q u i c o n t i n u o u s ( g e n e r a l i z e d ) s p e c t r a

(see ( 3 . 4 ) , i n c o n n e c t i o n w i t h ( 3 . 1 1 ) ) - Thus, by h y p o t h e s i s ( c f . a l s o t h e a f o r e - m e n t i o n e d e x t e n s i o n o f P r o p o s i t i o n V:3.1) vex a l g e b r a l i m ( E +

t h e l o c a l l y con-

@ F A ) h a s a l o c a l l y e q u i c o n t i n u o u s g e n e r a l i z e d speca, A

Q T

t r u m , so t h a t o n e f i n a l l y o b t a i n s t h e f o l l o w i n g ( c f . a l s o ( 4 . 1 8 ) )

(4.20) = l i m ffun? (E 8 F A

s

a T

, G / 5 lim( Hums ( E a ,

GI x Homs(FA, G))

.

a,A

The l a s t r e l a t i o n d e n o t e s a b i c o n t i n u o u s i n j e c t i o n . Now on t h e b a s i s

of

( 4 . 1 8 ) and ( 4 . 1 9 1 , t h i s a c t u a l l y p r o v e s o u r a s s e r t i o n c o n c e r n i n g

t h e map ( 4 . 1 4 )

,

and t h e p r o o f i s f u l l y e s t a b l i s h e d . I

Now, i f t h e a l g e b r a G I common r a n g e o f t h e g e n e r a l i z e d s p e c t r a

5.

429

"POINT EVALUATIONS"

i s i n p a r t i c u l a r c o m t a t i v e (hence w e f r e e ours e l v e s from t h e r e s t r i c t i o n s c a u s e d by ( 3 . 6 ) ) , one o b t a i n s more comp l e t e e x p r e s s i o n s f o r a l l t h e p r e c e d i n g f o r m u l a s (see e . g . t h e p r e v i ous C o r o l l a r y 4 . 2 ) . So, more p r e c i s e l y , w e have t h e n e x t . under c o n s i d e r a t i o n ,

Corollary 4.4. Suppose t h a t a l l t h e conditions of Theorem 9.1 are s a t i s f i e d and, moreover, Zet T be a (Zoca22y convex) compatib2e topo2ogg on the tensor prod-

uct aZgebra EC3F s a t i s f g i n g 1 3 . 2 ) and (3.41. Furthermore, suppose t h a t the algebra G i s commutative. Then, one has

& t h i n (canonical) homeomorphisms of the topo2ogica2 spaces invo2ved ( g i v e n by (4.12) and (3.5); c f . Lemma 3.1).

In particuZar, i f t h e conditions of Corollary 4.3 are s a t i s f i e d , concerning t h e genera2ized spectra considered, then t h e previous formula (4.211 i s f u r t h e r enriched a s f02lows

Ham (l?m(Ea6 s -

T

FA I , G) = Horn ( E G F , G) = Hum S

T

(i,GI x

Horns($, G )

a,

(4.22)

= Hoet (E, G)

X

Hum (F, G) = Hums(EQF, G) T

.

Here the aZgebras E , F are given b y ( 4 . 1 5 ) , whiZe the e q u a l i t i e s involved are t r u e within (canonica2) homeomorphisms of t h e respective topoloyicaL spaces ( g i v e n by (4.14) and (3.17); c f . a l s o t h e p r o o f of t h e p r e v i o u s C o r o l l a r y

4.3). I

5. General ized spectra and "point evaluations" W e c o n s i d e r n e x t a p a r t i c u l a r a p p l i c a t i o n of t h e f o r e g o i n g which

p r o v i d e s " p o i n t e v a l u a t i o n s " of c e r t a i n p a r t i c u l a r a l g e b r a morphisms ( c f . , f o r i n s t a n c e , ( 5 . 2 0 ) b e l o w ) . Such f o r m u l a e h a v e found

past

in

the

a p p l i c a t i o n s i n t h e r e p r e s e n t a t i o n t h e o r y of a l g e b r a s o f vec-

t o r - v a l u e d holomorphic maps o r t e n s o r p r o d u c t s of s u c h . See e . g . H . PORTA - J . T . SCHWARTZ [ I ]

and/or

A . MALLIOS [16 ;171. However, w e s h a l l need

f i r s t t h e following supplementary terminology. Thus, g i v e n a n a l g e b r a E ( n o t o p o l o g y i s c o n s i d e r e d ) , one def i n e s t h e c e n t e r of E , d e n o t e d by C f E I , by t h e r e l a t i o n (5.1)

C(E):={z€Z:xg= y x V yeE).

T h a t i s , one c o n s i d e r s t h o s e e l e m e n t s of E which commute w i t h a l l e l e ments i n E . I t i s c l e a r t h a t i n c a s e t h e a l g e b r a E h a s an i d e n t i t y

-

e l e m e n t l E , t h e n it b e l o n g s t o C ( Z ) , and i n e f f e c t t h e whole f i e l d of coefficients too, i.e.

,

C 5 C(El ( v i a o f c o u r s e t h e c o r r e s p o n d e n c e X

430

X.lE

XI1 SPECTRA

,

with A€@ )

.

I n p a r t i c u l a r , i f t h e c e n t e r of E i s r e d u c e d t o t h e

image o f C, u n d e r t h e p r e v i o u s i n j e c t i o n , w e s a y t h a t t h e a l g e b r a

E

i s c e n t r a l , i.e., one h a s t h e n , b y d e f i n i t i o n , t h e r e l a t i o n

C(El = C * I E

(5.2)

( I n t h e l a t t e r c a s e w e a l s o r e f e r t o E a s an a l g e b r a w i t h a t r i v i a l

center; we n o t e t h a t t h e a p p l i e d terminology h e r e i s r a t h e r a k i n t o t h e “ a l g e b r a i c “ o n e . Cf

.

e . g . P.M . COHN [l :p. 3631 ; however

,

see a l s o C .

E . RICKART [l: p. 86, D e f i n i t i o n ( 2 . 7 . 6 ) ] ) .

I t i s c l e a r from ( 5 . 1 ) t h a t the c e n t e r o f a given algebra E i s a com-

mutative subalgebra of E. I n p a r t i c u l a r , i f E i s a topological algebra ( D e f i n i t i o n I; 1 . 1 )

t h e n i t s c e n t e r i s a closed commutative subalgebra

3f

E. ( T h i s

i s s t r a i g h t f o r w a r d from ( 5 . 1 ) a n d t h e s e p a r a t e c o n t i n u i t y o f t h e m u l t i plication in E ) . NOW, f o r a n y s u b s e t M of a g i v e n a l g e b r a E l w e d e f i n e t h e com-

mutant

of

M , w e d e n o t e it by M‘, M’=

(5.3)

t o b e t h e set

{ z e E : a x = xa

V aB

M1,

i . e . , t h e s e t of t h o s e e l e m e n t s o f E which commute w i t h a l l e l e m e n t s of M . Furthermore, one d e f i n e s t h e b i c o m t a n t

of

M,

d e n o t e d by M ” ,

by t h e r e l a t i o n M”=

(5.4)

(M‘)’.

,

so t h a t o n e h a s M C M ” . Moreover, it i s c l e a r from ( 5 . 3 ) ( a n d t h e a s s o c i a t i v i t y of E )

T h a t i s , a s t h e commutant of M‘ g i v e n b y ( 5 . 3 )

t h a t M’ i s a stibalgebra of E, f o r a n y M G E , w h i l e i f E i s a t o p o l o g i c a l a l g e b r a t h e n M’ i s a closed subalgebra of E. So w e are now i n t h e p o s i t i o n t o s t a t e t h e r e s u l t , w e h a v e prom-

i s e d a l r e a d y a t t h e b e g i n n i n g of t h i s s e c t i o n . T h a t i s , w e h a v e .

Theorem 5.1. Suppose t h a t t h e c o n d i t i o n s of t h e previous Theorem 4 . 1 are s a t i s f i e d . Thus, f o r any element (5.5)

l e t h = f a g be t h e canonical decomposition o f h provided

If,

(5.6)

g ) E Homs(E, G) x ffomsfF, G)

hi/

( d . 1 2 ) such t h a t

.

Moreover, l e t

= I m (g) 9 be t h e subalgebras o f G , images of t h e maps (algebra morphismsi h , f (5.7)

Mh = I m i h ) ,

M

f

=Im

if),

M

and g , respec-

5.

431

"POINT EVALUATIONS"

t i v el y . Then, concerning t h e r e s p e c t i v e conunutants of t h e s e subalgebras of G , one

has t h e r e l a t i o n (5.8)

Hence, i f t h e algebra E

I :=

~i n M;

= ~ ; n M' n M ' .

limEcl)

is c o m u t a t i v e , one g e t s

f

s

.

M' n M" = M" n M' h

(5.9)

h

h

g

In p a r t i c u l a r , under t h e preceding c o n d i t i o n s f o r E, suppose f u r t h e r t h a t I m I h ) = M i s a c e n t r a l subalgebra of G ( c f . ( 5 . 2 ) ) . Then, I m ( g ) z M i s a cenh tra2 subalgebra of G too such t h a t one has t h e r e l a t i o n

=M

Im(f)

(5.10)

f

c*lE

g

*

Therefore, if

f(xI = Ax .lG ,

(5.11)

with Axe@ and f o r any x e E , denotes t h e map f map (5.12)

thus reduced by 1 5 . 1 0 1 , then t h e

A5

,y:E--t@:x",y(x):=

d e f i n e s a continuous character of E ; i . e . , ue have

x

(5.13)

E m(E).

Proof. W e f i r s t remark t h a t from t h e r e l a t i o n s c o r r e s p o n d i n g t o t h o s e i n ( 3 . 7 ) a n d which a r e p r o v i d e d f r o m ( 4 . 1 2 )

are contained i n M h .

both M

So

I

M i E M j n M'g '

f and M g i n c l u s i o n i s s t i l l v a l i d due t o t h e r e l a t i o n

one c o n c l u d e s t h a t while t h e r e v e r s e

h = j % g (see (3.9) a n d

( 3 . 1 0 ) ) . Hence, o n e a c t u a l l y h a s t h e r e l a t i o n

M' = M'

(5.14)

h

f

n M'

g

which of c o u r s e e n t a i l s ( 5 . 8 ) . O n t h e o t h e r h a n d , i f the algebra E = ( b y ( 4 . 1 0 ) ) l i m E u is c o m u t a t i v e ,

t h e n a p p l y i n g t h e p r e v i o u s argument ( c f .

Mf S Mi

(5.15) Hence,

( 3 . 7 ) ) one e a s i l y g e t s t h a t

.

(Mi)'= M" C M' which b e c a u s e o f ( 5 . 8 ) c e r t a i n l y y i e l d s ( 5 . 9 ) h - f F u r t h e r m o r e , u n d e r t h e same a s s u m p t i o n f o r E , one h a s M

(5.16)

n M'

c_

M~

g g = (by (5.9))

n M'

g

Mhn

=

~h

.

n M; n M'

Mifi M i =

g MhflMi

.

That i s , (5.17)

(cf. ( 5 . 1 ) ) I so t h a t i f MFL i s c e n t r a l , then t h e same i s

.

t r u e f o r M Conseg

432

XI1

q u e n t l y , s i n c e Mf E M h

,

SPECTRA

one c o n c l u d e s from ( 5 . 1 5 ) t h a t

n M i = C(Mh) = (by ( 5 . 2 ) ) C * Z G , h t h a t i s , w e g e t t h e d e s i r e d r e l a t i o n ( 5 . 1 0 ) . So from t h e l a s t r e l a (5.18)

M

C

f-

M

t i o n o n e now o b t a i n s , f o r e v e r y x e E , f(x) =

(5.19)

T h e r e f o r e , d e n o t i n g by j : C + G ( h H

xX . l G E C . l G ' X.1

G

) t h e r e s p e c t i v e canonical (bi-

continuous) i n j e c t i o n , t h e correspondence

x :x++AX: E +

C,

where

d e f i n e s i n f a c t a non-zero continuous complex algebra morphism of E an element theorem. I

xe

,

that is

n ( E ) , a s d e s i r e d , a n d t h i s c o m p l e t e s t h e p r o o f of t h e

433

P r o p e r t i e s o f Permanence o f T o p o l o g i c a l Tensor Product Algebras

CHAPTER

XI11

W e consider i n t h i s chapter s e v e r a l p r o p e r t i e s of topological

a l g e b r a s which a r e p r e s e r v e d when t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s , a s w e l l as c o n d i t i o n s under which t h e s i t u a t i o n i s r e v e r s e d , i . e . , " h e r e d i t y p r o p e r t i e s " of t h e r e s p e c t i v e t o p o l o g i c a l t e n s o r product a l g e b r a s . T h i s h a s been e n c o u n t e r e d a l r e a d y i n t h e f o r e g o i n g by considering, f o r instance, preservation

( a n d h e r e d i t y a s w e l l ) of t h e

local equicontinuity ( o r y e t e q u i c o n t i n u i t y ) of t h e

s p e c t r a of t h e f a c t o r

a l g e b r a s i n a given ( f i n i t e o r i n f i n i t e ) t o p o l o g i c a l tensor product a l g e b r a ; f u r t h e r m o r e , t h e s i m i l a r s i t u a t i o n when t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d were 1.1,

4-algebras. (See Chapt. XII: Lemma l .2 and Remark

a s w e l l a s Theorem 2.2; and, r e s p e c t i v e l y , Lemma 1 . 3 and Corol-

l a r y 2.2). The r e l e v a n t c o n c l u s i o n s , w e are l e d t o i n t h e s e q u e l a r e

of p a r t i c u l a r s i g n i f i c a n c e f o r t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s of t h e t y p e c o n s i d e r e d i n Chapter XI.

1. Boundaries o f topological tensor product algebras On t h e b a s i s of o u r p r e v i o u s c o n s i d e r a t i o n s i n Chapt. V1;Sect i o n 2, w e f i r s t examine t h e e x t e n t t o which t h e ( s i l o v ) boundary can be d e f i n e d f o r a s u i t a b l e t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a i n case t h i s boundary e x i s t s f o r t h e i n d i v i d u a l f a c t o r a l g e b r a s . So o u r f i r s t remark i s t h e f o l l o w i n g seemingly s t r o n g e r v e r s i o n

of p r o p o s i t i o n 2 ) i n Lemma VI;2.1;

it i s more a p p r o p r i a t e t o t h e n e x t

d i s c u s s i o n ( c f . Lemma 1.2 b e l o w ) . That i s , w e have.

Remark 1 . 1 . -

Applying t h e n o t a t i o n of Lemma VI;

2.1, one c o n c l u d e s t h a t : An element f E aE C m(EI if, and on2y i f , f o r every open neighborhood U of f i n miE) there e x i s t s an element x E E , w i t h

+ 0,

and such t h a t M ( G )

U. ( The c r u c i a l f a c t t h a t G f 0 f o l l o w s from t h e a r b i t r a r i n e s s of U and t h e f a c t t h a t M(2) = m(E), f o r e v e r y x E E w i t h 2 = O ; c f . V I ; (2.4)).

434

XI11 PROPERTIES OF PERMANENCE

S i n c e t h e t o p o l o g i c a l a l g e b r a s w e a r e g o i n g t o be d e a l t w i t h

w i l l be m a i n l y bounded o n e s ( c f . D e f i n i t i o n V 1 ; l . l ;

see a l s o t h e com-

ment b e f o r e D e f i n i t i o n V I ; 2 . 2 ) , w e c o n s i d e r f i r s t t h e i r b e h a v i o u r i n t a k i n g ( t o p o l o g i c a l ) t e n s o r p r o d u c t s . So w e h a v e t h e n e x t .

Lemma 1 . 1 . Let E , F be two bounded topological algebras and E C3F the corresponding t e n s o r product algebra endowed w i t h a compatible topology tion Xi4.1).

T

T ( cf

. Def i n i -

Then, E Q F i s a bounded topological algebra t o o . T

Proof. I n v i e w of D e f i n i t i o n V I i 1 . 1 ,

it s u f f i c e s t o prove t h a t

t h e r e a l - v a l u e d map

/z^(hll = / h ( z l l

(1 -1)

with h E m ( E Y F l ,

n

i s a bounded f u n c t i o n f o r e v e r y z = t x i @ y i € E E 8 F . T h i s r e d u c e s of c o u r s e i=l

t o an a n a l o g o u s p r o o f , f o r e v e r y decomposable t e n s o r x ~ y EE@ F . So one obtains

n

s u p / !my) ( h ) I = ( b e c a u s e o f C h a p t . XII; (1.5) ( 1 . 6 ) )

hem(E8FI

( s e e also N . BOUREMI [ 4 : Chap. 4 ; p. 2 6 , r e l . ( 2 5 ) ] ) . T h u s , a c c o r d i n g t o t h e hypothesis f o r the algebras E,F

t h i s i m p l i e s t h e a s s e r t i o n , and t h e

proof i s f i n i s h e d . I A c o n v e r s e t o t h e p r e v i o u s Lemma 1 . 1 i s c e r t a i n l y v a l i d when

e a c h one o f t h e g i v e n t o p o l o g i c a l a l g e b r a s E , F h a p p e n s t o p o s s e s s a t l e a s t o n e bounded element

( t h e c o r r e s p o n d i n g G e l f a n d t r a n s f o r m of t h e

e l e m e n t i n q u e s t i o n i s a bounded f u n c t i o n ; see ( 1 . 2 )

a b o v e ) . For

ex-

ample, t h i s w i l l be t h e ease when t h e given algebras E and F have i d e n t i t y e l e -

ments. Now,

t h e f o l l o w i n g r e s u l t a l s o w i l l b e needed i n t h e s e q u e l t h a t

r e f e r s t o a " t e n s o r p r o d u c t a n a l o g o n " of t h e r e l . V I i ( 2 . 4 ) .

So one h a s .

Lemma 1 . 2 . Let E , F be topological algebras and E B F t h e corresponding tenT sor product algebra equipped w i t h a compatible topology T. Then, f o r every decomposable tensor x a y e E Q F , one g e t s (1.3)

M(&I

x

Mlil

n

E M(x@y)

(see V I ; (2.4) ) . I n p a r t i c u l a r , f o r every decomposable t e n s o r x@ Y E E O F

with

z e y # 0 , one o b t a i n s

(1.4)

M(zcy) = M(2)

x

M(y^)

.

The preceding two r e l a t i o n s a r e considered i n the sense of t h e (canonical) b i j e c -

BOUNDARIES OF TENSOR

1.

4 35

PRODUCTS

t i o n XII;( 1 . 1 ) . Proof. For e v e r y if, 3 ) E M i ? )

X

f@ g e

M($), o n e c o n c l u d e s t h a t h

M ( z @ y l ( c f . X I I ; ( 1 . 7 ) ) . T h i s f o l l o w s from a n argument a n a l o g o u s t o t h a t

a p p l i e d i n ( 1 . 2 ) and a l s o from Theorem X I 1 ; l . l . n zmy

On t h e o t h e r h a n d , i f G $ O

as well

a n d (f,g l e

+ 0,

one g e t s ( e q u i v a l e n t l y ) 3 # O and A

( c f . X I I ; ( 1 . 6 ) ) ; t h u s , f o r every h e M i x o y ) with

m(E)X n l F l

h’e miE

(Lemma X I I ; l . I ) ,

F)

f’E

Now, by h y p o t h e s i s f o r $,-$

h=fmg

o n e g e t s from ( 1 - 2 )

m(E)

t h i s implies t h a t

I?(f)

I

g’ e

M(FI

=

sup

/?(f‘)I

and

f’e m(E)

.

s u p 1 y^(g’) 1 , so if, g ) e MIL?) x M(y^) Thus , one o b t a i n s t h e i n v e r s e r;‘e m(F) r e l a t i o n t o ( 1 . 3 ) t h a t p r o v e s ( 1 . 4 ) , and t h i s t e r m i n a t e s t h e p r o o f . I

(Glg)1 =

A s a b y p r o d u c t of t h e p r e c e d i n g w e a l s o h a v e t h e f o l l o w i n g .

Corollary 1 . I . Le: E, F be c o m u t a t i v e l o c a l l y m-convex &-algebras and l e t E 8 F be t h e r e s p e c t i v e Locally m-convex t e n s o r product algebra of E , F equipped w i t h a compatible topology T ( D e f i n i t i o n X ; 3 . 1 ) . Then, for every (x,y ) C E X F , one g e t s t h e f o l l o w i n g reLation concerxing t h e corresponding s p e c t r a l r a d i i : r

(1.6)

(E

Proof.

Q7 F)

i x a y ) = r (x) ’ r i y ) E F

The a s s e r t i o n f o l l o w s i m m e d i a t e l y from a n a p p l i c a t i o n o f i n c o n j u n c t i o n w i t h C o r o l l a r y I I I ; 6 . 5 ( s e e 111;

t h e reasoning i n (1.2)

( 6 . 1 6 ) a n d Theorem I ; 6 . 4 ) , Lemma X I I ; 1 . 3 and C o r o l l a r y I I I ; 6 . 6 .

I

B e f o r e w e come t o o u r f i r s t main r e s u l t i n t h i s s e c t i o n , w e s t i l l need t h e f o l l o w i n g u s e f u l

Lemma 1.3. Let E , F be t o p o l o g i c a l algebras and E @ F t h e r e s p e c t i v e t o p o l o T

gicaL t e n s o r product algebra under a compatible topology Then, f o r everti p a i r

(2,

g) e (E@F) O ( z , g)

(1.7)

is given by an

-

:???‘(El --tC

:f

E

E such t h a t

2

= Olz, g)

.

SimiZarZy, one g e t s t h e r c l a t i o n (1.9)

T

(Definition X;4.1 )

m(F),t h e map

;If og)

element of E A ( G e l ’ f a n d t r a n s f o r m a l g e b r a o f E ) . That is,

t h c x e x i s t s a n element x (1 .8)

x

;= Y l z , f),

.

4 36

XI11

P R O P E R T I E S OF PERMANENCE

f o r some element y E F , where t h e map '?(z, f) :

m(F/

-+

C is given by the r e l a t i o n

f ) ] fg) = ; ( f o g ) , with g E m ( F ) , and f o r every ( 2 , f)€ ( E 8 F ) X m ( E ) . In p a r t i c u l a r , for complete topological algebras E , F if E 8 F i s s t i l l a (complete) topological algebra, then ( 1 . 8 ) and ( 1 . 9 ) are t r u e for every element z [Y(z,

A

e E$F. Proof.

n ?. x i @ yi E E Q F and g e m(FI, one g e t s ,

Indeed , f o r e v e r y z

i =1

f o r every f e ??Z(E/, t h e r e l a t i o n

A

2(fag) = f(zg(yi)xil = (;gfyi)xil(f) z z

(1.lo)

NOW, t h i s p r o v e s o u r a s s e r r t i o n , c o n c e r n i n g ( 1 . 8 )

E ; similarly

,

. g ( y .)x. e

taking z 7,

for (1-9).

On t h e o t h e r h a n d , f o r e v e r y g e m ( F ) , t h e c o r r e s p o n d e n c e (1.11)

Ci

g

:EQF-E: T

zxi@yi+7g(yi)xi 7,

d e f i n e s a continuous linear map ( c f . Lemma 4 . I below)

. SO

i f E i s complete

( 1 . 1 1 ) c a n b e e x t e n d e d t o E 8 F , w e d e n o t e t h i s e x t e n d e d map s t i l l by T

B g , s u c h t h a t one o b t a i n s from ( 1 . 8 ) t h e r e l a t i o n A

(1.12)

wz,

f o r e v e r y ( z , gl e ( E 8 F ) ( Z c 1 ) a EI relation

in

x

gi =

m(F). I n d e e d ,

E8F, with z = l i m za

eg i z i for every

Z E

E6F T

and

a

net

one o b t a i n s , f o r e v e r y f e W(E/, t h e

,

n = limffag)(zcl) = (fag)(limza)= (fag)(z)= ;(fag),

a

c1

which p r o v e s ( 1 . 1 2 ) . A s i m i l a r argument i s o b v i o u s l y v a l i d f o r t h e map ( 1 . 9 ) , a n d t h i s c o m p l e t e s t h e p r o o f . I So w e come now t o t h e n e x t main r e s u l t of t h i s s e c t i o n . Namely,

w e have.

Theorem 1.1. Let E , F be bounded topological aZgebras and E 8 F the corresponding (bounded ; c f . Lemma 1 . l ) topological tensor product algebra of E , F i n a compatible topoZogy T. Then, concerning the respective ( S i l o v l boundaries of the topological algebras i n question ( c f . D e f i n i t i o n V 1 ; 2 . 2 ) , one concludes t h a t a ( E @ F ) e x i s t s i f t h i s i s the ease f o r b o t h aE, aF. Moreover, one has then T

:he r e l a t i o n (1.14)

a ( E 8 F ) = aF x aF, T

within a homeomorphism provided by t h e r e s t r i c t i o n o f the (canonical) homeomorphism X I I ; ( l . l l ) t o the spaces appeared i n ( 1 . 1 4 ) .

1.

437

BOUNDmIES OF TENSOR PRODUCTS

Proof. W e f i r s t p r o v e t h a t ( 1 .15)

aE x aF

5

a(E@FI, ?

i n t h e s e n s e t h a t t h e image o f t h e r e s t r i c t i o n of X I I ; ( l . l 2 ) ( “ c a n o n i -

c a l homeomorphism”) t o aE

X

aF

_C

mfE)

m ( F ) is contained i n

X

afEBF,JC T

m ( E 8 F I . T h i s w i l l a l s o prove t h e a s s e r t i o n t h a t a ( E @ F ) e x i s t s : Thus, T

T

if ff,g)eaExaF

and W i s a n e i g h b o r h o o d o f

b e a n e i g h b o r h o o d of If, g l i n XI1;l.l).

f o g = h e ??ZIEfF), let U x V

m(E)x m ( F ) s u c h t h a t U 8 V C W (Theorem a n d Lemma VI;2.1 ( c f .

T h e r e f o r e , by h y p o t h e s i s f o r ( f , g )

a l s o t h e above Remark 1 . 1 1 , t h e r e e x i s t s a decomposable t e n s o r x o y e A

E @ F with x o y

+0

and s u c h t h a t M ( P ) C U a n d M f Q ) E I’ A

M l z o y ) = ( b y ( 1 . 4 ) ) M(2) x M($) C U X Y :

( 1 .16)

.

Hence

one g e t s

W

h e f o g e a ( E B F ) , which p r o v e s ( 1 . 5 ) .

t h e r e f o r e (Lemma V I ; 2 . 1 ) ,

T

F u r t h e r m o r e , w e a l s o p r o v e now t h a t

( 1 .17)

afE@F) T

5

T h a t i s , one a c t u a l l y p r o v e s t h a t aE E @ F , s o that T

2.2.

aE

x

X

aF

.

aF is a

( c l o s e d ) boundary s e t f o r

( 1 . 1 7 ) f o l l o w s i m m e d i a t e l y from t h e same D e f i n i t i o n V I ;

Indeed, f o r e v e r y element z e E B F , t h e p a i r

Consequently ( D e f i n i t i o n V I ; Z . l ) ,

t h e r e e x i s t s an element f ‘ e a E

(2(f’@J!ll( =

(1.18)

g e m ( F ) deg l s E A (Lemma 1 . 3 ) .

(z, g) with

f i n e s a map l i k e ( 1 . 7 ) and h e n c e a n e l e m e n t ;=@fz,

sup/2(fog)I m(E)

f

.

E

S i m i l a r l y , c o n s i d e r i n g t h e map c o r r e s p o n d i n g t o ( 1 . 9 ) t h e r c o n c l u d e s t h e e x i s t e n c e of a n e l e m e n t g ’ e a F

I2if

(1.19)

og’)

I= g

f o r every f

E

with

s u p 12lf o g l € m(F)

1

,

one f u r -

such t h a t I

M ( E ) a n d z e E @ F . T h u s , t h e l a s t two r e l a t i o n s y i e l d now

t h e following s u p )2(f o g ) J = s u p ( suplz^(f o g l l h=f@gem(EBF) g e m ( F ) fem(E) (1.20)

= ( b y (1.18))

s u p l i ( f ’ o g ) l = (by ( 1 . 1 9 ) ) [ ; ( f ’ @ g ’ ) \ g e m(F)

with (f‘,g’)e

aE x a F . S o o n e g e t s V 1 ; ( 2 . 1 ) f o r t h e e l e m e n t z e E O F

which t h u s p r o v e s ( 1 . 1 7 ) NOW,

,

,

and t h i s c o m p l e t e s t h e p r o o f . I

i f one c o n s i d e r s complete t o p o l o g i c a l a l g e b r a s , t h e n on t h e

b a s i s o f t h e c o r r e s p o n d i n g p a r t o f t h e p r e v i o u s Lemma 1 . 3 a n d i n conj u n c t i o n w i t h Theorem X I I ; l . 2 ,

o n e e s t a b l i s h e s a n argument a n a l o g o u s

4 38

XI11

PROPERTIES OF PERMANENCE

t o t h a t of t h e p r e v i o u s p r o o f , which i n t u r n y i e l d s t h e f o l l o w i n g basic r e s u l t . Theorem 1.2. Let E , F be complete bounded topological algebras such t h a t t h e

conditions of Theorem X I I ; l . 2 are s a t i s f i e d . Then, concerning

the respective (Silovl boundaries of the topological algebras considered, the existence of a E , aF

implies that of a ( E @ F ) . Furthermore, i n t h a t e a s e , one has the r e l a t i o n T

a(E6FI = aExaF,

(1.21)

within a homeomorphism, which i s provided by XII;(1.20). I I n view of Remark V I ; 2 . 1

,

w e c o u l d c o n s i d e r above topological a l -

gebras with i d e n t i t y elements and compact spectra (see C o r o l l a r y V I ; 2 . 1 )

. In

case o f t h e p r e v i o u s Theorem 1 . 2 , t h i s would imply t h e a l g e b r a s cons i d e r e d t o have, i n f a c t , equieontinuous spectra ( c f . C o r o l l a r y VI; 1 . 3 and Theorem V I ; 1 . 1 )

. So

i n c a s e o f commutative a l g e b r a s it would be

e q u i v a l e n t t o c o n s i d e r &-algebras ( w i t h i d e n t i t y e l e m e n t s . Cf. e v e r , Example V1;2.1).

,

how-

In t h i s respect, a straightforward application

of t h e p r e v i o u s d i s c u s s i o n p r o v i d e s now t h e n e x t . C o r o l l a r y 1.2.

Let E , F be two ( c o m u t a t i v e ) Banach algebras ( w i t h i d e n t i t y

elements) and S i l o v boundaries aE, a F , r e s p e c t i v e l y . Furthermore, l e t

T

be a ( l o c a l -

l y convex algebra) topology on E B F s a t i s f y i n g the conditions of CoroZZary XII;1.2. Then, t h e respective S i l o v boundary a ( E G F 1 of the IBanachl algebra E 8 F i s given T

T

by t h e r e l a t i o n a(E6F) = aExaF.

(1-22)

This i s v a l i d within a homeomorphism of the spaces involved, and i s provided by XII;

(1.24). I

2 . C o n t i n u i t y o f t h e Gel'fand map Concerning t h e p r o p e r t y i n t i t l e o f t h i s s e c t i o n ( c f . Chapt.VI; Section l ) , i f w e consider topological tensor products, w e actually have t h e f o l l o w i n g r e s u l t . Theorem 2.1. Let E , F be topological algebras w i t h spectra mlEl, m l F ) , re-

s p e c t i v e l y , and l e t E B F be the corresponding topological tensor produet algebra T of E , F under a compatible topology T ( c f . D e f i n i t i o n X ; 4.1: 1 ) - 3 a ) ) . Then, the algebras E , F have the respective Gel'fand maps continuous i f , and only i f , t h i s i s the case f o r t h e algebra E B F . T

Proof. Suppose t h a t t h e a l g e b r a s E , F have c o n t i n u o u s G e l ' f a n d

3.

439

SPECTRALLY BARRELLED ALGEBRAS

maps, a n d c o n s i d e r a compact set K

_C

( E % F ) . Thus w e h a v e

K $ prl(KI x pr2(Kl,

(2.1)

where p r i

( i = l , 2 ) d e n o t e t h e c a n o n i c a l p r o j e c t i o n maps o f m ( E 8 F ) = T

m ( E ) x 1 ? Z ( F ) (Theorem X I 1 ; l . l )

t h e sets

pr.lK) ( i = Z , 2 )

o n t o t h e r e s p e c t i v e f a c t o r s . Hence,both

a r e compact s u b s e t s o f t h e c o r r e s p o n d i n g s p a c e s

a n d h e n c e e q u i c o n t i n u o u s , by h y p o t h e s i s a n d Theorem V 1 ; l . l .

Thus, t h e

set p r ( K ) @ p r ( K ) i s an equicontinuous subset of m ( E 8 F ) ( D e f i n i t i o n 1

2

T

XII;4.1,cond.3a)); f o r t h e set K .

so a f o r t i o r i , d u e t o ( 2 . 1 ) , t h e same i s t r u e

But t h i s i m p l i e s t h e c o n t i n u i t y of t h e G e l ’ f a n d map o f

E B F ( c f . Theorem V I ; 1 . 1 ) . T

Assuming now t h e l a s t c o n c l u s i o n t o b e i n f o r c e , i f c o m p a c t , t h e set K @ t g }

,

K C m ( E / is

w i t h g e m ( F I , i s a compact s u b s e t of ? ? Y ( E Y F )

h e n c e , from h y p o t h e s i s , e q u i c o n t i n u o u s . Moreover, a p p l y i n g a s i m i l a r r e a s o n i n g t o t h a t i n X I I ; ( l . l 9 ) , one e a s i l y c o n c l u d e s t h a t K i s i n f a c t a n e q u i c o n t i n u o u s s u b s e t of ? ? Z ( E ) ; now t h i s p r o v e s t h a t t h e G e l t f a n d map o f E i s c o n t i n u o u s , w h i l e a s i m i l a r a r g u m e n t i s v a l i d f o r F

a s w e l l . T h i s c o m p l e t e s t h e proof o f t h e theorem. B

Scholium 2.1.- I t becomes c l e a r from t h e p r e v i o u s p r o o f t h a t a s i m i l a r c o n c l u s i o n t o t h a t of Theorem 2 . 1 c a n b e d e r i v e d , i f o n e c o n s i d e r s a com-

p Z e t s t o p o l o g i c a l t e n s o r product algebra o f t h e form EGF,

f o r s u i t a b l e E , F a n d T ; t h i s , of c o u r s e ( c o n -

c e r n i n g t h e p r e c e d i n g p r o o f ) ,i n s o f a r as XII; 1 1 . 2 4 ) is

v a l i d . But t h e l a t t e r may b e i n f o r c e w i t h o u t t h e a l g e b r a E & F t o have n e c e s s a r i l y a l o c a l l y equiconT

t i n u o u s s p e c t r u m ( o r , e q u i v a l e n t l y (Lemma X I I ; 1 . 2 ) , t h e same f o r t h e a l g e b r a s E , F). S e e , f o r i n s t a n c e , Lemma X I I ; 1 . 4 .

3. Spectrally barrelled algebras W e examine n e x t t h e b e h a v i o u r of s p e c t r a l l y b a r r e l l e d a l g e b r a s

( c f . D e f i n i t i o n V; 1.3)

w h i l e t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s . So

w e have.

Theorem 3.1. Let E , F be topological algebras whose spectra are m ( E ) and m ( F ) , r e s p e c t i v e l y , and l e t E B F be t h e corresponding t e n s o r product algebra o f T

E , F endowed w i t h a compatible topology T ( D e f i n i t i o n X ; 4 . 1 :1 )

- 3a) ) .

Then,

t h e f o l l o w i n g two a s s e r t i o n s a r e equivalent: 1 ) Each one o f t h e given algebras E , F is a s p e c t r a l l y b a r r e l l e d algebra.

440

XI11 P R O P E R T I E S OF PERMANENCE

2) The topological ( t e n s o r p r o d u c t ) algebra

i s spectral13 barrel-

E8F T

led. Proof. Suppose t h a t 1 ) i s v a l i d , and l e t B

C_

1 ? T ( E O F ) be a (weakT

l y ) bounded s u b s e t o f t h e spectrum o f E 8 F . Now t h e r e l a t i o n T

(3.1)

B

5 prl ( B )

X

pr2 ( B ) E m ( E )

x

m ( F ) E E,’ x F i

(see t h e n o t a t i o n i n (2.1)) i m p l i e s t h a t B i s an equicontinuous subset of m(E8Fl

t o o . Indeed, t h e s e t s p r i ( B ) ( i = 1 , 2 ) a r e bounded s u b s e t s o f

m ( E ) , m ( F ) , r e s p e c t i v e l y , and hence by h y p o t h e s i s e q u i c o n t i n u o u s . So

i m p l i e s now t h a t prl ( B ) 8 p r 2 ( B )

i s an e q u i c o n t i n u o u s s u b s e t of ??Z(E8 F ) . Thus, due t o (3.1 1 , t h i s i s a f o r T t i o r i t r u e f o r B , which p r o v e s p r o p o s i t i o n 2) ; so w e proved t h a t cond. 3 a ) of D e f i n i t i o n X;4.1

1 ) implies 2 ) .

On t h e o t h e r hand, i f 2) i s v a l i d and B C ? ? Z ( E ) i s bounded, t h e n a l s o B x { g } C m ( E ) x m ( F ) C E’,” F‘, i s a bounded s e t , f o r e v e r y g e m ( F ) . S o from t h e h y p o t h e s i s f o r

T

(see cond. 3) of

-

D e f i n i t i o n X ; 4.1)

and

x‘@ y ’ : E i x F i t h e r e f o r e the c o n t i n u i t y of t h e b i l i n e a r map (x’, y ’ ) -((Ef$F)i , t h e set B 8 { g } i s a bounded s u b s e t of m ( E f $ F ) G ( E f $ F / i , hence by h y p o t h e s i s e q u i c o n t i n u o u s t o o . Thus, a p p l y i n g now a s i m i l a r argument t o t h a t i n X I I ; ( l . l 9 ) , one c o n c l u d e s t h a t B i s a l s o an e q u i c o n t i n u o u s s u b s e t of ? ? Z ( E ) , which a c t u a l l y p r o v e s t h a t E i s a s p e c t r a l l y b a r r e l l e d a l g e b r a . Now an analogous argument can be a p p l i e d t o F ,

s o 2) * l )

as w e l l , and t h i s f i n i s h e s t h e proof o f t h e theorem. I Schol ium 3 . 1 . - By c o n s i d e r i n g complete t e n s o r p r o d u c t a l g e b r a s of t h e form E 6 F ( f o r a p p r o p r i a t e

E, F and

T )

,

T

one a c t u a l l y g e t s a s i m i l a r c o n c l u s i o n

t o Theorem 3.1; t h i s i s e a s i l y u n d e r s t o o d when app l y i n g t h e argument of t h e p r e v i o u s p r o o f , a t l e a s t , i n s o f a r a s a r e l a t i o n l i k e XII;l1.24) holds t r u e . Thus, one can c o n s i d e r , f o r example, spectrally barrelzed algebras w i t h l o c a l l y compact spectra ( h e n c e , e q u i v a l e n t -

l y , l o c a l l y e q u i c o n t i n u o u s ; see Theorem V ; 1.1 )

.

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

would b e equivalent w i t h the a s s e r t i o n the previous two conditions t o be v a l i d for t h e algebra E $ F ( c f . Lemma V ; 2.2 a s w e l l a s Lemma X I I ; 1.2). The p r e v i o u s a r g u -

ment should a l s o be compared w i t h t h e r e a s o n i n g i n Lemma V I I I ; 1.2.

4. SEMI-SIMPLICITY

441

4 . Semi-simplicity W e continue i n t h i s s e c t i o n t h e study of topological t e n s o r

p r o d u c t a l g e b r a s , a s it c o n c e r n s t h e p r e s e r v a t i o n a n d / o r i n h e r i t a n c e o f b e i n g t h e a l g e b r a s under d i s c u s s i o n semi-simple. Among o t h e r t h i n g s , t h i s a l s o p r e s e n t s a c e r t a i n p a r t i c u l a r i n t e r e s t i n i t s e l f connected w i t h t h e t y p e o f “ c o m p a t i b i l i t y ” of t h e t o p o l o g y t h a t one may cons i d e r on t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a , when t a k i n g i t s comp l e t i o n ( c f . Theorem 4.3 b e l o w ) . W e have c o n s i d e r e d a l r e a d y semi-simple t o p o l o g i c a l a l g e b r a s i n

C h a p t e r V I I I (see D e f i n i t i o n V I I I ; 3 . 2 ) h a v i n g t h e u n d e r l y i n g t o p o l o g i c a l v e c t o r s p a c e s n o t n e c e s s a r i l y l o c a l l y convex. However, i n o r d e r t o a v o i d n o t a t i o n a l c o m p l e x i t i e s , h a v i n g t o d o , mainly, w i t h t h e “comp a t i b i l i t y ” of t h e t e n s o r i a l t o p o l o g i e s w e a r e g o i n g t o c o n s i d e r , w e

r e s t r i c t o u r s e l v e s i n t h e s e q u e l , e x c l u s i v e l y , t o locally convex algebras ( w i t h o r w i t h o u t c o n t i n u o u s m u l t i p l i c a t i o n ) , and i n p a r t i c u l a r t o locally rn-convex o n e s ( e . g . s i l o v a l g e b r a s ) . So w e s t a r t w i t h t h e f o l l o w ing useful r e s u l t .

Lemna 4.1. Let E , F be two locally convex spaces whose topological duals are F’, respectively. Moreover, consider a compatible (locally convex) tensorial topology T on E 7 F ( D e f i n i t i o n X; 2.1). Then, f o r every x‘EE’(resp., y’EF’), there e x i s t s a continuous linear mzp

E‘,

n

ex, : E ? F - F : ( r e s p . , 8 , : E @ F+E Y

:z - 0

T

Y

E=

,(z)

I xiayi-eX.(z)

:= 5 y’(yi)xi)

such t h a t , concerning the correspond-

2

ing transpose m p s , one has the relation te , f y ’ ) = t 0 ,(x‘) = (4.2)

Y

X

for every y’ E p‘

:= i

i=l

6’8

y’

(resp. , x’ E E’).

Proof. W e prove

t h e c o n t i n u i t y o f ( 4 . 1 ) w i t h r e s p e c t t o t h e re-

l a t i v e t o p o l o g y induced on E @ F from t h e c a n o n i c a l i n j e c t i o n s (4.3)

E@F

5 &(E;,

Fs’) 5 Ss(E;, F )

(see C h a p t . X ; ( 2 . 2 4 ) and ( 2 . 4 1 ) ) ; t h e l a s t s p a c e i n ( 4 . 3 ) c a r r i e s t h e t o p o l o g y s of s i m p l e convergence. So t h e r e l a t i o n (4.4)

s < e = e

IB@F

=

E
( c f . X ; (2.28) and ( 2 . 3 5 ) ) would imply t h e n t h e c o n t i n u i t y o f t h e map with x’EE’, r e l a t i v e t o t h e g i v e n t o p o l o g y T. ( A n a n a l o g o u s a r -

ex.,

gument i s v a l i d , o f c o u r s e , f o r t h e map B y , ,

y’E F ’ ) :

442

XI11

P R O P E R T I E S OF PERMANENCE

Now, t h e c o n t i n u i t y of

i n t h e t o p o l o g y s i s e a s i l y checked

(4.1)

from t h e form of t h e c a n o n i c a l i n j e c t i o n , s a y , a : E B F 4 f s { E i , F): t h e l a t t e r map i s g i v e n by

f o r any z net

z6

7z

n

.I xi e y .

EE

B F and x ' E E ' .

Therefore, t h e convergence of a

2=1

i n E 8 F C Ls(Ei

i n F , f o r every x ' e E ' .

,F/

Thus,

i s e q u i v a l e n t w i t h [ a f z 6 11 ( x ' ) - + [ a ( z l l ( x ' ) &

one g e t s from ( 4 . 5 ) t h a t ex,(z )-+8 6

i n F , which p r o v e s t h e c o n t i n u i t y o f t h e map 0

X"

f x ' , y') E E ' x F ' ,

F u r t h e r m o r e , for a n y p a i r

&

,(z) X

x'eE'. one g e t s t h e f o l l o w -

i n g r e l a t i o n , c o n c e r n i n g t h e r e s p e c t i v e t r a n s p o s e map o f

(4.1):

n f o r e v e r y e l e m e n t z z . 1 x . @ y . E E 8 F . S o w e h a v e p r o v e d ( 4 . 2 ) , and t h i s 2

c o m p l e t e s t h e p r o o f of

2

t h e lemma. I

I f w e a r e g i v e n complete l o c a l l y convex spaces E , F ing r e l a t i o n s (4.1)

,t h e n

t h e preced-

a n d ( 4 . 2 ) c a n b e e x t e n d e d by c o n t i n u i t y t o t h e re-

s p e c t i v e complete ( l o c a l l y convex) t e n s o r p r o d u c t space E 6 F T

.

W e come n e x t t o t h e f i r s t main r e s u l t o f t h i s s e c t i o n . So one

has.

Theorem 4.1. Let E , F be l o c a l l y convex algebras and E F t h e r e s p e c t i v e l o c a l l y convex tensor product algebra of E , F i n a compatible topology

(cf. Defi-

n i t i o n X i 3 . 1 ) . Then, E B F i s semi-simple if, and only if, t h i s i s t h e case f o r T

each one of the given algebras E , F .

i s semi-simple,

Proof. Suppose t h a t E 8 F T

and l e t x E E w i t h

2=

0.

T h u s , f o r any y e F and k e m ( E 8 F l , one o b t a i n s T

A

( x e y J ( k l = h ( x e y I = (by Chapt. X I I ; (1.6), (1.7)) (4.7)

(f e g l ( x C 9 y ) = f l x l g l y ) = g ( y l 2 l f ) = 0 T h e r e f o r e , due t o t h e h y p o t h e s i s f o r y

E

F , which i m p l i e s t h a t x = 0 ( c f

semi-simple,

.

E B F , one g e t s T

Lemma X ; 1 . 1 )

. So

. xmpy=O f o r every the algebra E is

a n d a n a n a l o g o u s p r o o f c a n be g i v e n f o r F . T h i s e s t a b -

l i s h e s t h e "only i f " p a r t of t h e a s s e r t i o n . NOW, assume t h a t b o t h

t h e g i v e n a l g e b r a s E , F are s e m i - s i m p l e ,

and c o n s i d e r a n element z e E Q F

with z f 0 .

Thus, on t h e b a s i s o f t h e

f i r s t of t h e c a n o n i c a l i n j e c t i o n s ( 4 . 3 ) , one p r o v e s t h e e x i s t e n c e of an element ( x ' , y ' l e E ' x

F'

such t h a t

4. SEMI-SIMPLICITY

(4.8)

(a'ay'J(zi

443

,(z)) # 0. Y so by t h e s e m i - s i m p l i c i t y of E l t h e r e e x i s t s an

= (applying t h e n o t a t i o n of (4.6)) 2'18

Therefore,

8 ,(z)#O; Y element f e m ( E ) such t h a t

n

(4.9)

eY , ( z i ( f i

(4.10)

ef ( z i ( g i

.

= f i e , i z i ) = ( c f . (4.6)) y ' ( e ( z i i # u Y f A c c o r d i n g l y , 0 ( z ) # 0 so t h a t from t h e s e m i - s i m p l i c i t y of F , one g e t s f an element g e m ( F ) w i t h A

fag

where

= g ( e ( z i i = (by ( 4 . 6 ) ) ( f a g ) ( z ) = $ ( f a g )#

f

o,

.

h e n T ( E 8 F ) (Theorem X I I ; 1 . 1 ) T h e r e f o r e , 2 # 0 which p r o v e s T E 8 F (see D e f i n i t i o n V I I I ; 3 . 2 ) . I

t h e s e m i - s i m p l i c i t y of

Scholium 4.1

.- A s

T

f o l l o w s from t h e p r e v i o u s proof

, we

made u s e

t h e r e o f Lemma 4.1 o n l y " a l g e b r a i c a l l y " ; namely, no u s e was made of any c o n t i n u i t y of t h e maps ( 4 . 1 ) . tion E 8 F S d ( E ; , F i )

applied

Furthermore, t h e canonical i n j e c -

i n ( 4 . 8 ) i s d e r i v e d a l r e a d y from t h e

semi-simplicity of t h e a l g e b r a s E and F (assumed i n t h a t p a r t of t h e f o r any

p r o o f ) . T h e r e f o r e , one c o u l d g e t t h e p r e c e d i n g Theorem 4 . 1

topological a l g e b m s E , F a s w e l l and a compatible topology

T

, not

n e c e s s a r i l y l o c a l l y convex o n e s ,

on E 8 F ( c f . D e f i n i t i o n X ; 4 . 1 ) . O f c o u r s e t h e

s i t u a t i o n i s more c o m p l i c a t e d , i f one wants t o c o n s i d e r complete t e n s o r p r o d u c t a l g e b r a s o f t h e form E & F , a s w e s h a l l see i n t h e s e q u e l . T So t o f a c i l i t a t e arguments, w e r e s t r i c t o u r a t t e n t i o n t o l o c a l l y convex a l g e b r a s .

( I n t h i s r e s p e c t , c f . a l s o t h e comment a f t e r t h e end o f

t h e proof o f t h e above Lemma 4 . 1 ) . Before w e proceed f u r t h e r , w e comment a b i t more on t h e t e r m i nology w e a r e going t o apply i n t h e s e q u e l . Thus, suppose t h a t w e a r e g i v e n two l o c a l l y convex a l g e b r a s E l F i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n i s f u l f i l l e d : EGF T

, i.e. ,

t h e completion of

E 8 F with respect T

t o a c o m p a t i b l e t e n s o r i a l topology

(4.11)

T

on E @ F

(see D e f i n i t i o n X; 3 . 1 ) i s a ( c o m p l e t e ) locaZly convex algebra. One e n c o u n t e r s t h e s i t u a t i o n d e s c r i b e d by t h e p r e v i o u s c o n d i t i o n i n c a s e , o f c o u r s e , t h e g i v e n l o c a l l y convex a l g e b r a s E , F have

(4.11)

( j o i n t l y ) continuous m u l t i p l i c a t i o n s . On t h e o t h e r hand, one may s t i l l look a t t h e s i t u a t i o n p r o v i d e d by Lemma X I ; 1 . 1 i n c o n n e c t i o n w i t h Theorem XI; 1 . 1 . NOW,

i f t h e u n d e r l y i n g l o c a l l y convex s p a c e s E and F

(of t h e

XI11 PROPERTIES OF PERMANENCE

444

above a l g e b r a s ) a r e compZete, t h e r e s p e c t i v e l o c a l l y convex space F i ) i s complete t o o ( c f . X ; ( 2 . 4 1 ) ) .

&e(E;,

So i f one l o o k s a t t h e ( c a -

n o n i c a l ) continuous Zinear i n j e c t i o n (4.12)

p r o v i d e d by X ; ( 2 . 2 4 ) and t h e r e l a t i o n t i o n X; 3 . 1

, then

E<

"extending it by continuity"

, w i t h T g i v e n by D e f i n i , one g e t s a ( c a n o n i c a l ) con-

T

tinuous l i n e a r map (4.13) NOW,

i f w e a s k f o r t h e e x t e n t t o which t h e k t t e r map i s aZso one-to-

one, w e a r e t h u s l e d t o a well-known problem posed by A . Grothendieck ("ProbZdme de b i u n i v o c i t d " ; see A . GROTHENDIECK [3: Chap. I ; p . 35 f f .] ) , a s it c o n c e r n s t h e r e s p e c t i v e complete l o c a l l y convex s p a c e s o c c u r r e d i n ( 4 . 1 3 ) . On t h e o t h e r hand, a s w e s h a l l p r e s e n t l y see, t h e same probl e n h a s an i n t i m a t e r e l a t i o n ( i t i s , i n e f f e c t , e q u i v a l e n t , see Theo-

r e m 4 . 3 below) w i t h t h e q u e s t i o n of b e i n g t h e a l g e b r a E 6 F semi-simple T (cf. also ( 4 . 1 1 ) ) . Now, l o o k i n g a t t h e s i t u a t i o n which one h a s i f l o c a l l y convex s p a c e s are i n v o l v e d , w e conclude t h a t f o r e v e r y nuclear ( l o c a l l y convex) algebra E ( c f . D e f i n i t i o n V I I I ; 9 . 2 ) t h e map ( 4 . 1 3 ) i s i n j e c t i v e ; h e r e one c o n s i d e r s t h e c o m p l e t i o n of E and t h e c o r r e s p o n d i n g t o p o l o g i c a l t e n s o r p r o d u c t w i t h any complete l o c a l l y convex a l g e b r a F i n t h e topo-

logy

F (Lemma X i 2 . 1 ) . ( W e s t i l l

(2.41 )

,

(2.42) )

. Moreover,

assume h e r e ( 4 . 7 7 ) ; see a l s o Chapt. X ;

one comes t o t h e same c o n c l u s i o n f o r a cer-

t a i n p a r t i c u l a r c l a s s of l o c a l l y convex a l g e b r a s , namely of t h o s e s a t i s f y i n g t h e approximation property (cf. e . g . A . GROTHENDIECK [3:Chap. I ; p . 169, Lemme 1 9 1 ) ; y e t f o r e v e r y Banach algebra having t h e same property ("Ba-

.

nach approximation p r o p e r t y " : I b i d . ; p . 168, C o r o l l a i r e 3 ) Of c o u r s e e v e r y n u c l e a r ( l o c a l l y convex) a l g e b r a h a s t h e approximation p r o p e r t y (see e . g . F . T R E V E S [ I : p. 520, P r o p o s i t i o n 5 0 . 3 1 ) . T

On t h e o t h e r hand, l o c a l l y convex t e n s o r i a l ( a l g e b r a ) t o p o l o g i e s f o r which t h e p r e c e d i n g map ( 4 . 1 3 ) i s one-to-one a r e now c l a s s i f i e d

by t h e f o l l o w i n g . D e f i n i t i o n 4.1.

Suppose t h a t w e a r e g i v e n two complete l o c a l l y

i s a ( c o m p l e t e ) l o c a l l y convex a l Then, w e s h a l l say t h a t T i s a f a i t h f u Z topology on E Q F , whenever t h e c o r r e s p o n d i n g ( c a convex a l g e b r a s E l F such t h a t E 6 F T

g e b r a , t h e topology

T

s a t i s f y i n g Definition X; 3 . 1 .

n o n i c a l c o n t i n u o u s l i n e a r ) map ( 4 . 1 3 )

i s one-to-one.

W e come now t o t h e f o l l o w i n g r e s u l t , an analogon i n c a s e o f com-

4. SEMI-SIMPLICITY

p l e t e a l g e b r a s of o u r p r e v i o u s Theorem 4 . 1 .

Theorem 4.2.

445

Namely, w e have.

Let E , F be two complete l o c a l l y convex algebras w i t h l o c a l l y

equicontinuous spectra m(E), ? ? Z ( F ) , r e s p e c t i v e l y , such t h a t E $ F t o be a ( c o v p l e t e ) l o c a l l y convex a l g e b m w i t h respect t o a compatible ( l o c a l l y conved tensorz h Z topology

that

T

T

on E 8 F ( D e f i n i t i o n X;3.1:

i s a f a i t h f u l topology on E 8 F ( cf

X ; (2.11, (2.3)). Moreover, suppose

. Definition

4 . 1 ) . Then, the fo22ow-

i n g two a s s e r t i o n s are e q u i v a l e n t : h

1 ) E 8 F i s semi-simple. T

21 Each one of t h e given algebras E , F are semi-simple.

proof. Suppose t h a t 1 ) i s v a l i d , and l e t x e E w i t h

?=a.

Thus,

f o r e v e r y p a i r (y, ~ 2 ) B F x M ( E % F ) ,one g e t s a r e l a t i o n a n a l o g o u s t o T

( 4 . 7 ) with

h = f e g and ( f , g ) e ? R ( E ) x Z T ( F ) , p r o v i d e d by Theorem XII;1.2.

This proves t h e semi-simplicity of t h e algebra E , while a s i m i l a r a r gument c a n be a p p l i e d t o F , which p r o v e s t h a t 1 ) * 2 ) .

Remark 4.1. - It becomes c l e a r from t h e above f i r s t p a r t of t h e proof of Theorem 4 . 2 t h a t t h e applied argument i s i n f o r c e even i n t h e context of Theorem XII; 1 . 2 . Thus, no u s e h a s been made of t h e p r e v i o u s D e f i n i t i o n 4.1 ( i .e , , of t h e " f a i t h f u l n e s s " of t h e topology T ) . End of t h e proof of 1"heorem 4.2. A s s u m e now t h a t t h e a l g e b r a s E , F a r e semi-simple,

and l e t z e E 6 F with z # 0 . Then, from t h e h y p o t h e s i s f o r

7

T

( D e f i n i t i o n 4 . 1 ) t h e ( c a n o n i c a l ) image o f z i n & ( E i , F ) i s non-zero t o o ; hence, t h e r e e x i s t s a p a i r ( x ' , y ' ) € E ' x (4.14)

[j^(Z)l(S',

y 7 = ( x ' e y')(z)

( c f . ( 4 . 1 3 ) ) . Thus, on t h e b a s i s of

F'

such t h a t

+0

(4.141, t h e argument now p r o c e e d s

i n a s i m i l a r way t o t h a t a p p l i e d t o t h e r e l a t i o n s ( 4 . 8 ) - ( 4 . 1 0 ) d e r t o f i n d an element

g)e

($,

m(E) x m(F),and

i n or-

hence (Theorem XII;l.2)

an element h = f o g B ~ x ( E $ F ) h(z)

(4.15)

E

(f o g ) ( z J =

z^(f o g ) #

0 .

Now, t h i s p r o v e s t h e s e m i - s i m p l i c i t y of t h e a l g e b r a

E6F 7

, so

2)

+.

1)

a s w e l l , and t h i s c o m p l e t e s t h e proof o f t h e theorem. I As a c l a r i f i c a t i o n ,

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

p r e c e d i n g m o t i v a t e d by t h e t y p e of t o p o l o g i c a l a l g e b r a s appeared i n t h e comment b e f o r e D e f i n i t i o n 4 . 1 .

But f i r s t w e have t o e s t a b l i s h a

b i t more t e r m i n o l o g y . So w e have t h e n e x t .

Definition 4.2.

W e say t h a t a g i v e n l o c a l l y m-convex a l g e b r a

E

446

XITI P R O P E R T I E S OF PERMANENCE

h a s t h e approximation property, whenever t h e r e e x i s t s an Arens-Michael decomposition of E , s a y , ( E a , f aB ) ClCI ( c f . 111; ( 3 . 9 ) ) , i n such a way ka , C I E It ,o have t h e (Banach) ap-

t h a t t h e r e s p e c t i v e Banach a l g e b r a s proximation p r o p e r t y .

For t h e l a s t p r o p e r t y see e . g . P r o p o s i t i o n 35, ( A 5 )

1.

A . GROTHENDIECK 13: Chap.

I ; p - 164,

Now f o l l o w i n g t h e same a u t h o r ( i b i d . ; p . 1 6 9 ,

Lemme 1 9 ) , one c o n c l u d e s t h a t i f a l o c a l l y m-convex a l g e b r a E h a s t h e

approximation p r o p e r t y , t h e n the respective l o c a l l y convex space E has t h e

same property ( s e e D e f i n i t i o n X ; 2 . 4 above) ; moreover, f o r every l o c a l l y A

h

convex space F , one g e t s f o r t h e r e s p e c t i v e c o m p l e t i o n s P , F t h a t t h e canonical continuous l i n e a r m p

,$88

(4.16)

4 Se

(E;

,F i )

i s i n j e c t i v e . Thus, g i v e n t h e a l g e b r a E a s above, and f o r any l o c a l l y m-convex a l g e b r a F , t h e p r o j e c t i v e t e n s o r i a l t o p o l o g y

TI

i s always a

f a i t h f u l topology ( D e f i n i t i o n 4 . 1 ) . The promised c l a r i f i c a t i o n i s now a d i r e c t consequence of t h e p r e v i o u s d i s c u s s i o n . So w e have t h e n e x t .

Corollary 4.1. Let E , F be l o c a l l y m-convex algebras such t h a t one of them has t h e approximation property ( D e f i n i t i o n 4 . 2 ) , while t h e i r completions are Q-aZgebras. Then, t h e (complete) l o c a l l y m-convex algebra

28:

(E

%, is

semi-simple i f , and only i f , t h i s i s t h e case f o r each one of t h e complete algebras and

2.

I

W e come n e x t t o t h e c o n n e c t i o n of t h e " f a i t h f u l n e s s " of a g i v e n

c o m p a t i b l e t e n s o r i a l t o p o l o g y T on E ~ (FD e f i n i t i o n 4 . 1 ) witk, t h e semiT s i m p l i c i t y of t h e a l g e b r a E 6 F . I n d e e d , assuming f u r t h e r t h a t t h e alT

g e b r a s E and F a r e semi-simple,

t h e p r e c e d i n g two c o n d i t i o n s a r e , i n

f a c t , e q u i v a l e n t . So one h a s t h e f o l l o w i n g .

Theorem 4.3. Suppose t h a t a l l t h e conditions, except of t h e f a i t h f u l n e s s of t h e topology T , i n Theorem 4 . 2 are s a t i s f i e d . Then, the following two a s s e r t i o n s are equivalent: A

1 ) t h e algebra

E8F

i s semi-simple.

T 2 ) The given compatible ( l o c a l l y convex) t e n s o r i a l topology T on E 8 F i s

f a i t h f u l ( D e f i n i t i o n 4.1) and each one of t h e algebras E , F i s semi-simpZ.e:. Proof. Suppose t h a t t h e a l g e b r a E6F T

with z + U ,

m(E6FI T

hence

T

2 # 0.

such t h a t z^(h)# a ,

with h = f m g

,.

E B F i s semi-simple,

and l e t Z E

T h e r e f o r e , t h e r e e x i s t s an element h e and hence a p a i r ( f , g ) B m l E l X M ( F ) C E i X F i

(Theorem X I I ; 1 . 2 ) such t h a t one h a s

5.

(4.17)

447

IDENTITY ELEMENTS

264) = 2 ( f a g l = ( f a g ) ( z l # O .

T h i s p r o v e s t h a t ( 4 . 1 3 ) i s one-to-one

so t h a t t h e t o p o l o g y T i s , by

d e f i n i t i o n , f a i t h f u l . F u r t h e r m o r e , Theorem 4 . 2 t o g e t h e r w i t h Remark 4 . 1 g u a r a n t e e now t h a t t h e a l g e b r a s

E , F a r e semi-simple,

so t h a t

1) *2).

On t h e o t h e r hand, i f 2 ) i s v a l i d , t h e n a l l t h e c o n d i t i o n s of t h e p r e v i o u s Theorem 4 . 2 a r e s a t i s f i e d , hence by t h e same theorem t h e algebra

E6F T

is s e m i - s i m p l e ,

so 2 ) = > I ) a s w e l l , and t h i s com-

p l e t e s t h e proof. I An

analogous conclusion with Corollary 4.1,

within the context

o f t h e p r e c e d i n g Theorem 4 . 1 , i s now q u i t e c l e a r , so w e o m i t t h e d e t a i l s . ( T h u s one could o b t a i n a m o r e c o n c r e t e p i c t u r e t h a n t h e previous one). A s a n a p p l i c a t i o n of t h e f o r e g o i n g , w e f u r t h e r examine i n t h e

s u b s e q u e n t s e c t i o n s h e r e d i t a r y p r o p e r t i e s which a t o p o l o g i c a l t e n s o r product algebra

E4F T

may h a v e ( i m p l i e d from a n a l o g o u s p r o p e r t i e s of

t h e f a c t o r s ) , d e p e n d i n g i n p a r t i c u l a r on t h e s e m i - s i m p l i c i t y of t h e f a c t o r a l g e b r a s E,F. 5 . I d e n t i t y elements I t i s c l e a r of c o u r s e t h a t i n c a s e of two u n i t a l a l g e b r a s E , F

whose i d e n t i t y e l e m e n t s a r e l E , I F , r e s p e c t i v e l y , t h e e l e m e n t l E a I F i s a n i d e n t i t y o f t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a E@F ( c f . X ; ( 1 . 1 2 ) ) . Thus, t h e n o n - t r i v i a l

p a r t o f t h e n e x t theorem i s e x a c t l y

a converse conclusion i n t h e presence of semi-simplicity of t h e algeb r a s i n v o l v e d . But f i r s t w e comment on a n a u x i l i a r y u s e f u l r e s u l t ( i n f a c t , a " m u l t i p l i c a t i v e analogon" o f Lemma 4 . 1 ) , a s i s t h e n e x t . Lemma 5.1. L e t E , F be l o c a l l y convex aLgebras w i t h continuous m u l t i l i c a t i o n s and spectra ???(El, T R ( F ) , r e s p e c t i v e l y . Moreover, Let F @ F be t h e corresponding comT pZete l o c a l l y convex tensor product aZgebra o f E , F i n a compatible topoLogy T h

( D e f i n i t i o n X;3. 1 ) . Then, f o r every f e m(E) ( r e s p . , g € m(F)),the map (5.1) ( z ) = E g ( y .)x.) d e f i n e s a continuous (algebra) morE :2-8 i r e s p . , 6 :E@F+ 9 T 9 7 2 % phism of the topological algebras invoZved. Thus, one has an "extension by continu-

i t y " of ( 5 . 1 ) t o a map

448

XI11 PROPERTIES OF PERMANENCE

hence 5 E Horn ( E F ,2) Iresp., E Horn ( E 6 F , f 9 T spective transpose m p s of (5.11, one g e t s

te

(5.3) for every (f, g ) 6

m(E) X

Furthermore, concerning t h e re-

te cf)

= fege ~ ( E ~ F I 9 m(F) (and a l s o t h e analogous one t o t h e m p s (5.2)).

f

(gi =

E") I.

Proof. I n view o f Lemma 4.1

,

t h e rels. (5.1) define continuous

l i n e a r maps o f t h e r e s p e c t i v e l o c a l l y convex s p a c e s , so t h e y can b e extended by c o n t i n u i t y t o t h e c o r r e s p o n d i n g maps ( 5 . 2 ) . T h e r e f o r e , we have o n l y t o show t h a t (5.1) a r e , i n f a c t , m u l t i p l i c a t i v e ( l i n e a r ) maps, due t o o u r h y p o t h e s i s f o r f, g , and hence t h e same would be t r u e f o r t h e extended maps (5.21, because o f t h e c o n t i n u i t y of t h e m u l t i plication in E , F: Thus, f o r e v e r y f e ? ? ? ( E )

and any s , t i n E 8 F g i v e n by X;(l.ll),

one g e t s from (5.1) and X ; (1.12)

(5.4)

t h a t w a s t o b e proved. On t h e o t h e r hand, one g e t s (5.3) on t h e b a s i s of (4.2) and t h e f a c t t h a t f @ g E m ( E @ F ) ( = ( w i t h i n a bijection) m ( E $ F ) ) , f o r e v e r y (f, g ) T

e ~ T ( E ) x ~ ? ? ( F ) ( c f . X; (2.2) and XII;(l.lO) i n t h e proof of Lemma XII; 1.1). I Remark 5.1. - Concerning t h e p r e c e d i n g proof , w e n o t e t h a t t h e t o p o l o g i c a l p a r t o f t h e proof o f Lemma 4.1, and hence t h e local convexity 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 s i n v o l v e d , h a s been a p p l i e d o n l y i n c o n n e c t i o n w i t h t h e c o n t i n u i t y of t h e maps (5.1). T h e r e f o r e , i f t h e l a t t e r i s i r r e l e v a n t , i n some p a r t i c u l a r c o n t e x t , one could o b t a i n t h e previous Lemma 5.1 f o r any t o p o l o g i c a l a l g e b r a s ( n o t necess a r i l y l o c a l l y convex o n e s ) w i t h c o n t i n u o u s m u l t i p l i c a t i o n and any t e n s o r i a l t o p o l o g y T on E 8 F "comp a t i b l e i n t h e s e n s e o f D e f i n i t i o n X;4.1. Now t h e c o n t i n u i t y of (5.1) i s i n d i s p e n s a b l e , of c o u r s e , t o have ( 5 . 2 ) ; t h i s i s t h e case i n t h e n e x t theorem. So w e r e s t r i c t o u r s e l v e s t o l o c a l l y convex a l g e b r a s ; ( a l s o t o t h e e x t e n t t h a t w e do n o t assume a g i v e n i d e n t i t y element o f an a l g e b r a t o be a l s o s u c h i n a l l i t s s u b a l g e b r a s ! In t h i s r e s p e c t , see a l s o C o r o l l a r y 5.1 b e l o w ) . W e come n e x t t o o u r main r e s u l t o f t h i s s e c t i o n . So w e have.

Theorem 5.1.

Let E , F be complete semi-simple l o c a l l y convex algebras w i t h

5.

IDENTITY ELEMENTS

449

continuous m l t i p l i c a t i o n s and spectra %Z ( E l , ? ? Z ( F ) , r e s p e c t i v e l y . Moreover, l e t A

E B F be t h e corresponding complete loca2ly conuex tensor producb algebra o f E , F T

having a continuous rnultipZication i n a compatible t e n s o r i a l topology T (Def i n i t i o n Xi3.1). Then, t h e algebra E @ F has an i d e n t i t y element i-f,and only i f , t h i s 7

i s the case for each one of t h e given aZgebras E , F . Proof. A s w e remarked a t t h e b e g i n n i n g of t h i s s e c t i o n , i t suf-

f i c e s t o p r o v e t h e " o n l y i f " p a r t o f t h e a s s e r t i o n : So l e t e be a n i d e n t i t y e l e m e n t o f t h e a l g e b r a B g F : t h u s a p p l y i n g Lemma 5.1 one T

-

g e t s , f o r e v e r y g e ? ? Z ( F ) , an element e ( e l E E . NOW we p r o v e t h a t G ( e ) 9 9 i s an i d e n t i t y e2ement o f t h e algebra E. Indeed, f o r any x e B and f e m(E) , one g e t s

f(5

(5.5)

9

(e)x) =

f(eY ( e ) ) . f ( x l

= ( b y (5.3); see a l s o (4.10))

I f @ g ) ( e l . f ( x l = (by ( 5 . 3 ) ) h ( e ) . f ( x l = f ( x )

,

s o t h a t one h a s f(B (ejx-x) =

f o r every f

e9 ( F I X =

E

o ,

9 m ( E ) . H e n c e , by t h e s e m i - s i m p l i c i t y of E l one o b t a i n s

x f o r e v e r y x e E , which p r o v e s t h e a s s e r t i o n . A s i m i l a r a r g u -

ment i s v a l i d f o r F , and t h i s c o m p l e t e s t h e p r o o f . I

Remark 5.2.- A s it becomes clear from t h e _ p r e c e d i n g proof , c o n c e r n i n g t h e element e g ( O e E = E , one a c t u a l l y a p p l i e s t h e " e x t e n s i o n by c o n t i n u i t y " o f t h e a l r e a d y c o n t i n u o u s l i n e a r map B g : E Y F + F ( c f . Lemma 4 . 1 ) . A c c o r d i n g l y , t h e proof i s v a l i d f o r any completeh ZocalZy convex ( s e m i - s i m p l e ) algebras E , F for which E 8 F i s a ( c o m p l e t e ) l o c a l l y convex algebra i n a c o m p a h b l e ( l o c a l l y convex) t e n s o r i a l t o pology T on E B F . T h i s happens o f c o u r s e w i t h l o c a l l y convex a l g e b r a s h a v i n g c o n t i n u o u s m u l t i p l i c a t i o n s , a s w e assumed i n t h e p r e c e d i n g . (However , c f . a l s o Theorem XI; 1 . 1

.

A s an a p p l i c a t i o n of t h e p r e c e d i n g d i s c u s s i o n , w e s t i l l have

t h e following c l a r i f y i n g r e s u l t .

Corollary 5.1. L e t E , F be Fre'chet semi-simple l o c a l l y convex algebras and h

h

l e t E 8 F I = E 8 F ) be t h e corresponding Frdchet l o c a l l y convex aZgebra, complete TI

p r o j e c t i v e tensor product algebra of E , F (see Chapt

. X; Lemmas

2.1 and 3.1).

Then, t h e algebra E & F has an i d e n t i t y element i f , and only i f , t h i s i s t h e case f o r each one of t h e algebras E , F . I Schol ium 5.1.

-

The p r e v i o u s C o r o l l a r y 5.1 i s

o f c o u r s e a d i r e c t a p p l i c a t i o n of Theorem 5.1. Howe v e r , one c o u l d o b t a i n a more c o n c r e t e form of t h e

450

XI11 PROPERTIES OF PERMANENCE

r e s p e c t i v e i d e n t i t y e l e m e n t s by u s i n g a c l a s s i c a l r e a s o n i n g , p r o v i d e d by t h e p a r t i c u l a r h y p o t h e s i s

w e made: Thus, s i n c e t h e a l g e b r a

lar, a

Fre'chet l o c a l l y

E 6 F is, i n particu-

convex space, one may a p p l y

h e r e a s t a n d a r d r e s u l t due t o A . GROTHENDIECK [3: Chap. I ; p. 51, Th6orZme

I].

So i f e i s a n i d e n t i t y e l e -

ment o f t h e a l g e b r a E 6 F

,

t h e n e admits an expression

of t h e form m

( s e e t h u s t h e above Remark 5.1 )

.

Here

(2:

I , (y,)

are

n u l l sequences i n E , F , r e s p e c t i v e l y , and ( A n )

a

m

2 lA,l
s e q u e n c e o f complex numbers w i t h

l o c a l l y convex s p a c e E G F ( i b i d . , a n d / o r F . TREVES

[I: p. 459, Theorem 45.11). T h e r e f o r e , f o r e v e r y g E ( F ) , one g e t s from ( 5 . 2 ) a w e l l - d e f i n e d e l e m e n t

of E g i v e n by t h e r e l a t i o n (2)

( s e e a l s o F . T R E W S [I: p. 459, D e f i n i t i o n 45.1 and t h e s u b s e q u e n t comment] ) i n g of

. Now

applying t h e reason-

( 5 . 5 ) , o n e p r o v e s f u r t h e r on t h a t t h i s i s ,

i n e f f e c t , an i d e n t i t y element of t h e algebra E .

6. Regularity. s i l o v algebras W e c o n s i d e r n e x t " h e r e d i t a r y " p r o p e r t i e s of t o p o l o g i c a l t e n s o r

p r o d u c t s r e f e r r i n g t o regular topological algebras tion 2.1).

(see C h a p t . I X ; Def i n i -

So w e f i r s t h a v e t h e f o l l o w i n g p r e l i m i n a r y r e s u l t .

Lemma 6.1. Let E , F be topological algebras w i t h spectra m(E),? ? Z ( F ) , r e s p e c t i v e l y , and l e t E 8 F be t h e r e s p e c t i v e topological t e n s o r product algebra o f E , ? F i n a compatible topology T. Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) Each one o f t h e algebras E , F i s regular.

2) The algebra E 8 F i s regular. proof.

Suppose t h a t 1 ) i s v a l i d , and l e t

s u b s e t t o g e t h e r w i t h an e l e m e n t h

E

B C m I E 8 F ) be a c l o s e d

??Y(EYF) with h

T

f?

B . Thus, C B i s an

o p e n n e i g h b o r h o o d of h i n m ( E O F ) ; so l o o k i n g a t t h e r e s p e c t i v e " c a ?

6.

REGULARITY.

SILOV ALGEBRAS

451

m(F) w i t h

(f,g) e m(E)x

n o n i c a l d e c o m p o s i t i o n ” o f h by an e l e m e n t

= f @ g (Lemma X I I ; 1 . 1 ) , one g e t s open s e t s U

h

m ( E ) and V .G m(F/ w i t h

C

(f,g ) e U X V 5 C B

(6.1)

(Theorem X I I ; 1 . 1 ) ; so

f

# CU and

(see D e f i n i t i o n I X ; 2 . 1 1 , (6.2)

=I,

;(f)

g # CV. Hence by h y p o t h e s i s f o r E , F

t h e r e e x i s t e l e m e n t s X E E and Y E F

2

Accordingly, looking a t t h e element

= (by X I I ; ( l . 7 ) )

,

x a y eEQF

Icv.

=O

one h a s

f l x l g l y l = P(fly^(g)= 1 .

zay = 0

from ( 6 . 2 )

one g e t s

B=C(UxV), A

t h a t is the algebra EQF

,

y^

= ( x \ y l ( f o g ) =(fag)(xcy)

(XG&)(hl

Moreover, s i n c e by ( 6 . 1 )

y^(g) = I ,

and

=0lcU

such t h a t

IB

,

i s r e g u l a r , so t h a t 1 ) + 2 ) .

On t h e o t h e r h a n d , assuming t h a t 2 ) i s t r u e , l e t A C V ? ( E ) c l o s e d a n d a l s o f e m(E) w i t h f # A .

Thus, c o n c e r n i n g t h e

be

cZosed s e t

A x { g } G??Z(E) x m ( F ) = (Theorem X I I ; 1 . 1 ) m f E T F 1 , one h a s ( f , g ) # A x { g ) . S o h ~f Q g E c (A@{ g } ) E 5’2 ( E 8 F ) , a n d h e n c e t h e r e e x i s t s , by h y p o t h e s i s , an element

z EE8F

T such t h a t

z^(h) = z^(f @ g )= 1 and

(6.3)

z^ = O

IA@{g).

T h e r e f o r e , f r o m ( 6 . 3 ) and a p p l y i n g t h e n o t a t i o n of (4.1011,

one o b t a i n s

= $(fog) = ifagi(zi = fre

(6.4) with z s 0

g

map 0

9

(5.l)(seealso

,g E

E E

(2)

g

(2))

=

A

eg ( z i ( f )

= 1

( c f . (5.1) ; t h e c o n t i n u i t y of t h e r e s p e c t i v e l i n e a r

m(F) i s h e r e i r r e l e v a n t )

previous r e l a t i o n s (6.3)

,

. Moreover,

from t h e s e c o n d o f t h e

o n e now o b t a i n s

(6.5) which t o g e t h e r w i t h ( 6 . 4 ) p r o v e s t h e r e g u l a r i t y of t h e a l g e b r a E . A

s i m i l a r argument c a n a l s o b e a p p l i e d t o F , a n d t h i s c o m p l e t e s t h e p r o o f o f t h e lemma. I Now s p e c i a l i z i n g t o s u i t a b l e l o c a l l y convex a l g e b r a s , namely, t o t h o s e t h a t Theorem X I I ; 1 . 2 m i g h t be a p p l i e d , w e o b t a i n a n a n a l o gon of t h e p r e v i o u s Lemma 6 . 1 f o r complete algebras. I n t h i s r e s p e c t , o n e a p p l i e s Lemma 4.1 t o t h e a n a l o g o u s r e l a t i o n o f s i d e r i n g t h e continuous extension l i n e a r map 8

g

8 ,gem‘(F),

( 6 . 4 ) , when con-

of t h e c o n t i n u o u s (now)

9 d e f i n e d by ( 4 . l ) ( s e e a l s o ( 5 . 2 ) ) . So w e h a v e t h e n e x t .

452

XI11 PROPERTIES OF PERMANENCE

Theorem 6.1. Let E , F be complete l o c a l l y convex algebras with l o c a l l y equi-

continuous spectra

m(E>, m l F l , r e s p e c t i v e l y . Furthermore, l e t E G F be t h e comT

p l e t i o n of t h e respective ( l o c a l l y convex) tensor product algebra of E , F so a s t o be a (complete) locally convex algebra ( e . g . t h e a l g e b r a s E , F m i g h t h a v e continuous multiplications)

.

Then, the algebra E 8 F i s regular i f , and only i f , t h i s i s the case f o r each one of t h e given algebras E , F. I We come now t o o u r s e c o n d s u b j e c t i n t h i s s e c t i o n , namely, t h e a n a l o g o u s s t u d y a s b e f o r e c o n c e r n i n g S i l o v algebras; i.e. Definition 2.3)

,

(see Chapt. I X ;

commutative c o m p l e t e regular semi-simple l o c a l l y m-convex

algebras. Thus, s i n c e s e m i - s i m p l i c i t y a s w e l l a s c o m p l e t e n e s s o f t h e t o p o l o g i c a l a l g e b r a s under d i s c u s s i o n e n t e r t h e s t a g e a l r e a d y a b i n i -

t i o , " f a i t h f u l n e s s " o f t h e t e n s o r i a l t o p o l o g i e s t h a t m i g h t b e cons i d e r e d i s of p a r t i c u l a r i m p o r t a n c e , when d e a l i n g w i t h q u e s t i o n s l i k e t h o s e i n t h e p r e c e d i n g (see t h e p r e v i o u s Theorem 4 . 3 ,

for instance).

I n p a r t i c u l a r , by c o n s i d e r i n g &lov algebras which f u r t h e r pos-

sess t h e approximation property

i n t h e s e n s e o f D e f i n i t i o n 4.2,

one g e t s

as a n o t h e r a p p l i c a t i o n of t h e f o r e g o i n g t h e n e x t r e s u l t .

Theorem 6.2. Let E , F be c o m u t a t i v e complete l o c a l l y m-convex algebras with

l o c a l l y equicontinuous spectra

m(E1 and ??t(F),

r e s p e c t i v e l y . Furthermore, Zet

E 6 F be t h e (commutative complete l o c a l l y m-eonvzz d g e b r a ) completion of the tenT

sor product algebra of E , F w i t h respect t o a compatible ( l o c a l l y m-eonvexl tensor1211 topology T ( D e f i n i t i o n X ; 3.1). Then, the following two a s s e r t i o n s are equivalent: 1 ) The topoZogy

T

on E O F i s f a i t h f u l ( D e f i n i t i o n 4.1) and each one of E ,

F i s a S i l o v algebra.

2 1 E 6 F i s a S i l o v algebra. T

In p a r t i c u l a r , t h e previous a s s e r t i o n 1 ) is t r u e , i f e i t h e r one of t h e s i l o v

algebras E , F has t h e approximation property ( D e f i n i t i o n 4.2) and one takes

n ( P r o p o s i t i o n X ; 3.1 Proof.

x; 3 . 1

T=

).

F i r s t , w e remark t h a t by h y p o t h e s i s f o r T

a n d t h e comment a f t e r i t ), E & F T

(see D e f i n i t i o n

i s a c o m m u t a t i v e c o m p l e t e 10-

c a l l y m-convex a l g e b r a , h e n c e o n e may a p p l y , d u e t o h y p o t h e s i s , Theor e m XII; 1 . 2 . Thus, t h e a s s e r t i o n t h a t 1 ) - 2 ) i s now a d i r e c t c o n s e q u e n c e of t h e above Theorems 4 . 3 and 6 . 1 . F u r t h e r m o r e , t h e l a s t p a r t o f t h e t h e o r e m i s o b t a i n e d from t h e corrment f o l l o w i n g D e f i n i t i o n 4 . 2 , t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . I

and

7.

WIENER-TAUBER C O N D I T I O N

453

S i l o v a l g e b r a s i n c o n n e c t i o n w i t h t h e p r e v i o u s Theorem 6.2 have

a p a r t i c u l a r s i g n i f i c a n c e i n a p p l i c a t i o n s of t o p o l o g i c a l a l g e b r a s t h e o r y i n some o t h e r c o n t e x t t o o . ( CohomoZogy of topoZogica2 aZgebras; see e . g . A . MALLIOS [21]

a n d a l s o S . A . SELESNICK [I] ) .

W e conclude t h i s s e c t i o n w i t h s t i l l remarking t h a t t h e condition

s e t f o r t h i n t h e p r e v i o u s Theorem 6 . 2 , p r o p e r t y " f o r o n e of t h e a l g e b r a s E , F

concerning t h e "approximation c o n s i d e r e d t h e r e i s , of c o u r s e ,

not t h e " l e a s t c o n d i t i o n " t h a t w e m i g h t r e q u i r e . Thus, t h e t h e o r e m works of c o u r s e e q u a l l y w e l l i f o n e c o n s i d e r s , i n s t e a d o f t h e a f o r e mentioned p r o p e r t y ( D e f i n i t i o n 4 . 2 )

,

j u s t a LocalZy m-convex algebra for

which t h e underZying localZy convex space s a t i s f i e s t h e approximation property ( c f . D e f i n i t i o n X ; 2 . 4 ) . (This i s s t i l l i n f o r c e concerning a l s o our rele v a n t c o n s i d e r a t i o n s i n S e c t i o n 4 . However, it was n a t u r a l t o s e t D e f i n i t i o n 4 . 2 i n t h e c o n t e x t of t h e t h e o r y of l o c a l l y m-convex a l g e b r a s , d u e a l s o t o t h e r e s u l t of A . G r o t h e n d i e c k q u o t e d t h e r e . A s a matter of f a c t , t h e c r u c i a l p o i n t h e r e i s a g a i n t h e map ( 4 . 1 3 ) t o be i n j e c t i v e ;

see e . g . A . WILLIUS [ 2 ] ) .

7. Wiener-Tauber condition T o p o l o g i c a l a l g e b r a s s a t i s f y i n g t h e c o n d i t i o n i n t i t l e of t h i s s e c t i o n have b e e n t e r m e d a l r e a d y a s

Wiener-Tauber aZgebras ( c f . C h a p t .

I X ; D e f i n i t i o n 6 . 2 ) . T h u s , w e n e x t examine how f a r t h e s a i d c o n d i t i o n

i s p r e s e r v e d when t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s of s u c h a l g e b r a s . I n f a c t , w e p r e s e n t l y show t h a t t h i s p r o p e r t y t o o i s a l s o "tensor product preserving" u n d e r s u i t a b l e c o n d i t i o n s on t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d and on t h e r e s p e c t i v e t e n s o r i a l t o p o l o g i e s . The f o l l o w i n g comment w i l l b e u s e f u l i n t h e s e q u e l . Thus, supp o s e w e a r e g i v e n two t o p o l o g i c a l a l g e b r a s E , F w i t h s p e c t r a m(F/

,

m(E),

r e s p e c t i v e l y . Then, c o n c e r n i n g t h e s u p p o r t s of t h e G e l ' f a n d

transforms of t h e elements i n E , F

c o n s i d e r e d below, one h a s ( s e e a l s o

I V ; (4.1) )

(7.1)

n 2 x . @ y E E 8 F , w h i l e T i s a comi = ~i 7 p a t i b l e t o p o l o g y o n E 8 F . I n p a r t i c u l a r , f o r e v e r y decomposable t e n s o r x o y e E 8 F , one h a s t h e r e l a t i o n Here o n e c o n s i d e r s a n y e l e m e n t z a

(7.2)

S u p p ( x ! ) = Supp(i3 x Supp($l

.

So w e come n e x t t o t h e f o l l o w i n g a u x i l i a r y r e s u l t .

4 54

XI11 PROPERTIES OF PERMANENCE

Lemma 7.1. Let E , F be Wiener-Tauber algebras ( D e f i n i t i o n I X ; 6 . 2 ) w i t h spectra Z??'(EI, m(F),r e s p e c t i v e l y , and l e t E @ F be t h e t e n s o r product algebra of E , F w i t h r e s p e c t t o a compatible topology T ( D e f i n i t i o n X ; 3.2) making, i n par-

t i c u l a r , t h e canonical b i l n e a r map $ : E x F - +E B F ( c f . X ; ( 1 . 4 ) ) continuous. Then, t h e topological algebra E 8 F i s a Wiener-Tauber algebra. Proof. I n d e e d , f o r e v e r y x 8 y E E B F o f x@?

in E B F , there

and f o r e v e r y n e i g h b o r h o o d

e x i s t , by t h e c o n t i n u i t y o f @ ,

of x i n E a n d V of y i n F , s u c h t h a t @(U, V / = U BV C W

U

W

neighborhoods

.

SO by hypo-

t h e s i s f o r t h e a l g e b r a s E , F and a p p l y i n g t h e n o t a t i o n of I X ; ( 6 . 3 ) , o n e obtains

@ (7.3) = (U8V1

+

(U n J E ( m ) ) B ( V n J F i W ) )

n ( J E ( m ) SJ,(-))

( b y X ; ( 2 .2 9 ) ) h ' n J E Q F ( m i . T

Thus, one g e t s (7.4) which p r o v e s t h e a s s e r t i o n (cf. I X ; ( 6 . 3 ) )

,

s i n c e one c e r t a i n l y h a s

I m $ ) = (by (7.4) )

E Q F = [Im$l ( l i n e a r h u l l of (7.5)

[wl C_ J a 8 F l m ) T

T

( c f . a l s o I X ; ( 4 . 3 3 ) ) , and t h i s c o m p l e t e s t h e p r o o f . I A d i r e c t consequence o f

C o r o l l a r y 7.1.

( 7 . 5 ) i s now t h e f o l l o w i n g .

Let a l l t h e c o n d i t i o n s of t h e preceding Lemma 7 . 1 be s a t i s f i e d

and, moreover, suppose t h a t ( t h e g i v e n a l g e b r a s E , F a n d t h e t e n s o r i a l t o p o logy T on E 8 F

a r e s u c h t h a t ) E 6 F i s a (complete) topological algebra. Then,

E G F i s a Wiener-Tauber algebra. That is, one has t h e r e l a t i o n T

J E $ F ( = == ) E 6 F . I

(7.6)

T

By s p e c i a l i z i n g t o l o c a l l y m-convex

a l g e b r a s , w e g i v e now t h e

f o l l o w i n g r e s u l t which may b e v i e w e d a l s o a s a c l a r i f i c a t i o n o f t h e previous discussion,

i n c o n n e c t i o n w i t h Theorem I X ; 6 . 1 . ( I n

t h i s re-

s p e c t , w e n o t e t h a t F r e c h e t ( l o c a l l y convex) a l g e b r a s are Ptbk; c f . J . HORVATH [ I : p. 299, P r o p o s i t i o n 3 1 ) . So w e h a v e . Theorem 7.1.

Let E , F be Fr&het-&lov

compact spectra ???(E),

m(F),r e s p e c t i v e l y .

Wiener-Tauber algebras w i t h l o c a l l y a faithful (local-

Moreover, consider

l y rn-convex) metrizable topology T on t h e r e s p e c t i v e tensor product algebra E d F

( D e f i n i t i o n 4 . 1 ) . Then, E g F i s a Fre'chet-Silov Wiener-Tauber algebra t o o , i n such a way t h a t e v e i y proper closed i d e a l i n i t has a non-empty h u l l . That i s , eqviv-

7.

455

WIENER-TAUBER C O N D I T I O N

h

a l e n t l y , any such i d e a l i n E 8 F is a l i ~ a g scontained i n a c l o s e d regular maxima2 i d e a l of t h e algebra. Proof. First we remark that, by hypothesis for the algebras E , F , one has n = i (cf. X; (2.21) and the comment following it). So since by definition T is a locally convex (in fact, locally m-convex) compatible topology on E 8 F (Definition 4.1), one gets E < T ~ T (cf. X ; (2.35)), so that the conditions of Lemma 7.1, concerning T too, are satisfied

(see also the comment on Xi(2.18)). Therefore, since by hypothesis E 6 F is a (complete) locally m-convex algebra, Co-

(Definition X;3.1)

7

rollary 7.1 entails, by hypothesis, that E g F is a Wiener-Tauber algebra. Furthermore, still by hypothesis and Theorem 6.2, one has already that E 6 F is a Frschet-silov algebra; hence, since the same algebra, as we proved above, is also a Wiener-Tauber algebra, the last part of the assertion is now a consequence of Theorem IX;6.1 (cf. also Theorem XIIi1.2

and Theorem Vil.1). This terminates the proof. I

Scholium 7 . 1 . -

In this respect, we remark that

the conditions o f the preceding Theorem 7.1, with T = T , are satisfied in case one considers any two Fr&het-&'lou Wiener-Tauber &-algebras such t h a t e i t h e r one of them t o have t h e approximation propert3 (see also Theorem 6.2) . Therefore , E 6 F i s a Fre'chet-&lov Wiener-Tade r &-algebra (cf. also Lemma XII; 1-31, and hence (Theo-

rem IX; 6 .l ) eve? proper closed i d e a l i n E 6 F i s contained in a ( c l o s e d ) regular maximal i d e a l of t h i s algebra (see also Theorem 11; 6.1). Now, the usual group algebra L ' I G )

of a given locally compact

abelian group G (see Chapt. VI1;Section 4 ) is a commutative Banach (and hence a f o r t io r i a &-)algebra and, moreover, regular semisimple as well as a Wiener-Tauber (thus a Silov Wiener-Tauber) algebra.(We refer e.9. to L . H . LOOMIS [I] for these properties of the alge-

.

I

bra L 1 ( G j ) . Furthermore, L ( G ) has t h e approximation property (cf A . GROTHENDIECK [3: Chap. I; p. 185, Proposition 411). Accordingly, the preceding 1 argument is applied, in particular, to a generalized group aZgebra L ( G , E ) 1

= L ( G ) G E (cf. Theorem X I ; 5 . 1

and XI; (5.10)) where E i s a Fr6chet-Silov

Wiener-Tauber algebra w i t h a l o c a l l y compact speczrwn (hence,in particular, if

.

E is a (Frgchet-Silov Wiener-Tauber) &-algebra) Thus, every generalized 1 group algebra L ( G , E ) , a s above, s a t i s f i e s t h e c l a s s i c a l Wiener-Tauber c o n d i t i o n ,

in the sense of Theorem IX; 6.1 1

.

That is, every proper closed i d e a l

i n the

algebra L (G, E ) has a nun-empty h u l l ; equivalently, every such ideal is

4 56

X-I; PROPERTIES OF PERMANENCE

c o n t a i n e d i n a c l o s e d r e g u l a r maximal i d e a l of L ' ( G , E ) .

8. Appendix: Generalized spectra (contn'd.).

Canonical decomposition

The f o l l o w i n g l i n e s c a n be viewed a s a n o t h e r a p p l i c a t i o n of o u r i n connection w i t h our consider-

d i s c u s s i o n i n Chapt. XII; S e c t i o n 3 ,

a t i o n s i n t h e p r e c e d i n g S e c t i o n 4 . Thus, w e s t i l l c o n s i d e r h e r e cont i n u o u s a l g e b r a morphisms whose domains of d e f i n i t i o n a r e t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s l o o k i n g f o r "canonical decompositions" of such morphisms of t h e t y p e g i v e n by XIIi(3.11). B e s i d e s , t h e r a n g e s now o f t h e morphisms u n d e r d i s c u s s i o n a r e , i n p a r t i c u l a r , a p p r o p r i a t e t o p o l o g i c a l tensor product algebras too. W e s t a r t with t h e following useful r e s u l t .

Lemma 8.1. Let E , F be complete l o c a l l y eonvex algebras, i n such a way t h a t E 6 F i s a (complete) l o c a l l y convex algebra in a compatible ( l o c a l l y convex) tenT sorial topology T on E B F . Furthermore, suppose t h a t for a given element z e EBF h

one has the following reZation A

where

,

.

z = x . p , with

(8.1) IJ

2 #

0 ; x e E,

i s a complex-valued fwzction on m(F1 ( i n the sense t h a t ^z(hl = ; ( f a g ) =

z ( f ) p ( g l ; c f . Theorem XII; 1 . 2 ) . Then, t h e r e e x i s t s an element y e F such t h a t

u = i,

(8.2)

so that one has (8.3)

. . . A

z=xcey ( = 2 . i ) .

I n p a r t i c u l a r , if t h e algebra E g F i s semi-simple (see Theorem 4 . 3 ) , then one T

gets the r e h t i o n (8.4)

z=zay.

I n o t h e r words, i f t h e a l g e b r a E $ F i s s e m i s i m p l e , a n e l e m e n t 2 E E ~ Fr e d u c e s t o a decomposT a b l e t e n s o r x e y e % @ F i f f a n d o n l y i f , t h e above r e l a t i o n ( 8 . 1 ) ( o r , o f c o u r s e , i t s a n a l o g u e $=A;) holds true. Procf of L e m 8.1.

Applying t h e p r e c e d i n g r e l a t i o n ( 4 . 5 ) ,

one g e t s

f o r any z e E & F and S E m(E) a u n i q u e l y d e f i n e d e l e m e n t o f F , g i v e n by T

(8.5)

(See a l s o (4.1)

a n d / o r ( 5 . 2 ) , i n c o n n e c t i o n w i t h Remark 5 . 2 ;

one a p p l i e s h e r e o n l y t h e c o m p l e t e n e s s o f F ) f o r some f o e m(El, t h e n t h e element

. NOW,

if

thus,

?(So I = k-' # 0 ,

8.

A P P E N D I X : G E N E R A L I Z E D SPECTRA ( CONTN’D. )

457

So one g e t s u = c w i t h y e F g i v e n b y ( 8 . 6 1 , which p r o v e s t h e a s s e r t i o n c o n c e r n i n g ( 8 . 2 ) . The r e s t i s now s t r a i g h t f o r w a r d by t h e s a m e d e f i n i -

t i o n s , and t h i s c o m p l e t e s t h e proof

of

t h e lemma. I

W e come now t o o u r f i r s t main r e s u l t i n t h i s s e c t i o n . So w e have t h e n e x t .

Suppose t h a t we are given t h e l o c a l l y convex algebras E , F , G , H

Theorem 8.1.

such that G and H are complete, t h e algebras F , H are infra-barrelled ( a s ZocaLly convex spaces) w i t h i d e n t i t y elements 1F , l H ,r e s p e c t i v e l y , while t h e algebra F is a l s o semi-simple. Moreover, asswne t h a t a l l t h e algebras given have l o c a l l y equicontinuous spectra, with t h e spectra ? ? Y ( E l ,m(G)being, i n p a r t i c u l a r , connected and ? W F I

t o t a l l y disconnected. F i n a l l y , suppose t h a t E G F , G S H are (complete) T

l o c a l l y convex algebras w i t h respect t o ( l o c a l l y eonvex)

o

compatible t e n s o r i a l to-

pologies Y and o, a s i n d i c a t e d , and such t h a t t h e algebra

G 6G H

t o be, i n particu-

lur, semi-simple.

Then, f o r every p a i r ( h , p l e t l ~ m ( E $ F ,G $ H I y H u m ( E , G I s a t i s f y i n g the reZation (8.a)

h ( x a l P ) = u(x) @ l H x, e E ,

there e x i s t s an element vEtlom(F, H ) such t h a t one has h = I.~@v

(8.9)

I”canonical decomposition“ o f h ) . Proof. I t i s c l e a r t h a t t h e r a n g e of t h e r e s t r i c t i o n of t h e t r a n s pose of h t o ??Z(G$HI

is ??Z(E$FI;

h e n c e , one g e t s

th(aa%l= f @ g ,

(8.10)

for every ( a , B )

€

m(G1 x

( c f . Theorem X I I ; l . 2 ) .

m(H)= m ( G 60H ) , where If,

g i E m(El X m(F)=m ( E $ F )

F u r t h e r m o r e , one g e t s f o r e v e r y x E E

t

[ h ( a e B ) l ( x e I F I = ( a @ B ) ( h ( xo Z F l l = (by (8.8)) (cL&)(~(x)

elH) = c ~ ( ~ ( x=) (J~ o u ) ( x = ) (by (8.10))

If e g l ( x a l F I = f ( x l T h e r e f o r e , one h a s

.

XI11 PROPERTIES OF PERMANENCE

458

(8.11)

a

w i t h a , f a s i n (8.10). Now,

3

t h e s e t m(GI x { B

C m(GI

X

t

0

p ='p(a) = f

i s connected, i s a l s o c o n n e c t e d and i s

s i n c e by h y p o t h e s i s T T Z I G I

m(H) = m(G6HI t '

mapped by th o n t o t h e c o n n e c t e d set

h ( m ( G ) x { } I E ???(El x m(F)=m(E$F); T t h e r e f o r e , it i n t e r s e c t s only one connected component of t h e r a n g e o f t h , so by h y p o t h e s i s f o r t h e s p e c t r a o f t h e a l g e b r a s E , F , a s e t o f t h e form ??Z(E)x{g},w i t h g e m ( F l ( s e e , f o r

3.11). Thus, f o r a n y

= rtu(a), g) €

i n s t a n c e , J . MUNCRES [l:

,

-

t h a t one a c t u a l l y obtains a map

SO

I

t ~ m(H) : -m(F)

(8.12)

one h a s (8.10) f o r some ( f , g )

(a,B)e?TZ(GIxm(H),

M-(E) x W(F) ( c f . (8.11

:

B

t

p. 160, Theorem

'v(B)

=g

,

~ ( a land a € m ( G / ; i n f a c t , t h e l a t t e r map i s i n d e p e n d e n t of t h e p a r t i c u l a r e l e m e n t a e ( G ) and c e r t a i n -

w i t h t h ( a o B ) = fog

f o r every f =

l y continuous. F u r t h e r m o r e , a s f o l l o w s from (8.10) t h e map (8.12) i s l i n e a r on [ m ( H l ] ( l i n e a r h u l l o f ? ? Z ( H ) )and h e n c e it c a n b e extended b y

c o n t i n u i t y t o t h e ( r e s p e c t i v e t r a n s p o s e ) map t ~ [ ?:7 Y ( H ) ] =

(8.13)

Hi-+

Fi =

[m(F)I

( t a k e t h e h y p o t h e s i s f o r F I B i n t o a c c o u n t , a n d a l s o Lemma VIII;3.1). Thus, one g e t s from t h e p r e v i o u s r e l s . (8.10), (8.111, a n d (8.12)

t h(CLOB, = f o g = t u(alo t v(BI = I t uo t V ) ( c x c Q R I

(8.14)

,

f o r every ~ ~ o @ e m ( G 6 H h ) e; n c e , 0

t h = tU

(8.15)

t

@ V .

On t h e o t h e r h a n d , f o r a n y x o y e EBF a n d a@Be??Z(G&Hl,one h a s 0

n

h ( x o y ) ( a c ~ B=) ( a @ B l ( h ( x @ y y l=I [th(ataB)l ( x : y l = ( b y (8.14)) ( t p ( a l o t ~ ( $ ) ) ( x @ y ![=t p ( a ) l ( x )@ [ t V ( B I 1 ( y ) A t fi = p(x)(cx)a;( v(B)) = [ u ( x 1 o (y^ o t v l l ( a @ B )

.

T h e r e f o r e , one o b t a i n s

(8.16) w h i l e t h e s e c o n d member o f t h e l a s t r e l a t i o n i s , e s s e n t i a l l y , of t h e form (8.1). C o n s e q u e n t l y , b y h y p o t h e s i s f o r t h e a l g e b r a s G I H , t h e

z e H such t h a t

above Lemma 8.1 y i e l d s now an e l e m e n t h

(8.17)

2

Thus, f o r e v e r y B f

(8.18)

;(@I =

(Go

m(H),one t

tv)(B)= $( v(BII 0

A

t

= y o v. has

t

/-

= [ v t B , l ( y l = B ( v ( y l ) = v(yl(BI

I

so t h a t one g e t s z ^ = v ( y l . So from t h e h y p o t h e s i s c o n c e r n i n g t h e a l g e -

8. APPENDIX: GENZRALIZED SPECTRA

bra

GSH U

( C O N T 'ND. )

459

and from Theorem 4 . 3 , o n e now c o n c l u d e s t h a t

(8.19)

z =

vlyl

. v : F+H

Thus one a c t u a l l y o b t a i n s a l i n e a r map t o o , i n v i e w of

which i s m u l t i p l i c a t i v e

( 8 . 1 2 ) and ( 8 . 1 8 ) . F u r t h e r m o r e , from t h e c o n t i n u i t y

o f t h e map ( 8 . 1 3 ) and from t h e i n f r a - b a r r e l l e d n e s s of t h e a l g e b r a s F , H , one a l s o c o n c l u d e s t h e c o n t i n u i t y of v (see e . g . J . HORVLTH [ I : p. 218,

P r o p o s i t i o n 8 , and p . 258, C o r o l l a r y ] )

.

So we a c t u a l l y get an element v e

H u m ( F , HI. NOW,

from ( 8 . 1 6 ) , ( 8 . 1 7 )

h(xC3yi = p ( x ) 0

and ( 8 . 1 9 ) , one o b t a i n s

(i 0t vl=

I?(xl a v l y ) = p(x)0 v l y l

so t h a t , by t h e s e m i - s i m p l i c i t y o f t h e a l g e b r a (8.20)

I

G G H , one h a s 0

h ( x 0 y l = p ( x ) aov(yl = ( p C 3 v i ( x o y ) ,

f o r e v e r y decomposable t e n s o r x @ y e E Q F .

This, in turn, implies ( 8 . 9 )

o f c o u r s e , and t h i s c o m p l e t e s t h e proof o f t h e theorem. I A s it becomes c l e a r from t h e p r e c e d i n g p r o o f , t h e t o p o l o g i c a l

a l g e b r a E n e e d n o t n e c e s s a r i l y b e l o c a l l y convex. I n t h a t c a s e , o f

i s a (complete) topological algebra i n

c o u r s e , one assumes t h a t E&F T

a compatible t e n s o r i a l topology

T

on

E Q F ( D e f i n i t i o n X ; 4.1

).

B e f o r e w e come t o o u r f i n a l main r e s u l t o f t h i s s e c t i o n , w e comment a b i t more on t h e t e r m i n o l o g y which w e a r e g o i n g t o a p p l y b e l o w . So w e f i r s t s e t t h e f o l l o w i n g .

Definition 8.1. Given a t o p o l o g i c a l a l g e b r a E , a d i r e c t e d n e t ( u7 ,. % ) .e l i n E i s s a i d t o b e an approximate i d e n t i t y x E E , one h a s t h e r e l a t i o n (8.21)

of E , i f

,

f o r every element

l i m ux. = l + m xu 2. z ?, i = x '

(2-sided approximate i d e n t i t y o f

E 1.

T h u s , w e h a v e now t h e f o l l o w i n g .

Theorem 8.2. Let B, F be topological algebras such t h a t E has an appuwzirnatc i d e n t i t y luilie I , the algebra F an i d e n t i t y element I F , while E 6 F i s a (complete) T

topological algebra i n a compatible t e n s o r i a l topology

T

on EQF. Furthermore, l e t

be complete l o c a l l y convex algebras w i t h continuous m u l t i p l i c a t i o n s , the algebra H hawing, moreover, an i d e n t i t y element l H and such t h a t G S H i s a (complete) 0 semi-simple l o c a l l y convex algebra ( w i t h continuous m u l t i p l i c a t i o n ) w i t h respect t o G,H

a compatible ( l o c a l l y convex) t e n s o r i a l topology a on G 8 H . F i n a l l y , assume t h a t the following condition is s a t i s f i e d :

460

XI11 PROPERTIES OF PERMANENCE

F o r every p a i r

( h a p ) € H o m ( E 6 F , G) x H a m l E , G) ‘I

s a t i s f y i n g the r e l a t i o n

(8.22) (8.22a 1

hix@lpl=

~(3);

t h e r e e x i s t s an element g e M ( F I h = pog

(8.22b)

x f E, such t h a t

.

Then, for every p a i r ( h , p ) e H o m ( E $ F , G 2 H ) x H o m ( E , GI

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

(8.23)

h O elF) = w ( x I @l H ;x e E ,

there e x i s t s an element v e H o m ( F , H )

such t h a t

(8.24)

h = pev

.

Before embarking on the proof of the previous theorem,we do make some preliminary comments on the above condition (8.22): Thus, the said condition is certainly satisfied if the algebras E , G have identity elements. So arguing within the context of the preceding theorem, the relation (8.22b), with p E Hom(E, G) given by (8.22a) is then a consequence of X11;(3.11). Indeed, we consider first the restriction of the given h e H o m ( E 6 F , GI to E 8 F so as to get, from Lemma T T XIIi3.1 , an element f o g (cf. XII; (3.7)) which then is extended by continuity to the given element h = f o g = f @ g On the other hand, we still need in the sequel the following argumentation, which we better give into the form of the next.

.

Scholium 8.1.- We think always within the context of the previous Theorem 8.2: So denoting by idG the “identity map” on C;, one gets, for every element $ e m(H!, a continuous (algebra) morphism id o B : G@ H d H : G 0 s o t * @ ( t ) s . NOW, by hypothesis for the algebras G I H I the latter map can be extended by continuity to a similar one as follows (extending we still keep the same notation); i.e., we have (8.25)

idGo B e H o m ( G 6 H , GI CI

.

Therefore, for every h e H o m ( E S F , GGH), one obtains an element T

(8.26)

0

x = (idG @ B ) o h E H o m ( E $ F ,

GI.

Furthermore, for every pair ( h a p ) satisfying ( 8 . 2 3 ) for every X E E ,

, one

gets,

8. APPENDIX: GENERALIZED SPECTRA ( CONTN'D.

)

461

x(xolFI = (idGoB)IhIxc+lF))= ( b y (8.23)) IidGoB)Ip(x)eIHJ=uIx) So t h e p a i r

(x,u)

.

, a s d e f i n e d above, s a t i s f i e s ( 8 . 2 2 a ) ; hence, i f

( 8 . 2 2 ) i s f u l f i l l e d one o b t a i n s (8.27)

x z l i d @B)ok=pJaPg, G

f o r some g e m ( F ) . So w e come n e x t t o t h e

Proof of Theorem 8.2. F o r e v e r y p a i r (x, y ) e E

X F

, one

obtains

x o y = (by ( 8 . 2 l ) ) ( l i f n u . x l e y = ( s e e cond. 2 ) of Definition X i 4 . 1 ) (8.28)

z z lim(uixJaP y ) = lim[(uias y l ( x o Z p ) 1

i

i T h e r e f o r e , f o r any

( h , p l e HamlE~F,G 6 H ) T

X

U

.

Hom(E, G ) s a t i s f y i n g

(8.23) and

( a , B) e m(G) X m(H), w i t h ao B e m(G6H) ( c f . a l s o X I I ; (1 -12) and t h e hypot h e s i s f o r G 6 H ) and a(u(x)l = p i y f a ) # 0 , one o b t a i n s from ( 8 . 2 8 ) U

where Ba

i s t h u s a complex-valued map on m ' ( H ) .

T h e r e f o r e , due t o (8.29)

o n e now o b t a i n s (8.31) f o r every p a i r

(a, B ) e

m(G/x m(HI, w i t h

( 8 . 3 1 ) c a n f u r t h e r b e e x t e n d e d , f o r any

a(p(x) # 0. Now it i s c l e a r t h a t c1

E ? ~ Y ( G J w h a t s o e v e r (see a l s o

( 8 . 3 6 ) i n t h e s e q u e l ) . Hence, from Lemma 8 . 1 , w e c o n c l u d e now t h a t (8.32)

8,

P

,

e ( a , y ) = vCl(yie I I ^

f o r some e l e m e n t

v a f y l e F . A c c o r d i n g l y , b y t h e s e m i - s i m p l i c i t y of H

( c f . Theorem 4 . 3 )

,

o n e g e t s , i n d e e d , a map

v a : F-H

(8.33) s u c h t h a t v a ( y l = e ( a , y)

,y e F ,

,

g i v e n by ( 8 . 3 2 )

,

of c o u r s e , a c c o r d i n g t o t h e same d e f i n i t i o n s . Thus, from ( 8 . 3 1 ) a n d ( 8 . 3 2 ) , one g e t s

and which is continuous

462

XI11

P R O P E R T I E S OF PERMANENCE

Thus, due to the semi-simplicity of the algebra G G H , one finally ob0 tains (8.35)

h l x a y l = p(x1 w v,(yl

,

for every decomposable tensor x 8 y E E C3F. Now, we prove next that ( 8 . 3 3 ) i s , i n e f f e c t , independent of t h e parc1 E ??Y(G) involved by (8.31): That is, for any o1 , a2 in

t i c u l a r element

77ZlG) for which (8.31) is valid, one has (8.36) with y E F

. Indeed, for every

, one

has from (8.27)

= g ( y ) l J ( x l ( a l = M(idG@Bl(h(x@y)JJ

= (by (8.35))

c1 e W G i

N(idG @B) ( ~ ( x @v,(yll) ) =aiB(va(y).u(x)) = B(v,(yJl.

u(xJ ("l

Therefore, since a(1-1(x11#0, one has the relation (8.37)

B(V,(y))

A

= a(@(", y ) ) =v,(y)(B)

= g(yJ

which certainly implies (8.35).FurthermoreI (8.37) and the semi-simplicity of the algebra H yield now that the continuous map (8.33) is in fact multiplicative, so that one finally gets an element (8.38)

v e Hom(F, H J .

Hence, one concludes from (8.35) that (8.39)

h f x a y l = ~ ~ x l o v ~ y l : ~ ~ 8 u i ~, x o y J

for every decomposable tensor x @ y € E @ F . So the last relation, extended further by "linearity and continuity", entails (8.24) of course, and this completes the proof of Theorem 8.2.4 We conclude with considering one further application of the previous discussion, in particular, concerning the last relation (8.39). Thus, we have next the following.

Corollary 8.1. Suppose t h a t t h e c o n d i t i o n s of t h e p r e v i o u s Theorem 8 . 2 a r e s a t i s f i e d , and l e t ( h , p , v ) b e c t r i a d of maps a s i n (8.24). Then, t h e r e l a t i o n (8.40)

Im(hlEBF) = G B H

holds t r u e i f , and only if, each one of t h e maps p , v i s an o n t o map. On t h e o t h e r hand, t h e r e s t r i c t i o n of h t o E 8 F , h l E B F , i s one-to-one and o n l y i f , t h i s i s t h e case f o r each one of t h e maps u,v

.

if,

8. APPENDIX : GENERALIZED SPECTRA ( CONTN 'D. )

463

Proof. First we remark that (8.40) is a direct consequence of if p,v are onto maps. Conversely, let us assume that (8.40)

(8.39),

is valid; then, for every pair I s , t I e G x H with t # O E H , there exists n an element z = I xi ayi E E B F such that i=I

h(zI= sat.

(8.41)

Therefore, for any ( a , B I E m(GI X m ( H I with B(t) # 0 (apply the semi-simplicity of the algebra H ; cf. Theorem 4.3), one has

So by the semi-simplicity of G(Theorem 4.31, one concludes that

(8.43)

s

=u(

1

D(v(tII

z

B(vlyiIIxi)

This shows that p is an onto map, while an analogous argument is also applicab1.e to V , which thus proves the necessity of the condition in study

.

Now, suppose that h is 1 - 1 ly, one has by ( 8 . 3 9 )

, and let

x E E with ulxI = 0 . According-

h l x e y l = (1~0vI(xayI = u ( x I e ~ ( y = i 0. Therefore, according to the hypothesis for h , x e y = 0 for every y E F , which implies (Lemma X; 1 . 3 ) that x = 0 ; namely, the map u is 1 - 1 , and a similar reasoning can be applied to the case of v. On the other hand, the converse of our last assertion in the statement, is certainly valid for the restricted map h (8.39)

and of Lemma X ;1 .I

,

IEBF

after a direct application of

and this completes the proof. I

Scholium 8.2.- The previous discussion seems to have a particular interest in the special (however, important!) case where one considers automorphisms of topological tensor product algebras. By the last term one means of course a topological-algebraic isomorp h i s m of a given topological algebra onto itself.

Thus, the preceding Corollary 8.1 may be viewed as providing a criterion of having certain particular automorphisms of a given topological tensor product algebra E 6 F (of the type considered above) as (canonically) deT composable in corresponding automorphisms of the individual factor algebras of the product in question. (Thus, one considers in the last part of this corollary the particular case that G = E and H = F )

.

464

XI11 PROPERTIES OF PERMANENCE

In this respect, let us also assume, for simplicity, that the algebras under consideration have identity elements, so as the previous cond. (8.22) is then automatically fulfilled. (See the comment before the preceding Scholium 8.1). Thus , by considering ( t o p o l o g i c a l ) algebras w i t h i d e n t i t y elements , the rest of the conditions of Theorem 8.1 being valid (with G = E and H=F), one concludes the following: Any autornorphism of EG F t h a t "leaves the subalgebra T

(8.44)

E Z E G F invariant" (cf. (8.22a) or yet (8.231, i s (canoniT c a l l y ) decomposable in a tensor product of autornorphisms of E and F . (The converse 5s c e r t a i n l y t r u e ) .

The " invariance property" of the automorphisms required in (8.44) can be formulated of course (symmetrically) for the algebra F instead, so that one should actually say in (8.44), more precisely, any autornorphism of EGF t h a t leavcs e i t h e r one o f the f a c t o r algebras E or F invariant. (It T is still clear that all of our previous discussion also admits this " s y m e t r y " with respect to the factor algebras E, F ) . The previous conclusion hasa special bearing on some recent considerations by R.D. MEHTA-M.H. VASAVADA [l] , given in the context of the theory of Banach algebras (commutative unital and semi-simple!). Indeed, it was the last paper which, mainly, motivated our comment on this possible application of our considerations in this section. In this respect, we still note with the above authors that not every autornorpkism of a given topological tensor product algebra i s (canonically) decomposable, as above. S o this does not happen even in the very special (and important, as well) case of (Banach) function algebras (seeibid.; p . 16, Remarks 3) . Furthermore, following the afore-mentioned authors within the present more general context, we still remark the following: Namely, arguing in the framework of Theorem 8.1, we observe that the hypothesis set forth by that theorem leads actually to the rel. (8.15),i.e.,to a canonical decomposition of t h e transpose of h , restricted to m(E$F) (SO, in facttan automorphism of the latter space), i n t o s i m i l a r transposes (in fact automorphisms)of ZYEI, ??Z(F), r e s p e c t i v e l y . On the other hand, t h i s is proved t o b e , i n e f f e c t (see also R.D. MEHTA -M.H. VASAVADA [l: p. 15,Theorem 21 )

, a necessary and s u f f i c i e n t condition f o r a s i m i l a r icanonicaZ) decomposition

,

of h , as above, in the sense of (8.44) which thus is f u r t h e r equivalent

to the conditions given above by our theorems.(However, we may leave at this point furthertechnicaldetails to the interested reader) -

465

G e n e r a l i z e d S p e c t r a in t h e P r e s e n c e o f Approximate Identities. Representation Theory

CHAPTER X I V

We examine in this chapter possible extensions of our previous results in Chapt.XI1; Section 3 and, in particular, of the basic Theorem XII;3.1. So we consider below (cf. Section 1 ) that same familiar framework, but where now the topological algebras under discussion have instead only approximate i d e n t i t i e s (cf. Definition XIII: 8.1 ) In compensation, the range of the algebra morphisms under consideration is now, in particular, a topological algebra of operators on a suitable topological vector space: the situation here reminds, of course, that one which one encounters in the classical representation theory (in Hilbert spaces, for instance. See Theorem 1.1 below).

.

1. Topological algebras with approximate identities. Representation theory We start with establishing the relevant terminology. So we have already encountered in the last section of the previous chapter the notion of an approximate identity in a given topological algebra. NOW, if B (ui I i e I is an approximate identity of a given topological algebra E (Definition XIII; 8.1 ) , we say that it is a bounded approximate i d e n t i t y , if B (considered as the image in E of the function defining the net in question) is a bounded subset of the underlying to the algebra topological vector space E. So we first have the following result which is also useful in some other particular instances. Namely, one has.

Lemma 1.1. and B = ( u2. )Z. E

(1. I )

I

Let E be a topological algebra w i t h a continuous m u l t i p l i c a t i o n a bounded upproximate i d e n t i t y i n E . Then, t h e farniLy B 2 -(u2 )

i i a I

i s a l s o a bounded approximate i d e n t i t y i n E . Proof. We first remark that, due to the continuity of the multiplication in E , the product of two bounded subsets of E is a bounded

466

XIV

SPECTRA AND APPROXIMATE I D E N T I T I E S

s u b s e t of E t o o ( c f . t h e d i s c u s s i o n i n C h a p t . I ; S e c t i o n 4 , l a r , C o r o l l a r y 4.5) ; hence, a

f or tior i

in particu-

t h e set B z i n (1.1) i s a

bounded s u b s e t o f E . F u r t h e r m o r e , Eor e v e r y n e i g h b o r h o o d U o f z e r o i n

E, t h e r e e x i s t by same h y p o t h e s i s n e i g h b o r h o o d s V , W o f z e r o i n s u c h t h a t 8. WE U ; h e n c e , by h y p o t h e s i s f o r B , t h e r e e x i s t s

E

X > O such

t h a t one h a s

~ ~ e X . 8i ,e I .

(1.2)

On t h e o t h e r h a n d , from XI11;(8.21) w e g e t li.m(x-ui3:l

(1.3)

= 0 , f o r every

2.

X E E , so t h a t t h e r e e x i s t s i o ( A , W I

ioc I s u c h t h a t

2 - U ' X

Z

e - w1 X

,

f o r e v e r y i > i oi n I. Thus, from t h e l a s t two r e l a t i o n s , one h a s 1

=v.wC_u,

u.(x--U.x)eAI'.--W

A

f o r every i > i , ; i . e . ,

the net

( u i ( X - u .zx ) ) i e I

(1.4)

c o n v e r g e s t o z e r o i n E , so t h a t o n e h a s

T h e r e f o r e , one g e t s

1imuZz = 1imu.x = i t i t

(1.5)

2

which e s s e n t i a l l y p r o v e s t h e a s s e r t i o n c o n c e r n i n g t h e f a m i l y (1.1) and t h i s completes t h e p r o o f . I Remark.I t i s c l e a r from t h e p r e c e d i n g p r o o f ( c f . ( 1 . 2 ) ) t h a t one a c t u a l l y needs t h e n e t (uil t o b e "eventuaZZy bounded" i n t h e s e n s e t h a t ; f o r e v e r y n e i g h b o r h o o d V of z e r o i n E l t h e r e e x i s t s an i n d e x i0 = 0iI V l E I , s u c h t h a t u

i e AV

with

i>iol

f o r some A > O . I n t h a t c a s e o f c o u r s e one p r o v e s t h a t t h e n e t (1 .I ) is an eventuazly bounded approximate identity of E t o o . Furthermore, w e a l s o have t h e f o l l o w i n g . Lemma 1.2. Let E be a topological algebra w i t h continuous muZtipZication and

l U i ) iE I

a bounded approximate i d e n t i t y of E. Then,

approximate i d e n t i t y i n the completion o f E ,

l U i ) ie I is a l s o a bounded ( s e e a l s o C h a p t . I; S e c t i o n 4 )

.

Proof. I t i s c l e a r f r o m h y p o t h e s i s t h a t ( u i l i e 1 i s a bounded n e t i n 6 ( c f . t h e r e l e v a n t d e f i n i t i o n s and S e c t i o n I ; 4 ) . So i f x e E^, t h e r e

exists a net

(xglg

E E , with x

= limrc6 6

; hence,

f o r e v e r y neighborhood

N of C E i , t h e r e e x i s t

467

REPRESENTATION THEORY

1.

neighborhoods U , V

of 0 i n

such t h a t

U.V+U+Vr N h

( d u e t o t h e c o n t i n u i t y of t h e a l g e b r a i c o p e r a t i o n s i n E ) and u i € U, f o r e v e r y i > i o( f o r some i o e I ; t h e l a t t e r i s e s s e n t i a l l y t r u e f o r any

i e I. B u t see a l s o t h e p r e v i o u s Remark)

.

Moreover, x6 - x e V f o r e v e r y

6260 ( f o r some 6 o e J ) ; t h e r e f o r e , one g e t s UiX6'

- X6'

e u,

f o r e v e r y i>il, f o r some i €1,and 6'> 1

u

60. So one f i n a l l y g e t s

.x - x = u .(x - x 6 A + I U i X 6 , -xg,i+(x6'

f o r e v e r y i>i'>

-21

e 51.

v + u + If = N ,

io, il , w i t l i (some f i x e d b u t a r b i t r a r y ! ) 6'>6,

,

and

t h i s terminates the proof.1 W e come now t o c o n s i d e r t h e f o l l o w i n g g e n e r a l c o n c e p t whosesome

p a r t i c u l a r i n s t a n c e s w i l l be a p p l i e d i n t h e e n s u i n g d i s c u s s i o n . So we have t h e n e x t .

Definition 1.1.

Suppose w e a r e g i v e n a t o p o l o g i c a l a l g e b r a E t o -

g e t h e r w i t h a t o p o l o g i c a l v e c t o r s p a c e H. Moreover, l e t L,(HJ

be t h e

a l g e b r a of c o n t i n u o u s l i n e a r endomorphisms of H ( " l i n e a r o p e r a t o r s " on H , w i t h r i n g m u l t i p l i c a t i o n t h e " c o m p o s i t i o n " of o p e r a t o r s ) , endowed w i t h a v e c t o r s p a c e t o p o l o g y G m a k i n g S G ( H ) i n t o a t o p o l o g i c a l algebra.

w e c a l l a continuous representation a l g e b r a E i n H , any e l e m e n t NOW,

of t h e g i v e n ( t o p o l o g i c a l )

@ e HomfE, S 6 f H I ) +

(1.6)

( " e x t e n d e d g e n e r a l i z e d spectrum" of t h e t o p o l o g i c a l a l g e b r a s c o n s i d e r e d ; c f . Chapt. V ; ( 8 . 2 ) ) . U s u a l l y , w e c a l l H t h e representation space of the algebra E

(with r e s p e c t t o t h e given element @ i n ( 1 . 6 ) ) .

So s t i l l arguing within t h e preceding context,

6 =s

if i n particular

( t o p o l o g y of s i m p l e convergence i n H ) , i t i s r e a d i l y s e e n from

t h e same d e f i n i t i o n s t h a t BOURBAKI

t h e topological vector space

L s ( H ) (see e . g . N.

[7: Chap. 3 ; § 3 , p . 181) becomes a topological algebra.

S o it i s t h i s l o c a l l y conuex algebra

f ( H i which w e s h a l l m a i n l y be

d e a l t w i t h i n t h e s e q u e l , when i n t h e p a r t i c u l a r c a s e c o n s i d e r e d H i s a s u i t a b l e l o c a l l y convex s p a c e ( i b i d . ; p . 1 7 , C o r o l l a i r e ) . S i m i l a r l y , i f H i s a l o c a l l y convex s p a c e and

G =b

( t o p o l o g y of

bounded convergence i n H ) , it i s e a s y t o v e r i f y , s t i l l on t h e b a s i s

of t h e r e s p e c t i v e d e f i n i t i o n s , t h a t t h e l o c a l l y convex space

Sb(HI ( i b i d . ;

p . 1 8 ) is, i n f a c t , a l o c a l l y convex algebra. The l a t t e r i s f u r t h e r quasi-com-

468

XIV

SPECTRA AND APPROXIMATE IDENTITIES

p l e t e , if the locally convex space H i s barrelled and quasi-complete (ibid.;

p. 31, Corollaire 2). Now under the same hypothesis for H and from the same result, as before (loc. cit.) one concludes that t h i s i s a l s o t h e case f o r t h e algebra S s ( H ) . A particular instance of the preceding is provided, of course, by considering the l e f t (resp., r i g h t ) representations of a given topological algebra E . That is, by considering the elements

1 (resp., r ) e ffam ( E , S s ( E l )

(1.7)

given by the relation [ l ( x l l ( y l = x y (resp., [ r ( x l l ( y ) = y x ) ,

(1 .8)

for any x , y in E . (We also write, as we did it in the foregoing, 1X Z ( x ) , and r I r ( x l , respectively, for x a E l . It is readily seen by the X respective definitions that (1.8) provides, really, elements in the set figured on ( 1 . 7 ) . Furthermore, we still observe that (1 .7) y i e l d s a continuous (algebra) isomorphism of E i n t o S ( E l , i n case the topological algebra E hasanapproxim t e i d e n t i t y ( cf . Definition XIII: 8.1 ) So we come now to the following basic result of this section.

.

Theorem 1 . 1 . Let E , F be l o c a l l y convex algebras w i t h bounded approximate I and ( v j l x , r e s p e c t i v e l y . Moreover, l e t E &5 F be t h e complet i o n of t h e tensor product algebra E B F of E , F w i t h respect t o a ( l o c a l l y conaex!

i d e n t i t i e s (uiJi

compatible t e n s o r i a l topology T on E B F , making E 6 F i n t o a (complete) l o c a l l y convex algebra and t h e canonical b i l i n e a r map @ : E x F -

E O F continuous. F i n a l l y , l e t T H be a complete barrelled l o c a l l y convex space and S s ( H ) the (quasi-complete) local-

l y convex algebra of continuous l i n e a r endomorphisms of H endowed w i t h the topology s of simple convergence i n H .

Then, f o r every element

(1.9)

hatfom(E, S s ( H ) )

s a t i s f y i n g the r e l a t i o n (1 .lo)

[ Im(h)

(HI 1 = H

("non-degenerate" r e p r e s e n t a t i o n ) , there e x i s t uniqueZy defined elements

(1.11)

f e H o r n ( E , S s ( H I ) and

gaffam(F,Ss(H))

such t h a t one has

(1.12) I n p a r t i c u l a r , one g e t s the r e l a t i o n

(1.13)

h(xeyl

I

( f m g l ( x m yl = f ( x ) g ( y l = g ( y l f ( x )

469

REPRESENTATION THEORY

1.

( b o r n t a t i o n r e t a t i o n " for the respective operators on H ) , f o r every decomposabZe tensor xay e E 6 P .

Proof. Suppose we are given an element h as in ( 1 . 9 ) . Then, by hypothesis for the net f v . ) G F , a s well as for the topology T (cf. 3 jeK Definition X;2.1rcond.(2.1);separate continuity of 9 is here enough), t h e family (1.14)

,

for every x e E. Accordingly, due to the hypothesis for H, an equicontinuous family i n Ls(H) (cf., for example, N . BOURBAKI [7:Chap. 3; p. 27, ThBorSme 21). NOW, for every element ~ = [ h ( x ' s y ' ) ] ( 2 ) E [Im(hI ] ( H I , the net ( 1 . 1 4 ) converges sinrply (namely, with respect to the topology s ) in Lls(HI (see also the subsequent Scholium 1 . 1 ; 1 ) 1 . S o one actually gets, for every xeE, the following:

yieZds a bounded n e t i n Ls(HI

(1.15)

lim h (x o v .I ( h (x'a y ' I (XI 3

3

= (by hypothesis for

)=

1im h ( x x 'o v .y ' I 3

(v.),~,T)

3

3

(2)

h(xx'o y ' ) l z I .

Now this can be extended by linearity to the "linear hull" of A E [Im(h)l (HI, i.e., the subspace, say, M a [ A l G H I while ( 1 . 1 5 ) is obviously linear with respect to x e E . Thus , for every x B E , one gets a linear map , say, f ( x ) , of M into H given by ( 1 . 1 5 ) , for every S e H. S o l o n the basis of the preceding, one g e t s , i n effect, a Zinear map f : E+

(1.16)

L(M,fi)

which is of course independent of t h e p a r t i c u l a r approximate i d e n t i t y ( v . ) i n F 3 appeared in ( 1 . 1 5 ) Furthermore, since ( 1 . 1 4 ) is still an equicontinuous family in the space SfM,HI, one gets in fact, from ( 1 . 1 5 ) , a map f(x/€.C(M,H) for every x e E (see N . BOURBAKI [7: Chap. 3 ; p. 25, Corollaire]). So in view of ( 1 . 1 0 ) it can extended by continuity (uniquely) to an element f ( x ) e LIH) for every X E E (we retain the previous notation for the extended map too). Consequently, one thus obtains a Zinear map

.

( 1 .17)

f :E-+Ll(H)

and this is in fact the desired element in (1.1 1 ) , as we next prove. Thus, because of ( 1 . 1 5 ) , we may write f ( x ) = limh(xoo.), for any 3 3 X B E ,where the respective limit is taken in S(HI with respect to the simple convergence in M % [ A ] . However, since ( 1 . 1 4 ) is an equicontinuous family in S s ( H ) , the topology of simple convergence in H coincides with that of simple convergence in the total subset A of H(cf. ( 1 . 1 0 ) ;

470

XIV SPECTRA AND APPROXIMATE IDENTITIES

also N. BOURBAKLT [7: Chap. 3; p. 23, Proposition 5 1 ) . So one finally defines (1.17) by the relation

fix) = lim h(2ov.j 1 L ~ ( H ,I

(1.18)

3

3

for every xeE. We prove next that the map f :E --+ Ss(Hl , given by (1.18), is continuous. Indeed, for any continuous semi-norm N on f s ( H I , one gets the following, on the basis of (1.18), standard argumentation, and in conjunction with the hypothesis for @ and ( v . 1 ; 3

N ( f (x)) = N(lim hlx o v .) ) = lim N(h(x

j

=

J

j

Q

v . ) I C sup N(h(x av .I J

j

3

)

5 ~*P(x, )

I k*X.supq(v.))*p(x) 3

for some a > 0, p a continuous semi-norm on E , and r a continuous seminorm of the topology T , and for any XEE. This proves our assertion. On the other hand, we prove further that (1.17) defines, in effect, a morphism of the respective algebras, so that by the foregoing one thus obtains the following relation (1.20)

f E HornIE, L s ( H I ) .

Indeed, from a repeated application of (1.181, one gets for any elements x, 1c' in E. flxx') = limh(xx'@v2) = limh(xx'o 1imv.v I j' 3 j ' j " j = limIlimh(xx'ov.v I ) =lim(lim (h(xov.)-h(x'ov )I) j 3' 3 j ' j j' 3 j' = lim Ih(xov.J-limh(x'o v I ) = lim(h(xapv.I.f(x'I) 3 j 3 j' j' j = lim Ih(xsv.I).f(x'l = f(xI-f(x'i , Li 3

which proves our assertion for f and hence finally (1.20). NOW, a similar argument to the preceding and use of any (bounded) approximate identity luili I in E , yields a map (1.21)

g e Horn(F, L s ( H ) )

such that one has, for every y e F , (1.22)

g(yI = lim h ( u i o y l 2

I

Ls(H)

.

Finally, for every decomposable tensor x@ y E E@ F, one obtains hlx o yj = h((limxui) o y ) = hf lim (xuiQ y I ) i 2

1.

= limh(xu.my) = l + m h ( x u i o l i m v . 2

i

(1.23)

471

REPRESENTATION THEORY

j

2

.

2-

I =lim(limh(xu.m v . y I ) i

j

"

3

= 1i.m(1ifn(ib(x o v .j h(u,o y ) ] ) = Iim ((1i.m PI (x Q v . i I . h(ui 3

2.

3

z

3

3

Q

= (1 imh(x o v .I ) . (lim h (uia y / I = f (XI .g(yI = (fa g i ( x m y i j 3 i

y))

.

NOW, this can be extended further by linearity to give the desired relation ( 1 . 1 2 ) Thus, ( 1 . 2 3 ) already provides one half of ( 1 . 1 3 ) : the rest of it can be obtained of course similarly to the last relation, and this completes the proof of the the0rem.I

.

Scholium 1.1.-

1 ) Concerning the form of the

element 5 =_ h(x'o y ' I ( z I E A = [Im(h)](HI applied in the preceding proof (see ( 1 . 1 5 ) ) , one may consider a simpler (equivalent) form of ( 1 . l o ) , as follows from ( 1 . 2 4 ) below. This was applied, in fact, in ( 1 . 1 5 ) . So, if E , H are topological vector spaces with the completion of E l then f o r every element h E I: (E, C. (HI) one g e t s the r e l a t i o n

-

(1.24)

[ h ( z ) ( H ) ]=

-.

Indeed, by hypothesis for h , one has the relation

which entails one half of ( 1 . 2 4 ) , the rest being obvious. 2 ) The hypothesis for the (joint) continuityof @ has been applied, in fact, to obtain ( 1 . 1 9 ) . So by considering suitable locally convex algebras E and F(cf. e.g. Chapt. 1;Lemma 4 . 2 ) , one might assume instead on E8F any (locally convex) compatible tensorial topology T , for which E 8 F is a (comT plete) locally convex algebra. 3 ) Arguing still in the context of the previous Theorem 1 . 1 , let us assume that h

where c denotes the topology of precompact convergence in H . Then, one proves t h a t f 8 Ham(E, Sc(HI) (and a similar conclusion for 9 ) : Indeed, one has (1.25)

fix) = lim h i x m v . ) I I:,(HI , j

3

4 72

XIV SPECTRA AND APPROXIMATE IDENTITIES f o r e v e r y x 6 E (and s i m i l a r l y f o r g ) , s i n c e on t h e equicontinuous family (h(xta v.) I C C(H) one h a s c = s 3

( c f . N. BUURBAKI [7:Chap. 3; p . 23, P r o p o s i t i o n 5 1 ) . Thus, a n obvious i n t e r p r e t a t i o n of

(1.19) s u f f i c e s

now t o y i e l d ( 1 . 2 5 ) . I n t h i s r e s p e c t , t h e case where one c o n s i d e r s ,

Monte2 space i n t h e p l a c e of H, h a s c e r t a i n l y a s p e c i a l i n t e r e s t : So i n t h a t c a s e one h a s i n LlH), b = c = s (Banach-Steinhaus; c f . a l s o e . g . J . H O R V h [l: p. 229, C o r o l l a r y ] ) . I n compensat i o n , w e t h e n g e t s t r o n g e r v e r s i o n s of t h e p r e v i o u s r e s u l t s t o t h e e x t e n t t h a t t h i s c o n c e r n s rep r e s e n t a t i o n t h e o r y of t o p o l o g i c a l a l g e b r a s i n more g e n e r a l (non-normed!) l o c a l l y convex s p a c e s . ( S e e , f o r i n s t a n c e , A . MALLIOS [ll] and R . A . HIRSCHFELD [1;2]). i n particular, a

W e come now t o a u s e f u l consequence of

( 1 . 1 8 ) , a s happens w i t h

t h e f o l l o w i n g r e s u l t ( i n d e e d , an e x t e n s i o n t o t h e p r e s e n t framework of a f a m i l i a r f a c t i n c l a s s i c a l C * - a l g e b r a s t h e o r y ) . So w e have.

Corollary 1.1.

Suppose that the conditions of the preceding Theorem 1.1 are

satisfied. Then ( a p p l y i n g t h e c o r r e s p o n d i n g n o t a t i o n ) , one gets

l i m f ( u . l = l i m g ( u . ) = i d H I Ss(H) ,

(1.26)

i

z

j

3

where i d H denotes the identity opemtor on H .

Proof. I n view of ( 1 . 1 8 ) , one h a s , f o r e v e r y element 5

h(x'oy')(zl

EA

f (ui) (h(x'ta y ')

(2))

= l i m h(ui ov . I ( h l x ' o y '1 ( 2 ) ) j

3

= limh(u.x'ov.y')(z) = h(u.x'm1imv.y ) ( z l j

2

3

=

h(Ui

x'o y ')

(2)

.

j

3

T h e r e f o r e . one o b t a i n s

( l i m f (ui))(h(x' i

0y

') ( z ) )

= l i m (f(uil(h(x'w y ' l l z ) ) ) i

= limh(uix'@ y')(.zl = h ( l i m u . x ' e y')(z) = h(x'oy')(z) i i 2

.

Now, t h i s p r o v e s i n f a c t t h e f o l l o w i n g r e l a t i o n (1 -27)

limf(ui) i

= idH,

i f w e r e s t r i c t o u r s e l v e s t o t h e t o t a l s u b s e t A of H ;

so one c o n c l u d e s

473

REPRESENTATION THEORY

1.

t h e l a s t r e l a t i o n f o r t h e whole H ( w i t h r e s p e c t t o L s ( H ) ) , b e c a u s e by t h e hypothesis f o r t h e n e t ( u i l ,

,

t h e s p a c e H , and by t h e c o n t i n u i t y

t h e n e t (f ( u i l Ii

I d e f i n e s a n e q u i c o n t i n u o u s s u b s e t of S f H ) ( c f . a l s o N. BOURBAKI [7: Chap. 3; p. 23, P r o p o s i t i o n 5 1 ) . F u r t h e r m o r e , a s i m i l a r argument i s i n f o r c e c o n c e r n i n g t h e n e t ( g ( v j I ) j , K , w h i c h t h u s i m p l i e s t h e rest of ( 1 . 2 6 ) , a n d w i t h it a l s o t h e end of t h e p r o o f . I

of f

Keeping s t i l l t h e n o t a t i o n o f t h e above Theorem 1 . 1 , l e t u s den o t e by Horn IE

(1.28)

$ F,

LsfH)

J0

t h e subspace of H o m ( E 6 F , S s ( H ) ) , defined by the condition ( 1 . 2 0 ) . T

Furthermore,

d e n o t e by

(HomlE, Cs ( H ) )

(1.29)

X

HomlF, Ss ( H I I lo

t h e subspace of t h e Cartesian product space Hom(E, L s ( H ) I i z e d by ( 1 . 1 3 ) and the r e l a t i o n

X

H o m ( F , f s ( H ) I c ham c te r -

[Im(fogl(H)] = H .

(1.30)

Thus, w e h a v e now t h e f o l l o w i n g .

Lemma 1.3. Suppose t h a t t h e conditions of the previous Theorem 1.1 are sat-

i s f i e d . Moreover, assume t h a t the l o c a l l y convex aZgebras E , F have continuousrnuZtip l i c a t i o n s such t h a t E S F i s a (complete) l o c a l l y convex aZgebm having a l s o a conT tinuous mul t i pl i cation. Finally, suppose that t h e topology ‘I s a t i s f i e s the respect i v e condition t o XII;(3.3) with G = L s ( H J . Then, one g e t s the reLation Hom(E! F , Ss(HIJo = (Hom(E, S8(H)J

(1.31)

x

Hom(F, CsfH(Hl)IO,

which i s v a l i d wi thin a b i j e c t i o n . Proof. O n t h e b a s i s of (1 . 1 3 )

h*(f,gJ i s 1-1.

, one

c o n c l u d e s t h a t t h e correspondence

between t h e s p a c e s ( 1 . 2 8 ) a n d ( 1 . 2 9 1 , O n t h e o t h e r h a n d , b e c a u s e of X I I ; ( 3 . 3 )

p r o v i d e d by Theorem 1.1,

, the

above rel. (1.13)

,

a n d t h e h y p o t h e s i s f o r t h e a l g e b r a E g F , o n e g e t s f o r e v e r y p a i r (f,g! T belonging t o ( 1 . 2 9 ) , an element f o g = h e H o m ( E $ F , L s ( H ) J which by (1.30) ‘I

i s t h u s a n e l e m e n t of t h e s p a c e ( 1 . 2 8 ) . Moreover, i f (f‘, g’)

p a i r i n ( 1 . 2 9 ) c o r r e s p o n d i n g t o f o g (Theorem 1 . 1 )

, th en one

is that

obtains

from ( 1 . 1 8 ) a n d ( 1 . 2 6 ) (1.32)

= l + m f ( x ) g f v . ) = f ( x l ( l + m g ( v . ) I = f f x c ) . i d H = f(x) 3

3

3

3

f o r e v e r y x e E . So f ‘ = f , a n d s i m i l a r l y o n e o b t a i n s g’= g. Consequently,

4 74

XIV SPECTRA AND APPROXIMATE IDENTITIES

the correspondence provided by Theorem 1.1 between the spaces appeared in (1 -31) (which is 1-1 , as proved above) is in fact an onto map too, and this terminates the proof of the 1emma.I

Scholium 1.2.-

We remark that (1.311 is actuaZly t r u e w i t h i n a continu-

ous b i j e c t i o n of the respective topological spaces, when these spaces

are endowed with the corresponding "weak topologies". Indeed, this is a direct consequence of Lemma XII; 3.1. On the other hand, considering the space t e ; ( H ) as a topological a l gebra w i t h continuous muZtipZication , with appropriate and H (see, for instance, the previous Scholium 1.1 ; 3 ) ) and admitting f u r t h e r t h a t (the generalized spectra) (1 - 3 3 )

Hams ( E , S G ( H ) )

and

Horns (F, SG (HI)

are ZocaZZy equicontinuous, we could also obtain the c o n t i n u i t y of the inverse map of ( 1 . 3 1 ) from an application now of Lemma XII; 3 . 2 .

In this respect, we still note that in several particular instances, yet important in the applications, a "weakly continuous representation" @ of a given topological (non-normed!) algebra E in a Hilbert space H is actually "strongly continuous"; i.e., an element (1 - 3 4 )

@ E Horn

(E, S s ( H ) )

.

So this happens, for instance, if E is a barrelzed l o c a l l y convex aZgebm. (See e.g. R.M. BROOKS [51 , G . LASSNER [l] , H.J. BORCHERS- J . YNGVASON [I]). 2. Elementary measures o f representations

We start with the following extension of the classical Riesz representation Theorem to the case of completely regular spaces. (For its

classical version see e.g. W . RUDIN [l: p. 139, Theorem 6.191. We also refer to N. DUNFORD- J.T. SCHWARTZ [l] for details of the terminology applied in the next lemma). This is, indeed, a basic tool throughout the sequel. So we have.

Lemma 2 . 1 , Let X be a completely regular space and

CJX) t h e

ZocaZly con-

vex space of complex-valued continuous f u n c t i o n s on X endowed w i t h the topoZogy of compact convergence in X (cf. Chapt. I; Example 3.1).

Furthermore, l e t M c ( X ) be

t h e (vector)space o f countably a d d i t i v e Bore1 measures on X whose t o t a l a t i o n s are regular measures w i t h compact supports.Znnen, (2.1)

(

one g e t s

e c ( x ) ) ' = pi ,

w i t h i n an isomorphism of vector spaces; the l a t t e r is given by the reZation

vari-

2.

MEASURES OF REPRESENTATIONS

475

lifl = / f ( z ) d p t ( x ) ,

(2.2)

f o r every f e CJXI. Here p l e M c ( X I

corresponds t o 1 e ( c c ( X I 1' (topological

dual space). Proof. See R.M. BROOKS [2: p. 1 1 , Theorem 5 . 1 1 and/or W. DIETRICH,J r . [l : p. 203, Theorem 1 , iii)]

.I

Now, suppose we axe given a topological algebra E and a topological vector space H , and let S s ( H ) be the topological algebra of continuous linear endomorphisms of H in the topology s of simple convergence in H (see Chapt. I; Example 2 , ( 1 ) 1. Thus, consider now an element T e Horn (Z, S s ( B ) )

(2.3)

(cf. Chapt. V ; Definition 8 . 1 ; "weakly continuous representation" of E in H . But see also ( 1 . 3 4 ) ) . Moreover, for every a e H , consider the associated continuous linear map

a^

(2.4)

: I~(H)+

H :u + + Q ~ ( u:= ) u(a),

.

and finally let 1 eH' (topological dual of H) So, for any triad ( T , a , l ) as before, one gets an element of E', i.e., a continuous linear form on E by the relation loa^oT.

(2.5)

Thus, by Lemma 2 . 1 ) , we e ( C c ( X ) ) ' . That p T ( a , 1 I E Me(Xl,

restricting ourselves to the case that E i e c ( X ) (cf. get for every triad IT, a, 1 ) , as above, an element 2 0 2 0 T is, according to the same lemma, a measure on X, say, so that one has by ( 2 . 2 )

( l o a " o T ) ( f I = l([T(f)l(aI)= [ u T ( a , Ll

(2.6)

for every f e

cc(X) . Therefore, f o r

]If)

= / f ( ; c l d p T ( a , 1)

,

every representation

T : ec(X)+Ls(H/,

(2.7)

one g e t s a map

DT

(2.8)

:H x H ' -

Mc(X)

such t h a t (2.6) i s t r u e .

So one is led now to the following. D e f i n i t i o n 2.1.

The image of ( 2 . 8 ) ,

Im(pT),

is said to be the s e t

of elementary measures of t h e representation T I while an element u T ( a , Z)eIm(pT)

is called the elementary measure of T associated to the pair ( a , 1 ) E H

x

as above. Thus, it is now our main objective in the following lines to

H',

XIV SPECTRA AND APPROXIMATE IDENTITIES

476

point out that the s e t of elementary measures of a given representation T I as in (2.7) , m y be viewed as a s e t of "spectral measures" on X (cf. Definition 2.2 below, and also Lemma 2.3). So it is just from the point of view of the last statement, and in connection with the applications considered in the next chapter, that the content of this section fits the previous framework, and, of course, that of the following Chapter X V (cf. e.g. XV;(1.7)). Thus, we first have the following.

Lemma 2.2. Let E

Cc ( X )

(cf. Lemma 2.1)¶ H a topologicaZ vector space and a e H , as well as a representatLon 2

T

(2.9)

Then, f o r every f E c c ( X ) ,

B Horn(

Cc(xi, s S ( x ) ) .

t h e map

-

[ l J T ( a , 1 I ( f ) : "H

(2.10)

C

given by the r e l a t i o n (2.11)

[ v T ( a , 1 ) 1 ffl = ( cf . (2.6) 1 l( [ T ( f ) ]( a ) ) ,

w i t h 1 e H; d e f i n e s a l i n e a r form on H ' . In p a r t i c u l a r , t h e map 12.10) provides an eZement of (Hil'(where Hs' denotes the weak topological dual space of H ) . Furthermore, i f H i s a Hausdorff l o c a l l y convex space, then ( 2 . 1 0 ) d e f i n e s an element of H , say, (2.12)

such that (considering (2.12) as a function of a € H ) one g e t s b(., f ) = Tlfl

(2.13) f o r every f e

e S(H) ,

Cc(x).

It is clear that (2.1 1) defines a linear form on H ' for any given triad ( T , a , I ) , as above. This is also continuous in the weak topology of H' as follows easily from the respective definitions and (2.6). So it does yield an element of (Hi)'. Therefore, in the particular case when H is a Hausdorff locally convex space, one has ( H ' I ' = H (within a (canonical) linear isomorphism S provided by the "evaluation map"; see e . g . J. H O R V h [l: p.187, Proposition 21). So one obtains (uniquely) an element of H I by the relation Proof.

(2.14)

[uT(a, .ll(f)

b l a , f ) = b e H.

Consequently, for every leH', one has (2.15)

477

2. MEASURES OF REPRESENTATIONS

Moreover, the last relation yields also (Hahn-Banach) the following b(a,f ) = [T(f)l(a)

(2.16)

for every aEH,and hence the relation ( 2 . 1 3 ) as well which completes the proof. I On the other hand, if u T ( a , 1 ) E Mc(Xl is the (elementary) measure corresponding to a given triad ( T , a , 1 ) (see ( 2 . 5 ) ) , one can consider it as a (complex-valued) function on the o-algebra B(Xl of Borel sets of X. Thus, in a similar way as in Lemma 2.2 one obtains an operatorvalued function b(.,-c)&tfi7)l

(2.17)

for every Borel set

T E

U X ) ; moreover, one has

l ( b ( a , r)1= [u,(a, 1 ) 1 ( ~ 1= j X , d u T ( a , 1 )

(2.18)

.

So we classify the previous notion through the following.

Definition 2.2. Suppose that we have the context (of the first part) of the previous Lemma 2.2. Then, ( 2 . 1 7 ) is said to define an idempotent-valued measure on X I if the next two conditions are satisfied: b ( . , p n - r l = b(-,plob(.,rl

1)

,

for any Borel sets p , ~in B ( X ) . (2.19)

l(b(a, .)1

2)

for any (a, 2 )

E Hx

H’

E

Mc(Xl

,

(cf. (2.8)).

(For the notion in question, the terms projection-vaZued measure are also in use).

, or

yet

spectmi! measure, on X

Thus, we have now the following.

Lemma 2.3. Let us assume t h a t t h e conditions of t h e f i r s t part of t h e preceding L e m 2. 2 are s a t i s f i e d . Then, t h e n a p 12.171 y i e l d s a spectmZ measure on X (Definition 2.2).

(2.17) (2.20)

Proof. First we remark that from the very definition of the map one concludes that l ( b ( a , . ) I = b T ( a , 1) E M e ( X ) ,

for any ( a , 1 ) E H x H’ , which of course proves ( 2 . 1 9 ) ; 2). On the other hand, one has (2.21)

Z(T(fgl(a)l =

f ( x ) g ( x i d v T ( c z , 1) = / f ( d d v , ( a , 1 )

for any f ,g in Cc(Xl.Here we set

I

XIV

478

SPECTRA AND APPROXIMATE I D E N T I T I E S

v ( a , 1 ) := g.pT(a, 1 )

T

Thus, w e s t i l l have a n e l e m e n t of M c I X l , t h e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n

a s f o l l o w s from (2.1) and

cc(X) (see

a l s o N. BOURBAKI [8:

one c o n c l u d e s from (2.6)

Chap. 3; p. 50, 51, no 42). Moreover,

l(T(fgl(al) =

.

Z( T ( f I T ( g ) ( a l l = l ( T ( f ) ( T ( g ) ( a l ) J

(2.22) = f 1x1 dpT ( T ( g )( a ) , 1)

I

CctX).

f o r any f,g i n

Now, i n v i r t u e o f

(2.21) and (2.22). one o b t a i n s

/ g ( x ) d p T ( a , 0 n - r ) = (by (2.18))/ g ( x ) d Z ( b ( a ,

pnT)

= (by (2.18)) 1 ( b (Tlgl ( a ) , P I ) = [ b ( - ,P ) * ( l ) 1 (Tlgl ( a ) ) = / g ( x l d ( b ( . , P I * ( Z l ) ! b ( a , T ) l = / g ( z ) d l ( b ( b ( a , T), PI)I f o r every

g e Cc(Xl

,

where b ( . , ' r l *

s t a n d s f o r t h e a d j o i n t o p e r a t o r of

(2.17). Thus, one h a s Z ( b ( a , pn'rl) = l ( b f b ( a , T J , P), f o r every l e H ' ,

so t h a t (Hahn-Banach) one o b t a i n s b(a, pn'rl = b(b(a,

f o r every a e H .

TI,

P),

S o w e have proved i n f a c t (2.19) ; 1 )

,

and t h i s com-

p l e t e s t h e proof. I W e summarize t h e p r e c e d i n g i n t o t h e f o l l o w i n g more c o n c l u s i v e

r e s u l t . That i s , w e have.

Theorem 2.1

. Let

E

c (X) ( c f .

Lemma 2.1) and H a Hausdorff l o c a l l y convex

space. Then, denoting by M c ( X , PlHll

(2.23)

t h e s e t of s p e c t r a l measures on X ( c f . D e f i n i t i o n

Ham( Cc(Xl,f s ( H ) )

(2.24)

2.2), one g e t s t h e r e l a t i o n

= Mc(X, P ( H I ) ,

w i t h i n a b i j e c t i o n , given by ( 2 . 2 0 ) . Furthermore, one has t h e r e l a t i o n l ( [ T ( f ) l ( a l )= ] f ( x ) d l ( b ( a , .I)

(2.25) f o r every ( a , 1 ) e H x H ' ,

with f e

Proof. For e v e r y t r i a d an e l e m e n t of

,

C'(X). ( T , a , 2)

as i n (2.5) , one g e t s from (2.I71

(2.23)(see Lemma 2.3). Now, it i s an e a s y consequence o f

(2.12) , (2.13) t h a t (2.20) d e f i n e s , i n e f f e c t , a one-to-one between t h e two s e t s a p p e a r e d i n (2.24).

correspondence

MEASURES OF REPRESENTATIONS

479

W e p r o v e now t h a t (2.20) d e f i n e s an o n t o map:

Thus, by c o n s i d e r i n g

2.

-

a n i d e m p o t e n t - v a l u e d measure on X ( c f . ( 2 . 1 7 ) ) , t h e map

(a, 2,f)

(2.26)

p ( a , 2, f):= / f ( x ) d l ( b i a , - 1 )

c

d e f i n e s a c o n t i n u o u s 3 - l i n e a r form on H x P ’ x (X). The a s s e r t i o n f o l l o w s from ( 2 . 2 0 ) , ( 2 . 2 1 ) c o n c e r n i n g t h e v a r i a b f e f E cc(Xfrom I; (2.17) f o r a e H , and f i n a l l y , c o n c e r n i n g l € H ’ , from t h e h y p o t h e s i s f o r Lemma 2 . 2 ) .

H I

(see a l s o t h e p r o o f o f t h e above

namely, from t h e r e l a t i o n Ei =(H‘)’

T h u s , o n e d e f i n e s a n e l e m e n t T e .C( c c I X l , - C s ( H ) )

by

2 ( T ( f ) ( a ) ) = p ( a , 1, f) = j f ( x ) d Z ( b l a , . ) ) ,

(2.27)

so t h a t one g e t s T = Ula, 2 ,

(2.28) f o r every p a i r

(a,

Z)

E H ~ H ‘ ; m o r e o v e r , t h e c o n t i n u i t y of t h e map T i s

h e r e deduced from t h a t o f

(2.26).

Furthermore, w e prove t h a t , a n e l e m e n t i n Horn( q ( X ) , l , ( H ) ) .

(a, l i e HxH‘,

f o r any

Now t h i s i s a c t u a l l y d e r i v e d from t h e

f a c t t h a t T commutes w i t h t h e s p e c t r a l measure b ( . ,T ) ; i . e .

,

one h a s

T ( f ) b ( - ,P/ = b ( . , P ) T ( f ) ,

(2.29) f o r any

(2.28) y i e l d s

f e Cc(Xl

and

p E B(X).

I n d e e d , f o r a n y 1, E H ‘ ,

one g e t s from

(2.27) t h e following Z ( [ T l f ) l ( b ( a , p ) ) ) = / f ( x ) d l ( b ( b ( a , p),

-1))

= ( c f . (2.19); 1 ) ) j f ( x l d l ( b ( b ( a , p n r ) ) = ! f ( x ) d Z ( b ( b ( a , T l , P) = / f ( x ) d ( b ( . , P ) * ( l l ) ( b ( a , r ) ) = [ b ( . , P)*(Z)I(T(f)(a)) = 2 ( b ( . , p ) ( T ( f ) ( a ) ) J= Z ( b ( T ( f l ( a l , P I )

.

T h e r e f o r e (Hahn-Banach), one o b t a i n s T ( f / 1 ( b ( a , 0 ) ) = b f T ( f )( a ) , P/

I

f o r e v e r y a e H , which i n f a c t p r o v e s ( 2 . 2 9 ) . On t h e o t h e r h a n d , o n e has

Z ( b ( T ( g l ( a l , P)) = ( b y (2.29)) l ( T ( g l ( b ( a , 0 ) ) )

= ( b y (2.27)) j g ( d d Z ( b ( b ( a , p), Consequently,

f o r a n y f, g i n

c,(X) , and

T))

= j g ( x ) d l ( b ( a ,r l ) .

f o r e v e r y 2 e H’

2 ( T ( f ! ( T ( g l ( a ) I ) = 1 f l x ) d Z ( b ( T ( g )( a ) , T ) )

= / f ( x ) g ( x l d Z ( b ( a , T ) ) = t(T(fgl(all. T h e r e f o r e ( s t i l l by Hahn-Banach)

,

one o b t a i n s

( T ( f ) T ( g ) f a ) = T ( f g l ( a ),

, one

has

480

XIV

SPECTRA AND APPROXIMATE I D E N T I T I E S

f o r e v e r y a e H , which i s j u s t o u r a s s e r t i o n f o r

T and t h i s completes

t h e proof of t h e theorem. I I n t h i s r e s p e c t , w e s t i l l note t h e following, being a d i r e c t consequence of D e f i n i t i o n 2.2. Namely, (2.30)

Spectra2 measures commute with themselues and are a l s o idempotent operators.

Indeed, one h a s from (2.19); 1 ) (2.31)

b l . , p n r l = b l . , p ) . b ( . , ~ )= b ( . , r n p l = b l . , T l b l * , P )

,

f o r any p I T i n B ( X l . Moreover, i n view o f t h e same c o n d i t i o n , one gets

f o r every p e B ( X )

(2.32)

b l . , p ) = b ( . , pnpl = b l . , p ) . b ( . , ~ ) = b l . , p )2 .

R e c a p i t u l a t i n g t h e p r e v i o u s d i s c u s s i o n , w e n o t e t h a t w e have t h u s e n c o u n t e r e d a method o f i d e n t i f y i n g " g e n e r a l i z e d c h a r a c t e r s " ( i . e . , r e p r e s e n t a t i o n s ) of c e r t a i n p a r t i c u l a r t o p o l o g i c a l ( f u n c t i o n ! ) a l g e b r a s through " i n t e g r a l r e p r e s e n t a t i o n s " ( c f . (2.25)). Now t h i s , i n t u r n , w a s accomplished by d e f i n i n g t h e c h a r a c t e r s i n q u e s t i o n a s " s p e c t r a l measures", i n f a c t , on t h e s p e c t r a of t h e i n i t i a l l y g i v e n a l g e b r a s ( c f . Theorem V I I ; 1.2). So t h i s t e c h n i q u e w i l l be a p p l i e d f u r t h e r i n the next chapter, systematically, i n order t o give us, within t h e p r e v i o u s e x t e n d e d framework, c e r t a i n fundamental c l a s s i c a l r e s u l t s

(see e . g . S e c t i o n s 5, 6 o f t h e n e x t c h a p t e r ) .

481

Topological Algebras with Involution. R e p r e s e n t a t i o n T h e o r y (contn'd.)

CHAPTER XV

W e d i g r e s s a b i t i n t h i s f i n a l c h a p t e r from o u r g e n e r a l p l a n n o t t o i n c l u d e , namely, i n o u r s t u d y ( t o p o l o g i c a l ) *-algebras, a s u b j e c t , howe v e r , which d e s e r v e s even a s e p a r a t e t r e a t m e n t i n i t s e l f . Thus, i n t h e s e q u e l w e a r e mainly concerned w i t h commutative ( t o p o l o g i c a l ) a l g e b r a s endowed w i t h a n i n v o l u t i o n . Our main o b j e c t i v e h e r e i s t o a p p l y Theor e m 2 . 1 of t h e p r e v i o u s c h a p t e r , w i t h i n such a framework, t h a t i s , t o c o n s i d e r t h e spectrum of an a l g e b r a , a s above, a s t h e ( c o m p l e t e l y reg u l a r ) s p a c e on which s p e c t r a l measures a r e d e f i n e d and t h e n f i 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 of such measures. The r e s u l t s t h u s d e r i v e d a r e f u r t h e r a p p l i e d , i n p a r t i c u l a r , t o g e n e r a l i z e d group a l g e b r a s (see Chapter X I f o r t h e l a s t t e r m ) .

1. Preliminaries Suppose t h a t w e have a (complex) a l g e b r a E w i t h an i n v o l u t i o n d e n o t e d by

" * ' I .

That

i s , w e have w i t h E an i n v o l u t i v e h e r m i t i a n s n t i -

(auto)morphism o f E , o r y e t a self-map o f E , x ~ x * x, E E , such t h a t t h e following conditions a r e s a t i s f i e d :

x** = X Ax*+ iiy* iii) (xu)* = Y*Z*, i) (x*)*

(1.1)

ii) ( h x + ~ ! y ) = *

f o r any x, y i n E , and h , p i n C. H e r e 'I - " d e n o t e s "complex con jugat i o n " ( i . e . , t h e i n v o l u t i o n i n C!). I n p a r t i c u l a r , such an a l g e b r a E i s c a l l e d a *-algebra. A *-morphism between two g i v e n * - a l g e b r a s E , F i s an element $ e Hom(E, F ) which " p r e s e r v e s i n v o l u t i o n " , i .e , such t h a t

.

@(x*) = (@(x) )*,

(1.2)

S E E ,

( w e simply w r i t e @ ( x ) * u s i n g , by an o b v i o u s abuse of n o t a t i o n , t h e

same symbol f o r b o t h i n v o l u t i o n s i n t h e a l g e b r a s E and F).We d e n o t e t h e r e s p e c t i v e set of *-morphisms o f E i n t o F by Hom*(E, F ) . by a topoZogicaZ *-algebra E , w e mean a t o p o l o g i c a l a l g e b r a E which i s a l s o a * - a l g e b r a ; w e make no assumption of any c o n t i n u i t y NOW,

f o r t h e i n v o l u t i o n i n E , u n l e s s it i s i n d i c a t e d o t h e r w i s e . So a t o p o l o g i c a l * - a l g e b r a E i s s a i d t o be s e l f - a d j o i n t , i f

the

482

ALGEBRAS WITH INVOLUTION

XV

r e s p e c t i v e G e l ' f a n d map of

E

9:E

(1.3)

Cc(m(E))

+

i n t h i s respect, we consider

( c f . C h a p t . V; (1.3)) i s a *-morphism; t h e r a n g e of $ a s

a * - a l g e b r a by complex c o n j u g a t i o n of f u n c t i o n s .

Thus, e q u i v a l e n t l y ,

3 satisfies

s

(1.4)

the relation (x*i = $ ( X I I

t h e Gel'fand t r a n s f o r m a l g e b r a of E (see

f o r every xeE. Therefore,

C h a p t . V; (1.6) i s t h u s a * - s u b a l g e b r a o f t h e r a n g e o f under (complex) c o n j u g a t i o n . On t h e o t h e r hand.,

by a *-representation

of

g,i..e.

I

"closed"

a given *-algebra

E

i n a H i l b e r t s p a c e H , i s meant a n e l e m e n t @

(1.5)

€

Hom*(E, L ( H ) )

I

where L ( H ) i s t h e * - a l g e b r a of bounded l i n e a r o p e r a t o r s on H ( " r e p r e -

.

s e n t a t i o n space of @ ) I n p a r t i c u l a r , i f E i s a t o p o l o g i c a l * - a l g e b r a , by a n ( G i l c o n tinuous *-representation of E i n H I w e mean a n e l e m e n t @ e Hom*(E, LI'H)) n ffom(E,

(1.6)

SG(II))

where G d e n o t e s a n y o f t h e s t a n d a r d " o p e r a t o r t o p o l o g i e s " i n S ( H ) (see, f o r i n s t a n c e , M . A . NAYMARK [1: p. 4491). The s e t d e f i n e d by (1.6) i s den o t e d by Ham*(E, L a ( H ) ) . N o w by c o n s i d e r i n g t h e ( l o c a l l y rn-convex)

m a XIV; 2 . 1 )

*-algebra

c c f X ) (Lem-

a n d a l s o an e l e m e n t T

(1.7)

E

Ham*

o n e g e t s a s p e c t r a l measure on X

(

cc(X), Ls(HII

(Theorem X I V ; 2 . 1 )

,say,

P T I which i n

f a c t i s a (bounded) s e l f - a d j o i n t o p e r a t o r on H I namely a projection

( i . e . , a seZf-adjoint idempotent e l e m e n t ) i n L ( H ) : T h i s i s a n e a s y c o n s e quence of t h e h y p o t h e s i s t h a t T i s a *-morphism a s w e l l a s o f t h e f a c t t h a t it c a n b e e x t e n d e d a s s u c h ( i n v o l u t i o n p r e s e r v i n g e x t e n s i o n ) t o 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 s of t h e elements o f LOOMIS [ l : p. 93, Theorem] )

B ( X ) ( c f . also L . H .

.

Thus, l e t u s c o n s i d e r a semi-simple s e l f - a d j o i n t

topological

* - a l g e b r a E a n d an e l e m e n t T i n t h e s e t (1.5); moreover, assume t h a t T i s continuous with respect t o t h e i n i t i a l topology, say,

on E by t h e G e l ' f a n d map t h e map (1.8) y i e l d s a n element of

3 (cf.

T',

defined

(1.3)), and t h e t o p o l o g y s i n S ( H ) . Then

$ = T 0 F - I : EA-

ffam * ( P A , L s ( H / I

Ls(HI

,

where E A = $ I E I

carries the

2 . TOPOLOGICAL *-ALGEBRAS. REPRESENTATIONS

r e l a t i v e t o p o l o g y from t h e r a n g e o f

483

3.

F u r t h e r m o r e , by h y p o t h e s i s f o r E , E n becomes a s e l f - a d j o i n t

sub-

a l g e b r a of c c ( n 2 ' ( E ) ) ; it i s a l s o " s e p a r a t i n g " and " n o n - v a n i s h i n g " on ?YY(El. ( T h e l a t t e r i s a d i r e c t consequence of t h e d e f i n i t i o n s of Enand

.

M ( E ) I r e s p e c t i v e l y ) C o n s e q u e n t l y , E" i s dense i n C,(??Z(E)) (Stone-Weiers t r a s s Theorem; see e . g . L . NACHBIN [4: p. 48, C o r o l l a i r e 2 3 ) . T h a t i s , one o b t a i n s (1.9)

so t h a t t h e map @ c a n be e x t e n d e d t o t h e whole r a n g e o f

g.

According-

l y , one c a n a p p l y t o t h e e x t e n d e d map @ and h e n c e , i n p a r t i c u l a r , t o t h e g i v e n r e p r e s e n t a t i o n T of E , t h e t e c h n i q u e d e v e l o p e d i n S e c t i o n 2

of t h e p r e v i o u s c h a p t e r . I n t h i s r e s p e c t , i f < , > denotes t h e inner-product i n t h e H i l b e r t s p a c e H , w e a p p l y t h e n o t a t i o n < T ( f l ( a l ,b > , w i t h a , b i n H , f o r a n e l e m e n t of t h e form X I V ; ( 2 . 1 5 ) (Fre'chet-Riesz Theorem; c f . , f o r i n s t a n c e , J. HORVhH [I: p. 42, P r o p o s i t i o n I ] )

.

The f o l l o w i n g r e s u l t i s now an immediate consequence o f t h e p r e c e d i n g d i s c u s s i o n and t h e t h e o r y d e v e l o p e d i n S e c t i o n X I V ; 2 . Thus,

w e have.

Lemma 1.1. Let E be a semi-simple self-adjoin-t topological *-algebra w i t h s p e c t m m(El, and H a Hilbert space. Then, every element ( 1- 1 0 )

T E tbrn*(E[

~ ' 1 , Ls(H))

gives r i s e t o a p r o j e c t i o n v a l u e d ( s p e c t r a l ) measure on ??Z(El, in such a manner t h a t one has t h e r e l a t i o n (1.11)

for any a , b in H and

XB

E. Moreover, i f

M ( m ( E l , P(H)) denotes the preceding

s e t of measures ( c f . a l s o XIV; (2.23)), one g e t s (1.12)

w i t h i n a b i j e c t i o n implied from ( 1 . 9 1 . I W e c o n s i d e r i n t h e n e x t s e c t i o n c e r t a i n p a r t i c u l a r examples where e l e m e n t s of t h e s e t ( 1 . 6 ) ( w i t h 6 = s ) belong, i n d e e d , t o ( 1 . l o ) .

2. Certain particular (commutative) topological *-algebras and their representations VJe f i r s t comment on t h e r e l e v a n t t e r m i n o l o g y c o n c e r n i n g l o c a l l y

484

XV

ALGEBRAS WITH INVOLUTION

rr-convex *-algebras satisfying an additional condition. Thus, suppose that we are given a locally m-convex algebra E , and let r = ( p a I, I be a fundamental defining family of (continuous) submultiplicative semi-norms for the topology of E (see Chapt. I; Proposition 3.2).Moreover, if the involution in E is continuous, it is easily seen that r may be supposed to satisfy the condition P,(X*l

(2.1)

= p,lxl

I

with X E E , and for every a € I . Thus, otherwise, the family r ’ = ( q a j a E I where one defines qa Ix)

(2.2)

supc PJX), p,(x*I

1,

E

I,

with x e E , is equivalent to the given family r , satisfying moreover the desired condition. (In this respect, see also R.M. BROOKS [2: p. 7, Theorem 3.31). A similar argument is certainly valid for any locally convex *-algebra (not necessarily locally m-convex one) having a continuous multiplication. Accordingly , c o n t i n u i t y of t h e i n v o l u t i o n i n any l o c a l l y convex ( resp. , l o c a l l y m-convex ) *-algebra is equivalent t o the condit i o n ( 2 . 1 ) , while suitable families are considered, as above. NOW, by a b c a l Z y nrconvex C*-aZgebra we mean a topological *-algebra E whose topology can be defined by a fundamental family r = of submultiplicative semi-norms (Chapt. 1;Theorem 3.1) in such a way that one has p,(x*xl

(2.3)

= (p,lxll

2 I

with X C E , and for every a f I . Furthermore, in virtue of the submultiplicativity of p a ’ s , one can prove (applying a similar argumentation to that in the “normed case“) that (2.3) is in fact equivalent with the seemingly weaker condition ipa(x)12

(2.3’)

5 p,(x*x)

.

In particular, the previous r e l a t i o n implies (2.1) and hence the continuity of the involution in E (cf., for instance, J. DIXMIERCI: p . 9,1.3.4]). On the other hand, given a locally m-convex C*-algebra E , if we consider the corresponding Arens-Michael decomposition of E , namely, E 5 lAm ia (cf. Chapt. 111; Theorem 3.1 ) , we conclude that the individual Banach algebras Ea are, in fact, C*-algebras, the involution in being transferred from the given one in E because of (2.1). Therefore, on the basis of the respective reasoning in the normed case (cf. e.g. C . E . RICKART [l: p. 187, Lemma 4.1.141 and/or V . P T k [l: p. 284, Theorem 10.111, one can prove that ( 2 . 3 ) (the so-called “C*-condit i o n “ , in the more general case considered) i s equivaZent t o h

2.

TOPOLOGICAL *-ALGEBRAS.

485

REPRESENTATIONS

pa(x*xl = pJxl .p,(x*),

(2.4)

f o r any x E E , and a E I. W e a r e now i n t h e p o s i t i o n t o p r o v e t h e f o l l o w i n g .

Lemma 2 . 1 . Let ( E ,

r= (pa)aEI I

be an advertibly complete locally m-convex

r’ = (qh I heK ) a locally rn-convex C*-aZgebra. Moreover, l e t Hom*(E, F ) . Then, for every h E K, one gets the relation

*-algebra and (F,

qh(T(x))

(2.5)

rE(x)

T

E

,

for every hermitian element x E E (i.e., x = x*), where r denotes the spectral E radius i n E ( c f . C h a p t . 11; ( 1 . 1 0 ) ) . Furthermore, one has the relation (qA(T(x)))‘

(2.6)

s

rE(x*x),

f o r any h e K and x e E .

proof.

aeI

F o r e v e r y A E K , t h e r e e x i s t s by h y p o t h e s i s f o r T , k > 0 and

s u c h t h a t one h a s qA(T(x)) 5 k-p,(x)

(2.7)

f o r every x e E (cf., f o r instance

,

,

.

HORVATH

[I : p. 97, P r o p o s i t i o n 21 ) Moreover, T(x) i s a h e r m i t i a n e l e m e n t of F i n c a s e x E E i s , so t h a t by c o n s i d e r i n g t h e Arens-Michael d e c o m p o s i t i o n o f F , i f FA i s t h e C * - a l J.

h

gebra corresponding t o he K

, one

KART [I: p. 187, Lemma 4.1.141,as

qh(T(x)l =

g e t s t h e following;

(see a l s o C. E . RIC-

w e l l a s C h a p t . 111; ( 4 . 6 ) ) .

)I [TIx)lAllh

r~~LT(x)lA)

= lim((1 [ T ( X ) I ~ / ~ = ~lim(qh(T(xn)))l’n ) ~ / ~ 5 (by (2.7))

n

f o r e v e r y h e r m i t i a n e l e m e n t x e E , so t h a t t h e d e s i r e d r e l a t i o n ( 2 . 5 )

i s now a d i r e c t c o n s e q u e n c e o f 1 1 1 : ( 6 . 1 ) . On t h e o t h e r h a n d , s i n c e x*x i s a h e r m i t i a n e l e m e n t o f E , e v e r y X E E , o n e g e t s , due t o t h e h y p o t h e s i s f o r F a n d ( 2 . 3 ) , (q,(T(x))) 2 = qX(T(x)*T(x))= q,(T(x*x)) 5 ( b y (2.5)) rE(x*x) f o r e v e r y x E E . So w e v e r i f i e d

(2.6)

,

for

,

and t h i s c o m p l e t e s t h e p r o o f o f

t h e lemma. I I n t h i s respect, we still note t h a t (2.6)

is n o n - t r i v i a l ,

c o u r s e , just f o r t h o s e ( h e r m i t i a n ) e l e m e n t s x e E, f o r which one rE(xI

<+wj,

i.e., bounded elements o f E (see C h a p t . 111; ( 6 . 4 )

,

and a l s o

. . Thus , c o n s i d e r i n g

C h a p t . V I ; D e f i n i t i o n 1 . 1 a n d Remark 1 1 )

of has

i n par-

t i c u l a r commutative ( b o u n d e d ) a l g e b r a s , w e g e t t h e f o l l o w i n g u s e f u l

486

XV

ALGEBRAS WITH INVOLUTION

c o n s e q u e n c e of t h e p r e v i o u s d i s c u s s i o n ( s e e a l s o Theorem 2 . 1

in the

sequel).

Corollary 2 . 1 . Let E be a c o m t a t i v e a d v e r t i b l y complete bounded ( E might be e.g.

a & - a l g e b r a ; c f . Theorem VI;1.3) l o c a l l y m-convex *-algebra w i t h a

continuous i n v o l u t i o n . Moreover, l e t F b e a l o c a l l y m-convex C*-algebra and

T

e

Hom*(E, F). Then, one g e t s

(2.8)

T

e Hom*(E[

Tml,

F),

where T~ denotes t h e (semi-normed a l g e b r a ) topology defined on E by t h e spec-

tr;L radius r proof.

E '

By h y p o t h e s i s and C o r o l l a r y 111;6 . 2 ,

the spectral radius

r E d e f i n e s a s u b m u l t i p l i c a t i v e semi-norm on t h e a l g e b r a E ( = B ( E ) ; c f . a l s o t h e p r e c e d i n g comment). Hence, E [ T , ]

i s a semi-normed

(see C h a p t . I ; S e c t i o n 3 ) , s o t h a t , i n v i e w of

algebra

( 2 . 5 ) , t h e map T re-

s t r i c t e d t o t h e ( r e a l ) s u b s p a c e Eh of t h e h e r m i t i a n e l e m e n t s of E i s a c o n t i n u o u s l i n e a r map of E C E [ T ~ ]i n t o P. T h e r e f o r e , t h e d e s i r e d hr e l a t i o n ( 2 . 8 ) f o l l o w s now from t h e c a n o n i c a l d e c o m p o s i t i o n of any z E E i n t o i t s hermitian components ( c f . , f o r i n s t a n c e , C . E . RICKART [ I : p. 1791 )

.I

2.1,

w e s t i l l remark t h a t i f E i s , i n p a r t i c u l a r , a &-algebra, then ( 2 . 8 ) i s

N o w , a r g u i n g w i t h i n t h e same c o n t e x t a s i n t h e above C o r o l l a r y

i n f a c t equivalent t o i" E Hom*IE, F ) (see P r o p o s i t i o n 111; 6.2). On t h e o t h e r h a n d , a p p l y i n g now a n a n a l o g o u s argument t o t h a t

cf t h e p r o o f of Lemma 2 . 1 ( s e e a l s o C . E . RICKART [ I : p. 11, Lemma 1 . 4 . 2 ] ) , w e c o n c l u d e t h e f o l l o w i n g : For any a d v e r t i b z y complete l o c a l l y m-convex algebra E and a uniform ( l o c a l l y m-convex) algebra F ( c f . D e f i n i t i o n V I I I ; 5.1), T E H o m ( E , F ) i m p l i e s ( o r y e t i s e q u i v a l e n t , i n case of a commutative &-algebra E , t o ) 2'f H o m ( E [ - r m ] , F). The p r e c e d i n g C o r o l l a r y 2 . 1 w i l l b e a p p l i e d i n t h e s e q u e l when i n t h e p a r t i c u l a r case c o n s i d e r e d F = L t i ( H ) , t h e C*-algebra of bounded l i n -

ear operators on a given H i l b e r t space H , endooed w i t h t h e "uniform operator topology" u i n H . T h u s , s u p p o s e t h a t B i s a semi-simple a d v e r t i b l y complete s e l f - a d j o i n t l o c a l l y m-convex *-algebra w i t h continuous i n v o l u t i o n and spectrum m(E).SO if b ( m ( E ) ) d e n o t e s t h e s u b a l g e b r a of C ( m ( E ) ) c o n s i s t i n g of bounded f u n c t i o n s one h a s , by h y p o t h e s i s ,

( S e e a l s o C h a p t . V I I 1 ; t h e comment f o l l o w i n g D e f i n i t i o n s 3.1 and 3 . 2 ) . On t h e o t h e r h a n d , a p p l y i n g C o r o l l a r y 2 . 1 ,

w e have

TOPOLOGICAL *-ALGEBRAS. REPRESENTATIONS

2.

(2.10)

Hom*(E, L u l H ) ) C Hom*(E[ T,],

T h e r e f o r e , g i v e n an element T

E

fu(H))

487

I

Hom*(E, L (H)), t h e r e e x i s t s a > C s u c h U

t h a t one h a s , f o r e v e r y x e E ,

a . 11 T(Z) jl 5

:=

PE(d

= ( C o r o l l a r y 111;6.5)

f ( " sup-norm''

of

2

E

EA C

E

sup I A 1 x E SPE(2)

s u p I2(f) I (El

11 2 /Irn

eb ( m( E ) ) ) .

A c c o r d i n g l y , T i s continuous f o r t h e r e l a t i v e topology on EA induced on

it by t h e u n , i f o m topology i n ? Z ( E ) . Hence, T admits an i n t e g r a l representation of t h e form ( 1 . 1 1 ) ( c f . a l s o Lemma 2 . 1 ) . Now,

it i s i n t h i s form (more c l a s s i c a l , i n f a c t ! ) t h a t

(2.11)

we are going t o apply ( 1 . 1 2 ) i n t h e sequel. Furthermore,

i f t h e a l g e b r a E l a s a b o v e , i s i n p a r t i c u l a r a Frb-

c h e t ( l o c a l l y m-convex * - ) algebra, t h e n o n e h a s t h e r e l a t i o n Hom*(E, L u ( H ) ) = Hom*(E, L u ( H ) )

(2.12)

,

a f a c t t h a t w i l l a l s o b e u s e f u l i n t h e s e q u e l . ( I n t h i s r e s p e c t , see a l s o t h e n e x t Remark 2 . 1 , a s w e l l a s Scholium 2 . 1 ) . Remark 2 . 1 . - The p r e c e d i n g r e l a t i o n ( 2 . 1 2 ) i s a c o n s e q u e n c e of a r e s u l t o f R.M. BROOKS [5: p. 61, Lemma 3.11 f o r F r g c h e t * - a l g e b r a s w i t h i d e n t i t y elem e n t s . However, i t c a n b e e x t e n d e d t o n o n - u n i t a l algebras, reducing t h e latter case t o u n i t a l algeb r a s ( s e e , f o r i n s t a n c e , M. FRAGOULOPOULOU [5: p. 127, Scholium 3 . 1 , (ii)]) W .e s t i l l remark t h a t t h e above r e s u l t of Brooks i s b a s e d on t h e "automatic continui t y " of p o s i t i v e l i n e a r forms defined on a Fre'cket topologic a l ( n o t n e c e s s a r i l y l o c a l l y in-convex) *-algebra havi n g an i d e n t i t y element and continuous i n v o l u t i o n : Sya DoBin Theorem; cf. H . G . DALES [I: p. 178, Theorem l l . 1 1 o r y e t X I A DAO-XING ( see SYA DO-SIN) [2: p. 62 , Lemma 2 . 2 .I] a n d / o r SYA DO-SIN [I: p . 510, Theorem I ] .

Scholium 2 . 1 . -

A n o t h e r i n s t a n c e where t h e p r e c e d i n g may b e a p p l i -

e d i s , of c o u r s e , t h e c a s e t h a t t h e t o p o l o g i c a l * - a l g e b r a u n d e r cons i d e r a t i o n " a d m i t s a f u n c t i o n a l r e p r e s e n t a t i o n " , i n t h e s e n s e of Chapt. V I I I ; S e c t i o n 3. I n

c a l l y m-convex

t h i s r e s p e c t , o n e h a s i n c a s e of commutative lo-

C * - a l g e b r a s a n e x t e n d e d form o f t h e c l a s s i c a l GeZ'fand-

NaZmrk Theorem ( c f . e.5. M . A .

f l A h 4 R K [I: p. 230, Theorem I ] ) . A v e r s i o n of

t h e l a t t e r r e s u l t w i t h i n t h e p r e s e n t c o n t e x t might be r e a d lows :

as fol-

488

XV ALGEBRAS WITH INVOLUTION

A c o m t a t i v e compZete l o c a l l y m-convex C*-algebra E

with an i d e n t i t y element " a h i t s a functional representa(2.13)

tion" (i.e., it i s i s o m o r p h i c , a s a t o p o l o g i c a l a l g e b r a , t o C c ( ? 7 Z ( E ) ) ) i f ,and only i f , t h e r e s p e c t i v e

Cel'fand map o f E i s continuous. The a s s e r t i o n f o l l o w s f r o m E . A . MICHAEL [I: p. 3 6 , Theorem 8.41 and Chapt. V I I I ; Theorem 3 . 2 ,

i n c o n j u n c t i o n w i t h Chapt. V I I ; ( 1 . 2 9 ) . It would be v e r s i o n of t h e same re-

p o s s i b l e , of c o u r s e , t o g i v e a " n o n - u n i t a l " s u l t by c o n s i d e r i n g i n s t e a d t h e a l g e b r a ing a t infinity". 1.11)

Co(m(E)) (functions

"vanish-

I n t h i s r e s p e c t , see a l s o K . SCHMUDGEN [l: p. 168, S a t z

. W i t h i n t h e same v e i n o f i d e a s , we s t i l l remark t h a t a c o m t a t i v e

l o c a l l y m-convex C*-algebra E i s always a uniform

aZgebra. (Apply a n Arens-Mi-

c h a e l d e c o m p o s i t i o n of E i n c o n j u n c t i o n w i t h t h e "normed case a n a l o -

.

gon" ; c f . e . g . M. A . NAFMARK [1: p. 231, p r o o f of Theorem 13 ) S O E i s semisimple ( c f . Lemma V I I I ; 5 . 1 ) Therefore, i f E has a l s o a continuous Gel'fand

.

, x : p , t h e n it may be considered ( w i t h i n a t o p o l o g i c a l a l g e b r a isomorphism)

a s a topological subalgebra of C c I ? 7 Z ( E ) ) ( c f . Lemma V I I I ; 5 . 2 ) . Thus, o n e c a n apply t h e p r e v i o u s argumentation i n connection w i t h ( 1 . 9 ) .

3. SNAG Theorem (the classical case) The r e a s o n i n g i n t h e p r e v i o u s s e c t i o n i s a p p l i e d , i n p a r t i c u l a r , 1

when one c o n s i d e r s t h e group algebra L ( G ) , where G i s a Zocally compact abel1

ian group. S o t h e ( c o m m u t a t i v e ) Banach a l g e b r a L ( G I ( c f . Chapt. V I I ; Sect i o n 4 ) i s always semi-simple

( see

I . E . SEGAL [l: p. 77, Theorem 1 - 5 1 o r y e t

L . H . LOOMIS [l: p. 146, C o r o l l a r y ] ) ; a l s o it i s s e l f - a d j o i n t

t h e involution f

c--t f

*,

f

E

L2(G),

-

f * ( x ) := f ( - x J

(3.1)

with respect t o

d e f i n e d by r

x eE

( c f . e . g . L . H . LOOMS [ l : p. 1181). Thus by c o n s i d e r i n g t h e r e s p e c t i v e

Fourier-GeZ'fand

transform algebra

1

of L ( G I

, one

has

( S t o n e - W e i e r s t r a s s Theorem ( f o r l o c a l l y compact s p a c e s ) ; c f . e . 9 .

W.

PAGE [I: p. 370, T h e o r e m ] ) . The l a s t set i n ( 3 . 2 ) d e n o t e s t h e s u b a l g e b r a

of

cb ( $ ) c o n s i s t i n g

i n f i n i t y " of

2,

of t h o s e f u n c t i o n s which " v a n i s h a t t h e p o i n t a t

endowed w i t h t h e r e l a t i v e "sup-norm t o p o l o g y " from

t h e a m b i e n t (Banach, hence a f o r t i o r i Q as indicated.

3.3.11).

,

and t h u s bounded ) a l g e b r a

( S e e a l s o Theorem V ; 1 . 3 a n d R . LARSEN [I: p. 74, Theorem

3.

NOW,

489

SNAG THEOREY ( C L A S S I C A L CASE)

1

i f TeHom*(L ( G ) , f u ( H ) ) ,

t h e n by a p p l y i n g f i r s t ( 2 . 1 2 ) ( s e e

a l s o Remark 2 . 1 ) and t h e n ( 2 . 1 1 ) , one c o n c l u d e s t h e e x i s t e n c e of a 1 p r o j e c t i o n - v a l u e d m e a s u r e , s a y , P T I on m(L (GlI = ( c h a r a c t e r g r o u p o f G ; c f . C h a p t . VI1;Theorem 4 . 1 ) ,

i n s u c h a way t h a t one h a s

< T ( f ) ( a ) , b > = l f ^ d ,

(3.3) 1

.

w i t h f E L ( G ) , and f o r a n y a , b i n H ( c f . a l s o ( 1 . I 1 ) ) N o w , i f G i s any t o p o l o g i c a l g r o u p , by a weakly continuous unitary

representation of G I o n e means a ( g r o u p ) morphism

0 of G

i n t o t h e group

o f u n i t a r y e l e m e n t s o f t h e * - a l g e b r a LlH), f o r some H i l b e r t s p a c e H I which i s c o n t i n u o u s w i t h r e s p e c t t o t h e s p a c e l s ( H s ( i H i l ) ; i . e . , s u c h t h a t t h e map x - < @ ( x ) ( a ) , b > , X E G , i s c o n t i n u o u s , f o r any a , b i n H.So if

Mo&(G, U ( H I I d e n o t e s t h e l a t t e r s e t , t h e n f o r every @ E MoMG, UlHl) ,w i t h

G b e i n g now a l o c a l l y compact abelian group

, one

1 g e t s an element T eHom*(L (GI,

S(H)l by t h e r e l a t i o n < T ( f l f a i , b > = J f ( x ) < + ( z , ) ( ab) > , dx 1 w i t h f E L ( G ) , and f o r a n y a , b i n H . ( S e e e . g . G . W . MACKEY [I: p. 7393 a n d / (3.4)

o r L . H . LOOMIS [I: p. 146, 3 6 E ] ) . Conversely, t o a n y ( n o n - d e g e n e r a t e ) T there corresponds a uniquely defined element

t h i s correspondence i s , i n f a c t , a b i j e c t i o n

0E

Mo&(G,

such

U(HlI , w h i l e

( c f . J . DIXMIER [ I : p. 284, P r o p o s i -

t i o n 13.3.41). T h u s , i n v i r t u e of

( 3 . 3 ) , o n e now o b t a i n s T

< ~ ( f ) ( a )b> , = J f ( a ) d < P (a),b > = ( c f . VII;

(4.12))

1

f o r e v e r y f E L ( G ) , so t h a t one c o n c l u d e s t h e r e l a t i o n

< + f x ) f a l b, > = j m d c P T ( a I , b > I

(3.5) f o r any a,b i n H .

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

Theorem 3.1

.

(Stone-Naimark-Ambrose-Godement ( = S N A G ) Theorem I . Let G be a

Locally compact abelian group and @ a weakly continuous unitary representation o f G h

i n a Hilbert space H . Then, t h e r e e x i s t s a projection-valued measure P on G in such a way t h a t one has

(3.6)

Liith x

@(XI

E

G.

I

= /clizldP,

490

ALGEBRAS WITH INVOLUTION

XV

I n t h e n e x t s e c t i o n w e a r e g o i n g t o see a n e x t e n s i o n of t h e p r e v i o u s r e s u l t , by c o n s i d e r i n g a g e n e r a l i z e d g r o u p a l g e b r a L 1( G , E ) ( c f . C h a p t . X I : S e c t i o n 5 ) . F u r t h e r m o r e , w e a l s o o b t a i n an " a b s t r a c t form" o f it t h r o u g h t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s ( s e e S e c t i o n 5 below).

4. Representations of generalized group algebras. SNAG Theorem (extended form) I n t h e s e q u e l w e c o n s i d e r g r o u p a l g e b r a s of t h e form L 1 ( G , E ) (see C h a p t . X I ; S e c t i o n 5 1 , where now E i s a ( c o m m u t a t i v e ) l o c a l l y rnconvex * - a l g e b r a w i t h a bounded a p p r o x i m a t e i d e n t i t y ( s e e Theorem X I : 5 . 1 , and C h a p t e r X I V )

.

So c o n s i d e r a n e l e m e n t 1 1 1 1 ~ = Z ~ . ~ ~ . E L ( G I L @( GE IG~ E= L I G I ~ EE L I G , E )

,&z

z

TI

( c f . X 1 : ( 5 . 1 5 ) ) . Then, one d e f i n e s o f c o u r s e t h e i n v o l u t i o n i n t h e a l g e b r a L1(GI B E

by t h e r e l a t i o n

(4.1) 1

where t h e f u n c t i o n f * E L ( G I

.

i s g i v e n by ( 3 . 1 )

Thus, i f t h e a l g e b r a E

h a s a c o n t i n u o u s i n v o l u t i o n , t h e n one e x t e n d s ( 4 . 1 ) by " c o n t i n u i t y 1

and ( a n t i )l i n e a r i t y " t o L ( G I $ E :

moreover, t h e i n v o l u t i o n t h u s e x t e n d -

, due t o e d s t i l l r e t a i n s i t s " m u l t i p l i c a t i v e p r o p e r t y ' ' ( c f . ( 1 . 1 ) : iii)) 1

t h e c o n t i n u i t y of t h e m u l t i p l i c a t i o n i n L ( G ) B E ( c f . a l s o P r o p o s i t i o n X ; 3.1

and P r o p o s i t i o n I ; 1 . 6 ) Now,

7T

.

1

i n t h i s c o n t e x t , w e r e c a l l t h a t L (G) has always a bounded ap-

proximate i d e n t i t y c o n s i s t i n g of hermitian elemnents ( i n f a c t , p o s i t i v e o n e s ; cf.

,

f o r i n s t a n c e , L . H . LOOMIS [I: p. 124, Theorem 31E])

.

On t h e o t h e r

hand , every complete l o c a l l y m-convex C*-algebra has a bounded approximate i d e n t -

i t y c o n s i s t i n g o f p o s i t i v e elements ( i . e . , h e r m i t i a n e l e m e n t s h a v i n g nonn e g a t i v e ( r e a l ) s p e c t r a : c f . A . INOUE [I: p. 208, Theorem 2 . 6 1 )

.

I n d e e d , s o m e t h i n g l e s s i s g o i n g t o be a p p l i e d i n t h e s e q u e l , which w e t h u s p r e s e n t w i t h t h e f o l l o w i n g

Lemma 4.1. Let E be a t o p o l o g i c a l *-algebra w i t h a continuous m u l t i p l i c a t i o n having a l s o a continuous i n v o l u t i o n and a bounded approximate i d e n t i t y . Then, E has a bounded approximate i d e n t i t y c o n s i s t i n g o f hermitian elements. Proof. Suppose t h a t B P ( u ) . C E i s a bounded a p p r o x i m a t e i d e n t i z e I i t y i n E ( c f . D e f i n i t i o n X I I I ; 8 . 1 and C h a p t . X I V ) So t h e same h o l d s

.

t r u e f o r t h e n e t a'* = (u:IieI

,a s

f o l l o w s from t h e c o n t i n u i t y and ( a n t i )

4. SNAG THEOREM

(EXTENDED FORM)

491

l i n e a r i t y o f t h e i n v o l u t i o n i n E , and a l s o from t h e r e l a t i o n

1 i m u T ; r = ( l i m x * u . ) = (x*)* = x

i

z

i

z

,

w i t h Z E E ; (a s i m i l a r r e l a t i o n i s c e r t a i n l y v a l i d f o r t h e n e t ( x u ? ) , x E E ) . S o a p p l y i n g now an argument s i m i l a r t o t h a t i n t h e p r o o f of Lem-

m a XIV; 1 . 1 , we prove t h a t the n e t iu.u*.) z z i r I has the desired prope r t i e s . ( S e e a l s o A . INOUE [ I : p. 2 0 5 , P r o p o s i t i o n 2 . 1 1 ) . I NOW,

i n v i e w o f Theorem X I V ; 1 . 1

which w e a r e g o i n g t o a p p l y i n

t h e s e q u e l , w e remark t h e f o l l o w i n g :

Suppose t h a t t h e ( l o c a l l y convex) t o p o l o g i c a l algebras of Theorem X I V ; 1.1 a r e endowed w i t h continuous i n v o l u t i o n s . Moreo v e r l e t S (Hi be t h e C*-algebra of a g i v e n H i l b e r t space

(see e . g .

(4.2)

H

( 2 . 1 0 ) ) . Then, f o r every element

h E Hani*(EfF, Lu(H))

(4.2a)

s a t i s f y i n g XIV; ( 1 . l o ) , t h e r e e x i s t uniquely d e f i n e d elements (4.2b)

f e ffom*(E, f u ( H I )

and

g e Honi*(F, Su(H1l

,

which f u l f i l t h e r e l a t i o n s XIV; ( 1 . 1 2 1 , ( 1 . 1 3 ) . I n d e e d , t h e a s s e r t i o n c o n c e r n i n g ( 4 . 2 b ) f o l l o w s from t h e c o n t i n u i t y o f t h e i n v o l u t i o n s i n E , F , t h e r e l a t i o n s X I V ; ( 1 . 1 8 ) , ( 1 . 2 2 ) which d e f i n e f ,g

,r e s p e c t i v e l y ,

and from t h e p r e v i o u s Lemma 4 . 1

.

C o n v e r s e l y , t h e r e l a t i o n XIV; ( 1 . 1 2 ) d e f i n e s u n i q u e l y t h e element (4.201,

i n case we are g i v e n t h e elements ( 4 . 2 b ) which a l s o s a t i s f y XIV; ( 1 . 1 3 1 , ( 1 . 3 0 ) . ( A p p l y w i t h i n t h e p r e v i o u s c o n t e x t Lemma X I V ; 1 . 3 1 . So w e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g . ( I n f a c t ,

a " v e c t o r i z a t i o n " of Theorem 3.1 )

.

Theorem 4.1. Let E be a commutative complete l o c a l l y m-convex *-algebra w i t h a bounded approximate i d e n t i t y and continuous i n v o Z u t i o n . Moreover, l e t G be a l o 1

c a l l y compact a b e l i a n group and L iG, E l t h e corresponding g e n e r a l i z e d group a l g e bra. Furthermore, suppose we a r c given a H i l b e r t space Hand a non-degenerate e l e ment ( c f . XIV;(l.lO)) (4.3)

A ~ H u ~ ~ * ( L ' I E), G,

s,(HI).

Then, t h e r e e x i s t s a uniquely d e f i n e d p a i r (@, T )

(4.4)

e

Muh(G, U i H ) ) x

Hum*(E, S u ( H ) )

,

i n such a way t h a t t h e f o l l o w i n g r e l a t i o n s t o b e s a t i s f i e d :

i)

@ and T " c o m t e

w i t h each other"; i. e . , one has t h e r e l a t i o n

492

XV

(4.5)

ALGEBRAS WITH INVOLWl'ION

@(XI

f o r any x

EG

and

0

T(a) = T(aJ o @ ( x ),

a e E.

ii) For any a, b in H, one has (4.6)

< [ A ( f ) l I a ) ,b> = / < [ T f f f ' x ) )o @ ( x ) ] ( ~b)>, d x

.

Proof. By h y p o t h e s i s and t h e p r e c e d i n g comment on ( 4 . 2 ) , o n e h a s a s i n ( 4 . 3 ) , a u n i q u e d e c o m p o s i t i o n of t h e form

f o r every given A , (4.7) with

A = $ ~ T , $E

1

Hom*(L ( G ) , E u ( H ) ) and T

E

Hum*(E, Eu(H)I

, "commuting

with eachother";

so t h e y s a t i s f y t h e r e l a t i o n

l ( f )0 T(aJ

(4.8) 1

= T(aJ

0

mcf) ,

( c f . XIV;(1.13)).Furtherrnorel

f o r any f e L ( G ) a n d a c '

from ( 3 . 3 ) , (3.4)

one o b t a i n s a u n i q u e l y d e f i n e d 9 E iUotr(G, U ( H J ) s u c h t h a t < [ $ ( f l l ( a l , b ; = ~ f ( x ) < [ @ ( x ) I ( ba>) d ,x

(4.9)

,

1

f o r any f e L ( G I and a , b i n H. T h i s p r o v e s our a s s e r t i o n c o n c e r n i n g t h e e x i s t e n c e of t h e p a i r i n ( 4 . 4 ) . On t h e o t h e r h a n d , from ( 4 . 9 ) and ( 4 . 5 1 , one g e t s

( f ( x . i < @ ( r c l ( T ( a l ( a l lb, > d x = < $ f f l ( T f a l ( a l l ,b >

f o r e v e r y f € L 1 ( G ) . T h e r e f o r e , one has t h e r e l a t i o n (4.10)

< ( @ ( x ) T ( a ) ) ( a b> ) , = < ( T ( a ) @ ( x ) ) ( a )b, >

f o r any a,b i n H , with

a ) € G x E ; t h u s , we have proved

(2,

, (4.51.

1 NOW, on t h e b a s i s of t h e r e l a t i o n Li(GI 8 E = i ( G , E l

c o n t i n u i t y of t h e o p e r a t i o n s i n v o l v e d i n ( 4 . 6 ) j u s t for elements of t h e form

,

and o f t h e

it s u f f i c e s t o prove ( 4 . 6 )

1

f @ a e L ( G ) 8 E : Thus, w e h a v e

< [ A f f @ a a ) ] ( a )b, > = ( b y ( 4 . 7 ) ) c [ ~ l f ) T ( a l l ( a bl ,>

( w e s e t of c o u r s e (fe a ) ( x ) = f f x ) a ; see e . g . X I : ( 1 . 4 ) ) . NOW, t h i s comp l e t e s t h e proof of t h e t h e o r e m . 1

5.

493

ABSTRACT FORMS OF SNAG THEOREM

equivalence between (4.2a) and ( 4 . 2 ~ )

Our p r e v i o u s d i s c u s s i o n o n t h e

a c t u a l l y y i e l d s a l s o a converse t o t h e preceding Theorem 4 . 1 : Thus I one g e t s t h e r e s u l t i n q u e s t i o n by s p e c i a l i z i n g t h e a f o r e - m e n t i o n e d

relations

i n t h e c o n t e x t o f t h e above t h e o r e m and by c o n s i d e r i n g f u r t h e r t h e r e l a t i o n b e t w e e n t h e r e p r e s e n t a t i o n s @ a n d $ ( c f . (3.3) # (3.4), and a l s o

( 4 . 7 ) ) . But w e o m i t t h e d e t a i l s of t h i s s t a t e m e n t . C o n s e q u e n t l y , o n e t h u s c o n c l u d e s t h a t (4.31 and ( 4 . 4 ) are equivalent

when, of course, t h e corresponding r e l a t i o n s t o C h a p t . XIV; ( 1 . 1 0 ) , (1.131, and ( 1 . 3 0 ) are s a t i s f i e d ( s e e , i n p a r t i c u l a r

(4.5) )

I

.

5. A b s t r a c t forms o f "SNAG Theorem'' t y p e W e p r o v e i n t h i s s e c t i o n t h e f o l l o w i n g e x t e n s i o n o f Theorem ( " S N A G Theorem" )

Theorem 5.1.

. Thus

I

3.1

w e have.

Let E be a (commutative) complete semi-simple s e l f - a d j o i n t spec-

t r a l l y b a r r e l l e d l o c a l l y in-convex *-algebra w i t h continuous i n v o l u t i o n and spectrum m(E).We assume f u r t h e r t h a t

$(El = E A

(5.1)

Co(??(E)I ;

( t h e l a s t s e t d e n o t e s t h e (Banach) a l g e b r a of

(complex-valued) con -

t i n u o u s f u n c t i o n s o n m(E) which " v a n i s h a t i n f i n i t y " , endowed w i t h t h e t o p o l o g y u of u n i f o r m c o n v e r g e n c e i n m(E)).Moreover, l e t G be a l o -

c a l l y compact abelian group w i t h dual group pose t h a t we are given a p a i r ($, T j e Mo'r(G,

(5.2)

U(H))

;and x

R a H i l b e r t space. F i n a l l y , sup-

Hom*(E,

Lu(H))

such t h a t one has (5.3)

$(xi o T t a i = T l a ) o $ ( z ) ,

f o r e v e q (x,a I f G x E . Then, t h e r e e x i s t s a projection-valued measure, say, P(0, h i , on

5x

??'LYE)

i n such a way t h a t one has t h e r e l a t i o n (5.4)

< [ @ ( x ) T ( a ) l ( a6) , = / G ) s ( h j d < P ( a ,h i ( a l , b >

f o r any a , b i n H , w i t h (x,a )

E

G

X

E

.

T o make t h i n g s e a s i e r , b e f o r e e m b a r k i n g o n t h e p r o o f ,

we f i r s t

comment o n c e r t a i n i m m e d i a t e i m p l i c a t i o n s t h a t w e h a v e a l r e a d y f r o m o u r h y p o t h e s i s . Thus, w e remark t h e f o l l o w i n g : 1 1 1 ) Suppose t h a t 6 s € i o m * ( L (G), L ( H ) ) = uam*(L (G), L u ( H ) ) ( c f . ( 2 . 1 2 ) ) 1 i s t h e r e p r e s e n t a t i o n o f L (GI which c o r r e s p o n d s t o 4 E M u h l G , UIH)) (see

( 3 . 4 ) ) . T h u s , by a p p l y i n g a n a r g u m e n t s i m i l a r t o t h a t f o r t h e p r o o f

494

XV ALGEBRAS WITH INVOLUTION

of ( 4 . 1 0 )

,

o n e c o n c l u d e s t h a t the respective p a i r

(7,

T ) i s also " c o m t i n g " .

T h a t i s , one has the r e l a t i o n $ i f ) o T(ai = T ( a )o

(5.5)

5 cf)

,

2 ) The r e l a t i o n ( 5 . 1 ) i m p l i e s , i n e f f e c t , t h a t E i s a bounded aZ-

gebra ( c f . D e f i n i t i o n V I ; 1 .'I ),

so t h a t s i n c e i t i s a l s o s p e c t r a l l y

b a r r e l l e d , w e c o n c l u d e t h a t E has an equicontinuous spectrum

( E j . (Cf. D e f i -

.

n i t i o n V; 1 . 3 a s w e l l a s Remark V I ; 1 . 1 ) Hence, E i s a c t u a l l y a &-aZge1 bra (see Theorem V I ; 1 . 3 : 2 ) 3 1 ) ) . F u r t h e r m o r e , L (GI b e i n g a Banach a l l g e b r a , h e n c e a & - a l g e b r a , has a l s o an equicontinuous spectrum m(L ( G ) ) = 2 ( c f . P r o p o s i t i o n 11; 7 . 1

a n d Theorem V I I ; 4 . 1 ) . 1

Consequently, concern-

1

i n g now t h e s p e c t r u m of t h e a l g e b r a L ( G , El = L ( G ) $ E ( c f . X I ; ( 5 . 1 4 ) and t h e comment f o l l o w i n g i t ) , one may a p p l y Theorem X I I ; 1 . 2 . S o one g e t s (5.6)

~ I L ('G , E ) ) = ~ ( L I GI )

$ E ) = rnd (G))

x

WE)

=

E

x

M(E),

w i t h i n homeomorphisms.

3 ) S i n c e by t h e p r e v i o u s remark 2 ) m ( E l i s e q u i c o n t i n u o u s , it i s a f o r t i o r i l o c a l l y e q u i c o n t i n u o u s a n d hence

ZocaZZy compact ( c f . Theo-

r e m V; 1 . 1 ) . T h e r e f o r e l b y c o n s i d e r i n g t h e ( B a n a c h ) a l g e b r a s ("sup-norm and C,(G^)one g e t s t o p o l o g y " ) Co(??i?(El)

co(z) $ C o ( ? ? Z ( E ) ) = co(Ex m(E)),

(5.7)

within an isomorphism ( i n f a c t , isometry) of t h e topological algebras considered

(see A.GROTHENDIECK

[3: Chap. I ; p. 9 0 1 1

,

a s w e l l a s Corollary X I ; 1 . 1 ) .

1

4 ) The aZgebra L ( G , E ) = L ( G ) 2 E i s semi-simpZe: I n view o f t h e s e m i 1 n s i m p l i c i t y of t h e a l g e b r a s L ( G ) ( S e g a l ' s Theorem) a n d E l t h e a s s e r t i o n

i s a consequence of C o r o l l a r y X I I I ; 4 . 1 .

NOW, i n

connection with t h e

1

l a s t r e f e r e n c e , w e s t i l l r e c a l l t h a t L ( G ) enjoys the ( m e t r i c ) approximaD e f i n i t i o n X ; 2 . 4 ; b u t see A . G R ~ T ~ E N D ~ ~[3: C KChap.

t i o n property ( c f . e . g .

I ; p. 178, D B f i n i t i o n 10, and p. 185, P r o p o s i t i o n 411). Moreover, w e have

n o t e d a l r e a d y (see t h e above remark 2 ) ) t h a t E i s a & - a l g e b r a .

5.

495

ABSTRACT FORMS OF SNAG THEOREM

5 ) F i n a l l y , L 1 IG, E l i s a bounded algebra, b e i n g a c c o r d i n g t o t h e p r e c e d i n g d i s c u s s i o n a ( c o m m u t a t i v e ) c o m p l e t e l o c a l l y m-convex $ - a l g e b r a ( c f . Theorem V I ; 1 . 3 ) . T h u s , u n d e r t h e h y p o t h e s i s o f Theorem 5 . 1

,

t h e algebra L 1 (G, E i

,

c o n s i d e r e d i n t h a t t h e o r e m , f u l f i l s t h e c o n d i t i o n s o f Corollary 2 . 1 . So w e come n e x t t o t h e

Proof of Theorem 5 . 1 . S i n c e t h e a l g e b r a E i s , by h y p o t h e s i s , s e m i s i m p l e a n d s e l f - a d j o i n t , one g e t s from ( 5 . l ) ( a p p l y i n g Stone-Weiers t r a s s Theorem: see a l s o t h e r e s o n i n g a p p l i e d t o ( 1 . 9 ) )

-

$.(E) = E n

(5.8)

= co("?YE))

F u r t h e r m o r e , f r o m C o r o l l a r y 2 . l ( s e e t h e p r e v i o u s remark 2 ) ) c l u d e s t h a t T EHom*(E[rm

1,

S u ( H I ) , s o t h a t by

t h e whole a l g e b r a r a n g e of t o t h e whole a l g e b r a junction with ( 4 . 2 )

s. S i m i l a r l y ,

co(i'i :hence

, the

$ c a n be e x t e n d e d

by ( 3 . 2 )

one g e t s from ( 5 . 5 )

following

one con-

(5.8) T can be extended t o

,

and i n con-

( c o n t i n u o u s map)

= ( b y ( 5 . 7 ) ) Hom*fCo& x ~ ( E ) I sum)) ,

(5.9)

= (see (2.12) and Remark 2 . 1 ) Hom*(eoi2 ~ m i E ) ) , 2 ~ ( H ) )

( w e r e t a i n t h e same symbols f o r t h e e x t e n d e d maps

5

and T ) .

W e remark a t t h i s p o i n t t h a t by a p p l y i n g a s i m i l a r argument t o t h a t i n t h e above r e l a t i o n ( 5 . 9 ) , one concludes t h a t

$ @T E ?om*(L'(G)$ E , S u ( H ) )

(5.10)

.

i

C o n s e q u e n t l y , s i n c e L (G, E ) f u l f i l s t h e c o n d i t i o n s of C o r o l l a r y 2 . 1 , a s w e have a l r e a d y remarked i n t h e p r e c e d i n g , t h e map @ o T i s aZsolcon_tinuous w i t h _respect to t h e re2ative topology induced o n g ( L (G)? E ) by e o ( G x m(E)I

.

T h u s , a s a c o n s e q u e n c e of

( 5 . 9 ) ( o r y e t (5.10)

,

,t h u s

an a l t e r n a t i v e

,

w e c a n now a p p l y Lemma 1 . I t o g e t a p r o j e c t i o n - v a l u e d m e a s u r e , s a y , P l a , h ) , on t h e s p e c t r u m of i L (G, E ) , i . e . , on t h e s p a c e x m(E). T h e r e f o r e , i n v i r t u e of ( 1 . I 1 ) , 1 o n e h a s t h e f o l l o w i n g , f o r a n y f o a E L (G)@!? a n d a , b i n H : t o o u r a r g u m e n t a t i o n ) a n d of

(5.1 )

(3.2)

< [ ( ~ o T ) ( f o a a l ] ( abl>, = j f m a i a , h l d < P ( a , h ) ( a ) , b > =

f^(a,Ja(hid

= ( b y V I I ; ( 4 . 1 2 ) )

4 96

ALGEBRAS WITH INVOLUTION

XV

1

f o r e v e r y f e L (G). Thus, t h e l a s t r e l a t i o n s y i e l d now ( 5 . 4 1 , and t h i s t e r m i n a t e s t h e p r o o f o f t h e theorem. I W e c l o s e t h i s s e c t i o n w i t h t h e f o l l o w i n g commentary, where w e

d i s c u s s t h e g e n e r a l i d e a b e h i n d t h e p r e c e d i n g Theorem 5 . 1 , p o i n t i n g So w e have t h e n e x t .

t h u s o u t p o s s i b l e a b s t r a c t forms of it a s w e l l .

Scholium 5.1. - By a n a l y z i n g t h e p r o o f o f t h e above Theorem 5.1 , w e r e a l i z e t h a t t h e main i n g r e d i e n t i n it i s t h e c l a s s i c a l SNAG Theor e m (Theorem 3 . 1 ) . This i s c o n c l u d e d i f o n e a l s o c o n s i d e r s t h e comment f o l l o w i n g C o r o l l a r y 2 . 1 a s w e l l a s t h e g e n e r a l a r g u m e n t a t i o n i n C h a p t . X1V;Section 2 . So by t a k i n g f u r t h e r ( 1 . 9 ) i n t o a c c o u n t , one c a n a p p l y a s i m i -

l a r argument t o t h a t o f Lemma 1 . I

,

f o r i n s t a n c e ( o r y e t s e e a l s o (2.11)),

t o g e t a n " i n t e g r a l r e p r e s e n t a t i o n " of t h e a l g e b r a morphism ( r e p r e s e n t a t i o n ) of t h e t o p o l o g i c a l a l g e b r a c o n s i d e r e d . Then, it i s c e r t a i n l y c l e a r t h a t following an analogous reasoning t o t h e preceding one,

as w e l l a s t h e g e n e r a l l i n e o f t h o u g h t d e v e l o p e d h i t h e r t o , one i s l e d t o t h e following conclusion:

h e can consider a "spectral decomposition" of t h e form h

( 5 . 4 ) f o r a suitable topological tensor product algebra E Q F T

(5.11)

r e l a t i v e t o a given pair o f "eomuting representations" of t h e algebras E , F . N o w c o n c e r n i n g t h e t y p e of t o p o l o g i c a l a l g e b r a s which might b e

c o n s i d e r e d above, o n e c a n t a k e , f o r example, t h a t of t h e a l g e b r a E i n t h e p r e v i o u s Theorem 5 . 1 ;

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

T

under considera-

t i o n s h o u l d a l s o be a f a i t h f u l o n e ( c f . Theorem X I I I ; 4 . 1 ) . F u r t h e r m o r e , i n view o f Theorem X I V ; 2 . 1

,

the topological algebras

considered need not necessarily be bounded ones. ( A s i m i l a r r e l a t i o n t o ( 5 . 7 )

i s t r u e , of c o u r s e , f o r a l g e b r a s of t h e form C,(m(E)); c f . , f o r example, Lemma X I I ; 1.4).0n t h e o t h e r h a n d , t h e a l g e b r a s E , F c o u l d have a l s o l o c a l l y compact and o-compact ( h e n c e , hemicompact) spectra. W e c a n t h e n a p p l y Theorem X I I ; 1 . 2

Ccfm(F)) being

and b e s i d e s (2.12), t h e a l g e b r a s

c c C ? ? Y ( E ) )and

now F r g c h e t ( c f . S . WARNER [5: p. 267, Theorem 2 1 1 .

I n c o n c l u s i o n , one m i g h t c o n s i d e r t h e p r e c e d i n g l i n e s a s a h i n t t o a possible (finite!)

rem.

"topological tensor product anatogon"

of

SNAG

Tho-

6.

491

APPENDIX

6. Appendix: Enveloping locally m-convex C*-algebras W e c o l l e c t i n t h i s a p p e n d i x s e v e r a l r e m a r k s c o n c e r n i n g a cert a i n t y p e o f t o p o l o g i c a l * - a l g e b r a s which seems t o b e of p a r t i c u l a r i n t e r e s t i n connection with t h e t h e o r y of * - r e p r e s e n t a t i o n s

Of

toPo-

l o g i c a l * - a l g e b r a s . T h i s argument c o u l d a l s o be c o n s i d e r e d a s a n a t u r a l o u t g r o w t h of t h e machinery d e v e l o p e d so f a r . T h u s , w e f i r s t g i v e , i n p a r t i c u l a r , a n example of a ( c o m t a t i v e ) topological *-algebra whose en -

veloping ( l o c a l l y m-convex) C*-algebra is a barrelled &-algebra (and in f a c t a Banach algebra). T o make o u r e x p o s i t i o n more c o m p r e h e n s i b l e , w e f i r s t r e c a l l t h e corresponding d e f i n i t i o n s of t h e b a s i c n o t i o n s t h a t w e a r e going t o a p p l y b e l o w . ( H o w e v e r , f o r f u r t h e r d e t a i l s of t h e t e r m i n o l o g y a p p l i e d

w e r e f e r t o M. FRAGOULOPOULOU [ 2 ] ) . Thus , s u p p o s e w e h a v e a I c c a l l y m-eonvex *-algebra ( E ; r = pa la I) w i t h a continuous i n v o l u t i o n ( c f . S e c t i o n 2 ) , h a v i n g a l s o a bounded approxi-

mate i d e n t i t y (see C h a p t . X I V ; S e c t i o n 1 ) . NOW, o n e d e f i n e s on E a new family, say, W a e 1

(6.1)

o f *-preserving semi-norms s a t i s f y i n g ( 2 . 3 ) . ( C*-semi-norms" ; t h e s e a r e , i n f a c t , subtnultipZicative o n e s : See 2 . SEBESTeN [I: p. 2 , Theorem 21 ) T h u s ,

.

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

H e r e R,(E)

d e n o t e s t h e set o f c o n t i n u o u s r e p r e s e n t a t i o n s of E i n H i l -

bert s p a c e s ( w i t h r e s p e c t t o t h e uniform o p e r a t o r t o p o l o g i e s ) such t h a t one h a s

I / @(xl/I

(6.3)

k.pa(d,

xeE,

f o r some k > 0 (see M. FEAGOULOPOULOU [ 2 : p. 68, Lemma 4 . I ] ) ( 6 . 1 ) d e f i n e s E as a l o c a l l y m-convex C*-algebra,

.

So t h e f a m i l y

t h e t o p o l o g y o f which i s , i n

g e n e r a l , weaker t h a n t h e i n i t i a l l y g i v e n t o p o l o g y i n E . NOW, t h e envelaping ( l o c a l l y m-convex

n i t i o n , t h e "Hausdorff completion"

C*-l

aZgebra of E , i s , by d e f i -

( c f . N . BOURBAKI [ 4 : Chap. 2 ; p. 23,

D g f i n i t i o n 4 1 ) of t h e a l g e b r a E w i t h r e s p e c t t o t h e f a m i l y of s e m i norms ( 6 . 1 ) ; h e n c e , by i t s d e f i n i t i o n , a complete l o c a l l y m-convex C*-algebra. W e d e n o t e it by E I E ) . The l a t t e r a l g e b r a h a s f o r t h e r e p r e s e n t a t i o n t h e o r y o f E t h e

s a m e i m p o r t a n t s i g n i f i c a n c e a s t h e a n a l o g o u s one does f o r t h e c l a s s i c a l case o f t h e r e p r e s e n t a t i o n t h e o r y o f Bariach * - a l g e b r a s . S e e , f o r

498

XV

i n s t a n c e , J . D I X M I E R [I]

ALGEBRAS WITH I N V O L U T I O N

and

[2: p. 69,

M . FRAGOULOPOULOU

Theorem 4 . 1 1 . But

i n d e a l i n g w i t h r e p r e s e n t a t i o n s of t o p o l o g i c a l * - a l g e b r a s and of t h e a s s o c i a t e d a l g e b r a s E I E ) , a s a b o v e , a c e r t a i n s p e c i a l c l a s s of l o c a l l y in-convex a l g e b r a s seems t o p l a y a p a r t i c u l a r r61e. Namely, t h o s e l o c a l l y rn-convex

*-algebras

( w i t h c o n t i n u o u s i n v o l u t i o n and a bounded

a p p r o x i m a t e i d e n t i t y ) , f o r which t h e c o r r e s p o n d i n g ( e n v e l o p i n g ) a l q e bra E(E)

b a r r e l l e d l o c a l l y m-convex &-algebra ( s e e M . FRA-

is, i n e f f e c t , a

COULOPOULOU [2: p. 70, D e f i n i t i o n 4 . 2 1

and [5: p . 18 f f . ] ) . Now, a (com-

p l e t e ) a l g e b r a o f t h e l a t t e r t y p e i s made i n f a c t i n t o a ( " n o r m a b l e " ) C*-algebra

( L a s s n e r ' s Theorem : C f . G . LASSNER [l] a n d / o r

Theorems 2 . 2 , 2 . 3 1

f o r a s h o r t e r p r o o f o f t h e same r e s u l t ) . So w e a r e

l e d t o a t h e o r e t i c account theory ( ! )

M . FRAGOULOPOULOU [7:

( a t l e a s t ) f o r t h e bounds of t h e r e l e v a n t

.

W e come n e x t t o t h e d i s c u s s i o n of t h e ( c o m m u t a t i v e ) example a l -

r e a d y p r o m i s e d a t t h e b e g i n n i n g of t h i s s e c t i o n . Then, r e l y i n g o n t h i s ,

w e a l s o g i v e a n a n a l o g o u s non-commutative

example of t h e afore-men-

t i o n e d t y p e of t o p o l o g i c a l a l g e b r a s . So w e f i r s t h a v e t h e f o l l o w i n g . Example 6 . 1 . - Suppose t h a t w e a r e g i v e n a second countable Cm-manifold X . F u r t h e r m o r e ,

complex-valued

c -functions m

f i n i t e dimensional compact

l e t c"iX) b e t h e a l g e b r a of

on X c o n s i d e r e d a s a * - a l g e b r a by com-

e%

p l e x c o n j u g a t i o n of f u n c t i o n s . Thus,

is a

( c o m m u t a t i v e ) FQ ( F r 6 -

c h e t & ) - l o c a l l y m-convex *-algebra ( w i t h a n i d e n t i t y e l e m e n t ) : By Example I V ; 4. (2) ( c f . I V ; ( 4 . 1 9 ) )

,

e"(X) is a

F r g c h e t l o c a l l y m-convex a l g e b r a

whose s p e c t r u m i s (homeomorphic t o ) X

(Theorem V I I ; 2 . 1 ) .

Hence, b y

Lemma V I ; 1 . 3 ( s e e a l s o C o r o l l a r y V I ; 1.1 and P r o p o s i t i o n V ; l . l ) ,

c (X) m

i s a & - a l g e b r a t o o . Moreover it i s c l e a r from I V ; ( 4 . 1 3 ) t h a t t h e a l g e b r a i n q u e s t i o n h a s a continuous invoZution ( i n f a c t , t h e i n v o l u t i o n s a t i s f i e s ( 2 . 1 ) ) . So w e h a v e now t h e f o l l o w i n g : The enveloping ( l o c a z l y m-convex C*-) algebra of c m ( X ) is t h e (commutative P - ) algebra e J X ) ; i.e., one has t h e

re l a t i o n (6.4)

E(C-W

=

eu(X),

& t h i n an isomorphism of t o p o l o g i c a l algebr-as .

Of c o u r s e , h e r e cu(X) s t a n d s f o r t h e a l g e b r a of complex-valued cont i n u o u s f u n c t i o n s on X , endowed w i t h t h e t o p o l o g y u o f u n i f o r m c o n v e r g e n c e i n X. N o w t o p r o v e t h e p r e v i o u s a s s e r t i o n , w e make u s e o f t h e f o l l o w i n g r e s u l t which a l s o h a s a n i n d e p e n d e n t i n t e r e s t i n c o n n e c t i o n w i t h

6. APPENDIX

499

some other context (TopologicaZ Algebraic Geometry ; see, for instance, A . V. FERREIRA - G . T O M A S S I N I

Lemma 6.1.

[I: p. 473, Lemma 0.I]

)

. So

we have.

Let E b e a c o m u t a t i v e a d v e r t i b l y complete l o c a l l y m-convex alge-

bra w i t h an i d e n t i t y element and s p e c t m

m(E),having

a l s o a continuous Gel’fand

map. Then, the f o l l o w i n g t h r e e propositions are e q u i v a l e n t : 1 ) E i s a Q-algebra. 2) m ( E )

i s a weakly compact subset of E’.

3 ) The topology of E can be defined by a f a m i l y

=

of submulti-

p l i c a t i v e semi-norms i n such a way t h a t , f o r every U E I ,one has t h e r e l a t i o n

(6.5)

=m(gcl),

m ( E ) =???(Ea)

w i t h i n homeomorphisms

(cf.Chapt. V; Lemma 6.3 for the notation applied)

.

Proof. We first have that l ) - 2 ) , according to Lemma VI; 1.3. Furthermore, if (6.5) is true, since E is a (commutative) Banach alc1 gebra with an identity element, m(ga)2 m ( E l C Ei will be compact, so that by hypothesis for E an equicontinuous subset of E‘ (cf. Theorem VI; 1.1). Therefore, E is a Q-algebra as well (Lemma VI; 1.3)] i.e., h

.

3) 4 1) On the other hand, suppose that E is a Q-algebra; then (the same lemma, as above, or yet Proposition 11; 7.1)] m ( E ) is an equicontinu-

n=

ous subset of E‘. Consequently, if ( Uc1 )Cfer is a local basis of E corresponding to an Arens-Michael decomposition of it (see Theorem 111; 3.1), then there exists an index c1 E l such that

m(E)C

(6.6)

(Uc1i0

(cf. J . HORVATH [I: p. 2 0 0 , Proposition 61). So by Lemma V ; 6.3 (cf. V; (6.23)), one has (6.7) within homeomorphisms, while the analogous relation is obviously true for every C L ’ E I ,with a’>a .Therefore, by considering the set (6.8)

I

c1

a

i a ’ e l : a‘> a }

]

where c1 E I is determined by (6.6), one finds a c o f i n a l s u b s e t of I . S o we may restrict ourselves to the subsystem = (Ucl,)a,eIcl of n which

ncl

still defines the same topology in F. Accordingly, the same provides a corresponding family of submultiplicative semi-norms (see Proposition I; 3.2) for which (6.7) is in force. So we have proved that 1) implies 3) as well, and this completes the proof of the lemma. I

So we come next to the

500

XV ALGEBRAS WITH INVOLUTION

m(cw(X)) = X (Theorem V I I ;

Proof of ( 6 . 4 ) . S i n c e

2.1),

one g e t s

( S t o n e - W e i e r s t r a s s Theorem)

C-~XI

(6.9)

(see a l s o V I I ; (2.1)

ep)

=

and L . NACHBIN [4: p. 48, C o r o l l a r y 2 , a n d Remark I]).

Now l e t u s p u t , f o r c o n v e n i e n c e , E

= C m ( X ) , and l e t (;,IaeI

be

t h e f a m i l y of commutative Banach * - a l g e b r a s a r i s i n g from a n Arens-Mic h a e l d e c o m p o s i t i o n o f E . Moreover, d e n o t e by

R;(E)

t h e set of

(topo-

l o g i c a l l y ) i r r e d u c i b l e * - r e p r e s e n t a t i o n s $ of E ( i n H i l b e r t s p a c e s ) such t h a t

11 $ ( X I 11

2

palxi, x

t h e s u b s e q u e n t comment] )

.

E

E

.(See

M . FRAGOULOPOULOU [2: p. 67; (3.3)

T h u s , one g e t s

Rile) = R'(za/

,

and

within a bi-

j e c t i o n ( i b i d . ; p . 67, P r o p o s i t i o n 3 . 5 ) . Here t h e s e c o n d member o f t h e l a s t r e l a t i o n s t a n d s f o r t h e s e t o f a l l c o n t i n u o u s ( t o p o l o g i c a l l y ) irr e d u c i b l e * - r e p r e s e n t a t i o n s of t h e c o m m u t a t i v e Banach * - a l g e b r a C o n s e q u e n t l y ( c f . e . g . R.D. MOSM [1:p.70, = R*(i? I

R'IEI

(6.10)

c1

Corollary 6.4]),

=m(i? = (Lemma 6.1) a =m(Ea.)

kcu.

one g e t s

m(,Pi

( w i t h i n b i j e c t i o n s ) , f o r e v e r y a E I. Moreover, i f $ E m ( E ) and

i s the

one h a s

a s s o c i a t e d e l e m e n t i n ??Z($,),

$a(xa) = @(xi,

(6.11)

h

f o r e v e r y aEI, w i t h Z E E and zcl= [ z I a e E c 1 ( c f . V ; ( 6 . 1 6 ) ) . Thus, c o n s i d e r now t h e ( s u b m u l t i p l i c a t i v e ) C*-semi-norms i n g t h e C*-loealZy m-convex topology ( o f

defin-

t h e e n v e l o p i n g a l g e b r a ) of B ( c f .

( 6 . 2 ) a n d M. FRAGOULOPOULOU [2: p. 68, Lemma 4 . I ] ) ; t h e n w e h a v e (6.12)

sup

@ f o r every

c1

/ I mcx) I1

E R&(EI

=

s u p I1 @(dII = @ E ???(El

II 2 I l m

f

E I . T h e r e f o r e , t h e p r e v i o u s topology i s i n f a c t t h e "sup-

norm" t o p o l o g y i n

CtXl.

On t h e o t h e r h a n d , t h e e n v e l o p i n g l o c a l l y m-

convex C*-algebra of ew(X) i s , b y d e f i n i t i o n , t h e ( H a u s d o r f f ) complet i o n of t h e l a t t e r a l g e b r a w i t h r e s p e c t t o t h e t o p o l o g y d e f i n e d by t h e semi-norms

( 6 . 1 2 ) , h e n c e , a c t u a l l y by t h e "sup-norm", o n a c c o u n t

of t h e same r e l a t i o n . T h i s f i n a l l y p r o v e s t h e a s s e r t i o n , b e c a u s e of (6.9). I

Now r e l y i n g on t h e p r e c e d i n g Example 6 . 1 a s w e l l a s on t h e t e c h n i q u e of t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s , w e s t i l l o f f e r a n o t h e r example of t h e t y p e c o n s i d e r e d a b o v e , b u t where now ( a s w e a l s o p r o mised i t e a r l i e r ) t h e i n i t i a l ( l o c a l l y m-convex) bra is

non-commutative

t o p o l o g i c a l *-alge-

( a n d t h e same i s t r u e f o r t h e r e s u l t i n g e n v e l o p -

ing algebra). So w e have t h e n e x t

6.

L e t u s assume t h a t t h e c o n d i t i o n s i n t h e p r e v i o u s

Example 6 . 2 . Example 6 . 1 ,

50 1

APPENDIX

concerning t h e d i f f e r e n t i a l manifold X considered, a r e

s a t i s f i e d . Moreover, s u p p o s e t h a t G i s a l o c a l l y compact group C* ( G ) b e t h e

group C*-algebra

G; i.e.

of

c*(G) :=

(6.13)

,

and l e t

o n e h a s , by d e f i n i t i o n ,

E(L'(G))

.

I n o t h e r words, o n e c o n s i d e r s t h e e n v e l o p i n g (normed) C * - a l g e b r a

of

1

( t h e Banach * - a l g e b r a ) L - i G ) ( c f . C h a p t . V I I ; S e c t i o n 4, and J. DIXMIER [l : p. 2701 )

.

Furthermore, c o n s i d e r t h e f o l l o w i n g " g e n e r a l i z e d group

a l g e b r a " of G

(see Theorem X I ; 5 . 1 , a n d X I ; ( 5 . 1 4 )

t o g e t h e r w i t h t h e subsequent c o m -

ment t h e r e ) . NOW, by h y p o t h e s i s f o r X , t h e a l g e b r a ( 6 . 1 4 ) i s a Frdchet

l o c a l l y m-convex *-algebra ( w i t h a c o n t i n u o u s i n v o l u t i o n ; t h e l a t t e r a s s e r t i o n f o l l o w s e a s i l y from t h e r e l e v a n t comment on ( 4 . 1 ) ) . Moreover, t h i s algebra i s

non-comutative

( u n l e s s G i s a b e l i a n ! S e e a l s o L . H . LOO-

MIS [I: P. 123, Theorem 31C1). On t h e o t h e r h a n d , one has t h e r e l a t i o n E ~ L ~ ~G " , c

(6.15)

x i i i=

ep, c*(G)),

within an isomorphism of topological algebras. The s e c o n d member of t h e C * - a l g e b r a of c o n t i n u o u s C*(Gl-valued f u n c t i o n s on X: I n d e e d , by ( 6 . 4 )

4.21,

,

(6.15)

is

( 6 . 1 4 ) , a n d M. FRAGOULOPOULOU [5: p. 134, C o r o l l a r y

one h a s t h e f o l l o w i n g

= ( c f . M. TA'AKESAKIrl: p.211,

Theorem 4 . 1 4 1 )

C*(GIG q X ) = ( b y Theorem X I ; 1.1) c u ( X , E

C*(G)l

.

The p r e c e d i n g r e l a t i o n s a r e v a l i d , of c o u r s e , w i t h i n isomorphisms of t h e r e s p e c t i v e t o p o l o g i c a l a l g e b r a s . So w e h a v e p r o v e d ( 6 . 1 5 ) .

I n c o n c l u s i o n , ( 6 . 1 4 ) p r o v i d e s a n example of a inon-commutative) l o c a l l y m-convex *-algebra whose enveloping algebl-a ( g i v e n by ( 6 . 1 5 ) ) i s ( t o p o l o -

gizable a s ) a ( n o m e d ) C*-algebra.

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T h e o r y " . Marcel D e k k e r , N e w Y o r k , 1 9 8 3 .

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527

List

of

Symbols

a E - T ( E , m(E)),190

a ox 241 axi I

a ( E $ F / , 436 a ( E $ F ) , 436

2 (X) , a,(X)

132

I

a(x,E l ,

395

A=f.AeC:

/XIS1

1,

(o-completion)I

8,

36 30

21

E ^ = F ( E )74, E + = E B C , 256 E

1

= E @ C ( " u n i t i z a t i o n " of an algebra E )

E = & lm

Ba

,

,

83

E = l i m ( E a , fa(,) EQlF

I

E ~ F E,

,

83

360 Tl

~

EqF, E@F 2

32

PF E

~ F 366 ,

E8F

, 370

2

EBF, E

~

F E ~ F , 372

&

E

EF

= S E ( E L , F ) , 373

E ~ F , E ~ F 375 , T

T

E(c."(x)I

€'(x)= F O X

(dfl,

, 240

I

498

( e " ( X ) l t , I 132 I

45

F t X ) , 219

528

LIST OF SYMBOLS

SE(G/, 1

403

.t(E@F, G ) , 380 T

lAmEa

, lLm(Ea,

limi,

I

86

f a a l l 110

LIST OF SYMBOLS

u=

{ X E ~ :I h l = I

1 , 232

(UI (balanced h u l l ) , 6 < U > (convex h u l l ) , 6

2,

73

x o y , 43

x o F , 45 x e y , 360 $(7?7CE)I = S p E ( x ) , 104

( x , V , p I , 208

I

e n d of a proof in the text)

( o r y e t of a s t a t e m e n t n o t p r o v e d

529

This Page Intentionally Left Blank

531

INDEX

a-barrel ( =algebraic barrel) , 3 absolutely convex, 7

-,

algebra, 1 37

-, -,

32, 38

-, m-barrelled,

-, A-convex, -, A-normed,

-, -, -, -,

-,

advertibly complete, 44, 45

-,

Arens (Lw~[0,211), 2 Arens-Calderdn, 299 Baire, 24 Banach, 31

-, -,

-,

-, barrelled locally convex, - - - m-convex , 9

9

I

-, -,

bounded, 184 central, 430 complete locally convex, 21

-I

- - m-convex , 21

-,

-, - topological, 21 -, division, 61

-, -, -

-,

-,

Frgchet (topological), 9 - locally convex, 9 - - m-convex, 9 f u l l , 266 Gel’fand-Mazur, 308

- , group, 231

-, Hilbert, 303 -, inductive limit, 112 - - - locally m-convex,

-, -,

120

infinite (projective)topological tensor product ( = infinite tensor product locally convex), 385 - tensor product, 384

infra-Pt5k ( Br-complete), 308 - - I algebraically, 310

-, -

local, 351 locally bounded, 39 8

m-infrabarrelled, 306 Michael , 269 morphism, 14 Nachbin-Shirota, 262 normable, 13 normal, 334 nuclear, 302 of polynomials, 162

-, polynomial, 312 - , primary, 301 -, projective limit, 162 - - -, strictly dense, 174 -, Ptdk (fully complete) , - , - , algebraically, 310

-, -, -,

-, -, -, -, -, -, -, -I

-,

267

quasi-complete, 23 regular, 332 Riemann, 353 semi-normable, 13 semi-normed, 1 Silov, 334 simple, 338 spectrally barrelled, 142 Stein, 228 tensor product , 362 topologically simple, 338 uniform, 214

532

IXTDEX

c h a r t o f a m a n i f o l d (see l o c a l chart)

Waelbroeck, 54 Warner , 214

,

Wiener-Tauber

c e n t e r (of an a l g e b r a ) ,

349

commutant , 430

42

a-normed, r-complete

429

compact d i s c , 373

, 280

0-complete ( = s e q u e n t i a l l y c o m p l e t e ) , 63 a p p r o x i m a t e i d e n t i t y , 465

- - ,bounded, 465

-

S t e i n s e t , 161

c o m p a t i b l e t o p o l o g y , 364, 375, 382 c o n c a v i t y module, 41 C o n d i t i o n I N ) , 250

- - ,- , e v e n t u a l l y ,

continuous r e p r e s e n t a t i o n , 461

466

a p p r o x i m a t i o n p r o p e r t y , 312

c o n t r a c t i b l e s p a c e , 306

- _ ,Banach-Grothendieck,

c o n v o l u t i o n a l g e b r a of e m - f u n c t i o n s , 325 - m u l t i p l i c a t i o n ( L1 - a l g e b r a ) , 232

--

303,374

f o r a l o c a l l y m-convex a l g e b r a , 445

c o t a n g e n t s p a c e , 243

a t l a s , Nachbin, 245 a-convex h u l l , a b s o l u t e l y , 380

_ - ,b a l a n c e d a-norm,

-,

a n d , 380

40

submultiplicative

B a r r e l (see a - ,

,

41

I-, and rn-bar-

rel) bicommutant

,

Decomposition ( o f a l o c a l l y m-convex a l g e b r a ) , A r e n s - M i c h a e l , 91 d i v i s i o n a l g e b r a , l o c a l l y convex, 61

- -,t o p o l o g i c a l ,

61

E-modif i c a t i o n , 297 430

boundary , 189

e l e m e n t a r y measure of a r e p r e s e n t a t i o n , 415

- s e t , 189

e n v e l o p e o f holomorphy, 160 & - p r o d u c t , 373

C m - a t l a s , 245

-,

maximal,

245

Fourier-Gel’fand

C m - a n a l o g o n of Banach-Stone Theorem, 277

- of S t o n e - W e i e r s t r a s s Theorem, 240

CIX)-embedded, C*-semi-norm,

222

b i l i n e a r map, 360

character, generalized,

271

- g r o u p , 232 - - , s e c o n d , 240 - of a l o c a l l y compact g r o u p , c o n t i n u o u s , 232

--

an algebra, 61

f u n d a m e n t a l d e f i n i n g f a m i l y of subm u l t i p l i c a t i v e semi-norms ( f o r t h e t o p o l o g y o f a l o c a l l y m-convex a l g e b r a ) , 15 G a l o i s c o r r e s p o n d e n c e , 222

491

c a n o n i c a l b a s i s of t h e t a n g e n t s p a c e , 241

-

t r a n s f o r m , 239

G(=GSteaux)-holomorphic f u n c t i o n , 31 5 g a u g e f u n c t i o n (Minkowski f u n c t i o n al), 1 $? ( E l -convex , 354

5 (El-regular ,

355

S ( E ) - s e p a r a b l e , 354 G e l ’ f a n d map, 73

-

s p a c e , 139 t o p o l o g y , 139

INDEX

533

- transform algebra, 14

-, cw-analogon of Uryson’s,

- - - , generalized, 212

-,

- - of

t

(element of an algebra),

13

- - - - , generalized, 212

generators (of an algebra), algebraic , 1 4 1 -, topological, 141 group C*-algebra, 501 Hermitian element (of a *-algebra) , 485 hk-topology, 331 holomorpic set, 161 holomorphically convex set, 161 hypocontinuous bilinear map ( = ( & E , U3 I - hypocont inuous ) , 28 F Ideal, left (right), 63

-, closed maximal, -, 2-sided1 63

67

-,-,

maximal, 64 idempotent subset (of an algebra), 1 inductive system of algebras, 111

- - - sets, 109 - - - tensor product algebras, 36 3

intertwining operators, 77 inverse system of algebras ( = projective system of algebras) , 83 invertible element (of an algebra), k-covering family, 165 - sequence, 128

k-space, 166 k-topology, 166

225

Embedding (Rernmert-Bishop-Narasimhan) , 229

-, Grothendieck’s, -, Schur’s, 7 1

392

-,

Whitney‘s Imbedding , 241 local basis, 5 chart, 129 - coordinate (function), 130 expression of a tangent vector, 24 1 - global coordinates, 228 locally convex algebra ( = with separately continuous multiplication) , 4

-

--

--

- with continuous multiplication, 4

topological algebra, 9 equicontinuous (see spectrum)

m ( = multiplicatively)-convex algebra, 5

---

---

C*-algebra , 484 - , enveloping, 491 *-algebra , 484 topological algebra, 9

- uniformly, 148

m-barrel, 5 m-bornivorous , 307 m-bounded , 307 m-convex, 5 m-set, 1 measure , idempotent-valued, 477 -, spectral, 477 multiplication, bounded preserving, 29 -, (jointly) continuous, 4 -, separately continuous, 4 multiplicative distribution, 326

LFQ -algebra, 301

-

Z(=linear)-barrel, 3 (Zkl -algebra, 304 Lemma, Arens-Calderbn, 299

Nachbin atlas, 245

subset (of an algebra= m-set), 1

- chart, 245

534

INDEX

- Shirota space, 262

-,

normal coordinates , 244 - family of functions, 334

semi-simple (topological) algebra, 266 -, functionally, 267 -, strongly ( = functionally), 267 semi-topological group, 53 Silov-Arens-Calder6n-Waelbroeck theory, 295 - - - - functor, 296

O(X)-regular,

229

Oka s principle , 300 I-homogeneous norm, 40 order of homogenuity, 41 Polynomial, n-homogeneous, 311 continuous, 311

- ,_ ,

- boundary, 189

space of (Schwartz) distributions, 326

polynomially convex hull, 162 projective system of algebras, 83

submultiplicative, 1

spectral decomposition of a pair of " commuting r epre sent ations" , 496

--,

strictly dense, 174 Prolla-Machado ( = (PMI-) condition, 397

- map, 295 - - , weakly, 295

-

mapping theorem, polynomial, 198

&-algebra, 43

- synthesis, algebra of, 349

quasi-invertible ( = quasi-regular) element (of an algebra) , 47

- -,

Regular element, 47

-

family of functions, 332 representation, irreducible, 77

- space, 482

-,

weakly continuous unitary, 489

set of, 349

spectrum, algebraic , 67

-,-, extended,

68

-, generalized, 177 -,-, extended, 177 -, local, 47 -, locally equicontinuous, 142, -, topological (global), 69, 139

-,-,

178

extended, 141 -hypocontinuous, 68

,r)

resolvent equation, (first), 51

fG

- - , (second), 51 - function, 50

*-algebra, 481 -, topological , 481 -,-, self-adjoint, 482 *-morphism, 481 *-representation, 482 support (of a function), compact,

- set, 50

Riemann surface, 353 Runge type, 319

G -hypocontinuous,

Schwartz space, 132 S6-set, 161 semi-norm, absorbing , 37 -,-, left (right), 37 - ported by a compact s e t , 315

-,

127

28

projective tensor product, 368

z-hypocontinuous, 28 tangent space, 240 tensor, decomposable, 360 tensor product a-norm, 379,382 -

-

--

topology , inductive , 370 - , projective, 366

INDEX

535

t e s t f u n c t i o n s , 132

- I

Theorem, F r o b e n i u s , 61

- , i n v e r s e image, 80

-, -,

-,

G e l f and-Mazur

,

61

( c o m m u t a t i v e ) Gel'fand-NaTmark, 488

-, -,

Igusa-Remmert-Iwahashi,

230

-, L o c a l , 348 -, M a l l i a v i n ' s , 349 -, N a c h b i n ' s , 249 -,-, v e c t o r i z a t i o n o f , 395 -, Oka-Weil ( " i n f i n i t e - d i m e n 319

-,

P o n t r j a g i n D u a l i t y , 240

-, -, -,

Ptbk's,

-,

Sya Do-Sin, 487

268

R i e s z r e p r e s e n t a t i o n , 474 SNAG (= Stone-Nahark-Ambrose-Godement) , 489

topological algebra, division, 63 - - , f i n i t e l y g e n e r a t e d , 283

---

with continuous inversion, 51 - - multiplication, 4

- _ - - q u a s i - i n v e r s i o n , 51 - - - separately continuous

m u l t i p l i c a t i o n (= t o p o l o gical algebra), 4

-

s p a c e , compactly g e n e r a t e d ( = k - s p a c e ) , 166

topology, b i p r o j e c t i v e tensori a l , 311

-, ern-,131 - compatible

with t h e tensor p r o d u c t a l g e b r a s t r u c t u r e , 375

-----_

vector space struct u r e , 364

- ( o n H W ) ) , compact p o r t e d , 316

-,

compactly g e n e r a t e d (= k-toP o l o g y ) , 166 f a i t h f u l , 444

-, -, f i n a l l o c a l l y c o n v e x , - - - rn-convex, 120 -, - v e c t o r s p a c e , 113 I

,

331

J a c o b s o n , 331 l o c a l l y m-convex i n d u c t i v e

l i m i t , 120 -I

Nackey, 374

- , m e a s u r e , 129

K r a m m , 354

sional"),

-,

h u l l - k e r n e l (hk-topology)

113

-,

M i c h a e l , 269

-

of b i e q u i c o n t i n u o u s c o n v e r g e n c e , 371

-,

p r o j e c t i v e t e n s o r i a l , 366

-,

Stone-Jacobson,

-,

s t r o n g e s t l o c a l l y m-convex,

-, -, -,

t r i v i a l l o c a l l y rn-convex, van Hove,

331 8

8

137

Z a r i s k i , 331

t r i p l e t f o r a topological algeb r a , 208 Unit semi-ball, closed, 1 , 2 unitary group, (multiplicative) 232 Wiener-Tauber

condition,

349

,

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