Chapter XIII Properties of Permanence of Topological Tensor Product Algebras

Chapter XIII Properties of Permanence of Topological Tensor Product Algebras

433 P r o p e r t i e s o f Permanence o f T o p o l o g i c a l Tensor Product Algebras CHAPTER XI11 W e consider i n t h i s chapter s e v e r a...

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433

P r o p e r t i e s o f Permanence o f T o p o l o g i c a l Tensor Product Algebras

CHAPTER

XI11

W e consider i n t h i s chapter s e v e r a l p r o p e r t i e s of topological

a l g e b r a s which a r e p r e s e r v e d when t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s , a s w e l l as c o n d i t i o n s under which t h e s i t u a t i o n i s r e v e r s e d , i . e . , " h e r e d i t y p r o p e r t i e s " of t h e r e s p e c t i v e t o p o l o g i c a l t e n s o r product a l g e b r a s . T h i s h a s been e n c o u n t e r e d a l r e a d y i n t h e f o r e g o i n g by considering, f o r instance, preservation

( a n d h e r e d i t y a s w e l l ) of t h e

local equicontinuity ( o r y e t e q u i c o n t i n u i t y ) of t h e

s p e c t r a of t h e f a c t o r

a l g e b r a s i n a given ( f i n i t e o r i n f i n i t e ) t o p o l o g i c a l tensor product a l g e b r a ; f u r t h e r m o r e , t h e s i m i l a r s i t u a t i o n when t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d were 1.1,

4-algebras. (See Chapt. XII: Lemma l .2 and Remark

a s w e l l a s Theorem 2.2; and, r e s p e c t i v e l y , Lemma 1 . 3 and Corol-

l a r y 2.2). The r e l e v a n t c o n c l u s i o n s , w e are l e d t o i n t h e s e q u e l a r e

of p a r t i c u l a r s i g n i f i c a n c e f o r t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a s of t h e t y p e c o n s i d e r e d i n Chapter XI.

1. Boundaries o f topological tensor product algebras On t h e b a s i s of o u r p r e v i o u s c o n s i d e r a t i o n s i n Chapt. V1;Sect i o n 2, w e f i r s t examine t h e e x t e n t t o which t h e ( s i l o v ) boundary can be d e f i n e d f o r a s u i t a b l e t o p o l o g i c a l t e n s o r p r o d u c t a l g e b r a i n case t h i s boundary e x i s t s f o r t h e i n d i v i d u a l f a c t o r a l g e b r a s . So o u r f i r s t remark i s t h e f o l l o w i n g seemingly s t r o n g e r v e r s i o n

of p r o p o s i t i o n 2 ) i n Lemma VI;2.1;

it i s more a p p r o p r i a t e t o t h e n e x t

d i s c u s s i o n ( c f . Lemma 1.2 b e l o w ) . That i s , w e have.

Remark 1 . 1 . -

Applying t h e n o t a t i o n of Lemma VI;

2.1, one c o n c l u d e s t h a t : An element f E aE C m(EI if, and on2y i f , f o r every open neighborhood U of f i n miE) there e x i s t s an element x E E , w i t h

+ 0,

and such t h a t M ( G )

U. ( The c r u c i a l f a c t t h a t G f 0 f o l l o w s from t h e a r b i t r a r i n e s s of U and t h e f a c t t h a t M(2) = m(E), f o r e v e r y x E E w i t h 2 = O ; c f . V I ; (2.4)).

434

XI11 PROPERTIES OF PERMANENCE

S i n c e t h e t o p o l o g i c a l a l g e b r a s w e a r e g o i n g t o be d e a l t w i t h

w i l l be m a i n l y bounded o n e s ( c f . D e f i n i t i o n V 1 ; l . l ;

see a l s o t h e com-

ment b e f o r e D e f i n i t i o n V I ; 2 . 2 ) , w e c o n s i d e r f i r s t t h e i r b e h a v i o u r i n t a k i n g ( t o p o l o g i c a l ) t e n s o r p r o d u c t s . So w e h a v e t h e n e x t .

Lemma 1 . 1 . Let E , F be two bounded topological algebras and E C3F the corresponding t e n s o r product algebra endowed w i t h a compatible topology tion Xi4.1).

T

T ( cf

. Def i n i -

Then, E Q F i s a bounded topological algebra t o o . T

Proof. I n v i e w of D e f i n i t i o n V I i 1 . 1 ,

it s u f f i c e s t o prove t h a t

t h e r e a l - v a l u e d map

/z^(hll = / h ( z l l

(1 -1)

with h E m ( E Y F l ,

n

i s a bounded f u n c t i o n f o r e v e r y z = t x i @ y i € E E 8 F . T h i s r e d u c e s of c o u r s e i=l

t o an a n a l o g o u s p r o o f , f o r e v e r y decomposable t e n s o r x ~ y [email protected] F . So one obtains

n

s u p / !my) ( h ) I = ( b e c a u s e o f C h a p t . XII; (1.5) ( 1 . 6 ) )

hem(E8FI

( s e e also N . BOUREMI [ 4 : Chap. 4 ; p. 2 6 , r e l . ( 2 5 ) ] ) . T h u s , a c c o r d i n g t o t h e hypothesis f o r the algebras E,F

t h i s i m p l i e s t h e a s s e r t i o n , and t h e

proof i s f i n i s h e d . I A c o n v e r s e t o t h e p r e v i o u s Lemma 1 . 1 i s c e r t a i n l y v a l i d when

e a c h one o f t h e g i v e n t o p o l o g i c a l a l g e b r a s E , F h a p p e n s t o p o s s e s s a t l e a s t o n e bounded element

( t h e c o r r e s p o n d i n g G e l f a n d t r a n s f o r m of t h e

e l e m e n t i n q u e s t i o n i s a bounded f u n c t i o n ; see ( 1 . 2 )

a b o v e ) . For

ex-

ample, t h i s w i l l be t h e ease when t h e given algebras E and F have i d e n t i t y e l e -

ments. Now,

t h e f o l l o w i n g r e s u l t a l s o w i l l b e needed i n t h e s e q u e l t h a t

r e f e r s t o a " t e n s o r p r o d u c t a n a l o g o n " of t h e r e l . V I i ( 2 . 4 ) .

So one h a s .

Lemma 1 . 2 . Let E , F be topological algebras and E B F t h e corresponding tenT sor product algebra equipped w i t h a compatible topology T. Then, f o r every decomposable tensor x a y e E Q F , one g e t s (1.3)

M(&I

x

Mlil

n

E M([email protected])

(see V I ; (2.4) ) . I n p a r t i c u l a r , f o r every decomposable t e n s o r [email protected] Y E E O F

with

z e y # 0 , one o b t a i n s

(1.4)

M(zcy) = M(2)

x

M(y^)

.

The preceding two r e l a t i o n s a r e considered i n the sense of t h e (canonical) b i j e c -

BOUNDARIES OF TENSOR

1.

4 35

PRODUCTS

t i o n XII;( 1 . 1 ) . Proof. For e v e r y if, 3 ) E M i ? )

X

[email protected] g e

M($), o n e c o n c l u d e s t h a t h

M ( z @ y l ( c f . X I I ; ( 1 . 7 ) ) . T h i s f o l l o w s from a n argument a n a l o g o u s t o t h a t

a p p l i e d i n ( 1 . 2 ) and a l s o from Theorem X I 1 ; l . l . n zmy

On t h e o t h e r h a n d , i f G $ O

as well

a n d (f,g l e

+ 0,

one g e t s ( e q u i v a l e n t l y ) 3 # O and A

( c f . X I I ; ( 1 . 6 ) ) ; t h u s , f o r every h e M i x o y ) with

m(E)X n l F l

h’e miE

(Lemma X I I ; l . I ) ,

F)

f’E

Now, by h y p o t h e s i s f o r $,-$

h=fmg

o n e g e t s from ( 1 - 2 )

m(E)

t h i s implies t h a t

I?(f)

I

.

g’ e

M(FI

=

sup

/?(f‘)I

and

f’e m(E)

s u p 1 y^(g’) 1 , so if, g ) e MIL?) x M(y^) Thus , one o b t a i n s t h e i n v e r s e r;‘e m(F) r e l a t i o n t o ( 1 . 3 ) t h a t p r o v e s ( 1 . 4 ) , and t h i s t e r m i n a t e s t h e p r o o f . I

(Glg)1 =

A s a b y p r o d u c t of t h e p r e c e d i n g w e a l s o h a v e t h e f o l l o w i n g .

Corollary 1 . I . Le: E, F be c o m u t a t i v e l o c a l l y m-convex &-algebras and l e t E 8 F be t h e r e s p e c t i v e Locally m-convex t e n s o r product algebra of E , F equipped w i t h a compatible topology T ( D e f i n i t i o n X ; 3 . 1 ) . Then, for every (x,y ) C E X F , one g e t s t h e f o l l o w i n g reLation concerxing t h e corresponding s p e c t r a l r a d i i : r

(1.6)

Proof.

(E

Q7 F)

i x a y ) = r (x) ’ r i y ) E F

The a s s e r t i o n f o l l o w s i m m e d i a t e l y from a n a p p l i c a t i o n o f i n c o n j u n c t i o n w i t h C o r o l l a r y I I I ; 6 . 5 ( s e e 111;

t h e reasoning i n (1.2)

( 6 . 1 6 ) a n d Theorem I ; 6 . 4 ) , Lemma X I I ; 1 . 3 and C o r o l l a r y I I I ; 6 . 6 .

I

B e f o r e w e come t o o u r f i r s t main r e s u l t i n t h i s s e c t i o n , w e s t i l l need t h e f o l l o w i n g u s e f u l

Lemma 1.3. Let E , F be t o p o l o g i c a l algebras and E @ F t h e r e s p e c t i v e t o p o l o gicaL t e n s o r product algebra under a compatible topology Then, f o r everti p a i r

(2,

g) e ([email protected]) O ( z , g)

(1.7)

is given by an

-

:???‘(El --tC

:f

E

E such t h a t

2

= Olz, g)

.

SimiZarZy, one g e t s t h e r c l a t i o n (1.9)

T

(Definition X;4.1 )

m(F),t h e map

;If og)

element of E A ( G e l ’ f a n d t r a n s f o r m a l g e b r a o f E ) . That is,

t h c x e x i s t s a n element x (1 .8)

x

T

;= Y l z , f),

.

4 36

XI11

P R O P E R T I E S OF PERMANENCE

f o r some element y E F , where t h e map '?(z, f) :

m(F/

-+

C is given by the r e l a t i o n

f ) ] fg) = ; ( f o g ) , with g E m ( F ) , and f o r every ( 2 , f)€ ( E 8 F ) X m ( E ) . In p a r t i c u l a r , for complete topological algebras E , F if E 8 F i s s t i l l a (complete) topological algebra, then ( 1 . 8 ) and ( 1 . 9 ) are t r u e for every element z [Y(z,

A

e E$F. Proof.

n ?. x i @ yi E E Q F and g e m(FI, one g e t s ,

Indeed , f o r e v e r y z

i =1

f o r every f e ??Z(E/, t h e r e l a t i o n

A

2(fag) = f(zg(yi)xil = (;gfyi)xil(f) z z

(1.lo)

NOW, t h i s p r o v e s o u r a s s e r r t i o n , c o n c e r n i n g ( 1 . 8 )

E ; similarly

,

. g ( y .)x. e

taking z 7,

for (1-9).

On t h e o t h e r h a n d , f o r e v e r y g e m ( F ) , t h e c o r r e s p o n d e n c e (1.11)

Ci

g

:EQF-E: T

[email protected]+7g(yi)xi 7,

d e f i n e s a continuous linear map ( c f . Lemma 4 . I below)

. SO

i f E i s complete

( 1 . 1 1 ) c a n b e e x t e n d e d t o E 8 F , w e d e n o t e t h i s e x t e n d e d map s t i l l by T

B g , s u c h t h a t one o b t a i n s from ( 1 . 8 ) t h e r e l a t i o n (1.12)

wz,

f o r e v e r y ( z , gl e ( E 8 F ) ( Z c 1 ) a EI relation

in

x

gi =

m(F). I n d e e d ,

E8F, with z = l i m za

A

eg i z i for every

Z E

E6F T

and

a

net

one o b t a i n s , f o r e v e r y f e W(E/, t h e

,

n = limffag)(zcl) = (fag)(limza)= (fag)(z)= ;(fag),

a

c1

which p r o v e s ( 1 . 1 2 ) . A s i m i l a r argument i s o b v i o u s l y v a l i d f o r t h e map ( 1 . 9 ) , a n d t h i s c o m p l e t e s t h e p r o o f . I So w e come now t o t h e n e x t main r e s u l t of t h i s s e c t i o n . Namely,

w e have.

Theorem 1.1. Let E , F be bounded topological aZgebras and E 8 F the corresponding (bounded ; c f . Lemma 1 . l ) topological tensor product algebra of E , F i n a compatible topoZogy T. Then, concerning the respective ( S i l o v l boundaries of the topological algebras i n question ( c f . D e f i n i t i o n V 1 ; 2 . 2 ) , one concludes t h a t a ( E @ F ) e x i s t s i f t h i s i s the ease f o r b o t h aE, aF. Moreover, one has then T

:he r e l a t i o n (1.14)

a ( E 8 F ) = aF x aF, T

within a homeomorphism provided by t h e r e s t r i c t i o n o f the (canonical) homeomorphism X I I ; ( l . l l ) t o the spaces appeared i n ( 1 . 1 4 ) .

1.

437

BOUNDmIES OF TENSOR PRODUCTS

Proof. W e f i r s t p r o v e t h a t ( 1 .15)

aE x aF

5

a([email protected], ?

i n t h e s e n s e t h a t t h e image o f t h e r e s t r i c t i o n of X I I ; ( l . l 2 ) ( “ c a n o n i -

c a l homeomorphism”) t o aE

X

aF

_C

mfE)

m ( F ) is contained i n

X

afEBF,JC T

m ( E 8 F I . T h i s w i l l a l s o prove t h e a s s e r t i o n t h a t a ( E @ F ) e x i s t s : Thus, T

if ff,g)eaExaF

and W i s a n e i g h b o r h o o d o f

b e a n e i g h b o r h o o d of If, g l i n XI1;l.l).

T

f o g = h e ??ZIEfF), let U x V

m(E)x m ( F ) s u c h t h a t U 8 V C W (Theorem a n d Lemma VI;2.1 ( c f .

T h e r e f o r e , by h y p o t h e s i s f o r ( f , g )

a l s o t h e above Remark 1 . 1 1 , t h e r e e x i s t s a decomposable t e n s o r x o y e A

E @ F with x o y

+0

and s u c h t h a t M ( P ) C U a n d M f Q ) E I’ A

M l z o y ) = ( b y ( 1 . 4 ) ) M(2) x M($) C U X Y :

( 1 .16)

.

Hence

one g e t s

W

h e f o g e a ( E B F ) , which p r o v e s ( 1 . 5 ) .

t h e r e f o r e (Lemma V I ; 2 . 1 ) ,

T

F u r t h e r m o r e , w e a l s o p r o v e now t h a t

( 1 .17)

[email protected]) T

5

T h a t i s , one a c t u a l l y p r o v e s t h a t aE E @ F , s o that T

2.2.

aE

x

X

aF

.

aF is a

( c l o s e d ) boundary s e t f o r

( 1 . 1 7 ) f o l l o w s i m m e d i a t e l y from t h e same D e f i n i t i o n V I ;

Indeed, f o r e v e r y element z e E B F , t h e p a i r

Consequently ( D e f i n i t i o n V I ; Z . l ) ,

t h e r e e x i s t s an element f ‘ e a E

(2(f’@J!ll( =

(1.18)

g e m ( F ) deg l s E A (Lemma 1 . 3 ) .

(z, g) with

f i n e s a map l i k e ( 1 . 7 ) and h e n c e a n e l e m e n t ;[email protected],

sup/2(fog)I m(E)

f

.

E

S i m i l a r l y , c o n s i d e r i n g t h e map c o r r e s p o n d i n g t o ( 1 . 9 ) t h e r c o n c l u d e s t h e e x i s t e n c e of a n e l e m e n t g ’ e a F

I2if

(1.19)

og’)

I= g

f o r every f

E

with

s u p 12lf o g l € m(F)

1

,

one f u r -

such t h a t I

M ( E ) a n d z e E @ F . T h u s , t h e l a s t two r e l a t i o n s y i e l d now

t h e following s u p )2(f o g ) J = s u p ( suplz^(f o g l l [email protected](EBF) g e m ( F ) fem(E) (1.20)

= ( b y (1.18))

s u p l i ( f ’ o g ) l = (by ( 1 . 1 9 ) ) [ ; ( f ’ @ g ’ ) \ g e m(F)

with (f‘,g’)e

aE x a F . S o o n e g e t s V 1 ; ( 2 . 1 ) f o r t h e e l e m e n t z e E O F

which t h u s p r o v e s ( 1 . 1 7 ) NOW,

,

,

and t h i s c o m p l e t e s t h e p r o o f . I

i f one c o n s i d e r s complete t o p o l o g i c a l a l g e b r a s , t h e n on t h e

b a s i s o f t h e c o r r e s p o n d i n g p a r t o f t h e p r e v i o u s Lemma 1 . 3 a n d i n conj u n c t i o n w i t h Theorem X I I ; l . 2 ,

o n e e s t a b l i s h e s a n argument a n a l o g o u s

4 38

XI11

PROPERTIES OF PERMANENCE

t o t h a t of t h e p r e v i o u s p r o o f , which i n t u r n y i e l d s t h e f o l l o w i n g basic r e s u l t . Theorem 1.2. Let E , F be complete bounded topological algebras such t h a t t h e

conditions of Theorem X I I ; l . 2 are s a t i s f i e d . Then, concerning

the respective (Silovl boundaries of the topological algebras considered, the existence of a E , aF

implies that of a ( E @ F ) . Furthermore, i n t h a t e a s e , one has the r e l a t i o n T

a(E6FI = aExaF,

(1.21)

within a homeomorphism, which i s provided by XII;(1.20). I I n view of Remark V I ; 2 . 1

,

w e c o u l d c o n s i d e r above topological a l -

gebras with i d e n t i t y elements and compact spectra (see C o r o l l a r y V I ; 2 . 1 )

. In

case o f t h e p r e v i o u s Theorem 1 . 2 , t h i s would imply t h e a l g e b r a s cons i d e r e d t o have, i n f a c t , equieontinuous spectra ( c f . C o r o l l a r y VI; 1 . 3 and Theorem V I ; 1 . 1 )

. So

i n c a s e o f commutative a l g e b r a s it would be

e q u i v a l e n t t o c o n s i d e r &-algebras ( w i t h i d e n t i t y e l e m e n t s . Cf. e v e r , Example V1;2.1).

,

how-

In t h i s respect, a straightforward application

of t h e p r e v i o u s d i s c u s s i o n p r o v i d e s now t h e n e x t . C o r o l l a r y 1.2.

Let E , F be two ( c o m u t a t i v e ) Banach algebras ( w i t h i d e n t i t y

elements) and S i l o v boundaries aE, a F , r e s p e c t i v e l y . Furthermore, l e t

T

be a ( l o c a l -

l y convex algebra) topology on E B F s a t i s f y i n g the conditions of CoroZZary XII;1.2. Then, t h e respective S i l o v boundary a ( E G F 1 of the IBanachl algebra E 8 F i s given T

by t h e r e l a t i o n

T

a(E6F) = aExaF.

(1-22)

This i s v a l i d within a homeomorphism of the spaces involved, and i s provided by XII;

(1.24). I

2 . C o n t i n u i t y o f t h e Gel'fand map Concerning t h e p r o p e r t y i n t i t l e o f t h i s s e c t i o n ( c f . Chapt.VI; Section l ) , i f w e consider topological tensor products, w e actually have t h e f o l l o w i n g r e s u l t . Theorem 2.1. Let E , F be topological algebras w i t h spectra mlEl, m l F ) , re-

s p e c t i v e l y , and l e t E B F be the corresponding topological tensor produet algebra T of E , F under a compatible topology T ( c f . D e f i n i t i o n X ; 4.1: 1 ) - 3 a ) ) . Then, the algebras E , F have the respective Gel'fand maps continuous i f , and only i f , t h i s i s the case f o r t h e algebra E B F . T

Proof. Suppose t h a t t h e a l g e b r a s E , F have c o n t i n u o u s G e l ' f a n d

3.

439

SPECTRALLY BARRELLED ALGEBRAS

maps, a n d c o n s i d e r a compact set K

_C

( E % F ) . Thus w e h a v e

K $ prl(KI x pr2(Kl,

(2.1)

where p r i

( i = l , 2 ) d e n o t e t h e c a n o n i c a l p r o j e c t i o n maps o f m ( E 8 F ) = T

m ( E ) x 1 ? Z ( F ) (Theorem X I 1 ; l . l )

t h e sets

pr.lK) ( i = Z , 2 )

o n t o t h e r e s p e c t i v e f a c t o r s . Hence,both

a r e compact s u b s e t s o f t h e c o r r e s p o n d i n g s p a c e s

a n d h e n c e e q u i c o n t i n u o u s , by h y p o t h e s i s a n d Theorem V 1 ; l . l .

Thus, t h e

set p r ( K ) @ p r ( K ) i s an equicontinuous subset of m ( E 8 F ) ( D e f i n i t i o n 1

2

XII;4.1,cond.3a)); f o r t h e set K .

T

so a f o r t i o r i , d u e t o ( 2 . 1 ) , t h e same i s t r u e

But t h i s i m p l i e s t h e c o n t i n u i t y of t h e G e l ’ f a n d map o f

E B F ( c f . Theorem V I ; 1 . 1 ) . T

Assuming now t h e l a s t c o n c l u s i o n t o b e i n f o r c e , i f

c o m p a c t , t h e set K @ t g }

,

K C m ( E / is

w i t h g e m ( F I , i s a compact s u b s e t of ? ? Y ( E Y F )

h e n c e , from h y p o t h e s i s , e q u i c o n t i n u o u s . Moreover, a p p l y i n g a s i m i l a r r e a s o n i n g t o t h a t i n X I I ; ( l . l 9 ) , one e a s i l y c o n c l u d e s t h a t K i s i n f a c t a n e q u i c o n t i n u o u s s u b s e t of ? ? Z ( E ) ; now t h i s p r o v e s t h a t t h e G e l t f a n d map o f E i s c o n t i n u o u s , w h i l e a s i m i l a r a r g u m e n t i s v a l i d f o r F

a s w e l l . T h i s c o m p l e t e s t h e proof o f t h e theorem. B

Scholium 2.1.- I t becomes c l e a r from t h e p r e v i o u s p r o o f t h a t a s i m i l a r c o n c l u s i o n t o t h a t of Theorem 2 . 1 c a n b e d e r i v e d , i f o n e c o n s i d e r s a com-

p Z e t s t o p o l o g i c a l t e n s o r product algebra o f t h e form EGF,

f o r s u i t a b l e E , F a n d T ; t h i s , of c o u r s e ( c o n -

c e r n i n g t h e p r e c e d i n g p r o o f ) ,i n s o f a r as XII; 1 1 . 2 4 ) is

v a l i d . But t h e l a t t e r may b e i n f o r c e w i t h o u t t h e a l g e b r a E & F t o have n e c e s s a r i l y a l o c a l l y equiconT

t i n u o u s s p e c t r u m ( o r , e q u i v a l e n t l y (Lemma X I I ; 1 . 2 ) , t h e same f o r t h e a l g e b r a s E , F). S e e , f o r i n s t a n c e , Lemma X I I ; 1 . 4 .

3. Spectrally barrelled algebras W e examine n e x t t h e b e h a v i o u r of s p e c t r a l l y b a r r e l l e d a l g e b r a s

( c f . D e f i n i t i o n V; 1.3)

w h i l e t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s . So

w e have.

Theorem 3.1. Let E , F be topological algebras whose spectra are m ( E ) and m ( F ) , r e s p e c t i v e l y , and l e t E B F be t h e corresponding t e n s o r product algebra o f T

E , F endowed w i t h a compatible topology T ( D e f i n i t i o n X ; 4 . 1 :1 )

- 3a) ) .

Then,

t h e f o l l o w i n g two a s s e r t i o n s a r e equivalent: 1 ) Each one o f t h e given algebras E , F is a s p e c t r a l l y b a r r e l l e d algebra.

440

XI11 P R O P E R T I E S OF PERMANENCE 2) The topological ( t e n s o r p r o d u c t ) algebra

i s spectral13 barrel-

E8F T

led. Proof. Suppose t h a t 1 ) i s v a l i d , and l e t B

C_

1 ? T ( E O F ) be a (weakT

l y ) bounded s u b s e t o f t h e spectrum o f E 8 F . Now t h e r e l a t i o n T

(3.1)

B

5 prl ( B )

X

pr2 ( B ) E m ( E )

x

m ( F ) E E,’ x F i

(see t h e n o t a t i o n i n (2.1)) i m p l i e s t h a t B i s an equicontinuous subset of m(E8Fl

t o o . Indeed, t h e s e t s p r i ( B ) ( i = 1 , 2 ) a r e bounded s u b s e t s o f

m ( E ) , m ( F ) , r e s p e c t i v e l y , and hence by h y p o t h e s i s e q u i c o n t i n u o u s . So

i m p l i e s now t h a t prl ( B ) 8 p r 2 ( B )

i s an e q u i c o n t i n u o u s s u b s e t of ??Z(E8 F ) . Thus, due t o (3.1 1 , t h i s i s a f o r T t i o r i t r u e f o r B , which p r o v e s p r o p o s i t i o n 2) ; so w e proved t h a t cond. 3 a ) of D e f i n i t i o n X;4.1

1 ) implies 2 ) .

On t h e o t h e r hand, i f 2) i s v a l i d and B C ? ? Z ( E ) i s bounded, t h e n a l s o B x { g } C m ( E ) x m ( F ) C E’,” F‘, i s a bounded s e t , f o r e v e r y g e m ( F ) . S o from t h e h y p o t h e s i s f o r

T

(see cond. 3) of

-

D e f i n i t i o n X ; 4.1)

and

x‘@ y ’ : E i x F i t h e r e f o r e the c o n t i n u i t y of t h e b i l i n e a r map (x’, y ’ ) -((Ef$F)i , t h e set B 8 { g } i s a bounded s u b s e t of m ( E f $ F ) G ( E f $ F / i , hence by h y p o t h e s i s e q u i c o n t i n u o u s t o o . Thus, a p p l y i n g now a s i m i l a r argument t o t h a t i n X I I ; ( l . l 9 ) , one c o n c l u d e s t h a t B i s a l s o an e q u i c o n t i n u o u s s u b s e t of ? ? Z ( E ) , which a c t u a l l y p r o v e s t h a t E i s a s p e c t r a l l y b a r r e l l e d a l g e b r a . Now an analogous argument can be a p p l i e d t o F ,

s o 2) * l )

as w e l l , and t h i s f i n i s h e s t h e proof o f t h e theorem. I Schol ium 3 . 1 . - By c o n s i d e r i n g complete t e n s o r p r o d u c t a l g e b r a s of t h e form E 6 F ( f o r a p p r o p r i a t e

E, F and

T )

,

T

one a c t u a l l y g e t s a s i m i l a r c o n c l u s i o n

t o Theorem 3.1; t h i s i s e a s i l y u n d e r s t o o d when app l y i n g t h e argument of t h e p r e v i o u s p r o o f , a t l e a s t , i n s o f a r a s a r e l a t i o n l i k e XII;l1.24) holds t r u e . Thus, one can c o n s i d e r , f o r example, spectrally barrelzed algebras w i t h l o c a l l y compact spectra ( h e n c e , e q u i v a l e n t -

l y , l o c a l l y e q u i c o n t i n u o u s ; see Theorem V ; 1.1 )

.

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

would b e equivalent w i t h the a s s e r t i o n the previous two conditions t o be v a l i d for t h e algebra E $ F ( c f . Lemma V ; 2.2 a s w e l l a s Lemma X I I ; 1.2). The p r e v i o u s a r g u -

ment should a l s o be compared w i t h t h e r e a s o n i n g i n Lemma V I I I ; 1.2.

4. SEMI-SIMPLICITY

441

4 . Semi-simplicity W e continue i n t h i s s e c t i o n t h e study of topological t e n s o r

p r o d u c t a l g e b r a s , a s it c o n c e r n s t h e p r e s e r v a t i o n a n d / o r i n h e r i t a n c e o f b e i n g t h e a l g e b r a s under d i s c u s s i o n semi-simple. Among o t h e r t h i n g s , t h i s a l s o p r e s e n t s a c e r t a i n p a r t i c u l a r i n t e r e s t i n i t s e l f connected w i t h t h e t y p e o f “ c o m p a t i b i l i t y ” of t h e t o p o l o g y t h a t one may cons i d e r on t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a , when t a k i n g i t s comp l e t i o n ( c f . Theorem 4.3 b e l o w ) . W e have c o n s i d e r e d a l r e a d y semi-simple t o p o l o g i c a l a l g e b r a s i n

C h a p t e r V I I I (see D e f i n i t i o n V I I I ; 3 . 2 ) h a v i n g t h e u n d e r l y i n g t o p o l o g i c a l v e c t o r s p a c e s n o t n e c e s s a r i l y l o c a l l y convex. However, i n o r d e r t o a v o i d n o t a t i o n a l c o m p l e x i t i e s , h a v i n g t o d o , mainly, w i t h t h e “comp a t i b i l i t y ” of t h e t e n s o r i a l t o p o l o g i e s w e a r e g o i n g t o c o n s i d e r , w e

r e s t r i c t o u r s e l v e s i n t h e s e q u e l , e x c l u s i v e l y , t o locally convex algebras ( w i t h o r w i t h o u t c o n t i n u o u s m u l t i p l i c a t i o n ) , and i n p a r t i c u l a r t o locally rn-convex o n e s ( e . g . s i l o v a l g e b r a s ) . So w e s t a r t w i t h t h e f o l l o w ing useful r e s u l t .

Lemna 4.1. Let E , F be two locally convex spaces whose topological duals are F’, respectively. Moreover, consider a compatible (locally convex) tensorial topology T on E 7 F ( D e f i n i t i o n X; 2.1). Then, f o r every x‘EE’(resp., y’EF’), there e x i s t s a continuous linear mzp

E‘,

n

ex, : E ? F - F : ( r e s p . , 8 , : E @ F+E Y

T

:z - 0

Y

E=

,(z)

I xiayi-eX.(z)

:= 5 y’(yi)xi) 2

such t h a t , concerning the correspond-

ing transpose m p s , one has the relation te , f y ’ ) = t 0 ,(x‘) = (4.2) X

for every y’ E p‘

(resp. , x’ E E’).

Proof. W e prove

:= i

i=l

Y

6’8

y’

t h e c o n t i n u i t y o f ( 4 . 1 ) w i t h r e s p e c t t o t h e re-

l a t i v e t o p o l o g y induced on E @ F from t h e c a n o n i c a l i n j e c t i o n s (4.3)

[email protected]

5 &(E;,

Fs’) 5 Ss(E;, F )

(see C h a p t . X ; ( 2 . 2 4 ) and ( 2 . 4 1 ) ) ; t h e l a s t s p a c e i n ( 4 . 3 ) c a r r i e s t h e t o p o l o g y s of s i m p l e convergence. So t h e r e l a t i o n (4.4)

s < e = e

[email protected]

=

E
( c f . X ; (2.28) and ( 2 . 3 5 ) ) would imply t h e n t h e c o n t i n u i t y o f t h e map with x’EE’, r e l a t i v e t o t h e g i v e n t o p o l o g y T. ( A n a n a l o g o u s a r -

ex.,

gument i s v a l i d , o f c o u r s e , f o r t h e map B y , ,

y’E F ’ ) :

442

XI11

P R O P E R T I E S OF PERMANENCE

Now, t h e c o n t i n u i t y of

i n t h e t o p o l o g y s i s e a s i l y checked

(4.1)

from t h e form of t h e c a n o n i c a l i n j e c t i o n , s a y , a : E B F 4 f s { E i , F): t h e l a t t e r map i s g i v e n by

f o r any z net

z6

7z

n

.I xi e y .

2=1

EE

B F and x ' E E ' .

i n E 8 F C Ls(Ei

i n F , f o r every x ' e E ' .

,F/

Thus,

Therefore, t h e convergence of a

i s e q u i v a l e n t w i t h [ a f z 6 11 ( x ' ) - + [ a ( z l l ( x ' ) &

one g e t s from ( 4 . 5 ) t h a t ex,(z )-+8

i n F , which p r o v e s t h e c o n t i n u i t y o f t h e map 0

X"

f x ' , y') E E ' x F ' ,

F u r t h e r m o r e , for a n y p a i r

6

x'eE'.

&

X

,(z)

one g e t s t h e f o l l o w -

i n g r e l a t i o n , c o n c e r n i n g t h e r e s p e c t i v e t r a n s p o s e map o f

(4.1):

n f o r e v e r y e l e m e n t z z . 1 x . @ y . E E 8 F . S o w e h a v e p r o v e d ( 4 . 2 ) , and t h i s

c o m p l e t e s t h e p r o o f of

2

2

t h e lemma. I

I f w e a r e g i v e n complete l o c a l l y convex spaces E , F ing r e l a t i o n s (4.1)

,t h e n

t h e preced-

a n d ( 4 . 2 ) c a n b e e x t e n d e d by c o n t i n u i t y t o t h e re-

s p e c t i v e complete ( l o c a l l y convex) t e n s o r p r o d u c t space E 6 F T

.

W e come n e x t t o t h e f i r s t main r e s u l t o f t h i s s e c t i o n . So one

has.

Theorem 4.1. Let E , F be l o c a l l y convex algebras and E F t h e r e s p e c t i v e l o c a l l y convex tensor product algebra of E , F i n a compatible topology

(cf. Defi-

n i t i o n X i 3 . 1 ) . Then, E B F i s semi-simple if, and only if, t h i s i s t h e case f o r T

each one of the given algebras E , F .

i s semi-simple,

Proof. Suppose t h a t E 8 F T

and l e t x E E w i t h

2=

0.

T h u s , f o r any y e F and k e m ( E 8 F l , one o b t a i n s T

A

( x e y J ( k l = h ( x e y I = (by Chapt. X I I ; (1.6), (1.7))

(4.7)

(f e g l ( x C 9 y ) = f l x l g l y ) = g ( y l 2 l f ) = 0

T h e r e f o r e , due t o t h e h y p o t h e s i s f o r y

E

F , which i m p l i e s t h a t x = 0 ( c f

semi-simple,

.

E B F , one g e t s T

Lemma X ; 1 . 1 )

. So

. xmpy=O f o r every the algebra E is

a n d a n a n a l o g o u s p r o o f c a n be g i v e n f o r F . T h i s e s t a b -

l i s h e s t h e "only i f " p a r t of t h e a s s e r t i o n . NOW, assume t h a t b o t h

t h e g i v e n a l g e b r a s E , F are s e m i - s i m p l e ,

and c o n s i d e r a n element z e E Q F

with z f 0 .

Thus, on t h e b a s i s o f t h e

f i r s t of t h e c a n o n i c a l i n j e c t i o n s ( 4 . 3 ) , one p r o v e s t h e e x i s t e n c e of an element ( x ' , y ' l e E ' x

F'

such t h a t

4. SEMI-SIMPLICITY (4.8)

(a'ay'J(zi

443

,(z)) # 0. Y so by t h e s e m i - s i m p l i c i t y of E l t h e r e e x i s t s an

= (applying t h e n o t a t i o n of (4.6)) 2'18

Therefore,

8 ,(z)#O; Y element f e m ( E ) such t h a t

n

(4.9)

eY , ( z i ( f i

(4.10)

ef ( z i ( g i

.

= f i e , i z i ) = ( c f . (4.6)) y ' ( e ( z i i # u Y f A c c o r d i n g l y , 0 ( z ) # 0 so t h a t from t h e s e m i - s i m p l i c i t y of F , one g e t s f an element g e m ( F ) w i t h A

where

fag

= g ( e ( z i i = (by ( 4 . 6 ) ) ( f a g ) ( z ) = $ ( f a g )#

f

o,

.

h e n T ( E 8 F ) (Theorem X I I ; 1 . 1 ) T h e r e f o r e , 2 # 0 which p r o v e s T E 8 F (see D e f i n i t i o n V I I I ; 3 . 2 ) . I

t h e s e m i - s i m p l i c i t y of

Scholium 4.1

.- A s

T

f o l l o w s from t h e p r e v i o u s proof

, we

made u s e

t h e r e o f Lemma 4.1 o n l y " a l g e b r a i c a l l y " ; namely, no u s e was made of any c o n t i n u i t y of t h e maps ( 4 . 1 ) . tion E 8 F S d ( E ; , F i )

applied

Furthermore, t h e canonical i n j e c -

i n ( 4 . 8 ) i s d e r i v e d a l r e a d y from t h e

semi-simplicity of t h e a l g e b r a s E and F (assumed i n t h a t p a r t of t h e f o r any

p r o o f ) . T h e r e f o r e , one c o u l d g e t t h e p r e c e d i n g Theorem 4 . 1

topological a l g e b m s E , F a s w e l l and a compatible topology

T

, not

n e c e s s a r i l y l o c a l l y convex o n e s ,

on E 8 F ( c f . D e f i n i t i o n X ; 4 . 1 ) . O f c o u r s e t h e

s i t u a t i o n i s more c o m p l i c a t e d , i f one wants t o c o n s i d e r complete t e n s o r p r o d u c t a l g e b r a s o f t h e form E & F , a s w e s h a l l see i n t h e s e q u e l . T So t o f a c i l i t a t e arguments, w e r e s t r i c t o u r a t t e n t i o n t o l o c a l l y convex a l g e b r a s .

( I n t h i s r e s p e c t , c f . a l s o t h e comment a f t e r t h e end o f

t h e proof o f t h e above Lemma 4 . 1 ) . Before w e proceed f u r t h e r , w e comment a b i t more on t h e t e r m i nology w e a r e going t o apply i n t h e s e q u e l . Thus, suppose t h a t w e a r e g i v e n two l o c a l l y convex a l g e b r a s E l F i n such a way t h a t t h e f o l l o w i n g c o n d i t i o n i s f u l f i l l e d : EGF T

, i.e. ,

t h e completion of

E 8 F with respect T

t o a c o m p a t i b l e t e n s o r i a l topology

(4.11)

T

on E @ F

(see D e f i n i t i o n X; 3 . 1 ) i s a ( c o m p l e t e ) locaZly convex algebra. One e n c o u n t e r s t h e s i t u a t i o n d e s c r i b e d by t h e p r e v i o u s c o n d i t i o n i n c a s e , o f c o u r s e , t h e g i v e n l o c a l l y convex a l g e b r a s E , F have

(4.11)

( j o i n t l y ) continuous m u l t i p l i c a t i o n s . On t h e o t h e r hand, one may s t i l l look a t t h e s i t u a t i o n p r o v i d e d by Lemma X I ; 1 . 1 i n c o n n e c t i o n w i t h Theorem XI; 1 . 1 . NOW,

i f t h e u n d e r l y i n g l o c a l l y convex s p a c e s E and F

(of t h e

444

XI11 PROPERTIES OF PERMANENCE

above a l g e b r a s ) a r e compZete, t h e r e s p e c t i v e l o c a l l y convex space F i ) i s complete t o o ( c f . X ; ( 2 . 4 1 ) ) .

&e(E;,

So i f one l o o k s a t t h e ( c a -

n o n i c a l ) continuous Zinear i n j e c t i o n (4.12)

p r o v i d e d by X ; ( 2 . 2 4 ) and t h e r e l a t i o n t i o n X; 3 . 1

, then

E<

"extending it by continuity"

, w i t h T g i v e n by D e f i n i , one g e t s a ( c a n o n i c a l ) con-

T

tinuous l i n e a r map (4.13) NOW,

i f w e a s k f o r t h e e x t e n t t o which t h e k t t e r map i s aZso one-to-

one, w e a r e t h u s l e d t o a well-known problem posed by A . Grothendieck ("ProbZdme de b i u n i v o c i t d " ; see A . GROTHENDIECK [3: Chap. I ; p . 35 f f .] ) , a s it c o n c e r n s t h e r e s p e c t i v e complete l o c a l l y convex s p a c e s o c c u r r e d i n ( 4 . 1 3 ) . On t h e o t h e r hand, a s w e s h a l l p r e s e n t l y see, t h e same probl e n h a s an i n t i m a t e r e l a t i o n ( i t i s , i n e f f e c t , e q u i v a l e n t , see Theo-

r e m 4 . 3 below) w i t h t h e q u e s t i o n of b e i n g t h e a l g e b r a E 6 F semi-simple T (cf. also ( 4 . 1 1 ) ) . Now, l o o k i n g a t t h e s i t u a t i o n which one h a s i f l o c a l l y convex s p a c e s are i n v o l v e d , w e conclude t h a t f o r e v e r y nuclear ( l o c a l l y convex) algebra E ( c f . D e f i n i t i o n V I I I ; 9 . 2 ) t h e map ( 4 . 1 3 ) i s i n j e c t i v e ; h e r e one c o n s i d e r s t h e c o m p l e t i o n of E and t h e c o r r e s p o n d i n g t o p o l o g i c a l t e n s o r p r o d u c t w i t h any complete l o c a l l y convex a l g e b r a F i n t h e topo-

logy

F (Lemma X i 2 . 1 ) . ( W e s t i l l

(2.41 )

,

(2.42) )

. Moreover,

assume h e r e ( 4 . 7 7 ) ; see a l s o Chapt. X ;

one comes t o t h e same c o n c l u s i o n f o r a cer-

t a i n p a r t i c u l a r c l a s s of l o c a l l y convex a l g e b r a s , namely of t h o s e s a t i s f y i n g t h e approximation property (cf. e . g . A . GROTHENDIECK [3:Chap. I ; p . 169, Lemme 1 9 1 ) ; y e t f o r e v e r y Banach algebra having t h e same property ("Ba-

.

nach approximation p r o p e r t y " : I b i d . ; p . 168, C o r o l l a i r e 3 ) Of c o u r s e e v e r y n u c l e a r ( l o c a l l y convex) a l g e b r a h a s t h e approximation p r o p e r t y (see e . g . F . T R E V E S [ I : p. 520, P r o p o s i t i o n 5 0 . 3 1 ) . T

On t h e o t h e r hand, l o c a l l y convex t e n s o r i a l ( a l g e b r a ) t o p o l o g i e s f o r which t h e p r e c e d i n g map ( 4 . 1 3 ) i s one-to-one a r e now c l a s s i f i e d

by t h e f o l l o w i n g . D e f i n i t i o n 4.1.

Suppose t h a t w e a r e g i v e n two complete l o c a l l y

i s a ( c o m p l e t e ) l o c a l l y convex a l Then, w e s h a l l say t h a t T i s a f a i t h f u Z topology on E Q F , whenever t h e c o r r e s p o n d i n g ( c a convex a l g e b r a s E l F such t h a t E 6 F

g e b r a , t h e topology

T

T

s a t i s f y i n g Definition X; 3 . 1 .

n o n i c a l c o n t i n u o u s l i n e a r ) map ( 4 . 1 3 )

i s one-to-one.

W e come now t o t h e f o l l o w i n g r e s u l t , an analogon i n c a s e o f com-

445

4. SEMI-SIMPLICITY

p l e t e a l g e b r a s of o u r p r e v i o u s Theorem 4 . 1 .

Theorem 4.2.

Namely, w e have.

Let E , F be two complete l o c a l l y convex algebras w i t h l o c a l l y

equicontinuous spectra m(E), ? ? Z ( F ) , r e s p e c t i v e l y , such t h a t E $ F t o be a ( c o v p l e t e ) l o c a l l y convex a l g e b m w i t h respect t o a compatible ( l o c a l l y conved tensorz h Z topology

that

T

T

on E 8 F ( D e f i n i t i o n X;3.1:

i s a f a i t h f u l topology on E 8 F ( cf

X ; (2.11, (2.3)). Moreover, suppose

. Definition

4 . 1 ) . Then, the fo22ow-

i n g two a s s e r t i o n s are e q u i v a l e n t : h

1 ) E 8 F i s semi-simple. T

21 Each one of t h e given algebras E , F are semi-simple.

proof. Suppose t h a t 1 ) i s v a l i d , and l e t x e E w i t h

?=a.

Thus,

f o r e v e r y p a i r (y, ~ 2 ) B F x M ( E % F ) ,one g e t s a r e l a t i o n a n a l o g o u s t o ( 4 . 7 ) with

T

h = f e g and ( f , g ) e ? R ( E ) x Z T ( F ) , p r o v i d e d by Theorem XII;1.2.

This proves t h e semi-simplicity of t h e algebra E , while a s i m i l a r a r gument c a n be a p p l i e d t o F , which p r o v e s t h a t 1 ) * 2 ) .

Remark 4.1. - It becomes c l e a r from t h e above f i r s t p a r t of t h e proof of Theorem 4 . 2 t h a t t h e applied argument i s i n f o r c e even i n t h e context of Theorem XII; 1 . 2 . Thus, no u s e h a s been made of t h e p r e v i o u s D e f i n i t i o n 4.1 ( i .e , , of t h e " f a i t h f u l n e s s " of t h e topology T ) . End of t h e proof of 1"heorem 4.2. A s s u m e now t h a t t h e a l g e b r a s E , F a r e semi-simple,

and l e t z e E 6 F with z # 0 . Then, from t h e h y p o t h e s i s f o r

7

T

( D e f i n i t i o n 4 . 1 ) t h e ( c a n o n i c a l ) image o f z i n & ( E i , F ) i s non-zero t o o ; hence, t h e r e e x i s t s a p a i r ( x ' , y ' ) € E ' x (4.14)

[j^(Z)l(S',

y 7 = ( x ' e y')(z)

( c f . ( 4 . 1 3 ) ) . Thus, on t h e b a s i s of

F'

such t h a t

+0

(4.141, t h e argument now p r o c e e d s

i n a s i m i l a r way t o t h a t a p p l i e d t o t h e r e l a t i o n s ( 4 . 8 ) - ( 4 . 1 0 ) d e r t o f i n d an element

g)e

($,

m(E) x m(F),and

i n or-

hence (Theorem XII;l.2)

an element h = f o g B ~ x ( E $ F ) h(z)

(4.15)

E

(f o g ) ( z J =

z^(f o g ) #

0 .

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

E6F 7

, so

2)

+.

1)

a s w e l l , and t h i s c o m p l e t e s t h e proof o f t h e theorem. I As a c l a r i f i c a t i o n ,

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

p r e c e d i n g m o t i v a t e d by t h e t y p e of t o p o l o g i c a l a l g e b r a s appeared i n t h e comment b e f o r e D e f i n i t i o n 4 . 1 .

But f i r s t w e have t o e s t a b l i s h a

b i t more t e r m i n o l o g y . So w e have t h e n e x t .

Definition 4.2.

W e say t h a t a g i v e n l o c a l l y m-convex a l g e b r a

E

446

XITI P R O P E R T I E S OF PERMANENCE

h a s t h e approximation property, whenever t h e r e e x i s t s an Arens-Michael decomposition of E , s a y , ( E a , f aB ) ClCI ( c f . 111; ( 3 . 9 ) ) , i n such a way t h a t t h e r e s p e c t i v e Banach a l g e b r a s ka , C I E It ,o have t h e (Banach) approximation p r o p e r t y . For t h e l a s t p r o p e r t y see e . g . P r o p o s i t i o n 35, ( A 5 )

1.

A . GROTHENDIECK 13: Chap.

I ; p - 164,

Now f o l l o w i n g t h e same a u t h o r ( i b i d . ; p . 1 6 9 ,

Lemme 1 9 ) , one c o n c l u d e s t h a t i f a l o c a l l y m-convex a l g e b r a E h a s t h e

approximation p r o p e r t y , t h e n the respective l o c a l l y convex space E has t h e

same property ( s e e D e f i n i t i o n X ; 2 . 4 above) ; moreover, f o r every l o c a l l y A

h

convex space F , one g e t s f o r t h e r e s p e c t i v e c o m p l e t i o n s P , F t h a t t h e canonical continuous l i n e a r m p

,$88

(4.16)

4 Se

(E;

,F i )

i s i n j e c t i v e . Thus, g i v e n t h e a l g e b r a E a s above, and f o r any l o c a l l y m-convex a l g e b r a F , t h e p r o j e c t i v e t e n s o r i a l t o p o l o g y

TI

i s always a

f a i t h f u l topology ( D e f i n i t i o n 4 . 1 ) . The promised c l a r i f i c a t i o n i s now a d i r e c t consequence of t h e p r e v i o u s d i s c u s s i o n . So w e have t h e n e x t .

Corollary 4.1. Let E , F be l o c a l l y m-convex algebras such t h a t one of them has t h e approximation property ( D e f i n i t i o n 4 . 2 ) , while t h e i r completions are Q-aZgebras. Then, t h e (complete) l o c a l l y m-convex algebra

28:

(E

%, is

semi-simple i f , and only i f , t h i s i s t h e case f o r each one of t h e complete algebras and

2.

I

W e come n e x t t o t h e c o n n e c t i o n of t h e " f a i t h f u l n e s s " of a g i v e n

c o m p a t i b l e t e n s o r i a l t o p o l o g y T on E ~ (FD e f i n i t i o n 4 . 1 ) witk, t h e semiT s i m p l i c i t y of t h e a l g e b r a E 6 F . I n d e e d , assuming f u r t h e r t h a t t h e alT

g e b r a s E and F a r e semi-simple,

t h e p r e c e d i n g two c o n d i t i o n s a r e , i n

f a c t , e q u i v a l e n t . So one h a s t h e f o l l o w i n g .

Theorem 4.3. Suppose t h a t a l l t h e conditions, except of t h e f a i t h f u l n e s s of t h e topology T , i n Theorem 4 . 2 are s a t i s f i e d . Then, the following two a s s e r t i o n s are equivalent: 1 ) t h e algebra

A

E8F

i s semi-simple.

T 2 ) The given compatible ( l o c a l l y convex) t e n s o r i a l topology T on E 8 F i s

f a i t h f u l ( D e f i n i t i o n 4.1) and each one of t h e algebras E , F i s semi-simpZ.e:. Proof. Suppose t h a t t h e a l g e b r a E6F T

with z + U ,

m(E6FI T

hence

2 # 0.

such t h a t z^(h)# a ,

with h = f m g

,.

E B F i s semi-simple, T

and l e t Z E

T h e r e f o r e , t h e r e e x i s t s an element h e and hence a p a i r ( f , g ) B m l E l X M ( F ) C E i X F i

(Theorem X I I ; 1 . 2 ) such t h a t one h a s

5.

(4.17)

447

IDENTITY ELEMENTS

264) = 2 ( f a g l = ( f a g ) ( z l # O .

T h i s p r o v e s t h a t ( 4 . 1 3 ) i s one-to-one

so t h a t t h e t o p o l o g y T i s , by

d e f i n i t i o n , f a i t h f u l . F u r t h e r m o r e , Theorem 4 . 2 t o g e t h e r w i t h Remark 4 . 1 g u a r a n t e e now t h a t t h e a l g e b r a s

E , F a r e semi-simple,

so t h a t

1) *2).

On t h e o t h e r hand, i f 2 ) i s v a l i d , t h e n a l l t h e c o n d i t i o n s of t h e p r e v i o u s Theorem 4 . 2 a r e s a t i s f i e d , hence by t h e same theorem t h e algebra

E6F T

is s e m i - s i m p l e ,

so 2 ) = > I ) a s w e l l , and t h i s com-

p l e t e s t h e proof. I An

analogous conclusion with Corollary 4.1,

within the context

o f t h e p r e c e d i n g Theorem 4 . 1 , i s now q u i t e c l e a r , so w e o m i t t h e d e t a i l s . ( T h u s one could o b t a i n a m o r e c o n c r e t e p i c t u r e t h a n t h e previous one). A s a n a p p l i c a t i o n of t h e f o r e g o i n g , w e f u r t h e r examine i n t h e

s u b s e q u e n t s e c t i o n s h e r e d i t a r y p r o p e r t i e s which a t o p o l o g i c a l t e n s o r product algebra

E4F T

may h a v e ( i m p l i e d from a n a l o g o u s p r o p e r t i e s of

t h e f a c t o r s ) , d e p e n d i n g i n p a r t i c u l a r on t h e s e m i - s i m p l i c i t y of t h e f a c t o r a l g e b r a s E,F. 5 . I d e n t i t y elements I t i s c l e a r of c o u r s e t h a t i n c a s e of two u n i t a l a l g e b r a s E , F

whose i d e n t i t y e l e m e n t s a r e l E , I F , r e s p e c t i v e l y , t h e e l e m e n t l E a I F i s a n i d e n t i t y o f t h e r e s p e c t i v e t e n s o r p r o d u c t a l g e b r a [email protected] ( c f . X ; ( 1 . 1 2 ) ) . Thus, t h e n o n - t r i v i a l

p a r t o f t h e n e x t theorem i s e x a c t l y

a converse conclusion i n t h e presence of semi-simplicity of t h e algeb r a s i n v o l v e d . But f i r s t w e comment on a n a u x i l i a r y u s e f u l r e s u l t ( i n f a c t , a " m u l t i p l i c a t i v e analogon" o f Lemma 4 . 1 ) , a s i s t h e n e x t . Lemma 5.1. L e t E , F be l o c a l l y convex aLgebras w i t h continuous m u l t i l i c a t i o n s and spectra ???(El, T R ( F ) , r e s p e c t i v e l y . Moreover, Let F @ F be t h e corresponding comT pZete l o c a l l y convex tensor product aZgebra o f E , F i n a compatible topoLogy T h

( D e f i n i t i o n X;3. 1 ) . Then, f o r every f e m(E) ( r e s p . , g € m(F)),the map (5.1) ( z ) = E g ( y .)x.) d e f i n e s a continuous (algebra) morE :2-8 i r e s p . , 6 :[email protected]+ 9 T 9 7 2 % phism of the topological algebras invoZved. Thus, one has an "extension by continu-

i t y " of ( 5 . 1 ) t o a map

448

XI11 PROPERTIES OF PERMANENCE

hence 5 E Horn ( E F ,2) Iresp., E Horn ( E 6 F , f 9 T spective transpose m p s of (5.11, one g e t s

te

(5.3) for every (f, g ) 6

m(E) X

Furthermore, concerning t h e re-

te cf)

= fege ~ ( E ~ F I 9 m(F) (and a l s o t h e analogous one t o t h e m p s (5.2)).

f

(gi =

E") I.

Proof. I n view o f Lemma 4.1

,

t h e rels. (5.1) define continuous

l i n e a r maps o f t h e r e s p e c t i v e l o c a l l y convex s p a c e s , so t h e y can b e extended by c o n t i n u i t y t o t h e c o r r e s p o n d i n g maps ( 5 . 2 ) . T h e r e f o r e , we have o n l y t o show t h a t (5.1) a r e , i n f a c t , m u l t i p l i c a t i v e ( l i n e a r ) maps, due t o o u r h y p o t h e s i s f o r f, g , and hence t h e same would be t r u e f o r t h e extended maps (5.21, because o f t h e c o n t i n u i t y of t h e m u l t i plication in E , F: Thus, f o r e v e r y f e ? ? ? ( E )

and any s , t i n E 8 F g i v e n by X;(l.ll),

one g e t s from (5.1) and X ; (1.12)

(5.4)

t h a t w a s t o b e proved. On t h e o t h e r hand, one g e t s (5.3) on t h e b a s i s of (4.2) and t h e f a c t t h a t f @ g E m ( E @ F ) ( = ( w i t h i n a bijection) m ( E $ F ) ) , f o r e v e r y (f, g ) T

e ~ T ( E ) x ~ ? ? ( F ) ( c f . X; (2.2) and XII;(l.lO) i n t h e proof of Lemma XII; 1.1). I Remark 5.1. - Concerning t h e p r e c e d i n g proof , w e n o t e t h a t t h e t o p o l o g i c a l p a r t o f t h e proof o f Lemma 4.1, and hence t h e local convexity of t h e t o p o l o g i c a l v e c t o r s p a c e s i n v o l v e d , h a s been a p p l i e d o n l y i n c o n n e c t i o n w i t h t h e c o n t i n u i t y of t h e maps (5.1). T h e r e f o r e , i f t h e l a t t e r i s i r r e l e v a n t , i n some p a r t i c u l a r c o n t e x t , one could o b t a i n t h e previous Lemma 5.1 f o r any t o p o l o g i c a l a l g e b r a s ( n o t necess a r i l y l o c a l l y convex o n e s ) w i t h c o n t i n u o u s m u l t i p l i c a t i o n and any t e n s o r i a l t o p o l o g y T on E 8 F "comp a t i b l e i n t h e s e n s e o f D e f i n i t i o n X;4.1. Now t h e c o n t i n u i t y of (5.1) i s i n d i s p e n s a b l e , of c o u r s e , t o have ( 5 . 2 ) ; t h i s i s t h e case i n t h e n e x t theorem. So w e r e s t r i c t o u r s e l v e s t o l o c a l l y convex a l g e b r a s ; ( a l s o t o t h e e x t e n t t h a t w e do n o t assume a g i v e n i d e n t i t y element o f an a l g e b r a t o be a l s o s u c h i n a l l i t s s u b a l g e b r a s ! In t h i s r e s p e c t , see a l s o C o r o l l a r y 5.1 b e l o w ) . W e come n e x t t o o u r main r e s u l t o f t h i s s e c t i o n . So w e have.

Theorem 5.1.

Let E , F be complete semi-simple l o c a l l y convex algebras w i t h

5.

449

IDENTITY ELEMENTS

continuous m l t i p l i c a t i o n s and spectra %Z ( E l , ? ? Z ( F ) , r e s p e c t i v e l y . Moreover, l e t A

E B F be t h e corresponding complete loca2ly conuex tensor producb algebra o f E , F T

having a continuous rnultipZication i n a compatible t e n s o r i a l topology T (Def i n i t i o n Xi3.1). Then, t h e algebra E @ F has an i d e n t i t y element i-f,and only i f , t h i s 7

i s the case for each one of t h e given aZgebras E , F . Proof. A s w e remarked a t t h e b e g i n n i n g of t h i s s e c t i o n , i t suf-

f i c e s t o p r o v e t h e " o n l y i f " p a r t o f t h e a s s e r t i o n : So l e t e be a n i d e n t i t y e l e m e n t o f t h e a l g e b r a B g F : t h u s a p p l y i n g Lemma 5.1 one T

-

g e t s , f o r e v e r y g e ? ? Z ( F ) , an element e ( e l E E . NOW we p r o v e t h a t G ( e ) 9 9 i s an i d e n t i t y e2ement o f t h e algebra E. Indeed, f o r any x e B and f e m(E) , one g e t s

f(5

(5.5)

9

(e)x) =

f(eY ( e ) ) . f ( x l

= ( b y (5.3); see a l s o (4.10))

I f @ g ) ( e l . f ( x l = (by ( 5 . 3 ) ) h ( e ) . f ( x l = f ( x )

,

s o t h a t one h a s f(B (ejx-x) =

f o r every f

e9 ( F I X =

E

o ,

9 m ( E ) . H e n c e , by t h e s e m i - s i m p l i c i t y of E l one o b t a i n s

x f o r e v e r y x e E , which p r o v e s t h e a s s e r t i o n . A s i m i l a r a r g u -

ment i s v a l i d f o r F , and t h i s c o m p l e t e s t h e p r o o f . I

Remark 5.2.- A s it becomes clear from t h e _ p r e c e d i n g proof , c o n c e r n i n g t h e element e g ( O e E = E , one a c t u a l l y a p p l i e s t h e " e x t e n s i o n by c o n t i n u i t y " o f t h e a l r e a d y c o n t i n u o u s l i n e a r map B g : E Y F + F ( c f . Lemma 4 . 1 ) . A c c o r d i n g l y , t h e proof i s v a l i d f o r any completeh ZocalZy convex ( s e m i - s i m p l e ) algebras E , F for which E 8 F i s a ( c o m p l e t e ) l o c a l l y convex algebra i n a c o m p a h b l e ( l o c a l l y convex) t e n s o r i a l t o pology T on E B F . T h i s happens o f c o u r s e w i t h l o c a l l y convex a l g e b r a s h a v i n g c o n t i n u o u s m u l t i p l i c a t i o n s , a s w e assumed i n t h e p r e c e d i n g . (However , c f . a l s o Theorem XI; 1 . 1

.

A s an a p p l i c a t i o n of t h e p r e c e d i n g d i s c u s s i o n , w e s t i l l have

t h e following c l a r i f y i n g r e s u l t .

Corollary 5.1. L e t E , F be Fre'chet semi-simple l o c a l l y convex algebras and h

h

l e t E 8 F I = E 8 F ) be t h e corresponding Frdchet l o c a l l y convex aZgebra, complete TI

p r o j e c t i v e tensor product algebra of E , F (see Chapt

. X; Lemmas

2.1 and 3.1).

Then, t h e algebra E & F has an i d e n t i t y element i f , and only i f , t h i s i s t h e case f o r each one of t h e algebras E , F . I Schol ium 5.1.

-

The p r e v i o u s C o r o l l a r y 5.1 i s

o f c o u r s e a d i r e c t a p p l i c a t i o n of Theorem 5.1. Howe v e r , one c o u l d o b t a i n a more c o n c r e t e form of t h e

450

XI11 PROPERTIES OF PERMANENCE

r e s p e c t i v e i d e n t i t y e l e m e n t s by u s i n g a c l a s s i c a l r e a s o n i n g , p r o v i d e d by t h e p a r t i c u l a r h y p o t h e s i s

w e made: Thus, s i n c e t h e a l g e b r a

lar, a

Fre'chet l o c a l l y

E 6 F is, i n particu-

convex space, one may a p p l y

h e r e a s t a n d a r d r e s u l t due t o A . GROTHENDIECK [3: Chap. I ; p. 51, Th6orZme

I].

So i f e i s a n i d e n t i t y e l e -

ment o f t h e a l g e b r a E 6 F

,

t h e n e admits an expression

of t h e form m

( s e e t h u s t h e above Remark 5.1 )

.

Here

(2:

I , (y,)

are

n u l l sequences i n E , F , r e s p e c t i v e l y , and ( A n ) m

a

2 lA,l
s e q u e n c e o f complex numbers w i t h

l o c a l l y convex s p a c e E G F ( i b i d . , a n d / o r F . TREVES

[I: p. 459, Theorem 45.11). T h e r e f o r e , f o r e v e r y g E ( F ) , one g e t s from ( 5 . 2 ) a w e l l - d e f i n e d e l e m e n t

of E g i v e n by t h e r e l a t i o n (2)

( s e e a l s o F . T R E W S [I: p. 459, D e f i n i t i o n 45.1 and t h e s u b s e q u e n t comment] ) i n g of

. Now

applying t h e reason-

( 5 . 5 ) , o n e p r o v e s f u r t h e r on t h a t t h i s i s ,

i n e f f e c t , an i d e n t i t y element of t h e algebra E .

6. Regularity. s i l o v algebras W e c o n s i d e r n e x t " h e r e d i t a r y " p r o p e r t i e s of t o p o l o g i c a l t e n s o r

p r o d u c t s r e f e r r i n g t o regular topological algebras tion 2.1).

(see C h a p t . I X ; Def i n i -

So w e f i r s t h a v e t h e f o l l o w i n g p r e l i m i n a r y r e s u l t .

Lemma 6.1. Let E , F be topological algebras w i t h spectra m(E),? ? Z ( F ) , r e s p e c t i v e l y , and l e t E 8 F be t h e r e s p e c t i v e topological t e n s o r product algebra o f E , ? F i n a compatible topology T. Then, t h e following two a s s e r t i o n s are e q u i v a l e n t : 1 ) Each one o f t h e algebras E , F i s regular.

2) The algebra E 8 F i s regular. proof.

Suppose t h a t 1 ) i s v a l i d , and l e t

s u b s e t t o g e t h e r w i t h an e l e m e n t h

E

B C m I E 8 F ) be a c l o s e d

??Y(EYF) with h

T

f?

B . Thus, C B i s an

o p e n n e i g h b o r h o o d of h i n m ( E O F ) ; so l o o k i n g a t t h e r e s p e c t i v e " c a ?

6.

REGULARITY.

SILOV ALGEBRAS

451

m(F) w i t h

(f,g) e m(E)x

n o n i c a l d e c o m p o s i t i o n ” o f h by an e l e m e n t

= f @ g (Lemma X I I ; 1 . 1 ) , one g e t s open s e t s U

h

m ( E ) and V .G m(F/ w i t h

C

(f,g ) e U X V 5 C B

(6.1)

(Theorem X I I ; 1 . 1 ) ; so

f

# CU and

(see D e f i n i t i o n I X ; 2 . 1 1 , (6.2)

=I,

;(f)

g # CV. Hence by h y p o t h e s i s f o r E , F

t h e r e e x i s t e l e m e n t s X E E and Y E F

2

Accordingly, looking a t t h e element

= (by X I I ; ( l . 7 ) )

,

x a y eEQF

Icv.

=O

one h a s

f l x l g l y l = P(fly^(g)= 1 .

zay = 0

from ( 6 . 2 )

one g e t s

B=C(UxV), A

t h a t is the algebra EQF

,

y^

= ( x \ y l ( f o g ) =(fag)(xcy)

(XG&)(hl

Moreover, s i n c e by ( 6 . 1 )

y^(g) = I ,

and

=0lcU

such t h a t

IB

,

i s r e g u l a r , so t h a t 1 ) + 2 ) .

On t h e o t h e r h a n d , assuming t h a t 2 ) i s t r u e , l e t A C V ? ( E ) c l o s e d a n d a l s o f e m(E) w i t h f # A .

Thus, c o n c e r n i n g t h e

be

cZosed s e t

A x { g } G??Z(E) x m ( F ) = (Theorem X I I ; 1 . 1 ) m f E T F 1 , one h a s ( f , g ) # A x { g ) . S o h ~f Q g E c ([email protected]{ g } ) E 5’2 ( E 8 F ) , a n d h e n c e t h e r e e x i s t s , by h y p o t h e s i s , an element

z EE8F

T such t h a t

z^(h) = z^(f @ g )= 1 and

(6.3)

z^ = O

[email protected]{g).

T h e r e f o r e , f r o m ( 6 . 3 ) and a p p l y i n g t h e n o t a t i o n of (4.1011,

one o b t a i n s

= $(fog) = ifagi(zi = fre

(6.4) with z s 0 map 0

g

9

(5.l)(seealso

,g E

E E

(2)

g

(2))

=

A

eg ( z i ( f )

= 1

( c f . (5.1) ; t h e c o n t i n u i t y of t h e r e s p e c t i v e l i n e a r

m(F) i s h e r e i r r e l e v a n t )

previous r e l a t i o n s (6.3)

,

. Moreover,

from t h e s e c o n d o f t h e

o n e now o b t a i n s

(6.5) which t o g e t h e r w i t h ( 6 . 4 ) p r o v e s t h e r e g u l a r i t y of t h e a l g e b r a E . A

s i m i l a r argument c a n a l s o b e a p p l i e d t o F , a n d t h i s c o m p l e t e s t h e p r o o f o f t h e lemma. I Now s p e c i a l i z i n g t o s u i t a b l e l o c a l l y convex a l g e b r a s , namely, t o t h o s e t h a t Theorem X I I ; 1 . 2 m i g h t be a p p l i e d , w e o b t a i n a n a n a l o gon of t h e p r e v i o u s Lemma 6 . 1 f o r complete algebras. I n t h i s r e s p e c t , o n e a p p l i e s Lemma 4.1 t o t h e a n a l o g o u s r e l a t i o n o f s i d e r i n g t h e continuous extension l i n e a r map 8

g

8 ,gem‘(F),

( 6 . 4 ) , when con-

of t h e c o n t i n u o u s (now)

9 d e f i n e d by ( 4 . l ) ( s e e a l s o ( 5 . 2 ) ) . So w e h a v e t h e n e x t .

452

XI11 PROPERTIES OF PERMANENCE Theorem 6.1. Let E , F be complete l o c a l l y convex algebras with l o c a l l y equi-

continuous spectra

m(E>, m l F l , r e s p e c t i v e l y . Furthermore, l e t E G F be t h e comT

p l e t i o n of t h e respective ( l o c a l l y convex) tensor product algebra of E , F so a s t o be a (complete) locally convex algebra ( e . g . t h e a l g e b r a s E , F m i g h t h a v e continuous multiplications)

.

Then, the algebra E 8 F i s regular i f , and only i f , t h i s i s the case f o r each one of t h e given algebras E , F. I We come now t o o u r s e c o n d s u b j e c t i n t h i s s e c t i o n , namely, t h e a n a l o g o u s s t u d y a s b e f o r e c o n c e r n i n g S i l o v algebras; i.e. Definition 2.3)

,

(see Chapt. I X ;

commutative c o m p l e t e regular semi-simple l o c a l l y m-convex

algebras. Thus, s i n c e s e m i - s i m p l i c i t y a s w e l l a s c o m p l e t e n e s s o f t h e t o p o l o g i c a l a l g e b r a s under d i s c u s s i o n e n t e r t h e s t a g e a l r e a d y a b i n i -

t i o , " f a i t h f u l n e s s " o f t h e t e n s o r i a l t o p o l o g i e s t h a t m i g h t b e cons i d e r e d i s of p a r t i c u l a r i m p o r t a n c e , when d e a l i n g w i t h q u e s t i o n s l i k e t h o s e i n t h e p r e c e d i n g (see t h e p r e v i o u s Theorem 4 . 3 ,

for instance).

I n p a r t i c u l a r , by c o n s i d e r i n g &lov algebras which f u r t h e r pos-

sess t h e approximation property

i n t h e s e n s e o f D e f i n i t i o n 4.2,

one g e t s

as a n o t h e r a p p l i c a t i o n of t h e f o r e g o i n g t h e n e x t r e s u l t .

Theorem 6.2. Let E , F be c o m u t a t i v e complete l o c a l l y m-convex algebras with

l o c a l l y equicontinuous spectra

m(E1 and ??t(F),

r e s p e c t i v e l y . Furthermore, Zet

E 6 F be t h e (commutative complete l o c a l l y m-eonvzz d g e b r a ) completion of the tenT

sor product algebra of E , F w i t h respect t o a compatible ( l o c a l l y m-eonvexl tensor1211 topology T ( D e f i n i t i o n X ; 3.1). Then, the following two a s s e r t i o n s are equivalent: 1 ) The topoZogy

T

on E O F i s f a i t h f u l ( D e f i n i t i o n 4.1) and each one of E ,

F i s a S i l o v algebra.

2 1 E 6 F i s a S i l o v algebra. T

In p a r t i c u l a r , t h e previous a s s e r t i o n 1 ) is t r u e , i f e i t h e r one of t h e s i l o v

algebras E , F has t h e approximation property ( D e f i n i t i o n 4.2) and one takes

n ( P r o p o s i t i o n X ; 3.1 Proof.

x; 3 . 1

T=

).

F i r s t , w e remark t h a t by h y p o t h e s i s f o r T

a n d t h e comment a f t e r i t ), E & F T

(see D e f i n i t i o n

i s a c o m m u t a t i v e c o m p l e t e 10-

c a l l y m-convex a l g e b r a , h e n c e o n e may a p p l y , d u e t o h y p o t h e s i s , Theor e m XII; 1 . 2 . Thus, t h e a s s e r t i o n t h a t 1 ) - 2 ) i s now a d i r e c t c o n s e q u e n c e of t h e above Theorems 4 . 3 and 6 . 1 . F u r t h e r m o r e , t h e l a s t p a r t o f t h e t h e o r e m i s o b t a i n e d from t h e corrment f o l l o w i n g D e f i n i t i o n 4 . 2 , t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . I

and

7.

WIENER-TAUBER C O N D I T I O N

453

S i l o v a l g e b r a s i n c o n n e c t i o n w i t h t h e p r e v i o u s Theorem 6.2 have

a p a r t i c u l a r s i g n i f i c a n c e i n a p p l i c a t i o n s of t o p o l o g i c a l a l g e b r a s t h e o r y i n some o t h e r c o n t e x t t o o . ( CohomoZogy of topoZogica2 aZgebras; see e . g . A . MALLIOS [21]

a n d a l s o S . A . SELESNICK [I] ) .

W e conclude t h i s s e c t i o n w i t h s t i l l remarking t h a t t h e condition

s e t f o r t h i n t h e p r e v i o u s Theorem 6 . 2 , p r o p e r t y " f o r o n e of t h e a l g e b r a s E , F

concerning t h e "approximation c o n s i d e r e d t h e r e i s , of c o u r s e ,

not t h e " l e a s t c o n d i t i o n " t h a t w e m i g h t r e q u i r e . Thus, t h e t h e o r e m works of c o u r s e e q u a l l y w e l l i f o n e c o n s i d e r s , i n s t e a d o f t h e a f o r e mentioned p r o p e r t y ( D e f i n i t i o n 4 . 2 )

,

j u s t a LocalZy m-convex algebra for

which t h e underZying localZy convex space s a t i s f i e s t h e approximation property ( c f . D e f i n i t i o n X ; 2 . 4 ) . (This i s s t i l l i n f o r c e concerning a l s o our rele v a n t c o n s i d e r a t i o n s i n S e c t i o n 4 . However, it was n a t u r a l t o s e t D e f i n i t i o n 4 . 2 i n t h e c o n t e x t of t h e t h e o r y of l o c a l l y m-convex a l g e b r a s , d u e a l s o t o t h e r e s u l t of A . G r o t h e n d i e c k q u o t e d t h e r e . A s a matter of f a c t , t h e c r u c i a l p o i n t h e r e i s a g a i n t h e map ( 4 . 1 3 ) t o be i n j e c t i v e ;

see e . g . A . WILLIUS [ 2 ] ) .

7. Wiener-Tauber condition T o p o l o g i c a l a l g e b r a s s a t i s f y i n g t h e c o n d i t i o n i n t i t l e of t h i s s e c t i o n have b e e n t e r m e d a l r e a d y a s

Wiener-Tauber aZgebras ( c f . C h a p t .

I X ; D e f i n i t i o n 6 . 2 ) . T h u s , w e n e x t examine how f a r t h e s a i d c o n d i t i o n

i s p r e s e r v e d when t a k i n g t o p o l o g i c a l t e n s o r p r o d u c t s of s u c h a l g e b r a s . I n f a c t , w e p r e s e n t l y show t h a t t h i s p r o p e r t y t o o i s a l s o "tensor product preserving" u n d e r s u i t a b l e c o n d i t i o n s on t h e t o p o l o g i c a l a l g e b r a s i n v o l v e d and on t h e r e s p e c t i v e t e n s o r i a l t o p o l o g i e s . The f o l l o w i n g comment w i l l b e u s e f u l i n t h e s e q u e l . Thus, supp o s e w e a r e g i v e n two t o p o l o g i c a l a l g e b r a s E , F w i t h s p e c t r a m(F/

,

m(E),

r e s p e c t i v e l y . Then, c o n c e r n i n g t h e s u p p o r t s of t h e G e l ' f a n d

transforms of t h e elements i n E , F

c o n s i d e r e d below, one h a s ( s e e a l s o

I V ; (4.1) )

(7.1)

n 2 x . @ y E E 8 F , w h i l e T i s a comi = ~i 7 p a t i b l e t o p o l o g y o n E 8 F . I n p a r t i c u l a r , f o r e v e r y decomposable t e n s o r x o y e E 8 F , one h a s t h e r e l a t i o n Here o n e c o n s i d e r s a n y e l e m e n t z a

(7.2)

S u p p ( x ! ) = Supp(i3 x Supp($l

.

So w e come n e x t t o t h e f o l l o w i n g a u x i l i a r y r e s u l t .

4 54

XI11 PROPERTIES OF PERMANENCE

Lemma 7.1. Let E , F be Wiener-Tauber algebras ( D e f i n i t i o n I X ; 6 . 2 ) w i t h spectra Z??'(EI, m(F),r e s p e c t i v e l y , and l e t E @ F be t h e t e n s o r product algebra of E , F w i t h r e s p e c t t o a compatible topology T ( D e f i n i t i o n X ; 3.2) making, i n par-

t i c u l a r , t h e canonical b i l n e a r map $ : E x F - +E B F ( c f . X ; ( 1 . 4 ) ) continuous. Then, t h e topological algebra E 8 F i s a Wiener-Tauber algebra. Proof. I n d e e d , f o r e v e r y x 8 y E E B F o f [email protected]?

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 z = x . p , with

A

(8.1)

where

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)

l o c a l l y convex algebras w i t h respect t o ( l o c a l l y eonvex)

T

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 lur, semi-simple.

G 6G H

t o be, i n particu-

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 = [email protected]

(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

0

t

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 '

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 mapped by th o n t o t h e c o n n e c t e d set

??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:

(a,B)e?TZ(GIxm(H),

M-(E) x W(F) ( c f . (8.11

t ~ m(H) : -m(F)

(8.12)

,

one h a s (8.10) f o r some ( f , g )

-

t h a t one a c t u a l l y obtains a map

SO

I

:

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 [email protected]??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

= y o v.

m(H),one t

tv)(B)= $( v(BII 0

A

t

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) topological algebra i n a compatible t e n s o r i a l topology

T

on EQF. Furthermore, l e t G , H 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 T

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

[email protected]=

~(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 : [email protected] 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

U

X

.

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,

f o r some e l e m e n t ( c f . Theorem 4 . 3 )

P

,

e ( a , y ) = vCl(yie I I ^

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

,

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 h l x a y l = p(x1 w v,(yl

(8.35)

,

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([email protected](h([email protected])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) -