Chapter VII Spectra of Certain Particular Topological Algebras

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)

f o r every

f

f

c ( X 1 . The a l g e b r a

:=

SuPlf(Z)I T.

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)

6 :x

-

-

SPECTRA OF PARTICULAR ALGEBRAS

VII

(ectx)I;

:

6 ( x ) = 6,

d e f i n e s a homeomorphism of X i n t o t h e range of

:f

-

&,if)

:= f ( x )

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

c ( X ) : f

cX

!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 e x i s t s a function g i n I E ( X i , g i v e n by ( 1 . 8 ) , which never v a n i s h e s on X . ( S o t h e r e e x i s t s t h e i n v e r s e f u n c t i o n of g and 1 = g . L E I). I t i s t h i s a u x i l i a r y re9 s u l t , d e r i v e d from t h e p r e c e d i n g p r o o f , which w i l l c o n s t a n t l y b e a p p l i e d i n t h e s e q u e l . I n t h i s re s p e c t , i t i s o f c o u r s e e q u i v a l e n t w i t h Lemma 1 . 2 t o say t h a t :

e

I i s a proper i d e a l of C t X ) i f , and o n l y if, t h e r e e x i s t s a p o i n t x f X , w i t h f ( x i = 0 , f o r every f E I.

Y e t a p p l y i n g t h e n o t a t i o n of ( 1 . 1 5 ) below, t h e previous statement is equivalent with t h e r e l a t i o n

I GI, f o r some x e X . ( I n t h i s c o n c e r n , see a l s o t h e n e x t Remark 1 . 2 )

.

Lemma 1.3. Let X be a compact space and I an i d e a l of t h e algebra e t X ) . Bes i d e s , consider the s e t (1.10)

and an element @ of C (x), w i t h t h e p r o p e r t y t h a t t h e r e e x i s t s an open neighborhood U of A in X on which @ v a n i s h e s ; t h a t i s , assume t h a t

(1.11)

A G U G Z(@).

Then, @ E I .

Proof. By ( 1 . 1 1 ) and t h e d e f i n i t i o n of A

,

one v e r i f i e s € o r t h e

compact s p a c e K a C l i c X and t h e " r e s t r i c t i o n o f t h e i d e a l I" t o K

(de-

IIK) t h a t t h e c o n d i t i o n s of t h e p r e v i o u s Lemma 1 . 2 a r e s a t i s f i e d . T h e r e f o r e , t h e r e e x i s t s a f u n c t i o n g e C ( K ) ( = I ) which n e v e r IK v a n i s h e s on K (see a l s o Remark 1.1 ) Thus a p p l y i n g l ' i e t z e ' s Extension n o t e d by

.

Theorem ( c f . , f o r i n s t a n c e , J . DUGUNDJI [1: p. 1 4 9 , Theorem 5.11) one ob1 t a i n s a n e l e m e n t h e C t X l e x t e n d i n g - e C I K ) s u c h t h a t one h a s ( s e e g

1.

219

SPECTRUM OF c ( X )

also ( 1 . 1 1 ) ) (1.12)

(we actually consider here the extension of g to the hole of X ) , and this finishes the proof. I Lemma 1.4. Let X be a compact space and I an i d e a l of t h e algebra

CtX)

.

Then, one g e t s t h e r e l a t i o n

(1.13)

" I A ,

where A C X is given by ( 2 . 1 0 ) and

7 denotes

t h e closure of I in the Banach alge-

bra e i x ) . Proof. Let @ $,

8 IA

, with

@

# 0 , and

0. Then, by hypothesis for

E

the sets

(1.14)

M = { z 8 X :

I@(zII5 E } and N = { r e X : ( @ ( rE)] ( >

define two non-empty closed subsets of X , with M n N = 0. Hence, since X is, in particular a normal space, there exists ( Uryson's L e m ) an element g E ClX),with 0 < g 6 1 , and in such a manner that M E Z ( g l and

(1.15)

g = l

on N .

Thus, by definition of M and (1.15), one obtains set u = C r e X : (qdrllc 1

$ g = 0 on the open

-$-

hence, in particular, A C U E Ziggl.

Therefore (Lemma 1.3) , $9 €I, so that for any and (1.15))

/ 1 ~ - s ~= /ll@(1-9)ll /

E

> 0 , one gets (cf. (1.14)

< E r

i.e., $E?, and hence I A C I . Moreover, I E I A (cf. (1.6) and (l.lO)), S O that since I A is a closed ideal of the preceding yields already the proof of the asserti0n.I

e(X),

Thus, the previous discussion provides already the following basic result. That is, we have

Corollary 1 . 1 . Let X be a compact space and C (XI t h e respective Banach a l gebra of comptex-vatued continuous functions on X i n t h e uniform (sup-norm) topology in X. Furthermore, l e t F t X ) be the s e t of a l l non-empty closed and proper subs e t s of X, and

J ( C t x ) ) t h a t of non-trivial ctosed and proper i d e a l s of t h e Ba-

nach algebra C i X l . Then, the map

(1.16)

8: F(X)-

J i C ( X ) ) : A -eiAl:=

'A '

220

VII

SPECTRA OF PARTICULAR ALGEBRAS

&ere IA i s given by ( 1 . 5 ) , y i e l d s a b i j e c t i o n between t h e r e s p e c t i v e s e t s . t h e range o f 8 i s J I

Proof. W e h a v e a l r e a d y n o t i c e d t h a t

f o r a completely regular space X )

. NOW,

,B

if A

CIX))(even

a r e any two members of F ( X )

with A # F , then t h e r e e x i s t s ( X is a f o r t i o r i a completely regular

C I X l i n s u c h a way t h a t o n e h a s , f o r i n s t a n c e , A G Z I f ) and f ( x ) = l , f o r some X E B ~ C A T . h u s , i n any c a s e , o n e o b t a i n s I A # IB, t h a t i s t h e map 8 i s o n e - t o - o n e . F u r t h e r m o r e , i f I e l J ( C ( X ) ) and I , i s t h e c o r r e s p o n d i n g i d e a l of C i X ) d e f i n e d by ( 1 . 5 ) and ( l . l O ) , s p a c e ) a n e l e m e n t f~

H

one h a s

Lemma I . 4 ) e(A)=I =I=i, A

(1.17)

and t h i s f i n i s h e s t h e p r o o f . 1

I n p a r t i c u l a r , w e now g e t t h e f o l l o w i n g .

Theorem 1.1. Suppose we have t h e c o n t e x t o f t h e preceding Corollary 1.1. Then, t h e r e e x i s t s a o n e - t o - o n e and onto CGrrespondenCe between t h e s e t o f a l L maximal i d e a l s of t h e Banach algebra C I X ) and t h e p o i n t s o f X ( d e r i v e d from t h e res p e c t i v e r e s t r i c t i o n of t h e above map ( 1 . 1 6 ) ) .

I I f I i s a maximal i d e a l of C I X l , t h e n (Lemma 1 . 4 ) , iA= i s g i v e n b y ( 1 . 1 0 ) ( e v e r y maximal i d e a l of a & - a l g e b r a w i t h a n i d e n t i t y e l e m e n t and h e n c e of t h e Banach a l g e b r a C l X ) , i s c l o s e d ; Proof.

where A E X

c f . Theorem I I ; 6 . 1 ) . N o w I I = I=~I ~ . (~s o ) since

e

f o r e v e r y x e A , one g e t s

Furthermore, f o r any x

€

I A = II x } I h e n c e

one g e t s A = { z > a s w e l l ) .

i s 1-1,

the set

X

I =I ={fEetxi:frxi=oi x tx}

(1.18)

d e f i n e s a maximal i d e a l of C i X l : I n d e e d s u p p o s e , o t h e r w i s e , t h a t I i s a maximal i d e a l of C i X ) w i t h I GI; t h e n

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

5

i G I = I X

f o r some p o i n t

Y

Y E X . Now, t h e l a s t r e l a t i o n e n t a i l s t h a t I GI x Ix,yl

Ix

'

s o t h a t one h a s 8 1 1 d ) = I, = 'i'hat

is (Corollary 1.1)

, x=y,

Yl =

e(b, ~ 1 ) .

a n d hence I = I which i s t h e a s s e r t i o n , X

and t h i s c o m p l e t e s t h e p r o o f of t h e t h e o r e m . 1 The f o l l o w i n g i s now a d i r e c t a p p l i c a t i o n o f Lemma 1.1 and t h e p r e v i o u s Theorem 1 . 1 , i n c o n j u n c t i o n w i t h C o r o l l a r y 11; 7 . 3 . T h u s I w e have.

221

Corollary 1 . 2 . L e t X be a compact space and C(X/ t h e r e s p e c t i v e Banach a l gebra, as above. Then, concerning t h e s p e c t r m of C(X/,one has t h e r e l a t i o n

m(e(x)) = X

(1.19)

,

w i t h i n a homeomorphism o f t h e r e s p e c t i v e topological spaces ( g i v e n by

(

.I

1.2) )

.-

Schol i u m 1 . I S i n c e e v e r y maximal i d e a l of a Banach a l g e b r a ( a n d y e t , more g e n e r a l l y , of a Q-alg e b r a ) w i t h a n i d e n t i t y e l e m e n t i s c l o s e d (Theorem 11; 6 . 1 ) , i t i s a c o n s e q u e n c e of t h e p r e v i o u s r e l a t i o n ( 1 . 1 9 ) t h a t t h e topology of a compact space X i s compZeteZy determined by t h e algebra ( a c t u a l l y r i n g ) structure of i t s r e s p e c t i v e “function algebra” C I X l ; t h i s e s s e n t i a l l y amounts t o t h e c l a s s i c a l Banach-Stone Pheorern, a c c o r d i n g t o which two compact spaces are homeomorp h i c if, and o n l y i f , t h e i r r e s p e c t i v e f u n c t i o n algebras are isomorphic ( a s r i n g s ) . Now, w i t h i n t h e p r e c e d i n g framework, t h e l a s t r e s u l t t u r n s o u t t o be t h e b e s t p o s s i b l e ; namely, t h e l o c a l l y m-convex algebra C,(Xl, w i t h X a c o m p l e t e l y r e g u l a r s p a c e , i s a &-algebra i f , and only i f , X i s a compact space ( c f . , f o r i n s t a n c e , W . DIETRICH, J r . [3: p. 58, Theorem 2.1.5, i)])

.

T h u s , w e come now t o t h e p r o m i s e d e x t e n s i o n of the p r e v i o u s C o r o l l a r y 1 . 2 t o t h e case one h a s a n a r b i t r a r y compZeteZy regular space

x; b u t

w e f i r s t g e t a s i m i l a r e x t e n s i o n of C o r o l l a r y 1 . 1 . T h a t i s , w e h a v e .

Lemma 1.5. Let X be a completely regular space and C (Xl t h e l o c a l l y m-conv e x algebra of complex-valued continuous f u n c t i o n s on X i n t h e compact-open topology. Then, there e x i s t s a one-to-one and onto correspondence between t h e s e t o f a l l non-empty closed and proper s u b s e t s o f X and t h a t o f n o n - t r i v i a l closed and proper i d e a l s o f c c I X 1 , g i v e n by t h e r e s p e c t i v e map t o (1.161. Proof. W e h a v e a l r e a d y remarked i n t h e p r o o f of C o r o l l a r y 1 . 1

t h a t t h e map 0 , g i v e n by ( 1 . 1 6 ) , d e f i n e s a n i n j e c t i o n f o r e v e r y comp l e t e l y r e g u l a r s p a c e X ; so it r e m a i n s a c t u a l l y t o p r o v e t h a t ‘8 i s an

onto map: Thus, a p p l y i n g t h e n o t a t i o n of C o r o l l a r y I . 1 , w e

must p r o v e t h a t

i s of t h e form IA , where t h e s e t A -C X i s g i v e n by ( 1 . l o ) . Moreover, s i n c e w e a l w a y s h a v e t h a t I C IA, we are a c t u a l l y led t o prove t h e every l e J l e c ( X I I relation

I r?=1 A

(1.20)

( t h a t i s , t h e r e s p e c t i v e r e l a t i o n t o ( 1 . 1 3 ) ) . Thus, by d e f i n i t i o n o f t h e topology i n

cc(Xl

,

i f I$e IA one h a s t o p r o v e t h a t , f o r any

and K a compact s u b s e t of X I t h e f o l l o w i n g r e l a t i o n i s v a l i d (1.21)

p K ( b- h / <

E

,

E

>0

222

VII

f o r some h e r , where

SPECTRA OF PARTICULAR ALGEBRAS

i s d e f i n e d by I ; ( 3 . 1 3 ) :

p,

t o t h e compact set

Thus, r e s t r i c t i n g t h e g i v e n i d e a l I e J ( C c ( X ) I K : c X ( c o n s i d e r e d by ( 1 . 2 1 ) ) , i . e . ,

transpose

t a k i n g t h e image of I u n d e r

j, = tiK o f t h e c a n o n i c a l i n j e c t i o n

set

fl,

j,(~) = { j K ( f ): f E I 3 = c

(1.22)

S

i, : KS X, one g e t s t h e

:f e

CIKI.

which, i n f a c t , i s an i d e a l o f the algebra

the

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 (1.26)

I@=

CIx)

c,(XI,

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)

.

X

x

) 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 -

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

Thus, t h e r e s p e c t i v e Gel'fand map of t h e algebra

-

defined

C J X i i s the

.

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.

c"(X)

SPECTRUM OF

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 g oiE

(2.10)

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:

, p r o v e s t h a t t h e image

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 ,

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

-

-.@ (fx)

one h a s

@(&,)

@if)

@if)

,

(P(f)

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

one o b t a i n s

(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 ,

la (z)l
Hence, la fz) I = I , f o r every z~G . Q, F i n a l l y , one e a s i l y c o n c l u d e s by ( 4 . 8 ) and t h e c o n t i n u i t y of t h e map ( 4 . 6 ) t h a t t h e map a : G+

@

U G C i s continuous a s w e l l , a n d t h i s com-

p l e t e s t h e p r o o f of t h e lemma. I W e come now t o p r o v e t h a t 1

(4.8) provides, i n e f f e c t , a b i j e c t i o n

b e t w e e n t h e s p e c t r u m of L ( G ) and t h e c h a r a c t e r g r o u p of G .

However, w e

n e e d f i r s t some more p r e l i m i n a r y m a t e r i a l . Thus, s i n c e e v e r y e l e m e n t

1

@ e??Z( L (G)) i s , i n p a r t i c u l a r , a con-

t i n u o u s l i n e a r form on L ‘ I G ) , one h a s t h e r e l a t i o n (4.11)

MIL’IGI

= ZWL’(G)

I c: ( L ~ ( G Ij

*

2 LYG) .

4.

SPECTRUM OF L ' ( G )

235

The last relation in ( 4 . 1 1 ) yields, within an i s o m e t r i c isomorphism 0 , the 1 topological dual of (the Banach space) L f G / (in the respective "strong topology") as the Banach space L c ' ( G ) of all e s s e n t i a l l y bounded measurable on G . In parti.cular, this is expressed, through the Radon-Nikodini Theorem, by the relation

functions

@ I f ) = Jflzin(ccl d z

(4.12)

,

1

with f e L 1G1 , for a uniquely defined u e L " ( G I , which corresponds, via o, 1 to a given @ € ( L ( G l i ' . (In this regard, cf. L . H . LOOMIS [I: 7 5 C , D ] , and/ or M. A . NA?MARK 11: p. 140, 5 1 6 , Theorem 3 1 ) In this respect, we still remark that ~ ~ L c 4 ( G an ) , element of ?

.

being, by definition, a bounded and continuous (therefore, measurable) h

function on G. Furthermore, t h e i n v e r s e map o f a in 14.111 r e s t r i c t e d t o G 1 1 has range i n t o MIL ( G l l c ( L f G ) ) ' : thus, ( 4 . 1 2 ) yields a (continuous) character Q, of L 1(G), for every u e 6 : Thus, for any elements f , g in L 1i G ) , one gets by ( 4 . 1 2 ) (through a repeated application of Fubini's Theorem, and the translation invariance of d z ) Q , ( f * g i = J ( f * g ) f z i m d z = (by (4.2)) J(Jf(z-y)giyidy =

JJ : i r , ' p ( g ) a i z

ZZdx =

JJ f i z - y i g i y i Z Z d z d d y ___

+yiJzdg = ~

~

;i.t.Jy ~

i

~

= J I J f ( x l a l c ! ? z i g ( y i m d y = I J f ( x l ~ d x ) l J g f y l a ( y / d y / = @1fi@(g)

which is the assertion. On the other hand, the converse of the last assertion concerning ( 4 . 1 2 ) is also true. First we shall need, however, some more comment on the respective terminology. So if i e 6 i is an approximate i d e n t 1 ity of L (G) one has, by definition, the relation l i m ( f * e 6 i = l i m i e * f i =f

(4.13)

6

6

6

1

,

1

for e ery f E L ( G l . Thus, f o r every @Em(L fG)), @if)=@llimif*e6!)= limr$lf*e61 6 6

(4.14

= l i m @ i f i @ ( e 6=1 @ffi.lim@1e6

6

Hence

one gets

6

.

since @ # 0 , one concludes t h e r e l a t i o n limQ,ie6 i = 1 6

(4.15)

,

1

f o r every @ E m ( L(GI). (In this respect, we remark that the last conclu-

sion concerning ( 4 . 1 5 ) is certainly valid for any topological algebra whatsoever, withintof course, the appropriate context).

~

236

VII

NOW,

SPECTRA OF PARTICULAR ALGEBRAS

a p p l y i n g ( 4 . 7 ) w i t h g = e 6 one g e t s , f o r e v e r y

1

$€mfL(G)),

$(fxl$(e6 I = $ifi$((e6lz I

so t h a t f o r

@(f)

# 0

one o b t a i n s

Hence, by ( 4 . 8 ) , w e have $ f ( e 6 1 x I = $ ( e ) a (x)

(4. 6)

6

$

f o r e v e r y x e G . I n p a r t i c u l a r , o n e h a s by ( 4 . 1 5 ) l i m $ f i e 6 1 z I = a (z),

(4. 7)

$

6

f o r every x E G .

Thus, w e come f i n a l l y t o o u r p r e v i o u s a s s e r t i o n t h a t , namely, t h e i n v e r s e map of 0-1 I

(4.18)

:

2

-

1

M(L'(GI ) = 17il ( L ( c i

,

which i s , i n d e e d , t h e a s s e r t i o n . ( I n t h e p r e v i o u s argument w e h a v e a l s o a p p l y t h e f a c t t h a t , f o r e v e r y a e ; , one h a s t h e r e l a t i o n d - x / = a(x) , w i t h X E G I a s f o l l o w s from t h e same d e f i n i t i o n s ) , Thus, w e c a n summarize t h e p r e c e d i n g i n t o t h e form o f t h e n e x t . Lemma 4.2.

map (4.20)

-

Suppose t h a t t h e c o n d i t i o n s of L e m a 4 . 1 are s a t i s f i e d . Then, t h e 0 :$

a$: ~T(L'(GII

d e f i n e d by (4.81, y i e l d s a one-to-one

-E,

and o n t o correspondence between t h e r e s p e c t i v e

4.

231

SPECTRUM OF L?G) 1

of the canonicaZ

spaces. I n f a c t , it i s t h e r e s t r i c t i o n t o the s p e c t r m of L (G) 1

( i s o m e t r i c ) isomorphism (L (GII '

Lm(GI.

Proof. I f e ( @ ) = a = 9(J,) = a

t h e n by ( 4 . 1 9 )

@ one g e t s @ =

i),

J,

with

@, J,

that is,

hand, f o r any a E ~ ~ L m f G Ione , g e t s by ( 4 . 1 2 ) 1

1

i n g it a n e l e m e n t 9 € M(L (G)) € f L (GI)', 9(6) = a i s related t o @ v i a (4.19) 9

+; theref ore,

.

elements

of

0 i s one-to-one.

,

mfL1(GI)

On t h e o t h e r

and t h e comment f o l l o w -

i n s u c h a way t h a t t h e r e s p e c t i v e Hence, one h a s o-'faI = o-"(a I =

@

= 0( @I = o ( @ ), t h a t i s 9 i s an onto map a s w e l l . The a =a+ l a s t a s s e r t i o n of t h e s t a t e m e n t i s c l e a r a l r e a d y by t h e p r e v i o u s d i s c u s s i o n , and t h i s t e r m i n a t e s t h e proof o f t h e 1emma.I W e come n e x t t o prove t h a t t h e p r e c e d i n g map 9 i s , i n f a c t , a

homeomorphism. Thus, w e h a v e . Theorem 4.1.

Let t h e conditions of Lemma 4 . 2 be s a t i s f i e d . Then, the map

e

(4.21)

: ~ ( L ' I G ) -Z, ) 1

defined by ( 4 . 8 1 , i s a homeomorphism of t h e spectmun of L ( G I

onto the character

group of G. Proof. According t o Lemma 4 . 2 ,

f i r s t that

6 - l i s continuous. Namely,

9 i s a b i j e c t i o n . Thus, w e prove w e must prove t h a t f o r any n e i g h -

borhood v ( @ O ; f , E= ) { cp

(4.22)

E

I f^rw -?(ao) I < E I

~ ( L ~ I G I ) :

1

from a fundamental system of such a t t h e p o i n t

$,=9-l(a0) e ?YZ(L I G ) ) ,

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

U(ao;K,61={af2:

(4.23)

of a. E (4.24)

ECcfGI

Ic~(T)--c~~(x)~<~

V T ~ K I

such t h a t

O-*(U(aO ; K, 6 1 ) E V ( @ o ; f ,

E

I.

Thus, one has t o prove t h a t (4.25)

l@(f)-@o(Jc)l

'€3

f o r every @ = €I-l(aI, w i t h a € U(ao;K , 6 1 . T h e r e f o r e , i n view of

obtains (4.26)

F u r t h e r m o r e , one h a s by d e f i n i t i o n t h e r e l a t i o n (4.27)

K(GI = L?GI

( 4 .19), one

238

VII SPECTRA OF PARTICULAR ALGEBRAS

with respect t o t h e since

L1 -norm

1

1 L (GI d e f i n e d by

(topology) of

(4.1 )

. So

f € L f G ) , o n e c o n c l u d e s t h a t f o r a n y E > O t h e r e e x i s t s an e l e m e n t such t h a t

g€KfG)

N1(f-g)

=!I

5.

f(s)-gfz)\dx<

Thus, f o r e v e r y compact K C G , o n e h a s t h e r e l a t i o n (see a l s o N . BOUR-

BAKI [8: Chap. 4 ; p. 185, C o r o l l a i r e ]

)

jIf(x)-g(z)ldx= JKIf(zi-g(xiIdr+j

CK

Consequently, t a k i n g obtains

/f(xl-g(xiIdz < + .

K = Supp ( : I ) , w i t h g a s b e f o r e , s i n c e g = O ( C K

Thus, from ( 4 . 2 6 ) a n d t a k i n g K a s b e f o r e a n d i n ( 4 . 2 3 ) 6 =

One

E

211fl11

one h a s f o r e v e r y a e t i o n 1)

c'(clo;

1 @(f)- 4Jo

K, 6) ( c f . a l s o

If) I S

I If K

I

t h e above R e f . ; p . 205, P r o p o s i -

- -

-

(sl I */adz)~1 (xi I d x $0

- -

If(x)I.la ( x i - a lxlldx 4J 40

( w e a l s o r e c a l l t h a t la ixll = 1 , w i t h z e G , f o r e v e r y 4 J € ?7?(L'(G)l). T h a t 4J i s , w e do h a v e ( 4 . 2 5 ) , which w a s t o b e p r o v e d . The p r e c e d i n g p a r t of t h e p r o o f e s t a b l i s h e s , i n f a c t , one h a l f o f t h e a s s e r t i o n . The rest w i l l b e d e r i v e d from t h e f o l l o w i n g two l e m -

mas.

Lemma 1 . L e t G be a ZocaZly compact a b e l i a n group, L ' f G )

-

1

t h e r e s p e c t i v e group algebra and ?Z(L ( G I ) Then, t h e f u n c t i o n (1)

h : (x,4 )

i t s spectrum.

h(x, $11:=a f x ) ,

4J

d e f i n e d b y 1 4 . 8 1 , is a (complex-valued) continuous f u n c t i o n Iroof. F o r e v e r y f e LI(G)

1

,

( w e assumed t h a t $ E mfL(G)) )

one o b t a i n s

,

which p r o v e s ( c f

.

4. SPECTRUM OF

d(G)

239

-

a l s o ( 4 . 6 ) ) the continuity of the function

(z,@ ) j p :) G x VY(L’(G)I

(2)

1

f o r every f e L ( G ) . Now, f o r e v e r y

( x ~ , @t ~ h e) r e e x i s t s

# 0 . Thus, o n e g e t s

an e l e m e n t f , e L’fG), w i t h t h e c o n t i n u i t y of

( 1 ) a t a given point

t i o n of a

$

FJ$)+t(@o)#

fxo ,@o) b y where w e

t($),

d i v i d i n g t h e map a p p e a r e d i n ( 2 ) b y a l s o have t h a t

c ,

0 (see a l s o t h e d e f i n i -

i n Lemma 4 . 1 ) . 1

Furthermore, w e a l s o need t h e f o l l o w i n g t o p o l o g i c a l f a c t .

Lemma 2 . Suppose we upG g i v e n t h e t o p o l o g i c a l spaces X, a s w e l l as a continuous f u n c t i o n

Y,Z

h :X x Y d Z . Then, f o r any compact s u b s e t K of X , and any open subset A o f 2, t h e s e t V=

y e Y :h f z , y )

E

A

V

z e K}

i s an open s u b s e t o f Y. Proof. C f . L . H . LOOMIS [ I : p. 12, Lemma

5 ~ 1 1.

End o f t h e proof o f Theorem 4.1. W e p r o v e now t h a t t h e map 0 - I , which i s g i v e n by ( 4 . 2 1 ) , i s an o p m map: Thus, g i v e n aoeG a n d any open n e i g h h

borhood

K,

U(clo;

t h a t E-l~i!!’ao;K,

ao= N$,i = a

)

$0

.

E)

E))

1

,

a s i n ( 4 . 2 3 ) , one p r o v e s

is an open s e t i n 5 W f L fG)) c o n t a i n i n g $o= 8 - l ( a 0 ) ( s o

Namely, w e h a v e

a-I(uia,;

NOW,

. from a l o c a l b a s i s a t a

K,

E

)I =

c + e V Z ( L1(GI) : ei4) =

t h i s i s a n open s e t i n

2

1 m ( L f G ) ) containing

u(a0;K, EJI

@o

,

a s t h i s follows

from t h e c o n t i n u i t y o f t h e map ( c f . t h e above Lemma 1 )

h :(z,ql)-hfz,f$,):=

ja f x ) - a

@

(.El $0

a n d Lemma 2 , which i s j u s t t h e a s s e r t i o n . T h i s c o m p l e t e s t h e proof of Theorem 4 . 1 . I A d i r e c t a p p l i c a t i o n of t h e i d e n t i f i c a t i o n provided by t h e p r e -

c e d i n g Theorem 4 . 1

i s t h a t t h e c l a s s i c a l Fcurier-Gel’fand

transform of t h e

1

g r o u p a l g e b r a L f X n l a c t s on t h e c h a r a c t e r g r o u p of IRn ( c o r r e s p o n d i n g

t o t h e c a n o n i c a l i n n e r product of I R n )

,

n a m e l y , t h e g r o u p IRn i t s e l f ( w i t h -

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

240

V I I SPECTRA OF PARTICULAR ALGEBRAS

(4.28) 1

with f e L ( E n ) . (See, for instance, E.M. S T E I N - G . WEISS [I:pp. 2 , 31). Thus, one also gets, straightforwardly, the classical fact (ibid.) that the Fatrier transform of t h e convolution of two functions i s the (pointwise) product of t h e i r Fourier transforms (the Gel'fand map is an (algebra) morphism). Now consider the given group G , as above, and its character group G ; thus the latter, being a locally compact (abelian) group,has its own character group, denoted by 2 (second character group of G ) Now this group is, within a homeomorphism, the initial group G itse1f;in l o c a l l y compact abelian group i s (within a homeomorphism) i t s other words, own second character group ( Pontrjagin Duality Theorem; cf . , for example , L . H. LOOMIS [I: p.151, 37D]). h

.

h

C"(X) (contn'd.). The Nachbin Theorem ( n e c e s s i t y )

5. The l o c a l l y m-convex algebra

We consider in this and the following section the Cm-analogon" of the classical Stone-Waierstrass Theorem (see, for instance, L . NACHBIN [4: p. 48, Corollary 21) : it is due to L . NACHBIN [l] . We need first, however, still more terminology from the theory of Differential Manifolds than the one already applied in Section IV;4.(2) and Section 2 above, which thus we are going next to supply. Thus suppose we are given a d i f f e r e n t i a l manifold X which we assume, thereinafter, to be connected second countable and Hausdorff, of course. Furthermore, we suppose henceforth that

cw

(5.11

C ;

(XI

(see also Section IV;4.(2)), that is, we consider exclusively r e a l functions on X . (The complex-case can be treated in a simivalued ( em-) lar way to the analogous one used for Stone-Weierstrass Theorem; namely, " s e l f - a d j o i n t " subalgebras of ( X I should be considered instead. Cf. Section 7.2 below) Thus, for every x e X , denote by T ( X , X I the tangent space of the manifold X at x ; this is (isomorphic to) the n-dimensional numerical space IRn , where n =dimX S o , for every element V E T ( X , x ) , tangent vector of X at x , one defines a (real) linear form on C " t X I , say l,.,, by the relation

ccm

.

.

(5.2)

5.

the notation applied i n (5.2) , ( A i )

CmIX).C o n c e r n i n g

f o r every f E

24 1

N A C H B I N THEOREM (NECESSITY)

w i t h l S i < n , d e n o t e s t h e s e t of

,

( l o c a l ) C o o r d i n a t e s of veT'IX,x:I a t t h e

p o i n t x € X ( c f . a l s o ( 5 . 7 ) b e l o w ) , which c o r r e s p o n d s t o t h e l o c a l c h a r t

iu, @) = (U;X I ,. . . , xni

(5.3)

of t h e m a n i f o l d X a t x , where (5.4)

xi=uio@, I S i S n ,

1SiSn

a n d ui : W n+iR,

,

are the canonical coordinates (projections

o n t o t h e r e s p e c t i v e a x e s ) of

so one h a s , by d e f i n i t i o n ,

IRn;

(5.5) with i = I ,

...,

Now,

eqression"

n , f o r e v e r y f &?Xi.

t h e expression of an element v E T ( X , x ) i n local coordina*es ( " l o c a l of v )

, which

c o r r e s p o n d s t o any g i v e n c h a r t of the manifold

X a t x , l i k e ( 5 . 3 ) , i s g i v e n by t h e r e l a t i o n n

(5.6)

1)

= t A i ( Za J x i=l 7,

.

Furthermore, one o b t a i n s A . = v(xiI = idxilz

(5.7)

(vl ,

( 5 . 4 ) ) . T h u s , t h e ( r e a l ) numbers X i , l S i 6 n ,

with 16i<,n ( c f .

i n (5.7)

a

Local coordinates of v , w i t h r e s p e c t t o t h e canonical b a s i s {C-) j ax-. s I S i S n , of T I X , x ) d e f i n e d , f o r i n s t a n c e , b y ( 5 . 5 ) and c o r r e s p o n d i n g " t o

are the

. ,snI

t h e g i v e n c h a r t (U;xI ,. .

of X a t z.

In p a r t i c u l a r , t h e r e l a t i o n ( 5 . 2 ) defines T(X, X I ,

as a (real-valued)

emlX) (viz.

,a

"Leibniz m a p " )

. Thus,

( 5 . 2 ) and I V ; ( 4 . 1 4 ) ) .

2

V

E

by c o n s i d e r i n g t h e l a t t e r a l g e b r a

equipped with t h e c a n o n i c a l em-topology ( c f readily verifies that

Zv , f o r e v e r y v

derivation ( a t t h e p o i n t x ) of t h e a l g e b r a

.

Section IV; 4 . ( 2 ) )

,

one

i s , i n f a c t , a continuous l i n e a r f o m on c m ( X ) ( c f .

W e a r e now i n t h e p o s i t i o n t o s t a t e t h e f o l l o w i n g lemma which

a c t u a l l y c o n s t i t u t e s t h e " o n l y i f " p a r t of o u r main a s s e r t i o n b e l o w (Theorem 6 . 1 ) . Lemma 5.1.

Thus, w e h a v e .

Let X be u ( n o n - t r i v i a l ) finite-dimensional

1 5 n =dimX < m ) and

em-munifoZd ( v i z . ,

~ " t x lt h e algebra o f (real-valued) C m - f u n c t i o n s on X en-

dowed w i t h t h e canonical

c -topology m

( c f . S e c t i o n IV; 4 . ( 2 ) ) . Moreover, l e t A

be a dense subalgebra o f c m i X I ; i . e . , we assume t h a t

Then, t h e following t h r e e c o n d i t i o n s are s a t i s f i e d :

242

VII SPECTIiA OF PARTICULAR ALGEBRAS 1 ) The algebra A i s %on-vanishing

on X"; namely, not a l l f u n c t k n s from A

vanish a t a l l p o i n t s o f X . ( E q u i v a l e n t l y , f o r every p o i n t ment f € A ,

X , t h e r e e x i s t s an e l e -

with f l x l # 0 ) .

2 ) The algebra A '!separates t h e p o i n t s o f X"; i . e . ,

x #y,

X E

there e x i s t s an element f

E A,

f o r any x , y i n X , w i t h

with f i x ) # f ( y l .

3 ) For any p o i n t X E X and tangent v e c t o r v E T l X , x l , w i t h v # 0, t h e r e i s f E A , such t h a t

( d f J X ( v l = v ( f I # 0.

(5.9)

Thus, t h e ( p o i n t ) d i f f e r e n t i a l s o f t h e f u n c t i o n s i n A , a t any p o i n t X E X , separate t h e F o i i t i s of t h e r e s p e c t i v e tangent space T ( X , x ) .

Proof. By c o n s i d e r i n g t h e t o p o l o g i c a l d u a l s o f t h e ( l o c a l l y convex) spaces appeared i n ( 5 . 8 ) , (5.10)

A'

o n e h a s , of c o u r s e , t h e r e l a t i o n

= (~OO(XI)',

w i t h i n a l i n e a r s p a c e isomorphism ( s e e , f o r i n s t a n c e , G . KOTHE [l: p. 158, ( I l ) ] ) . On t h e o t h e r h a n d , i n view o f Lemma 2 . 1 ,

one g e t s

(5.11) where t h e map

6:

i s , one o b t a i n s

f o r any x e X

and

is the Xs c"(X))'

e v a l u a t i o n map ( "Dirac map") ; t h a t

f 6 e m ( X ) . So t h e f i r s t r e a t i o n i n

(5.11) is valid

w i t h i n a homeomorphism ( Whitney's Imbedding Lemma; c f . t h e p r o o f o f t h e s a n e lemma above (Lemma l ) ) , w h i l e t h e p o i n t s of X c o r r e s p o n d t h r o u g h ( 5 . 1 2 ) t o non-zero c o n t i n u o u s l i n e a r f o r m s on

C"(X)( t h e

l a t t e r alge-

bra contains the constants). Thus, t h e f i r s t two of t h e s t a t e d c o n d i t i o n s f o r A a r e now a d i r e c t c o n s e q u e n c e of

( 5 . 1 0 ) and ( 5 . 1 1 ) . Furthermore,

f o r every v f 0

i n T ( X , x ) , one g e t s from ( 5 . 2 ) c o n c e r n i n g t h e r e s p e c t i v e d e r i v a t i o n , lv#O

( t h e map v ~ - + l ~ , v ~ T ( X , xd )e ,f i n e d by ( 5 . 2 ) , i s a b i j e c t i o n ) . So

s t i l l from ( 5 . 1 0 ) a n d t h e f a c t t h a t lv€ lC"lX)I'

( s e e t h e comment be-

f o r e Lemma 5 . 1 ) , o n e c o n c l u d e s t h a t t h e r e e x i s t s f € A , w i t h %,if) = v ( f )

= tdf),(v)

# 0 ; i.e., one o b t a i n s c o n d . 3 ) a s w e l l , a n d t h i s f i n i s h e s t h e

proof. The f o l l o w i n g i s a n e s s e n t i a l t e c h n i c a l i s s u e i n t h e p r o o f o f t h e c o n v e r s e of t h e p r e v i o u s Lemma 5 . 1 , which w e p r o v e i n t h e s e q u e l . Lemma 5 . 2 . Consider a Cm-rnanifot"oZd X and t h e r e s p e c t i v e ( r e a l ) algeb1.a Cm(X)

5. NACHBIN THEOREM

24 3

(NECESSITY)

a s in t h e above L e m a 5 . 1 . Moreover, l e t A be a subalgebra of

C m ( X ) satisfying

cond. 3 ) of t h e same lemma. FinaZZy, l e t T ( X , x ) be t h e tangent space of X a t a p o i n t z~X . Then, t h e r e e x i s t s a b a s i s

c v l , . . . , 'n'

(5.13)

of T(X, x i , t o g e t h e r w i t h a b a s i s of (T(X, xi)* = T*IX,2) ( a l g e b r a i c d u a l of TiX, x), o r y e t t h e cotangent space o f t h e m a n i f o l d X a t z),c o n s i s t i n g of d i f f e r e n t i a l s a t x of f u n c t i o n s from t h e algebra A S C " ( X 1 ;

i . e . , of t h e form

{(dfi)x)lsi gn

(5.14)

w i t h f. E A, 1 5 i 5 n , in such a manner t h a t one has

(df.) z xI v7,. )

(5.15)

a d

=1, 1 S i L n ,

(df.)(v.l=O, l S i < j S n .

(5.16)

Froof.

I f vl # 0 i n

zx 3 T(X, x ) , t h e r e e x i s t s b y h y p o t h e s i s ( c o n d . 3 )

o f Lemma 5 . 1 ) a n e l e m e n t g l f A s u c h t h a t

IdglIx(v,) = u l ( g l ) # 0

(5.17) Now,

-

i f d i m X = n > 2 , o n e g e t s from ( 5 . 1 7 ) t h a t dimiker idgl

(5.18)

Ix I

(see, f o r i n s t a n c e , J . HORVLTH [I: p.411);

21

hence, t h e r e e x i s t s v I E T / X , x )

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

C # u2e k e r

(5.19) Therefore,

(dgl)x )

.

s t i l l by h y p o t h e s i s , o n e f i n d s a n e l e m e n t g 2 e A , w i t h (dg,

(5.20)

Now,

(

).-

iv,) = u 2 ( g g I # 0 .

i f dimX23, then ( i b i d . ) dimfker(dgiiZ) 2 2 , i = l , 2

(5.21)

,

so t h a t o n e h a s (5.22)

dim(ker(dgl),n

Thus , t h e r e e x i s t s v3 E TIX, (5.23)

2)

ker(dg2Sc) t 1 .

such t h a t

0 # u3 E ker(dgI),

n ker(dg2ix ,

w h i l e o n e o b t a i n s , by h y p o t h e s i s ,

(dg3!,1v3) = v3(g3) # O r

(5.24)

f o r some g3 E A . So r e p e a t i n g t h i s a r g u m e n t one f i n a l l y o b t a i n s a ( f i n i t e ) s e q u e n c e { v2,. quence

.., v n }

i n T(X, x ) , t o g e t h e r w i t h a c o r r e s p o n d i n g se-

{g,, . . . , g n } i n A ; now by s e t t i n g

244

VII

SPECTRA OF PARTICULAR ALGEBRAS

(5.25)

fi

with a . = v . ( f . ) # 0 2

2

a sequence

2

{f,,

,1 S i S n

..., f n }

:=I gieA,

(cf. (5.17)

,

lziln, (5.20), e t c . )

,

one f i n a l l y o b t a i n s

i n A , which b y t h e p r e c e d i n g s a t i s f i e s t h e re-

lation ( d f .) ( v . ) = v , . ( f . ) = 6 . . , zx 3 d 2 23

(5.26)

w i t h l < i < j S n . T h a t i s , w e h a v e ( 5 . 1 5 ) and ( 5 . 1 6 ) . F u r t h e r m o r e , it i s r e a d i l y s e e n from ( 5 . 2 6 ) t h a t t h e two f a m i l i e s { v i } and

{(dfiIx 1

,

1 S i S n , a r e l i n e a r l y i n d e p e n d e n t i n T(X, x ) a n d i t s a l g e b r a i c d u a l , re-

s p e c t i v e l y , a n d t h i s c o m p l e t e s t h e p r o o f of t h e 1emma.i NOW, t h e f o l l o w i n g s t a n d a r d f a c t s from t h e r u d i m e n t s of

Rieman-

nian D i f f e r e n t i a l Geometry w i l l n e x t b e n e e d e d . T h u s , o u r m a n i f o l d X bei n g a l w a y s l o c a l l y compact a n d , by h y p o t h e s i s , 2nd c o u n t a b l e , it i s pa-

racompact ( c f . , f o r i n s t a n c e , J . Dugundji [l: p. 174, Theorem 6 . 5 1 ) . So :he manifold X admits a Cm-Riemannian metric o r , e q u i v a l e n t l y , it i s , i n f a c t , a Cm-Riemannian manifold (see S . KOEAYASHI -K. NOMIZU

[ 1 : p.

6 0 1 ) . Thus , a s a

c o n s e q u e n c e of t h e r e s p e c t i v e t h e o r y f o r t h e "exponential function" i n X I one g e t s t h e f o l l o w i n g l e m m a . ( C f . ,

f o r e x a m p l e , t h e l a s t R e f . : p. 148,

P r o p o s i t i o n 8 . 3 ; o r y e t F . BRICKELL- R.S. CLARK [I: p. I 8 O f l ) . Lemma. (Normal coordinates). Let X be a Cm-Riemannian

manifold and x a point of X . Then, f o r every b a s i s { v l , ...,

vn} of T ( X , x ) , there e x i s t s a local chart (17, $ ) = (U; x l , . .

.

, 3cn) (see ( 5 . 3 ) ) of the manifold X a t x such t h a t t h e basis ( v . ) coincides w i t h the Ncanonical basis" (-1 a oxi x ' 1 <= i S n , of T(X,x ) which corresponds t o the chart (U,@). i-kt i s , one has the r e l a t i o n (5.27)

vi =

a

(-)axi

x

w i t h 1 5 i S n. The f o l l o w i n g r e s u l t i s now a c o n s e q u e n c e o f t h e p r e v i o u s Lem-

m a and Lemma 5 . 2 . Namely, w e h a v e .

Corollary 5.1. Let the conditions of L e m 5.2 be s a t i s f i e d . Then, f o r every point x

X , there e x i s t s a local chart of the manifold X a t x , whose "coordinate

functions" are (appropriate r e s t r i c t i o n s of f u n c t i o n s ) from t h e given algebra A C

C"(X).Y e t

which, i n f a c t , amounts t o t h e same t h i n g , the algebra A provides "glo-

bal-local coordinates" a t every p o i n t of the manifold X . Woof. If ( V i ) l 5 i 6 n i s t h e b a s i s o f Ti'X, x ) , p r o v i d e d from Lemm a 5 . 2 , t h e n i n v i e w o f t h e p r e v i o u s Lemma t h e r e e x i s t s a l o c a l c h a r t

5.

245

NACHBIN THEOREM (NECESSITY)

vie T(X,x), I S i S n , as t h e c o r r e s p o n d i n g c a n o n i c a l b a s i s . Moreover, t h e f u n c t i o n s f . E A , 1 < i=C n , p r o (U, $1 a t x h a v i n g t h e g i v e n v e c t o r s

v i d e d by t h e same Lemma 5 . 2 ,

satisfy the relation

((=Is a

(djy,

(5.28)

)

= 6i

,

3 w i t h I < i l n ( c f . ( 5 . 2 6 ) and ( 5 . 2 7 ) ) . Now, by c o n s i d e r i n g t h e v e c t o r -

valued function f := (fi I I < i S n : x-f

(5.29) w i t h fi",

matrix o f

p

lRn,

a s a b o v e , o n e o b t a i n s , c o n c e r n i n g t h e r e s p e c t i v e Jacobian a t x, the relation

(5.30) w i t h I < i , j S n . So o n e g e t s (5.31)

from ( 5 . 2 8 ) ( s e e a l s o ( 5 . 2 ) ) clij

with 1 < iS j < n

.

,

Hence, d e t J z ( f I # O F so t h a t t h e map

id$),

(5.32)

= €iij

+

: T(X,x ) - T ( I R n ,

?(XI)

d e f i n e s a n isomorphism. T h e r e f o r e , b y t h e

3

IRn

Inverse Function Theorem ( f o r

-

m a n i f o l d s ( !) ; see , f o r i n s t a n c e , F. BRICKELL R. S. CLARK [l : p. 63, Prop+ c s i t i o n 4.4.11 ) , t h e map f i s a "local diffeomorphism": Namely, t h e r e ex-

i s t s a n open n e i g h b o r h o o d V of x , i n s u c h a way t h a t t h e r e s t r i c t i o n -+ t o V i s a d i f f e o m o r p h i s m o f V o n t o ( t h e open s e t ) f ( V l c IR'I T h e r e -

of

f'

fore, the pair

+

( V , f J , ) = (L';

(5.33)

(f.1z V ) l 5 i S n 1

y i e l d s a l o c a l c h a r t of t h e manifold X a t t h e p o i n t x , s a t i s f y i n g t h e r e q u i r e d c o n d i t i o n s ( c f . a l s o ( 5 . 3 ) ) ; moreover, t h e f u n c t i o n s

fiIv

(5.34)

, IsiSn,

correspond, of c o u r s e , t o t h e l a s t s t a t e m e n t of t h e c o r o l l a r y ( c f . a l s o t h e r e l . (5.35) i n t h e s e q u e l ) ,and t h e proof i s f i n i s h e d . I I n v i e w o f t h e p r e v i o u s lemma, o n e o b t a i n s a e m - a t l a s m a n i f o l d X which i s , of c o u r s e , c o m p a t i b l e w i t h t h e g i v e n

t u r e of X .

I t b e l o n g s , namely, t o t h e

maximal e m - a t l a s

of t h e

em-struc-

o f X . Thus, w e

are l e d t o t h e following. D e f i n i t i o n 5.1. Keeping t h e n o t a t i o n of C o r o l l a r y 5.1

,

the

em-

a t l a s o f t h e m a n i f o l d X which i s p r o v i d e d , a c c o r d i n g t o t h e p r e v i o u s comment, from t h i s c o r o l l a r y , i s c a l l e d t h e Nachbin a t l a s o f X associated

with +,he algebra A C C " ( X ) . Any c h a r t o f t h i s a t l a s i s a l s o c a l l e d a Nachof X , a t t h e p o i n t I C E X u n d e r c o n s i d e r a t i o n , corresponding t o the

b i n chart

246

VII

SPECTRA OF PARTICULAR

ALGEBRAS

algebra A . NOW, g i v e n a n y c h a r t (U, $I) = (U; x

,..., x

)

( c f . ( 5 . 3 ) ) o f t h e mani-

f o l d X a t x E X , i t s coordinate functions xi, 1 < i < = nmay , be considered as appro-

priate restrictions

( e v e n t u a l l y , however, t o an open s u b s e t o f V ) of func-

t i o n s from the algebra e m ( X ) ( c f . , f o r i n s t a n c e , S. HELGASON [ I : p . 6 , Lemma

1.2, a n d t h e comment a t t h e end o f p . 71 ) terminology ( c f . C o r o l l a r y 5.1)

,

. Thus,

applying t h e previous

one c o n c l u d e s t h a t , for every em-rizani-

C r n t X )provides global-lo-

f o l d X ( n o t n e c e s s a r i l y p a r a c o m p a c t ) , t h e algebra cal coordinates a t every point x e X .

The p r e c e d i n g c h a r a c t e r i z e s , i n f a c t , c o n d . 3 ) of Lemma 5 . 1 . So one h a s t h e following f a c t :

Cond. 3 ) of Lema 5 . 1 i s equivalent with t h e assumption t h a t the algebra A C e r n t X ) provides global-ZocaZ coor-

(5.35)

d i n a t e s , for every p o i n t x e X. Namely ( s e e a l s o C o r o l l a r y 5 . 1 ) , f o r e v e r y p o i n t x E X , t h e r e exi s t a n open n e i g h b o r h o o d U o f x a n d f u n c t i o n s C e A , w i t h 1 S i S n = dim X, such t h a t t h e p a i r

cu; (f:I u )l
(5.36)

d e f i n e s a l o c a l c h a r t of t h e m a n i f o l d X a t x

( t h e c h a r t (5.36) belongs

t o t h e maximal e m - a t l a s o f X). The r e s p e c t i v e by t h e f o l l o w i n g map ($: u-IRn:

(5.37)

x

-

$ ( x i := I

= ( f 1 ( x ) ,..., f

chart-map o f U i s g i v e n

(filU)(X)ll'

.<

-t=n

n ( X H .

Thus, o n e h a l f of t h e a s s e r t i o n i n ( 5 . 3 5 ) i s d e r i v e d a l r e a d y from C o r o l l a r y 5 . 1 .

F u r t h e r m o r e , s u p p o s e now t h a t t h i s c o n d i t i o n

is

t r u e , and l e t v e T ( X , x ) , w i t h v f 0 ; so f o r any c h a r t , l i k e (5.31, a t x , one a t l e a s t of t h e r e s p e c t i v e c o o r d i n a t e s of

1)

( c f . ( 5 . 7 ) ) i s non-

z e r o . Hence, c o n s i d e r i n g a t x t h e p r e v i o u s c h a r t ( 5 . 3 6 ) , o n e g e t s ( c f . a l s o (5.37))

v(xi)=v(filu) =v(fil =idfilx(vl# 0 ,

(5.38)

f o r o n e i n d e x i , a t l e a s t , a s above; i . e . ,

f o r some f .

E

A , which p r o v e s

the assertion.

6 . The Nachbin Theorem (sufficiency) W e prove f u r t h e r i n t h i s s e c t i o n t h a t t h e t h r e e c o n d i t i o n s f o r

a n a l g e b r a A S e " ( X i , s e t f o r t h b y Lemma 5 . 1 , a r e , i n d e e d , s u f f i c i e n t

6. NACHBIN THEOREM (SUFFICIENCY)

247

i n o r d e r t h a t ( 5 . 8 ) t o hold t r u e . W e f i r s t h a v e t h e f o l l o w i n g r e s u l t , which a l s o s u p p l e m e n t s t h e

p r e v i o u s Lemma 5 . 2 .

Lemma 6.1

. Let

X be an n-dimensional (paracompact) e m - m a n i f o I d and K a e m ( X ) , satisfying the

compact s u b s e t of X . Moreover, l e t A be a subalgebra of

-

t h r e e c o n d i t i o n s of Lermna 5 . 1 . Then, t h e r e e x i s t s a diffeomorphism

0: V

(6.1)

@(V)C B N ,

for an appropriate p o s i t i v e i n t e g e r N , where V i s an open neighborhood of K and @(V) c a r r i e s t h e submanifold s t r u c t u r e induced on it from B N under t h e n a t u r a l imbedding 0 ( V j E R N . In p a r t i c u l a r , t h e map 0 i s made of f u n c t i o n s from t h e g i v e n algebra A

C

c"(X!

( c f . a l s o t h e n e x t C o r o l l a r y 6.1 )

.

S i n c e t h e m a n i f o l d X i s a l o c a l l y compact s p a c e , t h e r e ex-

Prcof.

i s t s an open n e i g h b o r h o o d V of t h e g i v e n compact K E X which i s r e l a t i v e l y compact, i . e . ,

one h a s

(6.2)

K C V Cv,

w i t h V C X a compact s e t ( s e e , f o r i n s t a n c e , N . EOUREAKI [4: Chap. I ; p . 65, P r o p o s i t i o n

lo] ) . NOW, f o r e v e r y x

€ 7 ,t h e r e

e x i s t s by h y p o t h e s i s

, so

a f u n c t i o n f e A , w i t h f ( x l # O ( c f . c o n d . 1 ) of Lemma 5 . 1 ) f o r e v e r y y e N J I l w i t h Nx

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

a r e f i n i t e many s u c h n e i g h b o r h o o d s T d x . l l S i S k , i n g of t h e compact

v,

Thus, t h e r e

p r o v i d i n g a n open c o v e r -

7

i n s u c h a way t h a t t h e r e s p e c t i v e f u n c t i o n s f .

d e f i n e a map (6.3)

F = ( f i ) I S{$

which n e v e r v a n i s h e s on

7. T h a t

-

z: : v-

k

IR

,

i s , one h a s

F ( z l := ( f i (xt.))15i6k # OelR

(6.4)

2.

t h a t f(y)#O,

k

,

f o r every x E V . On t h e o t h e r h a n d , by C o r o l l a r y 5 . 1 , c o v e r i n g of

7,

t h e r e e x i s t s a f i n i t e open

c o n s i s t i n g o f Nachbin c h a r t s , s a y

I ( u ~ ;( $ ) l S j 5 n ) ~ , ~ ~ i z l ,

(6.5)

so t h a t , b y t a k i n g ( 6 . 4 )

and ( 6 . 5 ) i n t o a c c o u n t , w e f u r t h e r s e t

i

$

(6.6)

with 1 6 i S l

atd

= 'k+(i-I)i1+j '

ISj6n.

Furthermore, f o r any x, y i n T l w i t h x f y , t h e r e 2 ) o f Lemma 5 . 1 a f u n c t i o n f ~ A , w i t h f ( x l # f ( y ) .

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

(2,y

i

s u c h t h a t f(x7 # f ( y ' i ,

T h u s , t h e compact s e t ( c f . a l s o ( 6 . 5 ) )

e x i s t s by c o n d .

T h e r e f o r e , one f i n d s f o r any (x', y') e W.

248

VII SPECTRA OF PARTICULAR ALGEBRAS

n=~x~-(ru,xu1lv...uIUzxVz)~

(6.7)

d o e s n o t c o n t a i n t h e d i a g o n a l of 7 x 7 . T h u s , o n e c o n c l u d e s by t h e p r e c e d i n g t h a t t h e r e e x i s t f i n i t e many f u n c t i o n s , s a y h,,.. ., hm i n A , i n s u c h a way t h a t one h a s h . ( x f # h.(yl, Z < i < m

(6.8) f o r every

(2,y l

E R

. Thus,

,

setting

h . = fk+ln+i

(6.9)

, 1ai5m ,

we f u r t h e r consider t h e function =lR N

Q = ( f c i ) l < a < N: V-Q(V)

(6.10)

I

w i t h N = k + l n +m e m . NOW, a p p l y i n g ( 6 . 8 )

f o r e l e m e n t s of 0 , o r a n a p p r o p r i a t e Nach-

b i n c h a r t from ( 6 . 5 ) , o t h e r w i s e ( i . e . , f o r n o n - d i a g o n a l e l e m e n t s of - V x V b e l o n g i n g , however, t o t h e s e t ( U 1 x U 2 1 U . U (Ul x U,)) , o n e r e a d i l y

..

r e a l i z e s t h a t t h e map Q i s one-to-one. B e s i d e s , it i s a c L n - m a p definition ( c f . (6.6) inverse map CP-'

, (6.9)

by t h e same

and ( 6 . 1 0 ) ) . F u r t h e r m o r e , c o n s i d e r i n g t h e

o f Q , l e t x = @ - ' ( @ ( x ) ) GV ; t h u s t a k i n g a g a i n a Nachbin

c h a r t a t x , o n e c o n c l u d e s t h a t Q i s indeed a diffeomorphism. T h a t i s , i f

x e V S v b e l o n g s t o a Nachbin c h a r t , s a y ( U i ; f l '

,...,ft I , from

( 6 . 5 ) one

e s s e n t i a l l y c o n s i d e r s t h e diffeomorphism CP

(6.11)

1 ui

@(U.) c R n S R N

; ' iU

w i t h r e s p e c t t o t h e induced C " - s t r u c t u r e t h e " c a n o n i c a l imbedding" that

,

since f

ci

€

A Cc"(X)

r e s t r i c t i o n o f t h e map

lRnS I R N .

,1Sa 2 N ,

,

on lRn from R N I d e f i n e d by

( I n t h i s concern , w e f u r t h e r n o t e

one a c t u a l l y c o n s i d e r s b y ( 6 . 1 0 ) t h e

(fa):X+IRN

to

V;

a s i m i l a r remark h o l d s t r u e

f o r t h e map ( 6 . 1 1 ) 1 . I A s a c o n s e q u e n c e o f t h e p r e v i o u s Lemma 6 . 1 ,

i n g f a c t which, a s w e s h a l l see ( c f . Theorem 6 . 2 1 ,

w e have t h e f o l l o w characterizes, in

e f f e c t , the r e l a t i o n (5.8) concerning t h e given algebra A E e m f X ) .

So

one has

e

"-manifoZd and A Coroll ary 6.1. Let X be an n-dimensional (paracompact) a subalgebra of e " t X ) s a t i s f g i n g the three conditions of Lema 5 . 1 . Then, t h e foZlowing f a c t holds t r u e :

For every compact K G X , there e x i s t s a canonical imbedding (6.12.1 )

@:v-@(v)GlR

N- I a ) E n N

249

6. NACHBIN THEOREM (SUFFICIENCY)

of an open neighborhood V of K i n t o a ( f i n i t e dimensional) R N , whose coordinate functions

eucZidean space

0.= u . 0 O : V -W,

(6.12.2)

z

(see also ( 5 . 4 )

,

z

ISiSlV,

a r e r e s t r i c t i o n s t o V of ( r e a l -

v a l u e d ) C m - f u n c t i o n s on X which) belong t o the algebra A.

The p r e v i o u s a s s e r t i o n f o l l o w s , o f c o u r s e , d i r e c t l y from t h e p r o o f o f Lemma 6 . 1

c a l imbedding" NOW,

( c f . ( 6 . 1 0 ) ) and t h e same d e f i n i t i o n of a "canoni-

( s e e , f o r i n s t a n c e , F. BRICKELL- R.S. CLARK [ l : p. 811)

-

t a k i n g t h e d e f i n i t i o n o f t h e e m - t o p o l o g y i n t h e algebra c w ( X )

.

i n t o a c c o u n t ( s e e C h a p t e r Iv; S e c t i o n 4. ( 2 ) : t h e r e l s ( 4 . 1 4 ) a n d (4.191 ) , it i s on t h e image o f a compact K C X t h r o u g h t h e p r e v i o u s imbedding (6.12.1)

on which o n e a c t u a l l y c o n s i d e r s t h e a p p r o x i m a t i o n s e t f o r t h

by t h e r e l a t i o n ( 5 . 8 ) . So w e have now t h e f o l l o w i n g .

Theorem 6.1. (L. Nachbin). Consider a finite-dimensional 2nd countable Hausd o r f f e m - m a n i f o l d X , and l e t e " ( X ) be the algebra o f real-valued C m - f u n c t i o n s

ew(X). Then,

on X endowed w i t h t h e e m - t o p o l o g y . Moreover, l e t A be a subalgebra oJ the following two a s s e r t i o n s are equivalent: 1) A i s a dense subalgebra of i . e . , one has the r e l a t i o n (6.13) A = em(X).

e"(X),

2 ) A i s " s u f f i c i e n t l y separating"

ordinates a t every point of

on X providing, moreover, global-locaZ co-

x. Note.-

S a y i n g t h a t t h e a l g e b r a A i s suffici.enton X , w e mean t h a t it i s non-vanishi n g and s e p a r a t i n g ; c f . Lemma 5.1 f o r t h e relev a n t terminology.

ly separating

Proof of Theorem 6 . 1 , I f

( 6 . 1 3 ) i s v a l i d , t h e n it h a s a l r e a d y b e e n

p r o v e d i n t h e p r e v i o u s s e c t i o n (Lemma 5 . 1 ) t h a t A i s s u f f i c i e n t l y s e p a r a t i n g , y i e l d i n g a l s o global-local coordinates a t every p o i n t x e X ( c f . C o r o l l a r y 5.1 a n d a l s o ( 5 . 3 5 ) ) . Thus, 1 ) * 2 ) . On t h e o t h e r h a n d , from what h a s b e e n s a i d above, t o p r o v e (6.13) it s u f f i c e s t o t a k e any compact K C X on

which one c o n s i d e r s , i n t u r n ,

t h e a p p r o x i m a t i o n c l a i m e d by ( 6 . 1 3 ) . Thus,

by assuming 2), o n e g e t s from

C o r o l l a r y 6.1 t h a t , c o n c e r n i n g t h e g i v e n compact K C X ,

i s satisfied.

-

T h e r e f o r e , by c o n s i d e r i n g t h e i n v e r s e map o f (6.14)

,-I

:0 ( V l

V L X

Condition ( 6 . 1 2 )

(6.12.1), i.e.,

250

VII

SPECTRA OF PARTICULAR ALGEBRAS

one o b t a i n s t h a t f o @-I E C " ( @ ( V ) ) , f o r e v e r y t i o n t o t h e compact s e t

f E c"(X). Thus by r e s t r i c -

@ ( K I G @ ( V ) G i R None , f i n d s a function $ee"tiRN)

( c f . S e c t i o n 2 ; Lemma 2 ) s u c h t h a t (6.15) T h e r e f o r e , one h a s (6.16)

s o t h a t one g e t s f r o m ( 6 . 1 2 . 2 ) f(x) = @(O(xil = $4@l(x),...,@N(x)),

(6.17)

f o r e v e r y xEK.Furthermore, due t o (6.12.1),

one c o n c l u d e s t h a t O / Q . ( V )

and h e n c e 0 d @(K) S @ ( V ) .

(6.18)

Thus, b y c o n s i d e r i n g C O ( K ) a s a n open n e i g h b o r h o o d of 0 E X N , one may f i n a l l y assume t h a t

$to) = 0

(6.19)

( s e e a l s o , f o r example, S. KOBAYASHI-K. NOMIZU [ l : p. 272, Lemma 2 1 1 . NOW, a n a p p l i c a t i o n o f t h e c l a s s i c a l Weierstrass Theorem y i e l d s , i n view of ( 6 . 1 7 ) , t h a t t h e g i v e n f u n c t i o n f e c"(Xl i s approximated i n t h e Cm-topology of

ern(X) through

f u n c t i o n s from t h e g i v e n a l g e b r a A .

Namely, by p o l y n o m i a l s ( w i t h o u t c o n s t a n t terms; c f . (6.19) ) i n t h e v a r i ables

Q1 ix),.

(6.20)

.. , b N ( X ) ,

w i t h x e K , h e n c e i n t h e f u n c t i o n s Qi, 1 5 i S N . Thus, t h e f u n c t i o n s c"(X)

are a p p r o x i m a t e d i n t h e

C"-topo:ogy

in

by e l e m e n t s from t h e

g i v e n a l g e b r a A ( c f . ( 6 . 1 2 ) ; t h e a l g e b r a A d o e s n o t n e c e s s a r i l y cont a i n t h e i d e n t i t y e l e m e n t of r e l a t i o n (6.13), i.e.,

e m ( X ) ) . S o w e f i n a l l y have t h e d e s i r e d

2 ) = > 1 ) as w e l l ,

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

of t h e theorem. 4 The s i t u a t i o n set f o r t h by t h e p r e c e d i n g p r o o f i s f u r t h e r c l a r i f i e d b y t h e n e x t t h e o r e m . However, w e do s e t f i r s t t h e f o l l o w i n g .

Definition 6.1. L e t X b e a f i n i t e - d i m e n s i o n a l C " - m a n i f o l d a s u b a l g e b r a of

C"(X).Now,

and A

w e s h a l l say t h a t t h e algebra A s a t i s f i e s

Condition ( N l , i n c a s e t h e t h r e e c o n d i t i o n s i n Lemma 5 . 1 h o l d t r u e f o r A (Namely, A i s t h u s s u f f i c i e n t l y s e p a r a t i n g on X (Theorem 6 . 1 ) and a l s o y i e l d s g l o b a l - l o c a l c o o r d i n a t e s a t every p o i n t of X ( c f . ( 5 . 3 5 ) ) .

.

The f o l l o w i n g i l l u m i n a t e s t h e meaning o f

Cond. (6.12). So w e h a v e .

7.

251

V A R I A N T S OF N A C H B I N ’ S THEOREM

Theorem 6.2. Let X be an n-dimensional (paracompact) e m - m a n i f o l d and

C”(X).Then,

subalgebra o f

1 ) A is a dense subalgebra o f

.

C”-topotogy

A a

t h e following three a s s e r t i o n s are e q u i v a l e n t :

Cml.XI, where t h e l a t t e r algebra c a r r i e s t h e

21 A s a t i s f i e s Condition ( N I ( c f . D e f i n i t i o n 6.1 )

.

5) A s a t i s f i e s Condition ( 6 . 1 2 ) .

Proof. The f a c t t h a t 1 ) - 2 ) i s Nachbin’s Theorem ( i - e . , Theorem 2)+3) a s f o l l o w s from C o r o l l a r y 6 . 1 . On t h e o t h e r h a n d ,

6 . 1 ) . NOW,

b y a s s u m i n g 3 ) and a p p l y i n g t h e argument u s e d i n t h e s e c o n d h a l f o f t h e p r o o f of Theorem 6 . 1

,

one o b t a i n s t h a t

=

Cm(X).

T h a t i s , 3 ) => 1 )

a s w e l l , and t h i s f i n i s h e s t h e p r o o f . 1

7. Appendix: Variants o f Nachbin’s Theorem W e c o n s i d e r , i n b r i e f , i n t h i s a p p e n d i x some v a r i a n t s o f Nach-

b i n ’ s Theorem r e f e r r i n g t o t h e a l g e b r a s e c t i o n 7.1 below)

,

and t h e a l g e b r a

CgtX),

cl(X)

w i t h 15rn
o f complex-valued

em-

f u n c t i o n s on a C m - m a n i f o l d X (see S u b s e c t i o n 7 . 2 ) .

7.1. Differentiability of class t h e previous Sections 5, 6 e d a l r e a d y i n L . NACHBIN [ I ] .

t h a t t h e space X i n

is a (finite-dimensional) d i f f e r e n t i a l

d i f f e r e n t i a b i l i t y class

m a n i f o l d of

em.The case

ern ,w i t h

1 2 rn < 03,

h a s been c o n s i d e r -

Thus, a s f a r a s o n e c o n s i d e r s real-valued

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

C g l X ) of r e a l - v a l u e d d i f f e r e n t i a b l e f u n c on a ( p a r a c o m p a c t ) C”-rnanifoId X. So one g e t s t i o n s of “cZass t h e a n a l o g o n o f Theorem 6 . 1 i n t h i s c a s e too. Of c o u r s e , t h e a l g e b r a

used h e r e f o r t h e a l g e b r a

em

i s endowed, i n t u r n , w i t h t h e r e s p e c t i v e c a n o n i c a l

Cm-topology,i.e.,

t h e topology of uniform convergence on cornpacta of t h e ( d i f f e r e n t i a b l e ) f u n c t i o n s on X and of t h e i r d e r i v a t i v e s u p t o order rn (see, f o r i n s t a n c e , F. TREVES [ I :

p. 85 f f ] )

.

The t o p o l o g i c a l a l g e b r a

( r e a l ) Frgehet localZy m-eonuex algebra a l s o Corollary I;7.2 compact X )

.

CrnfX) thus

defined is s t i l l a

( w i t h an i d e n t i t y e l e m e n t ; i b i d . See

o r y e t C o r o l l a r y I ; 7 . 3 of t h i s book i n c a s e o f

I n t h i s r e s p e c t , w e s t i l l n o t e t h a t t h e r e s p e c t i v e m a t e r i a l of C h a p t e r IV, S e c t i o n 4 . ( 2 ) u s e d above h o l d s t r u e , of c o u r s e , f o r t h e ‘I

( r e a l ) Crn-caseIt as w e l l .

7.2. Comp1exification.-

W e consider next the case t h a t the differ-

252

VII

SPECTRA OF PARTICULAR ALGEBRAS

e n t i a b l e f u n c t i o n s i n v o l v e d i n t h e p r e c e d i n g d i s c u s s i o n a r e complexv a l u e d . That i s , w e s e t

c"(x) =

(7.2)

The c a s e of t h e a l g e b r a

e,"(X)

cCrncx).

is then treated similarly a s i n the

previous s e c t i o n 7 . 1 . Thus , c o n c e r n i n g Condition ( N l ( D e f i n i t i o n 6.1 ) f o r every f E e , " ( X )

,

, we

remark t h a t ,

t h e c o r r e s p o n d i n g d i f f e r e n t i a l of f a t some p o i n t i s now, i n g e n e r a l , a complex-valued map on T(X, x); i.e., one h a s

x EX,

( d f )x : F(X, X I

(7.3)

-

Q:

t h i s b e i n g , o f c o u r s e , IR-linear ( s a t i s f y i n g , moreover, t h e L e i b n i z c o n d i t i o n ; c f . ( 5 . 2 ) ) . So one may c o n s i d e r ( 7 . 3 ) a s a Leibniz map on t h e compZexified tangent space o f X a t x of the aZgebra 1 7 . 2 ) ) . T h a t i s , w e have

( o r y e t a co.nplex derivation a t X E X ,

,

( d f l , : T (X,x) = T I X , x) 8 C -C C IR i n s i c h a manner t h a t (7.4)

(dfl,

(7.5)

f o r a n y v e T ( X , x I and X E C . NOW, s i n c e C % I R + i l R imaginary part of

(7.4)

,

(V

8 A) =

, with

X. t d f l , ( v )

I

i = R , by c o n s i d e r i n g t h e r e a l and

namely, R e ( ( d f & ) and l r n ( ( d f ) x ) , r e s p e c t i v e l y ,

one g e t s t h e a n a l o g o u s r e s u l t to Lemma 5 . 2 , h e n c e , f i n a l l y , a Nachbin c h a r t a t x e x c o r r e s p o n d i n g t o R e ( i d f 1 , ) . [ I n t h i s r e s p e c t , see a l s o N . BOURBAKI [2:

Chap. 2 ; p . 1 4 , C o r o l l a i r e I] )

i n t h e proof of Theorem 6 . 1

. So

t h e argument a p p l i e d

i s s t i l l v a l i d , assuming moreover t h a t t h e

a l g e b r a A G C Z t X ) i s seZf-adjoint; t h a t i s , w e assume t h a t P E A , f o r e v e r y f e A

(7d e n o t i n g

t h e complex-conjugate of t h e (complex-valued) f u n c -

t i o n f e A ) . Thuslone obtains t h e following. Theorem 7.1. Let X be an n-dimensional paracompact Cm-manifoZd and c " ( X l the l o c a l l y m-convex algebra o f complex-valued e " - f u n c t i o n s

.

on X i n the C " - t o -

p o l o g y ( c f . S e c t i o n I V ; 4 . ( 2 ) ) Moreover, l e t A be a s e l f - a d j o i n t subalgebra of Then, one has the r e l a t i o n (7.6) A = em(X)

c"(X).

i f , and o n l y i f , the corresponding t h r e e conditions of Lema 5.1 are s a t i s f i e d . I