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 ] .
This Page Intentionally Left Blank
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 ) 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 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
. 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 =
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 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 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 )
.
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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 /
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 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
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 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 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 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) ?,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
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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
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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)' 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
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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
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 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)=
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( 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)~: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, 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
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.
This Page Intentionally Left Blank
503
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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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